An Anharmonic Solution to the Equation of Motion for the Simple Pendulum

ERIC Educational Resources Information Center

Johannessen, Kim

2011-01-01

An anharmonic solution to the differential equation describing the oscillations of a simple pendulum at large angles is discussed. The solution is expressed in terms of functions not involving the Jacobi elliptic functions. In the derivation, a sinusoidal expression, including a linear and a Fourier sine series in the argument, has been applied.…

A Simple Equation to Predict a Subscore's Value

ERIC Educational Resources Information Center

Feinberg, Richard A.; Wainer, Howard

2014-01-01

Subscores are often used to indicate test-takers' relative strengths and weaknesses and so help focus remediation. But a subscore is not worth reporting if it is too unreliable to believe or if it contains no information that is not already contained in the total score. It is possible, through the use of a simple linear equation provided in…

Catmull-Rom Curve Fitting and Interpolation Equations

ERIC Educational Resources Information Center

Jerome, Lawrence

2010-01-01

Computer graphics and animation experts have been using the Catmull-Rom smooth curve interpolation equations since 1974, but the vector and matrix equations can be derived and simplified using basic algebra, resulting in a simple set of linear equations with constant coefficients. A variety of uses of Catmull-Rom interpolation are demonstrated,…

Electromagnetic momentum and the energy–momentum tensor in a linear medium with magnetic and dielectric properties

DOE Office of Scientific and Technical Information (OSTI.GOV)

Crenshaw, Michael E., E-mail: michael.e.crenshaw4.civ@mail.mil

2014-04-15

In a continuum setting, the energy–momentum tensor embodies the relations between conservation of energy, conservation of linear momentum, and conservation of angular momentum. The well-defined total energy and the well-defined total momentum in a thermodynamically closed system with complete equations of motion are used to construct the total energy–momentum tensor for a stationary simple linear material with both magnetic and dielectric properties illuminated by a quasimonochromatic pulse of light through a gradient-index antireflection coating. The perplexing issues surrounding the Abraham and Minkowski momentums are bypassed by working entirely with conservation principles, the total energy, and the total momentum. We derivemore » electromagnetic continuity equations and equations of motion for the macroscopic fields based on the material four-divergence of the traceless, symmetric total energy–momentum tensor. We identify contradictions between the macroscopic Maxwell equations and the continuum form of the conservation principles. We resolve the contradictions, which are the actual fundamental issues underlying the Abraham–Minkowski controversy, by constructing a unified version of continuum electrodynamics that is based on establishing consistency between the three-dimensional Maxwell equations for macroscopic fields, the electromagnetic continuity equations, the four-divergence of the total energy–momentum tensor, and a four-dimensional tensor formulation of electrodynamics for macroscopic fields in a simple linear medium.« less

Code Samples Used for Complexity and Control

NASA Astrophysics Data System (ADS)

Ivancevic, Vladimir G.; Reid, Darryn J.

2015-11-01

The following sections are included: * MathematicaⓇ Code * Generic Chaotic Simulator * Vector Differential Operators * NLS Explorer * 2C++ Code * C++ Lambda Functions for Real Calculus * Accelerometer Data Processor * Simple Predictor-Corrector Integrator * Solving the BVP with the Shooting Method * Linear Hyperbolic PDE Solver * Linear Elliptic PDE Solver * Method of Lines for a Set of the NLS Equations * C# Code * Iterative Equation Solver * Simulated Annealing: A Function Minimum * Simple Nonlinear Dynamics * Nonlinear Pendulum Simulator * Lagrangian Dynamics Simulator * Complex-Valued Crowd Attractor Dynamics * Freeform Fortran Code * Lorenz Attractor Simulator * Complex Lorenz Attractor * Simple SGE Soliton * Complex Signal Presentation * Gaussian Wave Packet * Hermitian Matrices * Euclidean L2-Norm * Vector/Matrix Operations * Plain C-Code: Levenberg-Marquardt Optimizer * Free Basic Code: 2D Crowd Dynamics with 3000 Agents

DOE Office of Scientific and Technical Information (OSTI.GOV)

Dubrovsky, V. G.; Topovsky, A. V.

New exact solutions, nonstationary and stationary, of Veselov-Novikov (VN) equation in the forms of simple nonlinear and linear superpositions of arbitrary number N of exact special solutions u{sup (n)}, n= 1, Horizontal-Ellipsis , N are constructed via Zakharov and Manakov {partial_derivative}-dressing method. Simple nonlinear superpositions are represented up to a constant by the sums of solutions u{sup (n)} and calculated by {partial_derivative}-dressing on nonzero energy level of the first auxiliary linear problem, i.e., 2D stationary Schroedinger equation. It is remarkable that in the zero energy limit simple nonlinear superpositions convert to linear ones in the form of the sums ofmore » special solutions u{sup (n)}. It is shown that the sums u=u{sup (k{sub 1})}+...+u{sup (k{sub m})}, 1 Less-Than-Or-Slanted-Equal-To k{sub 1} < k{sub 2} < Horizontal-Ellipsis < k{sub m} Less-Than-Or-Slanted-Equal-To N of arbitrary subsets of these solutions are also exact solutions of VN equation. The presented exact solutions include as superpositions of special line solitons and also superpositions of plane wave type singular periodic solutions. By construction these exact solutions represent also new exact transparent potentials of 2D stationary Schroedinger equation and can serve as model potentials for electrons in planar structures of modern electronics.« less

Solving ay'' + by' + cy = 0 with a Simple Product Rule Approach

ERIC Educational Resources Information Center

Tolle, John

2011-01-01

When elementary ordinary differential equations (ODEs) of first and second order are included in the calculus curriculum, second-order linear constant coefficient ODEs are typically solved by a method more appropriate to differential equations courses. This method involves the characteristic equation and its roots, complex-valued solutions, and…

A Family of Ellipse Methods for Solving Non-Linear Equations

ERIC Educational Resources Information Center

Gupta, K. C.; Kanwar, V.; Kumar, Sanjeev

2009-01-01

This note presents a method for the numerical approximation of simple zeros of a non-linear equation in one variable. In order to do so, the method uses an ellipse rather than a tangent approach. The main advantage of our method is that it does not fail even if the derivative of the function is either zero or very small in the vicinity of the…

Local energy decay for linear wave equations with variable coefficients

NASA Astrophysics Data System (ADS)

Ikehata, Ryo

2005-06-01

A uniform local energy decay result is derived to the linear wave equation with spatial variable coefficients. We deal with this equation in an exterior domain with a star-shaped complement. Our advantage is that we do not assume any compactness of the support on the initial data, and its proof is quite simple. This generalizes a previous famous result due to Morawetz [The decay of solutions of the exterior initial-boundary value problem for the wave equation, Comm. Pure Appl. Math. 14 (1961) 561-568]. In order to prove local energy decay, we mainly apply two types of ideas due to Ikehata-Matsuyama [L2-behaviour of solutions to the linear heat and wave equations in exterior domains, Sci. Math. Japon. 55 (2002) 33-42] and Todorova-Yordanov [Critical exponent for a nonlinear wave equation with damping, J. Differential Equations 174 (2001) 464-489].

A Maple package for computing Gröbner bases for linear recurrence relations

NASA Astrophysics Data System (ADS)

Gerdt, Vladimir P.; Robertz, Daniel

2006-04-01

A Maple package for computing Gröbner bases of linear difference ideals is described. The underlying algorithm is based on Janet and Janet-like monomial divisions associated with finite difference operators. The package can be used, for example, for automatic generation of difference schemes for linear partial differential equations and for reduction of multiloop Feynman integrals. These two possible applications are illustrated by simple examples of the Laplace equation and a one-loop scalar integral of propagator type.

An alternative to the breeder's and Lande's equations.

PubMed

Houchmandzadeh, Bahram

2014-01-10

The breeder's equation is a cornerstone of quantitative genetics, widely used in evolutionary modeling. Noting the mean phenotype in parental, selected parents, and the progeny by E(Z0), E(ZW), and E(Z1), this equation relates response to selection R = E(Z1) - E(Z0) to the selection differential S = E(ZW) - E(Z0) through a simple proportionality relation R = h(2)S, where the heritability coefficient h(2) is a simple function of genotype and environment factors variance. The validity of this relation relies strongly on the normal (Gaussian) distribution of the parent genotype, which is an unobservable quantity and cannot be ascertained. In contrast, we show here that if the fitness (or selection) function is Gaussian with mean μ, an alternative, exact linear equation of the form R' = j(2)S' can be derived, regardless of the parental genotype distribution. Here R' = E(Z1) - μ and S' = E(ZW) - μ stand for the mean phenotypic lag with respect to the mean of the fitness function in the offspring and selected populations. The proportionality coefficient j(2) is a simple function of selection function and environment factors variance, but does not contain the genotype variance. To demonstrate this, we derive the exact functional relation between the mean phenotype in the selected and the offspring population and deduce all cases that lead to a linear relation between them. These results generalize naturally to the concept of G matrix and the multivariate Lande's equation Δ(z) = GP(-1)S. The linearity coefficient of the alternative equation are not changed by Gaussian selection.

Analytical Studies on the Synchronization of a Network of Linearly-Coupled Simple Chaotic Systems

NASA Astrophysics Data System (ADS)

Sivaganesh, G.; Arulgnanam, A.; Seethalakshmi, A. N.; Selvaraj, S.

2018-05-01

We present explicit generalized analytical solutions for a network of linearly-coupled simple chaotic systems. Analytical solutions are obtained for the normalized state equations of a network of linearly-coupled systems driven by a common chaotic drive system. Two parameter bifurcation diagrams revealing the various hidden synchronization regions, such as complete, phase and phase-lag synchronization are identified using the analytical results. The synchronization dynamics and their stability are studied using phase portraits and the master stability function, respectively. Further, experimental results for linearly-coupled simple chaotic systems are presented to confirm the analytical results. The synchronization dynamics of a network of chaotic systems studied analytically is reported for the first time.

Acceleration and Velocity Sensing from Measured Strain

NASA Technical Reports Server (NTRS)

Pak, Chan-Gi; Truax, Roger

2015-01-01

A simple approach for computing acceleration and velocity of a structure from the strain is proposed in this study. First, deflection and slope of the structure are computed from the strain using a two-step theory. Frequencies of the structure are computed from the time histories of strain using a parameter estimation technique together with an autoregressive moving average model. From deflection, slope, and frequencies of the structure, acceleration and velocity of the structure can be obtained using the proposed approach. Simple harmonic motion is assumed for the acceleration computations, and the central difference equation with a linear autoregressive model is used for the computations of velocity. A cantilevered rectangular wing model is used to validate the simple approach. Quality of the computed deflection, acceleration, and velocity values are independent of the number of fibers. The central difference equation with a linear autoregressive model proposed in this study follows the target response with reasonable accuracy. Therefore, the handicap of the backward difference equation, phase shift, is successfully overcome.

Linear stability analysis of collective neutrino oscillations without spurious modes

NASA Astrophysics Data System (ADS)

Morinaga, Taiki; Yamada, Shoichi

2018-01-01

Collective neutrino oscillations are induced by the presence of neutrinos themselves. As such, they are intrinsically nonlinear phenomena and are much more complex than linear counterparts such as the vacuum or Mikheyev-Smirnov-Wolfenstein oscillations. They obey integro-differential equations, for which it is also very challenging to obtain numerical solutions. If one focuses on the onset of collective oscillations, on the other hand, the equations can be linearized and the technique of linear analysis can be employed. Unfortunately, however, it is well known that such an analysis, when applied with discretizations of continuous angular distributions, suffers from the appearance of so-called spurious modes: unphysical eigenmodes of the discretized linear equations. In this paper, we analyze in detail the origin of these unphysical modes and present a simple solution to this annoying problem. We find that the spurious modes originate from the artificial production of pole singularities instead of a branch cut on the Riemann surface by the discretizations. The branching point singularities on the Riemann surface for the original nondiscretized equations can be recovered by approximating the angular distributions with polynomials and then performing the integrals analytically. We demonstrate for some examples that this simple prescription does remove the spurious modes. We also propose an even simpler method: a piecewise linear approximation to the angular distribution. It is shown that the same methodology is applicable to the multienergy case as well as to the dispersion relation approach that was proposed very recently.

FINITE ELEMENT MODEL FOR TIDAL AND RESIDUAL CIRCULATION.

USGS Publications Warehouse

Walters, Roy A.

1986-01-01

Harmonic decomposition is applied to the shallow water equations, thereby creating a system of equations for the amplitude of the various tidal constituents and for the residual motions. The resulting equations are elliptic in nature, are well posed and in practice are shown to be numerically well-behaved. There are a number of strategies for choosing elements: the two extremes are to use a few high-order elements with continuous derivatives, or to use a large number of simpler linear elements. In this paper simple linear elements are used and prove effective.

Coincidence degree and periodic solutions of neutral equations

NASA Technical Reports Server (NTRS)

Hale, J. K.; Mawhin, J.

1973-01-01

The problem of existence of periodic solutions for some nonautonomous neutral functional differential equations is examined. It is an application of a basic theorem on the Fredholm alternative for periodic solutions of some linear neutral equations and of a generalized Leray-Schauder theory. Although proofs are simple, the results are nontrivial extensions to the neutral case of existence theorems for periodic solutions of functional differential equations.

An Alternative to the Breeder’s and Lande’s Equations

PubMed Central

Houchmandzadeh, Bahram

2013-01-01

The breeder’s equation is a cornerstone of quantitative genetics, widely used in evolutionary modeling. Noting the mean phenotype in parental, selected parents, and the progeny by E(Z0), E(ZW), and E(Z1), this equation relates response to selection R = E(Z1) − E(Z0) to the selection differential S = E(ZW) − E(Z0) through a simple proportionality relation R = h2S, where the heritability coefficient h2 is a simple function of genotype and environment factors variance. The validity of this relation relies strongly on the normal (Gaussian) distribution of the parent genotype, which is an unobservable quantity and cannot be ascertained. In contrast, we show here that if the fitness (or selection) function is Gaussian with mean μ, an alternative, exact linear equation of the form R′ = j2S′ can be derived, regardless of the parental genotype distribution. Here R′ = E(Z1) − μ and S′ = E(ZW) − μ stand for the mean phenotypic lag with respect to the mean of the fitness function in the offspring and selected populations. The proportionality coefficient j2 is a simple function of selection function and environment factors variance, but does not contain the genotype variance. To demonstrate this, we derive the exact functional relation between the mean phenotype in the selected and the offspring population and deduce all cases that lead to a linear relation between them. These results generalize naturally to the concept of G matrix and the multivariate Lande’s equation Δz¯=GP−1S. The linearity coefficient of the alternative equation are not changed by Gaussian selection. PMID:24212080

Qualitative properties of large buckled states of spherical shells

NASA Technical Reports Server (NTRS)

Shih, K. G.; Antman, S. S.

1985-01-01

A system of 6th-order quasi-linear Ordinary Differential Equations is analyzed to show the global existence of axisymmetrically buckled states. A surprising nodal property is obtained which shows that everywhere along a branch of solutions that bifurcates from a simple eigenvalue of the linearized equation, the number of simultaneously vanishing points of both shear resultant and circumferential bending moment resultant remains invariant, provided that a certain auxiliary condition is satisfied.

Low-Dispersion Scheme for Nonlinear Acoustic Waves in Nonuniform Flow

NASA Technical Reports Server (NTRS)

Baysal, Oktay; Kaushik, Dinesh K.; Idres, Moumen

1997-01-01

The linear dispersion-relation-preserving scheme and its boundary conditions have been extended to the nonlinear Euler equations. This allowed computing, a nonuniform flowfield and a nonlinear acoustic wave propagation in such a medium, by the same scheme. By casting all the equations, boundary conditions, and the solution scheme in generalized curvilinear coordinates, the solutions were made possible for non-Cartesian domains and, for the better deployment of the grid points, nonuniform grid step sizes could be used. It has been tested for a number of simple initial-value and periodic-source problems. A simple demonstration of the difference between a linear and nonlinear propagation was conducted. The wall boundary condition, derived from the momentum equations and implemented through a pressure at a ghost point, and the radiation boundary condition, derived from the asymptotic solution to the Euler equations, have proven to be effective for the nonlinear equations and nonuniform flows. The nonreflective characteristic boundary conditions also have shown success but limited to the nonlinear waves in no mean flow, and failed for nonlinear waves in nonuniform flow.

Fokker-Planck equation for the non-Markovian Brownian motion in the presence of a magnetic field

NASA Astrophysics Data System (ADS)

Das, Joydip; Mondal, Shrabani; Bag, Bidhan Chandra

2017-10-01

In the present study, we have proposed the Fokker-Planck equation in a simple way for a Langevin equation of motion having ordinary derivative (OD), the Gaussian random force and a generalized frictional memory kernel. The equation may be associated with or without conservative force field from harmonic potential. We extend this method for a charged Brownian particle in the presence of a magnetic field. Thus, the present method is applicable for a Langevin equation of motion with OD, the Gaussian colored thermal noise and any kind of linear force field that may be conservative or not. It is also simple to apply this method for the colored Gaussian noise that is not related to the damping strength.

Fokker-Planck equation for the non-Markovian Brownian motion in the presence of a magnetic field.

PubMed

Das, Joydip; Mondal, Shrabani; Bag, Bidhan Chandra

2017-10-28

In the present study, we have proposed the Fokker-Planck equation in a simple way for a Langevin equation of motion having ordinary derivative (OD), the Gaussian random force and a generalized frictional memory kernel. The equation may be associated with or without conservative force field from harmonic potential. We extend this method for a charged Brownian particle in the presence of a magnetic field. Thus, the present method is applicable for a Langevin equation of motion with OD, the Gaussian colored thermal noise and any kind of linear force field that may be conservative or not. It is also simple to apply this method for the colored Gaussian noise that is not related to the damping strength.

Partial Fractions via Calculus

ERIC Educational Resources Information Center

Bauldry, William C.

2018-01-01

The standard technique taught in calculus courses for partial fraction expansions uses undetermined coefficients to generate a system of linear equations; we present a derivative-based technique that calculus and differential equations instructors can use to reinforce connections to calculus. Simple algebra shows that we can use the derivative to…

Tori and chaos in a simple C1-system

NASA Astrophysics Data System (ADS)

Roessler, O. E.; Kahiert, C.; Ughleke, B.

A piecewise-linear autonomous 3-variable ordinary differential equation is presented which permits analytical modeling of chaotic attractors. A once-differentiable system of equations is defined which consists of two linear half-systems which meet along a threshold plane. The trajectories described by each equation is thereby continuous along the divide, forming a one-parameter family of invariant tori. The addition of a damping term produces a system of equations for various chaotic attractors. Extension of the system by means of a 4-variable generalization yields hypertori and hyperchaos. It is noted that the hierarchy established is amenable to analysis by the use of Poincare half-maps. Applications of the systems of ordinary differential equations to modeling turbulent flows are discussed.

Theory of advection-driven long range biotic transport

USDA-ARS?s Scientific Manuscript database

We propose a simple mechanistic model to examine the effects of advective flow on the spread of fungal diseases spread by wind-blown spores. The model is defined by a set of two coupled non-linear partial differential equations for spore densities. One equation describes the long-distance advectiv...

Kinetics of DSB rejoining and formation of simple chromosome exchange aberrations

NASA Technical Reports Server (NTRS)

Cucinotta, F. A.; Nikjoo, H.; O'Neill, P.; Goodhead, D. T.

2000-01-01

PURPOSE: To investigate the role of kinetics in the processing of DNA double strand breaks (DSB), and the formation of simple chromosome exchange aberrations following X-ray exposures to mammalian cells based on an enzymatic approach. METHODS: Using computer simulations based on a biochemical approach, rate-equations that describe the processing of DSB through the formation of a DNA-enzyme complex were formulated. A second model that allows for competition between two processing pathways was also formulated. The formation of simple exchange aberrations was modelled as misrepair during the recombination of single DSB with undamaged DNA. Non-linear coupled differential equations corresponding to biochemical pathways were solved numerically by fitting to experimental data. RESULTS: When mediated by a DSB repair enzyme complex, the processing of single DSB showed a complex behaviour that gives the appearance of fast and slow components of rejoining. This is due to the time-delay caused by the action time of enzymes in biomolecular reactions. It is shown that the kinetic- and dose-responses of simple chromosome exchange aberrations are well described by a recombination model of DSB interacting with undamaged DNA when aberration formation increases with linear dose-dependence. Competition between two or more recombination processes is shown to lead to the formation of simple exchange aberrations with a dose-dependence similar to that of a linear quadratic model. CONCLUSIONS: Using a minimal number of assumptions, the kinetics and dose response observed experimentally for DSB rejoining and the formation of simple chromosome exchange aberrations are shown to be consistent with kinetic models based on enzymatic reaction approaches. A non-linear dose response for simple exchange aberrations is possible in a model of recombination of DNA containing a DSB with undamaged DNA when two or more pathways compete for DSB repair.

Investigation of ODE integrators using interactive graphics. [Ordinary Differential Equations

NASA Technical Reports Server (NTRS)

Brown, R. L.

1978-01-01

Two FORTRAN programs using an interactive graphic terminal to generate accuracy and stability plots for given multistep ordinary differential equation (ODE) integrators are described. The first treats the fixed stepsize linear case with complex variable solutions, and generates plots to show accuracy and error response to step driving function of a numerical solution, as well as the linear stability region. The second generates an analog to the stability region for classes of non-linear ODE's as well as accuracy plots. Both systems can compute method coefficients from a simple specification of the method. Example plots are given.

A Solution to the Fundamental Linear Fractional Order Differential Equation

NASA Technical Reports Server (NTRS)

Hartley, Tom T.; Lorenzo, Carl F.

1998-01-01

This paper provides a solution to the fundamental linear fractional order differential equation, namely, (sub c)d(sup q, sub t) + ax(t) = bu(t). The impulse response solution is shown to be a series, named the F-function, which generalizes the normal exponential function. The F-function provides the basis for a qth order "fractional pole". Complex plane behavior is elucidated and a simple example, the inductor terminated semi- infinite lossy line, is used to demonstrate the theory.

Iteration with Spreadsheets.

ERIC Educational Resources Information Center

Smith, Michael

1990-01-01

Presents several examples of the iteration method using computer spreadsheets. Examples included are simple iterative sequences and the solution of equations using the Newton-Raphson formula, linear interpolation, and interval bisection. (YP)

Whole stand volume tables for quaking aspen in the Rocky Mountains

Treesearch

Wayne D. Shepperd; H. Todd Mowrer

1984-01-01

Linear regression equations were developed to predict stand volumes for aspen given average stand basal area and average stand height. Tables constructed from these equations allow easy field estimation of gross merchantable cubic and board foot Scribner Rules per acre, and cubic meters per hectare using simple prism cruise data.

From Feynman rules to conserved quantum numbers, I

NASA Astrophysics Data System (ADS)

Nogueira, P.

2017-05-01

In the context of Quantum Field Theory (QFT) there is often the need to find sets of graph-like diagrams (the so-called Feynman diagrams) for a given physical model. If negative, the answer to the related problem 'Are there any diagrams with this set of external fields?' may settle certain physical questions at once. Here the latter problem is formulated in terms of a system of linear diophantine equations derived from the Lagrangian density, from which necessary conditions for the existence of the required diagrams may be obtained. Those conditions are equalities that look like either linear diophantine equations or linear modular (i.e. congruence) equations, and may be found by means of fairly simple algorithms that involve integer computations. The diophantine equations so obtained represent (particle) number conservation rules, and are related to the conserved (additive) quantum numbers that may be assigned to the fields of the model.

Well-posedness, linear perturbations, and mass conservation for the axisymmetric Einstein equations

DOE Office of Scientific and Technical Information (OSTI.GOV)

Dain, Sergio; Ortiz, Omar E.; Facultad de Matematica, Astronomia y Fisica, FaMAF, Universidad Nacional de Cordoba, Instituto de Fisica Enrique Gaviola, IFEG, CONICET, Ciudad Universitaria

2010-02-15

For axially symmetric solutions of Einstein equations there exists a gauge which has the remarkable property that the total mass can be written as a conserved, positive definite, integral on the spacelike slices. The mass integral provides a nonlinear control of the variables along the whole evolution. In this gauge, Einstein equations reduce to a coupled hyperbolic-elliptic system which is formally singular at the axis. As a first step in analyzing this system of equations we study linear perturbations on a flat background. We prove that the linear equations reduce to a very simple system of equations which provide, thoughmore » the mass formula, useful insight into the structure of the full system. However, the singular behavior of the coefficients at the axis makes the study of this linear system difficult from the analytical point of view. In order to understand the behavior of the solutions, we study the numerical evolution of them. We provide strong numerical evidence that the system is well-posed and that its solutions have the expected behavior. Finally, this linear system allows us to formulate a model problem which is physically interesting in itself, since it is connected with the linear stability of black hole solutions in axial symmetry. This model can contribute significantly to solve the nonlinear problem and at the same time it appears to be tractable.« less

Renormalization-group theory of plasma microturbulence

DOE Office of Scientific and Technical Information (OSTI.GOV)

Carati, D.; Chriaa, K.; Balescu, R.

1994-08-01

The dynamical renormalization-group methods are applied to the gyrokinetic equation describing drift-wave turbulence in plasmas. As in both magnetohydrodynamic and neutral turbulence, small-scale fluctuations appear to act as effective dissipative processes on large-scale phenomena. A linear renormalized gyrokinetic equation is derived. No artificial forcing is introduced into the equations and all the renormalized corrections are expressed in terms of the fluctuating electric potential. The link with the quasilinear limit and the direct interaction approximation is investigated. Simple analytical expressions for the anomalous transport coefficients are derived by using the linear renormalized gyrokinetic equation. Examples show that both quasilinear and Bohmmore » scalings can be recovered depending on the spectral amplitude of the electric potential fluctuations.« less

A diffuse-interface method for two-phase flows with soluble surfactants

PubMed Central

Teigen, Knut Erik; Song, Peng; Lowengrub, John; Voigt, Axel

2010-01-01

A method is presented to solve two-phase problems involving soluble surfactants. The incompressible Navier–Stokes equations are solved along with equations for the bulk and interfacial surfactant concentrations. A non-linear equation of state is used to relate the surface tension to the interfacial surfactant concentration. The method is based on the use of a diffuse interface, which allows a simple implementation using standard finite difference or finite element techniques. Here, finite difference methods on a block-structured adaptive grid are used, and the resulting equations are solved using a non-linear multigrid method. Results are presented for a drop in shear flow in both 2D and 3D, and the effect of solubility is discussed. PMID:21218125

Slackline dynamics and the Helmholtz-Duffing oscillator

NASA Astrophysics Data System (ADS)

Athanasiadis, Panos J.

2018-01-01

Slacklining is a new, rapidly expanding sport, and understanding its physics is paramount for maximizing fun and safety. Yet, compared to other sports, very little has been published so far on slackline dynamics. The equations of motion describing a slackline are fundamentally nonlinear, and assuming linear elasticity, they lead to a form of the Duffing equation. Following this approach, characteristic examples of slackline motion are simulated, including trickline bouncing, leash falls and longline surfing. The time-dependent solutions of the differential equations describing the system are acquired by numerical integration. A simple form of energy dissipation (linear drag) is added in some cases. It is recognized in this study that geometric nonlinearity is a fundamental aspect characterizing the dynamics of slacklines. Sports, and particularly slackline, is an excellent way of engaging young people with physics. A slackline is a simple yet insightful example of a nonlinear oscillator. It is very easy to model in the laboratory, as well as to rig and try on a university campus. For instructive purposes, its behaviour can be explored by numerically integrating the respective equations of motion. A form of the Duffing equation emerges naturally in the analysis and provides a powerful introduction to nonlinear dynamics. The material is suitable for graduate students and undergraduates with a background in classical mechanics and differential equations.

[Ultrasonic measurements of fetal thalamus, caudate nucleus and lenticular nucleus in prenatal diagnosis].

PubMed

Yang, Ruiqi; Wang, Fei; Zhang, Jialing; Zhu, Chonglei; Fan, Limei

2015-05-19

To establish the reference values of thalamus, caudate nucleus and lenticular nucleus diameters through fetal thalamic transverse section. A total of 265 fetuses at our hospital were randomly selected from November 2012 to August 2014. And the transverse and length diameters of thalamus, caudate nucleus and lenticular nucleus were measured. SPSS 19.0 statistical software was used to calculate the regression curve of fetal diameter changes and gestational weeks of pregnancy. P < 0.05 was considered as having statistical significance. The linear regression equation of fetal thalamic length diameter and gestational week was: Y = 0.051X+0.201, R = 0.876, linear regression equation of thalamic transverse diameter and fetal gestational week was: Y = 0.031X+0.229, R = 0.817, linear regression equation of fetal head of caudate nucleus length diameter and gestational age was: Y = 0.033X+0.101, R = 0.722, linear regression equation of fetal head of caudate nucleus transverse diameter and gestational week was: R = 0.025 - 0.046, R = 0.711, linear regression equation of fetal lentiform nucleus length diameter and gestational week was: Y = 0.046+0.229, R = 0.765, linear regression equation of fetal lentiform nucleus diameter and gestational week was: Y = 0.025 - 0.05, R = 0.772. Ultrasonic measurement of diameter of fetal thalamus caudate nucleus, and lenticular nucleus through thalamic transverse section is simple and convenient. And measurements increase with fetal gestational weeks and there is linear regression relationship between them.

Expressions of the fundamental equation of gradient elution and a numerical solution of these equations under any gradient profile.

PubMed

Nikitas, P; Pappa-Louisi, A

2005-09-01

The original work carried out by Freiling and Drake in gradient liquid chromatography is rewritten in the current language of reversed-phase liquid chromatography. This allows for the rigorous derivation of the fundamental equation for gradient elution and the development of two alternative expressions of this equation, one of which is free from the constraint that the holdup time must be constant. In addition, the above derivation results in a very simple numerical solution of the various equations of gradient elution under any gradient profile. The theory was tested using eight catechol-related solutes in mobile phases modified with methanol, acetonitrile, or 2-propanol. It was found to be a satisfactory prediction of solute gradient retention behavior even if we used a simple linear description for the isocratic elution of these solutes.

Metric versus observable operator representation, higher spin models

NASA Astrophysics Data System (ADS)

Fring, Andreas; Frith, Thomas

2018-02-01

We elaborate further on the metric representation that is obtained by transferring the time-dependence from a Hermitian Hamiltonian to the metric operator in a related non-Hermitian system. We provide further insight into the procedure on how to employ the time-dependent Dyson relation and the quasi-Hermiticity relation to solve time-dependent Hermitian Hamiltonian systems. By solving both equations separately we argue here that it is in general easier to solve the former. We solve the mutually related time-dependent Schrödinger equation for a Hermitian and non-Hermitian spin 1/2, 1 and 3/2 model with time-independent and time-dependent metric, respectively. In all models the overdetermined coupled system of equations for the Dyson map can be decoupled algebraic manipulations and reduces to simple linear differential equations and an equation that can be converted into the non-linear Ermakov-Pinney equation.

Individualized Math Problems in Simple Equations. Oregon Vo-Tech Mathematics Problem Sets.

ERIC Educational Resources Information Center

Cosler, Norma, Ed.

This is one of eighteen sets of individualized mathematics problems developed by the Oregon Vo-Tech Math Project. Each of these problem packages is organized around a mathematical topic and contains problems related to diverse vocations. Solutions are provided for all problems. Problems in this volume require solution of linear equations, systems…

Keep Your Distance! Using Second-Order Ordinary Differential Equations to Model Traffic Flow

ERIC Educational Resources Information Center

McCartney, Mark

2004-01-01

A simple mathematical model for how vehicles follow each other along a stretch of road is presented. The resulting linear second-order differential equation with constant coefficients is solved and interpreted. The model can be used as an application of solution techniques taught at first-year undergraduate level and as a motivator to encourage…

A new solution procedure for a nonlinear infinite beam equation of motion

NASA Astrophysics Data System (ADS)

Jang, T. S.

2016-10-01

Our goal of this paper is of a purely theoretical question, however which would be fundamental in computational partial differential equations: Can a linear solution-structure for the equation of motion for an infinite nonlinear beam be directly manipulated for constructing its nonlinear solution? Here, the equation of motion is modeled as mathematically a fourth-order nonlinear partial differential equation. To answer the question, a pseudo-parameter is firstly introduced to modify the equation of motion. And then, an integral formalism for the modified equation is found here, being taken as a linear solution-structure. It enables us to formulate a nonlinear integral equation of second kind, equivalent to the original equation of motion. The fixed point approach, applied to the integral equation, results in proposing a new iterative solution procedure for constructing the nonlinear solution of the original beam equation of motion, which consists luckily of just the simple regular numerical integration for its iterative process; i.e., it appears to be fairly simple as well as straightforward to apply. A mathematical analysis is carried out on both natures of convergence and uniqueness of the iterative procedure by proving a contractive character of a nonlinear operator. It follows conclusively,therefore, that it would be one of the useful nonlinear strategies for integrating the equation of motion for a nonlinear infinite beam, whereby the preceding question may be answered. In addition, it may be worth noticing that the pseudo-parameter introduced here has double roles; firstly, it connects the original beam equation of motion with the integral equation, second, it is related with the convergence of the iterative method proposed here.

Biological electric fields and rate equations for biophotons.

PubMed

Alvermann, M; Srivastava, Y N; Swain, J; Widom, A

2015-04-01

Biophoton intensities depend upon the squared modulus of the electric field. Hence, we first make some general estimates about the inherent electric fields within various biosystems. Generally, these intensities do not follow a simple exponential decay law. After a brief discussion on the inapplicability of a linear rate equation that leads to strict exponential decay, we study other, nonlinear rate equations that have been successfully used for biosystems along with their physical origins when available.

Macroscopic dielectric function within time-dependent density functional theory—Real time evolution versus the Casida approach

NASA Astrophysics Data System (ADS)

Sander, Tobias; Kresse, Georg

2017-02-01

Linear optical properties can be calculated by solving the time-dependent density functional theory equations. Linearization of the equation of motion around the ground state orbitals results in the so-called Casida equation, which is formally very similar to the Bethe-Salpeter equation. Alternatively one can determine the spectral functions by applying an infinitely short electric field in time and then following the evolution of the electron orbitals and the evolution of the dipole moments. The long wavelength response function is then given by the Fourier transformation of the evolution of the dipole moments in time. In this work, we compare the results and performance of these two approaches for the projector augmented wave method. To allow for large time steps and still rely on a simple difference scheme to solve the differential equation, we correct for the errors in the frequency domain, using a simple analytic equation. In general, we find that both approaches yield virtually indistinguishable results. For standard density functionals, the time evolution approach is, with respect to the computational performance, clearly superior compared to the solution of the Casida equation. However, for functionals including nonlocal exchange, the direct solution of the Casida equation is usually much more efficient, even though it scales less beneficial with the system size. We relate this to the large computational prefactors in evaluating the nonlocal exchange, which renders the time evolution algorithm fairly inefficient.

Rational solutions of CYBE for simple compact real Lie algebras

NASA Astrophysics Data System (ADS)

Pop, Iulia; Stolin, Alexander

2007-04-01

In [A.A. Stolin, On rational solutions of Yang-Baxter equation for sl(n), Math. Scand. 69 (1991) 57-80; A.A. Stolin, On rational solutions of Yang-Baxter equation. Maximal orders in loop algebra, Comm. Math. Phys. 141 (1991) 533-548; A. Stolin, A geometrical approach to rational solutions of the classical Yang-Baxter equation. Part I, in: Walter de Gruyter & Co. (Ed.), Symposia Gaussiana, Conf. Alg., Berlin, New York, 1995, pp. 347-357] a theory of rational solutions of the classical Yang-Baxter equation for a simple complex Lie algebra g was presented. We discuss this theory for simple compact real Lie algebras g. We prove that up to gauge equivalence all rational solutions have the form X(u,v)={Ω}/{u-v}+t1∧t2+⋯+t∧t2n, where Ω denotes the quadratic Casimir element of g and {ti} are linearly independent elements in a maximal torus t of g. The quantization of these solutions is also emphasized.

The Pendulum: A Paradigm for the Linear Oscillator

ERIC Educational Resources Information Center

Newburgh, Ronald

2004-01-01

The simple pendulum is a model for the linear oscillator. The usual mathematical treatment of the problem begins with a differential equation that one solves with the techniques of the differential calculus, a formal process that tends to obscure the physics. In this paper we begin with a kinematic description of the motion obtained by experiment…

Quantized Algebra I Texts

NASA Astrophysics Data System (ADS)

DeBuvitz, William

2014-03-01

I am a volunteer reader at the Princeton unit of "Learning Ally" (formerly "Recording for the Blind & Dyslexic") and I recently discovered that high school students are introduced to the concept of quantization well before they take chemistry and physics. For the past few months I have been reading onto computer files a popular Algebra I textbook, and I was surprised and dismayed by how it treated simultaneous equations and quadratic equations. The coefficients are always simple integers in examples and exercises, even when they are related to physics. This is probably a good idea when these topics are first presented to the students. It makes it easy to solve simultaneous equations by the method of elimination of a variable. And it makes it easy to solve some quadratic equations by factoring. The textbook also discusses the method of substitution for linear equations and the use of the quadratic formula, but only with simple integers.

Modeling of aircraft unsteady aerodynamic characteristics. Part 1: Postulated models

NASA Technical Reports Server (NTRS)

Klein, Vladislav; Noderer, Keith D.

1994-01-01

A short theoretical study of aircraft aerodynamic model equations with unsteady effects is presented. The aerodynamic forces and moments are expressed in terms of indicial functions or internal state variables. The first representation leads to aircraft integro-differential equations of motion; the second preserves the state-space form of the model equations. The formulations of unsteady aerodynamics is applied in two examples. The first example deals with a one-degree-of-freedom harmonic motion about one of the aircraft body axes. In the second example, the equations for longitudinal short-period motion are developed. In these examples, only linear aerodynamic terms are considered. The indicial functions are postulated as simple exponentials and the internal state variables are governed by linear, time-invariant, first-order differential equations. It is shown that both approaches to the modeling of unsteady aerodynamics lead to identical models.

Generalized Lagrange Jacobi Gauss-Lobatto (GLJGL) Collocation Method for Solving Linear and Nonlinear Fokker-Planck Equations

NASA Astrophysics Data System (ADS)

Parand, K.; Latifi, S.; Moayeri, M. M.; Delkhosh, M.

2018-05-01

In this study, we have constructed a new numerical approach for solving the time-dependent linear and nonlinear Fokker-Planck equations. In fact, we have discretized the time variable with Crank-Nicolson method and for the space variable, a numerical method based on Generalized Lagrange Jacobi Gauss-Lobatto (GLJGL) collocation method is applied. It leads to in solving the equation in a series of time steps and at each time step, the problem is reduced to a problem consisting of a system of algebraic equations that greatly simplifies the problem. One can observe that the proposed method is simple and accurate. Indeed, one of its merits is that it is derivative-free and by proposing a formula for derivative matrices, the difficulty aroused in calculation is overcome, along with that it does not need to calculate the General Lagrange basis and matrices; they have Kronecker property. Linear and nonlinear Fokker-Planck equations are given as examples and the results amply demonstrate that the presented method is very valid, effective, reliable and does not require any restrictive assumptions for nonlinear terms.

Finding all solutions of nonlinear equations using the dual simplex method

NASA Astrophysics Data System (ADS)

Yamamura, Kiyotaka; Fujioka, Tsuyoshi

2003-03-01

Recently, an efficient algorithm has been proposed for finding all solutions of systems of nonlinear equations using linear programming. This algorithm is based on a simple test (termed the LP test) for nonexistence of a solution to a system of nonlinear equations using the dual simplex method. In this letter, an improved version of the LP test algorithm is proposed. By numerical examples, it is shown that the proposed algorithm could find all solutions of a system of 300 nonlinear equations in practical computation time.

Simple derivation of the Lindblad equation

NASA Astrophysics Data System (ADS)

Pearle, Philip

2012-07-01

The Lindblad equation is an evolution equation for the density matrix in quantum theory. It is the general linear, Markovian, form which ensures that the density matrix is Hermitian, trace 1, positive and completely positive. Some elementary examples of the Lindblad equation are given. The derivation of the Lindblad equation presented here is ‘simple’ in that all it uses is the expression of a Hermitian matrix in terms of its orthonormal eigenvectors and real eigenvalues. Thus, it is appropriate for students who have learned the algebra of quantum theory. Where helpful, arguments are first given in a two-dimensional Hilbert space.

ΛCDM Cosmology for Astronomers

NASA Astrophysics Data System (ADS)

Condon, J. J.; Matthews, A. M.

2018-07-01

The homogeneous, isotropic, and flat ΛCDM universe favored by observations of the cosmic microwave background can be described using only Euclidean geometry, locally correct Newtonian mechanics, and the basic postulates of special and general relativity. We present simple derivations of the most useful equations connecting astronomical observables (redshift, flux density, angular diameter, brightness, local space density, ...) with the corresponding intrinsic properties of distant sources (lookback time, distance, spectral luminosity, linear size, specific intensity, source counts, ...). We also present an analytic equation for lookback time that is accurate within 0.1% for all redshifts z. The exact equation for comoving distance is an elliptic integral that must be evaluated numerically, but we found a simple approximation with errors <0.2% for all redshifts up to z ≈ 50.

Fluid equations with nonlinear wave-particle resonances^

NASA Astrophysics Data System (ADS)

Mattor, Nathan

1997-11-01

We have derived fluid equations that include linear and nonlinear wave-particle resonance effects. This greatly extends previous ``Landau-fluid'' closures, which include linear Landau damping. (G.W. Hammett and F.W. Perkins, Phys. Rev. Lett. 64,) 3019 (1990).^, (Z. Chang and J. D. Callen, Phys. Fluids B 4,) 1167 (1992). The new fluid equations are derived with no approximation regarding nonlinear kinetic interaction, and so additionally include numerous nonlinear kinetic effects. The derivation starts with the electrostatic drift kinetic equation for simplicity, with a Maxwellian distribution function. Fluid closure is accomplished through a simple integration trick applied to the drift kinetic equation, using the property that the nth moment of Maxwellian distribution is related to the nth derivative. The result is a compact closure term appearing in the highest moment equation, a term which involves a plasma dispersion function of the electrostatic field and its derivatives. The new term reduces to the linear closures in appropriate limits, so both approaches retain linear Landau damping. But the nonlinearly closed equations have additional desirable properties. Unlike linear closures, the nonlinear closure retains the time-reversibility of the original kinetic equation. We have shown directly that the nonlinear closure retains at least two nonlinear resonance effects: wave-particle trapping and Compton scattering. Other nonlinear kinetic effects are currently under investigation. The new equations correct two previous discrepancies between kinetic and Landau-fluid predictions, including a propagator discrepancy (N. Mattor, Phys. Fluids B 4,) 3952 (1992). and a numerical discrepancy for the 3-mode shearless bounded slab ITG problem. (S. E. Parker et al.), Phys. Plasmas 1, 1461 (1994). ^* In collaboration with S. E. Parker, Department of Physics, University of Colorado, Boulder. ^ Work performed at LLNL under DoE contract No. W7405-ENG-48.

The entrainment matrix of a superfluid nucleon mixture at finite temperatures

NASA Astrophysics Data System (ADS)

Leinson, Lev B.

2018-06-01

It is considered a closed system of non-linear equations for the entrainment matrix of a non-relativistic mixture of superfluid nucleons at arbitrary temperatures below the onset of neutron superfluidity, which takes into account the essential dependence of the superfluid energy gap in the nucleon spectra on the velocities of superfluid flows. It is assumed that the protons condense into the isotropic 1S0 state, and the neutrons are paired into the spin-triplet 3P2 state. It is derived an analytic solution to the non-linear equations for the entrainment matrix under temperatures just below the critical value for the neutron superfluidity onset. In general case of an arbitrary temperature of the superfluid mixture the non-linear equations are solved numerically and fitted by simple formulas convenient for a practical use with an arbitrary set of the Landau parameters.

On the derivation of linear irreversible thermodynamics for classical fluids

PubMed Central

Theodosopulu, M.; Grecos, A.; Prigogine, I.

1978-01-01

We consider the microscopic derivation of the linearized hydrodynamic equations for an arbitrary simple fluid. Our discussion is based on the concept of hydrodynamical modes, and use is made of the ideas and methods of the theory of subdynamics. We also show that this analysis leads to the Gibbs relation for the entropy of the system. PMID:16592516

Squeeze-film dampers for turbomachinery stabilization

NASA Technical Reports Server (NTRS)

Mclean, L. J.; Hahn, E. J.

1984-01-01

A technique for investigating the stability and damping present in centrally preloaded radially symmetric multi-mass flexible rotor bearing systems is presented. In general, one needs to find the eigenvalues of the linearized perturbation equations, though zero frequency stability maps may be found by solving as many simultaneous non-linear equations as there are dampers; and in the case of a single damper, such maps may be found directly, regardless of the number of degrees of freedom. The technique is illustrated for a simple symmetric four degree of freedom flexible rotor with an unpressurized damper. This example shows that whereas zero frequency stability maps are likely to prove to be a simple way to delineate multiple solution possibilities, they do not provide full stability information. Further, particularly for low bearing parameters, the introduction of an unpressurized squeeze film damper may promote instability in an otherwise stable system.

HESS Opinions: Linking Darcy's equation to the linear reservoir

NASA Astrophysics Data System (ADS)

Savenije, Hubert H. G.

2018-03-01

In groundwater hydrology, two simple linear equations exist describing the relation between groundwater flow and the gradient driving it: Darcy's equation and the linear reservoir. Both equations are empirical and straightforward, but work at different scales: Darcy's equation at the laboratory scale and the linear reservoir at the watershed scale. Although at first sight they appear similar, it is not trivial to upscale Darcy's equation to the watershed scale without detailed knowledge of the structure or shape of the underlying aquifers. This paper shows that these two equations, combined by the water balance, are indeed identical provided there is equal resistance in space for water entering the subsurface network. This implies that groundwater systems make use of an efficient drainage network, a mostly invisible pattern that has evolved over geological timescales. This drainage network provides equally distributed resistance for water to access the system, connecting the active groundwater body to the stream, much like a leaf is organized to provide all stomata access to moisture at equal resistance. As a result, the timescale of the linear reservoir appears to be inversely proportional to Darcy's conductance, the proportionality being the product of the porosity and the resistance to entering the drainage network. The main question remaining is which physical law lies behind pattern formation in groundwater systems, evolving in a way that resistance to drainage is constant in space. But that is a fundamental question that is equally relevant for understanding the hydraulic properties of leaf veins in plants or of blood veins in animals.

EnviroLand: A Simple Computer Program for Quantitative Stream Assessment.

ERIC Educational Resources Information Center

Dunnivant, Frank; Danowski, Dan; Timmens-Haroldson, Alice; Newman, Meredith

2002-01-01

Introduces the Enviroland computer program which features lab simulations of theoretical calculations for quantitative analysis and environmental chemistry, and fate and transport models. Uses the program to demonstrate the nature of linear and nonlinear equations. (Author/YDS)

A simple finite element method for linear hyperbolic problems

DOE PAGES

Mu, Lin; Ye, Xiu

2017-09-14

Here, we introduce a simple finite element method for solving first order hyperbolic equations with easy implementation and analysis. Our new method, with a symmetric, positive definite system, is designed to use discontinuous approximations on finite element partitions consisting of arbitrary shape of polygons/polyhedra. Error estimate is established. Extensive numerical examples are tested that demonstrate the robustness and flexibility of the method.

A simple finite element method for linear hyperbolic problems

DOE Office of Scientific and Technical Information (OSTI.GOV)

Mu, Lin; Ye, Xiu

Here, we introduce a simple finite element method for solving first order hyperbolic equations with easy implementation and analysis. Our new method, with a symmetric, positive definite system, is designed to use discontinuous approximations on finite element partitions consisting of arbitrary shape of polygons/polyhedra. Error estimate is established. Extensive numerical examples are tested that demonstrate the robustness and flexibility of the method.

On the Singularity Structure of WKB Solution of the Boosted Whittaker Equation: its Relevance to Resurgent Functions with Essential Singularities

NASA Astrophysics Data System (ADS)

Kamimoto, Shingo; Kawai, Takahiro; Koike, Tatsuya

2016-12-01

Inspired by the symbol calculus of linear differential operators of infinite order applied to the Borel transformed WKB solutions of simple-pole type equation [Kamimoto et al. (RIMS Kôkyûroku Bessatsu B 52:127-146, 2014)], which is summarized in Section 1, we introduce in Section 2 the space of simple resurgent functions depending on a parameter with an infra-exponential type growth order, and then we define the assigning operator A which acts on the space and produces resurgent functions with essential singularities. In Section 3, we apply the operator A to the Borel transforms of the Voros coefficient and its exponentiation for the Whittaker equation with a large parameter so that we may find the Borel transforms of the Voros coefficient and its exponentiation for the boosted Whittaker equation with a large parameter. In Section 4, we use these results to find the explicit form of the alien derivatives of the Borel transformed WKB solutions of the boosted Whittaker equation with a large parameter. The results in this paper manifest the importance of resurgent functions with essential singularities in developing the exact WKB analysis, the WKB analysis based on the resurgent function theory. It is also worth emphasizing that the concrete form of essential singularities we encounter is expressed by the linear differential operators of infinite order.

A simple mechanical system mimicking phase transitions in a one-dimensional medium

NASA Astrophysics Data System (ADS)

Charru, François

1997-11-01

We study a simple mechanical oscillator the bifurcations of which illustrate first- and second-order phase transitions. The phase diagram of the oscillator exhibits a coexistence curve. This curve ends at a critical point, where three critical exponents can be defined. A metronome may be used to illustrate the main results. We then consider a linear array of such oscillators with elastic coupling, which is governed by the damped Klein - Gordon equation. The classical solutions of this equation, such as fronts propagating in an unstable or in a metastable state, can be guessed at and discussed from the point of view of a mechanical model.

PHYSICS REQUIRES A SIMPLE LOW MACH NUMBER FLOW TO BE COMPRESSIBLE

EPA Science Inventory

Radial, laminar, plane, low velocity flow represents the simplest, non-linear fluid dynamics problem. Ostensibly this apparently trivial flow could be solved using the incompressible Navier-Stokes equations, universally believed to be adequate for such problems. Most researchers ...

The RC Circuit--A Multipurpose Laboratory Experiment.

ERIC Educational Resources Information Center

Wood, Herbert T.

1993-01-01

Describes an experiment that demonstrates the use of Kirchoff's rules in the analysis of electrical circuits. The experiment also involves the solution of a linear nonhomogeneous differential equation that is slightly different from the standard one for the simple RC circuit. (ZWH)

The Jeffcott equations in nonlinear rotordynamics

NASA Technical Reports Server (NTRS)

Zalik, R. A.

1987-01-01

The Jeffcott equations are a system of coupled differential equations representing the behavior of a rotating shaft. This is a simple model which allows investigation of the basic dynamic behavior of rotating machinery. Nolinearities can be introduced by taking into consideration deadband, side force, and rubbing, among others. The properties of the solutions of the Jeffcott equations with deadband are studied. In particular, it is shown how bounds for the solution of these equations can be obtained from bounds for the solutions of the linearized equations. By studying the behavior of the Fourier transforms of the solutions, we are also able to predict the onset of destructive vibrations. These conclusions are verified by means of numerical solutions of the equations, and of power spectrum density (PSD) plots. This study offers insight into a possible detection method to determine pump stability margins during flight and hot fire tests, and was motivated by the need to explain a phenomenon observed in the development phase of the cryogenic pumps of the Space Shuttle, during hot fire ground testing; namely, the appearance of vibrations at frequencies that could not be accounted for by means of linear models.

An interative solution of an integral equation for radiative transfer by using variational technique

NASA Technical Reports Server (NTRS)

Yoshikawa, K. K.

1973-01-01

An effective iterative technique is introduced to solve a nonlinear integral equation frequently associated with radiative transfer problems. The problem is formulated in such a way that each step of an iterative sequence requires the solution of a linear integral equation. The advantage of a previously introduced variational technique which utilizes a stepwise constant trial function is exploited to cope with the nonlinear problem. The method is simple and straightforward. Rapid convergence is obtained by employing a linear interpolation of the iterative solutions. Using absorption coefficients of the Milne-Eddington type, which are applicable to some planetary atmospheric radiation problems. Solutions are found in terms of temperature and radiative flux. These solutions are presented numerically and show excellent agreement with other numerical solutions.

Multiphysics modeling of non-linear laser-matter interactions for optically active semiconductors

NASA Astrophysics Data System (ADS)

Kraczek, Brent; Kanp, Jaroslaw

Development of photonic devices for sensors and communications devices has been significantly enhanced by computational modeling. We present a new computational method for modelling laser propagation in optically-active semiconductors within the paraxial wave approximation (PWA). Light propagation is modeled using the Streamline-upwind/Petrov-Galerkin finite element method (FEM). Material response enters through the non-linear polarization, which serves as the right-hand side of the FEM calculation. Maxwell's equations for classical light propagation within the PWA can be written solely in terms of the electric field, producing a wave equation that is a form of the advection-diffusion-reaction equations (ADREs). This allows adaptation of the computational machinery developed for solving ADREs in fluid dynamics to light-propagation modeling. The non-linear polarization is incorporated using a flexible framework to enable the use of multiple methods for carrier-carrier interactions (e.g. relaxation-time-based or Monte Carlo) to enter through the non-linear polarization, as appropriate to the material type. We demonstrate using a simple carrier-carrier model approximating the response of GaN. Supported by ARL Materials Enterprise.

The Form of the Solutions of the Linear Integro-Differential Equations of Subsonic Aeroelasticity.

DTIC Science & Technology

1979-09-01

coefficients w (0) are given in Table 3; it V follows that, for T > 0 and (E - K v2) non-singular, the inverse transform of M- ) has the form, using (B-I) V...degree of freedom system by expanding )M- I in the form of equation (35), obtaining its inverse transform using the v -1results of Appendix A and hence...obtaining the inverse transform of M- l . The two-dimensional case, when the characteristic equation has a zero root, is not as simple. * Assuming all

An efficient computer based wavelets approximation method to solve Fuzzy boundary value differential equations

NASA Astrophysics Data System (ADS)

Alam Khan, Najeeb; Razzaq, Oyoon Abdul

2016-03-01

In the present work a wavelets approximation method is employed to solve fuzzy boundary value differential equations (FBVDEs). Essentially, a truncated Legendre wavelets series together with the Legendre wavelets operational matrix of derivative are utilized to convert FB- VDE into a simple computational problem by reducing it into a system of fuzzy algebraic linear equations. The capability of scheme is investigated on second order FB- VDE considered under generalized H-differentiability. Solutions are represented graphically showing competency and accuracy of this method.

Homotopy approach to optimal, linear quadratic, fixed architecture compensation

NASA Technical Reports Server (NTRS)

Mercadal, Mathieu

1991-01-01

Optimal linear quadratic Gaussian compensators with constrained architecture are a sensible way to generate good multivariable feedback systems meeting strict implementation requirements. The optimality conditions obtained from the constrained linear quadratic Gaussian are a set of highly coupled matrix equations that cannot be solved algebraically except when the compensator is centralized and full order. An alternative to the use of general parameter optimization methods for solving the problem is to use homotopy. The benefit of the method is that it uses the solution to a simplified problem as a starting point and the final solution is then obtained by solving a simple differential equation. This paper investigates the convergence properties and the limitation of such an approach and sheds some light on the nature and the number of solutions of the constrained linear quadratic Gaussian problem. It also demonstrates the usefulness of homotopy on an example of an optimal decentralized compensator.

Nonlinear field equations for aligning self-propelled rods.

PubMed

Peshkov, Anton; Aranson, Igor S; Bertin, Eric; Chaté, Hugues; Ginelli, Francesco

2012-12-28

We derive a set of minimal and well-behaved nonlinear field equations describing the collective properties of self-propelled rods from a simple microscopic starting point, the Vicsek model with nematic alignment. Analysis of their linear and nonlinear dynamics shows good agreement with the original microscopic model. In particular, we derive an explicit expression for density-segregated, banded solutions, allowing us to develop a more complete analytic picture of the problem at the nonlinear level.

Observability of discretized partial differential equations

NASA Technical Reports Server (NTRS)

Cohn, Stephen E.; Dee, Dick P.

1988-01-01

It is shown that complete observability of the discrete model used to assimilate data from a linear partial differential equation (PDE) system is necessary and sufficient for asymptotic stability of the data assimilation process. The observability theory for discrete systems is reviewed and applied to obtain simple observability tests for discretized constant-coefficient PDEs. Examples are used to show how numerical dispersion can result in discrete dynamics with multiple eigenvalues, thereby detracting from observability.

On the solution of the complex eikonal equation in acoustic VTI media: A perturbation plus optimization scheme

NASA Astrophysics Data System (ADS)

Huang, Xingguo; Sun, Jianguo; Greenhalgh, Stewart

2018-04-01

We present methods for obtaining numerical and analytic solutions of the complex eikonal equation in inhomogeneous acoustic VTI media (transversely isotropic media with a vertical symmetry axis). The key and novel point of the method for obtaining numerical solutions is to transform the problem of solving the highly nonlinear acoustic VTI eikonal equation into one of solving the relatively simple eikonal equation for the background (isotropic) medium and a system of linear partial differential equations. Specifically, to obtain the real and imaginary parts of the complex traveltime in inhomogeneous acoustic VTI media, we generalize a perturbation theory, which was developed earlier for solving the conventional real eikonal equation in inhomogeneous anisotropic media, to the complex eikonal equation in such media. After the perturbation analysis, we obtain two types of equations. One is the complex eikonal equation for the background medium and the other is a system of linearized partial differential equations for the coefficients of the corresponding complex traveltime formulas. To solve the complex eikonal equation for the background medium, we employ an optimization scheme that we developed for solving the complex eikonal equation in isotropic media. Then, to solve the system of linearized partial differential equations for the coefficients of the complex traveltime formulas, we use the finite difference method based on the fast marching strategy. Furthermore, by applying the complex source point method and the paraxial approximation, we develop the analytic solutions of the complex eikonal equation in acoustic VTI media, both for the isotropic and elliptical anisotropic background medium. Our numerical results demonstrate the effectiveness of our derivations and illustrate the influence of the beam widths and the anisotropic parameters on the complex traveltimes.

Assessment and correction of skinfold thickness equations in estimating body fat in children with cerebral palsy.

PubMed

Gurka, Matthew J; Kuperminc, Michelle N; Busby, Marjorie G; Bennis, Jacey A; Grossberg, Richard I; Houlihan, Christine M; Stevenson, Richard D; Henderson, Richard C

2010-02-01

To assess the accuracy of skinfold equations in estimating percentage body fat in children with cerebral palsy (CP), compared with assessment of body fat from dual energy X-ray absorptiometry (DXA). Data were collected from 71 participants (30 females, 41 males) with CP (Gross Motor Function Classification System [GMFCS] levels I-V) between the ages of 8 and 18 years. Estimated percentage body fat was computed using established (Slaughter) equations based on the triceps and subscapular skinfolds. A linear model was fitted to assess the use of a simple correction to these equations for children with CP. Slaughter's equations consistently underestimated percentage body fat (mean difference compared with DXA percentage body fat -9.6/100 [SD 6.2]; 95% confidence interval [CI] -11.0 to -8.1). New equations were developed in which a correction factor was added to the existing equations based on sex, race, GMFCS level, size, and pubertal status. These corrected equations for children with CP agree better with DXA (mean difference 0.2/100 [SD=4.8]; 95% CI -1.0 to 1.3) than existing equations. A simple correction factor to commonly used equations substantially improves the ability to estimate percentage body fat from two skinfold measures in children with CP.

Cryptography: Cracking Codes.

ERIC Educational Resources Information Center

Myerscough, Don; And Others

1996-01-01

Describes an activity whose objectives are to encode and decode messages using linear functions and their inverses; to use modular arithmetic, including use of the reciprocal for simple equation solving; to analyze patterns and make and test conjectures; to communicate procedures and algorithms; and to use problem-solving strategies. (ASK)

A simple strategy for varying the restart parameter in GMRES(m)

DOE Office of Scientific and Technical Information (OSTI.GOV)

Baker, A H; Jessup, E R; Kolev, T V

2007-10-02

When solving a system of linear equations with the restarted GMRES method, a fixed restart parameter is typically chosen. We present numerical experiments that demonstrate the beneficial effects of changing the value of the restart parameter in each restart cycle on the total time to solution. We propose a simple strategy for varying the restart parameter and provide some heuristic explanations for its effectiveness based on analysis of the symmetric case.

The Quantum Arnold Transformation for the damped harmonic oscillator: from the Caldirola-Kanai model toward the Bateman model

NASA Astrophysics Data System (ADS)

López-Ruiz, F. F.; Guerrero, J.; Aldaya, V.; Cossío, F.

