Halton Sequences for Mixed Logit
Train, Kenneth
2000-01-01
Customers’ Choice Among Energy Supplier Simulation based oncustomers’ choice of energy supplier. Surveyed customerspreferences for energy suppliers, such that a mixed logit is
Customer-Specific Taste Parameters and Mixed Logit: Households' Choice of Electricity Supplier
Revelt, David; Train, Kenneth
2000-01-01
M a y 2000 Keywords: energy suppliers, mixed logit, tastecustomers' choice among energy suppliers in conjoint-typecustomers' choice o f energy supplier and estimate the value
Approximation Methods for Pricing Problems under the Nested Logit Model with Price Bounds
Rusmevichientong, Paat
Approximation Methods for Pricing Problems under the Nested Logit Model with Price Bounds W@orie.cornell.edu September 5, 2013 Abstract We consider two variants of a pricing problem under the nested logit model. In the first variant, the set of products offered to customers is fixed and we want to determine the prices
Estimating long-term world coal production with logit and probit transforms David Rutledge
Low, Steven H.
Estimating long-term world coal production with logit and probit transforms David Rutledge form 27 October 2010 Accepted 27 October 2010 Available online 4 November 2010 Keywords: Coal reserves Coal resources Coal production estimates IPCC Logistic model Cumulative normal model An estimate
Joint Stocking and Product Offer Decisions Under the Multinomial Logit Model
Topaloglu, Huseyin
Joint Stocking and Product Offer Decisions Under the Multinomial Logit Model Huseyin Topaloglu This paper studies a joint stocking and product offer problem. We have access to a number of products to satisfy the demand over a finite selling horizon. Given that customers choose among the set of offered
The equations Linear plate equation
Grunau, Hans-Christoph
The equations Linear plate equation Paneitz equation Willmore equation, one dimensional Some fourth order differential equations related to differential geometry Hans-Christoph Grunau OttovonGuerickeUniversit¨at Magdeburg Nice, January 26, 2006 Hans-Christoph Grunau Differential equations of fourth order #12;The
QG Equations QG Vorticity Equation
Hennon, Christopher C.
QG Equations QG Vorticity Equation The vorticity equation can be written in isobaric and vector Vorticity Equation: 1) Frictional effects are negligible 2) Tilting terms are negligible on the synoptic these assumptions are applied, the vorticity equation becomes: ( ) ( )Hgg g VffV t vv ·-+·-= (1) Furthermore, f
Phung, Kim-dang.- Le Laboratoire de MathÃ©matiques
I: Heat equation II: SchrÃ¶dinger equation III: Wave equation IV: Radiative transfer equation;I: Heat equation II: SchrÃ¶dinger equation III: Wave equation IV: Radiative transfer equation QUCP: Heat equation II: SchrÃ¶dinger equation III: Wave equation IV: Radiative transfer equation QUCP
Relativistic quaternionic wave equation
Schwartz, C
2006-01-01
Schrodinger ?time dependent? equation, ? 1 and ? 2 , then?TCP?. The current conservation equation ?3.2? is still truefor this extended wave equation ?8.1?, however, Eq. ?6.7?
Transport Equations Thomas Hillen
Hillen, Thomas
Transport Equations Thomas Hillen supported by NSERC University of Alberta, Edmonton Transport V , V compact and symmetric. Transport Equations p.2/33 #12;Directed Movement The equation pt(t, x of v. Transport Equations p.3/33 #12;With Directional Changes µ: turning rate. T(v, v ): probability
Einstein's Equation in Pictures
Matthew Frank
2002-03-28
This paper gives a self-contained, elementary, and largely pictorial statement of Einstein's equation.
Differential Equations: Page 1 Differential equations
Hogg, Andrew
) is a nth order differential equation. The aim is calculate the unknown function y(x). A linear differential First order differential equations 1.1 Direct integration If dy dx = g(x) subject to y(b) = y0 then y and is to be downloaded or copied for your private study only. #12;Differential Equations: Page 2 1.3 Integrating factor
Finite Temperature Schrödinger Equation
Xiang-Yao Wu; Bai-Jun Zhang; Xiao-Jing Liu; Nuo Ba; Yi-Heng Wu; Qing-Cai Wang; Yan Wang
2011-06-11
We know Schr\\"{o}dinger equation describes the dynamics of quantum systems, which don't include temperature. In this paper, we propose finite temperature Schr\\"{o}dinger equation, which can describe the quantum systems in an arbitrary temperature. When the temperature T=0, it become Shr\\"{o}dinger equation.
Relativistic Quaternionic Wave Equation II
Schwartz, Charles
2007-01-01
Relativistic quaternionic wave equation. II J. Math. Phys.Relativistic quaternionic wave equation. II Charles Schwartzcomponent quaternionic wave equation recently introduced. A
INTEGRAL EQUATION PRECONDITIONING FOR THE SOLUTION OF POISSON'S EQUATION ON
Ferguson, Thomas S.
INTEGRAL EQUATION PRECONDITIONING FOR THE SOLUTION OF POISSON'S EQUATION ON GEOMETRICALLY COMPLEX with the implementation and investigation of integral equation based solvers as preconditioners for finite difference discretizations of Poisson equations in geometrically complex domains. The target discretizations are those
Separable Differential Equations
PRETEX (Halifax NS) #1 1054 1999 Mar 05 10:59:16
2010-01-20
Feb 16, 2007 ... preceding differential equation and several mem- bers of the given family of curves. Describe the family of orthogonal trajectories. 34. Consider ...
Integrating the Jacobian equation
Airton von Sohsten de Medeiros; Ráderson Rodrigues da Silva
2014-09-16
We show essentially that the differential equation $\\frac{\\partial (P,Q)}{\\partial (x,y)} =c \\in {\\mathbb C}$, for $P,\\,Q \\in {\\mathbb C}[x,y]$, may be "integrated", in the sense that it is equivalent to an algebraic system of equations involving the homogeneous components of $P$ and $Q$. Furthermore, the first equations in this system give explicitly the homogeneous components of $Q$ in terms of those of $P$. The remaining equations involve only the homogeneous components of $P$.
Fractional Heisenberg Equation
Vasily E. Tarasov
2008-04-03
Fractional derivative can be defined as a fractional power of derivative. The commutator (i/h)[H, ], which is used in the Heisenberg equation, is a derivation on a set of observables. A derivation is a map that satisfies the Leibnitz rule. In this paper, we consider a fractional derivative on a set of quantum observables as a fractional power of the commutator (i/h)[H, ]. As a result, we obtain a fractional generalization of the Heisenberg equation. The fractional Heisenberg equation is exactly solved for the Hamiltonians of free particle and harmonic oscillator. The suggested Heisenberg equation generalize a notion of quantum Hamiltonian systems to describe quantum dissipative processes.
On the generalized continuity equation
Arbab I. Arbab; Hisham. M. Widatallah
2010-02-27
A generalized continuity equation extending the ordinary continuity equation has been found using quanternions. It is shown to be compatible with Dirac, Schrodinger, Klein-Gordon and diffusion equations. This generalized equation is Lorentz invariant. The transport properties of electrons are found to be governed by Schrodinger-like equation and not by the diffusion equation.
Sergei Kuksin; Alberto Maiocchi
2015-01-17
In this chapter we present a general method of constructing the effective equation which describes the behaviour 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 behaviour 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. In the case of the NLS equation we use next some heuristic approximation from the arsenal of wave turbulence to show that under the iterated limit "the volume goes to infinity", taken after the limit "the amplitude of oscillations goes to zero", the energy spectrum of solutions for the effective equation is described by a Zakharov-type kinetic equation. Evoking the Zakharov ansatz we show that stationary in time and homogeneous in space solutions for the latter equation have a power law form. Our method applies to various weakly nonlinear wave systems, appearing in plasma, meteorology and oceanology.
Natale, Michael J.
Standard second-order self-adjoint forms of the transport equation, such as the even-parity, odd-parity, and self-adjoint angular flux equation, cannot be used in voids. Perhaps more importantly, they experience numerical convergence difficulties...
The relativistic Pauli equation
David Delphenich
2012-07-24
After discussing the way that C2 and the algebra of complex 2x2 matrices can be used for the representation of both non-relativistic rotations and Lorentz transformations, we show that Dirac bispinors can be more advantageously represented as 2x2 complex matrices. One can then give the Dirac equation a form for such matrix-valued wave functions that no longer necessitates the introduction of gamma matrices or a choice for their representation. The minimally-coupled Dirac equation for a charged spinning particle in an external electromagnetic field then implies a second order equation in the matrix-valued wave functions that is of Klein-Gordon type and represents the relativistic analogue of the Pauli equation. We conclude by presenting the Lagrangian form for the relativistic Pauli equation.
TRANSFORMING THE HEUN EQUATION TO THE HYPERGEOMETRIC EQUATION
Maier, Robert S.
TRANSFORMING THE HEUN EQUATION TO THE HYPERGEOMETRIC EQUATION: I. POLYNOMIAL TRANSFORMATIONS ROBERT S. MAIER #3; Abstract. The reductions of the Heun equation to the hypergeometric equation parameter and normalized accessory parameter of the Heun equation are each restricted to take values
TRANSFORMING THE HEUN EQUATION TO THE HYPERGEOMETRIC EQUATION
Maier, Robert S.
TRANSFORMING THE HEUN EQUATION TO THE HYPERGEOMETRIC EQUATION: I. POLYNOMIAL TRANSFORMATIONS ROBERT S. MAIER Abstract. The reductions of the Heun equation to the hypergeometric equation by rational accessory parameter of the Heun equation are each restricted to take values in a discrete set. The possible
TRANSFORMING THE HEUN EQUATION TO THE HYPERGEOMETRIC EQUATION
TRANSFORMING THE HEUN EQUATION TO THE HYPERGEOMETRIC EQUATION: I. POLYNOMIAL TRANSFORMATIONS ROBERT S. MAIER #3; Abstract. The reductions of the Heun equation to the hypergeometric equation and normalized accessory parameter of the Heun equation are each restricted to take values in a discrete set
Preisendorfer, Rudolph W
1957-01-01
dealt vdth a pair of irradiance functions representing twoHjC^.n^)^ which i s the irradiance a t time t on a unit areaCalifornia UNIFIED IRRADIANCE EQUATIONS R. W. Preisendorfer
First order differential equations
Samy Tindel
2015-09-29
Logistic growth. Hypothesis: Growth rate depends on population. Related equation: dy dt. = h(y)y. Specifications for h: h(y) ? r > 0 for small values of y y ?
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.
Aggregation Equation with Degenerate Diffusion
Yao, Yao
2012-01-01
for Patlak-Keller-Segel Equation with Degenerate Dif-for the aggregation equation with degenerate di?usion,3 An Aggregation Equation with Di?usion in the Periodic
Solving Symbolic Equations with PRESS
Sterling, L.; Bundy, Alan; Byrd, L.; O'Keefe, R.; Silver, B.
1982-01-01
We outline a program, PRESS (PRolog Equation Solving System) for solving symbolic, transcendental, non-differential equations. The methods used for solving equations are described, together with the service facilities. The ...
Differential Equations of Mathematical Physics
Max Lein
2015-08-16
These lecture notes for the course APM 351 at the University of Toronto are aimed at mathematicians and physicists alike. It is not meant as an introductory course to PDEs, but rather gives an overview of how to view and solve differential equations that are common in physics. Among others, I cover Hamilton's equations, variations of the Schr\\"odinger equation, the heat equation, the wave equation and Maxwells equations.
Gauged Knizhnik-Zamolodchikov equation
I. I Kogan; A. Lewis; O. A. Soloviev
1996-11-25
Correlation functions of gauged WZNW models are shown to satisfy a differential equation, which is a gauge generalization of the Knizhnik-Zamolodchikov equation.
Diophantine Equations and Congruent Number Equation Solutions
Mamuka Meskhishvili
2015-04-16
By using pairs of nontrivial rational solutions of congruent number equation $$ C_N:\\;\\;y^2=x^3-N^2x, $$ constructed are pairs of rational right (Pythagorean) triangles with one common side and the other sides equal to the sum and difference of the squares of the same rational numbers. The parametrizations are found for following Diophantine systems: \\begin{align*} (p^2\\pm q^2)^2-a^2 & =\\square_{1,2}\\,, \\\\[0.2cm] c^2-(p^2\\pm q^2)^2 & =\\square_{1,2}\\,, \\\\[0.2cm] a^2+(p^2\\pm q^2)^2 & =\\square_{1,2}\\,, \\\\[0.2cm] (p^2\\pm q^2)^2-a^2 & =(r^2\\pm s^2)^2. \\end{align*}
Flavored quantum Boltzmann equations
Cirigliano, Vincenzo; Lee, Christopher; Ramsey-Musolf, Michael J.; Tulin, Sean [Theoretical Division, Los Alamos National Laboratory, Los Alamos, New Mexico, 87545 (United States); Center for Theoretical Physics, University of California, and Theoretical Physics Group, Lawrence Berkeley National Laboratory, Berkeley, California, 94720 (United States); Department of Physics, University of Wisconsin-Madison, 1150 University Avenue, Madison, Wisconsin, 53706 (United States) and Kellogg Radiation Laboratory, California Institute of Technology, Pasadena, California, 91125 (United States); Theory Group, TRIUMF, 4004 Wesbrook Mall, Vancouver, British Columbia, V6T 2A3 (Canada)
2010-05-15
We derive from first principles, using nonequilibrium field theory, the quantum Boltzmann equations that describe the dynamics of flavor oscillations, collisions, and a time-dependent mass matrix in the early universe. Working to leading nontrivial order in ratios of relevant time scales, we study in detail a toy model for weak-scale baryogenesis: two scalar species that mix through a slowly varying time-dependent and CP-violating mass matrix, and interact with a thermal bath. This model clearly illustrates how the CP asymmetry arises through coherent flavor oscillations in a nontrivial background. We solve the Boltzmann equations numerically for the density matrices, investigating the impact of collisions in various regimes.
Deviation differential equations. Jacobi fields
G. Sardanashvily
2013-04-02
Given a differential equation on a smooth fibre bundle Y, we consider its canonical vertical extension to that, called the deviation equation, on the vertical tangent bundle VY of Y. Its solutions are Jacobi fields treated in a very general setting. In particular, the deviation of Euler--Lagrange equations of a Lagrangian L on a fibre bundle Y are the Euler-Lagrange equations of the canonical vertical extension of L onto VY. Similarly, covariant Hamilton equations of a Hamiltonian form H are the Hamilton equations of the vertical extension VH of H onto VY.
Dirac Equation at Finite Temperature
Xiang-Yao Wu; Bo-Jun Zhang; Xiao-Jing Liu; Nuo Ba; Yi-Heng Wu; Si-Qi Zhang; Jing Wang; Chun-Hong Li
2012-12-01
In this paper, we propose finite temperature Dirac equation, which can describe the quantum systems in an arbitrary temperature for a relativistic particle of spin-1/2. When the temperature T=0, it become Dirac equation. With the equation, we can study the relativistic quantum systems in an arbitrary temperature.
Assignment II Saha & Boltzmann equations
Spoon, Henrik
Assignment II Saha & Boltzmann equations January 21, 2002 This assignment is meant to give you some practical experience in using the Saha and Boltzmann equations that govern the level populations in atoms;s =kT the partition function of ionization stage r. The Saha equation: N r+1 N r = 2U r+1 U r P e #18
Structural Equation Modeling For Travel Behavior Research
Golob, Thomas F.
2011-01-01
STREAMS (Structural Equation Modeling Made Simple) is aGerbing, 1988. Structural equation modeling in practice: aP.M. , 1989. EQS Structural Equations Program Manual. BMDP
SOURCE TERMS IN THE TRANSIENT SEEPAGE EQUATION
Narasimhan, T.N.
2013-01-01
IN THE TRANSIENT SEEPAGE EQUATION T.N. Narasimhan FebruaryIN THE TRANSIENT SEEPAGE EQUATION T. N. Narasimhan Earthan integral transient seepage equation that includes source
A Master Equation Approach to the `3 + 1' Dirac Equation
Keith A. Earle
2011-02-06
A derivation of the Dirac equation in `3+1' dimensions is presented based on a master equation approach originally developed for the `1+1' problem by McKeon and Ord. The method of derivation presented here suggests a mechanism by which the work of Knuth and Bahrenyi on causal sets may be extended to a derivation of the Dirac equation in the context of an inference problem.
On the generalized Jacobi equation
Volker Perlick
2007-10-14
The standard text-book Jacobi equation (equation of geodesic deviation) arises by linearizing the geodesic equation around some chosen geodesic, where the linearization is done with respect to the coordinates and the velocities. The generalized Jacobi equation, introduced by Hodgkinson in 1972 and further developed by Mashhoon and others, arises if the linearization is done only with respect to the coordinates, but not with respect to the velocities. The resulting equation has been studied by several authors in some detail for timelike geodesics in a Lorentzian manifold. Here we begin by briefly considering the generalized Jacobi equation on affine manifolds, without a metric; then we specify to lightlike geodesics in a Lorentzian manifold. We illustrate the latter case by considering particular lightlike geodesics (a) in Schwarzschild spacetime and (b) in a plane-wave spacetime.
S. C. Tiwari
2007-06-09
A generalized harmonic map equation is presented based on the proposed action functional in the Weyl space (PLA, 135, 315, 1989).
Schroeder's Equation in Several Variables
1910-10-20
2000 Mathematics Subject Classification: Primary: 32H50. Secondary: 30D05, 39B32, 47B33. Keywords: Schroeder's functional equation, iteration, composition
Heun equation, Teukolsky equation, and type-D metrics
D. Batic; H. Schmid
2007-01-15
Starting with the whole class of type-D vacuum backgrounds with cosmological constant we show that the separated Teukolsky equation for zero rest-mass fields with spin $s=\\pm 2$ (gravitational waves), $s=\\pm 1$ (electromagnetic waves) and $s=\\pm 1/2$ (neutrinos) is an Heun equation in disguise.
Saturation and linear transport equation
Krzysztof Kutak
2009-04-29
We show that the GBW saturation model provides an exact solution to the one dimensional linear transport equation. We also show that it is motivated by the BK equation considered in the saturated regime when the diffusion and the splitting term in the diffusive approximation are balanced by the nonlinear term.
Media with no Fresnel equation
Peinke, Joachim
Media with no Fresnel equation Alberto Favaro & Ismo V. Lindell Outline Part 1: Local linear media Part 2: Jump conditions Part 3: media with no G(q) Conclusions Electromagnetic media with no Fresnel with no Fresnel equation Alberto Favaro & Ismo V. Lindell Outline Part 1: Local linear media Part 2: Jump
Logit Models for Estimating Urban Area Through Travel
Talbot, Eric
2011-10-21
because the variable of interest is a proportion, rather than a continuous number. To improve on these previous models, the research described in this document developed a system of two models. Each model was developed using logistic regression... to develop through trip tables. For predictor variables, the models use results from a very simple gravity model; the average daily traffic (ADT) at each external station as a proportion of the total ADT at all available external stations; the number...
A Grassmann integral equation K. Scharnhorst a)
Scharnhorst, Klaus
A Grassmann integral equation K. Scharnhorst a) HumboldtÂUniversita Ë? t zu Berlin, Institut fu Ë? r Grassmann integral equation in analogy to integral equations studied in real analysis. A Grassmann integral equation is an equation which involves Grassmann #Berezin# integrations and which is to be obeyed
A Grassmann integral equation K. Scharnhorsta)
Scharnhorst, Klaus
A Grassmann integral equation K. Scharnhorsta) Humboldt-UniversitaÂ¨t zu Berlin, Institut fu Grassmann integral equation in analogy to integral equations studied in real analysis. A Grassmann integral equation is an equation which involves Grassmann Berezin integrations and which is to be obeyed
Energy stories, equations and transition
Ernst, Damien
Energy stories, equations and transition Une histoire d'énergie: équations et transition% - Transition #12;ERoEI · ERoEI for « Energy Sustainable Energy April 28th, 2015 Raphael Fonteneau, University of Liège, Belgium @R_Fonteneau #12;Energy
van de Walle, Axel
Equation (30) should read F (T2) T2 = F (T1) T1 + Z 1/T2 1/T1 E d (1/T) Equation (E1) should be the same as Equation (38). Accordingly, the inlined equation just below Equation (E11) should be: ¡ kAB/ kAAkBB - 1 ¢ ¿ 1. To facilitate comparisons, Equation (E14) gives the effective cluster interac- tion using
Entropic corrections to Einstein equations
Hendi, S. H. [Physics Department, College of Sciences, Yasouj University, Yasouj 75914 (Iran, Islamic Republic of); Research Institute for Astronomy and Astrophysics of Maragha (RIAAM), Maragha (Iran, Islamic Republic of); Sheykhi, A. [Research Institute for Astronomy and Astrophysics of Maragha (RIAAM), Maragha (Iran, Islamic Republic of); Department of Physics, Shahid Bahonar University, P.O. Box 76175-132, Kerman (Iran, Islamic Republic of)
2011-04-15
Considering the general quantum corrections to the area law of black hole entropy and adopting the viewpoint that gravity interprets as an entropic force, we derive the modified forms of Modified Newtonian dynamics (MOND) theory of gravitation and Einstein field equations. As two special cases we study the logarithmic and power-law corrections to entropy and find the explicit form of the obtained modified equations.
How accurate is Limber's equation?
P. Simon
2007-08-24
The so-called Limber equation is widely used in the literature to relate the projected angular clustering of galaxies to the spatial clustering of galaxies in an approximate way. This paper gives estimates of where the regime of applicability of Limber's equation stops. Limber's equation is accurate for small galaxy separations but breaks down beyond a certain separation that depends mainly on the ratio sigma/R and to some degree on the power-law index, gamma, of spatial clustering xi; sigma is the one-sigma width of the galaxy distribution in comoving distance, and R the mean comoving distance. As rule-of-thumb, a 10% relative error is reached at 260 sigma/R arcmin for gamma~1.6, if the spatial clustering is a power-law. More realistic xi are discussed in the paper. Limber's equation becomes increasingly inaccurate for larger angular separations. Ignoring this effect and blindly applying Limber's equation can possibly bias results for the inferred spatial correlation. It is suggested to use in cases of doubt, or maybe even in general, the exact equation that can easily be integrated numerically in the form given in the paper.
Schroedinger equation and classical physics
Milos V. Lokajicek
2012-05-30
Any time-dependent solution of Schr\\"{o}dinger equation may be always correlated to a solution of Hamilton equations or to a statistical combination of their solutions; only the set of corresponding solutions is somewhat smaller (due to existence of quantization). There is not any reason to the physical interpretation according to Copenhagen alternative as Bell's inequalities are valid in the classical physics only (and not in any alternative based on Schr\\"{o}dinger equation). The advantage of Schr\\"{o}dinger equation consists then in that it enables to represent directly the time evolution of a statistical distribution of classical initial states (which is usual in collision experiments). The Schr\\"{o}dinger equation (without assumptions added by Bohr) may then represent the common physical theory for microscopic as well as macroscopic physical systems. However, together with the last possibility the solutions of Schr\\"{o}dinger equation may be helpful also in analyzing the influence of other statistically distributed properties (e.g., spin orientations or space structures) of individual matter objects forming a corresponding physical system, which goes in principle beyond the classical physics. In any case, the contemporary quantum theory represents the phenomenological approximative description of some matter characteristics only, without providing any insight into quantum mechanism emergence. In such a case it is necessary to take into account more detailed properties at least of some involved objects.
4.3 Boundary integral equations
2010-10-18
62. CHAPTER 4. OBSTACLE SCATTERING. 4.3 Boundary integral equations. We introduce the equivalent sources for the Helmholtz equation and establish ...
A connection between the shallow-water equations and the Euler-Poincaré equations
Roberto Camassa; Long Lee
2014-04-18
The Euler-Poincar\\'e differential (EPDiff) equations and the shallow water (SW) equations share similar wave characteristics. Using the Hamiltonian structure of the SW equations with flat bottom topography, we establish a connection between the EPDiff equations and the SW equations in one and multi-dimensions. Additionally, we show that the EPDiff equations can be recast in a curl formulation.
Equation-Based Power Model Integration in ESESC
Sinha, Meeta
2013-01-01
3.1 Power Equation . . . . . . . . . . . . . . . . . .3.1.1for the power model equation . . . . . . . . . . . .cache energy equations . . . . . . . . . . . . vi Abstract
Chapter 2' First order Differential Equations I 2,] Linear Equations ...
1' _ _ , Draw a direction ?eld for the given differential equation. I- a - - - - "/4 ..... in solution. Water containing 1 lb of salt per gallon is entering at a rate of 3 g ..... A body falling in a relatively dense ?uid, oil for example, is acted on by three forces.
Acoustics Beyond the Wave Equation Paul Pereira
Pulfrey, David L.
Acoustics Beyond the Wave Equation Paul Pereira November 20, 2003 #12;2 1 Navier-Stokes Equation). The traditional study of acoustics concerns itself with the linearized equations of fluid mechanics, however. The fundamental equations of Nonlinear Acoustics are those of fluid dynamics, a mathematical description of which
Conservation of Mass The Continuity Equation
Hennon, Christopher C.
Conservation of Mass The Continuity Equation The equations of motion describe the "conservation. Holton derives the continuity equation in two ways: Eulerian and Lagrangian. We will consider to the sum of all the net mass flows coming from all 3 directions (equations 4,5, and 6): ( ) ( ) ( ) tzyx z
Exact Vacuum Solutions to the Einstein Equation
Ying-Qiu Gu
2007-06-17
In this paper, we present a framework for getting a series of exact vacuum solutions to the Einstein equation. This procedure of resolution is based on a canonical form of the metric. According to this procedure, the Einstein equation can be reduced to some 2-dimensional Laplace-like equations or rotation and divergence equations, which are much convenient for the resolution.
Commutative Relations for the Nonlinear Dirac Equation
Ying-Qiu Gu
2008-02-14
By constructing the commutative operators chain, we derive conditions for solving the eigenfunctions of Dirac equation and Schr\\"odinger type equation via separation of variables. Detailed calculation shows that, only a few cases can be completely reduced into ordinary differential equation system. So the effective perturbation or approximation methods for the resolution of the spinor equation are necessary, especially for the nonlinear cases.
Torque Equation (See section 4.9)
McCalley, James D.
1 Torque Equation (See section 4.9) Our goal is to combine the state-space voltage equations with the state-space torque equations. To achieve this, we need to do the following three things to the torque equation: 1. Address the difference in power bases. 2. Address the difference in speed (time) bases. 3
Differential-Equation Based Absorbing Boundary Conditions
Schneider, John B.
Chapter 6 Differential-Equation Based Absorbing Boundary Conditions 6.1 Introduction A simple in the analysis of a wide range of FDTD-related topics. 6.2 The Advection Equation The wave equation that governs.2) The second form represents the equation in terms of an operator operating on Ez where the operator
LAPLACE'S EQUATION FROM TWO PERSPECTIVES MICHAEL FOSCO
May, J. Peter
LAPLACE'S EQUATION FROM TWO PERSPECTIVES MICHAEL FOSCO Abstract. We study Laplace's equation from the perspectives of partial differential equations and probabil- ity theory. We formulate the problem using both. Laplace's Equation In Probability 10 Acknowledgments 14 References 14 1. Introduction A natural way
221A Miscellaneous Notes Continuity Equation
Murayama, Hitoshi
221A Miscellaneous Notes Continuity Equation 1 The Continuity Equation As I received questions equation. It appears in Sakurai, pp. 101102, but he does not go into the general discussions about what is meant by the one of the most famous equations in physics (Sakurai (2.4.15)), t + · = 0, (1) called
Fourier's Law from Closure Equations
Jean Bricmont; Antti Kupiainen
2006-09-01
We give a rigorous derivation of Fourier's law from a system of closure equations for a nonequilibrium stationary state of a Hamiltonian system of coupled oscillators subjected to heat baths on the boundary. The local heat flux is proportional to the temperature gradient with a temperature dependent heat conductivity and the stationary temperature exhibits a nonlinear profile.
Evolution equation for quantum entanglement
Loss, Daniel
LETTERS Evolution equation for quantum entanglement THOMAS KONRAD1 , FERNANDO DE MELO2,3 , MARKUS of the time evolution of this resource under realistic conditions--that is, when corrupted by environment describes the time evolution of entanglement on passage of either component through an arbitrary noisy
Blink, J.A.
1983-09-01
In 1977, Dave Young published an equation-of-state (EOS) for lithium. This EOS was used by Lew Glenn in his AFTON calculations of the HYLIFE inertial-fusion-reactor hydrodynamics. In this paper, I summarize Young's development of the EOS and demonstrate a computer program (MATHSY) that plots isotherms, isentropes and constant energy lines on a P-V diagram.
Lyapunov Exponents for Burgers' Equation
Alexei Kourbatov
2015-02-23
We establish the existence, uniqueness, and stability of the stationary solution of the one-dimensional viscous Burgers equation with the Dirichlet boundary conditions on a finite interval. We obtain explicit formulas for solutions and analytically determine the Lyapunov exponents characterizing the asymptotic behavior of arbitrary solutions approaching the stationary one.
Use of Regression Equations 1 Running head: Equations from summary data
Crawford, John R.
Use of Regression Equations 1 Running head: Equations from summary data Neuropsychology, in press the final version published in the APA journal. It is not the copy of record Using regression equations.crawford@abdn.ac.uk #12;Use of Regression Equations 2 Abstract Regression equations have many useful roles
Five-dimensional Monopole Equation with Hedge-Hog Ansatz and Abel's Differential Equation
Hironobu Kihara
2011-02-10
We review the generalized monopole in the five-dimensional Euclidean space. A numerical solution with the Hedge-Hog ansatz is studied. The Bogomol'nyi equation becomes a second order autonomous non-linear differential equation. The equation can be translated into the Abel's differential equation of the second kind and is an algebraic differential equation.
Estimation of saturation and coherence effects in the KGBJS equation - a non-linear CCFM equation
Michal Deak
2012-10-01
We solve the modified non-linear extension of the CCFM equation - KGBJS equation - numerically for certain initial conditions and compare the resulting gluon Green functions with those obtained from solving the original CCFM equation and the BFKL and BK equations for the same initial conditions. We improve the low transversal momentum behaviour of the KGBJS equation by a small modification.
Korteweg-de Vries equation with a forcing term Solitary waves of the Kawahara equation
Fominov, Yakov
Korteweg-de Vries equation with a forcing term Solitary waves of the Kawahara equation Conclusion;Korteweg-de Vries equation with a forcing term Solitary waves of the Kawahara equation Conclusion Overview 1 Korteweg-de Vries equation with a forcing term Model Stationary solutions Stability 2 Solitary
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.
Iterative solutions of simultaneous equations
Laycock, Guyron Brantley
1962-01-01
ITERATIVE SOLUTIONS OP SIKJLTANEOUS EQUATIONS G~cn Hrantlep I aycock Approved. as to style snd, content by& (Chairman of Committee) E. c. (Head. of Department August 1/62 ACKNOWLEDGEMENT The author wishes to thank Dr. Hi A. Luther for his time sn4.... . . . ~ ~ . . ~ III. JACOBI AND 6AUSS-SEIDEL METHODS I V ~ C ONCLUS I GN ~ ~ ~ a ~ ~ ~ t ~ ~ ~ ~ a ~ 1 ~ ~ ~ ~ ~ ~ 9 ~ . ~ 18 V BIBLIOGRAPHY ~ ~ ~ o ~ ~ t ~ ~ ~ ~ 1 ~ ~ ~ VI ~ APPENDIX ~ ~ o ~ ~ e ~ o ~ ~ o o ~ ~ ~ . 22 Px'ogl am Lisliiixlgs...
Mahouton Norbert Hounkonnou; André Ronveaux
2013-06-20
This paper addresses a general method of polynomial transformation of hypergeometric equations. Examples of some classical special equations of mathematical physics are generated. Heun's equation and exceptional Jacobi polynomials are also treated.
Optimization and Nonlinear Equations Gordon K. Smyth
Smyth, Gordon K.
Optimization and Nonlinear Equations Gordon K. Smyth Bioinformatics Division, Walter and Eliza Hall, University of Melbourne, Victoria, Australia 12 November 2014 Abstract Optimization means finding. Many optimization al- gorithms are derived from algorithms that solve the nonlinear equations defined
18.03 Differential Equations, Spring 2006
Miller, Haynes
Differential Equations are the language in which the laws of nature are expressed. Understanding properties of solutions of differential equations is fundamental to much of contemporary science and engineering. Ordinary ...
