Bäcklund transformations relating different Hamilton-Jacobi equations
NASA Astrophysics Data System (ADS)
Sozonov, A. P.; Tsiganov, A. V.
2015-06-01
We discuss one of the possible finite-dimensional analogues of the general Bäcklund transformation relating different partial differential equations. We show that different Hamilton-Jacobi equations can be obtained from the same Lax matrix. We consider Hénon-Heiles systems on the plane, Neumann and Chaplygin systems on the sphere, and two integrable systems with velocity-dependent potentials as examples.
Central Schemes for Multi-Dimensional Hamilton-Jacobi Equations
NASA Technical Reports Server (NTRS)
Bryson, Steve; Levy, Doron; Biegel, Bryan (Technical Monitor)
2002-01-01
We present new, efficient central schemes for multi-dimensional Hamilton-Jacobi equations. These non-oscillatory, non-staggered schemes are first- and second-order accurate and are designed to scale well with an increasing dimension. Efficiency is obtained by carefully choosing the location of the evolution points and by using a one-dimensional projection step. First-and second-order accuracy is verified for a variety of multi-dimensional, convex and non-convex problems.
Particle dynamics inside shocks in Hamilton-Jacobi equations.
Khanin, Konstantin; Sobolevski, Andrei
2010-04-13
The characteristic curves of a Hamilton-Jacobi equation can be seen as action-minimizing trajectories of fluid particles. For non-smooth 'viscosity' solutions, which give rise to discontinuous velocity fields, this description is usually pursued only up to the moment when trajectories hit a shock and cease to minimize the Lagrangian action. In this paper we show that, for any convex Hamiltonian, there exists a uniquely defined canonical global non-smooth coalescing flow that extends particle trajectories and determines the dynamics inside shocks. We also provide a variational description of the corresponding effective velocity field inside shocks, and discuss the relation to the 'dissipative anomaly' in the limit of vanishing viscosity. PMID:20211875
Quantitative Compactness Estimates for Hamilton-Jacobi Equations
NASA Astrophysics Data System (ADS)
Ancona, Fabio; Cannarsa, Piermarco; Nguyen, Khai T.
2016-02-01
We study quantitative compactness estimates in {W^{1,1}_{loc}} for the map {S_t}, {t > 0} that is associated with the given initial data {u_0in Lip (R^N)} for the corresponding solution {S_t u_0} of a Hamilton-Jacobi equation u_t+Hbig(nabla_{x} ubig)=0, qquad t≥ 0,quad xinR^N, with a uniformly convex Hamiltonian {H=H(p)}. We provide upper and lower estimates of order {1/\\varepsilon^N} on the Kolmogorov {\\varepsilon}-entropy in {W^{1,1}} of the image through the map S t of sets of bounded, compactly supported initial data. Estimates of this type are inspired by a question posed by Lax (Course on Hyperbolic Systems of Conservation Laws. XXVII Scuola Estiva di Fisica Matematica, Ravello, 2002) within the context of conservation laws, and could provide a measure of the order of "resolution" of a numerical method implemented for this equation.
On Dynamics of Lagrangian Trajectories for Hamilton-Jacobi Equations
NASA Astrophysics Data System (ADS)
Khanin, Konstantin; Sobolevski, Andrei
2016-02-01
Characteristic curves of a Hamilton-Jacobi equation can be seen as action minimizing trajectories of fluid particles. However this description is valid only for smooth solutions. For nonsmooth "viscosity" solutions, which give rise to discontinuous velocity fields, this picture holds only up to the moment when trajectories hit a shock and cease to minimize the Lagrangian action. In this paper we discuss two physically meaningful regularization procedures, one corresponding to vanishing viscosity and another to weak noise limit. We show that for any convex Hamiltonian, a viscous regularization allows us to construct a nonsmooth flow that extends particle trajectories and determines dynamics inside the shock manifolds. This flow consists of integral curves of a particular "effective" velocity field, which is uniquely defined everywhere in the flow domain and is discontinuous on shock manifolds. The effective velocity field arising in the weak noise limit is generally non-unique and different from the viscous one, but in both cases there is a fundamental self-consistency condition constraining the dynamics.
Stochastic homogenization of nonconvex Hamilton-Jacobi equations in one space dimension
NASA Astrophysics Data System (ADS)
Armstrong, Scott N.; Tran, Hung V.; Yu, Yifeng
2016-09-01
We prove stochastic homogenization for a general class of coercive, nonconvex Hamilton-Jacobi equations in one space dimension. Some properties of the effective Hamiltonian arising in the nonconvex case are also discussed.
NASA Technical Reports Server (NTRS)
Fitzpatrick, P. M.; Harmon, G. R.; Cochran, J. E.; Shaw, W. A.
1974-01-01
Some methods of approaching a solution to the Hamilton-Jacobi equation are outlined and examples are given to illustrate particular methods. These methods may be used for cases where the Hamilton-Jacobi equation is not separable and have been particularly useful in solving the rigid body motion of an earth satellite subjected to gravity torques. These general applications may also have usefulness in studying the motion of satellites with aerodynamic torque and in studying space vehicle motion where thrusting is involved.
Holographic Wilson loops, Hamilton-Jacobi equation, and regularizations
NASA Astrophysics Data System (ADS)
Pontello, Diego; Trinchero, Roberto
2016-04-01
The minimal area for surfaces whose borders are rectangular and circular loops are calculated using the Hamilton-Jacobi (HJ) equation. This amounts to solving the HJ equation for the value of the minimal area, without calculating the shape of the corresponding surface. This is done for bulk geometries that are asymptotically anti-de Sitter (AdS). For the rectangular contour, the HJ equation, which is separable, can be solved exactly. For the circular contour an expansion in powers of the radius is implemented. The HJ approach naturally leads to a regularization which consists in locating the contour away from the border. The results are compared with the ɛ -regularization which leaves the contour at the border and calculates the area of the corresponding minimal surface up to a diameter smaller than the one of the contour at the border. The results for the circular loop do not coincide if the expansion parameter is taken to be the radius of the contour at the border. It is shown that using this expansion parameter the ɛ -regularization leads to incorrect results for certain solvable non-AdS cases. However, if the expansion parameter is taken to be the radius of the minimal surface whose area is computed, then the results coincide with the HJ scheme. This is traced back to the fact that in the HJ case the expansion parameter for the area of a minimal surface is intrinsic to the surface; however, the radius of the contour at the border is related to the way one chooses to regularize in the ɛ -scheme the calculation of this area.
Game theory to characterize solutions of a discrete-time Hamilton-Jacobi equation
NASA Astrophysics Data System (ADS)
Toledo, Porfirio
2013-12-01
We study the behavior of solutions of a discrete-time Hamilton-Jacobi equation in a minimax framework of game theory. The solutions of this problem represent the optimal payoff of a zero-sum game of two players, where the number of moves between the players converges to infinity. A real number, called the critical value, plays a central role in this work; this number is the asymptotic average action of optimal trajectories. The aim of this paper is to show the existence and characterization of solutions of a Hamilton-Jacobi equation for this kind of games.
Chou, Chia-Chun
2014-03-14
The complex quantum Hamilton-Jacobi equation-Bohmian trajectories (CQHJE-BT) method is introduced as a synthetic trajectory method for integrating the complex quantum Hamilton-Jacobi equation for the complex action function by propagating an ensemble of real-valued correlated Bohmian trajectories. Substituting the wave function expressed in exponential form in terms of the complex action into the time-dependent Schrödinger equation yields the complex quantum Hamilton-Jacobi equation. We transform this equation into the arbitrary Lagrangian-Eulerian version with the grid velocity matching the flow velocity of the probability fluid. The resulting equation describing the rate of change in the complex action transported along Bohmian trajectories is simultaneously integrated with the guidance equation for Bohmian trajectories, and the time-dependent wave function is readily synthesized. The spatial derivatives of the complex action required for the integration scheme are obtained by solving one moving least squares matrix equation. In addition, the method is applied to the photodissociation of NOCl. The photodissociation dynamics of NOCl can be accurately described by propagating a small ensemble of trajectories. This study demonstrates that the CQHJE-BT method combines the considerable advantages of both the real and the complex quantum trajectory methods previously developed for wave packet dynamics.
Chou, Chia-Chun
2014-03-14
The complex quantum Hamilton-Jacobi equation-Bohmian trajectories (CQHJE-BT) method is introduced as a synthetic trajectory method for integrating the complex quantum Hamilton-Jacobi equation for the complex action function by propagating an ensemble of real-valued correlated Bohmian trajectories. Substituting the wave function expressed in exponential form in terms of the complex action into the time-dependent Schrödinger equation yields the complex quantum Hamilton-Jacobi equation. We transform this equation into the arbitrary Lagrangian-Eulerian version with the grid velocity matching the flow velocity of the probability fluid. The resulting equation describing the rate of change in the complex action transported along Bohmian trajectories is simultaneously integrated with the guidance equation for Bohmian trajectories, and the time-dependent wave function is readily synthesized. The spatial derivatives of the complex action required for the integration scheme are obtained by solving one moving least squares matrix equation. In addition, the method is applied to the photodissociation of NOCl. The photodissociation dynamics of NOCl can be accurately described by propagating a small ensemble of trajectories. This study demonstrates that the CQHJE-BT method combines the considerable advantages of both the real and the complex quantum trajectory methods previously developed for wave packet dynamics. PMID:24628169
High-Order Central WENO Schemes for 1D Hamilton-Jacobi Equations
NASA Technical Reports Server (NTRS)
Bryson, Steve; Levy, Doron; Biegel, Bryan A. (Technical Monitor)
2002-01-01
In this paper we derive fully-discrete Central WENO (CWENO) schemes for approximating solutions of one dimensional Hamilton-Jacobi (HJ) equations, which combine our previous works. We introduce third and fifth-order accurate schemes, which are the first central schemes for the HJ equations of order higher than two. The core ingredient is the derivation of our schemes is a high-order CWENO reconstructions in space.
A TVD-type method for 2D scalar Hamilton-Jacobi equations on unstructured meshes
NASA Astrophysics Data System (ADS)
Tang, Lingyan; Song, Songhe
2006-10-01
In this paper, a TVD-type difference scheme which satisfies maximum principle is developed for 2D scalar Hamilton-Jacobi equations on unstructured triangular meshes. The main ideas are node-based approximations and derivative-limited reconstruction with quadratic interpolation polynomial. The solution's slope satisfies maximum principle. Numerical experiments are performed to demonstrate high-order accuracy in smooth fields and good resolution of derivative singularities. The new method is simpler than WENO.
A Discontinuous Galerkin Finite Element Method for Hamilton-Jacobi Equations
NASA Technical Reports Server (NTRS)
Hu, Changqing; Shu, Chi-Wang
1998-01-01
In this paper, we present a discontinuous Galerkin finite element method for solving the nonlinear Hamilton-Jacobi equations. This method is based on the Runge-Kutta discontinuous Galerkin finite element method for solving conservation laws. The method has the flexibility of treating complicated geometry by using arbitrary triangulation, can achieve high order accuracy with a local, compact stencil, and are suited for efficient parallel implementation. One and two dimensional numerical examples are given to illustrate the capability of the method.
Compressed Semi-Discrete Central-Upwind Schemes for Hamilton-Jacobi Equations
NASA Technical Reports Server (NTRS)
Bryson, Steve; Kurganov, Alexander; Levy, Doron; Petrova, Guergana
2003-01-01
We introduce a new family of Godunov-type semi-discrete central schemes for multidimensional Hamilton-Jacobi equations. These schemes are a less dissipative generalization of the central-upwind schemes that have been recently proposed in series of works. We provide the details of the new family of methods in one, two, and three space dimensions, and then verify their expected low-dissipative property in a variety of examples.
The nonconvex multi-dimensional Riemann problem for Hamilton-Jacobi equations
NASA Technical Reports Server (NTRS)
Osher, Stanley
1989-01-01
Simple inequalities for the Riemann problem for a Hamilton-Jacobi equation in N space dimension when neither the initial data nor the Hamiltonian need be convex (or concave) are presented. The initial data is globally continuous, affine in each orthant, with a possible jump in normal derivative across each coordinate plane, x sub i = 0. The inequalities become equalities wherever a maxmin equals a minmax and thus an exact closed form solution to this problem is then obtained.
NASA Astrophysics Data System (ADS)
Zatloukal, Václav
2016-04-01
Classical field theory is considered as a theory of unparametrized surfaces embedded in a configuration space, which accommodates, in a symmetric way, spacetime positions and field values. Dynamics is defined by a (Hamiltonian) constraint between multivector-valued generalized momenta, and points in the configuration space. Starting from a variational principle, we derive local equations of motion, that is, differential equations that determine classical surfaces and momenta. A local Hamilton-Jacobi equation applicable in the field theory then follows readily. The general method is illustrated with three examples: non-relativistic Hamiltonian mechanics, De Donder-Weyl scalar field theory, and string theory.
ERIC Educational Resources Information Center
Field, J. H.
2011-01-01
It is shown how the time-dependent Schrodinger equation may be simply derived from the dynamical postulate of Feynman's path integral formulation of quantum mechanics and the Hamilton-Jacobi equation of classical mechanics. Schrodinger's own published derivations of quantum wave equations, the first of which was also based on the Hamilton-Jacobi…
Numerical Schemes for the Hamilton-Jacobi and Level Set Equations on Triangulated Domains
NASA Technical Reports Server (NTRS)
Barth, Timothy J.; Sethian, James A.
2006-01-01
Borrowing from techniques developed for conservation law equations, we have developed both monotone and higher order accurate numerical schemes which discretize the Hamilton-Jacobi and level set equations on triangulated domains. The use of unstructured meshes containing triangles (2D) and tetrahedra (3D) easily accommodates mesh adaptation to resolve disparate level set feature scales with a minimal number of solution unknowns. The minisymposium talk will discuss these algorithmic developments and present sample calculations using our adaptive triangulation algorithm applied to various moving interface problems such as etching, deposition, and curvature flow.
Fast methods for the Eikonal and related Hamilton- Jacobi equations on unstructured meshes.
Sethian, J A; Vladimirsky, A
2000-05-23
The Fast Marching Method is a numerical algorithm for solving the Eikonal equation on a rectangular orthogonal mesh in O(M log M) steps, where M is the total number of grid points. The scheme relies on an upwind finite difference approximation to the gradient and a resulting causality relationship that lends itself to a Dijkstra-like programming approach. In this paper, we discuss several extensions to this technique, including higher order versions on unstructured meshes in Rn and on manifolds and connections to more general static Hamilton-Jacobi equations. PMID:10811874
Fast methods for the eikonal and related Hamilton-Jacobi equations on unstructured meshes
Sethian, J.A.; Vladimirsky, A.
1999-12-01
The Fast Marching Method is a numerical algorithm for solving the Eikonal equation on a rectangular orthogonal mesh in O(M log M) steps, where M is the total number of grid points. The scheme relies on an upwind finite difference approximation to the gradient and a resulting causality relationship that lends itself to a Dijkstra-like programming approach. In this paper, we discuss several extensions to this technique, including higher order versions on unstructured meshes in R{sup n} and on manifolds and connections to more general static Hamilton-Jacobi equations.
Fast methods for the Eikonal and related Hamilton- Jacobi equations on unstructured meshes.
Sethian, J A; Vladimirsky, A
2000-05-23
The Fast Marching Method is a numerical algorithm for solving the Eikonal equation on a rectangular orthogonal mesh in O(M log M) steps, where M is the total number of grid points. The scheme relies on an upwind finite difference approximation to the gradient and a resulting causality relationship that lends itself to a Dijkstra-like programming approach. In this paper, we discuss several extensions to this technique, including higher order versions on unstructured meshes in Rn and on manifolds and connections to more general static Hamilton-Jacobi equations.
On the regularizing effect for unbounded solutions of first-order Hamilton-Jacobi equations
NASA Astrophysics Data System (ADS)
Barles, Guy; Chasseigne, Emmanuel
2016-05-01
We give a simplified proof of regularizing effects for first-order Hamilton-Jacobi Equations of the form ut + H (x , t , Du) = 0 in RN × (0 , + ∞) in the case where the idea is to first estimate ut. As a consequence, we have a Lipschitz regularity in space and time for coercive Hamiltonians and, for hypo-elliptic Hamiltonians, we also have an Hölder regularizing effect in space following a result of L.C. Evans and M.R. James.
The nonconvex multi-dimensional Riemann problem for Hamilton-Jacobi equations
NASA Technical Reports Server (NTRS)
Bardi, Martino; Osher, Stanley
1991-01-01
Simple inequalities are presented for the viscosity solution of a Hamilton-Jacobi equation in N space dimensions when neither the initial data nor the Hamiltonian need be convex (or concave). The initial data are uniformly Lipschitz and can be written as the sum of a convex function in a group of variables and a concave function in the remaining variables, therefore including the nonconvex Riemann problem. The inequalities become equalities wherever a 'maxmin' equals a 'minmax', and thus a representation formula for this problem is obtained, generalizing the classical Hopi formulas.
Hamilton-Jacobi-Bellman equations and approximate dynamic programming on time scales.
Seiffertt, John; Sanyal, Suman; Wunsch, Donald C
2008-08-01
The time scales calculus is a key emerging area of mathematics due to its potential use in a wide variety of multidisciplinary applications. We extend this calculus to approximate dynamic programming (ADP). The core backward induction algorithm of dynamic programming is extended from its traditional discrete case to all isolated time scales. Hamilton-Jacobi-Bellman equations, the solution of which is the fundamental problem in the field of dynamic programming, are motivated and proven on time scales. By drawing together the calculus of time scales and the applied area of stochastic control via ADP, we have connected two major fields of research. PMID:18632378
Dey, Bijoy K; Janicki, Marek R; Ayers, Paul W
2004-10-01
Classical dynamics can be described with Newton's equation of motion or, totally equivalently, using the Hamilton-Jacobi equation. Here, the possibility of using the Hamilton-Jacobi equation to describe chemical reaction dynamics is explored. This requires an efficient computational approach for constructing the physically and chemically relevant solutions to the Hamilton-Jacobi equation; here we solve Hamilton-Jacobi equations on a Cartesian grid using Sethian's fast marching method. Using this method, we can--starting from an arbitrary initial conformation--find reaction paths that minimize the action or the time. The method is demonstrated by computing the mechanism for two different systems: a model system with four different stationary configurations and the H+H(2)-->H(2)+H reaction. Least-time paths (termed brachistochrones in classical mechanics) seem to be a suitable chioce for the reaction coordinate, allowing one to determine the key intermediates and final product of a chemical reaction. For conservative systems the Hamilton-Jacobi equation does not depend on the time, so this approach may be useful for simulating systems where important motions occur on a variety of different time scales.
Massless Spin-Zero Particle and the Classical Action via Hamilton-Jacobi Equation in Gödel Universe
NASA Astrophysics Data System (ADS)
Bahrehbakhsh, A. F.; Momeni, D.; Myrzakulov, R.
2012-08-01
In this letter we investigate the separability of the Klein-Gordon and Hamilton-Jacobi equation in Gödel universe. We show that the Klein-Gordon eigen modes are quantized and the complete spectrum of the particle's energy is a mixture of an azimuthal quantum number, m and a principal quantum number, n and a continuous wave number k. We also show that the Hamilton-Jacobi equation gives a closed function for classical action. These results may be used to calculate the Casimir vacuum energy in Gödel universe.
NASA Astrophysics Data System (ADS)
Oberman, Adam M.; Salvador, Tiago
2015-03-01
We build a simple and general class of finite difference schemes for first order Hamilton-Jacobi (HJ) Partial Differential Equations. These filtered schemes are convergent to the unique viscosity solution of the equation. The schemes are accurate: we implement second, third and fourth order accurate schemes in one dimension and second order accurate schemes in two dimensions, indicating how to build higher order ones. They are also explicit, which means they can be solved using the fast sweeping method. The accuracy of the method is validated with computational results for the eikonal equation and other HJ equations in one and two dimensions, using filtered schemes made from standard centered differences, higher order upwinding and ENO interpolation.
Numerical Schemes for the Hamilton-Jacobi and Level Set Equations on Triangulated Domains
NASA Technical Reports Server (NTRS)
Barth, Timothy J.; Sethian, James A.
1997-01-01
Borrowing from techniques developed for conservation law equations, numerical schemes which discretize the Hamilton-Jacobi (H-J), level set, and Eikonal equations on triangulated domains are presented. The first scheme is a provably monotone discretization for certain forms of the H-J equations. Unfortunately, the basic scheme lacks proper Lipschitz continuity of the numerical Hamiltonian. By employing a virtual edge flipping technique, Lipschitz continuity of the numerical flux is restored on acute triangulations. Next, schemes are introduced and developed based on the weaker concept of positive coefficient approximations for homogeneous Hamiltonians. These schemes possess a discrete maximum principle on arbitrary triangulations and naturally exhibit proper Lipschitz continuity of the numerical Hamiltonian. Finally, a class of Petrov-Galerkin approximations are considered. These schemes are stabilized via a least-squares bilinear form. The Petrov-Galerkin schemes do not possess a discrete maximum principle but generalize to high order accuracy.
High-Order Central WENO Schemes for Multi-Dimensional Hamilton-Jacobi Equations
NASA Technical Reports Server (NTRS)
Bryson, Steve; Levy, Doron; Biegel, Bryan (Technical Monitor)
2002-01-01
We present new third- and fifth-order Godunov-type central schemes for approximating solutions of the Hamilton-Jacobi (HJ) equation in an arbitrary number of space dimensions. These are the first central schemes for approximating solutions of the HJ equations with an order of accuracy that is greater than two. In two space dimensions we present two versions for the third-order scheme: one scheme that is based on a genuinely two-dimensional Central WENO reconstruction, and another scheme that is based on a simpler dimension-by-dimension reconstruction. The simpler dimension-by-dimension variant is then extended to a multi-dimensional fifth-order scheme. Our numerical examples in one, two and three space dimensions verify the expected order of accuracy of the schemes.
Hybrid massively parallel fast sweeping method for static Hamilton-Jacobi equations
NASA Astrophysics Data System (ADS)
Detrixhe, Miles; Gibou, Frédéric
2016-10-01
The fast sweeping method is a popular algorithm for solving a variety of static Hamilton-Jacobi equations. Fast sweeping algorithms for parallel computing have been developed, but are severely limited. In this work, we present a multilevel, hybrid parallel algorithm that combines the desirable traits of two distinct parallel methods. The fine and coarse grained components of the algorithm take advantage of heterogeneous computer architecture common in high performance computing facilities. We present the algorithm and demonstrate its effectiveness on a set of example problems including optimal control, dynamic games, and seismic wave propagation. We give results for convergence, parallel scaling, and show state-of-the-art speedup values for the fast sweeping method.
A Penalty Method for the Numerical Solution of Hamilton-Jacobi-Bellman (HJB) Equations in Finance
NASA Astrophysics Data System (ADS)
Witte, J. H.; Reisinger, C.
2010-09-01
We present a simple and easy to implement method for the numerical solution of a rather general class of Hamilton-Jacobi-Bellman (HJB) equations. In many cases, the considered problems have only a viscosity solution, to which, fortunately, many intuitive (e.g. finite difference based) discretisations can be shown to converge. However, especially when using fully implicit time stepping schemes with their desireable stability properties, one is still faced with the considerable task of solving the resulting nonlinear discrete system. In this paper, we introduce a penalty method which approximates the nonlinear discrete system to an order of O(1/ρ), where ρ>0 is the penalty parameter, and we show that an iterative scheme can be used to solve the penalised discrete problem in finitely many steps. We include a number of examples from mathematical finance for which the described approach yields a rigorous numerical scheme and present numerical results.
NASA Technical Reports Server (NTRS)
Calise, Anthony J.; Melamed, Nahum
1993-01-01
In this paper we develop a general procedure for constructing a matched asymptotic expansion of the Hamilton-Jacobi-Bellman equation based on the method of characteristics. The development is for a class of perturbation problems whose solution exhibits two-time-scale behavior. A regular expansion for problems of this type is inappropriate since it is not uniformly valid over a narrow range of the independent variable. Of particular interest here is the manner in which matching and boundary conditions are enforced when the expansion is carried out to first order. Two cases are distinguished - one where the left boundary condition coincides with or lies to the right of the singular region and one where the left boundary condition lies to the left of the singular region. A simple example is used to illustrate the procedure, and its potential application to aeroassisted plane change is described.
NASA Astrophysics Data System (ADS)
Osetrin, Konstantin; Filippov, Altair; Osetrin, Evgeny
2016-01-01
The characteristics of dust matter in spacetime models, admitting the existence of privilege coordinate systems are given, where the single-particle Hamilton-Jacobi equation can be integrated by the method of complete separation of variables. The resulting functional form of the 4-velocity field and energy density of matter for all types of spaces under consideration is presented.
NASA Astrophysics Data System (ADS)
Feng, Zhongwen; Li, Guoping; Jiang, Pengying; Pan, Yang; Zu, Xiaotao
2016-07-01
In this paper, we derive the deformed Hamilton-Jacobi equations from the generalized Klein-Gordon equation and generalized Dirac equation. Then, we study the tunneling rate, Hawking temperature and entropy of the higher-dimensional Reissner-Nordström de Sitter black hole via the deformed Hamilton-Jacobi equation. Our results show that the deformed Hamilton-Jacobi equations for charged scalar particles and charged fermions have the same expressions. Besides, the modified Hawking temperatures and entropy are related to the mass and charge of the black hole, the cosmology constant, the quantum number of emitted particles, and the term of GUP effects β.
NASA Astrophysics Data System (ADS)
Hamamuki, Nao; Nakayasu, Atsushi; Namba, Tokinaga
2015-12-01
We study a cell problem arising in homogenization for a Hamilton-Jacobi equation whose Hamiltonian is not coercive. We introduce a generalized notion of effective Hamiltonians by approximating the equation and characterize the solvability of the cell problem in terms of the generalized effective Hamiltonian. Under some sufficient conditions, the result is applied to the associated homogenization problem. We also show that homogenization for non-coercive equations fails in general.
Integrating the quantum Hamilton-Jacobi equations by wavefront expansion and phase space analysis
NASA Astrophysics Data System (ADS)
Bittner, Eric R.; Wyatt, Robert E.
2000-11-01
In this paper we report upon our computational methodology for numerically integrating the quantum Hamilton-Jacobi equations using hydrodynamic trajectories. Our method builds upon the moving least squares method developed by Lopreore and Wyatt [Phys. Rev. Lett. 82, 5190 (1999)] in which Lagrangian fluid elements representing probability volume elements of the wave function evolve under Newtonian equations of motion which include a nonlocal quantum force. This quantum force, which depends upon the third derivative of the quantum density, ρ, can vary rapidly in x and become singular in the presence of nodal points. Here, we present a new approach for performing quantum trajectory calculations which does not involve calculating the quantum force directly, but uses the wavefront to calculate the velocity field using mv=∇S, where S/ℏ is the argument of the wave function ψ. Additional numerical stability is gained by performing local gauge transformations to remove oscillatory components of the wave function. Finally, we use a dynamical Rayleigh-Ritz approach to derive ancillary equations-of-motion for the spatial derivatives of ρ, S, and v. The methodologies described herein dramatically improve the long time stability and accuracy of the quantum trajectory approach even in the presence of nodes. The method is applied to both barrier crossing and tunneling systems. We also compare our results to semiclassical based descriptions of barrier tunneling.
NASA Astrophysics Data System (ADS)
Popova, E. P.
2015-08-01
A two-dimensional model for an αΩ-dynamo is constructed, taking into account meridional flows. A Hamilton-Jacobi equation for the resulting system of magnetic-field generation equatons is constructed using an asymptotic method analogous to the WKB method. This equation makes it possible to analytically study the influence of meridional flows on the duration of the solar magnetic-activity cycle and the evolution of magnetic waves.
Wave front-ray synthesis for solving the multidimensional quantum Hamilton-Jacobi equation.
Wyatt, Robert E; Chou, Chia-Chun
2011-08-21
A Cauchy initial-value approach to the complex-valued quantum Hamilton-Jacobi equation (QHJE) is investigated for multidimensional systems. In this approach, ray segments foliate configuration space which is laminated by surfaces of constant action. The QHJE incorporates all quantum effects through a term involving the divergence of the quantum momentum function (QMF). The divergence term may be expressed as a sum of two terms, one involving displacement along the ray and the other incorporating the local curvature of the action surface. It is shown that curvature of the wave front may be computed from coefficients of the first and second fundamental forms from differential geometry that are associated with the surface. Using the expression for the divergence, the QHJE becomes a Riccati-type ordinary differential equation (ODE) for the complex-valued QMF, which is parametrized by the arc length along the ray. In order to integrate over possible singularities in the QMF, a stable and accurate Möbius propagator is introduced. This method is then used to evolve rays and wave fronts for four systems in two and three dimensions. From the QMF along each ray, the wave function can be easily computed. Computational difficulties that may arise are described and some ways to circumvent them are presented.
Wave front-ray synthesis for solving the multidimensional quantum Hamilton-Jacobi equation.
Wyatt, Robert E; Chou, Chia-Chun
2011-08-21
A Cauchy initial-value approach to the complex-valued quantum Hamilton-Jacobi equation (QHJE) is investigated for multidimensional systems. In this approach, ray segments foliate configuration space which is laminated by surfaces of constant action. The QHJE incorporates all quantum effects through a term involving the divergence of the quantum momentum function (QMF). The divergence term may be expressed as a sum of two terms, one involving displacement along the ray and the other incorporating the local curvature of the action surface. It is shown that curvature of the wave front may be computed from coefficients of the first and second fundamental forms from differential geometry that are associated with the surface. Using the expression for the divergence, the QHJE becomes a Riccati-type ordinary differential equation (ODE) for the complex-valued QMF, which is parametrized by the arc length along the ray. In order to integrate over possible singularities in the QMF, a stable and accurate Möbius propagator is introduced. This method is then used to evolve rays and wave fronts for four systems in two and three dimensions. From the QMF along each ray, the wave function can be easily computed. Computational difficulties that may arise are described and some ways to circumvent them are presented. PMID:21861551
Wave front-ray synthesis for solving the multidimensional quantum Hamilton-Jacobi equation
Wyatt, Robert E.; Chou, Chia-Chun
2011-08-21
A Cauchy initial-value approach to the complex-valued quantum Hamilton-Jacobi equation (QHJE) is investigated for multidimensional systems. In this approach, ray segments foliate configuration space which is laminated by surfaces of constant action. The QHJE incorporates all quantum effects through a term involving the divergence of the quantum momentum function (QMF). The divergence term may be expressed as a sum of two terms, one involving displacement along the ray and the other incorporating the local curvature of the action surface. It is shown that curvature of the wave front may be computed from coefficients of the first and second fundamental forms from differential geometry that are associated with the surface. Using the expression for the divergence, the QHJE becomes a Riccati-type ordinary differential equation (ODE) for the complex-valued QMF, which is parametrized by the arc length along the ray. In order to integrate over possible singularities in the QMF, a stable and accurate Moebius propagator is introduced. This method is then used to evolve rays and wave fronts for four systems in two and three dimensions. From the QMF along each ray, the wave function can be easily computed. Computational difficulties that may arise are described and some ways to circumvent them are presented.
Directly solving the Hamilton-Jacobi equations by Hermite WENO Schemes
NASA Astrophysics Data System (ADS)
Zheng, Feng; Qiu, Jianxian
2016-02-01
In this paper, we present a class of new Hermite weighted essentially non-oscillatory (HWENO) schemes based on finite volume framework to directly solve the Hamilton-Jacobi (HJ) equations. For HWENO reconstruction, both the cell average and the first moment of the solution are evolved, and for two dimensional case, HWENO reconstruction is based on a dimension-by-dimension strategy which is the first used in HWENO reconstruction. For spatial discretization, one of key points for directly solving HJ equation is the reconstruction of numerical fluxes. We follow the idea put forward by Cheng and Wang (2014) [3] to reconstruct the values of solution at Gauss-Lobatto quadrature points and numerical fluxes at the interfaces of cells, and for neither the convex nor concave Hamiltonian case, the monotone modification of numerical fluxes is added, which can guarantee the precision in the smooth region and converge to the entropy solution when derivative discontinuities come up. The third order TVD Runge-Kutta method is used for the time discretization. Extensive numerical experiments in one dimensional and two dimensional cases are performed to verify the efficiency of the methods.
High-Order Semi-Discrete Central-Upwind Schemes for Multi-Dimensional Hamilton-Jacobi Equations
NASA Technical Reports Server (NTRS)
Bryson, Steve; Levy, Doron; Biegel, Bryan (Technical Monitor)
2002-01-01
We present the first fifth order, semi-discrete central upwind method for approximating solutions of multi-dimensional Hamilton-Jacobi equations. Unlike most of the commonly used high order upwind schemes, our scheme is formulated as a Godunov-type scheme. The scheme is based on the fluxes of Kurganov-Tadmor and Kurganov-Tadmor-Petrova, and is derived for an arbitrary number of space dimensions. A theorem establishing the monotonicity of these fluxes is provided. The spacial discretization is based on a weighted essentially non-oscillatory reconstruction of the derivative. The accuracy and stability properties of our scheme are demonstrated in a variety of examples. A comparison between our method and other fifth-order schemes for Hamilton-Jacobi equations shows that our method exhibits smaller errors without any increase in the complexity of the computations.
Killing tensors, warped products and the orthogonal separation of the Hamilton-Jacobi equation
Rajaratnam, Krishan McLenaghan, Raymond G.
2014-01-15
We study Killing tensors in the context of warped products and apply the results to the problem of orthogonal separation of the Hamilton-Jacobi equation. This work is motivated primarily by the case of spaces of constant curvature where warped products are abundant. We first characterize Killing tensors which have a natural algebraic decomposition in warped products. We then apply this result to show how one can obtain the Killing-Stäckel space (KS-space) for separable coordinate systems decomposable in warped products. This result in combination with Benenti's theory for constructing the KS-space of certain special separable coordinates can be used to obtain the KS-space for all orthogonal separable coordinates found by Kalnins and Miller in Riemannian spaces of constant curvature. Next we characterize when a natural Hamiltonian is separable in coordinates decomposable in a warped product by showing that the conditions originally given by Benenti can be reduced. Finally, we use this characterization and concircular tensors (a special type of torsionless conformal Killing tensor) to develop a general algorithm to determine when a natural Hamiltonian is separable in a special class of separable coordinates which include all orthogonal separable coordinates in spaces of constant curvature.
Computing tunneling paths with the Hamilton-Jacobi equation and the fast marching method
NASA Astrophysics Data System (ADS)
Dey, Bijoy K.; Ayers, Paul W.
We present a new method for computing the most probable tunneling paths based on the minimum imaginary action principle. Unlike many conventional methods, the paths are calculated without resorting to an optimization (minimization) scheme. Instead, a fast marching method coupled with a back-propagation scheme is used to efficiently compute the tunneling paths. The fast marching method solves a Hamilton-Jacobi equation for the imaginary action on a discrete grid where the action value at an initial point (usually the reactant state configuration) is known in the beginning. Subsequently, a back-propagation scheme uses a steepest descent method on the imaginary action surface to compute a path connecting an arbitrary point on the potential energy surface (usually a state in the product valley) to the initial state. The proposed method is demonstrated for the tunneling paths of two different systems: a model 2D potential surface and the collinear reaction. Unlike existing methods, where the tunneling path is based on a presumed reaction coordinate and a correction is made with respect to the reaction coordinate within an 'adiabatic' approximation, the proposed method is very general and makes no assumptions about the relationship between the reaction coordinate and tunneling path.
Quantum Hamilton-Jacobi theory.
Roncadelli, Marco; Schulman, L S
2007-10-26
Quantum canonical transformations have attracted interest since the beginning of quantum theory. Based on their classical analogues, one would expect them to provide a powerful quantum tool. However, the difficulty of solving a nonlinear operator partial differential equation such as the quantum Hamilton-Jacobi equation (QHJE) has hindered progress along this otherwise promising avenue. We overcome this difficulty. We show that solutions to the QHJE can be constructed by a simple prescription starting from the propagator of the associated Schrödinger equation. Our result opens the possibility of practical use of quantum Hamilton-Jacobi theory. As an application, we develop a surprising relation between operator ordering and the density of paths around a semiclassical trajectory. PMID:17995307
NASA Astrophysics Data System (ADS)
Ge, Hao; Qian, Hong
2012-09-01
Analytical (rational) mechanics is the mathematical structure of Newtonian deterministic dynamics developed by D'Alembert, Lagrange, Hamilton, Jacobi, and many other luminaries of applied mathematics. Diffusion as a stochastic process of an overdamped individual particle immersed in a fluid, initiated by Einstein, Smoluchowski, Langevin and Wiener, has no momentum since its path is nowhere differentiable. In this exposition, we illustrate how analytical mechanics arises in stochastic dynamics from a randomly perturbed ordinary differential equation dXt = b(Xt)dt+ɛdWt, where Wt is a Brownian motion. In the limit of vanishingly small ɛ, the solution to the stochastic differential equation other than ˙ {x} = b(x) are all rare events. However, conditioned on an occurrence of such an event, the most probable trajectory of the stochastic motion is the solution to Lagrangian mechanics with L = \\Vert ˙ {q}-b(q)\\Vert 2/4 and Hamiltonian equations with H(p, q) = \\dvbr p\\dvbr2+b(q)ṡp. Hamiltonian conservation law implies that the most probable trajectory for a "rare" event has a uniform "excess kinetic energy" along its path. Rare events can also be characterized by the principle of large deviations which expresses the probability density function for Xt as f(x, t) = e-u(x, t)/ɛ, where u(x, t) is called a large-deviation rate function which satisfies the corresponding Hamilton-Jacobi equation. An irreversible diffusion process with ∇×b≠0 corresponds to a Newtonian system with a Lorentz force ḋ {q} = (∇ × b)× ˙ {q}+({1}/{2})∇ \\Vert b\\Vert 2. The connection between stochastic motion and analytical mechanics can be explored in terms of various techniques of applied mathematics, for example, singular perturbations, viscosity solutions and integrable systems.
High-Order Semi-Discrete Central-Upwind Schemes for Multi-Dimensional Hamilton-Jacobi Equations
NASA Technical Reports Server (NTRS)
Bryson, Steve; Levy, Doron; Biegel, Bran R. (Technical Monitor)
2002-01-01
We present high-order semi-discrete central-upwind numerical schemes for approximating solutions of multi-dimensional Hamilton-Jacobi (HJ) equations. This scheme is based on the use of fifth-order central interpolants like those developed in [1], in fluxes presented in [3]. These interpolants use the weighted essentially nonoscillatory (WENO) approach to avoid spurious oscillations near singularities, and become "central-upwind" in the semi-discrete limit. This scheme provides numerical approximations whose error is as much as an order of magnitude smaller than those in previous WENO-based fifth-order methods [2, 1]. Thee results are discussed via examples in one, two and three dimensions. We also pregnant explicit N-dimensional formulas for the fluxes, discuss their monotonicity and tl!e connection between this method and that in [2].
NASA Technical Reports Server (NTRS)
Blanchard, D. L.; Chan, F. K.
1973-01-01
For a time-dependent, n-dimensional, special diagonal Hamilton-Jacobi equation a necessary and sufficient condition for the separation of variables to yield a complete integral of the form was established by specifying the admissible forms in terms of arbitrary functions. A complete integral was then expressed in terms of these arbitrary functions and also the n irreducible constants. As an application of the results obtained for the two-dimensional Hamilton-Jacobi equation, analysis was made for a comparatively wide class of dynamical problems involving a particle moving in Euclidean three-dimensional space under the action of external forces but constrained on a moving surface. All the possible cases in which this equation had a complete integral of the form were obtained and these are tubulated for reference.
Efficient High Order Central Schemes for Multi-Dimensional Hamilton-Jacobi Equations: Talk Slides
NASA Technical Reports Server (NTRS)
Bryson, Steve; Levy, Doron; Biegel, Brian R. (Technical Monitor)
2002-01-01
This viewgraph presentation presents information on the attempt to produce high-order, efficient, central methods that scale well to high dimension. The central philosophy is that the equations should evolve to the point where the data is smooth. This is accomplished by a cyclic pattern of reconstruction, evolution, and re-projection. One dimensional and two dimensional representational methods are detailed, as well.
NASA Astrophysics Data System (ADS)
Kraaij, Richard
2016-07-01
We prove the large deviation principle (LDP) for the trajectory of a broad class of finite state mean-field interacting Markov jump processes via a general analytic approach based on viscosity solutions. Examples include generalized Ehrenfest models as well as Curie-Weiss spin flip dynamics with singular jump rates. The main step in the proof of the LDP, which is of independent interest, is the proof of the comparison principle for an associated collection of Hamilton-Jacobi equations. Additionally, we show that the LDP provides a general method to identify a Lyapunov function for the associated McKean-Vlasov equation.
Hamilton-Jacobi approach to cosmology with nonlinear sigma model
NASA Astrophysics Data System (ADS)
Kerner, Richard; van Holten, Jan-Willem
2016-05-01
We start with a short introduction of the role that constraints and Lagrange multiplers play in variational calculus. After recalling briefly the properties of the nonlinear sigma model, we show how the Hamilton-Jacobi method can be applied to find its solutions. We discuss the importance of the Hamiltonian constraint in the standard cosmological model, and finally, apply the Hamilton-Jacobi method to the solution of coupled gravitational and sigma-field equations.
A Hamilton Jacobi formalism for thermodynamics
NASA Astrophysics Data System (ADS)
Rajeev, S. G.
2008-09-01
We show that classical thermodynamics has a formulation in terms of Hamilton-Jacobi theory, analogous to mechanics. Even though the thermodynamic variables come in conjugate pairs such as pressure/volume or temperature/entropy, the phase space is odd-dimensional. For a system with n thermodynamic degrees of freedom it is 2n+1-dimensional. The equations of state of a substance pick out an n-dimensional submanifold. A family of substances whose equations of state depend on n parameters define a hypersurface of co-dimension one. This can be described by the vanishing of a function which plays the role of a Hamiltonian. The ordinary differential equations (characteristic equations) defined by this function describe a dynamical system on the hypersurface. Its orbits can be used to reconstruct the equations of state. The 'time' variable associated to this dynamics is related to, but is not identical to, entropy. After developing this formalism on well-grounded systems such as the van der Waals gases and the Curie-Weiss magnets, we derive a Hamilton-Jacobi equation for black hole thermodynamics in General Relativity. The cosmological constant appears as a constant of integration in this picture.
Unified formalism for the generalized kth-order Hamilton-Jacobi problem
NASA Astrophysics Data System (ADS)
Colombo, Leonardo; de Léon, Manuel; Prieto-Martínez, Pedro Daniel; Román-Roy, Narciso
2014-08-01
The geometric formulation of the Hamilton-Jacobi theory enables us to generalize it to systems of higher-order ordinary differential equations. In this work we introduce the unified Lagrangian-Hamiltonian formalism for the geometric Hamilton-Jacobi theory on higher-order autonomous dynamical systems described by regular Lagrangian functions.
Lifted tensors and Hamilton-Jacobi separability
NASA Astrophysics Data System (ADS)
Waeyaert, G.; Sarlet, W.
2014-12-01
Starting from a bundle τ : E → R, the bundle π :J1τ∗ → E, which is the dual of the first jet bundle J1 τ and a sub-bundle of T∗ E, is the appropriate manifold for the geometric description of time-dependent Hamiltonian systems. Based on previous work, we recall properties of the complete lifts of a type (1 , 1) tensor R on E to both T∗ E and J1τ∗. We discuss how an interplay between both lifted tensors leads to the identification of related distributions on both manifolds. The integrability of these distributions, a coordinate free condition, is shown to produce exactly Forbat's conditions for separability of the time-dependent Hamilton-Jacobi equation in appropriate coordinates.
NASA Astrophysics Data System (ADS)
Gharbi, A.; Touloum, S.; Bouda, A.
2015-04-01
We study the Klein-Gordon equation with noncentral and separable potential under the condition of equal scalar and vector potentials and we obtain the corresponding relativistic quantum Hamilton-Jacobi equation. The application of the quantum Hamilton-Jacobi formalism to the double ring-shaped Kratzer potential leads to its relativistic energy spectrum as well as the corresponding eigenfunctions.
Conformal invariance and Hamilton Jacobi theory for dissipative systems
NASA Technical Reports Server (NTRS)
Kiehn, R. M.
1975-01-01
For certain dissipative systems, a comparison can be made between the Hamilton-Jacobi theory and the conformal invariance of action theory. The two concepts are not identical, but the conformal action theory covers the Hamilton-Jacobi theory.
Hamilton-Jacobi approach to non-slow-roll inflation
NASA Astrophysics Data System (ADS)
Kinney, William H.
1997-08-01
I describe a general approach to characterizing cosmological inflation outside the standard slow-roll approximation, based on the Hamilton-Jacobi formulation of scalar field dynamics. The basic idea is to view the equation of state of the scalar field matter as the fundamental dynamical variable, as opposed to the field value or the expansion rate. I discuss how to formulate the equations of motion for scalar and tensor fluctuations in situations where the assumption of slow roll is not valid. I apply the general results to the simple case of inflation from an ``inverted'' polynomial potential, and to the more complicated case of hybrid inflation.
Hamilton-Jacobi method for curved domain walls and cosmologies
NASA Astrophysics Data System (ADS)
Skenderis, Kostas; Townsend, Paul K.
2006-12-01
We use Hamiltonian methods to study curved domain walls and cosmologies. This leads naturally to first-order equations for all domain walls and cosmologies foliated by slices of maximal symmetry. For Minkowski and AdS-sliced domain walls (flat and closed FLRW cosmologies) we recover a recent result concerning their (pseudo)supersymmetry. We show how domain-wall stability is consistent with the instability of AdS vacua that violate the Breitenlohner-Freedman bound. We also explore the relationship to Hamilton-Jacobi theory and compute the wave-function of a 3-dimensional closed universe evolving towards de Sitter spacetime.