2012-08-01

Using a quantum version of the Arnold transformation of classical mechanics, all quantum dynamical systems whose classical equations of motion are non-homogeneous linear second-order ordinary differential equations (LSODE), including systems with friction linear in velocity such as the damped harmonic oscillator, can be related to the quantum free-particle dynamical system. This implies that symmetries and simple computations in the free particle can be exported to the LSODE-system. The quantum Arnold transformation is given explicitly for the damped harmonic oscillator, and an algebraic connection between the Caldirola-Kanai model for the damped harmonic oscillator and the Bateman system will be sketched out.

Stabilizing skateboard speed-wobble with reflex delay.

PubMed

Varszegi, Balazs; Takacs, Denes; Stepan, Gabor; Hogan, S John

2016-08-01

A simple mechanical model of the skateboard-skater system is analysed, in which the effect of human control is considered by means of a linear proportional-derivative (PD) controller with delay. The equations of motion of this non-holonomic system are neutral delay-differential equations. A linear stability analysis of the rectilinear motion is carried out analytically. It is shown how to vary the control gains with respect to the speed of the skateboard to stabilize the uniform motion. The critical reflex delay of the skater is determined as the function of the speed. Based on this analysis, we present an explanation for the linear instability of the skateboard-skater system at high speed. Moreover, the advantages of standing ahead of the centre of the board are demonstrated from the viewpoint of reflex delay and control gain sensitivity. © 2016 The Author(s).

Acoustic equations of state for simple lattice Boltzmann velocity sets.

PubMed

Viggen, Erlend Magnus

2014-07-01

The lattice Boltzmann (LB) method typically uses an isothermal equation of state. This is not sufficient to simulate a number of acoustic phenomena where the equation of state cannot be approximated as linear and constant. However, it is possible to implement variable equations of state by altering the LB equilibrium distribution. For simple velocity sets with velocity components ξ(iα)∈(-1,0,1) for all i, these equilibria necessarily cause error terms in the momentum equation. These error terms are shown to be either correctable or negligible at the cost of further weakening the compressibility. For the D1Q3 velocity set, such an equilibrium distribution is found and shown to be unique. Its sound propagation properties are found for both forced and free waves, with some generality beyond D1Q3. Finally, this equilibrium distribution is applied to a nonlinear acoustics simulation where both mechanisms of nonlinearity are simulated with good results. This represents an improvement on previous such simulations and proves that the compressibility of the method is still sufficiently strong even for nonlinear acoustics.

Phytoplankton productivity in relation to light intensity: A simple equation

USGS Publications Warehouse

Peterson, D.H.; Perry, M.J.; Bencala, K.E.; Talbot, M.C.

1987-01-01

A simple exponential equation is used to describe photosynthetic rate as a function of light intensity for a variety of unicellular algae and higher plants where photosynthesis is proportional to (1-e-??1). The parameter ?? (=Ik-1) is derived by a simultaneous curve-fitting method, where I is incident quantum-flux density. The exponential equation is tested against a wide range of data and is found to adequately describe P vs. I curves. The errors associated with photosynthetic parameters are calculated. A simplified statistical model (Poisson) of photon capture provides a biophysical basis for the equation and for its ability to fit a range of light intensities. The exponential equation provides a non-subjective simultaneous curve fitting estimate for photosynthetic efficiency (a) which is less ambiguous than subjective methods: subjective methods assume that a linear region of the P vs. I curve is readily identifiable. Photosynthetic parameters ?? and a are used widely in aquatic studies to define photosynthesis at low quantum flux. These parameters are particularly important in estuarine environments where high suspended-material concentrations and high diffuse-light extinction coefficients are commonly encountered. ?? 1987.

Asymptotic Bounds for Solutions to a System of Damped Integrodifferential Equations of Electromagnetic Theory.

DTIC Science & Technology

1979-05-28

of integrodiffereiTaT~- equations governs the evolution of the components of the electric displacement field in a simple class of rigid holohedral...vacuum so that j — C and J~ — c E , H = B. In0 [3] and [4] this author has treated the evolution equations associated with the Maxwell—Hopkinsofl...F .~~~~~~‘ r - — ~~~~~ ’~~~ “~ I I be viewed as a linearized version of a more general theory introduced by Volterra in 1912 [5] to treat the case

[Relation between Body Height and Combined Length of Manubrium and Mesosternum of Sternum Measured by CT-VRT in Southwest Han Population].

PubMed

Luo, Ying-zhen; Tu, Meng; Fan, Fei; Zheng, Jie-qian; Yang, Ming; Li, Tao; Zhang, Kui; Deng, Zhen-hua

2015-06-01

To establish the linear regression equation between body height and combined length of manubrium and mesostenum of sternum measured by CT volume rendering technique (CT-VRT) in southwest Han population. One hundred and sixty subjects, including 80 males and 80 females were selected from southwest Han population for routine CT-VRT (reconstruction thickness 1 mm) examination. The lengths of both manubrium and mesosternum were recorded, and the combined length of manubrium and mesosternum was equal to the algebraic sum of them. The sex-specific linear regression equations between the combined length of manubrium and mesosternum and the real body height of each subject were deduced. The sex-specific simple linear regression equations between the combined length of manubrium and mesostenum (x3) and body height (y) were established (male: y = 135.000+2.118 x3 and female: y = 120.790+2.808 x3). Both equations showed statistical significance (P < 0.05) with a 100% predictive accuracy. CT-VRT is an effective method for measurement of the index of sternum. The combined length of manubrium and mesosternum from CT-VRT can be used for body height estimation in southwest Han population.

A direct method of extracting surface recombination velocity from an electron beam induced current line scan

NASA Astrophysics Data System (ADS)

Ong, Vincent K. S.

1998-04-01

The extraction of diffusion length and surface recombination velocity in a semiconductor with the use of an electron beam induced current line scan has traditionally been done by fitting the line scan into complicated theoretical equations. It was recently shown that a much simpler equation is sufficient for the extraction of diffusion length. The linearization coefficient is the only variable that is needed to be adjusted in the curve fitting process. However, complicated equations are still necessary for the extraction of surface recombination velocity. It is shown in this article that it is indeed possible to extract surface recombination velocity with a simple equation, using only one variable, the linearization coefficient. An intuitive feel for the reason behind the method was discussed. The accuracy of the method was verified with the use of three-dimensional computer simulation, and was found to be even slightly better than that of the best existing method.

Interpolation problem for the solutions of linear elasticity equations based on monogenic functions

NASA Astrophysics Data System (ADS)

Grigor'ev, Yuri; Gürlebeck, Klaus; Legatiuk, Dmitrii

2017-11-01

Interpolation is an important tool for many practical applications, and very often it is beneficial to interpolate not only with a simple basis system, but rather with solutions of a certain differential equation, e.g. elasticity equation. A typical example for such type of interpolation are collocation methods widely used in practice. It is known, that interpolation theory is fully developed in the framework of the classical complex analysis. However, in quaternionic analysis, which shows a lot of analogies to complex analysis, the situation is more complicated due to the non-commutative multiplication. Thus, a fundamental theorem of algebra is not available, and standard tools from linear algebra cannot be applied in the usual way. To overcome these problems, a special system of monogenic polynomials the so-called Pseudo Complex Polynomials, sharing some properties of complex powers, is used. In this paper, we present an approach to deal with the interpolation problem, where solutions of elasticity equations in three dimensions are used as an interpolation basis.

A simple method to design non-collision relative orbits for close spacecraft formation flying

NASA Astrophysics Data System (ADS)

Jiang, Wei; Li, JunFeng; Jiang, FangHua; Bernelli-Zazzera, Franco

2018-05-01

A set of linearized relative motion equations of spacecraft flying on unperturbed elliptical orbits are specialized for particular cases, where the leader orbit is circular or equatorial. Based on these extended equations, we are able to analyze the relative motion regulation between a pair of spacecraft flying on arbitrary unperturbed orbits with the same semi-major axis in close formation. Given the initial orbital elements of the leader, this paper presents a simple way to design initial relative orbital elements of close spacecraft with the same semi-major axis, thus preventing collision under non-perturbed conditions. Considering the mean influence of J 2 perturbation, namely secular J 2 perturbation, we derive the mean derivatives of orbital element differences, and then expand them to first order. Thus the first order expansion of orbital element differences can be added to the relative motion equations for further analysis. For a pair of spacecraft that will never collide under non-perturbed situations, we present a simple method to determine whether a collision will occur when J 2 perturbation is considered. Examples are given to prove the validity of the extended relative motion equations and to illustrate how the methods presented can be used. The simple method for designing initial relative orbital elements proposed here could be helpful to the preliminary design of the relative orbital elements between spacecraft in a close formation, when collision avoidance is necessary.

Dirac delta representation by exact parametric equations.. Application to impulsive vibration systems

NASA Astrophysics Data System (ADS)

Chicurel-Uziel, Enrique

2007-08-01

A pair of closed parametric equations are proposed to represent the Heaviside unit step function. Differentiating the step equations results in two additional parametric equations, that are also hereby proposed, to represent the Dirac delta function. These equations are expressed in algebraic terms and are handled by means of elementary algebra and elementary calculus. The proposed delta representation complies exactly with the values of the definition. It complies also with the sifting property and the requisite unit area and its Laplace transform coincides with the most general form given in the tables. Furthermore, it leads to a very simple method of solution of impulsive vibrating systems either linear or belonging to a large class of nonlinear problems. Two example solutions are presented.

Bounding solutions of geometrically nonlinear viscoelastic problems

NASA Technical Reports Server (NTRS)

Stubstad, J. M.; Simitses, G. J.

1985-01-01

Integral transform techniques, such as the Laplace transform, provide simple and direct methods for solving viscoelastic problems formulated within a context of linear material response and using linear measures for deformation. Application of the transform operator reduces the governing linear integro-differential equations to a set of algebraic relations between the transforms of the unknown functions, the viscoelastic operators, and the initial and boundary conditions. Inversion either directly or through the use of the appropriate convolution theorem, provides the time domain response once the unknown functions have been expressed in terms of sums, products or ratios of known transforms. When exact inversion is not possible approximate techniques may provide accurate results. The overall problem becomes substantially more complex when nonlinear effects must be included. Situations where a linear material constitutive law can still be productively employed but where the magnitude of the resulting time dependent deformations warrants the use of a nonlinear kinematic analysis are considered. The governing equations will be nonlinear integro-differential equations for this class of problems. Thus traditional as well as approximate techniques, such as cited above, cannot be employed since the transform of a nonlinear function is not explicitly expressible.

Bounding solutions of geometrically nonlinear viscoelastic problems

NASA Technical Reports Server (NTRS)

Stubstad, J. M.; Simitses, G. J.

1986-01-01

Integral transform techniques, such as the Laplace transform, provide simple and direct methods for solving viscoelastic problems formulated within a context of linear material response and using linear measures for deformation. Application of the transform operator reduces the governing linear integro-differential equations to a set of algebraic relations between the transforms of the unknown functions, the viscoelastic operators, and the initial and boundary conditions. Inversion either directly or through the use of the appropriate convolution theorem, provides the time domain response once the unknown functions have been expressed in terms of sums, products or ratios of known transforms. When exact inversion is not possible approximate techniques may provide accurate results. The overall problem becomes substantially more complex when nonlinear effects must be included. Situations where a linear material constitutive law can still be productively employed but where the magnitude of the resulting time dependent deformations warrants the use of a nonlinear kinematic analysis are considered. The governing equations will be nonlinear integro-differential equations for this class of problems. Thus traditional as well as approximate techniques, such as cited above, cannot be employed since the transform of a nonlinear function is not explicitly expressible.

Experimentally validated multiphysics computational model of focusing and shock wave formation in an electromagnetic lithotripter.

PubMed

Fovargue, Daniel E; Mitran, Sorin; Smith, Nathan B; Sankin, Georgy N; Simmons, Walter N; Zhong, Pei

2013-08-01

A multiphysics computational model of the focusing of an acoustic pulse and subsequent shock wave formation that occurs during extracorporeal shock wave lithotripsy is presented. In the electromagnetic lithotripter modeled in this work the focusing is achieved via a polystyrene acoustic lens. The transition of the acoustic pulse through the solid lens is modeled by the linear elasticity equations and the subsequent shock wave formation in water is modeled by the Euler equations with a Tait equation of state. Both sets of equations are solved simultaneously in subsets of a single computational domain within the BEARCLAW framework which uses a finite-volume Riemann solver approach. This model is first validated against experimental measurements with a standard (or original) lens design. The model is then used to successfully predict the effects of a lens modification in the form of an annular ring cut. A second model which includes a kidney stone simulant in the domain is also presented. Within the stone the linear elasticity equations incorporate a simple damage model.

Experimentally validated multiphysics computational model of focusing and shock wave formation in an electromagnetic lithotripter

PubMed Central

Fovargue, Daniel E.; Mitran, Sorin; Smith, Nathan B.; Sankin, Georgy N.; Simmons, Walter N.; Zhong, Pei

2013-01-01

A multiphysics computational model of the focusing of an acoustic pulse and subsequent shock wave formation that occurs during extracorporeal shock wave lithotripsy is presented. In the electromagnetic lithotripter modeled in this work the focusing is achieved via a polystyrene acoustic lens. The transition of the acoustic pulse through the solid lens is modeled by the linear elasticity equations and the subsequent shock wave formation in water is modeled by the Euler equations with a Tait equation of state. Both sets of equations are solved simultaneously in subsets of a single computational domain within the BEARCLAW framework which uses a finite-volume Riemann solver approach. This model is first validated against experimental measurements with a standard (or original) lens design. The model is then used to successfully predict the effects of a lens modification in the form of an annular ring cut. A second model which includes a kidney stone simulant in the domain is also presented. Within the stone the linear elasticity equations incorporate a simple damage model. PMID:23927200

Unsteady Solution of Non-Linear Differential Equations Using Walsh Function Series

NASA Technical Reports Server (NTRS)

Gnoffo, Peter A.

2015-01-01

Walsh functions form an orthonormal basis set consisting of square waves. The discontinuous nature of square waves make the system well suited for representing functions with discontinuities. The product of any two Walsh functions is another Walsh function - a feature that can radically change an algorithm for solving non-linear partial differential equations (PDEs). The solution algorithm of non-linear differential equations using Walsh function series is unique in that integrals and derivatives may be computed using simple matrix multiplication of series representations of functions. Solutions to PDEs are derived as functions of wave component amplitude. Three sample problems are presented to illustrate the Walsh function series approach to solving unsteady PDEs. These include an advection equation, a Burgers equation, and a Riemann problem. The sample problems demonstrate the use of the Walsh function solution algorithms, exploiting Fast Walsh Transforms in multi-dimensions (O(Nlog(N))). Details of a Fast Walsh Reciprocal, defined here for the first time, enable inversion of aWalsh Symmetric Matrix in O(Nlog(N)) operations. Walsh functions have been derived using a fractal recursion algorithm and these fractal patterns are observed in the progression of pairs of wave number amplitudes in the solutions. These patterns are most easily observed in a remapping defined as a fractal fingerprint (FFP). A prolongation of existing solutions to the next highest order exploits these patterns. The algorithms presented here are considered a work in progress that provide new alternatives and new insights into the solution of non-linear PDEs.

Scattering of two spinning black holes in post-Minkowskian gravity, to all orders in spin, and effective-one-body mappings

NASA Astrophysics Data System (ADS)

Vines, Justin

2018-04-01

We demonstrate equivalences, under simple mappings, between the dynamics of three distinct systems—(i) an arbitrary-mass-ratio two-spinning-black-hole system, (ii) a spinning test black hole in a background Kerr spacetime, and (iii) geodesic motion in Kerr—when each is considered in the first post-Minkowskian (1PM) approximation to general relativity, i.e. to linear order G but to all orders in 1/c, and to all orders in the black holes’ spins, with all orders in the multipole expansions of their linearized gravitational fields. This is accomplished via computations of the net results of weak gravitational scattering encounters between two spinning black holes, namely the net O(G) changes in the holes’ momenta and spins as functions of the incoming state. The results are given in remarkably simple closed forms, found by solving effective Mathisson–Papapetrou–Dixon-type equations of motion for a spinning black hole in conjunction with the linearized Einstein equation, with appropriate matching to the Kerr solution. The scattering results fully encode the gauge-invariant content of a canonical Hamiltonian governing binary-black-hole dynamics at 1PM order, for generic (unbound and bound) orbits and spin orientations. We deduce one such Hamiltonian, which reproduces and resums the 1PM parts of all such previous post-Newtonian results, and which directly manifests the equivalences with the test-body limits via simple effective-one-body mappings.

Helicopter vibration suppression using simple pendulum absorbers on the rotor blade

NASA Technical Reports Server (NTRS)

Hamouda, M.-N. H.; Pierce, G. A.

1981-01-01

A design procedure is presented for the installation of simple pendulums on the blades of a helicopter rotor to suppress the root reactions. The procedure consists of a frequency response analysis for a hingeless rotor blade excited by a harmonic variation of spanwise airload distributions during forward flight, as well as a concentrated load at the tip. The structural modeling of the blade provides for elastic degrees of freedom in flap and lead-lag bending plus torsion. Simple flap and lead-lag pendulums are considered individually. Using a rational order scheme, the general nonlinear equations of motion are linearized. A quasi-steady aerodynamic representation is used in the formation of the airloads. The solution of the system equations derives from their representation as a transfer matrix. The results include the effect of pendulum tuning on the minimization of the hub reactions.

Numerical solutions to the time-dependent Bloch equations revisited.

PubMed

Murase, Kenya; Tanki, Nobuyoshi

2011-01-01

The purpose of this study was to demonstrate a simple and fast method for solving the time-dependent Bloch equations. First, the time-dependent Bloch equations were reduced to a homogeneous linear differential equation, and then a simple equation was derived to solve it using a matrix operation. The validity of this method was investigated by comparing with the analytical solutions in the case of constant radiofrequency irradiation. There was a good agreement between them, indicating the validity of this method. As a further example, this method was applied to the time-dependent Bloch equations in the two-pool exchange model for chemical exchange saturation transfer (CEST) or amide proton transfer (APT) magnetic resonance imaging (MRI), and the Z-spectra and asymmetry spectra were calculated from their solutions. They were also calculated using the fourth/fifth-order Runge-Kutta-Fehlberg (RKF) method for comparison. There was also a good agreement between them, and this method was much faster than the RKF method. In conclusion, this method will be useful for analyzing the complex CEST or APT contrast mechanism and/or investigating the optimal conditions for CEST or APT MRI. Copyright © 2011 Elsevier Inc. All rights reserved.

Time-Parallel Solutions to Ordinary Differential Equations on GPUs with a New Functional Optimization Approach Related to the Sobolev Gradient Method

DTIC Science & Technology

2012-10-01

black and approximations in cyan and magenta. The second ODE is the pendulum equation, given by: This ODE was also implemented using Crank...The drawback of approaches like the one proposed can be observed with a very simple example. Suppose vector is found by applying 4 linear...public release; distribution unlimited Figure 2. A phase space plot of the Pendulum example. Fine solution (black) contains 32768 time steps

Response of an oscillatory differential delay equation to a single stimulus.

PubMed

Mackey, Michael C; Tyran-Kamińska, Marta; Walther, Hans-Otto

2017-04-01

Here we analytically examine the response of a limit cycle solution to a simple differential delay equation to a single pulse perturbation of the piecewise linear nonlinearity. We construct the unperturbed limit cycle analytically, and are able to completely characterize the perturbed response to a pulse of positive amplitude and duration with onset at different points in the limit cycle. We determine the perturbed minima and maxima and period of the limit cycle and show how the pulse modifies these from the unperturbed case.

Determination of transport wind speed in the gaussian plume diffusion equation for low-lying point sources

NASA Astrophysics Data System (ADS)

Wang, I. T.

A general method for determining the effective transport wind speed, overlineu, in the Gaussian plume equation is discussed. Physical arguments are given for using the generalized overlineu instead of the often adopted release-level wind speed with the plume diffusion equation. Simple analytical expressions for overlineu applicable to low-level point releases and a wide range of atmospheric conditions are developed. A non-linear plume kinematic equation is derived using these expressions. Crosswind-integrated SF 6 concentration data from the 1983 PNL tracer experiment are used to evaluate the proposed analytical procedures along with the usual approach of using the release-level wind speed. Results of the evaluation are briefly discussed.

Assessment and correction of skinfold thickness equations in estimating body fat in children with cerebral palsy

PubMed Central

GURKA, MATTHEW J; KUPERMINC, MICHELLE N; BUSBY, MARJORIE G; BENNIS, JACEY A; GROSSBERG, RICHARD I; HOULIHAN, CHRISTINE M; STEVENSON, RICHARD D; HENDERSON, RICHARD C

2010-01-01

AIM To assess the accuracy of skinfold equations in estimating percentage body fat in children with cerebral palsy (CP), compared with assessment of body fat from dual energy X-ray absorptiometry (DXA). METHOD Data were collected from 71 participants (30 females, 41 males) with CP (Gross Motor Function Classification System [GMFCS] levels I–V) between the ages of 8 and 18 years. Estimated percentage body fat was computed using established (Slaughter) equations based on the triceps and subscapular skinfolds. A linear model was fitted to assess the use of a simple correction to these equations for children with CP. RESULTS Slaughter’s equations consistently underestimated percentage body fat (mean difference compared with DXA percentage body fat −9.6/100 [SD 6.2]; 95% confidence interval [CI] −11.0 to −8.1). New equations were developed in which a correction factor was added to the existing equations based on sex, race, GMFCS level, size, and pubertal status. These corrected equations for children with CP agree better with DXA (mean difference 0.2/100 [SD=4.8]; 95% CI −1.0 to 1.3) than existing equations. INTERPRETATION A simple correction factor to commonly used equations substantially improves the ability to estimate percentage body fat from two skinfold measures in children with CP. PMID:19811518

Variational method enabling simplified solutions to the linearized Boltzmann equation for oscillatory gas flows

NASA Astrophysics Data System (ADS)

Ladiges, Daniel R.; Sader, John E.

2018-05-01

Nanomechanical resonators and sensors, operated in ambient conditions, often generate low-Mach-number oscillating rarefied gas flows. Cercignani [C. Cercignani, J. Stat. Phys. 1, 297 (1969), 10.1007/BF01007482] proposed a variational principle for the linearized Boltzmann equation, which can be used to derive approximate analytical solutions of steady (time-independent) flows. Here we extend and generalize this principle to unsteady oscillatory rarefied flows and thus accommodate resonating nanomechanical devices. This includes a mathematical approach that facilitates its general use and allows for systematic improvements in accuracy. This formulation is demonstrated for two canonical flow problems: oscillatory Couette flow and Stokes' second problem. Approximate analytical formulas giving the bulk velocity and shear stress, valid for arbitrary oscillation frequency, are obtained for Couette flow. For Stokes' second problem, a simple system of ordinary differential equations is derived which may be solved to obtain the desired flow fields. Using this framework, a simple and accurate formula is provided for the shear stress at the oscillating boundary, again for arbitrary frequency, which may prove useful in application. These solutions are easily implemented on any symbolic or numerical package, such as Mathematica or matlab, facilitating the characterization of flows produced by nanomechanical devices and providing insight into the underlying flow physics.

Linear stiff string vibrations in musical acoustics: Assessment and comparison of models.

PubMed

Ducceschi, Michele; Bilbao, Stefan

2016-10-01

Strings are amongst the most common elements found in musical instruments and an appropriate physical description of string dynamics is essential to modelling, analysis, and simulation. For linear vibration in a single polarisation, the most common model is based on the Euler-Bernoulli beam equation under tension. In spite of its simple form, such a model gives unbounded phase and group velocities at large wavenumbers, and such behaviour may be interpreted as unphysical. The Timoshenko model has, therefore, been employed in more recent works to overcome such shortcoming. This paper presents a third model based on the shear beam equations. The three models are here assessed and compared with regard to the perceptual considerations in musical acoustics.

Exceptional point in a simple textbook example

NASA Astrophysics Data System (ADS)

Fernández, Francisco M.

2018-07-01

We propose to introduce the concept of exceptional points in intermediate courses on mathematics and classical mechanics by means of simple textbook examples. The first one is an ordinary second-order differential equation with constant coefficients. The second one is the well-known damped harmonic oscillator. From a strict mathematical viewpoint both are the same problem that enables one to connect the occurrence of linearly dependent exponential solutions with a defective matrix which cannot be diagonalized but can be transformed into a Jordan canonical form.

Products of multiple Fourier series with application to the multiblade transformation

NASA Technical Reports Server (NTRS)

Kunz, D. L.

1981-01-01

A relatively simple and systematic method for forming the products of multiple Fourier series using tensor like operations is demonstrated. This symbolic multiplication can be performed for any arbitrary number of series, and the coefficients of a set of linear differential equations with periodic coefficients from a rotating coordinate system to a nonrotating system is also demonstrated. It is shown that using Fourier operations to perform this transformation make it easily understood, simple to apply, and generally applicable.

Some simple solutions of Schrödinger's equation for a free particle or for an oscillator

NASA Astrophysics Data System (ADS)

Andrews, Mark

2018-05-01

For a non-relativistic free particle, we show that the evolution of some simple initial wave functions made up of linear segments can be expressed in terms of Fresnel integrals. Examples include the square wave function and the triangular wave function. The method is then extended to wave functions made from quadratic elements. The evolution of all these initial wave functions can also be found for the harmonic oscillator by a transformation of the free evolutions.

Development and validation of a predictive equation for lean body mass in children and adolescents.

PubMed

Foster, Bethany J; Platt, Robert W; Zemel, Babette S

2012-05-01

Lean body mass (LBM) is not easy to measure directly in the field or clinical setting. Equations to predict LBM from simple anthropometric measures, which account for the differing contributions of fat and lean to body weight at different ages and levels of adiposity, would be useful to both human biologists and clinicians. To develop and validate equations to predict LBM in children and adolescents across the entire range of the adiposity spectrum. Dual energy X-ray absorptiometry was used to measure LBM in 836 healthy children (437 females) and linear regression was used to develop sex-specific equations to estimate LBM from height, weight, age, body mass index (BMI) for age z-score and population ancestry. Equations were validated using bootstrapping methods and in a local independent sample of 332 children and in national data collected by NHANES. The mean difference between measured and predicted LBM was - 0.12% (95% limits of agreement - 11.3% to 8.5%) for males and - 0.14% ( - 11.9% to 10.9%) for females. Equations performed equally well across the entire adiposity spectrum, as estimated by BMI z-score. Validation indicated no over-fitting. LBM was predicted within 5% of measured LBM in the validation sample. The equations estimate LBM accurately from simple anthropometric measures.

A new approach to approximating the linear quadratic optimal control law for hereditary systems with control delays

NASA Technical Reports Server (NTRS)

Milman, M. H.

1985-01-01

A factorization approach is presented for deriving approximations to the optimal feedback gain for the linear regulator-quadratic cost problem associated with time-varying functional differential equations with control delays. The approach is based on a discretization of the state penalty which leads to a simple structure for the feedback control law. General properties of the Volterra factors of Hilbert-Schmidt operators are then used to obtain convergence results for the feedback kernels.

An Application of the H-Function to Curve-Fitting and Density Estimation.

DTIC Science & Technology

1983-12-01

equations into a model that is linear in its coefficients. Nonlinear least squares estimation is a relatively new area developed to accomodate models which...to converge on a solution (10:9-10). For the simple linear model and when general assump- tions are made, the Gauss-Markov theorem states that the...distribution. For example, if the analyst wants to model the time between arrivals to a queue for a computer simulation, he infers the true probability

Aeroelastic analysis of a troposkien-type wind turbine blade

NASA Technical Reports Server (NTRS)

Nitzsche, F.

1981-01-01

The linear aeroelastic equations for one curved blade of a vertical axis wind turbine in state vector form are presented. The method is based on a simple integrating matrix scheme together with the transfer matrix idea. The method is proposed as a convenient way of solving the associated eigenvalue problem for general support conditions.

Critical assessment of jet erosion test methodologies for cohesive soil and sediment

USDA-ARS?s Scientific Manuscript database

The submerged Jet Erosion Test (JET) is a commonly used technique to assess the erodibility of cohesive soil. Employing a linear excess shear stress equation and impinging jet theory, simple numerical methods have been developed to analyze data collected using a JET to determine the critical shear s...

Modelling a Simple Mechanical System.

ERIC Educational Resources Information Center

Morland, Tim

1999-01-01

Provides an example of the modeling power of Mathematics, demonstrated in a piece of A-Level student coursework which was undertaken as part of the MEI Structured Mathematics scheme. A system of two masses and two springs oscillating in one dimension is found to be accurately modeled by a system of linear differential equations. (Author/ASK)

Difference-Equation/Flow-Graph Circuit Analysis

NASA Technical Reports Server (NTRS)

Mcvey, I. M.

1988-01-01

Numerical technique enables rapid, approximate analyses of electronic circuits containing linear and nonlinear elements. Practiced in variety of computer languages on large and small computers; for circuits simple enough, programmable hand calculators used. Although some combinations of circuit elements make numerical solutions diverge, enables quick identification of divergence and correction of circuit models to make solutions converge.

Nonlinear modeling of wave-topography interactions, shear instabilities and shear induced wave breaking using vortex method

NASA Astrophysics Data System (ADS)

Guha, Anirban

2017-11-01

Theoretical studies on linear shear instabilities as well as different kinds of wave interactions often use simple velocity and/or density profiles (e.g. constant, piecewise) for obtaining good qualitative and quantitative predictions of the initial disturbances. Moreover, such simple profiles provide a minimal model to obtain a mechanistic understanding of shear instabilities. Here we have extended this minimal paradigm into nonlinear domain using vortex method. Making use of unsteady Bernoulli's equation in presence of linear shear, and extending Birkhoff-Rott equation to multiple interfaces, we have numerically simulated the interaction between multiple fully nonlinear waves. This methodology is quite general, and has allowed us to simulate diverse problems that can be essentially reduced to the minimal system with interacting waves, e.g. spilling and plunging breakers, stratified shear instabilities (Holmboe, Taylor-Caulfield, stratified Rayleigh), jet flows, and even wave-topography interaction problem like Bragg resonance. We found that the minimal models capture key nonlinear features (e.g. wave breaking features like cusp formation and roll-ups) which are observed in experiments and/or extensive simulations with smooth, realistic profiles.

Linearization of the bradford protein assay.

PubMed

Ernst, Orna; Zor, Tsaffrir

2010-04-12

Determination of microgram quantities of protein in the Bradford Coomassie brilliant blue assay is accomplished by measurement of absorbance at 590 nm. This most common assay enables rapid and simple protein quantification in cell lysates, cellular fractions, or recombinant protein samples, for the purpose of normalization of biochemical measurements. However, an intrinsic nonlinearity compromises the sensitivity and accuracy of this method. It is shown that under standard assay conditions, the ratio of the absorbance measurements at 590 nm and 450 nm is strictly linear with protein concentration. This simple procedure increases the accuracy and improves the sensitivity of the assay about 10-fold, permitting quantification down to 50 ng of bovine serum albumin. Furthermore, the interference commonly introduced by detergents that are used to create the cell lysates is greatly reduced by the new protocol. A linear equation developed on the basis of mass action and Beer's law perfectly fits the experimental data.

Real-Time Exponential Curve Fits Using Discrete Calculus

NASA Technical Reports Server (NTRS)

Rowe, Geoffrey

2010-01-01

An improved solution for curve fitting data to an exponential equation (y = Ae(exp Bt) + C) has been developed. This improvement is in four areas -- speed, stability, determinant processing time, and the removal of limits. The solution presented avoids iterative techniques and their stability errors by using three mathematical ideas: discrete calculus, a special relationship (be tween exponential curves and the Mean Value Theorem for Derivatives), and a simple linear curve fit algorithm. This method can also be applied to fitting data to the general power law equation y = Ax(exp B) + C and the general geometric growth equation y = Ak(exp Bt) + C.

An approximation of herd effect due to vaccinating children against seasonal influenza - a potential solution to the incorporation of indirect effects into static models.

PubMed

Van Vlaenderen, Ilse; Van Bellinghen, Laure-Anne; Meier, Genevieve; Nautrup, Barbara Poulsen

2013-01-22

Indirect herd effect from vaccination of children offers potential for improving the effectiveness of influenza prevention in the remaining unvaccinated population. Static models used in cost-effectiveness analyses cannot dynamically capture herd effects. The objective of this study was to develop a methodology to allow herd effect associated with vaccinating children against seasonal influenza to be incorporated into static models evaluating the cost-effectiveness of influenza vaccination. Two previously published linear equations for approximation of herd effects in general were compared with the results of a structured literature review undertaken using PubMed searches to identify data on herd effects specific to influenza vaccination. A linear function was fitted to point estimates from the literature using the sum of squared residuals. The literature review identified 21 publications on 20 studies for inclusion. Six studies provided data on a mathematical relationship between effective vaccine coverage in subgroups and reduction of influenza infection in a larger unvaccinated population. These supported a linear relationship when effective vaccine coverage in a subgroup population was between 20% and 80%. Three studies evaluating herd effect at a community level, specifically induced by vaccinating children, provided point estimates for fitting linear equations. The fitted linear equation for herd protection in the target population for vaccination (children) was slightly less conservative than a previously published equation for herd effects in general. The fitted linear equation for herd protection in the non-target population was considerably less conservative than the previously published equation. This method of approximating herd effect requires simple adjustments to the annual baseline risk of influenza in static models: (1) for the age group targeted by the childhood vaccination strategy (i.e. children); and (2) for other age groups not targeted (e.g. adults and/or elderly). Two approximations provide a linear relationship between effective coverage and reduction in the risk of infection. The first is a conservative approximation, recommended as a base-case for cost-effectiveness evaluations. The second, fitted to data extracted from a structured literature review, provides a less conservative estimate of herd effect, recommended for sensitivity analyses.

A second order discontinuous Galerkin fast sweeping method for Eikonal equations

NASA Astrophysics Data System (ADS)

Li, Fengyan; Shu, Chi-Wang; Zhang, Yong-Tao; Zhao, Hongkai

2008-09-01

In this paper, we construct a second order fast sweeping method with a discontinuous Galerkin (DG) local solver for computing viscosity solutions of a class of static Hamilton-Jacobi equations, namely the Eikonal equations. Our piecewise linear DG local solver is built on a DG method developed recently [Y. Cheng, C.-W. Shu, A discontinuous Galerkin finite element method for directly solving the Hamilton-Jacobi equations, Journal of Computational Physics 223 (2007) 398-415] for the time-dependent Hamilton-Jacobi equations. The causality property of Eikonal equations is incorporated into the design of this solver. The resulting local nonlinear system in the Gauss-Seidel iterations is a simple quadratic system and can be solved explicitly. The compactness of the DG method and the fast sweeping strategy lead to fast convergence of the new scheme for Eikonal equations. Extensive numerical examples verify efficiency, convergence and second order accuracy of the proposed method.

Effect of collision energy optimization on the measurement of peptides by selected reaction monitoring (SRM) mass spectrometry.

PubMed

Maclean, Brendan; Tomazela, Daniela M; Abbatiello, Susan E; Zhang, Shucha; Whiteaker, Jeffrey R; Paulovich, Amanda G; Carr, Steven A; Maccoss, Michael J

2010-12-15

Proteomics experiments based on Selected Reaction Monitoring (SRM, also referred to as Multiple Reaction Monitoring or MRM) are being used to target large numbers of protein candidates in complex mixtures. At present, instrument parameters are often optimized for each peptide, a time and resource intensive process. Large SRM experiments are greatly facilitated by having the ability to predict MS instrument parameters that work well with the broad diversity of peptides they target. For this reason, we investigated the impact of using simple linear equations to predict the collision energy (CE) on peptide signal intensity and compared it with the empirical optimization of the CE for each peptide and transition individually. Using optimized linear equations, the difference between predicted and empirically derived CE values was found to be an average gain of only 7.8% of total peak area. We also found that existing commonly used linear equations fall short of their potential, and should be recalculated for each charge state and when introducing new instrument platforms. We provide a fully automated pipeline for calculating these equations and individually optimizing CE of each transition on SRM instruments from Agilent, Applied Biosystems, Thermo-Scientific and Waters in the open source Skyline software tool ( http://proteome.gs.washington.edu/software/skyline ).

Virasoro constraints and polynomial recursion for the linear Hodge integrals

NASA Astrophysics Data System (ADS)

Guo, Shuai; Wang, Gehao

2017-04-01

The Hodge tau-function is a generating function for the linear Hodge integrals. It is also a tau-function of the KP hierarchy. In this paper, we first present the Virasoro constraints for the Hodge tau-function in the explicit form of the Virasoro equations. The expression of our Virasoro constraints is simply a linear combination of the Virasoro operators, where the coefficients are restored from a power series for the Lambert W function. Then, using this result, we deduce a simple version of the Virasoro constraints for the linear Hodge partition function, where the coefficients are restored from the Gamma function. Finally, we establish the equivalence relation between the Virasoro constraints and polynomial recursion formula for the linear Hodge integrals.

A moist Boussinesq shallow water equations set for testing atmospheric models

DOE Office of Scientific and Technical Information (OSTI.GOV)

Zerroukat, M., E-mail: mohamed.zerroukat@metoffice.gov.uk; Allen, T.

The shallow water equations have long been used as an initial test for numerical methods applied to atmospheric models with the test suite of Williamson et al. being used extensively for validating new schemes and assessing their accuracy. However the lack of physics forcing within this simplified framework often requires numerical techniques to be reworked when applied to fully three dimensional models. In this paper a novel two-dimensional shallow water equations system that retains moist processes is derived. This system is derived from three-dimensional Boussinesq approximation of the hydrostatic Euler equations where, unlike the classical shallow water set, we allowmore » the density to vary slightly with temperature. This results in extra (or buoyancy) terms for the momentum equations, through which a two-way moist-physics dynamics feedback is achieved. The temperature and moisture variables are advected as separate tracers with sources that interact with the mean-flow through a simplified yet realistic bulk moist-thermodynamic phase-change model. This moist shallow water system provides a unique tool to assess the usually complex and highly non-linear dynamics–physics interactions in atmospheric models in a simple yet realistic way. The full non-linear shallow water equations are solved numerically on several case studies and the results suggest quite realistic interaction between the dynamics and physics and in particular the generation of cloud and rain. - Highlights: • Novel shallow water equations which retains moist processes are derived from the three-dimensional hydrostatic Boussinesq equations. • The new shallow water set can be seen as a more general one, where the classical equations are a special case of these equations. • This moist shallow water system naturally allows a feedback mechanism from the moist physics increments to the momentum via buoyancy. • Like full models, temperature and moistures are advected as tracers that interact through a simplified yet realistic phase-change model. • This model is a unique tool to test numerical methods for atmospheric models, and physics–dynamics coupling, in a very realistic and simple way.« less

Comparison of some optimal control methods for the design of turbine blades

NASA Technical Reports Server (NTRS)

Desilva, B. M. E.; Grant, G. N. C.

1977-01-01

This paper attempts a comparative study of some numerical methods for the optimal control design of turbine blades whose vibration characteristics are approximated by Timoshenko beam idealizations with shear and incorporating simple boundary conditions. The blade was synthesized using the following methods: (1) conjugate gradient minimization of the system Hamiltonian in function space incorporating penalty function transformations, (2) projection operator methods in a function space which includes the frequencies of vibration and the control function, (3) epsilon-technique penalty function transformation resulting in a highly nonlinear programming problem, (4) finite difference discretization of the state equations again resulting in a nonlinear program, (5) second variation methods with complex state differential equations to include damping effects resulting in systems of inhomogeneous matrix Riccatti equations some of which are stiff, (6) quasi-linear methods based on iterative linearization of the state and adjoint equation. The paper includes a discussion of some substantial computational difficulties encountered in the implementation of these techniques together with a resume of work presently in progress using a differential dynamic programming approach.

Solvent and structural effects on the spectral shifts of 5-(substituted phenylazo)-3-cyano-6-hydroxy-1-(2-hydroxyethyl)-4-methyl-2-pyridones

NASA Astrophysics Data System (ADS)

Mirković, Jelena M.; Božić, Bojan Đ.; Mutavdžić, Dragosav R.; Ušćumlić, Gordana S.; Mijin, Dušan Ž.

2014-11-01

Spectral properties, solvatochromism and azo-hydrazone tautomerism of ten 5-(substituted phenylazo)-3-cyano-6-hydroxy-1-(2-hydroxyethyl)-4-methyl-2-pyridones in twenty-two solvents are investigated. For quantitative evaluation of the solvent effects on the UV-vis absorption maxima, the principles of the linear solvation energy relationships are used, i.e. models proposed by Kamlet-Taft and Catalán. Linear free energy relationships are applied to the UV-vis absorption spectra and correlation of absorption frequencies with Hammett substituent constants are performed. Furthermore, the influence of the electronic nature of the substituents on 1H and 13C NMR shifts is investigated by simple and extended Hammett equations, as well as by Swain-Lupton equation.

Localized light waves: Paraxial and exact solutions of the wave equation (a review)

NASA Astrophysics Data System (ADS)

Kiselev, A. P.

2007-04-01

Simple explicit localized solutions are systematized over the whole space of a linear wave equation, which models the propagation of optical radiation in a linear approximation. Much attention has been paid to exact solutions (which date back to the Bateman findings) that describe wave beams (including Bessel-Gauss beams) and wave packets with a Gaussian localization with respect to the spatial variables and time. Their asymptotics with respect to free parameters and at large distances are presented. A similarity between these exact solutions and harmonic in time fields obtained in the paraxial approximation based on the Leontovich-Fock parabolic equation has been studied. Higher-order modes are considered systematically using the separation of variables method. The application of the Bateman solutions of the wave equation to the construction of solutions to equations with dispersion and nonlinearity and their use in wavelet analysis, as well as the summation of Gaussian beams, are discussed. In addition, solutions localized at infinity known as the Moses-Prosser “acoustic bullets”, as well as their harmonic in time counterparts, “ X waves”, waves from complex sources, etc., have been considered. Everywhere possible, the most elementary mathematical formalism is used.

PROTEUS two-dimensional Navier-Stokes computer code, version 1.0. Volume 1: Analysis description

NASA Technical Reports Server (NTRS)

Towne, Charles E.; Schwab, John R.; Benson, Thomas J.; Suresh, Ambady

1990-01-01

A new computer code was developed to solve the two-dimensional or axisymmetric, Reynolds averaged, unsteady compressible Navier-Stokes equations in strong conservation law form. The thin-layer or Euler equations may also be solved. Turbulence is modeled using an algebraic eddy viscosity model. The objective was to develop a code for aerospace applications that is easy to use and easy to modify. Code readability, modularity, and documentation were emphasized. The equations are written in nonorthogonal body-fitted coordinates, and solved by marching in time using a fully-coupled alternating direction-implicit procedure with generalized first- or second-order time differencing. All terms are linearized using second-order Taylor series. The boundary conditions are treated implicitly, and may be steady, unsteady, or spatially periodic. Simple Cartesian or polar grids may be generated internally by the program. More complex geometries require an externally generated computational coordinate system. The documentation is divided into three volumes. Volume 1 is the Analysis Description, and describes in detail the governing equations, the turbulence model, the linearization of the equations and boundary conditions, the time and space differencing formulas, the ADI solution procedure, and the artificial viscosity models.

Inverse scattering transform for the KPI equation on the background of a one-line soliton*Inverse scattering transform for the KPI equation on the background of a one-line soliton

NASA Astrophysics Data System (ADS)

Fokas, A. S.; Pogrebkov, A. K.

2003-03-01

We study the initial value problem of the Kadomtsev-Petviashvili I (KPI) equation with initial data u(x1,x2,0) = u1(x1)+u2(x1,x2), where u1(x1) is the one-soliton solution of the Korteweg-de Vries equation evaluated at zero time and u2(x1,x2) decays sufficiently rapidly on the (x1,x2)-plane. This involves the analysis of the nonstationary Schrödinger equation (with time replaced by x2) with potential u(x1,x2,0). We introduce an appropriate sectionally analytic eigenfunction in the complex k-plane where k is the spectral parameter. This eigenfunction has the novelty that in addition to the usual jump across the real k-axis, it also has a jump across a segment of the imaginary k-axis. We show that this eigenfunction can be reconstructed through a linear integral equation uniquely defined in terms of appropriate scattering data. In turn, these scattering data are uniquely constructed in terms of u1(x1) and u2(x1,x2). This result implies that the solution of the KPI equation can be obtained through the above linear integral equation where the scattering data have a simple t-dependence.

Deriving the Regression Equation without Using Calculus

ERIC Educational Resources Information Center

Gordon, Sheldon P.; Gordon, Florence S.

2004-01-01

Probably the one "new" mathematical topic that is most responsible for modernizing courses in college algebra and precalculus over the last few years is the idea of fitting a function to a set of data in the sense of a least squares fit. Whether it be simple linear regression or nonlinear regression, this topic opens the door to applying the…

On the Debye-Hückel effect of electric screening

NASA Astrophysics Data System (ADS)

Campos, L. M. B. C.; Lau, F. J. P.

2014-07-01

The paper considers non-linear self-consistent electric potential equation (Sec. I), due to a cloud made of a single species of electric charges, satisfying a Boltzmann distribution law (Sec. II). Exact solutions are obtained in a simple logarithmic form, in three cases: (Sec. III) spherical radial symmetry; (Sec. IV) plane parallel symmetry; (Sec. V) a special case of azimuthal-cylindrical symmetry. All these solutions, and their transformations (Sec. VI), involve the Debye-Hückel radius; the latter was originally defined from a solution of the linearized self-consistent potential equation. Using an exact solution of the self-consistent potential equation, the distance at which the potential vanishes differs from the Debye-Hückel radius by a factor of √2 . The preceding (Secs. II-VI) simple logarithmic exact solutions of the self-consistent potential equations involve no arbitrary constants, and thus are special or singular integrals not the general integral. The general solution of the self-consistent potential equation is obtained in the plane parallel case (Sec. VII), and it involves two arbitrary constants that can be reduced to one via a translation (Sec. VIII). The plots of dimensionless potential (Figure 1), electric field (Figure 2), charge density (Figure 3), and total charge between ζ and infinity (Figure 4), versus distance normalized to Debye-Hückel radius ζ ≡ z/a, show that (Sec. IX) there is a continuum of solutions, ranging from a charge distribution concentrated inside the Debye-Hückel radius to one spread-out beyond it. The latter case leads to the limiting case of logarithmic potential, and stronger electric field; the former case, of very concentrated charge distribution, leads to a fratricide effect and weaker electric field.

On the Debye–Hückel effect of electric screening

DOE Office of Scientific and Technical Information (OSTI.GOV)

Campos, L. M. B. C.; Lau, F. J. P.

2014-07-15

The paper considers non-linear self-consistent electric potential equation (Sec. I), due to a cloud made of a single species of electric charges, satisfying a Boltzmann distribution law (Sec. II). Exact solutions are obtained in a simple logarithmic form, in three cases: (Sec. III) spherical radial symmetry; (Sec. IV) plane parallel symmetry; (Sec. V) a special case of azimuthal-cylindrical symmetry. All these solutions, and their transformations (Sec. VI), involve the Debye-Hückel radius; the latter was originally defined from a solution of the linearized self-consistent potential equation. Using an exact solution of the self-consistent potential equation, the distance at which the potentialmore » vanishes differs from the Debye-Hückel radius by a factor of √(2). The preceding (Secs. II–VI) simple logarithmic exact solutions of the self-consistent potential equations involve no arbitrary constants, and thus are special or singular integrals not the general integral. The general solution of the self-consistent potential equation is obtained in the plane parallel case (Sec. VII), and it involves two arbitrary constants that can be reduced to one via a translation (Sec. VIII). The plots of dimensionless potential (Figure 1), electric field (Figure 2), charge density (Figure 3), and total charge between ζ and infinity (Figure 4), versus distance normalized to Debye-Hückel radius ζ ≡ z/a, show that (Sec. IX) there is a continuum of solutions, ranging from a charge distribution concentrated inside the Debye-Hückel radius to one spread-out beyond it. The latter case leads to the limiting case of logarithmic potential, and stronger electric field; the former case, of very concentrated charge distribution, leads to a fratricide effect and weaker electric field.« less

Accurate electrostatic and van der Waals pull-in prediction for fully clamped nano/micro-beams using linear universal graphs of pull-in instability

NASA Astrophysics Data System (ADS)

Tahani, Masoud; Askari, Amir R.

2014-09-01

In spite of the fact that pull-in instability of electrically actuated nano/micro-beams has been investigated by many researchers to date, no explicit formula has been presented yet which can predict pull-in voltage based on a geometrically non-linear and distributed parameter model. The objective of present paper is to introduce a simple and accurate formula to predict this value for a fully clamped electrostatically actuated nano/micro-beam. To this end, a non-linear Euler-Bernoulli beam model is employed, which accounts for the axial residual stress, geometric non-linearity of mid-plane stretching, distributed electrostatic force and the van der Waals (vdW) attraction. The non-linear boundary value governing equation of equilibrium is non-dimensionalized and solved iteratively through single-term Galerkin based reduced order model (ROM). The solutions are validated thorough direct comparison with experimental and other existing results reported in previous studies. Pull-in instability under electrical and vdW loads are also investigated using universal graphs. Based on the results of these graphs, non-dimensional pull-in and vdW parameters, which are defined in the text, vary linearly versus the other dimensionless parameters of the problem. Using this fact, some linear equations are presented to predict pull-in voltage, the maximum allowable length, the so-called detachment length, and the minimum allowable gap for a nano/micro-system. These linear equations are also reduced to a couple of universal pull-in formulas for systems with small initial gap. The accuracy of the universal pull-in formulas are also validated by comparing its results with available experimental and some previous geometric linear and closed-form findings published in the literature.

Accurate radiative transfer calculations for layered media.

PubMed

Selden, Adrian C

2016-07-01

Simple yet accurate results for radiative transfer in layered media with discontinuous refractive index are obtained by the method of K-integrals. These are certain weighted integrals applied to the angular intensity distribution at the refracting boundaries. The radiative intensity is expressed as the sum of the asymptotic angular intensity distribution valid in the depth of the scattering medium and a transient term valid near the boundary. Integrated boundary equations are obtained, yielding simple linear equations for the intensity coefficients, enabling the angular emission intensity and the diffuse reflectance (albedo) and transmittance of the scattering layer to be calculated without solving the radiative transfer equation directly. Examples are given of half-space, slab, interface, and double-layer calculations, and extensions to multilayer systems are indicated. The K-integral method is orders of magnitude more accurate than diffusion theory and can be applied to layered scattering media with a wide range of scattering albedos, with potential applications to biomedical and ocean optics.

Density of Jatropha curcas Seed Oil and its Methyl Esters: Measurement and Estimations

NASA Astrophysics Data System (ADS)

Veny, Harumi; Baroutian, Saeid; Aroua, Mohamed Kheireddine; Hasan, Masitah; Raman, Abdul Aziz; Sulaiman, Nik Meriam Nik

2009-04-01

Density data as a function of temperature have been measured for Jatropha curcas seed oil, as well as biodiesel jatropha methyl esters at temperatures from above their melting points to 90 ° C. The data obtained were used to validate the method proposed by Spencer and Danner using a modified Rackett equation. The experimental and estimated density values using the modified Rackett equation gave almost identical values with average absolute percent deviations less than 0.03% for the jatropha oil and 0.04% for the jatropha methyl esters. The Janarthanan empirical equation was also employed to predict jatropha biodiesel densities. This equation performed equally well with average absolute percent deviations within 0.05%. Two simple linear equations for densities of jatropha oil and its methyl esters are also proposed in this study.

Calculation of the orientational linear and nonlinear correlation factors of polar liquids from the rotational Dean-Kawasaki equation.

PubMed

Déjardin, P M; Cornaton, Y; Ghesquière, P; Caliot, C; Brouzet, R

2018-01-28

A calculation of the Kirkwood and Piekara-Kielich correlation factors of polar liquids is presented using the forced rotational diffusion theory of Cugliandolo et al. [Phys. Rev. E 91, 032139 (2015)]. These correlation factors are obtained as a function of density and temperature. Our results compare reasonably well with the experimental temperature dependence of the linear dielectric constant of some simple polar liquids across a wide temperature range. A comparison of our results for the linear dielectric constant and the Kirkwood correlation factor with relevant numerical simulations of liquid water and methanol is given.

The large discretization step method for time-dependent partial differential equations

NASA Technical Reports Server (NTRS)

Haras, Zigo; Taasan, Shlomo

1995-01-01

A new method for the acceleration of linear and nonlinear time dependent calculations is presented. It is based on the Large Discretization Step (LDS) approximation, defined in this work, which employs an extended system of low accuracy schemes to approximate a high accuracy discrete approximation to a time dependent differential operator. Error bounds on such approximations are derived. These approximations are efficiently implemented in the LDS methods for linear and nonlinear hyperbolic equations, presented here. In these algorithms the high and low accuracy schemes are interpreted as the same discretization of a time dependent operator on fine and coarse grids, respectively. Thus, a system of correction terms and corresponding equations are derived and solved on the coarse grid to yield the fine grid accuracy. These terms are initialized by visiting the fine grid once in many coarse grid time steps. The resulting methods are very general, simple to implement and may be used to accelerate many existing time marching schemes.

Scalable Parallel Computation for Extended MHD Modeling of Fusion Plasmas

NASA Astrophysics Data System (ADS)

Glasser, Alan H.

2008-11-01

Parallel solution of a linear system is scalable if simultaneously doubling the number of dependent variables and the number of processors results in little or no increase in the computation time to solution. Two approaches have this property for parabolic systems: multigrid and domain decomposition. Since extended MHD is primarily a hyperbolic rather than a parabolic system, additional steps must be taken to parabolize the linear system to be solved by such a method. Such physics-based preconditioning (PBP) methods have been pioneered by Chac'on, using finite volumes for spatial discretization, multigrid for solution of the preconditioning equations, and matrix-free Newton-Krylov methods for the accurate solution of the full nonlinear preconditioned equations. The work described here is an extension of these methods using high-order spectral element methods and FETI-DP domain decomposition. Application of PBP to a flux-source representation of the physics equations is discussed. The resulting scalability will be demonstrated for simple wave and for ideal and Hall MHD waves.

Rosenzweig instability in a thin layer of a magnetic fluid

NASA Astrophysics Data System (ADS)

Korovin, V. M.

2013-12-01

A simple mathematical model of the initial stage of nonlinear evolution of the Rosenzweig instability in a thin layer of a nonlinearly magnetized viscous ferrofluid coating a horizontal nonmagnetizable plate is constructed on the basis of the system of equations and boundary conditions of ferrofluid dynamics. A dispersion relation is derived and analyzed using the linearized equations of this model. The critical magnetization of the initial layer with a flat free surface, the threshold wavenumber, and the characteristic time of evolution of the most rapidly growing mode are determined. The equation for the neutral stability curve, which is applicable for any physically admissible law of magnetization of a ferrofluid, is derived analytically.

Least-squares solution of incompressible Navier-Stokes equations with the p-version of finite elements

NASA Technical Reports Server (NTRS)

Jiang, Bo-Nan; Sonnad, Vijay

1991-01-01

A p-version of the least squares finite element method, based on the velocity-pressure-vorticity formulation, is developed for solving steady state incompressible viscous flow problems. The resulting system of symmetric and positive definite linear equations can be solved satisfactorily with the conjugate gradient method. In conjunction with the use of rapid operator application which avoids the formation of either element of global matrices, it is possible to achieve a highly compact and efficient solution scheme for the incompressible Navier-Stokes equations. Numerical results are presented for two-dimensional flow over a backward facing step. The effectiveness of simple outflow boundary conditions is also demonstrated.

Communication: Limitations of the stochastic quasi-steady-state approximation in open biochemical reaction networks

NASA Astrophysics Data System (ADS)

Thomas, Philipp; Straube, Arthur V.; Grima, Ramon

2011-11-01

It is commonly believed that, whenever timescale separation holds, the predictions of reduced chemical master equations obtained using the stochastic quasi-steady-state approximation are in very good agreement with the predictions of the full master equations. We use the linear noise approximation to obtain a simple formula for the relative error between the predictions of the two master equations for the Michaelis-Menten reaction with substrate input. The reduced approach is predicted to overestimate the variance of the substrate concentration fluctuations by as much as 30%. The theoretical results are validated by stochastic simulations using experimental parameter values for enzymes involved in proteolysis, gluconeogenesis, and fermentation.

Exact solution of a ratchet with switching sawtooth potential

NASA Astrophysics Data System (ADS)

Saakian, David B.; Klümper, Andreas

2018-01-01

We consider the flashing potential ratchet model with general asymmetric potential. Using Bloch functions, we derive equations which allow for the calculation of both the ratchet's flux and higher moments of distribution for rather general potentials. We indicate how to derive the optimal transition rates for maximal velocity of the ratchet. We calculate explicitly the exact velocity of a ratchet with simple sawtooth potential from the solution of a system of 8 linear algebraic equations. Using Bloch functions, we derive the equations for the ratchet with potentials changing periodically with time. We also consider the case of the ratchet with evolution with two different potentials acting for some random periods of time.

Constitutive error based parameter estimation technique for plate structures using free vibration signatures

NASA Astrophysics Data System (ADS)

Guchhait, Shyamal; Banerjee, Biswanath

2018-04-01

In this paper, a variant of constitutive equation error based material parameter estimation procedure for linear elastic plates is developed from partially measured free vibration sig-natures. It has been reported in many research articles that the mode shape curvatures are much more sensitive compared to mode shape themselves to localize inhomogeneity. Complying with this idea, an identification procedure is framed as an optimization problem where the proposed cost function measures the error in constitutive relation due to incompatible curvature/strain and moment/stress fields. Unlike standard constitutive equation error based procedure wherein a solution of a couple system is unavoidable in each iteration, we generate these incompatible fields via two linear solves. A simple, yet effective, penalty based approach is followed to incorporate measured data. The penalization parameter not only helps in incorporating corrupted measurement data weakly but also acts as a regularizer against the ill-posedness of the inverse problem. Explicit linear update formulas are then developed for anisotropic linear elastic material. Numerical examples are provided to show the applicability of the proposed technique. Finally, an experimental validation is also provided.

Mode Identification of High-Amplitude Pressure Waves in Liquid Rocket Engines

NASA Astrophysics Data System (ADS)

EBRAHIMI, R.; MAZAHERI, K.; GHAFOURIAN, A.

2000-01-01

Identification of existing instability modes from experimental pressure measurements of rocket engines is difficult, specially when steep waves are present. Actual pressure waves are often non-linear and include steep shocks followed by gradual expansions. It is generally believed that interaction of these non-linear waves is difficult to analyze. A method of mode identification is introduced. After presumption of constituent modes, they are superposed by using a standard finite difference scheme for solution of the classical wave equation. Waves are numerically produced at each end of the combustion tube with different wavelengths, amplitudes, and phases with respect to each other. Pressure amplitude histories and phase diagrams along the tube are computed. To determine the validity of the presented method for steep non-linear waves, the Euler equations are numerically solved for non-linear waves, and negligible interactions between these waves are observed. To show the applicability of this method, other's experimental results in which modes were identified are used. Results indicate that this simple method can be used in analyzing complicated pressure signal measurements.

Customized Steady-State Constraints for Parameter Estimation in Non-Linear Ordinary Differential Equation Models

PubMed Central

Rosenblatt, Marcus; Timmer, Jens; Kaschek, Daniel

2016-01-01

Ordinary differential equation models have become a wide-spread approach to analyze dynamical systems and understand underlying mechanisms. Model parameters are often unknown and have to be estimated from experimental data, e.g., by maximum-likelihood estimation. In particular, models of biological systems contain a large number of parameters. To reduce the dimensionality of the parameter space, steady-state information is incorporated in the parameter estimation process. For non-linear models, analytical steady-state calculation typically leads to higher-order polynomial equations for which no closed-form solutions can be obtained. This can be circumvented by solving the steady-state equations for kinetic parameters, which results in a linear equation system with comparatively simple solutions. At the same time multiplicity of steady-state solutions is avoided, which otherwise is problematic for optimization. When solved for kinetic parameters, however, steady-state constraints tend to become negative for particular model specifications, thus, generating new types of optimization problems. Here, we present an algorithm based on graph theory that derives non-negative, analytical steady-state expressions by stepwise removal of cyclic dependencies between dynamical variables. The algorithm avoids multiple steady-state solutions by construction. We show that our method is applicable to most common classes of biochemical reaction networks containing inhibition terms, mass-action and Hill-type kinetic equations. Comparing the performance of parameter estimation for different analytical and numerical methods of incorporating steady-state information, we show that our approach is especially well-tailored to guarantee a high success rate of optimization. PMID:27243005

Customized Steady-State Constraints for Parameter Estimation in Non-Linear Ordinary Differential Equation Models.