Ordinary Differential Equation: Chapter 2.5
2014-09-11
Suppose the maximum population allowed is 500. Simple choice of h(y) = 20(1 ? y/500) leads dy dt. = 20(1 ? y/500)y. This type equation called Logistic equation.
Padé interpolation for elliptic Painlevé equation
Masatoshi Noumi; Satoshi Tsujimoto; Yasuhiko Yamada
2012-08-08
An interpolation problem related to the elliptic Painlev\\'e equation is formulated and solved. A simple form of the elliptic Painlev\\'e equation and the Lax pair are obtained. Explicit determinant formulae of special solutions are also given.
Deriving Mathisson - Papapetrou equations from relativistic pseudomechanics
R. R. Lompay
2005-03-12
It is shown that the equations of motion of a test point particle with spin in a given gravitational field, so called Mathisson - Papapetrou equations, can be derived from Euler - Lagrange equations of the relativistic pseudomechanics -- relativistic mechanics, which side by side uses the conventional (commuting) and Grassmannian (anticommuting) variables. In this approach the known difficulties of the Mathisson - Papapetrou equations, namely, the problem of the choice of supplementary conditions and the problem of higher derivatives are not appear.
A Periodic Solution to Impulsive Logistic Equation
Gyong-Chol Kim; Hyong-Chol O; Sang-Mun Kim; Chol Kim
2014-03-28
In this paper is provided a new representation of periodic solution to the impulsive Logistic equation considered in [7].
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.
Equation determines pressure drop in coiled tubing
Yang, Y.S.
1995-12-04
A single equation can determine the pressure drop in wells with laminar, transitional, and turbulent incompressible fluid flow in coiled tubing or other steel tubulars. The single equation is useful, especially in computer-aided design and operations. The equation is derived and illustrated by an example.
The Schrodinger equation and negative energies
S. Bruce
2008-06-30
We present a nonrelativistic wave equation for the electron in (3+1)-dimensions which includes negative-energy eigenstates. We solve this equation for three well-known instances, reobtaining the corresponding Pauli equation (but including negative-energy eigenstates) in each case.
On non commutative sinh-Gordon Equation
U. Saleem; M. Siddiq; M. Hassan
2006-05-10
We give a noncommutative extension of sinh-Gordon equation. We generalize a linear system and Lax representation of the sinh-Gordon equation in noncommutative space. This generalization gives a noncommutative version of the sinh-Gordon equation with extra constraints, which can be expressed as global conserved currents.
Nonlinear extension of the CCFM equation
Krzysztof Kutak
2012-06-06
In order to study such effects like parton saturation in final states at the LHC one of the approaches is to combine physics of the BK and the CCFM evolution equations. We report on recently obtained resummed form of the BK equation and nonlinear extension of the CCFM equation.
Quantum-mechanical Landau-Lifshitz equation
D. Yearchuck; Y. Yerchak
2008-01-09
Quantum-mechanical analogue of Landau-Lifshitz equation has been derived. It has been established that Landau-Lifshitz equation is fundamental physical equation underlying the dynamics of spectroscopic transitions and transitional phenomena. New phenomenon is predicted: electrical spin wave resonance (ESWR) being to be electrical analogue of magnetic spin wave resonance.
Model solution State variable model: differential equation
Limburg, Karin E.
2/26/2014 1 Model solution State variable model: differential equation Models a rate of change equation General solution: the antiderivative Particular solution: require initial and boundary conditions up the general solution to a differential equation in a book Solve for initial and boundary
On a Modified Klein Gordon Equation
B. S. Lakshmi
2009-08-09
We consider a modified Klein-Gordon equation that arises at ultra high energies. In a suitable approximation it is shown that for the linear potential which is of interest in quark interactions, their confinement for example,we get solutions that mimic the Harmonic oscillator energy levels, surprisingly. An equation similar to the beam equation is obtained in the process.
Wave equations with energy dependent potentials
J. Formanek; R. J. Lombard; J. Mares
2003-09-22
We study wave equations with energy dependent potentials. Simple analytical models are found useful to illustrate difficulties encountered with the calculation and interpretation of observables. A formal analysis shows under which conditions such equations can be handled as evolution equation of quantum theory with an energy dependent potential. Once these conditions are met, such theory can be transformed into ordinary quantum theory.
Universal Equation for Efimov States
Eric Braaten; H. -W. Hammer; M. Kusunoki
2003-03-13
Efimov states are a sequence of shallow 3-body bound states that arise when the 2-body scattering length is large. Efimov showed that the binding energies of these states can be calculated in terms of the scattering length and a 3-body parameter by solving a transcendental equation involving a universal function of one variable. We calculate this universal function using effective field theory and use it to describe the three-body system of 4He atoms. We also extend Efimov's theory to include the effects of deep 2-body bound states, which give widths to the Efimov states.
Universal equation for Efimov states
Braaten, Eric; Hammer, H.-W.; Kusunoki, M.
2003-02-01
Efimov states are a sequence of shallow three-body bound states that arise when the two-body scattering length is large. Efimov showed that the binding energies of these states can be calculated in terms of the scattering length and a three-body parameter by solving a transcendental equation involving a universal function of one variable. We calculate this universal function using effective field theory and use it to describe the three-body system of {sup 4}He atoms. We also extend Efimov's theory to include the effects of deep two-body bound states, which give widths to the Efimov states.
The Square Root Depth Wave Equations
Colin C. Cotter; Darryl D. Holm; James R. Percival
2009-12-11
We introduce a set of coupled equations for multilayer water waves that removes the ill-posedness of the multilayer Green-Naghdi (MGN) equations in the presence of shear. The new well-posed equations are Hamiltonian and in the absence of imposed background shear they retain the same travelling wave solutions as MGN. We call the new model the Square Root Depth equations, from the modified form of their kinetic energy of vertical motion. Our numerical results show how the Square Root Depth equations model the effects of multilayer wave propagation and interaction, with and without shear.
Equator-S observations of He+ energization by EMIC waves in the
Carlson, Charles W.
Equator-S observations of He+ energization by EMIC waves in the dawnside equatorial magnetosphere C Equator-S observations of He+ energization by electromagnetic ion cyclotron (EMIC) waves in the dawn side. Introduction [2] Observations suggesting He+ energization by electromag- netic ion cyclotron (EMIC) waves
The properties of the first equation of the Vlasov chain of equations
E. E. Perepelkin; B. I. Sadovnikov; N. G. Inozemtseva
2015-02-06
A mathematically rigorous derivation of the first Vlasov equation as a well-known Schr\\"odinger equation for the probabilistic description of a system and families of the classic diffusion equations and heat conduction for the deterministic description of physical systems was inferred. A physical meaning of the phase of the wave function which is a scalar potential of the probabilistic flow velocity is demonstrated. Occurrence of the velocity potential vortex component leads to the Pauli equation for one of the spinar components. A scheme of the construction of the Schr\\"odinger equation solving from the Vlasov equation solving and vice-versa is shown. A process of introduction of the potential to the Schr\\"odinger equation and its interpretation are given. The analysis of the potential properties gives us the Maxwell equation, the equation of the kinematic point movement, and the movement of the medium within electromagnetic fields equation.
Time-periodic solutions of the Benjamin-Ono equation
Ambrose, D.M.
2009-01-01
application to Benjamin Ono equation. Chinese Physics, 14(solutions of Hamiltonian equations. In Dynamics and Pro-quelques generalisations de l’equation de Korteweg-deVries.
Monge equation of arbitrary degree in 1 + 1 space
A. N. Leznov; R. Torres-cordoba
2013-01-31
Solution of Monge equation of arbitrary degree (non linear differential equation n-orden) is connected with solution of functional equation for 4 functions with 4 different arguments. Some number solutions of this equation is represented in explicit form.
Darboux transformation for the NLS equation
Aktosun, Tuncay; Mee, Cornelis van der
2010-03-08
We analyze a certain class of integral equations associated with Marchenko equations and Gel'fand-Levitan equations. Such integral equations arise through a Fourier transformation on various ordinary differential equations involving a spectral parameter. When the integral operator is perturbed by a finite-rank perturbation, we explicitly evaluate the change in the solution in terms of the unperturbed quantities and the finite-rank perturbation. We show that this result provides a fundamental approach to derive Darboux transformations for various systems of ordinary differential operators. We illustrate our theory by providing the explicit Darboux transformation for the Zakharov-Shabat system and show how the potential and wave function change when a simple discrete eigenvalue is added to the spectrum, and thus we also provide a one-parameter family of Darboux transformations for the nonlinear Schroedinger equation.
Riemann-Liouville Fractional Einstein Field Equations
Joakim Munkhammar
2010-03-18
In this paper we establish a fractional generalization of Einstein field equations based on the Riemann-Liouville fractional generalization of the ordinary differential operator $\\partial_\\mu$. We show some elementary properties and prove that the field equations correspond to the regular Einstein field equations for the fractional order $\\alpha = 1$. In addition to this we show that the field theory is inherently non-local in this approach. We also derive the linear field equations and show that they are a generalized version of the time fractional diffusion-wave equation. We show that in the Newtonian limit a fractional version of Poisson's equation for gravity arises. Finally we conclude open problems such as the relation of the non-locality of this theory to quantum field theories and the possible relation to fractional mechanics.
Notes On The Klein-Gordon Equation
Fredrick Michael
2010-04-09
In this article, we derive the scalar parametrized Klein-Gordon equation from the formal information theory framework. The least biased probability distribution is obtained, and the scalar equation is recast in terms of a Fokker-Planck equation in terms of the imaginary time, or a Schroedinger equation for the proper time. This method yields the Green's function parametrized by an evolution parameter. The derivation can then allow the use of potentials as constraints along with the Hamiltonian or moments of the evolution. The information theoretic, analogously the maximum entropy method, also allows one to examine the possibility of utilizing generalized and non-extensive statistics in the derivation. This approach yields non-linear evolution in the parametrized Klein-Gordon partial differential equations. Furthermore, we examine the Klein-Gordon equation in curved space-time, and we compare our results to the results of Schwinger and Dewitt obtained from path integral approaches.
Some generalizations of the Raychaudhuri equation
Abreu, Gabriel
2010-01-01
The Raychaudhuri equation has seen extensive use in general relativity, most notably in the development of various singularity theorems. In this rather technical article we shall generalize the Raychaudhuri equation in several ways. First an improved version of the standard timelike Raychaudhuri equation is developed, where several key terms are lumped together as a divergence. This already has a number of interesting applications, both within the ADM formalism and elsewhere. Second, a spacelike version of the Raychaudhuri equation is briefly discussed. Third, a version of the Raychaudhuri equation is developed that does not depend on the use of normalized congruences. This leads to useful formulae for the "diagonal" part of the Ricci tensor. Fourth, a "two vector" version of the Raychaudhuri equation is developed that uses two congruences to effectively extract "off diagonal" information concerning the Ricci tensor.
Eigen Equation of the Nonlinear Spinor
Ying-Qiu Gu; Ta-tsien Li
2007-04-04
How to effectively solve the eigen solutions of the nonlinear spinor field equation coupling with some other interaction fields is important to understand the behavior of the elementary particles. In this paper, we derive a simplified form of the eigen equation of the nonlinear spinor, and then propose a scheme to solve their numerical solutions. This simplified equation has elegant and neat structure, which is more convenient for both theoretical analysis and numerical computation.
Nuclear Scissors with Pairing and Continuity Equation
E. B. Balbutsev; L. A. Malov; P. Schuck; M. Urban
2008-10-29
The coupled dynamics of the isovector and isoscalar giant quadrupole resonances and low lying modes (including scissors) are studied with the help of the Wigner Function Moments (WFM) method generalized to take into account pair correlations. Equations of motion for collective variables are derived on the basis of the Time Dependent Hartree-Fock-Bogoliubov (TDHFB) equations in the harmonic oscillator model with quadrupole-quadrupole (QQ) residual interaction and a Gaussian pairing force. Special care is taken of the continuity equation.
A MULTIDIMENSIONAL NONLINEAR SIXTH-ORDER QUANTUM DIFFUSION EQUATION
heat equation tn = n. The second one is the fourth-order DerridaLebowitzSpeerSpohn (DLSS) equation
A New Integral Equation for the Spheroidal equations in case of m equal 1
Guihua Tian; Shuquan Zhong
2012-01-05
The spheroidal wave functions are investigated in the case m=1. The integral equation is obtained for them. For the two kinds of eigenvalues in the differential and corresponding integral equations, the relation between them are given explicitly. Though there are already some integral equations for the spheroidal equations, the relation between their two kinds of eigenvalues is not known till now. This is the great advantage of our integral equation, which will provide useful information through the study of the integral equation. Also an example is given for the special case, which shows another way to study the eigenvalue problem.
Stochastic Master Equations in Thermal Environment
S Attal; C Pellegrini
2010-04-20
We derive the stochastic master equations which describe the evolution of open quantum systems in contact with a heat bath and undergoing indirect measurements. These equations are obtained as a limit of a quantum repeated measurement model where we consider a small system in contact with an infinite chain at positive temperature. At zero temperature it is well-known that one obtains stochastic differential equations of jump-diffusion type. At strictly positive temperature, we show that only pure diffusion type equations are relevant.
An acoustic wave equation based on viscoelasticity
Andrzej Hanyga
2014-01-30
An acoustic wave equation for pressure accounting for viscoelastic attenuation is derived from viscoelastic equations of motion. It is assumed that the relaxation moduli are completely monotonic. The acoustic equation differs significantly from the equations proposed by Szabo (1994) and in several other papers. Integral representations of dispersion and attenuation are derived. General properties and asymptotic behavior of attenuation and dispersion in the low and high frequency range are studied. The results are compatible with experiments. The relation between the asymptotic properties of attenuation and wavefront singularities is examined. The theory is applied to some classes of viscoelastic models and to the quasi-linear attenuation reported in seismology.
Electromagnetic Media with no Dispersion Equation
Ismo V. Lindell; Alberto Favaro
2013-03-25
It has been known through some examples that parameters of an electromagnetic medium can be so defined that there is no dispersion equation (Fresnel equation) to restrict the choice of the wave vector of a plane wave in such a medium, i.e., that the dispersion equation is satisfied identically for any wave vector. In the present paper, a more systematic study to define classes of media with no dispersion equation is attempted. The analysis makes use of coordinate-free four-dimensional formalism in terms of multivectors, multiforms and dyadics.
Localized Induction Equation for Stretched Vortex Filament
Kimiaki Konno; Hiroshi Kakuhata
2006-03-02
We study numerically the motion of the stretched vortex filaments by using the localized induction equation with the stretch and that without the stretch.
A counterexample against the Vlasov equation
C. Y. Chen
2009-04-19
A simple counterexample against the Vlasov equation is put forward, in which a magnetized plasma is perturbed by an electromagnetic standing wave.
Linear Equation in Finite Dimensional Algebra
Aleks Kleyn
2012-04-30
In the paper I considered methods for solving equations of the form axb+cxd=e in the algebra which is finite dimensional over the field.
Quadratic Equation over Associative D-Algebra
Aleks Kleyn
2015-05-30
In this paper, I treat quadratic equation over associative $D$-algebra. In quaternion algebra $H$, the equation $x^2=a$ has either $2$ roots, or infinitely many roots. Since $a\\in R$, $aequation has infinitely many roots. Otherwise, the equation has roots $x_1$, $x_2$, $x_2=-x_1$. I considered different forms of the Viete's theorem and a possibility to apply the method of completing the square. Assumed the hypothesis that, in noncommutative algebra, the equation $$(x-b)(x-a)+(x-a)(x-c)=0$$ $b\
The Fractional Kinetic Equation and Thermonuclear Functions
H. J. Haubold; A. M. Mathai
2000-01-16
The paper discusses the solution of a simple kinetic equation of the type used for the computation of the change of the chemical composition in stars like the Sun. Starting from the standard form of the kinetic equation it is generalized to a fractional kinetic equation and its solutions in terms of H-functions are obtained. The role of thermonuclear functions, which are also represented in terms of G- and H-functions, in such a fractional kinetic equation is emphasized. Results contained in this paper are related to recent investigations of possible astrophysical solutions of the solar neutrino problem.
Nonequilibrium Spin Magnetization Quantum Transport Equations
Buot, F A; Otadoy, R E S; Villarin, D L
2011-01-01
The classical Bloch equations of spin magnetization transport is extended to fully time-dependent and highly-nonlinear nonequilibrium quantum distribution function (QDF) transport equations. The leading terms consist of the Boltzmann kinetic equation with spin-orbit coupling in a magnetic field together with spin-dependent scattering terms which do not have any classical analogue, but should incorporate the spatio-temporal-dependent phase-space dynamics of Elliot-Yafet and D'yakonov-Perel scatterings. The resulting magnetization QDF transport equation serves as a foundation for computational spintronic and nanomagnetic device applications, in performing simulation of ultrafast-switching-speed/low-power performance and reliability analyses.
Super compact equation for water waves
Dyachenko, A I; Zakharov, V E
2015-01-01
We derive very simple compact equation for gravity water waves which includes nonlinear wave term (`a la NLSE) and advection term (may results in wave breaking).
Scalable Equation of State Capability
Epperly, T W; Fritsch, F N; Norquist, P D; Sanford, L A
2007-12-03
The purpose of this techbase project was to investigate the use of parallel array data types to reduce the memory footprint of the Livermore Equation Of State (LEOS) library. Addressing the memory scalability of LEOS is necessary to run large scientific simulations on IBM BG/L and future architectures with low memory per processing core. We considered using normal MPI, one-sided MPI, and Global Arrays to manage the distributed array and ended up choosing Global Arrays because it was the only communication library that provided the level of asynchronous access required. To reduce the runtime overhead using a parallel array data structure, a least recently used (LRU) caching algorithm was used to provide a local cache of commonly used parts of the parallel array. The approach was initially implemented in a isolated copy of LEOS and was later integrated into the main trunk of the LEOS Subversion repository. The approach was tested using a simple test. Testing indicated that the approach was feasible, and the simple LRU caching had a 86% hit rate.
Comment on ``Thermodynamically Admissible 13 Moment Equations from the Boltzmann Equation''
, they do not include classical hydrodynam- ics in the limit of small Knudsen numbers. The hydro- dynamic to the equations of hydrodynamics in the limit of small Knudsen numbers. Presently, the R13 equations have
Renormalized asymptotic solutions of the Burgers equation and the Korteweg-de Vries equation
Sergei V. Zakharov
2015-01-12
The Cauchy problem for the Burgers equation and the Korteweg-de Vries equation is considered. Uniform renormalized asymptotic solutions are constructed in cases of a large initial gradient and a perturbed initial weak discontinuity.
CONTROL VALVE TESTING PROCEDURES AND EQUATIONS
Rahmeyer, William J.
APPENDIX A CONTROL VALVE TESTING PROCEDURES AND EQUATIONS FOR LIQUID FLOWS #12;APPENDIX A CONTROL VALVE TESTING PROCEDURES AND EQUATIONS FOR LIQUID FLOWS 2 Cv Q P Sg net gpm net = / Cv = Q P / Sg 75 is used to relate the pressure loss of a valve to the discharge of the valve at a given valve opening
Optimization and Nonlinear Equations Gordon K. Smyth
Smyth, Gordon K.
Optimization and Nonlinear Equations Gordon K. Smyth May 1997 Optimization means to find that value of x which max imizes or minimizes a given function f(x). The idea of optimization goes to the heart with respect to the components of x. Except in linear cases, optimization and equation solving invariably
The Papapetrou equations and supplementary conditions
O. B. Karpov
2004-06-02
On the bases of the Papapetrou equations with various supplementary conditions and other approaches a comparative analysis of the equations of motion of rotating bodies in general relativity is made. The motion of a body with vertical spin in a circular orbit is considered. An expression for the spin-orbit force in a post-Newtonian approximation is investigated.
A Piece of Magic The Dirac Equation
Satija, Indu
Â·ss destruction nor generators of inexhaustible energy entered his ken. Of all the equations of physics, perhaps-five-year-old recent convert from electrical engineering to theoretical physics, produced a remarkable equation theoretical imperatives (some of which we now know to be wrong). Dirac sought to embody these principles
Switched differential algebraic equations Stephan Trenn
Trenn, Stephan
Switched differential algebraic equations Stephan Trenn Abstract In this chapter an electrical circuit with switches is modeled as a switched differential algebraic equation (switched DAE), i.e. each is the input. The resulting time-variance follows from the action of the switches present in the circuit
NOTE / NOTE Allometric equations for young northern
Battles, John
. Vadeboncoeur, Mary A. Arthur, Russell D. Briggs, and Carrie R. Levine Abstract: Estimates of aboveground-specific equations for estimating aboveground biomass Farrah R. Fatemi, Ruth D. Yanai, Steven P. Hamburg, Matthew A relationships. Despite the widespread use of this approach, there is little information about whether equations
Derivation of a Stochastic Neutron Transport Equation
Edward J. Allen
2010-04-14
Stochastic difference equations and a stochastic partial differential equation (SPDE) are simultaneously derived for the time-dependent neutron angular density in a general three-dimensional medium where the neutron angular density is a function of position, direction, energy, and time. Special cases of the equations are given such as transport in one-dimensional plane geometry with isotropic scattering and transport in a homogeneous medium. The stochastic equations are derived from basic principles, i.e., from the changes that occur in a small time interval. Stochastic difference equations of the neutron angular density are constructed, taking into account the inherent randomness in scatters, absorptions, and source neutrons. As the time interval decreases, the stochastic difference equations lead to a system of Ito stochastic differential equations (SDEs). As the energy, direction, and position intervals decrease, an SPDE is derived for the neutron angular density. Comparisons between numerical solutions of the stochastic difference equations and independently formulated Monte Carlo calculations support the accuracy of the derivations.
Hopf algebras and Dyson-Schwinger equations
Stefan Weinzierl
2015-06-30
In these lectures I discuss Hopf algebras and Dyson-Schwinger equations. The lectures start with an introduction to Hopf algebras, followed by a review where Hopf algebras occur in particles physics. The final part of these lectures is devoted to the relation between Hopf algebras and Dyson-Schwinger equations.
Proca Equation for Attosecond Electron Pulses
Magdalena Pelc; Janina Marciak-Kozlowska; Miroslaw Kozlowski
2008-03-03
In this paper the heat transport of attosecond electron pulses is investigated. It is shown that attosecond electrons can propagate as thermal waves or diffused as particle conglommerates, Proca equation as type equation for the thermal transport of the attosecond electron pulsem is formulated
JOURNAL OF INTEGRAL EQUATIONS AND APPLICATIONS
Atkinson, Kendall
of numerical methods for calculating fixed points of nonlinear integral operators. The emphasis is on general differential equations, and the methods used are very different than those used for Fredholm integral operatorsJOURNAL OF INTEGRAL EQUATIONS AND APPLICATIONS Volume 4, Number 1, Winter 1992 A SURVEY
DELAY DIFFERENTIAL EQUATIONS IN SINGLE SPECIES DYNAMICS
Ruan, Shigui
dynamics. Let x(t) denote the population size at time t; let b and d denote the birth rate and death rate Equations 10. Periodicity 11. State Dependent Delays 12. Diffusive Models with Delay References O. Arino et rate of the population. The solution of equation (1.1) with an initial population x(0) = x0 is given
Comment on ``Discrete Boltzmann Equation for Microfluidics''
Luo, Li-Shi
Comment on ``Discrete Boltzmann Equation for Microfluidics'' In a recent Letter [1], Li and Kwok use a lattice Boltzmann equation (LBE) for microfluidics. Their main claim is that an LBE model for microfluidics can be constructed based on the ``Bhatnagar-Gross-Kooky [sic]'' model by including ``the
Diffractive Nonlinear Geometrical Optics for Variational Wave Equations and the Einstein Equations
Giuseppe Ali; John K. Hunter
2005-11-02
We derive an asymptotic solution of the vacuum Einstein equations that describes the propagation and diffraction of a localized, large-amplitude, rapidly-varying gravitational wave. We compare and contrast the resulting theory of strongly nonlinear geometrical optics for the Einstein equations with nonlinear geometrical optics theories for variational wave equations.
The generalized Schrödinger–Langevin equation
Bargueño, Pedro; Miret-Artés, Salvador
2014-07-15
In this work, for a Brownian particle interacting with a heat bath, we derive a generalization of the so-called Schrödinger–Langevin or Kostin equation. This generalization is based on a nonlinear interaction model providing a state-dependent dissipation process exhibiting multiplicative noise. Two straightforward applications to the measurement process are then analyzed, continuous and weak measurements in terms of the quantum Bohmian trajectory formalism. Finally, it is also shown that the generalized uncertainty principle, which appears in some approaches to quantum gravity, can be expressed in terms of this generalized equation. -- Highlights: •We generalize the Kostin equation for arbitrary system–bath coupling. •This generalization is developed both in the Schrödinger and Bohmian formalisms. •We write the generalized Kostin equation for two measurement problems. •We reformulate the generalized uncertainty principle in terms of this equation.
Klein-Gordon Equation in Hydrodynamical Form
Cheuk-Yin Wong
2010-12-22
We follow and modify the Feshbach-Villars formalism by separating the Klein-Gordon equation into two coupled time-dependent Schroedinger equations for particle and antiparticle wave function components with positive probability densities. We find that the equation of motion for the probability densities is in the form of relativistic hydrodynamics where various forces have their classical counterparts, with the additional element of the quantum stress tensor that depends on the derivatives of the amplitude of the wave function. We derive the equation of motion for the Wigner function and we find that its approximate classical weak-field limit coincides with the equation of motion for the distribution function in the collisionless kinetic theory.
Decoupling vector wave equation, Proca and Maxwell equations in Petrov type N space-times
Koray Düzta?; ?brahim Semiz
2015-08-23
In this work we use Newman-Penrose (NP) two-spinor formalism to derive decoupled equations for vector fields in Petrov type N space-times. In the NP formalism, a four vector can be represented by one complex and two real scalars. Then, a decoupled second order differential equation for one of the real scalars can be derived from the vector wave equation if the space-time is of type N. The solution for this scalar can --in principle-- be used to derive decoupled equations for the other scalars. These results can be directly applies to Proca equation for massive vector fields. We also evaluate Maxwell equations in terms of NP complex scalars of electromagnetism. We derive a decoupled second order differential equation for $\\phi_0$, valid in type N space-times. Substituting any solution for $\\phi_0$ in Maxwell equations, leads to two first order differential equations for $\\phi_1$. We show that these first order equations identically satisfy integrability conditions. Thus, any solution for $\\phi_0$ guarantees the existence of a solution for $\\phi_1$, via either of the first order differential equations.
New wave equation for ultrarelativistic particles
Ginés R. Pérez Teruel
2014-12-15
Starting from first principles and general assumptions based on the energy-momentum relation of the Special Theory of Relativity we present a novel wave equation for ultrarelativistic matter. This wave equation arises when particles satisfy the condition, $p>>m$, i.e, when the energy-momentum relation can be approximated by, $E\\simeq p+\\frac{m^{2}}{2p}$. Interestingly enough, such as the Dirac equation, it is found that this wave equation includes spin in a natural way. Furthermore, the free solutions of this wave equation contain plane waves that are completely equivalent to those of the theory of neutrino oscillations. Therefore, the theory reproduces some standard results of the Dirac theory in the limit $p>>m$, but offers the possibility of an explicit Lorentz Invariance Violation of order, $\\mathcal{O}((mc)^{4}/p^{2})$. As a result, the theory could be useful to test small departures from Dirac equation and Lorentz Invariance at very high energies. On the other hand, the wave equation can also describe particles of spin 1 by a simple substitution of the spin operators, $\\boldsymbol{\\sigma}\\rightarrow\\boldsymbol{\\alpha}$. In addition, it naturally admits a Lagrangian formulation and a Hamiltonian formalism. We also discuss the associated conservation laws that arise through the symmetry transformations of the Lagrangian.
Gravitational instability via the Schrodinger equation
C. J. Short; P. Coles
2006-11-22
We explore a novel approach to the study of large-scale structure formation in which self-gravitating cold dark matter (CDM) is represented by a complex scalar field whose dynamics are governed by coupled Schrodinger and Poisson equations. We show that, in the quasi-linear regime, the Schrodinger equation can be reduced to the free-particle Schrodinger equation. We advocate using the free-particle Schrodinger equation as the basis of a new approximation method - the free-particle approximation - that is similar in spirit to the successful adhesion model. In this paper we test the free-particle approximation by appealing to a planar collapse scenario and find that our results are in excellent agreement with those of the Zeldovich approximation, provided care is taken when choosing a value for the effective Planck constant in the theory. We also discuss how extensions of the free-particle approximation are likely to require the inclusion of a time-dependent potential in the Schrodinger equation. Since the Schrodinger equation with a time-dependent potential is typically impossible to solve exactly, we investigate whether standard quantum-mechanical approximation techniques can be used, in a cosmological setting, to obtain useful solutions of the Schrodinger equation. In this paper we focus on one particular approximation method: time-dependent perturbation theory (TDPT). We elucidate the properties of perturbative solutions of the Schrodinger equation by considering a simple example: the gravitational evolution of a plane-symmetric density fluctuation. We use TDPT to calculate an approximate solution of the relevant Schrodinger equation and show that this perturbative solution can be used to successfully follow gravitational collapse beyond the linear regime, but there are several pitfalls to be avoided.
Uniqueness theorems for equations of Keldysh Type
Thomas H. Otway
2010-05-25
A fundamental result that characterizes elliptic-hyperbolic equations of Tricomi type, the uniqueness of classical solutions to the open Dirichlet problem, is extended to a large class of elliptic-hyperbolic equations of Keldysh type. The result implies the non-existence of classical solutions to the closed Dirichlet problem for this class of equations. A uniqueness theorem is also proven for a mixed Dirichlet-Neumann problem. A generalized uniqueness theorem for the adjoint operator leads to the existence of distribution solutions to the closed Dirichlet problem in a special case.
Integral equations of scattering in one dimension
Vania E. Barlette; Marcelo M. Leite; Sadhan K. Adhikari
2001-03-05
A self-contained discussion of integral equations of scattering is presented in the case of centrally-symmetric potentials in one dimension, which will facilitate the understanding of more complex scattering integral equations in two and three dimensions. The present discussion illustrates in a simple fashion the concept of partial-wave decomposition, Green's function, Lippmann-Schwinger integral equations of scattering for wave function and transition operator, optical theorem and unitarity relation. We illustrate the present approach with a Dirac delta potential.
Supersymmetric Ito equation: Bosonization and exact solutions
Ren Bo; Yu Jun [Institute of Nonlinear Science, Shaoxing University, Shaoxing 312000 (China); Lin Ji [Institute of Nonlinear Physics, ZheJiang Normal University, Jinhua, 321004 (China)
2013-04-15
Based on the bosonization approach, the N=1 supersymmetric Ito (sIto) system is changed to a system of coupled bosonic equations. The approach can effectively avoid difficulties caused by intractable fermionic fields which are anticommuting. By solving the coupled bosonic equations, the traveling wave solutions of the sIto system are obtained with the mapping and deformation method. Some novel types of exact solutions for the supersymmetric system are constructed with the solutions and symmetries of the usual Ito equation. In the meanwhile, the similarity reduction solutions of the model are also studied with the Lie point symmetry theory.
Wave Propagation Theory 2.1 The Wave Equation
2 Wave Propagation Theory 2.1 The Wave Equation The wave equation in an ideal fluid can be derived from hydrodynamics and the adia- batic relation between pressure and density. The equation for conservation of mass, Euler's equation (Newton's 2nd Law), and the adiabatic equation of state are respec
Integral representation of solutions to Fuchsian system and Heun's equation
Kouichi Takemura
2007-07-09
We obtain integral representations of solutions to special cases of the Fuchsian system of differential equations and Heun's differential equation. In particular, we calculate the monodromy of solutions to the Fuchsian equation that corresponds to Picard's solution of the sixth Painlev\\'e equation, and to Heun's equation.
Atilhan, Mert
2004-09-30
and thermophysical properties of natural gas for practical engineering applications. This thesis presents a new cubic EOS for pure argon. In this work, a theoretically based EOS represents the PVT behavior of pure fluids. The new equation has its basis...
Integral equations, fractional calculus and shift operator
D. Babusci; G. Dattoli; D. Sacchetti
2010-07-29
We present an extension of a previously developed method employing the formalism of the fractional derivatives to solve new classes of integral equations. This method uses different forms of integral operators that generalizes the exponential shift operator.