NASA Astrophysics Data System (ADS)
Faraggi, Alon E.; Matone, Marco
2015-07-01
Adaptation of the Hamilton-Jacobi formalism to quantum mechanics leads to a cocycle condition, which is invariant under D-dimensional Mobius transformations with Euclidean or Minkowski metrics. In this paper we aim to provide a pedagogical presentation of the proof of the Möbius symmetry underlying the cocycle condition. The Möbius symmetry implies energy quantization and undefinability of quantum trajectories, without assigning any prior interpretation to the wave function. As such, the Hamilton-Jacobi formalism, augmented with the global Möbius symmetry, provides an alternative starting point, to the axiomatic probability interpretation of the wave function, for the formulation of quantum mechanics and the quantum spacetime. The Möbius symmetry can only be implemented consistently if spatial space is compact, and correspondingly if there exist a finite ultraviolet length scale. Evidence for nontrivial space topology may exist in the cosmic microwave background radiation.
On the Hamilton-Jacobi method in classical and quantum nonconservative systems
NASA Astrophysics Data System (ADS)
Dutra, A. de Souza; Correa, R. A. C.; Moraes, P. H. R. S.
2016-08-01
In this work we show how to complete some Hamilton-Jacobi solutions of linear, nonconservative classical oscillatory systems which appeared in the literature, and we extend these complete solutions to the quantum mechanical case. In addition, we obtain the solution of the quantum Hamilton-Jacobi equation for an electric charge in an oscillating pulsing magnetic field. We also argue that for the case where a charged particle is under the action of an oscillating magnetic field, one can apply nuclear magnetic resonance techniques in order to find experimental results regarding this problem. We obtain all results analytically, showing that the quantum Hamilton-Jacobi formalism is a powerful tool to describe quantum mechanics.
Hamilton-Jacobi formalism for tachyon inflation
NASA Astrophysics Data System (ADS)
Aghamohammadi, A.; Mohammadi, A.; Golanbari, T.; Saaidi, Kh.
2014-10-01
Tachyon inflation is reconsidered by using the recent observational data obtained from Planck-2013 and BICEP2. The Hamilton-Jacobi formalism is picked out as a desirable approach in this work, which allows one to easily obtain the main parameters of the model. The Hubble parameter is supposed as a power-law and exponential function of the scalar field, and each case is considered separately. The constraints on the model, which come from observational data, are explained during the work. The results show a suitable value for the tensor spectral index and an appropriate form of the potential.
Hamilton-Jacobi Theory in Cauchy Data Space
NASA Astrophysics Data System (ADS)
Campos, CéAdric M.; de Leóan, Manuel; de Diego, David Martín; Vaquero, Miguel
2015-12-01
Recently, M. de LeóAn et al. [8] have developed a geometric Hamilton-Jacobi theory for classical fields in the setting of multisymplectic geometry. Our purpose in the current paper is to establish the corresponding Hamilton-Jacobi theory in the Cauchy data space, and relate both approaches.
Hamilton-Jacobi skeleton on cortical surfaces.
Shi, Y; Thompson, P M; Dinov, I; Toga, A W
2008-05-01
In this paper, we propose a new method to construct graphical representations of cortical folding patterns by computing skeletons on triangulated cortical surfaces. In our approach, a cortical surface is first partitioned into sulcal and gyral regions via the solution of a variational problem using graph cuts, which can guarantee global optimality. After that, we extend the method of Hamilton-Jacobi skeleton [1] to subsets of triangulated surfaces, together with a geometrically intuitive pruning process that can trade off between skeleton complexity and the completeness of representing folding patterns. Compared with previous work that uses skeletons of 3-D volumes to represent sulcal patterns, the skeletons on cortical surfaces can be easily decomposed into branches and provide a simpler way to construct graphical representations of cortical morphometry. In our experiments, we demonstrate our method on two different cortical surface models, its ability of capturing major sulcal patterns and its application to compute skeletons of gyral regions. PMID:18450539
Fronts propagating with curvature dependent speed: Algorithms based on Hamilton-Jacobi formulations
NASA Technical Reports Server (NTRS)
Osher, Stanley; Sethian, James A.
1987-01-01
New numerical algorithms are devised (PSC algorithms) for following fronts propagating with curvature-dependent speed. The speed may be an arbitrary function of curvature, and the front can also be passively advected by an underlying flow. These algorithms approximate the equations of motion, which resemble Hamilton-Jacobi equations with parabolic right-hand-sides, by using techniques from the hyperbolic conservation laws. Non-oscillatory schemes of various orders of accuracy are used to solve the equations, providing methods that accurately capture the formation of sharp gradients and cusps in the moving fronts. The algorithms handle topological merging and breaking naturally, work in any number of space dimensions, and do not require that the moving surface be written as a function. The methods can be used also for more general Hamilton-Jacobi-type problems. The algorithms are demonstrated by computing the solution to a variety of surface motion problems.
Scalar particles emission from black holes with topological defects using Hamilton-Jacobi method
NASA Astrophysics Data System (ADS)
Jusufi, Kimet
2015-11-01
We study quantum tunneling of charged and uncharged scalar particles from the event horizon of Schwarzschild-de Sitter and Reissner-Nordström-de Sitter black holes pierced by an infinitely long spinning cosmic string and a global monopole. In order to find the Hawking temperature and the tunneling probability we solve the Klein-Gordon equation by using the Hamilton-Jacobi method and WKB approximation. We show that Hawking temperature is independent of the presence of topological defects in both cases.
Hamilton-Jacobi formalism for string gas thermodynamics
NASA Astrophysics Data System (ADS)
Joseph, Anosh; Rajeev, Sarada G.
2009-03-01
We show that the thermodynamics of a system of strings at high energy densities under the ideal gas approximation has a formulation in terms of the Hamilton-Jacobi theory. The two parameters of the system, which have dimensions of energy density and number density, respectively, define a family of hypersurfaces of a codimension one, which can be described by the vanishing of a function F that plays the role of a Hamiltonian.
Hamilton-Jacobi approach to photon wave mechanics: near-field aspects.
Keller, O
2008-02-01
After having briefly reviewed the Hamilton-Jacobi theory of classical point-particle mechanics, its extension to the quantum regime and the formal identity between the Hamilton-Jacobi equation for Hamilton's characteristic function and the eikonal equation of geometrical optics, an eikonal theory for free photons is established. The space-time dynamics of the photon is described on the basis of the six-component Riemann-Silberstein energy wave function. Form-identical eikonal equations are obtained for the positive and negative helicity dynamics. Microscopic response theory is used to describe the linear photon-matter interaction. In the presence of matter the free-photon concept is replaced by a quasi-photon concept, and there is a quasi-photon for each of the two helicity states. After having established integro-differential equations for the wave functions of the two quasi-photons, the eikonal conditions for the quasi-photons are determined. It appears that the eikonal condition contains complicated space integrals of the gradient of the eikonal over volumes of near-field domain size. In these space integrals the dynamics of the electrons (matter particles) appears via transverse transition current densities between pairs of many-body states. Generalized microscopic polarization and magnetization fields are introduced to establish the connection between the quasi-photon and macroscopic eikonal theories. PMID:18304094
Hamilton-Jacobi-Bellman framework for optimal control in multistage energy systems
NASA Astrophysics Data System (ADS)
Sieniutycz, Stanislaw
2000-03-01
We enunciate parallelism for structures of variational principles in mechanics and thermodynamics in terms of the duality for thermoeconomic problems of maximizing of production profit and net profit which can be transferred to duality for least action and least abbreviated action which appear in mechanics. With the parallelism in mind, we review theory and macroscopic applications of a recently developed discrete formalism of Hamilton-Jacobi type which arises when Bellman's method of dynamic programming is applied to optimize active (work producing) and inactive (entropy generating) multistage energy systems with free intervals of an independent variable. Our original contribution develops a generalized theory for discrete processes in which these intervals can reside in the model inhomogeneously and can be constrained. We consider applications to multistage thermal machines, controlled unit operations, spontaneous relaxations, nonlinear heat conduction, and self-propagating reaction-diffusion fronts. They all satisfy a basic functional equation that leads to the Hamilton-Jacobi-Bellman equation (HJB equation) and a related discrete optimization algorithm with a maximum principle for a Hamiltonian. Correspondence is shown with the well-known HJB theory for continuous processes when the number of stages approaches an infinity. We show that a common unifying criterion, which is the criterion of a minimum generated entropy, can be proven to act locally in the majority of considered cases, although the related global statements can be invalid far from equilibrium. General limits are found which bound the consumption of the classical work potential (exergy) for finite durations.
A practical approach to the Hamilton-Jacobi formulation of holographic renormalization
NASA Astrophysics Data System (ADS)
Elvang, Henriette; Hadjiantonis, Marios
2016-06-01
We revisit the subject of holographic renormalization for asymptotically AdS spacetimes. For many applications of holography, one has to handle the divergences associated with the on-shell gravitational action. The brute force approach uses the Fefferman- Graham (FG) expansion near the AdS boundary to identify the divergences, but subsequent reversal of the expansion is needed to construct the infinite counterterms. While in principle straightforward, the method is cumbersome and application/reversal of FG is formally unsatisfactory. Various authors have proposed an alternative method based on the Hamilton-Jacobi equation. However, this approach may appear to be abstract, difficult to implement, and in some cases limited in applicability. In this paper, we clarify the Hamilton-Jacobi formulation of holographic renormalization and present a simple algorithm for its implementation to extract cleanly the infinite counterterms. While the derivation of the method relies on the Hamiltonian formulation of general relativity, the actual application of our algorithm does not. The work applies to any D-dimensional holographic dual with asymptotic AdS boundary, Euclidean or Lorentzian, and arbitrary slicing. We illustrate the method in several examples, including the FGPW model, a holographic model of 3d ABJM theory, and cases with marginal scalars such as a dilaton-axion system.
Hamilton-Jacobi solutions for strongly coupled gravity and matter
NASA Astrophysics Data System (ADS)
Salopek, D. S.
1998-05-01
A Green function method is developed for solving strongly coupled gravity and matter in the semiclassical limit. In the strong-coupling limit, one assumes that Newton's constant approaches infinity, 0264-9381/15/5/009/img1. As a result, one may neglect second-order spatial gradients, and each spatial point evolves like a homogeneous universe. After constructing the Green function solution to the Hamiltonian constraint, the momentum constraint is solved using functional methods in conjunction with the superposition principle for Hamilton-Jacobi theory. Exact and approximate solutions are given for a dust field or a scalar field interacting with gravity.
Classification of Hamilton-Jacobi separation in orthogonal coordinates with diagonal curvature
Rajaratnam, Krishan McLenaghan, Raymond G.
2014-08-15
We find all orthogonal metrics where the geodesic Hamilton-Jacobi equation separates and the Riemann curvature tensor satisfies a certain equation (called the diagonal curvature condition). All orthogonal metrics of constant curvature satisfy the diagonal curvature condition. The metrics we find either correspond to a Benenti system or are warped product metrics where the induced metric on the base manifold corresponds to a Benenti system. Furthermore, we show that most metrics we find are characterized by concircular tensors; these metrics, called Kalnins-Eisenhart-Miller metrics, have an intrinsic characterization which can be used to obtain them on a given space. In conjunction with other results, we show that the metrics we found constitute all separable metrics for Riemannian spaces of constant curvature and de Sitter space.
Quantum interference within the complex quantum Hamilton-Jacobi formalism
Chou, Chia-Chun; Sanz, Angel S.; Miret-Artes, Salvador; Wyatt, Robert E.
2010-10-15
Quantum interference is investigated within the complex quantum Hamilton-Jacobi formalism. As shown in a previous work [Phys. Rev. Lett. 102 (2009) 250401], complex quantum trajectories display helical wrapping around stagnation tubes and hyperbolic deflection near vortical tubes, these structures being prominent features of quantum caves in space-time Argand plots. Here, we further analyze the divergence and vorticity of the quantum momentum function along streamlines near poles, showing the intricacy of the complex dynamics. Nevertheless, despite this behavior, we show that the appearance of the well-known interference features (on the real axis) can be easily understood in terms of the rotation of the nodal line in the complex plane. This offers a unified description of interference as well as an elegant and practical method to compute the lifetime for interference features, defined in terms of the average wrapping time, i.e., considering such features as a resonant process.
Hamilton-Jacobi method for molecular distribution function in a chemical oscillator
NASA Astrophysics Data System (ADS)
Nakanishi, Hiizu; Sakaue, Takahiro; Wakou, Jun'ichi
2013-12-01
Using the Hamilton-Jacobi method, we solve chemical Fokker-Planck equations within the Gaussian approximation and obtain a simple and compact formula for a conditional probability distribution. The formula holds in general transient situations, and can be applied not only to a steady state but also to an oscillatory state. By analyzing the long time behavior of the solution in the oscillatory case, we obtain the phase diffusion constant along the periodic orbit and the steady distribution perpendicular to it. A simple method for numerical evaluation of these formulas are devised, and they are compared with Monte Carlo simulations in the case of Brusselator as an example. Some results are shown to be identical to previously obtained expressions.
Hamilton-Jacobi method for molecular distribution function in a chemical oscillator.
Nakanishi, Hiizu; Sakaue, Takahiro; Wakou, Jun'ichi
2013-12-01
Using the Hamilton-Jacobi method, we solve chemical Fokker-Planck equations within the Gaussian approximation and obtain a simple and compact formula for a conditional probability distribution. The formula holds in general transient situations, and can be applied not only to a steady state but also to an oscillatory state. By analyzing the long time behavior of the solution in the oscillatory case, we obtain the phase diffusion constant along the periodic orbit and the steady distribution perpendicular to it. A simple method for numerical evaluation of these formulas are devised, and they are compared with Monte Carlo simulations in the case of Brusselator as an example. Some results are shown to be identical to previously obtained expressions. PMID:24320362
Hamilton-Jacobi formalism for inflation with non-minimal derivative coupling
NASA Astrophysics Data System (ADS)
Sheikhahmadi, Haidar; Saridakis, Emmanuel N.; Aghamohammadi, Ali; Saaidi, Khaled
2016-10-01
In inflation with nonminimal derivative coupling there is not a conformal transformation to the Einstein frame where calculations are straightforward, and thus in order to extract inflationary observables one needs to perform a detailed and lengthy perturbation investigation. In this work we bypass this problem by performing a Hamilton-Jacobi analysis, namely rewriting the cosmological equations considering the scalar field to be the time variable. We apply the method to two specific models, namely the power-law and the exponential cases, and for each model we calculate various observables such as the tensor-to-scalar ratio, and the spectral index and its running. We compare them with 2013 and 2015 Planck data, and we show that they are in a very good agreement with observations.
The classical limit of minimal length uncertainty relation: revisit with the Hamilton-Jacobi method
NASA Astrophysics Data System (ADS)
Guo, Xiaobo; Wang, Peng; Yang, Haitang
2016-05-01
The existence of a minimum measurable length could deform not only the standard quantum mechanics but also classical physics. The effects of the minimal length on classical orbits of particles in a gravitation field have been investigated before, using the deformed Poisson bracket or Schwarzschild metric. In this paper, we first use the Hamilton-Jacobi method to derive the deformed equations of motion in the context of Newtonian mechanics and general relativity. We then employ them to study the precession of planetary orbits, deflection of light, and time delay in radar propagation. We also set limits on the deformation parameter by comparing our results with the observational measurements. Finally, comparison with results from previous papers is given at the end of this paper.
Alternative method for Hamilton-Jacobi PDEs in image processing
NASA Astrophysics Data System (ADS)
Lagoutte, A.; Salat, H.; Vachier, C.
2011-03-01
Multiscale signal analysis has been used since the early 1990s as a powerful tool for image processing, notably in the linear case. However, nonlinear PDEs and associated nonlinear operators have advantages over linear operators, notably preserving important features such as edges in images. In this paper, we focus on nonlinear Hamilton-Jacobi PDEs defined with adaptive speeds or, alternatively, on adaptive morphological fiters also called semi-flat morphological operators. Semi-flat morphology were instroduced by H. Heijmans and studied only in the case where the speed (or equivalently the filtering parameter) is a decreasing function of the luminance. It is proposed to extend the definition suggested by H. Heijmans in the case of non decreasing speeds. We also prove that a central property for defining morphological filters, that is the adjunction property, is preserved while dealing with our extended definitions. Finally experimental applications are presented on actual images, including connection of thin lines by semi-flat dilations and image filtering by semi-flat openings.
Quantum streamlines within the complex quantum Hamilton-Jacobi formalism
Chou, C.-C.; Wyatt, Robert E.
2008-09-28
Quantum streamlines are investigated in the framework of the quantum Hamilton-Jacobi formalism. The local structures of the quantum momentum function (QMF) and the Polya vector field near a stagnation point or a pole are analyzed. Streamlines near a stagnation point of the QMF may spiral into or away from it, or they may become circles centered on this point or straight lines. Additionally, streamlines near a pole display east-west and north-south opening hyperbolic structure. On the other hand, streamlines near a stagnation point of the Polya vector field for the QMF display general hyperbolic structure, and streamlines near a pole become circles enclosing the pole. Furthermore, the local structures of the QMF and the Polya vector field around a stagnation point are related to the first derivative of the QMF; however, the magnitude of the asymptotic structures for these two fields near a pole depends only on the order of the node in the wave function. Two nonstationary states constructed from the eigenstates of the harmonic oscillator are used to illustrate the local structures of these two fields and the dynamics of the streamlines near a stagnation point or a pole. This study presents the abundant dynamics of the streamlines in the complex space for one-dimensional time-dependent problems.
Quantum vortices within the complex quantum Hamilton-Jacobi formalism.
Chou, Chia-Chun; Wyatt, Robert E
2008-06-21
Quantum vortices are investigated in the framework of the quantum Hamilton-Jacobi formalism. A quantum vortex forms around a node in the wave function in the complex space, and the quantized circulation integral originates from the discontinuity in the real part of the complex action. Although the quantum momentum field displays hyperbolic flow around a node, the corresponding Polya vector field displays circular flow. It is shown that the Polya vector field of the quantum momentum function is parallel to contours of the probability density. A nonstationary state constructed from eigenstates of the harmonic oscillator is used to illustrate the formation of a transient excited state quantum vortex, and the coupled harmonic oscillator is used to illustrate quantization of the circulation integral in the multidimensional complex space. This study not only analyzes the formation of quantum vortices but also demonstrates the local structures for the quantum momentum field and for the Polya vector field near a node of the wave function. PMID:18570490
Quantum streamlines within the complex quantum Hamilton-Jacobi formalism.
Chou, Chia-Chun; Wyatt, Robert E
2008-09-28
Quantum streamlines are investigated in the framework of the quantum Hamilton-Jacobi formalism. The local structures of the quantum momentum function (QMF) and the Polya vector field near a stagnation point or a pole are analyzed. Streamlines near a stagnation point of the QMF may spiral into or away from it, or they may become circles centered on this point or straight lines. Additionally, streamlines near a pole display east-west and north-south opening hyperbolic structure. On the other hand, streamlines near a stagnation point of the Polya vector field for the QMF display general hyperbolic structure, and streamlines near a pole become circles enclosing the pole. Furthermore, the local structures of the QMF and the Polya vector field around a stagnation point are related to the first derivative of the QMF; however, the magnitude of the asymptotic structures for these two fields near a pole depends only on the order of the node in the wave function. Two nonstationary states constructed from the eigenstates of the harmonic oscillator are used to illustrate the local structures of these two fields and the dynamics of the streamlines near a stagnation point or a pole. This study presents the abundant dynamics of the streamlines in the complex space for one-dimensional time-dependent problems. PMID:19045012
Solving quantum trajectories in Coulomb potential by quantum Hamilton-Jacobi theory
NASA Astrophysics Data System (ADS)
Yang, Ciann-Dong
We show that the quantum central-force problems can be modeled and solved exactly by quantum Hamilton-Jacobi formulation, from which the quantum operators z, 2, and can be derived without using the quantization principle p ? (/i)?/?x. Quantum conservation laws expressed by the Poisson bracket show that the eigenvalues of these quantum operators are just equal to the constants of motion along the eigen-trajectories defined in a complex domain. The shell structure observed in bound systems, such as the hydrogen atom, is found to stem from the structure of the quantum potential, by which the quantum forces acting on the electron can be uniquely determined, the stability of atomic configuration can be justified, and the quantum trajectories of the electron can be obtained by integrating the related quantum Lagrange equations. On solving the quantum equations of motion, the solution of the Schrödinger equation serves as the first integration of the second-order quantum Lagrange equations. The stable equilibrium points of the derived first-order nonlinear quantum dynamics are shown to be identical to the positions with maximum probability predicted by standard quantum mechanics. The internal mechanism of how the quantum dynamics evolve continuously to classical dynamics and of how the quantum conservation laws transit continuously to the classical conservation laws as n ? ? are analyzed in detail. The construction of the quantum scattering trajectory by searching for an unbound solution for the Schrödinger equation is investigated.
NASA Astrophysics Data System (ADS)
Salisbury, Donald; Renn, Jürgen; Sundermeyer, Kurt
2016-02-01
Classical background independence is reflected in Lagrangian general relativity through covariance under the full diffeomorphism group. We show how this independence can be maintained in a Hamilton-Jacobi approach that does not accord special privilege to any geometric structure. Intrinsic space-time curvature-based coordinates grant equal status to all geometric backgrounds. They play an essential role as a starting point for inequivalent semiclassical quantizations. The scheme calls into question Wheeler’s geometrodynamical approach and the associated Wheeler-DeWitt equation in which 3-metrics are featured geometrical objects. The formalism deals with variables that are manifestly invariant under the full diffeomorphism group. Yet, perhaps paradoxically, the liberty in selecting intrinsic coordinates is precisely as broad as is the original diffeomorphism freedom. We show how various ideas from the past five decades concerning the true degrees of freedom of general relativity can be interpreted in light of this new constrained Hamiltonian description. In particular, we show how the Kuchař multi-fingered time approach can be understood as a means of introducing full four-dimensional diffeomorphism invariants. Every choice of new phase space variables yields new Einstein-Hamilton-Jacobi constraining relations, and corresponding intrinsic Schrödinger equations. We show how to implement this freedom by canonical transformation of the intrinsic Hamiltonian. We also reinterpret and rectify significant work by Dittrich on the construction of “Dirac observables.”
Hawking radiation of Kerr-de Sitter black holes using Hamilton-Jacobi method
NASA Astrophysics Data System (ADS)
Ibungochouba Singh, T.; Ablu Meitei, I.; Yugindro Singh, K.
2013-05-01
Hawking radiation of Kerr-de Sitter black hole is investigated using Hamilton-Jacobi method. When the well-behaved Painleve coordinate system and Eddington coordinate are used, we get the correct result of Bekenstein-Hawking entropy before and after radiation but a direct computation will lead to a wrong result via Hamilton-Jacobi method. Our results show that the tunneling probability is related to the change of Bekenstein-Hawking entropy and the derived emission spectrum deviates from the pure thermal but it is consistent with underlying unitary theory.
Gauge symmetry of the N-body problem in the Hamilton-Jacobi approach
NASA Astrophysics Data System (ADS)
Efroimsky, Michael; Goldreich, Peter
2003-12-01
In most books the Delaunay and Lagrange equations for the orbital elements are derived by the Hamilton-Jacobi method: one begins with the two-body Hamilton equations in spherical coordinates, performs a canonical transformation to the orbital elements, and obtains the Delaunay system. A standard trick is then used to generalize the approach to the N-body case. We reexamine this step and demonstrate that it contains an implicit condition which restricts the dynamics to a 9(N-1)-dimensional submanifold of the 12(N-1)-dimensional space spanned by the elements and their time derivatives. The tacit condition is equivalent to the constraint that Lagrange imposed ``by hand'' to remove the excessive freedom, when he was deriving his system of equations by variation of parameters. It is the condition of the orbital elements being osculating, i.e., of the instantaneous ellipse (or hyperbola) being always tangential to the physical velocity. Imposure of any supplementary condition different from the Lagrange constraint (but compatible with the equations of motion) is legitimate and will not alter the physical trajectory or velocity (though will alter the mathematical form of the planetary equations). This freedom of nomination of the supplementary constraint reveals a gauge-type internal symmetry instilled into the equations of celestial mechanics. Existence of this internal symmetry has consequences for the stability of numerical integrators. Another important aspect of this freedom is that any gauge different from that of Lagrange makes the Delaunay system noncanonical. In a more general setting, when the disturbance depends not only upon positions but also upon velocities, there is a ``generalized Lagrange gauge'' wherein the Delaunay system is symplectic. This special gauge renders orbital elements that are osculating in the phase space. It coincides with the regular Lagrange gauge when the perturbation is velocity independent.
NASA Astrophysics Data System (ADS)
Rahman, M. Atiqur; Hossain, M. Ilias
2013-06-01
The massive particles tunneling method has been used to investigate the Hawking non-thermal and purely thermal radiations of Schwarzschild Anti-de Sitter (SAdS) black hole. Considering the spacetime background to be dynamical, incorporate the self-gravitation effect of the emitted particles the imaginary part of the action has been derived from Hamilton-Jacobi equation. Using the conservation laws of energy and angular momentum we have showed that the non-thermal and purely thermal tunneling rates are related to the change of Bekenstein-Hawking entropy and the derived emission spectrum deviates from the pure thermal spectrum. The result obtained for SAdS black hole is also in accordance with Parikh and Wilczek's opinion and gives a correction to the Hawking radiation of SAdS black hole.
Hamilton-Jacobi Ansatz to Study the Hawking Radiation of Kerr-Newman Black Holes
NASA Astrophysics Data System (ADS)
Chen, Deyou; Yang, Shuzheng
Taking the self-gravitation interaction and unfixed background space-time into account, we study the Hawking radiation of Kerr-Newman-Kasuya black holes using Hamilton-Jacobi method. The result shows that the tunneling rate is related to the change of Bekenstein-Hawking entropy and the radiation spectrum deviates from the purely thermal one, which is accordant with that obtained using Parikh and Wilczek's method and gives a correction to the Hawking radiation of the black hole.
Chen, Zheng; Jagannathan, Sarangapani
2008-01-01
In this paper, we consider the use of nonlinear networks towards obtaining nearly optimal solutions to the control of nonlinear discrete-time (DT) systems. The method is based on least squares successive approximation solution of the generalized Hamilton-Jacobi-Bellman (GHJB) equation which appears in optimization problems. Successive approximation using the GHJB has not been applied for nonlinear DT systems. The proposed recursive method solves the GHJB equation in DT on a well-defined region of attraction. The definition of GHJB, pre-Hamiltonian function, HJB equation, and method of updating the control function for the affine nonlinear DT systems under small perturbation assumption are proposed. A neural network (NN) is used to approximate the GHJB solution. It is shown that the result is a closed-loop control based on an NN that has been tuned a priori in offline mode. Numerical examples show that, for the linear DT system, the updated control laws will converge to the optimal control, and for nonlinear DT systems, the updated control laws will converge to the suboptimal control. PMID:18269941
Coordinates Used in Derivation of Hawking Radiation via Hamilton-Jacobi Method
NASA Astrophysics Data System (ADS)
Liu, Bo; He, Xiaokai; Liu, Wenbiao
2009-05-01
Coordinates used in derivation of Hawking radiation via Hamilton-Jacobi method are investigated more deeply. In the case of a 4-dimensional Schwarzschild black hole, a direct computation leads to a wrong result. In the meantime, making use of the isotropic coordinate or invariant radial distance, we can get the correct conclusion. More coordinates including Painleve and Eddington-Finkelstein are tried to calculate the semi-classical Hawking emission rate. The reason of the discrepancy between naive coordinate and well-behaved coordinates is also discussed.
Jeong, Won-Ki; Fletcher, P Thomas; Tao, Ran; Whitaker, Ross
2007-01-01
In this paper we present a method to compute and visualize volumetric white matter connectivity in diffusion tensor magnetic resonance imaging (DT-MRI) using a Hamilton-Jacobi (H-J) solver on the GPU (Graphics Processing Unit). Paths through the volume are assigned costs that are lower if they are consistent with the preferred diffusion directions. The proposed method finds a set of voxels in the DTI volume that contain paths between two regions whose costs are within a threshold of the optimal path. The result is a volumetric optimal path analysis, which is driven by clinical and scientific questions relating to the connectivity between various known anatomical regions of the brain. To solve the minimal path problem quickly, we introduce a novel numerical algorithm for solving H-J equations, which we call the Fast Iterative Method (FIM). This algorithm is well-adapted to parallel architectures, and we present a GPU-based implementation, which runs roughly 50-100 times faster than traditional CPU-based solvers for anisotropic H-J equations. The proposed system allows users to freely change the endpoints of interesting pathways and to visualize the optimal volumetric path between them at an interactive rate. We demonstrate the proposed method on some synthetic and real DT-MRI datasets and compare the performance with existing methods. PMID:17968100
NASA Technical Reports Server (NTRS)
Fitzpatrick, P. M.; Harmon, G. R.; Liu, J. J. F.; Cochran, J. E.
1974-01-01
The formalism for studying perturbations of a triaxial rigid body within the Hamilton-Jacobi framework is developed. The motion of a triaxial artificial earth satellite about its center of mass is studied. Variables are found which permit separation, and the Euler angles and associated conjugate momenta are obtained as functions of canonical constants and time.
A Hamilton-Jacobi-Bellman approach for termination of seizure-like bursting.
Wilson, Dan; Moehlis, Jeff
2014-10-01
We use Hamilton-Jacobi-Bellman methods to find minimum-time and energy-optimal control strategies to terminate seizure-like bursting behavior in a conductance-based neural model. Averaging is used to eliminate fast variables from the model, and a target set is defined through bifurcation analysis of the slow variables of the model. This method is illustrated for a single neuron model and for a network model to illustrate its efficacy in terminating bursting once it begins. This work represents a numerical proof-of-concept that a new class of control strategies can be employed to mitigate bursting, and could ultimately be adapted to treat medically intractible epilepsy in patient-specific models. PMID:24965911
Hamilton-Jacobi tunneling method for dynamical horizons in different coordinate gauges
NASA Astrophysics Data System (ADS)
Di Criscienzo, Roberto; Hayward, Sean A.; Nadalini, Mario; Vanzo, Luciano; Zerbini, Sergio
2010-01-01
Previous work on dynamical black hole instability is further elucidated within the Hamilton-Jacobi method for horizon tunneling and the reconstruction of the classical action by means of the null expansion method. Everything is based on two natural requirements, namely that the tunneling rate is an observable and therefore it must be based on invariantly defined quantities, and that coordinate systems which do not cover the horizon should not be admitted. These simple observations can help to clarify some ambiguities, like the doubling of the temperature occurring in the static case when using singular coordinates and the role, if any, of the temporal contribution of the action to the emission rate. The formalism is also applied to FRW cosmological models, where it is observed that it predicts the positivity of the temperature naturally, without further assumptions on the sign of energy.
NASA Astrophysics Data System (ADS)
Ilias Hossain, M.; Atiqur Rahman, M.
2013-09-01
We have investigated Hawking non-thermal and purely thermal Radiations of Reissner Nordström anti-de Sitter (RNAdS) black hole by massive particles tunneling method. The spacetime background has taken as dynamical, incorporate the self-gravitation effect of the emitted particles the imaginary part of the action has derived from Hamilton-Jacobi equation. We have supposed that energy and angular momentum are conserved and have shown that the non-thermal and thermal tunneling rates are related to the change of Bekenstein-Hawking entropy and the derived emission spectrum deviates from the pure thermal spectrum. The results for RNAdS black hole is also in the same manner with Parikh and Wilczek's opinion and explored the new result for Hawking radiation of RNAdS black hole.
Self-gravitation interaction of IR deformed Hořava-Lifshitz gravity via new Hamilton-Jacobi method
NASA Astrophysics Data System (ADS)
Liu, Molin; Xu, Yin; Lu, Junwang; Yang, Yuling; Lu, Jianbo; Wu, Yabo
2014-06-01
The apparent discovery of logarithmic entropies has a significant impact on IR deformed Hořava-Lifshitz (IRDHL) gravity in which the original infrared (IR) property is improved by introducing three-geometry's Ricci scalar term "μ4 R" in action. Here, we reevaluate the Hawking radiation in IRDHL by using recent new Hamilton-Jacobi method (NHJM). In particular, a thorough analysis is considered both in asymptotically flat Kehagias-Sfetsos and asymptotically non-flat Park models in IRDHL. We find the NHJM offers simplifications on the technical side. The modification in the entropy expression is given by the physical interpretation of self-gravitation of the Hawking radiation in this new Hamilton-Jacobi (HJ) perspectives.
Ordonez-Miranda, J.; Alvarado-Gil, J. J.; Zambrano-Arjona, Miguel A.
2010-02-15
Dual-phase lagging model is one of the most promising approaches to generalize the Fourier heat conduction equation, and it can be reduced in the appropriate limits to the hyperbolic Cattaneo-Vernotte and to the parabolic equations. In this paper it is shown that the Hamilton-Jacobi and quantum theory formulations that have been developed to study the thermal-wave propagation in the Fourier framework can be extended to include the more general approach based on dual-phase lagging. It is shown that the problem of solving the heat conduction equation can be treated as a thermal harmonic oscillator. In the classical approach a formulation in canonical variables is presented. This formalism is used to introduce a quantum mechanical approach from which the expectation values of observables such as the temperature and heat flux are obtained. These formalisms permit to use a methodology that could provide a deeper insight into the phenomena of heat transport at different time scales in media with inhomogeneous thermophysical properties.
Mehraeen, Shahab; Jagannathan, Sarangapani
2011-11-01
In this paper, the direct neural dynamic programming technique is utilized to solve the Hamilton-Jacobi-Bellman equation forward-in-time for the decentralized near optimal regulation of a class of nonlinear interconnected discrete-time systems with unknown internal subsystem and interconnection dynamics, while the input gain matrix is considered known. Even though the unknown interconnection terms are considered weak and functions of the entire state vector, the decentralized control is attempted under the assumption that only the local state vector is measurable. The decentralized nearly optimal controller design for each subsystem consists of two neural networks (NNs), an action NN that is aimed to provide a nearly optimal control signal, and a critic NN which evaluates the performance of the overall system. All NN parameters are tuned online for both the NNs. By using Lyapunov techniques it is shown that all subsystems signals are uniformly ultimately bounded and that the synthesized subsystems inputs approach their corresponding nearly optimal control inputs with bounded error. Simulation results are included to show the effectiveness of the approach. PMID:21965197
Some Inverse Problems in Periodic Homogenization of Hamilton-Jacobi Equations
NASA Astrophysics Data System (ADS)
Luo, Songting; Tran, Hung V.; Yu, Yifeng
2016-09-01
We look at the effective Hamiltonian {overline{H}} associated with the Hamiltonian {H(p,x)=H(p)+V(x)} in the periodic homogenization theory. Our central goal is to understand the relation between {V} and {overline{H}}. We formulate some inverse problems concerning this relation. Such types of inverse problems are, in general, very challenging. In this paper, we discuss several special cases in both convex and nonconvex settings.
NASA Astrophysics Data System (ADS)
Sakalli, I.; Mirekhtiary, S. F.
2013-10-01
Hawking radiation of a non-asymptotically flat 4-dimensional spherically symmetric and static dilatonic black hole (BH) via the Hamilton-Jacobi (HJ) method is studied. In addition to the naive coordinates, we use four more different coordinate systems that are well-behaved at the horizon. Except for the isotropic coordinates, direct computation by the HJ method leads to the standard Hawking temperature for all coordinate systems. The isotropic coordinates allow extracting the index of refraction from the Fermat metric. It is explicitly shown that the index of refraction determines the value of the tunneling rate and its natural consequence, the Hawking temperature. The isotropic coordinates in the conventional HJ method produce a wrong result for the temperature of the linear dilaton. Here, we explain how this discrepancy can be resolved by regularizing the integral possessing a pole at the horizon.
Sakalli, I. Mirekhtiary, S. F.
2013-10-15
Hawking radiation of a non-asymptotically flat 4-dimensional spherically symmetric and static dilatonic black hole (BH) via the Hamilton-Jacobi (HJ) method is studied. In addition to the naive coordinates, we use four more different coordinate systems that are well-behaved at the horizon. Except for the isotropic coordinates, direct computation by the HJ method leads to the standard Hawking temperature for all coordinate systems. The isotropic coordinates allow extracting the index of refraction from the Fermat metric. It is explicitly shown that the index of refraction determines the value of the tunneling rate and its natural consequence, the Hawking temperature. The isotropic coordinates in the conventional HJ method produce a wrong result for the temperature of the linear dilaton. Here, we explain how this discrepancy can be resolved by regularizing the integral possessing a pole at the horizon.
On stochastic diffusion equations and stochastic Burgers' equations
NASA Astrophysics Data System (ADS)
Truman, A.; Zhao, H. Z.
1996-01-01
In this paper we construct a strong solution for the stochastic Hamilton Jacobi equation by using stochastic classical mechanics before the caustics. We thereby obtain the viscosity solution for a certain class of inviscid stochastic Burgers' equations. This viscosity solution is not continuous beyond the caustics of the corresponding Hamilton Jacobi equation. The Hopf-Cole transformation is used to identify the stochastic heat equation and the viscous stochastic Burgers' equation. The exact solutions for the above two equations are given in terms of the stochastic Hamilton Jacobi function under a no-caustic condition. We construct the heat kernel for the stochastic heat equation for zero potentials in hyperbolic space and for harmonic oscillator potentials in Euclidean space thereby obtaining the stochastic Mehler formula.
Existence of a solution to an equation arising from the theory of Mean Field Games
NASA Astrophysics Data System (ADS)
Gangbo, Wilfrid; Święch, Andrzej
2015-12-01
We construct a small time strong solution to a nonlocal Hamilton-Jacobi equation (1.1) introduced in [48], the so-called master equation, originating from the theory of Mean Field Games. We discover a link between metric viscosity solutions to local Hamilton-Jacobi equations studied in [2,19,20] and solutions to (1.1). As a consequence we recover the existence of solutions to the First Order Mean Field Games equations (1.2), first proved in [48], and make a more rigorous connection between the master equation (1.1) and the Mean Field Games equations (1.2).
NASA Astrophysics Data System (ADS)
Sakayanagi, Yoshihiro; Nakaura, Shigeki; Sampei, Mitsuji
The solvable condition of nonlinear H∞ control problems is given by the Hamilton Jacobi inequality (HJI). The state-dependent Riccati inequality (SDRI) is one of the approaches used to solve the HJI. The SDRI contains the state-dependent coefficient (SDC) form of a nonlinear system. The SDC form is not unique. If a poor SDC form is chosen, then there is no solution for the SDRI. In other words, there exist free parameters of the SDC form that affect the solvability of the SDRI. This study focuses on the free parameters of the SDC form. First, a representation of the free parameters of the SDC form is introduced. The solvability of an SDRI is a sufficient condition for that of the related HJI, and the free parameters affect the conservativeness of the SDRI approach. In addition, a new method for designing the free parameters that reduces the conservativeness of the SDRI approach is introduced. Finally, numerical examples to verify the effect of this method are presented.
Semilinear Kolmogorov Equations and Applications to Stochastic Optimal Control
Masiero, Federica
2005-03-15
Semilinear parabolic differential equations are solved in a mild sense in an infinite-dimensional Hilbert space. Applications to stochastic optimal control problems are studied by solving the associated Hamilton-Jacobi-Bellman equation. These results are applied to some controlled stochastic partial differential equations.
Invariant tori of the Poincare return map as solutions of functional difference equations
Warnock, R.L.
1991-01-01
Functional difference equations characterize the invariant surfaces of the Poincare return map of a general Hamiltonian system. Two different functional equations are derived. The first is analogous to the Hamilton-Jacobi equation and the second is a generalization of Moser's equation. Some properties of the equations, and schemes for solving them numerically, are discussed. 7 refs., 1 fig.
NASA Astrophysics Data System (ADS)
Villalba, Víctor M.; Catalá, Esteban Isasi
2002-10-01
We solve the Klein-Gordon and Dirac equations in an open cosmological universe with a partially horn topology in the presence of a time dependent magnetic field. Since the exact solution cannot be obtained explicitly for arbitrary time dependence of the field, we discuss the asymptotic behavior of the solutions with the help of the relativistic Hamilton-Jacobi equation.
Transport Equation Based Wall Distance Computations Aimed at Flows With Time-Dependent Geometry
NASA Technical Reports Server (NTRS)
Tucker, Paul G.; Rumsey, Christopher L.; Bartels, Robert E.; Biedron, Robert T.
2003-01-01
Eikonal, Hamilton-Jacobi and Poisson equations can be used for economical nearest wall distance computation and modification. Economical computations may be especially useful for aeroelastic and adaptive grid problems for which the grid deforms, and the nearest wall distance needs to be repeatedly computed. Modifications are directed at remedying turbulence model defects. For complex grid structures, implementation of the Eikonal and Hamilton-Jacobi approaches is not straightforward. This prohibits their use in industrial CFD solvers. However, both the Eikonal and Hamilton-Jacobi equations can be written in advection and advection-diffusion forms, respectively. These, like the Poisson s Laplacian, are commonly occurring industrial CFD solver elements. Use of the NASA CFL3D code to solve the Eikonal and Hamilton-Jacobi equations in advective-based forms is explored. The advection-based distance equations are found to have robust convergence. Geometries studied include single and two element airfoils, wing body and double delta configurations along with a complex electronics system. It is shown that for Eikonal accuracy, upwind metric differences are required. The Poisson approach is found effective and, since it does not require offset metric evaluations, easiest to implement. The sensitivity of flow solutions to wall distance assumptions is explored. Generally, results are not greatly affected by wall distance traits.
Transport Equation Based Wall Distance Computations Aimed at Flows With Time-Dependent Geometry
NASA Technical Reports Server (NTRS)
Tucker, Paul G.; Rumsey, Christopher L.; Bartels, Robert E.; Biedron, Robert T.
2003-01-01
Eikonal, Hamilton-Jacobi and Poisson equations can be used for economical nearest wall distance computation and modification. Economical computations may be especially useful for aeroelastic and adaptive grid problems for which the grid deforms, and the nearest wall distance needs to be repeatedly computed. Modifications are directed at remedying turbulence model defects. For complex grid structures, implementation of the Eikonal and Hamilton-Jacobi approaches is not straightforward. This prohibits their use in industrial CFD solvers. However, both the Eikonal and Hamilton-Jacobi equations can be written in advection and advection-diffusion forms, respectively. These, like the Poisson's Laplacian, are commonly occurring industrial CFD solver elements. Use of the NASA CFL3D code to solve the Eikonal and Hamilton-Jacobi equations in advective-based forms is explored. The advection-based distance equations are found to have robust convergence. Geometries studied include single and two element airfoils, wing body and double delta configurations along with a complex electronics system. It is shown that for Eikonal accuracy, upwind metric differences are required. The Poisson approach is found effective and, since it does not require offset metric evaluations, easiest to implement. The sensitivity of flow solutions to wall distance assumptions is explored. Generally, results are not greatly affected by wall distance traits.
Computations of Wall Distances Based on Differential Equations
NASA Technical Reports Server (NTRS)
Tucker, Paul G.; Rumsey, Chris L.; Spalart, Philippe R.; Bartels, Robert E.; Biedron, Robert T.
2004-01-01
The use of differential equations such as Eikonal, Hamilton-Jacobi and Poisson for the economical calculation of the nearest wall distance d, which is needed by some turbulence models, is explored. Modifications that could palliate some turbulence-modeling anomalies are also discussed. Economy is of especial value for deforming/adaptive grid problems. For these, ideally, d is repeatedly computed. It is shown that the Eikonal and Hamilton-Jacobi equations can be easy to implement when written in implicit (or iterated) advection and advection-diffusion equation analogous forms, respectively. These, like the Poisson Laplacian term, are commonly occurring in CFD solvers, allowing the re-use of efficient algorithms and code components. The use of the NASA CFL3D CFD program to solve the implicit Eikonal and Hamilton-Jacobi equations is explored. The re-formulated d equations are easy to implement, and are found to have robust convergence. For accurate Eikonal solutions, upwind metric differences are required. The Poisson approach is also found effective, and easiest to implement. Modified distances are not found to affect global outputs such as lift and drag significantly, at least in common situations such as airfoil flows.
Schrödinger equation revisited.
Schleich, Wolfgang P; Greenberger, Daniel M; Kobe, Donald H; Scully, Marlan O
2013-04-01
The time-dependent Schrödinger equation is a cornerstone of quantum physics and governs all phenomena of the microscopic world. However, despite its importance, its origin is still not widely appreciated and properly understood. We obtain the Schrödinger equation from a mathematical identity by a slight generalization of the formulation of classical statistical mechanics based on the Hamilton-Jacobi equation. This approach brings out most clearly the fact that the linearity of quantum mechanics is intimately connected to the strong coupling between the amplitude and phase of a quantum wave. PMID:23509260
Equations of motion in general relativity and quantum mechanics
NASA Astrophysics Data System (ADS)
O'Hara, Paul
2011-12-01
In a previous article a relationship was established between the linearized metrics of General Relativity associated with geodesics and the Dirac Equation of quantum mechanics. In this paper the extension of that result to arbitrary curves is investigated. A generalized Dirac equation is derived and shown to be related to the Lie derivative of the momentum along the curve. In addition,the equations of motion are derived from the Hamilton-Jacobi equation associated with the metric and the wave equation associated with the Hamiltonian is then shown not to commute with the Dirac operator. Finally, the Maxwell-Boltzmann distribution is shown to be a consequence of geodesic motion.
Dynamics of the chemical master equation, a strip of chains of equations in d-dimensional space.
Galstyan, Vahe; Saakian, David B
2012-07-01
We investigate the multichain version of the chemical master equation, when there are transitions between different states inside the long chains, as well as transitions between (a few) different chains. In the discrete version, such a model can describe the connected diffusion processes with jumps between different types. We apply the Hamilton-Jacobi equation to solve some aspects of the model. We derive exact (in the limit of infinite number of particles) results for the dynamic of the maximum of the distribution and the variance of distribution.
