Some Applications of Eliashberg Theory
NASA Astrophysics Data System (ADS)
Akis, Richard J.
Eliashberg theory, which was formulated assuming that the electron-phonon interaction is the mechanism for superconductivity, has been very successful in explaining the physical properties of most superconductors. Eliashberg theory is an extension of BCS theory, the original microscopic theory of superconductivity. BCS theory is recovered from Eliashberg theory in the weak electron-boson coupling limit. Recently, a new challenge to Eliashberg theory has been brought forth by the discovery of a new class of superconductors known as the high T_{c} oxides. As of this writing, the question of what is the superconducting mechanism for these materials is still unanswered. In this thesis, many superconducting properties have been calculated mainly in an effort to see if Eliashberg theory may still be applicable to these materials. The approach of this effort has depended on the property being studied. In this case of the critical temperature and the isotope effect, a great deal of work has been put in to fit actual experimental results, particularly for the isotope effect. We shall show that two distinct models, one with an additional electronic mechanism along with the phonons and the other with a very large coulomb repulsion, may be able to explain the experimental results. For the electronic specific heat, maxima that should not be exceeded by an Eliashberg superconductor are established for several quantities associated with this physical property. Unfortunately, some experimental values for these quantities appear to exceed these maxima. In the case of the the nuclear spin relaxation, which has not been very extensively studied in the past, we shall look at how the coherence peak in the relaxation rate can be reduced as a function of coupling strength and draw conclusions that are applicable to conventional superconductors. The behaviour of this property in the oxides is not ignored however, and some fitting of experiment including anisotrophy as well as
NASA Astrophysics Data System (ADS)
Bauer, Johannes; Sachdev, Subir
2015-08-01
We study charge-ordered solutions for fermions on a square lattice interacting with dynamic antiferromagnetic fluctuations. Our approach is based on real-space Eliashberg equations, which are solved self-consistently. We first show that the antiferromagnetic fluctuations can induce arc features in the spectral functions, as spectral weight is suppressed at the hot spots; however, no real pseudogap is generated. At low temperature, spontaneous charge order with a d -form factor can be stabilized for certain parameters. As long as the interacting Fermi surfaces possess hot spots, the ordering wave vector corresponds to the diagonal connection of the hot spots, similar to the non-self-consistent case. Tendencies towards observed axial order only appear in situations without hot spots.
Possible mixed coupling mechanism in FeTe1-x Se x within a multiband Eliashberg approach
NASA Astrophysics Data System (ADS)
Ummarino, G. A.; Daghero, D.
2015-11-01
We show that the phenomenology of the iron chalcogenide superconductor FeTe1-x Se x can be explained within an effective three-band s+/- -wave Eliashberg model. In particular, various experimental data reported in literature—the critical temperature, the energy gaps, the upper critical field, the superfluid density—can be reproduced by this model in a moderate strong-coupling regime provided that both an intraband phononic term and an interband antiferromagnetic spin-fluctuations term are included in the coupling matrix. The intraband coupling is unusual in Fe-based compounds and is required to explain the somehow anomalous association between gap amplitudes and Fermi surfaces, already evidenced by ARPES.
Possible mixed coupling mechanism in FeTe(1-x)Se(x) within a multiband Eliashberg approach.
Ummarino, G A; Daghero, D
2015-11-01
We show that the phenomenology of the iron chalcogenide superconductor FeTe(1-x)Se(x) can be explained within an effective three-band s±-wave Eliashberg model. In particular, various experimental data reported in literature-the critical temperature, the energy gaps, the upper critical field, the superfluid density-can be reproduced by this model in a moderate strong-coupling regime provided that both an intraband phononic term and an interband antiferromagnetic spin-fluctuations term are included in the coupling matrix. The intraband coupling is unusual in Fe-based compounds and is required to explain the somehow anomalous association between gap amplitudes and Fermi surfaces, already evidenced by ARPES. PMID:26445023
NASA Astrophysics Data System (ADS)
Zheng, Jing-Jing; Margine, E. R.
2016-08-01
The ab initio anisotropic Migdal-Eliashberg formalism has been used to examine the pairing mechanism and the nature of the superconducting gap in the recently discovered lithium-decorated monolayer graphene superconductor. Our results provide evidence that the superconducting transition in Li-decorated monolayer graphene can be explained within a standard phonon-mediated mechanism. We predict a single anisotropic superconducting gap and a critical temperature Tc=5.1 -7.6 K , in very good agreement with the experimental results.
ERIC Educational Resources Information Center
Nibbelink, William H.
1990-01-01
Proposed is a gradual transition from arithmetic to the idea of an equation with variables in the elementary grades. Vertical and horizontal formats of open sentences, the instructional sequence, vocabulary, and levels of understanding are discussed in this article. (KR)
NASA Astrophysics Data System (ADS)
Viljamaa, Panu; Jacobs, J. Richard; Chris; JamesHyman; Halma, Matthew; EricNolan; Coxon, Paul
2014-07-01
In reply to a Physics World infographic (part of which is given above) about a study showing that Euler's equation was deemed most beautiful by a group of mathematicians who had been hooked up to a functional magnetic-resonance image (fMRI) machine while viewing mathematical expressions (14 May, http://ow.ly/xHUFi).
1998-11-01
In the late 1950s to early 1960s Rudolph A. Marcus developed a theory for treating the rates of outer-sphere electron-transfer reactions. Outer-sphere reactions are reactions in which an electron is transferred from a donor to an acceptor without any chemical bonds being made or broken. (Electron-transfer reactions in which bonds are made or broken are referred to as inner-sphere reactions.) Marcus derived several very useful expressions, one of which has come to be known as the Marcus cross-relation or, more simply, as the Marcus equation. It is widely used for correlating and predicting electron-transfer rates. For his contributions to the understanding of electron-transfer reactions, Marcus received the 1992 Nobel Prize in Chemistry. This paper discusses the development and use of the Marcus equation. Topics include self-exchange reactions; net electron-transfer reactions; Marcus cross-relation; and proton, hydride, atom and group transfers.
DOE R&D Accomplishments Database
1998-09-21
In the late 1950s to early 1960s Rudolph A. Marcus developed a theory for treating the rates of outer-sphere electron-transfer reactions. Outer-sphere reactions are reactions in which an electron is transferred from a donor to an acceptor without any chemical bonds being made or broken. (Electron-transfer reactions in which bonds are made or broken are referred to as inner-sphere reactions.) Marcus derived several very useful expressions, one of which has come to be known as the Marcus cross-relation or, more simply, as the Marcus equation. It is widely used for correlating and predicting electron-transfer rates. For his contributions to the understanding of electron-transfer reactions, Marcus received the 1992 Nobel Prize in Chemistry. This paper discusses the development and use of the Marcus equation. Topics include self-exchange reactions; net electron-transfer reactions; Marcus cross-relation; and proton, hydride, atom and group transfers.
Shore, B.W.
1981-01-30
The equations of motion are discussed which describe time dependent population flows in an N-level system, reviewing the relationship between incoherent (rate) equations, coherent (Schrodinger) equations, and more general partially coherent (Bloch) equations. Approximations are discussed which replace the elaborate Bloch equations by simpler rate equations whose coefficients incorporate long-time consequences of coherence.
Spin field equations and Heun's equations
NASA Astrophysics Data System (ADS)
Jiang, Min; Wang, Xuejing; Li, Zhongheng
2015-06-01
The Kerr-Newman-(anti) de Sitter metric is the most general stationary black hole solution to the Einstein-Maxwell equation with a cosmological constant. We study the separability of the equations of the massless scalar (spin s=0), neutrino ( s=1/2), electromagnetic ( s=1), Rarita-Schwinger ( s=3/2), and gravitational ( s=2) fields propagating on this background. We obtain the angular and radial master equations, and show that the master equations are transformed to Heun's equation. Meanwhile, we give the condition of existence of event horizons for Kerr-Newman-(anti) de Sitter spacetime by using Sturm theorem.
NASA Technical Reports Server (NTRS)
Hamrock, B. J.; Dowson, D.
1981-01-01
Lubricants, usually Newtonian fluids, are assumed to experience laminar flow. The basic equations used to describe the flow are the Navier-Stokes equation of motion. The study of hydrodynamic lubrication is, from a mathematical standpoint, the application of a reduced form of these Navier-Stokes equations in association with the continuity equation. The Reynolds equation can also be derived from first principles, provided of course that the same basic assumptions are adopted in each case. Both methods are used in deriving the Reynolds equation, and the assumptions inherent in reducing the Navier-Stokes equations are specified. Because the Reynolds equation contains viscosity and density terms and these properties depend on temperature and pressure, it is often necessary to couple the Reynolds with energy equation. The lubricant properties and the energy equation are presented. Film thickness, a parameter of the Reynolds equation, is a function of the elastic behavior of the bearing surface. The governing elasticity equation is therefore presented.
NASA Technical Reports Server (NTRS)
1977-01-01
Basic differential equations governing compressible turbulent boundary layer flow are reviewed, including conservation of mass and energy, momentum equations derived from Navier-Stokes equations, and equations of state. Closure procedures were broken down into: (1) simple or zeroth-order methods, (2) first-order or mean field closure methods, and (3) second-order or mean turbulence field methods.
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)
Single wall penetration equations
NASA Technical Reports Server (NTRS)
Hayashida, K. B.; Robinson, J. H.
1991-01-01
Five single plate penetration equations are compared for accuracy and effectiveness. These five equations are two well-known equations (Fish-Summers and Schmidt-Holsapple), two equations developed by the Apollo project (Rockwell and Johnson Space Center (JSC), and one recently revised from JSC (Cour-Palais). They were derived from test results, with velocities ranging up to 8 km/s. Microsoft Excel software was used to construct a spreadsheet to calculate the diameters and masses of projectiles for various velocities, varying the material properties of both projectile and target for the five single plate penetration equations. The results were plotted on diameter versus velocity graphs for ballistic and spallation limits using Cricket Graph software, for velocities ranging from 2 to 15 km/s defined for the orbital debris. First, these equations were compared to each other, then each equation was compared with various aluminum projectile densities. Finally, these equations were compared with test results performed at JSC for the Marshall Space Flight Center. These equations predict a wide variety of projectile diameters at a given velocity. Thus, it is very difficult to choose the 'right' prediction equation. The thickness of a single plate could have a large variation by choosing a different penetration equation. Even though all five equations are empirically developed with various materials, especially for aluminum alloys, one cannot be confident in the shield design with the predictions obtained by the penetration equations without verifying by tests.
Interpretation of Bernoulli's Equation.
ERIC Educational Resources Information Center
Bauman, Robert P.; Schwaneberg, Rolf
1994-01-01
Discusses Bernoulli's equation with regards to: horizontal flow of incompressible fluids, change of height of incompressible fluids, gases, liquids and gases, and viscous fluids. Provides an interpretation, properties, terminology, and applications of Bernoulli's equation. (MVL)
Reflections on Chemical Equations.
ERIC Educational Resources Information Center
Gorman, Mel
1981-01-01
The issue of how much emphasis balancing chemical equations should have in an introductory chemistry course is discussed. The current heavy emphasis on finishing such equations is viewed as misplaced. (MP)
ERIC Educational Resources Information Center
Fay, Temple H.
2002-01-01
We investigate the pendulum equation [theta] + [lambda][squared] sin [theta] = 0 and two approximations for it. On the one hand, we suggest that the third and fifth-order Taylor series approximations for sin [theta] do not yield very good differential equations to approximate the solution of the pendulum equation unless the initial conditions are…
Solving Ordinary Differential Equations
NASA Technical Reports Server (NTRS)
Krogh, F. T.
1987-01-01
Initial-value ordinary differential equation solution via variable order Adams method (SIVA/DIVA) package is collection of subroutines for solution of nonstiff ordinary differential equations. There are versions for single-precision and double-precision arithmetic. Requires fewer evaluations of derivatives than other variable-order Adams predictor/ corrector methods. Option for direct integration of second-order equations makes integration of trajectory problems significantly more efficient. Written in FORTRAN 77.
Einstein equation at singularities
NASA Astrophysics Data System (ADS)
Stoica, Ovidiu-Cristinel
2014-02-01
Einstein's equation is rewritten in an equivalent form, which remains valid at the singularities in some major cases. These cases include the Schwarzschild singularity, the Friedmann-Lemaître-Robertson-Walker Big Bang singularity, isotropic singularities, and a class of warped product singularities. This equation is constructed in terms of the Ricci part of the Riemann curvature (as the Kulkarni-Nomizu product between Einstein's equation and the metric tensor).
What Makes a Chemical Equation an Equation?
ERIC Educational Resources Information Center
Fensham, Peter J.; Lui, Julia
2001-01-01
Explores how well chemistry graduates preparing for teaching can recognize the similarities and differences between the uses of the word "equation" in mathematics and in chemistry. Reports that the conservation similarities were much less frequently recognized than those involved in the creation of new entities. (Author/MM)
Octonic Gravitational Field Equations
NASA Astrophysics Data System (ADS)
Demir, Süleyman; Tanişli, Murat; Tolan, Tülay
2013-08-01
Generalized field equations of linear gravity are formulated on the basis of octons. When compared to the other eight-component noncommutative hypercomplex number systems, it is demonstrated that associative octons with scalar, pseudoscalar, pseudovector and vector values present a convenient and capable tool to describe the Maxwell-Proca-like field equations of gravitoelectromagnetism in a compact and simple way. Introducing massive graviton and gravitomagnetic monopole terms, the generalized gravitational wave equation and Klein-Gordon equation for linear gravity are also developed.
Octonic massless field equations
NASA Astrophysics Data System (ADS)
Demir, Süleyman; Tanişli, Murat; Kansu, Mustafa Emre
2015-05-01
In this paper, it is proven that the associative octons including scalar, pseudoscalar, pseudovector and vector values are convenient and capable tools to generalize the Maxwell-Dirac like field equations of electromagnetism and linear gravity in a compact and simple way. Although an attempt to describe the massless field equations of electromagnetism and linear gravity needs the sixteen real component mathematical structures, it is proved that these equations can be formulated in terms of eight components of octons. Furthermore, the generalized wave equation in terms of potentials is derived in the presence of electromagnetic and gravitational charges (masses). Finally, conservation of energy concept has also been investigated for massless fields.
Linear Equations: Equivalence = Success
ERIC Educational Resources Information Center
Baratta, Wendy
2011-01-01
The ability to solve linear equations sets students up for success in many areas of mathematics and other disciplines requiring formula manipulations. There are many reasons why solving linear equations is a challenging skill for students to master. One major barrier for students is the inability to interpret the equals sign as anything other than…
Octonic Massive Field Equations
NASA Astrophysics Data System (ADS)
Demir, Süleyman; Kekeç, Seray
2016-07-01
In the present paper we propose the octonic form of massive field equations based on the analogy with electromagnetism and linear gravity. Using the advantages of octon algebra the Maxwell-Dirac-Proca equations have been reformulated in compact and elegant way. The energy-momentum relations for massive field are discussed.
Octonic Massive Field Equations
NASA Astrophysics Data System (ADS)
Demir, Süleyman; Kekeç, Seray
2016-03-01
In the present paper we propose the octonic form of massive field equations based on the analogy with electromagnetism and linear gravity. Using the advantages of octon algebra the Maxwell-Dirac-Proca equations have been reformulated in compact and elegant way. The energy-momentum relations for massive field are discussed.
NASA Astrophysics Data System (ADS)
Zahari, N. M.; Sapar, S. H.; Mohd Atan, K. A.
2013-04-01
This paper discusses an integral solution (a, b, c) of the Diophantine equations x3n+y3n = 2z2n for n ≥ 2 and it is found that the integral solution of these equation are of the form a = b = t2, c = t3 for any integers t.
NASA Astrophysics Data System (ADS)
Molesini, Giuseppe
2005-02-01
Problems in the general validity of the lens equations are reported, requiring an assessment of the conditions for correct use. A discussion is given on critical behaviour of the lens equation, and a sign and meaning scheme is provided so that apparent inconsistencies are avoided.
Yagi, M.; Horton, W. )
1994-07-01
A set of reduced Braginskii equations is derived without assuming flute ordering and the Boussinesq approximation. These model equations conserve the physical energy. It is crucial at finite [beta] that the perpendicular component of Ohm's law be solved to ensure [del][center dot][bold j]=0 for energy conservation.
Nonlinear gyrokinetic equations
Dubin, D.H.E.; Krommes, J.A.; Oberman, C.; Lee, W.W.
1983-03-01
Nonlinear gyrokinetic equations are derived from a systematic Hamiltonian theory. The derivation employs Lie transforms and a noncanonical perturbation theory first used by Littlejohn for the simpler problem of asymptotically small gyroradius. For definiteness, we emphasize the limit of electrostatic fluctuations in slab geometry; however, there is a straight-forward generalization to arbitrary field geometry and electromagnetic perturbations. An energy invariant for the nonlinear system is derived, and various of its limits are considered. The weak turbulence theory of the equations is examined. In particular, the wave kinetic equation of Galeev and Sagdeev is derived from an asystematic truncation of the equations, implying that this equation fails to consider all gyrokinetic effects. The equations are simplified for the case of small but finite gyroradius and put in a form suitable for efficient computer simulation. Although it is possible to derive the Terry-Horton and Hasegawa-Mima equations as limiting cases of our theory, several new nonlinear terms absent from conventional theories appear and are discussed.
NASA Astrophysics Data System (ADS)
Kuksin, Sergei; Maiocchi, Alberto
In this chapter we present a general method of constructing the effective equation which describes the behavior of small-amplitude solutions for a nonlinear PDE in finite volume, provided that the linear part of the equation is a hamiltonian system with a pure imaginary discrete spectrum. The effective equation is obtained by retaining only the resonant terms of the nonlinearity (which may be hamiltonian, or may be not); the assertion that it describes the limiting behavior of small-amplitude solutions is a rigorous mathematical theorem. In particular, the method applies to the three- and four-wave systems. We demonstrate that different possible types of energy transport are covered by this method, depending on whether the set of resonances splits into finite clusters (this happens, e.g. in case of the Charney-Hasegawa-Mima equation), or is connected (this happens, e.g. in the case of the NLS equation if the space-dimension is at least two). For equations of the first type the energy transition to high frequencies does not hold, while for equations of the second type it may take place. Our method applies to various weakly nonlinear wave systems, appearing in plasma, meteorology and oceanography.
NASA Astrophysics Data System (ADS)
Sultana, Nasrin
This dissertation consists of five papers in which discrete Volterra equations of different types and orders are studied and results regarding the behavior of their solutions are established. The first paper presents some fundamental results about subexponential sequences. It also illustrates the subexponential solutions of scalar linear Volterra sum-difference equations are asymptotically stable. The exact value of the rate of convergence of asymptotically stable solutions is found by determining the asymptotic behavior of the transient renewal equations. The study of subexponential solutions is also continued in the second and third articles. The second paper investigates the same equation using the same process as considered in the first paper. The discussion focuses on a positive lower bound of the rate of convergence of the asymptotically stable solutions. The third paper addresses the rate of convergence of the solutions of scalar linear Volterra sum-difference equations with delay. The result is proved by developing the rate of convergence of transient renewal delay difference equations. The fourth paper discusses the existence of bounded solutions on an unbounded domain of more general nonlinear Volterra sum-difference equations using the Schaefer fixed point theorem and the Lyapunov direct method. The fifth paper examines the asymptotic behavior of nonoscillatory solutions of higher-order integro-dynamic equations and establishes some new criteria based on so-called time scales, which unifies and extends both discrete and continuous mathematical analysis. Beside these five research papers that focus on discrete Volterra equations, this dissertation also contains an introduction, a section on difference calculus, a section on time scales calculus, and a conclusion.
Nonlinear ordinary difference equations
NASA Technical Reports Server (NTRS)
Caughey, T. K.
1979-01-01
Future space vehicles will be relatively large and flexible, and active control will be necessary to maintain geometrical configuration. While the stresses and strains in these space vehicles are not expected to be excessively large, their cumulative effects will cause significant geometrical nonlinearities to appear in the equations of motion, in addition to the nonlinearities caused by material properties. Since the only effective tool for the analysis of such large complex structures is the digital computer, it will be necessary to gain a better understanding of the nonlinear ordinary difference equations which result from the time discretization of the semidiscrete equations of motion for such structures.
NASA Astrophysics Data System (ADS)
Pierret, Frédéric
2016-02-01
We derived the equations of Celestial Mechanics governing the variation of the orbital elements under a stochastic perturbation, thereby generalizing the classical Gauss equations. Explicit formulas are given for the semimajor axis, the eccentricity, the inclination, the longitude of the ascending node, the pericenter angle, and the mean anomaly, which are expressed in term of the angular momentum vector H per unit of mass and the energy E per unit of mass. Together, these formulas are called the stochastic Gauss equations, and they are illustrated numerically on an example from satellite dynamics.
A Comparison of IRT Equating and Beta 4 Equating.
