Amplitude modulation for the Swift-Hohenberg and Kuramoto-Sivashinski equations
NASA Astrophysics Data System (ADS)
Kirkinis, Eleftherios; O'Malley, Robert E.
2014-12-01
Employing a harmonic balance technique inspired from the methods of Renormalization Group and Multiple Scales [R. E. O'Malley, Jr. and E. Kirkinis. "A combined renormalization group-multiple scale method for singularly perturbed problems," Stud. Appl. Math. 124(4), 383-410, (2010)], we derive the amplitude equations for the Swift-Hohenberg and Kuramoto-Sivashinski equations to arbitrary order in the context of roll patterns. This new and straightforward derivation improves previous attempts and can be carried-out with symbolic computation that minimizes effort and avoids error.
NASA Astrophysics Data System (ADS)
Allouba, Hassan
2015-12-01
Generalizing the L-Kuramoto-Sivashinsky (L-KS) kernel from our earlier work, we give a novel explicit-kernel formulation useful for a large class of fourth order deterministic, stochastic, linear, and nonlinear PDEs in multispatial dimensions. These include pattern formation equations like the Swift-Hohenberg and many other prominent and new PDEs. We first establish existence, uniqueness, and sharp dimension-dependent spatio-temporal Hölder regularity for the canonical (zero drift) L-KS SPDE, driven by white noise on {R+×Rd} d = 1 3 . The spatio-temporal Hölder exponents are exactly the same as the striking ones we proved for our recently introduced Brownian-time Brownian motion (BTBM) stochastic integral equation, associated with time-fractional PDEs. The challenge here is that, unlike the positive BTBM density, the L-KS kernel is the Gaussian average of a modified, highly oscillatory, and complex Schrödinger propagator. We use a combination of harmonic and delicate analysis to get the necessary estimates. Second, attaching order parameters ε1 to the L-KS spatial operator and ε2 to the noise term, we show that the dimension-dependent critical ratio ε2 /ε1d/8 controls the limiting behavior of the L-KS SPDE, as ε1, ε2 ↘ 0; and we compare this behavior to that of the less regular second order heat SPDEs. Finally, we give a change-of-measure equivalence between the canonical L-KS SPDE and nonlinear L-KS SPDEs. In particular, we prove uniqueness in law for the Swift-Hohenberg and the law equivalence-and hence the same Hölder regularity-of the Swift-Hohenberg SPDE and the canonical L-KS SPDE on compacts in one-to-three dimensions.
Schüler, D.; Alonso, S.; Bär, M.; Torcini, A.
2014-12-15
Pattern formation often occurs in spatially extended physical, biological, and chemical systems due to an instability of the homogeneous steady state. The type of the instability usually prescribes the resulting spatio-temporal patterns and their characteristic length scales. However, patterns resulting from the simultaneous occurrence of instabilities cannot be expected to be simple superposition of the patterns associated with the considered instabilities. To address this issue, we design two simple models composed by two asymmetrically coupled equations of non-conserved (Swift-Hohenberg equations) or conserved (Cahn-Hilliard equations) order parameters with different characteristic wave lengths. The patterns arising in these systems range from coexisting static patterns of different wavelengths to traveling waves. A linear stability analysis allows to derive a two parameter phase diagram for the studied models, in particular, revealing for the Swift-Hohenberg equations, a co-dimension two bifurcation point of Turing and wave instability and a region of coexistence of stationary and traveling patterns. The nonlinear dynamics of the coupled evolution equations is investigated by performing accurate numerical simulations. These reveal more complex patterns, ranging from traveling waves with embedded Turing patterns domains to spatio-temporal chaos, and a wide hysteretic region, where waves or Turing patterns coexist. For the coupled Cahn-Hilliard equations the presence of a weak coupling is sufficient to arrest the coarsening process and to lead to the emergence of purely periodic patterns. The final states are characterized by domains with a characteristic length, which diverges logarithmically with the coupling amplitude.
Optimal Variational Asymptotic Method for Nonlinear Fractional Partial Differential Equations
Baranwal, Vipul K.; Pandey, Ram K.
2014-01-01
We propose optimal variational asymptotic method to solve time fractional nonlinear partial differential equations. In the proposed method, an arbitrary number of auxiliary parameters γ0, γ1, γ2,… and auxiliary functions H0(x), H1(x), H2(x),… are introduced in the correction functional of the standard variational iteration method. The optimal values of these parameters are obtained by minimizing the square residual error. To test the method, we apply it to solve two important classes of nonlinear partial differential equations: (1) the fractional advection-diffusion equation with nonlinear source term and (2) the fractional Swift-Hohenberg equation. Only few iterations are required to achieve fairly accurate solutions of both the first and second problems.
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.
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
On the solutions of some linear complex quaternionic equations.
Bolat, Cennet; İpek, Ahmet
2014-01-01
Some complex quaternionic equations in the type AX - XB = C are investigated. For convenience, these equations were called generalized Sylvester-quaternion equations, which include the Sylvester equation as special cases. By the real matrix representations of complex quaternions, the necessary and sufficient conditions for the solvability and the general expressions of the solutions are obtained. PMID:25101318
Early-warning signs for pattern-formation in stochastic partial differential equations
NASA Astrophysics Data System (ADS)
Gowda, Karna; Kuehn, Christian
2015-05-01
There have been significant recent advances in our understanding of the potential use and limitations of early-warning signs for predicting drastic changes, so called critical transitions or tipping points, in dynamical systems. A focus of mathematical modeling and analysis has been on stochastic ordinary differential equations, where generic statistical early-warning signs can be identified near bifurcation-induced tipping points. In this paper, we outline some basic steps to extend this theory to stochastic partial differential equations with a focus on analytically characterizing basic scaling laws for linear SPDEs and comparing the results to numerical simulations of fully nonlinear problems. In particular, we study stochastic versions of the Swift-Hohenberg and Ginzburg-Landau equations. We derive a scaling law of the covariance operator in a regime where linearization is expected to be a good approximation for the local fluctuations around deterministic steady states. We compare these results to direct numerical simulation, and study the influence of noise level, noise color, distance to bifurcation and domain size on early-warning signs.
A complex Noether approach for variational partial differential equations
NASA Astrophysics Data System (ADS)
Naz, R.; Mahomed, F. M.
2015-10-01
Scalar complex partial differential equations which admit variational formulations are studied. Such a complex partial differential equation, via a complex dependent variable, splits into a system of two real partial differential equations. The decomposition of the Lagrangian of the complex partial differential equation in the real domain is shown to yield two real Lagrangians for the split system. The complex Maxwellian distribution, transonic gas flow, Maxwellian tails, dissipative wave and Klein-Gordon equations are considered. The Noether symmetries and gauge terms of the split system that correspond to both the Lagrangians are constructed by the Noether approach. In the case of coupled split systems, the same Noether symmetries are obtained. The Noether symmetries for the uncoupled split systems are different. The conserved vectors of the split system which correspond to both the Lagrangians are compared to the split conserved vectors of the complex partial differential equation for the examples. The split conserved vectors of the complex partial differential equation are the same as the conserved vectors of the split system of real partial differential equations in the case of coupled systems. Moreover a Noether-like theorem for the split system is proved which provides the Noether-like conserved quantities of the split system from knowledge of the Noether-like operators. An interesting result on the split characteristics and the conservation laws is shown as well. The Noether symmetries and gauge terms of the Lagrangian of the split system with the split Noether-like operators and gauge terms of the Lagrangian of the given complex partial differential equation are compared. Folklore suggests that the split Noether-like operators of a Lagrangian of a complex Euler-Lagrange partial differential equation are symmetries of the Lagrangian of the split system of real partial differential equations. This is not the case. They are proved to be the same if the
Graphical Solution of the Monic Quadratic Equation with Complex Coefficients
ERIC Educational Resources Information Center
Laine, A. D.
2015-01-01
There are many geometrical approaches to the solution of the quadratic equation with real coefficients. In this article it is shown that the monic quadratic equation with complex coefficients can also be solved graphically, by the intersection of two hyperbolas; one hyperbola being derived from the real part of the quadratic equation and one from…
Bifurcating vortex solutions of the complex Ginzburg-Landau equation.
Kaper, H. G.; Takac, P.; Mathematics and Computer Science
1999-10-01
It is shown that the complex Ginzburg-Landau (CGL) equation on the real line admits nontrivial 2{pi}-periodic vortex solutions that have 2n simple zeros ('vortices') per period. The vortex solutions bifurcate from the trivial solution and inherit their zeros from the solution of the linearized equation. This result rules out the possibility that the vortices are determining nodes for vortex solutions of the CGL equation.
Stochastic analysis of complex reaction networks using binomial moment equations.
Barzel, Baruch; Biham, Ofer
2012-09-01
The stochastic analysis of complex reaction networks is a difficult problem because the number of microscopic states in such systems increases exponentially with the number of reactive species. Direct integration of the master equation is thus infeasible and is most often replaced by Monte Carlo simulations. While Monte Carlo simulations are a highly effective tool, equation-based formulations are more amenable to analytical treatment and may provide deeper insight into the dynamics of the network. Here, we present a highly efficient equation-based method for the analysis of stochastic reaction networks. The method is based on the recently introduced binomial moment equations [Barzel and Biham, Phys. Rev. Lett. 106, 150602 (2011)]. The binomial moments are linear combinations of the ordinary moments of the probability distribution function of the population sizes of the interacting species. They capture the essential combinatorics of the reaction processes reflecting their stoichiometric structure. This leads to a simple and transparent form of the equations, and allows a highly efficient and surprisingly simple truncation scheme. Unlike ordinary moment equations, in which the inclusion of high order moments is prohibitively complicated, the binomial moment equations can be easily constructed up to any desired order. The result is a set of equations that enables the stochastic analysis of complex reaction networks under a broad range of conditions. The number of equations is dramatically reduced from the exponential proliferation of the master equation to a polynomial (and often quadratic) dependence on the number of reactive species in the binomial moment equations. The aim of this paper is twofold: to present a complete derivation of the binomial moment equations; to demonstrate the applicability of the moment equations for a representative set of example networks, in which stochastic effects play an important role. PMID:23030885
Statistical complexity, virial expansion, and van der Waals equation
NASA Astrophysics Data System (ADS)
Pennini, F.; Plastino, A.
2016-09-01
We investigate the notion of LMC statistical complexity with regards to a real gas and in terms of the second virial coefficient. The ensuing results are applied to the van der Waals equation. Interestingly enough, one finds a complexity-interpretation for the associated phase transition.
Out-of-Core Solutions of Complex Sparse Linear Equations
NASA Technical Reports Server (NTRS)
Yip, E. L.
1982-01-01
ETCLIB is library of subroutines for obtaining out-of-core solutions of complex sparse linear equations. Routines apply to dense and sparse matrices too large to be stored in core. Useful for solving any set of linear equations, but particularly useful in cases where coefficient matrix has no special properties that guarantee convergence with any of interative processes. The only assumption made is that coefficient matrix is not singular.
Visualising the Complex Roots of Quadratic Equations with Real Coefficients
ERIC Educational Resources Information Center
Bardell, Nicholas S.
2012-01-01
The roots of the general quadratic equation y = ax[superscript 2] + bx + c (real a, b, c) are known to occur in the following sets: (i) real and distinct; (ii) real and coincident; and (iii) a complex conjugate pair. Case (iii), which provides the focus for this investigation, can only occur when the values of the real coefficients a, b, and c are…
Stochastic Schroedinger equations with general complex Gaussian noises
Bassi, Angelo
2003-06-01
Within the framework of non-Markovian stochastic Schroedinger equations, we generalize the results of [W. T. Strunz, Phys. Lett. A 224, 25 (1996)] to the case of general complex Gaussian noises; we analyze the two important cases of purely real and purely imaginary stochastic processes.
Generalizing the Boltzmann equation in complex phase space.
Zadehgol, Abed
2016-08-01
In this work, a generalized form of the BGK-Boltzmann equation is proposed, where the velocity, position, and time can be represented by real or complex variables. The real representation leads to the conventional BGK-Boltzmann equation, which can recover the continuity and Navier-Stokes equations. We show that the complex representation yields a different set of equations, and it can also recover the conservation and Navier-Stokes equations, at low Mach numbers, provided that the imaginary component of the macroscopic mass can be neglected. We briefly review the Constant Speed Kinetic Model (CSKM), which was introduced in Zadehgol and Ashrafizaadeh [J. Comp. Phys. 274, 803 (2014)JCTPAH0021-999110.1016/j.jcp.2014.06.053] and Zadehgol [Phys. Rev. E 91, 063311 (2015)PLEEE81539-375510.1103/PhysRevE.91.063311]. The CSKM is then used as a basis to show that the complex-valued equilibrium distribution function of the present model can be identified with a simple singularity in the complex phase space. The virtual particles, in the present work, are concentrated on virtual "branes" which surround the computational nodes. Employing the Cauchy integral formula, it is shown that certain variations of the "branes," in the complex phase space, do not affect the local kinetic states. This property of the new model, which is referred to as the "apparent jumps" in the present work, is used to construct new models. The theoretical findings have been tested by simulating three benchmark flows. The results of the present simulations are in excellent agreement with the previous results reported by others. PMID:27627421
Complex oscillator and Painlevé IV equation
Fernández C, David J. González, J.C.
2015-08-15
Supersymmetric quantum mechanics is a powerful tool for generating exactly solvable potentials departing from a given initial one. In this article the first- and second-order supersymmetric transformations will be used to obtain new exactly solvable potentials departing from the complex oscillator. The corresponding Hamiltonians turn out to be ruled by polynomial Heisenberg algebras. By applying a mechanism to reduce to second the order of these algebras, the connection with the Painlevé IV equation is achieved, thus giving place to new solutions for the Painlevé IV equation.
The Complexity of Relating Quantum Channels to Master Equations
NASA Astrophysics Data System (ADS)
Cubitt, Toby S.; Eisert, Jens; Wolf, Michael M.
2012-03-01
Completely positive, trace preserving (CPT) maps and Lindblad master equations are both widely used to describe the dynamics of open quantum systems. The connection between these two descriptions is a classic topic in mathematical physics. One direction was solved by the now famous result due to Lindblad, Kossakowski, Gorini and Sudarshan, who gave a complete characterisation of the master equations that generate completely positive semi-groups. However, the other direction has remained open: given a CPT map, is there a Lindblad master equation that generates it (and if so, can we find its form)? This is sometimes known as the Markovianity problem. Physically, it is asking how one can deduce underlying physical processes from experimental observations. We give a complexity theoretic answer to this problem: it is NP-hard. We also give an explicit algorithm that reduces the problem to integer semi-definite programming, a well-known NP problem. Together, these results imply that resolving the question of which CPT maps can be generated by master equations is tantamount to solving P = NP: any efficiently computable criterion for Markovianity would imply P = NP; whereas a proof that P = NP would imply that our algorithm already gives an efficiently computable criterion. Thus, unless P does equal NP, there cannot exist any simple criterion for determining when a CPT map has a master equation description. However, we also show that if the system dimension is fixed (relevant for current quantum process tomography experiments), then our algorithm scales efficiently in the required precision, allowing an underlying Lindblad master equation to be determined efficiently from even a single snapshot in this case. Our work also leads to similar complexity-theoretic answers to a related long-standing open problem in probability theory.
Bounded Error Schemes for the Wave Equation on Complex Domains
NASA Technical Reports Server (NTRS)
Abarbanel, Saul; Ditkowski, Adi; Yefet, Amir
1998-01-01
This paper considers the application of the method of boundary penalty terms ("SAT") to the numerical solution of the wave equation on complex shapes with Dirichlet boundary conditions. A theory is developed, in a semi-discrete setting, that allows the use of a Cartesian grid on complex geometries, yet maintains the order of accuracy with only a linear temporal error-bound. A numerical example, involving the solution of Maxwell's equations inside a 2-D circular wave-guide demonstrates the efficacy of this method in comparison to others (e.g. the staggered Yee scheme) - we achieve a decrease of two orders of magnitude in the level of the L2-error.
Computational complexities and storage requirements of some Riccati equation solvers
NASA Technical Reports Server (NTRS)
Utku, Senol; Garba, John A.; Ramesh, A. V.
1989-01-01
The linear optimal control problem of an nth-order time-invariant dynamic system with a quadratic performance functional is usually solved by the Hamilton-Jacobi approach. This leads to the solution of the differential matrix Riccati equation with a terminal condition. The bulk of the computation for the optimal control problem is related to the solution of this equation. There are various algorithms in the literature for solving the matrix Riccati equation. However, computational complexities and storage requirements as a function of numbers of state variables, control variables, and sensors are not available for all these algorithms. In this work, the computational complexities and storage requirements for some of these algorithms are given. These expressions show the immensity of the computational requirements of the algorithms in solving the Riccati equation for large-order systems such as the control of highly flexible space structures. The expressions are also needed to compute the speedup and efficiency of any implementation of these algorithms on concurrent machines.
Stability of the complex generalized Hartree-Fock equations
Goings, Joshua J.; Ding, Feizhi; Li, Xiaosong; Frisch, Michael J.
2015-04-21
For molecules with complex and competing magnetic interactions, it is often the case that the lowest energy Hartree-Fock solution may only be obtained by removing the spin and time-reversal symmetry constraints of the exact non-relativistic Hamiltonian. To do so results in the complex generalized Hartree-Fock (GHF) method. However, with the loss of variational constraints comes the greater possibility of converging to higher energy minima. Here, we report the implementation of stability test of the complex GHF equations, along with an orbital update scheme should an instability be found. We apply the methodology to finding the local minima of several spin-frustrated hydrogen rings, as well as the non-collinear molecular magnet Cr{sub 3}, illustrating the utility of the broken symmetry GHF method and some of its lesser-known nuances.
Probing Resonances of the Dirac Equation with Complex Momentum Representation.
Li, Niu; Shi, Min; Guo, Jian-You; Niu, Zhong-Ming; Liang, Haozhao
2016-08-01
Resonance plays critical roles in the formation of many physical phenomena, and several methods have been developed for the exploration of resonance. In this work, we propose a new scheme for resonance by solving the Dirac equation in the complex momentum representation, in which the resonant states are exposed clearly in the complex momentum plane and the resonance parameters can be determined precisely without imposing unphysical parameters. Combined with the relativistic mean-field theory, this method is applied to probe the resonances in ^{120}Sn with the energies, widths, and wave functions being obtained. Compared to other methods, this method is not only very effective for narrow resonances, but also can be reliably applied to broad resonances. PMID:27541464
Probing Resonances of the Dirac Equation with Complex Momentum Representation
NASA Astrophysics Data System (ADS)
Li, Niu; Shi, Min; Guo, Jian-You; Niu, Zhong-Ming; Liang, Haozhao
2016-08-01
Resonance plays critical roles in the formation of many physical phenomena, and several methods have been developed for the exploration of resonance. In this work, we propose a new scheme for resonance by solving the Dirac equation in the complex momentum representation, in which the resonant states are exposed clearly in the complex momentum plane and the resonance parameters can be determined precisely without imposing unphysical parameters. Combined with the relativistic mean-field theory, this method is applied to probe the resonances in 120120 with the energies, widths, and wave functions being obtained. Compared to other methods, this method is not only very effective for narrow resonances, but also can be reliably applied to broad resonances.
Oscillations and Chaos In The Periodically Forced, Complex Lorenz Equations
NASA Astrophysics Data System (ADS)
Eccles, F. J. R.; Read, P. L.; Moroz, I. M.; Haine, T. W. N.
A variety of numerical atmosphere-ocean models (both idealised and `realistic') have shown that cyclic forcing may have a strong influence on a number of oscillatory climatological phenomena over a range of timescales (e.g. ENSO and the annual cy- cle) with emerging features such as frequency entrainment, period doubling and phase locking. We study analogous phenomena on a laboratory scale by imposing cyclic forcing by varying the boundary conditions of a rotating differentially-heated annulus. Arguably the simplest possible representation of this system is a two layer model and Fowler et al. (1982) have shown that the quasi-geostrophic potential vorticity equa- tions governing this model reduce to the complex Lorenz equations in the weakly dis- persive, weakly dissipative case. As a complement to the laboratory work mentioned above, we will present a numerical analysis of the complex Lorenz equations. The work will include experiments both with and without the incorporation of a periodic forcing term on various timescales. Reference A.C. Fowler, J.D. Gibbon and M.J. McGuinness, Physica D, 7:139163, 1982
Modeling the respiratory chain complexes with biothermokinetic equations - the case of complex I.
Heiske, Margit; Nazaret, Christine; Mazat, Jean-Pierre
2014-10-01
The mitochondrial respiratory chain plays a crucial role in energy metabolism and its dysfunction is implicated in a wide range of human diseases. In order to understand the global expression of local mutations in the rate of oxygen consumption or in the production of adenosine triphosphate (ATP) it is useful to have a mathematical model in which the changes in a given respiratory complex are properly modeled. Our aim in this paper is to provide thermodynamics respecting and structurally simple equations to represent the kinetics of each isolated complexes which can, assembled in a dynamical system, also simulate the behavior of the respiratory chain, as a whole, under a large set of different physiological and pathological conditions. On the example of the reduced nicotinamide adenine dinucleotide (NADH)-ubiquinol-oxidoreductase (complex I) we analyze the suitability of different types of rate equations. Based on our kinetic experiments we show that very simple rate laws, as those often used in many respiratory chain models, fail to describe the kinetic behavior when applied to a wide concentration range. This led us to adapt rate equations containing the essential parameters of enzyme kinetic, maximal velocities and Henri-Michaelis-Menten like-constants (KM and KI) to satisfactorily simulate these data. PMID:25064016
Unpacking the Complexity of Linear Equations from a Cognitive Load Theory Perspective
ERIC Educational Resources Information Center
Ngu, Bing Hiong; Phan, Huy P.
2016-01-01
The degree of element interactivity determines the complexity and therefore the intrinsic cognitive load of linear equations. The unpacking of linear equations at the level of operational and relational lines allows the classification of linear equations in a hierarchical level of complexity. Mapping similar operational and relational lines across…
Finite dimensionality in the complex Ginzburg-Landau equation
Doering, C.R.; Gibbon, J.D.; Holm, D.D.; Nicolaenko, B.
1987-01-01
Finite dimensionality is shown to exist in the complex Ginzburg-Landau equation periodic on the interval (0,1). A cone condition is derived and explained which gives upper bounds on the number of Fourier modes required to span the universal attractor and hence upper bounds on the attractor dimension itself. In terms of the parameter R these bounds are not large. For instance, when vertical bar ..mu.. vertical bar less than or equal to ..sqrt..3, the Fourier spanning dimension is 0(R/sup 3/2/). Lower bounds are estimated from the number of unstable side-bands using ideas from work on the Eckhaus instability. Upper bounds on the dimension of the attractor itself are obtained by bounding (or, for vertical bar ..mu.. vertical bar less than or equal to ..sqrt..3, computing exactly) the Lyapunov dimension and invoking a recent theorem of Constantin and Foias, which asserts that the Lyapunov dimension, defined by the Kaplan-Yorke formula, is an upper bound on the Hausdorff dimension. 39 refs., 7 figs.
A note on the Dirichlet problem for model complex partial differential equations
NASA Astrophysics Data System (ADS)
Ashyralyev, Allaberen; Karaca, Bahriye
2016-08-01
Complex model partial differential equations of arbitrary order are considered. The uniqueness of the Dirichlet problem is studied. It is proved that the Dirichlet problem for higher order of complex partial differential equations with one complex variable has infinitely many solutions.
Inferring the time-dependent complex Ginzburg-Landau equation from modulus data
Yu, Rotha P.; Paganin, David M.; Morgan, Michael J.
2005-11-01
We present a formalism for inferring the equation of evolution of a complex wave field that is known to obey an otherwise unspecified (2+1)-dimensional time-dependent complex Ginzburg-Landau equation, given field moduli over various closely spaced planes. The phase of the complex wave field is retrieved via a noninterferometric method, and all terms in the equation of evolution are determined using only the magnitude of the complex wave field. The formalism is tested using simulated data for a generalized nonlinear system with a single-component complex wave field. The method can be generalized to multicomponent complex fields.
NASA Astrophysics Data System (ADS)
Mohammed, K. Elboree
2015-10-01
In this paper, we investigate the traveling wave solutions for the nonlinear dispersive equation, Korteweg-de Vries Zakharov-Kuznetsov (KdV-ZK) equation and complex coupled KdV system by using extended simplest equation method, and then derive the hyperbolic function solutions include soliton solutions, trigonometric function solutions include periodic solutions with special values for double parameters and rational solutions. The properties of such solutions are shown by figures. The results show that this method is an effective and a powerful tool for handling the solutions of nonlinear partial differential equations (NLEEs) in mathematical physics.
Visualising the Roots of Quadratic Equations with Complex Coefficients
ERIC Educational Resources Information Center
Bardell, Nicholas S.
2014-01-01
This paper is a natural extension of the root visualisation techniques first presented by Bardell (2012) for quadratic equations with real coefficients. Consideration is now given to the familiar quadratic equation "y = ax[superscript 2] + bx + c" in which the coefficients "a," "b," "c" are generally…
Local algorithm for computing complex travel time based on the complex eikonal equation
NASA Astrophysics Data System (ADS)
Huang, Xingguo; Sun, Jianguo; Sun, Zhangqing
2016-04-01
The traditional algorithm for computing the complex travel time, e.g., dynamic ray tracing method, is based on the paraxial ray approximation, which exploits the second-order Taylor expansion. Consequently, the computed results are strongly dependent on the width of the ray tube and, in regions with dramatic velocity variations, it is difficult for the method to account for the velocity variations. When solving the complex eikonal equation, the paraxial ray approximation can be avoided and no second-order Taylor expansion is required. However, this process is time consuming. In this case, we may replace the global computation of the whole model with local computation by taking both sides of the ray as curved boundaries of the evanescent wave. For a given ray, the imaginary part of the complex travel time should be zero on the central ray. To satisfy this condition, the central ray should be taken as a curved boundary. We propose a nonuniform grid-based finite difference scheme to solve the curved boundary problem. In addition, we apply the limited-memory Broyden-Fletcher-Goldfarb-Shanno technology for obtaining the imaginary slowness used to compute the complex travel time. The numerical experiments show that the proposed method is accurate. We examine the effectiveness of the algorithm for the complex travel time by comparing the results with those from the dynamic ray tracing method and the Gauss-Newton Conjugate Gradient fast marching method.
Gao, Yingjie; Zhang, Jinhai; Yao, Zhenxing
2015-12-01
The complex frequency shifted perfectly matched layer (CFS-PML) can improve the absorbing performance of PML for nearly grazing incident waves. However, traditional PML and CFS-PML are based on first-order wave equations; thus, they are not suitable for second-order wave equation. In this paper, an implementation of CFS-PML for second-order wave equation is presented using auxiliary differential equations. This method is free of both convolution calculations and third-order temporal derivatives. As an unsplit CFS-PML, it can reduce the nearly grazing incidence. Numerical experiments show that it has better absorption than typical PML implementations based on second-order wave equation. PMID:26723366
Symmetries and soliton solutions of the Galilean complex Sine-Gordon equation
NASA Astrophysics Data System (ADS)
de Melo, G. R.; de Montigny, M.; Pinfold, J.; Tuszynski, J. A.
2016-03-01
We discuss a new equation, the Galilean version of the complex Sine-Gordon equation in 1 + 1 dimensions, Ψxx (1 -Ψ* Ψ) + 2 imΨt +Ψ* Ψx2- Ψ(1 -Ψ* Ψ) 2 = 0, derived from its relativistic counterpart via Galilean covariance. We determine its Lie point symmetries, discuss some group-invariant solutions, and examine some soliton solutions. The reduction under Galilean symmetry leads to an equation similar to the stationary Gross-Pitaevskii equation. This work is motivated in part by recent applications of the relativistic complex Sine-Gordon equation to the dynamics of Q-balls.
Chou, Chia-Chun
2014-03-14
The complex quantum Hamilton-Jacobi equation-Bohmian trajectories (CQHJE-BT) method is introduced as a synthetic trajectory method for integrating the complex quantum Hamilton-Jacobi equation for the complex action function by propagating an ensemble of real-valued correlated Bohmian trajectories. Substituting the wave function expressed in exponential form in terms of the complex action into the time-dependent Schrödinger equation yields the complex quantum Hamilton-Jacobi equation. We transform this equation into the arbitrary Lagrangian-Eulerian version with the grid velocity matching the flow velocity of the probability fluid. The resulting equation describing the rate of change in the complex action transported along Bohmian trajectories is simultaneously integrated with the guidance equation for Bohmian trajectories, and the time-dependent wave function is readily synthesized. The spatial derivatives of the complex action required for the integration scheme are obtained by solving one moving least squares matrix equation. In addition, the method is applied to the photodissociation of NOCl. The photodissociation dynamics of NOCl can be accurately described by propagating a small ensemble of trajectories. This study demonstrates that the CQHJE-BT method combines the considerable advantages of both the real and the complex quantum trajectory methods previously developed for wave packet dynamics.
Chou, Chia-Chun
2014-03-14
The complex quantum Hamilton-Jacobi equation-Bohmian trajectories (CQHJE-BT) method is introduced as a synthetic trajectory method for integrating the complex quantum Hamilton-Jacobi equation for the complex action function by propagating an ensemble of real-valued correlated Bohmian trajectories. Substituting the wave function expressed in exponential form in terms of the complex action into the time-dependent Schrödinger equation yields the complex quantum Hamilton-Jacobi equation. We transform this equation into the arbitrary Lagrangian-Eulerian version with the grid velocity matching the flow velocity of the probability fluid. The resulting equation describing the rate of change in the complex action transported along Bohmian trajectories is simultaneously integrated with the guidance equation for Bohmian trajectories, and the time-dependent wave function is readily synthesized. The spatial derivatives of the complex action required for the integration scheme are obtained by solving one moving least squares matrix equation. In addition, the method is applied to the photodissociation of NOCl. The photodissociation dynamics of NOCl can be accurately described by propagating a small ensemble of trajectories. This study demonstrates that the CQHJE-BT method combines the considerable advantages of both the real and the complex quantum trajectory methods previously developed for wave packet dynamics. PMID:24628169
Spatial complexity of solutions of higher order partial differential equations
NASA Astrophysics Data System (ADS)
Kukavica, Igor
2004-03-01
We address spatial oscillation properties of solutions of higher order parabolic partial differential equations. In the case of the Kuramoto-Sivashinsky equation ut + uxxxx + uxx + u ux = 0, we prove that for solutions u on the global attractor, the quantity card {x epsi [0, L]:u(x, t) = lgr}, where L > 0 is the spatial period, can be bounded by a polynomial function of L for all \\lambda\\in{\\Bbb R} . A similar property is proven for a general higher order partial differential equation u_t+(-1)^{s}\\partial_x^{2s}u+ \\sum_{k=0}^{2s-1}v_k(x,t)\\partial_x^k u =0 .
NASA Astrophysics Data System (ADS)
Polyanin, Andrei D.; Zhurov, Alexei I.
2014-03-01
We propose a new method for constructing exact solutions to nonlinear delay reaction-diffusion equations of the form ut=kuxx+F(u,w), where u=u(x,t),w=u(x,t-τ), and τ is the delay time. The method is based on searching for solutions in the form u=∑n=1Nξn(x)ηn(t), where the functions ξn(x) and ηn(t) are determined from additional functional constraints (which are difference or functional equations) and the original delay partial differential equation. All of the equations considered contain one or two arbitrary functions of a single argument. We describe a considerable number of new exact generalized separable solutions and a few more complex solutions representing a nonlinear superposition of generalized separable and traveling wave solutions. All solutions involve free parameters (in some cases, infinitely many parameters) and so can be suitable for solving certain problems and testing approximate analytical and numerical methods for nonlinear delay PDEs. The results are extended to a wide class of nonlinear partial differential-difference equations involving arbitrary linear differential operators of any order with respect to the independent variables x and t (in particular, this class includes the nonlinear delay Klein-Gordon equation) as well as to some partial functional differential equations with time-varying delay.
Nuttall's integral equation and Bernshtein's asymptotic formula for a complex weight
NASA Astrophysics Data System (ADS)
Ikonomov, N. R.; Kovacheva, R. K.; Suetin, S. P.
2015-12-01
We obtain Nuttall's integral equation provided that the corresponding complex-valued function σ(x) does not vanish and belongs to the Dini-Lipschitz class. Using this equation, we obtain a complex analogue of Bernshtein's classical asymptotic formulae for polynomials orthogonal on the closed unit interval Δ= \\lbrack -1,1 \\rbrack with respect to a complex-valued weight h(x)=σ(x)/\\sqrt{1-x^2}.