Dynamics of the chemical master equation, a strip of chains of equations in d-dimensional space.
Galstyan, Vahe; Saakian, David B
2012-07-01
We investigate the multichain version of the chemical master equation, when there are transitions between different states inside the long chains, as well as transitions between (a few) different chains. In the discrete version, such a model can describe the connected diffusion processes with jumps between different types. We apply the Hamilton-Jacobi equation to solve some aspects of the model. We derive exact (in the limit of infinite number of particles) results for the dynamic of the maximum of the distribution and the variance of distribution. PMID:23005386
Modifications of the PCPT method for HJB equations
NASA Astrophysics Data System (ADS)
Kossaczký, I.; Ehrhardt, M.; Günther, M.
2016-10-01
In this paper we will revisit the modification of the piecewise constant policy timestepping (PCPT) method for solving Hamilton-Jacobi-Bellman (HJB) equations. This modification is called piecewise predicted policy timestepping (PPPT) method and if properly used, it may be significantly faster. We will quickly recapitulate the algorithms of PCPT, PPPT methods and of the classical implicit method and apply them on a passport option pricing problem with non-standard payoff. We will present modifications needed to solve this problem effectively with the PPPT method and compare the performance with the PCPT method and the classical implicit method.
State-Constrained Optimal Control Problems of Impulsive Differential Equations
Forcadel, Nicolas; Rao Zhiping Zidani, Hasnaa
2013-08-01
The present paper studies an optimal control problem governed by measure driven differential systems and in presence of state constraints. The first result shows that using the graph completion of the measure, the optimal solutions can be obtained by solving a reparametrized control problem of absolutely continuous trajectories but with time-dependent state-constraints. The second result shows that it is possible to characterize the epigraph of the reparametrized value function by a Hamilton-Jacobi equation without assuming any controllability assumption.
Fuhrman, Marco Tessitore, Gianmario
2005-05-15
We study a forward-backward system of stochastic differential equations in an infinite-dimensional framework and its relationships with a semilinear parabolic differential equation on a Hilbert space, in the spirit of the approach of Pardoux-Peng. We prove that the stochastic system allows us to construct a unique solution of the parabolic equation in a suitable class of locally Lipschitz real functions. The parabolic equation is understood in a mild sense which requires the notion of a generalized directional gradient, that we introduce by a probabilistic approach and prove to exist for locally Lipschitz functions.The use of the generalized directional gradient allows us to cover various applications to option pricing problems and to optimal stochastic control problems (including control of delay equations and reaction-diffusion equations),where the lack of differentiability of the coefficients precludes differentiability of solutions to the associated parabolic equations of Black-Scholes or Hamilton-Jacobi-Bellman type.
First-order partial differential equations in classical dynamics
NASA Astrophysics Data System (ADS)
Smith, B. R.
2009-12-01
Carathèodory's classic work on the calculus of variations explores in depth the connection between ordinary differential equations and first-order partial differential equations. The n second-order ordinary differential equations of a classical dynamical system reduce to a single first-order differential equation in 2n independent variables. The general solution of first-order partial differential equations touches on many concepts central to graduate-level courses in analytical dynamics including the Hamiltonian, Lagrange and Poisson brackets, and the Hamilton-Jacobi equation. For all but the simplest dynamical systems the solution requires one or more of these techniques. Three elementary dynamical problems (uniform acceleration, harmonic motion, and cyclotron motion) can be solved directly from the appropriate first-order partial differential equation without the use of advanced methods. The process offers an unusual perspective on classical dynamics, which is readily accessible to intermediate students who are not yet fully conversant with advanced approaches.
Sergyeyev, Artur; Krtous, Pavel
2008-02-15
We consider the Klein-Gordon equation in generalized higher-dimensional Kerr-NUT-(A)dS spacetime without imposing any restrictions on the functional parameters characterizing the metric. We establish commutativity of the second-order operators constructed from the Killing tensors found in [J. High Energy Phys. 02 (2007) 004] and show that these operators, along with the first-order operators originating from the Killing vectors, form a complete set of commuting symmetry operators (i.e., integrals of motion) for the Klein-Gordon equation. Moreover, we demonstrate that the separated solutions of the Klein-Gordon equation obtained in [J. High Energy Phys. 02 (2007) 005] are joint eigenfunctions for all of these operators. We also present an explicit form of the zero mode for the Klein-Gordon equation with zero mass. In the semiclassical approximation we find that the separated solutions of the Hamilton-Jacobi equation for geodesic motion are also solutions for a set of Hamilton-Jacobi-type equations which correspond to the quadratic conserved quantities arising from the above Killing tensors.
Trajectory approach to the Schrödinger-Langevin equation with linear dissipation for ground states
NASA Astrophysics Data System (ADS)
Chou, Chia-Chun
2015-11-01
The Schrödinger-Langevin equation with linear dissipation is integrated by propagating an ensemble of Bohmian trajectories for the ground state of quantum systems. Substituting the wave function expressed in terms of the complex action into the Schrödinger-Langevin equation yields the complex quantum Hamilton-Jacobi equation with linear dissipation. We transform this equation into the arbitrary Lagrangian-Eulerian version with the grid velocity matching the flow velocity of the probability fluid. The resulting equation is simultaneously integrated with the trajectory guidance equation. Then, the computational method is applied to the harmonic oscillator, the double well potential, and the ground vibrational state of methyl iodide. The excellent agreement between the computational and the exact results for the ground state energies and wave functions shows that this study provides a synthetic trajectory approach to the ground state of quantum systems.
Martirosyan, A; Saakian, David B
2011-08-01
We apply the Hamilton-Jacobi equation (HJE) formalism to solve the dynamics of the chemical master equation (CME). We found exact analytical expressions (in large system-size limit) for the probability distribution, including explicit expression for the dynamics of variance of distribution. We also give the solution for some simple cases of the model with time-dependent rates. We derived the results of the Van Kampen method from the HJE approach using a special ansatz. Using the Van Kampen method, we give a system of ordinary differential equations (ODEs) to define the variance in a two-dimensional case. We performed numerics for the CME with stationary noise. We give analytical criteria for the disappearance of bistability in the case of stationary noise in one-dimensional CMEs.
Martirosyan, A; Saakian, David B
2011-08-01
We apply the Hamilton-Jacobi equation (HJE) formalism to solve the dynamics of the chemical master equation (CME). We found exact analytical expressions (in large system-size limit) for the probability distribution, including explicit expression for the dynamics of variance of distribution. We also give the solution for some simple cases of the model with time-dependent rates. We derived the results of the Van Kampen method from the HJE approach using a special ansatz. Using the Van Kampen method, we give a system of ordinary differential equations (ODEs) to define the variance in a two-dimensional case. We performed numerics for the CME with stationary noise. We give analytical criteria for the disappearance of bistability in the case of stationary noise in one-dimensional CMEs. PMID:21928964
Reinforcement learning solution for HJB equation arising in constrained optimal control problem.
Luo, Biao; Wu, Huai-Ning; Huang, Tingwen; Liu, Derong
2015-11-01
The constrained optimal control problem depends on the solution of the complicated Hamilton-Jacobi-Bellman equation (HJBE). In this paper, a data-based off-policy reinforcement learning (RL) method is proposed, which learns the solution of the HJBE and the optimal control policy from real system data. One important feature of the off-policy RL is that its policy evaluation can be realized with data generated by other behavior policies, not necessarily the target policy, which solves the insufficient exploration problem. The convergence of the off-policy RL is proved by demonstrating its equivalence to the successive approximation approach. Its implementation procedure is based on the actor-critic neural networks structure, where the function approximation is conducted with linearly independent basis functions. Subsequently, the convergence of the implementation procedure with function approximation is also proved. Finally, its effectiveness is verified through computer simulations.
Reinforcement learning solution for HJB equation arising in constrained optimal control problem.
Luo, Biao; Wu, Huai-Ning; Huang, Tingwen; Liu, Derong
2015-11-01
The constrained optimal control problem depends on the solution of the complicated Hamilton-Jacobi-Bellman equation (HJBE). In this paper, a data-based off-policy reinforcement learning (RL) method is proposed, which learns the solution of the HJBE and the optimal control policy from real system data. One important feature of the off-policy RL is that its policy evaluation can be realized with data generated by other behavior policies, not necessarily the target policy, which solves the insufficient exploration problem. The convergence of the off-policy RL is proved by demonstrating its equivalence to the successive approximation approach. Its implementation procedure is based on the actor-critic neural networks structure, where the function approximation is conducted with linearly independent basis functions. Subsequently, the convergence of the implementation procedure with function approximation is also proved. Finally, its effectiveness is verified through computer simulations. PMID:26356598
A wave equation interpolating between classical and quantum mechanics
NASA Astrophysics Data System (ADS)
Schleich, W. P.; Greenberger, D. M.; Kobe, D. H.; Scully, M. O.
2015-10-01
We derive a ‘master’ wave equation for a family of complex-valued waves {{Φ }}\\equiv R{exp}[{{{i}}S}({cl)}/{{\\hbar }}] whose phase dynamics is dictated by the Hamilton-Jacobi equation for the classical action {S}({cl)}. For a special choice of the dynamics of the amplitude R which eliminates all remnants of classical mechanics associated with {S}({cl)} our wave equation reduces to the Schrödinger equation. In this case the amplitude satisfies a Schrödinger equation analogous to that of a charged particle in an electromagnetic field where the roles of the scalar and the vector potentials are played by the classical energy and the momentum, respectively. In general this amplitude is complex and thereby creates in addition to the classical phase {S}({cl)}/{{\\hbar }} a quantum phase. Classical statistical mechanics, as described by a classical matter wave, follows from our wave equation when we choose the dynamics of the amplitude such that it remains real for all times. Our analysis shows that classical and quantum matter waves are distinguished by two different choices of the dynamics of their amplitudes rather than two values of Planck’s constant. We dedicate this paper to the memory of Richard Lewis Arnowitt—a pioneer of many-body theory, a path finder at the interface of gravity and quantum mechanics, and a true leader in non-relativistic and relativistic quantum field theory.
Computational complexities and storage requirements of some Riccati equation solvers
NASA Technical Reports Server (NTRS)
Utku, Senol; Garba, John A.; Ramesh, A. V.
1989-01-01
The linear optimal control problem of an nth-order time-invariant dynamic system with a quadratic performance functional is usually solved by the Hamilton-Jacobi approach. This leads to the solution of the differential matrix Riccati equation with a terminal condition. The bulk of the computation for the optimal control problem is related to the solution of this equation. There are various algorithms in the literature for solving the matrix Riccati equation. However, computational complexities and storage requirements as a function of numbers of state variables, control variables, and sensors are not available for all these algorithms. In this work, the computational complexities and storage requirements for some of these algorithms are given. These expressions show the immensity of the computational requirements of the algorithms in solving the Riccati equation for large-order systems such as the control of highly flexible space structures. The expressions are also needed to compute the speedup and efficiency of any implementation of these algorithms on concurrent machines.
Symplectically invariant flow equations for N = 2, D = 4 gauged supergravity with hypermultiplets
NASA Astrophysics Data System (ADS)
Klemm, Dietmar; Petri, Nicolò; Rabbiosi, Marco
2016-04-01
We consider N = 2 supergravity in four dimensions, coupled to an arbitrary number of vector- and hypermultiplets, where abelian isometries of the quaternionic hyperscalar target manifold are gauged. Using a static and spherically or hyperbolically symmetric ansatz for the fields, a one-dimensional effective action is derived whose variation yields all the equations of motion. By imposing a sort of Dirac charge quantization condition, one can express the complete scalar potential in terms of a superpotential and write the action as a sum of squares. This leads to first-order flow equations, that imply the second-order equations of motion. The first-order flow turns out to be driven by Hamilton's characteristic function in the Hamilton-Jacobi formalism, and contains among other contributions the superpotential of the scalars. We then include also magnetic gaugings and generalize the flow equations to a symplectically covariant form. Moreover, by rotating the charges in an appropriate way, an alternative set of non-BPS first-order equations is obtained that corresponds to a different squaring of the action. Finally, we use our results to derive the attractor equations for near-horizon geometries of extremal black holes.
Front Quenching in the G-equation Model Induced by Straining of Cellular Flow
NASA Astrophysics Data System (ADS)
Xin, Jack; Yu, Yifeng
2014-10-01
We study homogenization of the G-equation with a flow straining term (or the strain G-equation) in two dimensional periodic cellular flow. The strain G-equation is a highly non-coercive and non-convex level set Hamilton-Jacobi equation. The main objective is to investigate how the flow induced straining (the nonconvex term) influences front propagation as the flow intensity A increases. Three distinct regimes are identified. When A is below the critical level, homogenization holds and the turbulent flame speed s T (effective Hamiltonian) is well-defined for any periodic flow with small divergence and is enhanced by the cellular flow as s T ≧ O( A/log A). In the second regime where A is slightly above the critical value, homogenization breaks down, and s T is not well-defined along any direction. Solutions become a mixture of a fast moving part and a stagnant part. When A is sufficiently large, the whole flame front ceases to propagate forward due to the flow induced straining. In particular, along directions p = (±1, 0) and (0, ±1), s T is well-defined again with a value of zero (trapping). A partial homogenization result is also proved. If we consider a similar but relatively simpler Hamiltonian, the trapping occurs along all directions. The analysis is based on the two-player differential game representation of solutions, selection of game strategies and trapping regions, and construction of connecting trajectories.
Hawking radiation of Schwarzschild-de Sitter black hole by Hamilton-Jacobi method
NASA Astrophysics Data System (ADS)
Rahman, M. Atiqur; Hossain, M. Ilias
2012-05-01
We investigate the Hawking radiation of Schwarzschild-de Sitter (SdS) black hole by massive particles tunneling method. We consider the spacetime background to be dynamical, incorporate the self-gravitation effect of the emitted particles and show that the tunneling rate is related to the change of Bekenstein-Hawking entropy and the derived emission spectrum deviates from the pure thermal spectrum when energy and angular momentum are conserved. Our result is also in accordance with Parikh and Wilczek's opinion and gives a correction to the Hawking radiation of SdS black hole.
Hassouna, M Sabry; Farag, A A
2007-09-01
A wide range of computer vision applications require an accurate solution of a particular Hamilton- Jacobi (HJ) equation, known as the Eikonal equation. In this paper, we propose an improved version of the fast marching method (FMM) that is highly accurate for both 2D and 3D Cartesian domains. The new method is called multi-stencils fast marching (MSFM), which computes the solution at each grid point by solving the Eikonal equation along several stencils and then picks the solution that satisfies the upwind condition. The stencils are centered at each grid point and cover its entire nearest neighbors. In 2D space, 2 stencils cover the 8-neighbors of the point, while in 3D space, 6 stencils cover its 26-neighbors. For those stencils that are not aligned with the natural coordinate system, the Eikonal equation is derived using directional derivatives and then solved using higher order finite difference schemes. The accuracy of the proposed method over the state-of-the-art FMM-based techniques has been demonstrated through comprehensive numerical experiments.
Least squares solutions of the HJB equation with neural network value-function approximators.
Tassa, Yuval; Erez, Tom
2007-07-01
In this paper, we present an empirical study of iterative least squares minimization of the Hamilton-Jacobi-Bellman (HJB) residual with a neural network (NN) approximation of the value function. Although the nonlinearities in the optimal control problem and NN approximator preclude theoretical guarantees and raise concerns of numerical instabilities, we present two simple methods for promoting convergence, the effectiveness of which is presented in a series of experiments. The first method involves the gradual increase of the horizon time scale, with a corresponding gradual increase in value function complexity. The second method involves the assumption of stochastic dynamics which introduces a regularizing second derivative term to the HJB equation. A gradual reduction of this term provides further stabilization of the convergence. We demonstrate the solution of several problems, including the 4-D inverted-pendulum system with bounded control. Our approach requires no initial stabilizing policy or any restrictive assumptions on the plant or cost function, only knowledge of the plant dynamics. In the Appendix, we provide the equations for first- and second-order differential backpropagation. PMID:17668659
From a Mechanical Lagrangian to the Schrödinger Equation
NASA Astrophysics Data System (ADS)
Bouda, A.
In the one-dimensional stationary case, we construct a mechanical Lagrangian describing the quantum motion of a nonrelativistic spinless system. This Lagrangian is written as a difference between a function T, which represents the quantum generalization of the kinetic energy and which depends on the coordinate x and the temporal derivatives of x up the third order, and the classical potential V(x). The Hamiltonian is then constructed and the corresponding canonical equations are deduced. The function T is first assumed to be arbitrary. The development of T in a power series together with the dimensional analysis allow us to fix univocally the series coefficients by requiring that the well-known quantum stationary Hamilton Jacobi equation be reproduced. As a consequence of this approach, we formulate the law of the quantum motion representing a new version of the quantum Newton law. We also analytically establish the famous Bohm relation μ ˙ {x}=∂ S0/∂ x outside the framework of the hydrodynamical approach and show that the well-known quantum potential, although it is a part of the kinetic term, plays really the role of an additional potential as assumed by Bohm.
Least squares solutions of the HJB equation with neural network value-function approximators.
Tassa, Yuval; Erez, Tom
2007-07-01
In this paper, we present an empirical study of iterative least squares minimization of the Hamilton-Jacobi-Bellman (HJB) residual with a neural network (NN) approximation of the value function. Although the nonlinearities in the optimal control problem and NN approximator preclude theoretical guarantees and raise concerns of numerical instabilities, we present two simple methods for promoting convergence, the effectiveness of which is presented in a series of experiments. The first method involves the gradual increase of the horizon time scale, with a corresponding gradual increase in value function complexity. The second method involves the assumption of stochastic dynamics which introduces a regularizing second derivative term to the HJB equation. A gradual reduction of this term provides further stabilization of the convergence. We demonstrate the solution of several problems, including the 4-D inverted-pendulum system with bounded control. Our approach requires no initial stabilizing policy or any restrictive assumptions on the plant or cost function, only knowledge of the plant dynamics. In the Appendix, we provide the equations for first- and second-order differential backpropagation.
Wu, Huai-Ning; Luo, Biao
2012-12-01
It is well known that the nonlinear H∞ state feedback control problem relies on the solution of the Hamilton-Jacobi-Isaacs (HJI) equation, which is a nonlinear partial differential equation that has proven to be impossible to solve analytically. In this paper, a neural network (NN)-based online simultaneous policy update algorithm (SPUA) is developed to solve the HJI equation, in which knowledge of internal system dynamics is not required. First, we propose an online SPUA which can be viewed as a reinforcement learning technique for two players to learn their optimal actions in an unknown environment. The proposed online SPUA updates control and disturbance policies simultaneously; thus, only one iterative loop is needed. Second, the convergence of the online SPUA is established by proving that it is mathematically equivalent to Newton's method for finding a fixed point in a Banach space. Third, we develop an actor-critic structure for the implementation of the online SPUA, in which only one critic NN is needed for approximating the cost function, and a least-square method is given for estimating the NN weight parameters. Finally, simulation studies are provided to demonstrate the effectiveness of the proposed algorithm. PMID:24808144
NASA Astrophysics Data System (ADS)
Ayissi, Raoul Domingo; Noutchegueme, Norbert; Etoua, Remy Magloire; Tchagna, Hugues Paulin Mbeutcha
2015-09-01
Recently in 2005, Briani and Rampazzo (Nonlinear Differ Equ Appl 12:71-91, 2005) gave, using results of Crandall and Lions (Ill J Math 31:665-688, 1987), Ishii (Indiana Univ Math J 33: 721-748, 1984, Bull Fac Sci Eng 28: 33-77, 1985) and Ley (Adv Diff Equ 6:547-576, 2001) a density approach to Hamilton-Jacobi equations with t-measurable Hamiltonians. In this paper we show, using an important result of Briani and Rampazzo (Nonlinear Differ Equ Appl 12:71-91, 2005) the existence and uniqueness of viscosity solutions to the one-body Liouville relativistic equation in Yang-Mills charged Bianchi space times with non-zero mass. To our knowledge, the method used here is original and thus, totally different from those used in Alves (C R Acad Sci Paris Sér A 278:1151-1154, 1975), Choquet-Bruhat and Noutchegueme (C R Acad Sci Paris Sér I 311, 1973), Choquet-Bruhat and Noutchegueme (Ann Inst Henri Poincaré 55:759-787, 1991), Choquet-Bruhat and Noutchegueme (Pitman Res Notes Math Ser 253:52-71, 1992), Noutchegueme and Noundjeu (Ann Inst Henri Poincaré 1:385-404, 2000), Wollman (J Math Anal Appl 127:103-121, 1987) and Choquet-Bruhat (Existence and uniqueness for the Einstein-Maxwell-Liouville system. Volume dedicated to Petrov, Moscow, 1971) who have studied the same equation.
NASA Astrophysics Data System (ADS)
Gondran, Michel; Gondran, Alexandre
2012-03-01
We study, in the semi-classical approximation, the convergence of the quantum density and the quantum action, solutions to the Madelung equations, when the Planck constant h tends to 0. We find two different solutions which depend on the initial density . In the first case where the initial quantum density is a classical density ρ0(X), the quantum density and the quantum action converge to a classical action and a classical density which satisfy the statistical Hamilton-Jacobi equations. These are the equations of a set of classical particles whose initial positions are known only by the density ρ0(X). In the second case, where initial density converges to a Dirac density, the density converges to the Dirac function which corresponds to a unique classical trajectory. Therefore we introduce into classical mechanics non-discerned particles (case 1), which explain the Gibbs paradox, and discerned particles (case 2). Finally, we deduce a quantum mechanics interpretation which depends on the initial conditions (preparation), the Broglie-Bohm interpretation in the first case and the Schrödinger interpretation in the second case.
A FAST ITERATIVE METHOD FOR SOLVING THE EIKONAL EQUATION ON TRIANGULATED SURFACES.
Fu, Zhisong; Jeong, Won-Ki; Pan, Yongsheng; Kirby, Robert M; Whitaker, Ross T
2011-01-01
This paper presents an efficient, fine-grained parallel algorithm for solving the Eikonal equation on triangular meshes. The Eikonal equation, and the broader class of Hamilton-Jacobi equations to which it belongs, have a wide range of applications from geometric optics and seismology to biological modeling and analysis of geometry and images. The ability to solve such equations accurately and efficiently provides new capabilities for exploring and visualizing parameter spaces and for solving inverse problems that rely on such equations in the forward model. Efficient solvers on state-of-the-art, parallel architectures require new algorithms that are not, in many cases, optimal, but are better suited to synchronous updates of the solution. In previous work [W. K. Jeong and R. T. Whitaker, SIAM J. Sci. Comput., 30 (2008), pp. 2512-2534], the authors proposed the fast iterative method (FIM) to efficiently solve the Eikonal equation on regular grids. In this paper we extend the fast iterative method to solve Eikonal equations efficiently on triangulated domains on the CPU and on parallel architectures, including graphics processors. We propose a new local update scheme that provides solutions of first-order accuracy for both architectures. We also propose a novel triangle-based update scheme and its corresponding data structure for efficient irregular data mapping to parallel single-instruction multiple-data (SIMD) processors. We provide detailed descriptions of the implementations on a single CPU, a multicore CPU with shared memory, and SIMD architectures with comparative results against state-of-the-art Eikonal solvers. PMID:22641200
Alarcón, Tomás
2014-05-14
In this paper, we propose two methods to carry out the quasi-steady state approximation in stochastic models of enzyme catalytic regulation, based on WKB asymptotics of the chemical master equation or of the corresponding partial differential equation for the generating function. The first of the methods we propose involves the development of multiscale generalisation of a WKB approximation of the solution of the master equation, where the separation of time scales is made explicit which allows us to apply the quasi-steady state approximation in a straightforward manner. To the lowest order, the multi-scale WKB method provides a quasi-steady state, Gaussian approximation of the probability distribution. The second method is based on the Hamilton-Jacobi representation of the stochastic process where, as predicted by large deviation theory, the solution of the partial differential equation for the corresponding characteristic function is given in terms of an effective action functional. The optimal transition paths between two states are then given by those paths that maximise the effective action. Such paths are the solutions of the Hamilton equations for the Hamiltonian associated to the effective action functional. The quasi-steady state approximation is applied to the Hamilton equations thus providing an approximation to the optimal transition paths and the transition time between two states. Using this approximation we predict that, unlike the mean-field quasi-steady approximation result, the rate of enzyme catalysis depends explicitly on the initial number of enzyme molecules. The accuracy and validity of our approximated results as well as that of our predictions regarding the behaviour of the stochastic enzyme catalytic models are verified by direct simulation of the stochastic model using Gillespie stochastic simulation algorithm.
Alarcón, Tomás
2014-05-14
In this paper, we propose two methods to carry out the quasi-steady state approximation in stochastic models of enzyme catalytic regulation, based on WKB asymptotics of the chemical master equation or of the corresponding partial differential equation for the generating function. The first of the methods we propose involves the development of multiscale generalisation of a WKB approximation of the solution of the master equation, where the separation of time scales is made explicit which allows us to apply the quasi-steady state approximation in a straightforward manner. To the lowest order, the multi-scale WKB method provides a quasi-steady state, Gaussian approximation of the probability distribution. The second method is based on the Hamilton-Jacobi representation of the stochastic process where, as predicted by large deviation theory, the solution of the partial differential equation for the corresponding characteristic function is given in terms of an effective action functional. The optimal transition paths between two states are then given by those paths that maximise the effective action. Such paths are the solutions of the Hamilton equations for the Hamiltonian associated to the effective action functional. The quasi-steady state approximation is applied to the Hamilton equations thus providing an approximation to the optimal transition paths and the transition time between two states. Using this approximation we predict that, unlike the mean-field quasi-steady approximation result, the rate of enzyme catalysis depends explicitly on the initial number of enzyme molecules. The accuracy and validity of our approximated results as well as that of our predictions regarding the behaviour of the stochastic enzyme catalytic models are verified by direct simulation of the stochastic model using Gillespie stochastic simulation algorithm. PMID:24832255
Alarcón, Tomás
2014-05-14
In this paper, we propose two methods to carry out the quasi-steady state approximation in stochastic models of enzyme catalytic regulation, based on WKB asymptotics of the chemical master equation or of the corresponding partial differential equation for the generating function. The first of the methods we propose involves the development of multiscale generalisation of a WKB approximation of the solution of the master equation, where the separation of time scales is made explicit which allows us to apply the quasi-steady state approximation in a straightforward manner. To the lowest order, the multi-scale WKB method provides a quasi-steady state, Gaussian approximation of the probability distribution. The second method is based on the Hamilton-Jacobi representation of the stochastic process where, as predicted by large deviation theory, the solution of the partial differential equation for the corresponding characteristic function is given in terms of an effective action functional. The optimal transition paths between two states are then given by those paths that maximise the effective action. Such paths are the solutions of the Hamilton equations for the Hamiltonian associated to the effective action functional. The quasi-steady state approximation is applied to the Hamilton equations thus providing an approximation to the optimal transition paths and the transition time between two states. Using this approximation we predict that, unlike the mean-field quasi-steady approximation result, the rate of enzyme catalysis depends explicitly on the initial number of enzyme molecules. The accuracy and validity of our approximated results as well as that of our predictions regarding the behaviour of the stochastic enzyme catalytic models are verified by direct simulation of the stochastic model using Gillespie stochastic simulation algorithm.
Relative motion of formation flying satellites in elliptic orbit
NASA Astrophysics Data System (ADS)
Owis, Ashraf
2012-07-01
The work presents a solution of the relative motion of formation flying satellites using the feedback optimal control approach. to obtain such a solution, the Taschauner-Hempel equations are used and techniques of feedback control via solving the Hamilton-Jacobi- Bellman equation. A generating function technique will be implemented to solvethe Hamilton-Jacobi- Bellman equation. we will find the solutions for both soft and hard constraints.
A Student's Guide to Lagrangians and Hamiltonians
NASA Astrophysics Data System (ADS)
Hamill, Patrick
2013-11-01
Part I. Lagrangian Mechanics: 1. Fundamental concepts; 2. The calculus of variations; 3. Lagrangian dynamics; Part II. Hamiltonian Mechanics: 4. Hamilton's equations; 5. Canonical transformations: Poisson brackets; 6. Hamilton-Jacobi theory; 7. Continuous systems; Further reading; Index.
A numerical study of turbulent flame speeds of curvature and strain G-equations in cellular flows
NASA Astrophysics Data System (ADS)
Liu, Yu-Yu; Xin, Jack; Yu, Yifeng
2013-01-01
We study front speeds of curvature and strain G-equations arising in turbulent combustion. These G-equations are Hamilton-Jacobi type level set partial differential equations (PDEs) with non-coercive Hamiltonians and degenerate nonlinear second order diffusion. The Hamiltonian of a strain G-equation is also non-convex. Numerical computation is performed based on monotone discretization and weighted essentially nonoscillatory (WENO) approximation of transformed G-equations on a fixed periodic domain. The advection field in the computation is a two dimensional Hamiltonian flow consisting of a periodic array of counter-rotating vortices, or cellular flows. Depending on whether the evolution is predominantly in the hyperbolic or parabolic regimes, suitable explicit and semi-implicit time stepping methods are chosen. The turbulent flame speeds are computed as the linear growth rates of large time solutions. A new nonlinear parabolic PDE is proposed for the reinitialization of level set functions to prevent piling up of multiple bundles of level sets on the periodic domain. We found that the turbulent flame speed sT of the curvature G-equation is enhanced as the intensity A of cellular flows increases, at a rate between those of the inviscid and viscous G-equations. The sT of the strain G-equation increases in small A, decreases in larger A, then drops down to zero at a large enough but finite value A∗. The flame front ceases to propagate at this critical intensity A∗, and is quenched by the cellular flow.
NASA Astrophysics Data System (ADS)
Liu, Yu-Yu; Xin, Jack; Yu, Yifeng
2011-11-01
G-equations are well-known front propagation models in turbulent combustion which describe the front motion law in the form of local normal velocity equal to a constant (laminar speed) plus the normal projection of fluid velocity. In level set formulation, G-equations are Hamilton-Jacobi equations with convex ( L 1 type) but non-coercive Hamiltonians. Viscous G-equations arise from either numerical approximations or regularizations by small diffusion. The nonlinear eigenvalue {bar H} from the cell problem of the viscous G-equation can be viewed as an approximation of the inviscid turbulent flame speed s T. An important problem in turbulent combustion theory is to study properties of s T, in particular how s T depends on the flow amplitude A. In this paper, we study the behavior of {bar H=bar H(A,d)} as A → + ∞ at any fixed diffusion constant d > 0. For cellular flow, we show that bar H(A,d)≤q C(d) quad text{for all} d >0 , where C( d) is a constant depending on d, but independent of A. Compared with {bar H(A,0)= O(A/log A), A≫ 1}, of the inviscid G-equation ( d = 0), presence of diffusion dramatically slows down front propagation. For shear flow, {lim_{Ato +infty}bar H(A,d)/A = λ (d) >0 } where λ ( d) is strictly decreasing in d, and has zero derivative at d = 0. The linear growth law is also valid for s T of the curvature dependent G-equation in shear flows.
Research on development of equations for performance trajectory computations
NASA Technical Reports Server (NTRS)
Harmon, G. R.
1974-01-01
The analytical foundation of the Hamilton-Jacobi theory was investigated for application to space flight problems. A specific problem area is defined as follows: (1) to attempt to use the first order perturbation theory, which has been developed for the motion of a uniaxial satellite in a gravitational field in studying the motion of a triaxial satellite in a gravity field; and (2) also to expand theory for the uniaxial case to higher order.
NASA Astrophysics Data System (ADS)
Feng, Zhong-wen; Li, Guo-ping; Zhang, Yan; Zu, Xiao-tao
2015-02-01
In this paper, we combine the Hamilton-Jacobi equation with a new general tortoise coordinate transformation to study quantum tunneling of scalar particles and fermions from the non-stationary higher dimensional Vaidya-de Sitter black hole. The results show that Hamilton-Jacobi equation is a semi-classical foundation equation which can easily derived from the particles' dynamic equations, it can helps us understand the origin of Hawking radiation. Besides, based on the dimensional analysis, we believed that the new general tortoise coordinate transformation is more reasonable than old ones.
Holographic trace anomaly and local renormalization group
NASA Astrophysics Data System (ADS)
Rajagopal, Srivatsan; Stergiou, Andreas; Zhu, Yechao
2015-11-01
The Hamilton-Jacobi method in holography has produced important results both at a renormalization group (RG) fixed point and away from it. In this paper we use the Hamilton-Jacobi method to compute the holographic trace anomaly for four- and six-dimensional boundary conformal field theories (CFTs), assuming higher-derivative gravity and interactions of scalar fields in the bulk. The scalar field contributions to the anomaly appear in CFTs with exactly marginal operators. Moving away from the fixed point, we show that the Hamilton-Jacobi formalism provides a deep connection between the holographic and the local RG. We derive the local RG equation holographically, and verify explicitly that it satisfies Weyl consistency conditions stemming from the commutativity of Weyl scalings. We also consider massive scalar fields in the bulk corresponding to boundary relevant operators, and comment on their effects to the local RG equation.
NASA Astrophysics Data System (ADS)
Lin, Kai; Yang, Shu-Zheng
2009-10-01
Fermions tunneling of the non-stationary Dilaton-Maxwell black hole is investigated with general tortoise coordinate transformation. The Dirac equation is simplified by semiclassical approximation so that the Hamilton-Jacobi equation is generated. Finally the tunneling rate and the Hawking temperature is calculated.
Multi-asset investment-consumption model with transaction costs
NASA Astrophysics Data System (ADS)
Zhao, Xiao-Yan; Nie, Zan-Kan
2005-09-01
In this paper, we consider the multi-asset optimal investment-consumption model: a riskless asset and d risky assets. when the initial time is t[greater-or-equal, slanted]0, for a proportional transaction costs and discount factors, we proof that the value function of the model is a unique viscosity solution of a Hamilton-Jacobi-Bellman (HJB) equations.
NASA Astrophysics Data System (ADS)
Feng, Zhong-Wen; Deng, Juan; Li, Guo-Ping; Yang, Shu-Zheng
2012-10-01
In this paper, the quantum tunneling of the non-stationary Kerr-Newman black hole is investigated via Hamilton-Jacobi equation and two types of general tortoise coordinate transformations. The tunneling rates, the Hawking temperatures and radiation spectrums are derived respectively. Our result shows that the new type of general tortoise coordinate transformation is more reasonable.
Stochastic Differential Games with Asymmetric Information
Cardaliaguet, Pierre Rainer, Catherine
2009-02-15
We investigate a two-player zero-sum stochastic differential game in which the players have an asymmetric information on the random payoff. We prove that the game has a value and characterize this value in terms of dual viscosity solutions of some second order Hamilton-Jacobi equation.
Quantum tunneling of massive spin-1 particles from non-stationary metrics
NASA Astrophysics Data System (ADS)
Sakalli, I.; Övgün, A.
2016-01-01
We focus on the HR of massive vector (spin-1) particles tunneling from Schwarzschild BH expressed in the Kruskal-Szekeres and dynamic Lemaitre coordinates. Using the Proca equation together with the Hamilton-Jacobi and the WKB methods, we show that the tunneling rate, and its consequence Hawking temperature are well recovered by the quantum tunneling of the massive vector particles.
Vector particles tunneling from BTZ black holes
NASA Astrophysics Data System (ADS)
Chen, Ge-Rui; Zhou, Shiwei; Huang, Yong-Chang
2015-11-01
In this paper we investigate vector particles' Hawking radiation from a Banados-Teitelboim-Zanelli (BTZ) black hole. By applying the Wentzel-Kramers-Brillouin (WKB) approximation and the Hamilton-Jacobi ansatz to the Proca equation, we obtain the tunneling spectrum of vector particles. The expected Hawking temperature is recovered.
Emission of scalar particles from cylindrical black holes
NASA Astrophysics Data System (ADS)
Gohar, H.; Saifullah, K.
2013-01-01
We study quantum tunneling of scalar particles from black strings. For this purpose we apply WKB approximation and Hamilton-Jacobi method to solve the Klein-Gordon equation for outgoing trajectories. We find the tunneling probability of outgoing charged and uncharged scalars from the event horizon of black strings, and hence the Hawking temperature for these black configurations.
Vector particles tunneling from four-dimensional Schwarzschild black holes
NASA Astrophysics Data System (ADS)
Chen, Ge-Rui; Zhou, Shiwei; Huang, Yong-Chang
2015-05-01
Vector particles' Hawking radiation from a four-dimensional Schwarzschild black hole is investigated. By applying the WKB approximation and the Hamilton-Jacobi ansatz to the Proca equation, we obtain the tunneling spectrum of vector particles and the expected Hawking temperature.
Scalar field radiation from dilatonic black holes
NASA Astrophysics Data System (ADS)
Gohar, H.; Saifullah, K.
2012-12-01
We study radiation of scalar particles from charged dilaton black holes. The Hamilton-Jacobi method has been used to work out the tunneling probability of outgoing particles from the event horizon of dilaton black holes. For this purpose we use WKB approximation to solve the charged Klein-Gordon equation. The procedure gives Hawking temperature for these black holes as well.
Phonon Emission from Acoustic Black Hole
NASA Astrophysics Data System (ADS)
Fang, Hengzhong; Zhou, Kaihu; Song, Yuming
2012-08-01
We study the phonon tunneling through the horizon of an acoustic black hole by solving the Hamilton-Jacobi equation. We also make use of the closed-path integral to calculate the tunneling probability, and an improved way to determine the temporal contribution is used. Both the results from the two methods agree with Hawking's initial analysis.
Canonical Transformation to the Free Particle
ERIC Educational Resources Information Center
Glass, E. N.; Scanio, Joseph J. G.
1977-01-01
Demonstrates how to find some canonical transformations without solving the Hamilton-Jacobi equation. Constructs the transformations from the harmonic oscillator to the free particle and uses these as examples of transformations that cannot be maintained when going from classical to quantum systems. (MLH)
Semiclassical Supersymmetric Quantum Gravity
NASA Astrophysics Data System (ADS)
Kiefer, Claus; Lück, Tobias; Vargas Moniz, Paulo
2008-09-01
We develop a semiclassical approximation scheme for the constraint equations of supersymmetric canonical quantum gravity. This is achieved by a Born-Oppenheimer type of expansion, in analogy to the case of the usual Wheeler-DeWitt equation. We recover at consecutive orders the Hamilton-Jacobi equation, the functional Schrödinger equation, and quantum gravitational correction terms to this Schrödinger equation. In particular, our work has the following implications: (i) the Hamilton-Jacobi equation and therefore the background spacetime must involve the gravitino, (ii) a (many fingered) local time parameter has to be present on Super Riem Σ (the space of all possible tetrad and gravitino fields), (iii) quantum supersymmetric gravitational corrections affect the evolution of the very early universe.
NASA Astrophysics Data System (ADS)
Christlieb, Andrew J.; Feng, Xiao; Seal, David C.; Tang, Qi
2016-07-01
We propose a high-order finite difference weighted ENO (WENO) method for the ideal magnetohydrodynamics (MHD) equations. The proposed method is single-stage (i.e., it has no internal stages to store), single-step (i.e., it has no time history that needs to be stored), maintains a discrete divergence-free condition on the magnetic field, and has the capacity to preserve the positivity of the density and pressure. To accomplish this, we use a Taylor discretization of the Picard integral formulation (PIF) of the finite difference WENO method proposed in Christlieb et al. (2015) [23], where the focus is on a high-order discretization of the fluxes (as opposed to the conserved variables). We use the version where fluxes are expanded to third-order accuracy in time, and for the fluid variables space is discretized using the classical fifth-order finite difference WENO discretization. We use constrained transport in order to obtain divergence-free magnetic fields, which means that we simultaneously evolve the magnetohydrodynamic (that has an evolution equation for the magnetic field) and magnetic potential equations alongside each other, and set the magnetic field to be the (discrete) curl of the magnetic potential after each time step. In this work, we compute these derivatives to fourth-order accuracy. In order to retain a single-stage, single-step method, we develop a novel Lax-Wendroff discretization for the evolution of the magnetic potential, where we start with technology used for Hamilton-Jacobi equations in order to construct a non-oscillatory magnetic field. The end result is an algorithm that is similar to our previous work Christlieb et al. (2014) [8], but this time the time stepping is replaced through a Taylor method with the addition of a positivity-preserving limiter. Finally, positivity preservation is realized by introducing a parameterized flux limiter that considers a linear combination of high and low-order numerical fluxes. The choice of the free
Multivariant function model generation
NASA Technical Reports Server (NTRS)
1974-01-01
The development of computer programs applicable to space vehicle guidance was conducted. The subjects discussed are as follows: (1) determination of optimum reentry trajectories, (2) development of equations for performance of trajectory computation, (3) vehicle control for fuel optimization, (4) development of equations for performance trajectory computations, (5) applications and solution of Hamilton-Jacobi equation, and (6) stresses in dome shaped shells with discontinuities at the apex.
Diploid biological evolution models with general smooth fitness landscapes and recombination.
Saakian, David B; Kirakosyan, Zara; Hu, Chin-Kun
2008-06-01
Using a Hamilton-Jacobi equation approach, we obtain analytic equations for steady-state population distributions and mean fitness functions for Crow-Kimura and Eigen-type diploid biological evolution models with general smooth hypergeometric fitness landscapes. Our numerical solutions of diploid biological evolution models confirm the analytic equations obtained. We also study the parallel diploid model for the simple case of recombination and calculate the variance of distribution, which is consistent with numerical results. PMID:18643300
Quantum mechanics from an equivalence principle
Faraggi, A.E.; Matone, M.
1997-05-15
The authors show that requiring diffeomorphic equivalence for one-dimensional stationary states implies that the reduced action S{sub 0} satisfies the quantum Hamilton-Jacobi equation with the Planck constant playing the role of a covariantizing parameter. The construction shows the existence of a fundamental initial condition which is strictly related to the Moebius symmetry of the Legendre transform and to its involutive character. The universal nature of the initial condition implies the Schroedinger equation in any dimension.
NASA Astrophysics Data System (ADS)
Lin, Kai; Yang, ShuZheng
2009-04-01
The 1/2 spin fermions tunneling at the horizon of n-dimensional Kerr-Anti-de Sitter black hole with one rotational parameter is researched via semi-classical approximation method, and the Hawking temperature and fermions tunneling rate are obtained in this Letter. Using a new method, the semi-classical Hamilton-Jacobi equation is gotten from the Dirac equation in this Letter, and the work makes several quantum tunneling theories more harmonious.
Hawking radiation of scalars from accelerating and rotating black holes with NUT parameter
NASA Astrophysics Data System (ADS)
Jan, Khush; Gohar, H.
2014-03-01
We study the quantum tunneling of scalars from charged accelerating and rotating black hole with NUT parameter. For this purpose we use the charged Klein-Gordon equation. We apply WKB approximation and the Hamilton-Jacobi method to solve charged Klein-Gordon equation. We find the tunneling probability of outgoing charged scalars from the event horizon of this black hole, and hence the Hawking temperature for this black hole
a Study of Kantowski-Sachs Model in Ashtekar Variables
NASA Astrophysics Data System (ADS)
Chakraborty, Subenoy; Chakravarty, Nabajit
In this paper we study classical and quantum cosmology in Kantowski-Sachs model using Ashtekar variables. Classical solutions are obtained for the above model with a cosmological term and Hamilton-Jacobi (HJ) equations have been studied to obtain inflationary solutions. In quantum cosmology, the wave function of the Universe is obtained using path integral formalism as well as by solving the Wheeler-DeWitt (WD) equation.
Measurement by phase severance
Noyes, H.P.
1987-03-01
It is claimed that the measurement process is more accurately described by ''quasi-local phase severance'' than by ''wave function collapse''. The approach starts from the observation that the usual route to quantum mechanics starting from the Hamilton-Jacobi equations throws away half the degrees of freedom, namely, the classical initial state parameters. To overcome this difficulty, the full set of Hamilton-Jacobi equations is interpreted as operator equations acting on a state vector. The measurement theory presented is based on the conventional S-matrix boundary condition of N/sub A/ free particles in the distant past and N/sub B/ free particles in the distant future and taking the usual free particle wave functions, multiplied by phase factors.
Gauge Invariance of Parametrized Systems and Path Integral Quantization
NASA Astrophysics Data System (ADS)
de Cicco, Hernán; Simeone, Claudio
Gauge invariance of systems whose Hamilton-Jacobi equation is separable is improved by adding surface terms to the action functional. The general form of these terms is given for some complete solutions of the Hamilton-Jacobi equation. The procedure is applied to the relativistic particle and toy universes, which are quantized by imposing canonical gauge conditions in the path integral; in the case of empty models, we first quantize the parametrized system called "ideal clock," and then we examine the possibility of obtaining the amplitude for the minisuperspaces by matching them with the ideal clock. The relation existing between the geometrical properties of the constraint surface and the variables identifying the quantum states in the path integral is discussed.
Swing-up and Stabilization of Inverted Pendulum by Nonlinear Optimal Control
NASA Astrophysics Data System (ADS)
Fujimoto, Ryu; Sakamoto, Noboru
In this paper, the problem of swing up and stabilization of an inverted pendulum by a single feedback control law is considered. The problem is formulated as an optimal control problem including input saturation and is solved via the stable manifold approach which is recently proposed for solving the Hamilton-Jacobi equation. In this approach, the problem is turned into the enhancement problem of the domain of validity to include the pending position. After a finite number of iterations, an optimal feedback control law is obtained and its effectiveness is verified by experiments. It is shown that the stable manifold approach can be applied for systems including practical nonlinearities such as saturation by directly deriving a controller satisfying the input limitation of the experimental setup. It is also reported that this system is an example in which non-unique solutions for the Hamilton-Jacobi equation exist.