ERIC Educational Resources Information Center
Kim, Dong-In; Brennan, Robert; Kolen, Michael
Four equating methods were compared using four equating criteria: first-order equity (FOE), second-order equity (SOE), conditional mean squared error (CMSE) difference, and the equipercentile equating property. The four methods were: (1) three parameter logistic (3PL) model true score equating; (2) 3PL observed score equating; (3) beta 4 true…
Equations For Rotary Transformers
NASA Technical Reports Server (NTRS)
Salomon, Phil M.; Wiktor, Peter J.; Marchetto, Carl A.
1988-01-01
Equations derived for input impedance, input power, and ratio of secondary current to primary current of rotary transformer. Used for quick analysis of transformer designs. Circuit model commonly used in textbooks on theory of ac circuits.
Diophantine Equations and Computation
NASA Astrophysics Data System (ADS)
Davis, Martin
Unless otherwise stated, we’ll work with the natural numbers: N = \\{0,1,2,3, dots\\}. Consider a Diophantine equation F(a1,a2,...,an,x1,x2,...,xm) = 0 with parameters a1,a2,...,an and unknowns x1,x2,...,xm For such a given equation, it is usual to ask: For which values of the parameters does the equation have a solution in the unknowns? In other words, find the set: \\{
Regularized Structural Equation Modeling
Jacobucci, Ross; Grimm, Kevin J.; McArdle, John J.
2016-01-01
A new method is proposed that extends the use of regularization in both lasso and ridge regression to structural equation models. The method is termed regularized structural equation modeling (RegSEM). RegSEM penalizes specific parameters in structural equation models, with the goal of creating easier to understand and simpler models. Although regularization has gained wide adoption in regression, very little has transferred to models with latent variables. By adding penalties to specific parameters in a structural equation model, researchers have a high level of flexibility in reducing model complexity, overcoming poor fitting models, and the creation of models that are more likely to generalize to new samples. The proposed method was evaluated through a simulation study, two illustrative examples involving a measurement model, and one empirical example involving the structural part of the model to demonstrate RegSEM’s utility. PMID:27398019
Nonlinear differential equations
Dresner, L.
1988-01-01
This report is the text of a graduate course on nonlinear differential equations given by the author at the University of Wisconsin-Madison during the summer of 1987. The topics covered are: direction fields of first-order differential equations; the Lie (group) theory of ordinary differential equations; similarity solutions of second-order partial differential equations; maximum principles and differential inequalities; monotone operators and iteration; complementary variational principles; and stability of numerical methods. The report should be of interest to graduate students, faculty, and practicing scientists and engineers. No prior knowledge is required beyond a good working knowledge of the calculus. The emphasis is on practical results. Most of the illustrative examples are taken from the fields of nonlinear diffusion, heat and mass transfer, applied superconductivity, and helium cryogenics.
Equating Training to Education.
ERIC Educational Resources Information Center
Davis, Lansing J.
1993-01-01
Distinguishes between education and employer-sponsored training in terms of process, purpose, and providers. Concludes that work-related training and postsecondary education are cognates within the classification education, and equating their learning outcomes is appropriate. (SK)
Relativistic Guiding Center Equations
White, R. B.; Gobbin, M.
2014-10-01
In toroidal fusion devices it is relatively easy that electrons achieve relativistic velocities, so to simulate runaway electrons and other high energy phenomena a nonrelativistic guiding center formalism is not sufficient. Relativistic guiding center equations including flute mode time dependent field perturbations are derived. The same variables as used in a previous nonrelativistic guiding center code are adopted, so that a straightforward modifications of those equations can produce a relativistic version.
SIMULTANEOUS DIFFERENTIAL EQUATION COMPUTER
Collier, D.M.; Meeks, L.A.; Palmer, J.P.
1960-05-10
A description is given for an electronic simulator for a system of simultaneous differential equations, including nonlinear equations. As a specific example, a homogeneous nuclear reactor system including a reactor fluid, heat exchanger, and a steam boiler may be simulated, with the nonlinearity resulting from a consideration of temperature effects taken into account. The simulator includes three operational amplifiers, a multiplier, appropriate potential sources, and interconnecting R-C networks.
Set Equation Transformation System.
Energy Science and Technology Software Center (ESTSC)
2002-03-22
Version 00 SETS is used for symbolic manipulation of Boolean equations, particularly the reduction of equations by the application of Boolean identities. It is a flexible and efficient tool for performing probabilistic risk analysis (PRA), vital area analysis, and common cause analysis. The equation manipulation capabilities of SETS can also be used to analyze noncoherent fault trees and determine prime implicants of Boolean functions, to verify circuit design implementation, to determine minimum cost fire protectionmore » requirements for nuclear reactor plants, to obtain solutions to combinatorial optimization problems with Boolean constraints, and to determine the susceptibility of a facility to unauthorized access through nullification of sensors in its protection system. Two auxiliary programs, SEP and FTD, are included. SEP performs the quantitative analysis of reduced Boolean equations (minimal cut sets) produced by SETS. The user can manipulate and evaluate the equations to find the probability of occurrence of any desired event and to produce an importance ranking of the terms and events in an equation. FTD is a fault tree drawing program which uses the proprietary ISSCO DISSPLA graphics software to produce an annotated drawing of a fault tree processed by SETS. The DISSPLA routines are not included.« less
The Bernoulli-Poiseuille Equation.
ERIC Educational Resources Information Center
Badeer, Henry S.; Synolakis, Costas E.
1989-01-01
Describes Bernoulli's equation and Poiseuille's equation for fluid dynamics. Discusses the application of the combined Bernoulli-Poiseuille equation in real flows, such as viscous flows under gravity and acceleration. (YP)
Introducing Chemical Formulae and Equations.
ERIC Educational Resources Information Center
Dawson, Chris; Rowell, Jack
1979-01-01
Discusses when the writing of chemical formula and equations can be introduced in the school science curriculum. Also presents ways in which formulae and equations learning can be aided and some examples for balancing and interpreting equations. (HM)
Parallel tridiagonal equation solvers
NASA Technical Reports Server (NTRS)
Stone, H. S.
1974-01-01
Three parallel algorithms were compared for the direct solution of tridiagonal linear systems of equations. The algorithms are suitable for computers such as ILLIAC 4 and CDC STAR. For array computers similar to ILLIAC 4, cyclic odd-even reduction has the least operation count for highly structured sets of equations, and recursive doubling has the least count for relatively unstructured sets of equations. Since the difference in operation counts for these two algorithms is not substantial, their relative running times may be more related to overhead operations, which are not measured in this paper. The third algorithm, based on Buneman's Poisson solver, has more arithmetic operations than the others, and appears to be the least favorable. For pipeline computers similar to CDC STAR, cyclic odd-even reduction appears to be the most preferable algorithm for all cases.
Nonlocal electrical diffusion equation
NASA Astrophysics Data System (ADS)
Gómez-Aguilar, J. F.; Escobar-Jiménez, R. F.; Olivares-Peregrino, V. H.; Benavides-Cruz, M.; Calderón-Ramón, C.
2016-07-01
In this paper, we present an analysis and modeling of the electrical diffusion equation using the fractional calculus approach. This alternative representation for the current density is expressed in terms of the Caputo derivatives, the order for the space domain is 0<β≤1 and for the time domain is 0<γ≤2. We present solutions for the full fractional equation involving space and time fractional derivatives using numerical methods based on Fourier variable separation. The case with spatial fractional derivatives leads to Levy flight type phenomena, while the time fractional equation is related to sub- or super diffusion. We show that the mathematical concept of fractional derivatives can be useful to understand the behavior of semiconductors, the design of solar panels, electrochemical phenomena and the description of anomalous complex processes.
Difference equation for superradiance
NASA Technical Reports Server (NTRS)
Lee, C. T.
1974-01-01
The evolution of a completely excited system of N two-level atoms, distributed over a large region and interacting with all modes of radiation field, is studied. The distinction between r-conserving (RC) and r-nonconserving (RNC) processes is emphasized. Considering the number of photons emitted as the discrete independent variable, the evolution is described by a partial difference equation. Numerical solution of this equation shows the transition from RNC dominance at the beginning to RC dominance later. This is also a transition from incoherent to coherent emission of radiation.
NASA Astrophysics Data System (ADS)
Biagetti, Matteo; Desjacques, Vincent; Kehagias, Alex; Racco, Davide; Riotto, Antonio
2016-04-01
Dark matter halos are the building blocks of the universe as they host galaxies and clusters. The knowledge of the clustering properties of halos is therefore essential for the understanding of the galaxy statistical properties. We derive an effective halo Boltzmann equation which can be used to describe the halo clustering statistics. In particular, we show how the halo Boltzmann equation encodes a statistically biased gravitational force which generates a bias in the peculiar velocities of virialized halos with respect to the underlying dark matter, as recently observed in N-body simulations.
NASA Technical Reports Server (NTRS)
Markley, F. Landis
1995-01-01
Kepler's Equation is solved over the entire range of elliptic motion by a fifth-order refinement of the solution of a cubic equation. This method is not iterative, and requires only four transcendental function evaluations: a square root, a cube root, and two trigonometric functions. The maximum relative error of the algorithm is less than one part in 10(exp 18), exceeding the capability of double-precision computer arithmetic. Roundoff errors in double-precision implementation of the algorithm are addressed, and procedures to avoid them are developed.
Obtaining Maxwell's equations heuristically
NASA Astrophysics Data System (ADS)
Diener, Gerhard; Weissbarth, Jürgen; Grossmann, Frank; Schmidt, Rüdiger
2013-02-01
Starting from the experimental fact that a moving charge experiences the Lorentz force and applying the fundamental principles of simplicity (first order derivatives only) and linearity (superposition principle), we show that the structure of the microscopic Maxwell equations for the electromagnetic fields can be deduced heuristically by using the transformation properties of the fields under space inversion and time reversal. Using the experimental facts of charge conservation and that electromagnetic waves propagate with the speed of light, together with Galilean invariance of the Lorentz force, allows us to finalize Maxwell's equations and to introduce arbitrary electrodynamics units naturally.
The Statistical Drake Equation
NASA Astrophysics Data System (ADS)
Maccone, Claudio
2010-12-01
We provide the statistical generalization of the Drake equation. From a simple product of seven positive numbers, the Drake equation is now turned into the product of seven positive random variables. We call this "the Statistical Drake Equation". The mathematical consequences of this transformation are then derived. The proof of our results is based on the Central Limit Theorem (CLT) of Statistics. In loose terms, the CLT states that the sum of any number of independent random variables, each of which may be ARBITRARILY distributed, approaches a Gaussian (i.e. normal) random variable. This is called the Lyapunov Form of the CLT, or the Lindeberg Form of the CLT, depending on the mathematical constraints assumed on the third moments of the various probability distributions. In conclusion, we show that: The new random variable N, yielding the number of communicating civilizations in the Galaxy, follows the LOGNORMAL distribution. Then, as a consequence, the mean value of this lognormal distribution is the ordinary N in the Drake equation. The standard deviation, mode, and all the moments of this lognormal N are also found. The seven factors in the ordinary Drake equation now become seven positive random variables. The probability distribution of each random variable may be ARBITRARY. The CLT in the so-called Lyapunov or Lindeberg forms (that both do not assume the factors to be identically distributed) allows for that. In other words, the CLT "translates" into our statistical Drake equation by allowing an arbitrary probability distribution for each factor. This is both physically realistic and practically very useful, of course. An application of our statistical Drake equation then follows. The (average) DISTANCE between any two neighboring and communicating civilizations in the Galaxy may be shown to be inversely proportional to the cubic root of N. Then, in our approach, this distance becomes a new random variable. We derive the relevant probability density
Comparison of Kernel Equating and Item Response Theory Equating Methods
ERIC Educational Resources Information Center
Meng, Yu
2012-01-01
The kernel method of test equating is a unified approach to test equating with some advantages over traditional equating methods. Therefore, it is important to evaluate in a comprehensive way the usefulness and appropriateness of the Kernel equating (KE) method, as well as its advantages and disadvantages compared with several popular item…
Structural Equation Model Trees
ERIC Educational Resources Information Center
Brandmaier, Andreas M.; von Oertzen, Timo; McArdle, John J.; Lindenberger, Ulman
2013-01-01
In the behavioral and social sciences, structural equation models (SEMs) have become widely accepted as a modeling tool for the relation between latent and observed variables. SEMs can be seen as a unification of several multivariate analysis techniques. SEM Trees combine the strengths of SEMs and the decision tree paradigm by building tree…
Parallel Multigrid Equation Solver
Energy Science and Technology Software Center (ESTSC)
2001-09-07
Prometheus is a fully parallel multigrid equation solver for matrices that arise in unstructured grid finite element applications. It includes a geometric and an algebraic multigrid method and has solved problems of up to 76 mullion degrees of feedom, problems in linear elasticity on the ASCI blue pacific and ASCI red machines.
ERIC Educational Resources Information Center
Fay, Temple H.
2010-01-01
Through numerical investigations, we study examples of the forced quadratic spring equation [image omitted]. By performing trial-and-error numerical experiments, we demonstrate the existence of stability boundaries in the phase plane indicating initial conditions yielding bounded solutions, investigate the resonance boundary in the [omega]…
ERIC Educational Resources Information Center
Savoy, L. G.
1988-01-01
Describes a study of students' ability to balance equations. Answers to a test on this topic were analyzed to determine the level of understanding and processes used by the students. Presented is a method to teach this skill to high school chemistry students. (CW)
Modelling by Differential Equations
ERIC Educational Resources Information Center
Chaachoua, Hamid; Saglam, Ayse
2006-01-01
This paper aims to show the close relation between physics and mathematics taking into account especially the theory of differential equations. By analysing the problems posed by scientists in the seventeenth century, we note that physics is very important for the emergence of this theory. Taking into account this analysis, we show the…
Do Differential Equations Swing?
ERIC Educational Resources Information Center
Maruszewski, Richard F., Jr.
2006-01-01
One of the units of in a standard differential equations course is a discussion of the oscillatory motion of a spring and the associated material on forcing functions and resonance. During the presentation on practical resonance, the instructor may tell students that it is similar to when they take their siblings to the playground and help them on…
Generalized reduced magnetohydrodynamic equations
Kruger, S.E.
1999-02-01
A new derivation of reduced magnetohydrodynamic (MHD) equations is presented. A multiple-time-scale expansion is employed. It has the advantage of clearly separating the three time scales of the problem associated with (1) MHD equilibrium, (2) fluctuations whose wave vector is aligned perpendicular to the magnetic field, and (3) those aligned parallel to the magnetic field. The derivation is carried out without relying on a large aspect ratio assumption; therefore this model can be applied to any general configuration. By accounting for the MHD equilibrium and constraints to eliminate the fast perpendicular waves, equations are derived to evolve scalar potential quantities on a time scale associated with the parallel wave vector (shear-Alfven wave time scale), which is the time scale of interest for MHD instability studies. Careful attention is given in the derivation to satisfy energy conservation and to have manifestly divergence-free magnetic fields to all orders in the expansion parameter. Additionally, neoclassical closures and equilibrium shear flow effects are easily accounted for in this model. Equations for the inner resistive layer are derived which reproduce the linear ideal and resistive stability criterion of Glasser, Greene, and Johnson. The equations have been programmed into a spectral initial value code and run with shear flow that is consistent with the equilibrium input into the code. Linear results of tearing modes with shear flow are presented which differentiate the effects of shear flow gradients in the layer with the effects of the shear flow decoupling multiple harmonics.
Brownian motion from Boltzmann's equation.
NASA Technical Reports Server (NTRS)
Montgomery, D.
1971-01-01
Two apparently disparate lines of inquiry in kinetic theory are shown to be equivalent: (1) Brownian motion as treated by the (stochastic) Langevin equation and Fokker-Planck equation; and (2) Boltzmann's equation. The method is to derive the kinetic equation for Brownian motion from the Boltzmann equation for a two-component neutral gas by a simultaneous expansion in the density and mass ratios.
Supersymmetric fifth order evolution equations
Tian, K.; Liu, Q. P.
2010-03-08
This paper considers supersymmetric fifth order evolution equations. Within the framework of symmetry approach, we give a list containing six equations, which are (potentially) integrable systems. Among these equations, the most interesting ones include a supersymmetric Sawada-Kotera equation and a novel supersymmetric fifth order KdV equation. For the latter, we supply some properties such as a Hamiltonian structures and a possible recursion operator.
Biaxial constitutive equation development
NASA Technical Reports Server (NTRS)
Jordan, E. H.; Walker, K. P.
1984-01-01
In developing the constitutive equations an interdisciplinary approach is being pursued. Specifically, both metallurgical and continuum mechanics considerations are recognized in the formulation. Experiments will be utilized to both explore general qualitative features of the material behavior that needs to be modeled and to provide a means of assessing the validity of the equations being developed. The model under development explicitly recognizes crystallographic slip on the individual slip systems. This makes possible direct representation of specific slip system phenomena. The present constitutive formulation takes the anisotropic creep theory and incorporates two state variables into the model to account for the effect of prior inelastic deformation history on the current rate-dependent response of the material.
Causal electromagnetic interaction equations
Zinoviev, Yury M.
2011-02-15
For the electromagnetic interaction of two particles the relativistic causal quantum mechanics equations are proposed. These equations are solved for the case when the second particle moves freely. The initial wave functions are supposed to be smooth and rapidly decreasing at the infinity. This condition is important for the convergence of the integrals similar to the integrals of quantum electrodynamics. We also consider the singular initial wave functions in the particular case when the second particle mass is equal to zero. The discrete energy spectrum of the first particle wave function is defined by the initial wave function of the free-moving second particle. Choosing the initial wave functions of the free-moving second particle it is possible to obtain a practically arbitrary discrete energy spectrum.
Nikolaevskiy equation with dispersion.
Simbawa, Eman; Matthews, Paul C; Cox, Stephen M
2010-03-01
The Nikolaevskiy equation was originally proposed as a model for seismic waves and is also a model for a wide variety of systems incorporating a neutral "Goldstone" mode, including electroconvection and reaction-diffusion systems. It is known to exhibit chaotic dynamics at the onset of pattern formation, at least when the dispersive terms in the equation are suppressed, as is commonly the practice in previous analyses. In this paper, the effects of reinstating the dispersive terms are examined. It is shown that such terms can stabilize some of the spatially periodic traveling waves; this allows us to study the loss of stability and transition to chaos of the waves. The secondary stability diagram ("Busse balloon") for the traveling waves can be remarkably complicated. PMID:20365845
NASA Astrophysics Data System (ADS)
Li, Wu
2015-08-01
We demonstrate the ab initio electrical transport calculation limited by electron-phonon coupling by using the full solution of the Boltzmann transport equation (BTE), which applies equally to metals and semiconductors. Numerical issues are emphasized in this work. We show that the simple linear interpolation of the electron-phonon coupling matrix elements from a relatively coarse grid to an extremely fine grid can ease the calculational burden, which makes the calculation feasible in practice. For the Brillouin zone (BZ) integration of the transition probabilities involving one δ function, the Gaussian smearing method with a physical choice of locally adaptive broadening parameters is employed. We validate the calculation in the cases of n -type Si and Al. The calculated conductivity and mobility are in good agreement with experiments. In the metal case we also demonstrate that the Gaussian smearing method with locally adaptive broadening parameters works excellently for the BZ integration with double δ functions involved in the Eliashberg spectral function and its transport variant. The simpler implementation is the advantage of the Gaussian smearing method over the tetrahedron method. The accuracy of the relaxation time approximation and the approximation made by Allen [Phys. Rev. B 17, 3725 (1978), 10.1103/PhysRevB.17.3725] has been examined by comparing with the exact solution of BTE. We also apply our method to n -type monolayer MoS2, for which a mobility of 150 cm2 v-1 s-1 is obtained at room temperature. Moreover, the mean free paths are less than 9 nm, indicating that in the presence of grain boundaries the mobilities should not be effectively affected if the grain boundary size is tens of nanometers or larger. The ab initio approach demonstrated in this paper can be directly applied to other materials without the need for any a priori knowledge about the electron-phonon scattering processes, and can be straightforwardly extended to study cases with
Multinomial diffusion equation
NASA Astrophysics Data System (ADS)
Balter, Ariel; Tartakovsky, Alexandre M.
2011-06-01
We describe a new, microscopic model for diffusion that captures diffusion induced fluctuations at scales where the concept of concentration gives way to discrete particles. We show that in the limit as the number of particles N→∞, our model is equivalent to the classical stochastic diffusion equation (SDE). We test our new model and the SDE against Langevin dynamics in numerical simulations, and show that our model successfully reproduces the correct ensemble statistics, while the classical model fails.
Multinomial diffusion equation
Balter, Ariel I.; Tartakovsky, Alexandre M.
2011-06-24
We describe a new, microscopic model for diffusion that captures diffusion induced uctuations at scales where the concept of concentration gives way to discrete par- ticles. We show that in the limit as the number of particles N ! 1, our model is equivalent to the classical stochastic diffusion equation (SDE). We test our new model and the SDE against Langevin dynamics in numerical simulations, and show that our model successfully reproduces the correct ensemble statistics, while the classical model fails.
Singularities for PRANDTL'S Equations
NASA Astrophysics Data System (ADS)
Lo Bosco, G.; Sammartino, M.; Sciacca, V.