Adaptive FE Methods for Conservation Equations
Hartmann, Ralf
Adaptive FE Methods for Conservation Equations Ralf Hartmann Abstract. We present an approach, University of Heidelberg. #12; 2 R. Hartmann by parts on each cell K results in X K2Th h (F (u); rv)K + (F (u
Generalized finite element method for Helmholtz equation
Hidajat, Realino Lulie
2009-05-15
This dissertation presents the Generalized Finite Element Method (GFEM) for the scalar Helmholtz equation, which describes the time harmonic acoustic wave propagation problem. We introduce several handbook functions for ...
MA262: Linear Algebra And Differential Equations
2015-02-23
Feb 23, 2015 ... carrying capacity, which models the maximal allowable population in an environment. A sketch. 9 ... mixig problems and electric circuits. 3.1 1st order ..... derivative, plug into the original equation and solve V . – A Bernoulli DE ...
Inverse Problems for Fractional Diffusion Equations
Zuo, Lihua
2013-06-21
; t > 0; (1.13) combined with the initial condition u(x; 0) = f(x); ?1 : u^t = ?s2u^; t > 0; u^(s; 0) = f^(s): Solving the above equation, we obtain u...
Electric-Magnetic Duality and WDVV Equations
B. de Wit; A. Marshakov
2001-06-11
We consider the associativity (or WDVV) equations in the form they appear in Seiberg-Witten theory and prove that they are covariant under generic electric-magnetic duality transformations. We discuss the consequences of this covariance from various perspectives.
ADAPTIVE DISCRETIZATION OF AN INTEGRODIFFERENTIAL EQUATION
Larsson, Stig
ADAPTIVE DISCRETIZATION OF AN INTEGROÂDIFFERENTIAL EQUATION MODELING QUASIÂSTATIC FRACTIONAL ORDER VISCOELASTICITY Klas Adolfsson # Mikael Enelund ## Stig Larsson ### # Department of Applied Mechanics, Chalmers Mechanics, Chalmers University of Technology, SE--412 96 GË?oteborg, Sweden, mikael
A Lagrangian for the quantionic field equation
Samir Lipovaca
2010-03-02
The purpose of this paper is to present a Lagrangian from which we can derive the quantionic field equation written in the Dirac gauge using the principle of stationary action.
Equator Appliance: ENERGY STAR Referral (EZ 3720)
Broader source: Energy.gov [DOE]
DOE referred Equator Appliance clothes washer EZ 3720 to EPA, brand manager of the ENERGY STAR program, for appropriate action after DOE testing revealed that the model does not meet ENERGY STAR requirements.
On the solutions to the string equation
A. Schwarz
1991-09-10
The set of solutions to the string equation $[P,Q]=1$ where $P$ and $Q$ are differential operators is described.It is shown that there exists one-to-one correspondence between this set and the set of pairs of commuting differential operators.This fact permits us to describe the set of solutions to the string equation in terms of moduli spa- ces of algebraic curves,however the direct description is much simpler. Some results are obtained for the superanalog to the string equation where $P$ and $Q$ are considered as superdifferential operators. It is proved that this equation is invariant with respect to Manin-Radul, Mulase-Rabin and Kac-van de Leur KP-hierarchies.
MATH 411 SPRING 2001 Ordinary Differential Equations
Alekseenko, Alexander
MATH 411 SPRING 2001 Ordinary Differential Equations Schedule # 749025 TR 01:00-02:15 316 Boucke Instructor: Alexander Alekseenko, 328 McAllister, 865-1984, alekseen@math.psu.edu The course
Charging Capacitors According to Maxwell's Equations: Impossible
Daniele Funaro
2014-11-02
The charge of an ideal parallel capacitor leads to the resolution of the wave equation for the electric field with prescribed initial conditions and boundary constraints. Independently of the capacitor's shape and the applied voltage, none of the corresponding solutions is compatible with the full set of Maxwell's equations. The paradoxical situation persists even by weakening boundary conditions, resulting in the impossibility to describe a trivial phenomenon such as the capacitor's charging process, by means of the standard Maxwellian theory.
Integration Rules for Loop Scattering Equations
Baadsgaard, Christian; Bourjaily, Jacob L; Damgaard, Poul H; Feng, Bo
2015-01-01
We formulate new integration rules for one-loop scattering equations analogous to those at tree-level, and test them in a number of non-trivial cases for amplitudes in scalar $\\phi^3$-theory. This formalism greatly facilitates the evaluation of amplitudes in the CHY representation at one-loop order, without the need to explicitly sum over the solutions to the loop-level scattering equations.
Finite Element Analysis of the Schroedinger Equation
Avtar S. Sehra
2007-04-17
The purpose of this work is to test the application of the finite element method to quantum mechanical problems, in particular for solving the Schroedinger equation. We begin with an overview of quantum mechanics, and standard numerical techniques. We then give an introduction to finite element analysis using the diffusion equation as an example. Three numerical time evolution methods are considered: the (tried and tested) Crank-Nicolson method, the continuous space-time method, and the discontinuous space-time method.
Lagrangian submanifolds and Hamilton-Jacobi equation
M. Barbero-Liñán; M. de León; D. Martín de Diego
2012-09-04
Lagrangian submanifolds are becoming a very essential tool to generalize and geometrically understand results and procedures in the area of mathematical physics. Here we use general Lagrangian submanifolds to provide a geometric version of the Hamilton-Jacobi equation. This interpretation allows us to study some interesting applications of Hamilton-Jacobi equation in holonomic, nonholonomic and time-dependent dynamics from a geometrical point of view.
Conformally Invariant Spinorial Equations in Six Dimensions
Carlos Batista
2015-06-04
This work deals with the conformal transformations in six-dimensional spinorial formalism. Several conformally invariant equations are obtained and their geometrical interpretation are worked out. Finally, the integrability conditions for some of these equations are established. Moreover, in the course of the article, some useful identities involving the curvature of the spinorial connection are attained and a digression about harmonic forms and more general massless fields is made.
Symmetric Instantons and Discrete Hitchin Equations
Ward, R S
2015-01-01
Self-dual Yang-Mills instantons on $R^4$ correspond to algebraic ADHM data. This paper describes how to specialize such ADHM data so that the instantons have a $T^2$ symmetry, and this in turn motivates an integrable discrete version of the 2-dimensional Hitchin equations. It is analogous to the way in which the ADHM data for $S^1$-symmetric instantons, or hyperbolic BPS monopoles, may be viewed as a discretization of the Nahm equations.
Painleve VI, Rigid Tops and Reflection Equation
A. Levin; M. Olshanetsky; A. Zotov
2006-06-01
We show that the Painlev{\\'e} VI equation has an equivalent form of the non-autonomous Zhukovsky-Volterra gyrostat. This system is a generalization of the Euler top in $C^3$ and include the additional constant gyrostat momentum. The quantization of its autonomous version is achieved by the reflection equation. The corresponding quadratic algebra generalizes the Sklyanin algebra. As by product we define integrable XYZ spin chain on a finite lattice with new boundary conditions.
A Reduced Basis Element Approach for the Reynolds Lubrication Equation
A Reduced Basis Element Approach for the Reynolds Lubrication Equation Lösen der Reynolds Reynolds Lubrication Equation 8 2.1 Introduction of the application, background setting . . . . . . . . . . . . . . . . . . . . . . 8 2.2 Reynolds Lubrication Equation
Pierantozzi, T.; Vazquez, L.
2005-11-01
Through fractional calculus and following the method used by Dirac to obtain his well-known equation from the Klein-Gordon equation, we analyze a possible interpolation between the Dirac and the diffusion equations in one space dimension. We study the transition between the hyperbolic and parabolic behaviors by means of the generalization of the D'Alembert formula for the classical wave equation and the invariance under space and time inversions of the interpolating fractional evolution equations Dirac like. Such invariance depends on the values of the fractional index and is related to the nonlocal property of the time fractional differential operator. For this system of fractional evolution equations, we also find an associated conserved quantity analogous to the Hamiltonian for the classical Dirac case.
Fairag, Faisal
in engines and drag reduction), aerodynamics (maneuvering flight of jet aircraft) and biophysical a practical engineering point of view, the study of Ladyzhenskaya equations and of proprieties
Combined Wronskian solutions to the 2D Toda molecule equation
Ma, Wen-Xiu
partic- ular solutions [14, 15], and thus possess linear subspaces of solutions. Therefore, though soliton equations are nonlinear, they are good neighbors to linear equations. However, given
From the Boltzmann equation to fluid mechanics on a manifold
Peter J. Love; Donato Cianci
2012-08-27
We apply the Chapman-Enskog procedure to derive hydrodynamic equations on an arbitrary surface from the Boltzmann equation on the surface.
A new nonlinear generalization of the Dirac equation
Nikolay Marchuk
2013-07-24
We postulate a new nonlinear generalization of the Dirac equation for an electron. Basic properties of the new equation are considered.
Two standard methods for solving the Ito equation
Alvaro Salas Salas
2008-05-21
In this paper we show some exact solutions for the Ito equation. These solutions are obtained by two methods: the tanh method and the projective Riccati equation method.
Hyperinstantons, the Beltrami Equation, and Triholomorphic Maps
Fré, P; Sorin, A S
2015-01-01
We consider the Beltrami equation for hydrodynamics and we show that its solutions can be viewed as instanton solutions of a more general system of equations. The latter are the equations of motion for an ${\\cal N}=2$ sigma model on 4-dimensional worldvolume (which is taken locally HyperK\\"ahler) with a 4-dimensional HyperK\\"ahler target space. By means of the 4D twisting procedure originally introduced by Witten for gauge theories and later generalized to 4D sigma-models by Anselmi and Fr\\'e, we show that the equations of motion describe triholomophic maps between the worldvolume and the target space. Therefore, the classification of the solutions to the 3-dimensional Beltrami equation can be performed by counting the triholomorphic maps. The counting is easily obtained by using several discrete symmetries. Finally, the similarity with holomorphic maps for ${\\cal N}=2$ sigma on Calabi-Yau space prompts us to reformulate the problem of the enumeration of triholomorphic maps in terms of a topological sigma mod...
Fourier transform of the 3d NS equations The 3d NS equations are
Salmon, Rick
1 Fourier transform of the 3d NS equations The 3d NS equations are (1) vi t + vj vi xj = - p xi easily add it in at the end. Our interest is in the advection and pressure terms. Introducing the Fourier transforms (2) vi x( ) = ui k( )eikx k p x( ) = p k( )eikx k we obtain the Fourier transform of (1
Differential Equations I Lab #8: Differential Equations and Linear Algebra with Mathematica
Peckham, Bruce B.
and NDSolve for differential equations, and LinearSolve, Eigenvector, Eigen- value, NullSpace, Inverse/instructor each output line generated by Mathematica. If you elect to write a report, your report should include an analytical solution to y + 3y + 2y = 3e4t. #12;2. The logistic differential equation (again). Consider
H. Kleinert; V. Zatloukal
2015-03-05
The statistics of rare events, the so-called black-swan events, is governed by non-Gaussian distributions with heavy power-like tails. We calculate the Green functions of the associated Fokker-Planck equations and solve the related stochastic differential equations. We also discuss the subject in the framework of path integration.
Electromagnetic field with constraints and Papapetrou equation
Z. Ya. Turakulov; A. T. Muminov
2006-01-12
It is shown that geometric optical description of electromagnetic wave with account of its polarization in curved space-time can be obtained straightforwardly from the classical variational principle for electromagnetic field. For this end the entire functional space of electromagnetic fields must be reduced to its subspace of locally plane monochromatic waves. We have formulated the constraints under which the entire functional space of electromagnetic fields reduces to its subspace of locally plane monochromatic waves. These constraints introduce variables of another kind which specify a field of local frames associated to the wave and contain some congruence of null-curves. The Lagrangian for constrained electromagnetic field contains variables of two kinds, namely, a congruence of null-curves and the field itself. This yields two kinds of Euler-Lagrange equations. Equations of first kind are trivial due to the constraints imposed. Variation of the curves yields the Papapetrou equations for a classical massless particle with helicity 1.
Chemical potential and the gap equation
Huan Chen; Wei Yuan; Lei Chang; Yu-Xin Liu; Thomas Klahn; Craig D. Roberts
2008-07-17
In general the kernel of QCD's gap equation possesses a domain of analyticity upon which the equation's solution at nonzero chemical potential is simply obtained from the in-vacuum result through analytic continuation. On this domain the single-quark number- and scalar-density distribution functions are mu-independent. This is illustrated via two models for the gap equation's kernel. The models are alike in concentrating support in the infrared. They differ in the form of the vertex but qualitatively the results are largely insensitive to the Ansatz. In vacuum both models realise chiral symmetry in the Nambu-Goldstone mode and in the chiral limit, with increasing chemical potential, exhibit a first-order chiral symmetry restoring transition at mu~M(0), where M(p^2) is the dressed-quark mass function. There is evidence to suggest that any associated deconfinement transition is coincident and also of first-order.
Bulk equations of motion from CFT correlators
Daniel Kabat; Gilad Lifschytz
2015-07-27
To O(1/N) we derive, purely from CFT data, the bulk equations of motion for interacting scalar fields and for scalars coupled to gauge fields and gravity. We first uplift CFT operators to mimic local AdS fields by imposing bulk microcausality. This requires adding an infinite tower of smeared higher-dimension double-trace operators to the CFT definition of a bulk field, with coefficients that we explicitly compute. By summing the contribution of the higher-dimension operators we derive the equations of motion satisfied by these uplifted CFT operators and show that we precisely recover the expected bulk equations of motion. We exhibit the freedom in the CFT construction which corresponds to bulk field redefinitions.
Bulk equations of motion from CFT correlators
Kabat, Daniel
2015-01-01
To O(1/N) we derive, purely from CFT data, the bulk equations of motion for interacting scalar fields and for scalars coupled to gauge fields and gravity. We first uplift CFT operators to mimic local AdS fields by imposing bulk microcausality. This requires adding an infinite tower of smeared higher-dimension double-trace operators to the CFT definition of a bulk field, with coefficients that we explicitly compute. By summing the contribution of the higher-dimension operators we derive the equations of motion satisfied by these uplifted CFT operators and show that we precisely recover the expected bulk equations of motion. We exhibit the freedom in the CFT construction which corresponds to bulk field redefinitions.
R. A. Soltz
2009-09-14
We present results from recent calculations of the QCD equation of state by the HotQCD Collaboration and review the implications for hydrodynamic modeling. The equation of state of QCD at zero baryon density was calculated on a lattice of dimensions $32^3 \\times 8$ with $m_l = 0.1 m_s$ (corresponding to a pion mass of $\\sim$220 MeV) using two improved staggered fermion actions, p4 and asqtad. C alculations were performed along lines of constant physics using more than 100M cpu-hours on BG/L supercomputers at LLNL, NYBlue, and SDSC. We present paramete rizations of the equation of state suitable for input into hydrodynamics models of heavy ion collisions.
Some Wave Equations for Electromagnetism and Gravitation
Zi-Hua Weng
2010-08-11
The paper studies the inferences of wave equations for electromagnetic fields when there are gravitational fields at the same time. In the description with the algebra of octonions, the inferences of wave equations are identical with that in conventional electromagnetic theory with vector terminology. By means of the octonion exponential function, we can draw out that the electromagnetic waves are transverse waves in a vacuum, and rephrase the law of reflection, Snell's law, Fresnel formula, and total internal reflection etc. The study claims that the theoretical results of wave equations for electromagnetic strength keep unchanged in the case for coexistence of gravitational and electromagnetic fields. Meanwhile the electric and magnetic components of electromagnetic waves can not be determined simultaneously in electromagnetic fields.
Interplay of Boltzmann equation and continuity equation for accelerated electrons in solar flares
Codispoti, Anna
2015-01-01
During solar flares a large amount of electrons are accelerated within the plasma present in the solar atmosphere. Accurate measurements of the motion of these electrons start becoming available from the analysis of hard X-ray imaging-spectroscopy observations. In this paper, we discuss the linearized perturbations of the Boltzmann kinetic equation describing an ensemble of electrons accelerated by the energy release occurring during solar flares. Either in the limit of high energy or at vanishing background temperature such an equation reduces to a continuity equation equipped with an extra force of stochastic nature. This stochastic force is actually described by the well known energy loss rate due to Coulomb collision with ambient particles, but, in order to match the collision kernel in the linearized Boltzmann equation it needs to be treated in a very specific manner. In the second part of the paper the derived continuity equation is solved with some hyperbolic techniques, and the obtained solution is wr...
Relativistic Wave Equations: An Operational Approach
G. Dattoli; E. Sabia; K. Górska; A. Horzela; K. A. Penson
2015-02-02
The use of operator methods of algebraic nature is shown to be a very powerful tool to deal with different forms of relativistic wave equations. The methods provide either exact or approximate solutions for various forms of differential equations, such as relativistic Schr\\"odinger, Klein-Gordon and Dirac. We discuss the free particle hypotheses and those relevant to particles subject to non-trivial potentials. In the latter case we will show how the proposed method leads to easily implementable numerical algorithms.
Changing the Equation in STEM Education
Broader source: Energy.gov [DOE]
Editor's Note: This is a cross post of an announcement that the White House featured on its blog last week. Check out the video below for Secretary Chu's thoughts on how an education in math and science helps students understand the world and deal with the pressing issues of our time. Today, President Obama announced the launch of Change the Equation, a CEO-led effort to dramatically improve education in science, technology, engineering, and math (STEM), as part of his “Educate to Innovate” campaign. Change the Equation is a non-profit organization dedicated to mobilizing the business community to improve the quality of STEM education in the United States.
Principle of Least Squares Regression Equations Residuals Correlation and Regression
Watkins, Joseph C.
Principle of Least Squares Regression Equations Residuals Topic 3 Correlation and Regression Linear Regression I 1 / 15 #12;Principle of Least Squares Regression Equations Residuals Outline Principle of Least Squares Regression Equations Residuals 2 / 15 #12;Principle of Least Squares Regression Equations
Kinetic equation for a soliton gas Chernogolovka, July 2009
Fominov, Yakov
Kinetic equation for a soliton gas Gennady El Chernogolovka, July 2009 Gennady El Kinetic equation, Kinetic equation for solitons, JETP (1971) Here we consider only strongly integrable systems (like KdV, NLS etc.) Gennady El Kinetic equation for a soliton gas #12;From N-solitons/N-gap potentials
Lecture by John F. Nash Jr. An Interesting Equation
Babu, G. Jogesh
Lecture by John F. Nash Jr. An Interesting Equation The equation that we have discovered is a 4th order covariant tensor partial differential equation applicable to the metric tensor of a spaceRp a s b - 1 2 gab Rps = 0 And this equation is formally divergence free in the same way
An energy estimate for a perturbed Hasegawa--Mima equation
Grauer, Rainer
An energy estimate for a perturbed Hasegawa--Mima equation Rainer Grauer Institut fÂ¨ur Theoretische transport at the plasma edge of a tokamak fusion reactor. A oneÂfield equation describing the electrostatic potential fluctuations in this regime is the soÂcalled Hasegawa--Mima equation. If this equation is driven
Evolution equations: Frobenius integrability, conservation laws and travelling waves
Geoff Prince; Naghmana Tehseen
2015-06-07
We give new results concerning the Frobenius integrability and solution of evolution equations admitting travelling wave solutions. In particular, we give a powerful result which explains the extraordinary integrability of some of these equations. We also discuss "local" conservations laws for evolution equations in general and demonstrate all the results for the Korteweg de Vries equation.
Cubic Nonlinear Schrodinger Equation with vorticity
Caliari, Marco
) Equation, plays a fundamental role in describing the hydrodynamics of a BoseEinstein condensate [4] (see Bose particles, as recently described within Stochastic Quantization by Lagrangian Variational the general one-particle Bose dynamics out of dynamical equilibrium. We observe that in the most simple
Pointwise Fourier Inversion: a Wave Equation Approach
Pointwise Fourier Inversion: a Wave Equation Approach Mark A. Pinsky1 Michael E. Taylor2. A general criterion for pointwise Fourier inversion 2. Pointwise Fourier inversion on Rn (n = 3) 3. Fourier inversion on R2 4. Fourier inversion on Rn (general n) 5. Fourier inversion on spheres 6. Fourier inversion
Differential Equations Math 125 Name Quiz Section
Burdzy, Krzysztof "Chris"
-trivial applications of Differential Equations. Forensic Mathematics A detective discovers a murder victim in a hotel the murder took place. Let u(t) be the temperature of the body after t hours. By Newton's Law of Cooling we = 98.6 F and solve for t. At what time did the murder take place? #12;Spread of a Rumor The Xylocom
The Kinematic Algebras from the Scattering Equations
Monteiro, Ricardo
2013-01-01
We study kinematic algebras associated to the recently proposed scattering equations, which arise in the description of the scattering of massless particles. In particular, we describe the role that these algebras play in the BCJ duality between colour and kinematics in gauge theory, and its relation to gravity. We find that the scattering equations are a consistency condition for a self-dual-type vertex which is associated to each solution of those equations. We also identify an extension of the anti-self-dual vertex, such that the two vertices are not conjugate in general. Both vertices correspond to the structure constants of Lie algebras. We give a prescription for the use of the generators of these Lie algebras in trivalent graphs that leads to a natural set of BCJ numerators. In particular, we write BCJ numerators for each contribution to the amplitude associated to a solution of the scattering equations. This leads to a decomposition of the determinant of a certain kinematic matrix, which appears natur...
Nonlocal kinetic equation: integrable hydrodynamic reductions, symmetries
, Troitsk, Moscow Region, Russia Lebedev Physical Institute, Russian Academy of Sciences, Moscow § SISSA, Trieste, Italy, and Institute of Metal Physics, Urals Division of Russian Academy of Sciences, Ekaterinburg, Russia We study a new class of nonlinear kinetic equations recently derived in the context
Evolution equations in QCD and QED
M. Slawinska
2008-05-12
Evolution equations of YFS and DGLAP types in leading order are considered. They are compared in terms of mathematical properties and solutions. In particular, it is discussed how the properties of evolution kernels affect solutions. Finally, comparison of solutions obtained numerically are presented.
SYSTEMS OF FUNCTIONAL EQUATIONS MICHAEL DRMOTA
Drmota, Michael
of planted plane trees. Hence the corresponding generating function y(x) satis#12;es the functional equation the asymptotic properties of the coeÃ?cients of generating functions which satisfy a system of functional a recursive description then the generating function y(x) = P o2Y x joj = P n#21;0 yn x n satis#12;es
Conservation of Energy Thermodynamic Energy Equation
Hennon, Christopher C.
, is derived beginning with an alternative form of the 1st Law of Thermodynamics, the internal energy formConservation of Energy Thermodynamic Energy Equation The previous two sections dealt addresses the conservation of energy. The first law of thermodynamics, of which you should be very familiar
Optimal polarisation equations in FLRW universes
Tram, Thomas; Lesgourgues, Julien, E-mail: thomas.tram@epfl.ch, E-mail: Julien.Lesgourgues@cern.ch [Institut de Théorie des Phénomènes Physiques, École Polytechnique Fédérale de Lausanne, CH-1015, Lausanne (Switzerland)
2013-10-01
This paper presents the linearised Boltzmann equation for photons for scalar, vector and tensor perturbations in flat, open and closed FLRW cosmologies. We show that E- and B-mode polarisation for all types can be computed using only a single hierarchy. This was previously shown explicitly for tensor modes in flat cosmologies but not for vectors, and not for non-flat cosmologies.
Thermodynamics of viscoelastic fluids: the temperature equation.
Wapperom, Peter
Thermodynamics of viscoelastic fluids: the temperature equation. Peter Wapperom Martien A. Hulsen and Hydrodynamics Rotterdamseweg 145 2628 AL Delft (The Netherlands) Abstract From the thermodynamics with internal. The well- known stress differential models that fit into the thermodynamic theory will be treated
MULTIVARIATE PUBLIC KEY CRYPTOSYSTEMS FROM DIOPHANTINE EQUATIONS
Gao, Shuhong
MULTIVARIATE PUBLIC KEY CRYPTOSYSTEMS FROM DIOPHANTINE EQUATIONS SHUHONG GAO AND RAYMOND HEINDL for multivariate public key cryptosystems, which combines ideas from both triangular and oil-vinegar schemes. We the framework. 1. Introduction 1.1. Multivariate Public Key Cryptography. Public key cryptography plays
The general solution of Schrodigers differential equation
Nikos Bagis
2009-10-31
In this note we solve theoretically the Schrodingers differential equation using results based on our previous work which concern semigroup operators. Our method does not use eigenvectors or eigenvalues and the solution depends only from the selected base of the Hilbert space.
NOTES ON THE JACOBI EQUATION ALEXANDER LYTCHAK
Lytchak, Alexander
NOTES ON THE JACOBI EQUATION ALEXANDER LYTCHAK Abstract. We discuss some properties of Jacobi spaces of Jacobi fields and give some applica- tions to Riemannian geometry. 1. Introduction This note is essentially a collection of results about conjugate points of Jacobi fields for which we could not find
Partial Differential Equations of Electrostatic MEMS
Fournier, John J.F.
Partial Differential Equations of Electrostatic MEMS by Yujin Guo B.Sc., China Three Gorges) The University of British Columbia July 2007 c Yujin Guo 2007 #12;Abstract Micro-Electromechanical Systems (MEMS their initial development in the 1980s, MEMS has revolutionized numerous branches of science and industry
Equation of State of Uranium and Plutonium
Barroso, Dalton Ellery Girão
2015-01-01
The objective of this work is to define the parameters of the three-term equation of state for uranium and plutonium, appropriate for conditions in which these materials are subjected to strong shock compressions, as in cylindrical and spherical implosions. The three-term equation of state takes into account the three components of the pressure that resist to compression in the solid: the elastic or "cold" pressure (coulombian repulsion between atoms), the thermal pressure due to vibratory motion of atoms in the lattice of the solid and the thermal pressure of electrons thermally excited. The equation of state defined here permits also to take into account the variation of the specific heat with the transition of the solid to the liquid or gaseous state due to continued growth of temperature in strong shock compressions. In the definition of uranium equation of state, experimental data on the uranium compression, available in the open scientific literature, are used. In the plutonium case, this element was co...
Equation of State of Uranium and Plutonium
Dalton Ellery Girão Barroso
2015-07-13
The objective of this work is to define the parameters of the three-term equation of state for uranium and plutonium, appropriate for conditions in which these materials are subjected to strong shock compressions, as in cylindrical and spherical implosions. The three-term equation of state takes into account the three components of the pressure that resist to compression in the solid: the elastic or "cold" pressure (coulombian repulsion between atoms), the thermal pressure due to vibratory motion of atoms in the lattice of the solid and the thermal pressure of electrons thermally excited. The equation of state defined here permits also to take into account the variation of the specific heat with the transition of the solid to the liquid or gaseous state due to continued growth of temperature in strong shock compressions. In the definition of uranium equation of state, experimental data on the uranium compression, available in the open scientific literature, are used. In the plutonium case, this element was considered initially in the alpha-phase or stabilized in the delta-phase. In the last case, an abrupt and instantaneous transition to the alpha-phase was considered when the delta-phase plutonium is submitted to strong compressions.
Wave function derivation of the JIMWLK equation
Alexey V. Popov
2008-12-16
Using the stationary lightcone perturbation theory, we propose the complete and careful derivation the JIMWLK equation. We show that the rigorous treatment requires the knowledge of a boosted wave function with second order accuracy. Previous wave function approaches are incomplete and implicitly used the time ordered perturbation theory, which requires a usage of an external target field.
Elementary Differential Equations with Boundary Value Problems
William F. Trench
2014-02-24
Dec 1, 2013 ... For more information, please contact jcostanz@trinity.edu. ..... (Laplace's Equation), the functions defining the boundary conditions on a given side of the rectangular .... change in heat of the object as its temperature changes from T0 to T is a(T ...... Let y be the angle measured from the rest position (vertically ...
Construction of tree volume tables from integration of taper equations
Coffman, Jerry Gale
1973-01-01
) were used as a basis for comparison. The integrated taper equation appears to be as accurate as the tradi- 2 tional volume equation V a + bD H, but somewhat less accurate than volume equations involving form class measurements. A computer program... and help throughout my graduate career. TABLE OF CONTENTS CHAPTER Page I INTRODUCTION AND OBJECTIVES II REVIEW OF LITERATURE III METHODS Sample Data Procedure Analysis 1 Analysis 2 13 Integration of Taper Equations to Volume Equations Tests...
Master equation for a quantum particle in a gas
Klaus Hornberger
2006-09-05
The equation for the quantum motion of a Brownian particle in a gaseous environment is derived by means of S-matrix theory. This quantum version of the linear Boltzmann equation accounts non-perturbatively for the quantum effects of the scattering dynamics and describes decoherence and dissipation in a unified framework. As a completely positive master equation it incorporates both the known equation for an infinitely massive Brownian particle and the classical linear Boltzmann equation as limiting cases.
Equation of state and singularities in FLRW cosmological models
L. Fernandez-Jambrina; R. Lazkoz
2010-01-18
We consider FLRW cosmological models with standard Friedmann equations, but leaving free the equation of state. We assume that the dark energy content of the universe is encoded in an equation of state $p=f(\\rho)$, which is expressed with most generality in the form of a power expansion. The inclusion of this expansion in Friedmann equations allows us to construct a perturbative solution and to relate the coefficients of the equation of state with the formation of singularities of different types.
The Swing Equation: Power Form, PerUnit, Error 1.0 Power Form of Swing Equation
McCalley, James D.
1 The Swing Equation: Power Form, PerUnit, Error 1.0 Power Form of Swing Equation Recall from when the swing equation is written in perunit, the numerical value of the torque version) to analyze error in the power form of the swing equation. But before we do that, we need to define pu speed
REDUCTION OF THE EQUATION FOR LOWER HYBRID WAVES IN A PLASMA TO A NONLINEAR SCH~DINGER EQUATION
Karney, Charles
REDUCTION OF THE EQUATION FOR LOWER HYBRID WAVES IN A PLASMA TO A NONLINEAR SCH~DINGER EQUATION C h Equation* by Charles F. F. Karney Research Laboratory of Electronics and Plasma Fusion Center and Development Administration (Contract E(l1-11-3070) #12;Reduction sf the Equation for Lower Hybrid Waves
The Semiclassical Einstein Equation on Cosmological Spacetimes
Daniel Siemssen
2015-03-06
The subject of this thesis is the coupling of quantum fields to a classical gravitational background in a semiclassical fashion. It contains a thorough introduction into quantum field theory on curved spacetime with a focus on the stress-energy tensor and the semiclassical Einstein equation. Basic notions of differential geometry, topology, functional and microlocal analysis, causality and general relativity will be summarised, and the algebraic approach to QFT on curved spacetime will be reviewed. Apart from these foundations, the original research of the author and his collaborators will be presented: Together with Fewster, the author studied the up and down structure of permutations using their decomposition into so-called atomic permutations. The relevance of these results to this thesis is their application in the calculation of the moments of quadratic quantum fields. In a work with Pinamonti, the author showed the local and global existence of solutions to the semiclassical Einstein equation in flat cosmological spacetimes coupled to a scalar field by solving simultaneously for the quantum state and the Hubble function in an integral-functional equation. The theorem is proved with a fixed-point theorem using the continuous functional differentiability and boundedness of the integral kernel of the integral-functional equation. In another work with Pinamonti the author proposed an extension of the semiclassical Einstein equations which couples the moments of a stochastic Einstein tensor to the moments of the quantum stress-energy tensor. In a toy model of a Newtonianly perturbed exponentially expanding spacetime it is shown that the quantum fluctuations of the stress-energy tensor induce an almost scale-invariant power spectrum for the perturbation potential and that non-Gaussianties arise naturally.
Kovalyov, Mikhail [Department of Mathematics, University of Alberta, Edmonton, Alberta, T6G 2G1 (Canada)
2010-06-15
In this article the sets of solutions of the sine-Gordon equation and its linearization the Klein-Gordon equation are discussed and compared. It is shown that the set of solutions of the sine-Gordon equation possesses a richer structure which partly disappears during linearization. Just like the solutions of the Klein-Gordon equation satisfy the linear superposition principle, the solutions of the sine-Gordon equation satisfy a nonlinear superposition principle.