PubMed

Rosenblatt, Marcus; Timmer, Jens; Kaschek, Daniel

2016-01-01

Ordinary differential equation models have become a wide-spread approach to analyze dynamical systems and understand underlying mechanisms. Model parameters are often unknown and have to be estimated from experimental data, e.g., by maximum-likelihood estimation. In particular, models of biological systems contain a large number of parameters. To reduce the dimensionality of the parameter space, steady-state information is incorporated in the parameter estimation process. For non-linear models, analytical steady-state calculation typically leads to higher-order polynomial equations for which no closed-form solutions can be obtained. This can be circumvented by solving the steady-state equations for kinetic parameters, which results in a linear equation system with comparatively simple solutions. At the same time multiplicity of steady-state solutions is avoided, which otherwise is problematic for optimization. When solved for kinetic parameters, however, steady-state constraints tend to become negative for particular model specifications, thus, generating new types of optimization problems. Here, we present an algorithm based on graph theory that derives non-negative, analytical steady-state expressions by stepwise removal of cyclic dependencies between dynamical variables. The algorithm avoids multiple steady-state solutions by construction. We show that our method is applicable to most common classes of biochemical reaction networks containing inhibition terms, mass-action and Hill-type kinetic equations. Comparing the performance of parameter estimation for different analytical and numerical methods of incorporating steady-state information, we show that our approach is especially well-tailored to guarantee a high success rate of optimization.

An overview of longitudinal data analysis methods for neurological research.

PubMed

Locascio, Joseph J; Atri, Alireza

2011-01-01

The purpose of this article is to provide a concise, broad and readily accessible overview of longitudinal data analysis methods, aimed to be a practical guide for clinical investigators in neurology. In general, we advise that older, traditional methods, including (1) simple regression of the dependent variable on a time measure, (2) analyzing a single summary subject level number that indexes changes for each subject and (3) a general linear model approach with a fixed-subject effect, should be reserved for quick, simple or preliminary analyses. We advocate the general use of mixed-random and fixed-effect regression models for analyses of most longitudinal clinical studies. Under restrictive situations or to provide validation, we recommend: (1) repeated-measure analysis of covariance (ANCOVA), (2) ANCOVA for two time points, (3) generalized estimating equations and (4) latent growth curve/structural equation models.

Exact linearized Coulomb collision operator in the moment expansion

DOE PAGES

Ji, Jeong -Young; Held, Eric D.

2006-10-05

In the moment expansion, the Rosenbluth potentials, the linearized Coulomb collision operators, and the moments of the collision operators are analytically calculated for any moment. The explicit calculation of Rosenbluth potentials converts the integro-differential form of the Coulomb collision operator into a differential operator, which enables one to express the collision operator in a simple closed form for any arbitrary mass and temperature ratios. In addition, it is shown that gyrophase averaging the collision operator acting on arbitrary distribution functions is the same as the collision operator acting on the corresponding gyrophase averaged distribution functions. The moments of the collisionmore » operator are linear combinations of the fluid moments with collision coefficients parametrized by mass and temperature ratios. Furthermore, useful forms involving the small mass-ratio approximation are easily found since the collision operators and their moments are expressed in terms of the mass ratio. As an application, the general moment equations are explicitly written and the higher order heat flux equation is derived.« less

Onset of Turbulence in a Pipe

NASA Astrophysics Data System (ADS)

Böberg, L.; Brösa, U.

1988-09-01

Turbulence in a pipe is derived directly from the Navier-Stokes equation. Analysis of numerical simulations revealed that small disturbances called 'mothers' induce other much stronger disturbances called 'daughters'. Daughters determine the look of turbulence, while mothers control the transfer of energy from the basic flow to the turbulent motion. From a practical point of view, ruling mothers means ruling turbulence. For theory, the mother-daughter process represents a mechanism permitting chaotic motion in a linearly stable system. The mechanism relies on a property of the linearized problem according to which the eigenfunctions become more and more collinear as the Reynolds number increases. The mathematical methods are described, comparisons with experiments are made, mothers and daughters are analyzed, also graphically, with full particulars, and the systematic construction of small systems of differential equations to mimic the non-linear process by means as simple as possible is explained. We suggest that more then 20 but less than 180 essential degrees of freedom take part in the onset of turbulence.

Numerical methods for solving moment equations in kinetic theory of neuronal network dynamics

NASA Astrophysics Data System (ADS)

Rangan, Aaditya V.; Cai, David; Tao, Louis

2007-02-01

Recently developed kinetic theory and related closures for neuronal network dynamics have been demonstrated to be a powerful theoretical framework for investigating coarse-grained dynamical properties of neuronal networks. The moment equations arising from the kinetic theory are a system of (1 + 1)-dimensional nonlinear partial differential equations (PDE) on a bounded domain with nonlinear boundary conditions. The PDEs themselves are self-consistently specified by parameters which are functions of the boundary values of the solution. The moment equations can be stiff in space and time. Numerical methods are presented here for efficiently and accurately solving these moment equations. The essential ingredients in our numerical methods include: (i) the system is discretized in time with an implicit Euler method within a spectral deferred correction framework, therefore, the PDEs of the kinetic theory are reduced to a sequence, in time, of boundary value problems (BVPs) with nonlinear boundary conditions; (ii) a set of auxiliary parameters is introduced to recast the original BVP with nonlinear boundary conditions as BVPs with linear boundary conditions - with additional algebraic constraints on the auxiliary parameters; (iii) a careful combination of two Newton's iterates for the nonlinear BVP with linear boundary condition, interlaced with a Newton's iterate for solving the associated algebraic constraints is constructed to achieve quadratic convergence for obtaining the solutions with self-consistent parameters. It is shown that a simple fixed-point iteration can only achieve a linear convergence for the self-consistent parameters. The practicability and efficiency of our numerical methods for solving the moment equations of the kinetic theory are illustrated with numerical examples. It is further demonstrated that the moment equations derived from the kinetic theory of neuronal network dynamics can very well capture the coarse-grained dynamical properties of integrate-and-fire neuronal networks.

Chemical association in simple models of molecular and ionic fluids. III. The cavity function

NASA Astrophysics Data System (ADS)

Zhou, Yaoqi; Stell, George

1992-01-01

Exact equations which relate the cavity function to excess solvation free energies and equilibrium association constants are rederived by using a thermodynamic cycle. A zeroth-order approximation, derived previously by us as a simple interpolation scheme, is found to be very accurate if the associative bonding occurs on or near the surface of the repulsive core of the interaction potential. If the bonding radius is substantially less than the core radius, the approximation overestimates the association degree and the association constant. For binary association, the zeroth-order approximation is equivalent to the first-order thermodynamic perturbation theory (TPT) of Wertheim. For n-particle association, the combination of the zeroth-order approximation with a ``linear'' approximation (for n-particle distribution functions in terms of the two-particle function) yields the first-order TPT result. Using our exact equations to go beyond TPT, near-exact analytic results for binary hard-sphere association are obtained. Solvent effects on binary hard-sphere association and ionic association are also investigated. A new rule which generalizes Le Chatelier's principle is used to describe the three distinct forms of behaviors involving solvent effects that we find. The replacement of the dielectric-continuum solvent model by a dipolar hard-sphere model leads to improved agreement with an experimental observation. Finally, equation of state for an n-particle flexible linear-chain fluid is derived on the basis of a one-parameter approximation that interpolates between the generalized Kirkwood superposition approximation and the linear approximation. A value of the parameter that appears to be near optimal in the context of this application is obtained from comparison with computer-simulation data.

MUSTA fluxes for systems of conservation laws

NASA Astrophysics Data System (ADS)

Toro, E. F.; Titarev, V. A.

2006-08-01

This paper is about numerical fluxes for hyperbolic systems and we first present a numerical flux, called GFORCE, that is a weighted average of the Lax-Friedrichs and Lax-Wendroff fluxes. For the linear advection equation with constant coefficient, the new flux reduces identically to that of the Godunov first-order upwind method. Then we incorporate GFORCE in the framework of the MUSTA approach [E.F. Toro, Multi-Stage Predictor-Corrector Fluxes for Hyperbolic Equations. Technical Report NI03037-NPA, Isaac Newton Institute for Mathematical Sciences, University of Cambridge, UK, 17th June, 2003], resulting in a version that we call GMUSTA. For non-linear systems this gives results that are comparable to those of the Godunov method in conjunction with the exact Riemann solver or complete approximate Riemann solvers, noting however that in our approach, the solution of the Riemann problem in the conventional sense is avoided. Both the GFORCE and GMUSTA fluxes are extended to multi-dimensional non-linear systems in a straightforward unsplit manner, resulting in linearly stable schemes that have the same stability regions as the straightforward multi-dimensional extension of Godunov's method. The methods are applicable to general meshes. The schemes of this paper share with the family of centred methods the common properties of being simple and applicable to a large class of hyperbolic systems, but the schemes of this paper are distinctly more accurate. Finally, we proceed to the practical implementation of our numerical fluxes in the framework of high-order finite volume WENO methods for multi-dimensional non-linear hyperbolic systems. Numerical results are presented for the Euler equations and for the equations of magnetohydrodynamics.

Designing with non-linear viscoelastic fluids

NASA Astrophysics Data System (ADS)

Schuh, Jonathon; Lee, Yong Hoon; Allison, James; Ewoldt, Randy

2017-11-01

Material design is typically limited to hard materials or simple fluids; however, design with more complex materials can provide ways to enhance performance. Using the Criminale-Ericksen-Filbey (CEF) constitutive model in the thin film lubrication limit, we derive a modified Reynolds Equation (based on asymptotic analysis) that includes shear thinning, first normal stress, and terminal regime viscoelastic effects. This allows for designing non-linear viscoelastic fluids in thin-film creeping flow scenarios, i.e. optimizing the shape of rheological material properties to achieve different design objectives. We solve the modified Reynolds equation using the pseudo-spectral method, and describe a case study in full-film lubricated sliding where optimal fluid properties are identified. These material-agnostic property targets can then guide formulation of complex fluids which may use polymeric, colloidal, or other creative approaches to achieve the desired non-Newtonian properties.

HCMM satellite follow-on investigation no. 25. Soil moisture and heat budget evalution in selected European zones of agricultural and environmental interest (TELLUS project)

NASA Technical Reports Server (NTRS)

1980-01-01

A simple procedure to evaluate actual evaporation was derived by linearizing the surface energy balance equation, using Taylor's expansion. The original multidimensional hypersurface could be reduced to a linear relationship between evaporation and surface temperature or to a surface relationship involving evaporation, surface temperature and albedo. This procedure permits a rapid sensitivity analysis of the surface energy balance equation as well as a speedy mapping of evaporation from remotely sensed surface temperatures and albedo. Comparison with experimental data yielded promising results. The validity of evapotranspiration and soil moisture models in semiarid conditions was tested. Wheat was the crop chosen for a continuous measurement campaign made in the south of Italy. Radiometric, micrometeorologic, agronomic and soil data were collected for processing and interpretation.

Exploring inductive linearization for pharmacokinetic-pharmacodynamic systems of nonlinear ordinary differential equations.

PubMed

Hasegawa, Chihiro; Duffull, Stephen B

2018-02-01

Pharmacokinetic-pharmacodynamic systems are often expressed with nonlinear ordinary differential equations (ODEs). While there are numerous methods to solve such ODEs these methods generally rely on time-stepping solutions (e.g. Runge-Kutta) which need to be matched to the characteristics of the problem at hand. The primary aim of this study was to explore the performance of an inductive approximation which iteratively converts nonlinear ODEs to linear time-varying systems which can then be solved algebraically or numerically. The inductive approximation is applied to three examples, a simple nonlinear pharmacokinetic model with Michaelis-Menten elimination (E1), an integrated glucose-insulin model and an HIV viral load model with recursive feedback systems (E2 and E3, respectively). The secondary aim of this study was to explore the potential advantages of analytically solving linearized ODEs with two examples, again E3 with stiff differential equations and a turnover model of luteinizing hormone with a surge function (E4). The inductive linearization coupled with a matrix exponential solution provided accurate predictions for all examples with comparable solution time to the matched time-stepping solutions for nonlinear ODEs. The time-stepping solutions however did not perform well for E4, particularly when the surge was approximated by a square wave. In circumstances when either a linear ODE is particularly desirable or the uncertainty in matching the integrator to the ODE system is of potential risk, then the inductive approximation method coupled with an analytical integration method would be an appropriate alternative.

Prediction equation for calculating fat mass in young Indian adults.

PubMed

Sandhu, Jaspal Singh; Gupta, Giniya; Shenoy, Shweta

2010-06-01

Accurate measurement or prediction of fat mass is useful in physiology, nutrition and clinical medicine. Most predictive equations currently used to assess percentage of body fat or fat mass, using simple anthropometric measurements were derived from people in western societies and they may not be appropriate for individuals with other genotypic and phenotypic characteristics. We developed equations to predict fat mass from anthropometric measurements in young Indian adults. Fat mass was measured in 60 females and 58 males, aged 20 to 29 yrs by using hydrostatic weighing and by simultaneous measurement of residual lung volume. Anthropometric measure included weight (kg), height (m) and 4 skinfold thickness [STs (mm)]. Sex specific linear regression model was developed with fat mass as the dependent variable and all anthropometric measures as independent variables. The prediction equation obtained for fat mass (kg) for males was 8.46+0.32 (weight) - 15.16 (height) + 9.54 (log of sum of 4 STs) (R2= 0. 53, SEE=3.42 kg) and - 20.22 + 0.33 (weight) + 3.44 (height) + 7.66 (log of sum of 4 STs) (R2=0.72, SEE=3.01kg) for females. A new prediction equation for the measurement of fat mass was derived and internally validated in young Indian adults using simple anthropometric measurements.

Algebraic approach to solve ttbar dilepton equations

NASA Astrophysics Data System (ADS)

Sonnenschein, Lars

2006-01-01

The set of non-linear equations describing the Standard Model kinematics of the top quark an- tiqark production system in the dilepton decay channel has at most a four-fold ambiguity due to two not fully reconstructed neutrinos. Its most precise and robust solution is of major importance for measurements of top quark properties like the top quark mass and t t spin correlations. Simple algebraic operations allow to transform the non-linear equations into a system of two polynomial equations with two unknowns. These two polynomials of multidegree eight can in turn be an- alytically reduced to one polynomial with one unknown by means of resultants. The obtained univariate polynomial is of degree sixteen and the coefficients are free of any singularity. The number of its real solutions is determined analytically by means of Sturm’s theorem, which is as well used to isolate each real solution into a unique pairwise disjoint interval. The solutions are polished by seeking the sign change of the polynomial in a given interval through binary brack- eting. Further a new Ansatz - exploiting an accidental cancelation in the process of transforming the equations - is presented. It permits to transform the initial system of equations into two poly- nomial equations with two unknowns. These two polynomials of multidegree two can be reduced to one univariate polynomial of degree four by means of resultants. The obtained quartic equation can be solved analytically. The analytical solution has singularities which can be circumvented by the algebraic approach described above.

Relation of the external mechanical stress to the properties of piezoelectric materials for energy harvesting

NASA Astrophysics Data System (ADS)

Jeong, Soon-Jong; Kim, Min-Soo; Lee, Dae-Su; Song, Jae-Sung; Cho, Kyung-Ho

2013-12-01

We investigated the piezoelectric properties and the generation of voltage and power under the mechanical compressive loads for three types of piezoelectric ceramics 0.2Pb(Mg1/3Nb2/3)O3-0.8Pb(Zr0.475Ti0.525)O3 (soft-PZT), 0.1Pb(Mg1/3Sb2/3)O3- 0.9Pb(Zr0.475Ti0.525)O3 (hard-PZT) and [0.675Pb(Mg1/3Nb2/3)O3-0.35PbTiO3]+5 wt% BaTiO3 (textured-PMNT). The piezoelectric d 33 coefficients of all specimens increased with increasing compressive load. The generated voltage and power showed a linear relation and square relation to the applied stress, respectively. These results were larger than those calculated using the simple piezoelectric equation due to the non-linear characteristics of the ceramics, so they were evaluated with a simple model based on a non-linear relation.

SU-E-T-75: A Simple Technique for Proton Beam Range Verification

DOE Office of Scientific and Technical Information (OSTI.GOV)

Burgdorf, B; Kassaee, A; Garver, E

2015-06-15

Purpose: To develop a measurement-based technique to verify the range of proton beams for quality assurance (QA). Methods: We developed a simple technique to verify the proton beam range with in-house fabricated devices. Two separate devices were fabricated; a clear acrylic rectangular cuboid and a solid polyvinyl chloride (PVC) step wedge. For efficiency in our clinic, we used the rectangular cuboid for double scattering (DS) beams and the step wedge for pencil beam scanning (PBS) beams. These devices were added to our QA phantom to measure dose points along the distal fall-off region (between 80% and 20%) in addition tomore » dose at mid-SOBP (spread out Bragg peak) using a two-dimensional parallel plate chamber array (MatriXX™, IBA Dosimetry, Schwarzenbruck, Germany). This method relies on the fact that the slope of the distal fall-off is linear and does not vary with small changes in energy. Using a multi-layer ionization chamber (Zebra™, IBA Dosimetry), percent depth dose (PDD) curves were measured for our standard daily QA beams. The range (energy) for each beam was then varied (i.e. ±2mm and ±5mm) and additional PDD curves were measured. The distal fall-off of all PDD curves was fit to a linear equation. The distal fall-off measured dose for a particular beam was used in our linear equation to determine the beam range. Results: The linear fit of the fall-off region for the PDD curves, when varying the range by a few millimeters for a specific QA beam, yielded identical slopes. The calculated range based on measured point dose(s) in the fall-off region using the slope resulted in agreement of ±1mm of the expected beam range. Conclusion: We developed a simple technique for accurately verifying the beam range for proton therapy QA programs.« less

A high-fidelity method to analyze perturbation evolution in turbulent flows

DOE Office of Scientific and Technical Information (OSTI.GOV)

Unnikrishnan, S., E-mail: sasidharannair.1@osu.edu; Gaitonde, Datta V., E-mail: gaitonde.3@osu.edu

2016-04-01

Small perturbation propagation in fluid flows is usually examined by linearizing the governing equations about a steady basic state. It is often useful, however, to study perturbation evolution in the unsteady evolving turbulent environment. Such analyses can elucidate the role of perturbations in the generation of coherent structures or the production of noise from jet turbulence. The appropriate equations are still the linearized Navier–Stokes equations, except that the linearization must be performed about the instantaneous evolving turbulent state, which forms the coefficients of the linearized equations. This is a far more difficult problem since in addition to the turbulent state,more » its rate of change and the perturbation field are all required at each instant. In this paper, we develop and use a novel technique for this problem by using a pair (denoted “baseline” and “twin”) of simultaneous synchronized Large-Eddy Simulations (LES). At each time-step, small disturbances whose propagation characteristics are to be studied, are introduced into the twin through a forcing term. At subsequent time steps, the difference between the two simulations is shown to be equivalent to solving the forced Navier–Stokes equations, linearized about the instantaneous turbulent state. The technique does not put constraints on the forcing, which could be arbitrary, e.g., white noise or other stochastic variants. We consider, however, “native” forcing having properties of disturbances that exist naturally in the turbulent environment. The method then isolates the effect of turbulence in a particular region on the rest of the field, which is useful in the study of noise source localization. The synchronized technique is relatively simple to implement into existing codes. In addition to minimizing the storage and retrieval of large time-varying datasets, it avoids the need to explicitly linearize the governing equations, which can be a very complicated task for viscous terms or turbulence closures. The method is illustrated by application to a well-validated Mach 1.3 jet. Specifically, the effects of turbulence on the jet lipline and core collapse regions on the near-acoustic field are isolated. The properties of the method, including linearity and effect of initial transients, are discussed. The results provide insight into how turbulence from different parts of the jet contribute to the observed dominance of low and high frequency content at shallow and sideline angles, respectively.« less

An approximation of herd effect due to vaccinating children against seasonal influenza – a potential solution to the incorporation of indirect effects into static models

PubMed Central

2013-01-01

Background Indirect herd effect from vaccination of children offers potential for improving the effectiveness of influenza prevention in the remaining unvaccinated population. Static models used in cost-effectiveness analyses cannot dynamically capture herd effects. The objective of this study was to develop a methodology to allow herd effect associated with vaccinating children against seasonal influenza to be incorporated into static models evaluating the cost-effectiveness of influenza vaccination. Methods Two previously published linear equations for approximation of herd effects in general were compared with the results of a structured literature review undertaken using PubMed searches to identify data on herd effects specific to influenza vaccination. A linear function was fitted to point estimates from the literature using the sum of squared residuals. Results The literature review identified 21 publications on 20 studies for inclusion. Six studies provided data on a mathematical relationship between effective vaccine coverage in subgroups and reduction of influenza infection in a larger unvaccinated population. These supported a linear relationship when effective vaccine coverage in a subgroup population was between 20% and 80%. Three studies evaluating herd effect at a community level, specifically induced by vaccinating children, provided point estimates for fitting linear equations. The fitted linear equation for herd protection in the target population for vaccination (children) was slightly less conservative than a previously published equation for herd effects in general. The fitted linear equation for herd protection in the non-target population was considerably less conservative than the previously published equation. Conclusions This method of approximating herd effect requires simple adjustments to the annual baseline risk of influenza in static models: (1) for the age group targeted by the childhood vaccination strategy (i.e. children); and (2) for other age groups not targeted (e.g. adults and/or elderly). Two approximations provide a linear relationship between effective coverage and reduction in the risk of infection. The first is a conservative approximation, recommended as a base-case for cost-effectiveness evaluations. The second, fitted to data extracted from a structured literature review, provides a less conservative estimate of herd effect, recommended for sensitivity analyses. PMID:23339290

A high-fidelity method to analyze perturbation evolution in turbulent flows

NASA Astrophysics Data System (ADS)

Unnikrishnan, S.; Gaitonde, Datta V.

2016-04-01

Small perturbation propagation in fluid flows is usually examined by linearizing the governing equations about a steady basic state. It is often useful, however, to study perturbation evolution in the unsteady evolving turbulent environment. Such analyses can elucidate the role of perturbations in the generation of coherent structures or the production of noise from jet turbulence. The appropriate equations are still the linearized Navier-Stokes equations, except that the linearization must be performed about the instantaneous evolving turbulent state, which forms the coefficients of the linearized equations. This is a far more difficult problem since in addition to the turbulent state, its rate of change and the perturbation field are all required at each instant. In this paper, we develop and use a novel technique for this problem by using a pair (denoted "baseline" and "twin") of simultaneous synchronized Large-Eddy Simulations (LES). At each time-step, small disturbances whose propagation characteristics are to be studied, are introduced into the twin through a forcing term. At subsequent time steps, the difference between the two simulations is shown to be equivalent to solving the forced Navier-Stokes equations, linearized about the instantaneous turbulent state. The technique does not put constraints on the forcing, which could be arbitrary, e.g., white noise or other stochastic variants. We consider, however, "native" forcing having properties of disturbances that exist naturally in the turbulent environment. The method then isolates the effect of turbulence in a particular region on the rest of the field, which is useful in the study of noise source localization. The synchronized technique is relatively simple to implement into existing codes. In addition to minimizing the storage and retrieval of large time-varying datasets, it avoids the need to explicitly linearize the governing equations, which can be a very complicated task for viscous terms or turbulence closures. The method is illustrated by application to a well-validated Mach 1.3 jet. Specifically, the effects of turbulence on the jet lipline and core collapse regions on the near-acoustic field are isolated. The properties of the method, including linearity and effect of initial transients, are discussed. The results provide insight into how turbulence from different parts of the jet contribute to the observed dominance of low and high frequency content at shallow and sideline angles, respectively.

Evaluation of geopotential and luni-solar perturbations by a recursive algorithm

NASA Technical Reports Server (NTRS)

Giacaglia, G. E. O.

1975-01-01

The disturbing functions due to the geopotential and Luni-solar attractions are linear and bilinear forms in spherical harmonics. Making use of recurrence relations for the solid spherical harmonics and their derivatives, recurrence formulas are obtained for high degree terms as function of lower degree for any term of those disturbing functions and their derivative with respect to any element. The equations obtained are effective when a numerical integration of the equations of motion is appropriate. In analytical theories, they provide a fast way of obtaining high degree terms starting from initial very simple functions.

Stochastic oscillations in models of epidemics on a network of cities

NASA Astrophysics Data System (ADS)

Rozhnova, G.; Nunes, A.; McKane, A. J.

2011-11-01

We carry out an analytic investigation of stochastic oscillations in a susceptible-infected-recovered model of disease spread on a network of n cities. In the model a fraction fjk of individuals from city k commute to city j, where they may infect, or be infected by, others. Starting from a continuous-time Markov description of the model the deterministic equations, which are valid in the limit when the population of each city is infinite, are recovered. The stochastic fluctuations about the fixed point of these equations are derived by use of the van Kampen system-size expansion. The fixed point structure of the deterministic equations is remarkably simple: A unique nontrivial fixed point always exists and has the feature that the fraction of susceptible, infected, and recovered individuals is the same for each city irrespective of its size. We find that the stochastic fluctuations have an analogously simple dynamics: All oscillations have a single frequency, equal to that found in the one-city case. We interpret this phenomenon in terms of the properties of the spectrum of the matrix of the linear approximation of the deterministic equations at the fixed point.

The Canopy Conductance of a Humid Grassland

NASA Astrophysics Data System (ADS)

Lu, C. T.; Hsieh, C. I.

2015-12-01

Penman-Monteith equation is widely used for estimating latent heat flux. The key parameter for implementing this equation is the canopy conductance (gc). Recent research (Blaken and Black, 2004) showed that gc could be well parameterized by a linear function of An/ (D0* X0c), where An represents net assimilation, D0 is leaf level saturation deficit, and X0c is CO2 mole fraction. In this study, we tried to use the same idea for estimating gcfor a humid grassland. The study site was located in County Cork, southwest Ireland (51o59''N 8o46''W), and perennial ryegrass (Lolium perenne L.) was the dominant grass species in this area. An eddy covariance system was used to measure the latent heat flux above this humid grassland. The measured gc was calculated by rearranging Penman-Monteith equation combined with the measured latent heat flux. Our data showed that the gc decreased as the vapor pressure deficit and temperature increased. And it increased as the net radiation increased. Therefore, we found out that the best parameterization of gc was a linear function of the product of the vapor deficit, temperature, and net radiation. Also, we used the gc which was estimated by this linear function to predict the latent heat flux by Penman-Monteith equation and compared the predictions with those where the gc was chosen to be a fixed value. Our analysis showed that this simple linear function for gc can improve the latent heat flux predictions (R square increased from 0.48 to 0.66).

Tensor-GMRES method for large sparse systems of nonlinear equations

NASA Technical Reports Server (NTRS)

Feng, Dan; Pulliam, Thomas H.

1994-01-01

This paper introduces a tensor-Krylov method, the tensor-GMRES method, for large sparse systems of nonlinear equations. This method is a coupling of tensor model formation and solution techniques for nonlinear equations with Krylov subspace projection techniques for unsymmetric systems of linear equations. Traditional tensor methods for nonlinear equations are based on a quadratic model of the nonlinear function, a standard linear model augmented by a simple second order term. These methods are shown to be significantly more efficient than standard methods both on nonsingular problems and on problems where the Jacobian matrix at the solution is singular. A major disadvantage of the traditional tensor methods is that the solution of the tensor model requires the factorization of the Jacobian matrix, which may not be suitable for problems where the Jacobian matrix is large and has a 'bad' sparsity structure for an efficient factorization. We overcome this difficulty by forming and solving the tensor model using an extension of a Newton-GMRES scheme. Like traditional tensor methods, we show that the new tensor method has significant computational advantages over the analogous Newton counterpart. Consistent with Krylov subspace based methods, the new tensor method does not depend on the factorization of the Jacobian matrix. As a matter of fact, the Jacobian matrix is never needed explicitly.

Consistent three-equation model for thin films

NASA Astrophysics Data System (ADS)

Richard, Gael; Gisclon, Marguerite; Ruyer-Quil, Christian; Vila, Jean-Paul

2017-11-01

Numerical simulations of thin films of newtonian fluids down an inclined plane use reduced models for computational cost reasons. These models are usually derived by averaging over the fluid depth the physical equations of fluid mechanics with an asymptotic method in the long-wave limit. Two-equation models are based on the mass conservation equation and either on the momentum balance equation or on the work-energy theorem. We show that there is no two-equation model that is both consistent and theoretically coherent and that a third variable and a three-equation model are required to solve all theoretical contradictions. The linear and nonlinear properties of two and three-equation models are tested on various practical problems. We present a new consistent three-equation model with a simple mathematical structure which allows an easy and reliable numerical resolution. The numerical calculations agree fairly well with experimental measurements or with direct numerical resolutions for neutral stability curves, speed of kinematic waves and of solitary waves and depth profiles of wavy films. The model can also predict the flow reversal at the first capillary trough ahead of the main wave hump.

Cosmological perturbations in mimetic Horndeski gravity

NASA Astrophysics Data System (ADS)

Arroja, Frederico; Bartolo, Nicola; Karmakar, Purnendu; Matarrese, Sabino

2016-04-01

We study linear scalar perturbations around a flat FLRW background in mimetic Horndeski gravity. In the absence of matter, we show that the Newtonian potential satisfies a second-order differential equation with no spatial derivatives. This implies that the sound speed for scalar perturbations is exactly zero on this background. We also show that in mimetic G3 theories the sound speed is equally zero. We obtain the equation of motion for the comoving curvature perturbation (first order differential equation) and solve it to find that the comoving curvature perturbation is constant on all scales in mimetic Horndeski gravity. We find solutions for the Newtonian potential evolution equation in two simple models. Finally we show that the sound speed is zero on all backgrounds and therefore the system does not have any wave-like scalar degrees of freedom.

Spatio-temporal dynamics induced by competing instabilities in two asymmetrically coupled nonlinear evolution equations.

PubMed

Schüler, D; Alonso, S; Torcini, A; Bär, M

2014-12-01

Pattern formation often occurs in spatially extended physical, biological, and chemical systems due to an instability of the homogeneous steady state. The type of the instability usually prescribes the resulting spatio-temporal patterns and their characteristic length scales. However, patterns resulting from the simultaneous occurrence of instabilities cannot be expected to be simple superposition of the patterns associated with the considered instabilities. To address this issue, we design two simple models composed by two asymmetrically coupled equations of non-conserved (Swift-Hohenberg equations) or conserved (Cahn-Hilliard equations) order parameters with different characteristic wave lengths. The patterns arising in these systems range from coexisting static patterns of different wavelengths to traveling waves. A linear stability analysis allows to derive a two parameter phase diagram for the studied models, in particular, revealing for the Swift-Hohenberg equations, a co-dimension two bifurcation point of Turing and wave instability and a region of coexistence of stationary and traveling patterns. The nonlinear dynamics of the coupled evolution equations is investigated by performing accurate numerical simulations. These reveal more complex patterns, ranging from traveling waves with embedded Turing patterns domains to spatio-temporal chaos, and a wide hysteretic region, where waves or Turing patterns coexist. For the coupled Cahn-Hilliard equations the presence of a weak coupling is sufficient to arrest the coarsening process and to lead to the emergence of purely periodic patterns. The final states are characterized by domains with a characteristic length, which diverges logarithmically with the coupling amplitude.

Vibration based algorithm for crack detection in cantilever beam containing two different types of cracks

NASA Astrophysics Data System (ADS)

Behzad, Mehdi; Ghadami, Amin; Maghsoodi, Ameneh; Michael Hale, Jack

2013-11-01

In this paper, a simple method for detection of multiple edge cracks in Euler-Bernoulli beams having two different types of cracks is presented based on energy equations. Each crack is modeled as a massless rotational spring using Linear Elastic Fracture Mechanics (LEFM) theory, and a relationship among natural frequencies, crack locations and stiffness of equivalent springs is demonstrated. In the procedure, for detection of m cracks in a beam, 3m equations and natural frequencies of healthy and cracked beam in two different directions are needed as input to the algorithm. The main accomplishment of the presented algorithm is the capability to detect the location, severity and type of each crack in a multi-cracked beam. Concise and simple calculations along with accuracy are other advantages of this method. A number of numerical examples for cantilever beams including one and two cracks are presented to validate the method.

An Overview of Longitudinal Data Analysis Methods for Neurological Research

PubMed Central

Locascio, Joseph J.; Atri, Alireza

2011-01-01

The purpose of this article is to provide a concise, broad and readily accessible overview of longitudinal data analysis methods, aimed to be a practical guide for clinical investigators in neurology. In general, we advise that older, traditional methods, including (1) simple regression of the dependent variable on a time measure, (2) analyzing a single summary subject level number that indexes changes for each subject and (3) a general linear model approach with a fixed-subject effect, should be reserved for quick, simple or preliminary analyses. We advocate the general use of mixed-random and fixed-effect regression models for analyses of most longitudinal clinical studies. Under restrictive situations or to provide validation, we recommend: (1) repeated-measure analysis of covariance (ANCOVA), (2) ANCOVA for two time points, (3) generalized estimating equations and (4) latent growth curve/structural equation models. PMID:22203825

Simple method for quick estimation of aquifer hydrogeological parameters

NASA Astrophysics Data System (ADS)

Ma, C.; Li, Y. Y.

2017-08-01

Development of simple and accurate methods to determine the aquifer hydrogeological parameters was of importance for groundwater resources assessment and management. Aiming at the present issue of estimating aquifer parameters based on some data of the unsteady pumping test, a fitting function of Theis well function was proposed using fitting optimization method and then a unitary linear regression equation was established. The aquifer parameters could be obtained by solving coefficients of the regression equation. The application of the proposed method was illustrated, using two published data sets. By the error statistics and analysis on the pumping drawdown, it showed that the method proposed in this paper yielded quick and accurate estimates of the aquifer parameters. The proposed method could reliably identify the aquifer parameters from long distance observed drawdowns and early drawdowns. It was hoped that the proposed method in this paper would be helpful for practicing hydrogeologists and hydrologists.

Deriving the Cost of Software Maintenance for Software Intensive Systems

DTIC Science & Technology

2011-08-29

more of software maintenance). Figure 4. SEER-SEM Maintenance Effort by Year Report (Reifer, Allen, Fersch, Hitchings, Judy , & Rosa, 2010...understand the linear relationship between two variables. The formula for the simple Pearson product-moment correlation is represented in Equation 5...standardization is required across the software maintenance community in order to ensure that the data being recorded can be employed beyond the agency or

A sliding mode control proposal for open-loop unstable processes.

PubMed

Rojas, Rubén; Camacho, Oscar; González, Luis

2004-04-01

This papers presents a sliding mode controller based on a first-order-plus-dead-time model of the process for controlling open-loop unstable systems. The proposed controller has a simple and fixed structure with a set of tuning equations as a function of the desired performance. Both linear and nonlinear models were used to study the controller performance by computer simulations.

On the Rate of Relaxation for the Landau Kinetic Equation and Related Models

NASA Astrophysics Data System (ADS)

Bobylev, Alexander; Gamba, Irene M.; Zhang, Chenglong

2017-08-01

We study the rate of relaxation to equilibrium for Landau kinetic equation and some related models by considering the relatively simple case of radial solutions of the linear Landau-type equations. The well-known difficulty is that the evolution operator has no spectral gap, i.e. its spectrum is not separated from zero. Hence we do not expect purely exponential relaxation for large values of time t>0. One of the main goals of our work is to numerically identify the large time asymptotics for the relaxation to equilibrium. We recall the work of Strain and Guo (Arch Rat Mech Anal 187:287-339 2008, Commun Partial Differ Equ 31:17-429 2006), who rigorously show that the expected law of relaxation is \\exp (-ct^{2/3}) with some c > 0. In this manuscript, we find an heuristic way, performed by asymptotic methods, that finds this "law of two thirds", and then study this question numerically. More specifically, the linear Landau equation is approximated by a set of ODEs based on expansions in generalized Laguerre polynomials. We analyze the corresponding quadratic form and the solution of these ODEs in detail. It is shown that the solution has two different asymptotic stages for large values of time t and maximal order of polynomials N: the first one focus on intermediate asymptotics which agrees with the "law of two thirds" for moderately large values of time t and then the second one on absolute, purely exponential asymptotics for very large t, as expected for linear ODEs. We believe that appearance of intermediate asymptotics in finite dimensional approximations must be a generic behavior for different classes of equations in functional spaces (some PDEs, Boltzmann equations for soft potentials, etc.) and that our methods can be applied to related problems.

Bounded Linear Stability Analysis - A Time Delay Margin Estimation Approach for Adaptive Control

NASA Technical Reports Server (NTRS)

Nguyen, Nhan T.; Ishihara, Abraham K.; Krishnakumar, Kalmanje Srinlvas; Bakhtiari-Nejad, Maryam

2009-01-01

This paper presents a method for estimating time delay margin for model-reference adaptive control of systems with almost linear structured uncertainty. The bounded linear stability analysis method seeks to represent the conventional model-reference adaptive law by a locally bounded linear approximation within a small time window using the comparison lemma. The locally bounded linear approximation of the combined adaptive system is cast in a form of an input-time-delay differential equation over a small time window. The time delay margin of this system represents a local stability measure and is computed analytically by a matrix measure method, which provides a simple analytical technique for estimating an upper bound of time delay margin. Based on simulation results for a scalar model-reference adaptive control system, both the bounded linear stability method and the matrix measure method are seen to provide a reasonably accurate and yet not too conservative time delay margin estimation.

Experimental and analytical investigation of inertial propulsion mechanisms and motion simulation of rigid multi-body mechanical systems

NASA Astrophysics Data System (ADS)

Almesallmy, Mohammed

Methodologies are developed for dynamic analysis of mechanical systems with emphasis on inertial propulsion systems. This work adopted the Lagrangian methodology. Lagrangian methodology is the most efficient classical computational technique, which we call Equations of Motion Code (EOMC). The EOMC is applied to several simple dynamic mechanical systems for easier understanding of the method and to aid other investigators in developing equations of motion of any dynamic system. In addition, it is applied to a rigid multibody system, such as Thomson IPS [Thomson 1986]. Furthermore, a simple symbolic algorithm is developed using Maple software, which can be used to convert any nonlinear n-order ordinary differential equation (ODE) systems into 1st-order ODE system in ready format to be used in Matlab software. A side issue, but equally important, we have started corresponding with the U.S. Patent office to persuade them that patent applications, claiming gross linear motion based on inertial propulsion systems should be automatically rejected. The precedent is rejection of patent applications involving perpetual motion machines.

Raney Distributions and Random Matrix Theory

NASA Astrophysics Data System (ADS)

Forrester, Peter J.; Liu, Dang-Zheng

2015-03-01

Recent works have shown that the family of probability distributions with moments given by the Fuss-Catalan numbers permit a simple parameterized form for their density. We extend this result to the Raney distribution which by definition has its moments given by a generalization of the Fuss-Catalan numbers. Such computations begin with an algebraic equation satisfied by the Stieltjes transform, which we show can be derived from the linear differential equation satisfied by the characteristic polynomial of random matrix realizations of the Raney distribution. For the Fuss-Catalan distribution, an equilibrium problem characterizing the density is identified. The Stieltjes transform for the limiting spectral density of the singular values squared of the matrix product formed from inverse standard Gaussian matrices, and standard Gaussian matrices, is shown to satisfy a variant of the algebraic equation relating to the Raney distribution. Supported on , we show that it too permits a simple functional form upon the introduction of an appropriate choice of parameterization. As an application, the leading asymptotic form of the density as the endpoints of the support are approached is computed, and is shown to have some universal features.

Effect of capillary forces on the nonstationary fall of a drop in an infinite fluid

NASA Astrophysics Data System (ADS)

Antanovskii, L. K.

1991-12-01

An explicit solution is presented for the linear problem concerning the motion of a drop in an infinite fluid in the presence of any number of surfactants (chemical reactions are not considered in the first approximation). It is shown that the behavior of the system considered is consistent with the Le Chatelier principle. The reactivity of the capillary forces is directly related to the fundamental principles of thermodynamics, which makes it possible to write equations of surfactant thermodiffusion in symmetric form and obtain a relatively simple solution to the linearized problem.

Triangle based TVD schemes for hyperbolic conservation laws

NASA Technical Reports Server (NTRS)

Durlofsky, Louis J.; Osher, Stanley; Engquist, Bjorn

1990-01-01

A triangle based total variation diminishing (TVD) scheme for the numerical approximation of hyperbolic conservation laws in two space dimensions is constructed. The novelty of the scheme lies in the nature of the preprocessing of the cell averaged data, which is accomplished via a nearest neighbor linear interpolation followed by a slope limiting procedures. Two such limiting procedures are suggested. The resulting method is considerably more simple than other triangle based non-oscillatory approximations which, like this scheme, approximate the flux up to second order accuracy. Numerical results for linear advection and Burgers' equation are presented.

Interpreting experimental data on egg production--applications of dynamic differential equations.

PubMed

France, J; Lopez, S; Kebreab, E; Dijkstra, J

2013-09-01

This contribution focuses on applying mathematical models based on systems of ordinary first-order differential equations to synthesize and interpret data from egg production experiments. Models based on linear systems of differential equations are contrasted with those based on nonlinear systems. Regression equations arising from analytical solutions to linear compartmental schemes are considered as candidate functions for describing egg production curves, together with aspects of parameter estimation. Extant candidate functions are reviewed, a role for growth functions such as the Gompertz equation suggested, and a function based on a simple new model outlined. Structurally, the new model comprises a single pool with an inflow and an outflow. Compartmental simulation models based on nonlinear systems of differential equations, and thus requiring numerical solution, are next discussed, and aspects of parameter estimation considered. This type of model is illustrated in relation to development and evaluation of a dynamic model of calcium and phosphorus flows in layers. The model consists of 8 state variables representing calcium and phosphorus pools in the crop, stomachs, plasma, and bone. The flow equations are described by Michaelis-Menten or mass action forms. Experiments that measure Ca and P uptake in layers fed different calcium concentrations during shell-forming days are used to evaluate the model. In addition to providing a useful management tool, such a simulation model also provides a means to evaluate feeding strategies aimed at reducing excretion of potential pollutants in poultry manure to the environment.

Modeling of active transmembrane transport in a mixture theory framework.

PubMed

Ateshian, Gerard A; Morrison, Barclay; Hung, Clark T

2010-05-01

This study formulates governing equations for active transport across semi-permeable membranes within the framework of the theory of mixtures. In mixture theory, which models the interactions of any number of fluid and solid constituents, a supply term appears in the conservation of linear momentum to describe momentum exchanges among the constituents. In past applications, this momentum supply was used to model frictional interactions only, thereby describing passive transport processes. In this study, it is shown that active transport processes, which impart momentum to solutes or solvent, may also be incorporated in this term. By projecting the equation of conservation of linear momentum along the normal to the membrane, a jump condition is formulated for the mechano-electrochemical potential of fluid constituents which is generally applicable to nonequilibrium processes involving active transport. The resulting relations are simple and easy to use, and address an important need in the membrane transport literature.

Integrodifference equations in patchy landscapes : II: population level consequences.

PubMed

Musgrave, Jeffrey; Lutscher, Frithjof

2014-09-01

We analyze integrodifference equations (IDEs) in patchy landscapes. Movement is described by a dispersal kernel that arises from a random walk model with patch dependent diffusion, settling, and mortality rates, and it incorporates individual behavior at an interface between two patch types. Growth follows a simple Beverton-Holt growth or linear decay. We obtain explicit formulae for the critical domain-size problem, and we illustrate how different individual behavior at the boundary between two patch types affects this quantity. We also study persistence conditions on an infinite, periodic, patchy landscape. We observe that if the population can persist on the landscape, the spatial profile of the invasion evolves into a discontinuous traveling periodic wave that moves with constant speed. Assuming linear determinacy, we calculate the dispersion relation and illustrate how movement behavior affects invasion speed. Numerical simulations justify our approach by showing a close correspondence between the spread rate obtained from the dispersion relation and from numerical simulations.

A dimensionally split Cartesian cut cell method for hyperbolic conservation laws

NASA Astrophysics Data System (ADS)

Gokhale, Nandan; Nikiforakis, Nikos; Klein, Rupert

2018-07-01

We present a dimensionally split method for solving hyperbolic conservation laws on Cartesian cut cell meshes. The approach combines local geometric and wave speed information to determine a novel stabilised cut cell flux, and we provide a full description of its three-dimensional implementation in the dimensionally split framework of Klein et al. [1]. The convergence and stability of the method are proved for the one-dimensional linear advection equation, while its multi-dimensional numerical performance is investigated through the computation of solutions to a number of test problems for the linear advection and Euler equations. When compared to the cut cell flux of Klein et al., it was found that the new flux alleviates the problem of oscillatory boundary solutions produced by the former at higher Courant numbers, and also enables the computation of more accurate solutions near stagnation points. Being dimensionally split, the method is simple to implement and extends readily to multiple dimensions.

Prediction of fat-free body mass from bioelectrical impedance and anthropometry among 3-year-old children using DXA

PubMed Central

Ejlerskov, Katrine T.; Jensen, Signe M.; Christensen, Line B.; Ritz, Christian; Michaelsen, Kim F.; Mølgaard, Christian

2014-01-01

For 3-year-old children suitable methods to estimate body composition are sparse. We aimed to develop predictive equations for estimating fat-free mass (FFM) from bioelectrical impedance (BIA) and anthropometry using dual-energy X-ray absorptiometry (DXA) as reference method using data from 99 healthy 3-year-old Danish children. Predictive equations were derived from two multiple linear regression models, a comprehensive model (height2/resistance (RI), six anthropometric measurements) and a simple model (RI, height, weight). Their uncertainty was quantified by means of 10-fold cross-validation approach. Prediction error of FFM was 3.0% for both equations (root mean square error: 360 and 356 g, respectively). The derived equations produced BIA-based prediction of FFM and FM near DXA scan results. We suggest that the predictive equations can be applied in similar population samples aged 2–4 years. The derived equations may prove useful for studies linking body composition to early risk factors and early onset of obesity. PMID:24463487

Prediction of fat-free body mass from bioelectrical impedance and anthropometry among 3-year-old children using DXA.

PubMed

Ejlerskov, Katrine T; Jensen, Signe M; Christensen, Line B; Ritz, Christian; Michaelsen, Kim F; Mølgaard, Christian

2014-01-27

For 3-year-old children suitable methods to estimate body composition are sparse. We aimed to develop predictive equations for estimating fat-free mass (FFM) from bioelectrical impedance (BIA) and anthropometry using dual-energy X-ray absorptiometry (DXA) as reference method using data from 99 healthy 3-year-old Danish children. Predictive equations were derived from two multiple linear regression models, a comprehensive model (height(2)/resistance (RI), six anthropometric measurements) and a simple model (RI, height, weight). Their uncertainty was quantified by means of 10-fold cross-validation approach. Prediction error of FFM was 3.0% for both equations (root mean square error: 360 and 356 g, respectively). The derived equations produced BIA-based prediction of FFM and FM near DXA scan results. We suggest that the predictive equations can be applied in similar population samples aged 2-4 years. The derived equations may prove useful for studies linking body composition to early risk factors and early onset of obesity.

Assessment of the effects of azimuthal mode number perturbations upon the implosion processes of fluids in cylinders

NASA Astrophysics Data System (ADS)

Lindstrom, Michael

2017-06-01

Fluid instabilities arise in a variety of contexts and are often unwanted results of engineering imperfections. In one particular model for a magnetized target fusion reactor, a pressure wave is propagated in a cylindrical annulus comprised of a dense fluid before impinging upon a plasma and imploding it. Part of the success of the apparatus is a function of how axially-symmetric the final pressure pulse is upon impacting the plasma. We study a simple model for the implosion of the system to study how imperfections in the pressure imparted on the outer circumference grow due to geometric focusing. Our methodology entails linearizing the compressible Euler equations for mass and momentum conservation about a cylindrically symmetric problem and analysing the perturbed profiles at different mode numbers. The linearized system gives rise to singular shocks and through analysing the perturbation profiles at various times, we infer that high mode numbers are dampened through the propagation. We also study the Linear Klein-Gordon equation in the context of stability of linear cylindrical wave formation whereby highly oscillatory, bounded behaviour is observed in a far field solution.

Comparison of Linear and Non-linear Regression Analysis to Determine Pulmonary Pressure in Hyperthyroidism.

PubMed

Scarneciu, Camelia C; Sangeorzan, Livia; Rus, Horatiu; Scarneciu, Vlad D; Varciu, Mihai S; Andreescu, Oana; Scarneciu, Ioan

2017-01-01

This study aimed at assessing the incidence of pulmonary hypertension (PH) at newly diagnosed hyperthyroid patients and at finding a simple model showing the complex functional relation between pulmonary hypertension in hyperthyroidism and the factors causing it. The 53 hyperthyroid patients (H-group) were evaluated mainly by using an echocardiographical method and compared with 35 euthyroid (E-group) and 25 healthy people (C-group). In order to identify the factors causing pulmonary hypertension the statistical method of comparing the values of arithmetical means is used. The functional relation between the two random variables (PAPs and each of the factors determining it within our research study) can be expressed by linear or non-linear function. By applying the linear regression method described by a first-degree equation the line of regression (linear model) has been determined; by applying the non-linear regression method described by a second degree equation, a parabola-type curve of regression (non-linear or polynomial model) has been determined. We made the comparison and the validation of these two models by calculating the determination coefficient (criterion 1), the comparison of residuals (criterion 2), application of AIC criterion (criterion 3) and use of F-test (criterion 4). From the H-group, 47% have pulmonary hypertension completely reversible when obtaining euthyroidism. The factors causing pulmonary hypertension were identified: previously known- level of free thyroxin, pulmonary vascular resistance, cardiac output; new factors identified in this study- pretreatment period, age, systolic blood pressure. According to the four criteria and to the clinical judgment, we consider that the polynomial model (graphically parabola- type) is better than the linear one. The better model showing the functional relation between the pulmonary hypertension in hyperthyroidism and the factors identified in this study is given by a polynomial equation of second degree where the parabola is its graphical representation.

Hydrodynamic representation of the Klein-Gordon-Einstein equations in the weak field limit: General formalism and perturbations analysis

NASA Astrophysics Data System (ADS)

Suárez, Abril; Chavanis, Pierre-Henri

2015-07-01

Using a generalization of the Madelung transformation, we derive the hydrodynamic representation of the Klein-Gordon-Einstein equations in the weak field limit. We consider a complex self-interacting scalar field with a λ |φ |4 potential. We study the evolution of the spatially homogeneous background in the fluid representation and derive the linearized equations describing the evolution of small perturbations in a static and in an expanding Universe. We compare the results with simplified models in which the gravitational potential is introduced by hand in the Klein-Gordon equation, and assumed to satisfy a (generalized) Poisson equation. Nonrelativistic hydrodynamic equations based on the Schrödinger-Poisson equations or on the Gross-Pitaevskii-Poisson equations are recovered in the limit c →+∞. We study the evolution of the perturbations in the matter era using the nonrelativistic limit of our formalism. Perturbations whose wavelength is below the Jeans length oscillate in time while perturbations whose wavelength is above the Jeans length grow linearly with the scale factor as in the cold dark matter model. The growth of perturbations in the scalar field model is substantially faster than in the cold dark matter model. When the wavelength of the perturbations approaches the cosmological horizon (Hubble length), a relativistic treatment is mandatory. In that case, we find that relativistic effects attenuate or even prevent the growth of perturbations. This paper exposes the general formalism and provides illustrations in simple cases. Other applications of our formalism will be considered in companion papers.

Simple and accurate theory for strong shock waves in a dense hard-sphere fluid.

PubMed

Montanero, J M; López de Haro, M; Santos, A; Garzó, V

1999-12-01

Following an earlier work by Holian et al. [Phys. Rev. E 47, R24 (1993)] for a dilute gas, we present a theory for strong shock waves in a hard-sphere fluid described by the Enskog equation. The idea is to use the Navier-Stokes hydrodynamic equations but taking the temperature in the direction of shock propagation rather than the actual temperature in the computation of the transport coefficients. In general, for finite densities, this theory agrees much better with Monte Carlo simulations than the Navier-Stokes and (linear) Burnett theories, in contrast to the well-known superiority of the Burnett theory for dilute gases.

Statistical analysis of water-level, springflow, and streamflow data for the Edwards Aquifer in south-central Texas

USGS Publications Warehouse

Puente, Celso

1976-01-01

Water-level, springflow, and streamflow data were used to develop simple and multiple linear-regression equations for use in estimating water levels in wells and the flow of three major springs in the Edwards aquifer in the eastern San Antonio area. The equations provide daily, monthly, and annual estimates that compare very favorably with observed data. Analyses of geologic and hydrologic data indicate that the water discharged by the major springs is supplied primarily by regional underflow from the west and southwest and by local recharge in the infiltration area in northern Bexar, Comal, and Hays Counties.

Anderson acceleration of the Jacobi iterative method: An efficient alternative to Krylov methods for large, sparse linear systems

DOE PAGES

Pratapa, Phanisri P.; Suryanarayana, Phanish; Pask, John E.

2015-12-01

We employ Anderson extrapolation to accelerate the classical Jacobi iterative method for large, sparse linear systems. Specifically, we utilize extrapolation at periodic intervals within the Jacobi iteration to develop the Alternating Anderson–Jacobi (AAJ) method. We verify the accuracy and efficacy of AAJ in a range of test cases, including nonsymmetric systems of equations. We demonstrate that AAJ possesses a favorable scaling with system size that is accompanied by a small prefactor, even in the absence of a preconditioner. In particular, we show that AAJ is able to accelerate the classical Jacobi iteration by over four orders of magnitude, with speed-upsmore » that increase as the system gets larger. Moreover, we find that AAJ significantly outperforms the Generalized Minimal Residual (GMRES) method in the range of problems considered here, with the relative performance again improving with size of the system. As a result, the proposed method represents a simple yet efficient technique that is particularly attractive for large-scale parallel solutions of linear systems of equations.« less

Anderson acceleration of the Jacobi iterative method: An efficient alternative to Krylov methods for large, sparse linear systems

DOE Office of Scientific and Technical Information (OSTI.GOV)

Pratapa, Phanisri P.; Suryanarayana, Phanish; Pask, John E.

We employ Anderson extrapolation to accelerate the classical Jacobi iterative method for large, sparse linear systems. Specifically, we utilize extrapolation at periodic intervals within the Jacobi iteration to develop the Alternating Anderson–Jacobi (AAJ) method. We verify the accuracy and efficacy of AAJ in a range of test cases, including nonsymmetric systems of equations. We demonstrate that AAJ possesses a favorable scaling with system size that is accompanied by a small prefactor, even in the absence of a preconditioner. In particular, we show that AAJ is able to accelerate the classical Jacobi iteration by over four orders of magnitude, with speed-upsmore » that increase as the system gets larger. Moreover, we find that AAJ significantly outperforms the Generalized Minimal Residual (GMRES) method in the range of problems considered here, with the relative performance again improving with size of the system. As a result, the proposed method represents a simple yet efficient technique that is particularly attractive for large-scale parallel solutions of linear systems of equations.« less

Prediction Equation for Calculating Fat Mass in Young Indian Adults

PubMed Central

Sandhu, Jaspal Singh; Gupta, Giniya; Shenoy, Shweta

2010-01-01

Purpose Accurate measurement or prediction of fat mass is useful in physiology, nutrition and clinical medicine. Most predictive equations currently used to assess percentage of body fat or fat mass, using simple anthropometric measurements were derived from people in western societies and they may not be appropriate for individuals with other genotypic and phenotypic characteristics. We developed equations to predict fat mass from anthropometric measurements in young Indian adults. Methods Fat mass was measured in 60 females and 58 males, aged 20 to 29 yrs by using hydrostatic weighing and by simultaneous measurement of residual lung volume. Anthropometric measure included weight (kg), height (m) and 4 skinfold thickness [STs (mm)]. Sex specific linear regression model was developed with fat mass as the dependent variable and all anthropometric measures as independent variables. Results The prediction equation obtained for fat mass (kg) for males was 8.46+0.32 (weight) − 15.16 (height) + 9.54 (log of sum of 4 STs) (R2= 0. 53, SEE=3.42 kg) and − 20.22 + 0.33 (weight) + 3.44 (height) + 7.66 (log of sum of 4 STs) (R2=0.72, SEE=3.01kg) for females. Conclusion A new prediction equation for the measurement of fat mass was derived and internally validated in young Indian adults using simple anthropometric measurements. PMID:22375197

Graphical Tools for Linear Structural Equation Modeling

DTIC Science & Technology

2014-06-01

others. 4Kenny and Milan (2011) write, “Identification is perhaps the most difficult concept for SEM researchers to understand. We have seen SEM...model to using typical SEM software to determine model identifia- bility. Kenny and Milan (2011) list the following drawbacks: (i) If poor starting...the well known recursive and null rules (Bollen, 1989) and the regression rule (Kenny and Milan , 2011). A Simple Criterion for Identifying Individual

Reversed inverse regression for the univariate linear calibration and its statistical properties derived using a new methodology

NASA Astrophysics Data System (ADS)

Kang, Pilsang; Koo, Changhoi; Roh, Hokyu

2017-11-01

Since simple linear regression theory was established at the beginning of the 1900s, it has been used in a variety of fields. Unfortunately, it cannot be used directly for calibration. In practical calibrations, the observed measurements (the inputs) are subject to errors, and hence they vary, thus violating the assumption that the inputs are fixed. Therefore, in the case of calibration, the regression line fitted using the method of least squares is not consistent with the statistical properties of simple linear regression as already established based on this assumption. To resolve this problem, "classical regression" and "inverse regression" have been proposed. However, they do not completely resolve the problem. As a fundamental solution, we introduce "reversed inverse regression" along with a new methodology for deriving its statistical properties. In this study, the statistical properties of this regression are derived using the "error propagation rule" and the "method of simultaneous error equations" and are compared with those of the existing regression approaches. The accuracy of the statistical properties thus derived is investigated in a simulation study. We conclude that the newly proposed regression and methodology constitute the complete regression approach for univariate linear calibrations.

Searching fundamental information in ordinary differential equations. Nondimensionalization technique.

PubMed

Sánchez Pérez, J F; Conesa, M; Alhama, I; Alhama, F; Cánovas, M

2017-01-01

Classical dimensional analysis and nondimensionalization are assumed to be two similar approaches in the search for dimensionless groups. Both techniques, simplify the study of many problems. The first approach does not need to know the mathematical model, being sufficient a deep understanding of the physical phenomenon involved, while the second one begins with the governing equations and reduces them to their dimensionless form by simple mathematical manipulations. In this work, a formal protocol is proposed for applying the nondimensionalization process to ordinary differential equations, linear or not, leading to dimensionless normalized equations from which the resulting dimensionless groups have two inherent properties: In one hand, they are physically interpreted as balances between counteracting quantities in the problem, and on the other hand, they are of the order of magnitude unity. The solutions provided by nondimensionalization are more precise in every case than those from dimensional analysis, as it is illustrated by the applications studied in this work.

Nonlinear Krylov and moving nodes in the method of lines

NASA Astrophysics Data System (ADS)

Miller, Keith

2005-11-01

We report on some successes and problem areas in the Method of Lines from our work with moving node finite element methods. First, we report on our "nonlinear Krylov accelerator" for the modified Newton's method on the nonlinear equations of our stiff ODE solver. Since 1990 it has been robust, simple, cheap, and automatic on all our moving node computations. We publicize further trials with it here because it should be of great general usefulness to all those solving evolutionary equations. Second, we discuss the need for reliable automatic choice of spatially variable time steps. Third, we discuss the need for robust and efficient iterative solvers for the difficult linearized equations (Jx=b) of our stiff ODE solver. Here, the 1997 thesis of Zulu Xaba has made significant progress.

A Simple and Accurate Rate-Driven Infiltration Model

NASA Astrophysics Data System (ADS)

Cui, G.; Zhu, J.

2017-12-01

In this study, we develop a novel Rate-Driven Infiltration Model (RDIMOD) for simulating infiltration into soils. Unlike traditional methods, RDIMOD avoids numerically solving the highly non-linear Richards equation or simply modeling with empirical parameters. RDIMOD employs infiltration rate as model input to simulate one-dimensional infiltration process by solving an ordinary differential equation. The model can simulate the evolutions of wetting front, infiltration rate, and cumulative infiltration on any surface slope including vertical and horizontal directions. Comparing to the results from the Richards equation for both vertical infiltration and horizontal infiltration, RDIMOD simply and accurately predicts infiltration processes for any type of soils and soil hydraulic models without numerical difficulty. Taking into account the accuracy, capability, and computational effectiveness and stability, RDIMOD can be used in large-scale hydrologic and land-atmosphere modeling.

Large-scale computation of incompressible viscous flow by least-squares finite element method

NASA Technical Reports Server (NTRS)

Jiang, Bo-Nan; Lin, T. L.; Povinelli, Louis A.