2006-03-01
We use a mixed spectral/finite-difference numerical method to investigate the possibility of a finite time blow-up of the solutions of Prandtl's equations for the case of the impulsively started cylinder. Our tool is the complex singularity tracking method. We show that a cubic root singularity seems to develop, in a time that can be made arbitrarily short, from a class of data uniformly bounded in H1.
ERIC Educational Resources Information Center
Vasile, Daniela
2012-01-01
We are frequently told that Hong Kong has a model system for learning mathematics. In this article Daniela Vasile notes one short-coming in that the pupils are not taught to problem-solve. She begins with a new class by asking them to write down the craziest equation they can come up with and bases her whole lesson, and the following homework,…
NASA Astrophysics Data System (ADS)
Konesky, Gregory
2009-08-01
In the almost half century since the Drake Equation was first conceived, a number of profound discoveries have been made that require each of the seven variables of this equation to be reconsidered. The discovery of hydrothermal vents on the ocean floor, for example, as well as the ever-increasing extreme conditions in which life is found on Earth, suggest a much wider range of possible extraterrestrial habitats. The growing consensus that life originated very early in Earth's history also supports this suggestion. The discovery of exoplanets with a wide range of host star types, and attendant habitable zones, suggests that life may be possible in planetary systems with stars quite unlike our Sun. Stellar evolution also plays an important part in that habitable zones are mobile. The increasing brightness of our Sun over the next few billion years, will place the Earth well outside the present habitable zone, but will then encompass Mars, giving rise to the notion that some Drake Equation variables, such as the fraction of planets on which life emerges, may have multiple values.
Generalized reduced MHD equations
Kruger, S.E.; Hegna, C.C.; Callen, J.D.
1998-07-01
A new derivation of reduced magnetohydrodynamic (MHD) equations is presented. A multiple-time-scale expansion is employed. It has the advantage of clearly separating the three time scales of the problem associated with (1) MHD equilibrium, (2) fluctuations whose wave vector is aligned perpendicular to the magnetic field, and (3) those aligned parallel to the magnetic field. The derivation is carried out without relying on a large aspect ratio assumption; therefore this model can be applied to any general toroidal configuration. By accounting for the MHD equilibrium and constraints to eliminate the fast perpendicular waves, equations are derived to evolve scalar potential quantities on a time scale associated with the parallel wave vector (shear-alfven wave time scale), which is the time scale of interest for MHD instability studies. Careful attention is given in the derivation to satisfy energy conservation and to have manifestly divergence-free magnetic fields to all orders in the expansion parameter. Additionally, neoclassical closures and equilibrium shear flow effects are easily accounted for in this model. Equations for the inner resistive layer are derived which reproduce the linear ideal and resistive stability criterion of Glasser, Greene, and Johnson.
The compressible adjoint equations in geodynamics: equations and numerical assessment
NASA Astrophysics Data System (ADS)
Ghelichkhan, Siavash; Bunge, Hans-Peter
2016-04-01
The adjoint method is a powerful means to obtain gradient information in a mantle convection model relative to past flow structure. While the adjoint equations in geodynamics have been derived for the conservation equations of mantle flow in their incompressible form, the applicability of this approximation to Earth is limited, because density increases by almost a factor of two from the surface to the Core Mantle Boundary. Here we introduce the compressible adjoint equations for the conservation equations in the anelastic-liquid approximation. Our derivation applies an operator formulation in Hilbert spaces, to connect to recent work in seismology (Fichtner et al (2006)) and geodynamics (Horbach et al (2014)), where the approach was used to derive the adjoint equations for the wave equation and incompressible mantle flow. We present numerical tests of the newly derived equations based on twin experiments, focusing on three simulations. A first, termed Compressible, assumes the compressible forward and adjoint equations, and represents the consistent means of including compressibility effects. A second, termed Mixed, applies the compressible forward equation, but ignores compressibility effects in the adjoint equations, where the incompressible equations are used instead. A third simulation, termed Incompressible, neglects compressibility effects entirely in the forward and adjoint equations relative to the reference twin. The compressible and mixed formulations successfully restore earlier mantle flow structure, while the incompressible formulation yields noticeable artifacts. Our results suggest the use of a compressible formulation, when applying the adjoint method to seismically derived mantle heterogeneity structure.
Differential Equations Compatible with Boundary Rational qKZ Equation
NASA Astrophysics Data System (ADS)
Takeyama, Yoshihiro
2011-10-01
We give diffierential equations compatible with the rational qKZ equation with boundary reflection. The total system contains the trigonometric degeneration of the bispectral qKZ equation of type (Cěen, Cn) which in the case of type GLn was studied by van Meer and Stokman. We construct an integral formula for solutions to our compatible system in a special case.
Estimating Equating Error in Observed-Score Equating. Research Report.
ERIC Educational Resources Information Center
van der Linden, Wim J.
Traditionally, error in equating observed scores on two versions of a test is defined as the difference between the transformations that equate the quantiles of their distributions in the sample and in the population of examinees. This definition underlies, for example, the well-known approximation to the standard error of equating by Lord (1982).…
Makkonen, Lasse
2016-04-01
Young's construction for a contact angle at a three-phase intersection forms the basis of all fields of science that involve wetting and capillary action. We find compelling evidence from recent experimental results on the deformation of a soft solid at the contact line, and displacement of an elastic wire immersed in a liquid, that Young's equation can only be interpreted by surface energies, and not as a balance of surface tensions. It follows that the a priori variable in finding equilibrium is not the position of the contact line, but the contact angle. This finding provides the explanation for the pinning of a contact line. PMID:26940644
Solitons and nonlinear wave equations
Dodd, Roger K.; Eilbeck, J. Chris; Gibbon, John D.; Morris, Hedley C.
1982-01-01
A discussion of the theory and applications of classical solitons is presented with a brief treatment of quantum mechanical effects which occur in particle physics and quantum field theory. The subjects addressed include: solitary waves and solitons, scattering transforms, the Schroedinger equation and the Korteweg-de Vries equation, and the inverse method for the isospectral Schroedinger equation and the general solution of the solvable nonlinear equations. Also considered are: isolation of the Korteweg-de Vries equation in some physical examples, the Zakharov-Shabat/AKNS inverse method, kinks and the sine-Gordon equation, the nonlinear Schroedinger equation and wave resonance interactions, amplitude equations in unstable systems, and numerical studies of solitons. 45 references.
Noncommutativity and the Friedmann Equations
NASA Astrophysics Data System (ADS)
Sabido, M.; Guzmán, W.; Socorro, J.
2010-07-01
In this paper we study noncommutative scalar field cosmology, we find the noncommutative Friedmann equations as well as the noncommutative Klein-Gordon equation, interestingly the noncommutative contributions are only present up to second order in the noncommutitive parameter.
Conservational PDF Equations of Turbulence
NASA Technical Reports Server (NTRS)
Shih, Tsan-Hsing; Liu, Nan-Suey
2010-01-01
Recently we have revisited the traditional probability density function (PDF) equations for the velocity and species in turbulent incompressible flows. They are all unclosed due to the appearance of various conditional means which are modeled empirically. However, we have observed that it is possible to establish a closed velocity PDF equation and a closed joint velocity and species PDF equation through conditions derived from the integral form of the Navier-Stokes equations. Although, in theory, the resulted PDF equations are neither general nor unique, they nevertheless lead to the exact transport equations for the first moment as well as all higher order moments. We refer these PDF equations as the conservational PDF equations. This observation is worth further exploration for its validity and CFD application
Thaller, B.
1992-01-01
This monograph treats most of the usual material to be found in texts on the Dirac equation such as the basic formalism of quantum mechanics, representations of Dirac matrices, covariant realization of the Dirac equation, interpretation of negative energies, Foldy-Wouthuysen transformation, Klein's paradox, spherically symmetric interactions and a treatment of the relativistic hydrogen atom, etc., and also provides excellent additional treatments of a variety of other relevant topics. The monograph contains an extensive treatment of the Lorentz and Poincare groups and their representations. The author discusses in depth Lie algebaic and projective representations, covering groups, and Mackey's theory and Wigner's realization of induced representations. A careful classification of external fields with respect to their behavior under Poincare transformations is supplemented by a basic account of self-adjointness and spectral properties of Dirac operators. A state-of-the-art treatment of relativistic scattering theory based on a time-dependent approach originally due to Enss is presented. An excellent introduction to quantum electrodynamics in external fields is provided. Various appendices containing further details, notes on each chapter commenting on the history involved and referring to original research papers and further developments in the literature, and a bibliography covering all relevant monographs and over 500 articles on the subject, complete this text. This book should satisfy the needs of a wide audience, ranging from graduate students in theoretical physics and mathematics to researchers interested in mathematical physics.
Inequivalence between the Schroedinger equation and the Madelung hydrodynamic equations
Wallstrom, T.C.
1994-03-01
By differentiating the Schroedinger equation and separating the real amd imaginary parts, one obtains the Madelung hydrodynamic equations, which have inspired numerous classical interpretations of quantum mechanics. Such interpretations frequently assume that these equations are equivalent to the Schroedinger equation, and thus provide an alternative basis for quantum mechanics. This paper proves that this is incorrect: to recover the Schroedinger equation, one must add by hand a quantization condition, as in the old quantum theory. The implications for various alternative interpretations of quantum mechanics are discussed.
``Riemann equations'' in bidifferential calculus
NASA Astrophysics Data System (ADS)
Chvartatskyi, O.; Müller-Hoissen, F.; Stoilov, N.
2015-10-01
We consider equations that formally resemble a matrix Riemann (or Hopf) equation in the framework of bidifferential calculus. With different choices of a first-order bidifferential calculus, we obtain a variety of equations, including a semi-discrete and a fully discrete version of the matrix Riemann equation. A corresponding universal solution-generating method then either yields a (continuous or discrete) Cole-Hopf transformation, or leaves us with the problem of solving Riemann equations (hence an application of the hodograph method). If the bidifferential calculus extends to second order, solutions of a system of "Riemann equations" are also solutions of an equation that arises, on the universal level of bidifferential calculus, as an integrability condition. Depending on the choice of bidifferential calculus, the latter can represent a number of prominent integrable equations, like self-dual Yang-Mills, as well as matrix versions of the two-dimensional Toda lattice, Hirota's bilinear difference equation, (2+1)-dimensional Nonlinear Schrödinger (NLS), Kadomtsev-Petviashvili (KP) equation, and Davey-Stewartson equations. For all of them, a recent (non-isospectral) binary Darboux transformation result in bidifferential calculus applies, which can be specialized to generate solutions of the associated "Riemann equations." For the latter, we clarify the relation between these specialized binary Darboux transformations and the aforementioned solution-generating method. From (arbitrary size) matrix versions of the "Riemann equations" associated with an integrable equation, possessing a bidifferential calculus formulation, multi-soliton-type solutions of the latter can be generated. This includes "breaking" multi-soliton-type solutions of the self-dual Yang-Mills and the (2+1)-dimensional NLS equation, which are parametrized by solutions of Riemann equations.
Solving Nonlinear Coupled Differential Equations
NASA Technical Reports Server (NTRS)
Mitchell, L.; David, J.
1986-01-01
Harmonic balance method developed to obtain approximate steady-state solutions for nonlinear coupled ordinary differential equations. Method usable with transfer matrices commonly used to analyze shaft systems. Solution to nonlinear equation, with periodic forcing function represented as sum of series similar to Fourier series but with form of terms suggested by equation itself.
The Forced Hard Spring Equation
ERIC Educational Resources Information Center
Fay, Temple H.
2006-01-01
Through numerical investigations, various examples of the Duffing type forced spring equation with epsilon positive, are studied. Since [epsilon] is positive, all solutions to the associated homogeneous equation are periodic and the same is true with the forcing applied. The damped equation exhibits steady state trajectories with the interesting…
Evaluating Cross-Lingual Equating.
ERIC Educational Resources Information Center
Rapp, Joel; Allalouf, Avi
This study examined the cross-lingual equating process adopted by a large scale testing system in which target language (TL) forms are equated to the source language (SL) forms using a set of translated items. The focus was on evaluating the degree of error inherent in the routine cross-lingual equating of the Verbal Reasoning subtest of the…
Successfully Transitioning to Linear Equations
ERIC Educational Resources Information Center
Colton, Connie; Smith, Wendy M.
2014-01-01
The Common Core State Standards for Mathematics (CCSSI 2010) asks students in as early as fourth grade to solve word problems using equations with variables. Equations studied at this level generate a single solution, such as the equation x + 10 = 25. For students in fifth grade, the Common Core standard for algebraic thinking expects them to…
Equating with Miditests Using IRT
ERIC Educational Resources Information Center
Fitzpatrick, Joseph; Skorupski, William P.
2016-01-01
The equating performance of two internal anchor test structures--miditests and minitests--is studied for four IRT equating methods using simulated data. Originally proposed by Sinharay and Holland, miditests are anchors that have the same mean difficulty as the overall test but less variance in item difficulties. Four popular IRT equating methods…
Generalized Klein-Kramers equations
NASA Astrophysics Data System (ADS)
Fa, Kwok Sau
2012-12-01
A generalized Klein-Kramers equation for a particle interacting with an external field is proposed. The equation generalizes the fractional Klein-Kramers equation introduced by Barkai and Silbey [J. Phys. Chem. B 104, 3866 (2000), 10.1021/jp993491m]. Besides, the generalized Klein-Kramers equation can also recover the integro-differential Klein-Kramers equation for continuous-time random walk; this means that it can describe the subdiffusive and superdiffusive regimes in the long-time limit. Moreover, analytic solutions for first two moments both in velocity and displacement (for force-free case) are obtained, and their dynamic behaviors are investigated.
Structural Equation Model Trees
Brandmaier, Andreas M.; von Oertzen, Timo; McArdle, John J.; Lindenberger, Ulman
2015-01-01
In the behavioral and social sciences, structural equation models (SEMs) have become widely accepted as a modeling tool for the relation between latent and observed variables. SEMs can be seen as a unification of several multivariate analysis techniques. SEM Trees combine the strengths of SEMs and the decision tree paradigm by building tree structures that separate a data set recursively into subsets with significantly different parameter estimates in a SEM. SEM Trees provide means for finding covariates and covariate interactions that predict differences in structural parameters in observed as well as in latent space and facilitate theory-guided exploration of empirical data. We describe the methodology, discuss theoretical and practical implications, and demonstrate applications to a factor model and a linear growth curve model. PMID:22984789
NASA Astrophysics Data System (ADS)
Cardona, Carlos; Gomez, Humberto
2016-06-01
Recently the CHY approach has been extended to one loop level using elliptic functions and modular forms over a Jacobian variety. Due to the difficulty in manipulating these kind of functions, we propose an alternative prescription that is totally algebraic. This new proposal is based on an elliptic algebraic curve embedded in a mathbb{C}{P}^2 space. We show that for the simplest integrand, namely the n - gon, our proposal indeed reproduces the expected result. By using the recently formulated Λ-algorithm, we found a novel recurrence relation expansion in terms of tree level off-shell amplitudes. Our results connect nicely with recent results on the one-loop formulation of the scattering equations. In addition, this new proposal can be easily stretched out to hyperelliptic curves in order to compute higher genus.
NASA Astrophysics Data System (ADS)
Gomez, Humberto
2016-06-01
The CHY representation of scattering amplitudes is based on integrals over the moduli space of a punctured sphere. We replace the punctured sphere by a double-cover version. The resulting scattering equations depend on a parameter Λ controlling the opening of a branch cut. The new representation of scattering amplitudes possesses an enhanced redundancy which can be used to fix, modulo branches, the location of four punctures while promoting Λ to a variable. Via residue theorems we show how CHY formulas break up into sums of products of smaller (off-shell) ones times a propagator. This leads to a powerful way of evaluating CHY integrals of generic rational functions, which we call the Λ algorithm.
Multinomial Diffusion Equation
Balter, Ariel I.; Tartakovsky, Alexandre M.
2011-06-01
We have developed a novel stochastic, space/time discrete representation of particle diffusion (e.g. Brownian motion) based on discrete probability distributions. We show that in the limit of both very small time step and large concentration, our description is equivalent to the space/time continuous stochastic diffusion equation. Being discrete in both time and space, our model can be used as an extremely accurate, efficient, and stable stochastic finite-difference diffusion algorithm when concentrations are so small that computationally expensive particle-based methods are usually needed. Through numerical simulations, we show that our method can generate realizations that capture the statistical properties of particle simulations. While our method converges converges to both the correct ensemble mean and ensemble variance very quickly with decreasing time step, but for small concentration, the stochastic diffusion PDE does not, even for very small time steps.
On nonautonomous Dirac equation
Hovhannisyan, Gro; Liu Wen
2009-12-15
We construct the fundamental solution of time dependent linear ordinary Dirac system in terms of unknown phase functions. This construction gives approximate representation of solutions which is useful for the study of asymptotic behavior. Introducing analog of Rayleigh quotient for differential equations we generalize Hartman-Wintner asymptotic integration theorems with the error estimates for applications to the Dirac system. We also introduce the adiabatic invariants for the Dirac system, which are similar to the adiabatic invariant of Lorentz's pendulum. Using a small parameter method it is shown that the change in the adiabatic invariants approaches zero with the power speed as a small parameter approaches zero. As another application we calculate the transition probabilities for the Dirac system. We show that for the special choice of electromagnetic field, the only transition of an electron to the positron with the opposite spin orientation is possible.
Parabolized stability equations
NASA Astrophysics Data System (ADS)
Herbert, Thorwald
1994-04-01
The parabolized stability equations (PSE) are a new approach to analyze the streamwise evolution of single or interacting Fourier modes in weakly nonparallel flows such as boundary layers. The concept rests on the decomposition of every mode into a slowly varying amplitude function and a wave function with slowly varying wave number. The neglect of the small second derivatives of the slowly varying functions with respect to the streamwise variable leads to an initial boundary-value problem that can be solved by numerical marching procedures. The PSE approach is valid in convectively unstable flows. The equations for a single mode are closely related to those of the traditional eigenvalue problems for linear stability analysis. However, the PSE approach does not exploit the homogeneity of the problem and, therefore, can be utilized to analyze forced modes and the nonlinear growth and interaction of an initial disturbance field. In contrast to the traditional patching of local solutions, the PSE provide the spatial evolution of modes with proper account for their history. The PSE approach allows studies of secondary instabilities without the constraints of the Floquet analysis and reproduces the established experimental, theoretical, and computational benchmark results on transition up to the breakdown stage. The method matches or exceeds the demonstrated capabilities of current spatial Navier-Stokes solvers at a small fraction of their computational cost. Recent applications include studies on localized or distributed receptivity and prediction of transition in model environments for realistic engineering problems. This report describes the basis, intricacies, and some applications of the PSE methodology.
Mode decomposition evolution equations
Wang, Yang; Wei, Guo-Wei; Yang, Siyang
2011-01-01
Partial differential equation (PDE) based methods have become some of the most powerful tools for exploring the fundamental problems in signal processing, image processing, computer vision, machine vision and artificial intelligence in the past two decades. The advantages of PDE based approaches are that they can be made fully automatic, robust for the analysis of images, videos and high dimensional data. A fundamental question is whether one can use PDEs to perform all the basic tasks in the image processing. If one can devise PDEs to perform full-scale mode decomposition for signals and images, the modes thus generated would be very useful for secondary processing to meet the needs in various types of signal and image processing. Despite of great progress in PDE based image analysis in the past two decades, the basic roles of PDEs in image/signal analysis are only limited to PDE based low-pass filters, and their applications to noise removal, edge detection, segmentation, etc. At present, it is not clear how to construct PDE based methods for full-scale mode decomposition. The above-mentioned limitation of most current PDE based image/signal processing methods is addressed in the proposed work, in which we introduce a family of mode decomposition evolution equations (MoDEEs) for a vast variety of applications. The MoDEEs are constructed as an extension of a PDE based high-pass filter (Europhys. Lett., 59(6): 814, 2002) by using arbitrarily high order PDE based low-pass filters introduced by Wei (IEEE Signal Process. Lett., 6(7): 165, 1999). The use of arbitrarily high order PDEs is essential to the frequency localization in the mode decomposition. Similar to the wavelet transform, the present MoDEEs have a controllable time-frequency localization and allow a perfect reconstruction of the original function. Therefore, the MoDEE operation is also called a PDE transform. However, modes generated from the present approach are in the spatial or time domain and can be
Langevin equation approach to reactor noise analysis: stochastic transport equation
Akcasu, A.Z. ); Stolle, A.M. )
1993-01-01
The application of the Langevin equation method to the study of fluctuations in the space- and velocity-dependent neutron density as well as in the detector outputs in nuclear reactors is presented. In this case, the Langevin equation is the stochastic linear neutron transport equation with a space- and velocity-dependent random neutron source, often referred to as the noise equivalent source (NES). The power spectral densities (PSDs) of the NESs in the transport equation, as well as in the accompanying detection rate equations, are obtained, and the cross- and auto-power spectral densities of the outputs of pairs of detectors are explicitly calculated. The transport-level expression for the R([omega]) ratio measured in the [sup 252]Cf source-driven noise analysis method is also derived. Finally, the implementation of the Langevin equation approach at different levels of approximation is discussed, and the stochastic one-speed transport and one-group P[sub 1] equations are derived by first integrating the stochastic transport equation over speed and then eliminating the angular dependence by a spherical harmonics expansion. By taking the large transport rate limit in the P[sub 1] description, the stochastic diffusion equation is obtained as well as the PSD of the NES in it. This procedure also leads directly to the stochastic Fick's law.