E. V. Shiryaeva; M. Yu. Zhukov
2014-10-10
In paper [S.I. Senashov, A. Yakhno. 2012. SIGMA. Vol.8. 071] the variant of the hodograph method based on the conservation laws for two hyperbolic quasilinear equations of the first order is described. Using these results we propose a method which allows to reduce the Cauchy problem for the two quasilinear PDE's to the Cauchy problem for ODE's. The proposed method is actually some similar method of characteristics for a system of two hyperbolic quasilinear equations. The method can be used effectively in all cases, when the linear hyperbolic equation in partial derivatives of the second order with variable coefficients, resulting from the application of the hodograph method, has an explicit expression for the Riemann-Green function. One of the method's features is the possibility to construct a multi-valued solutions. In this paper we present examples of method application for solving the classical shallow water equations.
Celso de Araujo Duarte
2015-10-15
Traditionally, the electromagnetic theory dictates the well-known second order differential equation for the components of the scalar and the vector potentials, or in other words, for the four-vector electromagnetic potential $\\phi^{\\mu}$. But the second order is not obligatory at least with respect to the electromagnetic radiation fields: actually, a heuristic first order differential equation can be constructed to describe the electromagnetic radiation, supported on the phenomenology of its electric and magnetic fields. Due to a formal similarity, such an equation suggests a direct comparative analysis with Dirac's equation for half spin fermions, conducting to the finding that the Dirac's spinor field $\\Psi$ for massive or massless fermions is equivalent to a set of two potential-like four vector fields $\\psi^{\\mu}$ and $\\chi^{\\mu}$. Under this point of view, striking similarities with the electromagnetic theory emerge with a category of "pseudo electric'' and "pseudo magnetic'' vector fermionic fields.
Summation formula for solutions of Riccati-Abel equation
Robert M. Yamaleev
2012-10-08
The generalized Riccati equation defined as an equation between first order derivative and the cubic polynomial is named Riccati-Abel equation. Unlike solutions of ordinary Riccati equation, the solutions of Riccati-Abel equation do not admit an addition formula. In the present paper we explain a nature of this fault and elaborate a method of solution of this problem. We show that the addition formula for Riccati-Abel equation can be established only for pair of solutions. Furthermore, it is shown that analogously with ordinary Riccati equation, the relationships with linear differential equations and the general complex algebra of third order can be established only for the pair of solutions of Riccati-Abel equation.
Fundamental Equation of State for Deuterium
Richardson, I. A.; Leachman, J. W., E-mail: jacob.leachman@wsu.edu [HYdrogen Properties for Energy Research (HYPER) Laboratory, School of Mechanical and Materials Engineering, Washington State University, P.O. Box 642920, Pullman, Washington 99164 (United States); Lemmon, E. W. [Applied Chemicals and Materials Division, National Institute of Standards and Technology, 325 Broadway, Boulder, Colorado 80305 (United States)] [Applied Chemicals and Materials Division, National Institute of Standards and Technology, 325 Broadway, Boulder, Colorado 80305 (United States)
2014-03-15
World utilization of deuterium is anticipated to increase with the rise of fusion-energy machines such as ITER and NIF. We present a new fundamental equation of state for the thermodynamic properties of fluid deuterium. Differences between thermodynamic properties of orthodeuterium, normal deuterium, and paradeuterium are described. Separate ideal-gas functions were fitted for these separable forms together with a single real-fluid residual function. The equation of state is valid from the melting line to a maximum pressure of 2000 MPa and an upper temperature limit of 600 K, corresponding to available experimental measurements. The uncertainty in predicted density is 0.5% over the valid temperature range and pressures up to 300 MPa. The uncertainties of vapor pressures and saturated liquid densities are 2% and 3%, respectively, while speed-of-sound values are accurate to within 1% in the liquid phase.
Euler's fluid equations: Optimal Control vs Optimization
Darryl D. Holm
2009-09-28
An optimization method used in image-processing (metamorphosis) is found to imply Euler's equations for incompressible flow of an inviscid fluid, without requiring that the Lagrangian particle labels exactly follow the flow lines of the Eulerian velocity vector field. Thus, an optimal control problem and an optimization problem for incompressible ideal fluid flow both yield the \\emph {same} Euler fluid equations, although their Lagrangian parcel dynamics are \\emph{different}. This is a result of the \\emph{gauge freedom} in the definition of the fluid pressure for an incompressible flow, in combination with the symmetry of fluid dynamics under relabeling of their Lagrangian coordinates. Similar ideas are also illustrated for SO(N) rigid body motion.
Ultrarelativistic Decoupling Transformation for Generalized Dirac Equations
Noble, J H
2015-01-01
The Foldy--Wouthuysen transformation is known to uncover the nonrelativistic limit of a generalized Dirac Hamiltonian, lending an intuitive physical interpretation to the effective operators within Schr\\"{o}dinger--Pauli theory. We here discuss the opposite, ultrarelativistic limit which requires the use of a fundamentally different expansion where the leading kinetic term in the Dirac equation is perturbed by the mass of the particle and other interaction (potential) terms, rather than vice versa. The ultrarelativistic decoupling transformation is applied to free Dirac particles (in the Weyl basis) and to high-energy tachyons, which are faster-than-light particles described by a fully Lorentz-covariant equation. The effective gravitational interactions are found. For tachyons, the dominant gravitational interaction term in the high-energy limit is shown to be attractive, and equal to the leading term for subluminal Dirac particles (tardyons) in the high-energy limit.
Ultrarelativistic Decoupling Transformation for Generalized Dirac Equations
J. H. Noble; U. D. Jentschura
2015-06-05
The Foldy--Wouthuysen transformation is known to uncover the nonrelativistic limit of a generalized Dirac Hamiltonian, lending an intuitive physical interpretation to the effective operators within Schr\\"{o}dinger--Pauli theory. We here discuss the opposite, ultrarelativistic limit which requires the use of a fundamentally different expansion where the leading kinetic term in the Dirac equation is perturbed by the mass of the particle and other interaction (potential) terms, rather than vice versa. The ultrarelativistic decoupling transformation is applied to free Dirac particles (in the Weyl basis) and to high-energy tachyons, which are faster-than-light particles described by a fully Lorentz-covariant equation. The effective gravitational interactions are found. For tachyons, the dominant gravitational interaction term in the high-energy limit is shown to be attractive, and equal to the leading term for subluminal Dirac particles (tardyons) in the high-energy limit.
Correction to Solution of Dirac Equation
Rui Chen
2015-10-11
Using the Chen unitary principle to test the Dirac theoryfor the hydrogen atomic spectrum shows that the standard Dirac function withthe Dirac energy levels is only one the formal solutions of theDirac-Coulomb equation, which conceals some pivotal mathematicalcontradictions. The theorem of existence of solution of the Dirac equationrequires an important modification to the Dirac angular momentum constantthat was defined by Dirac's algebra. It derives the modified radial Diracequation which has the consistency solution involving the quantum neutronradius and the neutron binding energy. The inevitable solution for otheratomic energy states is only equivalent to the Bohr solution. It concludesthat the Dirac equation is more suitable to describe the structure ofneutron. How to treat the difference between the unitary energy levels andthe result of the experimental observation of the atomic spectrums for thehydrogen atom needs to be solved urgently.
Correction to Solution of Dirac Equation
Rui Chen
2015-10-13
Using the China unitary principle to test the Dirac theoryfor the hydrogen atomic spectrum shows that the standard Dirac function withthe Dirac energy levels is only one the formal solutions of theDirac-Coulomb equation, which conceals some pivotal mathematicalcontradictions. The theorem of existence of solution of the Dirac equationrequires an important modification to the Dirac angular momentum constantthat was defined by Dirac's algebra. It derives the modified radial Diracequation which has the consistency solution involving the quantum neutronradius and the neutron binding energy. The inevitable solution for otheratomic energy states is only equivalent to the Bohr solution. It concludesthat the Dirac equation is more suitable to describe the structure ofneutron. How to treat the difference between the unitary energy levels andthe result of the experimental observation of the atomic spectrums for thehydrogen atom needs to be solved urgently.
Equation of State for Parallel Rigid Spherocylinders
Masashi Torikai
2012-08-06
The pair distribution function of monodisperse rigid spherocylinders is calculated by Shinomoto's method, which was originally proposed for hard spheres. The equation of state is derived by two different routes: Shinomoto's original route, in which a hard wall is introduced to estimate the pressure exerted on it, and the virial route. The pressure from Shinomoto's original route is valid only when the length-to-width ratio is less than or equal to 0.25 (i.e., when the spherocylinders are nearly spherical). The virial equation of state is shown to agree very well with the results of numerical simulations of spherocylinders with length-to-width ratio greater than or equal to 2.
Semirelativistic Bound-State Equations: Trivial Considerations
Wolfgang Lucha; Franz F. Schöberl
2014-07-17
Observing renewed interest in long-standing (semi-) relativistic descriptions of bound states, we would like to make a few comments on the eigenvalue problem posed by the spinless Salpeter equation and, illustrated by the examples of the nonsingular Woods-Saxon potential and the singular Hulth\\'en potential, recall elementary tools that practitioners looking for analytic albeit approximate solutions might find useful in their quest.
Weakly nonlocal fluid mechanics - the Schrodinger equation
P. Van; T. Fulop
2004-06-09
A weakly nonlocal extension of ideal fluid dynamics is derived from the Second Law of thermodynamics. It is proved that in the reversible limit the additional pressure term can be derived from a potential. The requirement of the additivity of the specific entropy function determines the quantum potential uniquely. The relation to other known derivations of Schr\\"odinger equation (stochastic, Fisher information, exact uncertainty) is clarified.
Bosonic Fradkin-Tseytlin equations unfolded
Oleg Shaynkman
2014-12-30
We test series of infinite-dimensional algebras as the candidates for higher spin extension of su(k,k). Adjoint and twisted-adjoint representations of su(k,k) on spaces of these algebras are carefully explored. For k=2 corresponding unfolded systems are analyzed and they shown to encode Fradkin-Tseytlin equations for some set of integer spins. In each case spectrum of spins is found.
High order difference methods for parabolic equations
Matuska, Daniel Alan
1971-01-01
HIGH ORDER DIFFERENCE METHODS FOR PARABOLIC E(PATIONS A Thesis by Daniel Alan Matuska Submitted to the Graduate College of Texas A&M University in partial fulfillment of the requirement for the degree of MASTER OF SCIENCE May 1971 Ma...)or Sub]ect: Mathematics HIGH ORDER DIFFERENCE METHODS FOR PARABOLIC EQUATIONS A Thesis by Daniel Alan Matuska Approved as to style and content by: Pc~ &~ (Chairman of Committee) (Head of Department) (Member) C . (Member) (Member) (Member...
Commuting Matrix Solutions of PQCD Evolution Equations
Mehrdad Goshtasbpour; Seyed Ali Shafiei
2013-03-16
A method of obtaining parton distributions directly from data is revealed in this series. In the process, the first step would be developing appropriate matrix solutions of the evolution equations in $x$ space. A division into commuting and non-commuting matrix solutions has been made. Here, well-developed commuting matrix solutions are presented. Results for finite LO evolution match those of standard LO sets. There is a real potential of doing non-parametric data analysis.
Global evolution of random vortex filament equation
Z. Brze?niak; M. Gubinelli; M. Neklyudov
2013-07-04
We prove the existence of a global solution for the filament equation with inital condition given by a geometric rough path in the sense of Lyons (1998).Our work gives a positive answer to a question left open in recent publications: Berselli and Gubinelli (2007) showed the existence of global solution for a smooth initial condition while Bessaih, Gubinelli, Russo (2005) proved the existence of a local solution for a general initial condition given by a rough path.
Quantum Potential Via General Hamilton - Jacobi Equation
Maedeh Mollai; Mohammad Razavi; Safa Jami; Ali Ahanj
2011-10-29
In this paper, we sketch and emphasize the automatic emergence of a quantum potential (QP) in general Hamilton-Jacobi equation via commuting relations, quantum canonical transformations and without the straight effect of wave function. The interpretation of QP in terms of independent entity is discussed along with the introduction of quantum kinetic energy. The method has been extended to relativistic regime, and same results have been concluded.
Lyapunov Functionals for the Enskog Equation
Zhenglu Jiang
2006-08-27
Two Lyapunov functionals are presented for the Enskog equation. One is to describe interactions between particles with various velocities and another is to measure the $L^1$ distance between two classical solutions. The former yields the time-asymptotic convergence of global classical solutions to the collision free motion while the latter is applied into the verification of the $L^1$ stability of global classical solutions.
Freeze Out and the Boltzmann Transport Equation
L. P. Csernai; V. K. Magas; E. Molnar; A. Nyiri; K. Tamosiunas
2005-02-20
Recently several works have appeared in the literature that addressed the problem of Freeze Out in energetic heavy ion reaction and aimed for a description based on the Boltzmann Transport Equation (BTE). In this paper we develop a dynamical Freeze-Out description, starting from the BTE, pointing out the basic limitations of the BTE approach, and the points where the BTE approach should be modified.
Modified Boltzmann Transport Equation and Freeze Out
Csernai, L P; Molnár, E; Nyiri, A; Tamosiunas, K
2005-01-01
We study Freeze Out process in high energy heavy ion reaction. The description of the process is based on the Boltzmann Transport Equation (BTE). We point out the basic limitations of the BTE approach and introduce Modified BTE. The Freeze Out dynamics is presented in the 4-dimensional space-time in a layer of finite thickness, and we employ Modified BTE for the realistic Freeze Out description.
Modified Boltzmann Transport Equation and Freeze Out
L. P. Csernai; V. K. Magas; E. Molnar; A. Nyiri; K. Tamosiunas
2005-05-26
We study Freeze Out process in high energy heavy ion reaction. The description of the process is based on the Boltzmann Transport Equation (BTE). We point out the basic limitations of the BTE approach and introduce Modified BTE. The Freeze Out dynamics is presented in the 4-dimensional space-time in a layer of finite thickness, and we employ Modified BTE for the realistic Freeze Out description.
Dilatonic Equation of Hydrostatic Equilibrium and Neutron Star Structure
S. H. Hendi; G. H. Bordbar; B. Eslam Panah; M. Najafi
2015-06-30
In this paper, we present a new hydrostatic equilibrium equation related to dilaton gravity. We consider a spherical symmetric metric to obtain the hydrostatic equilibrium equation of stars in $4$-dimensions, and generalize TOV equation to the case of regarding a dilaton field. Then, we calculate the structure properties of neutron star using our obtained hydrostatic equilibrium equation employing the modern equations of state of neutron star matter derived from microscopic calculations. We show that the maximum mass of neutron star depends on the parameters of dilaton field and cosmological constant. In other words, by setting the parameters of new hydrostatic equilibrium equation, we calculate the maximum mass of neutron star.
Stochastic semiclassical equations for weakly inhomogeneous cosmologies
Antonio Campos; Enric Verdaguer
1995-11-28
Semiclassical Einstein-Langevin equations for arbitrary small metric perturbations conformally coupled to a massless quantum scalar field in a spatially flat cosmological background are derived. Use is made of the fact that for this problem the in-in or closed time path effective action is simply related to the Feynman and Vernon influence functional which describes the effect of the ``environment'', the quantum field which is coarse grained here, on the ``system'', the gravitational field which is the field of interest. This leads to identify the dissipation and noise kernels in the in-in effective action, and to derive a fluctuation-dissipation relation. A tensorial Gaussian stochastic source which couples to the Weyl tensor of the spacetime metric is seen to modify the usual semiclassical equations which can be viewed now as mean field equations. As a simple application we derive the correlation functions of the stochastic metric fluctuations produced in a flat spacetime with small metric perturbations due to the quantum fluctuations of the matter field coupled to these perturbations.
Nonholonomic Hamilton-Jacobi equation and Integrability
Tomoki Ohsawa; Anthony M. Bloch
2009-12-18
We discuss an extension of the Hamilton-Jacobi theory to nonholonomic mechanics with a particular interest in its application to exactly integrating the equations of motion. We give an intrinsic proof of a nonholonomic analogue of the Hamilton--Jacobi theorem. Our intrinsic proof clarifies the difference from the conventional Hamilton-Jacobi theory for unconstrained systems. The proof also helps us identify a geometric meaning of the conditions on the solutions of the Hamilton-Jacobi equation that arise from nonholonomic constraints. The major advantage of our result is that it provides us with a method of integrating the equations of motion just as the unconstrained Hamilton--Jacobi theory does. In particular, we build on the work by Iglesias-Ponte, de Leon, and Martin de Diego so that the conventional method of separation of variables applies to some nonholonomic mechanical systems. We also show a way to apply our result to systems to which separation of variables does not apply.
Total Operators and Inhomogeneous Proper Values Equations
Jose G. Vargas
2015-07-09
Kaehler's two-sided angular momentum operator, K + 1, is neither vector-valued nor bivector-valued. It is total in the sense that it involves terms for all three dimensions. Constant idempotents that are "proper functions" of K+1's components are not proper functions of K+1. They rather satisfy "inhomogeneous proper-value equations", i.e. of the form (K + 1)U = {\\mu}U + {\\pi}, where {\\pi} is a scalar. We consider an equation of that type with K+1 replaced with operators T that comprise K + 1 as a factor, but also containing factors for both space and spacetime translations. We study the action of those T's on linear combinations of constant idempotents, so that only the algebraic (spin) part of K +1 has to be considered. {\\pi} is now, in general, a non-scalar member of a Kaehler algebra. We develop the system of equations to be satisfied by the combinations of those idempotents for which {\\pi} becomes a scalar. We solve for its solutions with {\\mu} = 0, which actually also makes {\\pi} = 0: The solutions with {\\mu} = {\\pi} = 0 all have three constituent parts, 36 of them being different in the ensemble of all such solutions. That set of different constituents is structured in such a way that we might as well be speaking of an algebraic representation of quarks. In this paper, however, we refrain from pursuing this identification in order to emphasize the purely mathematical nature of the argument.
Guiding Center Equations for Ideal Magnetohydrodynamic Modes
Roscoe B. White
2013-02-21
Guiding center simulations are routinely used for the discovery of mode-particle resonances in tokamaks, for both resistive and ideal instabilities and to find modifications of particle distributions caused by a given spectrum of modes, including large scale avalanches during events with a number of large amplitude modes. One of the most fundamental properties of ideal magnetohydrodynamics is the condition that plasma motion cannot change magnetic topology. The conventional representation of ideal magnetohydrodynamic modes by perturbing a toroidal equilibrium field through ?~B = ? X (? X B) however perturbs the magnetic topology, introducing extraneous magnetic islands in the field. A proper treatment of an ideal perturbation involves a full Lagrangian displacement of the field due to the perturbation and conserves magnetic topology as it should. In order to examine the effect of ideal magnetohydrodynamic modes on particle trajectories the guiding center equations should include a correct Lagrangian treatment. Guiding center equations for an ideal displacement ? are derived which perserve the magnetic topology and are used to examine mode particle resonances in toroidal confinement devices. These simulations are compared to others which are identical in all respects except that they use the linear representation for the field. Unlike the case for the magnetic field, the use of the linear field perturbation in the guiding center equations does not result in extraneous mode particle resonances.
Guiding center equations for ideal magnetohydrodynamic modes
White, R. B. [Plasma Physics Laboratory, Princeton University, P.O. Box 451, Princeton, New Jersey 08543 (United States)
2013-04-15
Guiding center simulations are routinely used for the discovery of mode-particle resonances in tokamaks, for both resistive and ideal instabilities and to find modifications of particle distributions caused by a given spectrum of modes, including large scale avalanches during events with a number of large amplitude modes. One of the most fundamental properties of ideal magnetohydrodynamics is the condition that plasma motion cannot change magnetic topology. The conventional representation of ideal magnetohydrodynamic modes by perturbing a toroidal equilibrium field through {delta}B-vector={nabla} Multiplication-Sign ({xi}-vector Multiplication-Sign B-vector), however, perturbs the magnetic topology, introducing extraneous magnetic islands in the field. A proper treatment of an ideal perturbation involves a full Lagrangian displacement of the field due to the perturbation and conserves magnetic topology as it should. In order to examine the effect of ideal magnetohydrodynamic modes on particle trajectories, the guiding center equations should include a correct Lagrangian treatment. Guiding center equations for an ideal displacement {xi}-vector are derived which preserve the magnetic topology and are used to examine mode particle resonances in toroidal confinement devices. These simulations are compared to others which are identical in all respects except that they use the linear representation for the field. Unlike the case for the magnetic field, the use of the linear field perturbation in the guiding center equations does not result in extraneous mode particle resonances.
Solution generating theorems for the TOV equation
Petarpa Boonserm; Matt Visser; Silke Weinfurtner
2007-07-17
The Tolman-Oppenheimer-Volkov [TOV] equation constrains the internal structure of general relativistic static perfect fluid spheres. We develop several "solution generating" theorems for the TOV, whereby any given solution can be "deformed" to a new solution. Because the theorems we develop work directly in terms of the physical observables -- pressure profile and density profile -- it is relatively easy to check the density and pressure profiles for physical reasonableness. This work complements our previous article [Phys. Rev. D71 (2005) 124307; gr-qc/0503007] wherein a similar "algorithmic" analysis of the general relativistic static perfect fluid sphere was presented in terms of the spacetime geometry -- in the present analysis the pressure and density are primary and the spacetime geometry is secondary. In particular, our "deformed" solutions to the TOV equation are conveniently parameterized in terms of delta rho_c and delta p_c, the finite shift in the central density and central pressure. We conclude by presenting a new physical and mathematical interpretation of the TOV equation -- as an integrability condition on the density and pressure profiles.
Measuring the dark matter equation of state
Ana Laura Serra; Mariano Javier de León Domínguez Romero
2011-05-30
The nature of the dominant component of galaxies and clusters remains unknown. While the astrophysics community supports the cold dark matter (CDM) paradigm as a clue factor in the current cosmological model, no direct CDM detections have been performed. Faber and Visser 2006 have suggested a simple method for measuring the dark matter equation of state that combines kinematic and gravitational lensing data to test the widely adopted assumption of pressureless dark matter. Following this formalism, we have measured the dark matter equation of state for first time using improved techniques. We have found that the value of the equation of state parameter is consistent with pressureless dark matter within the errors. Nevertheless, the measured value is lower than expected because typically the masses determined with lensing are larger than those obtained through kinematic methods. We have tested our techniques using simulations and we have also analyzed possible sources of error that could invalidate or mimic our results. In the light of this result, we can now suggest that the understanding of the nature of dark matter requires a complete general relativistic analysis.
A characterization of causal automorphisms by wave equations
Do-Hyung Kim
2011-11-07
A characterization of causal automorphism on Minkowski spacetime is given by use of wave equation. The result shows that causal analysis of spacetime may be replaced by studies of wave equation on manifolds.
Solutions of Lattice Differential Equations over Inhomogeneous Media
Brucal Hallare, Maila
2012-12-31
nontrivial examples of lattice differential equations (LDEs) on Z that are related to the (homogeneous) lattice Nagumo equation. The LDEs that we consider are used to model natural phenomena defined over an inhomogeneous medium, namely: (1) a lattice Nagumo...
Weighted Energy Decay for 1D Dirac Equation
E. Kopylova
2011-02-10
We obtain a dispersive long-time decay in weighted energy norms for solutions of the 1D Dirac equation with generic potential. The decay extends the results obtained by Jensen, Kato and Murata for the Schr\\"odinger equations.
On the invariant thermal Proca - Klein - Gordon equation
Magdalena Pelc
2007-10-14
In this paper we discuss the invariant thermal Proca - Klein - Gordon equation (PKG). We argue that for the thermal PKG equation the absolute velocity is equal v = alpha*c, where alpha is the fine stucture constant for electromagnetic interaction.
Self-adjointness of a generalized Camassa-Holm equation
N. H. Ibragimov; R. Khamitova; A. Valenti
2011-04-01
It is well known that the Camassa-Holm equation possesses numerous remarkable properties characteristic for KdV type equations. In this paper we show that it shares one more property with the KdV equation. Namely, Ibragimov has shown that the KdV and the modified KdV equations are self-adjoint. Starting from the generalization of the Camassa-Holm equation, we prove that the Camassa-Holm equation is self-adjoint. This property is important, e.g. for constructing conservation laws associated with symmetries of the equation in question. Accordingly, we construct conservation laws for the generalized Camassa-Holm equation using its symmetries.
Outline for Linear Equations and Inequalities of 2 variables
charlotb
2010-04-15
Outline for Linear Equations and Inequalities of 2 variables. A. 1. Substitute any value for x in the equation and solve for y. This results in a point (x, y). OR.
Stochastic partial differential equations with singular terminal condition
Popier, Alexandre
Stochastic partial differential equations with singular terminal condition A Matoussi, Lambert Piozin, A Popier To cite this version: A Matoussi, Lambert Piozin, A Popier. Stochastic partial differential equations with singular terminal condition. 2015. HAL Id: hal-01152687 https
Pad\\'e interpolation for elliptic Painlev\\'e equation
Noumi, Masatoshi; Yamada, Yasuhiko
2012-01-01
An interpolation problem related to the elliptic Painlev\\'e equation is formulated and solved. A simple form of the elliptic Painlev\\'e equation and the Lax pair are obtained. Explicit determinant formulae of special solutions are also given.
Propagation of ultra-short solitons in stochastic Maxwell's equations
Kurt, Levent; Schäfer, Tobias
2014-01-15
We study the propagation of ultra-short short solitons in a cubic nonlinear medium modeled by nonlinear Maxwell's equations with stochastic variations of media. We consider three cases: variations of (a) the dispersion, (b) the phase velocity, (c) the nonlinear coefficient. Using a modified multi-scale expansion for stochastic systems, we derive new stochastic generalizations of the short pulse equation that approximate the solutions of stochastic nonlinear Maxwell's equations. Numerical simulations show that soliton solutions of the short pulse equation propagate stably in stochastic nonlinear Maxwell's equations and that the generalized stochastic short pulse equations approximate the solutions to the stochastic Maxwell's equations over the distances under consideration. This holds for both a pathwise comparison of the stochastic equations as well as for a comparison of the resulting probability densities.
EFFECTIVE MACROSCOPIC DYNAMICS OF STOCHASTIC PARTIAL DIFFERENTIAL EQUATIONS IN PERFORATED
Duan, Jinqiao
EFFECTIVE MACROSCOPIC DYNAMICS OF STOCHASTIC PARTIAL DIFFERENTIAL EQUATIONS IN PERFORATED DOMAINS equation defined on a domain perforated with small holes or heterogeneities. The homogenized effective, effective macroscopic model, stochastic homogenization, white noise, probability distribution, perforated
Derivation of the Camassa-Holm equations for elastic waves
H. A. Erbay; S. Erbay; A. Erkip
2015-02-10
In this paper we provide a formal derivation of both the Camassa-Holm equation and the fractional Camassa-Holm equation for the propagation of small-but-finite amplitude long waves in a nonlocally and nonlinearly elastic medium. We first show that the equation of motion for the nonlocally and nonlinearly elastic medium reduces to the improved Boussinesq equation for a particular choice of the kernel function appearing in the integral-type constitutive relation. We then derive the Camassa-Holm equation from the improved Boussinesq equation using an asymptotic expansion valid as nonlinearity and dispersion parameters tend to zero independently. Our approach follows mainly the standard techniques used widely in the literature to derive the Camassa-Holm equation for shallow water waves. The case where the Fourier transform of the kernel function has fractional powers is also considered and the fractional Camassa-Holm equation is derived using the asymptotic expansion technique.
Nonlinear Integral Equations for the Inverse Problem in Corrosion ...
2012-06-15
Nonlinear Integral Equations for the Inverse. Problem in Corrosion Detection from Partial. Cauchy Data. Fioralba Cakoni. Department of Mathematical Sciences, ...
Soliton Solutions of Fractional order KdV-Burger's Equation
Muhammad Younis
2013-08-31
In this article, the new exact travelling wave solutions of the time-and space-fractional KdV-Burgers equation has been found. For this the fractional complex transformation have been implemented to convert nonlinear partial fractional differential equations to nonlinear ordinary differential equations, in the sense of the Jumarie's modified Riemann-Liouville derivative. Afterwards, the improved (G'/G)-expansion method can be implemented to celebrate the soliton solutions of KdV-Burger's equation of fractional order.
Direct Experimental Simulation of the Yang-Baxter Equation
Chao Zheng; Jun-lin Li; Si-yu Song; Gui Lu Long
2013-05-27
Introduced in the field of many-body statistical mechanics, Yang-Baxter equation has become an important tool in a variety fields of physics. In this work, we report the first direct experimental simulation of the Yang-Baxter equation using linear quantum optics. The equality between the two sides of the Yang-Baxter equation in two dimension has been demonstrated directly, and the spectral parameter transformation in the Yang-Baxter equation is explicitly confirmed.
221A Lecture Notes 1 HamiltonJacobi Equation
Murayama, Hitoshi
221A Lecture Notes WKB Method 1 HamiltonJacobi Equation We start from the Schr¨odinger equation this equation by using (x, t) = eiS(x,t)/¯h : - S t = 1 2m ( S)2 - i¯h 2m ( 2 S) + V . (2) Assuming = 0, this leads to an equation - S t = 1 2m ( S)2 - i¯h 2m ( 2 S) + V. (3) Now taking the formal limit ¯h 0
Transformations of Heun's equation and its integral relations
Léa Jaccoud El-Jaick; Bartolomeu D. B. Figueiredo
2011-01-26
We find transformations of variables which preserve the form of the equation for the kernels of integral relations among solutions of the Heun equation. These transformations lead to new kernels for the Heun equation, given by single hypergeometric functions (Lambe-Ward-type kernels) and by products of two hypergeometric functions (Erd\\'elyi-type). Such kernels, by a limiting process, also afford new kernels for the confluent Heun equation.
ARTICULATORY SYNTHESIS: NUMERICAL SOLUTION OF A HYPERBOLIC DIFFERENTIAL EQUATION
into the context of finite-difference approximations to a differential equation describing acoustic wave-that of solving the differ- ential equation describing acoustic (small amplitude), one-dimensional propagationARTICULATORY SYNTHESIS: NUMERICAL SOLUTION OF A HYPERBOLIC DIFFERENTIAL EQUATION Richard S. Mc
Theory Revision in Equation Discovery Ljupco Todorovski and Saso Dzeroski
Dzeroski, Saso
.Dzeroski@ijs.si Abstract. State of the art equation discovery systems start the discov- ery process from scratch, rather the accuracy of the model. 1 Introduction Most of the existing equation discovery systems make use of a very neglected by the equation discovery systems are the existing models in the domain. Rather than starting
The Whitham Equation as a Model for Surface Water Waves
Daulet Moldabayev; Henrik Kalisch; Denys Dutykh
2014-10-30
The Whitham equation was proposed as an alternate model equation for the simplified description of uni-directional wave motion at the surface of an inviscid fluid. As the Whitham equation incorporates the full linear dispersion relation of the water wave problem, it is thought to provide a more faithful description of shorter waves of small amplitude than traditional long wave models such as the KdV equation. In this work, we identify a scaling regime in which the Whitham equation can be derived from the Hamiltonian theory of surface water waves. The Whitham equation is integrated numerically, and it is shown that the equation gives a close approximation of inviscid free surface dynamics as described by the Euler equations. The performance of the Whitham equation as a model for free surface dynamics is also compared to two standard free surface models: the KdV and the BBM equation. It is found that in a wide parameter range of amplitudes and wavelengths, the Whitham equation performs on par with or better than both the KdV and BBM equations.
Two-component equations modelling water waves with constant vorticity
Joachim Escher; David Henry; Boris Kolev; Tony Lyons
2014-09-30
In this paper we derive a two-component system of nonlinear equations which model two-dimensional shallow water waves with constant vorticity. Then we prove well-posedness of this equation using a geometrical framework which allows us to recast this equation as a geodesic flow on an infinite dimensional manifold. Finally, we provide a criteria for global existence.
French-Chinese School on Differential and Functional Equations
Pouyanne, Nicolas
French-Chinese School on Differential and Functional Equations Wuhan University Wuhan, China, April 16th-27th, 2012 #12;French-Chinese School on Differential and Functional Equations, Wuhan, China, April 16th-27th, 2012 French-Chinese School on Differential and Functional Equations Wuhan, China, April
Radon transform and kinetic equations in tomographic representation
V. N. Chernega; V. I. Man'ko; B. I. Sadovnikov
2009-11-01
Statistical properties of classical random process are considered in tomographic representation. The Radon integral transform is used to construct the tomographic form of kinetic equations. Relation of probability density on phase space for classical systems with tomographic probability distributions is elucidated. Examples of simple kinetic equations like Liouville equations for one and many particles are studied in detail.