1993-01-01

The least-squares finite element method (LSFEM) based on the velocity-pressure-vorticity formulation is applied to large-scale/three-dimensional steady incompressible Navier-Stokes problems. This method can accommodate equal-order interpolations and results in symmetric, positive definite algebraic system which can be solved effectively by simple iterative methods. The first-order velocity-Bernoulli function-vorticity formulation for incompressible viscous flows is also tested. For three-dimensional cases, an additional compatibility equation, i.e., the divergence of the vorticity vector should be zero, is included to make the first-order system elliptic. The simple substitution of the Newton's method is employed to linearize the partial differential equations, the LSFEM is used to obtain discretized equations, and the system of algebraic equations is solved using the Jacobi preconditioned conjugate gradient method which avoids formation of either element or global matrices (matrix-free) to achieve high efficiency. To show the validity of this scheme for large-scale computation, we give numerical results for 2D driven cavity problem at Re = 10000 with 408 x 400 bilinear elements. The flow in a 3D cavity is calculated at Re = 100, 400, and 1,000 with 50 x 50 x 50 trilinear elements. The Taylor-Goertler-like vortices are observed for Re = 1,000.

Constructing general partial differential equations using polynomial and neural networks.

PubMed

Zjavka, Ladislav; Pedrycz, Witold

2016-01-01

Sum fraction terms can approximate multi-variable functions on the basis of discrete observations, replacing a partial differential equation definition with polynomial elementary data relation descriptions. Artificial neural networks commonly transform the weighted sum of inputs to describe overall similarity relationships of trained and new testing input patterns. Differential polynomial neural networks form a new class of neural networks, which construct and solve an unknown general partial differential equation of a function of interest with selected substitution relative terms using non-linear multi-variable composite polynomials. The layers of the network generate simple and composite relative substitution terms whose convergent series combinations can describe partial dependent derivative changes of the input variables. This regression is based on trained generalized partial derivative data relations, decomposed into a multi-layer polynomial network structure. The sigmoidal function, commonly used as a nonlinear activation of artificial neurons, may transform some polynomial items together with the parameters with the aim to improve the polynomial derivative term series ability to approximate complicated periodic functions, as simple low order polynomials are not able to fully make up for the complete cycles. The similarity analysis facilitates substitutions for differential equations or can form dimensional units from data samples to describe real-world problems. Copyright © 2015 Elsevier Ltd. All rights reserved.

Conservative, unconditionally stable discretization methods for Hamiltonian equations, applied to wave motion in lattice equations modeling protein molecules

NASA Astrophysics Data System (ADS)

LeMesurier, Brenton

2012-01-01

A new approach is described for generating exactly energy-momentum conserving time discretizations for a wide class of Hamiltonian systems of DEs with quadratic momenta, including mechanical systems with central forces; it is well-suited in particular to the large systems that arise in both spatial discretizations of nonlinear wave equations and lattice equations such as the Davydov System modeling energetic pulse propagation in protein molecules. The method is unconditionally stable, making it well-suited to equations of broadly “Discrete NLS form”, including many arising in nonlinear optics. Key features of the resulting discretizations are exact conservation of both the Hamiltonian and quadratic conserved quantities related to continuous linear symmetries, preservation of time reversal symmetry, unconditional stability, and respecting the linearity of certain terms. The last feature allows a simple, efficient iterative solution of the resulting nonlinear algebraic systems that retain unconditional stability, avoiding the need for full Newton-type solvers. One distinction from earlier work on conservative discretizations is a new and more straightforward nearly canonical procedure for constructing the discretizations, based on a “discrete gradient calculus with product rule” that mimics the essential properties of partial derivatives. This numerical method is then used to study the Davydov system, revealing that previously conjectured continuum limit approximations by NLS do not hold, but that sech-like pulses related to NLS solitons can nevertheless sometimes arise.

Stochastic Mixing Model with Power Law Decay of Variance

NASA Technical Reports Server (NTRS)

Fedotov, S.; Ihme, M.; Pitsch, H.

2003-01-01

Here we present a simple stochastic mixing model based on the law of large numbers (LLN). The reason why the LLN is involved in our formulation of the mixing problem is that the random conserved scalar c = c(t,x(t)) appears to behave as a sample mean. It converges to the mean value mu, while the variance sigma(sup 2)(sub c) (t) decays approximately as t(exp -1). Since the variance of the scalar decays faster than a sample mean (typically is greater than unity), we will introduce some non-linear modifications into the corresponding pdf-equation. The main idea is to develop a robust model which is independent from restrictive assumptions about the shape of the pdf. The remainder of this paper is organized as follows. In Section 2 we derive the integral equation from a stochastic difference equation describing the evolution of the pdf of a passive scalar in time. The stochastic difference equation introduces an exchange rate gamma(sub n) which we model in a first step as a deterministic function. In a second step, we generalize gamma(sub n) as a stochastic variable taking fluctuations in the inhomogeneous environment into account. In Section 3 we solve the non-linear integral equation numerically and analyze the influence of the different parameters on the decay rate. The paper finishes with a conclusion.

Angular and linear fields of view of Galilean telescopes and telemicroscopes.

PubMed

Katz, Milton

2007-06-01

The calculation of the angular fields of view (FOVs) of Galilean telescopes generally necessitates the calculation of the pupils and ports. This, in turn, requires knowledge of the optical design of the telescope, in particular, the focal lengths or powers of the objective and ocular lenses. Equations for finding the FOV that obviate the need to calculate pupils and ports, or even to know the lens powers of the telescope, are presented in this article. The equations can be used to find the FOVs in image space of real Galilean telescopes of known magnification, merely by measuring the distance between the objective and ocular lenses and the diameter of the objective lens. The equations include the effects of eye pupil diameter and eye relief. Linear FOVs (LFOVs) of Galilean telemicroscopes are similarly determined. Two image space angular FOV equations were derived: (1) an equation to determine the angular FOVs of a telescope with various amounts of vignetting and eye relief; and (2) an equivalent equation for the LFOVs of telescopes fitted with lens caps for near vision. The FOV increases linearly with increasing vignetting. Increasing the eye relief results in a nonlinear decrease in the FOV, shown as a fraction of the normalized value for zero eye relief. Decrements in the FOVs with increasing eye relief as a fraction of the normalized field angle when the eye relief = 0 are shown to be constant regardless of the vignetting level. A transition of the objective lens from field stop to aperture stop occurs when the eye pupil diameter exceeds the diameter of the objective lens divided by the magnification. Equations have been derived for Galilean telescopes and telemicroscopes that make it unnecessary to find pupils and ports, or to know the powers of the lenses. They provide a direct and simple evaluation of angular and LFOVs as functions of magnification, objective lens diameter, eye pupil diameter, eye relief, and vignetting, and enable comparisons of actual telescopes.

Cotton-type and joint invariants for linear elliptic systems.

PubMed

Aslam, A; Mahomed, F M

2013-01-01

Cotton-type invariants for a subclass of a system of two linear elliptic equations, obtainable from a complex base linear elliptic equation, are derived both by spliting of the corresponding complex Cotton invariants of the base complex equation and from the Laplace-type invariants of the system of linear hyperbolic equations equivalent to the system of linear elliptic equations via linear complex transformations of the independent variables. It is shown that Cotton-type invariants derived from these two approaches are identical. Furthermore, Cotton-type and joint invariants for a general system of two linear elliptic equations are also obtained from the Laplace-type and joint invariants for a system of two linear hyperbolic equations equivalent to the system of linear elliptic equations by complex changes of the independent variables. Examples are presented to illustrate the results.

Cotton-Type and Joint Invariants for Linear Elliptic Systems

PubMed Central

Aslam, A.; Mahomed, F. M.

2013-01-01

Cotton-type invariants for a subclass of a system of two linear elliptic equations, obtainable from a complex base linear elliptic equation, are derived both by spliting of the corresponding complex Cotton invariants of the base complex equation and from the Laplace-type invariants of the system of linear hyperbolic equations equivalent to the system of linear elliptic equations via linear complex transformations of the independent variables. It is shown that Cotton-type invariants derived from these two approaches are identical. Furthermore, Cotton-type and joint invariants for a general system of two linear elliptic equations are also obtained from the Laplace-type and joint invariants for a system of two linear hyperbolic equations equivalent to the system of linear elliptic equations by complex changes of the independent variables. Examples are presented to illustrate the results. PMID:24453871

Spatio-temporal dynamics induced by competing instabilities in two asymmetrically coupled nonlinear evolution equations

DOE Office of Scientific and Technical Information (OSTI.GOV)

Schüler, D.; Alonso, S.; Bär, M.

2014-12-15

Pattern formation often occurs in spatially extended physical, biological, and chemical systems due to an instability of the homogeneous steady state. The type of the instability usually prescribes the resulting spatio-temporal patterns and their characteristic length scales. However, patterns resulting from the simultaneous occurrence of instabilities cannot be expected to be simple superposition of the patterns associated with the considered instabilities. To address this issue, we design two simple models composed by two asymmetrically coupled equations of non-conserved (Swift-Hohenberg equations) or conserved (Cahn-Hilliard equations) order parameters with different characteristic wave lengths. The patterns arising in these systems range from coexistingmore » static patterns of different wavelengths to traveling waves. A linear stability analysis allows to derive a two parameter phase diagram for the studied models, in particular, revealing for the Swift-Hohenberg equations, a co-dimension two bifurcation point of Turing and wave instability and a region of coexistence of stationary and traveling patterns. The nonlinear dynamics of the coupled evolution equations is investigated by performing accurate numerical simulations. These reveal more complex patterns, ranging from traveling waves with embedded Turing patterns domains to spatio-temporal chaos, and a wide hysteretic region, where waves or Turing patterns coexist. For the coupled Cahn-Hilliard equations the presence of a weak coupling is sufficient to arrest the coarsening process and to lead to the emergence of purely periodic patterns. The final states are characterized by domains with a characteristic length, which diverges logarithmically with the coupling amplitude.« less

Comparison of heaving buoy and oscillating flap wave energy converters

NASA Astrophysics Data System (ADS)

Abu Bakar, Mohd Aftar; Green, David A.; Metcalfe, Andrew V.; Najafian, G.

2013-04-01

Waves offer an attractive source of renewable energy, with relatively low environmental impact, for communities reasonably close to the sea. Two types of simple wave energy converters (WEC), the heaving buoy WEC and the oscillating flap WEC, are studied. Both WECs are considered as simple energy converters because they can be modelled, to a first approximation, as single degree of freedom linear dynamic systems. In this study, we estimate the response of both WECs to typical wave inputs; wave height for the buoy and corresponding wave surge for the flap, using spectral methods. A nonlinear model of the oscillating flap WEC that includes the drag force, modelled by the Morison equation is also considered. The response to a surge input is estimated by discrete time simulation (DTS), using central difference approximations to derivatives. This is compared with the response of the linear model obtained by DTS and also validated using the spectral method. Bendat's nonlinear system identification (BNLSI) technique was used to analyze the nonlinear dynamic system since the spectral analysis was only suitable for linear dynamic system. The effects of including the nonlinear term are quantified.

Validated modified Lycopodium spore method development for standardisation of ingredients of an ayurvedic powdered formulation Shatavaryadi churna.

PubMed

Kumar, Puspendra; Jha, Shivesh; Naved, Tanveer

2013-01-01

Validated modified lycopodium spore method has been developed for simple and rapid quantification of herbal powdered drugs. Lycopodium spore method was performed on ingredients of Shatavaryadi churna, an ayurvedic formulation used as immunomodulator, galactagogue, aphrodisiac and rejuvenator. Estimation of diagnostic characters of each ingredient of Shatavaryadi churna individually was carried out. Microscopic determination, counting of identifying number, measurement of area, length and breadth of identifying characters were performed using Leica DMLS-2 microscope. The method was validated for intraday precision, linearity, specificity, repeatability, accuracy and system suitability, respectively. The method is simple, precise, sensitive, and accurate, and can be used for routine standardisation of raw materials of herbal drugs. This method gives the ratio of individual ingredients in the powdered drug so that any adulteration of genuine drug with its adulterant can be found out. The method shows very good linearity value between 0.988-0.999 for number of identifying character and area of identifying character. Percentage purity of the sample drug can be determined by using the linear equation of standard genuine drug.

Unified Theory for Decoding the Signals from X-Ray Florescence and X-Ray Diffraction of Mixtures.

PubMed

Chung, Frank H

2017-05-01

For research and development or for solving technical problems, we often need to know the chemical composition of an unknown mixture, which is coded and stored in the signals of its X-ray fluorescence (XRF) and X-ray diffraction (XRD). X-ray fluorescence gives chemical elements, whereas XRD gives chemical compounds. The major problem in XRF and XRD analyses is the complex matrix effect. The conventional technique to deal with the matrix effect is to construct empirical calibration lines with standards for each element or compound sought, which is tedious and time-consuming. A unified theory of quantitative XRF analysis is presented here. The idea is to cancel the matrix effect mathematically. It turns out that the decoding equation for quantitative XRF analysis is identical to that for quantitative XRD analysis although the physics of XRD and XRF are fundamentally different. The XRD work has been published and practiced worldwide. The unified theory derives a new intensity-concentration equation of XRF, which is free from the matrix effect and valid for a wide range of concentrations. The linear decoding equation establishes a constant slope for each element sought, hence eliminating the work on calibration lines. The simple linear decoding equation has been verified by 18 experiments.

Advantages of formulating an evolution equation directly for elastic distortional deformation in finite deformation plasticity

NASA Astrophysics Data System (ADS)

Rubin, M. B.; Cardiff, P.

2017-11-01

Simo (Comput Methods Appl Mech Eng 66:199-219, 1988) proposed an evolution equation for elastic deformation together with a constitutive equation for inelastic deformation rate in plasticity. The numerical algorithm (Simo in Comput Methods Appl Mech Eng 68:1-31, 1988) for determining elastic distortional deformation was simple. However, the proposed inelastic deformation rate caused plastic compaction. The corrected formulation (Simo in Comput Methods Appl Mech Eng 99:61-112, 1992) preserves isochoric plasticity but the numerical integration algorithm is complicated and needs special methods for calculation of the exponential map of a tensor. Alternatively, an evolution equation for elastic distortional deformation can be proposed directly with a simplified constitutive equation for inelastic distortional deformation rate. This has the advantage that the physics of inelastic distortional deformation is separated from that of dilatation. The example of finite deformation J2 plasticity with linear isotropic hardening is used to demonstrate the simplicity of the numerical algorithm.

Dynamics in a Maximally Symmetric Universe

NASA Astrophysics Data System (ADS)

Bewketu, Asnakew

2016-03-01

Our present understanding of the evolution of the universe relies upon the Friedmann- Robertson- Walker cosmological models. This model is so successful that it is now being considered as the Standard Model of Cosmology. So in this work we derive the Fried- mann equations using the Friedmann-Robertson-Walker metric together with Einstein field equation and then we give a simple method to reduce Friedmann equations to a second order linear differential equation when it is supplemented with a time dependent equation of state. Furthermore, as illustrative examples, we solve this equation for some specific time dependent equation of states. And also by using the Friedmann equations with some time dependent equation of state we try to determine the cosmic scale factor(the rate at which the universe expands) and age of the Friedmann universe, for the matter dominated era, radiation dominated era and for both matter and radiation dominated era by considering different cases. We have finally discussed the observable quantities that can be evidences for the accelerated expansion of the Friedmann universe. I would like to acknowledge Addis Ababa University for its financial and material support to my work on the title mentioned above.

Approximate Equation of State for Overdriven and Reflected Detonation Products

NASA Astrophysics Data System (ADS)

Liu, Zhi-Yue; Itoh, Shigeru

2001-06-01

Approximate Equation of State for Overdriven and Reflected Detonation Products Zhi-Yue Liu and Shigeru Itoh Shock Wave and Condensed Matter Research Center, Kumamoto University, 2-39-1 Kurokami, Kumamoto, 860-8555, Japan ABSTRACT There are several types of equations of state (EOS) to describe the isentropic expansion behavior of the detonation products after the explosive is exploded. Of which, Jones-Wilkins-Lee (or JWL) equation and polytropic gas (or gamma-law) equation are most popularly employed owing to either the completely experimental calibration or simple form in expression. However, both equations fail to correctly describe the behavior of detonation products above the Chapman-Jouguet (C-J) state. For this reason, a new form of equation of state is proposed for overcoming this deficiency. The new equation of state is simply a linear combination of JWL and gamma-law equations. The coefficient appeared in the EOS can be obtained by fitting to the data from the overdriven detonation experiments. Results show that this form of EOS not only gives the satisfactory description to the variation of the detonation products in overdriven or reflected state also can simply be incorporated into the hydrodynamic computer codes.

Instability of 2D Flows to Hydrostatic 3D Perturbations.

NASA Astrophysics Data System (ADS)

Straub, David N.

2003-01-01

Considered here is the evolution of three-dimensional perturbations to the hydrostatic equations linearized about a two-dimensional base state U. Motivated by an argument by T. Warn, this study begins with the nonrotating, unstratified case, and draws analogies between the perturbation equations and equations describing evolution of material line elements and scalar gradients embedded in the same 2D flow. When U is chaotic, both scalar gradients and line elements are characterized by rapid growth, and this leads one to suspect that the perturbations behave similarly. A generalized Okubo-Weiss parameter is proposed, and it is argued that this gives a reasonable litmus test for identifying regions where growth is most probable. Rotation modifies the generalized Okubo-Weiss parameter and tends to curb growth of the perturbation fields, as expected. It is also pointed out that, in realistic geophysical settings, the stability parameter can be suggestive of growth locally, even when a globally defined Rossby number is small.Also considered is the effect of a constant stratification. The perturbation equations can then be separated into vertical modes that have simple sinusoidal structures. The equations describing the evolution of a given mode take a form analogous to the shallow water equations, linearized about U. Numerical simulations of these, assuming a simple but chaotic prescription of U, are carried out. For sufficiently strong stratification, a balance dynamics similar to that suggested by Riley, Metcalfe, and Weissman is recovered. For a given value of the buoyancy frequency N, however, this balance breaks down at high vertical wavenumbers. For high vertical wavenumbers, the modified Okubo-Weiss parameter once again appears to give a potentially useful indication of when growth should be expected. When the Rossby number is small, this criterion predicts stability, and growth occurs only when stratification effects are comparable to or larger than rotational effects. More specifically, growth is seen when the relevant Rossby radius is comparable to or larger than the characteristic length scale of U. It is also found in this limit that approximate geostrophic adjustment occurs prior to growth.

EIT Intensity Noise Spectroscopy

NASA Astrophysics Data System (ADS)

Crescimanno, Michael; Xiao, Yanhong; Baryakhtar, Maria; Hohensee, Michael; Phillips, David; Walsworth, Ron

2008-10-01

Intensity noise correlations in coherently-prepared media can reveal underlying spectroscopic detail, such as power broadening-free resonances. We analyze recent experimental results using very simple theory: The intensity noise correlation spectra can be quantitatively understood entirely in terms of static ensemble averages of the medium's steady state response. This is significantly simpler than stochastic integration of the Bloch equations, and leads to physical insights we apply to non-linear Faraday rotation and noise spectra in optically thick media.

Schrödinger and Dirac solutions to few-body problems

NASA Astrophysics Data System (ADS)

Muolo, Andrea; Reiher, Markus

We elaborate on the variational solution of the Schrödinger and Dirac equations for small atomic and molecular systems without relying on the Born-Oppenheimer approximation. The all-particle equations of motion are solved in a numerical procedure that relies on the variational principle, Cartesian coordinates and parametrized explicitly correlated Gaussians functions. A stochastic optimization of the variational parameters allows the calculation of accurate wave functions for ground and excited states. Expectation values such as the radial and angular distribution functions or the dipole moment can be calculated. We developed a simple strategy for the elimination of the global translation that allows to generally adopt laboratory-fixed cartesian coordinates. Simple expressions for the coordinates and operators are then preserved throughout the formalism. For relativistic calculations we devised a kinetic-balance condition for explicitly correlated basis functions. We demonstrate that the kinetic-balance condition can be obtained from the row reduction process commonly applied to solve systems of linear equations. The resulting form of kinetic balance establishes a relation between all components of the spinor of an N-fermion system. ETH Zürich, Laboratorium für Physikalische Chemie, CH-8093 Zürich, Switzerland.

Nonlinear and dissipative constitutive equations for coupled first-order acoustic field equations that are consistent with the generalized Westervelt equation

NASA Astrophysics Data System (ADS)

Verweij, Martin D.; Huijssen, Jacob

2006-05-01

In diagnostic medical ultrasound, it has become increasingly important to evaluate the nonlinear field of an acoustic beam that propagates in a weakly nonlinear, dissipative medium and that is steered off-axis up to very wide angles. In this case, computations cannot be based on the widely used KZK equation since it applies only to small angles. To benefit from successful computational schemes from elastodynamics and electromagnetics, we propose to use two first-order acoustic field equations, accompanied by two constitutive equations, as an alternative basis. This formulation quite naturally results in the contrast source formalism, makes a clear distinction between fundamental conservation laws and medium behavior, and allows for a straightforward inclusion of any medium inhomogenities. This paper is concerned with the derivation of relevant constitutive equations. We take a pragmatic approach and aim to find those constitutive equations that represent the same medium as implicitly described by the recognized, full wave, nonlinear equations such as the generalized Westervelt equation. We will show how this is achieved by considering the nonlinear case without attenuation, the linear case with attenuation, and the nonlinear case with attenuation. As a result we will obtain surprisingly simple constitutive equations for the full wave case.

Numerical solution of the Hele-Shaw equations

DOE Office of Scientific and Technical Information (OSTI.GOV)

Whitaker, N.

1987-04-01

An algorithm is presented for approximating the motion of the interface between two immiscible fluids in a Hele-Shaw cell. The interface is represented by a set of volume fractions. We use the Simple Line Interface Calculation method along with the method of fractional steps to transport the interface. The equation of continuity leads to a Poisson equation for the pressure. The Poisson equation is discretized. Near the interface where the velocity field is discontinuous, the discretization is based on a weak formulation of the continuity equation. Interpolation is used on each side of the interface to increase the accuracy ofmore » the algorithm. The weak formulation as well as the interpolation are based on the computed volume fractions. This treatment of the interface is new. The discretized equations are solved by a modified conjugate gradient method. Surface tension is included and the curvature is computed through the use of osculating circles. For perturbations of small amplitude, a surprisingly good agreement is found between the numerical results and linearized perturbation theory. Numerical results are presented for the finite amplitude growth of unstable fingers. 62 refs., 13 figs.« less

A new fractional nonlocal model and its application in free vibration of Timoshenko and Euler-Bernoulli beams

NASA Astrophysics Data System (ADS)

Rahimi, Zaher; Sumelka, Wojciech; Yang, Xiao-Jun

2017-11-01

The application of fractional calculus in fractional models (FMs) makes them more flexible than integer models inasmuch they can conclude all of integer and non-integer operators. In other words FMs let us use more potential of mathematics to modeling physical phenomena due to the use of both integer and fractional operators to present a better modeling of problems, which makes them more flexible and powerful. In the present work, a new fractional nonlocal model has been proposed, which has a simple form and can be used in different problems due to the simple form of numerical solutions. Then the model has been used to govern equations of the motion of the Timoshenko beam theory (TBT) and Euler-Bernoulli beam theory (EBT). Next, free vibration of the Timoshenko and Euler-Bernoulli simply-supported (S-S) beam has been investigated. The Galerkin weighted residual method has been used to solve the non-linear governing equations.

Parallels between control PDE's (Partial Differential Equations) and systems of ODE's (Ordinary Differential Equations)

NASA Technical Reports Server (NTRS)

Hunt, L. R.; Villarreal, Ramiro

1987-01-01

System theorists understand that the same mathematical objects which determine controllability for nonlinear control systems of ordinary differential equations (ODEs) also determine hypoellipticity for linear partial differentail equations (PDEs). Moreover, almost any study of ODE systems begins with linear systems. It is remarkable that Hormander's paper on hypoellipticity of second order linear p.d.e.'s starts with equations due to Kolmogorov, which are shown to be analogous to the linear PDEs. Eigenvalue placement by state feedback for a controllable linear system can be paralleled for a Kolmogorov equation if an appropriate type of feedback is introduced. Results concerning transformations of nonlinear systems to linear systems are similar to results for transforming a linear PDE to a Kolmogorov equation.

A heat transfer model for a hot helium airship

NASA Astrophysics Data System (ADS)

Rapert, R. M.

1987-06-01

Basic heat transfer empirical and analytic equations are applied to a double envelope airship concept which uses heated Helium in the inner envelope to augment and control gross lift. The convective and conductive terms lead to a linear system of five equations for the concept airship, with the nonlinear radiation terms included by an iterative solution process. The graphed results from FORTRAN program solutions are presented for the variables of interest. These indicate that a simple use of airship engine exhaust heat gives more than a 30 percent increase in gross airship lift. Possibly more than 100 percent increase can be achieved if a 'stream injection' heating system, with associated design problems, is used.

Computational and spectroscopic data correlation study of N,N'-bisarylmalonamides (Part II).

PubMed

Arsovski, Violeta M; Božić, Bojan Đ; Mirković, Jelena M; Vitnik, Vesna D; Vitnik, Željko J; Petrović, Slobodan D; Ušćumlić, Gordana S; Mijin, Dušan Ž

2015-09-01

To complement a previous UV study, we present a quantitative evaluation of substituent effects on spectroscopic data ((1)H and (13)C NMR chemical shifts as well as FT-IR absorption frequency) applied to N,N'-bisarylmalonamides, using simple and extended Hammett equations as well as the Swain-Lupton equation. Furthermore, the DFT CAM-B3LYP/6-311+G(d,p) method was applied to study the impact of different solvents on the geometry of the molecules and their spectral data. Additionally, experimental data are correlated with theoretical results; excellent linear dependence was obtained. The overall results presented in this paper show that N,N'-bisarylmalonamides are prominent candidates for model molecules.

Application of stepwise multiple regression techniques to inversion of Nimbus 'IRIS' observations.

NASA Technical Reports Server (NTRS)

Ohring, G.

1972-01-01

Exploratory studies with Nimbus-3 infrared interferometer-spectrometer (IRIS) data indicate that, in addition to temperature, such meteorological parameters as geopotential heights of pressure surfaces, tropopause pressure, and tropopause temperature can be inferred from the observed spectra with the use of simple regression equations. The technique of screening the IRIS spectral data by means of stepwise regression to obtain the best radiation predictors of meteorological parameters is validated. The simplicity of application of the technique and the simplicity of the derived linear regression equations - which contain only a few terms - suggest usefulness for this approach. Based upon the results obtained, suggestions are made for further development and exploitation of the stepwise regression analysis technique.

Using Linear Equating to Map PROMIS(®) Global Health Items and the PROMIS-29 V2.0 Profile Measure to the Health Utilities Index Mark 3.

PubMed

Hays, Ron D; Revicki, Dennis A; Feeny, David; Fayers, Peter; Spritzer, Karen L; Cella, David

2016-10-01

Preference-based health-related quality of life (HR-QOL) scores are useful as outcome measures in clinical studies, for monitoring the health of populations, and for estimating quality-adjusted life-years. This was a secondary analysis of data collected in an internet survey as part of the Patient-Reported Outcomes Measurement Information System (PROMIS(®)) project. To estimate Health Utilities Index Mark 3 (HUI-3) preference scores, we used the ten PROMIS(®) global health items, the PROMIS-29 V2.0 single pain intensity item and seven multi-item scales (physical functioning, fatigue, pain interference, depressive symptoms, anxiety, ability to participate in social roles and activities, sleep disturbance), and the PROMIS-29 V2.0 items. Linear regression analyses were used to identify significant predictors, followed by simple linear equating to avoid regression to the mean. The regression models explained 48 % (global health items), 61 % (PROMIS-29 V2.0 scales), and 64 % (PROMIS-29 V2.0 items) of the variance in the HUI-3 preference score. Linear equated scores were similar to observed scores, although differences tended to be larger for older study participants. HUI-3 preference scores can be estimated from the PROMIS(®) global health items or PROMIS-29 V2.0. The estimated HUI-3 scores from the PROMIS(®) health measures can be used for economic applications and as a measure of overall HR-QOL in research.

Efficient Determination of Free Energy Landscapes in Multiple Dimensions from Biased Umbrella Sampling Simulations Using Linear Regression.

PubMed

Meng, Yilin; Roux, Benoît

2015-08-11

The weighted histogram analysis method (WHAM) is a standard protocol for postprocessing the information from biased umbrella sampling simulations to construct the potential of mean force with respect to a set of order parameters. By virtue of the WHAM equations, the unbiased density of state is determined by satisfying a self-consistent condition through an iterative procedure. While the method works very effectively when the number of order parameters is small, its computational cost grows rapidly in higher dimension. Here, we present a simple and efficient alternative strategy, which avoids solving the self-consistent WHAM equations iteratively. An efficient multivariate linear regression framework is utilized to link the biased probability densities of individual umbrella windows and yield an unbiased global free energy landscape in the space of order parameters. It is demonstrated with practical examples that free energy landscapes that are comparable in accuracy to WHAM can be generated at a small fraction of the cost.

Efficient Determination of Free Energy Landscapes in Multiple Dimensions from Biased Umbrella Sampling Simulations Using Linear Regression

PubMed Central

2015-01-01

The weighted histogram analysis method (WHAM) is a standard protocol for postprocessing the information from biased umbrella sampling simulations to construct the potential of mean force with respect to a set of order parameters. By virtue of the WHAM equations, the unbiased density of state is determined by satisfying a self-consistent condition through an iterative procedure. While the method works very effectively when the number of order parameters is small, its computational cost grows rapidly in higher dimension. Here, we present a simple and efficient alternative strategy, which avoids solving the self-consistent WHAM equations iteratively. An efficient multivariate linear regression framework is utilized to link the biased probability densities of individual umbrella windows and yield an unbiased global free energy landscape in the space of order parameters. It is demonstrated with practical examples that free energy landscapes that are comparable in accuracy to WHAM can be generated at a small fraction of the cost. PMID:26574437

Integral method for transient He II heat transfer in a semi-infinite domain

NASA Astrophysics Data System (ADS)

Baudouy, B.

2002-05-01

Integral methods are suited to solve a non-linear system of differential equations where the non-linearity can be found either in the differential equations or in the boundary conditions. Though they are approximate methods, they have proven to give simple solutions with acceptable accuracy for transient heat transfer in He II. Taking in account the temperature dependence of thermal properties, direct solutions are found without the need of adjusting a parameter. Previously, we have presented a solution for the clamped heat flux and in the present study this method is used to accommodate the clamped-temperature problem. In the case of constant thermal properties, this method yields results that are within a few percent of the exact solution for the heat flux at the axis origin. We applied this solution to analyze recovery from burnout and find an agreement within 10% at low heat flux, whereas at high heat flux the model deviates from the experimental data suggesting the need for a more refined thermal model.

Stable two-dimensional solitary pulses in linearly coupled dissipative Kadomtsev-Petviashvili equations.

PubMed

Feng, Bao-Feng; Malomed, Boris A; Kawahara, Takuji

2002-11-01

We present a two-dimensional (2D) generalization of the stabilized Kuramoto-Sivashinsky system, based on the Kadomtsev-Petviashvili (KP) equation including dissipation of the generic [Newell-Whitehead-Segel (NWS)] type and gain. The system directly applies to the description of gravity-capillary waves on the surface of a liquid layer flowing down an inclined plane, with a surfactant diffusing along the layer's surface. Actually, the model is quite general, offering a simple way to stabilize nonlinear media, combining the weakly 2D dispersion of the KP type with gain and NWS dissipation. Other applications are internal waves in multilayer fluids flowing down an inclined plane, double-front flames in gaseous mixtures, etc. Parallel to this weakly 2D model, we also introduce and study a semiphenomenological one, whose dissipative terms are isotropic, rather than of the NWS type, in order to check if qualitative results are sensitive to the exact form of the lossy terms. The models include an additional linear equation of the advection-diffusion type, linearly coupled to the main KP-NWS equation. The extra equation provides for stability of the zero background in the system, thus opening a way for the existence of stable localized pulses. We focus on the most interesting case, when the dispersive part of the system is of the KP-I type, which corresponds, e.g., to capillary waves, and makes the existence of completely localized 2D pulses possible. Treating the losses and gain as small perturbations and making use of the balance equation for the field momentum, we find that the equilibrium between the gain and losses may select two steady-state solitons from their continuous family existing in the absence of the dissipative terms (the latter family is found in an exact analytical form, and is numerically demonstrated to be stable). The selected soliton with the larger amplitude is expected to be stable. Direct simulations completely corroborate the analytical predictions, for both the physical and phenomenological models.

Modular forms, Schwarzian conditions, and symmetries of differential equations in physics

NASA Astrophysics Data System (ADS)

Abdelaziz, Y.; Maillard, J.-M.

2017-05-01

We give examples of infinite order rational transformations that leave linear differential equations covariant. These examples are non-trivial yet simple enough illustrations of exact representations of the renormalization group. We first illustrate covariance properties on order-two linear differential operators associated with identities relating the same {}_2F1 hypergeometric function with different rational pullbacks. These rational transformations are solutions of a differentially algebraic equation that already emerged in a paper by Casale on the Galoisian envelopes. We provide two new and more general results of the previous covariance by rational functions: a new Heun function example and a higher genus {}_2F1 hypergeometric function example. We then focus on identities relating the same {}_2F1 hypergeometric function with two different algebraic pullback transformations: such remarkable identities correspond to modular forms, the algebraic transformations being solution of another differentially algebraic Schwarzian equation that also emerged in Casale’s paper. Further, we show that the first differentially algebraic equation can be seen as a subcase of the last Schwarzian differential condition, the restriction corresponding to a factorization condition of some associated order-two linear differential operator. Finally, we also explore generalizations of these results, for instance, to {}_3F2 , hypergeometric functions, and show that one just reduces to the previous {}_2F1 cases through a Clausen identity. The question of the reduction of these Schwarzian conditions to modular correspondences remains an open question. In a _2F1 hypergeometric framework the Schwarzian condition encapsulates all the modular forms and modular equations of the theory of elliptic curves, but these two conditions are actually richer than elliptic curves or {}_2F1 hypergeometric functions, as can be seen on the Heun and higher genus example. This work is a strong incentive to develop more differentially algebraic symmetry analysis in physics.

Theory of linear sweep voltammetry with diffuse charge: Unsupported electrolytes, thin films, and leaky membranes

NASA Astrophysics Data System (ADS)

Yan, David; Bazant, Martin Z.; Biesheuvel, P. M.; Pugh, Mary C.; Dawson, Francis P.

2017-03-01

Linear sweep and cyclic voltammetry techniques are important tools for electrochemists and have a variety of applications in engineering. Voltammetry has classically been treated with the Randles-Sevcik equation, which assumes an electroneutral supported electrolyte. In this paper, we provide a comprehensive mathematical theory of voltammetry in electrochemical cells with unsupported electrolytes and for other situations where diffuse charge effects play a role, and present analytical and simulated solutions of the time-dependent Poisson-Nernst-Planck equations with generalized Frumkin-Butler-Volmer boundary conditions for a 1:1 electrolyte and a simple reaction. Using these solutions, we construct theoretical and simulated current-voltage curves for liquid and solid thin films, membranes with fixed background charge, and cells with blocking electrodes. The full range of dimensionless parameters is considered, including the dimensionless Debye screening length (scaled to the electrode separation), Damkohler number (ratio of characteristic diffusion and reaction times), and dimensionless sweep rate (scaled to the thermal voltage per diffusion time). The analysis focuses on the coupling of Faradaic reactions and diffuse charge dynamics, although capacitive charging of the electrical double layers is also studied, for early time transients at reactive electrodes and for nonreactive blocking electrodes. Our work highlights cases where diffuse charge effects are important in the context of voltammetry, and illustrates which regimes can be approximated using simple analytical expressions and which require more careful consideration.

New formulations for tsunami runup estimation

NASA Astrophysics Data System (ADS)

Kanoglu, U.; Aydin, B.; Ceylan, N.

2017-12-01

We evaluate shoreline motion and maximum runup in two folds: One, we use linear shallow water-wave equations over a sloping beach and solve as initial-boundary value problem similar to the nonlinear solution of Aydın and Kanoglu (2017, Pure Appl. Geophys., https://doi.org/10.1007/s00024-017-1508-z). Methodology we present here is simple; it involves eigenfunction expansion and, hence, avoids integral transform techniques. We then use several different types of initial wave profiles with and without initial velocity, estimate shoreline properties and confirm classical runup invariance between linear and nonlinear theories. Two, we use the nonlinear shallow water-wave solution of Kanoglu (2004, J. Fluid Mech. 513, 363-372) to estimate maximum runup. Kanoglu (2004) presented a simple integral solution for the nonlinear shallow water-wave equations using the classical Carrier and Greenspan transformation, and further extended shoreline position and velocity to a simpler integral formulation. In addition, Tinti and Tonini (2005, J. Fluid Mech. 535, 33-64) defined initial condition in a very convenient form for near-shore events. We use Tinti and Tonini (2005) type initial condition in Kanoglu's (2004) shoreline integral solution, which leads further simplified estimates for shoreline position and velocity, i.e. algebraic relation. We then use this algebraic runup estimate to investigate effect of earthquake source parameters on maximum runup and present results similar to Sepulveda and Liu (2016, Coast. Eng. 112, 57-68).

Simplified, inverse, ejector design tool

NASA Technical Reports Server (NTRS)

Dechant, Lawrence J.

1993-01-01

A simple lumped parameter based inverse design tool has been developed which provides flow path geometry and entrainment estimates subject to operational, acoustic, and design constraints. These constraints are manifested through specification of primary mass flow rate or ejector thrust, fully-mixed exit velocity, and static pressure matching. Fundamentally, integral forms of the conservation equations coupled with the specified design constraints are combined to yield an easily invertible linear system in terms of the flow path cross-sectional areas. Entrainment is computed by back substitution. Initial comparison with experimental and analogous one-dimensional methods show good agreement. Thus, this simple inverse design code provides an analytically based, preliminary design tool with direct application to High Speed Civil Transport (HSCT) design studies.

A Simple Model for Nonlinear Confocal Ultrasonic Beams

NASA Astrophysics Data System (ADS)

Zhang, Dong; Zhou, Lin; Si, Li-Sheng; Gong, Xiu-Fen

2007-01-01

A confocally and coaxially arranged pair of focused transmitter and receiver represents one of the best geometries for medical ultrasonic imaging and non-invasive detection. We develop a simple theoretical model for describing the nonlinear propagation of a confocal ultrasonic beam in biological tissues. On the basis of the parabolic approximation and quasi-linear approximation, the nonlinear Khokhlov-Zabolotskaya-Kuznetsov (KZK) equation is solved by using the angular spectrum approach. Gaussian superposition technique is applied to simplify the solution, and an analytical solution for the second harmonics in the confocal ultrasonic beam is presented. Measurements are performed to examine the validity of the theoretical model. This model provides a preliminary model for acoustic nonlinear microscopy.

Computing eigenfunctions and eigenvalues of boundary-value problems with the orthogonal spectral renormalization method

NASA Astrophysics Data System (ADS)

Cartarius, Holger; Musslimani, Ziad H.; Schwarz, Lukas; Wunner, Günter

2018-03-01

The spectral renormalization method was introduced in 2005 as an effective way to compute ground states of nonlinear Schrödinger and Gross-Pitaevskii type equations. In this paper, we introduce an orthogonal spectral renormalization (OSR) method to compute ground and excited states (and their respective eigenvalues) of linear and nonlinear eigenvalue problems. The implementation of the algorithm follows four simple steps: (i) reformulate the underlying eigenvalue problem as a fixed-point equation, (ii) introduce a renormalization factor that controls the convergence properties of the iteration, (iii) perform a Gram-Schmidt orthogonalization process in order to prevent the iteration from converging to an unwanted mode, and (iv) compute the solution sought using a fixed-point iteration. The advantages of the OSR scheme over other known methods (such as Newton's and self-consistency) are (i) it allows the flexibility to choose large varieties of initial guesses without diverging, (ii) it is easy to implement especially at higher dimensions, and (iii) it can easily handle problems with complex and random potentials. The OSR method is implemented on benchmark Hermitian linear and nonlinear eigenvalue problems as well as linear and nonlinear non-Hermitian PT -symmetric models.

Effects from Unsaturated Zone Flow during Oscillatory Hydraulic Testing

NASA Astrophysics Data System (ADS)

Lim, D.; Zhou, Y.; Cardiff, M. A.; Barrash, W.

2014-12-01

In analyzing pumping tests on unconfined aquifers, the impact of the unsaturated zone is often neglected. Instead, desaturation at the water table is often treated as a free-surface boundary, which is simple and allows for relatively fast computation. Richards' equation models, which account for unsaturated flow, can be compared with saturated flow models to validate the use of Darcy's Law. In this presentation, we examine the appropriateness of using fast linear steady-periodic models based on linearized water table conditions in order to simulate oscillatory pumping tests in phreatic aquifers. We compare oscillatory pumping test models including: 1) a 2-D radially-symmetric phreatic aquifer model with a partially penetrating well, simulated using both Darcy's Law and Richards' Equation in COMSOL; and 2) a linear phase-domain numerical model developed in MATLAB. Both COMSOL and MATLAB models are calibrated to match oscillatory pumping test data collected in the summer of 2013 at the Boise Hydrogeophysical Research Site (BHRS), and we examine the effect of model type on the associated parameter estimates. The results of this research will aid unconfined aquifer characterization efforts and help to constrain the impact of the simplifying physical assumptions often employed during test analysis.

Maximal liquid bridges between horizontal cylinders

NASA Astrophysics Data System (ADS)

Cooray, Himantha; Huppert, Herbert E.; Neufeld, Jerome A.

2016-08-01

We investigate two-dimensional liquid bridges trapped between pairs of identical horizontal cylinders. The cylinders support forces owing to surface tension and hydrostatic pressure that balance the weight of the liquid. The shape of the liquid bridge is determined by analytically solving the nonlinear Laplace-Young equation. Parameters that maximize the trapping capacity (defined as the cross-sectional area of the liquid bridge) are then determined. The results show that these parameters can be approximated with simple relationships when the radius of the cylinders is small compared with the capillary length. For such small cylinders, liquid bridges with the largest cross-sectional area occur when the centre-to-centre distance between the cylinders is approximately twice the capillary length. The maximum trapping capacity for a pair of cylinders at a given separation is linearly related to the separation when it is small compared with the capillary length. The meniscus slope angle of the largest liquid bridge produced in this regime is also a linear function of the separation. We additionally derive approximate solutions for the profile of a liquid bridge, using the linearized Laplace-Young equation. These solutions analytically verify the above-mentioned relationships obtained for the maximization of the trapping capacity.

A novel technique to solve nonlinear higher-index Hessenberg differential-algebraic equations by Adomian decomposition method.

PubMed

Benhammouda, Brahim

2016-01-01

Since 1980, the Adomian decomposition method (ADM) has been extensively used as a simple powerful tool that applies directly to solve different kinds of nonlinear equations including functional, differential, integro-differential and algebraic equations. However, for differential-algebraic equations (DAEs) the ADM is applied only in four earlier works. There, the DAEs are first pre-processed by some transformations like index reductions before applying the ADM. The drawback of such transformations is that they can involve complex algorithms, can be computationally expensive and may lead to non-physical solutions. The purpose of this paper is to propose a novel technique that applies the ADM directly to solve a class of nonlinear higher-index Hessenberg DAEs systems efficiently. The main advantage of this technique is that; firstly it avoids complex transformations like index reductions and leads to a simple general algorithm. Secondly, it reduces the computational work by solving only linear algebraic systems with a constant coefficient matrix at each iteration, except for the first iteration where the algebraic system is nonlinear (if the DAE is nonlinear with respect to the algebraic variable). To demonstrate the effectiveness of the proposed technique, we apply it to a nonlinear index-three Hessenberg DAEs system with nonlinear algebraic constraints. This technique is straightforward and can be programmed in Maple or Mathematica to simulate real application problems.

Non-Linear Dynamics of Saturn's Rings

NASA Astrophysics Data System (ADS)

Esposito, L. W.

2016-12-01

Non-linear processes can explain why Saturn's rings are so active and dynamic. Ring systems differ from simple linear systems in two significant ways: 1. They are systems of granular material: where particle-to-particle collisions dominate; thus a kinetic, not a fluid description needed. Stresses are strikingly inhomogeneous and fluctuations are large compared to equilibrium. 2. They are strongly forced by resonances: which drive a non-linear response, that push the system across thresholds that lead to persistent states. Some of this non-linearity is captured in a simple Predator-Prey Model: Periodic forcing from the moon causes streamline crowding; This damps the relative velocity. About a quarter phase later, the aggregates stir the system to higher relative velocity and the limit cycle repeats each orbit, with relative velocity ranging from nearly zero to a multiple of the orbit average. Summary of Halo Results: A predator-prey model for ring dynamics produces transient structures like `straw' that can explain the halo morphology and spectroscopy: Cyclic velocity changes cause perturbed regions to reach higher collision speeds at some orbital phases, which preferentially removes small regolith particles; surrounding particles diffuse back too slowly to erase the effect: this gives the halo morphology; this requires energetic collisions (v ≈ 10m/sec, with throw distances about 200km, implying objects of scale R ≈ 20km).Transform to Duffing Eqn : With the coordinate transformation, z = M2/3, the Predator-Prey equations can be combined to form a single second-order differential equation with harmonic resonance forcing.Ring dynamics and history implications: Moon-triggered clumping explains both small and large particles at resonances. We calculate the stationary size distribution using a cell-to-cell mapping procedure that converts the phase-plane trajectories to a Markov chain. Approximating it as an asymmetric random walk with reflecting boundaries determines the power law index, using results of numerical simulations in the tidal environment. Aggregates can explain many dynamic aspects of the rings and can renew rings by shielding and recycling the material within them, depending on how long the mass is sequestered. We can ask: Are Saturn's rings a chaotic non-linear driven system?

Non-Linear Dynamics of Saturn’s Rings

NASA Astrophysics Data System (ADS)

Esposito, Larry W.

2015-11-01

Non-linear processes can explain why Saturn’s rings are so active and dynamic. Ring systems differ from simple linear systems in two significant ways: 1. They are systems of granular material: where particle-to-particle collisions dominate; thus a kinetic, not a fluid description needed. We find that stresses are strikingly inhomogeneous and fluctuations are large compared to equilibrium. 2. They are strongly forced by resonances: which drive a non-linear response, pushing the system across thresholds that lead to persistent states.Some of this non-linearity is captured in a simple Predator-Prey Model: Periodic forcing from the moon causes streamline crowding; This damps the relative velocity, and allows aggregates to grow. About a quarter phase later, the aggregates stir the system to higher relative velocity and the limit cycle repeats each orbit.Summary of Halo Results: A predator-prey model for ring dynamics produces transient structures like ‘straw’ that can explain the halo structure and spectroscopy: This requires energetic collisions (v ≈ 10m/sec, with throw distances about 200km, implying objects of scale R ≈ 20km).Transform to Duffing Eqn : With the coordinate transformation, z = M2/3, the Predator-Prey equations can be combined to form a single second-order differential equation with harmonic resonance forcing.Ring dynamics and history implications: Moon-triggered clumping at perturbed regions in Saturn’s rings creates both high velocity dispersion and large aggregates at these distances, explaining both small and large particles observed there. We calculate the stationary size distribution using a cell-to-cell mapping procedure that converts the phase-plane trajectories to a Markov chain. Approximating the Markov chain as an asymmetric random walk with reflecting boundaries allows us to determine the power law index from results of numerical simulations in the tidal environment surrounding Saturn. Aggregates can explain many dynamic aspects of the rings and can renew rings by shielding and recycling the material within them, depending on how long the mass is sequestered. We can ask: Are Saturn’s rings a chaotic non-linear driven system?

Effective Surfactants Blend Concentration Determination for O/W Emulsion Stabilization by Two Nonionic Surfactants by Simple Linear Regression.

PubMed

Hassan, A K

2015-01-01

In this work, O/W emulsion sets were prepared by using different concentrations of two nonionic surfactants. The two surfactants, tween 80(HLB=15.0) and span 80(HLB=4.3) were used in a fixed proportions equal to 0.55:0.45 respectively. HLB value of the surfactants blends were fixed at 10.185. The surfactants blend concentration is starting from 3% up to 19%. For each O/W emulsion set the conductivity was measured at room temperature (25±2°), 40, 50, 60, 70 and 80°. Applying the simple linear regression least squares method statistical analysis to the temperature-conductivity obtained data determines the effective surfactants blend concentration required for preparing the most stable O/W emulsion. These results were confirmed by applying the physical stability centrifugation testing and the phase inversion temperature range measurements. The results indicated that, the relation which represents the most stable O/W emulsion has the strongest direct linear relationship between temperature and conductivity. This relationship is linear up to 80°. This work proves that, the most stable O/W emulsion is determined via the determination of the maximum R² value by applying of the simple linear regression least squares method to the temperature-conductivity obtained data up to 80°, in addition to, the true maximum slope is represented by the equation which has the maximum R² value. Because the conditions would be changed in a more complex formulation, the method of the determination of the effective surfactants blend concentration was verified by applying it for more complex formulations of 2% O/W miconazole nitrate cream and the results indicate its reproducibility.

Consensus Algorithms for Networks of Systems with Second- and Higher-Order Dynamics

NASA Astrophysics Data System (ADS)

Fruhnert, Michael

This thesis considers homogeneous networks of linear systems. We consider linear feedback controllers and require that the directed graph associated with the network contains a spanning tree and systems are stabilizable. We show that, in continuous-time, consensus with a guaranteed rate of convergence can always be achieved using linear state feedback. For networks of continuous-time second-order systems, we provide a new and simple derivation of the conditions for a second-order polynomials with complex coefficients to be Hurwitz. We apply this result to obtain necessary and sufficient conditions to achieve consensus with networks whose graph Laplacian matrix may have complex eigenvalues. Based on the conditions found, methods to compute feedback gains are proposed. We show that gains can be chosen such that consensus is achieved robustly over a variety of communication structures and system dynamics. We also consider the use of static output feedback. For networks of discrete-time second-order systems, we provide a new and simple derivation of the conditions for a second-order polynomials with complex coefficients to be Schur. We apply this result to obtain necessary and sufficient conditions to achieve consensus with networks whose graph Laplacian matrix may have complex eigenvalues. We show that consensus can always be achieved for marginally stable systems and discretized systems. Simple conditions for consensus achieving controllers are obtained when the Laplacian eigenvalues are all real. For networks of continuous-time time-variant higher-order systems, we show that uniform consensus can always be achieved if systems are quadratically stabilizable. In this case, we provide a simple condition to obtain a linear feedback control. For networks of discrete-time higher-order systems, we show that constant gains can be chosen such that consensus is achieved for a variety of network topologies. First, we develop simple results for networks of time-invariant systems and networks of time-variant systems that are given in controllable canonical form. Second, we formulate the problem in terms of Linear Matrix Inequalities (LMIs). The condition found simplifies the design process and avoids the parallel solution of multiple LMIs. The result yields a modified Algebraic Riccati Equation (ARE) for which we present an equivalent LMI condition.

Solving ordinary differential equations by electrical analogy: a multidisciplinary teaching tool

NASA Astrophysics Data System (ADS)

Sanchez Perez, J. F.; Conesa, M.; Alhama, I.

2016-11-01

Ordinary differential equations are the mathematical formulation for a great variety of problems in science and engineering, and frequently, two different problems are equivalent from a mathematical point of view when they are formulated by the same equations. Students acquire the knowledge of how to solve these equations (at least some types of them) using protocols and strict algorithms of mathematical calculation without thinking about the meaning of the equation. The aim of this work is that students learn to design network models or circuits in this way; with simple knowledge of them, students can establish the association of electric circuits and differential equations and their equivalences, from a formal point of view, that allows them to associate knowledge of two disciplines and promote the use of this interdisciplinary approach to address complex problems. Therefore, they learn to use a multidisciplinary tool that allows them to solve these kinds of equations, even students of first course of engineering, whatever the order, grade or type of non-linearity. This methodology has been implemented in numerous final degree projects in engineering and science, e.g., chemical engineering, building engineering, industrial engineering, mechanical engineering, architecture, etc. Applications are presented to illustrate the subject of this manuscript.

Time-dependent spectral renormalization method

NASA Astrophysics Data System (ADS)

Cole, Justin T.; Musslimani, Ziad H.

2017-11-01

The spectral renormalization method was introduced by Ablowitz and Musslimani (2005) as an effective way to numerically compute (time-independent) bound states for certain nonlinear boundary value problems. In this paper, we extend those ideas to the time domain and introduce a time-dependent spectral renormalization method as a numerical means to simulate linear and nonlinear evolution equations. The essence of the method is to convert the underlying evolution equation from its partial or ordinary differential form (using Duhamel's principle) into an integral equation. The solution sought is then viewed as a fixed point in both space and time. The resulting integral equation is then numerically solved using a simple renormalized fixed-point iteration method. Convergence is achieved by introducing a time-dependent renormalization factor which is numerically computed from the physical properties of the governing evolution equation. The proposed method has the ability to incorporate physics into the simulations in the form of conservation laws or dissipation rates. This novel scheme is implemented on benchmark evolution equations: the classical nonlinear Schrödinger (NLS), integrable PT symmetric nonlocal NLS and the viscous Burgers' equations, each of which being a prototypical example of a conservative and dissipative dynamical system. Numerical implementation and algorithm performance are also discussed.

Technological pedagogical content knowledge of junior high school mathematics teachers in teaching linear equation

NASA Astrophysics Data System (ADS)

Wati, S.; Fitriana, L.; Mardiyana

2018-04-01

Linear equation is one of the topics in mathematics that are considered difficult. Student difficulties of understanding linear equation can be caused by lack of understanding this concept and the way of teachers teach. TPACK is a way to understand the complex relationships between teaching and content taught through the use of specific teaching approaches and supported by the right technology tools. This study aims to identify TPACK of junior high school mathematics teachers in teaching linear equation. The method used in the study was descriptive. In the first phase, a survey using a questionnaire was carried out on 45 junior high school mathematics teachers in teaching linear equation. While in the second phase, the interview involved three teachers. The analysis of data used were quantitative and qualitative technique. The result PCK revealed teachers emphasized developing procedural and conceptual knowledge through reliance on traditional in teaching linear equation. The result of TPK revealed teachers’ lower capacity to deal with the general information and communications technologies goals across the curriculum in teaching linear equation. The result indicated that PowerPoint constitutes TCK modal technological capability in teaching linear equation. The result of TPACK seems to suggest a low standard in teachers’ technological skills across a variety of mathematics education goals in teaching linear equation. This means that the ability of teachers’ TPACK in teaching linear equation still needs to be improved.

Demonstration of Detection and Ranging Using Solvable Chaos

NASA Technical Reports Server (NTRS)

Corron, Ned J.; Stahl, Mark T.; Blakely, Jonathan N.

2013-01-01

Acoustic experiments demonstrate a novel approach to ranging and detection that exploits the properties of a solvable chaotic oscillator. This nonlinear oscillator includes an ordinary differential equation and a discrete switching condition. The chaotic waveform generated by this hybrid system is used as the transmitted waveform. The oscillator admits an exact analytic solution that can be written as the linear convolution of binary symbols and a single basis function. This linear representation enables coherent reception using a simple analog matched filter and without need for digital sampling or signal processing. An audio frequency implementation of the transmitter and receiver is described. Successful acoustic ranging measurements are presented to demonstrate the viability of the approach.

Bayesian parameter estimation for nonlinear modelling of biological pathways.

PubMed

Ghasemi, Omid; Lindsey, Merry L; Yang, Tianyi; Nguyen, Nguyen; Huang, Yufei; Jin, Yu-Fang

2011-01-01

The availability of temporal measurements on biological experiments has significantly promoted research areas in systems biology. To gain insight into the interaction and regulation of biological systems, mathematical frameworks such as ordinary differential equations have been widely applied to model biological pathways and interpret the temporal data. Hill equations are the preferred formats to represent the reaction rate in differential equation frameworks, due to their simple structures and their capabilities for easy fitting to saturated experimental measurements. However, Hill equations are highly nonlinearly parameterized functions, and parameters in these functions cannot be measured easily. Additionally, because of its high nonlinearity, adaptive parameter estimation algorithms developed for linear parameterized differential equations cannot be applied. Therefore, parameter estimation in nonlinearly parameterized differential equation models for biological pathways is both challenging and rewarding. In this study, we propose a Bayesian parameter estimation algorithm to estimate parameters in nonlinear mathematical models for biological pathways using time series data. We used the Runge-Kutta method to transform differential equations to difference equations assuming a known structure of the differential equations. This transformation allowed us to generate predictions dependent on previous states and to apply a Bayesian approach, namely, the Markov chain Monte Carlo (MCMC) method. We applied this approach to the biological pathways involved in the left ventricle (LV) response to myocardial infarction (MI) and verified our algorithm by estimating two parameters in a Hill equation embedded in the nonlinear model. We further evaluated our estimation performance with different parameter settings and signal to noise ratios. Our results demonstrated the effectiveness of the algorithm for both linearly and nonlinearly parameterized dynamic systems. Our proposed Bayesian algorithm successfully estimated parameters in nonlinear mathematical models for biological pathways. This method can be further extended to high order systems and thus provides a useful tool to analyze biological dynamics and extract information using temporal data.

Nonlinear to Linear Elastic Code Coupling in 2-D Axisymmetric Media.

DOE Office of Scientific and Technical Information (OSTI.GOV)

Preston, Leiph

Explosions within the earth nonlinearly deform the local media, but at typical seismological observation distances, the seismic waves can be considered linear. Although nonlinear algorithms can simulate explosions in the very near field well, these codes are computationally expensive and inaccurate at propagating these signals to great distances. A linearized wave propagation code, coupled to a nonlinear code, provides an efficient mechanism to both accurately simulate the explosion itself and to propagate these signals to distant receivers. To this end we have coupled Sandia's nonlinear simulation algorithm CTH to a linearized elastic wave propagation code for 2-D axisymmetric media (axiElasti)more » by passing information from the nonlinear to the linear code via time-varying boundary conditions. In this report, we first develop the 2-D axisymmetric elastic wave equations in cylindrical coordinates. Next we show how we design the time-varying boundary conditions passing information from CTH to axiElasti, and finally we demonstrate the coupling code via a simple study of the elastic radius.« less

Solution of fractional kinetic equation by a class of integral transform of pathway type

NASA Astrophysics Data System (ADS)

Kumar, Dilip

2013-04-01

Solutions of fractional kinetic equations are obtained through an integral transform named Pα-transform introduced in this paper. The Pα-transform is a binomial type transform containing many class of transforms including the well known Laplace transform. The paper is motivated by the idea of pathway model introduced by Mathai [Linear Algebra Appl. 396, 317-328 (2005), 10.1016/j.laa.2004.09.022]. The composition of the transform with differential and integral operators are proved along with convolution theorem. As an illustration of applications to the general theory of differential equations, a simple differential equation is solved by the new transform. Being a new transform, the Pα-transform of some elementary functions as well as some generalized special functions such as H-function, G-function, Wright generalized hypergeometric function, generalized hypergeometric function, and Mittag-Leffler function are also obtained. The results for the classical Laplace transform is retrieved by letting α → 1.

A pressure-based semi-implicit space-time discontinuous Galerkin method on staggered unstructured meshes for the solution of the compressible Navier-Stokes equations at all Mach numbers

NASA Astrophysics Data System (ADS)

Tavelli, Maurizio; Dumbser, Michael

2017-07-01

We propose a new arbitrary high order accurate semi-implicit space-time discontinuous Galerkin (DG) method for the solution of the two and three dimensional compressible Euler and Navier-Stokes equations on staggered unstructured curved meshes. The method is pressure-based and semi-implicit and is able to deal with all Mach number flows. The new DG scheme extends the seminal ideas outlined in [1], where a second order semi-implicit finite volume method for the solution of the compressible Navier-Stokes equations with a general equation of state was introduced on staggered Cartesian grids. Regarding the high order extension we follow [2], where a staggered space-time DG scheme for the incompressible Navier-Stokes equations was presented. In our scheme, the discrete pressure is defined on the primal grid, while the discrete velocity field and the density are defined on a face-based staggered dual grid. Then, the mass conservation equation, as well as the nonlinear convective terms in the momentum equation and the transport of kinetic energy in the energy equation are discretized explicitly, while the pressure terms appearing in the momentum and energy equation are discretized implicitly. Formal substitution of the discrete momentum equation into the total energy conservation equation yields a linear system for only one unknown, namely the scalar pressure. Here the equation of state is assumed linear with respect to the pressure. The enthalpy and the kinetic energy are taken explicitly and are then updated using a simple Picard procedure. Thanks to the use of a staggered grid, the final pressure system is a very sparse block five-point system for three dimensional problems and it is a block four-point system in the two dimensional case. Furthermore, for high order in space and piecewise constant polynomials in time, the system is observed to be symmetric and positive definite. This allows to use fast linear solvers such as the conjugate gradient (CG) method. In addition, all the volume and surface integrals needed by the scheme depend only on the geometry and the polynomial degree of the basis and test functions and can therefore be precomputed and stored in a preprocessing stage. This leads to significant savings in terms of computational effort for the time evolution part. In this way also the extension to a fully curved isoparametric approach becomes natural and affects only the preprocessing step. The viscous terms and the heat flux are also discretized making use of the staggered grid by defining the viscous stress tensor and the heat flux vector on the dual grid, which corresponds to the use of a lifting operator, but on the dual grid. The time step of our new numerical method is limited by a CFL condition based only on the fluid velocity and not on the sound speed. This makes the method particularly interesting for low Mach number flows. Finally, a very simple combination of artificial viscosity and the a posteriori MOOD technique allows to deal with shock waves and thus permits also to simulate high Mach number flows. We show computational results for a large set of two and three-dimensional benchmark problems, including both low and high Mach number flows and using polynomial approximation degrees up to p = 4.