Stochastic Computational Approach for Complex Nonlinear Ordinary Differential Equations
NASA Astrophysics Data System (ADS)
Junaid, Ali Khan; Muhammad, Asif Zahoor Raja; Ijaz Mansoor, Qureshi
2011-02-01
We present an evolutionary computational approach for the solution of nonlinear ordinary differential equations (NLODEs). The mathematical modeling is performed by a feed-forward artificial neural network that defines an unsupervised error. The training of these networks is achieved by a hybrid intelligent algorithm, a combination of global search with genetic algorithm and local search by pattern search technique. The applicability of this approach ranges from single order NLODEs, to systems of coupled differential equations. We illustrate the method by solving a variety of model problems and present comparisons with solutions obtained by exact methods and classical numerical methods. The solution is provided on a continuous finite time interval unlike the other numerical techniques with comparable accuracy. With the advent of neuroprocessors and digital signal processors the method becomes particularly interesting due to the expected essential gains in the execution speed.
The coquaternion algebra and complex partial differential equations
NASA Astrophysics Data System (ADS)
Dimiev, Stancho; Konstantinov, Mihail; Todorov, Vladimir
2009-11-01
In this paper we consider the problem of differentiation of coquaternionic functions. Let us recall that coquaternions are elements of an associative non-commutative real algebra with zero divisor, introduced by James Cockle (1849) under the name of split-quaternions or coquaternions. Developing two type complex representations for Cockle algebra (complex and paracomplex ones) we present the problem in a non-commutative form of the δ¯-type holomorphy. We prove that corresponding differentiable coquaternionic functions, smooth and analytic, satisfy PDE of complex, and respectively of real variables. Applications for coquaternionic polynomials are sketched.
Solution of nonlinear flow equations for complex aerodynamic shapes
NASA Technical Reports Server (NTRS)
Djomehri, M. Jahed
1992-01-01
Solution-adaptive CFD codes based on unstructured methods for 3-D complex geometries in subsonic to supersonic regimes were investigated, and the computed solution data were analyzed in conjunction with experimental data obtained from wind tunnel measurements in order to assess and validate the predictability of the code. Specifically, the FELISA code was assessed and improved in cooperation with NASA Langley and Imperial College, Swansea, U.K.
Solution of coupled integral equations for quantum scattering in the presence of complex potentials
Franz, Jan
2015-01-15
In this paper, we present a method to compute solutions of coupled integral equations for quantum scattering problems in the presence of a complex potential. We show how the elastic and absorption cross sections can be obtained from the numerical solution of these equations in the asymptotic region at large radial distances.
Defocusing complex short-pulse equation and its multi-dark-soliton solution
NASA Astrophysics Data System (ADS)
Feng, Bao-Feng; Ling, Liming; Zhu, Zuonong
2016-05-01
In this paper, we propose a complex short-pulse equation of both focusing and defocusing types, which governs the propagation of ultrashort pulses in nonlinear optical fibers. It can be viewed as an analog of the nonlinear Schrödinger (NLS) equation in the ultrashort-pulse regime. Furthermore, we construct the multi-dark-soliton solution for the defocusing complex short-pulse equation through the Darboux transformation and reciprocal (hodograph) transformation. One- and two-dark-soliton solutions are given explicitly, whose properties and dynamics are analyzed and illustrated.
NASA Astrophysics Data System (ADS)
Koide, T.; Kodama, T.
2015-09-01
The stochastic variational method (SVM) is the generalization of the variational approach to systems described by stochastic variables. In this paper, we investigate the applicability of SVM as an alternative field-quantization scheme, by considering the complex Klein-Gordon equation. There, the Euler-Lagrangian equation for the stochastic field variables leads to the functional Schrödinger equation, which can be interpreted as the Euler (ideal fluid) equation in the functional space. The present formulation is a quantization scheme based on commutable variables, so that there appears no ambiguity associated with the ordering of operators, e.g., in the definition of Noether charges.
Complex and singular solutions of KdV and MKdV equations
NASA Technical Reports Server (NTRS)
Buti, B.; Rao, N. N.; Khadkikar, S. B.
1986-01-01
The Korteweg-de Vries (KdV) and the modified Korteweg-de Vries (MKdV) equations are shown to have, besides the regular real solutions, exact regular complex as well as singular solutions. The singular solution for the KdV is real but for the MKdV it is pure imaginary. Implications of the complex solutions are discussed.
Graphical Representation of Complex Solutions of the Quadratic Equation in the "xy" Plane
ERIC Educational Resources Information Center
McDonald, Todd
2006-01-01
This paper presents a visual representation of complex solutions of quadratic equations in the xy plane. Rather than moving to the complex plane, students are able to experience a geometric interpretation of the solutions in the xy plane. I am also working on these types of representations with higher order polynomials with some success.
Complex solitary waves and soliton trains in KdV and mKdV equations
NASA Astrophysics Data System (ADS)
Modak, Subhrajit; Singh, Akhil Pratap; Panigrahi, Prasanta Kumar
2016-06-01
We demonstrate the existence of complex solitary wave and periodic solutions of the Korteweg-de Vries (KdV) and modified Korteweg-de Vries (mKdV) equations. The solutions of the KdV (mKdV) equation appear in complex-conjugate pairs and are even (odd) under the simultaneous actions of parity (𝓟) and time-reversal (𝓣) operations. The corresponding localized solitons are hydrodynamic analogs of Bloch soliton in magnetic system, with asymptotically vanishing intensity. The 𝓟𝓣-odd complex soliton solution is shown to be iso-spectrally connected to the fundamental sech2 solution through supersymmetry. Physically, these complex solutions are analogous to the experimentally observed grey solitons of non-liner Schödinger equation, governing the dynamics of shallow water waves and hence may also find physical verification.
Quantifying Spatiotemporal Chaos Using Dimensions, Defects, and the Local Wavevector Field
NASA Astrophysics Data System (ADS)
Egolf, David A.; Melnikov, Ilarion V.; Bodenschatz, Eberhard
1997-11-01
We have studied the relationships among a number of dynamical and static properties of spatiotemporally chaotic systems. Using a new method to obtain local measurements of wavevectors, we have computed for Rayleigh-Bènard convection (experiment and simulation) a variety of order parameters (including various lengthscales, distributions, and defect densities) and studied dynamical processes such as the invasion of spatiotemporal chaos into regions of stationary behavior. Using finite-time Lyapunov dimensions, we have determined that for the 2D complex Ginzburg-Landau equation the dimension per defect D_defect = 2 over a wide range of parameters, while for a 2D reaction-diffusion model D_defect = 3. However, for a 2D generalized Swift-Hohenberg model the number of defects does not appear to be linearly related to the dimension. This work is supported by NSF-ASC-9503963, NSF-DMR-9320124, and the Cornell Theory Center.
Estimation of the Quality Factor (q) with Tomography Using the Complex Eikonal Equation.
NASA Astrophysics Data System (ADS)
Espinosa, T.; Piedrahita, C.; Cabrera, F.; Fernandez, J. P.
2015-12-01
The propagation of seismic wave through viscoelastic media is affected by the attenuation that is caused by the quality factor Q, resulting significant loss of signal strength and bandwidth. Gas trapped in sediment is a example of these media. Seismic images of geological structures underneath shallow gas often suffer from resolution degradation and effect of amplitude dimming, making their identification and interpretation difficult. This affects the ability to accurately predict reservoir properties. Thus, there is a need to compensate the attenuation due to Q, to be estimated using tomography seismic. This work takes place in a viscoelastic medium in the frequency domain, where is incorporated the attenuation to replace the elastic real parameters by parameters visco-elastic complex frequency dependent, consequently the equations must work complex and thus solution should be sought in the complex space. Complex eikonal equation is obtained from the equation of motion in a viscoelastic medium in the frequency domain. The objective is to apply a tomography method to estimate a model of complex velocity and Q model to achieve an improvement in seismic imaging in areas where there are strong attenuation factors or fractured media. To achieve calculating Q, first complex eikonal equation is solved in a medium viscoelastic using ray tracing. The resulting travel time is complex; its real part describes the wave propagation and its imaginary part describes the effects of attenuation. A process of tomography is then performed, the initial models of complex velocity and Q are determined; the models are smoothly inhomogeneous, with a constant gradient of the square of slowness. For such models, an exact solution of the complex eikonal equation can be found analytically by using complex ray tracing. given the initial complex velocity models and Q, calculate theoretical travel times and and finally making the inversion using the Gauss-Newton method, fit the initial velocity
Svinarenko, A. A.; Loboda, A. V.; Sukharev, D. E.; Dubrovskaya, Yu. V.; Mudraya, N. V.; Serga, I. N.; Glushkov, A. V.
2010-05-04
We report the further development of an effective approach to construction of the electron Green's function (GF) for the Dirac equation with a complex energy and non-singular central nuclear potential. The nuclear charge distribution and the corresponding nuclear potential are received within the relativistic mean field (RMF) model. The Green's function is usually represented as a combination of two fundamental solutions of the Dirac equation. In the numerical procedure we use the Ivanov-Ivanova effective numerical algorithm and reduce a definition of the Dirac equation fundamental solutions to solving the system of differential equations, which includes the differential equations for the RMF nuclear potential too. As an application, we estimate the self-energy shift correction to atomic levels energies within the Mohr covariant procedure and presented GF approach and calculate the transitions energies for some heavy Li-like multi-charged ions within the QED many-body perturbation theory formalism.
Higher dimensional systems of differential equations obtainable by iterative use of complex methods
NASA Astrophysics Data System (ADS)
Qadir, Asghar; Mahomed, Fazal M.
2015-04-01
A procedure had been developed to solve systems of two ordinary and partial differential equations (ODEs and PDEs) that could be obtained from scalar complex ODEs by splitting into their real and imaginary parts. The procedure was extended to four dimensional systems obtainable by splitting complex systems of two ODEs into their real and imaginary parts. As it stood, this procedure could be extended to any even dimension but not to odd dimensional systems. In this paper, the complex splitting is used iteratively to obtain three and four dimensional systems of ODEs and four dimensional systems of PDEs for four functions of two and four variables that correspond to a scalar base equation. We also provide characterization criteria for such systems to correspond to the base equation and a clear procedure to construct the base equation. The new systems of four ODEs are distinct from the class obtained by the single split of a two dimensional system. The previous complex methods split each infinitesimal symmetry generator into a pair of operators such that the entire set of operators do not form a Lie algebra. The iterative procedure sheds some light on the emergence of these "Lie-like" operators. In this procedure the higher dimensional system may not have any or the required symmetry for being directly solvable by symmetry and other methods although the base equation can have sufficient symmetry properties. Illustrative examples are provided.
Non-equilibrium dynamics of the complex Ginzburg-Landau equation
NASA Astrophysics Data System (ADS)
Liu, Weigang; Tauber, Uwe
The complex Ginzburg-Landau equation combines the quantum many-particle nonlinear Schrödinger equation with the time-dependent Ginzburg-Landau equation or model A relaxational dynamics. It arises in quite diverse contexts that include spontaneous pattern formation out of equilibrium, chemical oscillations, multi-mode lasers, thermal convection in binary fluids, cyclic population dynamics, and driven-dissipative Bose-Einstein condensates. Indeed, the complex Ginzburg-Landau equation exhibits a remarkably rich phase diagram with intriguing dynamics. We employ detailed numerical studies as well as analytical tools such as the perturbative renormalization group and the spherical model limit to study the non-equilibrium coarsening and critical aging scaling for the complex Ginzburg-Landau equation following quenches from an initial disordered configuration to either one of the ordered phases or the critical point. This research is supported by the U. S. Department of Energy, Office of Basic Energy Sciences, Division of Materials Science and Engineering under Award DE-FG02-09ER46613.
Time periodic spatial disorder in a complex Ginzburg{endash}Landau equation
Bazhenov, M.; Rabinovich, M.; Rubchinsky, L.
1996-06-01
The phenomenon of time-periodic evolution of spatial chaos (1) is investigated in the frame of a one and two-dimensional complex Ginzburg{endash}Landau equation. It is found that there exists a region of the parameters at which a disordered spatial distribution of the field behaves periodically in time; the boundaries of this region are determined. A system of ordinary differential equations describing spatial disorder is derived. The effect of the size of the system on the shape and period of oscillations is investigated. It is established that in a two-dimensional case the regime of time periodic spatial disorder arises only in the narrow band and the critical width of the band is estimated. The phenomenon investigated in this paper indicates that a family of limit cycles with finite basins may exist in the functional phase space of complex Ginzburg{endash}Landau equation in finite regions of the parameters. {copyright} {ital 1996 American Institute of Physics.}
A Local Discontinuous Galerkin Method for the Complex Modified KdV Equation
Li Wenting; Jiang Kun
2010-09-30
In this paper, we develop a local discontinuous Galerkin(LDG) method for solving complex modified KdV(CMKdV) equation. The LDG method has the flexibility for arbitrary h and p adaptivity. We prove the L{sup 2} stability for general solutions.
Complex structures and zero-curvature equations for σ-models
NASA Astrophysics Data System (ADS)
Bykov, Dmitri
2016-09-01
We construct zero-curvature representations for the equations of motion of a class of σ-models with complex homogeneous target spaces, not necessarily symmetric. We show that in the symmetric case the proposed flat connection is gauge-equivalent to the conventional one.
The dynamics of scroll wave filaments in the complex Ginzburg-Landau equation
NASA Astrophysics Data System (ADS)
Gabbay, Michael; Ott, Edward; Guzdar, Parvez N.
1998-07-01
An analytical treatment is presented for scroll waves in the complex Ginzburg-Landau equation in the limit of small filament curvature, torsion, and phase twist. Explicit expressions for the filament velocity and frequency shift are found. The theoretical results are verified numerically in the case of circular untwisted scroll rings and for straight and sinusoidal scroll filaments with phase twist.
Motion of Scroll Wave Filaments in the Complex Ginzburg-Landau Equation
NASA Astrophysics Data System (ADS)
Gabbay, Michael; Ott, Edward; Guzdar, Parvez N.
1997-03-01
Explicit asymptotic analytical results are derived for the motion of scroll wave filaments in the complex Ginzburg-Landau equation. Good agreement with numerical tests is obtained. The analysis highlights the necessity of allowing for previously ignored small wave-number shifts in the propagation of the waves away from the filament.
NASA Astrophysics Data System (ADS)
Sagis, Leonard M. C.; Öttinger, Hans Christian
2013-08-01
In this paper we present a general model for the dynamic behavior of multiphase systems in which the bulk phases and interfaces have a complex microstructure (for example, immiscible polymer blends with added compatibilizers, or polymer stabilized emulsions with thickening agents dispersed in the continuous phase). The model is developed in the context of the GENERIC framework (general equation for the nonequilibrium reversible irreversible coupling). We incorporate scalar and tensorial structural variables in the set of independent bulk and surface excess variables, and these structural variables allow us to link the highly nonlinear rheological response typically observed in complex multiphase systems, directly to the time evolution of the microstructure of the bulk phases and phase interfaces. We present a general form of the Poisson and dissipative brackets for the chosen set of bulk and surface excess variables, and show that to satisfy the entropy degeneracy property, we need to add several contributions to the moving interface normal transfer term, involving the tensorial bulk and interfacial structural variables. We present the full set of balance equations, constitutive equations, and boundary conditions for the calculation of the time evolution of the bulk and interfacial variables, and this general set of equations can be used to develop specific models for a wide range of complex multiphase systems.
The dynamics of vortex filaments in the complex Ginzburg- Landau equation
NASA Astrophysics Data System (ADS)
Gabbay, Michael
1997-09-01
An asymptotic theory is developed for scroll waves in the complex Ginzburg-Landau equation. The theory is valid in the limit of small vortex filament curvature, torsion and phase twist and in the absence of filament-filament interaction. Explicit expressions for the filament velocity and phase evolution are found. The theoretical results are verified numerically in the case of circular untwisted vortex rings and for straight and sinusoidal filaments with phase twist. Numerical evidence for the reconnection of vortex filaments in the complex Ginzburg- Landau equation is shown. An estimate is given for the maximum intervortex separation beyond which coplanar filaments of locally opposite charge will not reconnect. This is done by balancing the motion of the filaments toward each other that would result if they were straight (a two-dimensional effect) with the opposing motion due to the filament curvature. The estimated vortex separation is in good agreement with numerical experiment.
Adjoint equations and analysis of complex systems: Application to virus infection modelling
NASA Astrophysics Data System (ADS)
Marchuk, G. I.; Shutyaev, V.; Bocharov, G.
2005-12-01
Recent development of applied mathematics is characterized by ever increasing attempts to apply the modelling and computational approaches across various areas of the life sciences. The need for a rigorous analysis of the complex system dynamics in immunology has been recognized since more than three decades ago. The aim of the present paper is to draw attention to the method of adjoint equations. The methodology enables to obtain information about physical processes and examine the sensitivity of complex dynamical systems. This provides a basis for a better understanding of the causal relationships between the immune system's performance and its parameters and helps to improve the experimental design in the solution of applied problems. We show how the adjoint equations can be used to explain the changes in hepatitis B virus infection dynamics between individual patients.
Birth and death master equation for the evolution of complex networks
NASA Astrophysics Data System (ADS)
Alvarez-Martínez, R.; Cocho, G.; Rodríguez, R. F.; Martínez-Mekler, G.
2014-05-01
Master equations for the evolution of complex networks with positive (birth) and negative (death) transition probabilities per unit time are analyzed. Explicit equations for the time evolution of the total number of nodes and for the relative node frequencies are given. It is shown that, in the continuous limit, the master equation reduces to a Fokker-Planck equation (FPE). The basic dynamical function for its stationary solution is the ratio between its drift and diffusion coefficients. When this ratio is approximated by partial fractions (Padé's approximants), a hierarchy of stationary solutions of the FPE is obtained analytically, which are expressed as an exponential times the product of powers of monomials and binomials. It is also shown that if the difference between birth and death transition probabilities goes asymptotically to zero, the exponential factor in the solution is absent. Fits to real complex network probability distribution functions are shown. Comparison with rank-ordered data shows that, in general, the value of this exponential factor is close to unity, evidencing crossovers among power-law scale invariant regimes which might be associated to an underlying criticality and are related to a generalization of the beta distribution. The time dependent solution is also obtained analytically in terms of hyper-geometric functions. It is also shown that the FPE has similarity solutions. The limitations of the approach here presented are also discussed.
Exact Lyapunov dimension of the universal attractor for the complex Ginzburg-Landau equation
Doering, C.R.; Gibbon, J.D.; Holm, D.D.; Nicolaenko, B.
1987-12-28
We present an exact analytic computation of the Lyapunov dimension of the universal attractor of the complex Ginzburg-Landau partial differential equation for a finite range of its parameter values. We obtain upper bounds on the attractor's dimension when the parameters do not permit an exact evaluation by our methods. The exact Lyapunov dimension agrees with an estimate of the number of degrees of freedom based on a simple linear stability analysis and mode-counting argument.
NASA Technical Reports Server (NTRS)
Desmarais, R. N.; Rowe, W. S.
1984-01-01
For the design of active controls to stabilize flight vehicles, which requires the use of unsteady aerodynamics that are valid for arbitrary complex frequencies, algorithms are derived for evaluating the nonelementary part of the kernel of the integral equation that relates unsteady pressure to downwash. This part of the kernel is separated into an infinite limit integral that is evaluated using Bessel and Struve functions and into a finite limit integral that is expanded in series and integrated termwise in closed form. The developed series expansions gave reliable answers for all complex reduced frequencies and executed faster than exponential approximations for many pressure stations.
Superlattice Patterns in the Complex Ginzburg--Landau Equation with Multiresonant Forcing
NASA Astrophysics Data System (ADS)
Conway, Jessica M.; Riecke, Hermann
2009-01-01
Motivated by the rich variety of complex patterns observed on the surface of fluid layers that are vibrated at multiple frequencies, we investigate the effect of such resonant forcing on systems undergoing a Hopf bifurcation to spatially homogeneous oscillations. We use an extension of the complex Ginzburg-Landau equation that systematically captures weak forcing functions with a spectrum consisting of frequencies close to the 1:1-, the 1:2-, and the 1:3-resonance. By slowly modulating the amplitude of the 1:2-forcing component, we render the bifurcation to subharmonic patterns supercritical despite the quadratic interaction introduced by the 1:3-forcing. Our weakly nonlinear analysis shows that quite generally the forcing function can be tuned such that resonant triad interactions with weakly damped modes stabilize subharmonic superlattice patterns comprising four or five Fourier modes. Using direct simulations of the extended complex Ginzburg-Landau equation, we confirm our weakly nonlinear analysis. In sufficiently large systems domains of different complex patterns compete with each other on a slow time scale. As expected from leading-order energy arguments, with increasing strength of the triad interaction the more complex patterns eventually win out against the simpler patterns. We characterize these ordering dynamics using the spectral entropy of the patterns. For system parameters reported for experiments on the oscillatory Belousov-Zhabotinsky reaction we explicitly show that the forcing parameters can be tuned such that 4-mode patterns are the preferred patterns.
Clemens, M.; Weiland, T.
1996-12-31
In the field of computational electrodynamics the discretization of Maxwell`s equations using the Finite Integration Theory (FIT) yields very large, sparse, complex symmetric linear systems of equations. For this class of complex non-Hermitian systems a number of conjugate gradient-type algorithms is considered. The complex version of the biconjugate gradient (BiCG) method by Jacobs can be extended to a whole class of methods for complex-symmetric algorithms SCBiCG(T, n), which only require one matrix vector multiplication per iteration step. In this class the well-known conjugate orthogonal conjugate gradient (COCG) method for complex-symmetric systems corresponds to the case n = 0. The case n = 1 yields the BiCGCR method which corresponds to the conjugate residual algorithm for the real-valued case. These methods in combination with a minimal residual smoothing process are applied separately to practical 3D electro-quasistatical and eddy-current problems in electrodynamics. The practical performance of the SCBiCG methods is compared with other methods such as QMR and TFQMR.
Soliton interactions and complexes for coupled nonlinear Schrödinger equations.
Jiang, Yan; Tian, Bo; Liu, Wen-Jun; Sun, Kun; Li, Min; Wang, Pan
2012-03-01
Under investigation in this paper are the coupled nonlinear Schrödinger (CNLS) equations, which can be used to govern the optical-soliton propagation and interaction in such optical media as the multimode fibers, fiber arrays, and birefringent fibers. By taking the 3-CNLS equations as an example for the N-CNLS ones (N≥3), we derive the analytic mixed-type two- and three-soliton solutions in more general forms than those obtained in the previous studies with the Hirota method and symbolic computation. With the choice of parameters for those soliton solutions, soliton interactions and complexes are investigated through the asymptotic and graphic analysis. Soliton interactions and complexes with the bound dark solitons in a mode or two modes are observed, including that (i) the two bright solitons display the breatherlike structures while the two dark ones stay parallel, (ii) the two bright and dark solitons all stay parallel, and (iii) the states of the bound solitons change from the breatherlike structures to the parallel one even with the distance between those solitons smaller than that before the interaction with the regular one soliton. Asymptotic analysis is also used to investigate the elastic and inelastic interactions between the bound solitons and the regular one soliton. Furthermore, some discussions are extended to the N-CNLS equations (N>3). Our results might be helpful in such applications as the soliton switch, optical computing, and soliton amplification in the nonlinear optics. PMID:22587200
Wound-up phase turbulence in the complex Ginzburg-Landau equation
NASA Astrophysics Data System (ADS)
Montagne, R.; Hernández-García, E.; Amengual, A.; San Miguel, M.
1997-07-01
We consider phase turbulent regimes with nonzero winding number in the one-dimensional complex Ginzburg-Landau equation. We find that phase turbulent states with winding number larger than a critical one are only transients and decay to states within a range of allowed winding numbers. The analogy with the Eckhaus instability for nonturbulent waves is stressed. The transition from phase to defect turbulence is interpreted as an ergodicity breaking transition that occurs when the range of allowed winding numbers vanishes. We explain the states reached at long times in terms of three basic states, namely, quasiperiodic states, frozen turbulence states, and riding turbulence states. Justification and some insight into them are obtained from an analysis of a phase equation for nonzero winding number: Rigidly moving solutions of this equation, which correspond to quasiperiodic and frozen turbulence states, are understood in terms of periodic and chaotic solutions of an associated system of ordinary differential equations. A short report of some of our results has already been published [R. Montagne et al., Phys. Rev. Lett. 77, 267 (1996)].
Latchio Tiofack, Camus G; Mohamadou, Alidou; Kofané, Timoléon C; Moubissi, Alain B
2009-12-01
We consider a higher-order complex Ginzburg-Landau equation, with the fourth-order dispersion and cubic-quintic nonlinear terms, which can describe the propagation of an ultrashort subpicosecond or femtosecond optical pulse in an optical fiber system. We investigate the modulational instability (MI) of continuous wave solution of this equation. Several types of modulational instability gains are shown to exist in both the anomalous and normal dispersion regimes. We find that depending on the sign of the fourth-order dispersion coefficient, the MI appears for normal and anomalous dispersion regime. Simulations of the full system demonstrate that the development of the MI leads to establishment of a regular or chaotic array of pulses, a chain of well-separated peaks with continuously growing or decaying amplitudes depending on the sign of the loss/gain coefficient and higher-order dispersions terms. Comparison of the calculations with reported numerical results shows a satisfactory agreement. PMID:20365291
Longwave oscillatory patterns in liquids: outside the world of the complex Ginzburg-Landau equation
NASA Astrophysics Data System (ADS)
Nepomnyashchy, Alexander; Shklyaev, Sergey
2016-02-01
The main subject of the present review is longwave oscillatory patterns in systems with conservation laws, that cannot be described by the complex Ginzburg-Landau equation. As basic examples, we consider nonlinear patterns created by Marangoni and buoyancy instabilities in pure and binary liquids, where the longwave nature of instabilities is related to conservation of the liquid volume, conservation of mass or approximate conservation of the mean temperature. Also, we discuss the excitation of longwave instabilities by a time-periodic parameter modulation.
Reconnection of vortex filaments in the complex Ginzburg-Landau equation
NASA Astrophysics Data System (ADS)
Gabbay, Michael; Ott, Edward; Guzdar, Parvez N.
1998-08-01
A criterion for the reconnection of vortex filaments in the complex Ginzburg-Landau equation is presented. In particular, we give an estimate of the maximum intervortex separation beyond which coplanar filaments of locally opposite charge will not reconnect. This is done by balancing the motion of the filaments toward each other that would result if they were straight (a two-dimensional effect) with the opposing motion due to the filament curvature. Numerical experiments are in good agreement with the estimated vortex separation.
Lu, Dingjie; Xie, Yi Min; Huang, Xiaodong; Zhou, Shiwei; Li, Qing
2015-11-28
Analytical studies on the size effects of a simply-shaped beam fixed at both ends have successfully explained the sudden changes of effective Young's modulus as its diameter decreases below 100 nm. Yet they are invalid for complex nanostructures ubiquitously existing in nature. In accordance with a generalized Young-Laplace equation, one of the representative size effects is transferred to non-uniformly distributed pressure against an external surface due to the imbalance of inward and outward loads. Because the magnitude of pressure depends on the principal curvatures, iterative steps have to be adopted to gradually stabilize the structure in finite element analysis. Computational results are in good agreement with both experiment data and theoretical prediction. Furthermore, the investigation on strengthened and softened Young's modulus for two complex nanostructures demonstrates that the proposed computational method provides a general and effective approach to analyze the size effects for nanostructures in arbitrary shape.
Wong, Pring; Pang, Li-Hui; Huang, Long-Gang; Li, Yan-Qing; Lei, Ming; Liu, Wen-Jun
2015-09-15
The study of the complex Ginzburg–Landau equation, which can describe the fiber laser system, is of significance for ultra-fast laser. In this paper, dromion-like structures for the complex Ginzburg–Landau equation are considered due to their abundant nonlinear dynamics. Via the modified Hirota method and simplified assumption, the analytic dromion-like solution is obtained. The partial asymmetry of structure is particularly discussed, which arises from asymmetry of nonlinear and dispersion terms. Furthermore, the stability of dromion-like structures is analyzed. Oscillation structure emerges to exhibit strong interference when the dispersion loss is perturbed. Through the appropriate modulation of modified exponent parameter, the oscillation structure is transformed into two dromion-like structures. It indicates that the dromion-like structure is unstable, and the coherence intensity is affected by the modified exponent parameter. Results in this paper may be useful in accounting for some nonlinear phenomena in fiber laser systems, and understanding the essential role of modified Hirota method.
Non-Abelian Gauge Groups for Real and Complex Amended Maxwell's Equations
NASA Astrophysics Data System (ADS)
Rauscher, E. A.
2002-04-01
We have developed an eight dimensional complex Minkowski space M4, compiled of four real dimensions and four imaginary dimensions, which is constant with Lorentz invariance and analytic continuation in the complex plane(1). Complexification, of Maxwell's equations requires a non-Abelian gauge group, which amends the usual theory which utilizes the usual unimodular Weyl U1 group. We have examined the modification of gauge conditions using higher symmetry groups such as SU2, SUn and other groups such as the SL(2,c) double cover group of the rotational group SO(3,1). The mappability of the twistor algebra and the spinor calculus is analyzed in the context of the electromagnetic theory. Thus we are led to new and interesting physics involving extended metrical space constraints, the usual transverse and also longitudinal, non Hertzian electric and magnetic field solutions to Maxwell's equations, possibly leading to new communications systems and antennae theory, non-zero solutions to Ñ·B, and a possible finite but small rest mass of the photon. Comparison of our theoretical approach is made to the work of T.W. Barrett and H.F. Hermuth?s work on amended Maxwell's theories. (1) C. Ramon and E. A. Rauscher, Found. of Phys. 10, 661 (1980)
Spiral defect chaos in an advection-reaction-diffusion system
NASA Astrophysics Data System (ADS)
Affan, H.; Friedrich, R.
2014-06-01
This paper comprises numerical and theoretical studies of spatiotemporal patterns in advection-reaction-diffusion systems in which the chemical species interact with the hydrodynamic fluid. Due to the interplay between the two, we obtained the spiral defect chaos in the activator-inhibitor-type model. We formulated the generalized Swift-Hohenberg-type model for this system. Then the evolution of fractal boundaries due to the effect of the strong nonlinearity at the interface of the two chemical species is studied numerically. The purpose of the present paper is to point out that spiral defect chaos, observed in model equations of the extended Swift-Hohenberg equation for low Prandtl number convection, may actually be obtained also in certain advection-reaction-diffusion systems.
A multiscale asymptotic analysis of time evolution equations on the complex plane
NASA Astrophysics Data System (ADS)
Braga, Gastão A.; Conti, William R. P.
2016-07-01
Using an appropriate norm on the space of entire functions, we extend to the complex plane the renormalization group method as developed by Bricmont et al. The method is based upon a multiscale approach that allows for a detailed description of the long time asymptotics of solutions to initial value problems. The time evolution equation considered here arises in the study of iterations of the block spin renormalization group transformation for the hierarchical N-vector model. We show that, for initial conditions belonging to a certain Fréchet space of entire functions of exponential type, the asymptotics is universal in the sense that it is dictated by the fixed point of a certain operator acting on the space of initial conditions.
A low-complexity Reed-Solomon decoder using new key equation solver
NASA Astrophysics Data System (ADS)
Xie, Jun; Yuan, Songxin; Tu, Xiaodong; Zhang, Chongfu
2006-09-01
This paper presents a low-complexity parallel Reed-Solomon (RS) (255,239) decoder architecture using a novel pipelined variable stages recursive Modified Euclidean (ME) algorithm for optical communication. The pipelined four-parallel syndrome generator is proposed. The time multiplexing and resource sharing schemes are used in the novel recursive ME algorithm to reduce the logic gate count. The new key equation solver can be shared by two decoder macro. A new Chien search cell which doesn't need initialization is proposed in the paper. The proposed decoder can be used for 2.5Gb/s data rates device. The decoder is implemented in Altera' Stratixll device. The resource utilization is reduced about 40% comparing to the conventional method.
Spiral wave dynamics in the complex Ginzburg--Landau equation with broken chiral symmetry
NASA Astrophysics Data System (ADS)
Nam, Keeyeol; Ott, Edward; Gabbay, Michael; Guzdar, Parvez N.
1998-07-01
The effect of adding a chiral symmetry breaking term to the two-dimensional complex Ginzburg-Landau equation is investigated. We find that this term causes a shift in the frequency of the spiral wave solutions and that the sign of this shift depends on the topological charge (handedness) of the spiral. For parameters such that nearly stationary spiral domains form (called a “frozen” state), we find that, due to this charge-dependent frequency shift, the boundary between oppositely charged spiral domains moves, resulting in the domination of one domain of charge over the other. In addition, we introduce a quantity which measures the chirality of patterns and use it to characterize the transition between frozen and turbulent states. We also find that, depending on parameters, this transition occurs in two qualitatively distinct ways.