Ahmedov, Haji; Aliev, Alikram N.
2008-09-15
We examine the separability properties of the equation of motion for a stationary string near a rotating charged black hole with two independent angular momenta in five-dimensional minimal gauged supergravity. It is known that the separability problem for the stationary string in a general stationary spacetime is reduced to that for the usual Hamilton-Jacobi equation for geodesics of its quotient space with one dimension fewer. Using this fact, we show that the 'effective metric' of the quotient space does not allow the complete separability for the Hamilton-Jacobi equation, albeit such a separability occurs in the original spacetime of the black hole. We also show that only for two special cases of interest the Hamilton-Jacobi equation admits the complete separation of variables and therefore the integrability for the stationary string motion in the original background, namely, when the black hole has zero electric charge or it has an arbitrary electric charge but two equal angular momenta. We give the explicit expressions for the Killing tensors corresponding to these cases. However, for the general black hole spacetime the effective metric of the quotient space admits a conformal Killing tensor. We construct the explicit expression for this tensor.
Quantum nonthermal effect of the Vaidya-Bonner-de Sitter black hole
NASA Astrophysics Data System (ADS)
Pan, Wei-Zhen; Yang, Xue-Jun; Yu, Guo-Xiang
2014-02-01
Using the Hamilton-Jacobi equation of a scalar particle in the curve space-time and a correct-dimension new tortoise coordinate transformation, the quantum nonthermal radiation of the Vaidya-Bonner-de Sitter black hole is investigated. The energy condition for the occurrence of the Starobinsky-Unruh process is obtained. The event horizon surface gravity and the Hawking temperature on the event horizon are also given.
Nonlinear longitudinal control of a supermaneuverable aircraft
NASA Technical Reports Server (NTRS)
Garrard, William L.; Snell, Anthony; Enns, Dale F.
1989-01-01
A technique is described which can be used for design of feedback controllers for high-performance aircraft operating in flight conditions in which nonlinearities significantly affect performance. Designs are performed on a mathematical model of the longitudinal dynamics of a hypothetical aircraft similar to proposed supermaneuverable flight test vehicles. Nonlinear controller designs are performed using truncated solutions of the Hamilton-Jacobi-Bellman equation. Preliminary results show that the method yields promising results.
Tunneling Radiation of Vector Particles in Four and Five Dimensional Black Holes
NASA Astrophysics Data System (ADS)
Chen, Bingbing
2016-07-01
Recent research shows that the WKB approximation and the Hamilton-Jacobi method has been succeed in studying the tunneling radiation of vector particles. In view of this, our main aim in this letter is to study the Proca equation and the vector particles tunneling radiation in the 4-dimensional and 5-dimensional black holes. And finally, the results here show that the temperature of vector particle is the same as Dirac particle's and other particle's.
Hawking radiation of spin-1 particles from a three-dimensional rotating hairy black hole
NASA Astrophysics Data System (ADS)
Sakalli, I.; Ovgun, A.
2015-09-01
We study the Hawking radiation of spin-1 particles (so-called vector particles) from a three-dimensional rotating black hole with scalar hair using a Hamilton-Jacobi ansatz. Using the Proca equation in the WKB approximation, we obtain the tunneling spectrum of vector particles. We recover the standard Hawking temperature corresponding to the emission of these particles from a rotating black hole with scalar hair.
General Nth order integrals of motion in the Euclidean plane
NASA Astrophysics Data System (ADS)
Post, S.; Winternitz, P.
2015-10-01
The general form of an integral of motion that is a polynomial of order N in the momenta is presented for a Hamiltonian system in two-dimensional Euclidean space. The classical and the quantum cases are treated separately, emphasizing both the similarities and the differences between the two. The main application will be to study Nth order superintegrable systems that allow separation of variables in the Hamilton-Jacobi and Schrödinger equations, respectively.
Massive vector particles tunneling from Kerr and Kerr-Newman black holes
NASA Astrophysics Data System (ADS)
Li, Xiang-Qian; Chen, Ge-Rui
2015-12-01
In this paper, we investigate the Hawking radiation of massive spin-1 particles from 4-dimensional Kerr and Kerr-Newman black holes. By applying the Hamilton-Jacobi ansatz and the WKB approximation to the field equations of the massive bosons in Kerr and Kerr-Newman space-time, the quantum tunneling method is successfully implemented. As a result, we obtain the tunneling rate of the emitted vector particles and recover the standard Hawking temperature of both the two black holes.
Quantum tunneling from scalar fields in rotating black strings
NASA Astrophysics Data System (ADS)
Gohar, H.; Saifullah, K.
2013-08-01
Using the Hamilton-Jacobi method of quantum tunneling and complex path integration, we study Hawking radiation of scalar particles from rotating black strings. We discuss tunneling of both charged and uncharged scalar particles from the event horizons. For this purpose, we use the Klein-Gordon equation and find the tunneling probability of outgoing scalar particles. The procedure gives Hawking temperature for rotating charged black strings as well.
Gauge fixation and global phase time for minisuperspaces
NASA Astrophysics Data System (ADS)
Simeone, Claudio
1999-09-01
Homogeneous and isotropic cosmological models whose Hamilton-Jacobi equation is separable are deparametrized by turning their action functional into that of an ordinary gauge system. Canonical gauge conditions imposed on the gauge system are used to define a global phase time in terms of the canonical coordinates and momenta of the minisuperspaces. The procedure clearly shows how the geometry of the constraint surface restricts the choice of time; the consequences that this has on the path integral quantization are discussed.
Viscosity Solutions of Systems of PDEs with Interconnected Obstacles and Switching Problem
Hamadene, S. Morlais, M. A.
2013-04-15
This paper deals with existence and uniqueness of a solution in viscosity sense, for a system of m variational partial differential inequalities with inter-connected obstacles. A particular case is the Hamilton-Jacobi-Bellmann system of the Markovian stochastic optimal m-states switching problem. The switching cost functions depend on (t,x). The main tool is the notion of systems of reflected backward stochastic differential equations with oblique reflection.
Hawking Radiation of Mass Generating Particles from Dyonic Reissner-Nordström Black Hole
NASA Astrophysics Data System (ADS)
Sakalli, I.; Övgün, A.
2016-09-01
The Hawking radiation is considered as a quantum tunneling process, which can be studied in the framework of the Hamilton-Jacobi method. In this study, we present the wave equation for a mass generating massive and charged scalar particle (boson). In sequel, we analyse the quantum tunneling of these bosons from a generic 4-dimensional spherically symmetric black hole. We apply the Hamilton-Jacobi formalism to derive the radial integral solution for the classically forbidden action which leads to the tunneling probability. To support our arguments, we take the dyonic Reissner-Nordström black hole as a test background. Comparing the tunneling probability obtained with the Boltzmann formula, we succeed in reading the standard Hawking temperature of the dyonic Reissner-Nordström black hole.
NASA Astrophysics Data System (ADS)
Vanzo, L.
2011-07-01
The tunneling method for stationary black holes in the Hamilton-Jacobi variant is reconsidered in the light of some critiques that have been moved against. It is shown that once the tunneling trajectories have been correctly identified the method is free from internal inconsistencies, it is manifestly covariant, it allows for the extension to spinning particles and it can even be used without solving the Hamilton-Jacobi equation. These conclusions borrow support on a simple analytic continuation of the classical action of a pointlike particle, made possible by the unique assumption that it should be analytic in the complexified Schwarzschild or Kerr-Newman space-time. A more general version of the Parikh-Wilczek method will also be proposed along these lines.
Hypersurface-invariant approach to cosmological perturbations
NASA Astrophysics Data System (ADS)
Salopek, D. S.; Stewart, J. M.
1995-01-01
Using Hamilton-Jacobi theory, we develop a formalism for solving semiclassical cosmological perturbations which does not require an explicit choice of time hypersurface. The Hamilton-Jacobi equation for gravity interacting with matter (either a scalar or dust field) is solved by making an ansatz which includes all terms quadratic in the spatial curvature. Gravitational radiation and scalar perturbations are treated on an equal footing. Our technique encompasses linear perturbation theory and it also describes some mild nonlinear effects. As a concrete example of the method, we compute the galaxy-galaxy correlation function as well as large-angle microwave background fluctuations for power-law inflation, and we compare with recent observations.
Nonlinear evolutions and non-Gaussianity in generalized gravity
NASA Astrophysics Data System (ADS)
Koh, Seoktae; Kim, Sang Pyo; Song, Doo Jong
2005-06-01
We use the Hamilton-Jacobi theory to study the nonlinear evolutions of inhomogeneous spacetimes during inflation in generalized gravity. We find the exact solutions to the lowest order Hamilton-Jacobi equation for special scalar potentials and introduce an approximation method for general potentials. The conserved quantity invariant under a change of timelike hypersurfaces proves useful in dealing with gravitational perturbations. In the long-wavelength approximation, we find a conserved quantity related to the new canonical variable that makes the Hamiltonian density vanish, and calculate the non-Gaussianity in generalized gravity. The slow-roll inflation models with a single scalar field in generalized gravity predict too small non-Gaussianity to be detected by future CMB experiments.
NASA Astrophysics Data System (ADS)
Schrödinger, E.
2011-02-01
Preface; Part I. The de Sitter Universe: 1. Synthetic construction; 2. The reduced model: geodesics; 3. The elliptic interpretation; 4. The static frame; 5. The determination of parallaxes; 6. The Lemaître-Robertson frame; Part II. The Theory of Geodesics: 7. On null geodesics; i. Determination of the parameter for null lines in special cases; ii. Frequency shift; 8. Free particles and light rays in general expanding spaces, flat or hyperspherical; i. Flat spaces; ii. Spherical spaces; iii. The red shift for spherical spaces; Part III. Waves in General Riemannian Space-Time: 9. The nature of our approximation; 10. The Hamilton-Jacobi theory in a gravitational field; 11. Procuring approximate solutions of the Hamilton-Jacobi equation from wave theory; Part IV. Waves in an Expanding Universe: 12. General considerations; 13. Proper vibrations and wave parcels; Bibliography.
Young, C.W.
1997-10-01
In 1967, Sandia National Laboratories published empirical equations to predict penetration into natural earth materials and concrete. Since that time there have been several small changes to the basic equations, and several more additions to the overall technique for predicting penetration into soil, rock, concrete, ice, and frozen soil. The most recent update to the equations was published in 1988, and since that time there have been changes in the equations to better match the expanding data base, especially in concrete penetration. This is a standalone report documenting the latest version of the Young/Sandia penetration equations and related analytical techniques to predict penetration into natural earth materials and concrete. 11 refs., 6 tabs.
NASA Astrophysics Data System (ADS)
Viljamaa, Panu; Jacobs, J. Richard; Chris; JamesHyman; Halma, Matthew; EricNolan; Coxon, Paul
2014-07-01
In reply to a Physics World infographic (part of which is given above) about a study showing that Euler's equation was deemed most beautiful by a group of mathematicians who had been hooked up to a functional magnetic-resonance image (fMRI) machine while viewing mathematical expressions (14 May, http://ow.ly/xHUFi).
DOE R&D Accomplishments Database
1998-09-21
In the late 1950s to early 1960s Rudolph A. Marcus developed a theory for treating the rates of outer-sphere electron-transfer reactions. Outer-sphere reactions are reactions in which an electron is transferred from a donor to an acceptor without any chemical bonds being made or broken. (Electron-transfer reactions in which bonds are made or broken are referred to as inner-sphere reactions.) Marcus derived several very useful expressions, one of which has come to be known as the Marcus cross-relation or, more simply, as the Marcus equation. It is widely used for correlating and predicting electron-transfer rates. For his contributions to the understanding of electron-transfer reactions, Marcus received the 1992 Nobel Prize in Chemistry. This paper discusses the development and use of the Marcus equation. Topics include self-exchange reactions; net electron-transfer reactions; Marcus cross-relation; and proton, hydride, atom and group transfers.
"Spring theory of relativity" originating from quantum mechanics
NASA Astrophysics Data System (ADS)
Yefremov, Alexander P.
Compact derivation of mathematical equations similar to those of quantum and classical mechanics is given on the base of fractal decomposition of a three-dimensional space. In physical units the equations become Shrödinger and Hamilton-Jacobi equations, the wave function of a free particle associated with a virtual ring. Locally uniform motion of the ring in the physical space provides an original helix (or regular cylindrical spring) model of a relativistic theory equivalent in results with special relativity, the free particle's relativistic Lagrangian emerging automatically. Irregular spring model generates theory similar to general relativity.
Liouville properties and critical value of fully nonlinear elliptic operators
NASA Astrophysics Data System (ADS)
Bardi, Martino; Cesaroni, Annalisa
2016-10-01
We prove some Liouville properties for sub- and supersolutions of fully nonlinear degenerate elliptic equations in the whole space. Our assumptions allow the coefficients of the first order terms to be large at infinity, provided they have an appropriate sign, as in Ornstein-Uhlenbeck operators. We give two applications. The first is a stabilization property for large times of solutions to fully nonlinear parabolic equations. The second is the solvability of an ergodic Hamilton-Jacobi-Bellman equation that identifies a unique critical value of the operator.
On some integrable systems in the extended lobachevsky space
Kurochkin, Yu. A. Otchik, V. S.; Ovsiyuk, E. M.; Shoukavy, Dz. V.
2011-06-15
Some classical and quantum-mechanical problems previously studied in Lobachevsky space are generalized to the extended Lobachevsky space (unification of the real, imaginary Lobachevsky spaces and absolute). Solutions of the Schroedinger equation with Coulomb potential in two coordinate systems of the imaginary Lobachevsky space are considered. The problem of motion of a charged particle in the homogeneous magnetic field in the imaginary Lobachevsky space is treated both classically and quantum mechanically. In the classical case, Hamilton-Jacoby equation is solved by separation of variables, and constraints for integrals of motion are derived. In the quantum case, solutions of Klein-Fock-Gordon equation are found.
FAST TRACK COMMUNICATION: Hořava-Lifshitz holography
NASA Astrophysics Data System (ADS)
Nishioka, Tatsuma
2009-12-01
We derive the detailed balance condition as a solution to the Hamilton-Jacobi equation in the Hořava-Lifshitz gravity. This result leads us to propose the existence of the d-dimensional quantum field theory on the future boundary of the (d + 1)-dimensional Hořava-Lifshitz gravity from the viewpoint of the holographic renormalization group. We also obtain a Ricci flow equation of the boundary theory as the holographic RG flow, which is the Hamilton equation in the bulk gravity, by tuning parameters in the theory.
An approximate atmospheric guidance law for aeroassisted plane change maneuvers
NASA Astrophysics Data System (ADS)
Speyer, Jason L.; Crues, Edwin Z.
An approximate optimal guidance law for the aeroassisted plane change problem is presented which is based upon an expansion of the Hamilton-Jacobi-Bellman equation with respect to the small parameter of Breakwell et al. (1985). The present law maximizes the final velocity of the reentry vehicle while meeting terminal constraints on altitude, flight path angle, and heading angle. The integrable zeroth-order solution found when the small parameter is set to zero corresponds to a solution of the problem where the aerodynamic forces dominate the inertial forces. Higher order solutions in the expansion are obtained from the solution of linear partial differential equations requiring only quadrature integration.
Accurate Analytic Results for the Steady State Distribution of the Eigen Model
NASA Astrophysics Data System (ADS)
Huang, Guan-Rong; Saakian, David B.; Hu, Chin-Kun
2016-04-01
Eigen model of molecular evolution is popular in studying complex biological and biomedical systems. Using the Hamilton-Jacobi equation method, we have calculated analytic equations for the steady state distribution of the Eigen model with a relative accuracy of O(1/N), where N is the length of genome. Our results can be applied for the case of small genome length N, as well as the cases where the direct numerics can not give accurate result, e.g., the tail of distribution.
GENERAL: On particles tunneling from the Taub-NUT-AdS black hole
NASA Astrophysics Data System (ADS)
Zeng, Xiao-Xiong; Li, Qiang
2009-11-01
This paper discusses tunneling of scalar particles and Dirac particles from the Taub-NUT-AdS black hole by the Hamilton-Jacobi equation, initially used by Angheben et al, and the Dirac equation, recently proposed by Kerner and Mann. This is performed in the dragging coordinate frame so as to avoid the ergosphere dragging effect. A general form is obtained for the temperature of scalar and Dirac particles tunneling from the Taub-NUT-Ads black hole, which is commensurate with other methods as expected.
An approximate atmospheric guidance law for aeroassisted plane change maneuvers
NASA Technical Reports Server (NTRS)
Speyer, Jason L.; Crues, Edwin Z.
1988-01-01
An approximate optimal guidance law for the aeroassisted plane change problem is presented which is based upon an expansion of the Hamilton-Jacobi-Bellman equation with respect to the small parameter of Breakwell et al. (1985). The present law maximizes the final velocity of the reentry vehicle while meeting terminal constraints on altitude, flight path angle, and heading angle. The integrable zeroth-order solution found when the small parameter is set to zero corresponds to a solution of the problem where the aerodynamic forces dominate the inertial forces. Higher order solutions in the expansion are obtained from the solution of linear partial differential equations requiring only quadrature integration.
Cesaroni, Annalisa
2006-01-15
We prove optimality principles for semicontinuous bounded viscosity solutions of Hamilton-Jacobi-Bellman equations. In particular, we provide a representation formula for viscosity supersolutions as value functions of suitable obstacle control problems. This result is applied to extend the Lyapunov direct method for stability to controlled Ito stochastic differential equations. We define the appropriate concept of the Lyapunov function to study stochastic open loop stabilizability in probability and local and global asymptotic stabilizability (or asymptotic controllability). Finally, we illustrate the theory with some examples.
A minimax approach to mean field games
NASA Astrophysics Data System (ADS)
Averboukh, Yu V.
2015-07-01
An initial boundary value problem for the system of equations of a determined mean field game is considered. The proposed definition of a generalized solution is based on the minimax approach to the Hamilton-Jacobi equation. We prove the existence of the generalized (minimax) solution using the Nash equilibrium in the auxiliary differential game with infinitely many identical players. We show that the minimax solution of the original system provides the \\varepsilon-Nash equilibrium in the differential game with a finite number of players. Bibliography: 34 titles.
Neutrino Tunneling from NUT Kerr Newman de Sitter Black Hole
NASA Astrophysics Data System (ADS)
Yang, Nan; Yang, Juan; Li, Jin
2013-08-01
In this paper, the method of semi-classical is applied to explore the Hawking radiation of a NUT-Kerr-Newman de Sitter Black Hole from tunneling point of view. The Hamilton-Jacobi equation in NUT-Kerr-Newman de Sitter space time is derived by the method presented by Lin and Yang (Chin. Phys. B, 20:110403, 2011). We obtain the Hawking temperatures at the event horizon and cosmological horizon and we also obtain the tunneling probability of neutrino following the semi-classical quantum equation. The results show the common features of NUT-Kerr-Newman de Sitter Black Hole.
Fermions tunneling from rotating stationary Kerr black hole with electric charge and magnetic charge
NASA Astrophysics Data System (ADS)
Yang, Juan; Yang, Shu-Zheng
2010-06-01
In this paper, the method of semi-classical fermion tunneling is extended to explore the fermion tunneling behavior of a Kerr-Newman-Kasuya black hole. Thus, the Hamilton-Jacobi equation in Kerr-Newman-Kasuya space-time is derived by the method presented in Refs. Lin and Yang (2009) [24-26], the Hawking temperature at the horizon and the tunneling probability of spin- 1/2 fermions are finally obtained following the semi-classical quantum equation. The results indicate the common features of this black hole.
On the Motion of a Free Particle in Kinematic Relativity
NASA Astrophysics Data System (ADS)
Popescu, Ioan Antoniu
From the viewpoint that Milne's Kinematic Relativity is a fundamental theory of matter in all domain, from elementary particle physics tocosmology, we study the problem of a free particle in the presence of the Universe at large, in the classical and quantum pictures. We formulate the Hamilton-Jacobi, Dirac, and Kelin-Gordon equations for a free particle in the presence of the Universe at large. The form of these equations suggests that Kinematic Relativity is more suitable to serve as a basis for a description of the Early Universe than the ordinary relativistic and quantum theories of contemporary physics.
NASA Technical Reports Server (NTRS)
Lombaerts, Thomas; Schuet, Stefan R.; Wheeler, Kevin; Acosta, Diana; Kaneshige, John
2013-01-01
This paper discusses an algorithm for estimating the safe maneuvering envelope of damaged aircraft. The algorithm performs a robust reachability analysis through an optimal control formulation while making use of time scale separation and taking into account uncertainties in the aerodynamic derivatives. Starting with an optimal control formulation, the optimization problem can be rewritten as a Hamilton- Jacobi-Bellman equation. This equation can be solved by level set methods. This approach has been applied on an aircraft example involving structural airframe damage. Monte Carlo validation tests have confirmed that this approach is successful in estimating the safe maneuvering envelope for damaged aircraft.
Tunnelling of relativistic particles from new type black hole in new massive gravity
Gecim, Ganim; Sucu, Yusuf E-mail: ysucu@akdeniz.edu.tr
2013-02-01
In the framework of the three dimensional New Massive Gravity theory introduced by Bergshoeff, Hohm and Townsend, we analyze the behavior of relativistic spin-1/2 and spin-0 particles in the New-type Black Hole backgroud, solution of the New Massive Gravity.We solve Dirac equation for spin-1/2 and Klein-Gordon equation for spin-0. Using Hamilton-Jacobi method, we discuss tunnelling probability and Hawking temperature of the spin-1/2 and spin-0 particles for the black hole. We observe that the tunnelling probability and Hawking temperature are same for the spin-1/2 and spin-0.
Tunneling of massive vector particles from rotating charged black strings
NASA Astrophysics Data System (ADS)
Jusufi, Kimet; Övgün, Ali
2016-07-01
We study the quantum tunneling of charged massive vector bosons from a charged static and a rotating black string. We apply the standard methods, first we use the WKB approximation and the Hamilton-Jacobi equation, and then we end up with a set of four linear equations. Finally, solving for the radial part by using the determinant of the metric equals zero, the corresponding tunneling rate and the Hawking temperature is recovered in both cases. The tunneling rate deviates from pure thermality and is consistent with an underlying unitary theory.
Tunnelling of scalar and Dirac particles from squashed charged rotating Kaluza-Klein black holes
NASA Astrophysics Data System (ADS)
Stetsko, M. M.
2016-02-01
The thermal radiation of scalar particles and Dirac fermions from squashed charged rotating five-dimensional black holes is considered. To obtain the temperature of the black holes we use the tunnelling method. In the case of scalar particles we make use of the Hamilton-Jacobi equation. To consider tunnelling of fermions the Dirac equation was investigated. The examination shows that the radial parts of the action for scalar particles and fermions in the quasi-classical limit in the vicinity of horizon are almost the same and as a consequence it gives rise to identical expressions for the temperature in the two cases.
Shore, B.W.
1981-01-30
The equations of motion are discussed which describe time dependent population flows in an N-level system, reviewing the relationship between incoherent (rate) equations, coherent (Schrodinger) equations, and more general partially coherent (Bloch) equations. Approximations are discussed which replace the elaborate Bloch equations by simpler rate equations whose coefficients incorporate long-time consequences of coherence.
Semiclassical approximation to supersymmetric quantum gravity
NASA Astrophysics Data System (ADS)
Kiefer, Claus; Lück, Tobias; Moniz, Paulo
2005-08-01
We develop a semiclassical approximation scheme for the constraint equations of supersymmetric canonical quantum gravity. This is achieved by a Born-Oppenheimer type of expansion, in analogy to the case of the usual Wheeler-DeWitt equation. The formalism is only consistent if the states at each order depend on the gravitino field. We recover at consecutive orders the Hamilton-Jacobi equation, the functional Schrödinger equation, and quantum gravitational correction terms to this Schrödinger equation. In particular, the following consequences are found: (i) the Hamilton-Jacobi equation and therefore the background spacetime must involve the gravitino, (ii) a (many-fingered) local time parameter has to be present on super Riem Σ (the space of all possible tetrad and gravitino fields), (iii) quantum supersymmetric gravitational corrections affect the evolution of the very early Universe. The physical meaning of these equations and results, in particular, the similarities to and differences from the pure bosonic case, are discussed.
The Lagrangian theory of Staeckel Systems
NASA Astrophysics Data System (ADS)
Broucke, R.
1981-10-01
A purely Lagrangian formulation and a direct proof of the separation of variables theorem is given for what is called Staeckel Systems in dynamics and celestial mechanics. The proof is essentially based on some properties of determinants and minors (given in Appendix A). In contrast with the standard literature on the subject, the use of the Hamiltonian, canonical transformations or the Hamilton-Jacobi equation is avoided by using instead a more elementary approach based on the Lagrangian. In Appendix B we use the Kepler Problem as an illustration of the Lagrangian theory of Staeckel Systems.
Algorithm For Optimal Control Of Large Structures
NASA Technical Reports Server (NTRS)
Salama, Moktar A.; Garba, John A..; Utku, Senol
1989-01-01
Cost of computation appears competitive with other methods. Problem to compute optimal control of forced response of structure with n degrees of freedom identified in terms of smaller number, r, of vibrational modes. Article begins with Hamilton-Jacobi formulation of mechanics and use of quadratic cost functional. Complexity reduced by alternative approach in which quadratic cost functional expressed in terms of control variables only. Leads to iterative solution of second-order time-integral matrix Volterra equation of second kind containing optimal control vector. Cost of algorithm, measured in terms of number of computations required, is of order of, or less than, cost of prior algoritms applied to similar problems.
Optimal Quantum Feedback for Canonical Observables
NASA Astrophysics Data System (ADS)
Gough, John
2008-08-01
We consider the problem of optimal feedback control of a quantum system with linear dynamics undergoing continual non-demolition measurement of position and/or momentum, or both together. Specifically, we show that a stable domain of solutions for the filtered state of the system will be given by a class of randomized squeezed states and we exercise the control problem amongst these states. Bellman's principle is then applied directly to optimal feedback control of such dynamical systems and the Hamilton Jacobi Bellman equation for the minimum cost is derived. The situation of quadratic performance criteria is treated as the important special case and solved exactly for the class of relaxed states.
Lasry-Lions, Lax-Oleinik and generalized characteristics
NASA Astrophysics Data System (ADS)
Chen, Cui; Cheng, Wei
2016-09-01
In the recent works \\cite{Cannarsa-Chen-Cheng} and \\cite{Cannarsa-Cheng3}, an intrinsic approach of the propagation of singularities along the generalized characteristics was obtained, even in global case, by a procedure of sup-convolution with the kernel the fundamental solutions of the associated Hamilton-Jacobi equations. In the present paper, we exploit the relations among Lasry-Lions regularization, Lax-Oleinik operators (or inf/sup-convolution) and generalized characteristics, which are discussed in the context of the variational setting of Tonelli Hamiltonian dynamics, such as Mather theory and weak KAM theory.
Computing interface motion in compressible gas dynamics
NASA Technical Reports Server (NTRS)
Mulder, W.; Osher, S.; Sethan, James A.
1992-01-01
An analysis is conducted of the coupling of Osher and Sethian's (1988) 'Hamilton-Jacobi' level set formulation of the equations of motion for propagating interfaces to a system of conservation laws for compressible gas dynamics, giving attention to both the conservative and nonconservative differencing of the level set function. The capabilities of the method are illustrated in view of the results of numerical convergence studies of the compressible Rayleigh-Taylor and Kelvin-Helmholtz instabilities for air-air and air-helium boundaries.
Time dependent optimal switching controls in online selling models
Bradonjic, Milan; Cohen, Albert
2010-01-01
We present a method to incorporate dishonesty in online selling via a stochastic optimal control problem. In our framework, the seller wishes to maximize her average wealth level W at a fixed time T of her choosing. The corresponding Hamilton-Jacobi-Bellmann (HJB) equation is analyzed for a basic case. For more general models, the admissible control set is restricted to a jump process that switches between extreme values. We propose a new approach, where the optimal control problem is reduced to a multivariable optimization problem.
Numerical solution of continuous-time mean-variance portfolio selection with nonlinear constraints
NASA Astrophysics Data System (ADS)
Yan, Wei; Li, Shurong
2010-03-01
An investment problem is considered with dynamic mean-variance (M-V) portfolio criterion under discontinuous prices described by jump-diffusion processes. Some investment strategies are restricted in the study. This M-V portfolio with restrictions can lead to a stochastic optimal control model. The corresponding stochastic Hamilton-Jacobi-Bellman equation of the problem with linear and nonlinear constraints is derived. Numerical algorithms are presented for finding the optimal solution in this article. Finally, a computational experiment is to illustrate the proposed methods by comparing with M-V portfolio problem which does not have any constraints.
Numerical Computation of Diffusion on a Surface
Schwartz, Peter; Adalsteinsson, David; Colella, Phillip; Arkin, Adam Paul; Onsum, Matthew
2005-02-24
We present a numerical method for computing diffusive transport on a surface derived from image data. Our underlying discretization method uses a Cartesian grid embedded boundary method for computing the volume transport in region consisting of all points a small distance from the surface. We obtain a representation of this region from image data using a front propagation computation based on level set methods for solving the Hamilton-Jacobi and eikonal equations. We demonstrate that the method is second-order accurate in space and time, and is capable of computing solutions on complex surface geometries obtained from image data of cells.
Dirac particles tunneling from black holes with topological defects
NASA Astrophysics Data System (ADS)
Jusufi, Kimet
2016-08-01
We study Hawking radiation of Dirac particles with spin-1 / 2 as a tunneling process from Schwarzschild-de Sitter and Reissner-Nordström-de Sitter black holes in background spacetimes with a spinning cosmic string and a global monopole. Solving Dirac's equation by employing the Hamilton-Jacobi method and WKB approximation we find the corresponding tunneling probabilities and the Hawking temperature. Furthermore, we show that the Hawking temperature of those black holes remains unchanged in presence of topological defects in both cases.
EPR & Klein Paradoxes in Complex Hamiltonian Dynamics and Krein Space Quantization
NASA Astrophysics Data System (ADS)
Payandeh, Farrin
2015-07-01
Negative energy states are applied in Krein space quantization approach to achieve a naturally renormalized theory. For example, this theory by taking the full set of Dirac solutions, could be able to remove the propagator Green function's divergences and automatically without any normal ordering, to vanish the expected value for vacuum state energy. However, since it is a purely mathematical theory, the results are under debate and some efforts are devoted to include more physics in the concept. Whereas Krein quantization is a pure mathematical approach, complex quantum Hamiltonian dynamics is based on strong foundations of Hamilton-Jacobi (H-J) equations and therefore on classical dynamics. Based on complex quantum Hamilton-Jacobi theory, complex spacetime is a natural consequence of including quantum effects in the relativistic mechanics, and is a bridge connecting the causality in special relativity and the non-locality in quantum mechanics, i.e. extending special relativity to the complex domain leads to relativistic quantum mechanics. So that, considering both relativistic and quantum effects, the Klein-Gordon equation could be derived as a special form of the Hamilton-Jacobi equation. Characterizing the complex time involved in an entangled energy state and writing the general form of energy considering quantum potential, two sets of positive and negative energies will be realized. The new states enable us to study the spacetime in a relativistic entangled “space-time” state leading to 12 extra wave functions than the four solutions of Dirac equation for a free particle. Arguing the entanglement of particle and antiparticle leads to a contradiction with experiments. So, in order to correct the results, along with a previous investigation [1], we realize particles and antiparticles as physical entities with positive energy instead of considering antiparticles with negative energy. As an application of modified descriptions for entangled (space
Redundancy of constraints in the classical and quantum theories of gravitation.
NASA Technical Reports Server (NTRS)
Moncrief, V.
1972-01-01
It is shown that in Dirac's version of the quantum theory of gravitation, the Hamiltonian constraints are greatly redundant. If the Hamiltonian constraint condition is satisfied at one point on the underlying, closed three-dimensional manifold, then it is automatically satisfied at every point, provided only that the momentum constraints are everywhere satisfied. This permits one to replace the usual infinity of Hamiltonian constraints by a single condition which may be taken in the form of an integral over the manifold. Analogous theorems are given for the classical Einstein Hamilton-Jacobi equations.
NASA Technical Reports Server (NTRS)
Henriksen, R. N.; Nelson, L. A.
1985-01-01
Clock synchronization in an arbitrarily accelerated observer congruence is considered. A general solution is obtained that maintains the isotropy and coordinate independence of the one-way speed of light. Attention is also given to various particular cases including, rotating disk congruence or ring congruence. An explicit, congruence-based spacetime metric is constructed according to Einstein's clock synchronization procedure and the equation for the geodesics of the space-time was derived using Hamilton-Jacobi method. The application of interferometric techniques (absolute phase radio interferometry, VLBI) to the detection of the 'global Sagnac effect' is also discussed.
NASA Technical Reports Server (NTRS)
Mishne, D.; Speyer, J. L.
1986-01-01
A stochastic feedback control law for a space vehicle performing an aeroassisted plane-change maneuver is developed. The stochastic control law is designed to minimize the energy loss while taking into consideration the uncertainty in the atmospheric density. The solution is based on expansion of the stochastic Hamilton-Jacobi-Bellman equation (or dynamic programming) about a zeroth-order known integrable solution. The resulting guidance law is expressed as a series expansion in the noise power spectral densities. A numerical example indicates the potential improvement of this method.
Geodesic Monte Carlo on Embedded Manifolds.
Byrne, Simon; Girolami, Mark
2013-12-01
Markov chain Monte Carlo methods explicitly defined on the manifold of probability distributions have recently been established. These methods are constructed from diffusions across the manifold and the solution of the equations describing geodesic flows in the Hamilton-Jacobi representation. This paper takes the differential geometric basis of Markov chain Monte Carlo further by considering methods to simulate from probability distributions that themselves are defined on a manifold, with common examples being classes of distributions describing directional statistics. Proposal mechanisms are developed based on the geodesic flows over the manifolds of support for the distributions, and illustrative examples are provided for the hypersphere and Stiefel manifold of orthonormal matrices. PMID:25309024
Tunnelling of vector particles from Lorentzian wormholes in 3+1 dimensions
NASA Astrophysics Data System (ADS)
Sakalli, I.; Ovgun, A.
2015-06-01
In this article, we consider the Hawking radiation (HR) of vector (massive spin-1) particles from the traversable Lorentzian wormholes (TLWH) in 3+1 dimensions. We start by providing the Proca equations for the TLWH. Using the Hamilton-Jacobi (HJ) ansatz with the WKB approximation in the quantum tunneling method, we obtain the probabilities of the emission/absorption modes. Then, we derive the tunneling rate of the emitted vector particles and manage to read the standard Hawking temperature of the TLWH. The result obtained represents a negative temperature, which is also discussed.
Wu, Shuang-Qing
2008-03-28
I present the general exact solutions for nonextremal rotating charged black holes in the Gödel universe of five-dimensional minimal supergravity theory. They are uniquely characterized by four nontrivial parameters: namely, the mass m, the charge q, the Kerr equal rotation parameter a, and the Gödel parameter j. I calculate the conserved energy, angular momenta, and charge for the solutions and show that they completely satisfy the first law of black hole thermodynamics. I also study the symmetry and separability of the Hamilton-Jacobi and the massive Klein-Gordon equations in these Einstein-Maxwell-Chern-Simons-Gödel black hole backgrounds. PMID:18517852
Discrete-time neural inverse optimal control for nonlinear systems via passivation.
Ornelas-Tellez, Fernando; Sanchez, Edgar N; Loukianov, Alexander G
2012-08-01
This paper presents a discrete-time inverse optimal neural controller, which is constituted by combination of two techniques: 1) inverse optimal control to avoid solving the Hamilton-Jacobi-Bellman equation associated with nonlinear system optimal control and 2) on-line neural identification, using a recurrent neural network trained with an extended Kalman filter, in order to build a model of the assumed unknown nonlinear system. The inverse optimal controller is based on passivity theory. The applicability of the proposed approach is illustrated via simulations for an unstable nonlinear system and a planar robot. PMID:24807528
Inflationary solutions in the nonminimally coupled scalar field theory
NASA Astrophysics Data System (ADS)
Koh, Seoktae; Kim, Sang Pyo; Song, Doo Jong
2005-08-01
We study analytically and numerically the inflationary solutions for various type scalar potentials in the nonminimally coupled scalar field theory. The Hamilton-Jacobi equation is used to deal with nonlinear evolutions of inhomogeneous spacetimes and the long-wavelength approximation is employed to find the homogeneous solutions during an inflation period. The constraints that lead to a sufficient number of e-folds, a necessary condition for inflation, are found for the nonminimal coupling constant and initial conditions of the scalar field for inflation potentials. In particular, we numerically find an inflationary solution in the new inflation model of a nonminimal scalar field.
NASA Astrophysics Data System (ADS)
Wu, Shuang-Qing
2008-03-01
I present the general exact solutions for nonextremal rotating charged black holes in the Gödel universe of five-dimensional minimal supergravity theory. They are uniquely characterized by four nontrivial parameters: namely, the mass m, the charge q, the Kerr equal rotation parameter a, and the Gödel parameter j. I calculate the conserved energy, angular momenta, and charge for the solutions and show that they completely satisfy the first law of black hole thermodynamics. I also study the symmetry and separability of the Hamilton-Jacobi and the massive Klein-Gordon equations in these Einstein-Maxwell-Chern-Simons-Gödel black hole backgrounds.
Complex solutions for the scalar field model of the Universe
NASA Astrophysics Data System (ADS)
Lyons, Glenn W.
1992-08-01
The Hartle-Hawking proposal is implemented for Hawking's scalar field model of the Universe. For this model the complex saddle-point geometries required by the semiclassical approximation to the path integral cannot simply be deformed into real Euclidean and real Lorentzian sections. Approximate saddle points are constructed which are fully complex and have contours of real Lorentzian evolution. The semiclassical wave function is found to give rise to classical spacetimes at late times and extra terms in the Hamilton-Jacobi equation do not contribute significantly to the potential.
Nonsingularity in the no-boundary Universe.
NASA Astrophysics Data System (ADS)
Wu, Zhongchao
1997-05-01
In the no-boundary Universe of Hartle and Hawking (1982), the path integral for the quantum state of the Universe must be summed only over nonsingular histories. If the quantum corrections to the Hamilton-Jacobi equation in the interpretation of the wave packet is taken into account, then all classical trajectories should be nonsingular. The quantum behaviour of the classical singularity in the S1×Sm model (m ≥ 2) is also clarified. It is argued that the Universe should evolve from the zero momentum state, instead from a zero volume state, to a 3-geometry state.
Equatorial gravitational lensing by accelerating and rotating black hole with NUT parameter
NASA Astrophysics Data System (ADS)
Sharif, M.; Iftikhar, Sehrish
2016-01-01
This paper is devoted to study equatorial gravitational lensing in accelerating and rotating black hole with a NUT parameter in the strong field limit. For this purpose, we first calculate null geodesic equation using the Hamilton-Jacobi separation method. We then numerically obtain deflection angle and deflection coefficients which depend on acceleration and spin parameter of the black hole. We also investigate observables in the strong field limit by taking the example of a black hole in the center of galaxy. It is concluded that acceleration parameter has a significant effect on the strong field lensing in the equatorial plane.
Cosmic censorship of rotating Anti-de Sitter black hole
NASA Astrophysics Data System (ADS)
Gwak, Bogeun; Lee, Bum-Hoon
2016-02-01
We test the validity of cosmic censorship in the rotating anti-de Sitter black hole. For this purpose, we investigate whether the extremal black hole can be overspun by the particle absorption. The particle absorption will change the mass and angular momentum of the black hole, which is analyzed using the Hamilton-Jacobi equations consistent with the laws of thermodynamics. We have found that the mass of the extremal black hole increases more than the angular momentum. Therefore, the outer horizon of the black hole still exists, and cosmic censorship is valid.
NASA Astrophysics Data System (ADS)
Övgün, Ali; Jusufi, Kimet
2016-05-01
In this paper, we investigate the tunneling process of charged massive bosons W^{±} (spin-1 particles) from noncommutative charged black holes such as charged RN black holes and charged BTZ black holes. By applying the WKB approximation and by using the Hamilton-Jacobi equation we derive the tunneling rate and the corresponding Hawking temperature for those black holes configuration. Furthermore, we show the quantum gravity effects using the GUP on the Hawking temperature for the noncommutative RN black holes. The tunneling rate shows that the radiation deviates from pure thermality and is consistent with an underlying unitary theory.
NASA Technical Reports Server (NTRS)
Hamrock, B. J.; Dowson, D.
1981-01-01
Lubricants, usually Newtonian fluids, are assumed to experience laminar flow. The basic equations used to describe the flow are the Navier-Stokes equation of motion. The study of hydrodynamic lubrication is, from a mathematical standpoint, the application of a reduced form of these Navier-Stokes equations in association with the continuity equation. The Reynolds equation can also be derived from first principles, provided of course that the same basic assumptions are adopted in each case. Both methods are used in deriving the Reynolds equation, and the assumptions inherent in reducing the Navier-Stokes equations are specified. Because the Reynolds equation contains viscosity and density terms and these properties depend on temperature and pressure, it is often necessary to couple the Reynolds with energy equation. The lubricant properties and the energy equation are presented. Film thickness, a parameter of the Reynolds equation, is a function of the elastic behavior of the bearing surface. The governing elasticity equation is therefore presented.
ERIC Educational Resources Information Center
Blakley, G. R.
1982-01-01
Reviews mathematical techniques for solving systems of homogeneous linear equations and demonstrates that the algebraic method of balancing chemical equations is a matter of solving a system of homogeneous linear equations. FORTRAN programs using this matrix method to chemical equation balancing are available from the author. (JN)
Polymer quantization and the saddle point approximation of partition functions
NASA Astrophysics Data System (ADS)
Morales-Técotl, Hugo A.; Orozco-Borunda, Daniel H.; Rastgoo, Saeed
2015-11-01
The saddle point approximation of the path integral partition functions is an important way of deriving the thermodynamical properties of black holes. However, there are certain black hole models and some mathematically analog mechanical models for which this method cannot be applied directly. This is due to the fact that their action evaluated on a classical solution is not finite and its first variation does not vanish for all consistent boundary conditions. These problems can be dealt with by adding a counterterm to the classical action, which is a solution of the corresponding Hamilton-Jacobi equation. In this work we study the effects of polymer quantization on a mechanical model presenting the aforementioned difficulties and contrast it with the above counterterm method. This type of quantization for mechanical models is motivated by the loop quantization of gravity, which is known to play a role in the thermodynamics of black hole systems. The model we consider is a nonrelativistic particle in an inverse square potential, and we analyze two polarizations of the polymer quantization in which either the position or the momentum is discrete. In the former case, Thiemann's regularization is applied to represent the inverse power potential, but we still need to incorporate the Hamilton-Jacobi counterterm, which is now modified by polymer corrections. In the latter, momentum discrete case, however, such regularization could not be implemented. Yet, remarkably, owing to the fact that the position is bounded, we do not need a Hamilton-Jacobi counterterm in order to have a well-defined saddle point approximation. Further developments and extensions are commented upon in the discussion.
Kinetic energy equations for the average-passage equation system
NASA Technical Reports Server (NTRS)
Johnson, Richard W.; Adamczyk, John J.
1989-01-01
Important kinetic energy equations derived from the average-passage equation sets are documented, with a view to their interrelationships. These kinetic equations may be used for closing the average-passage equations. The turbulent kinetic energy transport equation used is formed by subtracting the mean kinetic energy equation from the averaged total instantaneous kinetic energy equation. The aperiodic kinetic energy equation, averaged steady kinetic energy equation, averaged unsteady kinetic energy equation, and periodic kinetic energy equation, are also treated.
Quantum demolition filtering and optimal control of unstable systems.
Belavkin, V P
2012-11-28
A brief account of the quantum information dynamics and dynamical programming methods for optimal control of quantum unstable systems is given to both open loop and feedback control schemes corresponding respectively to deterministic and stochastic semi-Markov dynamics of stable or unstable systems. For the quantum feedback control scheme, we exploit the separation theorem of filtering and control aspects as in the usual case of quantum stable systems with non-demolition observation. This allows us to start with the Belavkin quantum filtering equation generalized to demolition observations and derive the generalized Hamilton-Jacobi-Bellman equation using standard arguments of classical control theory. This is equivalent to a Hamilton-Jacobi equation with an extra linear dissipative term if the control is restricted to Hamiltonian terms in the filtering equation. An unstable controlled qubit is considered as an example throughout the development of the formalism. Finally, we discuss optimum observation strategies to obtain a pure quantum qubit state from a mixed one. PMID:23091216
Towards Quantum Cybernetics:. Optimal Feedback Control in Quantum Bio Informatics
NASA Astrophysics Data System (ADS)
Belavkin, V. P.
2009-02-01
A brief account of the quantum information dynamics and dynamical programming methods for the purpose of optimal control in quantum cybernetics with convex constraints and cońcave cost and bequest functions of the quantum state is given. Consideration is given to both open loop and feedback control schemes corresponding respectively to deterministic and stochastic semi-Markov dynamics of stable or unstable systems. For the quantum feedback control scheme with continuous observations we exploit the separation theorem of filtering and control aspects for quantum stochastic micro-dynamics of the total system. This allows to start with the Belavkin quantum filtering equation and derive the generalized Hamilton-Jacobi-Bellman equation using standard arguments of classical control theory. This is equivalent to a Hamilton-Jacobi equation with an extra linear dissipative term if the control is restricted to only Hamiltonian terms in the filtering equation. A controlled qubit is considered as an example throughout the development of the formalism. Finally, we discuss optimum observation strategies to obtain a pure quantum qubit state from a mixed one.
Reflections on Chemical Equations.
ERIC Educational Resources Information Center
Gorman, Mel
1981-01-01
The issue of how much emphasis balancing chemical equations should have in an introductory chemistry course is discussed. The current heavy emphasis on finishing such equations is viewed as misplaced. (MP)
Interpretation of Bernoulli's Equation.