Long-wave instabilities and saturation in thin film equations
Pugh, Mary
to shorter wavelengths which then dissipate the energy. The nonlinearity in the KS equation is advective.2) The equation arises as an interface model in bio-fluids [15], solar convec- tion [19], and binary alloys [48Long-wave instabilities and saturation in thin film equations A. L. Bertozzi Department
Longwave instabilities and saturation in thin film equations
Pugh, Mary
then dissipate the energy. The nonlinearity in the KS equation is advective, and a#ects the dyÂ namics di.2) The equation arises as an interface model in bioÂfluids [15], solar convecÂ tion [19], and binary alloys [48LongÂwave instabilities and saturation in thin film equations A. L. Bertozzi Department
Partitioning Multivariate Polynomial Equations via Vertex Separators for Algebraic Cryptanal-
International Association for Cryptologic Research (IACR)
Partitioning Multivariate Polynomial Equations via Vertex Separators for Algebraic Cryptanal- ysis. In this paper, we apply similar graph theory techniques to systems of multivariate polynomial equations to a system of multivariate polynomial equations is an NP-complete problem [7, Ch. 3.9]. A variety of solution
EXISTENCE OF INSENSITIZING CONTROLS FOR A SEMILINEAR HEAT EQUATION WITH
González Burgos, Manuel
EXISTENCE OF INSENSITIZING CONTROLS FOR A SEMILINEAR HEAT EQUATION WITH A SUPERLINEAR NONLINEARITY system of heat equations, the first one of semilinear type. In addition, the control enters on the second by D.G.E.S. (Spain), Grant PB981134. Abstract In this paper we consider a semilinear heat equation (in
September 2011 Discrete Wheeler-DeWitt Equation
Hamber, Herbert W.
September 2011 Discrete Wheeler-DeWitt Equation Herbert W. Hamber 1 Institut des Hautes Etudes, Cambridge CB3 0JG, United Kingdom. ABSTRACT We present a discrete form of the Wheeler-DeWitt equation, with the solutions to the lattice equations providing a suitable approximation to the continuum wave functional
On linearization of super sine-Gordon equation
M. Siddiq; M. Hassan
2006-05-09
Two sets of super Riccati equations are presented which result in two linear problems of super sine-Gordon equation. The linear problems are then shown to be related to each other by a super gauge transformation and to the super B\\"{a}cklund transformation of the equation.
Differential Equations Lectures INF2320 p. 1/6
Stølen, Ketil
of Calculus, we get the solution r(t) = r(0)+ t 0 f(s)ds (3) · The integral can then be calculated as accurateDifferential Equations Lectures INF2320 p. 1/6 #12;Differential equations · A differential be determined · In practice, differential equations typically describe quantities that changes in relation
UNCONDITIONALLY STABLE METHODS FOR HAMILTON-JACOBI EQUATIONS
UNCONDITIONALLY STABLE METHODS FOR HAMILTON-JACOBI EQUATIONS KENNETH HVISTENDAHL KARLSEN AND NILS to the Cauchy problem for Hamilton-Jacobi equations of the form u t + H(Dxu) = 0. The methods are based stable numerical methods for the Cauchy problem for multi-dimensional Hamilton-Jacobi equations ( u t +H
Reasoning About Systems of Physics Equations Chun Wai Liew1
Liew, Chun Wai
Reasoning About Systems of Physics Equations Chun Wai Liew1 and Donald E. Smith2 1 Department Physics require the student to enter a system of algebraic equations as the answer. Tutoring systems must presents an approach that accepts from the student a system of equations describing the physics
Developments of the Price equation and natural selection under uncertainty
Grafen, Alan
success, following Darwin (1859). Here, this project is pursued by developing the Price equation, ¢rstDevelopments of the Price equation and natural selection under uncertainty Alan Grafen Department to employ these approaches. Here, a new theore- tical development arising from the Price equation provides
Heavy tailed K distributions imply a fractional advection dispersion equation
Meerschaert, Mark M.
Dispersion Equation (FADE) to model contaminant transport in porous media. This equation characterizes, and Particle Jumps Equations of contaminant transport in porous media are based on assumptions about hydraulic governing groundwater flow (e.g., Freeze and Cherry, 1979): h K v - = (1) where v is average velocity
Lagrangian Reduction, the EulerPoincare Equations, and Semidirect Products
Marsden, Jerrold
reduction for semidirect products, which applies to examples such as the heavy top, com- pressible fluids equations for a fluid or a rigid body, namely Lie-Poisson systems on the dual of a Lie algebra and their Lagrangian counterpart, the "pure" Euler-Poincar´e equations on a Lie algebra. The Lie-Poisson Equations
On Gaussian Beams Described by Jacobi's Equation
Steven Thomas Smith
2014-04-18
Gaussian beams describe the amplitude and phase of rays and are widely used to model acoustic propagation. This paper describes four new results in the theory of Gaussian beams. (1) A new version of the \\v{C}erven\\'y equations for the amplitude and phase of Gaussian beams is developed by applying the equivalence of Hamilton-Jacobi theory with Jacobi's equation that connects Riemannian curvature to geodesic flow. Thus the paper makes a fundamental connection between Gaussian beams and an acoustic channel's so-called intrinsic Gaussian curvature from differential geometry. (2) A new formula $\\pi(c/c")^{1/2}$ for the distance between convergence zones is derived and applied to several well-known profiles. (3) A class of "model spaces" are introduced that connect the acoustics of ducting/divergence zones with the channel's Gaussian curvature $K=cc"-(c')^2$. The "model" SSPs yield constant Gaussian curvature in which the geometry of ducts corresponds to great circles on a sphere and convergence zones correspond to antipodes. The distance between caustics $\\pi(c/c")^{1/2}$ is equated with an ideal hyperbolic cosine SSP duct. (4) An "intrinsic" version of \\v{C}erven\\'y's formulae for the amplitude and phase of Gaussian beams is derived that does not depend on an "extrinsic" arbitrary choice of coordinates such as range and depth. Direct comparisons are made between the computational frameworks used by the three different approaches to Gaussian beams: Snell's law, the extrinsic Frenet-Serret formulae, and the intrinsic Jacobi methods presented here. The relationship of Gaussian beams to Riemannian curvature is explained with an overview of the modern covariant geometric methods that provide a general framework for application to other special cases.
Stochastic evolution equations with random generators
Leon, Jorge A.; Nualart, David
1998-05-01
maximal inequality for the Skorohod integral deduced from the It ˆ o’s formula for this anticipating stochastic integral. 1. Introduction. In this paper we study nonlinear stochastic evolution equations of the form X t = ? + ? t 0 #3;A#3;s#4;X s +F#3;s#7;X.... The functions F#3;s#7;?#7; x#4; and B#3;s#7;?#7; x#4; are predictable processes satisfying suitable Lipschitz–type conditions and taking values in H and L 2 #3;U#7;H#4;, respectively. We will assume that A#3;s#7;?#4; is a random family of unbounded operators...
Canonical equations of ideal magnetic hydrodynamics
Gorskii, V.B.
1987-07-01
Ideal magnetohydrodynamics is used to consider a general class of adiabatic flow in magnetic liquids. Two invariants of the canonical equations of motion--Hamiltonian and Lagrangian--are determined in terms of the canonical variables by using the approximate variational formulations. The resulting model describes adiabatic three-dimensional flow of a nonviscous compressible liquid with ideal electric conductivity and zero heat conductivity. A Clebsch transformation is used to arrive at a form of the Lagrange-Cauchy integral for a vortex flow.
An Interesting Class of Partial Differential Equations
Wen-an Yong
2007-08-28
This paper presents an observation that under reasonable conditions, many partial differential equations from mathematical physics possess three structural properties. One of them can be understand as a variant of the celebrated Onsager reciprocal relation in Modern Thermodynamics. It displays a direct relation of irreversible processes to the entropy change. We show that the properties imply various entropy dissipation conditions for hyperbolic relaxation problems. As an application of the observation, we propose an approximation method to solve relaxation problems. Moreover, the observation is interpreted physically and verified with eight (sets of) systems from different fields.
Solving the Schroedinger equation using Smolyak interpolants
Avila, Gustavo; Carrington, Tucker Jr.
2013-10-07
In this paper, we present a new collocation method for solving the Schroedinger equation. Collocation has the advantage that it obviates integrals. All previous collocation methods have, however, the crucial disadvantage that they require solving a generalized eigenvalue problem. By combining Lagrange-like functions with a Smolyak interpolant, we device a collocation method that does not require solving a generalized eigenvalue problem. We exploit the structure of the grid to develop an efficient algorithm for evaluating the matrix-vector products required to compute energy levels and wavefunctions. Energies systematically converge as the number of points and basis functions are increased.
The Schrodinger Equation as a Volterra Problem
Mera, Fernando Daniel
2011-08-08
is obtained from Chapter IV of Lawrence C. Evans?s book on partial di erential equations [5]. Let y = x+ z, where 2 = 2~tm ; then we can rewrite the Poisson integral as u(x; t) = 1 i n=2 Z Rn eijzj 2 f(x+ z) dz (II.18) where jzj = jx yj . Let... implies that 8 > 0 9 > 0 such that 8x 2 Rn with jx yj 0, then there exists a t so small such that jf(x + z) f(x)j < for all z...
Efficient Solution of the Simplified PN Equations
Hamilton, Steven P [ORNL; Evans, Thomas M [ORNL
2015-01-01
In this paper we show new solver strategies for the multigroup SPN equations for nuclear reactor analysis. By forming the complete matrix over space, moments, and energy a robust set of solution strategies may be applied. Power iteration, shifted power iteration, Rayleigh quotient iteration, Arnoldi's method, and a generalized Davidson method, each using algebraic and physics-based multigrid preconditioners, have been compared on C5G7 MOX test problem as well as an operational PWR model. Our results show that the most ecient approach is the generalized Davidson method, that is 30{40 times faster than traditional power iteration and 6{10 times faster than Arnoldi's method.
Noncommutative Algebraic Equations and Noncommutative Eigenvalue Problem
Albert Schwarz
2000-04-27
We analyze the perturbation series for noncommutative eigenvalue problem $AX=X\\lambda$ where $\\lambda$ is an element of a noncommutative ring, $ A$ is a matrix and $X$ is a column vector with entries from this ring. As a corollary we obtain a theorem about the structure of perturbation series for Tr $x^r$ where $x$ is a solution of noncommutative algebraic equation (for $r=1$ this theorem was proved by Aschieri, Brace, Morariu, and Zumino, hep-th/0003228, and used to study Born-Infeld lagrangian for the gauge group $U(1)^k$).
Reducing differential equations for multiloop master integrals
Roman N. Lee
2015-04-21
We present an algorithm of the reduction of the differential equations for master integrals the Fuchsian form with the right-hand side matrix linearly depending on dimensional regularization parameter $\\epsilon$. We consider linear transformations of the functions column which are rational in the variable and in $\\epsilon$. Apart from some degenerate cases described below, the algorithm allows one to obtain the required transformation or to ascertain irreducibility to the form required. Degenerate cases are quite anticipated and likely to correspond to irreducible systems.
Generalized Ideal Gas Equations for Structureful Universe
Shahid N. Afridi; Khalid Khan
2006-09-04
We have derived generalized ideal gas equations for a structureful universe consisting of all forms of matters. We have assumed a universe that contains superclusters. Superclusters are then made of clusters. Each cluster can be further divided into smaller ones and so on. We have derived an expression for the entropy of such a universe. Our model is rather independent of the geometry of the intermediate clusters. Our calculations are valid for a non-interacting universe within non-relativistic limits. We suggest that structure formation can reduce the expansion rate of the universe.
On the multivariate Burgers equation and the incompressible Navier-Stokes equation (Part I)
Joerg Kampen
2011-03-14
We provide a constructive global existence proof for the multivariate viscous Burgers equation system defined on the whole space or on a domain isomorphic to the n-torus and with time horizon up to infinity and C^{\\infty}- data (satisfying some growth conditions if the problem is posed on the whole space). The proof is by a time discretized semiexplicit perturbative expansion in transformed coordinates where the convergence is guaranteed by certain a priori estimates. The scheme is useful in order to define computation for related equation systems of fluid dynamics.
Schrödinger-Pauli Equation for the Standard Model Extension CPT-Violating Dirac Equation
Thomas D. Gutierrez
2015-04-06
It is instructive to investigate the non-relativistic limit of the simplest Standard Model Extension (SME) CPT-violating Dirac-like equation but with minimal coupling to the electromagnetic fields. In this limit, it becomes an intuitive Schr\\"odinger-Pauli-like equation. This is comparable to the free particle treatment as explored by Kostelecky and Lane, but this exercise only considers the $a$ and $b$ CPT-violating terms and $\\vec{p}/m$ terms to first order. Several toy systems are discussed.
Calculating work in weakly driving quantum master equations: backward and forward equations
Fei Liu
2015-06-28
We present a technical report that the two methods of calculating characteristic functions for the work distribution in the weakly driven quantum master equations are equivalent. One is obtained by the notion of quantum jump trajectory [Phys. Rev. E 89, 042122 (2014)], while the other is based on the two time energy measurements on the combined system and reservoir [Silaev, et al., Phys. Rev. E 90, 022103 (2014)]. They are indeed the backward and forward methods, respectively, which is very similar to the case of the Kolmogorov backward and forward equations in classical stochastic theory. The microscopic basis of the former method is also clarified.
Craig, Walter
The Boltzmann equation Global existence results Uniqueness Properties of propagation Main ideas of the proof On the Boltzmann equation: global solutions in one spatial dimension Walter Craig Department 11, 2005 Walter Craig McMaster University Global solutions of the Boltzmann equation #12;The
Integral equations for the H- X- and Y-functions
B. Rutily; L. Chevallier; J. Bergeat
2006-01-16
We come back to a non linear integral equation satisfied by the function H, which is distinct from the classical H-equation. Established for the first time by Busbridge (1955), it appeared occasionally in the literature since then. First of all, this equation is generalized over the whole complex plane using the method of residues. Then its counterpart in a finite slab is derived; it consists in two series of integral equations for the X- and Y-functions. These integral equations are finally applied to the solution of the albedo problem in a slab.
Topography influence on the Lake equations in bounded domains
Christophe Lacave; Toan T. Nguyen; Benoit Pausader
2013-06-10
We investigate the influence of the topography on the lake equations which describe the two-dimensional horizontal velocity of a three-dimensional incompressible flow. We show that the lake equations are structurally stable under Hausdorff approximations of the fluid domain and $L^p$ perturbations of the depth. As a byproduct, we obtain the existence of a weak solution to the lake equations in the case of singular domains and rough bottoms. Our result thus extends earlier works by Bresch and M\\'etivier treating the lake equations with a fixed topography and by G\\'erard-Varet and Lacave treating the Euler equations in singular domains.
Integer Algorithms to Solver Diophantine Linear Equations and Systems
Florentin Smarandache
2007-11-28
The present work includes some of the author's original researches on integer solutions of Diophantine liner equations and systems. The notion of "general integer solution" of a Diophantine linear equation with two unknowns is extended to Diophantine linear equations with $n$ unknowns and then to Diophantine linear systems. The proprieties of the general integer solution are determined (both for a Diophantine linear equation and for a Diophantine linear system). Seven original integer algorithms (two for Diophantine linear equations, and five for Diophantine linear systems) are exposed. The algorithms are strictly proved and an example for each of them is given. These algorithms can be easily implemented on the computer.
Modulated wave trains in generalized Kuramoto-Sivashinksi equations
Noble, Pascal
2010-01-01
This paper is concerned with the stability of periodic wave trains in a generalized Kuramoto-Sivashinski (gKS) equation. This equation is useful to describe the weak instability of low frequency perturbations for thin film flows down an inclined ramp. We provide a set of equations, namely Whitham's modulation equations, that determines the behaviour of low frequency perturbations of periodic wave trains. As a byproduct, we relate the spectral stability in the small wavenumber regime to properties of the modulation equations. This stability is always critical since 0 is a 0-Floquet number eigenvalue associated to translational invariance.
Generating functionals and Lagrangian partial differential equations
Vankerschaver, Joris; Liao, Cuicui; Leok, Melvin [Department of Mathematics, University of California, San Diego, 9500 Gilman Drive, Dept. 0112, La Jolla, California 92093-0112 (United States)] [Department of Mathematics, University of California, San Diego, 9500 Gilman Drive, Dept. 0112, La Jolla, California 92093-0112 (United States)
2013-08-15
The main goal of this paper is to derive an alternative characterization of the multisymplectic form formula for classical field theories using the geometry of the space of boundary values. We review the concept of Type-I/II generating functionals defined on the space of boundary data of a Lagrangian field theory. On the Lagrangian side, we define an analogue of Jacobi's solution to the Hamilton–Jacobi equation for field theories, and we show that by taking variational derivatives of this functional, we obtain an isotropic submanifold of the space of Cauchy data, described by the so-called multisymplectic form formula. As an example of the latter, we show that Lorentz's reciprocity principle in electromagnetism is a particular instance of the multisymplectic form formula. We also define a Hamiltonian analogue of Jacobi's solution, and we show that this functional is a Type-II generating functional. We finish the paper by defining a similar framework of generating functions for discrete field theories, and we show that for the linear wave equation, we recover the multisymplectic conservation law of Bridges.
Using the scalable nonlinear equations solvers package
Gropp, W.D.; McInnes, L.C.; Smith, B.F.
1995-02-01
SNES (Scalable Nonlinear Equations Solvers) is a software package for the numerical solution of large-scale systems of nonlinear equations on both uniprocessors and parallel architectures. SNES also contains a component for the solution of unconstrained minimization problems, called SUMS (Scalable Unconstrained Minimization Solvers). Newton-like methods, which are known for their efficiency and robustness, constitute the core of the package. As part of the multilevel PETSc library, SNES incorporates many features and options from other parts of PETSc. In keeping with the spirit of the PETSc library, the nonlinear solution routines are data-structure-neutral, making them flexible and easily extensible. This users guide contains a detailed description of uniprocessor usage of SNES, with some added comments regarding multiprocessor usage. At this time the parallel version is undergoing refinement and extension, as we work toward a common interface for the uniprocessor and parallel cases. Thus, forthcoming versions of the software will contain additional features, and changes to parallel interface may result at any time. The new parallel version will employ the MPI (Message Passing Interface) standard for interprocessor communication. Since most of these details will be hidden, users will need to perform only minimal message-passing programming.
Wave Equations for Discrete Quantum Gravity
Gudder, Stan
2015-01-01
This article is based on the covariant causal set ($c$-causet) approach to discrete quantum gravity. A $c$-causet $x$ is a finite partially ordered set that has a unique labeling of its vertices. A rate of change on $x$ is described by a covariant difference operator and this operator acting on a wave function forms the left side of the wave equation. The right side is given by an energy term acting on the wave function. Solutions to the wave equation corresponding to certain pairs of paths in $x$ are added and normalized to form a unique state. The modulus squared of the state gives probabilities that a pair of interacting particles is at various locations given by pairs of vertices in $x$. We illustrate this model for a few of the simplest nontrivial examples of $c$-causets. Three forces are considered, the attractive and repulsive electric forces and the strong nuclear force. Large models get much more complicated and will probably require a computer to analyze.
Wave Equations for Discrete Quantum Gravity
Stan Gudder
2015-08-29
This article is based on the covariant causal set ($c$-causet) approach to discrete quantum gravity. A $c$-causet $x$ is a finite partially ordered set that has a unique labeling of its vertices. A rate of change on $x$ is described by a covariant difference operator and this operator acting on a wave function forms the left side of the wave equation. The right side is given by an energy term acting on the wave function. Solutions to the wave equation corresponding to certain pairs of paths in $x$ are added and normalized to form a unique state. The modulus squared of the state gives probabilities that a pair of interacting particles is at various locations given by pairs of vertices in $x$. We illustrate this model for a few of the simplest nontrivial examples of $c$-causets. Three forces are considered, the attractive and repulsive electric forces and the strong nuclear force. Large models get much more complicated and will probably require a computer to analyze.
Multiscale functions, Scale dynamics and Applications to partial differential equations
Jacky Cresson; Frédéric Pierret
2015-09-03
Modeling phenomena from experimental data, always begin with a \\emph{choice of hypothesis} on the observed dynamics such as \\emph{determinism}, \\emph{randomness}, \\emph{derivability} etc. Depending on these choices, different behaviors can be observed. The natural question associated to the modeling problem is the following : \\emph{"With a finite set of data concerning a phenomenon, can we recover its underlying nature ?} From this problem, we introduce in this paper the definition of \\emph{multi-scale functions}, \\emph{scale calculus} and \\emph{scale dynamics} based on the \\emph{time-scale calculus} (see \\cite{bohn}). These definitions will be illustrated on the \\emph{multi-scale Okamoto's functions}. The introduced formalism explains why there exists different continuous models associated to an equation with different \\emph{scale regimes} whereas the equation is \\emph{scale invariant}. A typical example of such an equation, is the \\emph{Euler-Lagrange equation} and particularly the \\emph{Newton's equation} which will be discussed. Notably, we obtain a \\emph{non-linear diffusion equation} via the \\emph{scale Newton's equation} and also the \\emph{non-linear Schr\\"odinger equation} via the \\emph{scale Newton's equation}. Under special assumptions, we recover the classical \\emph{diffusion} equation and the \\emph{Schr\\"odinger equation}.
Poincare-invariant equations with a rising mass spectrum
Wilhelm I. Fushchych
2002-06-21
In this note we shall construct, in the framework of relativistic quantum mechanics, the Poincare-invariant motion equations with realistic mass spectra. These equations describe a system with mass spectra of the form $m^2=a^2+b^2 s(s+1)$, where a and b are arbitrary parameters. Such equations are obtained by a reduction of the motion equation for two particles to a one-particle equation which describes the particle in various mass and spin states. It we impose a certain condition on the wave function of the derived equation, such an equation describes the free motion of a fixed-mass particle with arbitrary (but fixed) spin s.
Nonlinear Integral-Equation Formulation of Orthogonal Polynomials
Carl M. Bender; E. Ben-Naim
2006-11-15
The nonlinear integral equation P(x)=\\int_alpha^beta dy w(y) P(y) P(x+y) is investigated. It is shown that for a given function w(x) the equation admits an infinite set of polynomial solutions P(x). For polynomial solutions, this nonlinear integral equation reduces to a finite set of coupled linear algebraic equations for the coefficients of the polynomials. Interestingly, the set of polynomial solutions is orthogonal with respect to the measure x w(x). The nonlinear integral equation can be used to specify all orthogonal polynomials in a simple and compact way. This integral equation provides a natural vehicle for extending the theory of orthogonal polynomials into the complex domain. Generalizations of the integral equation are discussed.
A Modified Equation for Neural Conductance and Resonance
M. Robert Showalter
1999-05-06
A modified equation, the S-K equation, fits data that the current neural conduction equation, the K-R equation, does not. The S-K equation is a modified Heaviside equation, based on a new interpretation of cross terms. Elements of neural anatomy and function are reviewed to put the S-K equation into context. The fit between S-K and resonance-like neural data is then shown. Appendix 1: Derivation of crossterms that represent combinations of physical laws for a line conductor of finite length. Appendix 2: Evaluation of crossterms that represent combinations of physical laws according to consistency arguments. Appendix 3: Some background on resonance. Appendix 4: Web access to some brain modeling, correspondence with NATURE, and discussion of the work in George Johnson's New York Times forums.
Jimack, Peter
-Stokes equations Ren´e Schneider1, and Peter K. Jimack1 1 School of Computing, University of Leeds, LS2 9JT, UK: rschneid@comp.leeds.ac.uk #12;PRECOND. DISCR. ADJOINT INCOMP. NAVIER-STOKES EQS. 1 of the steady state size parameter h. We briefly review some of the most important issues associated with the application
Fairag, Faisal
and energy related applications (global change, mixing of fuel and oxidizer in engines and drag reduction), aerodynamics (maneuvering flight of jet aircraft) and biophysical applications (blood flow in the heart equations considered here are identical to Smagorinsky model [21]. Thus, from a practical engineering point
Continuity Equation in Nonlinear Quantum Mechanics and the Galilei Relativity Principle
Wilhelm I. Fushchych; Vyacheslav M. Boyko
2002-08-13
Classes of the nonlinear Schrodinger-type equations compatible with the Galilei relativity principle are described. Solutions of these equations satisfy the continuity equation.
Um, E.S.
2013-01-01
mod- eling of the acoustic wave equation: Geophysics, 39,solution analysis of acoustic wave equation in the Laplace-solutions to the acoustic wave equation in the Laplace-
On the radius of analyticity of solutions to the cubic Szego{double acute} equation
Gérard, P; Guo, Y; Titi, ES
2015-01-01
nonlinear Schr¨odinger equation on surfaces. Invent. Math.for nonlinear analytic parabolic equations, Commu- nicationsin Partial Differential Equations 23 (1998), 1-16. [5] C.
Equations of motion for a (non-linear) scalar field model as derived from the field equations
Shmuel Kaniel; Yakov Itin
2006-08-02
The problem of derivation of the equations of motion from the field equations is considered. Einstein's field equations have a specific analytical form: They are linear in the second order derivatives and quadratic in the first order derivatives of the field variables. We utilize this particular form and propose a novel algorithm for thederivation of the equations of motion from the field equations. It is based on the condition of the balance between the singular terms of the field equation. We apply the algorithm to a nonlinear Lorentz invariant scalar field model. We show that it results in the Newton law of attraction between the singularities of the field moved on approximately geodesic curves. The algorithm is applicable to the $N$-body problem of the Lorentz invariant field equations.
The Raychaudhuri equation in homogeneous cosmologies
Albareti, F.D.; Cembranos, J.A.R.; Cruz-Dombriz, A. de la; Dobado, A. E-mail: cembra@fis.ucm.es E-mail: dobado@fis.ucm.es
2014-03-01
In this work we address the issue of studying the conditions required to guarantee the Focusing Theorem for both null and timelike geodesic congruences by using the Raychaudhuri equation. In particular we study the case of Friedmann-Robertson-Walker as well as more general Bianchi Type I spacetimes. The fulfillment of the Focusing Theorem is mandatory in small scales since it accounts for the attractive character of gravity. However, the Focusing Theorem is not satisfied at cosmological scales due to the measured negative deceleration parameter. The study of the conditions needed for congruences convergence is not only relevant at the fundamental level but also to derive the viability conditions to be imposed on extended theories of gravity describing the different expansion regimes of the universe. We illustrate this idea for f(R) gravity theories.
Equations determine coiled tubing collapse pressure
Avakov, V.; Taliaferro, W.
1995-07-24
A set of equations has been developed for calculating pipe collapse pressure for oval tubing such as coiled tubing. When coiled tubing is placed onto a reel, the tubing is forced into an oval shape and never again returns to perfect roundness because the coiling process exceeds the plasticity limits of the tubing. Straightening the tubing for the trip into the well does not restore roundness. The consequence of this physical property is that all coiled tubing collapse pressure calculations should be made considering oval tubing, not round tubing. Tubing collapse can occur when formation pressure against the coiled tubing exceeds the collapse resistance inherent in the coiled tubing. As coiled tubing becomes more oval in shape, it becomes more oval in shape, it becomes more susceptible to collapse from outside pressure.
Equation of State from Lattice QCD Calculations
Rajan Gupta
2011-04-01
We provide a status report on the calculation of the Equation of State (EoS) of QCD at finite temperature using lattice QCD. Most of the discussion will focus on comparison of recent results obtained by the HotQCD and Wuppertal-Budapest (W-B) collaborations. We will show that very significant progress has been made towards obtaining high precision results over the temperature range of T=150-700 MeV. The various sources of systematic uncertainties will be discussed and the differences between the two calculations highlighted. Our final conclusion is that the lattice results of EoS are getting precise enough to justify being used in the phenomenological analysis of heavy ion experiments at RHIC and LHC.
Bounding biomass in the Fisher equation
Birch, Daniel A; Young, William R
2007-01-01
The FKPP equation with a variable growth rate and advection by an incompressible velocity field is considered as a model for plankton dispersed by ocean currents. If the average growth rate is negative then the model has a survival-extinction transition; the location of this transition in the parameter space is constrained using variational arguments and delimited by simulations. The statistical steady state reached when the system is in the survival region of parameter space is characterized by integral constraints and upper and lower bounds on the biomass and productivity that follow from variational arguments and direct inequalities. In the limit of zero-decorrelation time the velocity field is shown to act as Fickian diffusion with an eddy diffusivity much larger than the molecular diffusivity and this allows a one-dimensional model to predict the biomass, productivity and extinction transitions. All results are illustrated with a simple growth and stirring model.
Bounding biomass in the Fisher equation
Daniel A. Birch; Yue-Kin Tsang; William R. Young
2007-03-17
The FKPP equation with a variable growth rate and advection by an incompressible velocity field is considered as a model for plankton dispersed by ocean currents. If the average growth rate is negative then the model has a survival-extinction transition; the location of this transition in the parameter space is constrained using variational arguments and delimited by simulations. The statistical steady state reached when the system is in the survival region of parameter space is characterized by integral constraints and upper and lower bounds on the biomass and productivity that follow from variational arguments and direct inequalities. In the limit of zero-decorrelation time the velocity field is shown to act as Fickian diffusion with an eddy diffusivity much larger than the molecular diffusivity and this allows a one-dimensional model to predict the biomass, productivity and extinction transitions. All results are illustrated with a simple growth and stirring model.
Assessment of UF6 Equation of State
Brady, P; Chand, K; Warren, D; Vandersall, J
2009-02-11
A common assumption in the mathematical analysis of flows of compressible fluids is to treat the fluid as a perfect gas. This is an approximation, as no real fluid obeys the perfect gas relationships over all temperature and pressure conditions. An assessment of the validity of treating the UF{sub 6} gas flow field within a gas centrifuge with perfect gas relationships has been conducted. The definition of a perfect gas is commonly stated in two parts: (1) the gas obeys the thermal equation of state, p = {rho}RT (thermally perfect), and, (2) the gas specific heats are constant (calorically perfect). Analysis indicates the thermally perfect assumption is valid for all flow conditions within the gas centrifuge, including shock fields. The low operating gas pressure is the primary factor in the suitability of the thermally perfect equation of state for gas centrifuge computations. UF{sub 6} is not calorically perfect, as the specific heats vary as a function of temperature. This effect is insignificant within the bulk of the centrifuge gas field, as gas temperatures vary over a narrow range. The exception is in the vicinity of shock fields, where temperature, pressure, and density gradients are large, and the variation of specific heats with temperature should be included in the technically detailed analyses. Results from a normal shock analysis incorporating variable specific heats is included herein, presented in the conventional form of shock parameters as a function of inlet Mach Number. The error introduced by assuming constant specific heats is small for a nominal UF{sub 6} shock field, such that calorically perfect shock relationships can be used for scaling and initial analyses. The more rigorous imperfect gas analysis should be used for detailed analyses.
Variational Principles for Constrained Electromagnetic Field and Papapetrou Equation
A. T. Muminov
2007-06-28
In our previous article [4] an approach to derive Papapetrou equations for constrained electromagnetic field was demonstrated by use of field variational principles. The aim of current work is to present more universal technique of deduction of the equations which could be applied to another types of non-scalar fields. It is based on Noether theorem formulated in terms of Cartan' formalism of orthonormal frames. Under infinitesimal coordinate transformation the one leads to equation which includes volume force of spin-gravitational interaction. Papapetrou equation for vector of propagation of the wave is derived on base of the equation. Such manner of deduction allows to formulate more accurately the constraints and clarify equations for the potential and for spin.
On the additional invariance of the Dirac and Maxwell equations
Wilhelm I. Fushchych
2002-06-21
In this note we show that there exists a new set of operators {Q} (this set is different from the operators which satisfy the Lie algebra of the Poincare group P(1,3) with respect to which the Dirac and Maxwell equations are invariant. We shall give the detailed proof of our assertions only for the Dirac equation, since for the Maxwell equations all the assertions are proved analogously.
Power-law Spatial Dispersion from Fractional Liouville Equation
Vasily E. Tarasov
2013-07-18
A microscopic model in the framework of fractional kinetics to describe spatial dispersion of power-law type is suggested. The Liouville equation with the Caputo fractional derivatives is used to obtain the power-law dependence of the absolute permittivity on the wave vector. The fractional differential equations for electrostatic potential in the media with power-law spatial dispersion are derived. The particular solutions of these equations for the electric potential of point charge in this media are considered.