Tetrahedral finite-volume solutions to the Navier-Stokes equations on complex configurations
NASA Astrophysics Data System (ADS)
Frink, N. T.; Pirzadeh, S. Z.
1999-09-01
A review of the algorithmic features and capabilities of the unstructured-grid flow solver USM3Dns is presented. This code, along with the tetrahedral grid generator, VGRIDns, is being extensively used throughout the USA for solving the Euler and Navier-Stokes equations on complex aerodynamic problems. Spatial discretization is accomplished by a tetrahedral cell-centered finite-volume formulation using Roe's upwind flux difference splitting. The fluxes are limited by either a Superbee or MinMod limiter. Solution reconstruction within the tetrahedral cells is accomplished with a simple, but novel, multidimensional analytical formula. Time is advanced by an implicit backward-Euler time-stepping scheme. Flow turbulence effects are modeled by the Spalart-Allmaras one-equation model, which is coupled with a wall function to reduce the number of cells in the near-wall region of the boundary layer. The issues of accuracy and robustness of USM3Dns Navier-Stokes capabilities are addressed for a flat-plate boundary layer, and a full F-16 aircraft with external stores at transonic speed.
General features and master equations for structurization in complex dusty plasmas
Tsytovich, V. N.; Morfill, G. E.
2012-02-15
Dust structurization is considered to be typical for complex plasmas. Homogeneous dusty plasmas are shown to be universally unstable. The dusty plasma structurization instability is similar to the gravitational instability and can results in creation of different compact dust structures. A general approach for investigation of the nonlinear stage of structurization in dusty plasmas is proposed and master equations for the description of self-organized structures are formulated in the general form that can be used for any nonlinear model of dust screening. New effects due to the scattering of ions on the nonlinearly screened grains are calculated: nonlinear ion dust drag force and nonlinear ion diffusion. The physics of confinement of dust and plasma components in the equilibria of compact dust structures is presented and is supported by numerical calculations of master equations. The necessary conditions for the existence of equilibrium structures are found for an arbitrary nonlinearity in dust screening. Features of compact dust structures observed in recent experiments agree with the numerically calculated ones. Some proposals for future experiments in spherical chamber are given.
Tetrahedral Finite-Volume Solutions to the Navier-Stokes Equations on Complex Configurations
NASA Technical Reports Server (NTRS)
Frink, Neal T.; Pirzadeh, Shahyar Z.
1998-01-01
A review of the algorithmic features and capabilities of the unstructured-grid flow solver USM3Dns is presented. This code, along with the tetrahedral grid generator, VGRIDns, is being extensively used throughout the U.S. for solving the Euler and Navier-Stokes equations on complex aerodynamic problems. Spatial discretization is accomplished by a tetrahedral cell-centered finite-volume formulation using Roe's upwind flux difference splitting. The fluxes are limited by either a Superbee or MinMod limiter. Solution reconstruction within the tetrahedral cells is accomplished with a simple, but novel, multidimensional analytical formula. Time is advanced by an implicit backward-Euler time-stepping scheme. Flow turbulence effects are modeled by the Spalart-Allmaras one-equation model, which is coupled with a wall function to reduce the number of cells in the near-wall region of the boundary layer. The issues of accuracy and robustness of USM3Dns Navier-Stokes capabilities are addressed for a flat-plate boundary layer, and a full F-16 aircraft with external stores at transonic speed.
Dislocation dynamics in Rayleigh-Bénard convection.
Walter, Th; Pesch, W; Bodenschatz, E
2004-09-01
Theoretical results on the dynamics of dislocations in Rayleigh-Bénard convection are reported both for a Swift-Hohenberg model and the Oberbeck-Boussinesq equations. For intermediate Prandtl numbers the motion of dislocations is found to be driven by the superposition of two independent contributions: (i) the Peach-Koehler force and (ii) an advection force on the dislocation core by its self-generated mean flow. Their competition allows to explain the experimentally observed bound dislocation pairs. PMID:15447003
Linear complexity integral-equation based methods for large-scale electromagnetic analysis
NASA Astrophysics Data System (ADS)
Chai, Wenwen
In general, to solve problems with N parameters, the optimal computational complexity is linear complexity O( N). However, for most computational electromagnetic methods, the complexity is higher than O(N). In this work, we introduced and further developed the H - and H2 -matrix based mathematical framework to break the computational barrier of existing integral-equation (IE)-based methods for large-scale electromagnetic analysis. Our significant contributions include the first-time dense matrix inversion and LU factorization of O(N) complexity for large-scale 3-D circuit extraction and a fast direct integral equation solver that outperforms existing direct solvers for large-scale electrodynamic analysis having millions of unknowns and ˜100 wavelengths. The major contributions of this work are: (1) Direct Matrix Solution of Linear Complexity for 3-D Integrated Circuit (IC) and Package Extraction • O(N) complexity dense matrix inversion and LU factorization algorithms and their applications to capacitance extraction and impedance extraction of large-scale 3-D circuits • O(N) direct matrix solution of highly irregular matrices consisting of both dense and sparse matrix blocks arising from full-wave analysis of general 3-D circuits with lossy conductors in multiple dielectrics. (2) Fast H - and H2 -Based IE Solvers for Large-Scale Electrodynamic Analysis • theoretical proof on the error bounded low-rank representation of electrodynamic integral operators • fast H2 -based iterative solver with O(N) computational cost and controlled accuracy from small to tens of wavelengths • fast H -based direct solver with computational cost minimized based on accuracy • Findings on how to reduce the complexity of H - and H2 -based methods for electrodynamic analysis, which are also applicable to many other fast IE solvers. (3) Fast Algorithms for Accelerating H - and H2 -Based Iterative and Direct Solvers • Optimal H -based representation and its applications from
Plastic Instability in Complex Strain Paths Predicted by Advanced Constitutive Equations
NASA Astrophysics Data System (ADS)
Butuc, Marilena C.; Barlat, Frédéric; Gracio, José J.; Vincze, Gabriela
2011-08-01
The present paper aims at predicting plastic instabilities under complex loading histories using an advanced sheet metal forming limit model. The onset of localized necking is computed using the Marciniak-Kuczinsky (MK) analysis [1] with a physically-based hardening model and the phenomenological anisotropic yield criterion Yld2000-2d [2]. The hardening model accounts for anisotropic work-hardening induced by the microstructural evolution at large strains, which was proposed by Teodosiu and Hu [3]. Simulations are carried out for linear and complex strain paths. Experimentally, two deep-drawing quality sheet metals are selected: a bake-hardening steel (BH) and a DC06 steel sheet. The validity of the model is assessed by comparing the predicted and experimental forming limits. The remarkable accuracy of the developed software to predict the forming limits under linear and non-linear strain path is obviously due to the performance of the advanced constitutive equations to describe with great detail the material behavior. The effect of strain-induced anisotropy on formability evolution under strain path changes, as predicted by the microstructural hardening model, is particularly well captured by the model.
Mrugalla, Florian; Kast, Stefan M
2016-09-01
Complex formation between molecules in solution is the key process by which molecular interactions are translated into functional systems. These processes are governed by the binding or free energy of association which depends on both direct molecular interactions and the solvation contribution. A design goal frequently addressed in pharmaceutical sciences is the optimization of chemical properties of the complex partners in the sense of minimizing their binding free energy with respect to a change in chemical structure. Here, we demonstrate that liquid-state theory in the form of the solute-solute equation of the reference interaction site model provides all necessary information for such a task with high efficiency. In particular, computing derivatives of the potential of mean force (PMF), which defines the free-energy surface of complex formation, with respect to potential parameters can be viewed as a means to define a direction in chemical space toward better binders. We illustrate the methodology in the benchmark case of alkali ion binding to the crown ether 18-crown-6 in aqueous solution. In order to examine the validity of the underlying solute-solute theory, we first compare PMFs computed by different approaches, including explicit free-energy molecular dynamics simulations as a reference. Predictions of an optimally binding ion radius based on free-energy derivatives are then shown to yield consistent results for different ion parameter sets and to compare well with earlier, orders-of-magnitude more costly explicit simulation results. This proof-of-principle study, therefore, demonstrates the potential of liquid-state theory for molecular design problems. PMID:27366935
NASA Astrophysics Data System (ADS)
Mrugalla, Florian; Kast, Stefan M.
2016-09-01
Complex formation between molecules in solution is the key process by which molecular interactions are translated into functional systems. These processes are governed by the binding or free energy of association which depends on both direct molecular interactions and the solvation contribution. A design goal frequently addressed in pharmaceutical sciences is the optimization of chemical properties of the complex partners in the sense of minimizing their binding free energy with respect to a change in chemical structure. Here, we demonstrate that liquid-state theory in the form of the solute–solute equation of the reference interaction site model provides all necessary information for such a task with high efficiency. In particular, computing derivatives of the potential of mean force (PMF), which defines the free-energy surface of complex formation, with respect to potential parameters can be viewed as a means to define a direction in chemical space toward better binders. We illustrate the methodology in the benchmark case of alkali ion binding to the crown ether 18-crown-6 in aqueous solution. In order to examine the validity of the underlying solute–solute theory, we first compare PMFs computed by different approaches, including explicit free-energy molecular dynamics simulations as a reference. Predictions of an optimally binding ion radius based on free-energy derivatives are then shown to yield consistent results for different ion parameter sets and to compare well with earlier, orders-of-magnitude more costly explicit simulation results. This proof-of-principle study, therefore, demonstrates the potential of liquid-state theory for molecular design problems.
NASA Astrophysics Data System (ADS)
Cheng, Hongyu; Si, Jianguo
2013-08-01
In this paper, we discuss the existence of time quasi-periodic solutions for quasi-periodically forced cubic complex Ginzburg-Landau equation of higher spatial dimension with basic frequency vector ω = (ω1, ω2, …, ωm). By constructing a KAM (Kolmogorov-Arnold-Moser) theorem for a dissipative system which depends on time in a quasi-periodic way, we obtain a Cantorian branch of m + 2-dimensional invariant tori for the equation.
NASA Astrophysics Data System (ADS)
Schuch, Dieter
2012-08-01
Quantum mechanics is essentially described in terms of complex quantities like wave functions. The interesting point is that phase and amplitude of the complex wave function are not independent of each other, but coupled by some kind of conservation law. This coupling exists in time-independent quantum mechanics and has a counterpart in its time-dependent form. It can be traced back to a reformulation of quantum mechanics in terms of nonlinear real Ermakov equations or equivalent complex nonlinear Riccati equations, where the quadratic term in the latter equation explains the origin of the phase-amplitude coupling. Since realistic physical systems are always in contact with some kind of environment this aspect is also taken into account. In this context, different approaches for describing open quantum systems, particularly effective ones, are discussed and compared. Certain kinds of nonlinear modifications of the Schrödinger equation are discussed as well as their interrelations and their relations to linear approaches via non-unitary transformations. The modifications of the aforementioned Ermakov and Riccati equations when environmental effects are included can be determined in the time-dependent case. From formal similarities conclusions can be drawn how the equations of time-independent quantum mechanics can be modified to also incluce the enviromental aspects.
NASA Astrophysics Data System (ADS)
Zhang, Jianying; Yan, Guangwu
2016-04-01
A lattice Boltzmann model for solving the (2+1) dimensional cubic-quintic complex Ginzburg-Landau equation (CQCGLE) is proposed. Different from the classic lattice Boltzmann models, this lattice Boltzmann model is based on uniformly distributed lattice points in a two-dimensional space, and the evolution of the model is about a spatial axis rather than time. The algorithm provides advantages similar to the lattice Boltzmann method in that it is easily adapted to complex Ginzburg-Landau equations. Numerical results reproduce the phenomena of the fusion of necklace-ring pattern and the effect of non-linearity on the soliton in the CQCGLE.
NASA Astrophysics Data System (ADS)
Zhang, Jianying; Yan, Guangwu
2015-12-01
A spatiotemporal lattice Boltzmann model for solving the three-dimensional cubic-quintic complex Ginzburg-Landau equation (CQCGLE) is proposed. Different from the classic lattice Boltzmann models, this lattice Boltzmann model is based on uniformly distributed lattice points in a three-dimensional spatiotemporal space, and the evolution of the model is about a spatial axis rather than time. The algorithm possesses advantages similar to the lattice Boltzmann method in that it is easily adapted to complex Ginzburg-Landau equations. Examples show that the model reproduces the phenomena in the CQCGLE accurately.
Dynamics on the attractor for the complex Ginzburg-Landau equation
Takac, P.
1994-08-01
We present a numerical study of the large-time asymptotic behavior of solutions to the one-dimensional complex Ginzburg-Landau equation with periodic boundary conditions. Our parameters belong to the Benjamin-Feir unstable region. Our solutions start near a pure-mode rotating wave that is stable under sideband perturbations for the Reynolds number R ranging over an interval (R{sub sub},R{sub sup}). We find sub- and super-critical bifurcations from this stable rotating wave to a stable 2-torus as the parameter R is decreased or increased past the critical value R{sub sub} or R{sub sup}. As R > R{sub sup} further increases, we observe a variety of dynamical phenomena, such as a local attractor consisting of three unstable manifolds of periodic orbits or 2-tori cyclically connected by manifolds of connection orbits. We compare our numerical simulations to both rigorous mathematical results and experimental observations for binary fluid mixtures.
Modeling of dielectric properties of complex fluids with an equation of state.
Maribo-Mogensen, Bjørn; Kontogeorgis, Georgios M; Thomsen, Kaj
2013-03-28
The static permittivity is a key property for describing solutions containing polar and hydrogen bonding compounds. However, the precise relationship between the molecular and dielectric properties is not well-established. Here we show that the relative permittivity at zero frequency (static permittivity) can be modeled simultaneously with thermodynamic properties. The static permittivity is calculated from an extension of the framework developed by Onsager, Kirkwood, and Fröhlich to associating mixtures. The thermodynamic properties are calculated from the cubic-plus-association (CPA) equation of state that includes the Wertheim association model as formulated in the statistical associating fluid theory (SAFT) to account for hydrogen bonding molecules. We show that, by using a simple description of the geometry of the association, we may calculate the Kirkwood g-factor as a function of the probability of hydrogen bond formation. The results show that it is possible to predict the static permittivity of complex mixtures over wide temperature and pressure ranges from simple extensions of well-established theories simultaneously with the calculation of thermodynamic properties. PMID:23458349
NASA Astrophysics Data System (ADS)
Baskonus, Haci Mehmet; Bulut, Hasan; Atangana, Abdon
2016-03-01
In this study, we improve a new analytical method called the ‘Modified exp(-{{Ω }}(ξ )) expansion function method’. This method is based on the exp(-{{Ω }}(ξ )) expansion function method. We obtain new analytical solutions expressed by hyperbolic, complex and complex hyperbolic function solutions to the nonlinear longitudinal wave equation in a magneto-electro-elastic circular rod. We plot two- and three-dimensional surfaces of analytical solutions by using Wolfram Mathematica 9.
ERIC Educational Resources Information Center
Bardell, Nicholas S.
2014-01-01
This paper describes how a simple application of de Moivre's theorem may be used to not only find the roots of a quadratic equation with real or generally complex coefficients but also to pinpoint their location in the Argand plane. This approach is much simpler than the comprehensive analysis presented by Bardell (2012, 2014), but it does not…
Yu, Rotha P.; Paganin, David M.; Morgan, Michael J.
2008-04-01
We develop a means to 'measure' the generalized 2+1-dimensional time-dependent complex Ginzburg-Landau equation, given both the wave-function modulus and gauge-field information over a series of five planes that are closely spaced in time. The methodology is tested using simulated data for a thin-film high-temperature superconductor in the Meissner state.
Cruz, Hans; Schuch, Dieter; Castaños, Octavio; Rosas-Ortiz, Oscar
2015-09-15
The sensitivity of the evolution of quantum uncertainties to the choice of the initial conditions is shown via a complex nonlinear Riccati equation leading to a reformulation of quantum dynamics. This sensitivity is demonstrated for systems with exact analytic solutions with the form of Gaussian wave packets. In particular, one-dimensional conservative systems with at most quadratic Hamiltonians are studied.
NASA Astrophysics Data System (ADS)
Manafian, Jalil
2015-12-01
We apply the Exp-function method (EFM) to the Biswas-Milovic equation and derive the exact solutions. This paper studies the Biswas-Milovic equation with power law, parabolic law and dual parabolic law nonlinearities by the aid of the Exp-function method. The obtained solutions not only constitute a novel analytical viewpoint in nonlinear complex phenomena, but they also form a new stand alone basis from which physical applications in this arena can be comprehended further, and, moreover, investigated. Furthermore, to concretely enrich this research production, we explain all cases, namely m=1 and m≥ 2. This method is developed for searching exact travelling-wave solutions of nonlinear partial differential equations. It is shown that this methods, with the help of symbolic computation, provides a straightforward and powerful mathematical tool for solving nonlinear partial differential equations in mathematical physics.
NASA Astrophysics Data System (ADS)
Ge, Liang; Sotiropoulos, Fotis
2007-08-01
A novel numerical method is developed that integrates boundary-conforming grids with a sharp interface, immersed boundary methodology. The method is intended for simulating internal flows containing complex, moving immersed boundaries such as those encountered in several cardiovascular applications. The background domain (e.g. the empty aorta) is discretized efficiently with a curvilinear boundary-fitted mesh while the complex moving immersed boundary (say a prosthetic heart valve) is treated with the sharp-interface, hybrid Cartesian/immersed-boundary approach of Gilmanov and Sotiropoulos [A. Gilmanov, F. Sotiropoulos, A hybrid cartesian/immersed boundary method for simulating flows with 3d, geometrically complex, moving bodies, Journal of Computational Physics 207 (2005) 457-492.]. To facilitate the implementation of this novel modeling paradigm in complex flow simulations, an accurate and efficient numerical method is developed for solving the unsteady, incompressible Navier-Stokes equations in generalized curvilinear coordinates. The method employs a novel, fully-curvilinear staggered grid discretization approach, which does not require either the explicit evaluation of the Christoffel symbols or the discretization of all three momentum equations at cell interfaces as done in previous formulations. The equations are integrated in time using an efficient, second-order accurate fractional step methodology coupled with a Jacobian-free, Newton-Krylov solver for the momentum equations and a GMRES solver enhanced with multigrid as preconditioner for the Poisson equation. Several numerical experiments are carried out on fine computational meshes to demonstrate the accuracy and efficiency of the proposed method for standard benchmark problems as well as for unsteady, pulsatile flow through a curved, pipe bend. To demonstrate the ability of the method to simulate flows with complex, moving immersed boundaries we apply it to calculate pulsatile, physiological flow
Swope, Sarah M; Parker, Ingrid M
2012-12-01
Herbivores, seed predators, and pollinators can exert strong impacts on their host plants. They can also affect the strength of each other's impact by modifying traits in their shared host, producing super- or sub-additive outcomes. This phenomenon is especially relevant to biological control of invasive plants because most invaders are attacked by multiple agents. Unfortunately, complex interactions among agents are rarely studied. We used structural equation modeling (SEM) to quantify the effect of two biocontrol agents and generalist pollinators on the invasive weed Centaurea solstitialis, and to identify and quantify the direct and indirect interaction pathways among them. The weevil Eustenopus villosus is both a bud herbivore and a predispersal seed predator; the fly Chaetorellia succinea is also a predispersal seed predator; Apis mellifera is the primary pollinator. We conducted this work at three sites spanning the longitudinal range of C. solstitialis in California (USA) from the coast to the Sierra Nevada Mountains. SEM revealed that bud herbivory had the largest total effect on the weed's fecundity. The direct effect of bud herbivory on final seed set was 2-4 times larger in magnitude than the direct effect of seed predation by both agents combined. SEM also revealed important indirect interactions; by reducing the number of inflorescences plants produced, bud herbivory indirectly reduced the plant's attractiveness to ovipositing seed predators. This indirect, positive pathway reduced bud herbivory's direct negative effect by 11-25%. In the same way, bud herbivory also reduced pollinator visitation, although the magnitude of this pathway was relatively small. E. villosus oviposition deterred C. succinea oviposition, which is unfortunate because C. succinea is the more voracious of the seed predators. Finally, C. succinea oviposition indirectly deterred pollinator visitation, thereby enhancing its net effect on the plant. This study demonstrates the
NASA Astrophysics Data System (ADS)
Ling, Liming; Feng, Bao-Feng; Zhu, Zuonong
2016-07-01
In the present paper, we are concerned with the general analytic solutions to the complex short pulse (CSP) equation including soliton, breather and rogue wave solutions. With the aid of a generalized Darboux transformation, we construct the N-bright soliton solution in a compact determinant form, the N-breather solution including the Akhmediev breather and a general higher order rogue wave solution. The first and second order rogue wave solutions are given explicitly and analyzed. The asymptotic analysis is performed rigorously for both the N-soliton and the N-breather solutions. All three forms of the analytical solutions admit either smoothed-, cusped- or looped-type ones for the CSP equation depending on the parameters. It is noted that, due to the reciprocal (hodograph) transformation, the rogue wave solution to the CSP equation can be a smoothed, cusponed or a looped one, which is different from the rogue wave solution found so far.
Richter, Marten; Renger, Thomas; Knorr, Andreas
2008-01-01
On the basis of the recent progress in the resolution of the structure of the antenna light harvesting complex II (LHC II) of the photosystem II, we propose a microscopically motivated theory to predict excitation intensity-dependent spectra. We show that optical Bloch equations provide the means to include all 2( N ) excited states of an oligomer complex of N coupled two-level systems and analyze the effects of Pauli Blocking and exciton-exciton annihilation on pump-probe spectra. We use LHC Bloch equations for 14 Coulomb coupled two-level systems, which describe the S (0) and S (1) level of every chlorophyll molecule. All parameter introduced into the Hamiltonian are based on microscopic structure and a quantum chemical model. The derived Bloch equations describe not only linear absorption but also the intensity dependence of optical spectra in a regime where the interplay of Pauli Blocking effects as well as exciton-exciton annihilation effects are important. As an example, pump-probe spectra are discussed. The observed saturation of the spectra for high intensities can be viewed as a relaxation channel blockade on short time scales due to Pauli blocking. The theoretical investigation is useful for the interpretation of the experimental data, if the experimental conditions exceed the low intensity pump limit and effects like strong Pauli Blocking and exciton-exciton annihilation need to be considered. These effects become important when multiple excitations are generated by the pump pulse in the complex. PMID:17924202
Cherne, Frank J; Jensen, Brian J; Elkin, Vyacheslav M
2009-01-01
The complexity of cerium combined with its interesting material properties makes it a desirable material to examine dynamically. Characteristics such as the softening of the material before the phase change, low pressure solid-solid phase change, predicted low pressure melt boundary, and the solid-solid critical point add complexity to the construction of its equation of state. Currently, we are incorporating a feedback loop between a theoretical understanding of the material and an experimental understanding. Using a model equation of state for cerium we compare calculated wave profiles with experimental wave profiles for a number of front surface impact (cerium impacting a plated window) experiments. Using the calculated release isentrope we predict the temperature of the observed rarefaction shock. These experiments showed that the release state occurs at different magnitudes, thus allowing us to infer where dynamic {gamma} - {alpha} phase boundary is.
NASA Astrophysics Data System (ADS)
Cisternas, Jaime; Descalzi, Orazio; Albers, Tony; Radons, Günter
2016-05-01
We demonstrate the occurrence of anomalous diffusion of dissipative solitons in a "simple" and deterministic prototype model: the cubic-quintic complex Ginzburg-Landau equation in two spatial dimensions. The main features of their dynamics, induced by symmetric-asymmetric explosions, can be modeled by a subdiffusive continuous-time random walk, while in the case dominated by only asymmetric explosions, it becomes characterized by normal diffusion.
Cisternas, Jaime; Descalzi, Orazio; Albers, Tony; Radons, Günter
2016-05-20
We demonstrate the occurrence of anomalous diffusion of dissipative solitons in a "simple" and deterministic prototype model: the cubic-quintic complex Ginzburg-Landau equation in two spatial dimensions. The main features of their dynamics, induced by symmetric-asymmetric explosions, can be modeled by a subdiffusive continuous-time random walk, while in the case dominated by only asymmetric explosions, it becomes characterized by normal diffusion. PMID:27258868
NASA Astrophysics Data System (ADS)
Dorkin, S. M.; Kaptari, L. P.; Hilger, T.; Kämpfer, B.
2014-03-01
In view of the mass spectrum of heavy mesons in vacuum, the analytical properties of the solutions of the truncated Dyson-Schwinger equation for the quark propagator within the rainbow approximation are analyzed in some detail. In Euclidean space, the quark propagator is not an analytical function possessing, in general, an infinite number of singularities (poles) which hamper solving the Bethe-Salpeter equation. However, for light mesons (with masses Mqq ¯≤1 GeV) all singularities are located outside the region within which the Bethe-Salpeter equation is defined. With an increase of the considered meson masses this region enlarges and already at masses ≥1 GeV, the poles of propagators of u, d, and s quarks fall within the integration domain of the Bethe-Salpeter equation. Nevertheless, it is established that for meson masses up to Mqq ¯≃3 GeV only the first, mutually complex conjugated poles contribute to the solution. We argue that, by knowing the position of the poles and their residues, a reliable parametrization of the quark propagators can be found and used in numerical procedures of solving the Bethe-Salpeter equation. Our analysis is directly related to the future physics program at FAIR with respect to open charm degrees of freedom.
Derivation of the equations of motion for complex structures by symbolic manipulation
NASA Technical Reports Server (NTRS)
Hale, A. L.; Meirovitch, L.
1978-01-01
This paper outlines a computer program especially tailored to the task of deriving explicit equations of motion for structures with point-connected substructures. The special purpose program is written in FORTRAN and is designed for performing the specific algebraic operations encountered in the derivation of explicit equations of motion. The derivation is by the Lagrangian approach. Using an orderly kinematical procedure and a discretization and/or truncation scheme, it is possible to write the kinetic and potential energy of each substructure in a compact vector-matrix form. Then, if each element of the matrices and vectors encountered in the kinetic and potential energy is a known algebraic expression, the computer program performs the necessary operations to evaluate the kinetic and potential energy of the system explicitly. Lagrange's equations for small motions about equilibrium can be deduced directly from the explicit form of the system kinetic and potential energy.
Solving Second-Order Ordinary Differential Equations without Using Complex Numbers
ERIC Educational Resources Information Center
Kougias, Ioannis E.
2009-01-01
Ordinary differential equations (ODEs) is a subject with a wide range of applications and the need of introducing it to students often arises in the last year of high school, as well as in the early stages of tertiary education. The usual methods of solving second-order ODEs with constant coefficients, among others, rely upon the use of complex…
Optical pulse propagation in fibers with random dispersion
NASA Astrophysics Data System (ADS)
Abdullaev, F. Kh.; Navotny, D. V.; Baizakov, B. B.
2004-05-01
The propagation of optical pulses in two types of fibers with randomly varying dispersion is investigated. The first type refers to a uniform fiber dispersion superimposed by random modulations with a zero mean. The second type is the dispersion-managed fiber line with fluctuating parameters of the dispersion map. Application of the mean field method leads to the nonlinear Schrödinger equation (NLSE) with a dissipation term, expressed by a fourth-order derivative of the wave envelope. The prediction of the mean field approach regarding the decay rate of a soliton is compared with that of the perturbation theory based on the inverse scattering transform (IST). A good agreement between these two approaches is found. Possible ways of compensation of the radiative decay of solitons using the linear and nonlinear amplification are explored. The corresponding mean field equation coincides with the complex Swift-Hohenberg equation. The condition for the autosolitonic regime in propagation of optical pulses along a fiber line with fluctuating dispersion is derived and the existence of autosoliton (dissipative soliton) is confirmed by direct numerical simulation of the stochastic NLSE. The dynamics of solitons in optical communication systems with random dispersion-management is further studied applying the variational principle to the mean field NLSE, which results in a system of ODEs for soliton parameters. Extensive numerical simulations of the stochastic NLSE, mean field equation and corresponding set of ODEs are performed to verify the predictions of the developed theory.
Peng, Sun; Jin, Guo; Tingfeng, Wang
2013-07-01
Based on the generalized Huygens-Fresnel diffraction integral (Collins' formula), the propagation equation of Hermite-Gauss beams through a complex optical system with a limiting aperture is derived. The elements of the optical system may be all those characterized by an ABCD ray-transfer matrix, as well as any kind of apertures represented by complex transmittance functions. To obtain the analytical expression, we expand the aperture transmittance function into a finite sum of complex Gaussian functions. Thus the limiting aperture is expressed as a superposition of a series of Gaussian-shaped limiting apertures. The advantage of this treatment is that we can treat almost all kinds of apertures in theory. As application, we define the width of the beam and the focal plane using an encircled-energy criterion and calculate the intensity distribution of Hermite-Gauss beams at the actual focus of an aperture lens. PMID:24323153
Li, Yunqi; Zhao, Qin; Huang, Qingrong
2014-01-30
A combination of turbidimetric titration, a sigmoidal Boltzmann equation approach and Monte Carlo simulation has been used to study the complex coacervation in serum albumin and pectin mixtures. The effects of the mass ratio of protein to polysaccharide on the critical pH values, the probability of complex coacervation and the electrostatic interaction from charge patches in serum albumin were investigated. Turbidimetric titration results showed an optimum pH for complex coacervation (pHm), which corresponded to the maximum turbidity in the protein/polysaccharide mixture. The pHm monotonically decreased as the ratio decreased, and could be fitted using the sigmoidal Boltzmann equation. It suggests that pHm could be a good ordering parameter to characterize the phase behavior associated with protein/polysaccharide complex coacervation. Qualitative understanding of pHm by taking into account the minimization of electrostatic interaction, as well as quantitative matching of pHm according to the concept of charge neutralization were both achieved. Our results suggest that the serum albumin/pectin complexes were ultimately neutralized by the partial charges originated from the titratable residues in protein and polysaccharide chains at pHm. The Monte Carlo simulation provided consistent phase boundaries for complex coacervation in the same system, and the intermolecular association strength was determined to be several kBT below the given ionic strength. The strongest binding site in the protein is convergent to the largest positive charge patch if pure electrostatic interaction was considered. Further inclusion of contribution from excluded volume resulted in the binding site distribution over five different positive charge patches at different protein/polysaccharide ratios and pH values. PMID:24299810
NASA Astrophysics Data System (ADS)
Shokri, Ali; Afshari, Fatemeh
2015-12-01
In this article, a high-order compact alternating direction implicit (HOC-ADI) finite difference scheme is applied to numerical solution of the complex Ginzburg-Landau (GL) equation in two spatial dimensions with periodical boundary conditions. The GL equation has been used as a mathematical model for various pattern formation systems in mechanics, physics, and chemistry. The proposed HOC-ADI method has fourth-order accuracy in space and second-order accuracy in time. To avoid solving the nonlinear system and to increase the accuracy and efficiency of the method, we proposed the predictor-corrector scheme. Validation of the present numerical solutions has been conducted by comparing with the exact and other methods results and evidenced a good agreement.
Validation of a zero-equation turbulence model for complex indoor airflow simulation
Srebric, J.; Chen, Q.; Glicksman, L.R.
1999-07-01
The design of an indoor environment requires a tool that can quickly predict the three-dimensional distributions of air velocity, temperature, and contaminant concentrations in the room on a desktop computer. This investigation has tested a zero-equation turbulence model for the prediction of the indoor environment in an office with displacement ventilation, with a heater and infiltration and with forced convection and a partition wall. The computed air velocity and temperature distributions agree well with the measured data. The computing time for each case is less than seven minutes on a PC Pentium II, 350 MHz.
NASA Astrophysics Data System (ADS)
Xu, Si-Liu; Belić, Milivoj R.
2014-11-01
We investigate the existence of spatiotemporal necklace vortex solitons or light bullets in the complex Ginzburg-Landau equation with the modulated Kummer-Gauss (KG) external lattice potential and the spiraling phase of vorticities S=0,1 , and 2. We find localized vortex necklaces in a three-dimensional nonlinear medium, trapped by the KG external potential with different orders of vorticity. Stable and quasi-stable solitons form from input pulses with embedded vorticity. The stability is established by calculating growth rates of the perturbed eigenmodes. We establish that spatiotemporal necklace solitons may coexist in a large domain of the parameter space.
NASA Astrophysics Data System (ADS)
Liang, Yingjie; Chen, Wen; Magin, Richard L.