ERIC Educational Resources Information Center
Bauman, Robert P.; Schwaneberg, Rolf
1994-01-01
Discusses Bernoulli's equation with regards to: horizontal flow of incompressible fluids, change of height of incompressible fluids, gases, liquids and gases, and viscous fluids. Provides an interpretation, properties, terminology, and applications of Bernoulli's equation. (MVL)
Trajectory-based modeling of fluid transport in a medium with smoothly varying heterogeneity
NASA Astrophysics Data System (ADS)
Vasco, D. W.; Pride, Steven R.; Commer, Michael
2016-04-01
Using an asymptotic methodology, valid in the presence of smoothly varying heterogeneity and prescribed boundaries, we derive a trajectory-based solution for tracer transport. The analysis produces a Hamilton-Jacobi partial differential equation for the phase of the propagating tracer front. The trajectories follow from the characteristic equations that are equivalent to the Hamilton-Jacobi equation. The paths are determined by the fluid velocity field, the total porosity, and the dispersion tensor. Due to their dependence upon the local hydrodynamic dispersion, they differ from conventional streamlines. This difference is borne out in numerical calculations for both uniform and dipole flow fields. In an application to the computational X-ray imaging of a saline tracer test, we illustrate that the trajectories may serve as the basis for a form of tracer tomography. In particular, we use the onset time of a change in attenuation for each volume element of the X-ray image as a measure of the arrival time of the saline tracer. The arrival times are used to image the spatial variation of the effective hydraulic conductivity within the laboratory sample.
Lifshitz holography: the whole shebang
NASA Astrophysics Data System (ADS)
Chemissany, Wissam; Papadimitriou, Ioannis
2015-01-01
We provide a general algorithm for constructing the holographic dictionary for any asymptotically locally Lifshitz background, with or without hyperscaling violation, and for any values of the dynamical exponents z and θ, as well as the vector hyperscaling violating exponent [1, 2], that are compatible with the null energy condition. The analysis is carried out for a very general bottom up model of gravity coupled to a massive vector field and a dilaton with arbitrary scalar couplings. The solution of the radial Hamilton-Jacobi equation is obtained recursively in the form of a graded expansion in eigenfunctions of two commuting operators [3], which are the appropriate generalization of the dilatation operator for non scale invariant and Lorentz violating boundary conditions. The Fefferman-Graham expansions, the sources and 1-point functions of the dual operators, the Ward identities, as well as the local counterterms required for holographic renormalization all follow from this asymptotic solution of the radial Hamilton-Jacobi equation. We also find a family of exact backgrounds with z > 1 and θ > 0 corresponding to a marginal deformation shifting the vector hyperscaling violating parameter and we present an example where the conformal anomaly contains the only z = 2 conformal invariant in d = 2 with four spatial derivatives.
Period of orbit as a test of the general theory of relativity
Preston, H.G.
1984-01-01
The formalism of the general relativity Hamilton-Jacobi equation suggests four types of corrections due to general relativity that might be the basis of a test of the predictions of general relativity. Tests has been conducted on only three of these corrections. The fourth correction is to period of orbit of a test mass. The period of orbit is derived here for the circular case using the geodesic equation. Using the Hamilton-Jacobi formalism and phase-integral canonical transforms, the general relativity form of Kepler's third law is derived and is valid for arbitrary eccentricities. The corrections due to general relativity on the period of the planets are calculated. For Mercury, the correction is 1.4 seconds. The time-delay experiment conducted at JPL and MIT numerically integrated the motion of six planets and least-square adjusted the orbital constants over a period of 80 years. This experiment did not detect the period of orbit because of the particular formalism used in the least square adjustment procedure. A modification of the least-square data-reduction program was made and programmed by the author. The modified program, as designed, will accurately measure the period of orbit.
NASA Astrophysics Data System (ADS)
Kostov, Ivan; Serban, Didina; Volin, Dmytro
2008-08-01
We give a realization of the Beisert, Eden and Staudacher equation for the planar Script N = 4 supersymetric gauge theory which seems to be particularly useful to study the strong coupling limit. We are using a linearized version of the BES equation as two coupled equations involving an auxiliary density function. We write these equations in terms of the resolvents and we transform them into a system of functional, instead of integral, equations. We solve the functional equations perturbatively in the strong coupling limit and reproduce the recursive solution obtained by Basso, Korchemsky and Kotański. The coefficients of the strong coupling expansion are fixed by the analyticity properties obeyed by the resolvents.
Einstein equation at singularities
NASA Astrophysics Data System (ADS)
Stoica, Ovidiu-Cristinel
2014-02-01
Einstein's equation is rewritten in an equivalent form, which remains valid at the singularities in some major cases. These cases include the Schwarzschild singularity, the Friedmann-Lemaître-Robertson-Walker Big Bang singularity, isotropic singularities, and a class of warped product singularities. This equation is constructed in terms of the Ricci part of the Riemann curvature (as the Kulkarni-Nomizu product between Einstein's equation and the metric tensor).
Solving Ordinary Differential Equations
NASA Technical Reports Server (NTRS)
Krogh, F. T.
1987-01-01
Initial-value ordinary differential equation solution via variable order Adams method (SIVA/DIVA) package is collection of subroutines for solution of nonstiff ordinary differential equations. There are versions for single-precision and double-precision arithmetic. Requires fewer evaluations of derivatives than other variable-order Adams predictor/ corrector methods. Option for direct integration of second-order equations makes integration of trajectory problems significantly more efficient. Written in FORTRAN 77.
Exact Approach to Inflationary Universe Models
NASA Astrophysics Data System (ADS)
del Campo, Sergio
In this chapter we introduce a study of inflationary universe models that are characterized by a single scalar inflation field . The study of these models is based on two dynamical equations: one corresponding to the Klein-Gordon equation for the inflaton field and the other to a generalized Friedmann equation. After describing the kinematics and dynamics of the models under the Hamilton-Jacobi scheme, we determine in some detail scalar density perturbations and relic gravitational waves. We also introduce the study of inflation under the hierarchy of the slow-roll parameters together with the flow equations. We apply this approach to the modified Friedmann equation that we call the Friedmann-Chern-Simons equation, characterized by F(H) = H^2- α H4, and the brane-world inflationary models expressed by the modified Friedmann equation.
Yagi, M.; Horton, W. )
1994-07-01
A set of reduced Braginskii equations is derived without assuming flute ordering and the Boussinesq approximation. These model equations conserve the physical energy. It is crucial at finite [beta] that the perpendicular component of Ohm's law be solved to ensure [del][center dot][bold j]=0 for energy conservation.
Uniqueness of Maxwell's Equations.
ERIC Educational Resources Information Center
Cohn, Jack
1978-01-01
Shows that, as a consequence of two feasible assumptions and when due attention is given to the definition of charge and the fields E and B, the lowest-order equations that these two fields must satisfy are Maxwell's equations. (Author/GA)
Octonic Massive Field Equations
NASA Astrophysics Data System (ADS)
Demir, Süleyman; Kekeç, Seray
2016-07-01
In the present paper we propose the octonic form of massive field equations based on the analogy with electromagnetism and linear gravity. Using the advantages of octon algebra the Maxwell-Dirac-Proca equations have been reformulated in compact and elegant way. The energy-momentum relations for massive field are discussed.
Adshead, Peter; Easther, Richard E-mail: richard.easther@yale.edu
2008-10-15
We analyze the theoretical limits on slow roll reconstruction, an optimal algorithm for recovering the inflaton potential (assuming a single-field slow roll scenario) from observational data. Slow roll reconstruction is based upon the Hamilton-Jacobi formulation of the inflationary dynamics. We show that at low inflationary scales the Hamilton-Jacobi equations simplify considerably. We provide a new classification scheme for inflationary models, based solely on the number of parameters needed to specify the potential, and provide forecasts for the bounds on the slow roll parameters from future data sets. A minimal running of the spectral index, induced solely by the first two slow roll parameters ({epsilon} and {eta}), appears to be effectively undetectable by realistic cosmic microwave background (CMB) experiments. However, since the ability to detect any running increases with the lever arm in comoving wavenumber, we conjecture that high redshift 21 cm data may allow tests of second-order consistency conditions on inflation. Finally, we point out that the second-order corrections to the spectral index are correlated with the inflationary scale, and thus the amplitude of the CMB B mode.
NASA Astrophysics Data System (ADS)
Kuksin, Sergei; Maiocchi, Alberto
In this chapter we present a general method of constructing the effective equation which describes the behavior of small-amplitude solutions for a nonlinear PDE in finite volume, provided that the linear part of the equation is a hamiltonian system with a pure imaginary discrete spectrum. The effective equation is obtained by retaining only the resonant terms of the nonlinearity (which may be hamiltonian, or may be not); the assertion that it describes the limiting behavior of small-amplitude solutions is a rigorous mathematical theorem. In particular, the method applies to the three- and four-wave systems. We demonstrate that different possible types of energy transport are covered by this method, depending on whether the set of resonances splits into finite clusters (this happens, e.g. in case of the Charney-Hasegawa-Mima equation), or is connected (this happens, e.g. in the case of the NLS equation if the space-dimension is at least two). For equations of the first type the energy transition to high frequencies does not hold, while for equations of the second type it may take place. Our method applies to various weakly nonlinear wave systems, appearing in plasma, meteorology and oceanography.
Nonlinear gyrokinetic equations
Dubin, D.H.E.; Krommes, J.A.; Oberman, C.; Lee, W.W.
1983-03-01
Nonlinear gyrokinetic equations are derived from a systematic Hamiltonian theory. The derivation employs Lie transforms and a noncanonical perturbation theory first used by Littlejohn for the simpler problem of asymptotically small gyroradius. For definiteness, we emphasize the limit of electrostatic fluctuations in slab geometry; however, there is a straight-forward generalization to arbitrary field geometry and electromagnetic perturbations. An energy invariant for the nonlinear system is derived, and various of its limits are considered. The weak turbulence theory of the equations is examined. In particular, the wave kinetic equation of Galeev and Sagdeev is derived from an asystematic truncation of the equations, implying that this equation fails to consider all gyrokinetic effects. The equations are simplified for the case of small but finite gyroradius and put in a form suitable for efficient computer simulation. Although it is possible to derive the Terry-Horton and Hasegawa-Mima equations as limiting cases of our theory, several new nonlinear terms absent from conventional theories appear and are discussed.
Nonlinear ordinary difference equations
NASA Technical Reports Server (NTRS)
Caughey, T. K.
1979-01-01
Future space vehicles will be relatively large and flexible, and active control will be necessary to maintain geometrical configuration. While the stresses and strains in these space vehicles are not expected to be excessively large, their cumulative effects will cause significant geometrical nonlinearities to appear in the equations of motion, in addition to the nonlinearities caused by material properties. Since the only effective tool for the analysis of such large complex structures is the digital computer, it will be necessary to gain a better understanding of the nonlinear ordinary difference equations which result from the time discretization of the semidiscrete equations of motion for such structures.
NASA Astrophysics Data System (ADS)
Pierret, Frédéric
2016-02-01
We derived the equations of Celestial Mechanics governing the variation of the orbital elements under a stochastic perturbation, thereby generalizing the classical Gauss equations. Explicit formulas are given for the semimajor axis, the eccentricity, the inclination, the longitude of the ascending node, the pericenter angle, and the mean anomaly, which are expressed in term of the angular momentum vector H per unit of mass and the energy E per unit of mass. Together, these formulas are called the stochastic Gauss equations, and they are illustrated numerically on an example from satellite dynamics.
Nonlinear differential equations
Dresner, L.
1988-01-01
This report is the text of a graduate course on nonlinear differential equations given by the author at the University of Wisconsin-Madison during the summer of 1987. The topics covered are: direction fields of first-order differential equations; the Lie (group) theory of ordinary differential equations; similarity solutions of second-order partial differential equations; maximum principles and differential inequalities; monotone operators and iteration; complementary variational principles; and stability of numerical methods. The report should be of interest to graduate students, faculty, and practicing scientists and engineers. No prior knowledge is required beyond a good working knowledge of the calculus. The emphasis is on practical results. Most of the illustrative examples are taken from the fields of nonlinear diffusion, heat and mass transfer, applied superconductivity, and helium cryogenics.
Relativistic Guiding Center Equations
White, R. B.; Gobbin, M.
2014-10-01
In toroidal fusion devices it is relatively easy that electrons achieve relativistic velocities, so to simulate runaway electrons and other high energy phenomena a nonrelativistic guiding center formalism is not sufficient. Relativistic guiding center equations including flute mode time dependent field perturbations are derived. The same variables as used in a previous nonrelativistic guiding center code are adopted, so that a straightforward modifications of those equations can produce a relativistic version.
SIMULTANEOUS DIFFERENTIAL EQUATION COMPUTER
Collier, D.M.; Meeks, L.A.; Palmer, J.P.
1960-05-10
A description is given for an electronic simulator for a system of simultaneous differential equations, including nonlinear equations. As a specific example, a homogeneous nuclear reactor system including a reactor fluid, heat exchanger, and a steam boiler may be simulated, with the nonlinearity resulting from a consideration of temperature effects taken into account. The simulator includes three operational amplifiers, a multiplier, appropriate potential sources, and interconnecting R-C networks.
Set Equation Transformation System.
2002-03-22
Version 00 SETS is used for symbolic manipulation of Boolean equations, particularly the reduction of equations by the application of Boolean identities. It is a flexible and efficient tool for performing probabilistic risk analysis (PRA), vital area analysis, and common cause analysis. The equation manipulation capabilities of SETS can also be used to analyze noncoherent fault trees and determine prime implicants of Boolean functions, to verify circuit design implementation, to determine minimum cost fire protectionmore » requirements for nuclear reactor plants, to obtain solutions to combinatorial optimization problems with Boolean constraints, and to determine the susceptibility of a facility to unauthorized access through nullification of sensors in its protection system. Two auxiliary programs, SEP and FTD, are included. SEP performs the quantitative analysis of reduced Boolean equations (minimal cut sets) produced by SETS. The user can manipulate and evaluate the equations to find the probability of occurrence of any desired event and to produce an importance ranking of the terms and events in an equation. FTD is a fault tree drawing program which uses the proprietary ISSCO DISSPLA graphics software to produce an annotated drawing of a fault tree processed by SETS. The DISSPLA routines are not included.« less
New class of cosmological solutions for a self-interacting scalar field
NASA Astrophysics Data System (ADS)
Chaadaev, A. A.; Chervon, S. V.
2013-12-01
New cosmological solutions are found to the system of Einstein scalar field equations using the scalar field φ as the argument. For a homogeneous and isotropic Universe, the system of equations is reduced to two equations, one of which is an equation of Hamilton-Jacobi type. Using the hyperbolically parameterized representation of this equation together with the consistency condition, explicit dependences of the potential V of the scalar field and the Hubble parameter H on φ are obtained. The dependences of the scalar field and the scale factor a on cosmic time t have also been found. It is shown that this scenario corresponds to the evolution of the Universe with accelerated expansion out to times distant from the initial singularity.
Introducing Chemical Formulae and Equations.
ERIC Educational Resources Information Center
Dawson, Chris; Rowell, Jack
1979-01-01
Discusses when the writing of chemical formula and equations can be introduced in the school science curriculum. Also presents ways in which formulae and equations learning can be aided and some examples for balancing and interpreting equations. (HM)
The Bernoulli-Poiseuille Equation.
ERIC Educational Resources Information Center
Badeer, Henry S.; Synolakis, Costas E.
1989-01-01
Describes Bernoulli's equation and Poiseuille's equation for fluid dynamics. Discusses the application of the combined Bernoulli-Poiseuille equation in real flows, such as viscous flows under gravity and acceleration. (YP)
Parallel tridiagonal equation solvers
NASA Technical Reports Server (NTRS)
Stone, H. S.
1974-01-01
Three parallel algorithms were compared for the direct solution of tridiagonal linear systems of equations. The algorithms are suitable for computers such as ILLIAC 4 and CDC STAR. For array computers similar to ILLIAC 4, cyclic odd-even reduction has the least operation count for highly structured sets of equations, and recursive doubling has the least count for relatively unstructured sets of equations. Since the difference in operation counts for these two algorithms is not substantial, their relative running times may be more related to overhead operations, which are not measured in this paper. The third algorithm, based on Buneman's Poisson solver, has more arithmetic operations than the others, and appears to be the least favorable. For pipeline computers similar to CDC STAR, cyclic odd-even reduction appears to be the most preferable algorithm for all cases.
Nonlocal electrical diffusion equation
NASA Astrophysics Data System (ADS)
Gómez-Aguilar, J. F.; Escobar-Jiménez, R. F.; Olivares-Peregrino, V. H.; Benavides-Cruz, M.; Calderón-Ramón, C.
2016-07-01
In this paper, we present an analysis and modeling of the electrical diffusion equation using the fractional calculus approach. This alternative representation for the current density is expressed in terms of the Caputo derivatives, the order for the space domain is 0<β≤1 and for the time domain is 0<γ≤2. We present solutions for the full fractional equation involving space and time fractional derivatives using numerical methods based on Fourier variable separation. The case with spatial fractional derivatives leads to Levy flight type phenomena, while the time fractional equation is related to sub- or super diffusion. We show that the mathematical concept of fractional derivatives can be useful to understand the behavior of semiconductors, the design of solar panels, electrochemical phenomena and the description of anomalous complex processes.
NASA Astrophysics Data System (ADS)
Popescu, Mihai; Dumitrache, Alexandru
2011-05-01
This study refers to minimization of quadratic functionals in infinite time. The coefficients of the quadratic form are quadratic matrix, function of the state variable. Dynamic constraints are represented by bilinear differential systems of the form x˙=A(x)x+B(x)u,x(0)=x0. One selects an adequate factorization of A( x) such that the analyzed system should be controllable. Employing the Hamilton-Jacobi equation it results the matrix algebraic equation of Riccati associated to the optimum problem. The necessary extremum conditions determine the adjoint variables λ and the control variables u as functions of state variable, as well as the adjoint system corresponding to those functions. Thus one obtains a matrix differential equation where the solution representing the positive defined symmetric matrix P( x), verifies the Riccati algebraic equation. The stability analysis for the autonomous systems solution resulting for the determined feedback control is performed using the Liapunov function method. Finally we present certain significant cases.
NASA Astrophysics Data System (ADS)
Chou, Chia-Chun
2015-08-01
The complex quantum Hamilton-Jacobi equation for the complex action is approximately solved by propagating individual Bohmian trajectories in real space. Equations of motion for the complex action and its spatial derivatives are derived through use of the derivative propagation method. We transform these equations into the arbitrary Lagrangian-Eulerian version with the grid velocity matching the flow velocity of the probability fluid. Setting higher-order derivatives equal to zero, we obtain a truncated system of equations of motion describing the rate of change in the complex action and its spatial derivatives transported along approximate Bohmian trajectories. A set of test trajectories is propagated to determine appropriate initial positions for transmitted trajectories. Computational results for transmitted wave packets and transmission probabilities are presented and analyzed for a one-dimensional Eckart barrier and a two-dimensional system involving either a thick or thin Eckart barrier along the reaction coordinate coupled to a harmonic oscillator.
Stochastic differential equations
Sobczyk, K. )
1990-01-01
This book provides a unified treatment of both regular (or random) and Ito stochastic differential equations. It focuses on solution methods, including some developed only recently. Applications are discussed, in particular an insight is given into both the mathematical structure, and the most efficient solution methods (analytical as well as numerical). Starting from basic notions and results of the theory of stochastic processes and stochastic calculus (including Ito's stochastic integral), many principal mathematical problems and results related to stochastic differential equations are expounded here for the first time. Applications treated include those relating to road vehicles, earthquake excitations and offshore structures.
NASA Technical Reports Server (NTRS)
Markley, F. Landis
1995-01-01
Kepler's Equation is solved over the entire range of elliptic motion by a fifth-order refinement of the solution of a cubic equation. This method is not iterative, and requires only four transcendental function evaluations: a square root, a cube root, and two trigonometric functions. The maximum relative error of the algorithm is less than one part in 10(exp 18), exceeding the capability of double-precision computer arithmetic. Roundoff errors in double-precision implementation of the algorithm are addressed, and procedures to avoid them are developed.
Five-dimensional Hamiltonian-Jacobi approach to relativistic quantum mechanics
Rose, Harald
2003-12-11
A novel theory is outlined for describing the dynamics of relativistic electrons and positrons. By introducing the Lorentz-invariant universal time as a fifth independent variable, the Hamilton-Jacobi formalism of classical mechanics is extended from three to four spatial dimensions. This approach allows one to incorporate gravitation and spin interactions in the extended five-dimensional Lagrangian in a covariant form. The universal time has the function of a hidden Bell parameter. By employing the method of variation with respect to the four coordinates of the particle and the components of the electromagnetic field, the path equation and the electromagnetic field produced by the charge and the spin of the moving particle are derived. In addition the covariant equations for the dynamics of the components of the spin tensor are obtained. These equations can be transformed to the familiar BMT equation in the case of homogeneous electromagnetic fields. The quantization of the five-dimensional Hamilton-Jacobi equation yields a five-dimensional spinor wave equation, which degenerates to the Dirac equation in the stationary case if we neglect gravitation. The quantity which corresponds to the probability density of standard quantum mechanics is the four-dimensional mass density which has a real physical meaning. By means of the Green method the wave equation is transformed into an integral equation enabling a covariant relativistic path integral formulation. Using this approach a very accurate approximation for the four-dimensional propagator is derived. The proposed formalism makes Dirac's hole theory obsolete and can readily be extended to many particles.
The Statistical Drake Equation
NASA Astrophysics Data System (ADS)
Maccone, Claudio
2010-12-01
We provide the statistical generalization of the Drake equation. From a simple product of seven positive numbers, the Drake equation is now turned into the product of seven positive random variables. We call this "the Statistical Drake Equation". The mathematical consequences of this transformation are then derived. The proof of our results is based on the Central Limit Theorem (CLT) of Statistics. In loose terms, the CLT states that the sum of any number of independent random variables, each of which may be ARBITRARILY distributed, approaches a Gaussian (i.e. normal) random variable. This is called the Lyapunov Form of the CLT, or the Lindeberg Form of the CLT, depending on the mathematical constraints assumed on the third moments of the various probability distributions. In conclusion, we show that: The new random variable N, yielding the number of communicating civilizations in the Galaxy, follows the LOGNORMAL distribution. Then, as a consequence, the mean value of this lognormal distribution is the ordinary N in the Drake equation. The standard deviation, mode, and all the moments of this lognormal N are also found. The seven factors in the ordinary Drake equation now become seven positive random variables. The probability distribution of each random variable may be ARBITRARY. The CLT in the so-called Lyapunov or Lindeberg forms (that both do not assume the factors to be identically distributed) allows for that. In other words, the CLT "translates" into our statistical Drake equation by allowing an arbitrary probability distribution for each factor. This is both physically realistic and practically very useful, of course. An application of our statistical Drake equation then follows. The (average) DISTANCE between any two neighboring and communicating civilizations in the Galaxy may be shown to be inversely proportional to the cubic root of N. Then, in our approach, this distance becomes a new random variable. We derive the relevant probability density
Comparison of Kernel Equating and Item Response Theory Equating Methods
ERIC Educational Resources Information Center
Meng, Yu
2012-01-01
The kernel method of test equating is a unified approach to test equating with some advantages over traditional equating methods. Therefore, it is important to evaluate in a comprehensive way the usefulness and appropriateness of the Kernel equating (KE) method, as well as its advantages and disadvantages compared with several popular item…
Accumulative Equating Error after a Chain of Linear Equatings
ERIC Educational Resources Information Center
Guo, Hongwen
2010-01-01
After many equatings have been conducted in a testing program, equating errors can accumulate to a degree that is not negligible compared to the standard error of measurement. In this paper, the author investigates the asymptotic accumulative standard error of equating (ASEE) for linear equating methods, including chained linear, Tucker, and…
Parallel Multigrid Equation Solver
2001-09-07
Prometheus is a fully parallel multigrid equation solver for matrices that arise in unstructured grid finite element applications. It includes a geometric and an algebraic multigrid method and has solved problems of up to 76 mullion degrees of feedom, problems in linear elasticity on the ASCI blue pacific and ASCI red machines.
Do Differential Equations Swing?
ERIC Educational Resources Information Center
Maruszewski, Richard F., Jr.
2006-01-01
One of the units of in a standard differential equations course is a discussion of the oscillatory motion of a spring and the associated material on forcing functions and resonance. During the presentation on practical resonance, the instructor may tell students that it is similar to when they take their siblings to the playground and help them on…
Modelling by Differential Equations
ERIC Educational Resources Information Center
Chaachoua, Hamid; Saglam, Ayse
2006-01-01
This paper aims to show the close relation between physics and mathematics taking into account especially the theory of differential equations. By analysing the problems posed by scientists in the seventeenth century, we note that physics is very important for the emergence of this theory. Taking into account this analysis, we show the…
ERIC Educational Resources Information Center
Fay, Temple H.
2010-01-01
Through numerical investigations, we study examples of the forced quadratic spring equation [image omitted]. By performing trial-and-error numerical experiments, we demonstrate the existence of stability boundaries in the phase plane indicating initial conditions yielding bounded solutions, investigate the resonance boundary in the [omega]…
Generalized reduced magnetohydrodynamic equations
Kruger, S.E.
1999-02-01
A new derivation of reduced magnetohydrodynamic (MHD) equations is presented. A multiple-time-scale expansion is employed. It has the advantage of clearly separating the three time scales of the problem associated with (1) MHD equilibrium, (2) fluctuations whose wave vector is aligned perpendicular to the magnetic field, and (3) those aligned parallel to the magnetic field. The derivation is carried out without relying on a large aspect ratio assumption; therefore this model can be applied to any general configuration. By accounting for the MHD equilibrium and constraints to eliminate the fast perpendicular waves, equations are derived to evolve scalar potential quantities on a time scale associated with the parallel wave vector (shear-Alfven wave time scale), which is the time scale of interest for MHD instability studies. Careful attention is given in the derivation to satisfy energy conservation and to have manifestly divergence-free magnetic fields to all orders in the expansion parameter. Additionally, neoclassical closures and equilibrium shear flow effects are easily accounted for in this model. Equations for the inner resistive layer are derived which reproduce the linear ideal and resistive stability criterion of Glasser, Greene, and Johnson. The equations have been programmed into a spectral initial value code and run with shear flow that is consistent with the equilibrium input into the code. Linear results of tearing modes with shear flow are presented which differentiate the effects of shear flow gradients in the layer with the effects of the shear flow decoupling multiple harmonics.
Structural Equation Model Trees
ERIC Educational Resources Information Center
Brandmaier, Andreas M.; von Oertzen, Timo; McArdle, John J.; Lindenberger, Ulman
2013-01-01
In the behavioral and social sciences, structural equation models (SEMs) have become widely accepted as a modeling tool for the relation between latent and observed variables. SEMs can be seen as a unification of several multivariate analysis techniques. SEM Trees combine the strengths of SEMs and the decision tree paradigm by building tree…
Brownian motion from Boltzmann's equation.
NASA Technical Reports Server (NTRS)
Montgomery, D.
1971-01-01
Two apparently disparate lines of inquiry in kinetic theory are shown to be equivalent: (1) Brownian motion as treated by the (stochastic) Langevin equation and Fokker-Planck equation; and (2) Boltzmann's equation. The method is to derive the kinetic equation for Brownian motion from the Boltzmann equation for a two-component neutral gas by a simultaneous expansion in the density and mass ratios.
Supersymmetric fifth order evolution equations
Tian, K.; Liu, Q. P.
2010-03-08
This paper considers supersymmetric fifth order evolution equations. Within the framework of symmetry approach, we give a list containing six equations, which are (potentially) integrable systems. Among these equations, the most interesting ones include a supersymmetric Sawada-Kotera equation and a novel supersymmetric fifth order KdV equation. For the latter, we supply some properties such as a Hamiltonian structures and a possible recursion operator.
Charged particle in higher dimensional weakly charged rotating black hole spacetime
Frolov, Valeri P.; Krtous, Pavel
2011-01-15
We study charged particle motion in weakly charged higher dimensional black holes. To describe the electromagnetic field we use a test field approximation and the higher dimensional Kerr-NUT-(A)dS metric as a background geometry. It is shown that for a special configuration of the electromagnetic field, the equations of motion of charged particles are completely integrable. The vector potential of such a field is proportional to one of the Killing vectors (called a primary Killing vector) from the 'Killing tower' of symmetry generating objects which exists in the background geometry. A free constant in the definition of the adopted electromagnetic potential is proportional to the electric charge of the higher dimensional black hole. The full set of independent conserved quantities in involution is found. We demonstrate that Hamilton-Jacobi equations are separable, as is the corresponding Klein-Gordon equation and its symmetry operators.
NASA Technical Reports Server (NTRS)
Oliger, Joseph
1997-01-01
Topics considered include: high-performance computing; cognitive and perceptual prostheses (computational aids designed to leverage human abilities); autonomous systems. Also included: development of a 3D unstructured grid code based on a finite volume formulation and applied to the Navier-stokes equations; Cartesian grid methods for complex geometry; multigrid methods for solving elliptic problems on unstructured grids; algebraic non-overlapping domain decomposition methods for compressible fluid flow problems on unstructured meshes; numerical methods for the compressible navier-stokes equations with application to aerodynamic flows; research in aerodynamic shape optimization; S-HARP: a parallel dynamic spectral partitioner; numerical schemes for the Hamilton-Jacobi and level set equations on triangulated domains; application of high-order shock capturing schemes to direct simulation of turbulence; multicast technology; network testbeds; supercomputer consolidation project.
A Sliding Mode Control with Optimized Sliding Surface for Aircraft Pitch Axis Control System
NASA Astrophysics Data System (ADS)
Lee, Sangchul; Kim, Kwangjin; Kim, Youdan
A sliding mode controller with an optimized sliding surface is proposed for an aircraft control system. The quadratic type of performance index for minimizing the angle of attack and the angular rate of the aircraft in the longitudinal motion is used to design the sliding surface. For optimization of the sliding surface, a Hamilton-Jacobi-Bellman (HJB) equation is formulated and it is solved through a numerical algorithm using a Generalized HJB (GHJB) equation and the Galerkin spectral method. The solution of this equation denotes a nonlinear sliding surface, on which the trajectory of the system approximately satisfies the optimality condition. Numerical simulation is performed for a nonlinear aircraft model with an optimized sliding surface and a simple linear sliding surface. The simulation result demonstrates that the proposed controller can be effectively applied to the longitudinal maneuver of an aircraft.
Coherent distributions for the rigid rotator
NASA Astrophysics Data System (ADS)
Grigorescu, Marius
2016-06-01
Coherent solutions of the classical Liouville equation for the rigid rotator are presented as positive phase-space distributions localized on the Lagrangian submanifolds of Hamilton-Jacobi theory. These solutions become Wigner-type quasiprobability distributions by a formal discretization of the left-invariant vector fields from their Fourier transform in angular momentum. The results are consistent with the usual quantization of the anisotropic rotator, but the expected value of the Hamiltonian contains a finite "zero point" energy term. It is shown that during the time when a quasiprobability distribution evolves according to the Liouville equation, the related quantum wave function should satisfy the time-dependent Schrödinger equation.
Method to describe stochastic dynamics using an optimal coordinate.
Krivov, Sergei V
2013-12-01
A general method to describe the stochastic dynamics of Markov processes is suggested. The method aims to solve three related problems: the determination of an optimal coordinate for the description of stochastic dynamics; the reconstruction of time from an ensemble of stochastic trajectories; and the decomposition of stationary stochastic dynamics into eigenmodes which do not decay exponentially with time. The problems are solved by introducing additive eigenvectors which are transformed by a stochastic matrix in a simple way - every component is translated by a constant distance. Such solutions have peculiar properties. For example, an optimal coordinate for stochastic dynamics with detailed balance is a multivalued function. An optimal coordinate for a random walk on a line corresponds to the conventional eigenvector of the one-dimensional Dirac equation. The equation for the optimal coordinate in a slowly varying potential reduces to the Hamilton-Jacobi equation for the action function. PMID:24483410
Unstructured grid generation using the distance function
NASA Technical Reports Server (NTRS)
Bihari, Barna L.; Chakravarthy, Sukumar R.
1991-01-01
A new class of methods for obtaining level sets to generate unstructured grids is presented. The consecutive grid levels are computed using the distance functions, which corresponds to solving the Hamilton-Jacobi equations representing the equations of motion of fronts propagating with curvature-dependent speed. The relationship between the distance function and the governing equations will be discussed as well as its application to generating grids. Multi-ply connected domains and complex geometries are handled naturally, with a straightforward generalization to several space dimensions. The grid points for the unstructured grid are obtained simultaneously with the grid levels. The search involved in checking for overlapping triangles is minimized by triangulating the entire domain one level at a time.
Crystal growth inside an octant.
Olejarz, Jason; Krapivsky, P L
2013-08-01
We study crystal growth inside an infinite octant on a cubic lattice. The growth proceeds through the deposition of elementary cubes into inner corners. After rescaling by the characteristic size, the interface becomes progressively more deterministic in the long-time limit. Utilizing known results for the crystal growth inside a two-dimensional corner, we propose a hyperbolic partial differential equation for the evolution of the limiting shape. This equation is interpreted as a Hamilton-Jacobi equation, which helps in finding an analytical solution. Simulations of the growth process are in excellent agreement with analytical predictions. We then study the evolution of the subleading correction to the volume of the crystal, the asymptotic growth of the variance of the volume of the crystal, and the total number of inner and outer corners. We also show how to generalize the results to arbitrary spatial dimension. PMID:24032777
Integrability of some charged rotating supergravity black hole solutions in four and five dimensions
NASA Astrophysics Data System (ADS)
Vasudevan, Muraari
2005-09-01
We study the integrability of geodesic flow in the background of some recently discovered charged rotating solutions of supergravity in four and five dimensions. Specifically, we work with the gauged multicharge Taub-NUT-Kerr-(anti-)de Sitter metric in four dimensions, and the U(1) 3 gauged charged-Kerr-(anti-)de Sitter black hole solution of N = 2 supergravity in five dimensions. We explicitly construct the nontrivial irreducible Killing tensors that permit separation of the Hamilton-Jacobi equation in these spacetimes. These results prove integrability for a large class of previously known supergravity solutions, including several BPS solitonic states. We also derive first-order equations of motion for particles in these backgrounds and examine some of their properties. Finally, we also examine the Klein-Gordon equation for a scalar field in these spacetimes and demonstrate separability.
NASA Astrophysics Data System (ADS)
Vasudevan, Muraari; Stevens, Kory A.
2005-12-01
We study the Hamilton-Jacobi and massive Klein-Gordon equations in the general Kerr-(Anti) de Sitter black hole background in all dimensions. Complete separation of both equations is carried out in cases when there are two sets of equal black hole rotation parameters. We analyze explicitly the symmetry properties of these backgrounds that allow for this Liouville integrability and construct a nontrivial irreducible Killing tensor associated with the enlarged symmetry group which permits separation. We also derive first-order equations of motion for particles in these backgrounds and examine some of their properties. This work greatly generalizes previously known results for both the Myers-Perry metrics, and the Kerr-(Anti) de Sitter metrics in higher dimensions.
Wave model for conservative bound systems
Popa, Alexandru
2005-06-22
In the hidden variable theory, Bohm proved a connection between the Schroedinger and Hamilton-Jacobi equations and showed the existence of classical paths, for which the generalized Bohr quantization condition is valid. In this paper we prove similar properties, starting from the equivalence between the Schroedinger and wave equations in the case of the conservative bound systems. Our approach is based on the equations and postulates of quantum mechanics without using any additional postulate. Like in the hidden variable theory, the above properties are proven without using the approximation of geometrical optics or the semiclassical approximation. Since the classical paths have only a mathematical significance in our analysis, our approach is consistent with the postulates of quantum mechanics.
The equivalence principle of quantum mechanics: Uniqueness theorem
Faraggi, A.E.; Matone, M.
1997-10-28
Recently the authors showed that the postulated diffeomorphic equivalence of states implies quantum mechanics. This approach takes the canonical variables to be dependent by the relation p = {partial_derivative}{sub q}S{sub 0} and exploits a basic GL(2,C)-symmetry which underlies the canonical formalism. In particular, they looked for the special transformations leading to the free system with vanishing energy. Furthermore, they saw that while on the one hand the equivalence principle cannot be consistently implemented in classical mechanics, on the other it naturally led to the quantum analogue of the Hamilton-Jacobi equation, thus implying the Schroedinger equation. In this letter they show that actually the principle uniquely leads to this solution. The authors also express the canonical and Schroedinger equations by means of the brackets recently introduced in the framework of N = 2 SYM. These brackets are the analogue of the Poisson brackets with the canonical variables taken as dependent.
Nikolaevskiy equation with dispersion.
Simbawa, Eman; Matthews, Paul C; Cox, Stephen M
2010-03-01
The Nikolaevskiy equation was originally proposed as a model for seismic waves and is also a model for a wide variety of systems incorporating a neutral "Goldstone" mode, including electroconvection and reaction-diffusion systems. It is known to exhibit chaotic dynamics at the onset of pattern formation, at least when the dispersive terms in the equation are suppressed, as is commonly the practice in previous analyses. In this paper, the effects of reinstating the dispersive terms are examined. It is shown that such terms can stabilize some of the spatially periodic traveling waves; this allows us to study the loss of stability and transition to chaos of the waves. The secondary stability diagram ("Busse balloon") for the traveling waves can be remarkably complicated. PMID:20365845
Causal electromagnetic interaction equations
Zinoviev, Yury M.
2011-02-15
For the electromagnetic interaction of two particles the relativistic causal quantum mechanics equations are proposed. These equations are solved for the case when the second particle moves freely. The initial wave functions are supposed to be smooth and rapidly decreasing at the infinity. This condition is important for the convergence of the integrals similar to the integrals of quantum electrodynamics. We also consider the singular initial wave functions in the particular case when the second particle mass is equal to zero. The discrete energy spectrum of the first particle wave function is defined by the initial wave function of the free-moving second particle. Choosing the initial wave functions of the free-moving second particle it is possible to obtain a practically arbitrary discrete energy spectrum.
Generalized reduced MHD equations
Kruger, S.E.; Hegna, C.C.; Callen, J.D.
1998-07-01
A new derivation of reduced magnetohydrodynamic (MHD) equations is presented. A multiple-time-scale expansion is employed. It has the advantage of clearly separating the three time scales of the problem associated with (1) MHD equilibrium, (2) fluctuations whose wave vector is aligned perpendicular to the magnetic field, and (3) those aligned parallel to the magnetic field. The derivation is carried out without relying on a large aspect ratio assumption; therefore this model can be applied to any general toroidal configuration. By accounting for the MHD equilibrium and constraints to eliminate the fast perpendicular waves, equations are derived to evolve scalar potential quantities on a time scale associated with the parallel wave vector (shear-alfven wave time scale), which is the time scale of interest for MHD instability studies. Careful attention is given in the derivation to satisfy energy conservation and to have manifestly divergence-free magnetic fields to all orders in the expansion parameter. Additionally, neoclassical closures and equilibrium shear flow effects are easily accounted for in this model. Equations for the inner resistive layer are derived which reproduce the linear ideal and resistive stability criterion of Glasser, Greene, and Johnson.
NASA Astrophysics Data System (ADS)
Konesky, Gregory
2009-08-01
In the almost half century since the Drake Equation was first conceived, a number of profound discoveries have been made that require each of the seven variables of this equation to be reconsidered. The discovery of hydrothermal vents on the ocean floor, for example, as well as the ever-increasing extreme conditions in which life is found on Earth, suggest a much wider range of possible extraterrestrial habitats. The growing consensus that life originated very early in Earth's history also supports this suggestion. The discovery of exoplanets with a wide range of host star types, and attendant habitable zones, suggests that life may be possible in planetary systems with stars quite unlike our Sun. Stellar evolution also plays an important part in that habitable zones are mobile. The increasing brightness of our Sun over the next few billion years, will place the Earth well outside the present habitable zone, but will then encompass Mars, giving rise to the notion that some Drake Equation variables, such as the fraction of planets on which life emerges, may have multiple values.
Double-Plate Penetration Equations
NASA Technical Reports Server (NTRS)
Hayashida, K. B.; Robinson, J. H.
2000-01-01
This report compares seven double-plate penetration predictor equations for accuracy and effectiveness of a shield design. Three of the seven are the Johnson Space Center original, modified, and new Cour-Palais equations. The other four are the Nysmith, Lundeberg-Stern-Bristow, Burch, and Wilkinson equations. These equations, except the Wilkinson equation, were derived from test results, with the velocities ranging up to 8 km/sec. Spreadsheet software calculated the projectile diameters for various velocities for the different equations. The results were plotted on projectile diameter versus velocity graphs for the expected orbital debris impact velocities ranging from 2 to 15 km/sec. The new Cour-Palais double-plate penetration equation was compared to the modified Cour-Palais single-plate penetration equation. Then the predictions from each of the seven double-plate penetration equations were compared to each other for a chosen shield design. Finally, these results from the equations were compared with test results performed at the NASA Marshall Space Flight Center. Because the different equations predict a wide range of projectile diameters at any given velocity, it is very difficult to choose the "right" prediction equation for shield configurations other than those exactly used in the equations' development. Although developed for various materials, the penetration equations alone cannot be relied upon to accurately predict the effectiveness of a shield without using hypervelocity impact tests to verify the design.
Reduction operators of Burgers equation
Pocheketa, Oleksandr A.; Popovych, Roman O.
2013-01-01
The solution of the problem on reduction operators and nonclassical reductions of the Burgers equation is systematically treated and completed. A new proof of the theorem on the special “no-go” case of regular reduction operators is presented, and the representation of the coefficients of operators in terms of solutions of the initial equation is constructed for this case. All possible nonclassical reductions of the Burgers equation to single ordinary differential equations are exhaustively described. Any Lie reduction of the Burgers equation proves to be equivalent via the Hopf–Cole transformation to a parameterized family of Lie reductions of the linear heat equation. PMID:23576819
New application to Riccati equation
NASA Astrophysics Data System (ADS)
Taogetusang; Sirendaoerji; Li, Shu-Min
2010-08-01
To seek new infinite sequence of exact solutions to nonlinear evolution equations, this paper gives the formula of nonlinear superposition of the solutions and Bäcklund transformation of Riccati equation. Based on the tanh-function expansion method and homogenous balance method, new infinite sequence of exact solutions to Zakharov-Kuznetsov equation, Karamoto-Sivashinsky equation and the set of (2+1)-dimensional asymmetric Nizhnik-Novikov-Veselov equations are obtained with the aid of symbolic computation system Mathematica. The method is of significance to construct infinite sequence exact solutions to other nonlinear evolution equations.
Evaluating Equating Results: Percent Relative Error for Chained Kernel Equating
ERIC Educational Resources Information Center
Jiang, Yanlin; von Davier, Alina A.; Chen, Haiwen
2012-01-01
This article presents a method for evaluating equating results. Within the kernel equating framework, the percent relative error (PRE) for chained equipercentile equating was computed under the nonequivalent groups with anchor test (NEAT) design. The method was applied to two data sets to obtain the PRE, which can be used to measure equating…
Differential Equations Compatible with Boundary Rational qKZ Equation
NASA Astrophysics Data System (ADS)
Takeyama, Yoshihiro
2011-10-01
We give diffierential equations compatible with the rational qKZ equation with boundary reflection. The total system contains the trigonometric degeneration of the bispectral qKZ equation of type (Cěen, Cn) which in the case of type GLn was studied by van Meer and Stokman. We construct an integral formula for solutions to our compatible system in a special case.
The compressible adjoint equations in geodynamics: equations and numerical assessment
NASA Astrophysics Data System (ADS)
Ghelichkhan, Siavash; Bunge, Hans-Peter
2016-04-01
The adjoint method is a powerful means to obtain gradient information in a mantle convection model relative to past flow structure. While the adjoint equations in geodynamics have been derived for the conservation equations of mantle flow in their incompressible form, the applicability of this approximation to Earth is limited, because density increases by almost a factor of two from the surface to the Core Mantle Boundary. Here we introduce the compressible adjoint equations for the conservation equations in the anelastic-liquid approximation. Our derivation applies an operator formulation in Hilbert spaces, to connect to recent work in seismology (Fichtner et al (2006)) and geodynamics (Horbach et al (2014)), where the approach was used to derive the adjoint equations for the wave equation and incompressible mantle flow. We present numerical tests of the newly derived equations based on twin experiments, focusing on three simulations. A first, termed Compressible, assumes the compressible forward and adjoint equations, and represents the consistent means of including compressibility effects. A second, termed Mixed, applies the compressible forward equation, but ignores compressibility effects in the adjoint equations, where the incompressible equations are used instead. A third simulation, termed Incompressible, neglects compressibility effects entirely in the forward and adjoint equations relative to the reference twin. The compressible and mixed formulations successfully restore earlier mantle flow structure, while the incompressible formulation yields noticeable artifacts. Our results suggest the use of a compressible formulation, when applying the adjoint method to seismically derived mantle heterogeneity structure.
Estimating Equating Error in Observed-Score Equating. Research Report.
ERIC Educational Resources Information Center
van der Linden, Wim J.
Traditionally, error in equating observed scores on two versions of a test is defined as the difference between the transformations that equate the quantiles of their distributions in the sample and in the population of examinees. This definition underlies, for example, the well-known approximation to the standard error of equating by Lord (1982).…
NASA Astrophysics Data System (ADS)
Taff, L. G.; Brennan, T. A.
1989-06-01
Intrigued by the recent advances in research on solving Kepler's equation, we have attacked the problem too. Our contributions emphasize the unified derivation of all known bounds and several starting values, a proof of the optimality of these bounds, a very thorough numerical exploration of a large variety of starting values and solution techniques in both mean anomaly/eccentricity space and eccentric anomaly/eccentricity space, and finally the best and simplest starting value/solution algorithm: M + e and Wegstein's secant modification of the method of successive substitutions. The very close second is Broucke's bounds coupled with Newton's second-order scheme.