A spinor representation of Maxwell equations and Dirac equation
Vaz, J. Jr.; Rodrigues, W.A. Jr.
1993-02-01
Using the Clifford bundle formalism and starting from the free Maxwell equations dF = {delta}F = 0 we show by writing F = b{psi}{gamma}{sup 1}{gamma}{sup 2}{psi}{sup *}, where {psi} is a Dirac-Hestenes spinor field, that the Dirac-Hestenes equation (which is the representative of the standard Dirac equation in the Clifford bundle over Minkowski spacetime) is equivalent under general assumptions to those free Maxwell equations. We briefly discuss the implications of our findings for the interpretation of quantum mechanics. 15 refs.
Menikoff, Ralph
2015-12-15
The JWL equation of state (EOS) is frequently used for the products (and sometimes reactants) of a high explosive (HE). Here we review and systematically derive important properties. The JWL EOS is of the Mie-Grueneisen form with a constant Grueneisen coefficient and a constants specific heat. It is thermodynamically consistent to specify the temperature at a reference state. However, increasing the reference state temperature restricts the EOS domain in the (V, e)-plane of phase space. The restrictions are due to the conditions that P ≥ 0, T ≥ 0, and the isothermal bulk modulus is positive. Typically, this limits the low temperature regime in expansion. The domain restrictions can result in the P-T equilibrium EOS of a partly burned HE failing to have a solution in some cases. For application to HE, the heat of detonation is discussed. Example JWL parameters for an HE, both products and reactions, are used to illustrate the restrictions on the domain of the EOS.
A note on "Kepler's equation".
NASA Astrophysics Data System (ADS)
Dutka, J.
1997-07-01
This note briefly points out the formal similarity between Kepler's equation and equations developed in Hindu and Islamic astronomy for describing the lunar parallax. Specifically, an iterative method for calculating the lunar parallax has been developed by the astronomer Habash al-Hasib al-Marwazi (about 850 A.D., Turkestan), which is surprisingly similar to the iterative method for solving Kepler's equation invented by Leonhard Euler (1707 - 1783).
Electronic representation of wave equation
NASA Astrophysics Data System (ADS)
Veigend, Petr; Kunovský, Jiří; Kocina, Filip; Nečasová, Gabriela; Šátek, Václav; Valenta, Václav
2016-06-01
The Taylor series method for solving differential equations represents a non-traditional way of a numerical solution. Even though this method is not much preferred in the literature, experimental calculations done at the Department of Intelligent Systems of the Faculty of Information Technology of TU Brno have verified that the accuracy and stability of the Taylor series method exceeds the currently used algorithms for numerically solving differential equations. This paper deals with solution of Telegraph equation using modelling of a series small pieces of the wire. Corresponding differential equations are solved by the Modern Taylor Series Method.
Drug Levels and Difference Equations
ERIC Educational Resources Information Center
Acker, Kathleen A.
2004-01-01
American university offers a course in finite mathematics whose focus is difference equation with emphasis on real world applications. The conclusion states that students learned to look for growth and decay patterns in raw data, to recognize both arithmetic and geometric growth, and to model both scenarios with graphs and difference equations.
Uncertainty of empirical correlation equations
NASA Astrophysics Data System (ADS)
Feistel, R.; Lovell-Smith, J. W.; Saunders, P.; Seitz, S.
2016-08-01
The International Association for the Properties of Water and Steam (IAPWS) has published a set of empirical reference equations of state, forming the basis of the 2010 Thermodynamic Equation of Seawater (TEOS-10), from which all thermodynamic properties of seawater, ice, and humid air can be derived in a thermodynamically consistent manner. For each of the equations of state, the parameters have been found by simultaneously fitting equations for a range of different derived quantities using large sets of measurements of these quantities. In some cases, uncertainties in these fitted equations have been assigned based on the uncertainties of the measurement results. However, because uncertainties in the parameter values have not been determined, it is not possible to estimate the uncertainty in many of the useful quantities that can be calculated using the parameters. In this paper we demonstrate how the method of generalised least squares (GLS), in which the covariance of the input data is propagated into the values calculated by the fitted equation, and in particular into the covariance matrix of the fitted parameters, can be applied to one of the TEOS-10 equations of state, namely IAPWS-95 for fluid pure water. Using the calculated parameter covariance matrix, we provide some preliminary estimates of the uncertainties in derived quantities, namely the second and third virial coefficients for water. We recommend further investigation of the GLS method for use as a standard method for calculating and propagating the uncertainties of values computed from empirical equations.
Generalized Multilevel Structural Equation Modeling
ERIC Educational Resources Information Center
Rabe-Hesketh, Sophia; Skrondal, Anders; Pickles, Andrew
2004-01-01
A unifying framework for generalized multilevel structural equation modeling is introduced. The models in the framework, called generalized linear latent and mixed models (GLLAMM), combine features of generalized linear mixed models (GLMM) and structural equation models (SEM) and consist of a response model and a structural model for the latent…
Students' Understanding of Quadratic Equations
ERIC Educational Resources Information Center
López, Jonathan; Robles, Izraim; Martínez-Planell, Rafael
2016-01-01
Action-Process-Object-Schema theory (APOS) was applied to study student understanding of quadratic equations in one variable. This required proposing a detailed conjecture (called a genetic decomposition) of mental constructions students may do to understand quadratic equations. The genetic decomposition which was proposed can contribute to help…
Graphical Solution of Polynomial Equations
ERIC Educational Resources Information Center
Grishin, Anatole
2009-01-01
Graphing utilities, such as the ubiquitous graphing calculator, are often used in finding the approximate real roots of polynomial equations. In this paper the author offers a simple graphing technique that allows one to find all solutions of a polynomial equation (1) of arbitrary degree; (2) with real or complex coefficients; and (3) possessing…
Complete solution of Boolean equations
NASA Technical Reports Server (NTRS)
Tapia, M. A.; Tucker, J. H.
1980-01-01
A method is presented for generating a single formula involving arbitary Boolean parameters, which includes in it each and every possible solution of a system of Boolean equations. An alternate condition equivalent to a known necessary and sufficient condition for solving a system of Boolean equations is given.
The Equations of Oceanic Motions
NASA Astrophysics Data System (ADS)
Müller, Peter
2006-10-01
Modeling and prediction of oceanographic phenomena and climate is based on the integration of dynamic equations. The Equations of Oceanic Motions derives and systematically classifies the most common dynamic equations used in physical oceanography, from large scale thermohaline circulations to those governing small scale motions and turbulence. After establishing the basic dynamical equations that describe all oceanic motions, M|ller then derives approximate equations, emphasizing the assumptions made and physical processes eliminated. He distinguishes between geometric, thermodynamic and dynamic approximations and between the acoustic, gravity, vortical and temperature-salinity modes of motion. Basic concepts and formulae of equilibrium thermodynamics, vector and tensor calculus, curvilinear coordinate systems, and the kinematics of fluid motion and wave propagation are covered in appendices. Providing the basic theoretical background for graduate students and researchers of physical oceanography and climate science, this book will serve as both a comprehensive text and an essential reference.
Extended Trial Equation Method for Nonlinear Partial Differential Equations
NASA Astrophysics Data System (ADS)
Gepreel, Khaled A.; Nofal, Taher A.
2015-04-01
The main objective of this paper is to use the extended trial equation method to construct a series of some new solutions for some nonlinear partial differential equations (PDEs) in mathematical physics. We will construct the solutions in many different functions such as hyperbolic function solutions, trigonometric function solutions, Jacobi elliptic function solutions, and rational functional solutions for the nonlinear PDEs when the balance number is a real number via the Zhiber-Shabat nonlinear differential equation. The balance number of this method is not constant as we shown in other methods, but it is changed by changing the trial equation derivative definition. This method allowed us to construct many new types of solutions. It is shown by using the Maple software package that all obtained solutions satisfy the original PDEs.
Higher derivative gravity: Field equation as the equation of state
NASA Astrophysics Data System (ADS)
Dey, Ramit; Liberati, Stefano; Mohd, Arif
2016-08-01
One of the striking features of general relativity is that the Einstein equation is implied by the Clausius relation imposed on a small patch of locally constructed causal horizon. The extension of this thermodynamic derivation of the field equation to more general theories of gravity has been attempted many times in the last two decades. In particular, equations of motion for minimally coupled higher-curvature theories of gravity, but without the derivatives of curvature, have previously been derived using a thermodynamic reasoning. In that derivation the horizon slices were endowed with an entropy density whose form resembles that of the Noether charge for diffeomorphisms, and was dubbed the Noetheresque entropy. In this paper, we propose a new entropy density, closely related to the Noetheresque form, such that the field equation of any diffeomorphism-invariant metric theory of gravity can be derived by imposing the Clausius relation on a small patch of local causal horizon.
Primordial equation of state transitions
NASA Astrophysics Data System (ADS)
Aravind, Aditya; Lorshbough, Dustin; Paban, Sonia
2016-06-01
We revisit the physics of transitions from a general equation of state parameter to the final stage of slow-roll inflation. We show that it is unlikely for the modes comprising the cosmic microwave background to contain imprints from a preinflationary equation of state transition and still be consistent with observations. We accomplish this by considering observational consistency bounds on the amplitude of excitations resulting from such a transition. As a result, the physics which initially led to inflation likely cannot be probed with observations of the cosmic microwave background. Furthermore, we show that it is unlikely that equation of state transitions may explain the observed low multipole power suppression anomaly.
Sedeonic Equations of Massive Fields
NASA Astrophysics Data System (ADS)
Mironov, Sergey V.; Mironov, Victor L.
2015-01-01
Prior work on space-time sedeon algebra models relativistic quantum mechanical equation of motion with corresponding field equations, mediated by massive or massless spin-1 or spin-1/2 particles. In the massless spin-1 case, such exchange particles mediate fields in analogy to Maxwell's equations in Lorentz gauge. This paper demonstrates fundamental aspects of massive field's theory, such as gauge invariance, charge conservation, Poynting's theorem, potential of a stationary scalar point source, plane wave solution, and interaction between point sources. We briefly discuss some aspects of sedeonic algebra and their potential physical applications.
SETS. Set Equation Transformation System
Worrel, R.B.
1992-01-13
SETS is used for symbolic manipulation of Boolean equations, particularly the reduction of equations by the application of Boolean identities. It is a flexible and efficient tool for performing probabilistic risk analysis (PRA), vital area analysis, and common cause analysis. The equation manipulation capabilities of SETS can also be used to analyze noncoherent fault trees and determine prime implicants of Boolean functions, to verify circuit design implementation, to determine minimum cost fire protection requirements for nuclear reactor plants, to obtain solutions to combinatorial optimization problems with Boolean constraints, and to determine the susceptibility of a facility to unauthorized access through nullification of sensors in its protection system.
Parametric Equations, Maple, and Tubeplots.
ERIC Educational Resources Information Center
Feicht, Louis
1997-01-01
Presents an activity that establishes a graphical foundation for parametric equations by using a graphing output form called tubeplots from the computer program Maple. Provides a comprehensive review and exploration of many previously learned topics. (ASK)
NASA Technical Reports Server (NTRS)
Shebalin, John V.
1987-01-01
The Boussinesq approximation is extended so as to explicitly account for the transfer of fluid energy through viscous action into thermal energy. Ideal and dissipative integral invariants are discussed, in addition to the general equations for thermal-fluid motion.
Bogoliubov equations and functional mechanics
NASA Astrophysics Data System (ADS)
Volovich, I. V.
2010-09-01
The functional classical mechanics based on the probability approach, where a particle is described not by a trajectory in the phase space but by a probability distribution, was recently proposed for solving the irreversibility problem, i.e., the problem of matching the time reversibility of microscopic dynamics equations and the irreversibility of macrosystem dynamics. In the framework of functional mechanics, we derive Bogoliubov-Boltzmann-type equations for finitely many particles. We show that a closed equation for a one-particle distribution function can be rigorously derived in functional mechanics without any additional assumptions required in the Bogoliubov method. We consider the possibility of using diffusion processes and the Fokker-Planck-Kolmogorov equation to describe isolated particles.
Improved beam propagation method equations.
Nichelatti, E; Pozzi, G
1998-01-01
Improved beam propagation method (BPM) equations are derived for the general case of arbitrary refractive-index spatial distributions. It is shown that in the paraxial approximation the discrete equations admit an analytical solution for the propagation of a paraxial spherical wave, which converges to the analytical solution of the paraxial Helmholtz equation. The generalized Kirchhoff-Fresnel diffraction integral between the object and the image planes can be derived, with its coefficients expressed in terms of the standard ABCD matrix. This result allows the substitution, in the case of an unaberrated system, of the many numerical steps with a single analytical step. We compared the predictions of the standard and improved BPM equations by considering the cases of a Maxwell fish-eye and of a Luneburg lens. PMID:18268554
Solving Differential Equations in R
Although R is still predominantly applied for statistical analysis and graphical representation, it is rapidly becoming more suitable for mathematical computing. One of the fields where considerable progress has been made recently is the solution of differential equations. Here w...
Friedmann equation with quantum potential
Siong, Ch'ng Han; Radiman, Shahidan; Nikouravan, Bijan
2013-11-27
Friedmann equations are used to describe the evolution of the universe. Solving Friedmann equations for the scale factor indicates that the universe starts from an initial singularity where all the physical laws break down. However, the Friedmann equations are well describing the late-time or large scale universe. Hence now, many physicists try to find an alternative theory to avoid this initial singularity. In this paper, we generate a version of first Friedmann equation which is added with an additional term. This additional term contains the quantum potential energy which is believed to play an important role at small scale. However, it will gradually become negligible when the universe evolves to large scale.
Evolutions equations in computational anatomy.
Younes, Laurent; Arrate, Felipe; Miller, Michael I
2009-03-01
One of the main purposes in computational anatomy is the measurement and statistical study of anatomical variations in organs, notably in the brain or the heart. Over the last decade, our group has progressively developed several approaches for this problem, all related to the Riemannian geometry of groups of diffeomorphisms and the shape spaces on which these groups act. Several important shape evolution equations that are now used routinely in applications have emerged over time. Our goal in this paper is to provide an overview of these equations, placing them in their theoretical context, and giving examples of applications in which they can be used. We introduce the required theoretical background before discussing several classes of equations of increasingly complexity. These equations include energy minimizing evolutions deriving from Riemannian gradient descent, geodesics, parallel transport and Jacobi fields. PMID:19059343
Overdetermined Systems of Linear Equations.
ERIC Educational Resources Information Center
Williams, Gareth
1990-01-01
Explored is an overdetermined system of linear equations to find an appropriate least squares solution. A geometrical interpretation of this solution is given. Included is a least squares point discussion. (KR)
Hidden Statistics of Schroedinger Equation
NASA Technical Reports Server (NTRS)
Zak, Michail
2011-01-01
Work was carried out in determination of the mathematical origin of randomness in quantum mechanics and creating a hidden statistics of Schr dinger equation; i.e., to expose the transitional stochastic process as a "bridge" to the quantum world. The governing equations of hidden statistics would preserve such properties of quantum physics as superposition, entanglement, and direct-product decomposability while allowing one to measure its state variables using classical methods.
Boltzmann equation and hydrodynamic fluctuations.
Colangeli, Matteo; Kröger, Martin; Ottinger, Hans Christian
2009-11-01
We apply the method of invariant manifolds to derive equations of generalized hydrodynamics from the linearized Boltzmann equation and determine exact transport coefficients, obeying Green-Kubo formulas. Numerical calculations are performed in the special case of Maxwell molecules. We investigate, through the comparison with experimental data and former approaches, the spectrum of density fluctuations and address the regime of finite Knudsen numbers and finite frequencies hydrodynamics. PMID:20364972
New determination equation for visibility
Fang Qiwan; Rao Jionghui; Ying Zhixiang; Tang Haijun; Jiang Chuanfu
1996-12-31
Range is an important tactical hard index in designing and manufacturing military laser rangefinders. But in practice it is also a soft index which is influenced by target characteristic and atmospheric visibility. In this article the problems in the range index are analyzed. The way to determine visibility is put forward. Extinction determination equation for visibility is derived. And it is applied in practice, which verifies the determination equation is functional and effective.
Revisiting the Simplified Bernoulli Equation
Heys, Jeffrey J; Holyoak, Nicole; Calleja, Anna M; Belohlavek, Marek; Chaliki, Hari P
2010-01-01
Background: The assessment of the severity of aortic valve stenosis is done by either invasive catheterization or non-invasive Doppler Echocardiography in conjunction with the simplified Bernoulli equation. The catheter measurement is generally considered more accurate, but the procedure is also more likely to have dangerous complications. Objective: The focus here is on examining computational fluid dynamics as an alternative method for analyzing the echo data and determining whether it can provide results similar to the catheter measurement. Methods: An in vitro heart model with a rigid orifice is used as a first step in comparing echocardiographic data, which uses the simplified Bernoulli equation, catheterization, and echocardiographic data, which uses computational fluid dynamics (i.e., the Navier-Stokes equations). Results: For a 0.93cm2 orifice, the maximum pressure gradient predicted by either the simplified Bernoulli equation or computational fluid dynamics was not significantly different from the experimental catheter measurement (p > 0.01). For a smaller 0.52cm2 orifice, there was a small but significant difference (p < 0.01) between the simplified Bernoulli equation and the computational fluid dynamics simulation, with the computational fluid dynamics simulation giving better agreement with experimental data for some turbulence models. Conclusion: For this simplified, in vitro system, the use of computational fluid dynamics provides an improvement over the simplified Bernoulli equation with the biggest improvement being seen at higher valvular stenosis levels. PMID:21625471
An Exact Mapping from Navier-Stokes Equation to Schr"odinger Equation via Riccati Equation
NASA Astrophysics Data System (ADS)
Christianto, Vic; Smarandache, Florentin
2010-03-01
In the present article we argue that it is possible to write down Schr"odinger representation of Navier-Stokes equation via Riccati equation. The proposed approach, while differs appreciably from other method such as what is proposed by R. M. Kiehn, has an advantage, i.e. it enables us extend further to quaternionic and biquaternionic version of Navier-Stokes equation, for instance via Kravchenko's and Gibbon's route. Further observation is of course recommended in order to refute or verify this proposition.
Allidina, A.Y.; Malinowski, K.; Singh, M.G.
1982-12-01
The possibilities were explored for enhancing parallelism in the simulation of systems described by algebraic equations, ordinary differential equations and partial differential equations. These techniques, using multiprocessors, were developed to speed up simulations, e.g. for nuclear accidents. Issues involved in their design included suitable approximations to bring the problem into a numerically manageable form and a numerical procedure to perform the computations necessary to solve the problem accurately. Parallel processing techniques used as simulation procedures, and a design of a simulation scheme and simulation procedure employing parallel computer facilities, were both considered.
Solving Parker's transport equation with stochastic differential equations on GPUs
NASA Astrophysics Data System (ADS)
Dunzlaff, P.; Strauss, R. D.; Potgieter, M. S.
2015-07-01
The numerical solution of transport equations for energetic charged particles in space is generally very costly in terms of time. Besides the use of multi-core CPUs and computer clusters in order to decrease the computation times, high performance calculations on graphics processing units (GPUs) have become available during the last years. In this work we introduce and describe a GPU-accelerated implementation of Parker's equation using Stochastic Differential Equations (SDEs) for the simulation of the transport of energetic charged particles with the CUDA toolkit, which is the focus of this work. We briefly discuss the set of SDEs arising from Parker's transport equation and their application to boundary value problems such as that of the Jovian magnetosphere. We compare the runtimes of the GPU code with a CPU version of the same algorithm. Compared to the CPU implementation (using OpenMP and eight threads) we find a performance increase of about a factor of 10-60, depending on the assumed set of parameters. Furthermore, we benchmark our simulation using the results of an existing SDE implementation of Parker's transport equation.
Turbulent fluid motion 3: Basic continuum equations
NASA Technical Reports Server (NTRS)
Deissler, Robert G.
1991-01-01
A derivation of the continuum equations used for the analysis of turbulence is given. These equations include the continuity equation, the Navier-Stokes equations, and the heat transfer or energy equation. An experimental justification for using a continuum approach for the study of turbulence is given.
Optimization of one-way wave equations.
Lee, M.W.; Suh, S.Y.
1985-01-01
The theory of wave extrapolation is based on the square-root equation or one-way equation. The full wave equation represents waves which propagate in both directions. On the contrary, the square-root equation represents waves propagating in one direction only. A new optimization method presented here improves the dispersion relation of the one-way wave equation. -from Authors
How to Obtain the Covariant Form of Maxwell's Equations from the Continuity Equation
ERIC Educational Resources Information Center
Heras, Jose A.