2016-07-01
Analytical solutions to the fractional diffusion equation are often obtained by using Laplace and Fourier transforms, which conveniently encode the order of the time and the space derivatives (α and β) as non-integer powers of the conjugate transform variables (s, and k) for the spectral and the spatial frequencies, respectively. This study presents a new solution to the fractional diffusion equation obtained using the Laplace transform and expressed as a Fox's H-function. This result clearly illustrates the kinetics of the underlying stochastic process in terms of the Laplace spectral frequency and entropy. The spectral entropy is numerically calculated by using the direct integration method and the adaptive Gauss-Kronrod quadrature algorithm. Here, the properties of spectral entropy are investigated for the cases of sub-diffusion and super-diffusion. We find that the overall spectral entropy decreases with the increasing α and β, and that the normal or Gaussian case with α = 1 and β = 2, has the lowest spectral entropy (i.e., less information is needed to describe the state of a Gaussian process). In addition, as the neighborhood over which the entropy is calculated increases, the spectral entropy decreases, which implies a spatial averaging or coarse graining of the material properties. Consequently, the spectral entropy is shown to provide a new way to characterize the temporal correlation of anomalous diffusion. Future studies should be designed to examine changes of spectral entropy in physical, chemical and biological systems undergoing phase changes, chemical reactions and tissue regeneration.
ERIC Educational Resources Information Center
Stapleton, Laura M.
2008-01-01
This article discusses replication sampling variance estimation techniques that are often applied in analyses using data from complex sampling designs: jackknife repeated replication, balanced repeated replication, and bootstrapping. These techniques are used with traditional analyses such as regression, but are currently not used with structural…
A node-centered local refinement algorithm for poisson's equation in complex geometries
McCorquodale, Peter; Colella, Phillip; Grote, David P.; Vay, Jean-Luc
2004-05-04
This paper presents a method for solving Poisson's equation with Dirichlet boundary conditions on an irregular bounded three-dimensional region. The method uses a nodal-point discretization and adaptive mesh refinement (AMR) on Cartesian grids, and the AMR multigrid solver of Almgren. The discrete Laplacian operator at internal boundaries comes from either linear or quadratic (Shortley-Weller) extrapolation, and the two methods are compared. It is shown that either way, solution error is second order in the mesh spacing. Error in the gradient of the solution is first order with linear extrapolation, but second order with Shortley-Weller. Examples are given with comparison with the exact solution. The method is also applied to a heavy-ion fusion accelerator problem, showing the advantage of adaptivity.
Appelo, D; Petersson, N A
2007-12-17
The isotropic elastic wave equation governs the propagation of seismic waves caused by earthquakes and other seismic events. It also governs the propagation of waves in solid material structures and devices, such as gas pipes, wave guides, railroad rails and disc brakes. In the vast majority of wave propagation problems arising in seismology and solid mechanics there are free surfaces. These free surfaces have, in general, complicated shapes and are rarely flat. Another feature, characterizing problems arising in these areas, is the strong heterogeneity of the media, in which the problems are posed. For example, on the characteristic length scales of seismological problems, the geological structures of the earth can be considered piecewise constant, leading to models where the values of the elastic properties are also piecewise constant. Large spatial contrasts are also found in solid mechanics devices composed of different materials welded together. The presence of curved free surfaces, together with the typical strong material heterogeneity, makes the design of stable, efficient and accurate numerical methods for the elastic wave equation challenging. Today, many different classes of numerical methods are used for the simulation of elastic waves. Early on, most of the methods were based on finite difference approximations of space and time derivatives of the equations in second order differential form (displacement formulation), see for example [1, 2]. The main problem with these early discretizations were their inability to approximate free surface boundary conditions in a stable and fully explicit manner, see e.g. [10, 11, 18, 20]. The instabilities of these early methods were especially bad for problems with materials with high ratios between the P-wave (C{sub p}) and S-wave (C{sub s}) velocities. For rectangular domains, a stable and explicit discretization of the free surface boundary conditions is presented in the paper [17] by Nilsson et al. In summary
Thermal equation of state of polarized fermions in one dimension via complex chemical potentials
NASA Astrophysics Data System (ADS)
Loheac, Andrew C.; Braun, Jens; Drut, Joaquín E.; Roscher, Dietrich
2015-12-01
We present a nonperturbative computation of the equation of state of polarized, attractively interacting, nonrelativistic fermions in one spatial dimension at finite temperature. We show results for the density, spin magnetization, magnetic susceptibility, and Tan's contact. We compare with the second-order virial expansion, a next-to-leading-order lattice perturbation theory calculation, and interpret our results in terms of pairing correlations. Our lattice Monte Carlo calculations implement an imaginary chemical potential difference to avoid the sign problem. The thermodynamic results on the imaginary side are analytically continued to obtain results on the real axis. We focus on an intermediate- to strong-coupling regime, and cover a wide range of temperatures and spin imbalances.
NASA Astrophysics Data System (ADS)
Yabunaka, Shunsuke
2014-10-01
We study interface and vortex motion in the two-component dissipative Ginzburg-Landau equation in two-dimensional space. We consider cases where the whole system is divided into several domains, and we assume that these domains are separated by interfaces and each domain contains quantized vortices. The equations for interface and vortex motion will be derived by means of a variational approach by Kawasaki. These equations indicate that, along an interface, the phase gradient fields of the complex order parameters is parallel to the interface. They also indicate that the interface motion is driven by the curvature and the phase gradient fields along the interface, and vortex motion is driven by the phase gradient field around the vortex. With respect to the static interactions between defects, we find an analogy between quantized vortices in a domain and electric charges in a vacuum domain surrounded by a metallic object in electrostatic. This analogy indicates that there is an attractive interaction between an interface and a quantized vortex with any charge. We also discuss several examples of interface and vortex motion.
A Complex Approach to Estimate Shadow Economy: The Structural Equation Modelling
NASA Astrophysics Data System (ADS)
Dell'Anno, Roberto; Schneider, Friedrich
This article develops some ideas of the application of the "complexity" approach in economics. The complexity approach criticizes the scientific method by distrusting sample reductionism and proposes a multidisciplinary approach. Hence, it abolishes old paradigms by arguing to build up another one with the endowment of greater realism. We argue that one should promote the sharing of knowledge and/or methodologies among disciplines and, for economics, limiting the "autistic" (or autarchy) process, which is critically discussed in economics already. Remembering (1936, p. viii) words, the problem for economics seems to be not so much to develop new ideas but to have the difficulties of "escaping from old ideas" and from "habitual modes of thought and expression".
Drake Equation for the Multiverse:. from the String Landscape to Complex Life
NASA Astrophysics Data System (ADS)
Gleiser, M.
It is argued that the selection criteria usually referred to as "anthropic conditions" for the existence of intelligent (typical) observers widely adopted in cosmology amount only to preconditions for primitive life. The existence of life does not imply in the existence of intelligent life. On the contrary, the transition from single-celled to complex, multicellular organisms is far from trivial, requiring stringent additional conditions on planetary platforms. An attempt is made to disentangle the necessary steps leading from a selection of universes out of a hypothetical multiverse to the existence of life and of complex life. It is suggested that what is currently called the "anthropic principle" should instead be named the "prebiotic principle."
ERIC Educational Resources Information Center
Uebelacker, James W.
This module considers ordinary linear differential equations with constant coefficients. The "complex method" used to find solutions is discussed, with numerous examples. The unit includes both problem sets and an exam, with answers provided for both. (MP)
NASA Astrophysics Data System (ADS)
Charalampidis, E. G.; Kevrekidis, P. G.; Frantzeskakis, D. J.; Malomed, B. A.
2016-08-01
We consider a two-component, two-dimensional nonlinear Schrödinger system with unequal dispersion coefficients and self-defocusing nonlinearities, chiefly with equal strengths of the self- and cross-interactions. In this setting, a natural waveform with a nonvanishing background in one component is a vortex, which induces an effective potential well in the second component, via the nonlinear coupling of the two components. We show that the potential well may support not only the fundamental bound state, but also multiring excited radial state complexes for suitable ranges of values of the dispersion coefficient of the second component. We systematically explore the existence, stability, and nonlinear dynamics of these states. The complexes involving the excited radial states are weakly unstable, with a growth rate depending on the dispersion of the second component. Their evolution leads to transformation of the multiring complexes into stable vortex-bright solitons ones with the fundamental state in the second component. The excited states may be stabilized by a harmonic-oscillator trapping potential, as well as by unequal strengths of the self- and cross-repulsive nonlinearities.
Moreira, Pedro S.; Sotiropoulos, Ioannis; Silva, Joana; Takashima, Akihiko; Sousa, Nuno; Leite-Almeida, Hugo; Costa, Patrício S.
2016-01-01
Background: Cognitive performance is a complex process influenced by multiple factors. Cognitive assessment in experimental animals is often based on longitudinal datasets analyzed using uni- and multi-variate analyses, that do not account for the temporal dimension of cognitive performance and also do not adequately quantify the relative contribution of individual factors onto the overall behavioral outcome. To circumvent these limitations, we applied an Autoregressive Latent Trajectory (ALT) to analyze the Morris water maze (MWM) test in a complex experimental design involving four factors: stress, age, sex, and genotype. Outcomes were compared with a traditional Mixed-Design Factorial ANOVA (MDF ANOVA). Results: In both the MDF ANOVA and ALT models, sex, and stress had a significant effect on learning throughout the 9 days. However, on the ALT approach, the effects of sex were restricted to the learning growth. Unlike the MDF ANOVA, the ALT model revealed the influence of single factors at each specific learning stage and quantified the cross interactions among them. In addition, ALT allows us to consider the influence of baseline performance, a critical and unsolved problem that frequently yields inaccurate interpretations in the classical ANOVA model. Discussion: Our findings suggest the beneficial use of ALT models in the analysis of complex longitudinal datasets offering a better biological interpretation of the interrelationship of the factors that may influence cognitive performance. PMID:26955327
NASA Astrophysics Data System (ADS)
Tauriello, Gerardo; Koumoutsakos, Petros
2015-02-01
We present a comparative study of penalization and phase field methods for the solution of the diffusion equation in complex geometries embedded using simple Cartesian meshes. The two methods have been widely employed to solve partial differential equations in complex and moving geometries for applications ranging from solid and fluid mechanics to biology and geophysics. Their popularity is largely due to their discretization on Cartesian meshes thus avoiding the need to create body-fitted grids. At the same time, there are questions regarding their accuracy and it appears that the use of each one is confined by disciplinary boundaries. Here, we compare penalization and phase field methods to handle problems with Neumann and Robin boundary conditions. We discuss extensions for Dirichlet boundary conditions and in turn compare with methods that have been explicitly designed to handle Dirichlet boundary conditions. The accuracy of all methods is analyzed using one and two dimensional benchmark problems such as the flow induced by an oscillating wall and by a cylinder performing rotary oscillations. This comparative study provides information to decide which methods to consider for a given application and their incorporation in broader computational frameworks. We demonstrate that phase field methods are more accurate than penalization methods on problems with Neumann boundary conditions and we present an error analysis explaining this result.
Scattering mean free path in continuous complex media: beyond the Helmholtz equation.
Baydoun, Ibrahim; Baresch, Diego; Pierrat, Romain; Derode, Arnaud
2015-09-01
We present theoretical calculations of the ensemble-averaged (or effective or coherent) wave field propagating in a heterogeneous medium considered as one realization of a random process. In the literature, it is usually assumed that heterogeneity can be accounted for by a random scalar function of the space coordinates, termed the potential. Physically, this amounts to replacing the constant wave speed in Helmholtz' equation by a space-dependent speed. In the case of acoustic waves, we show that this approach leads to incorrect results for the scattering mean free path, no matter how weak the fluctuations. The detailed calculation of the coherent wave field must take into account both a scalar and an operator part in the random potential. When both terms have identical amplitudes, the correct value for the scattering mean free paths is shown to be more than 4 times smaller (13/3, precisely) in the low-frequency limit, whatever the shape of the correlation function. Based on the diagrammatic approach of multiple scattering, theoretical results are obtained for the self-energy and mean free path within Bourret's and on-shell approximations. They are confirmed by numerical experiments. PMID:26465578
Classical irregular blocks, Hill's equation and PT-symmetric periodic complex potentials
NASA Astrophysics Data System (ADS)
Piatek, Marcin; Pietrykowski, Artur R.
2016-07-01
The Schrödinger eigenvalue problems for the Whittaker-Hill potential {Q}_2(x) = 1/2{h}^2 cos 4x + 4hμ cos 2x and the periodic complex potential {Q}_1(x)=1/4{h}^2{e}^{-} 4ix} + 2{h}^2 cos 2x are studied using their realizations in two-dimensional conformal field theory (2dCFT). It is shown that for the weak coupling (small) h ∈ ℝ and non-integer Floquet parameter ν ∉ ℤ spectra of hamiltonians ℋi = - d2/d x 2 + Q i( x), i = 1, 2 and corresponding two linearly independent eigenfunctions are given by the classical limit of the "single flavor" and "two flavors" ( N f = 1 , 2) irregular conformal blocks. It is known that complex nonhermitian hamiltonians which are PT-symmetric (= invariant under simultaneous parity P and time reversal T transformations) can have real eigenvalues. The hamiltonian ℋ1 is PT-symmetric for h, x ∈ ℝ. It is found that ℋ1 has a real spectrum in the weak coupling region for ν ∈ ℝ ℤ. This fact in an elementary way follows from a definition of the N f = 1 classical irregular block. Thus, ℋ1 can serve as yet another new model for testing postulates of PT-symmetric quantum mechanics.
Zhu, J.; Kais, S.; Rebentrost, P.; Aspuru-Guzik, Alan
2011-02-17
We present a detailed theoretical study of the transfer of electronic excitation energy through the Fenna-Matthews-Olson (FMO) pigment-protein complex, using the newly developed modified scaled hierarchical approach (Shi, Q.; et al. J. Chem. Phys.2009, 130, 084105). We show that this approach is computationally more efficient than the original hierarchical approach. The modified approach reduces the truncation levels of the auxiliary density operators and the correlation function. We provide a systematic study of how the number of auxiliary density operators and the higher-order correlation functions affect the exciton dynamics. The time scales of the coherent beating are consistent with experimental observations. Furthermore, our theoretical results exhibit population beating at physiological temperature. Additionally, the method does not require a low-temperature correction to obtain the correct thermal equilibrium at long times.
Thermostatted kinetic equations as models for complex systems in physics and life sciences
NASA Astrophysics Data System (ADS)
Bianca, Carlo
2012-12-01
Statistical mechanics is a powerful method for understanding equilibrium thermodynamics. An equivalent theoretical framework for nonequilibrium systems has remained elusive. The thermodynamic forces driving the system away from equilibrium introduce energy that must be dissipated if nonequilibrium steady states are to be obtained. Historically, further terms were introduced, collectively called a thermostat, whose original application was to generate constant-temperature equilibrium ensembles. This review surveys kinetic models coupled with time-reversible deterministic thermostats for the modeling of large systems composed both by inert matter particles and living entities. The introduction of deterministic thermostats allows to model the onset of nonequilibrium stationary states that are typical of most real-world complex systems. The first part of the paper is focused on a general presentation of the main physical and mathematical definitions and tools: nonequilibrium phenomena, Gauss least constraint principle and Gaussian thermostats. The second part provides a review of a variety of thermostatted mathematical models in physics and life sciences, including Kac, Boltzmann, Jager-Segel and the thermostatted (continuous and discrete) kinetic for active particles models. Applications refer to semiconductor devices, nanosciences, biological phenomena, vehicular traffic, social and economics systems, crowds and swarms dynamics.
Makowitz, H.
1992-10-01
We have studied various formulations of the concept of pressure, in the context of the usual Six-Equation Model of thermal-hydraulics. A different concept of pressure, than the usual one, has been used. This new pressure concept is Galilean Invariant, and results for the One-Pressure Model with the same complex characteristic roots as the Basic III-Posed Model,'' discussed in the literature for the cases we have investigated. We have also examined several Two-Pressure formulations and shown that two pressures are a necessary but not sufficient condition for obtaining a Well-Posed system. Several counter examples are presented. We have shown that the standard theory is not Galilean Invariant and suggested that the origin of III-Posedness is due to our closure relationships. We also question whether the current theory can satisfy conservation principles for mass, energy, and momentum.
Haghtalab, Mohammad; Faraji-Dana, Reza
2012-05-01
Analysis and optimization of diffraction effects in nanolithography through multilayered media with a fast and accurate field-theoretical approach is presented. The scattered field through an arbitrary two-dimensional (2D) mask pattern in multilayered media illuminated by a TM-polarized incident wave is determined by using an electric field integral equation formulation. In this formulation the electric field is represented in terms of complex images Green's functions. The method of moments is then employed to solve the resulting integral equation. In this way an accurate and computationally efficient approximate method is achieved. The accuracy of the proposed method is vindicated through comparison with direct numerical integration results. Moreover, the comparison is made between the results obtained by the proposed method and those obtained by the full-wave finite-element method. The ray tracing method is combined with the proposed method to describe the imaging process in the lithography. The simulated annealing algorithm is then employed to solve the inverse problem, i.e., to design an optimized mask pattern to improve the resolution. Two binary mask patterns under normal incident coherent illumination are designed by this method, where it is shown that the subresolution features improve the critical dimension significantly. PMID:22561933
NASA Astrophysics Data System (ADS)
Anderson, Justin Q.; Ryan, Rachel A.; Wu, Mingzhong; Carr, Lincoln D.
2014-02-01
A numerical exploration of a gain-loss nonlinear Schrödinger equation was carried out utilizing over 180 000 core hours to conduct more than 10 000 unique simulations in an effort to characterize the model's six dimensional parameter space. The study treated the problem in full generality, spanning a minimum of eight orders of magnitude for each of three linear and nonlinear gain terms and five orders of magnitude for higher order nonlinearities. The gain-loss nonlinear Schrödinger equation is of interest as a model for spin wave envelopes in magnetic thin film active feedback rings and analogous driven damped nonlinear physical systems. Bright soliton trains were spontaneously driven out of equilibrium and behaviors stable for tens of thousands of round trip times were numerically identified. Nine distinct complex dynamical behaviors with lifetimes on the order of ms were isolated as part of six identified solution classes. Numerically located dynamical behaviors include: (i) low dimensional chaotic modulations of bright soliton trains; (ii) spatially symmetric/asymmetric interactions of solitary wave peaks; (iii) dynamical pattern formation and recurrence; (iv) steady state solutions; and (v) intermittency. Simulations exhibiting chaotically modulating bright soliton trains were found to qualitatively match previous experimental observations. Ten new dynamical behaviors, eight demonstrating long lifetimes, are predicted to be observable in future experiments.
The Cauchy Problem in Local Spaces for the Complex Ginzburg-Landau EquationII. Contraction Methods
NASA Astrophysics Data System (ADS)
Ginibre, J.; Velo, G.
We continue the study of the initial value problem for the complex Ginzburg-Landau equation
On-off convection: Noise-induced intermittency near the convection threshold.
Fujisaka, H; Ouchi, K; Ohara, H
2001-09-01
A phenomenological nonlinear stochastic model of intermittency experimentally observed by Behn, Lange, and John [Phys. Rev. E 58, 2047 (1998)] in the electrohydrodynamic convection in nematics under dichotomous noise is proposed. This has the structure of the two-dimensional Swift-Hohenberg equation for local convection variable with fluctuating threshold. Numerical integration of the model equation shows intermittent emergence of convective pattern. Its statistics are found to obey those known, so far, for on-off intermittency. In the course of time, although the pattern intensity changes intermittently, no evident pattern change is observed. Adding additive noise, we observe an intermittent change of convective pattern. PMID:11580416
NASA Astrophysics Data System (ADS)
Zhang, Jianying; Yan, Guangwu; Wang, Moran
2016-02-01
A lattice Boltzmann model for solving the three-dimensional cubic-quintic complex Ginzburg-Landau equation (CQCGLE) is proposed. Differently from the classic lattice Boltzmann models, this lattice Boltzmann model is based on uniformly distributed lattice points in a three-dimensional space, and the evolution of the model is about a spatial axis rather than time. The algorithm provides advantages similar to the lattice Boltzmann method in that it is easily adapted to complex Ginzburg-Landau equations. Examples show that the model accurately reproduces the vortex tori pattern in the CQCGLE.
Lebedev, M E; Alfimov, G L; Malomed, Boris A
2016-07-01
We develop a general classification of the infinite number of families of solitons and soliton complexes in the one-dimensional Gross-Pitaevskii/nonlinear Schrödinger equation with a nonlinear lattice pseudopotential, i.e., periodically modulated coefficient in front of the cubic term, which takes both positive and negative local values. This model finds direct implementations in atomic Bose-Einstein condensates and nonlinear optics. The most essential finding is the existence of two branches of dipole solitons (DSs), which feature an antisymmetric shape, being essentially squeezed into a single cell of the nonlinear lattice. This soliton species was not previously considered in nonlinear lattices. We demonstrate that one branch of the DS family (namely, which obeys the Vakhitov-Kolokolov criterion) is stable, while unstable DSs spontaneously transform into stable fundamental solitons (FSs). The results are obtained in numerical and approximate analytical forms, the latter based on the variational approximation. Some stable bound states of FSs are found too. PMID:27475070
Misra, Amar P.; Roy Chowdhury, K.; Roy Chowdhury, A.
2007-01-15
Using the standard reductive perturbation technique, a nonlinear Schroedinger equation (NLSE) with complex coefficients is derived in a dusty plasma consisting of positive ions, nonthermal electrons, and charged dust grains. The effect of ion kinematic viscosity is taken into consideration, which makes the coefficients of NLSE complex. By means of a matching approach, the appearance mechanism of static pulses through a saddle-node bifurcation in the complex nonlinear Schroedinger equation is studied analytically. The analytical results are in good agreement with the direct numerical simulation. The modulational instability analysis is carried out for the dust ion-acoustic envelope solitary waves. The important role of the real part of the complex group velocity in the propagation of the one-dimensional wave packets in homogeneous active medium is predicted.
NASA Astrophysics Data System (ADS)
Kauczor, Joanna; Norman, Patrick; Christiansen, Ove; Coriani, Sonia
2013-12-01
We present a reduced-space algorithm for solving the complex (damped) linear response equations required to compute the complex linear response function for the hierarchy of methods: coupled cluster singles, coupled cluster singles and iterative approximate doubles, and coupled cluster singles and doubles. The solver is the keystone element for the development of damped coupled cluster response methods for linear and nonlinear effects in resonant frequency regions.
Relative stability of multipeak localized patterns of cavity solitons
Vladimirov, A. G.; Lefever, R.; Tlidi, M.
2011-10-15
We study the relative stability of different one-dimensional (1D) and two-dimensional (2D) clusters of closely packed localized peaks of the Swift-Hohenberg equation. In the 1D case, we demonstrate numerically the existence of a spatial Maxwell transition point where all clusters involving up to 15 peaks are equally stable. Above (below) this point, clusters become more (less) stable when their number of peaks increases. In the 2D case, since clusters involving more than two peaks may exhibit distinct spatial arrangements, this point splits into a set of Maxwell transition points.
Crystallization kinetics and self-induced pinning in cellular patterns
Aranson, Igor S.; Malomed, Boris A.; Pismen, Len M.; Tsimring, Lev S.
2000-07-01
Within the framework of the Swift-Hohenberg model it is shown numerically and analytically that the front propagation between cellular and uniform states is determined by periodic nucleation events triggered by the explosive growth of the localized zero-eigenvalue mode of the corresponding linear problem. We derive an evolution equation for this mode using asymptotic analysis, and evaluate the time interval between nucleation events, and hence the front speed. In the presence of noise, we find the velocity exponent of ''thermally activated'' front propagation (creep) beyond the pinning threshold. (c) 2000 The American Physical Society.
Mine, Makoto Okumura, Masahiko Sunaga, Tomoka Yamanaka, Yoshiya
2007-10-15
The Bogoliubov-de Gennes equations are used for a number of theoretical works on the trapped Bose-Einstein condensates. These equations are known to give the energies of the quasi-particles when all the eigenvalues are real. We consider the case in which these equations have complex eigenvalues. We give the complete set including those modes whose eigenvalues are complex. The quantum fields which represent neutral atoms are expanded in terms of the complete set. It is shown that the state space is an indefinite metric one and that the free Hamiltonian is not diagonalizable in the conventional bosonic representation. We introduce a criterion to select quantum states describing the metastablity of the condensate, called the physical state conditions. In order to study the instability, we formulate the linear response of the density against the time-dependent external perturbation within the regime of Kubo's linear response theory. Some states, satisfying all the physical state conditions, give the blow-up and damping behavior of the density distributions corresponding to the complex eigenmodes. It is qualitatively consistent with the result of the recent analyses using the time-dependent Gross-Pitaevskii equation.
Ge, Liang; Sotiropoulos, Fotis
2008-01-01
A novel numerical method is developed that integrates boundary-conforming grids with a sharp interface, immersed boundary methodology. The method is intended for simulating internal flows containing complex, moving immersed boundaries such as those encountered in several cardiovascular applications. The background domain (e.g the empty aorta) is discretized efficiently with a curvilinear boundary-fitted mesh while the complex moving immersed boundary (say a prosthetic heart valve) is treated with the sharp-interface, hybrid Cartesian/immersed-boundary approach of Gilmanov and Sotiropoulos [1]. To facilitate the implementation of this novel modeling paradigm in complex flow simulations, an accurate and efficient numerical method is developed for solving the unsteady, incompressible Navier-Stokes equations in generalized curvilinear coordinates. The method employs a novel, fully-curvilinear staggered grid discretization approach, which does not require either the explicit evaluation of the Christoffel symbols or the discretization of all three momentum equations at cell interfaces as done in previous formulations. The equations are integrated in time using an efficient, second-order accurate fractional step methodology coupled with a Jacobian-free, Newton-Krylov solver for the momentum equations and a GMRES solver enhanced with multigrid as preconditioner for the Poisson equation. Several numerical experiments are carried out on fine computational meshes to demonstrate the accuracy and efficiency of the proposed method for standard benchmark problems as well as for unsteady, pulsatile flow through a curved, pipe bend. To demonstrate the ability of the method to simulate flows with complex, moving immersed boundaries we apply it to calculate pulsatile, physiological flow through a mechanical, bileaflet heart valve mounted in a model straight aorta with an anatomical-like triple sinus. PMID:19194533
Ge, Liang; Sotiropoulos, Fotis
2007-08-01
A novel numerical method is developed that integrates boundary-conforming grids with a sharp interface, immersed boundary methodology. The method is intended for simulating internal flows containing complex, moving immersed boundaries such as those encountered in several cardiovascular applications. The background domain (e.g the empty aorta) is discretized efficiently with a curvilinear boundary-fitted mesh while the complex moving immersed boundary (say a prosthetic heart valve) is treated with the sharp-interface, hybrid Cartesian/immersed-boundary approach of Gilmanov and Sotiropoulos [1]. To facilitate the implementation of this novel modeling paradigm in complex flow simulations, an accurate and efficient numerical method is developed for solving the unsteady, incompressible Navier-Stokes equations in generalized curvilinear coordinates. The method employs a novel, fully-curvilinear staggered grid discretization approach, which does not require either the explicit evaluation of the Christoffel symbols or the discretization of all three momentum equations at cell interfaces as done in previous formulations. The equations are integrated in time using an efficient, second-order accurate fractional step methodology coupled with a Jacobian-free, Newton-Krylov solver for the momentum equations and a GMRES solver enhanced with multigrid as preconditioner for the Poisson equation. Several numerical experiments are carried out on fine computational meshes to demonstrate the accuracy and efficiency of the proposed method for standard benchmark problems as well as for unsteady, pulsatile flow through a curved, pipe bend. To demonstrate the ability of the method to simulate flows with complex, moving immersed boundaries we apply it to calculate pulsatile, physiological flow through a mechanical, bileaflet heart valve mounted in a model straight aorta with an anatomical-like triple sinus. PMID:19194533
NASA Astrophysics Data System (ADS)
Uchiyama, Yusuke; Konno, Hidetoshi
2014-04-01
Defect turbulence described by the one-dimensional complex Ginzburg-Landau equation is investigated and analyzed via a birth-death process of the local structures composed of defects, holes, and modulated amplitude waves (MAWs). All the number statistics of each local structure, in its stationary state, are subjected to Poisson statistics. In addition, the probability density functions of interarrival times of defects, lifetimes of holes, and MAWs show the existence of long-memory and some characteristic time scales caused by zigzag motions of oscillating traveling holes. The corresponding stochastic process for these observations is fully described by a non-Markovian master equation.
NASA Astrophysics Data System (ADS)
Kulikovskii, A. G.; Chugainova, A. P.; Shargatov, V. A.
2016-07-01
Solutions of the Riemann problem for a generalized Hopf equation are studied. The solutions are constructed using a sequence of non-overturning Riemann waves and shock waves with stable stationary and nonstationary structures.
Optical Rogue Waves in Vortex Turbulence
NASA Astrophysics Data System (ADS)
Gibson, Christopher J.; Yao, Alison M.; Oppo, Gian-Luca
2016-01-01
We present a spatiotemporal mechanism for producing 2D optical rogue waves in the presence of a turbulent state with creation, interaction, and annihilation of optical vortices. Spatially periodic structures with bound phase lose stability to phase unbound turbulent states in complex Ginzburg-Landau and Swift-Hohenberg models with external driving. When the pumping is high and the external driving is low, synchronized oscillations are unstable and lead to spatiotemporal vortex-mediated turbulence with high excursions in amplitude. Nonlinear amplification leads to rogue waves close to turbulent optical vortices, where the amplitude tends to zero, and to probability density functions (PDFs) with long tails typical of extreme optical events.
Martinho, Marlène; Dorlet, Pierre; Rivière, Eric; Thibon, Aurore; Ribal, Caroline; Banse, Frédéric; Girerd, Jean-Jacques
2008-01-01
The first example of a microcrystalline powder of a synthetic low-spin (LS) mononuclear Fe(III)(OOH) intermediate has been obtained by the precipitation of the [Fe(III)(L(5) (2))(OOH)](2+) complex at low temperature. The high purity of this thermally unstable powder is revealed by magnetic susceptibility measurements. EPR studies on this complex, in the solid state and also in frozen solution, are reported and reveal the coexistence of two related Fe(III)(OOH) species in both states. We also present a theoretical analysis of the g tensor for LS Fe(III) complexes, based on new perturbation equations. These simple equations provide distortion-energy parameters that are in good agreement with those obtained by a full-diagonalization calculation. PMID:18240118
ERIC Educational Resources Information Center
Jen, Tsung-Hau; Lee, Che-Di; Chien, Chin-Lung; Hsu, Ying-Shao; Chen, Kuan-Ming
2013-01-01
Based on the Trends in International Mathematics and Science Study 2007 study and a follow-up national survey, data for 3,901 Taiwanese grade 8 students were analyzed using structural equation modeling to confirm a social-relation-based affection-driven model (SRAM). SRAM hypothesized relationships among students' perceived social relationships in…
Pawlowski, Roger P.; Phipps, Eric T.; Salinger, Andrew G.; Owen, Steven J.; Siefert, Christopher M.; Staten, Matthew L.
2012-01-01
A template-based generic programming approach was presented in Part I of this series of papers [Sci. Program. 20 (2012), 197–219] that separates the development effort of programming a physical model from that of computing additional quantities, such as derivatives, needed for embedded analysis algorithms. In this paper, we describe the implementation details for using the template-based generic programming approach for simulation and analysis of partial differential equations (PDEs). We detail several of the hurdles that we have encountered, and some of the software infrastructure developed to overcome them. We end with a demonstration where we present shape optimization and uncertaintymore » quantification results for a 3D PDE application.« less
NASA Astrophysics Data System (ADS)
Wosiek, Jacek
2016-04-01
A positive representation for an arbitrary complex, gaussian weight is derived and used to construct a statistical formulation of gaussian path integrals directly in the Minkowski time. The positivity of Minkowski weights is achieved by doubling the number of real variables. The continuum limit of the new representation exists only if some of the additional couplings tend to infinity and are tuned in a specific way. The construction is then successfully applied to three quantum mechanical examples including a particle in a constant magnetic field — a simplest prototype of a Wilson line. Further generalizations are shortly discussed and an intriguing interpretation of new variables is alluded to.
Bravaya, Ksenia B.; Zuev, Dmitry; Epifanovsky, Evgeny; Krylov, Anna I.
2013-03-28
Theory and implementation of complex-scaled variant of equation-of-motion coupled-cluster method for excitation energies with single and double substitutions (EOM-EE-CCSD) is presented. The complex-scaling formalism extends the EOM-EE-CCSD model to resonance states, i.e., excited states that are metastable with respect to electron ejection. The method is applied to Feshbach resonances in atomic systems (He, H{sup -}, and Be). The dependence of the results on one-electron basis set is quantified and analyzed. Energy decomposition and wave function analysis reveal that the origin of the dependence is in electron correlation, which is essential for the lifetime of Feshbach resonances. It is found that one-electron basis should be sufficiently flexible to describe radial and angular electron correlation in a balanced fashion and at different values of the scaling parameter, {theta}. Standard basis sets that are optimized for not-complex-scaled calculations ({theta} = 0) are not sufficiently flexible to describe the {theta}-dependence of the wave functions even when heavily augmented by additional sets.