Validity of ELTB Equation for Suitable Description of BEC
Kim, Dooyoung; Kim, Jinguanghao; Yoon, Jin-Hee
2005-10-17
The Bose-Einstein condensation (BEC) has been found for various alkali-metal gases such as 7Li, 87Rb, Na, and H. For the description of atoms in this condensate state, the Gross-Pitaevskii (GP) equation has been widely used. However, the GP equation contains the nonlinear term, which makes this equation hard to solve. Therefore, physical quantities are usually obtained numerically, and sometimes it is difficult to extract a physical meaning from the calculated results. The nuclear theory group at Purdue University in the U.S. developed a new simple equation, the equivalent linear two-body (ELTB) equation, using the hyper-radius coordinates and tested it for one-dimensional BEC system. Their results are consistent with the numerical values from the GP equation within 4.5%.We test the validity of the ELTB equation for three-dimensional BEC system by calculating the energies per particle and the wave functions for 87Rb gas and for 7Li gas. We use the quantum-mechanical variational method for the BEC energy. Our result for 87Rb gas agrees with a numerical calculation based on the GP equation, with a relative error of 12% over a wide range of N from 100 to 10,000. The relative distances between particles for 7Li gas are consistent within a relative error of 17% for N {<=} 1300. The relatively simple form of the ELTB equation, compared with the GP equation, enables us to treat the N-body system easily and efficiently. We conclude that the ELTB equation is a powerful equation for describing BEC system because it is easy to treat.
Exact Anisotropic Solutions of the Generalized TOV Equation
Riazi, Nematollah; Sajadi, S Naseh; Assyaee, S Shahrokh
2015-01-01
We explore gravitating relativistic spheres composed of an anisotropic, barotropic uid. We assume a bi-polytropic equation of state which has a linear and a power-law terms. The generalized Tolman-Oppenheimer-Volkoff (TOV) equation which describes the hydrostatic equilibrium is obtained. The full system of equations are solved for solutions which are regular at the origin and asymptotically flat. Conditions for the appearance of horizon and a basic treatment of stability are also discussed.
An Equation of Motion with Quantum Effect in Spacetime
Jyh-Yang Wu
2009-05-26
In this paper, we shall present a new equation of motion with Quantum effect in spacetime. To do so, we propose a classical-quantum duality. We also generalize the Schordinger equation to the spacetime and obtain a relativistic wave equation. This will lead a generalization of Einstein's formula $E=m_0c^2$ in the spacetime. In general, we have $E=m_0c^2 + \\frac{\\hbar^2}{12m_0}R$ in a spacetime.
Power-law spatial dispersion from fractional Liouville equation
Tarasov, Vasily E. [Skobeltsyn Institute of Nuclear Physics, Lomonosov Moscow State University, Moscow 119991 (Russian Federation)] [Skobeltsyn Institute of Nuclear Physics, Lomonosov Moscow State University, Moscow 119991 (Russian Federation)
2013-10-15
A microscopic model in the framework of fractional kinetics to describe spatial dispersion of power-law type is suggested. The Liouville equation with the Caputo fractional derivatives is used to obtain the power-law dependence of the absolute permittivity on the wave vector. The fractional differential equations for electrostatic potential in the media with power-law spatial dispersion are derived. The particular solutions of these equations for the electric potential of point charge in this media are considered.
Generalized Harmonic Equations in 3+1 Form
J. David Brown
2011-11-29
The generalized harmonic equations of general relativity are written in 3+1 form. The result is a system of partial differential equations with first order time and second order space derivatives for the spatial metric, extrinsic curvature, lapse function and shift vector, plus fields that represent the time derivatives of the lapse and shift. This allows for a direct comparison between the generalized harmonic and the Arnowitt-Deser-Misner formulations. The 3+1 generalized harmonic equations are also written in terms of conformal variables and compared to the Baumgarte-Shapiro-Shibata-Nakamura equations with moving puncture gauge conditions.
Dirac equation in the Nonsymmetric Kaluza-Klein Theory
Kalinowski, M W
2015-01-01
We rederive Dirac equation in the Nonsymmetric Kaluza-Klein Theory gettig an electric dipole moment of fermion and CP violation.
Dirac equation in the Nonsymmetric Kaluza-Klein Theory
M. W. Kalinowski
2015-07-08
We rederive Dirac equation in the Nonsymmetric Kaluza-Klein Theory gettig an electric dipole moment of fermion and CP violation.
Penetration equations Young, C.W. [Applied Research Associates...
Office of Scientific and Technical Information (OSTI)
45 MILITARY TECHNOLOGY, WEAPONRY, AND NATIONAL DEFENSE; EARTH PENETRATORS; EQUATIONS; NUCLEAR WEAPONS; SOILS; ICE; ROCKS; CONCRETES; PERMAFROST; SCALING LAWS In 1967, Sandia...
Software for Numerical Methods for Partial Differential Equations
Software for Numerical Methods for Partial Differential Equations. This software was developed for and by the students in CS 615, Numerical Methods for Partial
SCIENCE ON SATURDAY- "Disastrous Equations: The Role of Mathematics...
Broader source: All U.S. Department of Energy (DOE) Office Webpages (Extended Search)
On Saturday MBG Auditorium SCIENCE ON SATURDAY- "Disastrous Equations: The Role of Mathematics in Understanding Tsunami" Professor J. Douglas Wright, Associate Professor...
Partial regularity for Monge-Ampère type equations
2013-11-07
Nov 16, 2013 ... standing their regularity is an important question. In particular, this kind of equations arises in the regularity theory of optimal transport maps.
Complex roots in the inhour equation of coupled reactors
Yeh, Elizabeth Ching
1970-01-01
Parameters. II Complex Roots s = x + iy of Complex Inhour Equation and Physical Parameters of Input Data. . . . . . . . . . . . . . . 39 III Analytical Bounds of Complex Roots. vii LIST OF FIGURES Figure Page Real roots of inhour equation for two... Roots of Inhour Equation To discuss the complex roots, we resolve equation (14-a) into its real ond imaginary parts by letting s = x + iy, then (x+iy) A(x+iy) = ( ? ? ? 6) +e e p+2g 2 R 2 -2r (x+iy) + [ ? + c e p 2 t l p+2 1Rx irqy (~-B)+c e e 2 R...
Evolution equation for 3-quark Wilson loop operator
R. E. Gerasimov; A. V. Grabovsky
2012-12-07
The evolution equation for the 3 quark Wilson loop operator has been derived in the leading logarithm approximation within Balitsky high energy operator expansion.
Massively parallel structured direct solver for equations describing ...
2011-10-17
need for the introduction of coupled systems of partial differential equations to lower ... complexity and interprocessor communication estimates of our algorithm.
An extension of the Derrida-Lebowitz-Speer-Spohn equation
Charles Bordenave; Pierre Germain; Thomas Trogdon
2015-04-19
Derrida, Lebowitz, Speer and Spohn have proposed a simplified model to describe the low temperature Glauber dynamics of an anchored Toom interface. We show how the derivation of the Derrida-Lebowitz-Speer-Spohn equation can be prolonged to obtain a new equation, generalizing the models obtained in the paper by these authors. We then investigate its properties from both an analytical and numerical perspective. Specifically, a numerical method is presented to approximate solutions of the prolonged equation. Using this method, we investigate the relationship between the solutions of the prolonged equation and the Tracy--Widom GOE distribution.
NumericalIntegratorforContinuum Equations ofSurfaceGrowthandErosion
Cuerno, Rodolfo
" ......................................................... 195 5.2.2 Solving Partial Differential Equations in MATLAB...................................................................................................................................... 212 #12;190 ComputationalNanotechnology:ModelingandApplicationswithMATLAB® In addition, self
Influence of Kaluza Scalar on the Raychaudhuri Equation
R. Parthasarathy
2013-11-01
It is shown that the influence of Kaluza scalar is to induce expansion in the Raychaudhuri equation for two representative solutions of the Kaluza theory.
Nonparametric reconstruction of the dark energy equation of state...
Office of Scientific and Technical Information (OSTI)
Journal Article: Nonparametric reconstruction of the dark energy equation of state from diverse data sets Citation Details In-Document Search Title: Nonparametric reconstruction of...
On Diophantine Equation $x^2 = 2y^4-1$
Florentin Smarandache
2007-03-16
In this short note we present a method of solving this Diophantine equation, method which is different from Ljunggren's, Mordell's, and R.K.Guy's.
Felix Bloch, Nuclear Induction, Bloch Equations, Bloch Theorem...
Office of Scientific and Technical Information (OSTI)
Felix Bloch, Nuclear Induction, and Bloch Equations Resources with Additional Information Stressing "the importance both of demonstrating the neutron's magnetic moment and of...
The fundamental solution of the unidirectional pulse propagation equation
Babushkin, I.; Bergé, L.
2014-03-15
The fundamental solution of a variant of the three-dimensional wave equation known as “unidirectional pulse propagation equation” (UPPE) and its paraxial approximation is obtained. It is shown that the fundamental solution can be presented as a projection of a fundamental solution of the wave equation to some functional subspace. We discuss the degree of equivalence of the UPPE and the wave equation in this respect. In particular, we show that the UPPE, in contrast to the common belief, describes wave propagation in both longitudinal and temporal directions, and, thereby, its fundamental solution possesses a non-causal character.
A Least-Squares Transport Equation Compatible with Voids
Hansen, Jon [Texas A & M Univ., College Station, TX (United States). Dept. of Nuclear Engineering; Peterson, Jacob [Texas A & M Univ., College Station, TX (United States). Dept. of Nuclear Engineering; Morel, Jim [Texas A & M Univ., College Station, TX (United States). Dept. of Nuclear Engineering; Ragusa, Jean [Texas A & M Univ., College Station, TX (United States). Dept. of Nuclear Engineering; Wang, Yaqi [Idaho National Lab. (INL), Idaho Falls, ID (United States)
2014-12-01
Standard second-order self-adjoint forms of the transport equation, such as the even-parity, odd-parity, and self-adjoint angular flux equation, cannot be used in voids. Perhaps more important, they experience numerical convergence difficulties in near-voids. Here we present a new form of a second-order self-adjoint transport equation that has an advantage relative to standard forms in that it can be used in voids or near-voids. Our equation is closely related to the standard least-squares form of the transport equation with both equations being applicable in a void and having a nonconservative analytic form. However, unlike the standard least-squares form of the transport equation, our least-squares equation is compatible with source iteration. It has been found that the standard least-squares form of the transport equation with a linear-continuous finite-element spatial discretization has difficulty in the thick diffusion limit. Here we extensively test the 1D slab-geometry version of our scheme with respect to void solutions, spatial convergence rate, and the intermediate and thick diffusion limits. We also define an effective diffusion synthetic acceleration scheme for our discretization. Our conclusion is that our least-squares S_{n} formulation represents an excellent alternative to existing second-order S_{n} transport formulations
MATH 262 Linear Algebra And Differential Equations Spring 2014 ...
2014-01-13
Instructor: Peijun Li, MATH 440, (765) 494-0846, lipeijun@math.purdue.edu ... Textbook: Differential Equations and Linear Algebra by Stephen Goode and Scott
Jerk, snap, and the cosmological equation of state
Matt Visser
2004-03-31
Taylor expanding the cosmological equation of state around the current epoch is the simplest model one can consider that does not make any a priori restrictions on the nature of the cosmological fluid. Most popular cosmological models attempt to be ``predictive'', in the sense that once somea priori equation of state is chosen the Friedmann equations are used to determine the evolution of the FRW scale factor a(t). In contrast, a retrodictive approach might usefully take observational dataconcerning the scale factor, and use the Friedmann equations to infer an observed cosmological equation of state. In particular, the value and derivatives of the scale factor determined at the current epoch place constraints on the value and derivatives of the cosmological equation of state at the current epoch. Determining the first three Taylor coefficients of the equation of state at the current epoch requires a measurement of the deceleration, jerk, and snap -- the second, third, and fourth derivatives of the scale factor with respect to time. Higher-order Taylor coefficients in the equation of state are related to higher-order time derivatives of the scale factor. Since the jerk and snap are rather difficult to measure, being related to the third and fourth terms in the Taylor series expansion of the Hubble law, it becomes clear why direct observational constraints on the cosmological equation of state are so relatively weak; and are likely to remain weak for the foreseeable future.
A six dimensional analysis of Maxwell's Field Equations
Ana Laura García-Perciante; Alfredo Sandoval-Villalbazo; L. S. García Colín
2002-02-08
A framework based on an extension of Kaluza's original idea of using a five dimensional space to unify gravity with electromagnetism is used to analyze Maxwell\\'{}s field equations. The extension consists in the use of a six dimensional space in which all equations of electromagnetism may be obtained using only Einstein's field equation. Two major advantages of this approach to electromagnetism are discussed, a full symmetric derivation for the wave equations for the potentials and a natural inclusion of magnetic monopoles without using any argument based on singularities.
Gravity and Zonal Flows of Giant Planets: From the Euler Equation to the Thermal Wind Equation
Cao, Hao
2015-01-01
Any non-spherical distribution of density inside planets and stars gives rise to a non-spherical external gravity and change of shape. If part or all of the observed zonal flows at the cloud deck of giant planets represent deep interior dynamics, then the density perturbations associated with the deep zonal flows could generate gravitational signals detectable by the planned Juno mission and the Cassini Proximal Orbits. It is currently debated whether the thermal wind equation (TWE) can be used to calculate the gravity field associated with deep zonal flows. Here we present a critical comparison between the Euler equation and the thermal wind equation. Our analysis shows that the applicability of the TWE in calculating the gravity moments depends crucially on retaining the non-sphericity of the background density and gravity. Only when the background non-sphericity of the planet is taken into account, the TWE makes accurate enough prediction (with a few tens of percent errors) for the high-degree gravity mome...
Uttam Ghosh; Srijan Sengupta; Susmita Sarkar; Shantanu Das
2015-09-07
The solution of non-linear differential equation, non-linear partial differential equation and non-linear fractional differential equation is current research in Applied Science. Here tanh-method and Fractional Sub-Equation methods are used to solve three non-linear differential equations and the corresponding fractional differential equation. The fractional differential equations here are composed with Jumarie fractional derivative. Both the solution is obtained in analytical traveling wave solution form. We have not come across solutions of these equations reported anywhere earlier.
A Convergent Reaction-Diffusion Master Equation
Samuel A Isaacson
2013-06-28
The reaction-diffusion master equation (RDME) is a lattice stochastic reaction-diffusion model that has been used to study spatially distributed cellular processes. The RDME is often interpreted as an approximation to spatially-continuous models in which molecules move by Brownian motion and react by one of several mechanisms when sufficiently close. In the limit that the lattice spacing approaches zero, in two or more dimensions, the RDME has been shown to lose bimolecular reactions. The RDME is therefore not a convergent approximation to any spatially-continuous model that incorporates bimolecular reactions. In this work we derive a new convergent RDME (CRDME) by finite volume discretization of a spatially-continuous stochastic reaction-diffusion model popularized by Doi. We demonstrate the numerical convergence of reaction time statistics associated with the CRDME. For sufficiently large lattice spacings or slow bimolecular reaction rates, we also show the reaction time statistics of the CRDME may be approximated by those from the RDME. The original RDME may therefore be interpreted as an approximation to the CRDME in several asymptotic limits.
Andrey Akhmeteli
2015-07-13
Previously (A. Akhmeteli, J. Math. Phys., v. 52, p. 082303 (2011)), the Dirac equation in an arbitrary electromagnetic field was shown to be generally equivalent to a fourth-order equation for just one component of the four-component Dirac spinor function. This was done for a specific (chiral) representation of gamma-matrices and for a specific component. In the current work, the result is generalized for a general representation of gamma-matrices and a general component (satisfying some conditions). The resulting equivalent of the Dirac equation is also manifestly relativistically covariant and should be useful in applications of the Dirac equation.
Slyusarchuk, V. E. E-mail: V.Ye.Slyusarchuk@NUWM.rv.ua
2014-06-01
The well-known theorems of Favard and Amerio on the existence of almost periodic solutions to linear and nonlinear almost periodic differential equations depend to a large extent on the H-classes and the requirement that the bounded solutions of these equations be separated. The present paper provides different conditions for the existence of almost periodic solutions. These conditions, which do not depend on the H-classes of the equations, are formulated in terms of a special functional on the set of bounded solutions of the equations under consideration. This functional is used, in particular, to test whether solutions are separated. Bibliography: 24 titles. (paper)
A. H. M. Kierkels; J. J. L. Velázquez
2015-11-04
We construct a family of self-similar solutions with fat tails to a quadratic kinetic equation. This equation describes the long time behaviour of weak solutions with finite mass to the weak turbulence equation associated to the nonlinear Schr\\"odinger equation. The solutions that we construct have finite mass, but infinite energy. In J. Stat. Phys. 159:668-712, self-similar solutions with finite mass and energy were constructed. Here we prove upper and lower exponential bounds on the tails of these solutions.
KH Computational Physics-2015 Basic Numerical Algorithms Ordinary differential equations
Gustafsson, Torgny
KH Computational Physics- 2015 Basic Numerical Algorithms Ordinary differential equations The set(xl) at certain points xl. Kristjan Haule, 2015 1 #12;KH Computational Physics- 2015 Basic Numerical Algorithms purpose routine · Numerov's algorithm: ¨y = f(t)y(t) ( for Schroedinger equation) · Verlet algorithm: ¨y
On two-component equations for zero mass particles
Wilhelm I. Fushchych; A. L. Grishchenko
2002-06-12
The paper presents a detailed theoretical-group analysis of three types of two-component equations of motion which describe the particle with zero mass and spin 1/2. There are studied P-, T- and C-propertias of the equations obtained.
Equisolvability of Series vs. Controller's Topology in Synchronous Language Equations
Brayton, Robert K.
Equisolvability of Series vs. Controller's Topology in Synchronous Language Equations Nina operators for abstract languages: synchronous composition, #15;, and parallel composition, #5;, and we studied the solutions of the equations defined over finite state machines (FSMs) of the type MA #15; MX
Volume equalized constitutive equations for foamed polymer solutions
Valkó, Peter
Volume equalized constitutive equations for foamed polymer solutions P. Valko and M. J. Economides; accepted 21 April 1992) Synopsis In order to develop a constitutive equation for foamed polymer solutions Hydroxypropylguar (HPG) solutions containing a small amount of foaming agent and foamed by nitrogen and/or carbon
Asymptotic Expansions of Defective Renewal Equations with Applications to Perturbed
Blanchet, Jose H.
Asymptotic Expansions of Defective Renewal Equations with Applications to Perturbed Risk Models. These expansions are applied to the analysis of Processor Sharing queues and perturbed risk models, and yield Introduction A defective renewal equation for a function ap (·) takes the form ap (t) = bp (t) + (1 - p) [0,t
From Petri Nets to Differential Equations -an Integrative Approach
Gilbert, David
From Petri Nets to Differential Equations - an Integrative Approach for Biochemical Network report on the results of an investigation into the inte- gration of Petri nets and ordinary differential equations (ODEs) for the modelling and analysis of biochemical networks, and the application of our approach
Boundary value problems for the onedimensional Willmore equation
Grunau, Hans-Christoph
Boundary value problems for the oneÂdimensional Willmore equation Klaus Deckelnick # and Hans--known that the corresponding surface # has to satisfy the Willmore equation #H + 2H(H 2 -K) = 0 on #, (1) # eÂmail: Klaus Willmore surfaces of prescribed genus has been proved by Simon [Sn] and Bauer & Kuwert [BK]. Also, local
Boundary value problems for the one-dimensional Willmore equation
Grunau, Hans-Christoph
Boundary value problems for the one-dimensional Willmore equation Klaus Deckelnick and HansÂknown that the corresponding surface has to satisfy the Willmore equation H + 2H(H2 - K) = 0 on , (1) e-mail: Klaus Willmore surfaces of prescribed genus has been proved by Simon [Sn] and Bauer & Kuwert [BK]. Also, local
FOURTH ORDER PARTIAL DIFFERENTIAL EQUATIONS ON GENERAL GEOMETRIES
FOURTH ORDER PARTIAL DIFFERENTIAL EQUATIONS ON GENERAL GEOMETRIES By John B. Greer Andrea L0436 Phone: 612/624-6066 Fax: 612/626-7370 URL: http://www.ima.umn.edu #12;Fourth Order Partial Differential (Bertalm´io, Cheng, Osher, and Sapiro 2001) to fourth order PDEs including the Cahn- Hilliard equation
A JUSTIFICATION OF EDDY CURRENTS MODEL FOR THE MAXWELL EQUATIONS
Buffa, Annalisa
A JUSTIFICATION OF EDDY CURRENTS MODEL FOR THE MAXWELL EQUATIONS H. AMMARI, A. BUFFA, AND J.-C. NÂ1823 Abstract. This paper is concerned with the approximation of the Maxwell equations by the eddy currents model, which appears as a correction of the quasi-static model. The eddy currents model is obtained
Exact null controllability of degenerate evolution equations with scalar control
Fedorov, Vladimir E; Shklyar, Benzion
2012-12-31
Necessary and sufficient conditions for the exact null controllability of a degenerate linear evolution equation with scalar control are obtained. These general results are used to examine the exact null controllability of the Dzektser equation in the theory of seepage. Bibliography: 13 titles.
The Lamb-Bateman integral equation and the fractional derivatives
D. Babusci; G. Dattoli; D. Sacchetti
2010-06-08
The Lamb-Bateman integral equation was introduced to study the solitary wave diffraction and its solution was written in terms of an integral transform. We prove that it is essentially the Abel integral equation and its solution can be obtained using the formalism of fractional calculus.
MATH 100 Introduction to the Profession Linear Equations in MATLAB
Fasshauer, Greg
's input-output model in economics, electric circuit problems, the steady-state analysis of a systemMATH 100 Â Introduction to the Profession Linear Equations in MATLAB Greg Fasshauer Department;Chapter 5 of Experiments with MATLAB Where do systems of linear equations come up? fasshauer@iit.edu MATH
Radio Interferometry & The Measurement Equation -1 School of Physics
Tittley, Eric
Radio Interferometry & The Measurement Equation - 1 School of Physics and Astronomy An Introduction to Radio Interferometry and The Measurement Equation Formalism Pedagogical Seminar Louise M. Ker March 2010 Abstract The next generation of radio telescopes, such as LOFAR, e-Merlin, ASKAP, MeerKat and even- tually
Time Domain Maxwell Equations Solved with Schwarz Waveform
Gander, Martin J.
with Dirichlet boundary conditions and was analyzed for the heat equation by Gander and Stuart [1998]. Giladi transmission conditions for the time domain Maxwell equations is given by -tEi,n + Ã? Hi,n - Ei,n = J, i is called the characteristic transmission condition, establishes how the subdomains communicate with each
Geometric Integration: Numerical Solution of Differential Equations on Manifolds
Scheichl, Robert
and the solar system. Conserved quantities of a Hamiltonian system, such as energy, linear and angular momentumGeometric Integration: Numerical Solution of Differential Equations on Manifolds C.J. Budd 1 & A riches. Psalms 104:24 Since their introduction by Sir Isaac Newton, differential equations have played
Global solutions of the Hunter-Saxton equation Alberto Bressan
- Introduction In this paper we investigate the Cauchy problem ut + u2 2 x = 1 4 x - - x u2 x dx , u(0, x) = ¯u-Saxton equation. In this paper we analyse various concepts of solutions for the above equations, and construct, bending, and splay (the latter being a fan-shaped spreading out from the original direction, bending being
Global solutions of the HunterSaxton equation Alberto Bressan
Introduction In this paper we investigate the Cauchy problem u t + # u 2 2 # x = 1 4 ## x -# - # # x # u 2 x dx as the HunterSaxton equation. In this paper we analyse various concepts of solutions for the above equations, bending, and splay (the latter being a fanshaped spreading out from the original direction, bending being
Simultaneous temperature and flux controllability for heat equations with memory
Ceragioli, Francesca
Simultaneous temperature and flux controllability for heat equations with memory S. Avdonin Torino -- Italy, luciano.pandolfi@polito.it June 14, 2010 Abstract It is known that, in the case of heat equation with memory, tem- perature can be controlled to an arbitrary square integrable target provided
Spinor equation for the $W^{\\pm}$ boson
Ruo Peng Wang
2011-09-20
I introduce spinor equations for the $W^{\\pm}$ fields. The properties of these spinor equations under space-time transformation and under charge conjugation are studied. The expressions for electric charge and current and densities of the $W^{\\pm}$ fields are obtained. Covariant quantization conditions are established, and the vacuum energy for the $W^{\\pm}$ fields is found to be zero.
Saturation effects in QCD from linear transport equation
Krzysztof Kutak
2010-09-09
We show that the GBW saturation model provides an exact solution to the one-dimensional linear transport equation. We also show that it is motivated by the BK equation considered in the saturated regime when the diffusion and the splitting term in the diffusive approximation are balanced by the nonlinear term.
Universality of the de Broglie-Einstein velocity equation
Yusuf Z. Umul
2007-12-06
The de Broglie-Einstein velocity equation is derived for a relativistic particle by using the energy and momentum relations in terms of wave and matter properties. It is shown that the velocity equation is independent from the relativistic effects and is valid also for the non-relativistic case. The results of this property is discussed.
Adaptive Calculation of Variable Coefficients Elliptic Differential Equations via Wavelets
Averbuch, Amir
Adaptive Calculation of Variable Coefficients Elliptic Differential Equations via Wavelets Amir in numerical solution of differential and integral equations. Classical methods for discretization lead-based multiplication is af- fected by different input parameters for the algorithm. We integrated a sparse
IDENTIFICATION OF MOBILITIES FOR THE BUCKLEYLEVERETT EQUATION BY FRONT TRACKING
is also used to solve the saturation equation in a commercial reservoir simulator [1]. 2. Equations mobility functions that are used as input for reservoir simulations. A new method has been developed is viscosity of phase i, k r i is relative permeability of phase i, s is the saturation of water, and p
Hamilton-Jacobi-Bellman Equations Analysis and Numerical Analysis
Flynn, E. Victor
Hamilton-Jacobi-Bellman Equations Analysis and Numerical Analysis Iain Smears #12;My deepest thanks at Durham University. Abstract. This work treats Hamilton-Jacobi-Bellman equations. Their relation and the inaugural papers on mean-field games. Original research on numerical methods for Hamilton-Jacobi
THE HAMILTON-JACOBI EQUATION, INTEGRABILITY, AND NONHOLONOMIC SYSTEMS
Fassò, Francesco
THE HAMILTON-JACOBI EQUATION, INTEGRABILITY, AND NONHOLONOMIC SYSTEMS LARRY BATES, FRANCESCO FASSÒ why it appears there should not be an analogue of a complete integral for the Hamilton-Jacobi equation for integrable nonholonomic systems. February 7, 2014 1. Introduction The Hamilton-Jacobi theory is at the heart
Hamilton-Jacobi equations with discontinuous source terms Nao Hamamuki
Ishii, Hitoshi
Hamilton-Jacobi equations with discontinuous source terms Nao Hamamuki We study the initial-value problem for the Hamilton-Jacobi equation of the form { tu(x, t) + H(x, xu(x, t)) = 0 in Rn × (0, T), u control problem with a semicontinuous running cost function. References [1] Y. Giga, N. Hamamuki, Hamilton-Jacobi
Generalized Input/Output Equations and Nonlinear Realizability
Wang, Yuan
operator satisfies a possibly high-order differential input/output equation, then it is locally realizable-9108250 and DMS-9403924 Keywords: Generating series, local realization of input/output operators, input/outputGeneralized Input/Output Equations and Nonlinear Realizability Yuan Wang Mathematics Department
Lyapunov control of bilinear Schrodinger equations Mazyar Mirrahimi a
Paris-Sud XI, Université de
Lyapunov control of bilinear Schr¨odinger equations Mazyar Mirrahimi a , Pierre Rouchon b , Gabriel´ee Cedex, France Abstract A Lyapunov-based approach for trajectory tracking of the Schr¨odinger equation Lyapunov function, Adiabatic invariants, Tracking. 1 Introduction Controllability of a finite dimensional
Stochastic differential equations and numerical simulation for pedestrians
Garrison, J.C.
1993-07-27
The mathematical foundation of the Ito interpretation of stochastic ordinary and partial differential equations is briefly explained. This provides the basis for a review of simple difference approximations to stochastic differential equations. An example arising in the theory of optical switching is discussed.
A Least-Squares Transport Equation Compatible with Voids
Hansen, Jon
2014-04-22
Standard second-order self-adjoint forms of the transport equation, such as the even-parity, odd-parity, and self-adjoint angular flux equation, cannot be used in voids. Perhaps more importantly, they experience numerical convergence difficulties...
MATH 225 Fall 2011 Differential Equations Course Syllabus
MATH 225 Fall 2011 Differential Equations Course Syllabus Instructor Info Instructor: Phone: Office: Email: Office Hours: Section Website: Course Website: http://mcs.mines.edu/Courses/math225/ Grading sciences. Prerequisites: MATH213 or MATH223 Text Differential Equations with Boundary-Value Problems, 7th
A NOTE ON VISCOUS SPLITTING OF DEGENERATE CONVECTIONDIFFUSION EQUATIONS
. It thus provides a suitable ``L 1 type'' framework for analyzing numerical schemes for convectiondiffusionA NOTE ON VISCOUS SPLITTING OF DEGENERATE CONVECTIONDIFFUSION EQUATIONS Steinar Evje, Kenneth possibly strongly degenerate convectiondiffusion problems. Since we allow the equations to be strongly
Parameter Estimation for the Heat Equation on Perforated Domains
Parameter Estimation for the Heat Equation on Perforated Domains H.T. Banks1 , D. Cioranescu2 , A: Inverse problems, parameter estimation, perforated domains, homogeniza- tion, thermal diffusion, ordinary porous samples by use of solutions of a heat equation on a randomly perforated domain. As noted
Heat Equations with Fractional White Noise Potentials
Hu, Y. [Department of Mathematics, University of Kansas, 405 Snow Hall, Lawrence, KS 66045-2142 (United States)], E-mail: hu@math.ukans.edu
2001-07-01
This paper is concerned with the following stochastic heat equations: ({partial_derivative}u{sub t}(x))/({partial_derivative}t=1/2 u{sub t}(x)+{omega}{sup H}.u{sub t}(x)), x element of {sup d}, t>0, where w{sup H} is a time independent fractional white noise with Hurst parameter H=(h{sub 1}, h{sub 2},..., h{sub d}) , or a time dependent fractional white noise with Hurst parameter H=(h{sub 0}, h{sub 1},..., h{sub d}) . Denote | H | =h{sub 1}+h{sub 2}+...+h{sub d} . When the noise is time independent, it is shown that if 1/2
Quantum master equation with balanced gain and loss
Dennis Dast; Daniel Haag; Holger Cartarius; Günter Wunner
2014-11-20
We present a quantum master equation describing a Bose-Einstein condensate with particle loss on one lattice site and particle gain on the other lattice site whose mean-field limit is a non-Hermitian PT-symmetric Gross-Pitaevskii equation. It is shown that the characteristic properties of PT-symmetric systems, such as the existence of stationary states and the phase shift of pulses between two lattice sites, are also found in the many-particle system. Visualizing the dynamics on a Bloch sphere allows us to compare the complete dynamics of the master equation with that of the Gross-Pitaevskii equation. We find that even for a relatively small number of particles the dynamics are in excellent agreement and the master equation with balanced gain and loss is indeed an appropriate many-particle description of a PT-symmetric Bose-Einstein condensate.
Solution of the Boussinesq equation using evolutionary vessels
Andrey Melnikov
2013-01-11
In this work we present a solution of the Boussinesq equation. The derived formulas include solitons, Schwartz class solutions and solutions, possessing singularities on a closed set Z of the (x,t) domain, obtained from the zeros of the tau function. The idea for solving the Boussinesq equation is identical to the (unified) idea of solving the KdV and the evolutionary NLS equations: we use a theory of evolutionary vessels. But a more powerful theory of non-symmetric evolutionary vessels is presented, inserting flexibility into the construction and allowing to deal with complex-valued solutions. A powerful scattering theory of Deift-Tomei-Trubowitz for a three dimensional operator, which is used to solve the Boussinesq equation, fits into our setting only in a particular case. On the other hand, we create a much wider class of solutions of the Boussinesq equation with singularities on a closed set $Z$.