2009-01-01
The covariant Maxwell equations are derived from the continuity equation for the electric charge. This result provides an axiomatic approach to Maxwell's equations in which charge conservation is emphasized as the fundamental axiom underlying these equations.
Fractional-calculus diffusion equation
2010-01-01
Background Sequel to the work on the quantization of nonconservative systems using fractional calculus and quantization of a system with Brownian motion, which aims to consider the dissipation effects in quantum-mechanical description of microscale systems. Results The canonical quantization of a system represented classically by one-dimensional Fick's law, and the diffusion equation is carried out according to the Dirac method. A suitable Lagrangian, and Hamiltonian, describing the diffusive system, are constructed and the Hamiltonian is transformed to Schrodinger's equation which is solved. An application regarding implementation of the developed mathematical method to the analysis of diffusion, osmosis, which is a biological application of the diffusion process, is carried out. Schrödinger's equation is solved. Conclusions The plot of the probability function represents clearly the dissipative and drift forces and hence the osmosis, which agrees totally with the macro-scale view, or the classical-version osmosis. PMID:20492677
Students' understanding of quadratic equations
NASA Astrophysics Data System (ADS)
López, Jonathan; Robles, Izraim; Martínez-Planell, Rafael
2016-05-01
Action-Process-Object-Schema theory (APOS) was applied to study student understanding of quadratic equations in one variable. This required proposing a detailed conjecture (called a genetic decomposition) of mental constructions students may do to understand quadratic equations. The genetic decomposition which was proposed can contribute to help students achieve an understanding of quadratic equations with improved interrelation of ideas and more flexible application of solution methods. Semi-structured interviews with eight beginning undergraduate students explored which of the mental constructions conjectured in the genetic decomposition students could do, and which they had difficulty doing. Two of the mental constructions that form part of the genetic decomposition are highlighted and corresponding further data were obtained from the written work of 121 undergraduate science and engineering students taking a multivariable calculus course. The results suggest the importance of explicitly considering these two highlighted mental constructions.
Maxwell's mixing equation revisited: characteristic impedance equations for ellipsoidal cells.
Stubbe, Marco; Gimsa, Jan
2015-07-21
We derived a series of, to our knowledge, new analytic expressions for the characteristic features of the impedance spectra of suspensions of homogeneous and single-shell spherical, spheroidal, and ellipsoidal objects, e.g., biological cells of the general ellipsoidal shape. In the derivation, we combined the Maxwell-Wagner mixing equation with our expression for the Clausius-Mossotti factor that had been originally derived to describe AC-electrokinetic effects such as dielectrophoresis, electrorotation, and electroorientation. The influential radius model was employed because it allows for a separation of the geometric and electric problems. For shelled objects, a special axial longitudinal element approach leads to a resistor-capacitor model, which can be used to simplify the mixing equation. Characteristic equations were derived for the plateau levels, peak heights, and characteristic frequencies of the impedance as well as the complex specific conductivities and permittivities of suspensions of axially and randomly oriented homogeneous and single-shell ellipsoidal objects. For membrane-covered spherical objects, most of the limiting cases are identical to-or improved with respect to-the known solutions given by researchers in the field. The characteristic equations were found to be quite precise (largest deviations typically <5% with respect to the full model) when tested with parameters relevant to biological cells. They can be used for the differentiation of orientation and the electric properties of cell suspensions or in the analysis of single cells in microfluidic systems. PMID:26200856
Explicit integration of Friedmann's equation with nonlinear equations of state
NASA Astrophysics Data System (ADS)
Chen, Shouxin; Gibbons, Gary W.; Yang, Yisong
2015-05-01
In this paper we study the integrability of the Friedmann equations, when the equation of state for the perfect-fluid universe is nonlinear, in the light of the Chebyshev theorem. A series of important, yet not previously touched, problems will be worked out which include the generalized Chaplygin gas, two-term energy density, trinomial Friedmann, Born-Infeld, two-fluid models, and Chern-Simons modified gravity theory models. With the explicit integration, we are able to understand exactly the roles of the physical parameters in various models play in the cosmological evolution which may also offer clues to a profound understanding of the problems in general settings. For example, in the Chaplygin gas universe, a few integrable cases lead us to derive a universal formula for the asymptotic exponential growth rate of the scale factor, of an explicit form, whether the Friedmann equation is integrable or not, which reveals the coupled roles played by various physical sectors and it is seen that, as far as there is a tiny presence of nonlinear matter, conventional linear matter makes contribution to the dark matter, which becomes significant near the phantom divide line. The Friedmann equations also arise in areas of physics not directly related to cosmology. We provide some examples ranging from geometric optics and central orbits to soap films and the shape of glaciated valleys to which our results may be applied.
Transport equations in tokamak plasmas
Callen, J. D.; Hegna, C. C.; Cole, A. J.
2010-05-15
Tokamak plasma transport equations are usually obtained by flux surface averaging the collisional Braginskii equations. However, tokamak plasmas are not in collisional regimes. Also, ad hoc terms are added for neoclassical effects on the parallel Ohm's law, fluctuation-induced transport, heating, current-drive and flow sources and sinks, small magnetic field nonaxisymmetries, magnetic field transients, etc. A set of self-consistent second order in gyroradius fluid-moment-based transport equations for nearly axisymmetric tokamak plasmas has been developed using a kinetic-based approach. The derivation uses neoclassical-based parallel viscous force closures, and includes all the effects noted above. Plasma processes on successive time scales and constraints they impose are considered sequentially: compressional Alfven waves (Grad-Shafranov equilibrium, ion radial force balance), sound waves (pressure constant along field lines, incompressible flows within a flux surface), and collisions (electrons, parallel Ohm's law; ions, damping of poloidal flow). Radial particle fluxes are driven by the many second order in gyroradius toroidal angular torques on a plasma species: seven ambipolar collision-based ones (classical, neoclassical, etc.) and eight nonambipolar ones (fluctuation-induced, polarization flows from toroidal rotation transients, etc.). The plasma toroidal rotation equation results from setting to zero the net radial current induced by the nonambipolar fluxes. The radial particle flux consists of the collision-based intrinsically ambipolar fluxes plus the nonambipolar fluxes evaluated at the ambipolarity-enforcing toroidal plasma rotation (radial electric field). The energy transport equations do not involve an ambipolar constraint and hence are more directly obtained. The 'mean field' effects of microturbulence on the parallel Ohm's law, poloidal ion flow, particle fluxes, and toroidal momentum and energy transport are all included self-consistently. The
Young's Equation at the Nanoscale
NASA Astrophysics Data System (ADS)
Seveno, David; Blake, Terence D.; De Coninck, Joël
2013-08-01
In 1805, Thomas Young was the first to propose an equation to predict the value of the equilibrium contact angle of a liquid on a solid. Today, the force exerted by a liquid on a solid, such as a flat plate or fiber, is routinely used to assess this angle. Moreover, it has recently become possible to study wetting at the nanoscale using an atomic force microscope. Here, we report the use of molecular-dynamics simulations to investigate the force distribution along a 15 nm fiber dipped into a liquid meniscus. We find very good agreement between the measured force and that predicted by Young’s equation.
Duffing's Equation and Nonlinear Resonance
ERIC Educational Resources Information Center
Fay, Temple H.
2003-01-01
The phenomenon of nonlinear resonance (sometimes called the "jump phenomenon") is examined and second-order van der Pol plane analysis is employed to indicate that this phenomenon is not a feature of the equation, but rather the result of accumulated round-off error, truncation error and algorithm error that distorts the true bounded solution onto…
Scale Shrinkage in Vertical Equating.
ERIC Educational Resources Information Center
Camilli, Gregory; And Others
1993-01-01
Three potential causes of scale shrinkage (measurement error, restriction of range, and multidimensionality) in item response theory vertical equating are discussed, and a more comprehensive model-based approach to establishing vertical scales is described. Test data from the National Assessment of Educational Progress are used to illustrate the…
Sonar equations for planetary exploration.
Ainslie, Michael A; Leighton, Timothy G
2016-08-01
The set of formulations commonly known as "the sonar equations" have for many decades been used to quantify the performance of sonar systems in terms of their ability to detect and localize objects submerged in seawater. The efficacy of the sonar equations, with individual terms evaluated in decibels, is well established in Earth's oceans. The sonar equations have been used in the past for missions to other planets and moons in the solar system, for which they are shown to be less suitable. While it would be preferable to undertake high-fidelity acoustical calculations to support planning, execution, and interpretation of acoustic data from planetary probes, to avoid possible errors for planned missions to such extraterrestrial bodies in future, doing so requires awareness of the pitfalls pointed out in this paper. There is a need to reexamine the assumptions, practices, and calibrations that work well for Earth to ensure that the sonar equations can be accurately applied in combination with the decibel to extraterrestrial scenarios. Examples are given for icy oceans such as exist on Europa and Ganymede, Titan's hydrocarbon lakes, and for the gaseous atmospheres of (for example) Jupiter and Venus. PMID:27586766
Perceptions of the Schrodinger equation
NASA Astrophysics Data System (ADS)
Efthimiades, Spyros
2014-03-01
The Schrodinger equation has been considered to be a postulate of quantum physics, but it is also perceived as the quantum equivalent of the non-relativistic classical energy relation. We argue that the Schrodinger equation cannot be a physical postulate, and we show explicitly that its second space derivative term is wrongly associated with the kinetic energy of the particle. The kinetic energy of a particle at a point is proportional to the square of the momentum, that is, to the square of the first space derivative of the wavefunction. Analyzing particle interactions, we realize that particles have multiple virtual motions and that each motion is accompanied by a wave that has constant amplitude. Accordingly, we define the wavefunction as the superposition of the virtual waves of the particle. In simple interaction settings we can tell what particle motions arise and can explain the outcomes in direct and tangible terms. Most importantly, the mathematical foundation of quantum mechanics becomes clear and justified, and we derive the Schrodinger, Dirac, etc. equations as the conditions the wavefunction must satisfy at each space-time point in order to fulfill the respective total energy equation.
Renaissance Learning Equating Study. Report
ERIC Educational Resources Information Center
Sewell, Julie; Sainsbury, Marian; Pyle, Katie; Keogh, Nikki; Styles, Ben
2007-01-01
An equating study was carried out in autumn 2006 by the National Foundation for Educational Research (NFER) on behalf of Renaissance Learning, to provide validation evidence for the use of the Renaissance Star Reading and Star Mathematics tests in English schools. The study investigated the correlation between the Star tests and established tests.…
The Forced Soft Spring Equation
ERIC Educational Resources Information Center
Fay, T. H.
2006-01-01
Through numerical investigations, this paper studies examples of the forced Duffing type spring equation with [epsilon] negative. By performing trial-and-error numerical experiments, the existence is demonstrated of stability boundaries in the phase plane indicating initial conditions yielding bounded solutions. Subharmonic boundaries are…
The Symbolism Of Chemical Equations
ERIC Educational Resources Information Center
Jensen, William B.
2005-01-01
A question about the historical origin of equal sign and double arrow symbolism in balanced chemical equation is raised. The study shows that Marshall proposed the symbolism in 1902, which includes the use of currently favored double barb for equilibrium reactions.
Mathematics and Reading Test Equating.
ERIC Educational Resources Information Center
Lee, Ong Kim; Wright, Benjamin D.
As part of a larger project to assess changes in student learning resulting from school reform, this study equates levels 6 through 14 of the mathematics and reading comprehension components of Form 7 of the Iowa Tests of Basic Skills (ITBS) with levels 7 through 14 of the mathematics and reading comprehension components of the CPS90 (another…
Optimized solution of Kepler's equation
NASA Technical Reports Server (NTRS)
Kohout, J. M.; Layton, L.
1972-01-01
A detailed description is presented of KEPLER, an IBM 360 computer program used for the solution of Kepler's equation for eccentric anomaly. The program KEPLER employs a second-order Newton-Raphson differential correction process, and it is faster than previously developed programs by an order of magnitude.
The solution of transcendental equations
NASA Technical Reports Server (NTRS)
Agrawal, K. M.; Outlaw, R.
1973-01-01
Some of the existing methods to globally approximate the roots of transcendental equations namely, Graeffe's method, are studied. Summation of the reciprocated roots, Whittaker-Bernoulli method, and the extension of Bernoulli's method via Koenig's theorem are presented. The Aitken's delta squared process is used to accelerate the convergence. Finally, the suitability of these methods is discussed in various cases.
Ordinary Differential Equation System Solver
Energy Science and Technology Software Center (ESTSC)
1992-03-05
LSODE is a package of subroutines for the numerical solution of the initial value problem for systems of first order ordinary differential equations. The package is suitable for either stiff or nonstiff systems. For stiff systems the Jacobian matrix may be treated in either full or banded form. LSODE can also be used when the Jacobian can be approximated by a band matrix.
Empirical equation estimates geothermal gradients
Kutasov, I.M. )
1995-01-02
An empirical equation can estimate geothermal (natural) temperature profiles in new exploration areas. These gradients are useful for cement slurry and mud design and for improving electrical and temperature log interpretation. Downhole circulating temperature logs and surface outlet temperatures are used for predicting the geothermal gradients.
Pendulum Motion and Differential Equations
ERIC Educational Resources Information Center
Reid, Thomas F.; King, Stephen C.
2009-01-01
A common example of real-world motion that can be modeled by a differential equation, and one easily understood by the student, is the simple pendulum. Simplifying assumptions are necessary for closed-form solutions to exist, and frequently there is little discussion of the impact if those assumptions are not met. This article presents a…
Relativistic equations with fractional and pseudodifferential operators
Babusci, D.; Dattoli, G.; Quattromini, M.
2011-06-15
In this paper we use different techniques from the fractional and pseudo-operators calculus to solve partial differential equations involving operators with noninteger exponents. We apply the method to equations resembling generalizations of the heat equations and discuss the possibility of extending the procedure to the relativistic Schroedinger and Dirac equations.
Simulated Equating Using Several Item Response Curves.
ERIC Educational Resources Information Center
Boldt, R. F.
The comparison of item response theory models for the Test of English as a Foreign Language (TOEFL) was extended to an equating context as simulation trials were used to "equate the test to itself." Equating sample data were generated from administration of identical item sets. Equatings that used procedures based on each model (simple item…
A Versatile Technique for Solving Quintic Equations
ERIC Educational Resources Information Center
Kulkarni, Raghavendra G.
2006-01-01
In this paper we present a versatile technique to solve several types of solvable quintic equations. In the technique described here, the given quintic is first converted to a sextic equation by adding a root, and the resulting sextic equation is decomposed into two cubic polynomials as factors in a novel fashion. The resultant cubic equations are…
Simple Derivation of the Lindblad Equation
ERIC Educational Resources Information Center
Pearle, Philip
2012-01-01
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…
Isothermal Equation Of State For Compressed Solids
NASA Technical Reports Server (NTRS)
Vinet, Pascal; Ferrante, John
1989-01-01
Same equation with three adjustable parameters applies to different materials. Improved equation of state describes pressure on solid as function of relative volume at constant temperature. Even though types of interatomic interactions differ from one substance to another, form of equation determined primarily by overlap of electron wave functions during compression. Consequently, equation universal in sense it applies to variety of substances, including ionic, metallic, covalent, and rare-gas solids. Only three parameters needed to describe equation for given material.
Integrable (2 k)-Dimensional Hitchin Equations
NASA Astrophysics Data System (ADS)
Ward, R. S.
2016-07-01
This letter describes a completely integrable system of Yang-Mills-Higgs equations which generalizes the Hitchin equations on a Riemann surface to arbitrary k-dimensional complex manifolds. The system arises as a dimensional reduction of a set of integrable Yang-Mills equations in 4 k real dimensions. Our integrable system implies other generalizations such as the Simpson equations and the non-abelian Seiberg-Witten equations. Some simple solutions in the k = 2 case are described.
Applications of film thickness equations
NASA Technical Reports Server (NTRS)
Hamrock, B. J.; Dowson, D.
1983-01-01
A number of applications of elastohydrodynamic film thickness expressions were considered. The motion of a steel ball over steel surfaces presenting varying degrees of conformity was examined. The equation for minimum film thickness in elliptical conjunctions under elastohydrodynamic conditions was applied to roller and ball bearings. An involute gear was also introduced, it was again found that the elliptical conjunction expression yielded a conservative estimate of the minimum film thickness. Continuously variable-speed drives like the Perbury gear, which present truly elliptical elastohydrodynamic conjunctions, are favored increasingly in mobile and static machinery. A representative elastohydrodynamic condition for this class of machinery is considered for power transmission equipment. The possibility of elastohydrodynamic films of water or oil forming between locomotive wheels and rails is examined. The important subject of traction on the railways is attracting considerable attention in various countries at the present time. The final example of a synovial joint introduced the equation developed for isoviscous-elastic regimes of lubrication.
Graviton corrections to Maxwell's equations
NASA Astrophysics Data System (ADS)
Leonard, Katie E.; Woodard, R. P.
2012-05-01
We use dimensional regularization to compute the one loop quantum gravitational contribution to the vacuum polarization on flat space background. Adding the appropriate Bogoliubov-Parsiuk-Hepp-Zimmermann counterterm gives a fully renormalized result which we employ to quantum correct Maxwell’s equations. These equations are solved to show that dynamical photons are unchanged, provided the free state wave functional is appropriately corrected. The response to the instantaneous appearance of a point dipole reveals a perturbative version of the long-conjectured, “smearing of the light cone”. There is no change in the far radiation field produced by an alternating dipole. However, the correction to the static electric field of a point charge shows strengthening at short distances, in contrast to expectations based on the renormalization group. We check for gauge dependence by working out the vacuum polarization in a general 3-parameter family of covariant gauges.
Renewal equations for option pricing
NASA Astrophysics Data System (ADS)
Montero, M.
2008-09-01
In this paper we will develop a methodology for obtaining pricing expressions for financial instruments whose underlying asset can be described through a simple continuous-time random walk (CTRW) market model. Our approach is very natural to the issue because it is based in the use of renewal equations, and therefore it enhances the potential use of CTRW techniques in finance. We solve these equations for typical contract specifications, in a particular but exemplifying case. We also show how a formal general solution can be found for more exotic derivatives, and we compare prices for alternative models of the underlying. Finally, we recover the celebrated results for the Wiener process under certain limits.
Fresnel Integral Equations: Numerical Properties
Adams, R J; Champagne, N J II; Davis, B A
2003-07-22
A spatial-domain solution to the problem of electromagnetic scattering from a dielectric half-space is outlined. The resulting half-space operators are referred to as Fresnel surface integral operators. When used as preconditioners for nonplanar geometries, the Fresnel operators yield surface Fresnel integral equations (FIEs) which are stable with respect to dielectric constant, discretization, and frequency. Numerical properties of the formulations are discussed.
Linear superposition in nonlinear equations.
Khare, Avinash; Sukhatme, Uday
2002-06-17
Several nonlinear systems such as the Korteweg-de Vries (KdV) and modified KdV equations and lambda phi(4) theory possess periodic traveling wave solutions involving Jacobi elliptic functions. We show that suitable linear combinations of these known periodic solutions yield many additional solutions with different periods and velocities. This linear superposition procedure works by virtue of some remarkable new identities involving elliptic functions. PMID:12059300
Equation of State Project Overview
Crockett, Scott
2015-09-11
A general overview of the Equation of State (EOS) Project will be presented. The goal is to provide the audience with an introduction of what our more advanced methods entail (DFT, QMD, etc.. ) and how these models are being utilized to better constrain the thermodynamic models. These models substantially reduce our regions of interpolation between the various thermodynamic limits. I will also present a variety example of recent EOS work.
Nonlocal Equations with Measure Data
NASA Astrophysics Data System (ADS)
Kuusi, Tuomo; Mingione, Giuseppe; Sire, Yannick
2015-08-01
We develop an existence, regularity and potential theory for nonlinear integrodifferential equations involving measure data. The nonlocal elliptic operators considered are possibly degenerate and cover the case of the fractional p-Laplacean operator with measurable coefficients. We introduce a natural function class where we solve the Dirichlet problem, and prove basic and optimal nonlinear Wolff potential estimates for solutions. These are the exact analogs of the results valid in the case of local quasilinear degenerate equations established by Boccardo and Gallouët (J Funct Anal 87:149-169, 1989, Partial Differ Equ 17:641-655, 1992) and Kilpeläinen and Malý (Ann Scuola Norm Sup Pisa Cl Sci (IV) 19:591-613, 1992, Acta Math 172:137-161, 1994). As a consequence, we establish a number of results that can be considered as basic building blocks for a nonlocal, nonlinear potential theory: fine properties of solutions, Calderón-Zygmund estimates, continuity and boundedness criteria are established via Wolff potentials. A main tool is the introduction of a global excess functional that allows us to prove a nonlocal analog of the classical theory due to Campanato (Ann Mat Pura Appl (IV) 69:321-381, 1965). Our results cover the case of linear nonlocal equations with measurable coefficients, and the one of the fractional Laplacean, and are new already in such cases.