Makowitz, H.
1992-10-01
We have studied various formulations of the concept of pressure, in the context of the usual Six-Equation Model of thermal-hydraulics. A different concept of pressure, than the usual one, has been used. This new pressure concept is Galilean Invariant, and results for the One-Pressure Model with the same complex characteristic roots as the ``Basic III-Posed Model,`` discussed in the literature for the cases we have investigated. We have also examined several Two-Pressure formulations and shown that two pressures are a necessary but not sufficient condition for obtaining a Well-Posed system. Several counter examples are presented. We have shown that the standard theory is not Galilean Invariant and suggested that the origin of III-Posedness is due to our closure relationships. We also question whether the current theory can satisfy conservation principles for mass, energy, and momentum.
Ghosh, Aryya; Vaval, Nayana; Pal, Sourav; Bartlett, Rodney J.
2014-10-28
The equation-of-motion coupled cluster method employing the complex absorbing potential has been used to investigate the low energy electron scattering by CO{sub 2}. We have studied the potential energy curve for the {sup 2}Π{sub u} resonance states of CO{sub 2}{sup −} upon bending as well as symmetric and asymmetric stretching of the molecule. Specifically, we have stretched the C−O bond length from 1.1 Å to 1.5 Å and the bending angles are changed between 180° and 132°. Upon bending, the low energy {sup 2}Π{sub u} resonance state is split into two components, i.e., {sup 2}A{sub 1}, {sup 2}B{sub 1} due to the Renner-Teller effect, which behave differently as the molecule is bent.
Tao, Liang; Vanroose, Wim; Reps, Brian; Rescigno, Thomas N.; McCurdy, C. William
2009-09-08
We demonstrate that exterior complex scaling (ECS) can be used to impose outgoing wave boundary conditions exactly on solutions of the time-dependent Schrodinger equation for atoms in intense electromagnetic pulses using finite grid methods. The procedure is formally exact when applied in the appropriate gauge and is demonstrated in a calculation of high harmonic generation in which multiphoton resonances are seen for long pulse durations. However, we also demonstrate that while the application of ECS in this way is formally exact, numerical error can appear for long time propagations that can only be controlled by extending the finite grid. A mathematical analysis of the origins of that numerical error, illustrated with an analytically solvable model, is also given.
NASA Astrophysics Data System (ADS)
Kumar, Praveen; Jang, Seogjoo
2013-04-01
The emission lineshape of the B850 band in the light harvesting complex 2 of purple bacteria is calculated by extending the approach of 2nd order time-nonlocal quantum master equation [S. Jang and R. J. Silbey, J. Chem. Phys. 118, 9312 (2003), 10.1063/1.1569239]. The initial condition for the emission process corresponds to the stationary excited state density where exciton states are entangled with the bath modes in equilibrium. This exciton-bath coupling, which is not diagonal in either site excitation or exciton basis, results in a new inhomogeneous term that is absent in the expression for the absorption lineshape. Careful treatment of all the 2nd order terms are made, and explicit expressions are derived for both full 2nd order lineshape expression and the one based on secular approximation that neglects off-diagonal components in the exciton basis. Numerical results are presented for a few representative cases of disorder and temperature. Comparison of emission line shape with the absorption line shape is also made. It is shown that the inhomogeneous term coming from the entanglement of the system and bath degrees of freedom makes significant contributions to the lineshape. It is also found that the perturbative nature of the theory can result in negative portion of lineshape in some situations, which can be removed significantly by inclusion of the inhomogeneous term and completely by using the secular approximation. Comparison of the emission and absorption lineshapes at different temperatures demonstrates the role of thermal population of different exciton states and exciton-phonon couplings.
ERIC Educational Resources Information Center
Taber, Keith S.; Bricheno, Pat
2009-01-01
The present paper discusses the conceptual demands of an apparently straightforward task set to secondary-level students--completing chemical word equations with a single omitted term. Chemical equations are of considerable importance in chemistry, and school students are expected to learn to be able to write and interpret them. However, it is…
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.
ERIC Educational Resources Information Center
Stanford Univ., CA. School Mathematics Study Group.
This text is the fourth of five in the Secondary School Advanced Mathematics (SSAM) series which was designed to meet the needs of students who have completed the Secondary School Mathematics (SSM) program, and wish to continue their study of mathematics. This text begins with a brief discussion of quadratic equations which motivates the…
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)
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
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…
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).
Liu, Ju; Gomez, Hector; Landis, Chad M.
2013-09-01
We propose a new methodology for the numerical solution of the isothermal Navier–Stokes–Korteweg equations. Our methodology is based on a semi-discrete Galerkin method invoking functional entropy variables, a generalization of classical entropy variables, and a new time integration scheme. We show that the resulting fully discrete scheme is unconditionally stable-in-energy, second-order time-accurate, and mass-conservative. We utilize isogeometric analysis for spatial discretization and verify the aforementioned properties by adopting the method of manufactured solutions and comparing coarse mesh solutions with overkill solutions. Various problems are simulated to show the capability of the method. Our methodology provides a means of constructing unconditionally stable numerical schemes for nonlinear non-convex hyperbolic systems of conservation laws.
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.
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.
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.
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.
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.
NASA Astrophysics Data System (ADS)
Abdel-Gawad, H. I.; Tantawy, M.; Abo Elkhair, R. E.
2016-07-01
Rogue waves are more precisely defined as waves whose height is more than twice the significant wave height. This remarkable height was measured (by Draupner in 1995). Thus, the need for constructing a mechanism for the rogue waves is of great utility. This motivated us to suggest a mechanism, in this work, that rogue waves may be constructed via nonlinear interactions of solitons and periodic waves. This suggestion is consolidated here, in an example, by studying the behavior of solutions of the complex (KdV). This is done here by the extending the solutions of its real version.
NASA Astrophysics Data System (ADS)
Fremier, A. K.; Estrada Carmona, N.; Harper, E.; DeClerck, F.
2011-12-01
Appropriate application of complex models to estimate system behavior requires understanding the influence of model structure and parameter estimates on model output. To date, most researchers perform local sensitivity analyses, rather than global, because of computational time and quantity of data produced. Local sensitivity analyses are limited in quantifying the higher order interactions among parameters, which could lead to incomplete analysis of model behavior. To address this concern, we performed a GSA on a commonly applied equation for soil loss - the Revised Universal Soil Loss Equation. USLE is an empirical model built on plot-scale data from the USA and the Revised version (RUSLE) includes improved equations for wider conditions, with 25 parameters grouped into six factors to estimate long-term plot and watershed scale soil loss. Despite RUSLE's widespread application, a complete sensitivity analysis has yet to be performed. In this research, we applied a GSA to plot and watershed scale data from the US and Costa Rica to parameterize the RUSLE in an effort to understand the relative importance of model factors and parameters across wide environmental space. We analyzed the GSA results using Random Forest, a statistical approach to evaluate parameter importance accounting for the higher order interactions, and used Classification and Regression Trees to show the dominant trends in complex interactions. In all GSA calculations the management of cover crops (C factor) ranks the highest among factors (compared to rain-runoff erosivity, topography, support practices, and soil erodibility). This is counter to previous sensitivity analyses where the topographic factor was determined to be the most important. The GSA finding is consistent across multiple model runs, including data from the US, Costa Rica, and a synthetic dataset of the widest theoretical space. The three most important parameters were: Mass density of live and dead roots found in the upper inch
The zero dispersion limits of nonlinear wave equations
Tso, T.
1992-01-01
In chapter 2 the author uses functional analytic methods and conservation laws to solve the initial-value problem for the Korteweg-de Vries equation, the Benjamin-Bona-Mahony equation, and the nonlinear Schroedinger equation for initial data that satisfy some suitable conditions. In chapter 3 the energy estimates are used to show that the strong convergence of the family of the solutions of the KdV equation obtained in chapter 2 in H[sup 3](R) as [epsilon] [yields] 0; also, it is shown that the strong L[sup 2](R)-limit of the solutions of the BBM equation as [epsilon] [yields] 0 before a critical time. In chapter 4 the author uses the Whitham modulation theory and averaging method to find the 2[pi]-periodic solutions and the modulation equations of the KdV equation, the BBM equation, the Klein-Gordon equation, the NLS equation, the mKdV equation, and the P-system. It is shown that the modulation equations of the KdV equation, the K-G equation, the NLS equation, and the mKdV equation are hyperbolic but those of the BBM equation and the P-system are not hyperbolic. Also, the relations are studied of the KdV equation and the mKdV equation. Finally, the author studies the complex mKdV equation to compare with the NLS equation, and then study the complex gKdV equation.
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.
The Pauli equation with complex boundary conditions
NASA Astrophysics Data System (ADS)
Kochan, D.; Krejčiřík, D.; Novák, R.; Siegl, P.
2012-11-01
We consider one-dimensional Pauli Hamiltonians in a bounded interval with possibly non-self-adjoint Robin-type boundary conditions. We study the influence of the spin-magnetic interaction on the interplay between the type of boundary conditions and the spectrum. Special attention is paid to {PT}-symmetric boundary conditions with the physical choice of the time-reversal operator {T}. This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to ‘Quantum physics with non-Hermitian operators’.
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
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
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.
Symbolic Solution of Linear Differential Equations
NASA Technical Reports Server (NTRS)
Feinberg, R. B.; Grooms, R. G.
1981-01-01
An algorithm for solving linear constant-coefficient ordinary differential equations is presented. The computational complexity of the algorithm is discussed and its implementation in the FORMAC system is described. A comparison is made between the algorithm and some classical algorithms for solving differential equations.
Lattice Boltzmann equation for relativistic quantum mechanics.
Succi, Sauro
2002-03-15
Relativistic versions of the quantum lattice Boltzmann equation are discussed. It is shown that the inclusion of nonlinear interactions requires the standard collision operator to be replaced by a pair of dynamic fields coupling to the relativistic wave function in a way which can be described by a multicomponent complex lattice Boltzmann equation. PMID:16210189
Communication complexity and information complexity
NASA Astrophysics Data System (ADS)
Pankratov, Denis
Information complexity enables the use of information-theoretic tools in communication complexity theory. Prior to the results presented in this thesis, information complexity was mainly used for proving lower bounds and direct-sum theorems in the setting of communication complexity. We present three results that demonstrate new connections between information complexity and communication complexity. In the first contribution we thoroughly study the information complexity of the smallest nontrivial two-party function: the AND function. While computing the communication complexity of AND is trivial, computing its exact information complexity presents a major technical challenge. In overcoming this challenge, we reveal that information complexity gives rise to rich geometrical structures. Our analysis of information complexity relies on new analytic techniques and new characterizations of communication protocols. We also uncover a connection of information complexity to the theory of elliptic partial differential equations. Once we compute the exact information complexity of AND, we can compute exact communication complexity of several related functions on n-bit inputs with some additional technical work. Previous combinatorial and algebraic techniques could only prove bounds of the form theta( n). Interestingly, this level of precision is typical in the area of information theory, so our result demonstrates that this meta-property of precise bounds carries over to information complexity and in certain cases even to communication complexity. Our result does not only strengthen the lower bound on communication complexity of disjointness by making it more exact, but it also shows that information complexity provides the exact upper bound on communication complexity. In fact, this result is more general and applies to a whole class of communication problems. In the second contribution, we use self-reduction methods to prove strong lower bounds on the information
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)
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.
Relations between nonlinear Riccati equations and other equations in fundamental physics
NASA Astrophysics Data System (ADS)
Schuch, Dieter
2014-10-01
Many phenomena in the observable macroscopic world obey nonlinear evolution equations while the microscopic world is governed by quantum mechanics, a fundamental theory that is supposedly linear. In order to combine these two worlds in a common formalism, at least one of them must sacrifice one of its dogmas. Linearizing nonlinear dynamics would destroy the fundamental property of this theory, however, it can be shown that quantum mechanics can be reformulated in terms of nonlinear Riccati equations. In a first step, it will be shown that the information about the dynamics of quantum systems with analytical solutions can not only be obtainable from the time-dependent Schrödinger equation but equally-well from a complex Riccati equation. Comparison with supersymmetric quantum mechanics shows that even additional information can be obtained from the nonlinear formulation. Furthermore, the time-independent Schrödinger equation can also be rewritten as a complex Riccati equation for any potential. Extension of the Riccati formulation to include irreversible dissipative effects is straightforward. Via (real and complex) Riccati equations, other fields of physics can also be treated within the same formalism, e.g., statistical thermodynamics, nonlinear dynamical systems like those obeying a logistic equation as well as wave equations in classical optics, Bose- Einstein condensates and cosmological models. Finally, the link to abstract "quantizations" such as the Pythagorean triples and Riccati equations connected with trigonometric and hyperbolic functions will be shown.
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…
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
Some Conceptual Issues in Observed-Score Equating
ERIC Educational Resources Information Center
van der Linden, Wim J.
2013-01-01
In spite of all of the technical progress in observed-score equating, several of the more conceptual aspects of the process still are not well understood. As a result, the equating literature struggles with rather complex criteria of equating, lack of a test-theoretic foundation, confusing terminology, and ad hoc analyses. A return to Lord's…
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 -∞
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.
Lump solutions of the BKP equation
NASA Astrophysics Data System (ADS)
Gilson, C. R.; Nimmo, J. J. C.
1990-07-01
Rational solutions of the BKP equation which decay to zero in all directions in the plane are obtained. These solutions are analogous to the lump solutions of the KPI equation. Properties of the single lump solution are described and the form of the N-lump solution is given. It is shown that single lump solutions are only non-singular for spectral parameters lying in certain regions of the complex plane.
Dynamics of the Kuramoto equation with spatially distributed control
NASA Astrophysics Data System (ADS)
Kashchenko, Ilia; Kaschenko, Sergey
2016-05-01
We consider the scalar complex equation with spatially distributed control. Its dynamical properties are studied by asymptotic methods when the control coefficient is either sufficiently large or sufficiently small and the function of distribution is either almost symmetric or significantly nonsymmetric relative to zero. In all cases we reduce original equation to quasinormal form - the family of special parabolic equations, which do not contain big and small parameters, which nonlocal dynamics determines the behaviour of solutions of the original equation.
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).
Solutions of the coupled Higgs field equations.
Talukdar, Benoy; Ghosh, Swapan K; Saha, Aparna; Pal, Debabrata
2013-07-01
By an appropriate choice for the phase of the complex nucleonic field and going over to the traveling coordinate, we reduce the coupled Higgs equations to the Hamiltonian form and treat the resulting equation using the dynamical system theory. We present a phase-space analysis of its stable points. The results of our study demonstrate that the equation can support both traveling- and standing-wave solutions. The traveling-wave solution appears in the form of a soliton and resides in the midst of doubly periodic standing-wave solutions. PMID:23944601
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)
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
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.
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.
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.
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.
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.
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…
Rate equations for sodium catalyzed quartz dissolution
NASA Astrophysics Data System (ADS)
Rimstidt, J. Donald
2015-10-01
Quartz dissolution rate data were fit to an equation that predicts the dissolution flux (J, mol/m2 sec) as a function of temperature (T, K), sodium concentration (mNa+, molal), and hydrogen ion activity (aH+). The same data fit equally well to an equation that expresses the rate as a function of temperature, sodium concentration, and hydroxide ion activity (aOH-) . These equations are more convenient to use than those given by Bickmore et al. (2008) because rates can be predicted without the implementation of a surface speciation model. They predict that at 25 °C quartz dissolves more than 200 times faster in seawater than in pure water. These two equations fit the data just as well as five other equations from Bickmore et al. (2008) that are based on surface species concentrations. All of these rate equations contain information about the reaction mechanism(s) for quartz dissolution but that information is ambiguous because the independent variables used to develop the equations are correlated. This means that rate equations alone cannot be used to infer the dissolution mechanism. Existing surface complexation, surface charge, terrace-ledge-kink, and Lewis acid-base models must be modified and amalgamated in order to develop a reliable model of the reaction mechanism(s).
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
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.
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.
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)
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.
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…
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: \\{
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.
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.
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)
Equations with Technology: Different Tools, Different Views
ERIC Educational Resources Information Center
Drijvers, Paul; Barzel, Barbel
2012-01-01
Has technology revolutionised the mathematics classroom, or is it still a device waiting to be exploited for the benefit of the learner? There are applets that will enable the user to solve complex equations at the push of a button. So, does this jeopardise other methods, make other methods redundant, or even diminish other methods in the mind of…
NASA Astrophysics Data System (ADS)
Thomson, Mark J.; McKellar, Bruce H. J.
1991-04-01
A simple, non-linear generalization of the MSW equation is presented and its analytic solution is outlined. The orbits of the polarization vector are shown to be periodic, and to lie on a sphere. Their non-trivial flow patterns fall into two topological categories, the more complex of which can become chaotic if perturbed.
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
On the thermo-acoustic Fant equation
NASA Astrophysics Data System (ADS)
Murray, P. R.; Howe, M. S.
2012-07-01
A 'reduced complexity' equation is derived to investigate combustion instabilities of a Rijke burner. The equation is nonlinear and furnishes limit cycle solutions for finite amplitude burner modes. It is a generalisation to combustion flows of the Fant equation used to investigate the production of voiced speech by unsteady throttling of flow by the vocal folds [G. Fant, Acoustic Theory of Speech Production. Mouton, The Hague, 1960]. In the thermo-acoustic problem the throttling occurs at the flame holder. The Fant equation governs the unsteady volume flow past the flame holder which, in turn, determines the acoustics of the entire system. The equation includes a fully determinate part that depends on the geometry of the flame holder and the thermo-acoustic system, and terms defined by integrals involving thermo-aerodynamic sources, such as a flame and vortex sound sources. These integrals provide a clear indication of what must be known about the flow to obtain a proper understanding of the dynamics of the thermo-acoustic system. Illustrative numerical results are presented for the linearised equation. This governs the growth rates of the natural acoustic modes, determined by system geometry, boundary conditions and mean temperature distribution, which are excited into instability by unsteady heat release from the flame and damped by large scale vorticity production and radiation losses into the environment. In addition, the equation supplies information about the 'combustion modes' excited by the local time-delay feedback dynamics of the flame.
Exact solution of some linear matrix equations using algebraic methods
NASA Technical Reports Server (NTRS)
Djaferis, T. E.; Mitter, S. K.
1977-01-01
A study is done of solution methods for Linear Matrix Equations including Lyapunov's equation, using methods of modern algebra. The emphasis is on the use of finite algebraic procedures which are easily implemented on a digital computer and which lead to an explicit solution to the problem. The action f sub BA is introduced a Basic Lemma is proven. The equation PA + BP = -C as well as the Lyapunov equation are analyzed. Algorithms are given for the solution of the Lyapunov and comment is given on its arithmetic complexity. The equation P - A'PA = Q is studied and numerical examples are given.
Singularities of the Euler equation and hydrodynamic stability
NASA Technical Reports Server (NTRS)
Tanveer, S.; Speziale, Charles G.
1993-01-01
Equations governing the motion of a specific class of singularities of the Euler equation in the extended complex spatial domain are derived. Under some assumptions, it is shown how this motion is dictated by the smooth part of the complex velocity at a singular point in the unphysical domain. These results are used to relate the motion of complex singularities to the stability of steady solutions of the Euler equation. A sufficient condition for instability is conjectured. Several examples are presented to demonstrate the efficacy of this sufficient condition which include the class of elliptical flows and the Kelvin-Stuart Cat's Eye.
Singularities of the Euler equation and hydrodynamic stability
NASA Technical Reports Server (NTRS)
Tanveer, S.; Speziale, Charles G.
1992-01-01
Equations governing the motion of a specific class of singularities of the Euler equation in the extended complex spatial domain are derived. Under some assumptions, it is shown how this motion is dictated by the smooth part of the complex velocity at a singular point in the unphysical domain. These results are used to relate the motion of complex singularities to the stability of steady solutions of the Euler equation. A sufficient condition for instability is conjectured. Several examples are presented to demonstrate the efficacy of this sufficient condition which include the class of elliptical flows and the Kelvin-Stuart Cat's Eye.
Damping filter method for obtaining spatially localized solutions.
Teramura, Toshiki; Toh, Sadayoshi
2014-05-01
Spatially localized structures are key components of turbulence and other spatiotemporally chaotic systems. From a dynamical systems viewpoint, it is desirable to obtain corresponding exact solutions, though their existence is not guaranteed. A damping filter method is introduced to obtain variously localized solutions and adapted in two typical cases. This method introduces a spatially selective damping effect to make a good guess at the exact solution, and we can obtain an exact solution through a continuation with the damping amplitude. The first target is a steady solution to the Swift-Hohenberg equation, which is a representative of bistable systems in which localized solutions coexist and a model for spanwise-localized cases. Not only solutions belonging to the well-known snaking branches but also those belonging to isolated branches known as "isolas" are found with continuation paths between them in phase space extended with the damping amplitude. This indicates that this spatially selective excitation mechanism has an advantage in searching spatially localized solutions. The second target is a spatially localized traveling-wave solution to the Kuramoto-Sivashinsky equation, which is a model for streamwise-localized cases. Since the spatially selective damping effect breaks Galilean and translational invariances, the propagation velocity cannot be determined uniquely while the damping is active, and a singularity arises when these invariances are recovered. We demonstrate that this singularity can be avoided by imposing a simple condition, and a localized traveling-wave solution is obtained with a specific propagation speed. PMID:25353864
Developing Simultaneous Linear Equations and Rational Equations
ERIC Educational Resources Information Center
Boss'e, Michael J.; Nandakumar, N. R.
2004-01-01
To demonstrate concepts or rapidly create quizzes, teachers commonly encounter the need to quickly create mathematical examples. Unfortunately, by producing undesirable or overly complex solutions, extemporaneously created examples can become problematic, create tense learning environments and become more confusing than they are worth. Experience…
Self-organization of network dynamics into local quantized states
Nicolaides, Christos; Juanes, Ruben; Cueto-Felgueroso, Luis
2016-02-17
Self-organization and pattern formation in network-organized systems emerges from the collective activation and interaction of many interconnected units. A striking feature of these non-equilibrium structures is that they are often localized and robust: only a small subset of the nodes, or cell assembly, is activated. Understanding the role of cell assemblies as basic functional units in neural networks and socio-technical systems emerges as a fundamental challenge in network theory. A key open question is how these elementary building blocks emerge, and how they operate, linking structure and function in complex networks. Here we show that a network analogue of themore » Swift-Hohenberg continuum model—a minimal-ingredients model of nodal activation and interaction within a complex network—is able to produce a complex suite of localized patterns. Thus, the spontaneous formation of robust operational cell assemblies in complex networks can be explained as the result of self-organization, even in the absence of synaptic reinforcements.« less
Self-organization of network dynamics into local quantized states
NASA Astrophysics Data System (ADS)
Nicolaides, Christos; Juanes, Ruben; Cueto-Felgueroso, Luis
2016-02-01
Self-organization and pattern formation in network-organized systems emerges from the collective activation and interaction of many interconnected units. A striking feature of these non-equilibrium structures is that they are often localized and robust: only a small subset of the nodes, or cell assembly, is activated. Understanding the role of cell assemblies as basic functional units in neural networks and socio-technical systems emerges as a fundamental challenge in network theory. A key open question is how these elementary building blocks emerge, and how they operate, linking structure and function in complex networks. Here we show that a network analogue of the Swift-Hohenberg continuum model—a minimal-ingredients model of nodal activation and interaction within a complex network—is able to produce a complex suite of localized patterns. Hence, the spontaneous formation of robust operational cell assemblies in complex networks can be explained as the result of self-organization, even in the absence of synaptic reinforcements.
Self-organization of network dynamics into local quantized states
Nicolaides, Christos; Juanes, Ruben; Cueto-Felgueroso, Luis
2016-01-01
Self-organization and pattern formation in network-organized systems emerges from the collective activation and interaction of many interconnected units. A striking feature of these non-equilibrium structures is that they are often localized and robust: only a small subset of the nodes, or cell assembly, is activated. Understanding the role of cell assemblies as basic functional units in neural networks and socio-technical systems emerges as a fundamental challenge in network theory. A key open question is how these elementary building blocks emerge, and how they operate, linking structure and function in complex networks. Here we show that a network analogue of the Swift-Hohenberg continuum model—a minimal-ingredients model of nodal activation and interaction within a complex network—is able to produce a complex suite of localized patterns. Hence, the spontaneous formation of robust operational cell assemblies in complex networks can be explained as the result of self-organization, even in the absence of synaptic reinforcements. PMID:26883170
Self-organization of network dynamics into local quantized states.
Nicolaides, Christos; Juanes, Ruben; Cueto-Felgueroso, Luis
2016-01-01
Self-organization and pattern formation in network-organized systems emerges from the collective activation and interaction of many interconnected units. A striking feature of these non-equilibrium structures is that they are often localized and robust: only a small subset of the nodes, or cell assembly, is activated. Understanding the role of cell assemblies as basic functional units in neural networks and socio-technical systems emerges as a fundamental challenge in network theory. A key open question is how these elementary building blocks emerge, and how they operate, linking structure and function in complex networks. Here we show that a network analogue of the Swift-Hohenberg continuum model-a minimal-ingredients model of nodal activation and interaction within a complex network-is able to produce a complex suite of localized patterns. Hence, the spontaneous formation of robust operational cell assemblies in complex networks can be explained as the result of self-organization, even in the absence of synaptic reinforcements. PMID:26883170
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.
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.
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.
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.
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.
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.
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…
Consistent lattice Boltzmann equations for phase transitions.
Siebert, D N; Philippi, P C; Mattila, K K
2014-11-01
Unlike conventional computational fluid dynamics methods, the lattice Boltzmann method (LBM) describes the dynamic behavior of fluids in a mesoscopic scale based on discrete forms of kinetic equations. In this scale, complex macroscopic phenomena like the formation and collapse of interfaces can be naturally described as related to source terms incorporated into the kinetic equations. In this context, a novel athermal lattice Boltzmann scheme for the simulation of phase transition is proposed. The continuous kinetic model obtained from the Liouville equation using the mean-field interaction force approach is shown to be consistent with diffuse interface model using the Helmholtz free energy. Density profiles, interface thickness, and surface tension are analytically derived for a plane liquid-vapor interface. A discrete form of the kinetic equation is then obtained by applying the quadrature method based on prescribed abscissas together with a third-order scheme for the discretization of the streaming or advection term in the Boltzmann equation. Spatial derivatives in the source terms are approximated with high-order schemes. The numerical validation of the method is performed by measuring the speed of sound as well as by retrieving the coexistence curve and the interface density profiles. The appearance of spurious currents near the interface is investigated. The simulations are performed with the equations of state of Van der Waals, Redlich-Kwong, Redlich-Kwong-Soave, Peng-Robinson, and Carnahan-Starling. PMID:25493907
A closure scheme for chemical master equations
Smadbeck, Patrick; Kaznessis, Yiannis N.
2013-01-01
Probability reigns in biology, with random molecular events dictating the fate of individual organisms, and propelling populations of species through evolution. In principle, the master probability equation provides the most complete model of probabilistic behavior in biomolecular networks. In practice, master equations describing complex reaction networks have remained unsolved for over 70 years. This practical challenge is a reason why master equations, for all their potential, have not inspired biological discovery. Herein, we present a closure scheme that solves the master probability equation of networks of chemical or biochemical reactions. We cast the master equation in terms of ordinary differential equations that describe the time evolution of probability distribution moments. We postulate that a finite number of moments capture all of the necessary information, and compute the probability distribution and higher-order moments by maximizing the information entropy of the system. An accurate order closure is selected, and the dynamic evolution of molecular populations is simulated. Comparison with kinetic Monte Carlo simulations, which merely sample the probability distribution, demonstrates this closure scheme is accurate for several small reaction networks. The importance of this result notwithstanding, a most striking finding is that the steady state of stochastic reaction networks can now be readily computed in a single-step calculation, without the need to simulate the evolution of the probability distribution in time. PMID:23940327
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)
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]…
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.
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…
Pattern dynamics associated with on-off convection in a one-dimensional system.
Ohara, Hidenori; Fujisaka, Hirokazu; Ouchi, Katsuya
2003-04-01
A numerical and theoretical analysis of the phenomenologically constructed nonlinear stochastic model of on-off intermittency experimentally observed by John et al. in the electrohydrodynamic convection in nematic liquid crystal under applied dichotomous electric field is carried out. The model has the structure of the one-dimensional Swift-Hohenberg equation with a fluctuating threshold which represents an applied electric field and either with or without additive noise which corresponds to thermal noise. It is found that the fundamental statistics of pattern dynamics without additive noise agree with those experimentally observed, and also with those reported previously in two-dimensional system. In contrast to that the presence of multiplicative noise generates an intermittent evolution of pattern intensity, whose statistics are in agreement with those of on-off intermittency so far known, the additive noise gives rise to the change of position of the convective pattern. It is found that the temporal evolution of the phase suitably introduced to describe the global convective pattern also shows an intermittent evolution. Its statistics are studied in a detailed way with numerical simulation and stochastic analysis. The comparison of these results turn out to be in good agreement with each other. PMID:12786476
Relaxation to ordered patterns in vertically oscillated granular layers
NASA Astrophysics Data System (ADS)
Goldman, D. I.; Shattuck, M. D.; Swinney, Harry L.; Gunaratne, G. H.; Kouri, D. J.; Hoffman, D. K.
1999-11-01
Standing wave patterns of squares or stripes form in vertically oscillated granular layers above a critical peak plate acceleration for a range of container drive frequencies. (F. Melo, P. B. Umbanhowar, and H. L. Swinney, Phys. Rev. Lett., 75), 3838 (1995) We examine pattern development when the container acceleration is abruptly increased from an oscillating flat layer, into the region where the patterns form (peak plate acceleration > 2.5g). We calculate the difference between the pattern and a uniform pattern of squares or stripes using a measure of pattern disorder. (G. H. Gunaratne, R. E. Jones, Q. Ouyang, H. L. Swinney, Phys. Rev. Lett., 75), 3281 (1995) The time evolution of the measured disorder parameter occurs in at least three distinct stages --- a fast decay associated with linear growth of modes, followed by a slower decay through a nonlinear selection stage, and finally a yet slower decay associated with finite container size effects. The relative time scales for these decays depend on the distance from pattern onset and noise level in the system. The results are consistent with pattern relaxation in a Swift-Hohenberg type equation with a subcritical bifurcation.
Homoclinic snaking in plane Couette flow: bending, skewing and finite-size effects
NASA Astrophysics Data System (ADS)
Gibson, J. F.; Schneider, T. M.
2016-05-01
Invariant solutions of shear flows have recently been extended from spatially periodic solutions in minimal flow units to spatially localized solutions on extended domains. One set of spanwise-localized solutions of plane Couette flow exhibits homoclinic snaking, a process by which steady-state solutions grow additional structure smoothly at their fronts when continued parametrically. Homoclinic snaking is well understood mathematically in the context of the one-dimensional Swift-Hohenberg equation. Consequently, the snaking solutions of plane Couette flow form a promising connection between the largely phenomenological study of laminar-turbulent patterns in viscous shear flows and the mathematically well-developed field of pattern-formation theory. In this paper we present a numerical study of the snaking solutions, generalizing beyond the fixed streamwise wavelength of previous studies. We find a number of new solution features, including bending, skewing, and finite-size effects. We show that the finite-size effects result from the shift-reflect symmetry of the traveling wave and establish the parameter regions over which snaking occurs. A new winding solution of plane Couette flow is derived from a strongly skewed localized equilibrium.
Competing resonances in spatially forced pattern-forming systems.
Mau, Yair; Haim, Lev; Hagberg, Aric; Meron, Ehud
2013-09-01
Spatial periodic forcing can entrain a pattern-forming system in the same way as temporal periodic forcing can entrain an oscillator. The forcing can lock the pattern's wave number to a fraction of the forcing wave number within tonguelike domains in the forcing parameter plane, it can increase the pattern's amplitude, and it can also create patterns below their onset. We derive these results using a multiple-scale analysis of a spatially forced Swift-Hohenberg equation in one spatial dimension. In two spatial dimensions the one-dimensional forcing can induce a symmetry-breaking instability that leads to two-dimensional (2D) patterns, rectangular or oblique. These patterns resonate with the forcing by locking their wave-vector component in the forcing direction to half the forcing wave number. The range of this type of 2:1 resonance overlaps with the 1:1 resonance tongue of stripe patterns. Using a multiple-scale analysis in the overlap region we show that the 2D patterns can destabilize the 1:1 resonant stripes even at exact resonance. This result sheds new light on the use of spatial periodic forcing for controlling patterns. PMID:24125335
Minimal continuum theories of structure formation in dense active fluids
NASA Astrophysics Data System (ADS)
Dunkel, Jörn; Heidenreich, Sebastian; Bär, Markus; Goldstein, Raymond E.