The Arrhenius equation revisited.
Peleg, Micha; Normand, Mark D; Corradini, Maria G
2012-01-01
The Arrhenius equation has been widely used as a model of the temperature effect on the rate of chemical reactions and biological processes in foods. Since the model requires that the rate increase monotonically with temperature, its applicability to enzymatic reactions and microbial growth, which have optimal temperature, is obviously limited. This is also true for microbial inactivation and chemical reactions that only start at an elevated temperature, and for complex processes and reactions that do not follow fixed order kinetics, that is, where the isothermal rate constant, however defined, is a function of both temperature and time. The linearity of the Arrhenius plot, that is, Ln[k(T)] vs. 1/T where T is in °K has been traditionally considered evidence of the model's validity. Consequently, the slope of the plot has been used to calculate the reaction or processes' "energy of activation," usually without independent verification. Many experimental and simulated rate constant vs. temperature relationships that yield linear Arrhenius plots can also be described by the simpler exponential model Ln[k(T)/k(T(reference))] = c(T-T(reference)). The use of the exponential model or similar empirical alternative would eliminate the confusing temperature axis inversion, the unnecessary compression of the temperature scale, and the need for kinetic assumptions that are hard to affirm in food systems. It would also eliminate the reference to the Universal gas constant in systems where a "mole" cannot be clearly identified. Unless proven otherwise by independent experiments, one cannot dismiss the notion that the apparent linearity of the Arrhenius plot in many food systems is due to a mathematical property of the model's equation rather than to the existence of a temperature independent "energy of activation." If T+273.16°C in the Arrhenius model's equation is replaced by T+b, where the numerical value of the arbitrary constant b is substantially larger than T and T
Makkonen, Lasse
2016-04-01
Young's construction for a contact angle at a three-phase intersection forms the basis of all fields of science that involve wetting and capillary action. We find compelling evidence from recent experimental results on the deformation of a soft solid at the contact line, and displacement of an elastic wire immersed in a liquid, that Young's equation can only be interpreted by surface energies, and not as a balance of surface tensions. It follows that the a priori variable in finding equilibrium is not the position of the contact line, but the contact angle. This finding provides the explanation for the pinning of a contact line. PMID:26940644
Conservational PDF Equations of Turbulence
NASA Technical Reports Server (NTRS)
Shih, Tsan-Hsing; Liu, Nan-Suey
2010-01-01
Recently we have revisited the traditional probability density function (PDF) equations for the velocity and species in turbulent incompressible flows. They are all unclosed due to the appearance of various conditional means which are modeled empirically. However, we have observed that it is possible to establish a closed velocity PDF equation and a closed joint velocity and species PDF equation through conditions derived from the integral form of the Navier-Stokes equations. Although, in theory, the resulted PDF equations are neither general nor unique, they nevertheless lead to the exact transport equations for the first moment as well as all higher order moments. We refer these PDF equations as the conservational PDF equations. This observation is worth further exploration for its validity and CFD application
Solitons and nonlinear wave equations
Dodd, Roger K.; Eilbeck, J. Chris; Gibbon, John D.; Morris, Hedley C.
1982-01-01
A discussion of the theory and applications of classical solitons is presented with a brief treatment of quantum mechanical effects which occur in particle physics and quantum field theory. The subjects addressed include: solitary waves and solitons, scattering transforms, the Schroedinger equation and the Korteweg-de Vries equation, and the inverse method for the isospectral Schroedinger equation and the general solution of the solvable nonlinear equations. Also considered are: isolation of the Korteweg-de Vries equation in some physical examples, the Zakharov-Shabat/AKNS inverse method, kinks and the sine-Gordon equation, the nonlinear Schroedinger equation and wave resonance interactions, amplitude equations in unstable systems, and numerical studies of solitons. 45 references.
``Riemann equations'' in bidifferential calculus
NASA Astrophysics Data System (ADS)
Chvartatskyi, O.; Müller-Hoissen, F.; Stoilov, N.
2015-10-01
We consider equations that formally resemble a matrix Riemann (or Hopf) equation in the framework of bidifferential calculus. With different choices of a first-order bidifferential calculus, we obtain a variety of equations, including a semi-discrete and a fully discrete version of the matrix Riemann equation. A corresponding universal solution-generating method then either yields a (continuous or discrete) Cole-Hopf transformation, or leaves us with the problem of solving Riemann equations (hence an application of the hodograph method). If the bidifferential calculus extends to second order, solutions of a system of "Riemann equations" are also solutions of an equation that arises, on the universal level of bidifferential calculus, as an integrability condition. Depending on the choice of bidifferential calculus, the latter can represent a number of prominent integrable equations, like self-dual Yang-Mills, as well as matrix versions of the two-dimensional Toda lattice, Hirota's bilinear difference equation, (2+1)-dimensional Nonlinear Schrödinger (NLS), Kadomtsev-Petviashvili (KP) equation, and Davey-Stewartson equations. For all of them, a recent (non-isospectral) binary Darboux transformation result in bidifferential calculus applies, which can be specialized to generate solutions of the associated "Riemann equations." For the latter, we clarify the relation between these specialized binary Darboux transformations and the aforementioned solution-generating method. From (arbitrary size) matrix versions of the "Riemann equations" associated with an integrable equation, possessing a bidifferential calculus formulation, multi-soliton-type solutions of the latter can be generated. This includes "breaking" multi-soliton-type solutions of the self-dual Yang-Mills and the (2+1)-dimensional NLS equation, which are parametrized by solutions of Riemann equations.
Continuous-time mean-variance portfolio selection with value-at-risk and no-shorting constraints
NASA Astrophysics Data System (ADS)
Yan, Wei
2012-01-01
An investment problem is considered with dynamic mean-variance(M-V) portfolio criterion under discontinuous prices which follow jump-diffusion processes according to the actual prices of stocks and the normality and stability of the financial market. The short-selling of stocks is prohibited in this mathematical model. Then, the corresponding stochastic Hamilton-Jacobi-Bellman(HJB) equation of the problem is presented and the solution of the stochastic HJB equation based on the theory of stochastic LQ control and viscosity solution is obtained. The efficient frontier and optimal strategies of the original dynamic M-V portfolio selection problem are also provided. And then, the effects on efficient frontier under the value-at-risk constraint are illustrated. Finally, an example illustrating the discontinuous prices based on M-V portfolio selection is presented.
Holographic renormalisation group flows and renormalisation from a Wilsonian perspective
NASA Astrophysics Data System (ADS)
Lizana, J. M.; Morris, T. R.; Pérez-Victoria, M.
2016-03-01
From the Wilsonian point of view, renormalisable theories are understood as submanifolds in theory space emanating from a particular fixed point under renormalisation group evolution. We show how this picture precisely applies to their gravity duals. We investigate the Hamilton-Jacobi equation satisfied by the Wilson action and find the corresponding fixed points and their eigendeformations, which have a diagonal evolution close to the fixed points. The relevant eigendeformations are used to construct renormalised theories. We explore the relation of this formalism with holographic renormalisation. We also discuss different renormalisation schemes and show that the solutions to the gravity equations of motion can be used as renormalised couplings that parametrise the renormalised theories. This provides a transparent connection between holographic renormalisation group flows in the Wilsonian and non-Wilsonian approaches. The general results are illustrated by explicit calculations in an interacting scalar theory in AdS space.
Mathematical Models of Quasi-Species Theory and Exact Results for the Dynamics.
Saakian, David B; Hu, Chin-Kun
2016-01-01
We formulate the Crow-Kimura, discrete-time Eigen model, and continuous-time Eigen model. These models are interrelated and we established an exact mapping between them. We consider the evolutionary dynamics for the single-peak fitness and symmetric smooth fitness. We applied the quantum mechanical methods to find the exact dynamics of the evolution model with a single-peak fitness. For the smooth symmetric fitness landscape, we map exactly the evolution equations into Hamilton-Jacobi equation (HJE). We apply the method to the Crow-Kimura (parallel) and Eigen models. We get simple formulas to calculate the dynamics of the maximum of distribution and the variance. We review the existing mathematical tools of quasi-species theory.
Fermion tunneling from higher-dimensional black holes
Lin Kai; Yang Shuzheng
2009-03-15
Via the semiclassical approximation method, we study the 1/2-spin fermion tunneling from a higher-dimensional black hole. In our work, the Dirac equations are transformed into a simple form, and then we simplify the fermion tunneling research to the study of the Hamilton-Jacobi equation in curved space-time. Finally, we get the fermion tunneling rates and the Hawking temperatures at the event horizon of higher-dimensional black holes. We study fermion tunneling of a higher-dimensional Schwarzschild black hole and a higher-dimensional spherically symmetric quintessence black hole. In fact, this method is also applicable to the study of fermion tunneling from four-dimensional or lower-dimensional black holes, and we will take the rainbow-Finsler black hole as an example in order to make the fact explicit.
Problems of Mathematical Finance by Stochastic Control Methods
NASA Astrophysics Data System (ADS)
Stettner, Łukasz
The purpose of this paper is to present main ideas of mathematics of finance using the stochastic control methods. There is an interplay between stochastic control and mathematics of finance. On the one hand stochastic control is a powerful tool to study financial problems. On the other hand financial applications have stimulated development in several research subareas of stochastic control in the last two decades. We start with pricing of financial derivatives and modeling of asset prices, studying the conditions for the absence of arbitrage. Then we consider pricing of defaultable contingent claims. Investments in bonds lead us to the term structure modeling problems. Special attention is devoted to historical static portfolio analysis called Markowitz theory. We also briefly sketch dynamic portfolio problems using viscosity solutions to Hamilton-Jacobi-Bellman equation, martingale-convex analysis method or stochastic maximum principle together with backward stochastic differential equation. Finally, long time portfolio analysis for both risk neutral and risk sensitive functionals is introduced.
Nonlinear feedback control of highly manoeuvrable aircraft
NASA Technical Reports Server (NTRS)
Garrard, William L.; Enns, Dale F.; Snell, S. A.
1992-01-01
This paper describes the application of nonlinear quadratic regulator (NLQR) theory to the design of control laws for a typical high-performance aircraft. The NLQR controller design is performed using truncated solutions of the Hamilton-Jacobi-Bellman equation of optimal control theory. The performance of the NLQR controller is compared with the performance of a conventional P + I gain scheduled controller designed by applying standard frequency response techniques to the equations of motion of the aircraft linearized at various angles of attack. Both techniques result in control laws which are very similar in structure to one another and which yield similar performance. The results of applying both control laws to a high-g vertical turn are illustrated by nonlinear simulation.
Evolutionary Games with Randomly Changing Payoff Matrices
NASA Astrophysics Data System (ADS)
Yakushkina, Tatiana; Saakian, David B.; Bratus, Alexander; Hu, Chin-Kun
2015-06-01
Evolutionary games are used in various fields stretching from economics to biology. In most of these games a constant payoff matrix is assumed, although some works also consider dynamic payoff matrices. In this article we assume a possibility of switching the system between two regimes with different sets of payoff matrices. Potentially such a model can qualitatively describe the development of bacterial or cancer cells with a mutator gene present. A finite population evolutionary game is studied. The model describes the simplest version of annealed disorder in the payoff matrix and is exactly solvable at the large population limit. We analyze the dynamics of the model, and derive the equations for both the maximum and the variance of the distribution using the Hamilton-Jacobi equation formalism.
Neuro-optimal control of helicopter UAVs
NASA Astrophysics Data System (ADS)
Nodland, David; Ghosh, Arpita; Zargarzadeh, H.; Jagannathan, S.
2011-05-01
Helicopter UAVs can be extensively used for military missions as well as in civil operations, ranging from multirole combat support and search and rescue, to border surveillance and forest fire monitoring. Helicopter UAVs are underactuated nonlinear mechanical systems with correspondingly challenging controller designs. This paper presents an optimal controller design for the regulation and vertical tracking of an underactuated helicopter using an adaptive critic neural network framework. The online approximator-based controller learns the infinite-horizon continuous-time Hamilton-Jacobi-Bellman (HJB) equation and then calculates the corresponding optimal control input that minimizes the HJB equation forward-in-time. In the proposed technique, optimal regulation and vertical tracking is accomplished by a single neural network (NN) with a second NN necessary for the virtual controller. Both of the NNs are tuned online using novel weight update laws. Simulation results are included to demonstrate the effectiveness of the proposed control design in hovering applications.
Optimal coordination and control of posture and movements.
Johansson, Rolf; Fransson, Per-Anders; Magnusson, Måns
2009-01-01
This paper presents a theoretical model of stability and coordination of posture and locomotion, together with algorithms for continuous-time quadratic optimization of motion control. Explicit solutions to the Hamilton-Jacobi equation for optimal control of rigid-body motion are obtained by solving an algebraic matrix equation. The stability is investigated with Lyapunov function theory and it is shown that global asymptotic stability holds. It is also shown how optimal control and adaptive control may act in concert in the case of unknown or uncertain system parameters. The solution describes motion strategies of minimum effort and variance. The proposed optimal control is formulated to be suitable as a posture and movement model for experimental validation and verification. The combination of adaptive and optimal control makes this algorithm a candidate for coordination and control of functional neuromuscular stimulation as well as of prostheses. Validation examples with experimental data are provided. PMID:19671443
Rowland, Brad A; Wyatt, Robert E
2007-10-28
One of the major obstacles in employing complex-valued trajectory methods for quantum barrier scattering calculations is the search for isochrones. In this study, complex-valued derivative propagation method trajectories in the arbitrary Lagrangian-Eulerian frame are employed to solve the complex Hamilton-Jacobi equation for quantum barrier scattering problems employing constant velocity trajectories moving along rectilinear paths whose initial points can be in the complex plane or even along the real axis. It is shown that this effectively removes the need for isochrones for barrier transmission problems. Model problems tested include the Eckart, Gaussian, and metastable quadratic+cubic potentials over a variety of wave packet energies. For comparison, the "exact" solution is computed from the time-dependent Schrodinger equation via pseudospectral methods. PMID:17979316
Complex-extended Bohmian mechanics.
Chou, Chia-Chun; Wyatt, Robert E
2010-04-01
Complex-extended Bohmian mechanics is investigated by analytically continuing the wave function in polar form into the complex plane. We derive the complex-extended version of the quantum Hamilton-Jacobi equation and the continuity equation in Bohmian mechanics. Complex-extended Bohmian mechanics recovers the standard real-valued Bohmian mechanics on the real axis. The trajectories on the real axis are in accord with the standard real-valued Bohmian trajectories. The trajectories launched away from the real axis never intersect the real axis, and they display symmetry with respect to the real axis. Trajectories display hyperbolic deflection around nodes of the wave function in the complex plane. PMID:20387916
Reaction-subdiffusion front propagation in a comblike model of spiny dendrites.
Iomin, A; Méndez, V
2013-07-01
Fractional reaction-diffusion equations are derived by exploiting the geometrical similarities between a comb structure and a spiny dendrite. In the framework of the obtained equations, two scenarios of reaction transport in spiny dendrites are explored, where both a linear reaction in spines and nonlinear Fisher-Kolmogorov-Petrovskii-Piskunov reactions along dendrites are considered. In the framework of fractional subdiffusive comb model, we develop a Hamilton-Jacobi approach to estimate the overall velocity of the reaction front propagation. One of the main effects observed is the failure of the front propagation for both scenarios due to either the reaction inside the spines or the interaction of the reaction with the spines. In the first case the spines are the source of reactions, while in the latter case, the spines are a source of a damping mechanism. PMID:23944491
Optimal control of aeroassisted plane change maneuver using feedback expansions
NASA Technical Reports Server (NTRS)
Mishne, D.; Speyer, J. L.
1986-01-01
A guidance law for an aeroassisted plane change maneuver is developed by an asymptotic expansion technique using a small parameter which essentially represents the ratio of the inertial forces to the atmospheric forces. This guidance law minimizes the energy loss while meeting terminal constraints on the altitude, flight path angle, and heading angle. By neglecting the inertial forces, the resulting optimization problem is integrable and can be determined in closed form. This zeroth-order solution is the first term in an asymptotic series solution of the Hamilton-Jacobi-Bellman equation. The remaining terms are determined from the solution of a first-order, linear partial differential equation whose solution requires only quadrature integration. Our initial results in using this guidance scheme are encouraging.
Mehraeen, Shahab; Dierks, Travis; Jagannathan, S; Crow, Mariesa L
2013-12-01
In this paper, the nearly optimal solution for discrete-time (DT) affine nonlinear control systems in the presence of partially unknown internal system dynamics and disturbances is considered. The approach is based on successive approximate solution of the Hamilton-Jacobi-Isaacs (HJI) equation, which appears in optimal control. Successive approximation approach for updating control and disturbance inputs for DT nonlinear affine systems are proposed. Moreover, sufficient conditions for the convergence of the approximate HJI solution to the saddle point are derived, and an iterative approach to approximate the HJI equation using a neural network (NN) is presented. Then, the requirement of full knowledge of the internal dynamics of the nonlinear DT system is relaxed by using a second NN online approximator. The result is a closed-loop optimal NN controller via offline learning. A numerical example is provided illustrating the effectiveness of the approach. PMID:24273142
Mathematical Models of Quasi-Species Theory and Exact Results for the Dynamics.
Saakian, David B; Hu, Chin-Kun
2016-01-01
We formulate the Crow-Kimura, discrete-time Eigen model, and continuous-time Eigen model. These models are interrelated and we established an exact mapping between them. We consider the evolutionary dynamics for the single-peak fitness and symmetric smooth fitness. We applied the quantum mechanical methods to find the exact dynamics of the evolution model with a single-peak fitness. For the smooth symmetric fitness landscape, we map exactly the evolution equations into Hamilton-Jacobi equation (HJE). We apply the method to the Crow-Kimura (parallel) and Eigen models. We get simple formulas to calculate the dynamics of the maximum of distribution and the variance. We review the existing mathematical tools of quasi-species theory. PMID:26342705
Fermion tunneling from higher-dimensional black holes
NASA Astrophysics Data System (ADS)
Lin, Kai; Yang, Shu-Zheng
2009-03-01
Via the semiclassical approximation method, we study the 1/2-spin fermion tunneling from a higher-dimensional black hole. In our work, the Dirac equations are transformed into a simple form, and then we simplify the fermion tunneling research to the study of the Hamilton-Jacobi equation in curved space-time. Finally, we get the fermion tunneling rates and the Hawking temperatures at the event horizon of higher-dimensional black holes. We study fermion tunneling of a higher-dimensional Schwarzschild black hole and a higher-dimensional spherically symmetric quintessence black hole. In fact, this method is also applicable to the study of fermion tunneling from four-dimensional or lower-dimensional black holes, and we will take the rainbow-Finsler black hole as an example in order to make the fact explicit.
Remnants by fermions' tunneling from a Hořava-Lifshitz black hole
NASA Astrophysics Data System (ADS)
Li, Guoping; Cheng, Tianhu; Li, Zhang; Feng, Zhongwen; Zu, Xiaotao
2015-01-01
Adopting the Hamilton-Jacobi method, we investigated the tunneling radiation of a deform Hořava-Lifshitz black hole, and the original tunneling rate and Hawking temperature are obtained. Based on the generalized uncertainty principle, recent researches imply that the quantum gravity corrected the Dirac equation exactly. Hence, the corrected Dirac equation can express the tunneling behavior of fermions may be more suitable, and meanwhile, the corrected Hawking temperature of the Hořava-Lifshitz black hole is obtained. Comparing with previous results, we find that the Hawking temperature is not only related to the mass of black hole, but also related to the mass and energy of outgoing fermions. Finally, we inferred that the Hawking radiation would stop by the reason of the quantum gravity, and the remnant of the black hole exists naturally, also the singularity of the black hole is avoided.
NASA Astrophysics Data System (ADS)
Villalba, Víctor M.
We compute the density of scalar and Dirac particles created by a cosmological anisotropic universe1,2 in the presence of a time dependent homogeneous electric field. In order to compute the rate of particles created we apply a quasiclassical approach that has been used successfully in different scenarios3,4. The idea behind the method is the following: First, we solve the relativistic Hamilton-Jacobi equation and, looking at its solutions, we identify positive and negative frequency modes. Second, after separating variables5,6, we solve the Klein-Gordon and Dirac equations and, after comparing with the results obtained for the quasiclassical limit, we identify the positive and negative frequency states. We show that the particle distribution becomes thermal when one neglects the electric interaction.
Hořava-Lifshitz quantum cosmology
NASA Astrophysics Data System (ADS)
Bertolami, Orfeu; Zarro, Carlos A. D.
2011-08-01
In this work, a minisuperspace model for the projectable Hořava-Lifshitz gravity without the detailed-balance condition is investigated. The Wheeler-DeWitt equation is derived and its solutions are studied and discussed for some particular cases where, due to Hořava-Lifshitz gravity, there is a “potential barrier” nearby a=0. For a vanishing cosmological constant, a normalizable wave function of the Universe is found. When the cosmological constant is nonvanishing, the WKB method is used to obtain solutions for the wave function of the Universe. Using the Hamilton-Jacobi equation, one discusses how the transition from quantum to classical regime occurs and, for the case of a positive cosmological constant, the scale factor is shown to grow exponentially, hence recovering the general relativity behavior for the late Universe.
Nonlinear solutions of long-wavelength gravitational radiation
NASA Astrophysics Data System (ADS)
Salopek, D. S.
1991-05-01
In a significant improvement over homogeneous minisuperspace models, it is shown that the classical nonlinear evolution of inhomogeneous scalar fields and the metric is tractable when the wavelength of the fluctuations is larger than the Hubble radius. Neglecting second-order spatial gradients, one can solve the energy constraint as well as the evolution equations by invoking a transformation to new canonical variables. The Hamilton-Jacobi equation is separable and complete solutions are given for gravitational radiation and multiple scalar fields interacting through an exponential potential. Although the time parameter is arbitrary, the natural choice is the determinant of the three-metric. The momentum constraint may be simply expressed in terms of the new canonical variables which define the spatial coordinates. The long-wavelength analysis is essential for a proper formulation of stochastic inflation which enables one to model non-Gaussian primordial fluctuations for structure formation.
From instantons to inflationary universe.
NASA Astrophysics Data System (ADS)
Khalatnikov, I. M.; Schiller, P.
1993-01-01
The present paper is based on a theory which includes gravity and a complex scalar field. In such a theory it is possible to analyze the evolution from instantons in the classically forbidden (Euclidean) region in minisuperspace to the inflationary universe in the classically allowed (Minkowski) region. The characteristics for the Hamilton-Jacobi equation, which define the action in the quasiclassical approximation, are described by four first-order differential equations. This four-dimensional dynamical system was integrated numerically. In the closed Euclidean region two types of instantons were found. It is shown that the instantons correspond to extremal trajectories. The existence of two types of instantons gives rise to different possibilities for tunneling from Euclidean region to Minkowski region and for creation of inflationary universes.
From instanton to inflationary universe
NASA Astrophysics Data System (ADS)
Khalatnikov, I. M.; Schiller, P.
1993-03-01
The present paper is based on a theory which includes gravity and a complex scalar field [I.M. Khalatnikov and A. Mezhlumian, Phys. Lett. A 169 (1992) 308]. It is shown that in such a theory we can proceed the evolution from instantons in the classically forbidden (euclidean) region in minisuperspace to the inflationary universe in the classically allowed (Minkowski) region. The characteristics for the Hamilton-Jacobi equation, which define the action in the quasiclassical approximation, are described by four differential equations of first order. This four-dimensional dynamical system was integrated numerically. In the euclidean compact region we found two types of instantons. It is shown that the instantons correspond to extremal trajectories. The existence of two types of instantons gives different possibilities for tunneling from euclidean to Minkowski regions and for creation of inflationary universes.
Solving Nonlinear Coupled Differential Equations
NASA Technical Reports Server (NTRS)
Mitchell, L.; David, J.
1986-01-01
Harmonic balance method developed to obtain approximate steady-state solutions for nonlinear coupled ordinary differential equations. Method usable with transfer matrices commonly used to analyze shaft systems. Solution to nonlinear equation, with periodic forcing function represented as sum of series similar to Fourier series but with form of terms suggested by equation itself.
Successfully Transitioning to Linear Equations
ERIC Educational Resources Information Center
Colton, Connie; Smith, Wendy M.
2014-01-01
The Common Core State Standards for Mathematics (CCSSI 2010) asks students in as early as fourth grade to solve word problems using equations with variables. Equations studied at this level generate a single solution, such as the equation x + 10 = 25. For students in fifth grade, the Common Core standard for algebraic thinking expects them to…
The Forced Hard Spring Equation
ERIC Educational Resources Information Center
Fay, Temple H.
2006-01-01
Through numerical investigations, various examples of the Duffing type forced spring equation with epsilon positive, are studied. Since [epsilon] is positive, all solutions to the associated homogeneous equation are periodic and the same is true with the forcing applied. The damped equation exhibits steady state trajectories with the interesting…
Equating with Miditests Using IRT
ERIC Educational Resources Information Center
Fitzpatrick, Joseph; Skorupski, William P.
2016-01-01
The equating performance of two internal anchor test structures--miditests and minitests--is studied for four IRT equating methods using simulated data. Originally proposed by Sinharay and Holland, miditests are anchors that have the same mean difficulty as the overall test but less variance in item difficulties. Four popular IRT equating methods…
Generalized Klein-Kramers equations
NASA Astrophysics Data System (ADS)
Fa, Kwok Sau
2012-12-01
A generalized Klein-Kramers equation for a particle interacting with an external field is proposed. The equation generalizes the fractional Klein-Kramers equation introduced by Barkai and Silbey [J. Phys. Chem. B 104, 3866 (2000), 10.1021/jp993491m]. Besides, the generalized Klein-Kramers equation can also recover the integro-differential Klein-Kramers equation for continuous-time random walk; this means that it can describe the subdiffusive and superdiffusive regimes in the long-time limit. Moreover, analytic solutions for first two moments both in velocity and displacement (for force-free case) are obtained, and their dynamic behaviors are investigated.
Multinomial Diffusion Equation
Balter, Ariel I.; Tartakovsky, Alexandre M.
2011-06-01
We have developed a novel stochastic, space/time discrete representation of particle diffusion (e.g. Brownian motion) based on discrete probability distributions. We show that in the limit of both very small time step and large concentration, our description is equivalent to the space/time continuous stochastic diffusion equation. Being discrete in both time and space, our model can be used as an extremely accurate, efficient, and stable stochastic finite-difference diffusion algorithm when concentrations are so small that computationally expensive particle-based methods are usually needed. Through numerical simulations, we show that our method can generate realizations that capture the statistical properties of particle simulations. While our method converges converges to both the correct ensemble mean and ensemble variance very quickly with decreasing time step, but for small concentration, the stochastic diffusion PDE does not, even for very small time steps.
NASA Astrophysics Data System (ADS)
Cardona, Carlos; Gomez, Humberto
2016-06-01
Recently the CHY approach has been extended to one loop level using elliptic functions and modular forms over a Jacobian variety. Due to the difficulty in manipulating these kind of functions, we propose an alternative prescription that is totally algebraic. This new proposal is based on an elliptic algebraic curve embedded in a mathbb{C}{P}^2 space. We show that for the simplest integrand, namely the n - gon, our proposal indeed reproduces the expected result. By using the recently formulated Λ-algorithm, we found a novel recurrence relation expansion in terms of tree level off-shell amplitudes. Our results connect nicely with recent results on the one-loop formulation of the scattering equations. In addition, this new proposal can be easily stretched out to hyperelliptic curves in order to compute higher genus.
On nonautonomous Dirac equation
Hovhannisyan, Gro; Liu Wen
2009-12-15
We construct the fundamental solution of time dependent linear ordinary Dirac system in terms of unknown phase functions. This construction gives approximate representation of solutions which is useful for the study of asymptotic behavior. Introducing analog of Rayleigh quotient for differential equations we generalize Hartman-Wintner asymptotic integration theorems with the error estimates for applications to the Dirac system. We also introduce the adiabatic invariants for the Dirac system, which are similar to the adiabatic invariant of Lorentz's pendulum. Using a small parameter method it is shown that the change in the adiabatic invariants approaches zero with the power speed as a small parameter approaches zero. As another application we calculate the transition probabilities for the Dirac system. We show that for the special choice of electromagnetic field, the only transition of an electron to the positron with the opposite spin orientation is possible.
Approximate optimal guidance for the advanced launch system
NASA Technical Reports Server (NTRS)
Feeley, T. S.; Speyer, J. L.
1993-01-01
A real-time guidance scheme for the problem of maximizing the payload into orbit subject to the equations of motion for a rocket over a spherical, non-rotating earth is presented. An approximate optimal launch guidance law is developed based upon an asymptotic expansion of the Hamilton - Jacobi - Bellman or dynamic programming equation. The expansion is performed in terms of a small parameter, which is used to separate the dynamics of the problem into primary and perturbation dynamics. For the zeroth-order problem the small parameter is set to zero and a closed-form solution to the zeroth-order expansion term of Hamilton - Jacobi - Bellman equation is obtained. Higher-order terms of the expansion include the effects of the neglected perturbation dynamics. These higher-order terms are determined from the solution of first-order linear partial differential equations requiring only the evaluation of quadratures. This technique is preferred as a real-time, on-line guidance scheme to alternative numerical iterative optimization schemes because of the unreliable convergence properties of these iterative guidance schemes and because the quadratures needed for the approximate optimal guidance law can be performed rapidly and by parallel processing. Even if the approximate solution is not nearly optimal, when using this technique the zeroth-order solution always provides a path which satisfies the terminal constraints. Results for two-degree-of-freedom simulations are presented for the simplified problem of flight in the equatorial plane and compared to the guidance scheme generated by the shooting method which is an iterative second-order technique.
Entwined paths, difference equations, and the Dirac equation
Ord, G.N.; Mann, R.B.
2003-02-01
Entwined space-time paths are bound pairs of trajectories which are traversed in opposite directions with respect to macroscopic time. In this paper, we show that ensembles of entwined paths on a discrete space-time lattice are simply described by coupled difference equations which are discrete versions of the Dirac equation. There is no analytic continuation, explicit or forced, involved in this description. The entwined paths are ''self-quantizing.'' We also show that simple classical stochastic processes that generate the difference equations as ensemble averages are stable numerically and converge at a rate governed by the details of the stochastic process. This result establishes the Dirac equation in one dimension as a phenomenological equation describing an underlying classical stochastic process, in the same sense that the diffusion and telegraph equations are phenomenological descriptions of stochastic processes.
Mode decomposition evolution equations
Wang, Yang; Wei, Guo-Wei; Yang, Siyang
2011-01-01
Partial differential equation (PDE) based methods have become some of the most powerful tools for exploring the fundamental problems in signal processing, image processing, computer vision, machine vision and artificial intelligence in the past two decades. The advantages of PDE based approaches are that they can be made fully automatic, robust for the analysis of images, videos and high dimensional data. A fundamental question is whether one can use PDEs to perform all the basic tasks in the image processing. If one can devise PDEs to perform full-scale mode decomposition for signals and images, the modes thus generated would be very useful for secondary processing to meet the needs in various types of signal and image processing. Despite of great progress in PDE based image analysis in the past two decades, the basic roles of PDEs in image/signal analysis are only limited to PDE based low-pass filters, and their applications to noise removal, edge detection, segmentation, etc. At present, it is not clear how to construct PDE based methods for full-scale mode decomposition. The above-mentioned limitation of most current PDE based image/signal processing methods is addressed in the proposed work, in which we introduce a family of mode decomposition evolution equations (MoDEEs) for a vast variety of applications. The MoDEEs are constructed as an extension of a PDE based high-pass filter (Europhys. Lett., 59(6): 814, 2002) by using arbitrarily high order PDE based low-pass filters introduced by Wei (IEEE Signal Process. Lett., 6(7): 165, 1999). The use of arbitrarily high order PDEs is essential to the frequency localization in the mode decomposition. Similar to the wavelet transform, the present MoDEEs have a controllable time-frequency localization and allow a perfect reconstruction of the original function. Therefore, the MoDEE operation is also called a PDE transform. However, modes generated from the present approach are in the spatial or time domain and can be
Menikoff, Ralph
2015-12-15
The JWL equation of state (EOS) is frequently used for the products (and sometimes reactants) of a high explosive (HE). Here we review and systematically derive important properties. The JWL EOS is of the Mie-Grueneisen form with a constant Grueneisen coefficient and a constants specific heat. It is thermodynamically consistent to specify the temperature at a reference state. However, increasing the reference state temperature restricts the EOS domain in the (V, e)-plane of phase space. The restrictions are due to the conditions that P ≥ 0, T ≥ 0, and the isothermal bulk modulus is positive. Typically, this limits the low temperature regime in expansion. The domain restrictions can result in the P-T equilibrium EOS of a partly burned HE failing to have a solution in some cases. For application to HE, the heat of detonation is discussed. Example JWL parameters for an HE, both products and reactions, are used to illustrate the restrictions on the domain of the EOS.
Identification of Uncertainties in the Geometry of Geophysical Objects
NASA Astrophysics Data System (ADS)
Papadopoulos, Dimitris; Herty, Michael; Rath, Volker; Behr, Marek
2010-05-01
A shape optimization method is used to reconstruct the unknown shape of geophysical layers from temperature measurements by the use of adjoint fields and level sets. The identification of the shape of the geophysical layers by temperature measurements is necessary for the efficient use of geothermal energy. Temperature is measured along thin vertical regions inside the domain, representing the boreholes. The method of speed is used to calculate the shape sensitivities, and the continuous adjoint approach is followed for the computation of the shape derivatives. Both the direct, and the adjoint problem are solved with a finite element method. The unknown shape is described with the help of the level set function; the advantage of the shape function is that no mesh movement or re-meshing is necessary but an additional Hamilton-Jacobi equation has to be solved. This equation is integrated in an artificial time, where the velocity represents the movement in the direction of the normal vector of the interface. The solution of the Hamilton-Jacobi equation is performed with essentially non-oscillatory (ENO) and weighted ENO (WENO) schemes. For large optimization steps re-initialization of the level set function is also necessary, in order to keep the magnitude of the level set function near unity, as well as to smooth the level set function. The dependence of the objective function and the quality of the recovered shape is studied with respect to the number, the position and the width of the boreholes. These studies represent a first step in the developments of corresponding methods for multiphysics investigations, e.g., in geothermal reservoirs, where multiphase fluid flow is an important component. The necessary generalizations of our method remain a challenging task for the future.
NASA Astrophysics Data System (ADS)
Arimoto, Suguru
An optimal regulator problem for endpoint position control of a robot arm with (or without) redundancy in its total degrees-of-freedom (DOF) is solved by combining Riemannian geometry with nonlinear control theory. Given a target point, within the task-space, that the arm endpoint should reach, a task-space position feedback with joint damping is shown to asymptotically stabilize reaching movements even if the number of DOF of the arm is greater than the dimension of the task space and thereby the inverse kinematics is ill-posed. Usually the speed of convergence of the endpoint trajectory is unsatisfactory, depending on the choice of feedback gains for joint damping. Hence, to speed up the convergence without incurring further energy consumption, an optimal control design for minimizing a performance index composed of an integral of joint dissipation energy plus a linear quadratic form of the task-space control input and output is introduced. It is then shown that the Hamilton-Jacobi-Bellman equation derived from the principle of optimality is solvable in control variables and the Hamilton-Jacobi equation itself has an explicit solution. Although the state of the original dynamics (the Euler-Lagrange equation) with DOF-redundancy contains uncontrollable and unobservable manifolds, the dynamics satisfies a nonlinear version of the Kalman-Yakubovich-Popov lemma and the task-space input-output passivity. An inverse problem of optimal regulator design for robotic arms under the effect of gravity is also tackled by combining Riemannian geometry with passivity-based control theory.
On the generalized Jacobi equation
NASA Astrophysics Data System (ADS)
Perlick, Volker
2008-05-01
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.
Equations of the Randomizer's Dynamics
NASA Astrophysics Data System (ADS)
Strzałko, Jarosław; Grabski, Juliusz; Perlikowski, Przemysław; Stefanski, Andrzej; Kapitaniak, Tomasz
Basing on the Newton-Euler laws of mechanics we derive the equations which describe the dynamics of the coin toss, the die throw, and roulette run. The equations for full 3D models and for lower dimensional simplifications are given. The influence of the air resistance and energy dissipation at the impacts is described. The obtained equations allow for the numerical simulation of the randomizer's dynamics and define the mapping of the initial conditions into the final outcome.
Solving Differential Equations in R
NASA Astrophysics Data System (ADS)
Soetaert, Karline; Meysman, Filip; Petzoldt, Thomas
2010-09-01
The open-source software R has become one of the most widely used systems for statistical data analysis and for making graphs, but it is also well suited for other disciplines in scientific computing. One of the fields where considerable progress has been made is the solution of differential equations. Here we first give an overview of the types of differential equations that R can solve, and then demonstrate how to use R for solving a 2-Dimensional partial differential equation.
A note on "Kepler's equation".
NASA Astrophysics Data System (ADS)
Dutka, J.
1997-07-01
This note briefly points out the formal similarity between Kepler's equation and equations developed in Hindu and Islamic astronomy for describing the lunar parallax. Specifically, an iterative method for calculating the lunar parallax has been developed by the astronomer Habash al-Hasib al-Marwazi (about 850 A.D., Turkestan), which is surprisingly similar to the iterative method for solving Kepler's equation invented by Leonhard Euler (1707 - 1783).
Deformation of the Dirac equation
NASA Astrophysics Data System (ADS)
Faizal, Mir; Kruglov, Sergey I.
2016-10-01
In this paper, we will first clarify the physical meaning of having a minimum measurable time. Then we will combine the deformation of the Dirac equation due to the existence of minimum measurable length and time scales with its deformation due to the doubly special relativity. We will also analyze this deformed Dirac equation in curved spacetime, and observe that this deformation of the Dirac equation also leads to a nontrivial modification of general relativity. Finally, we will analyze the stochastic quantization of this deformed Dirac equation on curved spacetime.
Quaternion Dirac Equation and Supersymmetry
NASA Astrophysics Data System (ADS)
Rawat, Seema; Negi, O. P. S.
2009-08-01
Quaternion Dirac equation has been analyzed and its supersymmetrization has been discussed consistently. It has been shown that the quaternion Dirac equation automatically describes the spin structure with its spin up and spin down components of two component quaternion Dirac spinors associated with positive and negative energies. It has also been shown that the supersymmetrization of quaternion Dirac equation works well for different cases associated with zero mass, nonzero mass, scalar potential and generalized electromagnetic potentials. Accordingly we have discussed the splitting of supersymmetrized Dirac equation in terms of electric and magnetic fields.
Electronic representation of wave equation
NASA Astrophysics Data System (ADS)
Veigend, Petr; Kunovský, Jiří; Kocina, Filip; Nečasová, Gabriela; Šátek, Václav; Valenta, Václav
2016-06-01
The Taylor series method for solving differential equations represents a non-traditional way of a numerical solution. Even though this method is not much preferred in the literature, experimental calculations done at the Department of Intelligent Systems of the Faculty of Information Technology of TU Brno have verified that the accuracy and stability of the Taylor series method exceeds the currently used algorithms for numerically solving differential equations. This paper deals with solution of Telegraph equation using modelling of a series small pieces of the wire. Corresponding differential equations are solved by the Modern Taylor Series Method.
Neural network-based optimal adaptive output feedback control of a helicopter UAV.
Nodland, David; Zargarzadeh, Hassan; Jagannathan, Sarangapani
2013-07-01
Helicopter unmanned aerial vehicles (UAVs) are widely used for both military and civilian operations. Because the helicopter UAVs are underactuated nonlinear mechanical systems, high-performance controller design for them presents a challenge. This paper introduces an optimal controller design via an output feedback for trajectory tracking of a helicopter UAV, using a neural network (NN). The output-feedback control system utilizes the backstepping methodology, employing kinematic and dynamic controllers and an NN observer. The online approximator-based dynamic controller learns the infinite-horizon Hamilton-Jacobi-Bellman equation in continuous time and calculates the corresponding optimal control input by minimizing a cost function, forward-in-time, without using the value and policy iterations. Optimal tracking is accomplished by using a single NN utilized for the cost function approximation. The overall closed-loop system stability is demonstrated using Lyapunov analysis. Finally, simulation results are provided to demonstrate the effectiveness of the proposed control design for trajectory tracking.
Techniques for developing approximate optimal advanced launch system guidance
NASA Technical Reports Server (NTRS)
Feeley, Timothy S.; Speyer, Jason L.
1991-01-01
An extension to the authors' previous technique used to develop a real-time guidance scheme for the Advanced Launch System is presented. The approach is to construct an optimal guidance law based upon an asymptotic expansion associated with small physical parameters, epsilon. The trajectory of a rocket modeled as a point mass is considered with the flight restricted to an equatorial plane while reaching an orbital altitude at orbital injection speeds. The dynamics of this problem can be separated into primary effects due to thrust and gravitational forces, and perturbation effects which include the aerodynamic forces and the remaining inertial forces. An analytic solution to the reduced-order problem represented by the primary dynamics is possible. The Hamilton-Jacobi-Bellman or dynamic programming equation is expanded in an asymptotic series where the zeroth-order term (epsilon = 0) can be obtained in closed form.
An investigation of nonlinear control of spacecraft attitude
NASA Astrophysics Data System (ADS)
Binette, Mark Richard
The design of controllers subject to the nonlinear H-infinity criterion is explored. The plants to be controlled are the attitude motion of spacecraft, subject to some disturbance torque. Two cases are considered: the regulation about an inertially-fixed direction, and an Earth-pointing spacecraft in a circular orbit, subject to the gravity-gradient torque. The spacecraft attitude is described using the modified Rodrigues parameters. A series of controllers are designed using the nonlinear H-infinity control criterion, and are subsequently generated using a Taylor series expansion to approximate solutions of the relevant Hamilton-Jacobi equations. The controllers are compared, using both input-output and initial condition simulations. A proof is used to demonstrate that the linearized controller solves the H-infinity control problem for the inertial pointing problem when describing the plant using the modified Rodrigues parameters.
Goreac, Dan; Serea, Oana-Silvia
2012-10-15
We aim at characterizing domains of attraction for controlled piecewise deterministic processes using an occupational measure formulation and Zubov's approach. Firstly, we provide linear programming (primal and dual) formulations of discounted, infinite horizon control problems for PDMPs. These formulations involve an infinite-dimensional set of probability measures and are obtained using viscosity solutions theory. Secondly, these tools allow to construct stabilizing measures and to avoid the assumption of stability under concatenation for controls. The domain of controllability is then characterized as some level set of a convenient solution of the associated Hamilton-Jacobi integral-differential equation. The theoretical results are applied to PDMPs associated to stochastic gene networks. Explicit computations are given for Cook's model for gene expression.
Nonlinear integrable ion traps
Nagaitsev, S.; Danilov, V.; /SNS Project, Oak Ridge
2011-10-01
Quadrupole ion traps can be transformed into nonlinear traps with integrable motion by adding special electrostatic potentials. This can be done with both stationary potentials (electrostatic plus a uniform magnetic field) and with time-dependent electric potentials. These potentials are chosen such that the single particle Hamilton-Jacobi equations of motion are separable in some coordinate systems. The electrostatic potentials have several free adjustable parameters allowing for a quadrupole trap to be transformed into, for example, a double-well or a toroidal-well system. The particle motion remains regular, non-chaotic, integrable in quadratures, and stable for a wide range of parameters. We present two examples of how to realize such a system in case of a time-independent (the Penning trap) as well as a time-dependent (the Paul trap) configuration.
Hidden symmetries, null geodesics, and photon capture in the Sen black hole
Hioki, Kenta; Miyamoto, Umpei
2008-08-15
Important classes of null geodesics and hidden symmetries in the Sen black hole are investigated. First, we obtain the principal null geodesics and circular photon orbits. Then, an irreducible rank-two Killing tensor and a conformal Killing tensor are derived, which represent the hidden symmetries. Analyzing the properties of Killing tensors, we clarify why the Hamilton-Jacobi and wave equations are separable in this spacetime. We also investigate the gravitational capture of photons by the Sen black hole and compare the result with those by the various charged/rotating black holes and naked singularities in the Kerr-Newman family. For these black holes and naked singularities, we show the capture regions in a two dimensional impact parameter space (or equivalently the 'shadows' observed at infinity) to form a variety of shapes such as the disk, circle, dot, arc, and their combinations.
NASA Astrophysics Data System (ADS)
Yang, Xiong; Liu, Derong; Wang, Ding
2014-03-01
In this paper, an adaptive reinforcement learning-based solution is developed for the infinite-horizon optimal control problem of constrained-input continuous-time nonlinear systems in the presence of nonlinearities with unknown structures. Two different types of neural networks (NNs) are employed to approximate the Hamilton-Jacobi-Bellman equation. That is, an recurrent NN is constructed to identify the unknown dynamical system, and two feedforward NNs are used as the actor and the critic to approximate the optimal control and the optimal cost, respectively. Based on this framework, the action NN and the critic NN are tuned simultaneously, without the requirement for the knowledge of system drift dynamics. Moreover, by using Lyapunov's direct method, the weights of the action NN and the critic NN are guaranteed to be uniformly ultimately bounded, while keeping the closed-loop system stable. To demonstrate the effectiveness of the present approach, simulation results are illustrated.