ON THE GENERALISED FANT EQUATION.
Howe, M S; McGowan, R S
2011-06-20
An analysis is made of the fluid-structure interactions involved in the production of voiced speech. It is usual to avoid time consuming numerical simulations of the aeroacoustics of the vocal tract and glottis by the introduction of Fant's 'reduced complexity' equation for the glottis volume velocity Q (G. Fant, Acoustic Theory of Speech Production, Mouton, The Hague 1960). A systematic derivation is given of Fant's equation based on the nominally exact equations of aerodynamic sound. This can be done with a degree of approximation that depends only on the accuracy with which the time-varying flow geometry and surface-acoustic boundary conditions can be specified, and replaces Fant's original 'lumped element' heuristic approach. The method determines all of the effective 'source terms' governing Q. It is illustrated by consideration of a simplified model of the vocal system involving a self-sustaining single-mass model of the vocal folds, that uses free streamline theory to account for surface friction and flow separation within the glottis. Identification is made of a new source term associated with the unsteady vocal fold drag produced by their oscillatory motion transverse to the mean flow. PMID:21603054
ON THE GENERALISED FANT EQUATION
Howe, M. S.; McGowan, R. S.
2011-01-01
An analysis is made of the fluid-structure interactions involved in the production of voiced speech. It is usual to avoid time consuming numerical simulations of the aeroacoustics of the vocal tract and glottis by the introduction of Fant’s ‘reduced complexity’ equation for the glottis volume velocity Q (G. Fant, Acoustic Theory of Speech Production, Mouton, The Hague 1960). A systematic derivation is given of Fant’s equation based on the nominally exact equations of aerodynamic sound. This can be done with a degree of approximation that depends only on the accuracy with which the time-varying flow geometry and surface-acoustic boundary conditions can be specified, and replaces Fant’s original ‘lumped element’ heuristic approach. The method determines all of the effective ‘source terms’ governing Q. It is illustrated by consideration of a simplified model of the vocal system involving a self-sustaining single-mass model of the vocal folds, that uses free streamline theory to account for surface friction and flow separation within the glottis. Identification is made of a new source term associated with the unsteady vocal fold drag produced by their oscillatory motion transverse to the mean flow. PMID:21603054
NASA Technical Reports Server (NTRS)
Brown, James L.; Naughton, Jonathan W.
1999-01-01
A thin film of oil on a surface responds primarily to the wall shear stress generated on that surface by a three-dimensional flow. The oil film is also subject to wall pressure gradients, surface tension effects and gravity. The partial differential equation governing the oil film flow is shown to be related to Burgers' equation. Analytical and numerical methods for solving the thin oil film equation are presented. A direct numerical solver is developed where the wall shear stress variation on the surface is known and which solves for the oil film thickness spatial and time variation on the surface. An inverse numerical solver is also developed where the oil film thickness spatial variation over the surface at two discrete times is known and which solves for the wall shear stress variation over the test surface. A One-Time-Level inverse solver is also demonstrated. The inverse numerical solver provides a mathematically rigorous basis for an improved form of a wall shear stress instrument suitable for application to complex three-dimensional flows. To demonstrate the complexity of flows for which these oil film methods are now suitable, extensive examination is accomplished for these analytical and numerical methods as applied to a thin oil film in the vicinity of a three-dimensional saddle of separation.
The complex chemical Langevin equation
Schnoerr, David; Sanguinetti, Guido; Grima, Ramon
2014-07-14
The chemical Langevin equation (CLE) is a popular simulation method to probe the stochastic dynamics of chemical systems. The CLE’s main disadvantage is its break down in finite time due to the problem of evaluating square roots of negative quantities whenever the molecule numbers become sufficiently small. We show that this issue is not a numerical integration problem, rather in many systems it is intrinsic to all representations of the CLE. Various methods of correcting the CLE have been proposed which avoid its break down. We show that these methods introduce undesirable artefacts in the CLE’s predictions. In particular, for unimolecular systems, these correction methods lead to CLE predictions for the mean concentrations and variance of fluctuations which disagree with those of the chemical master equation. We show that, by extending the domain of the CLE to complex space, break down is eliminated, and the CLE’s accuracy for unimolecular systems is restored. Although the molecule numbers are generally complex, we show that the “complex CLE” predicts real-valued quantities for the mean concentrations, the moments of intrinsic noise, power spectra, and first passage times, hence admitting a physical interpretation. It is also shown to provide a more accurate approximation of the chemical master equation of simple biochemical circuits involving bimolecular reactions than the various corrected forms of the real-valued CLE, the linear-noise approximation and a commonly used two moment-closure approximation.
On the generalised Fant equation
NASA Astrophysics Data System (ADS)
Howe, M. S.; McGowan, R. S.
2011-06-01
An analysis is made of the fluid-structure interactions involved in the production of voiced speech. It is usual to avoid time consuming numerical simulations of the aeroacoustics of the vocal tract and glottis by the introduction of Fant's 'reduced complexity' equation for the glottis volume velocity Q [G. Fant, Acoustic Theory of Speech Production, Mouton, The Hague 1960]. A systematic derivation is given of Fant's equation based on the nominally exact equations of aerodynamic sound. This can be done with a degree of approximation that depends only on the accuracy with which the time-varying flow geometry and surface-acoustic boundary conditions can be specified, and replaces Fant's original 'lumped element' heuristic approach. The method determines all of the effective 'source terms' governing Q. It is illustrated by consideration of a simplified model of the vocal system involving a self-sustaining single-mass model of the vocal folds, that uses free streamline theory to account for surface friction and flow separation within the glottis. Identification is made of a new source term associated with the unsteady vocal fold drag produced by their oscillatory motion transverse to the mean flow.
ADVANCED WAVE-EQUATION MIGRATION
L. HUANG; M. C. FEHLER
2000-12-01
Wave-equation migration methods can more accurately account for complex wave phenomena than ray-tracing-based Kirchhoff methods that are based on the high-frequency asymptotic approximation of waves. With steadily increasing speed of massively parallel computers, wave-equation migration methods are becoming more and more feasible and attractive for imaging complex 3D structures. We present an overview of several efficient and accurate wave-equation-based migration methods that we have recently developed. The methods are implemented in the frequency-space and frequency-wavenumber domains and hence they are called dual-domain methods. In the methods, we make use of different approximate solutions of the scalar-wave equation in heterogeneous media to recursively downward continue wavefields. The approximations used within each extrapolation interval include the Born, quasi-Born, and Rytov approximations. In one of our dual-domain methods, we use an optimized expansion of the square-root operator in the one-way wave equation to minimize the phase error for a given model. This leads to a globally optimized Fourier finite-difference method that is a hybrid split-step Fourier and finite-difference scheme. Migration examples demonstrate that our dual-domain migration methods provide more accurate images than those obtained using the split-step Fourier scheme. The Born-based, quasi-Born-based, and Rytov-based methods are suitable for imaging complex structures whose lateral variations are moderate, such as the Marmousi model. For this model, the computational cost of the Born-based method is almost the same as the split-step Fourier scheme, while other methods takes approximately 15-50% more computational time. The globally optimized Fourier finite-difference method significantly improves the accuracy of the split-step Fourier method for imaging structures having strong lateral velocity variations, such as the SEG/EAGE salt model, at an approximately 30% greater
NASA Astrophysics Data System (ADS)
Akter, Jesmin; Ali Akbar, M.
The modified simple equation (MSE) method is a competent and highly effective mathematical tool for extracting exact traveling wave solutions to nonlinear evolution equations (NLEEs) arising in science, engineering and mathematical physics. In this article, we implement the MSE method to find the exact solutions involving parameters to NLEEs via the Benney-Luke equation and the Phi-4 equations. The solitary wave solutions are derived from the exact traveling wave solutions when the parameters receive their special values.
ERIC Educational Resources Information Center
Savoye, Philippe
2009-01-01
In recent years, I started covering difference equations and z transform methods in my introductory differential equations course. This allowed my students to extend the "classical" methods for (ordinary differential equation) ODE's to discrete time problems arising in many applications.
Logistic equation of arbitrary order
NASA Astrophysics Data System (ADS)
Grabowski, Franciszek
2010-08-01
The paper is concerned with the new logistic equation of arbitrary order which describes the performance of complex executive systems X vs. number of tasks N, operating at limited resources K, at non-extensive, heterogeneous self-organization processes characterized by parameter f. In contrast to the classical logistic equation which exclusively relates to the special case of sub-extensive homogeneous self-organization processes at f=1, the proposed model concerns both homogeneous and heterogeneous processes in sub-extensive and super-extensive areas. The parameter of arbitrary order f, where -∞
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…
Research on two equation models
NASA Technical Reports Server (NTRS)
Yang, Z.
1993-01-01
The k-epsilon model is the most widely used turbulence model in engineering calculations. However, the model has several deficiencies that need to be fixed. This document presents improvements to the capabilities of the k-epsilon model in the following areas: a Galilean and tensorial invariant k-epsilon model for near wall turbulence; a new set of wall functions for attached flows; a new model equation for the dissipation rate, which has a better theoretical basis, contains the contribution of flow inhomogeneity, and captures the effect of the pressure gradient accurately; and a better model for bypass transition due to freestream turbulence.
Germanium multiphase equation of state
Crockett, Scott D.; Lorenzi-Venneri, Giulia De; Kress, Joel D.; Rudin, Sven P.
2014-05-07
A new SESAME multiphase germanium equation of state (EOS) has been developed using the best available experimental data and density functional theory (DFT) calculations. The equilibrium EOS includes the Ge I (diamond), the Ge II (β-Sn) and the liquid phases. The foundation of the EOS is based on density functional theory calculations which are used to determine the cold curve and the Debye temperature. Results are compared to Hugoniot data through the solid-solid and solid-liquid transitions. We propose some experiments to better understand the dynamics of this element
Advanced lab on Fresnel equations
NASA Astrophysics Data System (ADS)
Petrova-Mayor, Anna; Gimbal, Scott
2015-11-01
This experimental and theoretical exercise is designed to promote students' understanding of polarization and thin-film coatings for the practical case of a scanning protected-metal coated mirror. We present results obtained with a laboratory scanner and a polarimeter and propose an affordable and student-friendly experimental arrangement for the undergraduate laboratory. This experiment will allow students to apply basic knowledge of the polarization of light and thin-film coatings, develop hands-on skills with the use of phase retarders, apply the Fresnel equations for metallic coating with complex index of refraction, and compute the polarization state of the reflected light.
On the connection of the quadratic Lienard equation with an equation for the elliptic functions
NASA Astrophysics Data System (ADS)
Kudryashov, Nikolay A.; Sinelshchikov, Dmitry I.
2015-07-01
The quadratic Lienard equation is widely used in many applications. A connection between this equation and a linear second-order differential equation has been discussed. Here we show that the whole family of quadratic Lienard equations can be transformed into an equation for the elliptic functions. We demonstrate that this connection can be useful for finding explicit forms of general solutions of the quadratic Lienard equation. We provide several examples of application of our approach.
Solving Equations of Multibody Dynamics
NASA Technical Reports Server (NTRS)
Jain, Abhinandan; Lim, Christopher
2007-01-01
Darts++ is a computer program for solving the equations of motion of a multibody system or of a multibody model of a dynamic system. It is intended especially for use in dynamical simulations performed in designing and analyzing, and developing software for the control of, complex mechanical systems. Darts++ is based on the Spatial-Operator- Algebra formulation for multibody dynamics. This software reads a description of a multibody system from a model data file, then constructs and implements an efficient algorithm that solves the dynamical equations of the system. The efficiency and, hence, the computational speed is sufficient to make Darts++ suitable for use in realtime closed-loop simulations. Darts++ features an object-oriented software architecture that enables reconfiguration of system topology at run time; in contrast, in related prior software, system topology is fixed during initialization. Darts++ provides an interface to scripting languages, including Tcl and Python, that enable the user to configure and interact with simulation objects at run time.
Langevin Equation for DNA Dynamics
NASA Astrophysics Data System (ADS)
Grych, David; Copperman, Jeremy; Guenza, Marina
Under physiological conditions, DNA oligomers can contain well-ordered helical regions and also flexible single-stranded regions. We describe the site-specific motion of DNA with a modified Rouse-Zimm Langevin equation formalism that describes DNA as a coarse-grained polymeric chain with global structure and local flexibility. The approach has successfully described the protein dynamics in solution and has been extended to nucleic acids. Our approach provides diffusive mode analytical solutions for the dynamics of global rotational diffusion and internal motion. The internal DNA dynamics present a rich energy landscape that accounts for an interior where hydrogen bonds and base-stacking determine structure and experience limited solvent exposure. We have implemented several models incorporating different coarse-grained sites with anisotropic rotation, energy barrier crossing, and local friction coefficients that include a unique internal viscosity and our models reproduce dynamics predicted by atomistic simulations. The models reproduce bond autocorrelation along the sequence as compared to that directly calculated from atomistic molecular dynamics simulations. The Langevin equation approach captures the essence of DNA dynamics without a cumbersome atomistic representation.
Coupled rotor and fuselage equations of motion
NASA Technical Reports Server (NTRS)
Warmbrodt, W.
1979-01-01
The governing equations of motion of a helicopter rotor coupled to a rigid body fuselage are derived. A consistent formulation is used to derive nonlinear periodic coefficient equations of motion which are used to study coupled rotor/fuselage dynamics in forward flight. Rotor/fuselage coupling is documented and the importance of an ordering scheme in deriving nonlinear equations of motion is reviewed. The nature of the final equations and the use of multiblade coordinates are discussed.
Wave equation on spherically symmetric Lorentzian metrics
Bokhari, Ashfaque H.; Al-Dweik, Ahmad Y.; Zaman, F. D.; Kara, A. H.; Karim, M.
2011-06-15
Wave equation on a general spherically symmetric spacetime metric is constructed. Noether symmetries of the equation in terms of explicit functions of {theta} and {phi} are derived subject to certain differential constraints. By restricting the metric to flat Friedman case the Noether symmetries of the wave equation are presented. Invertible transformations are constructed from a specific subalgebra of these Noether symmetries to convert the wave equation with variable coefficients to the one with constant coefficients.
Higher spin versus renormalization group equations
NASA Astrophysics Data System (ADS)
Sachs, Ivo
2014-10-01
We present a variation of earlier attempts to relate renormalization group equations to higher spin equations. We work with a scalar field theory in 3 dimensions. In this case we show that the classical renormalization group equation is a variant of the Vasiliev higher spin equations with Kleinians on AdS4 for a certain subset of couplings. In the large N limit this equivalence extends to the quantum theory away from the conformal fixed points.
Nonlinear SCHRÖDINGER-PAULI Equations
NASA Astrophysics Data System (ADS)
Ng, Wei Khim; Parwani, Rajesh R.
2011-11-01
We obtain novel nonlinear Schrüdinger-Pauli equations through a formal non-relativistic limit of appropriately constructed nonlinear Dirac equations. This procedure automatically provides a physical regularisation of potential singularities brought forward by the nonlinear terms and suggests how to regularise previous equations studied in the literature. The enhancement of contributions coming from the regularised singularities suggests that the obtained equations might be useful for future precision tests of quantum nonlinearity.
One-Equation Algebraic Model Of Turbulence
NASA Technical Reports Server (NTRS)
Baldwin, B. S.; Barth, T. J.
1993-01-01
One-equation model of turbulence based on standard equations of k-epsilon model of turbulence, where k is turbulent energy and e is rate of dissipation of k. Derivation of one-equation model motivated partly by inaccuracies of flows computed by some Navier-Stokes-equations-solving algorithms incorporating algebraic models of turbulence. Satisfies need to avoid having to determine algebraic length scales.
Stability for a class of difference equations
NASA Astrophysics Data System (ADS)
Muroya, Yoshiaki; Ishiwata, Emiko
2009-06-01
We consider the following non-autonomous and nonlinear difference equations with unbounded delays: where 0equation to be globally asymptotically stable. These conditions improve the well known stability conditions for linear and nonlinear difference equations.
Local Observed-Score Kernel Equating
ERIC Educational Resources Information Center
Wiberg, Marie; van der Linden, Wim J.; von Davier, Alina A.
2014-01-01
Three local observed-score kernel equating methods that integrate methods from the local equating and kernel equating frameworks are proposed. The new methods were compared with their earlier counterparts with respect to such measures as bias--as defined by Lord's criterion of equity--and percent relative error. The local kernel item response…
More Issues in Observed-Score Equating
ERIC Educational Resources Information Center
van der Linden, Wim J.
2013-01-01
This article is a response to the commentaries on the position paper on observed-score equating by van der Linden (this issue). The response focuses on the more general issues in these commentaries, such as the nature of the observed scores that are equated, the importance of test-theory assumptions in equating, the necessity to use multiple…
Symmetry Breaking for Black-Scholes Equations
NASA Astrophysics Data System (ADS)
Yang, Xuan-Liu; Zhang, Shun-Li; Qu, Chang-Zheng
2007-06-01
Black-Scholes equation is used to model stock option pricing. In this paper, optimal systems with one to four parameters of Lie point symmetries for Black-Scholes equation and its extension are obtained. Their symmetry breaking interaction associated with the optimal systems is also studied. As a result, symmetry reductions and corresponding solutions for the resulting equations are obtained.
Boundary conditions for the subdiffusion equation
Shkilev, V. P.
2013-04-15
The boundary conditions for the subdiffusion equations are formulated using the continuous-time random walk model, as well as several versions of the random walk model on an irregular lattice. It is shown that the boundary conditions for the same equation in different models have different forms, and this difference considerably affects the solutions of this equation.
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…
On a Equation in Finite Algebraically Structures
ERIC Educational Resources Information Center
Valcan, Dumitru
2013-01-01
Solving equations in finite algebraically structures (semigroups with identity, groups, rings or fields) many times is not easy. Even the professionals can have trouble in such cases. Therefore, in this paper we proposed to solve in the various finite groups or fields, a binomial equation of the form (1). We specify that this equation has been…
Some new modular equations and their applications
NASA Astrophysics Data System (ADS)
Yi, Jinhee; Sim, Hyo Seob
2006-07-01
Ramanujan derived 23 beautiful eta-function identities, which are certain types of modular equations. We found more than 70 of certain types of modular equations by using Garvan's Maple q-series package. In this paper, we prove some new modular equations which we found by employing the theory of modular form and we give some applications for them.
The Effects of Repeaters on Test Equating.
ERIC Educational Resources Information Center
Andrulis, Richard S.; And Others
The purpose of this investigation was to establish the effects of repeaters on test equating. Since consideration was not given to repeaters in test equating, such as in the derivation of equations by Angoff (1971), the hypothetical effect needed to be established. A case study was examined which showed results on a test as expected; overall mean…
The Effect of Repeaters on Equating
ERIC Educational Resources Information Center
Kim, HeeKyoung; Kolen, Michael J.
2010-01-01
Test equating might be affected by including in the equating analyses examinees who have taken the test previously. This study evaluated the effect of including such repeaters on Medical College Admission Test (MCAT) equating using a population invariance approach. Three-parameter logistic (3-PL) item response theory (IRT) true score and…
Solving Absolute Value Equations Algebraically and Geometrically
ERIC Educational Resources Information Center
Shiyuan, Wei
2005-01-01
The way in which students can improve their comprehension by understanding the geometrical meaning of algebraic equations or solving algebraic equation geometrically is described. Students can experiment with the conditions of the absolute value equation presented, for an interesting way to form an overall understanding of the concept.
Multidimensional soliton equations in inhomogeneous media
NASA Astrophysics Data System (ADS)
Degasperis, A.; Manakov, S. V.; Zenchuk, A. I.
1998-12-01
We use the general formalism of the overline∂-problem to derive nonlinear PDEs that are soliton equations with coordinate-dependent coefficients. Examples of these novel equations are a reduction of the Darboux equations, and a NWRI-type system.
COST EQUATIONS FOR SMALL DRINKING WATER SYSTEMS
This report presents capital and operation/maintenance cost equations for 33 drinking water treatment processes as applied to small flows (2,500 gpd to 1 mgd). The equations are based on previous cost data development work performed under contract to EPA. These equations provide ...
Shaped cassegrain reflector antenna. [design equations
NASA Technical Reports Server (NTRS)
Rao, B. L. J.
1973-01-01
Design equations are developed to compute the reflector surfaces required to produce uniform illumination on the main reflector of a cassegrain system when the feed pattern is specified. The final equations are somewhat simple and straightforward to solve (using a computer) compared to the ones which exist already in the literature. Step by step procedure for solving the design equations is discussed in detail.
Effectiveness of Analytic Smoothing in Equipercentile Equating.
ERIC Educational Resources Information Center
Kolen, Michael J.