2013-04-01
Self-sustained dynamical phases of living matter can exhibit remarkable similarities over a wide range of scales, from mesoscopic vortex structures in microbial suspensions and motility assays of biopolymers to turbulent large-scale instabilities in flocks of birds or schools of fish. Here, we argue that, in many cases, the phenomenology of such active states can be efficiently described in terms of fourth- and higher-order partial differential equations. Structural transitions in these models can be interpreted as Landau-type kinematic transitions in Fourier (wavenumber) space, suggesting that microscopically different biological systems can share universal long-wavelength features. This general idea is illustrated through numerical simulations for two classes of continuum models for incompressible active fluids: a Swift-Hohenberg-type scalar field theory, and a minimal vector model that extends the classical Toner-Tu theory and appears to be a promising candidate for the quantitative description of dense bacterial suspensions. We discuss how microscopic symmetry-breaking mechanisms can enter macroscopic continuum descriptions of collective microbial motion near surfaces, and conclude by outlining future applications.
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.
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.
Propagating Qualitative Values Through Quantitative Equations
NASA Technical Reports Server (NTRS)
Kulkarni, Deepak
1992-01-01
In most practical problems where traditional numeric simulation is not adequate, one need to reason about a system with both qualitative and quantitative equations. In this paper, we address the problem of propagating qualitative values represented as interval values through quantitative equations. Previous research has produced exponential-time algorithms for approximate solution of the problem. These may not meet the stringent requirements of many real time applications. This paper advances the state of art by producing a linear-time algorithm that can propagate a qualitative value through a class of complex quantitative equations exactly and through arbitrary algebraic expressions approximately. The algorithm was found applicable to Space Shuttle Reaction Control System model.
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.
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.
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
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.
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,…
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.
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.
Solving Partial Differential Equations on Overlapping Grids
Henshaw, W D
2008-09-22
We discuss the solution of partial differential equations (PDEs) on overlapping grids. This is a powerful technique for efficiently solving problems in complex, possibly moving, geometry. An overlapping grid consists of a set of structured grids that overlap and cover the computational domain. By allowing the grids to overlap, grids for complex geometries can be more easily constructed. The overlapping grid approach can also be used to remove coordinate singularities by, for example, covering a sphere with two or more patches. We describe the application of the overlapping grid approach to a variety of different problems. These include the solution of incompressible fluid flows with moving and deforming geometry, the solution of high-speed compressible reactive flow with rigid bodies using adaptive mesh refinement (AMR), and the solution of the time-domain Maxwell's equations of electromagnetism.
Optimal active control for Burgers equations
NASA Technical Reports Server (NTRS)
Ikeda, Yutaka
1994-01-01
A method for active fluid flow control based on control theory is discussed. Dynamic programming and fixed point successive approximations are used to accommodate the nonlinear control problem. The long-term goal of this project is to establish an effective method applicable to complex flows such as turbulence and jets. However, in this report, the method is applied to stochastic Burgers equation as an intermediate step towards this goal. Numerical results are compared with those obtained by gradient search methods.
The quasicontinuum Fokker-Plank equation
Alexander, Francis J
2008-01-01
We present a regularized Fokker-Planck equation with more accurate short-time and high-frequency behavior for continuous-time, discrete-state systems. The regularization preserves crucial aspects of state-space discreteness lost in the standard Kramers-Moyal expansion. We apply the method to a simple example of biochemical reaction kinetics and to a two-dimensional symmetric random walk, and suggest its application to more complex systerns.
Unified viscoplastic constitutive equations and their applications
NASA Technical Reports Server (NTRS)
Lindholm, U. S.
1987-01-01
Unified constitutive equations for time- and temperature-dependent metallic plastic deformation have been applied in FEM simulations of forming processes; increasingly powerful computational tools and physical models are being used to numerically model complex engineering problems. Once confidence has been gained through adequate verification, these numerical models will increasingly replace experimental models. Attention is presently given to the contributions made by physical metallurgy, continuum mechanics, and computational mechanics.
Singularity formation for Prandtl’s equations
NASA Astrophysics Data System (ADS)
Gargano, F.; Sammartino, M.; Sciacca, V.
2009-10-01
We consider Prandtl’s equations for an impulsively started disk and follow the process of the formation of the singularity in the complex plane using the singularity tracking method. We classify Van Dommelen and Shen’s singularity as a cubic root singularity. We introduce a class of initial data, uniformly bounded in H1, which have a dipole singularity in the complex plane. These data lead to a solution blow-up whose time can be made arbitrarily short within the class. This is numerical evidence of the ill-posedness of the Prandtl equations in H1. The presence of a small viscosity in the streamwise direction changes the behavior of the singularities. They stabilize at a distance from the real axis which depends on the amount of viscosity. We show that the Van Dommelen and Shen singularity and the singularity predicted by E and Engquist in [W. E, B. Engquist, Blowup of the solutions to the unsteady Prandtl’s equations, Comm. Pure Appl. Math. 50 (1997) 1287-1293.] have different complex structures.
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
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.
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.
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).…
A Conceptual Approach to Absolute Value Equations and Inequalities
ERIC Educational Resources Information Center
Ellis, Mark W.; Bryson, Janet L.
2011-01-01
The absolute value learning objective in high school mathematics requires students to solve far more complex absolute value equations and inequalities. When absolute value problems become more complex, students often do not have sufficient conceptual understanding to make any sense of what is happening mathematically. The authors suggest that the…
A note on singularities of the 3-D Euler equation
NASA Technical Reports Server (NTRS)
Tanveer, S.
1994-01-01
In this paper, we consider analytic initial conditions with finite energy, whose complex spatial continuation is a superposition of a smooth background flow and a singular field. Through explicit calculation in the complex plane, we show that under some assumptions, the solution to the 3-D Euler equation ceases to be analytic in the real domain in finite time.
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
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.
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.
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.
Ultra Deep Wave Equation Imaging and Illumination
Alexander M. Popovici; Sergey Fomel; Paul Sava; Sean Crawley; Yining Li; Cristian Lupascu
2006-09-30
In this project we developed and tested a novel technology, designed to enhance seismic resolution and imaging of ultra-deep complex geologic structures by using state-of-the-art wave-equation depth migration and wave-equation velocity model building technology for deeper data penetration and recovery, steeper dip and ultra-deep structure imaging, accurate velocity estimation for imaging and pore pressure prediction and accurate illumination and amplitude processing for extending the AVO prediction window. Ultra-deep wave-equation imaging provides greater resolution and accuracy under complex geologic structures where energy multipathing occurs, than what can be accomplished today with standard imaging technology. The objective of the research effort was to examine the feasibility of imaging ultra-deep structures onshore and offshore, by using (1) wave-equation migration, (2) angle-gathers velocity model building, and (3) wave-equation illumination and amplitude compensation. The effort consisted of answering critical technical questions that determine the feasibility of the proposed methodology, testing the theory on synthetic data, and finally applying the technology for imaging ultra-deep real data. Some of the questions answered by this research addressed: (1) the handling of true amplitudes in the downward continuation and imaging algorithm and the preservation of the amplitude with offset or amplitude with angle information required for AVO studies, (2) the effect of several imaging conditions on amplitudes, (3) non-elastic attenuation and approaches for recovering the amplitude and frequency, (4) the effect of aperture and illumination on imaging steep dips and on discriminating the velocities in the ultra-deep structures. All these effects were incorporated in the final imaging step of a real data set acquired specifically to address ultra-deep imaging issues, with large offsets (12,500 m) and long recording time (20 s).
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…
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…
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…
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…
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.
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.
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.
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.
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.
String gravitational equations with Hermitian structure
NASA Astrophysics Data System (ADS)
Naderi, F.; Rezaei-Aghdam, A.; Darabi, F.
2016-02-01
We consider a string model at one-loop related to a σ-model whose antisymmetric tensor field is constructed as complex structure on the background manifold, especially on a manifold R × N where N is a complex manifold. As an example, we consider a homogeneous anisotropic (1 + 4)-dimensional σ-model where space part of the background is a four-dimensional complex manifold. By solving the related one-loop β-functions, we obtain a static solution so that by reduction of this solution to (1 + 3)-dimension, we obtain a static solution of Einstein equation where the matter sector is effectively interpreted as an inhomogeneous, anisotropic and barotropic matter satisfying all the energy conditions. Finally, the T-dual background of the solution is investigated and it is shown that the duality transformation and reduction processes commute with each other.
Doctoral Assistants = Critical Friends: A Simple yet Complex Equation
ERIC Educational Resources Information Center
Hay, John; Laguerre, Fabrice; Moore, Eric; Reedy, Katherine; Rose, Scott; Vickers, Jerome
2015-01-01
The Carnegie Project on the Education Doctorate (CPED) encourages doctoral candidates volunteering in order to give back and continue their relationship with the university after completing their dissertation. Volunteering can take on many forms, from acting as doctoral assistants to performing the role of critical friends on future doctoral…
Complex Oscillations in the Delayed FitzHugh-Nagumo Equation
NASA Astrophysics Data System (ADS)
Krupa, Maciej; Touboul, Jonathan D.
2016-02-01
Motivated by the dynamics of neuronal responses, we analyze the dynamics of the FitzHugh-Nagumo slow-fast system with delayed self-coupling. This system provides a canonical example of a canard explosion for sufficiently small delays. Beyond this regime, delays significantly enrich the dynamics, leading to mixed-mode oscillations, bursting and chaos. These behaviors emerge from a delay-induced subcritical Bogdanov-Takens instability arising at the fold points of the S-shaped critical manifold. Underlying the transition from canard-induced to delay-induced dynamics is an abrupt switch in the nature of the Hopf bifurcation.
Testing Complex Correlational Hypotheses with Structural Equation Models
ERIC Educational Resources Information Center
Preacher, Kristopher J.
2006-01-01
It is often of interest to estimate partial or semipartial correlation coefficients as indexes of the linear association between 2 variables after partialing one or both for the influence of covariates. Squaring these coefficients expresses the proportion of variance in 1 variable explained by the other variable after controlling for covariates.…
Synchronization with propagation - The functional differential equations
NASA Astrophysics Data System (ADS)
Rǎsvan, Vladimir
2016-06-01
The structure represented by one or several oscillators couple to a one-dimensional transmission environment (e.g. a vibrating string in the mechanical case or a lossless transmission line in the electrical case) turned to be attractive for the research in the field of complex structures and/or complex behavior. This is due to the fact that such a structure represents some generalization of various interconnection modes with lumped parameters for the oscillators. On the other hand the lossless and distortionless propagation along transmission lines has generated several research in electrical, thermal, hydro and control engineering leading to the association of some functional differential equations to the basic initial boundary value problems. The present research is performed at the crossroad of the aforementioned directions. We shall associate to the starting models some functional differential equations - in most cases of neutral type - and make use of the general theorems for existence and stability of forced oscillations for functional differential equations. The challenges introduced by the analyzed problems for the general theory are emphasized, together with the implication of the results for various applications.
Iterative solution of the Helmholtz equation
Larsson, E.; Otto, K.
1996-12-31
We have shown that the numerical solution of the two-dimensional Helmholtz equation can be obtained in a very efficient way by using a preconditioned iterative method. We discretize the equation with second-order accurate finite difference operators and take special care to obtain non-reflecting boundary conditions. We solve the large, sparse system of equations that arises with the preconditioned restarted GMRES iteration. The preconditioner is of {open_quotes}fast Poisson type{close_quotes}, and is derived as a direct solver for a modified PDE problem.The arithmetic complexity for the preconditioner is O(n log{sub 2} n), where n is the number of grid points. As a test problem we use the propagation of sound waves in water in a duct with curved bottom. Numerical experiments show that the preconditioned iterative method is very efficient for this type of problem. The convergence rate does not decrease dramatically when the frequency increases. Compared to banded Gaussian elimination, which is a standard solution method for this type of problems, the iterative method shows significant gain in both storage requirement and arithmetic complexity. Furthermore, the relative gain increases when the frequency increases.
Laplace's equation on convex polyhedra via the unified method
Ashton, A. C. L.
2015-01-01
We provide a new method to study the classical Dirichlet problem for Laplace's equation on a convex polyhedron. This new approach was motivated by Fokas’ unified method for boundary value problems. The central object in this approach is the global relation: an integral equation which couples the known boundary data and the unknown boundary values. This integral equation depends holomorphically on two complex parameters, and the resulting analysis takes place on a Banach space of complex analytic functions closely related to the classical Paley–Wiener space. We write the global relation in the form of an operator equation and prove that the relevant operator is bounded below using some novel integral identities. We give a new integral representation to the solution to the underlying boundary value problem which serves as a concrete realization of the fundamental principle of Ehrenpreis.
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.
Multigrid shallow water equations on an FPGA
NASA Astrophysics Data System (ADS)
Jeffress, Stephen; Duben, Peter; Palmer, Tim
2015-04-01
A novel computing technology for multigrid shallow water equations is investigated. As power consumption begins to constrain traditional supercomputing advances, weather and climate simulators are exploring alternative technologies that achieve efficiency gains through massively parallel and low power architectures. In recent years FPGA implementations of reduced complexity atmospheric models have shown accelerated speeds and reduced power consumption compared to multi-core CPU integrations. We continue this line of research by designing an FPGA dataflow engine for a mulitgrid version of the 2D shallow water equations. The multigrid algorithm couples grids of variable resolution to improve accuracy. We show that a significant reduction of precision in the floating point representation of the fine grid variables allows greater parallelism and thus improved overall peformance while maintaining accurate integrations. Preliminary designs have been constructed by software emulation. Results of the hardware implementation will be presented at the conference.
A New Low Dissipative High Order Schemes for MHD Equations
NASA Technical Reports Server (NTRS)
Yee, H. C.; Sjoegreen, Bjoern; Mansour, Nagi (Technical Monitor)
2002-01-01
The goal of this talk is to extend our recently developed highly parallelizable nonlinear stable high order schemes for complex multiscale hydrodynamic applications to the viscous MHD equations. These schemes employed multiresolution wavelets as adaptive numerical dissipation controls to limit the amount and to aid the selection and/or blending of the appropriate types of dissipation to be used. The new scheme is formulated for both the conservative and non-conservative form of the MHD equations in curvilinear grids.
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.
Complex-extended Bohmian mechanics.
Chou, Chia-Chun; Wyatt, Robert E
2010-04-01
Complex-extended Bohmian mechanics is investigated by analytically continuing the wave function in polar form into the complex plane. We derive the complex-extended version of the quantum Hamilton-Jacobi equation and the continuity equation in Bohmian mechanics. Complex-extended Bohmian mechanics recovers the standard real-valued Bohmian mechanics on the real axis. The trajectories on the real axis are in accord with the standard real-valued Bohmian trajectories. The trajectories launched away from the real axis never intersect the real axis, and they display symmetry with respect to the real axis. Trajectories display hyperbolic deflection around nodes of the wave function in the complex plane. PMID:20387916
Infinite hierarchy of nonlinear Schrödinger equations and their solutions
NASA Astrophysics Data System (ADS)
Ankiewicz, A.; Kedziora, D. J.; Chowdury, A.; Bandelow, U.; Akhmediev, N.
2016-01-01
We study the infinite integrable nonlinear Schrödinger equation hierarchy beyond the Lakshmanan-Porsezian-Daniel equation which is a particular (fourth-order) case of the hierarchy. In particular, we present the generalized Lax pair and generalized soliton solutions, plane wave solutions, Akhmediev breathers, Kuznetsov-Ma breathers, periodic solutions, and rogue wave solutions for this infinite-order hierarchy. We find that "even- order" equations in the set affect phase and "stretching factors" in the solutions, while "odd-order" equations affect the velocities. Hence odd-order equation solutions can be real functions, while even-order equation solutions are always complex.
Infinite hierarchy of nonlinear Schrödinger equations and their solutions.
Ankiewicz, A; Kedziora, D J; Chowdury, A; Bandelow, U; Akhmediev, N
2016-01-01
We study the infinite integrable nonlinear Schrödinger equation hierarchy beyond the Lakshmanan-Porsezian-Daniel equation which is a particular (fourth-order) case of the hierarchy. In particular, we present the generalized Lax pair and generalized soliton solutions, plane wave solutions, Akhmediev breathers, Kuznetsov-Ma breathers, periodic solutions, and rogue wave solutions for this infinite-order hierarchy. We find that "even- order" equations in the set affect phase and "stretching factors" in the solutions, while "odd-order" equations affect the velocities. Hence odd-order equation solutions can be real functions, while even-order equation solutions are always complex. PMID:26871072
Convergence of a random walk method for the Burgers equation
Roberts, S.
1985-10-01
In this paper we consider a random walk algorithm for the solution of Burgers' equation. The algorithm uses the method of fractional steps. The non-linear advection term of the equation is solved by advecting ''fluid'' particles in a velocity field induced by the particles. The diffusion term of the equation is approximated by adding an appropriate random perturbation to the positions of the particles. Though the algorithm is inefficient as a method for solving Burgers' equation, it does model a similar method, the random vortex method, which has been used extensively to solve the incompressible Navier-Stokes equations. The purpose of this paper is to demonstrate the strong convergence of our random walk method and so provide a model for the proof of convergence for more complex random walk algorithms; for instance, the random vortex method without boundaries.
Finite-difference solutions of the 3-D eikonal equation
Fei, Tong; Fehler, M.C.; Hildebrand, S.T.
1995-12-31
Prestack Kirchhoff depth migration requires the computation of traveltimes from surface source and receiver locations to subsurface image locations. In 3-D problems, computational efficiency becomes important. Finite-difference solutions of the eikonal equation provide computationally efficient methods for generating the traveltime information. Here, a novel finite-difference solutions of the eikonal equation provide computationally efficient methods for generating the traveltime information. Here, a novel finite-difference method for computing the first arrival traveltime by solving the eikonal equation has been developed in Cartesian coordinates. The method, which is unconditionally stable and computationally efficient, can handle instabilities due to caustics and provide information about head waves. The comparison of finite-difference solutions of the acoustic wave equation with the traveltime solutions from the eikonal equation in various structure models demonstrate that the method developed here can provide correct first arrival traveltime information even in areas of complex velocity structure.
NASA Astrophysics Data System (ADS)
Yao, Ruo-Xia; Wang, Wei; Chen, Ting-Hua
2014-11-01
Motivated by the widely used ansätz method and starting from the modified Riemann—Liouville derivative together with a fractional complex transformation that can be utilized to transform nonlinear fractional partial differential equations to nonlinear ordinary differential equations, new types of exact traveling wave solutions to three important nonlinear space- and time-fractional partial differential equations are obtained simultaneously in terms of solutions of a Riccati equation. The results are new and first reported in this paper.
NASA Technical Reports Server (NTRS)
Mather, John C.
2012-01-01
neutrons, liberating a little energy and creating complexity. Then, the expanding universe cooled some more, and neutrons and protons, no longer kept apart by immense temperatures, found themselves unstable and formed helium nuclei. Then, a little more cooling, and atomic nuclei and electrons were no longer kept apart, and the universe became transparent. Then a little more cooling, and the next instability began: gravitation pulled matter together across cosmic distances to form stars and galaxies. This instability is described as a "negative heat capadty" in which extracting energy from a gravitating system makes it hotter -- clearly the 2nd law of thermodynamics does not apply here! (This is the physicist's part of the answer to e e cummings' question: what is the wonder that's keeping the stars apart?) Then, the next instability is that hydrogen and helium nuclei can fuse together to release energy and make stars burn for billions of years. And then at the end of the fuel source, stars become unstable and explode and liberate the chemical elements back into space. And because of that, on planets like Earth, sustained energy flows support the development of additional instabilities and all kinds of complex patterns. Gravitational instability pulls the densest materials into the core of the Earth, leaving a thin skin of water and air, and makes the interior churn incessantly as heat flows outwards. And the heat from the sun, received mostly near the equator and flowing towards the poles, supports the complex atmospheric and oceanic circulations. And because or that, the physical Earth is full of natural chemical laboratories, concentrating elements here, mixing them there, raising and lowering temperatures, ceaselessly experimenting with uncountable events where new instabilities can arise. At least one of them was the new experiment called life. Now that we know that there are at least as many planets as there are stars, it is hard to imagine that nature's ceasess
NASA Technical Reports Server (NTRS)
Mather, John C.
2012-01-01
neutrons, liberating a little energy and creating complexity. Then, the expanding universe cooled some more, and neutrons and protons, no longer kept apart by immense temperatures, found themselves unstable and formed helium nuclei. Then, a little more cooling, and atomic nuclei and electrons were no longer kept apart, and the universe became transparent. Then a little more cooling, and the next instability began: gravitation pulled matter together across cosmic distances to form stars and galaxies. This instability is described as a "negative heat capadty" in which extracting energy from a gravitating system makes it hotter -- clearly the 2nd law of thermodynamics does not apply here! (This is the physicist's part of the answer to e e cummings' question: what is the wonder that's keeping the stars apart?) Then, the next instability is that hydrogen and helium nuclei can fuse together to release energy and make stars burn for billions of years. And then at the end of the fuel source, stars become unstable and explode and liberate the chemical elements back into space. And because of that, on planets like Earth, sustained energy flows support the development of additional instabilities and all kinds of complex patterns. Gravitational instability pulls the densest materials into the core of the Earth, leaving a thin skin of water and air, and makes the interior churn incessantly as heat flows outwards. And the heat from the sun, received mostly near the equator and flowing towards the poles, supports the complex atmospheric and oceanic circulations. And because or that, the physical Earth is full of natural chemical laboratories, concentrating elements here, mixing them there, raising and lowering temperatures, ceaselessly experimenting with uncountable events where new instabilities can arise. At least one of them was the new experiment called life. Now that we know that there are at least as many planets as there are stars, it is hard to imagine that nature's ceasess
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).
Variational principles for eigenvalues of the Klein-Gordon equation
Langer, Matthias; Tretter, Christiane
2006-10-15
In this paper variational principles for eigenvalues of an abstract model of the Klein-Gordon equation with electromagnetic potential are established. They are used to characterize and estimate eigenvalues in cases where the essential spectrum has a gap around 0, even in the presence of complex eigenvalues. As a consequence, a comparison between eigenvalues of the Klein-Gordon equation in R{sup d} and eigenvalues of certain Schroedinger operators is obtained. The results are illustrated on examples including the Klein-Gordon equation with Coulomb and square-well potential.
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.
A transport equation for eddy viscosity
NASA Technical Reports Server (NTRS)
Durbin, P. A.; Yang, Z.
1992-01-01
A transport equation for eddy viscosity is proposed for wall bounded turbulent flows. The proposed model reduces to a quasi-homogeneous form far from surfaces. Near to a surface, the nonhomogeneous effect of the wall is modeled by an elliptic relaxation model. All the model terms are expressed in local variables and are coordinate independent; the model is intended to be used in complex flows. Turbulent channel flow and turbulent boundary layer flows with/without pressure gradient are calculated using the present model. Comparisons between model calculations and direct numerical simulation or experimental data show good agreement.
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.
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.
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…
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.
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…
Difference methods for stiff delay differential equations. [DDESUB, in FORTRAN
Roth, Mitchell G.
1980-12-01
Delay differential equations of the form y'(t) = f(y(t), z(t)), where z(t) = (y/sub 1/(..cap alpha../sub 1/(y(t))),..., y/sub n/(..cap alpha../sub n/(y(t))))/sup T/ and ..cap alpha../sub i/(y(t)) less than or equal to t, arise in many scientific and engineering fields when transport lags and propagation times are physically significant in a dynamic process. Difference methods for approximating the solution of stiff delay systems require special stability properties that are generalizations of those employed for stiff ordinary differential equations. By use of the model equation y'(t) = py(t) + qy(t-1), with complex p and q, the definitions of A-stability, A( )-stability, and stiff stability have been generalize to delay equations. For linear multistep difference formulas, these properties extend directly from ordinary to delay equations. This straight forward extension is not true for implicit Runge-Kutta methods, as illustrated by the midpoint formula, which is A-stable for ordinary equations, but not for delay equations. A computer code for stiff delay equations was developed using the BDF. 24 figures, 5 tables.
Orbital stability of solitary waves for Kundu equation
NASA Astrophysics Data System (ADS)
Zhang, Weiguo; Qin, Yinghao; Zhao, Yan; Guo, Boling
In this paper, we consider the Kundu equation which is not a standard Hamiltonian system. The abstract orbital stability theory proposed by Grillakis et al. (1987, 1990) cannot be applied directly to study orbital stability of solitary waves for this equation. Motivated by the idea of Guo and Wu (1995), we construct three invariants of motion and use detailed spectral analysis to obtain orbital stability of solitary waves for Kundu equation. Since Kundu equation is more complex than the derivative Schrödinger equation, we utilize some techniques to overcome some difficulties in this paper. It should be pointed out that the results obtained in this paper are more general than those obtained by Guo and Wu (1995). We present a sufficient condition under which solitary waves are orbitally stable for 2c+sυ<0, while Guo and Wu (1995) only considered the case 2c+sυ>0. We obtain the results on orbital stability of solitary waves for the derivative Schrödinger equation given by Colin and Ohta (2006) as a corollary in this paper. Furthermore, we obtain orbital stability of solitary waves for Chen-Lee-Lin equation and Gerdjikov-Ivanov equation, respectively.
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.
A New Reynolds Stress Algebraic Equation Model
NASA Technical Reports Server (NTRS)
Shih, Tsan-Hsing; Zhu, Jiang; Lumley, John L.
1994-01-01
A general turbulent constitutive relation is directly applied to propose a new Reynolds stress algebraic equation model. In the development of this model, the constraints based on rapid distortion theory and realizability (i.e. the positivity of the normal Reynolds stresses and the Schwarz' inequality between turbulent velocity correlations) are imposed. Model coefficients are calibrated using well-studied basic flows such as homogeneous shear flow and the surface flow in the inertial sublayer. The performance of this model is then tested in complex turbulent flows including the separated flow over a backward-facing step and the flow in a confined jet. The calculation results are encouraging and point to the success of the present model in modeling turbulent flows with complex geometries.
Acoustic wave-equation-based earthquake location
NASA Astrophysics Data System (ADS)
Tong, Ping; Yang, Dinghui; Liu, Qinya; Yang, Xu; Harris, Jerry
2016-04-01
We present a novel earthquake location method using acoustic wave-equation-based traveltime inversion. The linear relationship between the location perturbation (δt0, δxs) and the resulting traveltime residual δt of a particular seismic phase, represented by the traveltime sensitivity kernel K(t0, xs) with respect to the earthquake location (t0, xs), is theoretically derived based on the adjoint method. Traveltime sensitivity kernel K(t0, xs) is formulated as a convolution between the forward and adjoint wavefields, which are calculated by numerically solving two acoustic wave equations. The advantage of this newly derived traveltime kernel is that it not only takes into account the earthquake-receiver geometry but also accurately honours the complexity of the velocity model. The earthquake location is obtained by solving a regularized least-squares problem. In 3-D realistic applications, it is computationally expensive to conduct full wave simulations. Therefore, we propose a 2.5-D approach which assumes the forward and adjoint wave simulations within a 2-D vertical plane passing through the earthquake and receiver. Various synthetic examples show the accuracy of this acoustic wave-equation-based earthquake location method. The accuracy and efficiency of the 2.5-D approach for 3-D earthquake location are further verified by its application to the 2004 Big Bear earthquake in Southern California.
Dispersion in tidally averaged transport equation
Cheng, R.T.; Casulli, V.
1992-01-01
A general governing inter-tidal transport equation for conservative solutes has been derived without invoking the weakly nonlinear approximation. The governing inter-tidal transport equation is a convection-dispersion equation in which the convective velocity is a mean Lagrangian residual current, and the inter-tidal dispersion coefficient is defined by a dispersion patch. When the weakly nonlinear condition is violated, the physical significance of the Stokes' drift, as used in tidal dynamics, becomes questionable. For nonlinear problems, analytical solutions for the mean Lagrangian residual current and for the inter-tidal dispersion coefficient do not exist, they must be determined numerically. A rectangular tidal inlet with a constriction is used in the first example. The solutions of the residual currents and the computed properties of the inter-tidal dispersion coefficient are used to illuminate the mechanisms of the inter-tidal transport processes. Then, the present formulation is tested in a geometrically complex tidal estuary – San Francisco Bay, California. The computed inter-tidal dispersion coefficients are in the range between 5×104 and 5×106 cm2/sec., which are consistent with the values reported in the literature
Linear stochastic degenerate Sobolev equations and applications†
NASA Astrophysics Data System (ADS)
Liaskos, Konstantinos B.; Pantelous, Athanasios A.; Stratis, Ioannis G.
2015-12-01
In this paper, a general class of linear stochastic degenerate Sobolev equations with additive noise is considered. This class of systems is the infinite-dimensional analogue of linear descriptor systems in finite dimensions. Under appropriate assumptions, the mild and strong well-posedness for the initial value problem are studied using elements of the semigroup theory and properties of the stochastic convolution. The final value problem is also examined and it is proved that this is uniquely strongly solvable and the solution is continuously dependent on the final data. Based on the results of the forward and backward problem, the conditions for the exact controllability are investigated for a special but important class of these equations. The abstract results are illustrated by applications in complex media electromagnetics, in the one-dimensional stochastic Dirac equation in the non-relativistic limit and in a potential application in input-output analysis in economics. Dedicated to Professor Grigoris Kalogeropoulos on the occasion of his seventieth birthday.
Mickens, R.E.
1997-12-12
The major thrust of this proposal was to continue our investigations of so-called non-standard finite-difference schemes as formulated by other authors. These schemes do not follow the standard rules used to model continuous differential equations by discrete difference equations. The two major aspects of this procedure consist of generalizing the definition of the discrete derivative and using a nonlocal model (on the computational grid or lattice) for nonlinear terms that may occur in the differential equations. Our aim was to investigate the construction of nonstandard finite-difference schemes for several classes of ordinary and partial differential equations. These equations are simple enough to be tractable, yet, have enough complexity to be both mathematically and scientifically interesting. It should be noted that all of these equations differential equations model some physical phenomena under an appropriate set of experimental conditions. The major goal of the project was to better understand the process of constructing finite-difference models for differential equations. In particular, it demonstrates the value of using nonstandard finite-difference procedures. A secondary goal was to construct and study a variety of analytical techniques that can be used to investigate the mathematical properties of the obtained difference equations. These mathematical procedures are of interest in their own right and should be a valuable contribution to the mathematics research literature in difference equations. All of the results obtained from the research done under this project have been published in the relevant research/technical journals or submitted for publication. Our expectation is that these results will lead to improved finite difference schemes for the numerical integration of both ordinary and partial differential equations. Section G of the Appendix gives a concise summary of the major results obtained under funding by the grant.
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.
ERIC Educational Resources Information Center
Gordon, Sandra L.; Anderson, Beth C.
To determine whether consensus existed among teachers about the complexity of common classroom materials, a survey was administered to 66 pre-service and in-service kindergarten and prekindergarten teachers. Participants were asked to rate 14 common classroom materials as simple, complex, or super-complex. Simple materials have one obvious part,…
A wave equation interpolating between classical and quantum mechanics
NASA Astrophysics Data System (ADS)
Schleich, W. P.; Greenberger, D. M.; Kobe, D. H.; Scully, M. O.
2015-10-01
We derive a ‘master’ wave equation for a family of complex-valued waves {{Φ }}\\equiv R{exp}[{{{i}}S}({cl)}/{{\\hbar }}] whose phase dynamics is dictated by the Hamilton-Jacobi equation for the classical action {S}({cl)}. For a special choice of the dynamics of the amplitude R which eliminates all remnants of classical mechanics associated with {S}({cl)} our wave equation reduces to the Schrödinger equation. In this case the amplitude satisfies a Schrödinger equation analogous to that of a charged particle in an electromagnetic field where the roles of the scalar and the vector potentials are played by the classical energy and the momentum, respectively. In general this amplitude is complex and thereby creates in addition to the classical phase {S}({cl)}/{{\\hbar }} a quantum phase. Classical statistical mechanics, as described by a classical matter wave, follows from our wave equation when we choose the dynamics of the amplitude such that it remains real for all times. Our analysis shows that classical and quantum matter waves are distinguished by two different choices of the dynamics of their amplitudes rather than two values of Planck’s constant. We dedicate this paper to the memory of Richard Lewis Arnowitt—a pioneer of many-body theory, a path finder at the interface of gravity and quantum mechanics, and a true leader in non-relativistic and relativistic quantum field theory.