Flow Equations for Hamiltonians from Continuous Unitary Transformations
Bruce Henry Bartlett
2003-05-19
This thesis presents an overview of the flow equations recently introduced by Wegner. The little known mathematical framework of the flow in the manifold of unitarily equivalent matrices, as discovered in the mathematical literature before Wegner's paper, is established in the initial chapter and used as a background for the entire presentation. The application of flow equations to the Foldy-Wouthuysen transformation and to the elimination of the electron-phonon coupling in a solid is reviewed. Recent flow equations approaches to the Lipkin model are examined thoroughly, paying special attention to their utility near the phase change boundary. We present more robust schemes by requiring that expectation values be flow dependent; either through a variational or self-consistent calculation. The similarity renormalization group equations recently developed by Glazek and Wilson are also reviewed. Their relationship to Wegner's flow equations is investigated through the aid of an instructive model.
The modified equation for spinless particles and superalgebra
Sadeghi, J.; Rostami, M.; Sadeghi, Z.
2013-09-15
In this paper we consider modified wave equations for spinless particles in an external magnetic field. We consider 4-potentials which guarantee Lorentz' and Coulomb's conditions. The new variable for modified wave equation leads us to consider the associated Laguerre differential equation. We take advantage of the factorization method in Laguerre differential equation and solve the modified equation. In order to obtain the wave function, energy spectrum and its quantization, we will establish conditions for the orbital quantum number. We account such orbital quantum number and obtain the raising and lowering operators. If we want to have supersymmetry partners, we need to apply the shape invariance condition. This condition for the partner potential will help us find the limit of ? as ?=±?(l)
Moment equations for chemical reactions on interstellar dust grains
Azi Lipshtat; Ofer Biham
2002-12-09
While most chemical reactions in the interstellar medium take place in the gas phase, those occurring on the surfaces of dust grains play an essential role. Chemical models based on rate equations including both gas phase and grain surface reactions have been used in order to simulate the formation of chemical complexity in interstellar clouds. For reactions in the gas phase and on large grains, rate equations, which are highly efficient to simulate, are an ideal tool. However, for small grains under low flux, the typical number of atoms or molecules of certain reactive species on a grain may go down to order one or less. In this case the discrete nature of the opulations of reactive species as well as the fluctuations become dominant, thus the mean-field approximation on which the rate equations are based does not apply. Recently, a master equation approach, that provides a good description of chemical reactions on interstellar dust grains, was proposed. Here we present a related approach based on moment equations that can be obtained from the master equation. These equations describe the time evolution of the moments of the distribution of the population of the various chemical species on the grain. An advantage of this approach is the fact that the production rates of molecular species are expressed directly in terms of these moments. Here we use the moment equations to calculate the rate of molecular hydrogen formation on small grains. It is shown that the moment equation approach is efficient in this case in which only a single reactive specie is involved. The set of equations for the case of two species is presented and the difficulties in implementing this approach for complex reaction networks involving multiple species are discussed.
Fournier, John J.F.
. It is called the specific heat of the body. · The rate at which heat energy crosses a surface is proportional), so the rate at which heat energy crosses the right hand end is AT x (x + dx, t). Similarly, the rateThe Heat Equation (One Space Dimension) In these notes we derive the heat equation for one space
Grossberg, Stephen
Appendix: Equations and Parameters This section describes BCS and FCS equations that incorporate data: Only a single scale is used, and hypercomplex and bipole cells in the BCS and monocular filling Simple Cells of the BCS Evensymmetric and oddsymmetric simple cell receptive fields centered
Grossberg, Stephen
Appendix: Equations and Parameters This section describes BCS and FCS equations that incorporate: Only a single scale is used, and hypercomplex and bipole cells in the BCS and monocular lling. This assures a positive response to ganzfelds. A2 Simple Cells of the BCS Even-symmetric and odd
The BMS Equation and c\\bar{c} Production; A Comparison of the BMS and BK Equations
Giuseppe Marchesini; A. H. Mueller
2015-10-29
We introduce two processes where the BMS equation appears in a context quite different from the original context of non-global jet observables. We note the strong similarities of the BMS equation to the BK and FKPP equations and argue that these, essentially identical equations, can be viewed either in terms of the probability, or amplitude, of something not happening or in terms of the nonlinear terms setting unitarity limits. Mostly analytic solutions are given for (i) the probability that no $c\\bar{c}$ pairs be produced in a jet decay and (ii) the probability that no-$c\\bar{c}$ pairs be produced in a high energy dipole nucleus scattering. Both these processes obey BMS equations, albeit with very different kernels.
QED for fields obeying a square root operator equation
Tobias Gleim
2008-02-25
Instead of using local field equations - like the Dirac equation for spin-1/2 and the Klein-Gordon equation for spin-0 particles - one could try to use non-local field equations in order to describe scattering processes. The latter equations can be obtained by means of the relativistic energy together with the correspondence principle, resulting in equations with a square root operator. By coupling them to an electromagnetic field and expanding the square root (and taking into account terms of quadratic order in the electromagnetic coupling constant e), it is possible to calculate scattering matrix elements within the framework of quantum electrodynamics, e.g. like those for Compton scattering or for the scattering of two identical particles. This will be done here for the scalar case. These results are then compared with the corresponding ones based on the Klein-Gordon equation. A proposal of how to transfer these reflections to the spin-1/2 case is also presented.
Equations, States, and Lattices of Infinite-Dimensional Hilbert Spaces
Norman D. Megill; Mladen Pavicic
2001-01-21
We provide several new results on quantum state space, on lattice of subspaces of an infinite dimensional Hilbert space, and on infinite dimensional Hilbert space equations as well as on connections between them. In particular we obtain an n-variable generalized orthoarguesian equation which holds in any infinite dimensional Hilbert space. Then we strengthen Godowski's result by showing that in an ortholattice on which strong states are defined Godowski's equations as well as the orthomodularity hold. We also prove that all 6- and 4-variable orthoarguesian equations presented in the literature can be reduced to new 4- and 3-variable ones, respectively and that Mayet's examples follow from Godowski's equations. To make a breakthrough in testing these massive equations we designed several novel algorithms for generating Greechie diagrams with an arbitrary number of blocks and atoms (currently testing with up to 50) and for automated checking of equations on them. A way of obtaining complex infinite dimensional Hilbert space from the Hilbert lattice equipped with several additional conditions and without invoking the notion of state is presented. Possible repercussions of the results to quantum computing problems are discussed.
Harmonic coordinates in the string and membrane equations
Chun-Lei He; Shou-Jun Huang
2010-04-16
In this note, we first show that the solutions to Cauchy problems for two versions of relativistic string and membrane equations are diffeomorphic. Then we investigate the coordinates transformation presented in Ref. [9] (see (2.20) in Ref. [9]) which plays an important role in the study on the dynamics of the motion of string in Minkowski space. This kind of transformed coordinates are harmonic coordinates, and the nonlinear relativistic string equations can be straightforwardly simplified into linear wave equations under this transformation.
Klein-Gordon and Dirac Equations with Thermodynamic Quantities
Altug Arda; Cevdet Tezcan; Ramazan Sever
2015-10-21
We study the thermodynamic quantities such as the Helmholtz free energy, the mean energy and the specific heat for both the Klein-Gordon, and Dirac equations. Our analyze includes two main subsections: ($i$) statistical functions for the Klein-Gordon equation with a linear potential having Lorentz vector, and Lorentz scalar parts ($ii$) thermodynamic functions for the Dirac equation with a Lorentz scalar, inverse-linear potential by assuming that the scalar potential field is strong ($A \\gg 1$). We restrict ourselves to the case where only the positive part of the spectrum gives a contribution to the sum in partition function. We give the analytical results for high temperatures.
New Dirac equation from the view point of particle
Ozaydin, Fatih; Altintas, Azmi Ali; Susam, Lidya Amon; Arik, Metin; Yarman, Tolga
2012-09-06
According to the classical approach, especially the Lorentz Invariant Dirac Equation, when particles are bound to each other, the interaction term appears as a quantity belonging to the 'field'. In this work, as a totally new approach, we propose to alter the rest masses of the particles due to their interaction, as much as their respective contributions to the static binding energy. Thus we re-write and solve the Dirac Equation for the hydrogen atom, and amazingly, obtain practically the same numerical results for the ground states, as those obtained from the Dirac Equation.
Skyrme models and nuclear matter equation of state
Adam, Christoph; Wereszczynski, Andrzej
2015-01-01
We investigate the role of pressure in a class of generalised Skyrme models. We introduce pressure as the trace of the spatial part of the energy-momentum tensor and show that it obeys the usual thermodynamical relation. Then, we compute analytically the mean-field equation of state in the high and medium pressure regimes by applying topological bounds on compact domains. The equation of state is further investigated numerically for the charge one skyrmions. We identify which term in a generalised Skyrme model is responsible for which part in the equation of state. Further, we compare our findings with the corresponding results in the Walecka model.
On equations of motion in twist-four evolution
Yao Ji; A. V. Belitsky
2014-10-23
Explicit diagrammatic calculation of evolution equations for high-twist correlation functions is a challenge already at one-loop order in QCD coupling. The main complication being quite involved mixing pattern of the so-called non-quasipartonic operators. Recently, this task was completed in the literature for twist-four nonsinglet sector. Presently, we elaborate on a particular component of renormalization corresponding to the mixing of gauge-invariant operators with QCD equations of motion. These provide an intrinsic contribution to evolution equations yielding total result in agreement with earlier computations that bypassed explicit analysis of Feynman graphs.
Solution of the Helmholtz equation for spin-2 fields
G. F. Torres del Castillo; J. E. Rojas Marcial
2003-05-01
The Helmholtz equation for symmetric, traceless, second-rank tensor fields in three-dimensional flat space is solved in spherical and cylindrical coordinates by separation of variables making use of the corresponding spin-weighted harmonics. It is shown that any symmetric, traceless, divergenceless second-rank tensor field that satisfies the Helmholtz equation can be expressed in terms of two scalar potentials that satisfy the Helmholtz equation. Two such expressions are given, which are adapted to the spherical or cylindrical coordinates. The application to the linearized Einstein theory is discussed.
Symmetries and exact solutions of the rotating shallow water equations
Alexander Chesnokov
2008-08-11
Lie symmetry analysis is applied to study the nonlinear rotating shallow water equations. The 9-dimensional Lie algebra of point symmetries admitted by the model is found. It is shown that the rotating shallow water equations are related with the classical shallow water model with the change of variables. The derived symmetries are used to generate new exact solutions of the rotating shallow equations. In particular, a new class of time-periodic solutions with quasi-closed particle trajectories is constructed and studied. The symmetry reduction method is also used to obtain some invariant solutions of the model. Examples of these solutions are presented with a brief physical interpretation.
Role of retardation in 3-D relativistic equations
A. D. Lahiff; I. R. Afnan
1997-08-20
Equal-time Green's function is used to derive a three-dimensional integral equation from the Bethe-Salpeter equation. The resultant equation, in the absence of anti-particles, is identical to the use of time-ordered diagrams, and has been used within the framework of $\\phi^2\\sigma$ coupling to study the role of energy dependence and non-locality when the two-body potential is the sum of $\\sigma$-exchange and crossed $\\sigma$ exchange. The results show that non-locality and energy dependence make a substantial contribution to both the on-shell and off-shell amplitudes.
Bifurcation in kinetic equation for interacting Fermi systems
Klaus Morawetz
2003-01-27
The finite duration of collisions appear as time-nonlocality in the kinetic equation. Analyzing the corresponding quantum kinetic equation for dense interacting Fermi systems a delay differential equation is obtained which combines time derivatives with finite time stepping known from the logistic mapping. The responsible delay time is explicitly calculated and discussed. As a novel feature oscillations in the time evolution of the distribution function itself appear and bifurcations up to chaotic behavior can occur. The temperature and density conditions are presented where such oscillations and bifurcations arise indicating an onset of phase transition.
A Master equation for force distributions in polydisperse frictional particles
Kuniyasu Saitoh; Vanessa Magnanimo; Stefan Luding
2015-05-28
An incremental evolution equation, i.e. a Master equation in statistical mechanics, is introduced for force distributions in polydisperse frictional particle packings. As basic ingredients of the Master equation, the conditional probability distributions of particle overlaps are determined by molecular dynamics simulations. Interestingly, tails of the distributions become much narrower in the case of frictional particles than frictionless particles, implying that correlations of overlaps are strongly reduced by microscopic friction. Comparing different size distributions, we find that the tails are wider for the wider distribution.
A new vapor pressure equation originating at the critical point
Nuckols, James William
1976-01-01
111 1v vi vi1 25 29 32 33 vi Table LIST OF TABLES Page 1. Hall-Eubank Coefficients for Ar, N2, and C2H6 2. HEN Equation Coefficients and Sources of Data 3. Comparison of the HEN, Frost-Kalkwarf, and Wagner Equations 17 27 vii LIST... in existence are discussed by Wagner (1973), Niiller (1964), and Reid and Sherwood (1966). These references indicate that the Frost-Kalkwarf (1953) and the Wagner (1973) equations achieve the best overall description of the coexistence curve, but they too...
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.
ON THE WELL-POSEDNESS OF THE DEGASPERIS-PROCESI EQUATION
ON THE WELL-POSEDNESS OF THE DEGASPERIS-PROCESI EQUATION GIUSEPPE M. COCLITE AND KENNETH H. KARLSEN order dispersive Degasperis-Procesi equation (DP) tu - 3 txxu + 4uxu = 3xu2 xxu + u3 xxxu. This equation as for the Camassa-Holm equation (one order more accurate than the KdV equation). We prove existence and L1 stability
Hanyk, Ladislav
and Differential-Algebraic Equations, SIAM, Philadelphia. Backus, G.E., 1967. Converting vector and tensor
Hanyk, Ladislav
and DifferentialAlgebraic Equations, SIAM, Philadelphia. Backus, G.E., 1967. Converting vector and tensor
Electrolux Gibson Air Conditioner and Equator Clothes Washer...
Broader source: Energy.gov (indexed) [DOE]
DOE testing in support of the ENERGY STAR program has revealed that an Electrolux Gibson air conditioner (model GAH105Q2T1) and an Equator clothes washer (model EZ 3720 CEE), both...
Exact Vacuum Solutions of Jordan, Brans-Dicke Field Equations
Sergey Kozyrev
2005-12-04
We present the static spherically symmetric vacuum solutions of the Jordan, Brans-Dicke field equations. The new solutions are obtained by considering a polar Gaussian, isothermal and radial hyperbolic metrics.
A Note on DeMoivre's Quintic Equation
M. L. Glasser
2009-07-18
The quintic equation with real coefficients $$x^5+5ax^3+5a^2x+b=0$$ is solved in terms of radicals and the results used to sum a hypergeometric series for several arguments.
Essential Differential Equations September 2013 Lecturer David Silvester
Silvester, David J.
Essential Differential Equations September 2013 Lecturer David Silvester Office Alan Turing 15 Classes Tues 34 Alan Turing G.207 Thur 13 Alan Turing G.205 Assessment Week 7 Test 20% Week 10
Symmetry and some exact solutions of nonlinear polywave equations
Zhdanov, Renat
Institute of Mathematics of the Academy of Sciences of Ukraine, Tereshchenkivska Str.3, 252004 Kiev, Ukraine of Ukraine, Tereshchenkivska Str.3, 252004 Kiev, Ukraine 1 #12; properties of equation (1) seems
Lattice Boltzmann equation simulations of turbulence, mixing, and combustion
Yu, Huidan
2006-04-12
We explore the capability of lattice Boltzmann equation (LBE) method for complex fluid flows involving turbulence, mixing, and reaction. In the first study, LBE schemes for binary scalar mixing and multi-component reacting flow with reactions...
Non-normality of Data in Structural Equation Models
Gao, Shengyi; Mokhtarian, Patricia L; Johnston, Robert A.
2008-01-01
J. F. Finch, and P. J. Curran. Structural equations models2, 1993, pp. 313-338. 16. Curran, P. J. , S. G. West, and J.appeared not to be useful. Curran et al. (16) compared the
June 12, 2007 Reduced MHD Equations with Coupled Alfven and
Lotko, William
that auroral phenomena such as parallel electric fields, accelerated electrons, and uplifted ionospheric ions of auroral Alfv´en waves. These equations include the parallel electric field, finite Larmor radius effects
Development of one-equation transition/turbulence models
Edwards, J.R.; Roy, C.J.; Blottner, F.G.; Hassan, H.A.
2000-01-14
This paper reports on the development of a unified one-equation model for the prediction of transitional and turbulent flows. An eddy viscosity--transport equation for nonturbulent fluctuation growth based on that proposed by Warren and Hassan is combined with the Spalart-Allmaras one-equation model for turbulent fluctuation growth. Blending of the two equations is accomplished through a multidimensional intermittency function based on the work of Dhawan and Narasimha. The model predicts both the onset and extent of transition. Low-speed test cases include transitional flow over a flat plate, a single element airfoil, and a multi-element airfoil in landing configuration. High-speed test cases include transitional Mach 3.5 flow over a 5{degree} cone and Mach 6 flow over a flared-cone configuration. Results are compared with experimental data, and the grid-dependence of selected predictions is analyzed.
Fit Index Sensitivity in Multilevel Structural Equation Modeling
Boulton, Aaron Jacob
2011-07-29
Multilevel Structural Equation Modeling (MSEM) is used to estimate latent variable models in the presence of multilevel data. A key feature of MSEM is its ability to quantify the extent to which a hypothesized model fits ...
Lorentz-Dirac equation in the delta-function pulse
Miroslav Pardy
2012-08-01
We formulate the Lorentz-Dirac equation in the plane wave and in the Dirac delta-function pulse. The discussion on the relation of the Dirac delta-function to the ultrashort laser pulse is involved.
Probing the softest region of the nuclear equation of state
Li, Ba; Ko, Che Ming.
1998-01-01
An attractive, energy-dependent mean-field potential for baryons is introduced in order to generate a soft region in the nuclear equation of state, as suggested by recent lattice QCD calculations of baryon-free matter at ...
M. J. Holst The Poisson-Boltzmann Equation
Holst, Michael J.
discontinuous coefficients representing material interfaces, rapid nonlinearities, and three spatial dimensions. Similar equations occur in various applications, including nuclear physics, semiconductor physics complex biomolecule lying in a solvent. We next study the theoretical properties of the linearized
The equation of state at high temperatures from lattice QCD
G. Endrodi; Z. Fodor; S. D. Katz; K. K. Szabo
2007-10-23
We present results for the equation of state upto previously unreachable, high temperatures. Since the temperature range is quite large, a comparison with perturbation theory can be done directly.
Equator Appliance: ENERGY STAR Referral (EZ 3720 CEE)
Broader source: Energy.gov [DOE]
DOE referred the matter of Equator clothes washer model EZ 3720 CEE to the EPA for appropriate action after DOE testing showed that the model does not meet the ENERGY STAR specification.
Fast methods for static Hamilton-Jacobi Partial Differential Equations
Vladimirsky, Alexander Boris
2001-01-01
k y , A . , Fast Methods for the Eikonal Jacobi Equations onMethod and "lifting-to-manifold" to solve an anisotropic static H a m i l t o n - Jacobi
Positive Lyapunov exponents for continuous quasiperiodic Schroedinger equations
Bjerkloev, Kristian [Department of Mathematics, University of Toronto, Toronto Ontario, M5S 3G3 (Canada)
2006-02-15
We prove that the continuous one-dimensional Schroedinger equation with an analytic quasi-periodic potential has positive Lyapunov exponents in the bottom of the spectrum for large couplings.
An inverse problem for the transmission wave equation.
2006-11-03
We consider a transmission wave equation in two embedded domains in. R. 2, where the speed is a1 > 0 in the inner domain and a2 > 0 in the outer domain.
Numerical Analysis and Partial Differential Equations March 12, 2009
Elliott, Charles
Equations . . . . . . . . . . . . . . . . . . . . . 71 6 Finite element error analysis 74 6.1 Galerkin.1 A Boundary Value Problem . . . . . . . . . . . . . . . . . . . . . . . 9 2.2 A Finite Element Method) . . . . . . . . . . . . . . . . . . . . 12 2.4 Finite Element Method . . . . . . . . . . . . . . . . . . . . . . . . . 12 II Finite Element
A rough analytic relation on partial differential equations
Kato, Tsuyoshi
2010-01-01
We introduce some analytic relations on the set of partial differential equations of two variables. It relies on a new comparison method to give rough asymptotic estimates for solutions which obey different partial differential equations. It uses a kind of scale transform called tropical geometry which connects automata with real rational dynamics. Two different solutions can be considered when their defining equations are transformed to the same automata at infinity. We have a systematic way to construct related pairs of different partial differential equations, and also construct some unrelated pairs concretely. These verify that the new relations are non trivial. We also make numerical calculations and compare the results for both related and unrelated pairs of PDEs.
Fast Explicit Operator Splitting Method for Convection-Diffusion Equations
Kurganov, Alexander
to the one- and two-dimensional systems of convection-diffusion equations which model the polymer flooding) processes in fluid mechanics, astrophysics, me- teorology, multiphase flow in oil reservoirs, polymer flow
MATH 411 SPRING 2001 Ordinary Di#erential Equations
Alekseenko, Alexander
MATH 411 SPRING 2001 Ordinary Di#erential Equations Schedule # 749025 TR 01:00Â02:15 316 Boucke Instructor: Alexander Alekseenko, 328 McAllister, 865Â1984, alekseen@math.psu.edu The course
New approach for solving master equation of open atomic system
Yi-Chong Ren; Hong-Yi Fan
2015-09-06
We describe a new approach called Ket-Bra Entangled State (KBES) Method which enables one convert master equations into Schr\\"odinger-like equation. In sharply contrast to the super-operator method, the KBES method is applicable for any master equation of finite-level system in theory, and the calculation can be completed by computer. With this method, we obtain the exact dynamic evolution of a radioactivity damped 2-level atom in time-dependent external field, and a 3-level atom coupled with bath; Moreover, the master equation of N-qubits Heisenberg chain each qubit coupled with a reservoir is also resolved in Sec.III; Besides, the paper briefly discuss the physical implications of the solution.
A fast enriched FEM for Poisson equations involving interfaces
Huynh, Thanh Le Ngoc
2008-01-01
We develop a fast enriched finite element method for solving Poisson equations involving complex geometry interfaces by using regular Cartesian grids. The presence of interfaces is accounted for by developing suitable jump ...
A Hamiltonian functional for the linearized Einstein vacuum field equations
R. Rosas-Rodriguez
2005-07-26
By considering the Einstein vacuum field equations linearized about the Minkowski metric, the evolution equations for the gauge-invariant quantities characterizing the gravitational field are written in a Hamiltonian form by using a conserved functional as Hamiltonian; this Hamiltonian is not the analog of the energy of the field. A Poisson bracket between functionals of the field, compatible with the constraints satisfied by the field variables, is obtained. The generator of spatial translations associated with such bracket is also obtained.
Multibump solutions for quasilinear elliptic equations with critical growth
Liu, Jiaquan; Wang, Zhi-Qiang; Wu, Xian
2013-12-15
The current paper is concerned with constructing multibump solutions for a class of quasilinear Schrödinger equations with critical growth. This extends the classical results of Coti Zelati and Rabinowitz [Commun. Pure Appl. Math. 45, 1217–1269 (1992)] for semilinear equations as well as recent work of Liu, Wang, and Guo [J. Funct. Anal. 262, 4040–4102 (2012)] for quasilinear problems with subcritical growth. The periodicity of the potentials is used to glue ground state solutions to construct multibump bound state solutions.
Simple derivation from postulates of generalized vacuum Maxwell equations
Chun Wa Wong
2013-05-11
The two postulates of special relativity plus the postulates of conserved charges, both electric and magnetic, and a resulting linear system are sufficient for the derivation of the generalized vacuum Maxwell equations with both charges. The derivative admits another set of Maxwell equations for charges that are the opposite-parity partners of the usual electric and magnetic charges. These new charges and their photons are parts of the parallel universe of dark matter.
Wave-Particle Duality and the Hamilton-Jacobi Equation
Gregory I. Sivashinsky
2009-12-28
The Hamilton-Jacobi equation of relativistic quantum mechanics is revisited. The equation is shown to permit solutions in the form of breathers (oscillating/spinning solitons), displaying simultaneous particle-like and wave-like behavior. The de Broglie wave thus acquires a clear deterministic meaning of a wave-like excitation of the classical action function. The problem of quantization in terms of the breathing action function and the double-slit experiment are discussed.
Optomechanical test of the Schr\\"odinger-Newton equation
Großardt, André; Ulbricht, Hendrik; Bassi, Angelo
2015-01-01
The Schr\\"odinger-Newton equation has been proposed as an experimentally testable alternative to quantum gravity, accessible at low energies. It contains self-gravitational terms, which slightly modify the quantum dynamics. Here we show that it distorts the spectrum of a harmonic system. Based on this effect, we propose an optomechanical experiment with a trapped microdisc to test the Schr\\"odinger-Newton equation, and we show that it can be realized with existing technology.
Laplace Operators on Fractals and Related Functional Equations
Gregory Derfel; Peter Grabner; Fritz Vogl
2012-06-06
We give an overview over the application of functional equations, namely the classical Poincar\\'e and renewal equations, to the study of the spectrum of Laplace operators on self-similar fractals. We compare the techniques used to those used in the euclidean situation. Furthermore, we use the obtained information on the spectral zeta function to define the Casimir energy of fractals. We give numerical values for this energy for the Sierpi\\'nski gasket.
Mpemba effect, Newton cooling law and heat transfer equation
Vladan Pankovic; Darko V. Kapor
2012-12-11
In this work we suggest a simple theoretical solution of the Mpemba effect in full agreement with known experimental data. This solution follows simply as an especial approximation (linearization) of the usual heat (transfer) equation, precisely linearization of the second derivation of the space part of the temperature function (as it is well-known Newton cooling law can be considered as the effective approximation of the heat (transfer) equation for constant space part of the temperature function).
The quantummechanical wave equations from a relativistic viewpoint
Engel Roza
2007-07-19
A derivation is presented of the quantummechanical wave equations based upon the Equity Principle of Einstein's General Relativity Theory. This is believed to be more generic than the common derivations based upon Einstein's energy relationship for moving particles. It is shown that Schrodinger's Equation, if properly formulated, is relativisticly covariant. This makes the critisized Klein-Gordon Equation for spinless massparticles obsolete. Therefore Dirac's Equation is presented from a different viewpoint and it is shown that the relativistic covariance of Schrodinger's Equation gives a natural explanation for the dual energy outcome of Dirac's derivation and for the nature of antiparticles. The propagation of wave functions in an energy field is studied in terms of propagation along geodesic lines in curved space-time, resulting in an equivalent formulation as with Feynman's path integral. It is shown that Maxwell's wave equation fits in the developed framework as the massless limit of moving particles. Finally the physical appearance of electrons is discussed including a quantitative calculation of the jitter phenomenon of a free moving electron.
Integral relations for solutions of confluent Heun equations
Léa Jaccoud El-Jaick; Bartolomeu D. B. Figueiredo
2015-03-02
Firstly, we construct kernels of integral relations among solutions of the confluent Heun equation (CHE) and its limit, the reduced CHE (RCHE). In both cases we generate additional kernels by systematically applying substitutions of variables. Secondly, we establish integral relations between known solutions of the CHE that are power series and solutions that are series of special functions; and similarly for solutions of the RCHE. Thirdly, by using one of the integral relations as an integral transformation we obtain a new series solution of the spheroidal wave equation. From this solution we construct new solutions of the general CHE, and show that these are suitable for solving the radial part of the two-center problem in quantum mechanics. Finally, by applying a limiting process to kernels for the CHEs we obtain kernels for {two} double-confluent Heun equations. As a result, we deal with kernels of four equations of the Heun family, each equation presenting a distinct structure of singularities. In addition, we find that the known kernels for the Mathieu equation are special instances of kernels of the RCHE.
Particle Number Fluctuations for van der Waals Equation of State
V. Vovchenko; D. V. Anchishkin; M. I. Gorenstein
2015-02-04
The van der Waals (VDW) equation of state describes a thermal equilibrium in system of particles, where both repulsive and attractive interactions between them are included. This equation predicts an existence of the 1st order liquid-gas phase transition and the critical point. The standard form of the VDW equation is given by the pressure function in the canonical ensemble (CE) with a fixed number of particles. In the present paper the VDW equation is transformed to the grand canonical ensemble (GCE). We argue that this procedure can be useful for new physical applications. Particularly, the fluctuations of number of particles, which are absent in the CE, can be studied in the GCE. For the VDW equation of state in the GCE the particle number fluctuations are calculated for the whole phase diagram, both outside and inside the liquid-gas mixed phase region. It is shown that the scaled variance of these fluctuations remains finite within the mixed phase and goes to infinity at the critical point. The GCE formulation of the VDW equation of state can be also an important step for its application to a statistical description of hadronic systems, where numbers of different particle species are usually not conserved.
Farquharson, Colin G.
Comparison of integral equation and physical scale modelling of the electromagnetic response history of EM numerical modelling in geophysics. Â· Another integral equation modelling program;Introduction: a brief history Â· Two main approaches to numerical modelling: integral equation; finite
A meshfree method for the Poisson equation with 3D wall-bounded flow application
Vasilyeva, Anna, S.M. Massachusetts Institute of Technology
2010-01-01
The numerical approximation of the Poisson equation can often be found as a subproblem to many more complex computations. In the case of Lagrangian approaches of flow equations, the Poisson equation often needs to be solved ...
The Cubic Dirac Equation: Small Initial Data in H1(R3)
Bejenaru, I; Herr, S
2015-01-01
for homogeneous wave-equations, Math. Z. 120 (1971), 93–106.semilinear Klein-Gordon equations with small weakly decayingComm. Partial Differential Equations 25 (2000), no. 11-12,
A determining form for the damped driven nonlinear Schrödinger equation-Fourier modes case
Jolly, MS; Sadigov, T; Titi, ES
2015-01-01
for the 2D Navier-Stokes equations - The general interpolantR. Temam, Navier-Stokes Equations and Turbulence, Cambridgenon-stationnaires des equations de Navier-Stokes en
Trefethen, Nick
membrane or drum (! ref ). In 3D, the most famous example is the propagation of sound waves in a gas=2 . In an unbounded domain, the wave equation is readily investigated by Fourier analysis. Separation of variables suitable technical assumptions, all solutions can be written this way. In a bounded domain , separation
A determining form for the damped driven nonlinear Schrödinger equation-Fourier modes case
Jolly, MS; Sadigov, T; Titi, ES
2015-01-01
1984) [5] J. Bourgain, Fourier transformation restrictionStokes equations - the Fourier modes case, Journal ofSCHR ODINGER EQUATION- FOURIER MODES CASE MICHAEL S. JOLLY,
Handbook of Industrial Engineering Equations, Formulas, and Calculations
Badiru, Adedeji B; Omitaomu, Olufemi A
2011-01-01
The first handbook to focus exclusively on industrial engineering calculations with a correlation to applications, Handbook of Industrial Engineering Equations, Formulas, and Calculations contains a general collection of the mathematical equations often used in the practice of industrial engineering. Many books cover individual areas of engineering and some cover all areas, but none covers industrial engineering specifically, nor do they highlight topics such as project management, materials, and systems engineering from an integrated viewpoint. Written by acclaimed researchers and authors, this concise reference marries theory and practice, making it a versatile and flexible resource. Succinctly formatted for functionality, the book presents: Basic Math Calculations; Engineering Math Calculations; Production Engineering Calculations; Engineering Economics Calculations; Ergonomics Calculations; Facility Layout Calculations; Production Sequencing and Scheduling Calculations; Systems Engineering Calculations; Data Engineering Calculations; Project Engineering Calculations; and Simulation and Statistical Equations. It has been said that engineers make things while industrial engineers make things better. To make something better requires an understanding of its basic characteristics and the underlying equations and calculations that facilitate that understanding. To do this, however, you do not have to be computational experts; you just have to know where to get the computational resources that are needed. This book elucidates the underlying equations that facilitate the understanding required to improve design processes, continuously improving the answer to the age-old question: What is the best way to do a job?