Finite-time H∞ filtering for non-linear stochastic systems
NASA Astrophysics Data System (ADS)
Hou, Mingzhe; Deng, Zongquan; Duan, Guangren
2016-09-01
This paper describes the robust H∞ filtering analysis and the synthesis of general non-linear stochastic systems with finite settling time. We assume that the system dynamic is modelled by Itô-type stochastic differential equations of which the state and the measurement are corrupted by state-dependent noises and exogenous disturbances. A sufficient condition for non-linear stochastic systems to have the finite-time H∞ performance with gain less than or equal to a prescribed positive number is established in terms of a certain Hamilton-Jacobi inequality. Based on this result, the existence of a finite-time H∞ filter is given for the general non-linear stochastic system by a second-order non-linear partial differential inequality, and the filter can be obtained by solving this inequality. The effectiveness of the obtained result is illustrated by a numerical example.
NASA Astrophysics Data System (ADS)
Xie, Zhi-Kun; Pan, Wei-Zhen; Yang, Xue-Jun
2013-03-01
Using a new tortoise coordinate transformation, we discuss the quantum nonthermal radiation characteristics near an event horizon by studying the Hamilton-Jacobi equation of a scalar particle in curved space-time, and obtain the event horizon surface gravity and the Hawking temperature on that event horizon. The results show that there is a crossing of particle energy near the event horizon. We derive the maximum overlap of the positive and negative energy levels. It is also found that the Hawking temperature of a black hole depends not only on the time, but also on the angle. There is a problem of dimension in the usual tortoise coordinate, so the present results obtained by using a correct-dimension new tortoise coordinate transformation may be more reasonable.
Zhang, Huaguang; Qin, Chunbin; Jiang, Bin; Luo, Yanhong
2014-12-01
The problem of H∞ state feedback control of affine nonlinear discrete-time systems with unknown dynamics is investigated in this paper. An online adaptive policy learning algorithm (APLA) based on adaptive dynamic programming (ADP) is proposed for learning in real-time the solution to the Hamilton-Jacobi-Isaacs (HJI) equation, which appears in the H∞ control problem. In the proposed algorithm, three neural networks (NNs) are utilized to find suitable approximations of the optimal value function and the saddle point feedback control and disturbance policies. Novel weight updating laws are given to tune the critic, actor, and disturbance NNs simultaneously by using data generated in real-time along the system trajectories. Considering NN approximation errors, we provide the stability analysis of the proposed algorithm with Lyapunov approach. Moreover, the need of the system input dynamics for the proposed algorithm is relaxed by using a NN identification scheme. Finally, simulation examples show the effectiveness of the proposed algorithm. PMID:25095274
Some reference formulas for the generating functions of canonical transformations
NASA Astrophysics Data System (ADS)
Anselmi, Damiano
2016-02-01
We study some properties of the canonical transformations in classical mechanics and quantum field theory and give a number of practical formulas concerning their generating functions. First, we give a diagrammatic formula for the perturbative expansion of the composition law around the identity map. Then we propose a standard way to express the generating function of a canonical transformation by means of a certain "componential" map, which obeys the Baker-Campbell-Hausdorff formula. We derive the diagrammatic interpretation of the componential map, work out its relation with the solution of the Hamilton-Jacobi equation and derive its time-ordered version. Finally, we generalize the results to the Batalin-Vilkovisky formalism, where the conjugate variables may have both bosonic and fermionic statistics, and describe applications to quantum field theory.
NASA Technical Reports Server (NTRS)
Shu, Chi-Wang
1997-01-01
In these lecture notes we describe the construction, analysis, and application of ENO (Essentially Non-Oscillatory) and WENO (Weighted Essentially Non-Oscillatory) schemes for hyperbolic conservation laws and related Hamilton- Jacobi equations. ENO and WENO schemes are high order accurate finite difference schemes designed for problems with piecewise smooth solutions containing discontinuities. The key idea lies at the approximation level, where a nonlinear adaptive procedure is used to automatically choose the locally smoothest stencil, hence avoiding crossing discontinuities in the interpolation procedure as much as possible. ENO and WENO schemes have been quite successful in applications, especially for problems containing both shocks and complicated smooth solution structures, such as compressible turbulence simulations and aeroacoustics. These lecture notes are basically self-contained. It is our hope that with these notes and with the help of the quoted references, the reader can understand the algorithms and code them up for applications.
Quantum and classical analysis of circular and sectorial billiards with central harmonic potential
NASA Astrophysics Data System (ADS)
Jurado-Taracena, Manuel; Plascencia, Alexis D.; Gutierrez-Vega, Julio C.
2012-06-01
We present the classical and quantum solutions to a particle confined in a circular sectorial billiard under a central harmonic potential, attractive and repulsive. The classical analysis is done by applying the Hamilton-Jacobi formalism; we derive the characteristic equations for periodic orbits, give expressions for the length of the trajectories in terms of elliptic integrals and study some geometrical constructions for the billiard. The quantum analysis leads to the study of the confluent hypergeometric function, from which we obtain the characteristic values for the energy spectra and the probability distributions inside the circular and sectorial billiard. As verification for the attractive case, as we increase the billiard radius our results approach the unbounded solutions. Finally we compare the classical probability distributions, obtained by assuming the probability as proportional to the time spent by the particle in each space interval, with the quantum ones.
Biological evolution in a multidimensional fitness landscape.
Saakian, David B; Kirakosyan, Zara; Hu, Chin-Kun
2012-09-01
We considered a multiblock molecular model of biological evolution, in which fitness is a function of the mean types of alleles located at different parts (blocks) of the genome. We formulated an infinite population model with selection and mutation, and calculated the mean fitness. For the case of recombination, we formulated a model with a multidimensional fitness landscape (the dimension of the space is equal to the number of blocks) and derived a theorem about the dynamics of initially narrow distribution. We also considered the case of lethal mutations. We also formulated the finite population version of the model in the case of lethal mutations. Our models, derived for the virus evolution, are interesting also for the statistical mechanics and the Hamilton-Jacobi equation as well.
Zhong, Xiangnan; He, Haibo; Zhang, Huaguang; Wang, Zhanshan
2014-12-01
In this paper, we develop and analyze an optimal control method for a class of discrete-time nonlinear Markov jump systems (MJSs) with unknown system dynamics. Specifically, an identifier is established for the unknown systems to approximate system states, and an optimal control approach for nonlinear MJSs is developed to solve the Hamilton-Jacobi-Bellman equation based on the adaptive dynamic programming technique. We also develop detailed stability analysis of the control approach, including the convergence of the performance index function for nonlinear MJSs and the existence of the corresponding admissible control. Neural network techniques are used to approximate the proposed performance index function and the control law. To demonstrate the effectiveness of our approach, three simulation studies, one linear case, one nonlinear case, and one single link robot arm case, are used to validate the performance of the proposed optimal control method.
Thermodynamic framework for discrete optimal control in multiphase flow systems
NASA Astrophysics Data System (ADS)
Sieniutycz, Stanislaw
1999-08-01
Bellman's method of dynamic programming is used to synthesize diverse optimization approaches to active (work producing) and inactive (entropy generating) multiphase flow systems. Thermal machines, optimally controlled unit operations, nonlinear heat conduction, spontaneous relaxation processes, and self-propagating wave fronts are all shown to satisfy a discrete Hamilton-Jacobi-Bellman equation and a corresponding discrete optimization algorithm of Pontryagin's type, with the maximum principle for a Hamiltonian. The extremal structures are always canonical. A common unifying criterion is set for all considered systems, which is the criterion of a minimum generated entropy. It is shown that constraints can modify the entropy functionals in a different way for each group of the processes considered; thus the resulting structures of these functionals may differ significantly. Practical conclusions are formulated regarding the energy savings and energy policy in optimally controlled systems.
Motion and high energy collision of magnetized particles around a Hořava-Lifshitz black hole
NASA Astrophysics Data System (ADS)
Abdujabbarov, Ahmadjon
2016-07-01
We have studied the orbits of magnetized particles around Hořava-Lifshitz black hole with mass M immersed in an asymptotically uniform magnetic field in the infrared approximation when ω M^2>> 1. It is shown that magnetized particle's orbit in Hořava-Lifshitz spacetime is different with compare to one in Schwarzschild spacetime due to the presence of additional terms related to the Kehagias-Sfetsos (KS) parameter ω. Using the Hamilton-Jacobi formalism, we have found the dependence of the area of stable circular orbits of the magnetized particle on dimensionless KS parameter tilde{ω} and have plotted them for several values of magnetic coupling parameter β as well as obtained the equations of motion of the magnetized particle. Moreover, we have studied the dependence of the collision of (magnetized, charged, non-charged) particles on KS parameter ω for some fixed values of the magnetic coupling parameter β.
NASA Astrophysics Data System (ADS)
Caplinger, J.; Sotnikov, V. I.; Wallerstein, A. J.
2014-12-01
A three dimensional numerical ray-tracing algorithm based on a Hamilton-Jacobi geometric optics approximation is used to analyze propagation of high frequency (HF) electromagnetic waves through a plasma with randomly distributed vortex structures having a spatial dependence in the plane perpendicular to earth's magnetic field. This spatial dependence in density is elongated and uniform along the magnetic field lines. Similar vortex structures may appear in the equatorial spread F region and in the Auroral zone of the ionosphere. The diffusion coefficient associated with wave vector deflection from a propagation path can be approximated by measuring the average deflection angle of the beam of rays. Then, the beam broadening can be described statistically using the Fokker-Planck equation. Visualizations of the ray propagation through generated density structures along with estimated and analytically calculated diffusion coefficients will be presented.
Hawking radiation as tunneling of vector particles from Kerr-Newman black hole
NASA Astrophysics Data System (ADS)
Ibungochouba Singh, T.; Ablu Meitei, I.; Yugindro Singh, K.
2016-03-01
In this paper, by applying the WKB approximation and Hamilton-Jacobi ansatz to the Proca equation, we investigate the tunneling of vector bosons across the event horizon of Kerr-Newman black hole and also the resulting vector particles' Hawking radiation. Universality of the properties of the emitted spectra of different types of particles is established for Kerr-Newman black hole. The coordinate problem for Hawking radiation of the vector particles is also investigated using three coordinate systems. The thermal spectrum of the radiated vector bosons determined using a direct computation corresponds to a temperature which is twice the Hawking temperature of Kerr-Newman black hole for scalar particles. If the well behaved Eddington coordinate system and Painleve coordinate system are used, the correct result of Hawking temperature is obtained. The reason for the discrepancy in the results of naive coordinate and well behaved coordinates is also discussed.
Inflation of small true vacuum bubble by quantization of Einstein-Hilbert action
NASA Astrophysics Data System (ADS)
He, DongShan; Cai, QingYu
2015-07-01
We study the quantization of the Einstein-Hilbert action for a small true vacuum bubble without matter or scalar field. The quantization of action induces an extra term of potential called quantum potential in Hamilton-Jacobi equation, which gives expanding solutions, including the exponential expansion solutions of the scalar factor a for the bubble. We show that exponential expansion of the bubble continues with a short period, no matter whether the bubble is closed, flat, or open. The exponential expansion ends spontaneously when the bubble becomes large, that is, the scalar factor a of the bubble approaches a Planck length l p. We show that it is the quantum potential of the small true vacuum bubble that plays the role of the scalar field potential suggested in the slow-roll inflation model. With the picture of quantum tunneling, we calculate particle creation rate during inflation, which shows that particles created by inflation have the capability of reheating the universe.
Motion and high energy collision of magnetized particles around a Hořava-Lifshitz black hole
NASA Astrophysics Data System (ADS)
Toshmatov, Bobir; Abdujabbarov, Ahmadjon; Ahmedov, Bobomurat; Stuchlík, Zdeněk
2015-11-01
We have studied the orbits of magnetized particles around Hořava-Lifshitz black hole with mass M immersed in an asymptotically uniform magnetic field in the infrared approximation when ω M2≫ 1. It is shown that magnetized particle's orbit in Hořava-Lifshitz spacetime is different with compare to one in Schwarzschild spacetime due to the presence of additional terms related to the Kehagias-Sfetsos (KS) parameter ω. Using the Hamilton-Jacobi formalism, we have found the dependence of the area of stable circular orbits of the magnetized particle on dimensionless KS parameter tilde{ω} and have plotted them for several values of magnetic coupling parameter β as well as obtained the equations of motion of the magnetized particle. Moreover, we have studied the dependence of the collision of (magnetized, charged, non-charged) particles on KS parameter ω for some fixed values of the magnetic coupling parameter β.
Zhong, Xiangnan; He, Haibo; Zhang, Huaguang; Wang, Zhanshan
2014-12-01
In this paper, we develop and analyze an optimal control method for a class of discrete-time nonlinear Markov jump systems (MJSs) with unknown system dynamics. Specifically, an identifier is established for the unknown systems to approximate system states, and an optimal control approach for nonlinear MJSs is developed to solve the Hamilton-Jacobi-Bellman equation based on the adaptive dynamic programming technique. We also develop detailed stability analysis of the control approach, including the convergence of the performance index function for nonlinear MJSs and the existence of the corresponding admissible control. Neural network techniques are used to approximate the proposed performance index function and the control law. To demonstrate the effectiveness of our approach, three simulation studies, one linear case, one nonlinear case, and one single link robot arm case, are used to validate the performance of the proposed optimal control method. PMID:25420238
Symplectic tracking through straight three dimensional fields by a method of generating functions
NASA Astrophysics Data System (ADS)
Titze, M.; Bahrdt, J.; Wüstefeld, G.
2016-01-01
For simulating single-particle trajectories, the derivation of final coordinates from known initial coordinates through arbitrary electromagnetic fields is of key interest in accelerator physics. We address this task in the case of straight stationary magnetic fields, using generating functions via a perturbative ansatz for the corresponding Hamilton-Jacobi equation. Such an approach is always symplectic, independent of the expansion order. We set up the Hamiltonian by static fields, represented by Fourier series, and outline this approach for the correct and complete set of 3D-multipole fields. Different types of multipoles can be treated with the same formalism, combining them with a specific table of Fourier coefficients characterizing their fields. The resulting particle-tracking routine maps the multipole in a single step. Results are compared with analytical estimations and high-resolution integration methods.
Weak electromagnetic field admitting integrability in Kerr-NUT-(A)dS spacetimes
NASA Astrophysics Data System (ADS)
Kolář, Ivan; Krtouš, Pavel
2015-06-01
We investigate properties of higher-dimensional generally rotating black-hole spacetimes, so-called Kerr-NUT-(anti)-de Sitter spacetimes, as well as a family of related spaces which share the same explicit and hidden symmetries. In these spaces, we study a particle motion in the presence of a weak electromagnetic field and compare it with its operator analogies. First, we find general commutativity conditions for classical observables and for their operator counterparts, then we investigate a fulfillment of these conditions in the Kerr-NUT-(anti)-de Sitter and related spaces. We find the most general form of the weak electromagnetic field compatible with the complete integrability of the particle motion and the comutativity of the field operators. For such a field we solve the charged Hamilton-Jacobi and Klein-Gordon equations by separation of variables.
Gravitinos tunneling from traversable Lorentzian wormholes
NASA Astrophysics Data System (ADS)
Sakalli, I.; Ovgun, A.
2015-09-01
Recent research shows that Hawking radiation (HR) is also possible around the trapping horizon of a wormhole. In this article, we show that the HR of gravitino (spin-) particles from the traversable Lorentzian wormholes (TLWH) reveals a negative Hawking temperature (HT). We first introduce the TLWH in the past outer trapping horizon geometry (POTHG). Next, we derive the Rarita-Schwinger equations (RSEs) for that geometry. Then, using both the Hamilton-Jacobi (HJ) ansätz and the WKB approximation in the quantum tunneling method, we obtain the probabilities of the emission/absorption modes. Finally, we derive the tunneling rate of the emitted gravitino particles, and succeed to read the HT of the TLWH.
Liu, Derong; Wang, Ding; Li, Hongliang
2014-02-01
In this paper, using a neural-network-based online learning optimal control approach, a novel decentralized control strategy is developed to stabilize a class of continuous-time nonlinear interconnected large-scale systems. First, optimal controllers of the isolated subsystems are designed with cost functions reflecting the bounds of interconnections. Then, it is proven that the decentralized control strategy of the overall system can be established by adding appropriate feedback gains to the optimal control policies of the isolated subsystems. Next, an online policy iteration algorithm is presented to solve the Hamilton-Jacobi-Bellman equations related to the optimal control problem. Through constructing a set of critic neural networks, the cost functions can be obtained approximately, followed by the control policies. Furthermore, the dynamics of the estimation errors of the critic networks are verified to be uniformly and ultimately bounded. Finally, a simulation example is provided to illustrate the effectiveness of the present decentralized control scheme. PMID:24807039
Biological evolution in a multidimensional fitness landscape.
Saakian, David B; Kirakosyan, Zara; Hu, Chin-Kun
2012-09-01
We considered a multiblock molecular model of biological evolution, in which fitness is a function of the mean types of alleles located at different parts (blocks) of the genome. We formulated an infinite population model with selection and mutation, and calculated the mean fitness. For the case of recombination, we formulated a model with a multidimensional fitness landscape (the dimension of the space is equal to the number of blocks) and derived a theorem about the dynamics of initially narrow distribution. We also considered the case of lethal mutations. We also formulated the finite population version of the model in the case of lethal mutations. Our models, derived for the virus evolution, are interesting also for the statistical mechanics and the Hamilton-Jacobi equation as well. PMID:23030957
On the stochastic SIS epidemic model in a periodic environment.
Bacaër, Nicolas
2015-08-01
In the stochastic SIS epidemic model with a contact rate a, a recovery rate b < a, and a population size N, the mean extinction time τ is such that (log τ)/N converges to c = b/a - 1 - log(b/a) as N grows to infinity. This article considers the more realistic case where the contact rate a(t) is a periodic function whose average is bigger than b. Then log τ/N converges to a new limit C, which is linked to a time-periodic Hamilton-Jacobi equation. When a(t) is a cosine function with small amplitude or high (resp. low) frequency, approximate formulas for C can be obtained analytically following the method used in Assaf et al. (Phys Rev E 78:041123, 2008). These results are illustrated by numerical simulations. PMID:25205518
Fermions tunneling from a general static Riemann black hole
NASA Astrophysics Data System (ADS)
Chen, Ge-Rui; Huang, Yong-Chang
2015-05-01
In this paper we investigate the tunneling of fermions from a general static Riemann black hole by following Kerner and Mann (Class Quantum Gravit 25:095014, 2008a; Phys Lett B 665:277-283, 2008b) methods. By applying the WKB approximation and the Hamilton-Jacobi ansatz to the Dirac equation, we obtain the standard Hawking temperature. Furthermore, Kerner and Mann (Class Quantum Gravit 25:095014, 2008a; Phys Lett B 665:277-283, 2008b) only calculated the tunneling spectrum of the Dirac particles with spin-up, and we extend the methods to investigate the tunneling of Dirac particles with arbitrary spin directions and also obtain the expected Hawking temperature. Our result provides further evidence for the universality of black hole radiation.
Hawking radiation from Elko particles tunnelling across black-strings horizon
NASA Astrophysics Data System (ADS)
da Rocha, R.; Hoff da Silva, J. M.
2014-09-01
We apply the tunnelling method for the emission and absorption of Elko particles in the event horizon of a black-string solution. We show that Elko particles are emitted at the expected Hawking temperature from black strings, but with a quite different signature with respect to the Dirac particles. We employ the Hamilton-Jacobi technique to black-hole tunnelling, by applying the WKB approximation to the coupled system of Dirac-like equations governing the Elko particle dynamics. As a typical signature, different Elko particles are shown to produce the same standard Hawking temperature for black strings. However, we prove that they present the same probability irrespectively of outgoing or ingoing the black-hole horizon. This provides a typical signature for mass-dimension-one fermions, that is different from the mass-dimension-three halves fermions inherent to Dirac particles, as different Dirac spinor fields have distinct inward and outward probability of tunnelling.
Massive mesons in Weyl-Dirac theory
NASA Astrophysics Data System (ADS)
Mirabotalebi, S.; Ahmadi, F.; Salehi, H.
2008-01-01
In order to study the mass generation of the vector fields in the framework of a conformal invariant gravitational model, the Weyl-Dirac theory is considered. The mass of the Weyl’s meson fields plays a principal role in this theory, it connects basically the conformal and gauge symmetries. We estimate this mass by using the large-scale characteristics of the observed universe. To do this we firstly specify a preferred conformal frame as a cosmological frame, then in this frame, we introduce an exact possible solution of the theory. We also study the dynamical effect of the massive vector meson fields on the trajectories of an elementary particle. We show that a local change of the cosmological frame leads to a Hamilton-Jacobi equation describing a particle with an adjustable mass. The dynamical effect of the massive vector meson field presents itself in the form of a correction term for the mass of the particle.
NASA Astrophysics Data System (ADS)
Liddle, Andrew R.
1994-01-01
The energy scale of inflation is of much interest, as it suggests the scale of grand unified physics, governs whether cosmological events such as topological defect formation can occur after inflation, and also determines the amplitude of gravitational waves which may be detectable using interferometers. The COBE results are used to limit the energy scale of inflation at the time large scale perturbations were imprinted. An exact dynamical treatment based on the Hamilton-Jacobi equations is then used to translate this into limits on the energy scale at the end of inflation. General constraints are given, and then tighter constraints based on physically motivated assumptions regarding the allowed forms of density perturbation and gravitational wave spectra. These are also compared with the values of familiar models.
Spherically symmetric nonlinear structures
NASA Astrophysics Data System (ADS)
Calzetta, Esteban A.; Kandus, Alejandra
1997-02-01
We present an analytical method to extract observational predictions about the nonlinear evolution of perturbations in a Tolman universe. We assume no a priori profile for them. We solve perturbatively a Hamilton-Jacobi equation for a timelike geodesic and obtain the null one as a limiting case in two situations: for an observer located in the center of symmetry and for a noncentered one. In the first case we find expressions to evaluate the density contrast and the number count and luminosity distance versus redshift relationships up to second order in the perturbations. In the second situation we calculate the CMBR anisotropies at large angular scales produced by the density contrast and by the asymmetry of the observer's location, up to first order in the perturbations. We develop our argument in such a way that the formulas are valid for any shape of the primordial spectrum.
NASA Astrophysics Data System (ADS)
Feng, Z. W.; Li, H. L.; Zu, X. T.; Yang, S. Z.
2016-04-01
We investigate the thermodynamics of Schwarzschild-Tangherlini black hole in the context of the generalized uncertainty principle (GUP). The corrections to the Hawking temperature, entropy and the heat capacity are obtained via the modified Hamilton-Jacobi equation. These modifications show that the GUP changes the evolution of the Schwarzschild-Tangherlini black hole. Specially, the GUP effect becomes susceptible when the radius or mass of the black hole approaches the order of Planck scale, it stops radiating and leads to a black hole remnant. Meanwhile, the Planck scale remnant can be confirmed through the analysis of the heat capacity. Those phenomena imply that the GUP may give a way to solve the information paradox. Besides, we also investigate the possibilities to observe the black hole at the Large Hadron Collider (LHC), and the results demonstrate that the black hole cannot be produced in the recent LHC.
Graphical Solution of Polynomial Equations
ERIC Educational Resources Information Center
Grishin, Anatole
2009-01-01
Graphing utilities, such as the ubiquitous graphing calculator, are often used in finding the approximate real roots of polynomial equations. In this paper the author offers a simple graphing technique that allows one to find all solutions of a polynomial equation (1) of arbitrary degree; (2) with real or complex coefficients; and (3) possessing…
Equations for nonbonded concrete overlays
NASA Astrophysics Data System (ADS)
Chou, Y. T.
1985-09-01
The nature of the design equations for the nonbonded concrete overlays currently used by the US Army Corps of Engineers was examined and the original source of the equation was also examined. Using simple mechanics, new overlay equations were developed which are suitable for different thicknesses and elastic properties in the overlay and base concrete slabs. The difference in the computed overlay thickness between the new and existing equations is not large when the overlay thickness is equal to or greater than the base slab. The difference can become excessive when the overlay thickness is much less than that of the base slab. The new equations were compared with the finite element computer program for concrete overlays with various combinations of slab thickness, elastic property, and subgrade modulus. The comparisons were very favorable, indicating that the overlay equations developed in this report are analytically correct. It was difficult to judge whether the new equations are superior to the existing equation. This conclusion was expected because for all the seven test sections analyzed, the overlay thicknesses were either equal to or greater than those of the base slabs.
Uncertainty of empirical correlation equations
NASA Astrophysics Data System (ADS)
Feistel, R.; Lovell-Smith, J. W.; Saunders, P.; Seitz, S.
2016-08-01
The International Association for the Properties of Water and Steam (IAPWS) has published a set of empirical reference equations of state, forming the basis of the 2010 Thermodynamic Equation of Seawater (TEOS-10), from which all thermodynamic properties of seawater, ice, and humid air can be derived in a thermodynamically consistent manner. For each of the equations of state, the parameters have been found by simultaneously fitting equations for a range of different derived quantities using large sets of measurements of these quantities. In some cases, uncertainties in these fitted equations have been assigned based on the uncertainties of the measurement results. However, because uncertainties in the parameter values have not been determined, it is not possible to estimate the uncertainty in many of the useful quantities that can be calculated using the parameters. In this paper we demonstrate how the method of generalised least squares (GLS), in which the covariance of the input data is propagated into the values calculated by the fitted equation, and in particular into the covariance matrix of the fitted parameters, can be applied to one of the TEOS-10 equations of state, namely IAPWS-95 for fluid pure water. Using the calculated parameter covariance matrix, we provide some preliminary estimates of the uncertainties in derived quantities, namely the second and third virial coefficients for water. We recommend further investigation of the GLS method for use as a standard method for calculating and propagating the uncertainties of values computed from empirical equations.
Students' Understanding of Quadratic Equations
ERIC Educational Resources Information Center
López, Jonathan; Robles, Izraim; Martínez-Planell, Rafael
2016-01-01
Action-Process-Object-Schema theory (APOS) was applied to study student understanding of quadratic equations in one variable. This required proposing a detailed conjecture (called a genetic decomposition) of mental constructions students may do to understand quadratic equations. The genetic decomposition which was proposed can contribute to help…
Generalized Multilevel Structural Equation Modeling
ERIC Educational Resources Information Center
Rabe-Hesketh, Sophia; Skrondal, Anders; Pickles, Andrew
2004-01-01
A unifying framework for generalized multilevel structural equation modeling is introduced. The models in the framework, called generalized linear latent and mixed models (GLLAMM), combine features of generalized linear mixed models (GLMM) and structural equation models (SEM) and consist of a response model and a structural model for the latent…
Simplified Relativistic Force Transformation Equation.
ERIC Educational Resources Information Center
Stewart, Benjamin U.
1979-01-01
A simplified relativistic force transformation equation is derived and then used to obtain the equation for the electromagnetic forces on a charged particle, calculate the electromagnetic fields due to a point charge with constant velocity, transform electromagnetic fields in general, derive the Biot-Savart law, and relate it to Coulomb's law.…
Complete solution of Boolean equations
NASA Technical Reports Server (NTRS)
Tapia, M. A.; Tucker, J. H.
1980-01-01
A method is presented for generating a single formula involving arbitary Boolean parameters, which includes in it each and every possible solution of a system of Boolean equations. An alternate condition equivalent to a known necessary and sufficient condition for solving a system of Boolean equations is given.
Transport equations for oscillating neutrinos
NASA Astrophysics Data System (ADS)
Zhang, Yunfan; Burrows, Adam
2013-11-01
We derive a suite of generalized Boltzmann equations, based on the density-matrix formalism, that incorporates the physics of neutrino oscillations for two- and three-flavor oscillations, matter refraction, and self-refraction. The resulting equations are straightforward extensions of the classical transport equations that nevertheless contain the full physics of quantum oscillation phenomena. In this way, our broadened formalism provides a bridge between the familiar neutrino transport algorithms employed by supernova modelers and the more quantum-heavy approaches frequently employed to illuminate the various neutrino oscillation effects. We also provide the corresponding angular-moment versions of this generalized equation set. Our goal is to make it easier for astrophysicists to address oscillation phenomena in a language with which they are familiar. The equations we derive are simple and practical, and are intended to facilitate progress concerning oscillation phenomena in the context of core-collapse supernova theory.
The Equations of Oceanic Motions
NASA Astrophysics Data System (ADS)
Müller, Peter
2006-10-01
Modeling and prediction of oceanographic phenomena and climate is based on the integration of dynamic equations. The Equations of Oceanic Motions derives and systematically classifies the most common dynamic equations used in physical oceanography, from large scale thermohaline circulations to those governing small scale motions and turbulence. After establishing the basic dynamical equations that describe all oceanic motions, M|ller then derives approximate equations, emphasizing the assumptions made and physical processes eliminated. He distinguishes between geometric, thermodynamic and dynamic approximations and between the acoustic, gravity, vortical and temperature-salinity modes of motion. Basic concepts and formulae of equilibrium thermodynamics, vector and tensor calculus, curvilinear coordinate systems, and the kinematics of fluid motion and wave propagation are covered in appendices. Providing the basic theoretical background for graduate students and researchers of physical oceanography and climate science, this book will serve as both a comprehensive text and an essential reference.
Equation predicts diesel cloud points
Tsang, C.Y.; Ker, V.S.F.; Miranda, R.D.; Wesch, J.C.
1988-03-28
Diesel fuel cloud points can be predicted by an empirical equation developed by NOCA/Husky Research Corp. The equation can accurately predict cloud points from feedstock and product data readily available in the refinery. The applicability of the equation to a full range of summer, winter, and arctic diesel blends was proven by studies conducted on data from four Canadian refineries that process a wide variety of conventional crude oils and synthetic crude from bitumen. Results of the studies show that the variance between equation predicted and measured cloud point values are within acceptable reproducibility of measured data. Considerable time can be saved in the refinery when the equation is used for optimizing diesel fuel blend formulations. Applicability ranges from daily blending calculations, to use in linear programs for long-term planning for distillate utilization.
Extended Trial Equation Method for Nonlinear Partial Differential Equations
NASA Astrophysics Data System (ADS)
Gepreel, Khaled A.; Nofal, Taher A.
2015-04-01
The main objective of this paper is to use the extended trial equation method to construct a series of some new solutions for some nonlinear partial differential equations (PDEs) in mathematical physics. We will construct the solutions in many different functions such as hyperbolic function solutions, trigonometric function solutions, Jacobi elliptic function solutions, and rational functional solutions for the nonlinear PDEs when the balance number is a real number via the Zhiber-Shabat nonlinear differential equation. The balance number of this method is not constant as we shown in other methods, but it is changed by changing the trial equation derivative definition. This method allowed us to construct many new types of solutions. It is shown by using the Maple software package that all obtained solutions satisfy the original PDEs.
Higher derivative gravity: Field equation as the equation of state
NASA Astrophysics Data System (ADS)
Dey, Ramit; Liberati, Stefano; Mohd, Arif
2016-08-01
One of the striking features of general relativity is that the Einstein equation is implied by the Clausius relation imposed on a small patch of locally constructed causal horizon. The extension of this thermodynamic derivation of the field equation to more general theories of gravity has been attempted many times in the last two decades. In particular, equations of motion for minimally coupled higher-curvature theories of gravity, but without the derivatives of curvature, have previously been derived using a thermodynamic reasoning. In that derivation the horizon slices were endowed with an entropy density whose form resembles that of the Noether charge for diffeomorphisms, and was dubbed the Noetheresque entropy. In this paper, we propose a new entropy density, closely related to the Noetheresque form, such that the field equation of any diffeomorphism-invariant metric theory of gravity can be derived by imposing the Clausius relation on a small patch of local causal horizon.
Wave equations for pulse propagation
NASA Astrophysics Data System (ADS)
Shore, B. W.
1987-06-01
Theoretical discussions of the propagation of pulses of laser radiation through atomic or molecular vapor rely on a number of traditional approximations for idealizing the radiation and the molecules, and for quantifying their mutual interaction by various equations of propagation (for the radiation) and excitation (for the molecules). In treating short-pulse phenomena it is essential to consider coherent excitation phenomena of the sort that is manifest in Rabi oscillations of atomic or molecular populations. Such processes are not adequately treated by rate equations for excitation nor by rate equations for radiation. As part of a more comprehensive treatment of the coupled equations that describe propagation of short pulses, this memo presents background discussion of the equations that describe the field. This memo discusses the origin, in Maxwell's equations, of the wave equation used in the description of pulse propagation. It notes the separation into lamellar and solenoidal (or longitudinal and transverse) and positive and negative frequency parts. It mentions the possibility of separating the polarization field into linear and nonlinear parts, in order to define a susceptibility or index of refraction and, from these, a phase and group velocity.
SETS. Set Equation Transformation System
Worrell, R.B.
1992-01-13
SETS is used for symbolic manipulation of Boolean equations, particularly the reduction of equations by the application of Boolean identities. It is a flexible and efficient tool for performing probabilistic risk analysis (PRA), vital area analysis, and common cause analysis. The equation manipulation capabilities of SETS can also be used to analyze noncoherent fault trees and determine prime implicants of Boolean functions, to verify circuit design implementation, to determine minimum cost fire protection requirements for nuclear reactor plants, to obtain solutions to combinatorial optimization problems with Boolean constraints, and to determine the susceptibility of a facility to unauthorized access through nullification of sensors in its protection system.
Pavement performance equations. Final report
Mahoney, J.P.; Kay, R.K.; Jackson, N.C.
1988-03-01
The WSDOT PMS data base was used to develop regression equations for three pavement surface types: bituminous surface treatments, asphalt concrete, and portland-cement concrete. The primary regression equations developed were to predict Pavement Condition Rating (PCR) which is a measure of the pavement surface distress (ranges from 100 (no distress) to below 0 (extensive distress)). Overall, the equations fit the data rather well given the expected variation of pavement performance information. The relative effects of age (time since construction or reconstruction) were illustrated for the three surface types.
Overdetermined Systems of Linear Equations.
ERIC Educational Resources Information Center
Williams, Gareth
1990-01-01
Explored is an overdetermined system of linear equations to find an appropriate least squares solution. A geometrical interpretation of this solution is given. Included is a least squares point discussion. (KR)
Solving Differential Equations in R
Although R is still predominantly applied for statistical analysis and graphical representation, it is rapidly becoming more suitable for mathematical computing. One of the fields where considerable progress has been made recently is the solution of differential equations. Here w...
NASA Technical Reports Server (NTRS)
Shebalin, John V.
1987-01-01
The Boussinesq approximation is extended so as to explicitly account for the transfer of fluid energy through viscous action into thermal energy. Ideal and dissipative integral invariants are discussed, in addition to the general equations for thermal-fluid motion.
Friedmann equation with quantum potential
Siong, Ch'ng Han; Radiman, Shahidan; Nikouravan, Bijan
2013-11-27
Friedmann equations are used to describe the evolution of the universe. Solving Friedmann equations for the scale factor indicates that the universe starts from an initial singularity where all the physical laws break down. However, the Friedmann equations are well describing the late-time or large scale universe. Hence now, many physicists try to find an alternative theory to avoid this initial singularity. In this paper, we generate a version of first Friedmann equation which is added with an additional term. This additional term contains the quantum potential energy which is believed to play an important role at small scale. However, it will gradually become negligible when the universe evolves to large scale.
Parametric Equations, Maple, and Tubeplots.
ERIC Educational Resources Information Center
Feicht, Louis
1997-01-01
Presents an activity that establishes a graphical foundation for parametric equations by using a graphing output form called tubeplots from the computer program Maple. Provides a comprehensive review and exploration of many previously learned topics. (ASK)
IKT for quantum hydrodynamic equations
NASA Astrophysics Data System (ADS)
Tessarotto, Massimo; Ellero, Marco; Nicolini, Piero
2007-11-01
A striking feature of standard quantum mechanics (SQM) is its analogy with classical fluid dynamics. In fact, it is well-known that the Schr"odinger equation is equivalent to a closed set of partial differential equations for suitable real-valued functions of position and time (denoted as quantum fluid fields) [Madelung, 1928]. In particular, the corresponding quantum hydrodynamic equations (QHE) can be viewed as the equations of a classical compressible and non-viscous fluid, endowed with potential velocity and quantized velocity circulation. In this reference, an interesting theoretical problem, in its own right, is the construction of an inverse kinetic theory (IKT) for such a type of fluids. In this note we intend to investigate consequences of the IKT recently formulated for QHE [M.Tessarotto et al., Phys. Rev. A 75, 012105 (2007)]. In particular a basic issue is related to the definition of the quantum fluid fields.
Hidden Statistics of Schroedinger Equation
NASA Technical Reports Server (NTRS)
Zak, Michail
2011-01-01
Work was carried out in determination of the mathematical origin of randomness in quantum mechanics and creating a hidden statistics of Schr dinger equation; i.e., to expose the transitional stochastic process as a "bridge" to the quantum world. The governing equations of hidden statistics would preserve such properties of quantum physics as superposition, entanglement, and direct-product decomposability while allowing one to measure its state variables using classical methods.
Geometrical Solutions of Quadratic Equations.
ERIC Educational Resources Information Center
Grewal, A. S.; Godloza, L.
1999-01-01
Demonstrates that the equation of a circle (x-h)2 + (y-k)2 = r2 with center (h; k) and radius r reduces to a quadratic equation x2-2xh + (h2 + k2 -r2) = O at the intersection with the x-axis. Illustrates how to determine the center of a circle as well as a point on a circle. (Author/ASK)
Allidina, A.Y.; Malinowski, K.; Singh, M.G.
1982-12-01
The possibilities were explored for enhancing parallelism in the simulation of systems described by algebraic equations, ordinary differential equations and partial differential equations. These techniques, using multiprocessors, were developed to speed up simulations, e.g. for nuclear accidents. Issues involved in their design included suitable approximations to bring the problem into a numerically manageable form and a numerical procedure to perform the computations necessary to solve the problem accurately. Parallel processing techniques used as simulation procedures, and a design of a simulation scheme and simulation procedure employing parallel computer facilities, were both considered.
An Exact Mapping from Navier-Stokes Equation to Schr"odinger Equation via Riccati Equation
NASA Astrophysics Data System (ADS)
Christianto, Vic; Smarandache, Florentin
2010-03-01
In the present article we argue that it is possible to write down Schr"odinger representation of Navier-Stokes equation via Riccati equation. The proposed approach, while differs appreciably from other method such as what is proposed by R. M. Kiehn, has an advantage, i.e. it enables us extend further to quaternionic and biquaternionic version of Navier-Stokes equation, for instance via Kravchenko's and Gibbon's route. Further observation is of course recommended in order to refute or verify this proposition.
Optimization of one-way wave equations.
Lee, M.W.; Suh, S.Y.
1985-01-01
The theory of wave extrapolation is based on the square-root equation or one-way equation. The full wave equation represents waves which propagate in both directions. On the contrary, the square-root equation represents waves propagating in one direction only. A new optimization method presented here improves the dispersion relation of the one-way wave equation. -from Authors
Turbulent fluid motion 3: Basic continuum equations
NASA Technical Reports Server (NTRS)
Deissler, Robert G.
1991-01-01
A derivation of the continuum equations used for the analysis of turbulence is given. These equations include the continuity equation, the Navier-Stokes equations, and the heat transfer or energy equation. An experimental justification for using a continuum approach for the study of turbulence is given.
A hyperbolic equation for turbulent diffusion
NASA Astrophysics Data System (ADS)
Ghosal, Sandip; Keller, Joseph B.
2000-09-01
A hyperbolic equation, analogous to the telegrapher's equation in one dimension, is introduced to describe turbulent diffusion of a passive additive in a turbulent flow. The predictions of this equation, and those of the usual advection-diffusion equation, are compared with data on smoke plumes in the atmosphere and on heat flow in a wind tunnel. The predictions of the hyperbolic equation fit the data at all distances from the source, whereas those of the advection-diffusion equation fit only at large distances. The hyperbolic equation is derived from an integrodifferential equation for the mean concentration which allows it to vary rapidly. If the mean concentration varies sufficiently slowly compared with the correlation time of the turbulence, the hyperbolic equation reduces to the advection-diffusion equation. However, if the mean concentration varies very rapidly, the hyperbolic equation should be replaced by the integrodifferential equation.
How to Obtain the Covariant Form of Maxwell's Equations from the Continuity Equation
ERIC Educational Resources Information Center
Heras, Jose A.
2009-01-01
The covariant Maxwell equations are derived from the continuity equation for the electric charge. This result provides an axiomatic approach to Maxwell's equations in which charge conservation is emphasized as the fundamental axiom underlying these equations.
Differential Equations for Morphological Amoebas
NASA Astrophysics Data System (ADS)
Welk, Martin; Breuß, Michael; Vogel, Oliver
This paper is concerned with amoeba median filtering, a structure-adaptive morphological image filter. It has been introduced by Lerallut et al. in a discrete formulation. Experimental evidence shows that iterated amoeba median filtering leads to segmentation-like results that are similar to those obtained by self-snakes, an image filter based on a partial differential equation. We investigate this correspondence by analysing a space-continuous formulation of iterated median filtering. We prove that in the limit of vanishing radius of the structuring elements, iterated amoeba median filtering indeed approximates a partial differential equation related to self-snakes and the well-known (mean) curvature motion equation. We present experiments with discrete iterated amoeba median filtering that confirm qualitative and quantitative predictions of our analysis.
Integration of quantum hydrodynamical equation
NASA Astrophysics Data System (ADS)
Ulyanova, Vera G.; Sanin, Andrey L.
2007-04-01
Quantum hydrodynamics equations describing the dynamics of quantum fluid are a subject of this report (QFD).These equations can be used to decide the wide class of problem. But there are the calculated difficulties for the equations, which take place for nonlinear hyperbolic systems. In this connection, It is necessary to impose the additional restrictions which assure the existence and unique of solutions. As test sample, we use the free wave packet and study its behavior at the different initial and boundary conditions. The calculations of wave packet propagation cause in numerical algorithm the division. In numerical algorithm at the calculations of wave packet propagation, there arises the problem of division by zero. To overcome this problem we have to sew together discrete numerical and analytical continuous solutions on the boundary. We demonstrate here for the free wave packet that the numerical solution corresponds to the analytical solution.
Students' understanding of quadratic equations
NASA Astrophysics Data System (ADS)
López, Jonathan; Robles, Izraim; Martínez-Planell, Rafael
2016-05-01
Action-Process-Object-Schema theory (APOS) was applied to study student understanding of quadratic equations in one variable. This required proposing a detailed conjecture (called a genetic decomposition) of mental constructions students may do to understand quadratic equations. The genetic decomposition which was proposed can contribute to help students achieve an understanding of quadratic equations with improved interrelation of ideas and more flexible application of solution methods. Semi-structured interviews with eight beginning undergraduate students explored which of the mental constructions conjectured in the genetic decomposition students could do, and which they had difficulty doing. Two of the mental constructions that form part of the genetic decomposition are highlighted and corresponding further data were obtained from the written work of 121 undergraduate science and engineering students taking a multivariable calculus course. The results suggest the importance of explicitly considering these two highlighted mental constructions.
Fractional-calculus diffusion equation
2010-01-01
Background Sequel to the work on the quantization of nonconservative systems using fractional calculus and quantization of a system with Brownian motion, which aims to consider the dissipation effects in quantum-mechanical description of microscale systems. Results The canonical quantization of a system represented classically by one-dimensional Fick's law, and the diffusion equation is carried out according to the Dirac method. A suitable Lagrangian, and Hamiltonian, describing the diffusive system, are constructed and the Hamiltonian is transformed to Schrodinger's equation which is solved. An application regarding implementation of the developed mathematical method to the analysis of diffusion, osmosis, which is a biological application of the diffusion process, is carried out. Schrödinger's equation is solved. Conclusions The plot of the probability function represents clearly the dissipative and drift forces and hence the osmosis, which agrees totally with the macro-scale view, or the classical-version osmosis. PMID:20492677
Maxwell's mixing equation revisited: characteristic impedance equations for ellipsoidal cells.
Stubbe, Marco; Gimsa, Jan
2015-07-21
We derived a series of, to our knowledge, new analytic expressions for the characteristic features of the impedance spectra of suspensions of homogeneous and single-shell spherical, spheroidal, and ellipsoidal objects, e.g., biological cells of the general ellipsoidal shape. In the derivation, we combined the Maxwell-Wagner mixing equation with our expression for the Clausius-Mossotti factor that had been originally derived to describe AC-electrokinetic effects such as dielectrophoresis, electrorotation, and electroorientation. The influential radius model was employed because it allows for a separation of the geometric and electric problems. For shelled objects, a special axial longitudinal element approach leads to a resistor-capacitor model, which can be used to simplify the mixing equation. Characteristic equations were derived for the plateau levels, peak heights, and characteristic frequencies of the impedance as well as the complex specific conductivities and permittivities of suspensions of axially and randomly oriented homogeneous and single-shell ellipsoidal objects. For membrane-covered spherical objects, most of the limiting cases are identical to-or improved with respect to-the known solutions given by researchers in the field. The characteristic equations were found to be quite precise (largest deviations typically <5% with respect to the full model) when tested with parameters relevant to biological cells. They can be used for the differentiation of orientation and the electric properties of cell suspensions or in the analysis of single cells in microfluidic systems. PMID:26200856
Explicit integration of Friedmann's equation with nonlinear equations of state
NASA Astrophysics Data System (ADS)
Chen, Shouxin; Gibbons, Gary W.; Yang, Yisong
2015-05-01
In this paper we study the integrability of the Friedmann equations, when the equation of state for the perfect-fluid universe is nonlinear, in the light of the Chebyshev theorem. A series of important, yet not previously touched, problems will be worked out which include the generalized Chaplygin gas, two-term energy density, trinomial Friedmann, Born-Infeld, two-fluid models, and Chern-Simons modified gravity theory models. With the explicit integration, we are able to understand exactly the roles of the physical parameters in various models play in the cosmological evolution which may also offer clues to a profound understanding of the problems in general settings. For example, in the Chaplygin gas universe, a few integrable cases lead us to derive a universal formula for the asymptotic exponential growth rate of the scale factor, of an explicit form, whether the Friedmann equation is integrable or not, which reveals the coupled roles played by various physical sectors and it is seen that, as far as there is a tiny presence of nonlinear matter, conventional linear matter makes contribution to the dark matter, which becomes significant near the phantom divide line. The Friedmann equations also arise in areas of physics not directly related to cosmology. We provide some examples ranging from geometric optics and central orbits to soap films and the shape of glaciated valleys to which our results may be applied.