1984-01-01
An analytic procedure for smoothing in equipercentile equating using cubic smoothing splines is described and illustrated. The effectiveness of the procedure is judged by comparing the results from smoothed equipercentile equating with those from other equating methods using multiple cross-validations for a variety of sample sizes. (Author/JKS)
Sparse dynamics for partial differential equations
Schaeffer, Hayden; Caflisch, Russel; Hauck, Cory D.; Osher, Stanley
2013-01-01
We investigate the approximate dynamics of several differential equations when the solutions are restricted to a sparse subset of a given basis. The restriction is enforced at every time step by simply applying soft thresholding to the coefficients of the basis approximation. By reducing or compressing the information needed to represent the solution at every step, only the essential dynamics are represented. In many cases, there are natural bases derived from the differential equations, which promote sparsity. We find that our method successfully reduces the dynamics of convection equations, diffusion equations, weak shocks, and vorticity equations with high-frequency source terms. PMID:23533273
The Riesz-Bessel Fractional Diffusion Equation
Anh, V.V. McVinish, R.
2004-05-15
This paper examines the properties of a fractional diffusion equation defined by the composition of the inverses of the Riesz potential and the Bessel potential. The first part determines the conditions under which the Green function of this equation is the transition probability density function of a Levy motion. This Levy motion is obtained by the subordination of Brownian motion, and the Levy representation of the subordinator is determined. The second part studies the semigroup formed by the Green function of the fractional diffusion equation. Applications of these results to certain evolution equations is considered. Some results on the numerical solution of the fractional diffusion equation are also provided.
Bogomol'nyi equations of classical solutions
NASA Astrophysics Data System (ADS)
Atmaja, Ardian N.; Ramadhan, Handhika S.
2014-11-01
We review the Bogomol'nyi equations and investigate an alternative route in obtaining it. It can be shown that the known Bogomol'nyi-Prasad-Sommerfield equations can be derived directly from the corresponding Euler-Lagrange equations via the separation of variables, without having to appeal to the Hamiltonian. We apply this technique to the Dirac-Born-Infeld solitons and obtain the corresponding equations and the potentials. This method is suitable for obtaining the first-order equations and determining the allowed potentials for noncanonical defects.
Spectrum Analysis of Some Kinetic Equations
NASA Astrophysics Data System (ADS)
Yang, Tong; Yu, Hongjun
2016-05-01
We analyze the spectrum structure of some kinetic equations qualitatively by using semigroup theory and linear operator perturbation theory. The models include the classical Boltzmann equation for hard potentials with or without angular cutoff and the Landau equation with {γ≥q-2} . As an application, we show that the solutions to these two fundamental equations are asymptotically equivalent (mod time decay rate {t^{-5/4}} ) as {tto∞} to that of the compressible Navier-Stokes equations for initial data around an equilibrium state.
Binomial moment equations for stochastic reaction systems.
Barzel, Baruch; Biham, Ofer
2011-04-15
A highly efficient formulation of moment equations for stochastic reaction networks is introduced. It is based on a set of binomial moments that capture the combinatorics of the reaction processes. The resulting set of equations can be easily truncated to include moments up to any desired order. The number of equations is dramatically reduced compared to the master equation. This formulation enables the simulation of complex reaction networks, involving a large number of reactive species much beyond the feasibility limit of any existing method. It provides an equation-based paradigm to the analysis of stochastic networks, complementing the commonly used Monte Carlo simulations. PMID:21568538
Evolution equation for quantum coherence
Hu, Ming-Liang; Fan, Heng
2016-01-01
The estimation of the decoherence process of an open quantum system is of both theoretical significance and experimental appealing. Practically, the decoherence can be easily estimated if the coherence evolution satisfies some simple relations. We introduce a framework for studying evolution equation of coherence. Based on this framework, we prove a simple factorization relation (FR) for the l1 norm of coherence, and identified the sets of quantum channels for which this FR holds. By using this FR, we further determine condition on the transformation matrix of the quantum channel which can support permanently freezing of the l1 norm of coherence. We finally reveal the universality of this FR by showing that it holds for many other related coherence and quantum correlation measures. PMID:27382933
Model Equations: "Black Box" Reconstruction
NASA Astrophysics Data System (ADS)
Bezruchko, Boris P.; Smirnov, Dmitry A.
Black box reconstruction is both the most difficult and the most tempting modelling problem when any prior information about an appropriate model structure is lacking. An intriguing thing is that a model capable of reproducing an observed behaviour or predicting further evolution should be obtained only from an observed time series, i.e. "from nothing" at first sight. Chances for a success are not large. Even more so, a "good" model would become a valuable tool to characterise an object and understand its dynamics. Lack of prior information causes one to utilise universal model structures, e.g. artificial neural networks, radial basis functions and algebraic polynomials are included in the right-hand sides of dynamical model equations. Such models are often multi-dimensional and involve quite many free parameters.
Evolution equation for quantum coherence
NASA Astrophysics Data System (ADS)
Hu, Ming-Liang; Fan, Heng
2016-07-01
The estimation of the decoherence process of an open quantum system is of both theoretical significance and experimental appealing. Practically, the decoherence can be easily estimated if the coherence evolution satisfies some simple relations. We introduce a framework for studying evolution equation of coherence. Based on this framework, we prove a simple factorization relation (FR) for the l1 norm of coherence, and identified the sets of quantum channels for which this FR holds. By using this FR, we further determine condition on the transformation matrix of the quantum channel which can support permanently freezing of the l1 norm of coherence. We finally reveal the universality of this FR by showing that it holds for many other related coherence and quantum correlation measures.
The equations of medieval cosmology
NASA Astrophysics Data System (ADS)
Buonanno, Roberto; Quercellini, Claudia
2009-04-01
In Dantean cosmography the Universe is described as a series of concentric spheres with all the known planets embedded in their rotation motion, the Earth located at the centre and Lucifer at the centre of the Earth. Beyond these "celestial spheres", Dante represents the "angelic choirs" as other nine spheres surrounding God. The rotation velocity increases with decreasing distance from God, that is with increasing Power (Virtù). We show that, adding Power as an additional fourth dimension to space, the modern equations governing the expansion of a closed Universe (i.e. with the density parameter Ω0 > 1) in the space-time, can be applied to the medieval Universe as imaged by Dante in his Divine Comedy. In this representation, the Cosmos acquires a unique description and Lucifer is not located at the centre of the hyperspheres.
Evolution equation for quantum coherence.
Hu, Ming-Liang; Fan, Heng
2016-01-01
The estimation of the decoherence process of an open quantum system is of both theoretical significance and experimental appealing. Practically, the decoherence can be easily estimated if the coherence evolution satisfies some simple relations. We introduce a framework for studying evolution equation of coherence. Based on this framework, we prove a simple factorization relation (FR) for the l1 norm of coherence, and identified the sets of quantum channels for which this FR holds. By using this FR, we further determine condition on the transformation matrix of the quantum channel which can support permanently freezing of the l1 norm of coherence. We finally reveal the universality of this FR by showing that it holds for many other related coherence and quantum correlation measures. PMID:27382933
Entropic corrections to Friedmann equations
NASA Astrophysics Data System (ADS)
Sheykhi, Ahmad
2010-05-01
Recently, Verlinde discussed that gravity can be understood as an entropic force caused by changes in the information associated with the positions of material bodies. In Verlinde’s argument, the area law of the black hole entropy plays a crucial role. However, the entropy-area relation can be modified from the inclusion of quantum effects, motivated from the loop quantum gravity. In this note, by employing this modified entropy-area relation, we derive corrections to Newton’s law of gravitation as well as modified Friedmann equations by adopting the viewpoint that gravity can be emerged as an entropic force. Our study further supports the universality of the log correction and provides a strong consistency check on Verlinde’s model.
Inferring Mathematical Equations Using Crowdsourcing
Wasik, Szymon
2015-01-01
Crowdsourcing, understood as outsourcing work to a large network of people in the form of an open call, has been utilized successfully many times, including a very interesting concept involving the implementation of computer games with the objective of solving a scientific problem by employing users to play a game—so-called crowdsourced serious games. Our main objective was to verify whether such an approach could be successfully applied to the discovery of mathematical equations that explain experimental data gathered during the observation of a given dynamic system. Moreover, we wanted to compare it with an approach based on artificial intelligence that uses symbolic regression to find such formulae automatically. To achieve this, we designed and implemented an Internet game in which players attempt to design a spaceship representing an equation that models the observed system. The game was designed while considering that it should be easy to use for people without strong mathematical backgrounds. Moreover, we tried to make use of the collective intelligence observed in crowdsourced systems by enabling many players to collaborate on a single solution. The idea was tested on several hundred players playing almost 10,000 games and conducting a user opinion survey. The results prove that the proposed solution has very high potential. The function generated during weeklong tests was almost as precise as the analytical solution of the model of the system and, up to a certain complexity level of the formulae, it explained data better than the solution generated automatically by Eureqa, the leading software application for the implementation of symbolic regression. Moreover, we observed benefits of using crowdsourcing; the chain of consecutive solutions that led to the best solution was obtained by the continuous collaboration of several players. PMID:26713846
Inferring Mathematical Equations Using Crowdsourcing.
Wasik, Szymon; Fratczak, Filip; Krzyskow, Jakub; Wulnikowski, Jaroslaw
2015-01-01
Crowdsourcing, understood as outsourcing work to a large network of people in the form of an open call, has been utilized successfully many times, including a very interesting concept involving the implementation of computer games with the objective of solving a scientific problem by employing users to play a game-so-called crowdsourced serious games. Our main objective was to verify whether such an approach could be successfully applied to the discovery of mathematical equations that explain experimental data gathered during the observation of a given dynamic system. Moreover, we wanted to compare it with an approach based on artificial intelligence that uses symbolic regression to find such formulae automatically. To achieve this, we designed and implemented an Internet game in which players attempt to design a spaceship representing an equation that models the observed system. The game was designed while considering that it should be easy to use for people without strong mathematical backgrounds. Moreover, we tried to make use of the collective intelligence observed in crowdsourced systems by enabling many players to collaborate on a single solution. The idea was tested on several hundred players playing almost 10,000 games and conducting a user opinion survey. The results prove that the proposed solution has very high potential. The function generated during weeklong tests was almost as precise as the analytical solution of the model of the system and, up to a certain complexity level of the formulae, it explained data better than the solution generated automatically by Eureqa, the leading software application for the implementation of symbolic regression. Moreover, we observed benefits of using crowdsourcing; the chain of consecutive solutions that led to the best solution was obtained by the continuous collaboration of several players. PMID:26713846
Exact solution to fractional logistic equation
NASA Astrophysics Data System (ADS)
West, Bruce J.
2015-07-01
The logistic equation is one of the most familiar nonlinear differential equations in the biological and social sciences. Herein we provide an exact solution to an extension of this equation to incorporate memory through the use of fractional derivatives in time. The solution to the fractional logistic equation (FLE) is obtained using the Carleman embedding technique that allows the nonlinear equation to be replaced by an infinite-order set of linear equations, which we then solve exactly. The formal series expansion for the initial value solution of the FLE is shown to be expressed in terms of a series of weighted Mittag-Leffler functions that reduces to the well known analytic solution in the limit where the fractional index for the derivative approaches unity. The numerical integration to the FLE provides an excellent fit to the analytic solution. We propose this approach as a general technique for solving a class of nonlinear fractional differential equations.
10. Exploring the Conformal Constraint Equations
NASA Astrophysics Data System (ADS)
Butscher, Adrian
One method of studying the asymptotic structure of spacetime is to apply Penrose's conformal rescaling technique. In this setting, the rescaled Einstein equations for the metric and the conformal factor in the unphysical spacetime degenerate where the conformal factor vanishes, namely at the boundary representing null infinity. This problem can be avoided by means of a technique of H. Friedrich, which replaces the Einstein equations in the unphysical spacetime by an equivalent system of equations which is regular at the boundary. The initial value problem for these equations produces a system of constraint equations known as the conformal constraint equations. This work describes some of the properties of the conformal constraint equations and develops a perturbative method of generating solutions near Euclidean space under certain simplifying assumptions.
Solving Space-Time Fractional Differential Equations by Using Modified Simple Equation Method
NASA Astrophysics Data System (ADS)
Kaplan, Melike; Akbulut, Arzu; Bekir, Ahmet
2016-05-01
In this article, we establish new and more general traveling wave solutions of space-time fractional Klein–Gordon equation with quadratic nonlinearity and the space-time fractional breaking soliton equations using the modified simple equation method. The proposed method is so powerful and effective to solve nonlinear space-time fractional differential equations by with modified Riemann–Liouville derivative.
Exact solutions of the time-fractional Fisher equation by using modified trial equation method
NASA Astrophysics Data System (ADS)
Tandogan, Yusuf Ali; Bildik, Necdet
2016-06-01
In this study, modified trial equation method has been proposed to obtain precise solutions of nonlinear fractional differential equation. Using the modified test equation method, we obtained some new exact solutions of the time fractional nonlinear Fisher equation. The obtained results are classified as a soliton solution, singular solutions, rational function solutions and periodic solutions.
The properties of the first equation of the Vlasov chain of equations
NASA Astrophysics Data System (ADS)
Perepelkin, E. E.; Sadovnikov, B. I.; Inozemtseva, N. G.
2015-05-01
A derivation of the first Vlasov equation as a well-known Schrödinger equation for the probabilistic description of a system and families of the classic diffusion equations and heat conduction for the deterministic description of physical systems was inferred. A physical meaning of the phase of the wave function which is a scalar potential of the probabilistic flow velocity is demonstrated. Occurrence of the velocity potential vortex component leads to the Pauli equation for one of the spinar components. A scheme for the construction of the Schrödinger equation solution from the Vlasov equation solution and vice-versa is shown. A process of introduction of the potential to the Schrödinger equation and its interpretation are given. The analysis of the potential properties gives us the Maxwell equation, the equation of the kinematic point movement, and the equation for movement of the medium within electromagnetic fields.
Complex PT-symmetric nonlinear Schrödinger equation and Burgers equation.
Yan, Zhenya
2013-04-28
The complex -symmetric nonlinear wave models have drawn much attention in recent years since the complex -symmetric extensions of the Korteweg-de Vries (KdV) equation were presented in 2007. In this review, we focus on the study of the complex -symmetric nonlinear Schrödinger equation and Burgers equation. First of all, we briefly introduce the basic property of complex symmetry. We then report on exact solutions of one- and two-dimensional nonlinear Schrödinger equations (known as the Gross-Pitaevskii equation in Bose-Einstein condensates) with several complex -symmetric potentials. Finally, some complex -symmetric extension principles are used to generate some complex -symmetric nonlinear wave equations starting from both -symmetric (e.g. the KdV equation) and non- -symmetric (e.g. the Burgers equation) nonlinear wave equations. In particular, we discuss exact solutions of some representative ones of the complex -symmetric Burgers equation in detail. PMID:23509385
Dust levitation about Itokawa's equator
NASA Astrophysics Data System (ADS)
Hartzell, C.; Zimmerman, M.; Takahashi, Y.
2014-07-01
levitation about Itokawa, we must include accurate plasma and gravity models. We use a 2D PIC code (described in [8]) to model the plasma environment about Itokawa's equator. The plasma model includes photoemission and shadowing. Thus, we model the plasma environment for various solar incidence angles. The plasma model gives us the 2D electric field components and the plasma potential. We model the gravity field around the equatorial cross-section using an Interior Gravity model [9]. The gravity model is based on the shape model acquired by the Hayabusa mission team and, unlike other models, is quick and accurate close to the surface of the body. Due to the nonspherical shape of Itokawa, the electrostatic force and the gravity may not be collinear. Given our accurate plasma and gravity environments, we are able to simulate the trajectories of dust grains about the equator of Itokawa. When modeling the trajectories of the grains, the current to the grains is calculated using Nitter et al.'s formulation [10] with the plasma sheath parameters provided by our PIC model (i.e., the potential minimum, the potential at the surface, and the sheath type). Additionally, we are able to numerically locate the equilibria about which dust grains may levitate. Interestingly, we observe that equilibria exist for grains up to 20 microns in radius about Itokawa's equator when the Sun is illuminating Itokawa's 'otter tail'. This grain size is significantly larger than the stably levitating grains we observed using our 1D plasma and gravity models. Conclusions and Future Work: The possibility of dust levitation above asteroids has implications both for our understanding of their evolution and for the design of future missions to these bodies. Using detailed gravity and plasma models, we are above to propagate the trajectories of dust particles about Itokawa's equator and identify the equilibria about which these grains will levitate. Using these simulations, we see that grains up to 20 microns
The telegraph equation in charged particle transport
NASA Technical Reports Server (NTRS)
Gombosi, T. I.; Jokipii, J. R.; Kota, J.; Lorencz, K.; Williams, L. L.
1993-01-01
We present a new derivation of the telegraph equation which modifies its coefficients. First, an infinite order partial differential equation is obtained for the velocity space solid angle-averaged phase-space distribution of particles which underwent at least a few collisions. It is shown that, in the lowest order asymptotic expansion, this equation simplifies to the well-known diffusion equation. The second-order asymptotic expansion for isotropic small-angle scattering results in a modified telegraph equation with a signal propagation speed of v(5/11) exp 1/2 instead of the usual v/3 exp 1/2. Our derivation of a modified telegraph equation follows from an expansion of the Boltzmann equation in the relevant smallness parameters and not from a truncation of an eigenfunction expansion. This equation is consistent with causality. It is shown that, under steady state conditions in a convecting plasma, the telegraph equation may be regarded as a diffusion equation with a modified transport coefficient, which describes a combination of diffusion and cosmic-ray inertia.
On some differential transformations of hypergeometric equations
NASA Astrophysics Data System (ADS)
Hounkonnou, M. N.; Ronveaux, A.
2015-04-01
Many algebraic transformations of the hypergeometric equation σ(x)z"(x) + τ(x)z'(x) + lz(x) = 0, where σ, τ, l are polynomial functions of degrees 2 (at most), 1, 0, respectively, are well known. Some of them involve x = x(t), a polynomial of degree r, in order to recover the Heun equation, extension of the hypergeometric equation by one more singularity. The case r = 2 was investigated by K. Kuiken (see 1979 SIAM J. Math. Anal. 10 (3) 655-657) and extended to r = 3,4, 5 by R. S. Maier (see 2005 J. Differ. Equat. 213 171 - 203). The transformations engendered by the function y(x) = A(x)z(x), also very popular in mathematics and physics, are used to get from the hypergeometric equation, for instance, the Schroedinger equation with appropriate potentials, as well as Heun and confluent Heun equations. This work addresses a generalization of Kimura's approach proposed in 1971, based on differential transformations of the hypergeometric equations involving y(x) = A(x)z(x) + B(x)z'(x). Appropriate choices of A(x) and B(x) permit to retrieve the Heun equations as well as equations for some exceptional polynomials. New relations are obtained for Laguerre and Hermite polynomials.
Exact and explicit solitary wave solutions to some nonlinear equations
Jiefang Zhang
1996-08-01
Exact and explicit solitary wave solutions are obtained for some physically interesting nonlinear evolutions and wave equations in physics and other fields by using a special transformation. These equations include the KdV-Burgers equation, the MKdV-Burgers equation, the combined KdV-MKdV equation, the Newell-Whitehead equation, the dissipative {Phi}{sup 4}-model equation, the generalized Fisher equation, and the elastic-medium wave equation.
Three-body equations for nuclear reactions
Chao, S.H.
1980-01-01
The problem of calculating three-body differential cross sections for two-body interactions which are local in configuration space is considered. The integral equations of Faddeev and recent modifications are reviewed. The difficulties both in solving and interpreting the various equations are discussed. An alternative set of exact operator equations is proposed. This new set of operator equations involve the two-body potentials directly rather than the two-body t-matrices. The solutions represent quantities different from those of previous equations. The formal structure of the operator equations is similar to but different from the Faddeev equations. It is demonstrated that the equations contain no terms which correspond to disconnected diagrams. The transformation of the equations to the Faddeev equations is given. Integral equations are obtained using the momentum representation. Only Cauchy-type singularities occur in the Green's function, and the equations have unique solutions. No off-energy shell t-matrices are required. Transition amplitudes are obtained from the solutions by quadrature, so that effects which result from the nature of the final state can be separated from the three-body effects. The angular momentum reduction of Omnes is used to obtain two-dimensional coupled integral equations. A method of numerical solution is developed using a generalization of the approximate product integration method due to Young. Examples corresponding to the reactions /sup 16/O(d,p)/sup 17/O g.s and /sup 16/O(d,d)/sup 16/O in two dimensions and /sup 16/O(d,p)/sup 17/O* (0.87 MeV) in three dimensions for a deuteron enegy of 5 MeV are considered using Gaussian potentials. The convergence of the summation over angular momentum is examined and a comparison is made with experiment to obtain the spectroscopic factor for the 0.87 MeV state of /sup 17/O.
Cosmic-Ray Modulation Equations
NASA Astrophysics Data System (ADS)
Moraal, H.