Development and validation of anthropometric equations to estimate appendicular muscle mass in elderly women.

PubMed

Pereira, Piettra Moura Galvão; da Silva, Giselma Alcântara; Santos, Gilberto Moreira; Petroski, Edio Luiz; Geraldes, Amandio Aristides Rihan

2013-07-02

This study aimed to examine the cross validity of two anthropometric equations commonly used and propose simple anthropometric equations to estimate appendicular muscle mass (AMM) in elderly women. Among 234 physically active and functionally independent elderly women, 101 (60 to 89 years) were selected through simple drawing to compose the study sample. The paired t test and the Pearson correlation coefficient were used to perform cross-validation and concordance was verified by intraclass correction coefficient (ICC) and by the Bland and Altman technique. To propose predictive models, multiple linear regression analysis, anthropometric measures of body mass (BM), height, girth, skinfolds, body mass index (BMI) were used, and muscle perimeters were included in the analysis as independent variables. Dual-Energy X-ray Absorptiometry (AMMDXA) was used as criterion measurement. The sample power calculations were carried out by Post Hoc Compute Achieved Power. Sample power values from 0.88 to 0.91 were observed. When compared, the two equations tested differed significantly from the AMMDXA (p <0.001 and p = 0.001). Ten population / specific anthropometric equations were developed to estimate AMM, among them, three equations achieved all validation criteria used: AMM (E2) = 4.150 +0.251 [bodymass (BM)] - 0.411 [bodymass index (BMI)] + 0.011 [Right forearm perimeter (PANTd) 2]; AMM (E3) = 4.087 + 0.255 (BM) - 0.371 (BMI) + 0.011 (PANTd) 2 - 0.035 [thigh skinfold (DCCO)]; MMA (E6) = 2.855 + 0.298 (BM) + 0.019 (Age) - 0,082 [hip circumference (PQUAD)] + 0.400 (PANTd) - 0.332 (BMI). The equations estimated the criterion method (p = 0.056 p = 0.158), and explained from 0.69% to 0.74% of variations observed in AMMDXA with low standard errors of the estimate (1.36 to 1.55 kg) and high concordance (ICC between 0,90 and 0.91 and concordance limits from -2,93 to 2,33 kg). The equations tested were not valid for use in physically active and functionally independent elderly women. The simple anthropometric equations developed in this study showed good practical applicability and high validity to estimate AMM in elderly women.

Development and validation of anthropometric equations to estimate appendicular muscle mass in elderly women

PubMed Central

2013-01-01

Objective This study aimed to examine the cross validity of two anthropometric equations commonly used and propose simple anthropometric equations to estimate appendicular muscle mass (AMM) in elderly women. Methods Among 234 physically active and functionally independent elderly women, 101 (60 to 89 years) were selected through simple drawing to compose the study sample. The paired t test and the Pearson correlation coefficient were used to perform cross-validation and concordance was verified by intraclass correction coefficient (ICC) and by the Bland and Altman technique. To propose predictive models, multiple linear regression analysis, anthropometric measures of body mass (BM), height, girth, skinfolds, body mass index (BMI) were used, and muscle perimeters were included in the analysis as independent variables. Dual-Energy X-ray Absorptiometry (AMMDXA) was used as criterion measurement. The sample power calculations were carried out by Post Hoc Compute Achieved Power. Sample power values from 0.88 to 0.91 were observed. Results When compared, the two equations tested differed significantly from the AMMDXA (p <0.001 and p = 0.001). Ten population / specific anthropometric equations were developed to estimate AMM, among them, three equations achieved all validation criteria used: AMM (E2) = 4.150 +0.251 [bodymass (BM)] - 0.411 [bodymass index (BMI)] + 0.011 [Right forearm perimeter (PANTd) 2]; AMM (E3) = 4.087 + 0.255 (BM) - 0.371 (BMI) + 0.011 (PANTd) 2 - 0.035 [thigh skinfold (DCCO)]; MMA (E6) = 2.855 + 0.298 (BM) + 0.019 (Age) - 0,082 [hip circumference (PQUAD)] + 0.400 (PANTd) - 0.332 (BMI). The equations estimated the criterion method (p = 0.056 p = 0.158), and explained from 0.69% to 0.74% of variations observed in AMMDXA with low standard errors of the estimate (1.36 to 1.55 kg) and high concordance (ICC between 0,90 and 0.91 and concordance limits from -2,93 to 2,33 kg). Conclusion The equations tested were not valid for use in physically active and functionally independent elderly women. The simple anthropometric equations developed in this study showed good practical applicability and high validity to estimate AMM in elderly women. PMID:23815948

2D instabilities of surface gravity waves on a linear shear current

NASA Astrophysics Data System (ADS)

Francius, Marc; Kharif, Christian

2016-04-01

Periodic 2D surface water waves propagating steadily on a rotational current have been studied by many authors (see [1] and references therein). Although the recent important theoretical developments have confirmed that periodic waves can exist over flows with arbitrary vorticity, their stability and their nonlinear evolution have not been much studied extensively so far. In fact, even in the rather simple case of uniform vorticity (linear shear), few papers have been published on the effect of a vertical shear current on the side-band instability of a uniform wave train over finite depth. In most of these studies [2-5], asymptotic expansions and multiple scales method have been used to obtain envelope evolution equations, which allow eventually to formulate a condition of (linear) instability to long modulational perturbations. It is noted here that this instability is often referred in the literature as the Benjamin-Feir or modulational instability. In the present study, we consider the linear stability of finite amplitude two-dimensional, periodic water waves propagating steadily on the free surface of a fluid with constant vorticity and finite depth. First, the steadily propagating surface waves are computed with steepness up to very close to the highest, using a Fourier series expansions and a collocation method, which constitutes a simple extension of Fenton's method [6] to the cases with a linear shear current. Then, the linear stability of these permanent waves to infinitesimal 2D perturbations is developed from the fully nonlinear equations in the framework of normal modes analysis. This linear stability analysis is an extension of [7] to the case of waves in the presence of a linear shear current and permits the determination of the dominant instability as a function of depth and vorticity for a given steepness. The numerical results are used to assess the accuracy of the vor-NLS equation derived in [5] for the characteristics of modulational instabilities due to resonant four-wave interactions, as well as to study the influence of vorticity and nonlinearity on the characteristics of linear instabilities due to resonant five-wave and six-wave interactions. Depending on the dimensionless depth, superharmonic instabilities due to five-wave interactions can become dominant with increasing positive vorticiy. Acknowledgments: This work was supported by the Direction Générale de l'Armement and funded by the ANR project n°. ANR-13-ASTR-0007. References [1] A. Constantin, Two-dimensionality of gravity water flows of constant non-zero vorticity beneath a surface wave train, Eur. J. Mech. B/Fluids, 2011, 30, 12-16. [2] R. S. Johnson, On the modulation of water waves on shear flows, Proc. Royal Soc. Lond. A., 1976, 347, 537-546. [3] M. Oikawa, K. Chow, D. J. Benney, The propagation of nonlinear wave packets in a shear flow with a free surface, Stud. Appl. Math., 1987, 76, 69-92. [4] A. I Baumstein, Modulation of gravity waves with shear in water, Stud. Appl. Math., 1998, 100, 365-90. [5] R. Thomas, C. Kharif, M. Manna, A nonlinear Schrödinger equation for water waves on finite depth with constant vorticity, Phys. Fluids, 2012, 24, 127102. [6] M. M Rienecker, J. D Fenton, A Fourier approximation method for steady water waves , J. Fluid Mech., 1981, 104, 119-137 [7] M. Francius, C. Kharif, Three-dimensional instabilities of periodic gravity waves in shallow water, J. Fluid Mech., 2006, 561, 417-437

On supporting students' understanding of solving linear equation by using flowchart

NASA Astrophysics Data System (ADS)

Toyib, Muhamad; Kusmayadi, Tri Atmojo; Riyadi

2017-05-01

The aim of this study was to support 7th graders to gradually understand the concepts and procedures of solving linear equation. Thirty-two 7th graders of a Junior High School in Surakarta, Indonesia were involved in this study. Design research was used as the research approach to achieve the aim. A set of learning activities in solving linear equation with one unknown were designed based on Realistic Mathematics Education (RME) approach. The activities were started by playing LEGO to find a linear equation then solve the equation by using flowchart. The results indicate that using the realistic problems, playing LEGO could stimulate students to construct linear equation. Furthermore, Flowchart used to encourage students' reasoning and understanding on the concepts and procedures of solving linear equation with one unknown.

Coupled bending-torsion steady-state response of pretwisted, nonuniform rotating beams using a transfer-matrix method

NASA Technical Reports Server (NTRS)

Gray, Carl E., Jr.

1988-01-01

Using the Newtonian method, the equations of motion are developed for the coupled bending-torsion steady-state response of beams rotating at constant angular velocity in a fixed plane. The resulting equations are valid to first order strain-displacement relationships for a long beam with all other nonlinear terms retained. In addition, the equations are valid for beams with the mass centroidal axis offset (eccentric) from the elastic axis, nonuniform mass and section properties, and variable twist. The solution of these coupled, nonlinear, nonhomogeneous, differential equations is obtained by modifying a Hunter linear second-order transfer-matrix solution procedure to solve the nonlinear differential equations and programming the solution for a desk-top personal computer. The modified transfer-matrix method was verified by comparing the solution for a rotating beam with a geometric, nonlinear, finite-element computer code solution; and for a simple rotating beam problem, the modified method demonstrated a significant advantage over the finite-element solution in accuracy, ease of solution, and actual computer processing time required to effect a solution.

An Edge-Based Method for the Incompressible Navier-Stokes Equations on Polygonal Meshes

NASA Astrophysics Data System (ADS)

Wright, Jeffrey A.; Smith, Richard W.

2001-05-01

A pressure-based method is presented for discretizing the unsteady incompressible Navier-Stokes equations using hybrid unstructured meshes. The edge-based data structure and assembly procedure adopted lead naturally to a strictly conservative discretization, which is valid for meshes composed of n-sided polygons. Particular attention is given to the construction of a pressure-velocity coupling procedure which is supported by edge data, resulting in a relatively simple numerical method that is consistent with the boundary and initial conditions required by the incompressible Navier-Stokes equations. Edge formulas are presented for assembling the momentum equations, which are based on an upwind-biased linear reconstruction of the velocity field. Similar formulas are presented for assembling the pressure equation. The method is demonstrated to be second-order accurate in space and time for two Navier-Stokes problems admitting an exact solution. Results for several other well-known problems are also presented, including lid-driven cavity flow, impulsively started cylinder flow, and unsteady vortex shedding from a circular cylinder. Although the method is by construction minimalist, it is shown to be accurate and robust for the problems considered.

On-orbit identifying the inertia parameters of space robotic systems using simple equivalent dynamics

NASA Astrophysics Data System (ADS)

Xu, Wenfu; Hu, Zhonghua; Zhang, Yu; Liang, Bin

2017-03-01

After being launched into space to perform some tasks, the inertia parameters of a space robotic system may change due to fuel consumption, hardware reconfiguration, target capturing, and so on. For precision control and simulation, it is required to identify these parameters on orbit. This paper proposes an effective method for identifying the complete inertia parameters (including the mass, inertia tensor and center of mass position) of a space robotic system. The key to the method is to identify two types of simple dynamics systems: equivalent single-body and two-body systems. For the former, all of the joints are locked into a designed configuration and the thrusters are used for orbital maneuvering. The object function for optimization is defined in terms of acceleration and velocity of the equivalent single body. For the latter, only one joint is unlocked and driven to move along a planned (exiting) trajectory in free-floating mode. The object function is defined based on the linear and angular momentum equations. Then, the parameter identification problems are transformed into non-linear optimization problems. The Particle Swarm Optimization (PSO) algorithm is applied to determine the optimal parameters, i.e. the complete dynamic parameters of the two equivalent systems. By sequentially unlocking the 1st to nth joints (or unlocking the nth to 1st joints), the mass properties of body 0 to n (or n to 0) are completely identified. For the proposed method, only simple dynamics equations are needed for identification. The excitation motion (orbit maneuvering and joint motion) is also easily realized. Moreover, the method does not require prior knowledge of the mass properties of any body. It is general and practical for identifying a space robotic system on-orbit.

An analysis of the extension of a ZnO piezoelectric semiconductor nanofiber under an axial force

NASA Astrophysics Data System (ADS)

Zhang, Chunli; Wang, Xiaoyuan; Chen, Weiqiu; Yang, Jiashi

2017-02-01

This paper presents a theoretical analysis on the axial extension of an n-type ZnO piezoelectric semiconductor nanofiber under an axial force. The phenomenological theory of piezoelectric semiconductors consisting of Newton’s second law of motion, the charge equation of electrostatics and the conservation of charge was used. The equations were linearized for small axial force and hence small electron concentration perturbation, and were reduced to one-dimensional equations for thin fibers. Simple and analytical expressions for the electromechanical fields and electron concentration in the fiber were obtained. The fields are either totally or partially described by hyperbolic functions relatively large near the ends of the fiber and change rapidly there. The behavior of the fields is sensitive to the initial electron concentration and the applied axial force. For higher initial electron concentrations the fields are larger near the ends and change more rapidly there.

DOE Office of Scientific and Technical Information (OSTI.GOV)

Pereira, S.H.; Pinho, A.S.S.; Silva, J.M. Hoff da

In this work the exact Friedmann-Robertson-Walker equations for an Elko spinor field coupled to gravity in an Einstein-Cartan framework are presented. The torsion functions coupling the Elko field spin-connection to gravity can be exactly solved and the FRW equations for the system assume a relatively simple form. In the limit of a slowly varying Elko spinor field there is a relevant contribution to the field equations acting exactly as a time varying cosmological model Λ( t )=Λ{sub *}+3β H {sup 2}, where Λ{sub *} and β are constants. Observational data using distance luminosity from magnitudes of supernovae constraint the parametersmore » Ω {sub m} and β, which leads to a lower limit to the Elko mass. Such model mimics, then, the effects of a dark energy fluid, here sourced by the Elko spinor field. The density perturbations in the linear regime were also studied in the pseudo-Newtonian formalism.« less

Fine Tuning the CJ Detonation Speed of a High Explosive products Equation of State

DOE Office of Scientific and Technical Information (OSTI.GOV)

Menikoff, Ralph

For high explosive (HE) simulations, inaccuracies of a per cent or two in the detonation wave speed can result from not suficiently resolving the reaction zone width or from small inaccuracies in calibrating the products equation of state (EOS) or from variation of HE lots. More accurate detonation speeds can be obtained by ne tuning the equation of state to compensate. Here we show that two simple EOS transformations can be used to adjust the CJ detonation speed by a couple of per cent with minimal effect on the CJ release isentrope. The two transformations are (1) a shift inmore » the energy origin and (2) a linear scaling of the speci c volume. The effectiveness of the transformations is demonstrated with simulations of the cylinder test for PBX 9502 starting with a products EOS for which the CJ detonation speed is 1 per cent too low.« less

Unpacking the Complexity of Linear Equations from a Cognitive Load Theory Perspective

ERIC Educational Resources Information Center

Ngu, Bing Hiong; Phan, Huy P.

2016-01-01

The degree of element interactivity determines the complexity and therefore the intrinsic cognitive load of linear equations. The unpacking of linear equations at the level of operational and relational lines allows the classification of linear equations in a hierarchical level of complexity. Mapping similar operational and relational lines across…

Modified Hyperspheres Algorithm to Trace Homotopy Curves of Nonlinear Circuits Composed by Piecewise Linear Modelled Devices

PubMed Central

Vazquez-Leal, H.; Jimenez-Fernandez, V. M.; Benhammouda, B.; Filobello-Nino, U.; Sarmiento-Reyes, A.; Ramirez-Pinero, A.; Marin-Hernandez, A.; Huerta-Chua, J.

2014-01-01

We present a homotopy continuation method (HCM) for finding multiple operating points of nonlinear circuits composed of devices modelled by using piecewise linear (PWL) representations. We propose an adaptation of the modified spheres path tracking algorithm to trace the homotopy trajectories of PWL circuits. In order to assess the benefits of this proposal, four nonlinear circuits composed of piecewise linear modelled devices are analysed to determine their multiple operating points. The results show that HCM can find multiple solutions within a single homotopy trajectory. Furthermore, we take advantage of the fact that homotopy trajectories are PWL curves meant to replace the multidimensional interpolation and fine tuning stages of the path tracking algorithm with a simple and highly accurate procedure based on the parametric straight line equation. PMID:25184157

A higher order panel method for linearized supersonic flow

NASA Technical Reports Server (NTRS)

Ehlers, F. E.; Epton, M. A.; Johnson, F. T.; Magnus, A. E.; Rubbert, P. E.

1979-01-01

The basic integral equations of linearized supersonic theory for an advanced supersonic panel method are derived. Methods using only linear varying source strength over each panel or only quadratic doublet strength over each panel gave good agreement with analytic solutions over cones and zero thickness cambered wings. For three dimensional bodies and wings of general shape, combined source and doublet panels with interior boundary conditions to eliminate the internal perturbations lead to a stable method providing good agreement experiment. A panel system with all edges contiguous resulted from dividing the basic four point non-planar panel into eight triangular subpanels, and the doublet strength was made continuous at all edges by a quadratic distribution over each subpanel. Superinclined panels were developed and tested on s simple nacelle and on an airplane model having engine inlets, with excellent results.

Schwarz maps of algebraic linear ordinary differential equations

NASA Astrophysics Data System (ADS)

Sanabria Malagón, Camilo

2017-12-01

A linear ordinary differential equation is called algebraic if all its solution are algebraic over its field of definition. In this paper we solve the problem of finding closed form solution to algebraic linear ordinary differential equations in terms of standard equations. Furthermore, we obtain a method to compute all algebraic linear ordinary differential equations with rational coefficients by studying their associated Schwarz map through the Picard-Vessiot Theory.

Development of helicopter attitude axes controlled hover flight without pilot assistance and vehicle crashes

NASA Astrophysics Data System (ADS)

Simon, Miguel

In this work, we show how to computerize a helicopter to fly attitude axes controlled hover flight without the assistance of a pilot and without ever crashing. We start by developing a helicopter research test bed system including all hardware, software, and means for testing and training the helicopter to fly by computer. We select a Remote Controlled helicopter with a 5 ft. diameter rotor and 2.2 hp engine. We equip the helicopter with a payload of sensors, computers, navigation and telemetry equipment, and batteries. We develop a differential GPS system with cm accuracy and a ground computerized navigation system for six degrees of freedom (6-DoF) free flight while tracking navigation commands. We design feedback control loops with yet-to-be-determined gains for the five control "knobs" available to a flying radio-controlled (RC) miniature helicopter: engine throttle, main rotor collective pitch, longitudinal cyclic pitch, lateral cyclic pitch, and tail rotor collective pitch. We develop helicopter flight equations using fundamental dynamics, helicopter momentum theory and blade element theory. The helicopter flight equations include helicopter rotor equations of motions, helicopter rotor forces and moments, helicopter trim equations, helicopter stability derivatives, and a coupled fuselage-rotor helicopter 6-DoF model. The helicopter simulation also includes helicopter engine control equations, a helicopter aerodynamic model, and finally helicopter stability and control equations. The derivation of a set of non-linear equations of motion for the main rotor is a contribution of this thesis work. We design and build two special test stands for training and testing the helicopter to fly attitude axes controlled hover flight, starting with one axis at a time and progressing to multiple axes. The first test stand is built for teaching and testing controlled flight of elevation and yaw (i.e., directional control). The second test stand is built for teaching and testing any one or combination of the following attitude axes controlled flight: (1) pitch, (2) roll and (3) yaw. The subsequent development of a novel method to decouple, stabilize and teach the helicopter hover flight is a primary contribution of this thesis. The novel method included the development of a non-linear modeling technique for linearizing the RPM state equation dynamics so that a simple but accurate transfer function is derivable between the "available torque of the engine" and RPM. Specifically, the main rotor and tail rotor torques are modeled accurately with a bias term plus a nonlinear term involving the product of RPM squared times the main rotor blade pitch angle raised to the three-halves power. Application of this non-linear modeling technique resulted in a simple, representative and accurate transfer function model of the open-loop plant for the entire helicopter system so that all the feedback control laws for autonomous flight purposes could be derived easily using classical control theory. This is one of the contributions of this dissertation work. After discussing the integration of hardware and software elements of our helicopter research test bed system, we perform a number of experiments and tests using the two specially built test stands. Feedback gains are derived for controlling the following: (1) engine throttle to maintain prescribed main rotor angular speed, (2) main rotor collective pitch to maintain constant elevation, (3) longitudinal cyclic pitch to maintain prescribed pitch angle, (4) lateral cyclic pitch to maintain prescribed roll angle, and (5) yaw axis to maintain prescribed compass direction. (Abstract shortened by UMI.)

Dynamical relationship between wind speed magnitude and meridional temperature contrast: Application to an interannual oscillation in Venusian middle atmosphere GCM

NASA Astrophysics Data System (ADS)

Yamamoto, Masaru; Takahashi, Masaaki

2018-03-01

We derive simple dynamical relationships between wind speed magnitude and meridional temperature contrast. The relationship explains scatter plot distributions of time series of three variables (maximum zonal wind speed UMAX, meridional wind speed VMAX, and equator-pole temperature contrast dTMAX), which are obtained from a Venus general circulation model with equatorial Kelvin-wave forcing. Along with VMAX and dTMAX, UMAX likely increases with the phase velocity and amplitude of a forced wave. In the scatter diagram of UMAX versus dTMAX, points are plotted along a linear equation obtained from a thermal-wind relationship in the cloud layer. In the scatter diagram of VMAX versus UMAX, the apparent slope is somewhat steep in the high UMAX regime, compared with the low UMAX regime. The scatter plot distributions are qualitatively consistent with a quadratic equation obtained from a diagnostic equation of the stream function above the cloud top. The plotted points in the scatter diagrams form a linear cluster for weak wave forcing, whereas they form a small cluster for strong wave forcing. An interannual oscillation of the general circulation forming the linear cluster in the scatter diagram is apparent in the experiment of weak 5.5-day wave forcing. Although a pair of equatorial Kelvin and high-latitude Rossby waves with a same period (Kelvin-Rossby wave) produces equatorward heat and momentum fluxes in the region below 60 km, the equatorial wave does not contribute to the long-period oscillation. The interannual fluctuation of the high-latitude jet core leading to the time variation of UMAX is produced by growth and decay of a polar mixed Rossby-gravity wave with a 14-day period.

Forecasting Air Force Logistics Command Second Destination Transportation: An Application of Multiple Regression Analysis and Neural Networks

DTIC Science & Technology

1990-09-01

without the help from the DSXR staff. William Lyons, Charles Ramsey , and Martin Meeks went above and beyond to help complete this research. Special...develop a valid forecasting model that is significantly more accurate than the one presently used by DSXR and suggested the development and testing of a...method, Strom tested DSXR’s iterative linear regression forecasting technique by examining P1 in the simple regression equation to determine whether

DOE Office of Scientific and Technical Information (OSTI.GOV)

Jing Yanfei, E-mail: yanfeijing@uestc.edu.c; Huang Tingzhu, E-mail: tzhuang@uestc.edu.c; Duan Yong, E-mail: duanyong@yahoo.c

This study is mainly focused on iterative solutions with simple diagonal preconditioning to two complex-valued nonsymmetric systems of linear equations arising from a computational chemistry model problem proposed by Sherry Li of NERSC. Numerical experiments show the feasibility of iterative methods to some extent when applied to the problems and reveal the competitiveness of our recently proposed Lanczos biconjugate A-orthonormalization methods to other classic and popular iterative methods. By the way, experiment results also indicate that application specific preconditioners may be mandatory and required for accelerating convergence.

Vibration analysis of rotor blades with pendulum absorbers

NASA Technical Reports Server (NTRS)

Murthy, V. R.; Hammond, C. E.

1979-01-01

A comprehensive vibration analysis of rotor blades with spherical pendulum absorbers is presented. Linearized equations of motion for small oscillations about the steady-state deflection of a spherical pendulum on elastic rotor blades undergoing coupled flapwise bending, chordwise bending, and torsional vibrations are obtained. A transmission matrix formulation is given to determine the natural vibrational characteristics of rotor blades with spherical or simple flapping pendulum absorbers. The natural frequencies and mode shapes of a hingeless rotor blade with a spherical pendulum are computed.

Indirect measurements of hydrogen: The deficit method for a many-component system

NASA Astrophysics Data System (ADS)

Levine, Timothy E.; Yu, Ning; Kodali, Padma; Walter, Kevin C.; Nastasi, Michael; Tesmer, Joseph R.; Maggiore, Carl J.; Mayer, James W.

We have developed a simple technique for determining hydrogen atomic fraction from the ion backscattering spectrometry (IBS) signals of the remaining species. This technique uses the surface heights of various IBS signals in the form of a linear matrix equation. We apply this technique to in situ analysis of ion-beam-induced densification of sol-gel zirconia thin films, where hydrogen is the most volatile species during irradiation. Attendant errors are discussed with an emphasis on stopping powers and Bragg's rule.

Anytime query-tuned kernel machine classifiers via Cholesky factorization

NASA Technical Reports Server (NTRS)

DeCoste, D.

2002-01-01

We recently demonstrated 2 to 64-fold query-time speedups of Support Vector Machine and Kernel Fisher classifiers via a new computational geometry method for anytime output bounds (DeCoste,2002). This new paper refines our approach in two key ways. First, we introduce a simple linear algebra formulation based on Cholesky factorization, yielding simpler equations and lower computational overhead. Second, this new formulation suggests new methods for achieving additional speedups, including tuning on query samples. We demonstrate effectiveness on benchmark datasets.

Quality-of-water data and statistical summary for selected coal-mined strip pits in Crawford and Cherokee counties, southeastern Kansas

USGS Publications Warehouse

Pope, Larry M.; Diaz, A.M.

1982-01-01

Quality-of-water data, collected October 21-23, 1980, and a statistical summary are presented for 42 coal-mined strip pits in Crawford and Cherokee Counties, Southeastern Kansas. The statistical summary includes minimum and maximum observed values , mean, and standard deviation. Simple linear regression equations relating specific conductance, dissolved solids, and acidity to concentrations of dissolved solids, sulfate, calcium, and magnesium, potassium, aluminum, and iron are also presented. (USGS)

Use of High Fidelity Methods in Multidisciplinary Optimization-A Preliminary Survey

NASA Technical Reports Server (NTRS)

Guruswamy, Guru P.; Kwak, Dochan (Technical Monitor)

2002-01-01

Multidisciplinary optimization is a key element of design process. To date multidiscipline optimization methods that use low fidelity methods are well advanced. Optimization methods based on simple linear aerodynamic equations and plate structural equations have been applied to complex aerospace configurations. However, use of high fidelity methods such as the Euler/ Navier-Stokes for fluids and 3-D (three dimensional) finite elements for structures has begun recently. As an activity of Multidiscipline Design Optimization Technical Committee (MDO TC) of AIAA (American Institute of Aeronautics and Astronautics), an effort was initiated to assess the status of the use of high fidelity methods in multidisciplinary optimization. Contributions were solicited through the members MDO TC committee. This paper provides a summary of that survey.

A volume-of-fluid method for simulation of compressible axisymmetric multi-material flow

NASA Astrophysics Data System (ADS)

de Niem, D.; Kührt, E.; Motschmann, U.

2007-02-01

A two-dimensional Eulerian hydrodynamic method for the numerical simulation of inviscid compressible axisymmetric multi-material flow in external force fields for the situation of pure fluids separated by macroscopic interfaces is presented. The method combines an implicit Lagrangian step with an explicit Eulerian advection step. Individual materials obey separate energy equations, fulfill general equations of state, and may possess different temperatures. Material volume is tracked using a piecewise linear volume-of-fluid method. An overshoot-free logically simple and economic material advection algorithm for cylinder coordinates is derived, in an algebraic formulation. New aspects arising in the case of more than two materials such as the material ordering strategy during transport are presented. One- and two-dimensional numerical examples are given.

Modified cable equation incorporating transverse polarization of neuronal membranes for accurate coupling of electric fields.

PubMed

Wang, Boshuo; Aberra, Aman S; Grill, Warren M; Peterchev, Angel V

2018-04-01

We present a theory and computational methods to incorporate transverse polarization of neuronal membranes into the cable equation to account for the secondary electric field generated by the membrane in response to transverse electric fields. The effect of transverse polarization on nonlinear neuronal activation thresholds is quantified and discussed in the context of previous studies using linear membrane models. The response of neuronal membranes to applied electric fields is derived under two time scales and a unified solution of transverse polarization is given for spherical and cylindrical cell geometries. The solution is incorporated into the cable equation re-derived using an asymptotic model that separates the longitudinal and transverse dimensions. Two numerical methods are proposed to implement the modified cable equation. Several common neural stimulation scenarios are tested using two nonlinear membrane models to compare thresholds of the conventional and modified cable equations. The implementations of the modified cable equation incorporating transverse polarization are validated against previous results in the literature. The test cases show that transverse polarization has limited effect on activation thresholds. The transverse field only affects thresholds of unmyelinated axons for short pulses and in low-gradient field distributions, whereas myelinated axons are mostly unaffected. The modified cable equation captures the membrane's behavior on different time scales and models more accurately the coupling between electric fields and neurons. It addresses the limitations of the conventional cable equation and allows sound theoretical interpretations. The implementation provides simple methods that are compatible with current simulation approaches to study the effect of transverse polarization on nonlinear membranes. The minimal influence by transverse polarization on axonal activation thresholds for the nonlinear membrane models indicates that predictions of stronger effects in linear membrane models with a fixed activation threshold are inaccurate. Thus, the conventional cable equation works well for most neuroengineering applications, and the presented modeling approach is well suited to address the exceptions.

New exact solutions of the Tzitzéica-type equations in non-linear optics using the expa function method

NASA Astrophysics Data System (ADS)

Hosseini, K.; Ayati, Z.; Ansari, R.

2018-04-01

One specific class of non-linear evolution equations, known as the Tzitzéica-type equations, has received great attention from a group of researchers involved in non-linear science. In this article, new exact solutions of the Tzitzéica-type equations arising in non-linear optics, including the Tzitzéica, Dodd-Bullough-Mikhailov and Tzitzéica-Dodd-Bullough equations, are obtained using the expa function method. The integration technique actually suggests a useful and reliable method to extract new exact solutions of a wide range of non-linear evolution equations.

Local Linear Observed-Score Equating

ERIC Educational Resources Information Center

Wiberg, Marie; van der Linden, Wim J.

2011-01-01

Two methods of local linear observed-score equating for use with anchor-test and single-group designs are introduced. In an empirical study, the two methods were compared with the current traditional linear methods for observed-score equating. As a criterion, the bias in the equated scores relative to true equating based on Lord's (1980)…

Aspects of porosity prediction using multivariate linear regression

DOE Office of Scientific and Technical Information (OSTI.GOV)

Byrnes, A.P.; Wilson, M.D.

1991-03-01

Highly accurate multiple linear regression models have been developed for sandstones of diverse compositions. Porosity reduction or enhancement processes are controlled by the fundamental variables, Pressure (P), Temperature (T), Time (t), and Composition (X), where composition includes mineralogy, size, sorting, fluid composition, etc. The multiple linear regression equation, of which all linear porosity prediction models are subsets, takes the generalized form: Porosity = C{sub 0} + C{sub 1}(P) + C{sub 2}(T) + C{sub 3}(X) + C{sub 4}(t) + C{sub 5}(PT) + C{sub 6}(PX) + C{sub 7}(Pt) + C{sub 8}(TX) + C{sub 9}(Tt) + C{sub 10}(Xt) + C{sub 11}(PTX) + C{submore » 12}(PXt) + C{sub 13}(PTt) + C{sub 14}(TXt) + C{sub 15}(PTXt). The first four primary variables are often interactive, thus requiring terms involving two or more primary variables (the form shown implies interaction and not necessarily multiplication). The final terms used may also involve simple mathematic transforms such as log X, e{sup T}, X{sup 2}, or more complex transformations such as the Time-Temperature Index (TTI). The X term in the equation above represents a suite of compositional variable and, therefore, a fully expanded equation may include a series of terms incorporating these variables. Numerous published bivariate porosity prediction models involving P (or depth) or Tt (TTI) are effective to a degree, largely because of the high degree of colinearity between p and TTI. However, all such bivariate models ignore the unique contributions of P and Tt, as well as various X terms. These simpler models become poor predictors in regions where colinear relations change, were important variables have been ignored, or where the database does not include a sufficient range or weight distribution for the critical variables.« less

A hybrid approach for nonlinear computational aeroacoustics predictions

NASA Astrophysics Data System (ADS)

Sassanis, Vasileios; Sescu, Adrian; Collins, Eric M.; Harris, Robert E.; Luke, Edward A.

2017-01-01

In many aeroacoustics applications involving nonlinear waves and obstructions in the far-field, approaches based on the classical acoustic analogy theory or the linearised Euler equations are unable to fully characterise the acoustic field. Therefore, computational aeroacoustics hybrid methods that incorporate nonlinear wave propagation have to be constructed. In this study, a hybrid approach coupling Navier-Stokes equations in the acoustic source region with nonlinear Euler equations in the acoustic propagation region is introduced and tested. The full Navier-Stokes equations are solved in the source region to identify the acoustic sources. The flow variables of interest are then transferred from the source region to the acoustic propagation region, where the full nonlinear Euler equations with source terms are solved. The transition between the two regions is made through a buffer zone where the flow variables are penalised via a source term added to the Euler equations. Tests were conducted on simple acoustic and vorticity disturbances, two-dimensional jets (Mach 0.9 and 2), and a three-dimensional jet (Mach 1.5), impinging on a wall. The method is proven to be effective and accurate in predicting sound pressure levels associated with the propagation of linear and nonlinear waves in the near- and far-field regions.

Frequency Preference Response to Oscillatory Inputs in Two-dimensional Neural Models: A Geometric Approach to Subthreshold Amplitude and Phase Resonance.

PubMed

Rotstein, Horacio G

2014-01-01

We investigate the dynamic mechanisms of generation of subthreshold and phase resonance in two-dimensional linear and linearized biophysical (conductance-based) models, and we extend our analysis to account for the effect of simple, but not necessarily weak, types of nonlinearities. Subthreshold resonance refers to the ability of neurons to exhibit a peak in their voltage amplitude response to oscillatory input currents at a preferred non-zero (resonant) frequency. Phase-resonance refers to the ability of neurons to exhibit a zero-phase (or zero-phase-shift) response to oscillatory input currents at a non-zero (phase-resonant) frequency. We adapt the classical phase-plane analysis approach to account for the dynamic effects of oscillatory inputs and develop a tool, the envelope-plane diagrams, that captures the role that conductances and time scales play in amplifying the voltage response at the resonant frequency band as compared to smaller and larger frequencies. We use envelope-plane diagrams in our analysis. We explain why the resonance phenomena do not necessarily arise from the presence of imaginary eigenvalues at rest, but rather they emerge from the interplay of the intrinsic and input time scales. We further explain why an increase in the time-scale separation causes an amplification of the voltage response in addition to shifting the resonant and phase-resonant frequencies. This is of fundamental importance for neural models since neurons typically exhibit a strong separation of time scales. We extend this approach to explain the effects of nonlinearities on both resonance and phase-resonance. We demonstrate that nonlinearities in the voltage equation cause amplifications of the voltage response and shifts in the resonant and phase-resonant frequencies that are not predicted by the corresponding linearized model. The differences between the nonlinear response and the linear prediction increase with increasing levels of the time scale separation between the voltage and the gating variable, and they almost disappear when both equations evolve at comparable rates. In contrast, voltage responses are almost insensitive to nonlinearities located in the gating variable equation. The method we develop provides a framework for the investigation of the preferred frequency responses in three-dimensional and nonlinear neuronal models as well as simple models of coupled neurons.

Students’ difficulties in solving linear equation problems

NASA Astrophysics Data System (ADS)

Wati, S.; Fitriana, L.; Mardiyana

2018-03-01

A linear equation is an algebra material that exists in junior high school to university. It is a very important material for students in order to learn more advanced mathematics topics. Therefore, linear equation material is essential to be mastered. However, the result of 2016 national examination in Indonesia showed that students’ achievement in solving linear equation problem was low. This fact became a background to investigate students’ difficulties in solving linear equation problems. This study used qualitative descriptive method. An individual written test on linear equation tasks was administered, followed by interviews. Twenty-one sample students of grade VIII of SMPIT Insan Kamil Karanganyar did the written test, and 6 of them were interviewed afterward. The result showed that students with high mathematics achievement donot have difficulties, students with medium mathematics achievement have factual difficulties, and students with low mathematics achievement have factual, conceptual, operational, and principle difficulties. Based on the result there is a need of meaningfulness teaching strategy to help students to overcome difficulties in solving linear equation problems.

Particle-size segregation and diffusive remixing in shallow granular avalanches

NASA Astrophysics Data System (ADS)

Gray, J. M. N. T.; Chugunov, V. A.

2006-12-01

Segregation and mixing of dissimilar grains is a problem in many industrial and pharmaceutical processes, as well as in hazardous geophysical flows, where the size-distribution can have a major impact on the local rheology and the overall run-out. In this paper, a simple binary mixture theory is used to formulate a model for particle-size segregation and diffusive remixing of large and small particles in shallow gravity-driven free-surface flows. This builds on a recent theory for the process of kinetic sieving, which is the dominant mechanism for segregation in granular avalanches provided the density-ratio and the size-ratio of the particles are not too large. The resulting nonlinear parabolic segregation remixing equation reduces to a quasi-linear hyperbolic equation in the no-remixing limit. It assumes that the bulk velocity is incompressible and that the bulk pressure is lithostatic, making it compatible with most theories used to compute the motion of shallow granular free-surface flows. In steady-state, the segregation remixing equation reduces to a logistic type equation and the ‘S’-shaped solutions are in very good agreement with existing particle dynamics simulations for both size and density segregation. Laterally uniform time-dependent solutions are constructed by mapping the segregation remixing equation to Burgers equation and using the Cole Hopf transformation to linearize the problem. It is then shown how solutions for arbitrary initial conditions can be constructed using standard methods. Three examples are investigated in which the initial concentration is (i) homogeneous, (ii) reverse graded with the coarse grains above the fines, and, (iii) normally graded with the fines above the coarse grains. Time-dependent two-dimensional solutions are also constructed for plug-flow in a semi-infinite chute.

Slip and barodiffusion phenomena in slow flows of a gas mixture

NASA Astrophysics Data System (ADS)

Zhdanov, V. M.

2017-03-01

The slip and barodiffusion problems for the slow flows of a gas mixture are investigated on the basis of the linearized moment equations following from the Boltzmann equation. We restrict ourselves to the set of the third-order moment equations and state two general relations (resembling conservation equations) for the moments of the distribution function similar to the conditions used by Loyalka [S. K. Loyalka, Phys. Fluids 14, 2291 (1971), 10.1063/1.1693331] in his approximation method (the modified Maxwell method). The expressions for the macroscopic velocities of the gas mixture species, the partial viscous stress tensors, and the reduced heat fluxes for the stationary slow flow of a gas mixture in the semi-infinite space over a plane wall are obtained as a result of the exact solution of the linearized moment equations in the 10- and 13-moment approximations. The general expression for the slip velocity and the simple and accurate expressions for the viscous, thermal, diffusion slip, and baroslip coefficients, which are given in terms of the basic transport coefficients, are derived by using the modified Maxwell method. The solutions of moment equations are also used for investigation of the flow and diffusion of a gas mixture in a channel formed by two infinite parallel plates. A fundamental result is that the barodiffusion factor in the cross-section-averaged expression for the diffusion flux contains contributions associated with the viscous transfer of momentum in the gas mixture and the effect of the Knudsen layer. Our study revealed that the barodiffusion factor is equal to the diffusion slip coefficient (correct to the opposite sign). This result is consistent with the Onsager's reciprocity relations for kinetic coefficients following from nonequilibrium thermodynamics of the discontinuous systems.

Determination of stress intensity factors for interface cracks under mixed-mode loading

NASA Technical Reports Server (NTRS)

Naik, Rajiv A.; Crews, John H., Jr.

1992-01-01

A simple technique was developed using conventional finite element analysis to determine stress intensity factors, K1 and K2, for interface cracks under mixed-mode loading. This technique involves the calculation of crack tip stresses using non-singular finite elements. These stresses are then combined and used in a linear regression procedure to calculate K1 and K2. The technique was demonstrated by calculating three different bimaterial combinations. For the normal loading case, the K's were within 2.6 percent of an exact solution. The normalized K's under shear loading were shown to be related to the normalized K's under normal loading. Based on these relations, a simple equation was derived for calculating K1 and K2 for mixed-mode loading from knowledge of the K's under normal loading. The equation was verified by computing the K's for a mixed-mode case with equal and normal shear loading. The correlation between exact and finite element solutions is within 3.7 percent. This study provides a simple procedure to compute K2/K1 ratio which has been used to characterize the stress state at the crack tip for various combinations of materials and loadings. Tests conducted over a range of K2/K1 ratios could be used to fully characterize interface fracture toughness.

Correlations for determining thermodynamic properties of hydrogen-helium gas mixtures at temperatures from 7,000 to 35,000 K

NASA Technical Reports Server (NTRS)

Zoby, E. V.; Gnoffo, P. A.; Graves, R. A., Jr.

1976-01-01

Simple relations for determining the enthalpy and temperature of hydrogen-helium gas mixtures were developed for hydrogen volumetric compositions from 1.0 to 0.7. These relations are expressed as a function of pressure and density and are valid for a range of temperatures from 7,000 to 35,000 K and pressures from 0.10 to 3.14 MPa. The proportionality constant and exponents in the correlation equations were determined for each gas composition by applying a linear least squares curve fit to a large number of thermodynamic calculations obtained from a detailed computer code. Although these simple relations yielded thermodynamic properties suitable for many engineering applications, their accuracy was improved significantly by evaluating the proportionality constants at postshock conditions and correlating these values as a function of the gas composition and the product of freestream velocity and shock angle. The resulting equations for the proportionality constants in terms of velocity and gas composition and the corresponding simple realtions for enthalpy and temperature were incorporated into a flow field computational scheme. Comparison was good between the thermodynamic properties determined from these relations and those obtained by using a detailed computer code to determine the properties. Thus, an appreciable savings in computer time was realized with no significant loss in accuracy.

Multiresolution quantum chemistry in multiwavelet bases: excited states from time-dependent Hartree–Fock and density functional theory via linear response

DOE PAGES

Yanai, Takeshi; Fann, George I.; Beylkin, Gregory; ...

2015-02-25

Using the fully numerical method for time-dependent Hartree–Fock and density functional theory (TD-HF/DFT) with the Tamm–Dancoff (TD) approximation we use a multiresolution analysis (MRA) approach to present our findings. From a reformulation with effective use of the density matrix operator, we obtain a general form of the HF/DFT linear response equation in the first quantization formalism. It can be readily rewritten as an integral equation with the bound-state Helmholtz (BSH) kernel for the Green's function. The MRA implementation of the resultant equation permits excited state calculations without virtual orbitals. Moreover, the integral equation is efficiently and adaptively solved using amore » numerical multiresolution solver with multiwavelet bases. Our implementation of the TD-HF/DFT methods is applied for calculating the excitation energies of H 2, Be, N 2, H 2O, and C 2H 4 molecules. The numerical errors of the calculated excitation energies converge in proportion to the residuals of the equation in the molecular orbitals and response functions. The energies of the excited states at a variety of length scales ranging from short-range valence excitations to long-range Rydberg-type ones are consistently accurate. It is shown that the multiresolution calculations yield the correct exponential asymptotic tails for the response functions, whereas those computed with Gaussian basis functions are too diffuse or decay too rapidly. Finally, we introduce a simple asymptotic correction to the local spin-density approximation (LSDA) so that in the TDDFT calculations, the excited states are correctly bound.« less

On solutions of the fifth-order dispersive equations with porous medium type non-linearity

NASA Astrophysics Data System (ADS)

Kocak, Huseyin; Pinar, Zehra

2018-07-01

In this work, we focus on obtaining the exact solutions of the fifth-order semi-linear and non-linear dispersive partial differential equations, which have the second-order diffusion-like (porous-type) non-linearity. The proposed equations were not studied in the literature in the sense of the exact solutions. We reveal solutions of the proposed equations using the classical Riccati equations method. The obtained exact solutions, which can play a key role to simulate non-linear waves in the medium with dispersion and diffusion, are illustrated and discussed in details.

A simple derivation of the exact quasiparticle theory and its extension to arbitrary initial excited eigenstates.

PubMed

Ohno, Kaoru; Ono, Shota; Isobe, Tomoharu

2017-02-28

The quasiparticle (QP) energies, which are minus of the energies required by removing or produced by adding one electron from/to the system, corresponding to the photoemission or inverse photoemission (PE/IPE) spectra, are determined together with the QP wave functions, which are not orthonormal and even not linearly independent but somewhat similar to the normal spin orbitals in the theory of the configuration interaction, by self-consistently solving the QP equation coupled with the equation for the self-energy. The electron density, kinetic, and all interaction energies can be calculated using the QP wave functions. We prove in a simple way that the PE/IPE spectroscopy and therefore this QP theory can be applied to an arbitrary initial excited eigenstate. In this proof, we show that the energy-dependence of the self-energy is not an essential difficulty, and the QP picture holds exactly if there is no relaxation mechanism in the system. The validity of the present theory for some initial excited eigenstates is tested using the one-shot GW approximation for several atoms and molecules.

Dynamics and thermodynamics of linear quantum open systems.

PubMed

Martinez, Esteban A; Paz, Juan Pablo

2013-03-29

We analyze the evolution of the quantum state of networks of quantum oscillators coupled with arbitrary external environments. We show that the reduced density matrix of the network always obeys a local master equation with a simple analytical solution. We use this to study the emergence of thermodynamical laws in the long time regime demonstrating two main results: First, we show that it is impossible to build a quantum absorption refrigerator using linear networks (thus, nonlinearity is an essential resource for such refrigerators recently studied by Levy and Kosloff [Phys. Rev. Lett. 108, 070604 (2012)] and Levy et al. [Phys. Rev. B 85, 061126 (2012)]). Then, we show that the third law imposes constraints on the low frequency behavior of the environmental spectral densities.

Closed-form solutions for linear regulator-design of mechanical systems including optimal weighting matrix selection

NASA Technical Reports Server (NTRS)

Hanks, Brantley R.; Skelton, Robert E.

1991-01-01

This paper addresses the restriction of Linear Quadratic Regulator (LQR) solutions to the algebraic Riccati Equation to design spaces which can be implemented as passive structural members and/or dampers. A general closed-form solution to the optimal free-decay control problem is presented which is tailored for structural-mechanical systems. The solution includes, as subsets, special cases such as the Rayleigh Dissipation Function and total energy. Weighting matrix selection is a constrained choice among several parameters to obtain desired physical relationships. The closed-form solution is also applicable to active control design for systems where perfect, collocated actuator-sensor pairs exist. Some examples of simple spring mass systems are shown to illustrate key points.

Understanding dynamics of Martian winter polar vortex with “improved” moist-convective shallow water model

NASA Astrophysics Data System (ADS)

Rostami, M.; Zeitlin, V.

2017-12-01

We show how the properties of the Mars polar vortex can be understood in the framework of a simple shallow-water type model obtained by vertical averaging of the adiabatic “primitive” equations, and “improved” by inclusion of thermal relaxation and convective fluxes due to the phase transitions of CO 2, the major constituent of the Martian atmosphere. We perform stability analysis of the vortex, show that corresponding mean zonal flow is unstable, and simulate numerically non-linear saturation of the instability. We show in this way that, while non-linear adiabatic saturation of the instability tends to reorganize the vortex, the diabatic effects prevent this, and thus provide an explanation of the vortex form and longevity.

Machining Chatter Analysis for High Speed Milling Operations

NASA Astrophysics Data System (ADS)

Sekar, M.; Kantharaj, I.; Amit Siddhappa, Savale

2017-10-01

Chatter in high speed milling is characterized by time delay differential equations (DDE). Since closed form solution exists only for simple cases, the governing non-linear DDEs of chatter problems are solved by various numerical methods. Custom codes to solve DDEs are tedious to build, implement and not error free and robust. On the other hand, software packages provide solution to DDEs, however they are not straight forward to implement. In this paper an easy way to solve DDE of chatter in milling is proposed and implemented with MATLAB. Time domain solution permits the study and model of non-linear effects of chatter vibration with ease. Time domain results are presented for various stable and unstable conditions of cut and compared with stability lobe diagrams.

Adaptive nonlinear control for autonomous ground vehicles

NASA Astrophysics Data System (ADS)

Black, William S.

We present the background and motivation for ground vehicle autonomy, and focus on uses for space-exploration. Using a simple design example of an autonomous ground vehicle we derive the equations of motion. After providing the mathematical background for nonlinear systems and control we present two common methods for exactly linearizing nonlinear systems, feedback linearization and backstepping. We use these in combination with three adaptive control methods: model reference adaptive control, adaptive sliding mode control, and extremum-seeking model reference adaptive control. We show the performances of each combination through several simulation results. We then consider disturbances in the system, and design nonlinear disturbance observers for both single-input-single-output and multi-input-multi-output systems. Finally, we show the performance of these observers with simulation results.

On differential operators generating iterative systems of linear ODEs of maximal symmetry algebra

NASA Astrophysics Data System (ADS)

Ndogmo, J. C.

2017-06-01

Although every iterative scalar linear ordinary differential equation is of maximal symmetry algebra, the situation is different and far more complex for systems of linear ordinary differential equations, and an iterative system of linear equations need not be of maximal symmetry algebra. We illustrate these facts by examples and derive families of vector differential operators whose iterations are all linear systems of equations of maximal symmetry algebra. Some consequences of these results are also discussed.

Evaluation of linearly solvable Markov decision process with dynamic model learning in a mobile robot navigation task.

PubMed

Kinjo, Ken; Uchibe, Eiji; Doya, Kenji

2013-01-01

Linearly solvable Markov Decision Process (LMDP) is a class of optimal control problem in which the Bellman's equation can be converted into a linear equation by an exponential transformation of the state value function (Todorov, 2009b). In an LMDP, the optimal value function and the corresponding control policy are obtained by solving an eigenvalue problem in a discrete state space or an eigenfunction problem in a continuous state using the knowledge of the system dynamics and the action, state, and terminal cost functions. In this study, we evaluate the effectiveness of the LMDP framework in real robot control, in which the dynamics of the body and the environment have to be learned from experience. We first perform a simulation study of a pole swing-up task to evaluate the effect of the accuracy of the learned dynamics model on the derived the action policy. The result shows that a crude linear approximation of the non-linear dynamics can still allow solution of the task, despite with a higher total cost. We then perform real robot experiments of a battery-catching task using our Spring Dog mobile robot platform. The state is given by the position and the size of a battery in its camera view and two neck joint angles. The action is the velocities of two wheels, while the neck joints were controlled by a visual servo controller. We test linear and bilinear dynamic models in tasks with quadratic and Guassian state cost functions. In the quadratic cost task, the LMDP controller derived from a learned linear dynamics model performed equivalently with the optimal linear quadratic regulator (LQR). In the non-quadratic task, the LMDP controller with a linear dynamics model showed the best performance. The results demonstrate the usefulness of the LMDP framework in real robot control even when simple linear models are used for dynamics learning.

Non-Linear Analysis of Mode II Fracture in the end Notched Flexure Beam

NASA Astrophysics Data System (ADS)

Rizov, V.

2016-03-01

Analysis is carried-out of fracture in the End Notched Flex- ure (ENF) beam configuration, taking into account the material nonlin- earity. For this purpose, the J-integral approach is applied. A non-linear model, based on the Classical beam theory is used. The mechanical be- haviour of the ENF configuration is described by the Ramberg-Osgood stress-strain curve. It is assumed that the material possesses the same properties in tension and compression. The influence is evaluated of the material constants in the Ramberg-Osgood stress-strain equation on the fracture behaviour. The effect of the crack length on the J-integral value is investigated, too. The analytical approach, developed in the present paper, is very useful for parametric analyses, since the simple formulae obtained capture the essentials of the non-linear fracture in the ENF con- figuration.

Angular-Rate Estimation Using Delayed Quaternion Measurements

NASA Technical Reports Server (NTRS)

Azor, R.; Bar-Itzhack, I. Y.; Harman, R. R.

1999-01-01

This paper presents algorithms for estimating the angular-rate vector of satellites using quaternion measurements. Two approaches are compared one that uses differentiated quaternion measurements to yield coarse rate measurements, which are then fed into two different estimators. In the other approach the raw quaternion measurements themselves are fed directly into the two estimators. The two estimators rely on the ability to decompose the non-linear part of the rotas rotational dynamics equation of a body into a product of an angular-rate dependent matrix and the angular-rate vector itself. This non unique decomposition, enables the treatment of the nonlinear spacecraft (SC) dynamics model as a linear one and, thus, the application of a PseudoLinear Kalman Filter (PSELIKA). It also enables the application of a special Kalman filter which is based on the use of the solution of the State Dependent Algebraic Riccati Equation (SDARE) in order to compute the gain matrix and thus eliminates the need to compute recursively the filter covariance matrix. The replacement of the rotational dynamics by a simple Markov model is also examined. In this paper special consideration is given to the problem of delayed quaternion measurements. Two solutions to this problem are suggested and tested. Real Rossi X-Ray Timing Explorer (RXTE) data is used to test these algorithms, and results are presented.

Collisionless kinetic theory of oblique tearing instabilities

DOE PAGES

Baalrud, S. D.; Bhattacharjee, A.; Daughton, W.

2018-02-15

The linear dispersion relation for collisionless kinetic tearing instabilities is calculated for the Harris equilibrium. In contrast to the conventional 2D geometry, which considers only modes at the center of the current sheet, modes can span the current sheet in 3D. Modes at each resonant surface have a unique angle with respect to the guide field direction. Both kinetic simulations and numerical eigenmode solutions of the linearized Vlasov-Maxwell equations have recently revealed that standard analytic theories vastly overestimate the growth rate of oblique modes. In this paper, we find that this stabilization is associated with the density-gradient-driven diamagnetic drift. Themore » analytic theories miss this drift stabilization because the inner tearing layer broadens at oblique angles sufficiently far that the assumption of scale separation between the inner and outer regions of boundary-layer theory breaks down. The dispersion relation obtained by numerically solving a single second order differential equation is found to approximately capture the drift stabilization predicted by solutions of the full integro-differential eigenvalue problem. Finally, a simple analytic estimate for the stability criterion is provided.« less

Collisionless kinetic theory of oblique tearing instabilities

DOE Office of Scientific and Technical Information (OSTI.GOV)

Baalrud, S. D.; Bhattacharjee, A.; Daughton, W.

The linear dispersion relation for collisionless kinetic tearing instabilities is calculated for the Harris equilibrium. In contrast to the conventional 2D geometry, which considers only modes at the center of the current sheet, modes can span the current sheet in 3D. Modes at each resonant surface have a unique angle with respect to the guide field direction. Both kinetic simulations and numerical eigenmode solutions of the linearized Vlasov-Maxwell equations have recently revealed that standard analytic theories vastly overestimate the growth rate of oblique modes. In this paper, we find that this stabilization is associated with the density-gradient-driven diamagnetic drift. Themore » analytic theories miss this drift stabilization because the inner tearing layer broadens at oblique angles sufficiently far that the assumption of scale separation between the inner and outer regions of boundary-layer theory breaks down. The dispersion relation obtained by numerically solving a single second order differential equation is found to approximately capture the drift stabilization predicted by solutions of the full integro-differential eigenvalue problem. Finally, a simple analytic estimate for the stability criterion is provided.« less

Estimation of Sonic Fatigue by Reduced-Order Finite Element Based Analyses

NASA Technical Reports Server (NTRS)

Rizzi, Stephen A.; Przekop, Adam

2006-01-01

A computationally efficient, reduced-order method is presented for prediction of sonic fatigue of structures exhibiting geometrically nonlinear response. A procedure to determine the nonlinear modal stiffness using commercial finite element codes allows the coupled nonlinear equations of motion in physical degrees of freedom to be transformed to a smaller coupled system of equations in modal coordinates. The nonlinear modal system is first solved using a computationally light equivalent linearization solution to determine if the structure responds to the applied loading in a nonlinear fashion. If so, a higher fidelity numerical simulation in modal coordinates is undertaken to more accurately determine the nonlinear response. Comparisons of displacement and stress response obtained from the reduced-order analyses are made with results obtained from numerical simulation in physical degrees-of-freedom. Fatigue life predictions from nonlinear modal and physical simulations are made using the rainflow cycle counting method in a linear cumulative damage analysis. Results computed for a simple beam structure under a random acoustic loading demonstrate the effectiveness of the approach and compare favorably with results obtained from the solution in physical degrees-of-freedom.

Collisionless kinetic theory of oblique tearing instabilities

NASA Astrophysics Data System (ADS)

Baalrud, S. D.; Bhattacharjee, A.; Daughton, W.

2018-02-01

The linear dispersion relation for collisionless kinetic tearing instabilities is calculated for the Harris equilibrium. In contrast to the conventional 2D geometry, which considers only modes at the center of the current sheet, modes can span the current sheet in 3D. Modes at each resonant surface have a unique angle with respect to the guide field direction. Both kinetic simulations and numerical eigenmode solutions of the linearized Vlasov-Maxwell equations have recently revealed that standard analytic theories vastly overestimate the growth rate of oblique modes. We find that this stabilization is associated with the density-gradient-driven diamagnetic drift. The analytic theories miss this drift stabilization because the inner tearing layer broadens at oblique angles sufficiently far that the assumption of scale separation between the inner and outer regions of boundary-layer theory breaks down. The dispersion relation obtained by numerically solving a single second order differential equation is found to approximately capture the drift stabilization predicted by solutions of the full integro-differential eigenvalue problem. A simple analytic estimate for the stability criterion is provided.

Numerical solution of a non-linear conservation law applicable to the interior dynamics of partially molten planets

NASA Astrophysics Data System (ADS)

Bower, Dan J.; Sanan, Patrick; Wolf, Aaron S.

2018-01-01

The energy balance of a partially molten rocky planet can be expressed as a non-linear diffusion equation using mixing length theory to quantify heat transport by both convection and mixing of the melt and solid phases. Crucially, in this formulation the effective or eddy diffusivity depends on the entropy gradient, ∂S / ∂r , as well as entropy itself. First we present a simplified model with semi-analytical solutions that highlights the large dynamic range of ∂S / ∂r -around 12 orders of magnitude-for physically-relevant parameters. It also elucidates the thermal structure of a magma ocean during the earliest stage of crystal formation. This motivates the development of a simple yet stable numerical scheme able to capture the large dynamic range of ∂S / ∂r and hence provide a flexible and robust method for time-integrating the energy equation. Using insight gained from the simplified model, we consider a full model, which includes energy fluxes associated with convection, mixing, gravitational separation, and conduction that all depend on the thermophysical properties of the melt and solid phases. This model is discretised and evolved by applying the finite volume method (FVM), allowing for extended precision calculations and using ∂S / ∂r as the solution variable. The FVM is well-suited to this problem since it is naturally energy conserving, flexible, and intuitive to incorporate arbitrary non-linear fluxes that rely on lookup data. Special attention is given to the numerically challenging scenario in which crystals first form in the centre of a magma ocean. The computational framework we devise is immediately applicable to modelling high melt fraction phenomena in Earth and planetary science research. Furthermore, it provides a template for solving similar non-linear diffusion equations that arise in other science and engineering disciplines, particularly for non-linear functional forms of the diffusion coefficient.

Transformation matrices between non-linear and linear differential equations

NASA Technical Reports Server (NTRS)

Sartain, R. L.

1983-01-01

In the linearization of systems of non-linear differential equations, those systems which can be exactly transformed into the second order linear differential equation Y"-AY'-BY=0 where Y, Y', and Y" are n x 1 vectors and A and B are constant n x n matrices of real numbers were considered. The 2n x 2n matrix was used to transform the above matrix equation into the first order matrix equation X' = MX. Specially the matrix M and the conditions which will diagonalize or triangularize M were studied. Transformation matrices P and P sub -1 were used to accomplish this diagonalization or triangularization to return to the solution of the second order matrix differential equation system from the first order system.

Optimisation of an idealised primitive equation ocean model using stochastic parameterization

NASA Astrophysics Data System (ADS)

Cooper, Fenwick C.