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.
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.
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.
NASA Astrophysics Data System (ADS)
Kravchenko, Vladislav V.
2005-01-01
Given a particular solution of a one-dimensional stationary Schrödinger equation this equation of second order can be reduced to a first-order linear ordinary differential equation. This is done with the aid of an auxiliary Riccati differential equation. In the present work we show that the same fact is true in a multidimensional situation also. For simplicity we consider the case of two or three independent variables. One particular solution of the stationary Schrödinger equation allows us to reduce this second-order equation to a linear first-order quaternionic differential equation. As in the one-dimensional case this is done with the aid of an auxiliary quaternionic Riccati equation. The resulting first-order quaternionic equation is equivalent to the static Maxwell system and is closely related to the Dirac equation. In the case of two independent variables it is the well-known Vekua equation from theory of pseudoanalytic (or generalized analytic) functions. Nevertheless, we show that even in this case it is very useful to consider not only complex valued functions, solutions of the Vekua equation, but complete quaternionic functions. In this way the first-order quaternionic equation represents two separate Vekua equations, one of which gives us solutions of the Schrödinger equation and the other one can be considered as an auxiliary equation of a simpler structure. Moreover for the auxiliary equation we always have the corresponding Bers generating pair (F, G), the base of the Bers theory of pseudoanalytic functions, and what is very important, the Bers derivatives of solutions of the auxiliary equation give us solutions of the main Vekua equation and as a consequence of the Schrödinger equation. Based on this fact we obtain an analogue of the Cauchy integral theorem for solutions of the stationary Schrödinger equation. Other results from theory of pseudoanalytic functions can be written for solutions of the Schrödinger equation. Moreover, for an ample
A multigrid preconditioner for the semiconductor equations
Meza, J.C.; Tuminaro, R.S.
1994-12-31
Currently, integrated circuits are primarily designed in a {open_quote}trial and error{close_quote} fashion. That is, prototypes are built and improved via experimentation and testing. In the near future, however, it may be possible to significantly reduce the time and cost of designing new devices by using computer simulations. To accurately perform these complex simulations in three dimensions, however, new algorithms and high performance computers are necessary. In this paper the authors discuss the use of multigrid preconditioning inside a semiconductor device modeling code, DANCIR. The DANCIR code is a full three-dimensional simulator capable of computing steady-state solutions of the drift-diffusion equations for a single semiconductor device and has been used to simulate a wide variety of different devices. At the inner core of DANCIR is a solver for the nonlinear equations that arise from the spatial discretization of the drift-diffusion equations on a rectangular grid. These nonlinear equations are resolved using Gummel`s method which requires three symmetric linear systems to be solved within each Gummel iteration. It is the resolution of these linear systems which comprises the dominant computational cost of this code. The original version of DANCIR uses a Cholesky preconditioned conjugate gradient algorithm to solve these linear systems. Unfortunately, this algorithm has a number of disadvantages: (1) it takes many iterations to converge (if it converges), (2) it can require a significant amount of computing time, and (3) it is not very parallelizable. To improve the situation, the authors consider a multigrid preconditioner. The multigrid method uses iterations on a hierarchy of grids to accelerate the convergence on the finest grid.
Maxwell’s Mixing Equation Revisited: Characteristic Impedance Equations for Ellipsoidal Cells
Stubbe, Marco; Gimsa, Jan
2015-01-01
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
Landscape evolution models: A review of their fundamental equations
NASA Astrophysics Data System (ADS)
Chen, Alex; Darbon, Jérôme; Morel, Jean-Michel
2014-08-01
This paper reviews the main physical laws proposed in landscape evolution models (LEMs). It discusses first the main partial differential equations involved in these models and their variants. These equations govern water runoff, stream incision, regolith-bedrock interaction, hillslope evolution, and sedimentation. A synthesis of existing LEMs is proposed. It proposes three models with growing complexity and with a growing number of components: two-equation models with only two components, governing water and bedrock evolution; three-equation models with three components where water, bedrock, and sediment interact; and finally models with four equations and four interacting components, namely water, bedrock, suspended sediment, and regolith. This analysis is not a mere compilation of existing LEMs. It attempts at giving the simplest and most general physically consistent set of equations, coping with all requirements stated in LEMs and LEM software. Three issues are in particular addressed and hopefully resolved. The first one is a correct formulation of the water transport equation down slopes. A general formulation for this equation is proposed, coping not only with the simplest form computing the drainage area but also with a sound energy dissipation argument associated with the Saint-Venant shallow water equations. The second issue arises from the coexistence of two competing modes, namely the detachment-limited erosion mode on hillslopes, and the transport-limited sediment transport on river beds. The third issue (linked to the second) is the fact that no conservation law is available for material in these two modes. A simple solution proposed to resolve these issues is the introduction, as suggested by several authors, of an additional variable for suspended sediment load in water. With only three variables and three equations, the above-mentioned contradictions seem to be eliminated. Several numerical experiments on real digital elevation models (DEMs
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.
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)
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...
Infinitesimal generators and the Loewner equation on complete hyperbolic manifolds
NASA Astrophysics Data System (ADS)
Arosio, Leandro; Bracci, Filippo
2011-12-01
We characterize infinitesimal generators on complete hyperbolic complex manifolds without any regularity assumption on the Kobayashi distance. This allows to prove a general Loewner type equation with regularity of any order {d in [1, +infty]} . Finally, based on these results, we focus on some open problems naturally arising.
Multigrid and cyclic reduction applied to the Helmholtz equation
NASA Technical Reports Server (NTRS)
Brackenridge, Kenneth
1993-01-01
We consider the Helmholtz equation with a discontinuous complex parameter and inhomogeneous Dirichlet boundary conditions in a rectangular domain. A variant of the direct method of cyclic reduction (CR) is employed to facilitate the design of improved multigrid (MG) components, resulting in the method of CR-MG. We demonstrate the improved convergence properties of this method.
People Are Variables Too: Multilevel Structural Equations Modeling
ERIC Educational Resources Information Center
Mehta, Paras D.; Neale, Michael C.
2005-01-01
The article uses confirmatory factor analysis (CFA) as a template to explain didactically multilevel structural equation models (ML-SEM) and to demonstrate the equivalence of general mixed-effects models and ML-SEM. An intuitively appealing graphical representation of complex ML-SEMs is introduced that succinctly describes the underlying model and…
Conservation laws, differential identities, and constraints of partial differential equations
NASA Astrophysics Data System (ADS)
Zharinov, V. V.
2015-11-01
We consider specific cohomological properties such as low-dimensional conservation laws and differential identities of systems of partial differential equations (PDEs). We show that such properties are inherent to complex systems such as evolution systems with constraints. The mathematical tools used here are the algebraic analysis of PDEs and cohomologies over differential algebras and modules.
State equations for an n-body spacecraft
NASA Technical Reports Server (NTRS)
Larson, V.
1974-01-01
Considerable attention, in the open literature, is being focused on the problem of developing a suitable set of deterministic dynamical equations for a complex spacecraft. The present paper addresses the problem of determining a set of state equations for an n-body spacecraft. The approach used in obtaining the state equations involves the application and interpretation of advanced dynamical principles. This set of state equations can be effectively used in the development of a stochastic controller for the spacecraft (in this latter development, the deterministic model developed in the present paper will be appropriately corrupted by plant noise). The major effort in the paper revolves around the determination of the plant matrices and the specification of the state vector and the control vector.
A differential equation for the Generalized Born radii.
Fogolari, Federico; Corazza, Alessandra; Esposito, Gennaro
2013-06-28
The Generalized Born (GB) model offers a convenient way of representing electrostatics in complex macromolecules like proteins or nucleic acids. The computation of atomic GB radii is currently performed by different non-local approaches involving volume or surface integrals. Here we obtain a non-linear second-order partial differential equation for the Generalized Born radius, which may be solved using local iterative algorithms. The equation is derived under the assumption that the usual GB approximation to the reaction field obeys Laplace's equation. The equation admits as particular solutions the correct GB radii for the sphere and the plane. The tests performed on a set of 55 different proteins show an overall agreement with other reference GB models and "perfect" Poisson-Boltzmann based values. PMID:23676843
Rate equations for ruby and alexandrite Q-switched lasers
NASA Astrophysics Data System (ADS)
Sulc, Jan; Jelinkova, Helena
2003-07-01
To have a complex view on giant pulse generation, a more precise computer model of the build up Q-switch pulse in solid-state laser was realized. As a time starting point of the rate equation calculation, the moment of a flashlamp trigger was chosen. A system of three or four main differential rate equations describes the energy transfer from a pumping source - capacitor to an output giant pulse. Two laser active media, i.e. ruby and alexandrite, were examined with this model. A passive Q-switch ruby laser needs solving of the system of four differential equations; three differential equations gave the computer results for electro-optically Q-switched alexandrite laser.
Spectral Decomposition of a Fokker-Planck Equation at Criticality
NASA Astrophysics Data System (ADS)
Bologna, M.; Beig, M. T.; Svenkeson, A.; Grigolini, P.; West, B. J.
2015-07-01
The mean field for a complex network consisting of a large but finite number of random two-state elements, , has been shown to satisfy a nonlinear Langevin equation. The noise intensity is inversely proportional to . In the limiting case , the solution to the Langevin equation exhibits a transition from exponential to inverse power law relaxation as criticality is approached from above or below the critical point. When , the inverse power law is truncated by an exponential decay with rate , the evaluation of which is the main purpose of this article. An analytic/numeric approach is used to obtain the lowest-order eigenvalues in the spectral decomposition of the solution to the corresponding Fokker-Planck equation and its equivalent Schrödinger equation representation.
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.
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
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.
NASA Technical Reports Server (NTRS)
Walthall, C. L.; Norman, J. M.; Blad, B. L.; Welles, J. M.; Campbell, G.
1985-01-01
A simple equation has been developed for describing the bidirectional reflectance of some vegetative canopies and bare soil surfaces. The equation describes directional reflectance as a function of zenith and azimuth view angles and solar azimuth angle. The equation works for simulated and field measured red and IR reflectance under clear sky conditions. Hemispherical reflectance can be calculated as a function of the simple equation coefficients by integrating the equation over the hemisphere of view angles. A single equation for estimating soil bidirectional reflectance was obtained using the relationships between solar zenith angles and the simple equation coefficients for medium and rough soil distributions. The equation has many useful applications such as providing a lower level boundary condition in complex plant canopy models and providing an additional tool for studying bidirectional effects on pointable sensors.
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.
Reck, Kasper; Thomsen, Erik V; Hansen, Ole
2011-01-31
The scalar wave equation, or Helmholtz equation, describes within a certain approximation the electromagnetic field distribution in a given system. In this paper we show how to solve the Helmholtz equation in complex geometries using conformal mapping and the homotopy perturbation method. The solution of the mapped Helmholtz equation is found by solving an infinite series of Poisson equations using two dimensional Fourier series. The solution is entirely based on analytical expressions and is not mesh dependent. The analytical results are compared to a numerical (finite element method) solution. PMID:21368995
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
Restricted Complexity Framework for Nonlinear Adaptive Control in Complex Systems
Williams, Rube B.
2004-02-04
Control law adaptation that includes implicit or explicit adaptive state estimation, can be a fundamental underpinning for the success of intelligent control in complex systems, particularly during subsystem failures, where vital system states and parameters can be impractical or impossible to measure directly. A practical algorithm is proposed for adaptive state filtering and control in nonlinear dynamic systems when the state equations are unknown or are too complex to model analytically. The state equations and inverse plant model are approximated by using neural networks. A framework for a neural network based nonlinear dynamic inversion control law is proposed, as an extrapolation of prior developed restricted complexity methodology used to formulate the adaptive state filter. Examples of adaptive filter performance are presented for an SSME simulation with high pressure turbine failure to support extrapolations to adaptive control problems.
Restricted Complexity Framework for Nonlinear Adaptive Control in Complex Systems
NASA Astrophysics Data System (ADS)
Williams, Rube B.
2004-02-01
Control law adaptation that includes implicit or explicit adaptive state estimation, can be a fundamental underpinning for the success of intelligent control in complex systems, particularly during subsystem failures, where vital system states and parameters can be impractical or impossible to measure directly. A practical algorithm is proposed for adaptive state filtering and control in nonlinear dynamic systems when the state equations are unknown or are too complex to model analytically. The state equations and inverse plant model are approximated by using neural networks. A framework for a neural network based nonlinear dynamic inversion control law is proposed, as an extrapolation of prior developed restricted complexity methodology used to formulate the adaptive state filter. Examples of adaptive filter performance are presented for an SSME simulation with high pressure turbine failure to support extrapolations to adaptive control problems.
Sibling Curves 3: Imaginary Siblings and Tracing Complex Roots
ERIC Educational Resources Information Center
Harding, Ansie; Engelbrecht, Johann
2009-01-01
Visualizing complex roots of a quadratic equation has been a quest since the inception of the Argand plane in the 1800s. Many algebraic and numerical methods exist for calculating complex roots of an equation, but few visual methods exist. Following on from papers by Harding and Engelbrecht (A. Harding and J. Engelbrecht, "Sibling curves and…
Trajectory approach to the Schrödinger-Langevin equation with linear dissipation for ground states
NASA Astrophysics Data System (ADS)
Chou, Chia-Chun
2015-11-01
The Schrödinger-Langevin equation with linear dissipation is integrated by propagating an ensemble of Bohmian trajectories for the ground state of quantum systems. Substituting the wave function expressed in terms of the complex action into the Schrödinger-Langevin equation yields the complex quantum Hamilton-Jacobi equation with linear dissipation. We transform this equation into the arbitrary Lagrangian-Eulerian version with the grid velocity matching the flow velocity of the probability fluid. The resulting equation is simultaneously integrated with the trajectory guidance equation. Then, the computational method is applied to the harmonic oscillator, the double well potential, and the ground vibrational state of methyl iodide. The excellent agreement between the computational and the exact results for the ground state energies and wave functions shows that this study provides a synthetic trajectory approach to the ground state of quantum systems.
On the Dynamics of Some Discretizations of Convection-Diffusion Equations
NASA Technical Reports Server (NTRS)
Sweby, Peter K.; Yee, H. C.; Rai, Man Mohan (Technical Monitor)
1995-01-01
Numerical discretizations of differential equations which model physical processes can possess dynamics quite different from that of the equations themselves. Recently the emphasis has been on the the dynamics of numerical discretizations for Ordinary Differential Equations (ODEs). For Partial Differential Equations (PDEs) using a method of lines approach the situation is more complex. First, the spatial discretisation may introduce dynamics not present in the original equations; second, the solution of the resulting system of ODEs is open to the modified dynamics of the ODE solver used. These two effects may interact in a complex manner. In this talk we present some results of our recent work on the dynamics of discretizations of convection-diffusion equations, including those produced using Total Variation Diminishing (TVD) schemes and adaptive grid techniques. A more general overview of the area may be found on our accompanying poster presentation.
Perturbations of linear delay differential equations at the verge of instability
NASA Astrophysics Data System (ADS)
Lingala, N.; Namachchivaya, N. Sri
2016-06-01
The characteristic equation for a linear delay differential equation (DDE) has countably infinite roots on the complex plane. This paper considers linear DDEs that are on the verge of instability, i.e., a pair of roots of the characteristic equation lies on the imaginary axis of the complex plane and all other roots have negative real parts. It is shown that when small noise perturbations are present, the probability distribution of the dynamics can be approximated by the probability distribution of a certain one-dimensional stochastic differential equation (SDE) without delay. This is advantageous because equations without delay are easier to simulate and one-dimensional SDEs are analytically tractable. When the perturbations are also linear, it is shown that the stability depends on a specific complex number. The theory is applied to study oscillators with delayed feedback. Some errors in other articles that use multiscale approach are pointed out.
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.
NASA Astrophysics Data System (ADS)
Battiston, Stefano; Caldarelli, Guido; Georg, Co-Pierre; May, Robert; Stiglitz, Joseph
2013-03-01
The intrinsic complexity of the financial derivatives market has emerged as both an incentive to engage in it, and a key source of its inherent instability. Regulators now faced with the challenge of taming this beast may find inspiration in the budding science of complex systems.
ERIC Educational Resources Information Center
Glanville, Ranulph
2007-01-01
This article considers the nature of complexity and design, as well as relationships between the two, and suggests that design may have much potential as an approach to improving human performance in situations seen as complex. It is developed against two backgrounds. The first is a world view that derives from second order cybernetics and radical…
Efficient traveltime solutions of the acoustic TI eikonal equation
NASA Astrophysics Data System (ADS)
Waheed, Umair bin; Alkhalifah, Tariq; Wang, Hui
2015-02-01
Numerical solutions of the eikonal (Hamilton-Jacobi) equation for transversely isotropic (TI) media are essential for imaging and traveltime tomography applications. Such solutions, however, suffer from the inherent higher-order nonlinearity of the TI eikonal equation, which requires solving a quartic polynomial for every grid point. Analytical solutions of the quartic polynomial yield numerically unstable formulations. Thus, it requires a numerical root finding algorithm, adding significantly to the computational load. Using perturbation theory we approximate, in a first order discretized form, the TI eikonal equation with a series of simpler equations for the coefficients of a polynomial expansion of the eikonal solution, in terms of the anellipticity anisotropy parameter. Such perturbation, applied to the discretized form of the eikonal equation, does not impose any restrictions on the complexity of the perturbed parameter field. Therefore, it provides accurate traveltime solutions even for models with complex distribution of velocity and anisotropic anellipticity parameter, such as that for the complicated Marmousi model. The formulation allows for large cost reduction compared to using the direct TI eikonal solver. Furthermore, comparative tests with previously developed approximations illustrate remarkable gain in accuracy in the proposed algorithm, without any addition to the computational cost.
Efficient traveltime solutions of the acoustic TI eikonal equation
Waheed, Umair bin Alkhalifah, Tariq Wang, Hui
2015-02-01
Numerical solutions of the eikonal (Hamilton–Jacobi) equation for transversely isotropic (TI) media are essential for imaging and traveltime tomography applications. Such solutions, however, suffer from the inherent higher-order nonlinearity of the TI eikonal equation, which requires solving a quartic polynomial for every grid point. Analytical solutions of the quartic polynomial yield numerically unstable formulations. Thus, it requires a numerical root finding algorithm, adding significantly to the computational load. Using perturbation theory we approximate, in a first order discretized form, the TI eikonal equation with a series of simpler equations for the coefficients of a polynomial expansion of the eikonal solution, in terms of the anellipticity anisotropy parameter. Such perturbation, applied to the discretized form of the eikonal equation, does not impose any restrictions on the complexity of the perturbed parameter field. Therefore, it provides accurate traveltime solutions even for models with complex distribution of velocity and anisotropic anellipticity parameter, such as that for the complicated Marmousi model. The formulation allows for large cost reduction compared to using the direct TI eikonal solver. Furthermore, comparative tests with previously developed approximations illustrate remarkable gain in accuracy in the proposed algorithm, without any addition to the computational cost.
Solving the Inverse-Square Problem with Complex Variables
ERIC Educational Resources Information Center
Gauthier, N.
2005-01-01
The equation of motion for a mass that moves under the influence of a central, inverse-square force is formulated and solved as a problem in complex variables. To find the solution, the constancy of angular momentum is first established using complex variables. Next, the complex position coordinate and complex velocity of the particle are assumed…
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.
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
Generalized Flip-Flop Input Equations Based on a Four-Valued Boolean Algebra
NASA Technical Reports Server (NTRS)
Tucker, Jerry H.; Tapia, Moiez A.
1996-01-01
A procedure is developed for obtaining generalized flip-flop input equations, and a concise method is presented for representing these equations. The procedure is based on solving a four-valued characteristic equation of the flip-flop, and can encompass flip-flops that are too complex to approach intuitively. The technique is presented using Karnaugh maps, but could easily be implemented in software.
Darboux transformation for a generalized Hirota-Satsuma coupled Korteweg-de Vries equation
Geng Xianguo; Ren Hongfeng; He Guoliang
2009-05-15
A Darboux transformation for the generalized Hirota-Satsuma coupled Korteweg-de Vries (KdV) equation is derived with the aid of the gauge transformation between the corresponding 4x4 matrix spectral problems with three potentials, by which some explicit solutions of the generalized Hirota-Satsuma coupled KdV equation are constructed. As a reduction, a Darboux transformation of the complex coupled KdV equation and its explicit solutions are obtained.
Darboux transformation for a generalized Hirota-Satsuma coupled Korteweg-de Vries equation.
Geng, Xianguo; Ren, Hongfeng; He, Guoliang
2009-05-01
A Darboux transformation for the generalized Hirota-Satsuma coupled Korteweg-de Vries (KdV) equation is derived with the aid of the gauge transformation between the corresponding 4x4 matrix spectral problems with three potentials, by which some explicit solutions of the generalized Hirota-Satsuma coupled KdV equation are constructed. As a reduction, a Darboux transformation of the complex coupled KdV equation and its explicit solutions are obtained. PMID:19518577
IDSOLVER: A general purpose solver for nth-order integro-differential equations
NASA Astrophysics Data System (ADS)
Gelmi, Claudio A.; Jorquera, Héctor
2014-01-01
Many mathematical models of complex processes may be posed as integro-differential equations (IDE). Many numerical methods have been proposed for solving those equations, but most of them are ad hoc thus new equations have to be solved from scratch for translating the IDE into the framework of the specific method chosen. Furthermore, there is a paucity of general-purpose numerical solvers that free the user from additional tasks.
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.
The Davey-Stewartson Equation on the Half-Plane
NASA Astrophysics Data System (ADS)
Fokas, A. S.
2009-08-01
The Davey-Stewartson (DS) equation is a nonlinear integrable evolution equation in two spatial dimensions. It provides a multidimensional generalisation of the celebrated nonlinear Schrödinger (NLS) equation and it appears in several physical situations. The implementation of the Inverse Scattering Transform (IST) to the solution of the initial-value problem of the NLS was presented in 1972, whereas the analogous problem for the DS equation was solved in 1983. These results are based on the formulation and solution of certain classical problems in complex analysis, namely of a Riemann Hilbert problem (RH) and of either a d-bar or a non-local RH problem respectively. A method for solving the mathematically more complicated but physically more relevant case of boundary-value problems for evolution equations in one spatial dimension, like the NLS, was finally presented in 1997, after interjecting several novel ideas to the panoply of the IST methodology. Here, this method is further extended so that it can be applied to evolution equations in two spatial dimensions, like the DS equation. This novel extension involves several new steps, including the formulation of a d-bar problem for a sectionally non-analytic function, i.e. for a function which has different non-analytic representations in different domains of the complex plane. This, in addition to the computation of a d-bar derivative, also requires the computation of the relevant jumps across the different domains. This latter step has certain similarities (but is more complicated) with the corresponding step for those initial-value problems in two dimensions which can be solved via a non-local RH problem, like KPI.
The deterministic and statistical Burgers equation
NASA Astrophysics Data System (ADS)
Fournier, J.-D.; Frisch, U.
Fourier-Lagrangian representations of the UV-region inviscid-limit solutions of the equations of Burgers (1939) are developed for deterministic and random initial conditions. The Fourier-mode amplitude behavior of the deterministic case is characterized by complex singularities with fast decrease, power-law preshocks with k indices of about -4/3, and shocks with k to the -1. In the random case, shocks are associated with a k to the -2 spectrum which overruns the smaller wavenumbers and appears immediately under Gaussian initial conditions. The use of the Hopf-Cole solution in the random case is illustrated in calculations of the law of energy decay by a modified Kida (1979) method. Graphs and diagrams of the results are provided.
A differential equation for specific catchment area
NASA Astrophysics Data System (ADS)
Gallant, John C.; Hutchinson, Michael F.
2011-05-01
Analysis of the behavior of specific catchment area in a stream tube leads to a simple nonlinear differential equation describing the rate of change of specific catchment area along a flow path. The differential equation can be integrated numerically along a flow path to calculate specific catchment area at any point on a digital elevation model without requiring the usual estimates of catchment area and width. The method is more computationally intensive than most grid-based methods for calculating specific catchment area, so its main application is as a reference against which conventional methods can be tested. This is the first method that provides a benchmark for more approximate methods in complex terrain with both convergent and divergent areas, not just on simple surfaces for which analytical solutions are known. Preliminary evaluation of the D8, M8, digital elevation model networks (DEMON), and D∞ methods indicate that the D∞ method is the best of those methods for estimating specific catchment area, but all methods overestimate in divergent terrain.
Use of Spreadsheets for Demonstrating the Solutions of Simple Differential Equations.
ERIC Educational Resources Information Center
Severn, John
1999-01-01
Explores simple equations such as the exponential decay function, a terminal velocity situation, and the more complex situations of the simple harmonic oscillator and damped oscillations. (Author/CCM)
Sharp spatiotemporal patterns in the diffusive time-periodic logistic equation
NASA Astrophysics Data System (ADS)
Du, Yihong; Peng, Rui
To reveal the complex influence of heterogeneous environment on population systems, we examine the asymptotic profile (as ɛ→0) of the positive solution to the perturbed periodic logistic equation {
Wavelets meet Burgulence: CVS-filtered Burgers equation
NASA Astrophysics Data System (ADS)
Nguyen van yen, Romain; Farge, Marie; Kolomenskiy, Dmitry; Schneider, Kai; Kingsbury, Nick
2008-08-01
Numerical experiments with the one-dimensional inviscid Burgers equation show that filtering the solution at each time step in a way similar to CVS (Coherent Vortex Simulation) gives the solution of the viscous Burgers equation. The CVS filter used here is based on a complex-valued translation-invariant wavelet representation of the velocity, from which one selects the wavelet coefficients having modulus larger than a threshold whose value is iteratively estimated. The flow evolution is computed from either deterministic or random initial conditions, considering both white noise and Brownian motion.
Novel Approach for Solving the Equation of Motion of a Simple Harmonic Oscillator. Classroom Notes
ERIC Educational Resources Information Center
Gauthier, N.
2004-01-01
An elementary method, based on the use of complex variables, is proposed for solving the equation of motion of a simple harmonic oscillator. The method is first applied to the equation of motion for an undamped oscillator and it is then extended to the more important case of a damped oscillator. It is finally shown that the method can readily be…
Modeling Noisy Data with Differential Equations Using Observed and Expected Matrices
ERIC Educational Resources Information Center
Deboeck, Pascal R.; Boker, Steven M.
2010-01-01
Complex intraindividual variability observed in psychology may be well described using differential equations. It is difficult, however, to apply differential equation models in psychological contexts, as time series are frequently short, poorly sampled, and have large proportions of measurement and dynamic error. Furthermore, current methods for…
A Case of Inconsistent Equatings: How the Man with Four Watches Decides What Time It Is
ERIC Educational Resources Information Center
Livingston, Samuel A.; Antal, Judit
2010-01-01
A simultaneous equating of four new test forms to each other and to one previous form was accomplished through a complex design incorporating seven separate equating links. Each new form was linked to the reference form by four different paths, and each path produced a different score conversion. The procedure used to resolve these inconsistencies…
New solutions for two integrable cases of a generalized fifth-order nonlinear equation
NASA Astrophysics Data System (ADS)
Wazwaz, Abdul-Majid
2015-05-01
Multiple-complexiton solutions for a new generalized fifth-order nonlinear integrable equation are constructed with the help of the Hirota's method and the simplified Hirota's method. By extending the real parameters into complex parameters, nonsingular complexiton solutions are obtained for two specific coefficients of the new generalized equation.
Some remarks on Li-Sinai-type solutions of the Burgers equation
NASA Astrophysics Data System (ADS)
Pauls, W.
2011-07-01
For a class of complex-valued initial conditions, we construct solutions for the inviscid Burgers equations of the same type as those recently introduced by Li and Sinai. Obtained by very simple considerations, these solutions correspond to stable distributions with 1 < α <= 2. We furthermore show that solutions of non-Gaussian type can be constructed for the viscous Burgers equation.
Truncation selection and payoff distributions applied to the replicator equation.
Morsky, B; Bauch, C T
2016-09-01
The replicator equation has been frequently used in the theoretical literature to explain a diverse array of biological phenomena. However, it makes several simplifying assumptions, namely complete mixing, an infinite population, asexual reproduction, proportional selection, and mean payoffs. Here, we relax the conditions of mean payoffs and proportional selection by incorporating payoff distributions and truncation selection into extensions of the replicator equation and agent-based models. In truncation selection, replicators with fitnesses above a threshold survive. The reproduction rate is equal for all survivors and is sufficient to replace the replicators that did not survive. We distinguish between two types of truncation: independent and dependent with respect to the fitness threshold. If the payoff variances from all strategy pairing are the same, then we recover the replicator equation from the independent truncation equation. However, if all payoff variances are not equal, then any boundary fixed point can be made stable (or unstable) if only the fitness threshold is chosen appropriately. We observed transient and complex dynamics in our models, which are not observed in replicator equations incorporating the same games. We conclude that the assumptions of mean payoffs and proportional selection in the replicator equation significantly impact replicator dynamics. PMID:27343031
A Multiple Equation Model of Demand for Health Care
Wirick, Grover C.
1966-01-01
Planning health care facilities for the future requires a means of estimating future consumption of services. Demand for medical care is looked upon as demand for separate components (hospital, doctor, dentist, medicine, other) rather than for a single, homogeneous product. A simultaneous equation model is proposed, and measures representing the forces thought to influence consumption (need, realization, motivation, resources, and availability of service) are fitted into the five equations. An optimized analysis variance method is employed on data from a sample survey of Michigan's population in 1958 to obtain single equation estimates of the five demand functions as a preliminary test of the model. The optimizing feature, which also includes an examination of complex interactions, retains variables in the equation on the basis of their estimating ability. The results indicate that a high degree of joint dependency exists among the components and that a simultaneous equation model is warranted. The study, intended as a research design, also reveals considerable variety in component equations, certain relevant and irrelevant variables, several important interactions, and a need for refining some measures in future studies. PMID:5971639
Studies of the Ginzburg-Landau equation
Rodriguez, J.D.
1988-01-01
The turbulence problem is the motivation for the study of reduction of phase space dimension in the Ginzburg-Landau equation. Chaotic solutions to this equation provide a turbulence analog. A basis set for the chaotic attractor is derived using the orthogonal decomposition of the correlation matrix. This matrix is computed explicitly at the point of maximal Liapunov dimension in the parameter range under study. The basis set is shown to be optimal in a least squares sense. Galerdin projection is then used to obtain a small set of O.D.E.'s. The case of spatially periodic, even initial data is studied first. Three complex O.D.E.'s were sufficient to reproduce the solution of the full system as given by a 16 point pseudo-spectral Fourier method. The case of homogeneous boundary conditions was studied next. Ten complex O.D.E.'s were required versus 128 for the pseudo-spectral solution. Using power spectra and Poincare sections the reduced systems were shown to reproduce the exact behavior over a wide parameter range. Savings in C.P.U. time of an order of magnitude were attained over pseudo-spectral algorithms. New results on the asymptotic behavior of limit cycle solutions were also obtained. Singular solutions, zero almost everywhere, with strong boundary layer character were found in the limit of large domain size. An infinite hierarchy of subharmonic solutions was shown to exist for the spatially periodic case, and a countable number of fixed point solutions was found for both spatially periodic and homogeneous cases.
The equations of relative motion in the orbital reference frame
NASA Astrophysics Data System (ADS)
Casotto, Stefano
2016-03-01
The analysis of relative motion of two spacecraft in Earth-bound orbits is usually carried out on the basis of simplifying assumptions. In particular, the reference spacecraft is assumed to follow a circular orbit, in which case the equations of relative motion are governed by the well-known Hill-Clohessy-Wiltshire equations. Circular motion is not, however, a solution when the Earth's flattening is accounted for, except for equatorial orbits, where in any case the acceleration term is not Newtonian. Several attempts have been made to account for the J_2 effects, either by ingeniously taking advantage of their differential effects, or by cleverly introducing ad-hoc terms in the equations of motion on the basis of geometrical analysis of the J_2 perturbing effects. Analysis of relative motion about an unperturbed elliptical orbit is the next step in complexity. Relative motion about a J_2-perturbed elliptic reference trajectory is clearly a challenging problem, which has received little attention. All these problems are based on either the Hill-Clohessy-Wiltshire equations for circular reference motion, or the de Vries/Tschauner-Hempel equations for elliptical reference motion, which are both approximate versions of the exact equations of relative motion. The main difference between the exact and approximate forms of these equations consists in the expression for the angular velocity and the angular acceleration of the rotating reference frame with respect to an inertial reference frame. The rotating reference frame is invariably taken as the local orbital frame, i.e., the RTN frame generated by the radial, the transverse, and the normal directions along the primary spacecraft orbit. Some authors have tried to account for the non-constant nature of the angular velocity vector, but have limited their correction to a mean motion value consistent with the J_2 perturbation terms. However, the angular velocity vector is also affected in direction, which causes precession
Heat Stress Equation Development and Usage for Dryden Flight Research Center (DFRC)
NASA Technical Reports Server (NTRS)
Houtas, Franzeska; Teets, Edward H., Jr.