Geodesic Deviation Equation in $f(T)$ gravity
F. Darabi; M. Mousavi; K. Atazadeh
2015-04-14
In this work, we show that it is possible to study the notion of geodesic deviation equation in $f(T)$ gravity, in spite of the fact that in teleparallel gravity there is no notion of geodesics, and the torsion is responsible for the appearance of gravitational interaction. In this regard, we obtain the GR equivalent equations for $f(T)$ gravity which are in the modified gravity form such as $f(R)$ gravity. Then, we obtain the GDE within the context of this modified gravity. In this way, the obtained geodesic deviation equation will correspond to the $f(T)$ gravity. Eventually, we extend the calculations to obtain the modification of Matting relation.
Invariant discretization schemes for the shallow-water equations
Alexander Bihlo; Roman O. Popovych
2013-01-03
Invariant discretization schemes are derived for the one- and two-dimensional shallow-water equations with periodic boundary conditions. While originally designed for constructing invariant finite difference schemes, we extend the usage of difference invariants to allow constructing of invariant finite volume methods as well. It is found that the classical invariant schemes converge to the Lagrangian formulation of the shallow-water equations. These schemes require to redistribute the grid points according to the physical fluid velocity, i.e., the mesh cannot remain fixed in the course of the numerical integration. Invariant Eulerian discretization schemes are proposed for the shallow-water equations in computational coordinates. Instead of using the fluid velocity as the grid velocity, an invariant moving mesh generator is invoked in order to determine the location of the grid points at the subsequent time level. The numerical conservation of energy, mass and momentum is evaluated for both the invariant and non-invariant schemes.
Consistency of equations of motion in conformal frames
J. R. Morris
2014-11-05
Four dimensional scalar-tensor theory is considered within two conformal frames, the Jordan frame (JF) and the Einstein frame (EF). The actions for the theory are equivalent and equations of motion can be obtained from each action. It is found that the JF equations of motion, expressed in terms of EF variables, translate directly into and agree with the EF equations of motion obtained from the EF action, provided that certain simple consistency conditions are satisfied, which is always the case. The implication is that a solution set obtained in one conformal frame can be reliably translated into a solution set for the other frame, and therefore the two frames are, at least, mathematically equivalent.
Dirac equation in low dimensions: The factorization method
J. A. Sanchez-Monroy; C. J. Quimbay
2014-09-30
We present a general approach to solve the (1+1) and (2+1)-dimensional Dirac equation in the presence of static scalar, pseudoscalar and gauge potentials, for the case in which the potentials have the same functional form and thus the factorization method can be applied. We show that the presence of electric potentials in the Dirac equation leads to a two Klein-Gordon equations including an energy-dependent potential. We then generalize the factorization method for the case of energy-dependent Hamiltonians. Additionally, the shape invariance is generalized for a specific class of energy-dependent Hamiltonians. We also present a condition for the absence of the Klein's paradox (stability of the Dirac sea), showing how Dirac particles in low dimensions can be confined for a wide family of potentials.
Where are the roots of the Bethe Ansatz equations?
R. S. Vieira; A. Lima-Santos
2015-07-13
Changing the variables in the Bethe Ansatz Equations (BAE) for the XXZ six-vertex model we had obtained a coupled system of polynomial equations. This provided a direct link between the BAE deduced from the Algebraic Bethe Ansatz (ABA) and the BAE arising from the Coordinate Bethe Ansatz (CBA). For two magnon states this polynomial system could be decoupled and the solutions given in terms of the roots of some self-inversive polynomials. From theorems concerning the distribution of the roots of self-inversive polynomials we made a thorough analysis of the two magnon states, which allowed us to find the location and multiplicity of the Bethe roots in the complex plane, to discuss the completeness and singularities of Bethe's equations, the ill-founded string-hypothesis concerning the location of their roots, as well as to find an interesting connection between the BAE with Salem's polynomials.
Fourth order gravity: equations, history, and applications to cosmology
H. -J. Schmidt
2006-03-25
The field equations following from a Lagrangian L(R) will be deduced and solved for special cases. If L is a non-linear function of the curvature scalar, then these equations are of fourth order in the metric. In the introduction we present the history of these equations beginning with the paper of H. Weyl from 1918, who first discussed them as alternative to Einstein's theory. In the third part, we give details about the cosmic no hair theorem, i.e., the details how within fourth order gravity with L= R + R^2 the inflationary phase of cosmic evolution turns out to be a transient attractor. Finally, the Bicknell theorem, i.e. the conformal relation from fourth order gravity to scalar-tensor theory, will be shortly presented.
Properties of the Boltzmann equation in the classical approximation
Tanji, Naoto; Epelbaum, Thomas; Gelis, Francois; Wu, Bin
2014-12-30
We study the Boltzmann equation with elastic point-like scalar interactions in two different versions of the the classical approximation. Although solving numerically the Boltzmann equation with the unapproximated collision term poses no problem, this allows one to study the effect of the ultraviolet cutoff in these approximations. This cutoff dependence in the classical approximations of the Boltzmann equation is closely related to the non-renormalizability of the classical statistical approximation of the underlying quantum field theory. The kinetic theory setup that we consider here allows one to study in a much simpler way the dependence on the ultraviolet cutoff, since one has also access to the non-approximated result for comparison.
Semi-Spectral Method for the Wigner equation
Oliver Furtmaier; Sauro Succi; Miller Mendoza
2015-06-29
We propose a numerical method to solve the Wigner equation in quantum systems of spinless, non-relativistic particles. The method uses a spectral decomposition into $L^2(\\mathbb{R}^d)$ basis functions in momentum-space to obtain a system of first-order advection-reaction equations. The resulting equations are solved by splitting the reaction and advection steps so as to allow the combination of numerical techniques from quantum mechanics and computational fluid dynamics by identifying the skew-hermitian reaction matrix as a generator of unitary rotations. The method is validated for the case of particles subject to a one-dimensional (an-)harmonic potential using finite-differences for the advection part. Thereby, we verify the second order of convergence and observe non-classical behavior in the evolution of the Wigner function.
Semi-Spectral Method for the Wigner equation
Furtmaier, Oliver; Mendoza, Miller
2015-01-01
We propose a numerical method to solve the Wigner equation in quantum systems of spinless, non-relativistic particles. The method uses a spectral decomposition into $L^2(\\mathbb{R}^d)$ basis functions in momentum-space to obtain a system of first-order advection-reaction equations. The resulting equations are solved by splitting the reaction and advection steps so as to allow the combination of numerical techniques from quantum mechanics and computational fluid dynamics by identifying the skew-hermitian reaction matrix as a generator of unitary rotations. The method is validated for the case of particles subject to a one-dimensional (an-)harmonic potential using finite-differences for the advection part. Thereby, we verify the second order of convergence and observe non-classical behavior in the evolution of the Wigner function.
Generalized linear Boltzmann equations for particle transport in polycrystals
Jens Marklof; Andreas Strömbergsson
2015-02-13
The linear Boltzmann equation describes the macroscopic transport of a gas of non-interacting point particles in low-density matter. It has wide-ranging applications, including neutron transport, radiative transfer, semiconductors and ocean wave scattering. Recent research shows that the equation fails in highly-correlated media, where the distribution of free path lengths is non-exponential. We investigate this phenomenon in the case of polycrystals whose typical grain size is comparable to the mean free path length. Our principal result is a new generalized linear Boltzmann equation that captures the long-range memory effects in this setting. A key feature is that the distribution of free path lengths has an exponential decay rate, as opposed to a power-law distribution observed in a single crystal.
The Schroedinger equation with friction from the quantum trajectory perspective
Garashchuk, Sophya; Dixit, Vaibhav; Gu Bing; Mazzuca, James [Department of Chemistry and Biochemistry, University of South Carolina, Columbia, South Carolina 29208 (United States)
2013-02-07
Similarity of equations of motion for the classical and quantum trajectories is used to introduce a friction term dependent on the wavefunction phase into the time-dependent Schroedinger equation. The term describes irreversible energy loss by the quantum system. The force of friction is proportional to the velocity of a quantum trajectory. The resulting Schroedinger equation is nonlinear, conserves wavefunction normalization, and evolves an arbitrary wavefunction into the ground state of the system (of appropriate symmetry if applicable). Decrease in energy is proportional to the average kinetic energy of the quantum trajectory ensemble. Dynamics in the high friction regime is suitable for simple models of reactions proceeding with energy transfer from the system to the environment. Examples of dynamics are given for single and symmetric and asymmetric double well potentials.
A decoupled system of hyperbolic equations for linearized cosmological perturbations
J. Ramirez; S. Kopeikin
2002-02-01
A decoupled system of hyperbolic partial differential equations for linear perturbations around any spatially flat FRW universe is obtained for a wide class of perturbations. The considered perturbing energy momentum-tensors can be expressed as the sum of the perturbation of a minimally coupled scalar field plus an arbitrary (weak) energy-momentum tensor which is covariantly conserved with respect to the background. The key ingredient in obtaining the decoupling of the equations is the introduction of a new covariant gauge which plays a similar role as harmonic gauge does for perturbations around Minkowski space-time. The case of universes satysfying a linear equation of state is discussed in particular, and closed analytic expressions for the retarded Green's functions solving the de Sitter, dust and radiation dominated cases are given.
Coherent states, vacuum structure and infinite component relativistic wave equations
Cirilo-Lombardo, Diego Julio
2015-01-01
It is commonly claimed in the recent literature that certain solutions to wave equations of positive energy of Dirac-type with internal variables are characterized by a non-thermal spectrum. As part of that statement, it was said that the transformations and symmetries involved in equations of such type correspond to a particular representation of the Lorentz group. In this paper we give the general solution to this problem emphasizing the interplay between the group structure, the corresponding algebra and the physical spectrum. This analysis is completed with a strong discussion and proving that: i) the physical states are represented by coherent states; ii) the solutions in previous references [1] are not general, ii) the symmetries of the considered physical system in [1] (equations and geometry) do not correspond to the Lorentz group but to the fourth covering: the Metaplectic group Mp(n).
Properties of the Boltzmann equation in the classical approximation
DOE Public Access Gateway for Energy & Science Beta (PAGES Beta)
Epelbaum, Thomas; Gelis, François; Tanji, Naoto; Wu, Bin
2014-12-30
We examine the Boltzmann equation with elastic point-like scalar interactions in two different versions of the the classical approximation. Although solving numerically the Boltzmann equation with the unapproximated collision term poses no problem, this allows one to study the effect of the ultraviolet cutoff in these approximations. This cutoff dependence in the classical approximations of the Boltzmann equation is closely related to the non-renormalizability of the classical statistical approximation of the underlying quantum field theory. The kinetic theory setup that we consider here allows one to study in a much simpler way the dependence on the ultraviolet cutoff, since onemore »has also access to the non-approximated result for comparison.« less
Master integrals for splitting functions from differential equations in QCD
O. Gituliar
2015-12-10
A method for calculating phase-space master integrals for the decay process $1 \\to n$ massless partons in QCD using integration-by-parts and differential equations techniques is discussed. The method is based on the appropriate choice of the basis for master integrals which leads to significant simplification of differential equations. We describe an algorithm how to construct the desirable basis, so that the resulting system of differential equations can be recursively solved in terms of (G)HPLs as a series in the dimensional regulator $\\epsilon$ to any order. We demonstrate its power by calculating master integrals for the NLO time-like splitting functions and discuss future applications of the proposed method at the NNLO precision.
Reformulating the Schrodinger equation as a Shabat-Zakharov system
Boonserm, Petarpa
2009-01-01
We reformulate the second-order Schrodinger equation as a set of two coupled first order differential equations, a so-called "Shabat-Zakharov system", (sometimes called a "Zakharov-Shabat" system). There is considerable flexibility in this approach, and we emphasise the utility of introducing an "auxiliary condition" or "gauge condition" that is used to cut down the degrees of freedom. Using this formalism, we derive the explicit (but formal) general solution to the Schrodinger equation. The general solution depends on three arbitrarily chosen functions, and a path-ordered exponential matrix. If one considers path ordering to be an "elementary" process, then this represents complete quadrature, albeit formal, of the second-order linear ODE.
Generic master equations for quasi-normal frequencies
Skakala, Jozef
2010-01-01
Generic master equations governing the highly-damped quasi-normal frequencies [QNFs] of one-horizon, two-horizon, and even three-horizon spacetimes can be obtained through either semi-analytic or monodromy techniques. While many technical details differ, both between the semi-analytic and monodromy approaches, and quite often among various authors seeking to apply the monodromy technique, there is nevertheless widespread agreement regarding the the general form of the QNF master equations. Within this class of generic master equations we can establish some rather general results, relating the existence of "families" of QNFs of the form omega_{a,n} = (offset)_a + i n (gap) to the question of whether or not certain ratios of parameters are rational or irrational.
Global regularity for the minimal surface equation in Minkowskian geometry
Stefanov, Atanas
2011-05-05
) the minimizers must satisfy the Euler–Lagrange equation @i #18; @L @ P#30;i #19; #0; @L @#30; D 0: Since @L@#30; D 0 and @L @ P#30;i D gji @j#30; 2 p 1C gkm.x/@k#30;@m#30; ; we get the minimal surface equation @i gji @j#30;p 1C gkm.x/@k#30;@m#30; D 0: (1....3) One should note that (1.3) is not guaranteed to be a hyperbolic equation, if kr#30;kL1 is not small. A local well-posedness theory for (1.3) (without smallness assumptions) can be built from the methods developed in [5], see also [13], [2]. Basically...
Multi-time Schrödinger equations cannot contain interaction potentials
Petrat, Sören; Tumulka, Roderich
2014-03-15
Multi-time wave functions are wave functions that have a time variable for every particle, such as ?(t{sub 1},x{sub 1},...,t{sub N},x{sub N}). They arise as a relativistic analog of the wave functions of quantum mechanics but can be applied also in quantum field theory. The evolution of a wave function with N time variables is governed by N Schrödinger equations, one for each time variable. These Schrödinger equations can be inconsistent with each other, i.e., they can fail to possess a joint solution for every initial condition; in fact, the N Hamiltonians need to satisfy a certain commutator condition in order to be consistent. While this condition is automatically satisfied for non-interacting particles, it is a challenge to set up consistent multi-time equations with interaction. We prove for a wide class of multi-time Schrödinger equations that the presence of interaction potentials (given by multiplication operators) leads to inconsistency. We conclude that interaction has to be implemented instead by creation and annihilation of particles, which, in fact, can be done consistently [S. Petrat and R. Tumulka, “Multi-time wave functions for quantum field theory,” Ann. Physics (to be published)]. We also prove the following result: When a cut-off length ? > 0 is introduced (in the sense that the multi-time wave function is defined only on a certain set of spacelike configurations, thereby breaking Lorentz invariance), then the multi-time Schrödinger equations with interaction potentials of range ? are consistent; however, in the desired limit ? ? 0 of removing the cut-off, the resulting multi-time equations are interaction-free, which supports the conclusion expressed in the title.
E. V. Shiryaeva; M. Yu. Zhukov
2014-10-17
The paper presents the solutions for the zonal electrophoresis equations are obtained by analytical and numerical methods. The method proposed by the authors is used. This method allows to reduce the Cauchy problem for two hyperbolic quasilinear PDE's to the Cauchy problem for ODE's. In some respect, this method is analogous to the method of characteristics for two hyperbolic equations. The method is effectively applicable in all cases when the explicit expression for the Riemann-Green function of some linear second order PDE, resulting from the use of the hodograph method for the original equations, is known. One of the method advantages is the possibility of constructing a multi-valued solutions. Compared with the previous authors paper, in which, in particular, the shallow water equations are studied, here we investigate the case when the Riemann-Green function can be represent as the sum of the terms each of them is a product of two multipliers depended on different variables. The numerical results for zonal electrophoresis equations are presented. For computing the different initial data (periodic, wave packet, the Gaussian distribution) are used.
Klein-Gordon equation from Maxwell-Lorentz dynamics
Ricardo J. Alonso-Blanco
2012-02-19
We consider Maxwell-Lorentz dynamics: that is to say, Newton's law under the action of a Lorentz's force which obeys the Maxwell equations. A natural class of solutions are those given by the Lagrangian submanifolds of the phase space when it is endowed with the symplectic structure modified by the electromagnetic field. We have found that the existence of this type of solution leads us directly to the Klein-Gordon equation as a compatibility condition. Therefore, surprisingly, quite natural assumptions on the classical theory involve a quantum condition without any process of limit. This result could be a partial response to the inquiries of Dirac.
Illite Dissolution Rates and Equation (100 to 280 dec C)
DOE Data Explorer [Office of Scientific and Technical Information (OSTI)]
Carroll, Susan
The objective of this suite of experiments was to develop a useful kinetic dissolution expression for illite applicable over an expanded range of solution pH and temperature conditions representative of subsurface conditions in natural and/or engineered geothermal reservoirs. Using our new data, the resulting rate equation is dependent on both pH and temperature and utilizes two specific dissolution mechanisms (a “neutral” and a “basic” mechanism). The form of this rate equation should be easily incorporated into most existing reactive transport codes for to predict rock-water interactions in EGS shear zones.
Equations of state and phase diagrams of hydrogen isotopes
Urlin, V. D.
2013-11-15
A new form of the semiempirical equation of state proposed for the liquid phase of hydrogen isotopes is based on the assumption that its structure is formed by cells some of which contain hydrogen molecules and others contain hydrogen atoms. The values of parameters in the equations of state of the solid (molecular and atomic) phases as well as of the liquid phase of hydrogen isotopes (protium and deuterium) are determined. Phase diagrams, shock adiabats, isentropes, isotherms, and the electrical conductivity of compressed hydrogen are calculated. Comparison of the results of calculations with available experimental data in a wide pressure range demonstrates satisfactory coincidence.
New elliptic solutions of the Yang-Baxter equation
D. Chicherin; S. E. Derkachov; V. P. Spiridonov
2015-05-13
We consider finite-dimensional reductions of the most general known solution of the Yang-Baxter equation with a rank 1 symmetry algebra, which is described by an integral operator with an elliptic hypergeometric kernel. The reduced R-operators reproduce at their bottom the standard Baxter's R-matrix for the 8-vertex model and Sklyanin's L-operator. The general formula has a remarkably compact form and yields new elliptic solutions of the Yang-Baxter equation based on the finite-dimensional representations of the elliptic modular double. The same result is reproduced using the fusion formalism.
Fluid equations in the presence of electron cyclotron current drive
Jenkins, Thomas G.; Kruger, Scott E. [Tech-X Corporation, 5621 Arapahoe Avenue, Boulder, Colorado 80303 (United States)
2012-12-15
Two-fluid equations, which include the physics imparted by an externally applied radiofrequency source near electron cyclotron resonance, are derived in their extended magnetohydrodynamic forms using the formalism of Hegna and Callen [Phys. Plasmas 16, 112501 (2009)]. The equations are compatible with the closed fluid/drift-kinetic model developed by Ramos [Phys. Plasmas 17, 082502 (2010); 18, 102506 (2011)] for fusion-relevant regimes with low collisionality and slow dynamics, and they facilitate the development of advanced computational models for electron cyclotron current drive-induced suppression of neoclassical tearing modes.
Evolution equation of entanglement for multi-qubit systems
Michael Siomau; Stephan Fritzsche
2010-11-24
We discuss entanglement evolution of a multi-qubit system when one of its qubits is subjected to a general noisy channel. For such a system, an evolution equation of entanglement for a lower bound for multi-qubit concurrence is derived. Using this evolution equation, the entanglement dynamics of an initially mixed three-qubit state composed of a GHZ and a W state is analyzed if one of the qubits is affected by a phase, an amplitude or a generalized amplitude damping channel.
Stable blowup for wave equations in odd space dimensions
Roland Donninger; Birgit Schörkhuber
2015-04-03
We consider semilinear wave equations with focusing power nonlinearities in odd space dimensions $d \\geq 5$. We prove that for every $p > \\frac{d+3}{d-1}$ there exists an open set of radial initial data in $H^{\\frac{d+1}{2}} \\times H^{\\frac{d-1}{2}}$ such that the corresponding solution exists in a backward lightcone and approaches the ODE blowup profile. The result covers the entire range of energy supercritical nonlinearities and extends our previous work for the three-dimensional radial wave equation to higher space dimensions.
Thermodynamically constrained correction to ab initio equations of state
French, Martin; Mattsson, Thomas R.
2014-07-07
We show how equations of state generated by density functional theory methods can be augmented to match experimental data without distorting the correct behavior in the high- and low-density limits. The technique is thermodynamically consistent and relies on knowledge of the density and bulk modulus at a reference state and an estimation of the critical density of the liquid phase. We apply the method to four materials representing different classes of solids: carbon, molybdenum, lithium, and lithium fluoride. It is demonstrated that the corrected equations of state for both the liquid and solid phases show a significantly reduced dependence of the exchange-correlation functional used.
A Simple Proof of the Ramsey Savings Equation
El-Hodiri, Mohamed
1978-01-01
SIAM REVIEW Vol. 20, No. 1, January 1978 A SIMPLE PROOF OF THE RAMSEY SAVINGS EQUATION* MOHAMED EL-HODIRI? In [2] Ramsey considers the problem of minimizing the accumulated differ- ence between bliss B and net utility U(x)- V(a), where U... no new "truths." It merely makes it easier to understand what is already a widely held belief. Ramsey considers the case where T . By taking the limit, and arguing around a bit, we can show that the Euler equations in our simple problem will still hold...
Non-Markovian Quantum Jump with Generalized Lindblad Master Equation
X. L. Huang; H. Y. Sun; X. X. Yi
2008-10-14
The Monte Carlo wave function method or the quantum trajectory/jump approach is a powerful tool to study dissipative dynamics governed by the Markovian master equation, in particular for high-dimensional systems and when it is difficult to simulate directly. In this paper, we extend this method to the non-Markovian case described by the generalized Lindblad master equation. Two examples to illustrate the method are presented and discussed. The results show that the method can correctly reproduce the dissipative dynamics for the system. The difference between this method and the traditional Markovian jump approach and the computational efficiency of this method are also discussed.
The random Schrödinger equation: homogenization in time-dependent potentials
Yu Gu; Lenya Ryzhik
2015-06-08
We analyze the solutions of the Schr\\"odinger equation with the low frequency initial data and a time-dependent weakly random potential. We prove a homogenization result for the low frequency component of the wave field. We also show that the dynamics generates a non-trivial energy in the high frequencies, which do not homogenize -- the high frequency component of the wave field remains random and the evolution of its energy is described by a kinetic equation. The transition from the homogenization of the low frequencies to the random limit of the high frequencies is illustrated by understanding the size of the small random fluctuations of the low frequency component.
Conditionally invariant solutions of the rotating shallow water wave equations
Benoit Huard
2010-05-11
This paper is devoted to the extension of the recently proposed conditional symmetry method to first order nonhomogeneous quasilinear systems which are equivalent to homogeneous systems through a locally invertible point transformation. We perform a systematic analysis of the rank-1 and rank-2 solutions admitted by the shallow water wave equations in (2 + 1) dimensions and construct the corresponding solutions of the rotating shallow water wave equations. These solutions involve in general arbitrary functions depending on Riemann invariants, which allow us to construct new interesting classes of solutions.
Electromagnetic interactions for the two-body spectator equations
J. Adam; Franz Gross; J.W. Van Orden
1997-10-01
This paper presents a new non-associative algebra which is used to (1) show how the spectator (or Gross) two-body equations and electromagnetic currents can be formally derived from the Bethe-Salpeter equation and currents if both are treated to all orders, (2) obtain explicit expressions for the Gross two-body electromagnetic currents valid to any order, and (3) prove that the currents so derived are exactly gauge invariant when truncated consistently to any finite order. In addition to presenting these new results, this work complements and extends previous treatments based largely on the analysis of sums of Feynman diagrams.
Ideal solar cell equation in the presence of photon recycling
Lan, Dongchen Green, Martin A.
2014-11-07
Previous derivations of the ideal solar cell equation based on Shockley's p-n junction diode theory implicitly assume negligible effects of photon recycling. This paper derives the equation in the presence of photon recycling that modifies the values of dark saturation and light-generated currents, using an approach applicable to arbitrary three-dimensional geometries with arbitrary doping profile and variable band gap. The work also corrects an error in previous work and proves the validity of the reciprocity theorem for charge collection in such a more general case with the previously neglected junction depletion region included.
The Unified Approach for Model Evaluation in Structural Equation Modeling
Pornprasertmanit, Sunthud
2014-08-31
hypothesized model M. The discrepancy (FM) between the two means and covariance matrices can be computed using Equation 1.1 (Kenny & McCoach, 2003): FM = tr { ? [?M] ?1 } ? log ? ? ?? [?M] ?1 ? ? ??N +[µ ?µM]? [?M]?1 [µ ?µM] , (1.1) where N is sample size. FM...- crepancy function is defined by Equation 1.2 (Browne & Cudeck, 1992; Satorra & Saris, 1985; Saris & Satorra, 1993): ?M = (N?1)FM. (1.2) That is, the discrepancy value can be calculated by dividing the chi-square value by sample size minus 1. Most fit...
Quantum optical master equation for solid-state quantum emitters
Ralf Betzholz; Juan Mauricio Torres; Marc Bienert
2014-12-15
We provide an elementary description of the dynamics of defect centers in crystals in terms of a quantum optical master equation which includes spontaneous decay and a simplified vibronic interaction with lattice phonons. We present the general solution of the dynamical equation by means of the eigensystem of the Liouville operator and exemplify the usage of this damping basis to calculate the dynamics of the electronic and vibrational degrees of freedom and to provide an analysis of the spectra of scattered light. The dynamics and spectral features are discussed with respect to the applicability for color centers, especially for negatively charged nitrogen-vacancy centers in diamond.
Efficiency of Carnot Cycle with Arbitrary Gas Equation of State
Tjiang, P C; Tjiang, Paulus C.; Sutanto, Sylvia H.
2006-01-01
The derivation of the efficiency of Carnot cycle is usually done by calculating the heats involved in two isothermal processes and making use of the associated adiabatic relation for a given working substance's equation of state, usually the ideal gas. We present a derivation of Carnot efficiency using the same procedure with Redlich-Kwong gas as working substance to answer the calculation difficulties raised by Agrawal and Menon. We also show that using the same procedure, the Carnot efficiency may be derived regardless of the functional form of the gas equation of state.
The modified Klein Gordon equation for neolithic population migration
M. Pelc; J. Marciak-Kozlowska; M. Kozlowski
2007-03-11
In this paper the model for the neolithic migration in Europe is developed. The new migration equation, the modified Klein Gordon equation is formulated and solved. It is shown that the migration process can be described as the hyperbolic diffusion with constant speed. In comparison to the existing models based on the generalization of the Fisher approach the present model describes the migration as the transport process with memory and offers the possibility to recover the initial state of migration which is the wave motion with finite velocity.
Efficiency of Carnot Cycle with Arbitrary Gas Equation of State
Paulus C. Tjiang; Sylvia H. Sutanto
2006-03-27
The derivation of the efficiency of Carnot cycle is usually done by calculating the heats involved in two isothermal processes and making use of the associated adiabatic relation for a given working substance's equation of state, usually the ideal gas. We present a derivation of Carnot efficiency using the same procedure with Redlich-Kwong gas as working substance to answer the calculation difficulties raised by Agrawal and Menon. We also show that using the same procedure, the Carnot efficiency may be derived regardless of the functional form of the gas equation of state.
Illite Dissolution Rates and Equation (100 to 280 dec C)
DOE Data Explorer [Office of Scientific and Technical Information (OSTI)]
Carroll, Susan
2014-10-17
The objective of this suite of experiments was to develop a useful kinetic dissolution expression for illite applicable over an expanded range of solution pH and temperature conditions representative of subsurface conditions in natural and/or engineered geothermal reservoirs. Using our new data, the resulting rate equation is dependent on both pH and temperature and utilizes two specific dissolution mechanisms (a “neutral” and a “basic” mechanism). The form of this rate equation should be easily incorporated into most existing reactive transport codes for to predict rock-water interactions in EGS shear zones.
The Jacobi Equation and Poisson Geometry on R4
Rubén Flores-Espinoza
2013-06-21
This paper is devoted to the study of solutions of the Jacobi equation in Euclidean four dimensional space R4. Each of such solutions define a Poisson tensor. Using the elementary vector calculus operations we give explicit formulas for the main geometric objets associated to the solutions of Jacobi equation, including its characteristic foliation, their symmetries and its generators, normal forms and some useful decomposition results for the solutions. In particular we study the classes of Poisson tensors of contant rank and those preserving a volume form.
Combined Field Integral Equation Based Theory of Characteristic Mode
Qi I. Dai; Qin S. Liu; Hui Gan; Weng Cho Chew
2015-03-04
Conventional electric field integral equation based theory is susceptible to the spurious internal resonance problem when the characteristic modes of closed perfectly conducting objects are computed iteratively. In this paper, we present a combined field integral equation based theory to remove the difficulty of internal resonances in characteristic mode analysis. The electric and magnetic field integral operators are shown to share a common set of non-trivial characteristic pairs (values and modes), leading to a generalized eigenvalue problem which is immune to the internal resonance corruption. Numerical results are presented to validate the proposed formulation. This work may offer efficient solutions to characteristic mode analysis which involves electrically large closed surfaces.
Equality: A tool for free-form equation editing
Cummins, Stephen; Davies, Ian; Rice, Andrew; Beresford, Alastair R.
2015-06-10
interface, the parser runs entirely in the web browser. This keeps the system self-contained and avoids all issues of 4http://facebook.github.io/react/ Fig. 1. The Equality equation editor. Note that the slightly jumbled symbols dropped onto the canvas have... . “Editing equations was easier because I could drag elements around With microsoft you had to put in the format (e.g. exponent) before actually putting in variables/numbers which was irritating” This reinforces our aim of providing a tool that supports...
A parallel multigrid-based preconditioner for the 3D heterogeneous high-frequency Helmholtz equation
Vuik, Kees
; Preconditioner; Multigrid method 1. Introduction Important applications of the acoustic wave equation can] with respect to time to the acoustic wave equation, the frequency- domain wave equation (also called Helmholtz and physical properties deep within the Earth's subsurface [25,26]. The numerical solution of the wave equation
Dispersion and attenuation for an acoustic wave equation consistent with viscoelasticity
Andrzej Hanyga
2014-01-30
An acoustic wave equation for pressure accounting for viscoelastic attenuation is derived from viscoelastic equations of motion. It differs significantly from the equations proposed by Szabo. Dispersion and attenuation associated with the viscoelastic wave equation is examined. The theory is applied to three classes of viscoelastic models and to the linear attenuation model.
Formal Proof of a Wave Equation Resolution Scheme: the Method Error
Mayero, Micaela
-dimensional acoustic wave equation are well-known to be convergent. We present a comprehensive formalization words: partial differential equation, acoustic wave equation, nu- merical scheme, Coq formal proofs 1 propagation models, the acoustic wave equation in a one-dimensional space domain, for it is a prototype model
Spokoiny, Vladimir
) Ural State University, Ekaterinburg, Russia 1 #12;2 Abstract The general reverse diffusion equations
A Block-Based Parallel Adaptive Scheme for Solving the 4D Vlasov Equation
Genaud, Stéphane
phenomena in plasma physics such as controlled thermonuclear fusion. This equation is defined in the phase
On the Lorentz invariance of the Square root Klein-Gordon Equation
Mohammad Javad Kazemi; Mohammad H. Barati; Jafar Khodagholizadeh; Alireza Babazadeh
2015-06-17
We show that the Born's rule is incompatible with Lorentz symmetry of the Square Root Klein-Gordon equation (SRKG equation). It has been demonstrated that the Born rule must be modified in relativistic regime if one wishes to keep the SRKG equation as the correct equation for describing quantum behavior.
I: Maxwell's equations with Ohm's law. II: Polynomial energy decay. Polynomial decay rate
Phung, Kim-dang.- Le Laboratoire de MathÃ©matiques
I: Maxwell's equations with Ohm's law. II: Polynomial energy decay. Polynomial decay rate: Maxwell's equations with Ohm's law. II: Polynomial energy decay. Maxwell's equation with Ohm's law Let@ = 0 2 L1 ( ) and 0 take "o = o = 1 #12;I: Maxwell's equations with Ohm's law. II: Polynomial energy
Astronomical Coordinate Systems \\What good are Mercator's North Poles and Equators
Walter, Frederick M.
CEN 511 Astronomical Coordinate Systems \\What good are Mercator's North Poles and Equators Tropics conversions. Celestial Based on the terrestrial globe. The Celestial equator and poles are projections of the terrestrial equator and poles. Celestial coordinates are measured in Right Ascension (RA; ), along the equator