2017-05-01

Using a simple parameterization, an idealised low resolution (biharmonic viscosity coefficient of 5 × 1012 m4s-1 , 128 × 128 grid) primitive equation baroclinic ocean gyre model is optimised to have a much more accurate climatological mean, variance and response to forcing, in all model variables, with respect to a high resolution (biharmonic viscosity coefficient of 8 × 1010 m4s-1 , 512 × 512 grid) equivalent. For example, the change in the climatological mean due to a small change in the boundary conditions is more accurate in the model with parameterization. Both the low resolution and high resolution models are strongly chaotic. We also find that long timescales in the model temperature auto-correlation at depth are controlled by the vertical temperature diffusion parameter and time mean vertical advection and are caused by short timescale random forcing near the surface. This paper extends earlier work that considered a shallow water barotropic gyre. Here the analysis is extended to a more turbulent multi-layer primitive equation model that includes temperature as a prognostic variable. The parameterization consists of a constant forcing, applied to the velocity and temperature equations at each grid point, which is optimised to obtain a model with an accurate climatological mean, and a linear stochastic forcing, that is optimised to also obtain an accurate climatological variance and 5 day lag auto-covariance. A linear relaxation (nudging) is not used. Conservation of energy and momentum is discussed in an appendix.

A coarse-grid projection method for accelerating incompressible flow computations

NASA Astrophysics Data System (ADS)

San, Omer; Staples, Anne E.

2013-01-01

We present a coarse-grid projection (CGP) method for accelerating incompressible flow computations, which is applicable to methods involving Poisson equations as incompressibility constraints. The CGP methodology is a modular approach that facilitates data transfer with simple interpolations and uses black-box solvers for the Poisson and advection-diffusion equations in the flow solver. After solving the Poisson equation on a coarsened grid, an interpolation scheme is used to obtain the fine data for subsequent time stepping on the full grid. A particular version of the method is applied here to the vorticity-stream function, primitive variable, and vorticity-velocity formulations of incompressible Navier-Stokes equations. We compute several benchmark flow problems on two-dimensional Cartesian and non-Cartesian grids, as well as a three-dimensional flow problem. The method is found to accelerate these computations while retaining a level of accuracy close to that of the fine resolution field, which is significantly better than the accuracy obtained for a similar computation performed solely using a coarse grid. A linear acceleration rate is obtained for all the cases we consider due to the linear-cost elliptic Poisson solver used, with reduction factors in computational time between 2 and 42. The computational savings are larger when a suboptimal Poisson solver is used. We also find that the computational savings increase with increasing distortion ratio on non-Cartesian grids, making the CGP method a useful tool for accelerating generalized curvilinear incompressible flow solvers.

A simple method for identifying parameter correlations in partially observed linear dynamic models.

PubMed

Li, Pu; Vu, Quoc Dong

2015-12-14

Parameter estimation represents one of the most significant challenges in systems biology. This is because biological models commonly contain a large number of parameters among which there may be functional interrelationships, thus leading to the problem of non-identifiability. Although identifiability analysis has been extensively studied by analytical as well as numerical approaches, systematic methods for remedying practically non-identifiable models have rarely been investigated. We propose a simple method for identifying pairwise correlations and higher order interrelationships of parameters in partially observed linear dynamic models. This is made by derivation of the output sensitivity matrix and analysis of the linear dependencies of its columns. Consequently, analytical relations between the identifiability of the model parameters and the initial conditions as well as the input functions can be achieved. In the case of structural non-identifiability, identifiable combinations can be obtained by solving the resulting homogenous linear equations. In the case of practical non-identifiability, experiment conditions (i.e. initial condition and constant control signals) can be provided which are necessary for remedying the non-identifiability and unique parameter estimation. It is noted that the approach does not consider noisy data. In this way, the practical non-identifiability issue, which is popular for linear biological models, can be remedied. Several linear compartment models including an insulin receptor dynamics model are taken to illustrate the application of the proposed approach. Both structural and practical identifiability of partially observed linear dynamic models can be clarified by the proposed method. The result of this method provides important information for experimental design to remedy the practical non-identifiability if applicable. The derivation of the method is straightforward and thus the algorithm can be easily implemented into a software packet.

Quasi-Newton methods for parameter estimation in functional differential equations

NASA Technical Reports Server (NTRS)

Brewer, Dennis W.

1988-01-01

A state-space approach to parameter estimation in linear functional differential equations is developed using the theory of linear evolution equations. A locally convergent quasi-Newton type algorithm is applied to distributed systems with particular emphasis on parameters that induce unbounded perturbations of the state. The algorithm is computationally implemented on several functional differential equations, including coefficient and delay estimation in linear delay-differential equations.

A Gaussian theory for fluctuations in simple liquids.

PubMed

Krüger, Matthias; Dean, David S

2017-04-07

Assuming an effective quadratic Hamiltonian, we derive an approximate, linear stochastic equation of motion for the density-fluctuations in liquids, composed of overdamped Brownian particles. From this approach, time dependent two point correlation functions (such as the intermediate scattering function) are derived. We show that this correlation function is exact at short times, for any interaction and, in particular, for arbitrary external potentials so that it applies to confined systems. Furthermore, we discuss the relation of this approach to previous ones, such as dynamical density functional theory as well as the formally exact treatment. This approach, inspired by the well known Landau-Ginzburg Hamiltonians, and the corresponding "Model B" equation of motion, may be seen as its microscopic version, containing information about the details on the particle level.

High-order local maximum principle preserving (MPP) discontinuous Galerkin finite element method for the transport equation

NASA Astrophysics Data System (ADS)

Anderson, R.; Dobrev, V.; Kolev, Tz.; Kuzmin, D.; Quezada de Luna, M.; Rieben, R.; Tomov, V.

2017-04-01

In this work we present a FCT-like Maximum-Principle Preserving (MPP) method to solve the transport equation. We use high-order polynomial spaces; in particular, we consider up to 5th order spaces in two and three dimensions and 23rd order spaces in one dimension. The method combines the concepts of positive basis functions for discontinuous Galerkin finite element spatial discretization, locally defined solution bounds, element-based flux correction, and non-linear local mass redistribution. We consider a simple 1D problem with non-smooth initial data to explain and understand the behavior of different parts of the method. Convergence tests in space indicate that high-order accuracy is achieved. Numerical results from several benchmarks in two and three dimensions are also reported.

A Gaussian theory for fluctuations in simple liquids

NASA Astrophysics Data System (ADS)

Krüger, Matthias; Dean, David S.

2017-04-01

Assuming an effective quadratic Hamiltonian, we derive an approximate, linear stochastic equation of motion for the density-fluctuations in liquids, composed of overdamped Brownian particles. From this approach, time dependent two point correlation functions (such as the intermediate scattering function) are derived. We show that this correlation function is exact at short times, for any interaction and, in particular, for arbitrary external potentials so that it applies to confined systems. Furthermore, we discuss the relation of this approach to previous ones, such as dynamical density functional theory as well as the formally exact treatment. This approach, inspired by the well known Landau-Ginzburg Hamiltonians, and the corresponding "Model B" equation of motion, may be seen as its microscopic version, containing information about the details on the particle level.

Highly Accurate Analytical Approximate Solution to a Nonlinear Pseudo-Oscillator

NASA Astrophysics Data System (ADS)

Wu, Baisheng; Liu, Weijia; Lim, C. W.

2017-07-01

A second-order Newton method is presented to construct analytical approximate solutions to a nonlinear pseudo-oscillator in which the restoring force is inversely proportional to the dependent variable. The nonlinear equation is first expressed in a specific form, and it is then solved in two steps, a predictor and a corrector step. In each step, the harmonic balance method is used in an appropriate manner to obtain a set of linear algebraic equations. With only one simple second-order Newton iteration step, a short, explicit, and highly accurate analytical approximate solution can be derived. The approximate solutions are valid for all amplitudes of the pseudo-oscillator. Furthermore, the method incorporates second-order Taylor expansion in a natural way, and it is of significant faster convergence rate.

Equating Scores from Adaptive to Linear Tests

ERIC Educational Resources Information Center

van der Linden, Wim J.

2006-01-01

Two local methods for observed-score equating are applied to the problem of equating an adaptive test to a linear test. In an empirical study, the methods were evaluated against a method based on the test characteristic function (TCF) of the linear test and traditional equipercentile equating applied to the ability estimates on the adaptive test…

Spacecraft Constrained Maneuver Planning Using Positively Invariant Constraint Admissible Sets (Postprint)

DTIC Science & Technology

2013-08-14

Connectivity Graph; Graph Search; Bounded Disturbances; Linear Time-Varying (LTV); Clohessy - Wiltshire -Hill (CWH) 16. SECURITY CLASSIFICATION OF: 17...the linearization of the relative motion model given by the Hill- Clohessy - Wiltshire (CWH) equations is used [14]. A. Nonlinear equations of motion...equations can be used to describe the motion of the debris. B. Linearized HCW equations in discrete-time For δr << R, the linearized Hill- Clohessy

Newton's method: A link between continuous and discrete solutions of nonlinear problems

NASA Technical Reports Server (NTRS)

Thurston, G. A.

1980-01-01

Newton's method for nonlinear mechanics problems replaces the governing nonlinear equations by an iterative sequence of linear equations. When the linear equations are linear differential equations, the equations are usually solved by numerical methods. The iterative sequence in Newton's method can exhibit poor convergence properties when the nonlinear problem has multiple solutions for a fixed set of parameters, unless the iterative sequences are aimed at solving for each solution separately. The theory of the linear differential operators is often a better guide for solution strategies in applying Newton's method than the theory of linear algebra associated with the numerical analogs of the differential operators. In fact, the theory for the differential operators can suggest the choice of numerical linear operators. In this paper the method of variation of parameters from the theory of linear ordinary differential equations is examined in detail in the context of Newton's method to demonstrate how it might be used as a guide for numerical solutions.

Practical implementation of the double linear damage rule and damage curve approach for treating cumulative fatigue damage

NASA Technical Reports Server (NTRS)

Manson, S. S.; Halford, G. R.

1980-01-01

Simple procedures are presented for treating cumulative fatigue damage under complex loading history using either the damage curve concept or the double linear damage rule. A single equation is provided for use with the damage curve approach; each loading event providing a fraction of damage until failure is presumed to occur when the damage sum becomes unity. For the double linear damage rule, analytical expressions are provided for determining the two phases of life. The procedure involves two steps, each similar to the conventional application of the commonly used linear damage rule. When the sum of cycle ratios based on phase 1 lives reaches unity, phase 1 is presumed complete, and further loadings are summed as cycle ratios on phase 2 lives. When the phase 2 sum reaches unity, failure is presumed to occur. No other physical properties or material constants than those normally used in a conventional linear damage rule analysis are required for application of either of the two cumulative damage methods described. Illustrations and comparisons of both methods are discussed.

Practical implementation of the double linear damage rule and damage curve approach for treating cumulative fatigue damage

NASA Technical Reports Server (NTRS)

Manson, S. S.; Halford, G. R.

1981-01-01

Simple procedures are given for treating cumulative fatigue damage under complex loading history using either the damage curve concept or the double linear damage rule. A single equation is given for use with the damage curve approach; each loading event providing a fraction of damage until failure is presumed to occur when the damage sum becomes unity. For the double linear damage rule, analytical expressions are given for determining the two phases of life. The procedure comprises two steps, each similar to the conventional application of the commonly used linear damage rule. Once the sum of cycle ratios based on Phase I lives reaches unity, Phase I is presumed complete, and further loadings are summed as cycle ratios based on Phase II lives. When the Phase II sum attains unity, failure is presumed to occur. It is noted that no physical properties or material constants other than those normally used in a conventional linear damage rule analysis are required for application of either of the two cumulative damage methods described. Illustrations and comparisons are discussed for both methods.

User's manual for interactive LINEAR: A FORTRAN program to derive linear aircraft models

NASA Technical Reports Server (NTRS)

Antoniewicz, Robert F.; Duke, Eugene L.; Patterson, Brian P.

1988-01-01

An interactive FORTRAN program that provides the user with a powerful and flexible tool for the linearization of aircraft aerodynamic models is documented in this report. The program LINEAR numerically determines a linear system model using nonlinear equations of motion and a user-supplied linear or nonlinear aerodynamic model. The nonlinear equations of motion used are six-degree-of-freedom equations with stationary atmosphere and flat, nonrotating earth assumptions. The system model determined by LINEAR consists of matrices for both the state and observation equations. The program has been designed to allow easy selection and definition of the state, control, and observation variables to be used in a particular model.

A high-order relaxation method with projective integration for solving nonlinear systems of hyperbolic conservation laws

NASA Astrophysics Data System (ADS)

Lafitte, Pauline; Melis, Ward; Samaey, Giovanni

2017-07-01

We present a general, high-order, fully explicit relaxation scheme which can be applied to any system of nonlinear hyperbolic conservation laws in multiple dimensions. The scheme consists of two steps. In a first (relaxation) step, the nonlinear hyperbolic conservation law is approximated by a kinetic equation with stiff BGK source term. Then, this kinetic equation is integrated in time using a projective integration method. After taking a few small (inner) steps with a simple, explicit method (such as direct forward Euler) to damp out the stiff components of the solution, the time derivative is estimated and used in an (outer) Runge-Kutta method of arbitrary order. We show that, with an appropriate choice of inner step size, the time step restriction on the outer time step is similar to the CFL condition for the hyperbolic conservation law. Moreover, the number of inner time steps is also independent of the stiffness of the BGK source term. We discuss stability and consistency, and illustrate with numerical results (linear advection, Burgers' equation and the shallow water and Euler equations) in one and two spatial dimensions.

Analytical and experimental study of the vibration of bonded beams with a lap joint

NASA Technical Reports Server (NTRS)

Rao, M. D.; Crocker, M. J.

1990-01-01

A theoretical model to study the flexural vibration of a bonded lap joint system is described in this paper. First, equations of motion at the joint region are derived using a differential element approach. The transverse displacements of the upper and lower beam are considered to be different. The adhesive is assumed to be linearly viscoelastic and the widely used Kelvin-Voight model is used to represent the viscoelastic behavior of the adhesive. The shear force at the interface between the adhesive and the beam is obtained from the simple bending motion equations of the two beams. The resulting equations of motion are combined with the equations of transverse vibration of the beams in the unjointed regions. These are later solved as a boundary value problem to obtain the eigenvalues and eigenvectors of the system. The model can be used to predict the natural frequencies, modal damping ratios, and mode shapes of the system for free vibration. Good agreement between numerical and experimental results was obtained for a system of graphite epoxy beams lap-jointed by an epoxy adhesive.

Supporting second grade lower secondary school students’ understanding of linear equation system in two variables using ethnomathematics

NASA Astrophysics Data System (ADS)

Nursyahidah, F.; Saputro, B. A.; Rubowo, M. R.

2018-03-01

The aim of this research is to know the students’ understanding of linear equation system in two variables using Ethnomathematics and to acquire learning trajectory of linear equation system in two variables for the second grade of lower secondary school students. This research used methodology of design research that consists of three phases, there are preliminary design, teaching experiment, and retrospective analysis. Subject of this study is 28 second grade students of Sekolah Menengah Pertama (SMP) 37 Semarang. The result of this research shows that the students’ understanding in linear equation system in two variables can be stimulated by using Ethnomathematics in selling buying tradition in Peterongan traditional market in Central Java as a context. All of strategies and model that was applied by students and also their result discussion shows how construction and contribution of students can help them to understand concept of linear equation system in two variables. All the activities that were done by students produce learning trajectory to gain the goal of learning. Each steps of learning trajectory of students have an important role in understanding the concept from informal to the formal level. Learning trajectory using Ethnomathematics that is produced consist of watching video of selling buying activity in Peterongan traditional market to construct linear equation in two variables, determine the solution of linear equation in two variables, construct model of linear equation system in two variables from contextual problem, and solving a contextual problem related to linear equation system in two variables.

Technical report. The application of probability-generating functions to linear-quadratic radiation survival curves.

PubMed

Kendal, W S

2000-04-01

To illustrate how probability-generating functions (PGFs) can be employed to derive a simple probabilistic model for clonogenic survival after exposure to ionizing irradiation. Both repairable and irreparable radiation damage to DNA were assumed to occur by independent (Poisson) processes, at intensities proportional to the irradiation dose. Also, repairable damage was assumed to be either repaired or further (lethally) injured according to a third (Bernoulli) process, with the probability of lethal conversion being directly proportional to dose. Using the algebra of PGFs, these three processes were combined to yield a composite PGF that described the distribution of lethal DNA lesions in irradiated cells. The composite PGF characterized a Poisson distribution with mean, chiD+betaD2, where D was dose and alpha and beta were radiobiological constants. This distribution yielded the conventional linear-quadratic survival equation. To test the composite model, the derived distribution was used to predict the frequencies of multiple chromosomal aberrations in irradiated human lymphocytes. The predictions agreed well with observation. This probabilistic model was consistent with single-hit mechanisms, but it was not consistent with binary misrepair mechanisms. A stochastic model for radiation survival has been constructed from elementary PGFs that exactly yields the linear-quadratic relationship. This approach can be used to investigate other simple probabilistic survival models.

Optimized nonorthogonal transforms for image compression.

PubMed

Guleryuz, O G; Orchard, M T

1997-01-01

The transform coding of images is analyzed from a common standpoint in order to generate a framework for the design of optimal transforms. It is argued that all transform coders are alike in the way they manipulate the data structure formed by transform coefficients. A general energy compaction measure is proposed to generate optimized transforms with desirable characteristics particularly suited to the simple transform coding operation of scalar quantization and entropy coding. It is shown that the optimal linear decoder (inverse transform) must be an optimal linear estimator, independent of the structure of the transform generating the coefficients. A formulation that sequentially optimizes the transforms is presented, and design equations and algorithms for its computation provided. The properties of the resulting transform systems are investigated. In particular, it is shown that the resulting basis are nonorthogonal and complete, producing energy compaction optimized, decorrelated transform coefficients. Quantization issues related to nonorthogonal expansion coefficients are addressed with a simple, efficient algorithm. Two implementations are discussed, and image coding examples are given. It is shown that the proposed design framework results in systems with superior energy compaction properties and excellent coding results.

JMOSFET: A MOSFET parameter extractor with geometry-dependent terms

NASA Technical Reports Server (NTRS)

Buehler, M. G.; Moore, B. T.

1985-01-01

The parameters from metal-oxide-silicon field-effect transistors (MOSFETs) that are included on the Combined Release and Radiation Effects Satellite (CRRES) test chips need to be extracted to have a simple but comprehensive method that can be used in wafer acceptance, and to have a method that is sufficiently accurate that it can be used in integrated circuits. A set of MOSFET parameter extraction procedures that are directly linked to the MOSFET model equations and that facilitate the use of simple, direct curve-fitting techniques are developed. In addition, the major physical effects that affect MOSFET operation in the linear and saturation regions of operation for devices fabricated in 1.2 to 3.0 mm CMOS technology are included. The fitting procedures were designed to establish single values for such parameters as threshold voltage and transconductance and to provide for slope matching between the linear and saturation regions of the MOSFET output current-voltage curves. Four different sizes of transistors that cover a rectangular-shaped region of the channel length-width plane are analyzed.

Localization of the eigenvalues of linear integral equations with applications to linear ordinary differential equations.

NASA Technical Reports Server (NTRS)

Sloss, J. M.; Kranzler, S. K.

1972-01-01

The equivalence of a considered integral equation form with an infinite system of linear equations is proved, and the localization of the eigenvalues of the infinite system is expressed. Error estimates are derived, and the problems of finding upper bounds and lower bounds for the eigenvalues are solved simultaneously.

Contractions and deformations of quasiclassical Lie algebras preserving a nondegenerate quadratic Casimir operator

DOE Office of Scientific and Technical Information (OSTI.GOV)

Campoamor-Stursberg, R., E-mail: rutwig@mat.ucm.e

2008-05-15

By means of contractions of Lie algebras, we obtain new classes of indecomposable quasiclassical Lie algebras that satisfy the Yang-Baxter equations in its reformulation in terms of triple products. These algebras are shown to arise naturally from noncompact real simple algebras with nonsimple complexification, where we impose that a nondegenerate quadratic Casimir operator is preserved by the limiting process. We further consider the converse problem and obtain sufficient conditions on integrable cocycles of quasiclassical Lie algebras in order to preserve nondegenerate quadratic Casimir operators by the associated linear deformations.

Accurate Monotonicity - Preserving Schemes With Runge-Kutta Time Stepping

NASA Technical Reports Server (NTRS)

Suresh, A.; Huynh, H. T.

1997-01-01

A new class of high-order monotonicity-preserving schemes for the numerical solution of conservation laws is presented. The interface value in these schemes is obtained by limiting a higher-order polynominal reconstruction. The limiting is designed to preserve accuracy near extrema and to work well with Runge-Kutta time stepping. Computational efficiency is enhanced by a simple test that determines whether the limiting procedure is needed. For linear advection in one dimension, these schemes are shown as well as the Euler equations also confirm their high accuracy, good shock resolution, and computational efficiency.

Prediction of free turbulent mixing using a turbulent kinetic energy method

NASA Technical Reports Server (NTRS)

Harsha, P. T.

1973-01-01

Free turbulent mixing of two-dimensional and axisymmetric one- and two-stream flows is analyzed by a relatively simple turbulent kinetic energy method. This method incorporates a linear relationship between the turbulent shear and the turbulent kinetic energy and an algebraic relationship for the length scale appearing in the turbulent kinetic energy equation. Good results are obtained for a wide variety of flows. The technique is shown to be especially applicable to flows with heat and mass transfer, for which nonunity Prandtl and Schmidt numbers may be assumed.

Linear Titration Curves of Acids and Bases.

PubMed

Joseph, N R

1959-05-29

The Henderson-Hasselbalch equation, by a simple transformation, becomes pH - pK = pA - pB, where pA and pB are the negative logarithms of acid and base concentrations. Sigmoid titration curves then reduce to straight lines; titration curves of polyelectrolytes, to families of straight lines. The method is applied to the titration of the dipeptide glycyl aminotricarballylic acid, with four titrable groups. Results are expressed as Cartesian and d'Ocagne nomograms. The latter is of a general form applicable to polyelectrolytes of any degree of complexity.

A Comparison between Multiple Regression Models and CUN-BAE Equation to Predict Body Fat in Adults

PubMed Central

Fuster-Parra, Pilar; Bennasar-Veny, Miquel; Tauler, Pedro; Yañez, Aina; López-González, Angel A.; Aguiló, Antoni

2015-01-01

Background Because the accurate measure of body fat (BF) is difficult, several prediction equations have been proposed. The aim of this study was to compare different multiple regression models to predict BF, including the recently reported CUN-BAE equation. Methods Multi regression models using body mass index (BMI) and body adiposity index (BAI) as predictors of BF will be compared. These models will be also compared with the CUN-BAE equation. For all the analysis a sample including all the participants and another one including only the overweight and obese subjects will be considered. The BF reference measure was made using Bioelectrical Impedance Analysis. Results The simplest models including only BMI or BAI as independent variables showed that BAI is a better predictor of BF. However, adding the variable sex to both models made BMI a better predictor than the BAI. For both the whole group of participants and the group of overweight and obese participants, using simple models (BMI, age and sex as variables) allowed obtaining similar correlations with BF as when the more complex CUN-BAE was used (ρ = 0:87 vs. ρ = 0:86 for the whole sample and ρ = 0:88 vs. ρ = 0:89 for overweight and obese subjects, being the second value the one for CUN-BAE). Conclusions There are simpler models than CUN-BAE equation that fits BF as well as CUN-BAE does. Therefore, it could be considered that CUN-BAE overfits. Using a simple linear regression model, the BAI, as the only variable, predicts BF better than BMI. However, when the sex variable is introduced, BMI becomes the indicator of choice to predict BF. PMID:25821960

A comparison between multiple regression models and CUN-BAE equation to predict body fat in adults.

PubMed

Fuster-Parra, Pilar; Bennasar-Veny, Miquel; Tauler, Pedro; Yañez, Aina; López-González, Angel A; Aguiló, Antoni

2015-01-01

Because the accurate measure of body fat (BF) is difficult, several prediction equations have been proposed. The aim of this study was to compare different multiple regression models to predict BF, including the recently reported CUN-BAE equation. Multi regression models using body mass index (BMI) and body adiposity index (BAI) as predictors of BF will be compared. These models will be also compared with the CUN-BAE equation. For all the analysis a sample including all the participants and another one including only the overweight and obese subjects will be considered. The BF reference measure was made using Bioelectrical Impedance Analysis. The simplest models including only BMI or BAI as independent variables showed that BAI is a better predictor of BF. However, adding the variable sex to both models made BMI a better predictor than the BAI. For both the whole group of participants and the group of overweight and obese participants, using simple models (BMI, age and sex as variables) allowed obtaining similar correlations with BF as when the more complex CUN-BAE was used (ρ = 0:87 vs. ρ = 0:86 for the whole sample and ρ = 0:88 vs. ρ = 0:89 for overweight and obese subjects, being the second value the one for CUN-BAE). There are simpler models than CUN-BAE equation that fits BF as well as CUN-BAE does. Therefore, it could be considered that CUN-BAE overfits. Using a simple linear regression model, the BAI, as the only variable, predicts BF better than BMI. However, when the sex variable is introduced, BMI becomes the indicator of choice to predict BF.

New Results on the Linear Equating Methods for the Non-Equivalent-Groups Design

ERIC Educational Resources Information Center

von Davier, Alina A.

2008-01-01

The two most common observed-score equating functions are the linear and equipercentile functions. These are often seen as different methods, but von Davier, Holland, and Thayer showed that any equipercentile equating function can be decomposed into linear and nonlinear parts. They emphasized the dominant role of the linear part of the nonlinear…

Transit-time and age distributions for nonlinear time-dependent compartmental systems.

PubMed

Metzler, Holger; Müller, Markus; Sierra, Carlos A

2018-02-06

Many processes in nature are modeled using compartmental systems (reservoir/pool/box systems). Usually, they are expressed as a set of first-order differential equations describing the transfer of matter across a network of compartments. The concepts of age of matter in compartments and the time required for particles to transit the system are important diagnostics of these models with applications to a wide range of scientific questions. Until now, explicit formulas for transit-time and age distributions of nonlinear time-dependent compartmental systems were not available. We compute densities for these types of systems under the assumption of well-mixed compartments. Assuming that a solution of the nonlinear system is available at least numerically, we show how to construct a linear time-dependent system with the same solution trajectory. We demonstrate how to exploit this solution to compute transit-time and age distributions in dependence on given start values and initial age distributions. Furthermore, we derive equations for the time evolution of quantiles and moments of the age distributions. Our results generalize available density formulas for the linear time-independent case and mean-age formulas for the linear time-dependent case. As an example, we apply our formulas to a nonlinear and a linear version of a simple global carbon cycle model driven by a time-dependent input signal which represents fossil fuel additions. We derive time-dependent age distributions for all compartments and calculate the time it takes to remove fossil carbon in a business-as-usual scenario.

Linear stability analysis of the Vlasov-Poisson equations in high density plasmas in the presence of crossed fields and density gradients

NASA Technical Reports Server (NTRS)

Kaup, D. J.; Hansen, P. J.; Choudhury, S. Roy; Thomas, Gary E.

1986-01-01

The equations for the single-particle orbits in a nonneutral high density plasma in the presence of inhomogeneous crossed fields are obtained. Using these orbits, the linearized Vlasov equation is solved as an expansion in the orbital radii in the presence of inhomogeneities and density gradients. A model distribution function is introduced whose cold-fluid limit is exactly the same as that used in many previous studies of the cold-fluid equations. This model function is used to reduce the linearized Vlasov-Poisson equations to a second-order ordinary differential equation for the linearized electrostatic potential whose eigenvalue is the perturbation frequency.

Chemical Equation Balancing.

ERIC Educational Resources Information Center

Blakley, G. R.

1982-01-01

Reviews mathematical techniques for solving systems of homogeneous linear equations and demonstrates that the algebraic method of balancing chemical equations is a matter of solving a system of homogeneous linear equations. FORTRAN programs using this matrix method to chemical equation balancing are available from the author. (JN)

A boundary integral approach in primitive variables for free surface flows

NASA Astrophysics Data System (ADS)

Casciola, C.; Piva, R.

The boundary integral formulation, very efficient for free surface potential flows, was considered for its possible extension to rotational flows either inviscid or viscous. We first analyze a general formulation for unsteady Navier-Stokes equations in primitive variables, which reduces to a representation for the Euler equations in the limiting case of Reynolds infinity. A first simplified model for rotational flows, obtained by decoupling kinematics and dynamics, reduces the integral equations to a known kinematical form whose mathematical and numerical properties have been studied. The dynamics equations to complete the model are obtained for the free surface and the wake. A simple and efficient scheme for the study of the non linear evolution of the wave system and its interaction with the body wake is presented. A steady state version for the calculation of the wave resistance is also reported. A second model was proposed for the simulation of rotational separated regions, by coupling the integral equations in velocity with an integral equation for the vorticity at the body boundary. The same procedure may be extended to include the diffusion of the vorticity in the flowfield. The vortex shedding from a cylindrical body in unsteady motion is discussed, as a first application of the model.

A Comparison between Linear IRT Observed-Score Equating and Levine Observed-Score Equating under the Generalized Kernel Equating Framework

ERIC Educational Resources Information Center

Chen, Haiwen

2012-01-01

In this article, linear item response theory (IRT) observed-score equating is compared under a generalized kernel equating framework with Levine observed-score equating for nonequivalent groups with anchor test design. Interestingly, these two equating methods are closely related despite being based on different methodologies. Specifically, when…

Linear ordinary differential equations with constant coefficients. Revisiting the impulsive response method using factorization

NASA Astrophysics Data System (ADS)

Camporesi, Roberto

2011-06-01

We present an approach to the impulsive response method for solving linear constant-coefficient ordinary differential equations based on the factorization of the differential operator. The approach is elementary, we only assume a basic knowledge of calculus and linear algebra. In particular, we avoid the use of distribution theory, as well as of the other more advanced approaches: Laplace transform, linear systems, the general theory of linear equations with variable coefficients and the variation of constants method. The approach presented here can be used in a first course on differential equations for science and engineering majors.

A new exact method for line radiative transfer

NASA Astrophysics Data System (ADS)

Elitzur, Moshe; Asensio Ramos, Andrés

2006-01-01

We present a new method, the coupled escape probability (CEP), for exact calculation of line emission from multi-level systems, solving only algebraic equations for the level populations. The CEP formulation of the classical two-level problem is a set of linear equations, and we uncover an exact analytic expression for the emission from two-level optically thick sources that holds as long as they are in the `effectively thin' regime. In a comparative study of a number of standard problems, the CEP method outperformed the leading line transfer methods by substantial margins. The algebraic equations employed by our new method are already incorporated in numerous codes based on the escape probability approximation. All that is required for an exact solution with these existing codes is to augment the expression for the escape probability with simple zone-coupling terms. As an application, we find that standard escape probability calculations generally produce the correct cooling emission by the CII 158-μm line but not by the 3P lines of OI.

Towards a universal method for calculating hydration free energies: a 3D reference interaction site model with partial molar volume correction.

PubMed

Palmer, David S; Frolov, Andrey I; Ratkova, Ekaterina L; Fedorov, Maxim V

2010-12-15

We report a simple universal method to systematically improve the accuracy of hydration free energies calculated using an integral equation theory of molecular liquids, the 3D reference interaction site model. A strong linear correlation is observed between the difference of the experimental and (uncorrected) calculated hydration free energies and the calculated partial molar volume for a data set of 185 neutral organic molecules from different chemical classes. By using the partial molar volume as a linear empirical correction to the calculated hydration free energy, we obtain predictions of hydration free energies in excellent agreement with experiment (R = 0.94, σ = 0.99 kcal mol (- 1) for a test set of 120 organic molecules).

Tree cover bimodality in savannas and forests emerging from the switching between two fire dynamics.

PubMed

De Michele, Carlo; Accatino, Francesco

2014-01-01

Moist savannas and tropical forests share the same climatic conditions and occur side by side. Experimental evidences show that the tree cover of these ecosystems exhibits a bimodal frequency distribution. This is considered as a proof of savanna-forest bistability, predicted by dynamic vegetation models based on non-linear differential equations. Here, we propose a change of perspective about the bimodality of tree cover distribution. We show, using a simple matrix model of tree dynamics, how the bimodality of tree cover can emerge from the switching between two linear dynamics of trees, one in presence and one in absence of fire, with a feedback between fire and trees. As consequence, we find that the transitions between moist savannas and tropical forests, if sharp, are not necessarily catastrophic.

Finite-element time-domain algorithms for modeling linear Debye and Lorentz dielectric dispersions at low frequencies.

PubMed

Stoykov, Nikolay S; Kuiken, Todd A; Lowery, Madeleine M; Taflove, Allen

2003-09-01

We present what we believe to be the first algorithms that use a simple scalar-potential formulation to model linear Debye and Lorentz dielectric dispersions at low frequencies in the context of finite-element time-domain (FETD) numerical solutions of electric potential. The new algorithms, which permit treatment of multiple-pole dielectric relaxations, are based on the auxiliary differential equation method and are unconditionally stable. We validate the algorithms by comparison with the results of a previously reported method based on the Fourier transform. The new algorithms should be useful in calculating the transient response of biological materials subject to impulsive excitation. Potential applications include FETD modeling of electromyography, functional electrical stimulation, defibrillation, and effects of lightning and impulsive electric shock.

Extending double modulation: combinatorial rules for identifying the modulations necessary for determining elasticities in metabolic pathways.

PubMed

Giersch, C; Cornish-Bowden, A

1996-10-07

The double modulation method for determining the elasticities of pathway enzymes, originally devised by Kacser & Burns (Biochem. Soc. Trans. 7, 1149-1160, 1979), is extended to pathways of complex topological structure, including branching and feedback loops. An explicit system of linear equations for the unknown elasticities is derived. The constraints imposed on this linear system imply that modulations of more than one enzyme are not necessarily independent. Simple combinatorial rules are described for identifying without using any algebra the set of independent modulations that allow the determination of the elasticities of any enzyme. By repeated application, the minimum numbers of modulations required to determine the elasticities of all enzymes of a given pathway can be determined. The procedure is illustrated with numerous examples.

Tight-binding study of stacking fault energies and the Rice criterion of ductility in the fcc metals

NASA Astrophysics Data System (ADS)

Mehl, Michael J.; Papaconstantopoulos, Dimitrios A.; Kioussis, Nicholas; Herbranson, M.

2000-02-01

We have used the Naval Research Laboratory (NRL) tight-binding (TB) method to calculate the generalized stacking fault energy and the Rice ductility criterion in the fcc metals Al, Cu, Rh, Pd, Ag, Ir, Pt, Au, and Pb. The method works well for all classes of metals, i.e., simple metals, noble metals, and transition metals. We compared our results with full potential linear-muffin-tin orbital and embedded atom method (EAM) calculations, as well as experiment, and found good agreement. This is impressive, since the NRL-TB approach only fits to first-principles full-potential linearized augmented plane-wave equations of state and band structures for cubic systems. Comparable accuracy with EAM potentials can be achieved only by fitting to the stacking fault energy.

Fast wavelet based algorithms for linear evolution equations

NASA Technical Reports Server (NTRS)

Engquist, Bjorn; Osher, Stanley; Zhong, Sifen

1992-01-01

A class was devised of fast wavelet based algorithms for linear evolution equations whose coefficients are time independent. The method draws on the work of Beylkin, Coifman, and Rokhlin which they applied to general Calderon-Zygmund type integral operators. A modification of their idea is applied to linear hyperbolic and parabolic equations, with spatially varying coefficients. A significant speedup over standard methods is obtained when applied to hyperbolic equations in one space dimension and parabolic equations in multidimensions.

Note on Solutions to a Class of Nonlinear Singular Integro-Differential Equations,

DTIC Science & Technology

1986-08-01

KdV) ut + 2uu x +Uxx x a 0, (1) the sine-Gordon equation Uxt a sin u, (2) and the Kadomtsev - Petviashvili (KP) equation (Ut + 2uu x + UXXx)x -3a 2u yy...SOUIN OA LSFNN ! /" / M.. \\boiz A.S ::-:- and ,M.O.. .- :1/1 / NOTE ON SOLUTIONS TO A CLASS OF NON \\ / LINEAR SINGULAR INTEGRO-DIFFERENTIA[ EQUATIONS by...important nonlinear evolution equations which can be linearized. Many of these equations fall into the category of linearization via soliton theory and

Eye Movements Reveal Students' Strategies in Simple Equation Solving

ERIC Educational Resources Information Center

Susac, Ana; Bubic, Andreja; Kaponja, Jurica; Planinic, Maja; Palmovic, Marijan

2014-01-01

Equation rearrangement is an important skill required for problem solving in mathematics and science. Eye movements of 40 university students were recorded while they were rearranging simple algebraic equations. The participants also reported on their strategies during equation solving in a separate questionnaire. The analysis of the behavioral…

Convergence Speed of a Dynamical System for Sparse Recovery

NASA Astrophysics Data System (ADS)

Balavoine, Aurele; Rozell, Christopher J.; Romberg, Justin

2013-09-01

This paper studies the convergence rate of a continuous-time dynamical system for L1-minimization, known as the Locally Competitive Algorithm (LCA). Solving L1-minimization} problems efficiently and rapidly is of great interest to the signal processing community, as these programs have been shown to recover sparse solutions to underdetermined systems of linear equations and come with strong performance guarantees. The LCA under study differs from the typical L1 solver in that it operates in continuous time: instead of being specified by discrete iterations, it evolves according to a system of nonlinear ordinary differential equations. The LCA is constructed from simple components, giving it the potential to be implemented as a large-scale analog circuit. The goal of this paper is to give guarantees on the convergence time of the LCA system. To do so, we analyze how the LCA evolves as it is recovering a sparse signal from underdetermined measurements. We show that under appropriate conditions on the measurement matrix and the problem parameters, the path the LCA follows can be described as a sequence of linear differential equations, each with a small number of active variables. This allows us to relate the convergence time of the system to the restricted isometry constant of the matrix. Interesting parallels to sparse-recovery digital solvers emerge from this study. Our analysis covers both the noisy and noiseless settings and is supported by simulation results.

Transport through a network of capillaries from ultrametric diffusion equation with quadratic nonlinearity

NASA Astrophysics Data System (ADS)

Oleschko, K.; Khrennikov, A.

2017-10-01

This paper is about a novel mathematical framework to model transport (of, e.g., fluid or gas) through networks of capillaries. This framework takes into account the tree structure of the networks of capillaries. (Roughly speaking, we use the tree-like system of coordinates.) As is well known, tree-geometry can be topologically described as the geometry of an ultrametric space, i.e., a metric space in which the metric satisfies the strong triangle inequality: in each triangle, the third side is less than or equal to the maximum of two other sides. Thus transport (e.g., of oil or emulsion of oil and water in porous media, or blood and air in biological organisms) through networks of capillaries can be mathematically modelled as ultrametric diffusion. Such modelling was performed in a series of recently published papers of the authors. However, the process of transport through capillaries can be only approximately described by the linear diffusion, because the concentration of, e.g., oil droplets, in a capillary can essentially modify the dynamics. Therefore nonlinear dynamical equations provide a more adequate model of transport in a network of capillaries. We consider a nonlinear ultrametric diffusion equation with quadratic nonlinearity - to model transport in such a network. Here, as in the linear case, we apply the theory of ultrametric wavelets. The paper also contains a simple introduction to theory of ultrametric spaces and analysis on them.

On order and chaos in the mergers of galaxies

NASA Astrophysics Data System (ADS)

Vandervoort, Peter O.

2018-03-01

This paper describes a low-dimensional model of the merger of two galaxies. The governing equations are the complete sets of moment equations of the first and second orders derived from the collisionless Boltzmann equations representing the galaxies. The moment equations reduce to an equation governing the relative motion of the galaxies, tensor virial equations, and equations governing the kinetic energy tensors. We represent the galaxies as heterogeneous ellipsoids with Gaussian stratifications of their densities, and we represent the mean stellar motions in terms of velocity fields that sustain those densities consistently with the equation of continuity. We reduce and solve the governing equations for a head-on encounter of a dwarf galaxy with a giant galaxy. That reduction includes the effect of dynamical friction on the relative motion of the galaxies. Our criterion for chaotic behaviour is sensitivity of the motion to small changes in the initial conditions. In a survey of encounters and mergers of a dwarf galaxy with a giant galaxy, chaotic behaviour arises mainly in non-linear oscillations of the dwarf galaxy. The encounter disrupts the dwarf, excites chaotic oscillations of the dwarf, or excites regular oscillations. Dynamical friction can drive a merger to completion within a Hubble time only if the dwarf is sufficiently massive. The survey of encounters and mergers is the basis for a simple model of the evolution of a `Local Group' consisting of a giant galaxy and a population of dwarf galaxies bound to the giant as satellites on radial orbits.

FAST TRACK COMMUNICATION: On the Liouvillian solution of second-order linear differential equations and algebraic invariant curves

NASA Astrophysics Data System (ADS)

Man, Yiu-Kwong

2010-10-01

In this communication, we present a method for computing the Liouvillian solution of second-order linear differential equations via algebraic invariant curves. The main idea is to integrate Kovacic's results on second-order linear differential equations with the Prelle-Singer method for computing first integrals of differential equations. Some examples on using this approach are provided.

Nonlinear Diophantine equation 11 x +13 y = z 2

NASA Astrophysics Data System (ADS)

Sugandha, A.; Tripena, A.; Prabowo, A.; Sukono, F.

2018-03-01

This research aims to obtaining the solutions (if any) from the Non Linear Diophantine equation of 11 x + 13 y = z 2. There are 3 possibilities to obtain the solutions (if any) from the Non Linear Diophantine equation, namely single, multiple, and no solution. This research is conducted in two stages: (1) by utilizing simulation to obtain the solutions (if any) from the Non Linear Diophantine equation of 11 x + 13 y = z 2 and (2) by utilizing congruency theory with its characteristics proven that the Non Linear Diophantine equation has no solution for non negative whole numbers (integers) of x, y, z.

Lie algebras and linear differential equations.

NASA Technical Reports Server (NTRS)

Brockett, R. W.; Rahimi, A.

1972-01-01

Certain symmetry properties possessed by the solutions of linear differential equations are examined. For this purpose, some basic ideas from the theory of finite dimensional linear systems are used together with the work of Wei and Norman on the use of Lie algebraic methods in differential equation theory.

A simple and general method for solving detailed chemical evolution with delayed production of iron and other chemical elements

NASA Astrophysics Data System (ADS)

Vincenzo, F.; Matteucci, F.; Spitoni, E.

2017-04-01

We present a theoretical method for solving the chemical evolution of galaxies by assuming an instantaneous recycling approximation for chemical elements restored by massive stars and the delay time distribution formalism for delayed chemical enrichment by Type Ia Supernovae. The galaxy gas mass assembly history, together with the assumed stellar yields and initial mass function, represents the starting point of this method. We derive a simple and general equation, which closely relates the Laplace transforms of the galaxy gas accretion history and star formation history, which can be used to simplify the problem of retrieving these quantities in the galaxy evolution models assuming a linear Schmidt-Kennicutt law. We find that - once the galaxy star formation history has been reconstructed from our assumptions - the differential equation for the evolution of the chemical element X can be suitably solved with classical methods. We apply our model to reproduce the [O/Fe] and [Si/Fe] versus [Fe/H] chemical abundance patterns as observed at the solar neighbourhood by assuming a decaying exponential infall rate of gas and different delay time distributions for Type Ia Supernovae; we also explore the effect of assuming a non-linear Schmidt-Kennicutt law, with the index of the power law being k = 1.4. Although approximate, we conclude that our model with the single-degenerate scenario for Type Ia Supernovae provides the best agreement with the observed set of data. Our method can be used by other complementary galaxy stellar population synthesis models to predict also the chemical evolution of galaxies.

Standard Errors of Equating for the Percentile Rank-Based Equipercentile Equating with Log-Linear Presmoothing

ERIC Educational Resources Information Center

Wang, Tianyou

2009-01-01

Holland and colleagues derived a formula for analytical standard error of equating using the delta-method for the kernel equating method. Extending their derivation, this article derives an analytical standard error of equating procedure for the conventional percentile rank-based equipercentile equating with log-linear smoothing. This procedure is…

A fresh look at linear ordinary differential equations with constant coefficients. Revisiting the impulsive response method using factorization

NASA Astrophysics Data System (ADS)

Camporesi, Roberto

2016-01-01

We present an approach to the impulsive response method for solving linear constant-coefficient ordinary differential equations of any order based on the factorization of the differential operator. The approach is elementary, we only assume a basic knowledge of calculus and linear algebra. In particular, we avoid the use of distribution theory, as well as of the other more advanced approaches: Laplace transform, linear systems, the general theory of linear equations with variable coefficients and variation of parameters. The approach presented here can be used in a first course on differential equations for science and engineering majors.

(N+1)-dimensional fractional reduced differential transform method for fractional order partial differential equations

NASA Astrophysics Data System (ADS)

Arshad, Muhammad; Lu, Dianchen; Wang, Jun

2017-07-01

In this paper, we pursue the general form of the fractional reduced differential transform method (DTM) to (N+1)-dimensional case, so that fractional order partial differential equations (PDEs) can be resolved effectively. The most distinct aspect of this method is that no prescribed assumptions are required, and the huge computational exertion is reduced and round-off errors are also evaded. We utilize the proposed scheme on some initial value problems and approximate numerical solutions of linear and nonlinear time fractional PDEs are obtained, which shows that the method is highly accurate and simple to apply. The proposed technique is thus an influential technique for solving the fractional PDEs and fractional order problems occurring in the field of engineering, physics etc. Numerical results are obtained for verification and demonstration purpose by using Mathematica software.

The spectral applications of Beer-Lambert law for some biological and dosimetric materials

NASA Astrophysics Data System (ADS)

Içelli, Orhan; Yalçin, Zeynel; Karakaya, Vatan; Ilgaz, Işıl P.

2014-08-01

The aim of this study is to conduct quantitative and qualitative analysis of biological and dosimetric materials which contain organic and inorganic materials and to make the determination by using the spectral theorem Beer-Lambert law. Beer-Lambert law is a system of linear equations for the spectral theory. It is possible to solve linear equations with a non-zero coefficient matrix determinant forming linear equations. Characteristic matrix of the linear equation with zero determinant is called point spectrum at the spectral theory.

Causal implications of viscous damping in compressible fluid flows

PubMed

Jordan; Meyer; Puri

2000-12-01

Classically, a compressible, isothermal, viscous fluid is regarded as a mathematical continuum and its motion is governed by the linearized continuity, Navier-Stokes, and state equations. Unfortunately, solutions of this system are of a diffusive nature and hence do not satisfy causality. However, in the case of a half-space of fluid set to motion by a harmonically vibrating plate the classical equation of motion can, under suitable conditions, be approximated by the damped wave equation. Since this equation is hyperbolic, the resulting solutions satisfy causal requirements. In this work the Laplace transform and other analytical and numerical tools are used to investigate this apparent contradiction. To this end the exact solutions, as well as their special and limiting cases, are found and compared for the two models. The effects of the physical parameters on the solutions and associated quantities are also studied. It is shown that propagating wave fronts are only possible under the hyperbolic model and that the concept of phase speed has different meanings in the two formulations. In addition, discontinuities and shock waves are noted and a physical system is modeled under both formulations. Overall, it is shown that the hyperbolic form gives a more realistic description of the physical problem than does the classical theory. Lastly, a simple mechanical analog is given and connections to viscoelastic fluids are noted. In particular, the research presented here supports the notion that linear compressible, isothermal, viscous fluids can, at least in terms of causality, be better characterized as a type of viscoelastic fluid.

Massive graviton on arbitrary background: derivation, syzygies, applications

DOE Office of Scientific and Technical Information (OSTI.GOV)

Bernard, Laura; Deffayet, Cédric; IHES, Institut des Hautes Études Scientifiques,Le Bois-Marie, 35 route de Chartres, F-91440 Bures-sur-Yvette

2015-06-23

We give the detailed derivation of the fully covariant form of the quadratic action and the derived linear equations of motion for a massive graviton in an arbitrary background metric (which were presented in arXiv:1410.8302 [hep-th]). Our starting point is the de Rham-Gabadadze-Tolley (dRGT) family of ghost free massive gravities and using a simple model of this family, we are able to express this action and these equations of motion in terms of a single metric in which the graviton propagates, hence removing in particular the need for a “reference metric' which is present in the non perturbative formulation. Wemore » show further how 5 covariant constraints can be obtained including one which leads to the tracelessness of the graviton on flat space-time and removes the Boulware-Deser ghost. This last constraint involves powers and combinations of the curvature of the background metric. The 5 constraints are obtained for a background metric which is unconstrained, i.e. which does not have to obey the background field equations. We then apply these results to the case of Einstein space-times, where we show that the 5 constraints become trivial, and Friedmann-Lemaître-Robertson-Walker space-times, for which we correct in particular some results that appeared elsewhere. To reach our results, we derive several non trivial identities, syzygies, involving the graviton fields, its derivatives and the background metric curvature. These identities have their own interest. We also discover that there exist backgrounds for which the dRGT equations cannot be unambiguously linearized.« less

Massive graviton on arbitrary background: derivation, syzygies, applications

DOE Office of Scientific and Technical Information (OSTI.GOV)

Bernard, Laura; Deffayet, Cédric; Strauss, Mikael von, E-mail: bernard@iap.fr, E-mail: deffayet@iap.fr, E-mail: strauss@iap.fr

2015-06-01

We give the detailed derivation of the fully covariant form of the quadratic action and the derived linear equations of motion for a massive graviton in an arbitrary background metric (which were presented in arXiv:1410.8302 [hep-th]). Our starting point is the de Rham-Gabadadze-Tolley (dRGT) family of ghost free massive gravities and using a simple model of this family, we are able to express this action and these equations of motion in terms of a single metric in which the graviton propagates, hence removing in particular the need for a ''reference metric' which is present in the non perturbative formulation. Wemore » show further how 5 covariant constraints can be obtained including one which leads to the tracelessness of the graviton on flat space-time and removes the Boulware-Deser ghost. This last constraint involves powers and combinations of the curvature of the background metric. The 5 constraints are obtained for a background metric which is unconstrained, i.e. which does not have to obey the background field equations. We then apply these results to the case of Einstein space-times, where we show that the 5 constraints become trivial, and Friedmann-Lemaître-Robertson-Walker space-times, for which we correct in particular some results that appeared elsewhere. To reach our results, we derive several non trivial identities, syzygies, involving the graviton fields, its derivatives and the background metric curvature. These identities have their own interest. We also discover that there exist backgrounds for which the dRGT equations cannot be unambiguously linearized.« less

On some Approximation Schemes for Steady Compressible Viscous Flow

NASA Astrophysics Data System (ADS)

Bause, M.; Heywood, J. G.; Novotny, A.; Padula, M.

This paper continues our development of approximation schemes for steady compressible viscous flow based on an iteration between a Stokes like problem for the velocity and a transport equation for the density, with the aim of improving their suitability for computations. Such schemes seem attractive for computations because they offer a reduction to standard problems for which there is already highly refined software, and because of the guidance that can be drawn from an existence theory based on them. Our objective here is to modify a recent scheme of Heywood and Padula [12], to improve its convergence properties. This scheme improved upon an earlier scheme of Padula [21], [23] through the use of a special ``effective pressure'' in linking the Stokes and transport problems. However, its convergence is limited for several reasons. Firstly, the steady transport equation itself is only solvable for general velocity fields if they satisfy certain smallness conditions. These conditions are met here by using a rescaled variant of the steady transport equation based on a pseudo time step for the equation of continuity. Another matter limiting the convergence of the scheme in [12] is that the Stokes linearization, which is a linearization about zero, has an inevitably small range of convergence. We replace it here with an Oseen or Newton linearization, either of which has a wider range of convergence, and converges more rapidly. The simplicity of the scheme offered in [12] was conducive to a relatively simple and clearly organized proof of its convergence. The proofs of convergence for the more complicated schemes proposed here are structured along the same lines. They strengthen the theorems of existence and uniqueness in [12] by weakening the smallness conditions that are needed. The expected improvement in the computational performance of the modified schemes has been confirmed by Bause [2], in an ongoing investigation.

Angular-Rate Estimation Using Star Tracker Measurements

NASA Technical Reports Server (NTRS)

Azor, R.; Bar-Itzhack, I.; Deutschmann, Julie K.; Harman, Richard R.

1999-01-01

This paper presents algorithms for estimating the angular-rate vector of satellites using quaternion measurements. Two approaches are compared, one that uses differentiated quatemion measurements to yield coarse rate measurements which are then fed into two different estimators. In the other approach the raw quatemion measurements themselves are fed directly into the two estimators. The two estimators rely on the ability to decompose the non-linear rate dependent part of the rotational dynamics equation of a rigid body into a product of an angular-rate dependent matrix and the angular-rate vector itself This decomposition, which is not unique, enables the treatment of the nonlinear spacecraft dynamics model as a linear one and, consequently, the application of a Pseudo-Linear Kalman Filter (PSELIKA). It also enables the application of a special Kalman filter which is based on the use of the solution of the State Dependent Algebraic Riccati Equation (SDARE) in order to compute the Kalman gain matrix and thus eliminates the need to propagate and update the filter covariance matrix. The replacement of the elaborate rotational dynamics by a simple first order Markov model is also examined. In this paper a special consideration is given to the problem of delayed quatemion measurements. Two solutions to this problem are suggested and tested. Real Rossi X-Ray Timing Explorer (RXTE) data is used to test these algorithms, and results of these tests are presented.

Angular-Rate Estimation using Star Tracker Measurements

NASA Technical Reports Server (NTRS)

Azor, R.; Bar-Itzhack, Itzhack Y.; Deutschmann, Julie K.; Harman, Richard R.

1999-01-01

This paper presents algorithms for estimating the angular-rate vector of satellites using quaternion measurements. Two approaches are compared, one that uses differentiated quaternion measurements to yield coarse rate measurements which are then fed into two different estimators. In the other approach the raw quaternion measurements themselves are fed directly into the two estimators. The two estimators rely on the ability to decompose the non-linear rate dependent part of the rotational dynamics equation of a rigid body into a product of an angular-rate dependent matrix and the angular-rate vector itself. This decomposition, which is not unique, enables the treatment of the nonlinear spacecraft dynamics model as a linear one and, consequently, the application of a Pseudo-Linear Kalman Filter (PSELIKA). It also enables the application of a special Kalman filter which is based on the use of the solution of the State Dependent Algebraic Riccati Equation (SDARE) in order to compute the Kalman gain matrix and thus eliminates the need to propagate and update the filter covariance matrix. The replacement of the elaborate rotational dynamics by a simple first order Markov model is also examined. In this paper a special consideration is given to the problem of delayed quaternion measurements. Two solutions to this problem are suggested and tested. Real Rossi X-Ray Timing Explorer (RXTE) data is used to test these algorithms, and results of these tests are presented.

Topology-induced bifurcations for the nonlinear Schrödinger equation on the tadpole graph.

PubMed

Cacciapuoti, Claudio; Finco, Domenico; Noja, Diego

2015-01-01

In this paper we give the complete classification of solitons for a cubic nonlinear Schrödinger equation on the simplest network with a nontrivial topology: the tadpole graph, i.e., a ring with a half line attached to it and free boundary conditions at the junction. This is a step toward the modelization of condensate propagation and confinement in quasi-one-dimensional traps. The model, although simple, exhibits a surprisingly rich behavior and in particular we show that it admits: (i) a denumerable family of continuous branches of embedded solitons vanishing on the half line and bifurcating from linear eigenstates and threshold resonances of the system; (ii) a continuous branch of edge solitons bifurcating from the previous families at the threshold of the continuous spectrum with a pitchfork bifurcation; and (iii) a finite family of continuous branches of solitons without linear analog. All the solutions are explicitly constructed in terms of elliptic Jacobian functions. Moreover we show that families of nonlinear bound states of the above kind continue to exist in the presence of a uniform magnetic field orthogonal to the plane of the ring when a well definite flux quantization condition holds true. In this sense the magnetic field acts as a control parameter. Finally we highlight the role of resonances in the linearization as a signature of the occurrence of bifurcations of solitons from the continuous spectrum.

New Equating Methods and Their Relationships with Levine Observed Score Linear Equating under the Kernel Equating Framework

ERIC Educational Resources Information Center

Chen, Haiwen; Holland, Paul

2010-01-01

In this paper, we develop a new curvilinear equating for the nonequivalent groups with anchor test (NEAT) design under the assumption of the classical test theory model, that we name curvilinear Levine observed score equating. In fact, by applying both the kernel equating framework and the mean preserving linear transformation of…

A canonical form of the equation of motion of linear dynamical systems

NASA Astrophysics Data System (ADS)

Kawano, Daniel T.; Salsa, Rubens Goncalves; Ma, Fai; Morzfeld, Matthias

2018-03-01

The equation of motion of a discrete linear system has the form of a second-order ordinary differential equation with three real and square coefficient matrices. It is shown that, for almost all linear systems, such an equation can always be converted by an invertible transformation into a canonical form specified by two diagonal coefficient matrices associated with the generalized acceleration and displacement. This canonical form of the equation of motion is unique up to an equivalence class for non-defective systems. As an important by-product, a damped linear system that possesses three symmetric and positive definite coefficients can always be recast as an undamped and decoupled system.

An application of the Maslov complex germ method to the one-dimensional nonlocal Fisher-KPP equation

NASA Astrophysics Data System (ADS)

Shapovalov, A. V.; Trifonov, A. Yu.

A semiclassical approximation approach based on the Maslov complex germ method is considered in detail for the one-dimensional nonlocal Fisher-Kolmogorov-Petrovskii-Piskunov (Fisher-KPP) equation under the supposition of weak diffusion. In terms of the semiclassical formalism developed, the original nonlinear equation is reduced to an associated linear partial differential equation and some algebraic equations for the coefficients of the linear equation with a given accuracy of the asymptotic parameter. The solutions of the nonlinear equation are constructed from the solutions of both the linear equation and the algebraic equations. The solutions of the linear problem are found with the use of symmetry operators. A countable family of the leading terms of the semiclassical asymptotics is constructed in explicit form. The semiclassical asymptotics are valid by construction in a finite time interval. We construct asymptotics which are different from the semiclassical ones and can describe evolution of the solutions of the Fisher-KPP equation at large times. In the example considered, an initial unimodal distribution becomes multimodal, which can be treated as an example of a space structure.

Accurate upwind methods for the Euler equations

NASA Technical Reports Server (NTRS)

Huynh, Hung T.

1993-01-01

A new class of piecewise linear methods for the numerical solution of the one-dimensional Euler equations of gas dynamics is presented. These methods are uniformly second-order accurate, and can be considered as extensions of Godunov's scheme. With an appropriate definition of monotonicity preservation for the case of linear convection, it can be shown that they preserve monotonicity. Similar to Van Leer's MUSCL scheme, they consist of two key steps: a reconstruction step followed by an upwind step. For the reconstruction step, a monotonicity constraint that preserves uniform second-order accuracy is introduced. Computational efficiency is enhanced by devising a criterion that detects the 'smooth' part of the data where the constraint is redundant. The concept and coding of the constraint are simplified by the use of the median function. A slope steepening technique, which has no effect at smooth regions and can resolve a contact discontinuity in four cells, is described. As for the upwind step, existing and new methods are applied in a manner slightly different from those in the literature. These methods are derived by approximating the Euler equations via linearization and diagonalization. At a 'smooth' interface, Harten, Lax, and Van Leer's one intermediate state model is employed. A modification for this model that can resolve contact discontinuities is presented. Near a discontinuity, either this modified model or a more accurate one, namely, Roe's flux-difference splitting. is used. The current presentation of Roe's method, via the conceptually simple flux-vector splitting, not only establishes a connection between the two splittings, but also leads to an admissibility correction with no conditional statement, and an efficient approximation to Osher's approximate Riemann solver. These reconstruction and upwind steps result in schemes that are uniformly second-order accurate and economical at smooth regions, and yield high resolution at discontinuities.

A simple model of ultrasound propagation in a cavitating liquid. Part I: Theory, nonlinear attenuation and traveling wave generation.

PubMed

Louisnard, O

2012-01-01

The bubbles involved in sonochemistry and other applications of cavitation oscillate inertially. A correct estimation of the wave attenuation in such bubbly media requires a realistic estimation of the power dissipated by the oscillation of each bubble, by thermal diffusion in the gas and viscous friction in the liquid. Both quantities and calculated numerically for a single inertial bubble driven at 20 kHz, and are found to be several orders of magnitude larger than the linear prediction. Viscous dissipation is found to be the predominant cause of energy loss for bubbles small enough. Then, the classical nonlinear Caflish equations describing the propagation of acoustic waves in a bubbly liquid are recast and simplified conveniently. The main harmonic part of the sound field is found to fulfill a nonlinear Helmholtz equation, where the imaginary part of the squared wave number is directly correlated with the energy lost by a single bubble. For low acoustic driving, linear theory is recovered, but for larger drivings, namely above the Blake threshold, the attenuation coefficient is found to be more than 3 orders of magnitude larger then the linear prediction. A huge attenuation of the wave is thus expected in regions where inertial bubbles are present, which is confirmed by numerical simulations of the nonlinear Helmholtz equation in a 1D standing wave configuration. The expected strong attenuation is not only observed but furthermore, the examination of the phase between the pressure field and its gradient clearly demonstrates that a traveling wave appears in the medium. Copyright © 2011 Elsevier B.V. All rights reserved.