2012-01-01
Heat Stress Indices are equations that integrate some or all variables (e.g. temperature, relative humidity, wind speed), directly or indirectly, to produce a number for thermal stress on humans for a particular environment. There are a large number of equations that have been developed which range from simple equations that may ignore basic factors (e.g. wind effects on thermal loading, fixed contribution from solar heating) to complex equations that attempt to incorporate all variables. Each equation is evaluated for a particular use, as well as considering the ease of use and reliability of the results. The meteorology group at the Dryden Flight Research Center has utilized and enhanced the American College of Sports Medicine equation to represent the specific environment of the Mojave Desert. The Dryden WBGT Heat Stress equation has been vetted and implemented as an automated notification to the entire facility for the safety of all personnel and visitors.
Group classification and conservation laws of anisotropic wave equations with a source
NASA Astrophysics Data System (ADS)
Ibragimov, N. H.; Gandarias, M. L.; Galiakberova, L. R.; Bruzon, M. S.; Avdonina, E. D.
2016-08-01
Linear and nonlinear waves in anisotropic media are useful in investigating complex materials in physics, biomechanics, biomedical acoustics, etc. The present paper is devoted to investigation of symmetries and conservation laws for nonlinear anisotropic wave equations with specific external sources when the equations in question are nonlinearly self-adjoint. These equations involve two arbitrary functions. Construction of conservation laws associated with symmetries is based on the generalized conservation theorem for nonlinearly self-adjoint partial differential equations. First we calculate the conservation laws for the basic equation without any restrictions on the arbitrary functions. Then we make the group classification of the basic equation in order to specify all possible values of the arbitrary functions when the equation has additional symmetries and construct the additional conservation laws.
Stability and periodicity of high-order Lorenz–Stenflo equations
NASA Astrophysics Data System (ADS)
Park, Junho; Han, Beom-Soon; Lee, Hyunho; Jeon, Ye-Lim; Baik, Jong-Jin
2016-06-01
In this paper, we derive high-order Lorenz–Stenflo equations with 6 variables and investigate periodic behaviors as well as stability of the equations. The stability of the high-order Lorenz–Stenflo equations is investigated by the linear stability analysis for various parameters. A periodicity diagram is also computed and it shows that the high-order Lorenz–Stenflo equations exhibit very different behaviors from the original Lorenz–Stenflo equations for both periodic and chaotic solutions. For example, period 3 regime for large parameters and scattered periodic regime are newly observed, and chaotic regimes exist for smaller values of r but for larger values of s than the original equations. In contrast, similarities such as the enclosure of the chaotic regime by the periodic regime or complex periodic regimes inside the chaotic regime are also observed for both the original and high-order Lorenz–Stenflo equations.
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.
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.…
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…
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.
... Screening guidelines may change over time as new technologies are developed and more is learned about Carney complex. It is important to talk with your doctor about appropriate screening tests. Learn more about what to expect when having ...
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…
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.
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…
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…
NASA Astrophysics Data System (ADS)
Levi, Decio; Olver, Peter; Thomova, Zora; Winternitz, Pavel
2009-11-01
presented at the SIDE 8 meeting were organized into the following special sessions: geometry of discrete and continuous Painlevé equations; continuous symmetries of discrete equations—theory and computational applications; algebraic aspects of discrete equations; singularity confinement, algebraic entropy and Nevanlinna theory; discrete differential geometry; discrete integrable systems and isomonodromy transformations; special functions as solutions of difference and q-difference equations. This special issue of the journal is organized along similar lines. The first three articles are topical review articles appearing in alphabetical order (by first author). The article by Doliwa and Nieszporski describes the Darboux transformations in a discrete setting, namely for the discrete second order linear problem. The article by Grammaticos, Halburd, Ramani and Viallet concentrates on the integrability of the discrete systems, in particular they describe integrability tests for difference equations such as singularity confinement, algebraic entropy (growth and complexity), and analytic and arithmetic approaches. The topical review by Konopelchenko explores the relationship between the discrete integrable systems and deformations of associative algebras. All other articles are presented in alphabetical order (by first author). The contributions were solicited from all participants as well as from the general scientific community. The contributions published in this special issue can be loosely grouped into several overlapping topics, namely: •Geometry of discrete and continuous Painlevé equations (articles by Spicer and Nijhoff and by Lobb and Nijhoff). •Continuous symmetries of discrete equations—theory and applications (articles by Dorodnitsyn and Kozlov; Levi, Petrera and Scimiterna; Scimiterna; Ste-Marie and Tremblay; Levi and Yamilov; Rebelo and Winternitz). •Yang--Baxter maps (article by Xenitidis and Papageorgiou). •Algebraic aspects of discrete equations
Complex order fractional derivatives in viscoelasticity
NASA Astrophysics Data System (ADS)
Atanacković, Teodor M.; Konjik, Sanja; Pilipović, Stevan; Zorica, Dušan
2016-06-01
We introduce complex order fractional derivatives in models that describe viscoelastic materials. This cannot be carried out unrestrictedly, and therefore we derive, for the first time, real valued compatibility constraints, as well as physical constraints that lead to acceptable models. As a result, we introduce a new form of complex order fractional derivative. Also, we consider a fractional differential equation with complex derivatives, and study its solvability. Results obtained for stress relaxation and creep are illustrated by several numerical examples.
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.
A perturbative solution to metadynamics ordinary differential equation
NASA Astrophysics Data System (ADS)
Tiwary, Pratyush; Dama, James F.; Parrinello, Michele
2015-12-01
Metadynamics is a popular enhanced sampling scheme wherein by periodic application of a repulsive bias, one can surmount high free energy barriers and explore complex landscapes. Recently, metadynamics was shown to be mathematically well founded, in the sense that the biasing procedure is guaranteed to converge to the true free energy surface in the long time limit irrespective of the precise choice of biasing parameters. A differential equation governing the post-transient convergence behavior of metadynamics was also derived. In this short communication, we revisit this differential equation, expressing it in a convenient and elegant Riccati-like form. A perturbative solution scheme is then developed for solving this differential equation, which is valid for any generic biasing kernel. The solution clearly demonstrates the robustness of metadynamics to choice of biasing parameters and gives further confidence in the widely used method.
ISDEP: Integrator of stochastic differential equations for plasmas
NASA Astrophysics Data System (ADS)
Velasco, J. L.; Bustos, A.; Castejón, F.; Fernández, L. A.; Martin-Mayor, V.; Tarancón, A.
2012-09-01
In this paper we present a general description of the ISDEP code (Integrator of Stochastic Differential Equations for Plasmas) and a brief overview of its physical results and applications so far. ISDEP is a Monte Carlo code that calculates the distribution function of a minority population of ions in a magnetized plasma. It solves the ion equations of motion taking into account the complex 3D structure of fusion devices, the confining electromagnetic field and collisions with other plasma species. The Monte Carlo method used is based on the equivalence between the Fokker-Planck and Langevin equations. This allows ISDEP to run in distributed computing platforms without communication between nodes with almost linear scaling. This paper intends to be a general description and a reference paper in ISDEP.
Control theory based airfoil design using the Euler equations
NASA Technical Reports Server (NTRS)
Jameson, Antony; Reuther, James
1994-01-01
This paper describes the implementation of optimization techniques based on control theory for airfoil design. In our previous work it was shown that control theory could be employed to devise effective optimization procedures for two-dimensional profiles by using the potential flow equation with either a conformal mapping or a general coordinate system. The goal of our present work is to extend the development to treat the Euler equations in two-dimensions by procedures that can readily be generalized to treat complex shapes in three-dimensions. Therefore, we have developed methods which can address airfoil design through either an analytic mapping or an arbitrary grid perturbation method applied to a finite volume discretization of the Euler equations. Here the control law serves to provide computationally inexpensive gradient information to a standard numerical optimization method. Results are presented for both the inverse problem and drag minimization problem.
A perturbative solution to metadynamics ordinary differential equation.
Tiwary, Pratyush; Dama, James F; Parrinello, Michele
2015-12-21
Metadynamics is a popular enhanced sampling scheme wherein by periodic application of a repulsive bias, one can surmount high free energy barriers and explore complex landscapes. Recently, metadynamics was shown to be mathematically well founded, in the sense that the biasing procedure is guaranteed to converge to the true free energy surface in the long time limit irrespective of the precise choice of biasing parameters. A differential equation governing the post-transient convergence behavior of metadynamics was also derived. In this short communication, we revisit this differential equation, expressing it in a convenient and elegant Riccati-like form. A perturbative solution scheme is then developed for solving this differential equation, which is valid for any generic biasing kernel. The solution clearly demonstrates the robustness of metadynamics to choice of biasing parameters and gives further confidence in the widely used method. PMID:26696051
An interpretation and solution of ill-conditioned linear equations
NASA Technical Reports Server (NTRS)
Ojalvo, I. U.; Ting, T.
1989-01-01
Data insufficiency, poorly conditioned matrices and singularities in equations occur regularly in complex optimization, correlation, and interdisciplinary model studies. This work concerns itself with two methods of obtaining certain physically realistic solutions to ill-conditioned or singular algebraic systems of linear equations arising from such studies. Two efficient computational solution procedures that generally lead to locally unique solutions are presented when there is insufficient data to completely define the model, or a least-squares error formulation of this system results in an ill-conditioned system of equations. If it is assumed that a reasonable estimate of the uncertain data is available in both cases cited above, then we shall show how to obtain realistic solutions efficiently, in spite of the insufficiency of independent data. The proposed methods of solution are more efficient than singular-value decomposition for dealing with such systems, since they do not require solutions for all the non-zero eigenvalues of the coefficient matrix.
Complex networks: Patterns of complexity
NASA Astrophysics Data System (ADS)
Pastor-Satorras, Romualdo; Vespignani, Alessandro
2010-07-01
The Turing mechanism provides a paradigm for the spontaneous generation of patterns in reaction-diffusion systems. A framework that describes Turing-pattern formation in the context of complex networks should provide a new basis for studying the phenomenon.
SOLVING PDES IN COMPLEX GEOMETRIES: A DIFFUSE DOMAIN APPROACH
LI, X.; LOWENGRUB, J.; RÄTZ, A.; VOIGT, A.
2011-01-01
We extend previous work and present a general approach for solving partial differential equations in complex, stationary, or moving geometries with Dirichlet, Neumann, and Robin boundary conditions. Using an implicit representation of the geometry through an auxilliary phase field function, which replaces the sharp boundary of the domain with a diffuse layer (e.g. diffuse domain), the equation is reformulated on a larger regular domain. The resulting partial differential equation is of the same order as the original equation, with additional lower order terms to approximate the boundary conditions. The reformulated equation can be solved by standard numerical techniques. We use the method of matched asymptotic expansions to show that solutions of the re-formulated equations converge to those of the original equations. We provide numerical simulations which confirm this analysis. We also present applications of the method to growing domains and complex three-dimensional structures and we discuss applications to cell biology and heteroepitaxy. PMID:21603084
Wave-equation Based Earthquake Location
NASA Astrophysics Data System (ADS)
Tong, P.; Yang, D.; Yang, X.; Chen, J.; Harris, J.
2014-12-01
Precisely locating earthquakes is fundamentally important for studying earthquake physics, fault orientations and Earth's deformation. In industry, accurately determining hypocenters of microseismic events triggered in the course of a hydraulic fracturing treatment can help improve the production of oil and gas from unconventional reservoirs. We develop a novel earthquake location method based on solving full wave equations to accurately locate earthquakes (including microseismic earthquakes) in complex and heterogeneous structures. Traveltime residuals or differential traveltime measurements with the waveform cross-correlation technique are iteratively inverted to obtain the locations of earthquakes. The inversion process involves the computation of the Fréchet derivative with respect to the source (earthquake) location via the interaction between a forward wavefield emitting from the source to the receiver and an adjoint wavefield reversely propagating from the receiver to the source. When there is a source perturbation, the Fréchet derivative not only measures the influence of source location but also the effects of heterogeneity, anisotropy and attenuation of the subsurface structure on the arrival of seismic wave at the receiver. This is essential for the accuracy of earthquake location in complex media. In addition, to reduce the computational cost, we can first assume that seismic wave only propagates in a vertical plane passing through the source and the receiver. The forward wavefield, adjoint wavefield and Fréchet derivative with respect to the source location are all computed in a 2D vertical plane. By transferring the Fréchet derivative along the horizontal direction of the 2D plane into the ones along Latitude and Longitude coordinates or local 3D Cartesian coordinates, the source location can be updated in a 3D geometry. The earthquake location obtained with this combined 2D-3D approach can then be used as the initial location for a true 3D wave-equation
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.
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.
Lattice Boltzmann method for the fractional advection-diffusion equation
NASA Astrophysics Data System (ADS)
Zhou, J. G.; Haygarth, P. M.; Withers, P. J. A.; Macleod, C. J. A.; Falloon, P. D.; Beven, K. J.; Ockenden, M. C.; Forber, K. J.; Hollaway, M. J.; Evans, R.; Collins, A. L.; Hiscock, K. M.; Wearing, C.; Kahana, R.; Villamizar Velez, M. L.
2016-04-01
Mass transport, such as movement of phosphorus in soils and solutes in rivers, is a natural phenomenon and its study plays an important role in science and engineering. It is found that there are numerous practical diffusion phenomena that do not obey the classical advection-diffusion equation (ADE). Such diffusion is called abnormal or superdiffusion, and it is well described using a fractional advection-diffusion equation (FADE). The FADE finds a wide range of applications in various areas with great potential for studying complex mass transport in real hydrological systems. However, solution to the FADE is difficult, and the existing numerical methods are complicated and inefficient. In this study, a fresh lattice Boltzmann method is developed for solving the fractional advection-diffusion equation (LabFADE). The FADE is transformed into an equation similar to an advection-diffusion equation and solved using the lattice Boltzmann method. The LabFADE has all the advantages of the conventional lattice Boltzmann method and avoids a complex solution procedure, unlike other existing numerical methods. The method has been validated through simulations of several benchmark tests: a point-source diffusion, a boundary-value problem of steady diffusion, and an initial-boundary-value problem of unsteady diffusion with the coexistence of source and sink terms. In addition, by including the effects of the skewness β , the fractional order α , and the single relaxation time τ , the accuracy and convergence of the method have been assessed. The numerical predictions are compared with the analytical solutions, and they indicate that the method is second-order accurate. The method presented will allow the FADE to be more widely applied to complex mass transport problems in science and engineering.
Lattice Boltzmann method for the fractional advection-diffusion equation.
Zhou, J G; Haygarth, P M; Withers, P J A; Macleod, C J A; Falloon, P D; Beven, K J; Ockenden, M C; Forber, K J; Hollaway, M J; Evans, R; Collins, A L; Hiscock, K M; Wearing, C; Kahana, R; Villamizar Velez, M L
2016-04-01
Mass transport, such as movement of phosphorus in soils and solutes in rivers, is a natural phenomenon and its study plays an important role in science and engineering. It is found that there are numerous practical diffusion phenomena that do not obey the classical advection-diffusion equation (ADE). Such diffusion is called abnormal or superdiffusion, and it is well described using a fractional advection-diffusion equation (FADE). The FADE finds a wide range of applications in various areas with great potential for studying complex mass transport in real hydrological systems. However, solution to the FADE is difficult, and the existing numerical methods are complicated and inefficient. In this study, a fresh lattice Boltzmann method is developed for solving the fractional advection-diffusion equation (LabFADE). The FADE is transformed into an equation similar to an advection-diffusion equation and solved using the lattice Boltzmann method. The LabFADE has all the advantages of the conventional lattice Boltzmann method and avoids a complex solution procedure, unlike other existing numerical methods. The method has been validated through simulations of several benchmark tests: a point-source diffusion, a boundary-value problem of steady diffusion, and an initial-boundary-value problem of unsteady diffusion with the coexistence of source and sink terms. In addition, by including the effects of the skewness β, the fractional order α, and the single relaxation time τ, the accuracy and convergence of the method have been assessed. The numerical predictions are compared with the analytical solutions, and they indicate that the method is second-order accurate. The method presented will allow the FADE to be more widely applied to complex mass transport problems in science and engineering. PMID:27176431
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.
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
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.
Exponential rational function method for space-time fractional differential equations
NASA Astrophysics Data System (ADS)
Aksoy, Esin; Kaplan, Melike; Bekir, Ahmet
2016-04-01
In this paper, exponential rational function method is applied to obtain analytical solutions of the space-time fractional Fokas equation, the space-time fractional Zakharov Kuznetsov Benjamin Bona Mahony, and the space-time fractional coupled Burgers' equations. As a result, some exact solutions for them are successfully established. These solutions are constructed in fractional complex transform to convert fractional differential equations into ordinary differential equations. The fractional derivatives are described in Jumarie's modified Riemann-Liouville sense. The exact solutions obtained by the proposed method indicate that the approach is easy to implement and effective.
NASA Astrophysics Data System (ADS)
Feng, Qing-Hua
2013-05-01
In this paper, an extended Riccati sub-ODE method is proposed to establish new exact solutions for fractional differential-difference equations in the sense of modified Riemann—Liouville derivative. By a fractional complex transformation, a given fractional differential-difference equation can be turned into another differential-difference equation of integer order. The validity of the method is illustrated by applying it to solve the fractional Hybrid lattice equation and the fractional relativistic Toda lattice system. As a result, some new exact solutions including hyperbolic function solutions, trigonometric function solutions and rational solutions are established.
Differential Complexes in Continuum Mechanics
NASA Astrophysics Data System (ADS)
Angoshtari, Arzhang; Yavari, Arash
2015-04-01
We study some differential complexes in continuum mechanics that involve both symmetric and non-symmetric second-order tensors. In particular, we show that the tensorial analogue of the standard grad-curl-div complex can simultaneously describe the kinematics and the kinetics of motion of a continuum. The relation between this complex and the de Rham complex allows one to readily derive the necessary and sufficient conditions for the compatibility of displacement gradient and the existence of stress functions on non-contractible bodies.We also derive the local compatibility equations in terms of the Green deformation tensor for motions of 2D and 3D bodies, and shells in curved ambient spaces with constant curvatures.
ERIC Educational Resources Information Center
Sumara, Dennis J.
2000-01-01
Discusses what Complexity Theory (presented as a rubric that collects theoretical understandings from a number of domains such as ecology, biology, neurology, and education) suggests about mind, selfhood, intelligence, and practices of reading, and the import of these reconceptualizations to reader-response researchers. Concludes that developing…
Atmospheric Science Data Center
2013-04-16
... View Larger Image The complex structure and beauty of polar clouds are highlighted by these images acquired ... Multi-angle Imaging SpectroRadiometer observes the daylit Earth continuously from pole to pole, and every 9 days views the entire globe ...
NASA Astrophysics Data System (ADS)
Fuhrmann, G.; Gröger, M.; Jäger, T.
2016-02-01
We introduce amorphic complexity as a new topological invariant that measures the complexity of dynamical systems in the regime of zero entropy. Its main purpose is to detect the very onset of disorder in the asymptotic behaviour. For instance, it gives positive value to Denjoy examples on the circle and Sturmian subshifts, while being zero for all isometries and Morse-Smale systems. After discussing basic properties and examples, we show that amorphic complexity and the underlying asymptotic separation numbers can be used to distinguish almost automorphic minimal systems from equicontinuous ones. For symbolic systems, amorphic complexity equals the box dimension of the associated Besicovitch space. In this context, we concentrate on regular Toeplitz flows and give a detailed description of the relation to the scaling behaviour of the densities of the p-skeletons. Finally, we take a look at strange non-chaotic attractors appearing in so-called pinched skew product systems. Continuous-time systems, more general group actions and the application to cut and project quasicrystals will be treated in subsequent work.
NASA Astrophysics Data System (ADS)
de Régules, Sergio
2016-04-01
Complexity science – which describes phenomena such as collective and emergent behaviour – is the focus of a new centre where researchers are examining everything from the spread of influenza to what a healthy heartbeat looks like. Sergio de Régules reports.
Optimization of High-order Wave Equations for Multicore CPUs
Energy Science and Technology Software Center (ESTSC)
2011-11-01
This is a simple benchmark to guage the performance of a high-order isotropic wave equation grid. The code is optimized for both SSE and AVX and is parallelized using OpenMP (see Optimization section). Structurally, the benchmark begins, reads a few command-line parameters, allocates and pads the four arrays (current, last, next wave fields, and the spatially varying but isotropic velocity), initializes these arrays, then runs the benchmark proper. The code then benchmarks the naive, SSEmore » (if supported), and AVX (if supported implementations) by applying the wave equation stencil 100 times and taking the average performance. Boundary conditions are ignored and would noiminally be implemented by the user. THus, the benchmark measures only the performance of the wave equation stencil and not a full simulation. The naive implementation is a quadruply (z,y,x, radius) nested loop that can handle arbitrarily order wave equations. The optimized (SSE/AVX) implentations are somewhat more complex as they operate on slabs and include a case statement to select an optimized inner loop depending on wave equation order.« less
Optimization of High-order Wave Equations for Multicore CPUs
Williams, Samuel
2011-11-01
This is a simple benchmark to guage the performance of a high-order isotropic wave equation grid. The code is optimized for both SSE and AVX and is parallelized using OpenMP (see Optimization section). Structurally, the benchmark begins, reads a few command-line parameters, allocates and pads the four arrays (current, last, next wave fields, and the spatially varying but isotropic velocity), initializes these arrays, then runs the benchmark proper. The code then benchmarks the naive, SSE (if supported), and AVX (if supported implementations) by applying the wave equation stencil 100 times and taking the average performance. Boundary conditions are ignored and would noiminally be implemented by the user. THus, the benchmark measures only the performance of the wave equation stencil and not a full simulation. The naive implementation is a quadruply (z,y,x, radius) nested loop that can handle arbitrarily order wave equations. The optimized (SSE/AVX) implentations are somewhat more complex as they operate on slabs and include a case statement to select an optimized inner loop depending on wave equation order.
A modified force-velocity equation for smooth muscle contraction.
Wang, J; Jiang, H; Stephens, N L
1994-01-01
It has been suggested that in skeletal muscle the force-velocity relationship may not be a simple hyperbolic one, as defined by Hill's equation. To determine whether smooth muscle demonstrated the same properties, quick-release force-velocity curves were obtained from canine tracheal smooth muscle. The results showed that the observed data points for tracheal smooth muscle systematically deviated from a hyperbola. Such deviation occurred at values of force (P) approaching maximum isometric force (Po) for curves elicited by quick release at 2 and 10 s in the course of isometric contractions. Shortening velocities under a given afterload were overestimated at the high-force end (P > 75% Po) by Hill's equation; this implied that a relationship more complex than a simple hyperbola was involved at high loads. We next focused on finding an equation to also fit those directly measured data points that did not conform to a hyperbola. Our rationale in developing the equation was that a plot of the linearized transform of Hill's equation should yield a straight line over the entire range of loads at which velocities were measured. The plot demonstrated that, in the low-load high-velocity portion of the curve, a peak value was reached at 70-80% Po, which decreased as load increased in the high-load low-velocity portion.(ABSTRACT TRUNCATED AT 250 WORDS) PMID:8175513
Solutions of differential equations in a Bernstein polynomial basis
NASA Astrophysics Data System (ADS)
Idrees Bhatti, M.; Bracken, P.
2007-08-01
An algorithm for approximating solutions to differential equations in a modified new Bernstein polynomial basis is introduced. The algorithm expands the desired solution in terms of a set of continuous polynomials over a closed interval and then makes use of the Galerkin method to determine the expansion coefficients to construct a solution. Matrix formulation is used throughout the entire procedure. However, accuracy and efficiency are dependent on the size of the set of Bernstein polynomials and the procedure is much simpler compared to the piecewise B spline method for solving differential equations. A recursive definition of the Bernstein polynomials and their derivatives are also presented. The current procedure is implemented to solve three linear equations and one nonlinear equation, and excellent agreement is found between the exact and approximate solutions. In addition, the algorithm improves the accuracy and efficiency of the traditional methods for solving differential equations that rely on much more complicated numerical techniques. This procedure has great potential to be implemented in more complex systems where there are no exact solutions available except approximations.
Bueyuekasik, Sirin A.; Pashaev, Oktay K.
2010-12-15
We construct a Madelung fluid model with time variable parameters as a dissipative quantum fluid and linearize it in terms of Schroedinger equation with time-dependent parameters. It allows us to find exact solutions of the nonlinear Madelung system in terms of solutions of the Schroedinger equation and the corresponding classical linear ordinary differential equation with variable frequency and damping. For the complex velocity field, the Madelung system takes the form of a nonlinear complex Schroedinger-Burgers equation, for which we obtain exact solutions using complex Cole-Hopf transformation. In particular, we give exact results for nonlinear Madelung systems related with Caldirola-Kanai-type dissipative harmonic oscillator. Collapse of the wave function in dissipative models and possible implications for the quantum cosmology are discussed.
Adaptive node techniques for Maxwell's equations
Hewett, D W
2000-04-01
The computational mesh in numerical simulation provides a framework on which to monitor the spatial dependence of function and their derivatives. Spatial mesh is therefore essential to the ability to integrate systems in time without loss of fidelity. Several philosophies have emerged to provide such fidelity (Eulerian, Lagrangian, Arbitrary Lagrangian Eulerian ALE, Adaptive Mesh Refinement AMR, and adaptive node generation/deletion). Regardless of the type of mesh, a major difficulty is in setting up the initial mesh. Clearly a high density of grid points is essential in regions of high geometric complexity and/or regions of intense, energetic activity. For some problems, mesh generation is such a crucial part of the problem that it can take as much computational effort as the run itself, and these tasks are now taking weeks of massively parallel CPU time. Mesh generation is no less crucial to electromagnetic calculations. In fact EM problem set up can be even more challenging without the clues given by fluid motion in hydrodynamic systems. When the mesh is advected with the fluid (Lagrangian), mesh points naturally congregate in regions of high activity. Similarly in AMR algorithms, strong gradients in the fluid flow are one of the triggers for mesh refinement. In the hyperbolic Maxwell's equations without advection, mesh point placement/motion is not so intuitive. In fixed geometry systems, it at least feasible to finely mesh high leverage, geometrically challenged areas. For other systems, where the action takes place far from the boundaries and, likely, changes position in time, the options are limited to either using a high resolution (expensive) mesh in all regions that could require such resolution or adaptively generating nodes to resolve the physics as it evolves. The authors have developed a new time of adaptive node technique for Maxwell's equations to deal with this set of issues.
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.
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…
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
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.
Chassin, David P.; Posse, Christian; Malard, Joel M.
2004-08-01
Physical analogs have shown considerable promise for understanding the behavior of complex adaptive systems, including macroeconomics, biological systems, social networks, and electric power markets. Many of today’s most challenging technical and policy questions can be reduced to a distributed economic control problem. Indeed, economically-based control of large-scale systems is founded on the conjecture that the price-based regulation (e.g., auctions, markets) results in an optimal allocation of resources and emergent optimal system control. This paper explores the state of the art in the use physical analogs for understanding the behavior of some econophysical systems and to deriving stable and robust control strategies for them. In particular we review and discussion applications of some analytic methods based on the thermodynamic metaphor according to which the interplay between system entropy and conservation laws gives rise to intuitive and governing global properties of complex systems that cannot be otherwise understood.
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.
Information Complexity and Biology
NASA Astrophysics Data System (ADS)
Bagnoli, Franco; Bignone, Franco A.; Cecconi, Fabio; Politi, Antonio
Kolmogorov contributed directly to Biology in essentially three problems: the analysis of population dynamics (Lotka-Volterra equations), the reaction-diffusion formulation of gene spreading (FKPP equation), and some discussions about Mendel's laws. However, the widely recognized importance of his contribution arises from his work on algorithmic complexity. In fact, the limited direct intervention in Biology reflects the generally slow growth of interest of mathematicians towards biological issues. From the early work of Vito Volterra on species competition, to the slow growth of dynamical systems theory, contributions to the study of matter and the physiology of the nervous system, the first 50-60 years have witnessed important contributions, but as scattered pieces apparently uncorrelated, and in branches often far away from Biology. Up to the 40' it is hard to see the initial loose build up of a convergence, for those theories that will become mainstream research by the end of the century, and connected by the study of biological systems per-se.
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.
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.
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.
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.
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.
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…
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)
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.
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.
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 ...
NASA Astrophysics Data System (ADS)
Schuch, Dieter
2014-04-01
Theoretical physics seems to be in a kind of schizophrenic state. Many phenomena in the observable macroscopic world obey nonlinear evolution equations, whereas the microscopic world is governed by quantum mechanics, a fundamental theory that is supposedly linear. In order to combine these two worlds in a common formalism, at least one of them must sacrifice one of its dogmas. I claim that linearity in quantum mechanics is not as essential as it apparently seems since quantum mechanics can be reformulated in terms of nonlinear Riccati equations. In a first step, it will be shown where complex Riccati equations appear in time-dependent quantum mechanics and how they can be treated and compared with similar space-dependent Riccati equations in supersymmetric quantum mechanics. Furthermore, the time-independent Schrödinger equation can also be rewritten as a complex Riccati equation. Finally, it will be shown that (real and complex) Riccati equations also appear in many other fields of physics, like statistical thermodynamics and cosmology.
Equations of explicitly-correlated coupled-cluster methods.
Shiozaki, Toru; Kamiya, Muneaki; Hirata, So; Valeev, Edward F
2008-06-21
The tensor contraction expressions defining a variety of high-rank coupled-cluster energies and wave functions that include the interelectronic distances (r(12)) explicitly (CC-R12) have been derived with the aid of a newly-developed computerized symbolic algebra smith. Efficient computational sequences to perform these tensor contractions have also been suggested, defining intermediate tensors-some reusable-as a sum of binary tensor contractions. smith can elucidate the index permutation symmetry of intermediate tensors that arise from a Slater-determinant expectation value of any number of excitation, deexcitation and other general second-quantized operators. smith also automates additional algebraic transformation steps specific to R12 methods, i.e. the identification and isolation of the special intermediates that need to be evaluated analytically and the resolution-of-the-identity insertion to facilitate high-dimensional molecular integral computation. The tensor contraction expressions defining the CC-R12 methods including through the connected quadruple excitation operator (CCSDTQ-R12) have been documented and efficient computational sequences have been suggested not just for the ground state but also for excited states via the equation-of-motion formalism (EOM-CC-R12) and for the so-called Lambda equation (Lambda-CC-R12) of the CC analytical gradient theory. Additional equations (the geminal amplitude equation) arise in CC-R12 that need to be solved to determine the coefficients multiplying the r(12)-dependent factors. The operation cost of solving the geminal amplitude equations of rank-k CC-R12 and EOM-CC-R12 (right-hand side) scales as O(n(6)) (k = 2) or O(n(7)) (k > or = 3) with the number of orbitals n and is surpassed by the cost of solving the usual amplitude equations O(n(2k+2)). While the complexity of the geminal amplitude equations of Lambda- and EOM-CC-R12 (left-hand side) nominally scales as O(n(2k+2)), it is less than that of the other O(n(2k
Wavelet and multiscale methods for operator equations
NASA Astrophysics Data System (ADS)
Dahmen, Wolfgang
More than anything else, the increase of computing power seems to stimulate the greed for tackling ever larger problems involving large-scale numerical simulation. As a consequence, the need for understanding something like the intrinsic complexity of a problem occupies a more and more pivotal position. Moreover, computability often only becomes feasible if an algorithm can be found that is asymptotically optimal. This means that storage and the number of floating point operations needed to resolve the problem with desired accuracy remain proportional to the problem size when the resolution of the discretization is refined. A significant reduction of complexity is indeed often possible, when the underlying problem admits a continuous model in terms of differential or integral equations. The physical phenomena behind such a model usually exhibit characteristic features over a wide range of scales. Accordingly, the most successful numerical schemes exploit in one way or another the interaction of different scales of discretization. A very prominent representative is the multigrid methodology; see, for instance, Hackbusch (1985) and Bramble (1993). In a way it has caused a breakthrough in numerical analysis since, in an important range of cases, it does indeed provide asymptotically optimal schemes. For closely related multilevel techniques and a unified treatment of several variants, such as multiplicative or additive subspace correction methods, see Bramble, Pasciak and Xu (1990), Oswald (1994), Xu (1992), and Yserentant (1993). Although there remain many unresolved problems, multigrid or multilevel schemes in the classical framework of finite difference and finite element discretizations exhibit by now a comparatively clear profile. They are particularly powerful for elliptic and parabolic problems.