2013-06-01
The temporal variation of the cosmic-ray intensity in the heliosphere is called cosmic-ray modulation. The main periodicity is the response to the 11-year solar activity cycle. Other variations include a 27-day solar rotation variation, a diurnal variation, and irregular variations such as Forbush decreases. General awareness of the importance of this cosmic-ray modulation has greatly increased in the last two decades, mainly in communities studying cosmogenic nuclides, upper atmospheric physics and climate, helio-climatology, and space weather, where corrections need to be made for these modulation effects. Parameterized descriptions of the modulation are even used in archeology and in planning the flight paths of commercial passenger jets. The qualitative, physical part of the modulation is generally well-understood in these communities. The mathematical formalism that is most often used to quantify it is the so-called Force-Field approach, but the origins of this approach are somewhat obscure and it is not always used correct. This is mainly because the theory was developed over more than 40 years, and all its aspects are not collated in a single document. This paper contains a formal mathematical description intended for these wider communities. It consists of four parts: (1) a description of the relations between four indicators of "energy", namely energy, speed, momentum and rigidity, (2) the various ways of how to count particles, (3) the description of particle motion with transport equations, and (4) the solution of such equations, and what these solutions mean. Part (4) was previously described in Caballero-Lopez and Moraal (J. Geophys. Res, 109: A05105, doi: 10.1029/2003JA010358, 2004). Therefore, the details are not all repeated here. The style of this paper is not to be rigorous. It rather tries to capture the relevant tools to do modulation studies, to show how seemingly unrelated results are, in fact, related to one another, and to point out the
Equation of state of polytetrafluoroethylene
NASA Astrophysics Data System (ADS)
Bourne, N. K.; Gray, G. T.
2003-06-01
The present drive to make munitions as safe as is feasible and to develop predictive models describing their constitutive response, has led to the development and production of plastic bonded explosives and propellants. There is a range of elastomers used as binder materials with the energetic components. One of these is known as Kel-F-800™ (poly-chloro-trifluroethylene) whose structure is in some ways analogous to that of poly-tetrafluoroethylene (PTFE or Teflon). Thus, it is of interest to assess the mechanical behavior of Teflon and to compare the response of five different production Teflon materials, two of which were produced in pedigree form, one as-received product, and two from previous in-depth literature studies. The equations of state of these variants were quantified by conducting a series of shock impact experiments in which both pressure-particle velocity and shock velocity-particle velocity dependencies were measured. The compressive behavior of Teflon, based upon the results of this study, appears to be independent of the production route and additives introduced.
Dirac equations with confining potentials
NASA Astrophysics Data System (ADS)
Noble, J. H.; Jentschura, U. D.
2015-01-01
This paper is devoted to a study of relativistic eigenstates of Dirac particles which are simultaneously bound by a static Coulomb potential and added linear confining potentials. Under certain conditions, despite the addition of radially symmetric, linear confining potentials, specific bound-state energies surprisingly preserve their exact Dirac-Coulomb values. The generality of the "preservation mechanism" is investigated. To this end, a Foldy-Wouthuysen transformation is used to calculate the corrections to the spin-orbit coupling induced by the linear confining potentials. We find that the matrix elements of the effective operators obtained from the scalar, and time-like confining potentials mutually cancel for specific ratios of the prefactors of the effective operators, which must be tailored to the preservation mechanism. The result of the Foldy-Wouthuysen transformation is used to verify that the preservation is restricted (for a given Hamiltonian) to only one reference state, rather than traceable to a more general relationship among the obtained effective low-energy operators. The results derived from the nonrelativistic effective operators are compared to the fully relativistic radial Dirac equations. Furthermore, we show that the preservation mechanism does not affect antiparticle (negative-energy) states.
Scalable Equation of State Capability
Epperly, T W; Fritsch, F N; Norquist, P D; Sanford, L A
2007-12-03
The purpose of this techbase project was to investigate the use of parallel array data types to reduce the memory footprint of the Livermore Equation Of State (LEOS) library. Addressing the memory scalability of LEOS is necessary to run large scientific simulations on IBM BG/L and future architectures with low memory per processing core. We considered using normal MPI, one-sided MPI, and Global Arrays to manage the distributed array and ended up choosing Global Arrays because it was the only communication library that provided the level of asynchronous access required. To reduce the runtime overhead using a parallel array data structure, a least recently used (LRU) caching algorithm was used to provide a local cache of commonly used parts of the parallel array. The approach was initially implemented in a isolated copy of LEOS and was later integrated into the main trunk of the LEOS Subversion repository. The approach was tested using a simple test. Testing indicated that the approach was feasible, and the simple LRU caching had a 86% hit rate.
Stability analysis of ecomorphodynamic equations
NASA Astrophysics Data System (ADS)
Bärenbold, F.; Crouzy, B.; Perona, P.
2016-02-01
In order to shed light on the influence of riverbed vegetation on river morphodynamics, we perform a linear stability analysis on a minimal model of vegetation dynamics coupled with classical one- and two-dimensional Saint-Venant-Exner equations of morphodynamics. Vegetation is modeled as a density field of rigid, nonsubmerged cylinders and affects flow via a roughness change. Furthermore, vegetation is assumed to develop following a logistic dependence and may be uprooted by flow. First, we perform the stability analysis of the reduced one-dimensional framework. As a result of the competitive interaction between vegetation growth and removal through uprooting, we find a domain in the parameter space where originally straight rivers are unstable toward periodic longitudinal patterns. For realistic values of the sediment transport parameter, the dominant longitudinal wavelength is determined by the parameters of the vegetation model. Bed topography is found to adjust to the spatial pattern fixed by vegetation. Subsequently, the stability analysis is repeated for the two-dimensional framework, where the system may evolve toward alternate or multiple bars. On a fixed bed, we find instability toward alternate bars due to flow-vegetation interaction, but no multiple bars. Both alternate and multiple bars are present on a movable, vegetated bed. Finally, we find that the addition of vegetation to a previously unvegetated riverbed favors instability toward alternate bars and thus the development of a single course rather than braiding.
Silicon Nitride Equation of State
NASA Astrophysics Data System (ADS)
Swaminathan, Pazhayannur; Brown, Robert
2015-06-01
This report presents the development a global, multi-phase equation of state (EOS) for the ceramic silicon nitride (Si3N4) . Structural forms include amorphous silicon nitride normally used as a thin film and three crystalline polymorphs. Crystalline phases include hexagonal α-Si3N4, hexagonalβ-Si3N4, and the cubic spinel c-Si3N4. Decomposition at about 1900 °C results in a liquid silicon phase and gas phase products such as molecular nitrogen, atomic nitrogen, and atomic silicon. The silicon nitride EOS was developed using EOSPro which is a new and extended version of the PANDA II code. Both codes are valuable tools and have been used successfully for a variety of material classes. Both PANDA II and EOSPro can generate a tabular EOS that can be used in conjunction with hydrocodes. The paper describes the development efforts for the component solid phases and presents results obtained using the EOSPro phase transition model to investigate the solid-solid phase transitions in relation to the available shock data. Furthermore, the EOSPro mixture model is used to develop a model for the decomposition products and then combined with the single component solid models to study the global phase diagram. Sponsored by the NASA Goddard Space Flight Center Living With a Star program office.
Double distributions and evolution equations
A.V. Radyushkin
1998-05-01
Applications of perturbative QCD to deeply virtual Compton scattering and hard exclusive meson electroproduction processes require a generalization of usual parton distributions for the case when long-distance information is accumulated in nonforward matrix elements < p{prime} {vert_bar}O(0,z){vert_bar}p > of quark and gluon light-cone operators. In their previous papers the authors used two types of nonperturbative functions parameterizing such matrix elements: double distributions F(x,y;t) and nonforward distribution functions F{sub {zeta}}(X;t). Here they discuss in more detail the double distributions (DD's) and evolution equations which they satisfy. They propose simple models for F(x,y;t=0) DD's with correct spectral and symmetry properties which also satisfy the reduction relations connecting them to the usual parton densities f(x). In this way, they obtain self-consistent models for the {zeta}-dependence of nonforward distributions. They show that, for small {zeta}, one can easily obtain nonforward distributions (in the X > {zeta} region) from the parton densities: F{sub {zeta}} (X;t=0) {approx} f(X{minus}{zeta}/2).
Comparison of logistic equations for population growth.
Jensen, A L
1975-12-01
Two different forms of the logistic equation for population growth appear in the ecological literature. In the form of the logistic equation that appears in recent ecology textbooks the parameters are the instantaneous rate of natural increase per individual and the carrying capacity of the environment. In the form of the logistic equation that appears in some older literature the parameters are the instantaneous birth rate per individual and the carrying capacity. The decision whether to use one form or the other depends on which form of the equation is biologically more realistic. In this study the form of the logistic equation in which the instantaneous birth rate per individual is a parameter is shown to be more realistic in terms of the birth and death processes of population growth. Application of the logistic equation to calculate yield from an exploited fish population also shows that the parameters must be the instantaneous birth rate per individual and the carrying capacity. PMID:1203427
Darboux transformation for the NLS equation
Aktosun, Tuncay; Mee, Cornelis van der
2010-03-08
We analyze a certain class of integral equations associated with Marchenko equations and Gel'fand-Levitan equations. Such integral equations arise through a Fourier transformation on various ordinary differential equations involving a spectral parameter. When the integral operator is perturbed by a finite-rank perturbation, we explicitly evaluate the change in the solution in terms of the unperturbed quantities and the finite-rank perturbation. We show that this result provides a fundamental approach to derive Darboux transformations for various systems of ordinary differential operators. We illustrate our theory by providing the explicit Darboux transformation for the Zakharov-Shabat system and show how the potential and wave function change when a simple discrete eigenvalue is added to the spectrum, and thus we also provide a one-parameter family of Darboux transformations for the nonlinear Schroedinger equation.
Stochastic differential equation model to Prendiville processes
Granita; Bahar, Arifah
2015-10-22
The Prendiville process is another variation of the logistic model which assumes linearly decreasing population growth rate. It is a continuous time Markov chain (CTMC) taking integer values in the finite interval. The continuous time Markov chain can be approximated by stochastic differential equation (SDE). This paper discusses the stochastic differential equation of Prendiville process. The work started with the forward Kolmogorov equation in continuous time Markov chain of Prendiville process. Then it was formulated in the form of a central-difference approximation. The approximation was then used in Fokker-Planck equation in relation to the stochastic differential equation of the Prendiville process. The explicit solution of the Prendiville process was obtained from the stochastic differential equation. Therefore, the mean and variance function of the Prendiville process could be easily found from the explicit solution.
Remarks on the Kuramoto-Sivashinsky equation
Nicolaenko, B.; Scheurer, B.
1983-01-01
We report here a joint work in progress on the Kuramoto-Sivashinsky equation. The question we address is the analytical study of a fourth order nonlinear evolution equation. This equation has been obtained by Sivashinsky in the context of combustion and independently by Kuramoto in the context of reaction diffusion-systems. Both were motivated by (nonlinear) stability of travelling waves. Numerical calculations have been done on this equation. All the results seem to indicate a chaotic behavior of the solution. Therefore, the analytical study is of interest in analogy with the Burger's and Navier-Stokes equations. Here we give some existence and uniqueness results for the equation in space dimension one, and we also study a fractional step method of numerical resolution. In a forthcoming joint paper with R. Temam, we will study the asymptotic behavior, as t approaches infinity, of the solution of (0.1) and give an estimate on the number of determining modes.
A state estimation of Liu equations
NASA Astrophysics Data System (ADS)
Ananyev, B. I.
2015-11-01
This paper is concerned with state estimation problems for so-called Liu equations. These equations are counterparts of well-known Ito ones and they were introduced by B. Liu under elaboration of his uncertain theory. The Liu equations may be solved backward and they represent a more convenient object for the state estimation problem solution especially for the case when distributions of disturbances are unknown. Using the dynamic programming principle, we derive an equation for the informational set consisting of all states that are compatible with measuring data. Special cases of Liu equations and constraints for disturbances are examined. Among them the linear equations with quadratic constraints are considered in most details. Some examples are also given.
Chandrasekhar equations for infinite dimensional systems
NASA Technical Reports Server (NTRS)
Ito, K.; Powers, R. K.
1985-01-01
Chandrasekhar equations are derived for linear time invariant systems defined on Hilbert spaces using a functional analytic technique. An important consequence of this is that the solution to the evolutional Riccati equation is strongly differentiable in time and one can define a strong solution of the Riccati differential equation. A detailed discussion on the linear quadratic optimal control problem for hereditary differential systems is also included.
Partitioning And Packing Equations For Parallel Processing
NASA Technical Reports Server (NTRS)
Arpasi, Dale J.; Milner, Edward J.
1989-01-01
Algorithm developed to identify parallelism in set of coupled ordinary differential equations that describe physical system and to divide set into parallel computational paths, along with parts of solution proceeds independently of others during at least part of time. Path-identifying algorithm creates number of paths consisting of equations that must be computed serially and table that gives dependent and independent arguments and "can start," "can end," and "must end" times of each equation. "Must end" time used subsequently by packing algorithm.
Dielectric polarization evolution equations and relaxation times
Baker-Jarvis, James; Riddle, Bill; Janezic, Michael D.
2007-05-15
In this paper we develop dielectric polarization evolution equations, and the resulting frequency-domain expressions, and relationships for the resulting frequency dependent relaxation times. The model is based on a previously developed equation that was derived using statistical-mechanical theory. We extract relaxation times from dielectric data and give illustrative examples for the harmonic oscillator and derive expressions for the frequency-dependent relaxation times and a time-domain integrodifferential equation for the Cole-Davidson model.
The Boltzmann equation in the difference formulation
Szoke, Abraham; Brooks III, Eugene D.
2015-05-06
First we recall the assumptions that are needed for the validity of the Boltzmann equation and for the validity of the compressible Euler equations. We then present the difference formulation of these equations and make a connection with the time-honored Chapman - Enskog expansion. We discuss the hydrodynamic limit and calculate the thermal conductivity of a monatomic gas, using a simplified approximation for the collision term. Our formulation is more consistent and simpler than the traditional derivation.
Switched electrical networks and bilinear equations
NASA Technical Reports Server (NTRS)
Wood, J. R.
1974-01-01
An investigation is conducted concerning the state equations which arise in the description of power processing systems. Attention is given to the role played by Lie groups and Lie algebras in the characterization of the dynamical features of the systems. The bilinear equations used for the representation of the network characteristics are discussed along with the nature of the solutions for the equations. The application of the described approaches is illustrated with the aid of a number of network examples.
Exact solutions of population balance equation
NASA Astrophysics Data System (ADS)
Lin, Fubiao; Flood, Adrian E.; Meleshko, Sergey V.
2016-07-01
Population balance equations have been used to model a wide range of processes including polymerization, crystallization, cloud formation, and cell dynamics, but the lack of analytical solutions necessitates the use of numerical techniques. The one-dimensional homogeneous population balance equation with time dependent but size independent growth rate and time dependent nucleation rate is investigated. The corresponding system of equations is solved analytically in this paper.
Geometrical and Graphical Solutions of Quadratic Equations.
ERIC Educational Resources Information Center
Hornsby, E. John, Jr.
1990-01-01
Presented are several geometrical and graphical methods of solving quadratic equations. Discussed are Greek origins, Carlyle's method, von Staudt's method, fixed graph methods and imaginary solutions. (CW)
Some remarks on unilateral matrix equations
Cerchiai, Bianca L.; Zumino, Bruno
2001-02-01
We briefly review the results of our paper LBNL-46775: We study certain solutions of left-unilateral matrix equations. These are algebraic equations where the coefficients and the unknown are square matrices of the same order, or, more abstractly, elements of an associative, but possibly noncommutative algebra, and all coefficients are on the left. Recently such equations have appeared in a discussion of generalized Born-Infeld theories. In particular, two equations, their perturbative solutions and the relation between them are studied, applying a unified approach based on the generalized Bezout theorem for matrix polynomials.
Analytic solutions of the relativistic Boltzmann equation
NASA Astrophysics Data System (ADS)
Hatta, Yoshitaka; Martinez, Mauricio; Xiao, Bo-Wen
2015-04-01
We present new analytic solutions to the relativistic Boltzmann equation within the relaxation time approximation. We first obtain spherically expanding solutions which are the kinetic counterparts of the exact solutions of the Israel-Stewart equation in the literature. This allows us to compare the solutions of the kinetic and hydrodynamic equations at an analytical level. We then derive a novel boost-invariant solution of the Boltzmann equation which has an unconventional dependence on the proper time. The existence of such a solution is also suggested in second-order hydrodynamics and fluid-gravity correspondence.
Non-Markovian stochastic evolution equations
NASA Astrophysics Data System (ADS)
Costanza, G.
2014-05-01
Non-Markovian continuum stochastic and deterministic equations are derived from a set of discrete stochastic and deterministic evolution equations. Examples are given of discrete evolution equations whose updating rules depend on two or more previous time steps. Among them, the continuum stochastic evolution equation of the Newton second law, the stochastic evolution equation of a wave equation, the stochastic evolution equation for the scalar meson field, etc. are obtained as special cases. Extension to systems of evolution equations and other extensions are considered and examples are given. The concept of isomorphism and almost isomorphism are introduced in order to compare the coefficients of the continuum evolution equations of two different smoothing procedures that arise from two different approaches. Usually these discrepancies arising from two sources: On the one hand, the use of different representations of the generalized functions appearing in the models and, on the other hand, the different approaches used to describe the models. These new concept allows to overcome controversies that were appearing during decades in the literature.
Integral equations for flows in wind tunnels
NASA Technical Reports Server (NTRS)
Fromme, J. A.; Golberg, M. A.
1979-01-01
This paper surveys recent work on the use of integral equations for the calculation of wind tunnel interference. Due to the large number of possible physical situations, the discussion is limited to two-dimensional subsonic and transonic flows. In the subsonic case, the governing boundary value problems are shown to reduce to a class of Cauchy singular equations generalizing the classical airfoil equation. The theory and numerical solution are developed in some detail. For transonic flows nonlinear singular equations result, and a brief discussion of the work of Kraft and Kraft and Lo on their numerical solution is given. Some typical numerical results are presented and directions for future research are indicated.
Integral equations for resonance and virtual states
Orlov, Y.V.; Turovtsev, V.V.
1984-05-01
Integral equations are derived for the resonance and virtual (antibound) states consisting of two or three bodies. The derivation is based on the analytic continuation of the integral equations of scattering theory to nonphysical energy sheets. The resulting equations can be used to exhibit the analytic properties of amplitudes that are necessary for practical calculations using the equations for the quasistationary levels and Gamov wave functions derived in this paper. The Fourier transformation and the normalization rule for the wave function are generalized to the case of nonstationary states. The energy of the antibound state of the tritium nucleus is calculated for a ''realistic'' local potential.
Exact Travelling Wave Solutions of the Nonlinear Evolution Equations by Auxiliary Equation Method
NASA Astrophysics Data System (ADS)
Kaplan, Melike; Akbulut, Arzu; Bekir, Ahmet
2015-10-01
The auxiliary equation method presents wide applicability to handling nonlinear wave equations. In this article, we establish new exact travelling wave solutions of the nonlinear Zoomeron equation, coupled Higgs equation, and equal width wave equation. The travelling wave solutions are expressed by the hyperbolic functions, trigonometric functions, and rational functions. It is shown that the proposed method provides a powerful mathematical tool for solving nonlinear wave equations in mathematical physics and engineering. Throughout the article, all calculations are made with the aid of the Maple packet program.
He, Xiaoyi; Lou, Li-Shi Lou, Li-Shi
1997-12-01
In this paper, the lattice Boltzmann equation is directly derived from the Boltzmann equation. It is shown that the lattice Boltzmann equation is a special discretized form of the Boltzmann equation. Various approximations for the discretization of the Boltzmann equation in both time and phase space are discussed in detail. A general procedure to derive the lattice Boltzmann model from the continuous Boltzmann equation is demonstrated explicitly. The lattice Boltzmann models derived include the two-dimensional 6-bit, 7-bit, and 9-bit, and three-dimensional 27-bit models. {copyright} {ital 1997} {ital The American Physical Society}
Cylindrical nonlinear Schroedinger equation versus cylindrical Korteweg-de Vries equation
Fedele, Renato; De Nicola, Sergio; Grecu, Dan; Visinescu, Anca; Shukla, Padma K.
2008-10-15
A correspondence between the family of cylindrical nonlinear Schroedinger (cNLS) equations and the one of cylindrical Korteweg-de Vries (cKdV) equations is constructed. It associates non stationary solutions of the first family with the ones of the second family. This is done by using a correspondence, recently found, between the families of generalized NLS equation and generalized KdV equation, and their solutions in the form of travelling waves, respectively. In particular, non-stationary soliton-like solutions of the cNLS equation can be associated with non-stationary soliton-like solutions of cKdV equation.
NASA Astrophysics Data System (ADS)
Zhuravlev, V. M.
2016-03-01
We discuss an extension of the theory of multidimensional second-order equations of the elliptic and hyperbolic types related to multidimensional quasilinear autonomous first-order partial differential equations. Calculating the general integrals of these equations allows constructing exact solutions in the form of implicit functions. We establish a connection with hydrodynamic equations. We calculate the number of free functional parameters of the constructed solutions. We especially construct and analyze implicit solutions of the Laplace and d'Alembert equations in a coordinate space of arbitrary finite dimension. In particular, we construct generalized Penrose-Rindler solutions of the d'Alembert equation in 3+1 dimensions.