Operator Factorization and the Solution of Second-Order Linear Ordinary Differential Equations

ERIC Educational Resources Information Center

Robin, W.

2007-01-01

The theory and application of second-order linear ordinary differential equations is reviewed from the standpoint of the operator factorization approach to the solution of ordinary differential equations (ODE). Using the operator factorization approach, the general second-order linear ODE is solved, exactly, in quadratures and the resulting…

Fixing convergence of Gaussian belief propagation

DOE Office of Scientific and Technical Information (OSTI.GOV)

Johnson, Jason K; Bickson, Danny; Dolev, Danny

Gaussian belief propagation (GaBP) is an iterative message-passing algorithm for inference in Gaussian graphical models. It is known that when GaBP converges it converges to the correct MAP estimate of the Gaussian random vector and simple sufficient conditions for its convergence have been established. In this paper we develop a double-loop algorithm for forcing convergence of GaBP. Our method computes the correct MAP estimate even in cases where standard GaBP would not have converged. We further extend this construction to compute least-squares solutions of over-constrained linear systems. We believe that our construction has numerous applications, since the GaBP algorithm ismore » linked to solution of linear systems of equations, which is a fundamental problem in computer science and engineering. As a case study, we discuss the linear detection problem. We show that using our new construction, we are able to force convergence of Montanari's linear detection algorithm, in cases where it would originally fail. As a consequence, we are able to increase significantly the number of users that can transmit concurrently.« less

A numerical technique for linear elliptic partial differential equations in polygonal domains.

PubMed

Hashemzadeh, P; Fokas, A S; Smitheman, S A

2015-03-08

Integral representations for the solution of linear elliptic partial differential equations (PDEs) can be obtained using Green's theorem. However, these representations involve both the Dirichlet and the Neumann values on the boundary, and for a well-posed boundary-value problem (BVPs) one of these functions is unknown. A new transform method for solving BVPs for linear and integrable nonlinear PDEs usually referred to as the unified transform ( or the Fokas transform ) was introduced by the second author in the late Nineties. For linear elliptic PDEs, this method can be considered as the analogue of Green's function approach but now it is formulated in the complex Fourier plane instead of the physical plane. It employs two global relations also formulated in the Fourier plane which couple the Dirichlet and the Neumann boundary values. These relations can be used to characterize the unknown boundary values in terms of the given boundary data, yielding an elegant approach for determining the Dirichlet to Neumann map . The numerical implementation of the unified transform can be considered as the counterpart in the Fourier plane of the well-known boundary integral method which is formulated in the physical plane. For this implementation, one must choose (i) a suitable basis for expanding the unknown functions and (ii) an appropriate set of complex values, which we refer to as collocation points, at which to evaluate the global relations. Here, by employing a variety of examples we present simple guidelines of how the above choices can be made. Furthermore, we provide concrete rules for choosing the collocation points so that the condition number of the matrix of the associated linear system remains low.

Kinetic modeling and fitting software for interconnected reaction schemes: VisKin.

PubMed

Zhang, Xuan; Andrews, Jared N; Pedersen, Steen E

2007-02-15

Reaction kinetics for complex, highly interconnected kinetic schemes are modeled using analytical solutions to a system of ordinary differential equations. The algorithm employs standard linear algebra methods that are implemented using MatLab functions in a Visual Basic interface. A graphical user interface for simple entry of reaction schemes facilitates comparison of a variety of reaction schemes. To ensure microscopic balance, graph theory algorithms are used to determine violations of thermodynamic cycle constraints. Analytical solutions based on linear differential equations result in fast comparisons of first order kinetic rates and amplitudes as a function of changing ligand concentrations. For analysis of higher order kinetics, we also implemented a solution using numerical integration. To determine rate constants from experimental data, fitting algorithms that adjust rate constants to fit the model to imported data were implemented using the Levenberg-Marquardt algorithm or using Broyden-Fletcher-Goldfarb-Shanno methods. We have included the ability to carry out global fitting of data sets obtained at varying ligand concentrations. These tools are combined in a single package, which we have dubbed VisKin, to guide and analyze kinetic experiments. The software is available online for use on PCs.

Preliminary study of the association between the elimination parameters of phenytoin and phenobarbital.

PubMed

Methaneethorn, Janthima; Panomvana, Duangchit; Vachirayonstien, Thaveechai

2017-09-26

Therapeutic drug monitoring is essential for both phenytoin and phenobarbital therapy given their narrow therapeutic indexes. Nevertheless, the measurement of either phenytoin or phenobarbital concentrations might not be available in some rural hospitals. Information assisting individualized phenytoin and phenobarbital combination therapy is important. This study's objective was to determine the relationship between the maximum rate of metabolism of phenytoin (Vmax) and phenobarbital clearance (CLPB), which can serve as a guide to individualized drug therapy. Data on phenytoin and phenobarbital concentrations of 19 epileptic patients concurrently receiving both drugs were obtained from medical records. Phenytoin and phenobarbital pharmacokinetic parameters were studied at steady-state conditions. The relationship between the elimination parameters of both drugs was determined using simple linear regression. A high correlation coefficient between Vmax and CLPB was found [r=0.744; p<0.001 for Vmax (mg/kg/day) vs. CLPB (L/kg/day)]. Such a relatively strong linear relationship between the elimination parameters of both drugs indicates that Vmax might be predicted from CLPB and vice versa. Regression equations were established for estimating Vmax from CLPB, and vice versa in patients treated with combination of phenytoin and phenobarbital. These proposed equations can be of use in aiding individualized drug therapy.

Linear free energy relationships of the 1H and 13C NMR chemical shifts of 3-methylene-2-substituted-1,4-pentadienes

NASA Astrophysics Data System (ADS)

Valentić, Nataša V.; Vitnik, Željko; Kozhushkov, Sergei I.; de Meijere, Armin; Ušćumlić, Gordana S.; Juranić, Ivan O.

2005-06-01

Linear free energy relationships (LFER) were applied to the 1H and 13C NMR chemical shifts ( δN, N= 1H and 13C, respectively) in the unsaturated backbone of cross-conjugated trienes 3-methylene-2-substituted-1,4-pentadienes. The NMR data were correlated using five different LFER models, based on the mono, the dual and the triple substituent parameter (MSP, DSP and TSP, respectively) treatment. The simple and extended Hammett equations, and the three postulated unconventional LFER models obtained by adaptation of the later, were used. The geometry data, which are needed in Karplus-type and McConnell-type analysis, were obtained using semi-empirical MNDO-PM3 calculations. In correlating the data the TSP approach was more successful than the MSP and DSP approaches. The fact that the calculated molecular geometries allow accurate prediction of the NMR data confirms the validity of unconventional LFER models used. These results suggest the s- cis conformation of the cross-conjugated triene as the preferred one. Postulated unconventional DSP and TSP equations enable the assessment of electronic substituent effects in the presence of other interfering influences.

A simple depth-averaged model for dry granular flow

NASA Astrophysics Data System (ADS)

Hung, Chi-Yao; Stark, Colin P.; Capart, Herve

Granular flow over an erodible bed is an important phenomenon in both industrial and geophysical settings. Here we develop a depth-averaged theory for dry erosive flows using balance equations for mass, momentum and (crucially) kinetic energy. We assume a linearized GDR-Midi rheology for granular deformation and Coulomb friction along the sidewalls. The theory predicts the kinematic behavior of channelized flows under a variety of conditions, which we test in two sets of experiments: (1) a linear chute, where abrupt changes in tilt drive unsteady uniform flows; (2) a rotating drum, to explore steady non-uniform flow. The theoretical predictions match the experimental results well in all cases, without the need to tune parameters or invoke an ad hoc equation for entrainment at the base of the flow. Here we focus on the drum problem. A dimensionless rotation rate (related to Froude number) characterizes flow geometry and accounts not just for spin rate, drum radius and gravity, but also for grain size, wall friction and channel width. By incorporating Coriolis force the theory can treat behavior under centrifuge-induced enhanced gravity. We identify asymptotic flow regimes at low and high dimensionless rotation rates that exhibit distinct power-law scaling behaviors.

A simple tool for estimating city-wide annual electrical energy savings from cooler surfaces

DOE PAGES

Pomerantz, Melvin; Rosado, Pablo J.; Levinson, Ronnen

2015-07-26

We present a simple method to estimate the maximum possible energy saving that might be achieved by increasing the albedo of surfaces in a large city. We restrict this to the "indirect effect", the cooling of outside air that lessens the demand for air conditioning (AC). Given the power demand of the electric utilities and data about the city, we can use a single linear equation to estimate the maximum saving. For example, the result for an albedo change of 0.2 of pavements in a typical warm city in California, such as Sacramento, is that the saving is less thanmore » about 2 kWh per m 2 per year. This may help decision makers choose which heat island mitigation techniques are economical from an energy-saving perspective.« less

Smith predictor based-sliding mode controller for integrating processes with elevated deadtime.

PubMed

Camacho, Oscar; De la Cruz, Francisco

2004-04-01

An approach to control integrating processes with elevated deadtime using a Smith predictor sliding mode controller is presented. A PID sliding surface and an integrating first-order plus deadtime model have been used to synthesize the controller. Since the performance of existing controllers with a Smith predictor decrease in the presence of modeling errors, this paper presents a simple approach to combining the Smith predictor with the sliding mode concept, which is a proven, simple, and robust procedure. The proposed scheme has a set of tuning equations as a function of the characteristic parameters of the model. For implementation of our proposed approach, computer based industrial controllers that execute PID algorithms can be used. The performance and robustness of the proposed controller are compared with the Matausek-Micić scheme for linear systems using simulations.

An efficient shooting algorithm for Evans function calculations in large systems

NASA Astrophysics Data System (ADS)

Humpherys, Jeffrey; Zumbrun, Kevin

2006-08-01

In Evans function computations of the spectra of asymptotically constant-coefficient linear operators, a basic issue is the efficient and numerically stable computation of subspaces evolving according to the associated eigenvalue ODE. For small systems, a fast, shooting algorithm may be obtained by representing subspaces as single exterior products [J.C. Alexander, R. Sachs, Linear instability of solitary waves of a Boussinesq-type equation: A computer assisted computation, Nonlinear World 2 (4) (1995) 471-507; L.Q. Brin, Numerical testing of the stability of viscous shock waves, Ph.D. Thesis, Indiana University, Bloomington, 1998; L.Q. Brin, Numerical testing of the stability of viscous shock waves, Math. Comp. 70 (235) (2001) 1071-1088; L.Q. Brin, K. Zumbrun, Analytically varying eigenvectors and the stability of viscous shock waves, in: Seventh Workshop on Partial Differential Equations, Part I, 2001, Rio de Janeiro, Mat. Contemp. 22 (2002) 19-32; T.J. Bridges, G. Derks, G. Gottwald, Stability and instability of solitary waves of the fifth-order KdV equation: A numerical framework, Physica D 172 (1-4) (2002) 190-216]. For large systems, however, the dimension of the exterior-product space quickly becomes prohibitive, growing as (n/k), where n is the dimension of the system written as a first-order ODE and k (typically ˜n/2) is the dimension of the subspace. We resolve this difficulty by the introduction of a simple polar coordinate algorithm representing “pure” (monomial) products as scalar multiples of orthonormal bases, for which the angular equation is a numerically optimized version of the continuous orthogonalization method of Drury-Davey [A. Davey, An automatic orthonormalization method for solving stiff boundary value problems, J. Comput. Phys. 51 (2) (1983) 343-356; L.O. Drury, Numerical solution of Orr-Sommerfeld-type equations, J. Comput. Phys. 37 (1) (1980) 133-139] and the radial equation is evaluable by quadrature. Notably, the polar-coordinate method preserves the important property of analyticity with respect to parameters.

Linear models for calculating digestibile energy for sheep diets.

PubMed

Fonnesbeck, P V; Christiansen, M L; Harris, L E

1981-05-01

Equations for estimating the digestible energy (DE) content of sheep diets were generated from the chemical contents and a factorial description of diets fed to lambs in digestion trials. The diet factors were two forages (alfalfa and grass hay), harvested at three stages of maturity (late vegetative, early bloom and full bloom), fed in two ingredient combinations (all hay or a 50:50 hay and corn grain mixture) and prepared by two forage texture processes (coarsely chopped or finely chopped and pelleted). The 2 x 3 x 2 x 2 factorial arrangement produced 24 diet treatments. These were replicated twice, for a total of 48 lamb digestion trials. In model 1 regression equations, DE was calculated directly from chemical composition of the diet. In model 2, regression equations predicted the percentage of digested nutrient from the chemical contents of the diet and then DE of the diet was calculated as the sum of the gross energy of the digested organic components. Expanded forms of model 1 and model 2 were also developed that included diet factors as qualitative indicator variables to adjust the regression constant and regression coefficients for the diet description. The expanded forms of the equations accounted for significantly more variation in DE than did the simple models and more accurately estimated DE of the diet. Information provided by the diet description proved as useful as chemical analyses for the prediction of digestibility of nutrients. The statistics indicate that, with model 1, neutral detergent fiber and plant cell wall analyses provided as much information for the estimation of DE as did model 2 with the combined information from crude protein, available carbohydrate, total lipid, cellulose and hemicellulose. Regression equations are presented for estimating DE with the most currently analyzed organic components, including linear and curvilinear variables and diet factors that significantly reduce the standard error of the estimate. To estimate De of a diet, the user utilizes the equation that uses the chemical analysis information and diet description most effectively.

Fast and local non-linear evolution of steep wave-groups on deep water: A comparison of approximate models to fully non-linear simulations

DOE Office of Scientific and Technical Information (OSTI.GOV)

Adcock, T. A. A.; Taylor, P. H.

2016-01-15

The non-linear Schrödinger equation and its higher order extensions are routinely used for analysis of extreme ocean waves. This paper compares the evolution of individual wave-packets modelled using non-linear Schrödinger type equations with packets modelled using fully non-linear potential flow models. The modified non-linear Schrödinger Equation accurately models the relatively large scale non-linear changes to the shape of wave-groups, with a dramatic contraction of the group along the mean propagation direction and a corresponding extension of the width of the wave-crests. In addition, as extreme wave form, there is a local non-linear contraction of the wave-group around the crest whichmore » leads to a localised broadening of the wave spectrum which the bandwidth limited non-linear Schrödinger Equations struggle to capture. This limitation occurs for waves of moderate steepness and a narrow underlying spectrum.« less

The condition of regular degeneration for singularly perturbed systems of linear differential-difference equations.

NASA Technical Reports Server (NTRS)

Cooke, K. L.; Meyer, K. R.

1966-01-01

Extension of problem of singular perturbation for linear scalar constant coefficient differential- difference equation with single retardation to several retardations, noting degenerate equation solution

Techniques for estimating selected streamflow characteristics of rural unregulated streams in Ohio

USGS Publications Warehouse

Koltun, G.F.; Whitehead, Matthew T.

2002-01-01

This report provides equations for estimating mean annual streamflow, mean monthly streamflows, harmonic mean streamflow, and streamflow quartiles (the 25th-, 50th-, and 75th-percentile streamflows) as a function of selected basin characteristics for rural, unregulated streams in Ohio. The equations were developed from streamflow statistics and basin-characteristics data for as many as 219 active or discontinued streamflow-gaging stations on rural, unregulated streams in Ohio with 10 or more years of homogenous daily streamflow record. Streamflow statistics and basin-characteristics data for the 219 stations are presented in this report. Simple equations (based on drainage area only) and best-fit equations (based on drainage area and at least two other basin characteristics) were developed by means of ordinary least-squares regression techniques. Application of the best-fit equations generally involves quantification of basin characteristics that require or are facilitated by use of a geographic information system. In contrast, the simple equations can be used with information that can be obtained without use of a geographic information system; however, the simple equations have larger prediction errors than the best-fit equations and exhibit geographic biases for most streamflow statistics. The best-fit equations should be used instead of the simple equations whenever possible.

Anticancer Agents Based on a New Class of Transition- State Analog Inhibitors for Serine and Cysteine Proteases

DTIC Science & Technology

1999-08-01

electrostatic repulsion between the het- eroatom and the ketone. Swain and Lupton31 have constructed a modified Hammett equation (eq 2) in which they...determined by nonlinear fit to the Michaelis-Menten equation for competitive inhibition using simple weighing. Competitive inhibition was confirmed... equation for competitive inhibition using simple weighing. Competitive inhibition was confirmed by Lineweaver - Burk analysis using simple

Linear stability analysis of the three-dimensional thermally-driven ocean circulation: application to interdecadal oscillations

NASA Astrophysics Data System (ADS)

Huck, Thierry; Vallis, Geoffrey K.

2001-08-01

What can we learn from performing a linear stability analysis of the large-scale ocean circulation? Can we predict from the basic state the occurrence of interdecadal oscillations, such as might be found in a forward integration of the full equations of motion? If so, do the structure and period of the linearly unstable modes resemble those found in a forward integration? We pursue here a preliminary study of these questions for a case in idealized geometry, in which the full nonlinear behavior can also be explored through forward integrations. Specifically, we perform a three-dimensional linear stability analysis of the thermally-driven circulation of the planetary geostrophic equations. We examine the resulting eigenvalues and eigenfunctions, comparing them with the structure of the interdecadal oscillations found in the fully nonlinear model in various parameter regimes. We obtain a steady state by running the time-dependent, nonlinear model to equilibrium using restoring boundary conditions on surface temperature. If the surface heat fluxes are then diagnosed, and these values applied as constant flux boundary conditions, the nonlinear model switches into a state of perpetual, finite amplitude, interdecadal oscillations. We construct a linearized version of the model by empirically evaluating the tangent linear matrix at the steady state, under both restoring and constant-flux boundary conditions. An eigen-analysis shows there are no unstable eigenmodes of the linearized model with restoring conditions. In contrast, under constant flux conditions, we find a single unstable eigenmode that shows a striking resemblance to the fully-developed oscillations in terms of three-dimensional structure, period and growth rate. The mode may be damped through either surface restoring boundary conditions or sufficiently large horizontal tracer diffusion. The success of this simple numerical method in idealized geometry suggests applications in the study of the stability of the ocean circulation in more realistic configurations, and the possibility of predicting potential oceanic modes, even weakly damped, that might be excited by stochastic atmospheric forcing or mesoscale ocean eddies.

A Few New 2+1-Dimensional Nonlinear Dynamics and the Representation of Riemann Curvature Tensors

NASA Astrophysics Data System (ADS)

Wang, Yan; Zhang, Yufeng; Zhang, Xiangzhi

2016-09-01

We first introduced a linear stationary equation with a quadratic operator in ∂x and ∂y, then a linear evolution equation is given by N-order polynomials of eigenfunctions. As applications, by taking N=2, we derived a (2+1)-dimensional generalized linear heat equation with two constant parameters associative with a symmetric space. When taking N=3, a pair of generalized Kadomtsev-Petviashvili equations with the same eigenvalues with the case of N=2 are generated. Similarly, a second-order flow associative with a homogeneous space is derived from the integrability condition of the two linear equations, which is a (2+1)-dimensional hyperbolic equation. When N=3, the third second flow associative with the homogeneous space is generated, which is a pair of new generalized Kadomtsev-Petviashvili equations. Finally, as an application of a Hermitian symmetric space, we established a pair of spectral problems to obtain a new (2+1)-dimensional generalized Schrödinger equation, which is expressed by the Riemann curvature tensors.

Solution of Volterra and Fredholm Classes of Equations via Triangular Orthogonal Function (A Combination of Right Hand Triangular Function and Left Hand Triangular Function) and Hybrid Orthogonal Function (A Combination of Sample Hold Function and Right Hand Triangular Function)

NASA Astrophysics Data System (ADS)

Mukhopadhyay, Anirban; Ganguly, Anindita; Chatterjee, Saumya Deep

2018-04-01

In this paper the authors have dealt with seven kinds of non-linear Volterra and Fredholm classes of equations. The authors have formulated an algorithm for solving the aforementioned equation types via Hybrid Function (HF) and Triangular Function (TF) piecewise-linear orthogonal approach. In this approach the authors have reduced integral equation or integro-differential equation into equivalent system of simultaneous non-linear equation and have employed either Newton's method or Broyden's method to solve the simultaneous non-linear equations. The authors have calculated the L2-norm error and the max-norm error for both HF and TF method for each kind of equations. Through the illustrated examples, the authors have shown that the HF based algorithm produces stable result, on the contrary TF-computational method yields either stable, anomalous or unstable results.

Linearized-moment analysis of the temperature jump and temperature defect in the Knudsen layer of a rarefied gas.

PubMed

Gu, Xiao-Jun; Emerson, David R

2014-06-01

Understanding the thermal behavior of a rarefied gas remains a fundamental problem. In the present study, we investigate the predictive capabilities of the regularized 13 and 26 moment equations. In this paper, we consider low-speed problems with small gradients, and to simplify the analysis, a linearized set of moment equations is derived to explore a classic temperature problem. Analytical solutions obtained for the linearized 26 moment equations are compared with available kinetic models and can reliably capture all qualitative trends for the temperature-jump coefficient and the associated temperature defect in the thermal Knudsen layer. In contrast, the linearized 13 moment equations lack the necessary physics to capture these effects and consistently underpredict kinetic theory. The deviation from kinetic theory for the 13 moment equations increases significantly for specular reflection of gas molecules, whereas the 26 moment equations compare well with results from kinetic theory. To improve engineering analyses, expressions for the effective thermal conductivity and Prandtl number in the Knudsen layer are derived with the linearized 26 moment equations.

Light propagation and the distance-redshift relation in a realistic inhomogeneous universe

NASA Technical Reports Server (NTRS)

Futamase, Toshifumi; Sasaki, Misao

1989-01-01

The propagation of light rays in a clumpy universe constructed by cosmological version of the post-Newtonian approximation was investigated. It is shown that linear approximation to the propagation equations is valid in the region where zeta is approximately less than 1 even if the density contrast is much larger than unity. Based on a gerneral order-of-magnitude statistical consideration, it is argued that the linear approximation is still valid where zeta is approximately greater than 1. A general formula for the distance-redshift relation in a clumpy universe is given. An explicit expression is derived for a simplified situation in which the effect of the gravitational potential of inhomogeneities dominates. In the light of the derived relation, the validity of the Dyer-Roeder distance is discussed. Also, statistical properties of light rays are investigated for a simple model of an inhomogeneous universe. The result of this example supports the validity of the linear approximation.

Linear Scaling of the Exciton Binding Energy versus the Band Gap of Two-Dimensional Materials

NASA Astrophysics Data System (ADS)

Choi, Jin-Ho; Cui, Ping; Lan, Haiping; Zhang, Zhenyu

2015-08-01

The exciton is one of the most crucial physical entities in the performance of optoelectronic and photonic devices, and widely varying exciton binding energies have been reported in different classes of materials. Using first-principles calculations within the G W -Bethe-Salpeter equation approach, here we investigate the excitonic properties of two recently discovered layered materials: phosphorene and graphene fluoride. We first confirm large exciton binding energies of, respectively, 0.85 and 2.03 eV in these systems. Next, by comparing these systems with several other representative two-dimensional materials, we discover a striking linear relationship between the exciton binding energy and the band gap and interpret the existence of the linear scaling law within a simple hydrogenic picture. The broad applicability of this novel scaling law is further demonstrated by using strained graphene fluoride. These findings are expected to stimulate related studies in higher and lower dimensions, potentially resulting in a deeper understanding of excitonic effects in materials of all dimensionalities.

An Efficient Spectral Method for Ordinary Differential Equations with Rational Function Coefficients

NASA Technical Reports Server (NTRS)

Coutsias, Evangelos A.; Torres, David; Hagstrom, Thomas

1994-01-01

We present some relations that allow the efficient approximate inversion of linear differential operators with rational function coefficients. We employ expansions in terms of a large class of orthogonal polynomial families, including all the classical orthogonal polynomials. These families obey a simple three-term recurrence relation for differentiation, which implies that on an appropriately restricted domain the differentiation operator has a unique banded inverse. The inverse is an integration operator for the family, and it is simply the tridiagonal coefficient matrix for the recurrence. Since in these families convolution operators (i.e. matrix representations of multiplication by a function) are banded for polynomials, we are able to obtain a banded representation for linear differential operators with rational coefficients. This leads to a method of solution of initial or boundary value problems that, besides having an operation count that scales linearly with the order of truncation N, is computationally well conditioned. Among the applications considered is the use of rational maps for the resolution of sharp interior layers.

A Simple and Specific Stability- Indicating RP-HPLC Method for Routine Assay of Adefovir Dipivoxil in Bulk and Tablet Dosage Form.

PubMed

Darsazan, Bahar; Shafaati, Alireza; Mortazavi, Seyed Alireza; Zarghi, Afshin

2017-01-01

A simple and reliable stability-indicating RP-HPLC method was developed and validated for analysis of adefovir dipivoxil (ADV).The chromatographic separation was performed on a C 18 column using a mixture of acetonitrile-citrate buffer (10 mM at pH 5.2) 36:64 (%v/v) as mobile phase, at a flow rate of 1.5 mL/min. Detection was carried out at 260 nm and a sharp peak was obtained for ADV at a retention time of 5.8 ± 0.01 min. No interferences were observed from its stress degradation products. The method was validated according to the international guidelines. Linear regression analysis of data for the calibration plot showed a linear relationship between peak area and concentration over the range of 0.5-16 μg/mL; the regression coefficient was 0.9999and the linear regression equation was y = 24844x-2941.3. The detection (LOD) and quantification (LOQ) limits were 0.12 and 0.35 μg/mL, respectively. The results proved the method was fast (analysis time less than 7 min), precise, reproducible, and accurate for analysis of ADV over a wide range of concentration. The proposed specific method was used for routine quantification of ADV in pharmaceutical bulk and a tablet dosage form.

Encouraging Students to Think Strategically when Learning to Solve Linear Equations

ERIC Educational Resources Information Center

Robson, Daphne; Abell, Walt; Boustead, Therese

2012-01-01

Students who are preparing to study science and engineering need to understand equation solving but adult students returning to study can find this difficult. In this paper, the design of an online resource, Equations2go, for helping students learn to solve linear equations is investigated. Students learning to solve equations need to consider…

FAST TRACK COMMUNICATION Quasi self-adjoint nonlinear wave equations

NASA Astrophysics Data System (ADS)

Ibragimov, N. H.; Torrisi, M.; Tracinà, R.

2010-11-01

In this paper we generalize the classification of self-adjoint second-order linear partial differential equation to a family of nonlinear wave equations with two independent variables. We find a class of quasi self-adjoint nonlinear equations which includes the self-adjoint linear equations as a particular case. The property of a differential equation to be quasi self-adjoint is important, e.g. for constructing conservation laws associated with symmetries of the differential equation.

ADM For Solving Linear Second-Order Fredholm Integro-Differential Equations

NASA Astrophysics Data System (ADS)

Karim, Mohd F.; Mohamad, Mahathir; Saifullah Rusiman, Mohd; Che-Him, Norziha; Roslan, Rozaini; Khalid, Kamil

2018-04-01

In this paper, we apply Adomian Decomposition Method (ADM) as numerically analyse linear second-order Fredholm Integro-differential Equations. The approximate solutions of the problems are calculated by Maple package. Some numerical examples have been considered to illustrate the ADM for solving this equation. The results are compared with the existing exact solution. Thus, the Adomian decomposition method can be the best alternative method for solving linear second-order Fredholm Integro-Differential equation. It converges to the exact solution quickly and in the same time reduces computational work for solving the equation. The result obtained by ADM shows the ability and efficiency for solving these equations.

Differential-Drive Mobile Robot Control Design based-on Linear Feedback Control Law

NASA Astrophysics Data System (ADS)

Nurmaini, Siti; Dewi, Kemala; Tutuko, Bambang

2017-04-01

This paper deals with the problem of how to control differential driven mobile robot with simple control law. When mobile robot moves from one position to another to achieve a position destination, it always produce some errors. Therefore, a mobile robot requires a certain control law to drive the robot’s movement to the position destination with a smallest possible error. In this paper, in order to reduce position error, a linear feedback control is proposed with pole placement approach to regulate the polynoms desired. The presented work leads to an improved understanding of differential-drive mobile robot (DDMR)-based kinematics equation, which will assist to design of suitable controllers for DDMR movement. The result show by using the linier feedback control method with pole placement approach the position error is reduced and fast convergence is achieved.

Improved Convergence and Robustness of USM3D Solutions on Mixed Element Grids (Invited)

NASA Technical Reports Server (NTRS)

Pandya, Mohagna J.; Diskin, Boris; Thomas, James L.; Frink, Neal T.

2015-01-01

Several improvements to the mixed-element USM3D discretization and defect-correction schemes have been made. A new methodology for nonlinear iterations, called the Hierarchical Adaptive Nonlinear Iteration Scheme (HANIS), has been developed and implemented. It provides two additional hierarchies around a simple and approximate preconditioner of USM3D. The hierarchies are a matrix-free linear solver for the exact linearization of Reynolds-averaged Navier Stokes (RANS) equations and a nonlinear control of the solution update. Two variants of the new methodology are assessed on four benchmark cases, namely, a zero-pressure gradient flat plate, a bump-in-channel configuration, the NACA 0012 airfoil, and a NASA Common Research Model configuration. The new methodology provides a convergence acceleration factor of 1.4 to 13 over the baseline solver technology.

Approximating the linear quadratic optimal control law for hereditary systems with delays in the control

NASA Technical Reports Server (NTRS)

Milman, Mark H.

1987-01-01

The fundamental control synthesis issue of establishing a priori convergence rates of approximation schemes for feedback controllers for a class of distributed parameter systems is addressed within the context of hereditary systems. Specifically, a factorization approach is presented for deriving approximations to the optimal feedback gains for the linear regulator-quadratic cost problem associated with time-varying functional differential equations with control delays. The approach is based on a discretization of the state penalty which leads to a simple structure for the feedback control law. General properties of the Volterra factors of Hilbert-Schmidt operators are then used to obtain convergence results for the controls, trajectories and feedback kernels. Two algorithms are derived from the basic approximation scheme, including a fast algorithm, in the time-invariant case. A numerical example is also considered.

Order Selection for General Expression of Nonlinear Autoregressive Model Based on Multivariate Stepwise Regression

NASA Astrophysics Data System (ADS)

Shi, Jinfei; Zhu, Songqing; Chen, Ruwen

2017-12-01

An order selection method based on multiple stepwise regressions is proposed for General Expression of Nonlinear Autoregressive model which converts the model order problem into the variable selection of multiple linear regression equation. The partial autocorrelation function is adopted to define the linear term in GNAR model. The result is set as the initial model, and then the nonlinear terms are introduced gradually. Statistics are chosen to study the improvements of both the new introduced and originally existed variables for the model characteristics, which are adopted to determine the model variables to retain or eliminate. So the optimal model is obtained through data fitting effect measurement or significance test. The simulation and classic time-series data experiment results show that the method proposed is simple, reliable and can be applied to practical engineering.

Approximating the linear quadratic optimal control law for hereditary systems with delays in the control

NASA Technical Reports Server (NTRS)

Milman, Mark H.

1988-01-01

The fundamental control synthesis issue of establishing a priori convergence rates of approximation schemes for feedback controllers for a class of distributed parameter systems is addressed within the context of hereditary schemes. Specifically, a factorization approach is presented for deriving approximations to the optimal feedback gains for the linear regulator-quadratic cost problem associated with time-varying functional differential equations with control delays. The approach is based on a discretization of the state penalty which leads to a simple structure for the feedback control law. General properties of the Volterra factors of Hilbert-Schmidt operators are then used to obtain convergence results for the controls, trajectories and feedback kernels. Two algorithms are derived from the basic approximation scheme, including a fast algorithm, in the time-invariant case. A numerical example is also considered.

S{sub 2}SA preconditioning for the S{sub n} equations with strictly non negative spatial discretization

DOE Office of Scientific and Technical Information (OSTI.GOV)

Bruss, D. E.; Morel, J. E.; Ragusa, J. C.

2013-07-01

Preconditioners based upon sweeps and diffusion-synthetic acceleration have been constructed and applied to the zeroth and first spatial moments of the 1-D S{sub n} transport equation using a strictly non negative nonlinear spatial closure. Linear and nonlinear preconditioners have been analyzed. The effectiveness of various combinations of these preconditioners are compared. In one dimension, nonlinear sweep preconditioning is shown to be superior to linear sweep preconditioning, and DSA preconditioning using nonlinear sweeps in conjunction with a linear diffusion equation is found to be essentially equivalent to nonlinear sweeps in conjunction with a nonlinear diffusion equation. The ability to use amore » linear diffusion equation has important implications for preconditioning the S{sub n} equations with a strictly non negative spatial discretization in multiple dimensions. (authors)« less

Time and frequency domain analysis of sampled data controllers via mixed operation equations

NASA Technical Reports Server (NTRS)

Frisch, H. P.

1981-01-01

Specification of the mathematical equations required to define the dynamic response of a linear continuous plant, subject to sampled data control, is complicated by the fact that the digital components of the control system cannot be modeled via linear ordinary differential equations. This complication can be overcome by introducing two new mathematical operations; namely, the operation of zero order hold and digial delay. It is shown that by direct utilization of these operations, a set of linear mixed operation equations can be written and used to define the dynamic response characteristics of the controlled system. It also is shown how these linear mixed operation equations lead, in an automatable manner, directly to a set of finite difference equations which are in a format compatible with follow on time and frequency domain analysis methods.

Quantum-field-theoretical approach to phase-space techniques: Generalizing the positive-P representation

NASA Astrophysics Data System (ADS)

Plimak, L. I.; Fleischhauer, M.; Olsen, M. K.; Collett, M. J.

2003-01-01

We present an introduction to phase-space techniques (PST) based on a quantum-field-theoretical (QFT) approach. In addition to bridging the gap between PST and QFT, our approach results in a number of generalizations of the PST. First, for problems where the usual PST do not result in a genuine Fokker-Planck equation (even after phase-space doubling) and hence fail to produce a stochastic differential equation (SDE), we show how the system in question may be approximated via stochastic difference equations (SΔE). Second, we show that introducing sources into the SDE’s (or SΔE’s) generalizes them to a full quantum nonlinear stochastic response problem (thus generalizing Kubo’s linear reaction theory to a quantum nonlinear stochastic response theory). Third, we establish general relations linking quantum response properties of the system in question to averages of operator products ordered in a way different from time normal. This extends PST to a much wider assemblage of operator products than are usually considered in phase-space approaches. In all cases, our approach yields a very simple and straightforward way of deriving stochastic equations in phase space.

The Dynamic Characteristic and Hysteresis Effect of an Air Spring

NASA Astrophysics Data System (ADS)

Löcken, F.; Welsch, M.

2015-02-01

In many applications of vibration technology, especially in chassis, air springs present a common alternative to steel spring concepts. A design-independent and therefore universal approach is presented to describe the dynamic characteristic of such springs. Differential and constitutive equations based on energy balances of the enclosed volume and the mountings are given to describe the nonlinear and dynamic characteristics. Therefore all parameters can be estimated directly from physical and geometrical properties, without parameter fitting. The numerically solved equations fit very well to measurements of a passenger car air spring. In a second step a simplification of this model leads to a pure mechanical equation. While in principle the same parameters are used, just an empirical correction of the effective heat transfer coefficient is needed to handle some simplification on this topic. Finally, a linearization of this equation leads to an analogous mechanical model that can be assembled from two common spring- and one dashpot elements in a specific arrangement. This transfer into "mechanical language" enables a system description with a simple force-displacement law and a consideration of the nonobvious hysteresis and stiffness increase of an air spring from a mechanical point of view.

PsiQuaSP-A library for efficient computation of symmetric open quantum systems.

PubMed

Gegg, Michael; Richter, Marten

2017-11-24

In a recent publication we showed that permutation symmetry reduces the numerical complexity of Lindblad quantum master equations for identical multi-level systems from exponential to polynomial scaling. This is important for open system dynamics including realistic system bath interactions and dephasing in, for instance, the Dicke model, multi-Λ system setups etc. Here we present an object-oriented C++ library that allows to setup and solve arbitrary quantum optical Lindblad master equations, especially those that are permutationally symmetric in the multi-level systems. PsiQuaSP (Permutation symmetry for identical Quantum Systems Package) uses the PETSc package for sparse linear algebra methods and differential equations as basis. The aim of PsiQuaSP is to provide flexible, storage efficient and scalable code while being as user friendly as possible. It is easily applied to many quantum optical or quantum information systems with more than one multi-level system. We first review the basics of the permutation symmetry for multi-level systems in quantum master equations. The application of PsiQuaSP to quantum dynamical problems is illustrated with several typical, simple examples of open quantum optical systems.

Simple equation for calculation of plasma clearance for evaluation of renal function without urine collection in rats.

PubMed

Liu, Xiang; Peng, Dejun; Tian, Hao; Lu, Chengyu

2017-01-01

To develop an equation for the evaluation of renal function in rats using three dilutions of plasma samples and to validate this method by comparison with a reference method. The investigation was conducted in Sprague-Dawley (SD) rats after delivery of three doses of iohexol, with blood samples collected before and after dosage using a quantitative blood collection method. Plasma iohexol concentrations were detected by high performance liquid chromatography (HPLC). The extraction recovery of iohexol from plasma was >97.30% and the calibration curve was linear (r 2  = 0.9997) over iohexol concentrations ranging from 10 to 1000 µg/mL. The method had an RE of <9.310 and intra- and inter-day RSD of <5.137% and <3.693%, respectively. The plasma clearance values obtained from the equation correlated closely (r = 0.763) with those obtained using the reference method. The relatively correlation in the results obtained using the method under investigation and the reference method indicate that this new equation can be used for preliminary assessment of renal function in rats. © 2016 Asian Pacific Society of Nephrology.

Linear Equating for the NEAT Design: Parameter Substitution Models and Chained Linear Relationship Models

ERIC Educational Resources Information Center

Kane, Michael T.; Mroch, Andrew A.; Suh, Youngsuk; Ripkey, Douglas R.

2009-01-01

This paper analyzes five linear equating models for the "nonequivalent groups with anchor test" (NEAT) design with internal anchors (i.e., the anchor test is part of the full test). The analysis employs a two-dimensional framework. The first dimension contrasts two general approaches to developing the equating relationship. Under a "parameter…

Supporting Students' Understanding of Linear Equations with One Variable Using Algebra Tiles

ERIC Educational Resources Information Center

Saraswati, Sari; Putri, Ratu Ilma Indra; Somakim

2016-01-01

This research aimed to describe how algebra tiles can support students' understanding of linear equations with one variable. This article is a part of a larger research on learning design of linear equations with one variable using algebra tiles combined with balancing method. Therefore, it will merely discuss one activity focused on how students…

Finite difference modelling of the temperature rise in non-linear medical ultrasound fields.

PubMed

Divall, S A; Humphrey, V F

2000-03-01

Non-linear propagation of ultrasound can lead to increased heat generation in medical diagnostic imaging due to the preferential absorption of harmonics of the original frequency. A numerical model has been developed and tested that is capable of predicting the temperature rise due to a high amplitude ultrasound field. The acoustic field is modelled using a numerical solution to the Khokhlov-Zabolotskaya-Kuznetsov (KZK) equation, known as the Bergen Code, which is implemented in cylindrical symmetric form. A finite difference representation of the thermal equations is used to calculate the resulting temperature rises. The model allows for the inclusion of a number of layers of tissue with different acoustic and thermal properties and accounts for the effects of non-linear propagation, direct heating by the transducer, thermal diffusion and perfusion in different tissues. The effect of temperature-dependent skin perfusion and variation in background temperature between the skin and deeper layers of the body are included. The model has been tested against analytic solutions for simple configurations and then used to estimate temperature rises in realistic obstetric situations. A pulsed 3 MHz transducer operating with an average acoustic power of 200 mW leads to a maximum steady state temperature rise inside the foetus of 1.25 degrees C compared with a 0.6 degree C rise for the same transmitted power under linear propagation conditions. The largest temperature rise occurs at the skin surface, with the temperature rise at the foetus limited to less than 2 degrees C for the range of conditions considered.

A new approach for modeling gravitational radiation from the inspiral of two neutron stars

NASA Astrophysics Data System (ADS)

Luke, Stephen A.

In this dissertation, a new method of applying the ADM formalism of general relativity to model the gravitational radiation emitted from the realistic inspiral of a neutron star binary is described. A description of the conformally flat condition (CFC) is summarized, and the ADM equations are solved by use of the CFC approach for a neutron star binary. The advantages and limitations of this approach are discussed, and the need for a more accurate improvement to this approach is described. To address this need, a linearized perturbation of the CFC spatial three metric is then introduced. The general relativistic hydrodynamic equations are then allowed to evolve against this basis under the assumption that the first-order corrections to the hydrodynamic variables are negligible compared to their CFC values. As a first approximation, the linear corrections to the conformal factor, lapse function, and shift vector are also assumed to be small compared to the extrinsic curvature and the three metric. A boundary matching method is then introduced as a way of computing the gravitational radiation of this relativistic system without use of the multipole expansion as employed by earlier applications of the CFC approach. It is assumed that at a location far from the source, the three metric is accurately described by a linear correction to Minkowski spacetime. The two polarizations of gravitational radiation can then be computed at that point in terms of the linearized correction to the metric. The evolution equations obtained from the linearized perturbative correction to the CFC approach and the method for recovery of the gravity wave signal are then tested by use of a three-dimensional numerical simulation. This code is used to compute the gravity wave signal emitted a pair of equal mass neutron stars in quasi-stable circular orbits at a point early in their inspiral phase. From this simple numerical analysis, the correct general trend of gravitational radiation is recovered. Comparisons with (5/2) post-Newtonian solutions show a similar gravitational waveform, although inaccuracies are still found to exist from this computation. Finally, several areas for improvement and potential future applications of this technique are discussed.

Approximate controllability of a system of parabolic equations with delay

NASA Astrophysics Data System (ADS)

Carrasco, Alexander; Leiva, Hugo

2008-09-01

In this paper we give necessary and sufficient conditions for the approximate controllability of the following system of parabolic equations with delay: where [Omega] is a bounded domain in , D is an n×n nondiagonal matrix whose eigenvalues are semi-simple with nonnegative real part, the control and B[set membership, variant]L(U,Z) with , . The standard notation zt(x) defines a function from [-[tau],0] to (with x fixed) by zt(x)(s)=z(t+s,x), -[tau][less-than-or-equals, slant]s[less-than-or-equals, slant]0. Here [tau][greater-or-equal, slanted]0 is the maximum delay, which is supposed to be finite. We assume that the operator is linear and bounded, and [phi]0[set membership, variant]Z, [phi][set membership, variant]L2([-[tau],0];Z). To this end: First, we reformulate this system into a standard first-order delay equation. Secondly, the semigroup associated with the first-order delay equation on an appropriate product space is expressed as a series of strongly continuous semigroups and orthogonal projections related with the eigenvalues of the Laplacian operator (); this representation allows us to reduce the controllability of this partial differential equation with delay to a family of ordinary delay equations. Finally, we use the well-known result on the rank condition for the approximate controllability of delay system to derive our main result.

LINEAR - DERIVATION AND DEFINITION OF A LINEAR AIRCRAFT MODEL

NASA Technical Reports Server (NTRS)

Duke, E. L.

1994-01-01

The Derivation and Definition of a Linear Model program, LINEAR, provides the user with a powerful and flexible tool for the linearization of aircraft aerodynamic models. LINEAR was developed to provide a standard, documented, and verified tool to derive linear models for aircraft stability analysis and control law design. Linear system models define the aircraft system in the neighborhood of an analysis point and are determined by the linearization of the nonlinear equations defining vehicle dynamics and sensors. LINEAR numerically determines a linear system model using nonlinear equations of motion and a user supplied linear or nonlinear aerodynamic model. The nonlinear equations of motion used are six-degree-of-freedom equations with stationary atmosphere and flat, nonrotating earth assumptions. LINEAR is capable of extracting both linearized engine effects, such as net thrust, torque, and gyroscopic effects and including these effects in the linear system model. The point at which this linear model is defined is determined either by completely specifying the state and control variables, or by specifying an analysis point on a trajectory and directing the program to determine the control variables and the remaining state variables. The system model determined by LINEAR consists of matrices for both the state and observation equations. The program has been designed to provide easy selection of state, control, and observation variables to be used in a particular model. Thus, the order of the system model is completely under user control. Further, the program provides the flexibility of allowing alternate formulations of both the state and observation equations. Data describing the aircraft and the test case is input to the program through a terminal or formatted data files. All data can be modified interactively from case to case. The aerodynamic model can be defined in two ways: a set of nondimensional stability and control derivatives for the flight point of interest, or a full non-linear aerodynamic model as used in simulations. LINEAR is written in FORTRAN and has been implemented on a DEC VAX computer operating under VMS with a virtual memory requirement of approximately 296K of 8 bit bytes. Both an interactive and batch version are included. LINEAR was developed in 1988.

Optical soliton solutions of the cubic-quintic non-linear Schrödinger's equation including an anti-cubic term

NASA Astrophysics Data System (ADS)

Kaplan, Melike; Hosseini, Kamyar; Samadani, Farzan; Raza, Nauman

2018-07-01

A wide range of problems in different fields of the applied sciences especially non-linear optics is described by non-linear Schrödinger's equations (NLSEs). In the present paper, a specific type of NLSEs known as the cubic-quintic non-linear Schrödinger's equation including an anti-cubic term has been studied. The generalized Kudryashov method along with symbolic computation package has been exerted to carry out this objective. As a consequence, a series of optical soliton solutions have formally been retrieved. It is corroborated that the generalized form of Kudryashov method is a direct, effectual, and reliable technique to deal with various types of non-linear Schrödinger's equations.

Prediction of Undsteady Flows in Turbomachinery Using the Linearized Euler Equations on Deforming Grids

NASA Technical Reports Server (NTRS)

Clark, William S.; Hall, Kenneth C.

1994-01-01

A linearized Euler solver for calculating unsteady flows in turbomachinery blade rows due to both incident gusts and blade motion is presented. The model accounts for blade loading, blade geometry, shock motion, and wake motion. Assuming that the unsteadiness in the flow is small relative to the nonlinear mean solution, the unsteady Euler equations can be linearized about the mean flow. This yields a set of linear variable coefficient equations that describe the small amplitude harmonic motion of the fluid. These linear equations are then discretized on a computational grid and solved using standard numerical techniques. For transonic flows, however, one must use a linear discretization which is a conservative linearization of the non-linear discretized Euler equations to ensure that shock impulse loads are accurately captured. Other important features of this analysis include a continuously deforming grid which eliminates extrapolation errors and hence, increases accuracy, and a new numerically exact, nonreflecting far-field boundary condition treatment based on an eigenanalysis of the discretized equations. Computational results are presented which demonstrate the computational accuracy and efficiency of the method and demonstrate the effectiveness of the deforming grid, far-field nonreflecting boundary conditions, and shock capturing techniques. A comparison of the present unsteady flow predictions to other numerical, semi-analytical, and experimental methods shows excellent agreement. In addition, the linearized Euler method presented requires one or two orders-of-magnitude less computational time than traditional time marching techniques making the present method a viable design tool for aeroelastic analyses.

Effect of cholesterol and triglycerides levels on the rheological behavior of human blood

NASA Astrophysics Data System (ADS)

Moreno, Leonardo; Calderas, Fausto; Sanchez-Olivares, Guadalupe; Medina-Torres, Luis; Sanchez-Solis, Antonio; Manero, Octavio

2015-02-01

Important public health problems worldwide such as obesity, diabetes, hyperlipidemia and coronary diseases are quite common. These problems arise from numerous factors, such as hyper-caloric diets, sedentary habits and other epigenetic factors. With respect to Mexico, the population reference values of total cholesterol in plasma are around 200 mg/dL. However, a large proportion has higher levels than this reference value. In this work, we analyze the rheological properties of human blood obtained from 20 donors, as a function of cholesterol and triglyceride levels, upon a protocol previously approved by the health authorities. Samples with high and low cholesterol and triglyceride levels were selected and analyzed by simple-continuous and linear-oscillatory shear flow. Rheometric properties were measured and related to the structure and composition of human blood. In addition, rheometric data were modeled by using several constitutive equations: Bautista-Manero-Puig (BMP) and the multimodal Maxwell equations to predict the flow behavior of human blood. Finally, a comparison was made among various models, namely, the BMP, Carreau and Quemada equations for simple shear rate flow. An important relationship was found between cholesterol, triglycerides and the structure of human blood. Results show that blood with high cholesterol levels (400 mg/dL) has flow properties fully different (higher viscosity and a more pseudo-plastic behavior) than blood with lower levels of cholesterol (tendency to Newtonian behavior or viscosity plateau at low shear rates).

Discrete integration of continuous Kalman filtering equations for time invariant second-order structural systems

NASA Technical Reports Server (NTRS)

Park, K. C.; Belvin, W. Keith

1990-01-01

A general form for the first-order representation of the continuous second-order linear structural-dynamics equations is introduced to derive a corresponding form of first-order continuous Kalman filtering equations. Time integration of the resulting equations is carried out via a set of linear multistep integration formulas. It is shown that a judicious combined selection of computational paths and the undetermined matrices introduced in the general form of the first-order linear structural systems leads to a class of second-order discrete Kalman filtering equations involving only symmetric sparse N x N solution matrices.

Application of variational and Galerkin equations to linear and nonlinear finite element analysis

NASA Technical Reports Server (NTRS)

Yu, Y.-Y.

1974-01-01

The paper discusses the application of the variational equation to nonlinear finite element analysis. The problem of beam vibration with large deflection is considered. The variational equation is shown to be flexible in both the solution of a general problem and in the finite element formulation. Difficulties are shown to arise when Galerkin's equations are used in the consideration of the finite element formulation of two-dimensional linear elasticity and of the linear classical beam.

A Comprehensive Analytical Model of Rotorcraft Aerodynamics and Dynamics. Part 1. Analysis Development

DTIC Science & Technology

1980-06-01

sufficient. Dropping the time lag terms, the equations for Xu, Xx’, and X reduce to linear algebraic equations.Y Hence in the quasistatic case the...quasistatic variables now are not described by differential equations but rather by linear algebraic equations. The solution for x0 then is simply -365...matrices for two-bladed rotor 414 7. LINEAR SYSTEM ANALYSIS 425 7,1 State Variable Form 425 7.2 Constant Coefficient System 426 7.2. 1 Eigen-analysis 426

An approach to predict the shape-memory behavior of amorphous polymers from Dynamic Mechanical Analysis (DMA) data

NASA Astrophysics Data System (ADS)

Kuki, Ákos; Czifrák, Katalin; Karger-Kocsis, József; Zsuga, Miklós; Kéki, Sándor

2015-02-01

The prediction of shape-memory behavior is essential regarding the design of a smart material for different applications. This paper proposes a simple and quick method for the prediction of shape-memory behavior of amorphous shape memory polymers (SMPs) on the basis of a single dynamic mechanical analysis (DMA) temperature sweep at constant frequency. All the parameters of the constitutive equations for linear viscoelasticity are obtained by fitting the DMA curves. The change with the temperature of the time-temperature superposition shift factor ( a T ) is expressed by the Williams-Landel-Ferry (WLF) model near and above the glass transition temperature ( T g ), and by the Arrhenius law below T g . The constants of the WLF and Arrhenius equations can also be determined. The results of our calculations agree satisfactorily with the experimental free recovery curves from shape-memory tests.

Least-squares Minimization Approaches to Interpret Total Magnetic Anomalies Due to Spheres

NASA Astrophysics Data System (ADS)

Abdelrahman, E. M.; El-Araby, T. M.; Soliman, K. S.; Essa, K. S.; Abo-Ezz, E. R.

2007-05-01

We have developed three different least-squares approaches to determine successively: the depth, magnetic angle, and amplitude coefficient of a buried sphere from a total magnetic anomaly. By defining the anomaly value at the origin and the nearest zero-anomaly distance from the origin on the profile, the problem of depth determination is transformed into the problem of finding a solution of a nonlinear equation of the form f(z)=0. Knowing the depth and applying the least-squares method, the magnetic angle and amplitude coefficient are determined using two simple linear equations. In this way, the depth, magnetic angle, and amplitude coefficient are determined individually from all observed total magnetic data. The method is applied to synthetic examples with and without random errors and tested on a field example from Senegal, West Africa. In all cases, the depth solutions are in good agreement with the actual ones.

Subsonic panel method for designing wing surfaces from pressure distribution

NASA Technical Reports Server (NTRS)

Bristow, D. R.; Hawk, J. D.

1983-01-01

An iterative method has been developed for designing wing section contours corresponding to a prescribed subcritical distribution of pressure. The calculations are initialized by using a surface panel method to analyze a baseline wing or wing-fuselage configuration. A first-order expansion to the baseline panel method equations is then used to calculate a matrix containing the partial derivative of potential at each control point with respect to each unknown geometry parameter. In every iteration cycle, the matrix is used both to calculate the geometry perturbation and to analyze the perturbed geometry. The distribution of potential on the perturbed geometry is established by simple linear extrapolation from the baseline solution. The extrapolated potential is converted to pressure by Bernoulli's equation. Not only is the accuracy of the approach good for very large perturbations, but the computing cost of each complete iteration cycle is substantially less than one analysis solution by a conventional panel method.

Autonomous orbital navigation using Kepler's equation

NASA Technical Reports Server (NTRS)

Boltz, F. W.

1974-01-01

A simple method of determining the six elements of elliptic satellite orbits has been developed for use aboard manned and unmanned spacecraft orbiting the earth, moon, or any planet. The system requires the use of a horizon sensor or other device for determining the local vertical, a precision clock or timing device, and Apollo-type navigation equipment including an inertial measurement unit (IMU), a digital computer, and a coupling data unit. The three elements defining the in-plane motion are obtained from simultaneous measurements of central angle traversed around the planet and elapsed flight time using a linearization of Kepler's equation about a reference orbit. It is shown how Kalman filter theory may also be used to determine the in-plane orbital elements. The three elements defining the orbit orientation are obtained from position angles in celestial coordinates derived from the IMU with the spacecraft vertically oriented after alignment of the IMU to a known inertial coordinate frame.

A multi-dimensional nonlinearly implicit, electromagnetic Vlasov-Darwin particle-in-cell (PIC) algorithm

NASA Astrophysics Data System (ADS)

Chen, Guangye; Chacón, Luis; CoCoMans Team

2014-10-01

For decades, the Vlasov-Darwin model has been recognized to be attractive for PIC simulations (to avoid radiative noise issues) in non-radiative electromagnetic regimes. However, the Darwin model results in elliptic field equations that renders explicit time integration unconditionally unstable. Improving on linearly implicit schemes, fully implicit PIC algorithms for both electrostatic and electromagnetic regimes, with exact discrete energy and charge conservation properties, have been recently developed in 1D. This study builds on these recent algorithms to develop an implicit, orbit-averaged, time-space-centered finite difference scheme for the particle-field equations in multiple dimensions. The algorithm conserves energy, charge, and canonical-momentum exactly, even with grid packing. A simple fluid preconditioner allows efficient use of large timesteps, O (√{mi/me}c/veT) larger than the explicit CFL. We demonstrate the accuracy and efficiency properties of the of the algorithm with various numerical experiments in 2D3V.

Mathematical modeling of spinning elastic bodies for modal analysis.

NASA Technical Reports Server (NTRS)

Likins, P. W.; Barbera, F. J.; Baddeley, V.

1973-01-01

The problem of modal analysis of an elastic appendage on a rotating base is examined to establish the relative advantages of various mathematical models of elastic structures and to extract general inferences concerning the magnitude and character of the influence of spin on the natural frequencies and mode shapes of rotating structures. In realization of the first objective, it is concluded that except for a small class of very special cases the elastic continuum model is devoid of useful results, while for constant nominal spin rate the distributed-mass finite-element model is quite generally tractable, since in the latter case the governing equations are always linear, constant-coefficient, ordinary differential equations. Although with both of these alternatives the details of the formulation generally obscure the essence of the problem and permit very little engineering insight to be gained without extensive computation, this difficulty is not encountered when dealing with simple concentrated mass models.

Dynamical patterns and regime shifts in the nonlinear model of soil microorganisms growth

NASA Astrophysics Data System (ADS)

Zaitseva, Maria; Vladimirov, Artem; Winter, Anna-Marie; Vasilyeva, Nadezda

2017-04-01

Dynamical model of soil microorganisms growth and turnover is formulated as a system of nonlinear partial differential equations of reaction-diffusion type. We consider spatial distributions of concentrations of several substrates and microorganisms. Biochemical reactions are modelled by chemical kinetic equations. Transport is modelled by simple linear diffusion for all chemical substances, while for microorganisms we use different transport functions, e.g. some of them can actively move along gradient of substrate concentration, while others cannot move. We solve our model in two dimensions, starting from uniform state with small initial perturbations for various parameters and find parameter range, where small initial perturbations grow and evolve. We search for bifurcation points and critical regime shifts in our model and analyze time-space profile and phase portraits of these solutions approaching critical regime shifts in the system, exploring possibility to detect such shifts in advance. This work is supported by NordForsk, project #81513.

Verification of low-Mach number combustion codes using the method of manufactured solutions

NASA Astrophysics Data System (ADS)

Shunn, Lee; Ham, Frank; Knupp, Patrick; Moin, Parviz

2007-11-01

Many computational combustion models rely on tabulated constitutive relations to close the system of equations. As these reactive state-equations are typically multi-dimensional and highly non-linear, their implications on the convergence and accuracy of simulation codes are not well understood. In this presentation, the effects of tabulated state-relationships on the computational performance of low-Mach number combustion codes are explored using the method of manufactured solutions (MMS). Several MMS examples are developed and applied, progressing from simple one-dimensional configurations to problems involving higher dimensionality and solution-complexity. The manufactured solutions are implemented in two multi-physics hydrodynamics codes: CDP developed at Stanford University and FUEGO developed at Sandia National Laboratories. In addition to verifying the order-of-accuracy of the codes, the MMS problems help highlight certain robustness issues in existing variable-density flow-solvers. Strategies to overcome these issues are briefly discussed.

Nucleophilic Participation in the Solvolyses of (Arylthio)methyl Chlorides and Derivatives: Application of Simple and Extended Forms of the Grunwald-Winstein Equations

PubMed Central

Kevill, Dennis N.; Park, Young Hoon; Park, Byoung-Chun; D’Souza, Malcolm J.

2012-01-01

The specific rates of solvolysis of chloromethyl phenyl sulfide [(phenylthio)methyl chloride] and its p-chloro-derivative have been determined at 0.0 °C in a wide range of hydroxylic solvents, including several containing a fluroalcohol. Treatment in terms of a two-term Grunwald-Winstein equation, incorporating terms based on solvent ionizing power (YCl) and solvent nucleophilicity (NT) suggest a mechanism similar to that for the solvolyses of tert-butyl chloride, involving in the rate-determining step a nucleophilic solvation of the incipient carbocation in an ionization process. A previous suggestion, that a third-term governed by the aromatic ring parameter (I) is required, is shown both for the new and for the previously studied related substrates to be an artifact, resulting from an appreciable degree of multicollinearity between I values and a linear combination of NT and YCl values. PMID:22711999

On a multigrid method for the coupled Stokes and porous media flow problem

NASA Astrophysics Data System (ADS)

Luo, P.; Rodrigo, C.; Gaspar, F. J.; Oosterlee, C. W.

2017-07-01

The multigrid solution of coupled porous media and Stokes flow problems is considered. The Darcy equation as the saturated porous medium model is coupled to the Stokes equations by means of appropriate interface conditions. We focus on an efficient multigrid solution technique for the coupled problem, which is discretized by finite volumes on staggered grids, giving rise to a saddle point linear system. Special treatment is required regarding the discretization at the interface. An Uzawa smoother is employed in multigrid, which is a decoupled procedure based on symmetric Gauss-Seidel smoothing for velocity components and a simple Richardson iteration for the pressure field. Since a relaxation parameter is part of a Richardson iteration, Local Fourier Analysis (LFA) is applied to determine the optimal parameters. Highly satisfactory multigrid convergence is reported, and, moreover, the algorithm performs well for small values of the hydraulic conductivity and fluid viscosity, that are relevant for applications.

SCBUCKLE user's manual: Buckling analysis program for simple supported and clamped panels

NASA Technical Reports Server (NTRS)

Cruz, Juan R.

1993-01-01

The program SCBUCKLE calculates the buckling loads and mode shapes of cylindrically curved, rectangular panels. The panel is assumed to have no imperfections. SCBUCKLE is capable of analyzing specially orthotropic symmetric panels (i.e., A(sub 16) = A(sub 26) = 0.0, D(sub 16) = D(sub 26) = 0.0, B(sub ij) = 0.0). The analysis includes first-order transverse shear theory and is capable of modeling sandwich panels. The analysis supports two types of boundary conditions: either simply supported or clamped on all four edges. The panel can be subjected to linearly varying normal loads N(sub x) and N(sub y) in addition to a constant shear load N(sub xy). The applied loads can be divided into two parts: a preload component; and a variable (eigenvalue-dependent) component. The analysis is based on the modified Donnell's equations for shallow shells. The governing equations are solved by Galerkin's method.

Constructive Development of the Solutions of Linear Equations in Introductory Ordinary Differential Equations

ERIC Educational Resources Information Center

Mallet, D. G.; McCue, S. W.

2009-01-01

The solution of linear ordinary differential equations (ODEs) is commonly taught in first-year undergraduate mathematics classrooms, but the understanding of the concept of a solution is not always grasped by students until much later. Recognizing what it is to be a solution of a linear ODE and how to postulate such solutions, without resorting to…

Universal GFR determination based on two time points during plasma iohexol disappearance.

PubMed

Ng, Derek K S; Schwartz, George J; Jacobson, Lisa P; Palella, Frank J; Margolick, Joseph B; Warady, Bradley A; Furth, Susan L; Muñoz, Alvaro

2011-08-01

An optimal measurement of glomerular filtration rate (GFR) should minimize the number of blood draws, and reduce procedural invasiveness and the burden to study personnel and cost, without sacrificing accuracy. Equations have been proposed to calculate GFR from the slow compartment separately for adults and children. To develop a universal equation, we used 1347 GFR measurements from two diverse groups consisting of 527 men in the Multicenter AIDS Cohort Study and 514 children in the Chronic Kidney Disease in Children cohort. Both studies used nearly identical two-compartment (fast and slow) protocols to measure GFR. To estimate the fast component from markers of body size and of the slow component, we used standard linear regression methods with the log-transformed fast area as the dependent variable. The fast area could be accurately estimated from body surface area by a simple parameter (6.4/body surface area) with no residual dependence on the slow area or other markers of body size. Our equation measures only the slow iohexol plasma disappearance curve with as few as two time points and was normalized to 1.73 m2 body surface area. It is of the form: GFR=slowGFR/[1+0.12(slowGFR/100)]. In a random sample utilizing a third of the patients for validation, there was excellent agreement between the calculated and measured GFR with low root mean square errors being 4.6 and 1.5 ml/min per 1.73 m2 for adults and children, respectively. Thus, our proposed simple equation, developed in a combined patient group with a broad range of GFRs, may be applied universally and is independent of the injected amount of iohexol.

A novel body circumferences-based estimation of percentage body fat.

PubMed

Lahav, Yair; Epstein, Yoram; Kedem, Ron; Schermann, Haggai

2018-03-01

Anthropometric measures of body composition are often used for rapid and cost-effective estimation of percentage body fat (%BF) in field research, serial measurements and screening. Our aim was to develop a validated estimate of %BF for the general population, based on simple body circumferences measures. The study cohort consisted of two consecutive samples of health club members, designated as 'development' (n 476, 61 % men, 39 % women) and 'validation' (n 224, 50 % men, 50 % women) groups. All subjects underwent anthropometric measurements as part of their registration to a health club. Dual-energy X-ray absorptiometry (DEXA) scan was used as the 'gold standard' estimate of %BF. Linear regressions where used to construct the predictive equation (%BFcal). Bland-Altman statistics, Lin concordance coefficients and percentage of subjects falling within 5 % of %BF estimate by DEXA were used to evaluate accuracy and precision of the equation. The variance inflation factor was used to check multicollinearity. Two distinct equations were developed for men and women: %BFcal (men)=10·1-0·239H+0·8A-0·5N; %BFcal (women)=19·2-0·239H+0·8A-0·5N (H, height; A, abdomen; N, neck, all in cm). Bland-Altman differences were randomly distributed and showed no fixed bias. Lin concordance coefficients of %BFcal were 0·89 in men and 0·86 in women. About 79·5 % of %BF predictions in both sexes were within ±5 % of the DEXA value. The Durnin-Womersley skinfolds equation was less accurate in our study group for prediction of %BF than %BFcal. We conclude that %BFcal offers the advantage of obtaining a reliable estimate of %BF from simple measurements that require no sophisticated tools and only a minimal prior training and experience.

Multigrid Methods for Fully Implicit Oil Reservoir Simulation

NASA Technical Reports Server (NTRS)

Molenaar, J.

1996-01-01

In this paper we consider the simultaneous flow of oil and water in reservoir rock. This displacement process is modeled by two basic equations: the material balance or continuity equations and the equation of motion (Darcy's law). For the numerical solution of this system of nonlinear partial differential equations there are two approaches: the fully implicit or simultaneous solution method and the sequential solution method. In the sequential solution method the system of partial differential equations is manipulated to give an elliptic pressure equation and a hyperbolic (or parabolic) saturation equation. In the IMPES approach the pressure equation is first solved, using values for the saturation from the previous time level. Next the saturations are updated by some explicit time stepping method; this implies that the method is only conditionally stable. For the numerical solution of the linear, elliptic pressure equation multigrid methods have become an accepted technique. On the other hand, the fully implicit method is unconditionally stable, but it has the disadvantage that in every time step a large system of nonlinear algebraic equations has to be solved. The most time-consuming part of any fully implicit reservoir simulator is the solution of this large system of equations. Usually this is done by Newton's method. The resulting systems of linear equations are then either solved by a direct method or by some conjugate gradient type method. In this paper we consider the possibility of applying multigrid methods for the iterative solution of the systems of nonlinear equations. There are two ways of using multigrid for this job: either we use a nonlinear multigrid method or we use a linear multigrid method to deal with the linear systems that arise in Newton's method. So far only a few authors have reported on the use of multigrid methods for fully implicit simulations. Two-level FAS algorithm is presented for the black-oil equations, and linear multigrid for two-phase flow problems with strong heterogeneities and anisotropies is studied. Here we consider both possibilities. Moreover we present a novel way for constructing the coarse grid correction operator in linear multigrid algorithms. This approach has the advantage in that it preserves the sparsity pattern of the fine grid matrix and it can be extended to systems of equations in a straightforward manner. We compare the linear and nonlinear multigrid algorithms by means of a numerical experiment.

Solving a mixture of many random linear equations by tensor decomposition and alternating minimization.

DOT National Transportation Integrated Search

2016-09-01

We consider the problem of solving mixed random linear equations with k components. This is the noiseless setting of mixed linear regression. The goal is to estimate multiple linear models from mixed samples in the case where the labels (which sample...

Unified Framework for Deriving Simultaneous Equation Algorithms for Water Distribution Networks

EPA Science Inventory

The known formulations for steady state hydraulics within looped water distribution networks are re-derived in terms of linear and non-linear transformations of the original set of partly linear and partly non-linear equations that express conservation of mass and energy. All of ...

Perturbations of linear delay differential equations at the verge of instability.

PubMed

Lingala, N; Namachchivaya, N Sri

2016-06-01

The characteristic equation for a linear delay differential equation (DDE) has countably infinite roots on the complex plane. This paper considers linear DDEs that are on the verge of instability, i.e., a pair of roots of the characteristic equation lies on the imaginary axis of the complex plane and all other roots have negative real parts. It is shown that when small noise perturbations are present, the probability distribution of the dynamics can be approximated by the probability distribution of a certain one-dimensional stochastic differential equation (SDE) without delay. This is advantageous because equations without delay are easier to simulate and one-dimensional SDEs are analytically tractable. When the perturbations are also linear, it is shown that the stability depends on a specific complex number. The theory is applied to study oscillators with delayed feedback. Some errors in other articles that use multiscale approach are pointed out.

Polynomial elimination theory and non-linear stability analysis for the Euler equations

NASA Technical Reports Server (NTRS)

Kennon, S. R.; Dulikravich, G. S.; Jespersen, D. C.

1986-01-01

Numerical methods are presented that exploit the polynomial properties of discretizations of the Euler equations. It is noted that most finite difference or finite volume discretizations of the steady-state Euler equations produce a polynomial system of equations to be solved. These equations are solved using classical polynomial elimination theory, with some innovative modifications. This paper also presents some preliminary results of a new non-linear stability analysis technique. This technique is applicable to determining the stability of polynomial iterative schemes. Results are presented for applying the elimination technique to a one-dimensional test case. For this test case, the exact solution is computed in three iterations. The non-linear stability analysis is applied to determine the optimal time step for solving Burgers' equation using the MacCormack scheme. The estimated optimal time step is very close to the time step that arises from a linear stability analysis.

Using "Tracker" to Prove the Simple Harmonic Motion Equation

ERIC Educational Resources Information Center

Kinchin, John

2016-01-01

Simple harmonic motion (SHM) is a common topic for many students to study. Using the free, though versatile, motion tracking software; "Tracker", we can extend the students experience and show that the general equation for SHM does lead to the correct period of a simple pendulum.

A three operator split-step method covering a larger set of non-linear partial differential equations

NASA Astrophysics Data System (ADS)

Zia, Haider

2017-06-01

This paper describes an updated exponential Fourier based split-step method that can be applied to a greater class of partial differential equations than previous methods would allow. These equations arise in physics and engineering, a notable example being the generalized derivative non-linear Schrödinger equation that arises in non-linear optics with self-steepening terms. These differential equations feature terms that were previously inaccessible to model accurately with low computational resources. The new method maintains a 3rd order error even with these additional terms and models the equation in all three spatial dimensions and time. The class of non-linear differential equations that this method applies to is shown. The method is fully derived and implementation of the method in the split-step architecture is shown. This paper lays the mathematical ground work for an upcoming paper employing this method in white-light generation simulations in bulk material.

An extended harmonic balance method based on incremental nonlinear control parameters

NASA Astrophysics Data System (ADS)

Khodaparast, Hamed Haddad; Madinei, Hadi; Friswell, Michael I.; Adhikari, Sondipon; Coggon, Simon; Cooper, Jonathan E.

2017-02-01

A new formulation for calculating the steady-state responses of multiple-degree-of-freedom (MDOF) non-linear dynamic systems due to harmonic excitation is developed. This is aimed at solving multi-dimensional nonlinear systems using linear equations. Nonlinearity is parameterised by a set of 'non-linear control parameters' such that the dynamic system is effectively linear for zero values of these parameters and nonlinearity increases with increasing values of these parameters. Two sets of linear equations which are formed from a first-order truncated Taylor series expansion are developed. The first set of linear equations provides the summation of sensitivities of linear system responses with respect to non-linear control parameters and the second set are recursive equations that use the previous responses to update the sensitivities. The obtained sensitivities of steady-state responses are then used to calculate the steady state responses of non-linear dynamic systems in an iterative process. The application and verification of the method are illustrated using a non-linear Micro-Electro-Mechanical System (MEMS) subject to a base harmonic excitation. The non-linear control parameters in these examples are the DC voltages that are applied to the electrodes of the MEMS devices.

Solving Simple Kinetics without Integrals

ERIC Educational Resources Information Center

de la Pen~a, Lisandro Herna´ndez

2016-01-01

The solution of simple kinetic equations is analyzed without referencing any topic from differential equations or integral calculus. Guided by the physical meaning of the rate equation, a systematic procedure is used to generate an approximate solution that converges uniformly to the exact solution in the case of zero, first, and second order…

Arterial blood gas reference values for sea level and an altitude of 1,400 meters.

PubMed

Crapo, R O; Jensen, R L; Hegewald, M; Tashkin, D P

1999-11-01

Blood gas measurements were collected on healthy lifetime nonsmokers at sea level (n = 96) and at an altitude of 1,400 meters (n = 243) to establish reference equations. At each study site, arterial blood samples were analyzed in duplicate on two separate blood gas analyzers and CO-oximeters. Arterial blood gas variables included Pa(O(2)), Pa(CO(2)), pH, and calculated alveolar-arterial PO(2) difference (AaPO(2)). CO-oximeter variables were Hb, COHb, MetHb, and Sa(O(2)). Subjects were 18 to 81 yr of age with 166 male and 173 female. Outlier data were excluded from multiple regression analysis, and reference equations were fitted to the data in two ways: (1) best fit using linear, squared, and cross-product terms; (2) simple equations, including only the variables that explained at least 3% of the variance. Two sets of equations were created: (1) using only the sea level data and (2) using the combined data with barometric pressure as an independent variable. Comparisons with earlier studies revealed small but significant differences; the decline in Pa(O(2)) with age at each altitude was consistent with most previous studies. At sea level, the equation that included barometric pressure predicted Pa(O(2)) slightly better than the sea level specific equation. The inclusion of barometric pressure in the equations allows better prediction of blood gas reference values at sea level and at altitudes as high as 1,400 meters.

DOE Office of Scientific and Technical Information (OSTI.GOV)

Dumbser, Michael, E-mail: michael.dumbser@unitn.it; Balsara, Dinshaw S., E-mail: dbalsara@nd.edu

In this paper a new, simple and universal formulation of the HLLEM Riemann solver (RS) is proposed that works for general conservative and non-conservative systems of hyperbolic equations. For non-conservative PDE, a path-conservative formulation of the HLLEM RS is presented for the first time in this paper. The HLLEM Riemann solver is built on top of a novel and very robust path-conservative HLL method. It thus naturally inherits the positivity properties and the entropy enforcement of the underlying HLL scheme. However, with just the slight additional cost of evaluating eigenvectors and eigenvalues of intermediate characteristic fields, we can represent linearlymore » degenerate intermediate waves with a minimum of smearing. For conservative systems, our paper provides the easiest and most seamless path for taking a pre-existing HLL RS and quickly and effortlessly converting it to a RS that provides improved results, comparable with those of an HLLC, HLLD, Osher or Roe-type RS. This is done with minimal additional computational complexity, making our variant of the HLLEM RS also a very fast RS that can accurately represent linearly degenerate discontinuities. Our present HLLEM RS also transparently extends these advantages to non-conservative systems. For shallow water-type systems, the resulting method is proven to be well-balanced. Several test problems are presented for shallow water-type equations and two-phase flow models, as well as for gas dynamics with real equation of state, magnetohydrodynamics (MHD & RMHD), and nonlinear elasticity. Since our new formulation accommodates multiple intermediate waves and has a broader applicability than the original HLLEM method, it could alternatively be called the HLLI Riemann solver, where the “I” stands for the intermediate characteristic fields that can be accounted for. -- Highlights: •New simple and general path-conservative formulation of the HLLEM Riemann solver. •Application to general conservative and non-conservative hyperbolic systems. •Inclusion of sub-structure and resolution of intermediate characteristic fields. •Well-balanced for single- and two-layer shallow water equations and multi-phase flows. •Euler equations with real equation of state, MHD equations, nonlinear elasticity.« less

Superfluidity in Strongly Interacting Fermi Systems with Applications to Neutron Stars

NASA Astrophysics Data System (ADS)

Khodel, Vladimir

The rotational dynamics and cooling history of neutron stars is influenced by the superfluid properties of nucleonic matter. In this thesis a novel separation technique is applied to the analysis of the gap equation for neutron matter. It is shown that the problem can be recast into two tasks: solving a simple system of linear integral equations for the shape functions of various components of the gap function and solving a system of non-linear algebraic equations for their scale factors. Important simplifications result from the fact that the ratio of the gap amplitude to the Fermi energy provides a small parameter in this problem. The relationship between the analytic structure of the shape functions and the density interval for the existence of superfluid gap is discussed. It is shown that in 1S0 channel the position of the first zero of the shape function gives an estimate of the upper critical density. The relation between the resonant behavior of the two-neutron interaction in this channel and the density dependence of the gap is established. The behavior of the gap in the limits of low and high densities is analyzed. Various approaches to calculation of the scale factors are considered: model cases, angular averaging, and perturbation theory. An optimization-based approach is proposed. The shape functions and scale factors for Argonne υ14 and υ18 potentials are determined in singlet and triplet channels. Dependence of the solution on the value of effective mass and medium polarization is studied.

Improved Simulation of Electrodiffusion in the Node of Ranvier by Mesh Adaptation.

PubMed

Dione, Ibrahima; Deteix, Jean; Briffard, Thomas; Chamberland, Eric; Doyon, Nicolas

2016-01-01

In neural structures with complex geometries, numerical resolution of the Poisson-Nernst-Planck (PNP) equations is necessary to accurately model electrodiffusion. This formalism allows one to describe ionic concentrations and the electric field (even away from the membrane) with arbitrary spatial and temporal resolution which is impossible to achieve with models relying on cable theory. However, solving the PNP equations on complex geometries involves handling intricate numerical difficulties related either to the spatial discretization, temporal discretization or the resolution of the linearized systems, often requiring large computational resources which have limited the use of this approach. In the present paper, we investigate the best ways to use the finite elements method (FEM) to solve the PNP equations on domains with discontinuous properties (such as occur at the membrane-cytoplasm interface). 1) Using a simple 2D geometry to allow comparison with analytical solution, we show that mesh adaptation is a very (if not the most) efficient way to obtain accurate solutions while limiting the computational efforts, 2) We use mesh adaptation in a 3D model of a node of Ranvier to reveal details of the solution which are nearly impossible to resolve with other modelling techniques. For instance, we exhibit a non linear distribution of the electric potential within the membrane due to the non uniform width of the myelin and investigate its impact on the spatial profile of the electric field in the Debye layer.

Global paths of time-periodic solutions of the Benjamin-Ono equation connecting arbitrary traveling waves

DOE Office of Scientific and Technical Information (OSTI.GOV)

Ambrose, David M.; Wilkening, Jon

2008-12-11

We classify all bifurcations from traveling waves to non-trivial time-periodic solutions of the Benjamin-Ono equation that are predicted by linearization. We use a spectrally accurate numerical continuation method to study several paths of non-trivial solutions beyond the realm of linear theory. These paths are found to either re-connect with a different traveling wave or to blow up. In the latter case, as the bifurcation parameter approaches a critical value, the amplitude of the initial condition grows without bound and the period approaches zero. We propose a conjecture that gives the mapping from one bifurcation to its counterpart on the othermore » side of the path of non-trivial solutions. By experimentation with data fitting, we identify the form of the exact solutions on the path connecting two traveling waves, which represents the Fourier coefficients of the solution as power sums of a finite number of particle positions whose elementary symmetric functions execute simple orbits in the complex plane (circles or epicycles). We then solve a system of algebraic equations to express the unknown constants in the new representation in terms of the mean, a spatial phase, a temporal phase, four integers (enumerating the bifurcation at each end of the path) and one additional bifurcation parameter. We also find examples of interior bifurcations from these paths of already non-trivial solutions, but we do not attempt to analyze their algebraic structure.« less

Piecewise linear emulator of the nonlinear Schrödinger equation and the resulting analytic solutions for Bose-Einstein condensates.

PubMed

Theodorakis, Stavros

2003-06-01

We emulate the cubic term Psi(3) in the nonlinear Schrödinger equation by a piecewise linear term, thus reducing the problem to a set of uncoupled linear inhomogeneous differential equations. The resulting analytic expressions constitute an excellent approximation to the exact solutions, as is explicitly shown in the case of the kink, the vortex, and a delta function trap. Such a piecewise linear emulation can be used for any differential equation where the only nonlinearity is a Psi(3) one. In particular, it can be used for the nonlinear Schrödinger equation in the presence of harmonic traps, giving analytic Bose-Einstein condensate solutions that reproduce very accurately the numerically calculated ones in one, two, and three dimensions.

Dissipative behavior of some fully non-linear KdV-type equations

NASA Astrophysics Data System (ADS)

Brenier, Yann; Levy, Doron

2000-03-01

The KdV equation can be considered as a special case of the general equation u t+f(u) x-δg(u xx) x=0, δ>0, where f is non-linear and g is linear, namely f( u)= u2/2 and g( v)= v. As the parameter δ tends to 0, the dispersive behavior of the KdV equation has been throughly investigated (see, e.g., [P.G. Drazin, Solitons, London Math. Soc. Lect. Note Ser. 85, Cambridge University Press, Cambridge, 1983; P.D. Lax, C.D. Levermore, The small dispersion limit of the Korteweg-de Vries equation, III, Commun. Pure Appl. Math. 36 (1983) 809-829; G.B. Whitham, Linear and Nonlinear Waves, Wiley/Interscience, New York, 1974] and the references therein). We show through numerical evidence that a completely different, dissipative behavior occurs when g is non-linear, namely when g is an even concave function such as g( v)=-∣ v∣ or g( v)=- v2. In particular, our numerical results hint that as δ→0 the solutions strongly converge to the unique entropy solution of the formal limit equation, in total contrast with the solutions of the KdV equation.

Symbolic computation of recurrence equations for the Chebyshev series solution of linear ODE's. [ordinary differential equations

NASA Technical Reports Server (NTRS)

Geddes, K. O.

1977-01-01

If a linear ordinary differential equation with polynomial coefficients is converted into integrated form then the formal substitution of a Chebyshev series leads to recurrence equations defining the Chebyshev coefficients of the solution function. An explicit formula is presented for the polynomial coefficients of the integrated form in terms of the polynomial coefficients of the differential form. The symmetries arising from multiplication and integration of Chebyshev polynomials are exploited in deriving a general recurrence equation from which can be derived all of the linear equations defining the Chebyshev coefficients. Procedures for deriving the general recurrence equation are specified in a precise algorithmic notation suitable for translation into any of the languages for symbolic computation. The method is algebraic and it can therefore be applied to differential equations containing indeterminates.

A General Linear Method for Equating with Small Samples

ERIC Educational Resources Information Center

Albano, Anthony D.

2015-01-01

Research on equating with small samples has shown that methods with stronger assumptions and fewer statistical estimates can lead to decreased error in the estimated equating function. This article introduces a new approach to linear observed-score equating, one which provides flexible control over how form difficulty is assumed versus estimated…

ESEA Title I Linking Project. Final Report.

ERIC Educational Resources Information Center

Holmes, Susan E.

The Rasch model for test score equating was compared with three other equating procedures as methods for implementing the norm referenced method (RMC Model A) of evaluating ESEA Title I projects. The Rasch model and its theoretical limitations were described. The three other equating methods used were: linear observed score equating, linear true…

A Factorization Approach to the Linear Regulator Quadratic Cost Problem

NASA Technical Reports Server (NTRS)

Milman, M. H.

1985-01-01

A factorization approach to the linear regulator quadratic cost problem is developed. This approach makes some new connections between optimal control, factorization, Riccati equations and certain Wiener-Hopf operator equations. Applications of the theory to systems describable by evolution equations in Hilbert space and differential delay equations in Euclidean space are presented.

Variations in the Solution of Linear First-Order Differential Equations. Classroom Notes

ERIC Educational Resources Information Center

Seaman, Brian; Osler, Thomas J.

2004-01-01

A special project which can be given to students of ordinary differential equations is described in detail. Students create new differential equations by changing the dependent variable in the familiar linear first-order equation (dv/dx)+p(x)v=q(x) by means of a substitution v=f(y). The student then creates a table of the new equations and…

The Equation, the Whole Equation, and Nothing but the Equation! One Approach to the Teaching of Linear Equations.

ERIC Educational Resources Information Center

Pirie, Susan E. B.; Martin, Lyndon

1997-01-01

Presents the results of a case study which looked at the mathematics classroom of one teacher trying to teach mathematics with meaning to pupils or lower ability at the secondary level. Contrasts methods of teaching linear equations to a variety of ability levels and uses the Pirie-Kieren model to account for the successful growth in understanding…

Algorithm refinement for stochastic partial differential equations: II. Correlated systems

DOE Office of Scientific and Technical Information (OSTI.GOV)

Alexander, Francis J.; Garcia, Alejandro L.; Tartakovsky, Daniel M.

2005-08-10

We analyze a hybrid particle/continuum algorithm for a hydrodynamic system with long ranged correlations. Specifically, we consider the so-called train model for viscous transport in gases, which is based on a generalization of the random walk process for the diffusion of momentum. This discrete model is coupled with its continuous counterpart, given by a pair of stochastic partial differential equations. At the interface between the particle and continuum computations the coupling is by flux matching, giving exact mass and momentum conservation. This methodology is an extension of our stochastic Algorithm Refinement (AR) hybrid for simple diffusion [F. Alexander, A. Garcia,more » D. Tartakovsky, Algorithm refinement for stochastic partial differential equations: I. Linear diffusion, J. Comput. Phys. 182 (2002) 47-66]. Results from a variety of numerical experiments are presented for steady-state scenarios. In all cases the mean and variance of density and velocity are captured correctly by the stochastic hybrid algorithm. For a non-stochastic version (i.e., using only deterministic continuum fluxes) the long-range correlations of velocity fluctuations are qualitatively preserved but at reduced magnitude.« less

PROTEUS two-dimensional Navier-Stokes computer code, version 1.0. Volume 3: Programmer's reference

NASA Technical Reports Server (NTRS)

Towne, Charles E.; Schwab, John R.; Benson, Thomas J.; Suresh, Ambady

1990-01-01

A new computer code was developed to solve the 2-D or axisymmetric, Reynolds-averaged, unsteady compressible Navier-Stokes equations in strong conservation law form. The thin-layer or Euler equations may also be solved. Turbulence is modeled using an algebraic eddy viscosity model. The objective was to develop a code for aerospace applications that is easy to use and easy to modify. Code readability, modularity, and documentation were emphasized. The equations are written in nonorthogonal body-fitted coordinates, and solved by marching in time using a fully-coupled alternating-direction-implicit procedure with generalized first- or second-order time differencing. All terms are linearized using second-order Taylor series. The boundary conditions are treated implicitly, and may be steady, unsteady, or spatially periodic. Simple Cartesian or polar grids may be generated internally by the program. More complex geometries require an externally generated computational coordinate system. The documentation is divided into three volumes. Volume 3 is the Programmer's Reference, and describes the program structure, the FORTRAN variables stored in common blocks, and the details of each subprogram.

Finite volume multigrid method of the planar contraction flow of a viscoelastic fluid

NASA Astrophysics Data System (ADS)

Moatssime, H. Al; Esselaoui, D.; Hakim, A.; Raghay, S.

2001-08-01

This paper reports on a numerical algorithm for the steady flow of viscoelastic fluid. The conservative and constitutive equations are solved using the finite volume method (FVM) with a hybrid scheme for the velocities and first-order upwind approximation for the viscoelastic stress. A non-uniform staggered grid system is used. The iterative SIMPLE algorithm is employed to relax the coupled momentum and continuity equations. The non-linear algebraic equations over the flow domain are solved iteratively by the symmetrical coupled Gauss-Seidel (SCGS) method. In both, the full approximation storage (FAS) multigrid algorithm is used. An Oldroyd-B fluid model was selected for the calculation. Results are reported for planar 4:1 abrupt contraction at various Weissenberg numbers. The solutions are found to be stable and smooth. The solutions show that at high Weissenberg number the domain must be long enough. The convergence of the method has been verified with grid refinement. All the calculations have been performed on a PC equipped with a Pentium III processor at 550 MHz. Copyright

Graphical Calculation of Estimated Energy Expenditure in Burn Patients.

PubMed

Egro, Francesco M; Manders, Ernest C; Manders, Ernest K

2018-03-01

Historically, estimated energy expenditure (EEE) has been related to the percent of body surface area burned. Subsequent evaluations of these estimates have indicated that the earlier formulas may overestimate the amount of caloric support necessary for burn-injured patients. Ireton-Jones et al derived 2 equations for determining the EEE required to support burn patients, 1 for ventilator-dependent patients and 1 for spontaneously breathing patients. Evidence has proved their reliability, but they remain challenging to apply in a clinical setting given the difficult and cumbersome mathematics involved. This study aims to introduce a graphical calculation of EEE in burn patients that can be easily used in the clinical setting. The multivariant linear regression analysis from Ireton-Jones et al yielded equations that were rearranged into the form of a simple linear equation of the type y = mx + b. By choosing an energy expenditure and the age of the subject, the weight was calculated. The endpoints were then calculated, and a graph was mapped by means of Adobe FrameMaker. A graphical representation of Ireton-Jones et al's equations was obtained by plotting the weight (kg) on the y axis, the age (years) on the x axis, and a series of parallel lines representing the EEE in burn patients. The EEE has been displayed graphically on a grid to allow rapid determination of the EEE needed for a given patient of a designated weight and age. Two graphs were plotted: 1 for ventilator-dependent patients and 1 for spontaneously breathing patients. Correction factors for sex, the presence of additional trauma, and obesity are indicated on the graphical calculators. We propose a graphical tool to calculate caloric requirements in a fast, easy, and portable manner.

Non-linear analysis of wave progagation using transform methods and plates and shells using integral equations

NASA Astrophysics Data System (ADS)

Pipkins, Daniel Scott

Two diverse topics of relevance in modern computational mechanics are treated. The first involves the modeling of linear and non-linear wave propagation in flexible, lattice structures. The technique used combines the Laplace Transform with the Finite Element Method (FEM). The procedure is to transform the governing differential equations and boundary conditions into the transform domain where the FEM formulation is carried out. For linear problems, the transformed differential equations can be solved exactly, hence the method is exact. As a result, each member of the lattice structure is modeled using only one element. In the non-linear problem, the method is no longer exact. The approximation introduced is a spatial discretization of the transformed non-linear terms. The non-linear terms are represented in the transform domain by making use of the complex convolution theorem. A weak formulation of the resulting transformed non-linear equations yields a set of element level matrix equations. The trial and test functions used in the weak formulation correspond to the exact solution of the linear part of the transformed governing differential equation. Numerical results are presented for both linear and non-linear systems. The linear systems modeled are longitudinal and torsional rods and Bernoulli-Euler and Timoshenko beams. For non-linear systems, a viscoelastic rod and Von Karman type beam are modeled. The second topic is the analysis of plates and shallow shells under-going finite deflections by the Field/Boundary Element Method. Numerical results are presented for two plate problems. The first is the bifurcation problem associated with a square plate having free boundaries which is loaded by four, self equilibrating corner forces. The results are compared to two existing numerical solutions of the problem which differ substantially.

A Novel Blast-mitigation Concept for Light Tactical Vehicles

DTIC Science & Technology

2013-01-01

analysis which utilizes the mass and energy (but not linear momentum ) conservation equations is provided. It should be noted that the identical final...results could be obtained using an analogous analysis which combines the mass and the linear momentum conservation equations. For a calorically...governing mass, linear momentum and energy conservation and heat conduction equations are solved within ABAQUS/ Explicit with a second-order accurate

Validation of a single-stage fixed-rate step test for the prediction of maximal oxygen uptake in healthy adults.

PubMed

Hansen, Dominique; Jacobs, Nele; Thijs, Herbert; Dendale, Paul; Claes, Neree

2016-09-01

Healthcare professionals with limited access to ergospirometry remain in need of valid and simple submaximal exercise tests to predict maximal oxygen uptake (VO2max ). Despite previous validation studies concerning fixed-rate step tests, accurate equations for the estimation of VO2max remain to be formulated from a large sample of healthy adults between age 18-75 years (n > 100). The aim of this study was to develop a valid equation to estimate VO2max from a fixed-rate step test in a larger sample of healthy adults. A maximal ergospirometry test, with assessment of cardiopulmonary parameters and VO2max , and a 5-min fixed-rate single-stage step test were executed in 112 healthy adults (age 18-75 years). During the step test and subsequent recovery, heart rate was monitored continuously. By linear regression analysis, an equation to predict VO2max from the step test was formulated. This equation was assessed for level of agreement by displaying Bland-Altman plots and calculation of intraclass correlations with measured VO2max . Validity further was assessed by employing a Jackknife procedure. The linear regression analysis generated the following equation to predict VO2max (l min(-1) ) from the step test: 0·054(BMI)+0·612(gender)+3·359(body height in m)+0·019(fitness index)-0·012(HRmax)-0·011(age)-3·475. This equation explained 78% of the variance in measured VO2max (F = 66·15, P<0·001). The level of agreement and intraclass correlation was high (ICC = 0·94, P<0·001) between measured and predicted VO2max . From this study, a valid fixed-rate single-stage step test equation has been developed to estimate VO2max in healthy adults. This tool could be employed by healthcare professionals with limited access to ergospirometry. © 2015 Scandinavian Society of Clinical Physiology and Nuclear Medicine. Published by John Wiley & Sons Ltd.

Spectrophotometric total reducing sugars assay based on cupric reduction.

PubMed

Başkan, Kevser Sözgen; Tütem, Esma; Akyüz, Esin; Özen, Seda; Apak, Reşat

2016-01-15

As the concentration of reducing sugars (RS) is controlled by European legislation for certain specific food and beverages, a simple and sensitive spectrophotometric method for the determination of RS in various food products is proposed. The method is based on the reduction of Cu(II) to Cu(I) with reducing sugars in alkaline medium in the presence of 2,9-dimethyl-1,10-phenanthroline (neocuproine: Nc), followed by the formation of a colored Cu(I)-Nc charge-transfer complex. All simple sugars tested had the linear regression equations with almost equal slope values. The proposed method was successfully applied to fresh apple juice, commercial fruit juices, milk, honey and onion juice. Interference effect of phenolic compounds in plant samples was eliminated by a solid phase extraction (SPE) clean-up process. The method was proven to have higher sensitivity and precision than the widely used dinitrosalicylic acid (DNS) colorimetric method. Copyright © 2015 Elsevier B.V. All rights reserved.

Meteorological adjustment of yearly mean values for air pollutant concentration comparison

NASA Technical Reports Server (NTRS)

Sidik, S. M.; Neustadter, H. E.

1976-01-01

Using multiple linear regression analysis, models which estimate mean concentrations of Total Suspended Particulate (TSP), sulfur dioxide, and nitrogen dioxide as a function of several meteorologic variables, two rough economic indicators, and a simple trend in time are studied. Meteorologic data were obtained and do not include inversion heights. The goodness of fit of the estimated models is partially reflected by the squared coefficient of multiple correlation which indicates that, at the various sampling stations, the models accounted for about 23 to 47 percent of the total variance of the observed TSP concentrations. If the resulting model equations are used in place of simple overall means of the observed concentrations, there is about a 20 percent improvement in either: (1) predicting mean concentrations for specified meteorological conditions; or (2) adjusting successive yearly averages to allow for comparisons devoid of meteorological effects. An application to source identification is presented using regression coefficients of wind velocity predictor variables.

A Complex-Valued Firing-Rate Model That Approximates the Dynamics of Spiking Networks

PubMed Central

Schaffer, Evan S.; Ostojic, Srdjan; Abbott, L. F.

2013-01-01

Firing-rate models provide an attractive approach for studying large neural networks because they can be simulated rapidly and are amenable to mathematical analysis. Traditional firing-rate models assume a simple form in which the dynamics are governed by a single time constant. These models fail to replicate certain dynamic features of populations of spiking neurons, especially those involving synchronization. We present a complex-valued firing-rate model derived from an eigenfunction expansion of the Fokker-Planck equation and apply it to the linear, quadratic and exponential integrate-and-fire models. Despite being almost as simple as a traditional firing-rate description, this model can reproduce firing-rate dynamics due to partial synchronization of the action potentials in a spiking model, and it successfully predicts the transition to spike synchronization in networks of coupled excitatory and inhibitory neurons. PMID:24204236

Two-phase flows within systems with ambient pressure

NASA Technical Reports Server (NTRS)

Hendricks, R. C.; Braun, M. J.; Wheeler, R. L., III; Mullen, R. L.

1985-01-01

In systems where the design inlet and outlet pressures are maintained above the thermodynamic critical pressure, it is often assumed that two phase flows within the system cannot occur. Designers rely on this simple rule of thumb to circumvent problems associated with a highly compressible two phase flow occurring within the supercritical pressure system along with the uncertainties in rotordynamics, load capacity, heat transfer, fluid mechanics, and thermophysical property variations. The simple rule of thumb is adequate in many low power designs but is inadequate for high performance turbomachines and linear systems, where two phase regions can exist even though outlet pressure is greater than critical pressure. Rotordynamic-fluid-mechanic restoring forces depend on momentum differences, and those for a two phase zone can differ significantly from those for a single-phase zone. Using the Reynolds equation the angular velocity, eccentricity, geometry, and ambient conditions are varied to determine the point of two phase flow incipience.

Estimation of sex and stature using anthropometry of the upper extremity in an Australian population.

PubMed

Howley, Donna; Howley, Peter; Oxenham, Marc F

2018-06-01

Stature and a further 8 anthropometric dimensions were recorded from the arms and hands of a sample of 96 staff and students from the Australian National University and The University of Newcastle, Australia. These dimensions were used to create simple and multiple logistic regression models for sex estimation and simple and multiple linear regression equations for stature estimation of a contemporary Australian population. Overall sex classification accuracies using the models created were comparable to similar studies. The stature estimation models achieved standard errors of estimates (SEE) which were comparable to and in many cases lower than those achieved in similar research. Generic, non sex-specific models achieved similar SEEs and R 2 values to the sex-specific models indicating stature may be accurately estimated when sex is unknown. Copyright © 2018 Elsevier B.V. All rights reserved.

Vacillations induced by interference of stationary and traveling planetary waves

NASA Technical Reports Server (NTRS)

Salby, Murry L.; Garcia, Rolando R.

1987-01-01

The interference pattern produced when a traveling planetary wave propagates over a stationary forced wave is explored, examining the interference signature in a variety of diagnostics. The wave field is first restricted to a diatomic spectrum consisting of two components: a single stationary wave and a single monochromatic traveling wave. A simple barotropic normal mode propagating over a simple stationary plane wave is considered, and closed form solutions are obtained. The wave fields are then restricted spatially, providing more realistic structures without sacrificing the advantages of an analytical solution. Both stationary and traveling wave fields are calculated numerically with the linearized Primitive Equations in a realistic basic state. The mean flow reaction to the fluctuating eddy forcing which results from interference is derived. Synoptic geopotential behavior corresponding to the combined wave and mean flow fields is presented, and the synoptic signature in potential vorticity on isentropic surfaces is examined.

Effect of rock rheology on fluid leak- off during hydraulic fracturing

NASA Astrophysics Data System (ADS)

Yarushina, V. M.; Bercovici, D.; Oristaglio, M. L.

2012-04-01

In this communication, we evaluate the effect of rock rheology on fluid leak­off during hydraulic fracturing of reservoirs. Fluid leak-off in hydraulic fracturing is often nonlinear. The simple linear model developed by Carter (1957) for flow of fracturing fluid into a reservoir has three different regions in the fractured zone: a filter cake on the fracture face, formed by solid additives from the fracturing fluid; a filtrate zone affected by invasion of the fracturing fluid; and a reservoir zone with the original formation fluid. The width of each zone, as well as its permeability and pressure drop, is assumed to remain constant. Physical intuition suggests some straightforward corrections to this classical theory to take into account the pressure dependence of permeability, the compressibility or non-Newtonian rheology of fracturing fluid, and the radial (versus linear) geometry of fluid leak­off from the borehole. All of these refinements, however, still assume that the reservoir rock adjacent to the fracture face is non­deformable. Although the effect of poroelastic stress changes on leak-off is usually thought to be negligible, at the very high fluid pressures used in hydraulic fracturing, where the stresses exceed the rock strength, elastic rheology may not be the best choice. For example, calculations show that perfectly elastic rock formations do not undergo the degree of compaction typically seen in sedimentary basins. Therefore, pseudo-elastic or elastoplastic models are used to fit observed porosity profiles with depth. Starting from balance equations for mass and momentum for fluid and rock, we derive a hydraulic flow equation coupled with a porosity equation describing rock compaction. The result resembles a pressure diffusion equation with the total compressibility being a sum of fluid, rock and pore-space compressibilities. With linear elastic rheology, the bulk formation compressibility is dominated by fluid compressibility. But the possibility of permanent, time-independent (plastic) rock deformation significantly increases the pore space compressibility (compaction), which becomes a leading term in the total compressibility. Inclusion of rock and fluid compressibilities in the model can explain both linear and nonlinear leak­off. In particular, inclusion of rock compaction and decompaction may be important for description of naturally fractured and tight gas reservoirs for which very strong dependence of permeability on porosity has been reported. Carter R.D. Derivation of the general equation for estimating the extent of the fractured area. Appendix I of "Optimum fluid characteristics for fracture extension", Drilling and Production Practice, G.C. Howard and C.R.Fast, New York, New York, USA, American Petroleum Institute (1957), 261-269.

Linear-time reconstruction of zero-recombinant Mendelian inheritance on pedigrees without mating loops.

PubMed

Liu, Lan; Jiang, Tao

2007-01-01

With the launch of the international HapMap project, the haplotype inference problem has attracted a great deal of attention in the computational biology community recently. In this paper, we study the question of how to efficiently infer haplotypes from genotypes of individuals related by a pedigree without mating loops, assuming that the hereditary process was free of mutations (i.e. the Mendelian law of inheritance) and recombinants. We model the haplotype inference problem as a system of linear equations as in [10] and present an (optimal) linear-time (i.e. O(mn) time) algorithm to generate a particular solution (A particular solution of any linear system is an assignment of numerical values to the variables in the system which satisfies the equations in the system.) to the haplotype inference problem, where m is the number of loci (or markers) in a genotype and n is the number of individuals in the pedigree. Moreover, the algorithm also provides a general solution (A general solution of any linear system is denoted by the span of a basis in the solution space to its associated homogeneous system, offset from the origin by a vector, namely by any particular solution. A general solution for ZRHC is very useful in practice because it allows the end user to efficiently enumerate all solutions for ZRHC and performs tasks such as random sampling.) in O(mn2) time, which is optimal because the size of a general solution could be as large as Theta(mn2). The key ingredients of our construction are (i) a fast consistency checking procedure for the system of linear equations introduced in [10] based on a careful investigation of the relationship between the equations (ii) a novel linear-time method for solving linear equations without invoking the Gaussian elimination method. Although such a fast method for solving equations is not known for general systems of linear equations, we take advantage of the underlying loop-free pedigree graph and some special properties of the linear equations.

Semigroup theory and numerical approximation for equations in linear viscoelasticity

NASA Technical Reports Server (NTRS)

Fabiano, R. H.; Ito, K.

1990-01-01

A class of abstract integrodifferential equations used to model linear viscoelastic beams is investigated analytically, applying a Hilbert-space approach. The basic equation is rewritten as a Cauchy problem, and its well-posedness is demonstrated. Finite-dimensional subspaces of the state space and an estimate of the state operator are obtained; approximation schemes for the equations are constructed; and the convergence is proved using the Trotter-Kato theorem of linear semigroup theory. The actual convergence behavior of different approximations is demonstrated in numerical computations, and the results are presented in tables.

Internal null controllability of a linear Schrödinger-KdV system on a bounded interval

NASA Astrophysics Data System (ADS)

Araruna, Fágner D.; Cerpa, Eduardo; Mercado, Alberto; Santos, Maurício C.

2016-01-01

The control of a linear dispersive system coupling a Schrödinger and a linear Korteweg-de Vries equation is studied in this paper. The system can be viewed as three coupled real-valued equations by taking real and imaginary parts in the Schrödinger equation. The internal null controllability is proven by using either one complex-valued control on the Schrödinger equation or two real-valued controls, one on each equation. Notice that the single Schrödinger equation is not known to be controllable with a real-valued control. The standard duality method is used to reduce the controllability property to an observability inequality, which is obtained by means of a Carleman estimates approach.

COMPARISON OF IMPLICIT SCHEMES TO SOLVE EQUATIONS OF RADIATION HYDRODYNAMICS WITH A FLUX-LIMITED DIFFUSION APPROXIMATION: NEWTON–RAPHSON, OPERATOR SPLITTING, AND LINEARIZATION

DOE Office of Scientific and Technical Information (OSTI.GOV)

Tetsu, Hiroyuki; Nakamoto, Taishi, E-mail: h.tetsu@geo.titech.ac.jp

Radiation is an important process of energy transport, a force, and a basis for synthetic observations, so radiation hydrodynamics (RHD) calculations have occupied an important place in astrophysics. However, although the progress in computational technology is remarkable, their high numerical cost is still a persistent problem. In this work, we compare the following schemes used to solve the nonlinear simultaneous equations of an RHD algorithm with the flux-limited diffusion approximation: the Newton–Raphson (NR) method, operator splitting, and linearization (LIN), from the perspective of the computational cost involved. For operator splitting, in addition to the traditional simple operator splitting (SOS) scheme,more » we examined the scheme developed by Douglas and Rachford (DROS). We solve three test problems (the thermal relaxation mode, the relaxation and the propagation of linear waves, and radiating shock) using these schemes and then compare their dependence on the time step size. As a result, we find the conditions of the time step size necessary for adopting each scheme. The LIN scheme is superior to other schemes if the ratio of radiation pressure to gas pressure is sufficiently low. On the other hand, DROS can be the most efficient scheme if the ratio is high. Although the NR scheme can be adopted independently of the regime, especially in a problem that involves optically thin regions, the convergence tends to be worse. In all cases, SOS is not practical.« less

Analysis of calibration data for the uranium active neutron coincidence counting collar with attention to errors in the measured neutron coincidence rate

DOE PAGES

Croft, Stephen; Burr, Thomas Lee; Favalli, Andrea; ...

2015-12-10

We report that the declared linear density of 238U and 235U in fresh low enriched uranium light water reactor fuel assemblies can be verified for nuclear safeguards purposes using a neutron coincidence counter collar in passive and active mode, respectively. The active mode calibration of the Uranium Neutron Collar – Light water reactor fuel (UNCL) instrument is normally performed using a non-linear fitting technique. The fitting technique relates the measured neutron coincidence rate (the predictor) to the linear density of 235U (the response) in order to estimate model parameters of the nonlinear Padé equation, which traditionally is used to modelmore » the calibration data. Alternatively, following a simple data transformation, the fitting can also be performed using standard linear fitting methods. This paper compares performance of the nonlinear technique to the linear technique, using a range of possible error variance magnitudes in the measured neutron coincidence rate. We develop the required formalism and then apply the traditional (nonlinear) and alternative approaches (linear) to the same experimental and corresponding simulated representative datasets. Lastly, we find that, in this context, because of the magnitude of the errors in the predictor, it is preferable not to transform to a linear model, and it is preferable not to adjust for the errors in the predictor when inferring the model parameters« less

A new analysis of the Fornberg-Whitham equation pertaining to a fractional derivative with Mittag-Leffler-type kernel

NASA Astrophysics Data System (ADS)

Kumar, Devendra; Singh, Jagdev; Baleanu, Dumitru

2018-02-01

The mathematical model of breaking of non-linear dispersive water waves with memory effect is very important in mathematical physics. In the present article, we examine a novel fractional extension of the non-linear Fornberg-Whitham equation occurring in wave breaking. We consider the most recent theory of differentiation involving the non-singular kernel based on the extended Mittag-Leffler-type function to modify the Fornberg-Whitham equation. We examine the existence of the solution of the non-linear Fornberg-Whitham equation of fractional order. Further, we show the uniqueness of the solution. We obtain the numerical solution of the new arbitrary order model of the non-linear Fornberg-Whitham equation with the aid of the Laplace decomposition technique. The numerical outcomes are displayed in the form of graphs and tables. The results indicate that the Laplace decomposition algorithm is a very user-friendly and reliable scheme for handling such type of non-linear problems of fractional order.

Nonlinear (time domain) and linearized (time and frequency domain) solutions to the compressible Euler equations in conservation law form

NASA Technical Reports Server (NTRS)

Sreenivas, Kidambi; Whitfield, David L.

1995-01-01

Two linearized solvers (time and frequency domain) based on a high resolution numerical scheme are presented. The basic approach is to linearize the flux vector by expressing it as a sum of a mean and a perturbation. This allows the governing equations to be maintained in conservation law form. A key difference between the time and frequency domain computations is that the frequency domain computations require only one grid block irrespective of the interblade phase angle for which the flow is being computed. As a result of this and due to the fact that the governing equations for this case are steady, frequency domain computations are substantially faster than the corresponding time domain computations. The linearized equations are used to compute flows in turbomachinery blade rows (cascades) arising due to blade vibrations. Numerical solutions are compared to linear theory (where available) and to numerical solutions of the nonlinear Euler equations.

Symbolic Solution of Linear Differential Equations

NASA Technical Reports Server (NTRS)

Feinberg, R. B.; Grooms, R. G.

1981-01-01

An algorithm for solving linear constant-coefficient ordinary differential equations is presented. The computational complexity of the algorithm is discussed and its implementation in the FORMAC system is described. A comparison is made between the algorithm and some classical algorithms for solving differential equations.

LQS_INVERSION v. 1.0

DOE Office of Scientific and Technical Information (OSTI.GOV)

Weiss, Chester J

FORTRAN90 codes for inversion of electrostatic geophysical data in terms of three subsurface parameters in a single-well, oilfield environment: the linear charge density of the steel well casing (L), the point charge associated with an induced fracture filled with a conductive contrast agent (Q) and the location of said fracture (s). Theory is described in detail in Weiss et al. (Geophysics, 2016). Inversion strategy is to loop over candidate fracture locations, and at each one minimize the squared Cartesian norm of the data misfit to arrive at L and Q. Solution method is to construct the 2x2 linear system ofmore » normal equations and compute L and Q algebraically. Practical Application: Oilfield environments where observed electrostatic geophysical data can reasonably be assumed by a simple L-Q-s model. This may include hydrofracking operations, as postulated in Weiss et al. (2016), but no field validation examples have so far been provided.« less

Linear spreading speeds from nonlinear resonant interaction

NASA Astrophysics Data System (ADS)

Faye, Grégory; Holzer, Matt; Scheel, Arnd

2017-06-01

We identify a new mechanism for propagation into unstable states in spatially extended systems, that is based on resonant interaction in the leading edge of invasion fronts. Such resonant invasion speeds can be determined solely based on the complex linear dispersion relation at the unstable equilibrium, but rely on the presence of a nonlinear term that facilitates the resonant coupling. We prove that these resonant speeds give the correct invasion speed in a simple example, we show that fronts with speeds slower than the resonant speed are unstable, and corroborate our speed criterion numerically in a variety of model equations, including a nonlocal scalar neural field model. GF received support from the project NONLOCAL (ANR-14-CE25-0013) funded by the French National Research Agency. MH was partially supported by the National Science Foundation through grant NSF-DMS-1516155. AS was partially supported by the National Science Foundation through grant NSF-DMS-1311740 and through a DAAD Fellowship.

Efficient parallel architecture for highly coupled real-time linear system applications

NASA Technical Reports Server (NTRS)

Carroll, Chester C.; Homaifar, Abdollah; Barua, Soumavo

1988-01-01

A systematic procedure is developed for exploiting the parallel constructs of computation in a highly coupled, linear system application. An overall top-down design approach is adopted. Differential equations governing the application under consideration are partitioned into subtasks on the basis of a data flow analysis. The interconnected task units constitute a task graph which has to be computed in every update interval. Multiprocessing concepts utilizing parallel integration algorithms are then applied for efficient task graph execution. A simple scheduling routine is developed to handle task allocation while in the multiprocessor mode. Results of simulation and scheduling are compared on the basis of standard performance indices. Processor timing diagrams are developed on the basis of program output accruing to an optimal set of processors. Basic architectural attributes for implementing the system are discussed together with suggestions for processing element design. Emphasis is placed on flexible architectures capable of accommodating widely varying application specifics.

On the correct use of stepped-sine excitations for the measurement of time-varying bioimpedance.

PubMed

Louarroudi, E; Sanchez, B

2017-02-01

When a linear time-varying (LTV) bioimpedance is measured using stepped-sine excitations, a compromise must be made: the temporal distortions affecting the data depend on the experimental time, which in turn sets the data accuracy and limits the temporal bandwidth of the system that needs to be measured. Here, the experimental time required to measure linear time-invariant bioimpedance with a specified accuracy is analyzed for different stepped-sine excitation setups. We provide simple equations that allow the reader to know whether LTV bioimpedance can be measured through repeated time- invariant stepped-sine experiments. Bioimpedance technology is on the rise thanks to a plethora of healthcare monitoring applications. The results presented can help to avoid distortions in the data while measuring accurately non-stationary physiological phenomena. The impact of the work presented is broad, including the potential of enhancing bioimpedance studies and healthcare devices using bioimpedance technology.

Improved Convergence and Robustness of USM3D Solutions on Mixed-Element Grids

NASA Technical Reports Server (NTRS)

Pandya, Mohagna J.; Diskin, Boris; Thomas, James L.; Frink, Neal T.

2016-01-01

Several improvements to the mixed-element USM3D discretization and defect-correction schemes have been made. A new methodology for nonlinear iterations, called the Hierarchical Adaptive Nonlinear Iteration Method, has been developed and implemented. The Hierarchical Adaptive Nonlinear Iteration Method provides two additional hierarchies around a simple and approximate preconditioner of USM3D. The hierarchies are a matrix-free linear solver for the exact linearization of Reynolds-averaged Navier-Stokes equations and a nonlinear control of the solution update. Two variants of the Hierarchical Adaptive Nonlinear Iteration Method are assessed on four benchmark cases, namely, a zero-pressure-gradient flat plate, a bump-in-channel configuration, the NACA 0012 airfoil, and a NASA Common Research Model configuration. The new methodology provides a convergence acceleration factor of 1.4 to 13 over the preconditioner-alone method representing the baseline solver technology.

Improved Convergence and Robustness of USM3D Solutions on Mixed-Element Grids

NASA Technical Reports Server (NTRS)

Pandya, Mohagna J.; Diskin, Boris; Thomas, James L.; Frinks, Neal T.

2016-01-01

Several improvements to the mixed-elementUSM3Ddiscretization and defect-correction schemes have been made. A new methodology for nonlinear iterations, called the Hierarchical Adaptive Nonlinear Iteration Method, has been developed and implemented. The Hierarchical Adaptive Nonlinear Iteration Method provides two additional hierarchies around a simple and approximate preconditioner of USM3D. The hierarchies are a matrix-free linear solver for the exact linearization of Reynolds-averaged Navier-Stokes equations and a nonlinear control of the solution update. Two variants of the Hierarchical Adaptive Nonlinear Iteration Method are assessed on four benchmark cases, namely, a zero-pressure-gradient flat plate, a bump-in-channel configuration, the NACA 0012 airfoil, and a NASA Common Research Model configuration. The new methodology provides a convergence acceleration factor of 1.4 to 13 over the preconditioner-alone method representing the baseline solver technology.

Efficient Craig Interpolation for Linear Diophantine (Dis)Equations and Linear Modular Equations

DTIC Science & Technology

2008-02-01

Craig interpolants has enabled the development of powerful hardware and software model checking techniques. Efficient algorithms are known for computing...interpolants in rational and real linear arithmetic. We focus on subsets of integer linear arithmetic. Our main results are polynomial time algorithms ...congruences), and linear diophantine disequations. We show the utility of the proposed interpolation algorithms for discovering modular/divisibility predicates

Wave induced density modification in RF sheaths and close to wave launchers

DOE Office of Scientific and Technical Information (OSTI.GOV)

Van Eester, D., E-mail: d.van.eester@fz-juelich.de; Crombé, K.; Department of Applied Physics, Ghent University, Ghent

2015-12-10

With the return to full metal walls - a necessary step towards viable fusion machines - and due to the high power densities of current-day ICRH (Ion Cyclotron Resonance Heating) or RF (radio frequency) antennas, there is ample renewed interest in exploring the reasons for wave-induced sputtering and formation of hot spots. Moreover, there is experimental evidence on various machines that RF waves influence the density profile close to the wave launchers so that waves indirectly influence their own coupling efficiency. The present study presents a return to first principles and describes the wave-particle interaction using a 2-time scale modelmore » involving the equation of motion, the continuity equation and the wave equation on each of the time scales. Through the changing density pattern, the fast time scale dynamics is affected by the slow time scale events. In turn, the slow time scale density and flows are modified by the presence of the RF waves through quasilinear terms. Although finite zero order flows are identified, the usual cold plasma dielectric tensor - ignoring such flows - is adopted as a first approximation to describe the wave response to the RF driver. The resulting set of equations is composed of linear and nonlinear equations and is tackled in 1D in the present paper. Whereas the former can be solved using standard numerical techniques, the latter require special handling. At the price of multiple iterations, a simple ’derivative switch-on’ procedure allows to reformulate the nonlinear problem as a sequence of linear problems. Analytical expressions allow a first crude assessment - revealing that the ponderomotive potential plays a role similar to that of the electrostatic potential arising from charge separation - but numerical implementation is required to get a feeling of the full dynamics. A few tentative examples are provided to illustrate the phenomena involved.« less

Time-dependent theoretical treatments of the dynamics of electrons and nuclei in molecular systems

DOE Office of Scientific and Technical Information (OSTI.GOV)

Deumens, E.; Diz, A.; Longo, R.

1994-07-01

An overview is presented of methods for time-dependent treatments of molecules as systems of electrons and nuclei. The theoretical details of these methods are reviewed and contrasted in the light of a recently developed time-dependent method called electron-nuclear dynamics. Electron-nuclear dynamics (END) is a formulation of the complete dynamics of electrons and nuclei of a molecular system that eliminates the necessity of constructing potential-energy surfaces. Because of its general formulation, it encompasses many aspects found in other formulations and can serve as a didactic device for clarifying many of the principles and approximations relevant in time-dependent treatments of molecular systems.more » The END equations are derived from the time-dependent variational principle applied to a chosen family of efficiently parametrized approximate state vectors. A detailed analysis of the END equations is given for the case of a single-determinantal state for the electrons and a classical treatment of the nuclei. The approach leads to a simple formulation of the fully nonlinear time-dependent Hartree-Fock theory including nuclear dynamics. The nonlinear END equations with the [ital ab] [ital initio] Coulomb Hamiltonian have been implemented at this level of theory in a computer program, ENDyne, and have been shown feasible for the study of small molecular systems. Implementation of the Austin Model 1 semiempirical Hamiltonian is discussed as a route to large molecular systems. The linearized END equations at this level of theory are shown to lead to the random-phase approximation for the coupled system of electrons and nuclei. The qualitative features of the general nonlinear solution are analyzed using the results of the linearized equations as a first approximation. Some specific applications of END are presented, and the comparison with experiment and other theoretical approaches is discussed.« less

Stress stiffening and approximate equations in flexible multibody dynamics

NASA Technical Reports Server (NTRS)

Padilla, Carlos E.; Vonflotow, Andreas H.

1993-01-01

A useful model for open chains of flexible bodies undergoing large rigid body motions, but small elastic deformations, is one in which the equations of motion are linearized in the small elastic deformations and deformation rates. For slow rigid body motions, the correctly linearized, or consistent, set of equations can be compared to prematurely linearized, or inconsistent, equations and to 'oversimplified,' or ruthless, equations through the use of open loop dynamic simulations. It has been shown that the inconsistent model should never be used, while the ruthless model should be used whenever possible. The consistent and inconsistent models differ by stress stiffening terms. These are due to zeroth-order stresses effecting virtual work via nonlinear strain-displacement terms. In this paper we examine in detail the nature of these stress stiffening terms and conclude that they are significant only when the associated zeroth-order stresses approach 'buckling' stresses. Finally it is emphasized that when the stress stiffening terms are negligible the ruthlessly linearized equations should be used.

A formulation of rotor-airframe coupling for design analysis of vibrations of helicopter airframes

NASA Technical Reports Server (NTRS)

Kvaternik, R. G.; Walton, W. C., Jr.

1982-01-01

A linear formulation of rotor airframe coupling intended for vibration analysis in airframe structural design is presented. The airframe is represented by a finite element analysis model; the rotor is represented by a general set of linear differential equations with periodic coefficients; and the connections between the rotor and airframe are specified through general linear equations of constraint. Coupling equations are applied to the rotor and airframe equations to produce one set of linear differential equations governing vibrations of the combined rotor airframe system. These equations are solved by the harmonic balance method for the system steady state vibrations. A feature of the solution process is the representation of the airframe in terms of forced responses calculated at the rotor harmonics of interest. A method based on matrix partitioning is worked out for quick recalculations of vibrations in design studies when only relatively few airframe members are varied. All relations are presented in forms suitable for direct computer implementation.

Symmetry groups of integro-differential equations for linear thermoviscoelastic materials with memory

NASA Astrophysics Data System (ADS)

Zhou, L.-Q.; Meleshko, S. V.

2017-07-01

The group analysis method is applied to a system of integro-differential equations corresponding to a linear thermoviscoelastic model. A recently developed approach for calculating the symmetry groups of such equations is used. The general solution of the determining equations for the system is obtained. Using subalgebras of the admitted Lie algebra, two classes of partially invariant solutions of the considered system of integro-differential equations are studied.

Chandrasekhar equations for infinite dimensional systems

NASA Technical Reports Server (NTRS)

Ito, K.; Powers, R.

1985-01-01

The existence of Chandrasekhar equations for linear time-invariant systems defined on Hilbert spaces is investigated. An important consequence is that the solution to the evolutional Riccati equation is strongly differentiable in time, and that a strong solution of the Riccati differential equation can be defined. A discussion of the linear-quadratic optimal-control problem for hereditary differential systems is also included.