Universal soil loss equation and revised universal soil loss equation
Technology Transfer Automated Retrieval System (TEKTRAN)
Soil erosion has long been recognized as a serious problem. Considerable efforts have been expended to address this problem. Thousands of plot years of data were summarized by ARS researchers in producing the Universal Soil Loss Equation (USLE). This technology has been used for conservation planni...
Jourdain's variational equation and Appell's equation of motion for nonholonomic dynamical systems
NASA Astrophysics Data System (ADS)
Wang, Li-Sheng; Pao, Yih-Hsing
2003-01-01
Based on Jourdain's variational equation proposed in 1909, we deduce a minimal set of general equations of motion for nonholomic dynamical systems of particles and rigid bodies. This equation of motion for the system, which differs slightly from the Gibbs-Appell equation, appears to be the same as the equation derived by Kane in 1961. Since the same equation was established by Appell in 1903 on the basis of D'Alembert's principle, the newly derived equation is named Appell's equation.
Transport equations in tokamak plasmas
Callen, J. D.; Hegna, C. C.; Cole, A. J.
2010-05-15
Tokamak plasma transport equations are usually obtained by flux surface averaging the collisional Braginskii equations. However, tokamak plasmas are not in collisional regimes. Also, ad hoc terms are added for neoclassical effects on the parallel Ohm's law, fluctuation-induced transport, heating, current-drive and flow sources and sinks, small magnetic field nonaxisymmetries, magnetic field transients, etc. A set of self-consistent second order in gyroradius fluid-moment-based transport equations for nearly axisymmetric tokamak plasmas has been developed using a kinetic-based approach. The derivation uses neoclassical-based parallel viscous force closures, and includes all the effects noted above. Plasma processes on successive time scales and constraints they impose are considered sequentially: compressional Alfven waves (Grad-Shafranov equilibrium, ion radial force balance), sound waves (pressure constant along field lines, incompressible flows within a flux surface), and collisions (electrons, parallel Ohm's law; ions, damping of poloidal flow). Radial particle fluxes are driven by the many second order in gyroradius toroidal angular torques on a plasma species: seven ambipolar collision-based ones (classical, neoclassical, etc.) and eight nonambipolar ones (fluctuation-induced, polarization flows from toroidal rotation transients, etc.). The plasma toroidal rotation equation results from setting to zero the net radial current induced by the nonambipolar fluxes. The radial particle flux consists of the collision-based intrinsically ambipolar fluxes plus the nonambipolar fluxes evaluated at the ambipolarity-enforcing toroidal plasma rotation (radial electric field). The energy transport equations do not involve an ambipolar constraint and hence are more directly obtained. The 'mean field' effects of microturbulence on the parallel Ohm's law, poloidal ion flow, particle fluxes, and toroidal momentum and energy transport are all included self-consistently. The
Transport Equations In Tokamak Plasmas
NASA Astrophysics Data System (ADS)
Callen, J. D.
2009-11-01
Tokamak plasma transport equations are usually obtained by flux surface averaging the collisional Braginskii equations. However, tokamak plasmas are not in collisional regimes. Also, ad hoc terms are added for: neoclassical effects on the parallel Ohm's law (trapped particle effects on resistivity, bootstrap current); fluctuation-induced transport; heating, current-drive and flow sources and sinks; small B field non-axisymmetries; magnetic field transients etc. A set of self-consistent second order in gyroradius fluid-moment-based transport equations for nearly axisymmetric tokamak plasmas has been developed recently using a kinetic-based framework. The derivation uses neoclassical-based parallel viscous force closures, and includes all the effects noted above. Plasma processes on successive time scales (and constraints they impose) are considered sequentially: compressional Alfv'en waves (Grad-Shafranov equilibrium, ion radial force balance); sound waves (pressure constant along field lines, incompressible flows within a flux surface); and ion collisions (damping of poloidal flow). Radial particle fluxes are driven by the many second order in gyroradius toroidal angular torques on the plasma fluid: 7 ambipolar collision-based ones (classical, neoclassical, etc.) and 8 non-ambipolar ones (fluctuation-induced, polarization flows from toroidal rotation transients etc.). The plasma toroidal rotation equation [1] results from setting to zero the net radial current induced by the non-ambipolar fluxes. The radial particle flux consists of the collision-based intrinsically ambipolar fluxes plus the non-ambipolar fluxes evaluated at the ambipolarity-enforcing toroidal plasma rotation (radial electric field). The energy transport equations do not involve an ambipolar constraint and hence are more directly obtained. The resultant transport equations will be presented and contrasted with the usual ones. [4pt] [1] J.D. Callen, A.J. Cole, C.C. Hegna, ``Toroidal Rotation In
Transport equations in tokamak plasmasa)
NASA Astrophysics Data System (ADS)
Callen, J. D.; Hegna, C. C.; Cole, A. J.
2010-05-01
Tokamak plasma transport equations are usually obtained by flux surface averaging the collisional Braginskii equations. However, tokamak plasmas are not in collisional regimes. Also, ad hoc terms are added for neoclassical effects on the parallel Ohm's law, fluctuation-induced transport, heating, current-drive and flow sources and sinks, small magnetic field nonaxisymmetries, magnetic field transients, etc. A set of self-consistent second order in gyroradius fluid-moment-based transport equations for nearly axisymmetric tokamak plasmas has been developed using a kinetic-based approach. The derivation uses neoclassical-based parallel viscous force closures, and includes all the effects noted above. Plasma processes on successive time scales and constraints they impose are considered sequentially: compressional Alfvén waves (Grad-Shafranov equilibrium, ion radial force balance), sound waves (pressure constant along field lines, incompressible flows within a flux surface), and collisions (electrons, parallel Ohm's law; ions, damping of poloidal flow). Radial particle fluxes are driven by the many second order in gyroradius toroidal angular torques on a plasma species: seven ambipolar collision-based ones (classical, neoclassical, etc.) and eight nonambipolar ones (fluctuation-induced, polarization flows from toroidal rotation transients, etc.). The plasma toroidal rotation equation results from setting to zero the net radial current induced by the nonambipolar fluxes. The radial particle flux consists of the collision-based intrinsically ambipolar fluxes plus the nonambipolar fluxes evaluated at the ambipolarity-enforcing toroidal plasma rotation (radial electric field). The energy transport equations do not involve an ambipolar constraint and hence are more directly obtained. The "mean field" effects of microturbulence on the parallel Ohm's law, poloidal ion flow, particle fluxes, and toroidal momentum and energy transport are all included self-consistently. The
Young's Equation at the Nanoscale
NASA Astrophysics Data System (ADS)
Seveno, David; Blake, Terence D.; De Coninck, Joël
2013-08-01
In 1805, Thomas Young was the first to propose an equation to predict the value of the equilibrium contact angle of a liquid on a solid. Today, the force exerted by a liquid on a solid, such as a flat plate or fiber, is routinely used to assess this angle. Moreover, it has recently become possible to study wetting at the nanoscale using an atomic force microscope. Here, we report the use of molecular-dynamics simulations to investigate the force distribution along a 15 nm fiber dipped into a liquid meniscus. We find very good agreement between the measured force and that predicted by Young’s equation.
Investigation of the kinetic model equations
NASA Astrophysics Data System (ADS)
Liu, Sha; Zhong, Chengwen
2014-03-01
Currently the Boltzmann equation and its model equations are widely used in numerical predictions for dilute gas flows. The nonlinear integro-differential Boltzmann equation is the fundamental equation in the kinetic theory of dilute monatomic gases. By replacing the nonlinear fivefold collision integral term by a nonlinear relaxation term, its model equations such as the famous Bhatnagar-Gross-Krook (BGK) equation are mathematically simple. Since the computational cost of solving model equations is much less than that of solving the full Boltzmann equation, the model equations are widely used in predicting rarefied flows, multiphase flows, chemical flows, and turbulent flows although their predictions are only qualitatively right for highly nonequilibrium flows in transitional regime. In this paper the differences between the Boltzmann equation and its model equations are investigated aiming at giving guidelines for the further development of kinetic models. By comparing the Boltzmann equation and its model equations using test cases with different nonequilibrium types, two factors (the information held by nonequilibrium moments and the different relaxation rates of high- and low-speed molecules) are found useful for adjusting the behaviors of modeled collision terms in kinetic regime. The usefulness of these two factors are confirmed by a generalized model collision term derived from a mathematical relation between the Boltzmann equation and BGK equation that is also derived in this paper. After the analysis of the difference between the Boltzmann equation and the BGK equation, an attempt at approximating the collision term is proposed.
Sonar equations for planetary exploration.
Ainslie, Michael A; Leighton, Timothy G
2016-08-01
The set of formulations commonly known as "the sonar equations" have for many decades been used to quantify the performance of sonar systems in terms of their ability to detect and localize objects submerged in seawater. The efficacy of the sonar equations, with individual terms evaluated in decibels, is well established in Earth's oceans. The sonar equations have been used in the past for missions to other planets and moons in the solar system, for which they are shown to be less suitable. While it would be preferable to undertake high-fidelity acoustical calculations to support planning, execution, and interpretation of acoustic data from planetary probes, to avoid possible errors for planned missions to such extraterrestrial bodies in future, doing so requires awareness of the pitfalls pointed out in this paper. There is a need to reexamine the assumptions, practices, and calibrations that work well for Earth to ensure that the sonar equations can be accurately applied in combination with the decibel to extraterrestrial scenarios. Examples are given for icy oceans such as exist on Europa and Ganymede, Titan's hydrocarbon lakes, and for the gaseous atmospheres of (for example) Jupiter and Venus.
Equations of motion for superfluids
Basile, A.G.; Elser, V.
1995-06-01
To the principles of least action and minimum error, for determining the time evolution of the parameters in a variational wave function, we add a third: continuous collapse dynamics. In this formulation, exact time evolution is applied for an infinitesimal time and is followed by projection of the state back into the variational manifold (``collapse``). All three principles lead to the same equations of motion when applied to complex parameters but take two distinct forms when the parameters are real. As an application of these principles, we study the time evolution of two variational wave functions for superfluids. The first wave function, containing real parameters, was considered by Kerman and Koonin [Ann. Phys. (N.Y.) 100, 332 (1976)] and leads to the Euler equation in the hydrodynamic limit. The equation for our second wave function, a coherent state of Feynman excitations with complex parameters, has essentially the same hydrodynamic limit. The latter wave function, however, has a significant advantage in that the equation it generates is useful and meaningful on a microscopic scale as well.
Duffing's Equation and Nonlinear Resonance
ERIC Educational Resources Information Center
Fay, Temple H.
2003-01-01
The phenomenon of nonlinear resonance (sometimes called the "jump phenomenon") is examined and second-order van der Pol plane analysis is employed to indicate that this phenomenon is not a feature of the equation, but rather the result of accumulated round-off error, truncation error and algorithm error that distorts the true bounded solution onto…
Pendulum Motion and Differential Equations
ERIC Educational Resources Information Center
Reid, Thomas F.; King, Stephen C.
2009-01-01
A common example of real-world motion that can be modeled by a differential equation, and one easily understood by the student, is the simple pendulum. Simplifying assumptions are necessary for closed-form solutions to exist, and frequently there is little discussion of the impact if those assumptions are not met. This article presents a…
The solution of transcendental equations
NASA Technical Reports Server (NTRS)
Agrawal, K. M.; Outlaw, R.
1973-01-01
Some of the existing methods to globally approximate the roots of transcendental equations namely, Graeffe's method, are studied. Summation of the reciprocated roots, Whittaker-Bernoulli method, and the extension of Bernoulli's method via Koenig's theorem are presented. The Aitken's delta squared process is used to accelerate the convergence. Finally, the suitability of these methods is discussed in various cases.
Renaissance Learning Equating Study. Report
ERIC Educational Resources Information Center
Sewell, Julie; Sainsbury, Marian; Pyle, Katie; Keogh, Nikki; Styles, Ben
2007-01-01
An equating study was carried out in autumn 2006 by the National Foundation for Educational Research (NFER) on behalf of Renaissance Learning, to provide validation evidence for the use of the Renaissance Star Reading and Star Mathematics tests in English schools. The study investigated the correlation between the Star tests and established tests.…
Ordinary Differential Equation System Solver
1992-03-05
LSODE is a package of subroutines for the numerical solution of the initial value problem for systems of first order ordinary differential equations. The package is suitable for either stiff or nonstiff systems. For stiff systems the Jacobian matrix may be treated in either full or banded form. LSODE can also be used when the Jacobian can be approximated by a band matrix.
Perceptions of the Schrodinger equation
NASA Astrophysics Data System (ADS)
Efthimiades, Spyros
2014-03-01
The Schrodinger equation has been considered to be a postulate of quantum physics, but it is also perceived as the quantum equivalent of the non-relativistic classical energy relation. We argue that the Schrodinger equation cannot be a physical postulate, and we show explicitly that its second space derivative term is wrongly associated with the kinetic energy of the particle. The kinetic energy of a particle at a point is proportional to the square of the momentum, that is, to the square of the first space derivative of the wavefunction. Analyzing particle interactions, we realize that particles have multiple virtual motions and that each motion is accompanied by a wave that has constant amplitude. Accordingly, we define the wavefunction as the superposition of the virtual waves of the particle. In simple interaction settings we can tell what particle motions arise and can explain the outcomes in direct and tangible terms. Most importantly, the mathematical foundation of quantum mechanics becomes clear and justified, and we derive the Schrodinger, Dirac, etc. equations as the conditions the wavefunction must satisfy at each space-time point in order to fulfill the respective total energy equation.
The Symbolism Of Chemical Equations
ERIC Educational Resources Information Center
Jensen, William B.
2005-01-01
A question about the historical origin of equal sign and double arrow symbolism in balanced chemical equation is raised. The study shows that Marshall proposed the symbolism in 1902, which includes the use of currently favored double barb for equilibrium reactions.
The Forced Soft Spring Equation
ERIC Educational Resources Information Center
Fay, T. H.
2006-01-01
Through numerical investigations, this paper studies examples of the forced Duffing type spring equation with [epsilon] negative. By performing trial-and-error numerical experiments, the existence is demonstrated of stability boundaries in the phase plane indicating initial conditions yielding bounded solutions. Subharmonic boundaries are…
Scale Shrinkage in Vertical Equating.
ERIC Educational Resources Information Center
Camilli, Gregory; And Others
1993-01-01
Three potential causes of scale shrinkage (measurement error, restriction of range, and multidimensionality) in item response theory vertical equating are discussed, and a more comprehensive model-based approach to establishing vertical scales is described. Test data from the National Assessment of Educational Progress are used to illustrate the…
Mathematics and Reading Test Equating.
ERIC Educational Resources Information Center
Lee, Ong Kim; Wright, Benjamin D.
As part of a larger project to assess changes in student learning resulting from school reform, this study equates levels 6 through 14 of the mathematics and reading comprehension components of Form 7 of the Iowa Tests of Basic Skills (ITBS) with levels 7 through 14 of the mathematics and reading comprehension components of the CPS90 (another…
Empirical equation estimates geothermal gradients
Kutasov, I.M. )
1995-01-02
An empirical equation can estimate geothermal (natural) temperature profiles in new exploration areas. These gradients are useful for cement slurry and mud design and for improving electrical and temperature log interpretation. Downhole circulating temperature logs and surface outlet temperatures are used for predicting the geothermal gradients.
Sonar equations for planetary exploration.
Ainslie, Michael A; Leighton, Timothy G
2016-08-01
The set of formulations commonly known as "the sonar equations" have for many decades been used to quantify the performance of sonar systems in terms of their ability to detect and localize objects submerged in seawater. The efficacy of the sonar equations, with individual terms evaluated in decibels, is well established in Earth's oceans. The sonar equations have been used in the past for missions to other planets and moons in the solar system, for which they are shown to be less suitable. While it would be preferable to undertake high-fidelity acoustical calculations to support planning, execution, and interpretation of acoustic data from planetary probes, to avoid possible errors for planned missions to such extraterrestrial bodies in future, doing so requires awareness of the pitfalls pointed out in this paper. There is a need to reexamine the assumptions, practices, and calibrations that work well for Earth to ensure that the sonar equations can be accurately applied in combination with the decibel to extraterrestrial scenarios. Examples are given for icy oceans such as exist on Europa and Ganymede, Titan's hydrocarbon lakes, and for the gaseous atmospheres of (for example) Jupiter and Venus. PMID:27586766
Optimized solution of Kepler's equation
NASA Technical Reports Server (NTRS)
Kohout, J. M.; Layton, L.
1972-01-01
A detailed description is presented of KEPLER, an IBM 360 computer program used for the solution of Kepler's equation for eccentric anomaly. The program KEPLER employs a second-order Newton-Raphson differential correction process, and it is faster than previously developed programs by an order of magnitude.
Lattice Boltzmann equation method for the Cahn-Hilliard equation.
Zheng, Lin; Zheng, Song; Zhai, Qinglan
2015-01-01
In this paper a lattice Boltzmann equation (LBE) method is designed that is different from the previous LBE for the Cahn-Hilliard equation (CHE). The starting point of the present CHE LBE model is from the kinetic theory and the work of Lee and Liu [T. Lee and L. Liu, J. Comput. Phys. 229, 8045 (2010)]; however, because the CHE does not conserve the mass locally, a modified equilibrium density distribution function is introduced to treat the diffusion term in the CHE. Numerical simulations including layered Poiseuille flow, static droplet, and Rayleigh-Taylor instability have been conducted to validate the model. The results show that the predictions of the present LBE agree well with the analytical solution and other numerical results. PMID:25679741
A Versatile Technique for Solving Quintic Equations
ERIC Educational Resources Information Center
Kulkarni, Raghavendra G.
2006-01-01
In this paper we present a versatile technique to solve several types of solvable quintic equations. In the technique described here, the given quintic is first converted to a sextic equation by adding a root, and the resulting sextic equation is decomposed into two cubic polynomials as factors in a novel fashion. The resultant cubic equations are…
A Bayesian Nonparametric Approach to Test Equating
ERIC Educational Resources Information Center
Karabatsos, George; Walker, Stephen G.
2009-01-01
A Bayesian nonparametric model is introduced for score equating. It is applicable to all major equating designs, and has advantages over previous equating models. Unlike the previous models, the Bayesian model accounts for positive dependence between distributions of scores from two tests. The Bayesian model and the previous equating models are…
Local Linear Observed-Score Equating
ERIC Educational Resources Information Center
Wiberg, Marie; van der Linden, Wim J.
2011-01-01
Two methods of local linear observed-score equating for use with anchor-test and single-group designs are introduced. In an empirical study, the two methods were compared with the current traditional linear methods for observed-score equating. As a criterion, the bias in the equated scores relative to true equating based on Lord's (1980)…
On abstract degenerate neutral differential equations
NASA Astrophysics Data System (ADS)
Hernández, Eduardo; O'Regan, Donal
2016-10-01
We introduce a new abstract model of functional differential equations, which we call abstract degenerate neutral differential equations, and we study the existence of strict solutions. The class of problems and the technical approach introduced in this paper allow us to generalize and extend recent results on abstract neutral differential equations. Some examples on nonlinear partial neutral differential equations are presented.
Simple Derivation of the Lindblad Equation
ERIC Educational Resources Information Center
Pearle, Philip
2012-01-01
The Lindblad equation is an evolution equation for the density matrix in quantum theory. It is the general linear, Markovian, form which ensures that the density matrix is Hermitian, trace 1, positive and completely positive. Some elementary examples of the Lindblad equation are given. The derivation of the Lindblad equation presented here is…
Multiple Test Equating Using the Rasch Model.
ERIC Educational Resources Information Center
Brigman, S. Leellen; Bashaw, W. L.
Procedures are presented for equating simultaneously several tests which have been calibrated by the Rasch Model. Three multiple test equating designs are described. A Full Matrix Design equates each test to all others. A Chain Design links tests sequentially. A Vector Design equates one test to each of the other tests. For each design, the Rasch…
Isothermal Equation Of State For Compressed Solids
NASA Technical Reports Server (NTRS)
Vinet, Pascal; Ferrante, John
1989-01-01
Same equation with three adjustable parameters applies to different materials. Improved equation of state describes pressure on solid as function of relative volume at constant temperature. Even though types of interatomic interactions differ from one substance to another, form of equation determined primarily by overlap of electron wave functions during compression. Consequently, equation universal in sense it applies to variety of substances, including ionic, metallic, covalent, and rare-gas solids. Only three parameters needed to describe equation for given material.
Implementing Parquet equations using HPX
NASA Astrophysics Data System (ADS)
Kellar, Samuel; Wagle, Bibek; Yang, Shuxiang; Tam, Ka-Ming; Kaiser, Hartmut; Moreno, Juana; Jarrell, Mark
A new C++ runtime system (HPX) enables simulations of complex systems to run more efficiently on parallel and heterogeneous systems. This increased efficiency allows for solutions to larger simulations of the parquet approximation for a system with impurities. The relevancy of the parquet equations depends upon the ability to solve systems which require long runs and large amounts of memory. These limitations, in addition to numerical complications arising from stability of the solutions, necessitate running on large distributed systems. As the computational resources trend towards the exascale and the limitations arising from computational resources vanish efficiency of large scale simulations becomes a focus. HPX facilitates efficient simulations through intelligent overlapping of computation and communication. Simulations such as the parquet equations which require the transfer of large amounts of data should benefit from HPX implementations. Supported by the the NSF EPSCoR Cooperative Agreement No. EPS-1003897 with additional support from the Louisiana Board of Regents.
Applications of film thickness equations
NASA Technical Reports Server (NTRS)
Hamrock, B. J.; Dowson, D.
1983-01-01
A number of applications of elastohydrodynamic film thickness expressions were considered. The motion of a steel ball over steel surfaces presenting varying degrees of conformity was examined. The equation for minimum film thickness in elliptical conjunctions under elastohydrodynamic conditions was applied to roller and ball bearings. An involute gear was also introduced, it was again found that the elliptical conjunction expression yielded a conservative estimate of the minimum film thickness. Continuously variable-speed drives like the Perbury gear, which present truly elliptical elastohydrodynamic conjunctions, are favored increasingly in mobile and static machinery. A representative elastohydrodynamic condition for this class of machinery is considered for power transmission equipment. The possibility of elastohydrodynamic films of water or oil forming between locomotive wheels and rails is examined. The important subject of traction on the railways is attracting considerable attention in various countries at the present time. The final example of a synovial joint introduced the equation developed for isoviscous-elastic regimes of lubrication.
Power equations in endurance sports.
van Ingen Schenau, G J; Cavanagh, P R
1990-01-01
This paper attempts to clarify the formulation of power equations applicable to a variety of endurance activities. An accurate accounting of the relationship between the metabolic power input and the mechanical power output is still elusive, due to such issues as storage and recovery of strain energy and the differing energy costs of concentric and eccentric muscle actions. Nevertheless, an instantaneous approach is presented which is based upon the application of conventional Newtonian mechanics to a rigid segment model of the body, and does not contain assumptions regarding the exact nature of segmental interactions--such as energy transfer, etc. The application of the equation to running, cycling, speed skating, swimming and rowing is discussed and definitions of power, efficiency, and economy are presented.
Differential equations in airplane mechanics
NASA Technical Reports Server (NTRS)
Carleman, M T
1922-01-01
In the following report, we will first draw some conclusions of purely theoretical interest, from the general equations of motion. At the end, we will consider the motion of an airplane, with the engine dead and with the assumption that the angle of attack remains constant. Thus we arrive at a simple result, which can be rendered practically utilizable for determining the trajectory of an airplane descending at a constant steering angle.
Langevin Equation on Fractal Curves
NASA Astrophysics Data System (ADS)
Satin, Seema; Gangal, A. D.
2016-07-01
We analyze random motion of a particle on a fractal curve, using Langevin approach. This involves defining a new velocity in terms of mass of the fractal curve, as defined in recent work. The geometry of the fractal curve, plays an important role in this analysis. A Langevin equation with a particular model of noise is proposed and solved using techniques of the Fα-Calculus.
Experimental determination of circuit equations
NASA Astrophysics Data System (ADS)
Shulman, Jason; Malatino, Frank; Widjaja, Matthew; Gunaratne, Gemunu H.
2015-01-01
Kirchhoff's laws offer a general, straightforward approach to circuit analysis. Unfortunately, their application becomes impractical for all but the simplest of circuits. This work presents an alternative procedure, based on an approach developed to analyze complex networks, thus making it appropriate for use on large, complicated circuits. The procedure is unusual in that it is not an analytic method but is based on experiment. Yet, this approach produces the same circuit equations obtained by more traditional means.
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.
Linear superposition in nonlinear equations.
Khare, Avinash; Sukhatme, Uday
2002-06-17
Several nonlinear systems such as the Korteweg-de Vries (KdV) and modified KdV equations and lambda phi(4) theory possess periodic traveling wave solutions involving Jacobi elliptic functions. We show that suitable linear combinations of these known periodic solutions yield many additional solutions with different periods and velocities. This linear superposition procedure works by virtue of some remarkable new identities involving elliptic functions. PMID:12059300
The complex chemical Langevin equation.
Schnoerr, David; Sanguinetti, Guido; Grima, Ramon
2014-07-14
The chemical Langevin equation (CLE) is a popular simulation method to probe the stochastic dynamics of chemical systems. The CLE's main disadvantage is its break down in finite time due to the problem of evaluating square roots of negative quantities whenever the molecule numbers become sufficiently small. We show that this issue is not a numerical integration problem, rather in many systems it is intrinsic to all representations of the CLE. Various methods of correcting the CLE have been proposed which avoid its break down. We show that these methods introduce undesirable artefacts in the CLE's predictions. In particular, for unimolecular systems, these correction methods lead to CLE predictions for the mean concentrations and variance of fluctuations which disagree with those of the chemical master equation. We show that, by extending the domain of the CLE to complex space, break down is eliminated, and the CLE's accuracy for unimolecular systems is restored. Although the molecule numbers are generally complex, we show that the "complex CLE" predicts real-valued quantities for the mean concentrations, the moments of intrinsic noise, power spectra, and first passage times, hence admitting a physical interpretation. It is also shown to provide a more accurate approximation of the chemical master equation of simple biochemical circuits involving bimolecular reactions than the various corrected forms of the real-valued CLE, the linear-noise approximation and a commonly used two moment-closure approximation.
The complex chemical Langevin equation
Schnoerr, David; Sanguinetti, Guido; Grima, Ramon
2014-07-14
The chemical Langevin equation (CLE) is a popular simulation method to probe the stochastic dynamics of chemical systems. The CLE’s main disadvantage is its break down in finite time due to the problem of evaluating square roots of negative quantities whenever the molecule numbers become sufficiently small. We show that this issue is not a numerical integration problem, rather in many systems it is intrinsic to all representations of the CLE. Various methods of correcting the CLE have been proposed which avoid its break down. We show that these methods introduce undesirable artefacts in the CLE’s predictions. In particular, for unimolecular systems, these correction methods lead to CLE predictions for the mean concentrations and variance of fluctuations which disagree with those of the chemical master equation. We show that, by extending the domain of the CLE to complex space, break down is eliminated, and the CLE’s accuracy for unimolecular systems is restored. Although the molecule numbers are generally complex, we show that the “complex CLE” predicts real-valued quantities for the mean concentrations, the moments of intrinsic noise, power spectra, and first passage times, hence admitting a physical interpretation. It is also shown to provide a more accurate approximation of the chemical master equation of simple biochemical circuits involving bimolecular reactions than the various corrected forms of the real-valued CLE, the linear-noise approximation and a commonly used two moment-closure approximation.
NASA Technical Reports Server (NTRS)
Brown, James L.; Naughton, Jonathan W.
1999-01-01
A thin film of oil on a surface responds primarily to the wall shear stress generated on that surface by a three-dimensional flow. The oil film is also subject to wall pressure gradients, surface tension effects and gravity. The partial differential equation governing the oil film flow is shown to be related to Burgers' equation. Analytical and numerical methods for solving the thin oil film equation are presented. A direct numerical solver is developed where the wall shear stress variation on the surface is known and which solves for the oil film thickness spatial and time variation on the surface. An inverse numerical solver is also developed where the oil film thickness spatial variation over the surface at two discrete times is known and which solves for the wall shear stress variation over the test surface. A One-Time-Level inverse solver is also demonstrated. The inverse numerical solver provides a mathematically rigorous basis for an improved form of a wall shear stress instrument suitable for application to complex three-dimensional flows. To demonstrate the complexity of flows for which these oil film methods are now suitable, extensive examination is accomplished for these analytical and numerical methods as applied to a thin oil film in the vicinity of a three-dimensional saddle of separation.
ON THE GENERALISED FANT EQUATION
Howe, M. S.; McGowan, R. S.
2011-01-01
An analysis is made of the fluid-structure interactions involved in the production of voiced speech. It is usual to avoid time consuming numerical simulations of the aeroacoustics of the vocal tract and glottis by the introduction of Fant’s ‘reduced complexity’ equation for the glottis volume velocity Q (G. Fant, Acoustic Theory of Speech Production, Mouton, The Hague 1960). A systematic derivation is given of Fant’s equation based on the nominally exact equations of aerodynamic sound. This can be done with a degree of approximation that depends only on the accuracy with which the time-varying flow geometry and surface-acoustic boundary conditions can be specified, and replaces Fant’s original ‘lumped element’ heuristic approach. The method determines all of the effective ‘source terms’ governing Q. It is illustrated by consideration of a simplified model of the vocal system involving a self-sustaining single-mass model of the vocal folds, that uses free streamline theory to account for surface friction and flow separation within the glottis. Identification is made of a new source term associated with the unsteady vocal fold drag produced by their oscillatory motion transverse to the mean flow. PMID:21603054
On the generalised Fant equation
NASA Astrophysics Data System (ADS)
Howe, M. S.; McGowan, R. S.
2011-06-01
An analysis is made of the fluid-structure interactions involved in the production of voiced speech. It is usual to avoid time consuming numerical simulations of the aeroacoustics of the vocal tract and glottis by the introduction of Fant's 'reduced complexity' equation for the glottis volume velocity Q [G. Fant, Acoustic Theory of Speech Production, Mouton, The Hague 1960]. A systematic derivation is given of Fant's equation based on the nominally exact equations of aerodynamic sound. This can be done with a degree of approximation that depends only on the accuracy with which the time-varying flow geometry and surface-acoustic boundary conditions can be specified, and replaces Fant's original 'lumped element' heuristic approach. The method determines all of the effective 'source terms' governing Q. It is illustrated by consideration of a simplified model of the vocal system involving a self-sustaining single-mass model of the vocal folds, that uses free streamline theory to account for surface friction and flow separation within the glottis. Identification is made of a new source term associated with the unsteady vocal fold drag produced by their oscillatory motion transverse to the mean flow.
The complex chemical Langevin equation
NASA Astrophysics Data System (ADS)
Schnoerr, David; Sanguinetti, Guido; Grima, Ramon
2014-07-01
The chemical Langevin equation (CLE) is a popular simulation method to probe the stochastic dynamics of chemical systems. The CLE's main disadvantage is its break down in finite time due to the problem of evaluating square roots of negative quantities whenever the molecule numbers become sufficiently small. We show that this issue is not a numerical integration problem, rather in many systems it is intrinsic to all representations of the CLE. Various methods of correcting the CLE have been proposed which avoid its break down. We show that these methods introduce undesirable artefacts in the CLE's predictions. In particular, for unimolecular systems, these correction methods lead to CLE predictions for the mean concentrations and variance of fluctuations which disagree with those of the chemical master equation. We show that, by extending the domain of the CLE to complex space, break down is eliminated, and the CLE's accuracy for unimolecular systems is restored. Although the molecule numbers are generally complex, we show that the "complex CLE" predicts real-valued quantities for the mean concentrations, the moments of intrinsic noise, power spectra, and first passage times, hence admitting a physical interpretation. It is also shown to provide a more accurate approximation of the chemical master equation of simple biochemical circuits involving bimolecular reactions than the various corrected forms of the real-valued CLE, the linear-noise approximation and a commonly used two moment-closure approximation.
Nonlocal Equations with Measure Data
NASA Astrophysics Data System (ADS)
Kuusi, Tuomo; Mingione, Giuseppe; Sire, Yannick
2015-08-01
We develop an existence, regularity and potential theory for nonlinear integrodifferential equations involving measure data. The nonlocal elliptic operators considered are possibly degenerate and cover the case of the fractional p-Laplacean operator with measurable coefficients. We introduce a natural function class where we solve the Dirichlet problem, and prove basic and optimal nonlinear Wolff potential estimates for solutions. These are the exact analogs of the results valid in the case of local quasilinear degenerate equations established by Boccardo and Gallouët (J Funct Anal 87:149-169, 1989, Partial Differ Equ 17:641-655, 1992) and Kilpeläinen and Malý (Ann Scuola Norm Sup Pisa Cl Sci (IV) 19:591-613, 1992, Acta Math 172:137-161, 1994). As a consequence, we establish a number of results that can be considered as basic building blocks for a nonlocal, nonlinear potential theory: fine properties of solutions, Calderón-Zygmund estimates, continuity and boundedness criteria are established via Wolff potentials. A main tool is the introduction of a global excess functional that allows us to prove a nonlocal analog of the classical theory due to Campanato (Ann Mat Pura Appl (IV) 69:321-381, 1965). Our results cover the case of linear nonlocal equations with measurable coefficients, and the one of the fractional Laplacean, and are new already in such cases.
a Multiple Riccati Equations Rational-Exponent Method and its Application to Whitham-Broer Equation
NASA Astrophysics Data System (ADS)
Liu, Qing; Wang, Zi-Hua; Jia, Dong-Li
2013-03-01
According to two dependent solutions to a generalized Riccati equation together with the equation itself, a multiple Riccati equations rational-exponent method is proposed and applied to Whitham-Broer-Kaup equation. It shows that this method is a more concise and efficient approach and can uniformly derive many types of combined solutions to nonlinear partial differential equations.
Hawking Temperature Calculation Using Tunneling Mechanism
NASA Astrophysics Data System (ADS)
Liu, Xianming; Liu, Wenbiao
2009-12-01
Based on the black hole tunneling mechanism in which the ingoing particles should be absorbed absolutely, Hamilton-Jacobi method is modified to investigate Hawking radiation from a Schwarzschild black hole once again. The results are independent on the coordinates and the standard Hawking temperature is obtained without the factor of 2 problem. Moreover, it is also pointed out that the modified Hamilton-Jacobi method can be applied to investigate Hawking radiation from the general horizons in dynamical spacetimes.
ERIC Educational Resources Information Center
Savoye, Philippe
2009-01-01
In recent years, I started covering difference equations and z transform methods in my introductory differential equations course. This allowed my students to extend the "classical" methods for (ordinary differential equation) ODE's to discrete time problems arising in many applications.
ERIC Educational Resources Information Center
Chen, Haiwen; Holland, Paul
2010-01-01
In this paper, we develop a new curvilinear equating for the nonequivalent groups with anchor test (NEAT) design under the assumption of the classical test theory model, that we name curvilinear Levine observed score equating. In fact, by applying both the kernel equating framework and the mean preserving linear transformation of…
NASA Astrophysics Data System (ADS)
Makkonen, Lasse
2016-04-01
Young’s construction for a contact angle at a three-phase intersection forms the basis of all fields of science that involve wetting and capillary action. We find compelling evidence from recent experimental results on the deformation of a soft solid at the contact line, and displacement of an elastic wire immersed in a liquid, that Young’s equation can only be interpreted by surface energies, and not as a balance of surface tensions. It follows that the a priori variable in finding equilibrium is not the position of the contact line, but the contact angle. This finding provides the explanation for the pinning of a contact line.
NASA Astrophysics Data System (ADS)
Ho, Choon-Lin; Hosotani, Yutaka
Starting from the quantum field theory of nonrelativistic matter on a torus interacting with Chern-Simons gauge fields, we derive the Schrödinger equation for an anyon system. The nonintegrable phases of the Wilson line integrals on a torus play an essential role. In addition to generating degenerate vacua, they enter in the definition of a many-body Schrödinger wave function in quantum mechanics, which can be defined as a regular function of the coordinates of anyons. It obeys a non-Abelian representation of the braid group algebra, being related to Einarsson’s wave function by a singular gauge transformation.
Germanium multiphase equation of state
Crockett, Scott D.; Lorenzi-Venneri, Giulia De; Kress, Joel D.; Rudin, Sven P.
2014-05-07
A new SESAME multiphase germanium equation of state (EOS) has been developed using the best available experimental data and density functional theory (DFT) calculations. The equilibrium EOS includes the Ge I (diamond), the Ge II (β-Sn) and the liquid phases. The foundation of the EOS is based on density functional theory calculations which are used to determine the cold curve and the Debye temperature. Results are compared to Hugoniot data through the solid-solid and solid-liquid transitions. We propose some experiments to better understand the dynamics of this element
Advanced lab on Fresnel equations
NASA Astrophysics Data System (ADS)
Petrova-Mayor, Anna; Gimbal, Scott
2015-11-01
This experimental and theoretical exercise is designed to promote students' understanding of polarization and thin-film coatings for the practical case of a scanning protected-metal coated mirror. We present results obtained with a laboratory scanner and a polarimeter and propose an affordable and student-friendly experimental arrangement for the undergraduate laboratory. This experiment will allow students to apply basic knowledge of the polarization of light and thin-film coatings, develop hands-on skills with the use of phase retarders, apply the Fresnel equations for metallic coating with complex index of refraction, and compute the polarization state of the reflected light.
Scattering equations and Feynman diagrams
NASA Astrophysics Data System (ADS)
Baadsgaard, Christian; Bjerrum-Bohr, N. E. J.; Bourjaily, Jacob L.; Damgaard, Poul H.
2015-09-01
We show a direct matching between individual Feynman diagrams and integration measures in the scattering equation formalism of Cachazo, He and Yuan. The connection is most easily explained in terms of triangular graphs associated with planar Feynman diagrams in φ 3-theory. We also discuss the generalization to general scalar field theories with φ p interactions, corresponding to polygonal graphs involving vertices of order p. Finally, we describe how the same graph-theoretic language can be used to provide the precise link between individual Feynman diagrams and string theory integrands.
Linear superposition solutions to nonlinear wave equations
NASA Astrophysics Data System (ADS)
Liu, Yu
2012-11-01
The solutions to a linear wave equation can satisfy the principle of superposition, i.e., the linear superposition of two or more known solutions is still a solution of the linear wave equation. We show in this article that many nonlinear wave equations possess exact traveling wave solutions involving hyperbolic, triangle, and exponential functions, and the suitable linear combinations of these known solutions can also constitute linear superposition solutions to some nonlinear wave equations with special structural characteristics. The linear superposition solutions to the generalized KdV equation K(2,2,1), the Oliver water wave equation, and the k(n, n) equation are given. The structure characteristic of the nonlinear wave equations having linear superposition solutions is analyzed, and the reason why the solutions with the forms of hyperbolic, triangle, and exponential functions can form the linear superposition solutions is also discussed.
Difference equations and some of their solutions
NASA Astrophysics Data System (ADS)
Atakishiyev, Natig M.
1996-04-01
Some methods for solving difference equations are discussed. As particular examples we consider in detail the difference equations corresponding to the Kravchuk functions, q-harmonic oscillator wavefunctions, and the Clebsch-Gordan coefficients for the quantum algebra suq(2).
Model equations for high current transport
Lee, E.P.
1985-06-01
The use of distribution functions to model transverse beam dynamics is discussed. Emphasis is placed on envelope equations, moments, the Vlasov equation, and the Kapchinski-Vladimirskij distribution. 10 refs.
Solving Equations of Multibody Dynamics
NASA Technical Reports Server (NTRS)
Jain, Abhinandan; Lim, Christopher
2007-01-01
Darts++ is a computer program for solving the equations of motion of a multibody system or of a multibody model of a dynamic system. It is intended especially for use in dynamical simulations performed in designing and analyzing, and developing software for the control of, complex mechanical systems. Darts++ is based on the Spatial-Operator- Algebra formulation for multibody dynamics. This software reads a description of a multibody system from a model data file, then constructs and implements an efficient algorithm that solves the dynamical equations of the system. The efficiency and, hence, the computational speed is sufficient to make Darts++ suitable for use in realtime closed-loop simulations. Darts++ features an object-oriented software architecture that enables reconfiguration of system topology at run time; in contrast, in related prior software, system topology is fixed during initialization. Darts++ provides an interface to scripting languages, including Tcl and Python, that enable the user to configure and interact with simulation objects at run time.
Langevin Equation for DNA Dynamics
NASA Astrophysics Data System (ADS)
Grych, David; Copperman, Jeremy; Guenza, Marina
Under physiological conditions, DNA oligomers can contain well-ordered helical regions and also flexible single-stranded regions. We describe the site-specific motion of DNA with a modified Rouse-Zimm Langevin equation formalism that describes DNA as a coarse-grained polymeric chain with global structure and local flexibility. The approach has successfully described the protein dynamics in solution and has been extended to nucleic acids. Our approach provides diffusive mode analytical solutions for the dynamics of global rotational diffusion and internal motion. The internal DNA dynamics present a rich energy landscape that accounts for an interior where hydrogen bonds and base-stacking determine structure and experience limited solvent exposure. We have implemented several models incorporating different coarse-grained sites with anisotropic rotation, energy barrier crossing, and local friction coefficients that include a unique internal viscosity and our models reproduce dynamics predicted by atomistic simulations. The models reproduce bond autocorrelation along the sequence as compared to that directly calculated from atomistic molecular dynamics simulations. The Langevin equation approach captures the essence of DNA dynamics without a cumbersome atomistic representation.
Equations of state for hydrocodes
NASA Astrophysics Data System (ADS)
Lomonosov, I.
2013-06-01
The equation of state (EOS) governing the system of gas dynamic equations defines significantly accuracy and reliability of results of numerical modeling. In our report, we will formulate main mathematical and physical demands to wide-range EOS for hydrocodes. Our semi-empirical EOS model fully assigns the free energy thermodynamic potential for metals over entire phase diagram region of practical interest. It accounts for solid, liquid, plasma states as well as two-phase regions of melting and evaporation. Available now are wide-range multi-phase EOS for 30 simple and transition metals of the most practical interest. Their direct usage in computer codes leads to complicated and not economy calculations, so they are usually involved in numerical modeling in tabular form. The EOS code for calculation of tables can produce the complete set of thermodynamic derivatives (such as pressure, sound velocity, heat capacity) using any one of input pairs: volume-temperature, volume-internal energy or volume-pressure. The input grid can be linear, logarithmic or arbitrary; each point in 2D output tables is marked by symbol which indicates the physical state, such as solid, liquid, gas, plasma or mesh. We also present in our talk estimations of shock melting and evaporating and importance of these effects for results of numerical modeling.
Wave equation on spherically symmetric Lorentzian metrics
Bokhari, Ashfaque H.; Al-Dweik, Ahmad Y.; Zaman, F. D.; Kara, A. H.; Karim, M.
2011-06-15
Wave equation on a general spherically symmetric spacetime metric is constructed. Noether symmetries of the equation in terms of explicit functions of {theta} and {phi} are derived subject to certain differential constraints. By restricting the metric to flat Friedman case the Noether symmetries of the wave equation are presented. Invertible transformations are constructed from a specific subalgebra of these Noether symmetries to convert the wave equation with variable coefficients to the one with constant coefficients.
Bilinear approach to the supersymmetric Gardner equation
NASA Astrophysics Data System (ADS)
Babalic, C. N.; Carstea, A. S.
2016-08-01
We study a supersymmetric version of the Gardner equation (both focusing and defocusing) using the superbilinear formalism. This equation is new and cannot be obtained from the supersymmetric modified Korteweg-de Vries equation with a nonzero boundary condition. We construct supersymmetric solitons and then by passing to the long-wave limit in the focusing case obtain rational nonsingular solutions. We also discuss the supersymmetric version of the defocusing equation and the dynamics of its solutions.
On a Equation in Finite Algebraically Structures
ERIC Educational Resources Information Center
Valcan, Dumitru
2013-01-01
Solving equations in finite algebraically structures (semigroups with identity, groups, rings or fields) many times is not easy. Even the professionals can have trouble in such cases. Therefore, in this paper we proposed to solve in the various finite groups or fields, a binomial equation of the form (1). We specify that this equation has been…
Multi-time equations, classical and quantum
Petrat, Sören; Tumulka, Roderich
2014-01-01
Multi-time equations are evolution equations involving several time variables, one for each particle. Such equations have been considered for the purpose of making theories manifestly Lorentz invariant. We compare their status and significance in classical and quantum physics. PMID:24711721
Solving Absolute Value Equations Algebraically and Geometrically
ERIC Educational Resources Information Center
Shiyuan, Wei
2005-01-01
The way in which students can improve their comprehension by understanding the geometrical meaning of algebraic equations or solving algebraic equation geometrically is described. Students can experiment with the conditions of the absolute value equation presented, for an interesting way to form an overall understanding of the concept.
Equating Scores from Adaptive to Linear Tests
ERIC Educational Resources Information Center
van der Linden, Wim J.
2006-01-01
Two local methods for observed-score equating are applied to the problem of equating an adaptive test to a linear test. In an empirical study, the methods were evaluated against a method based on the test characteristic function (TCF) of the linear test and traditional equipercentile equating applied to the ability estimates on the adaptive test…
Students' Equation Understanding and Solving in Iran
ERIC Educational Resources Information Center
Barahmand, Ali; Shahvarani, Ahmad
2014-01-01
The purpose of the present article is to investigate how 15-year-old Iranian students interpret the concept of equation, its solution, and studying the relation between the students' equation understanding and solving. Data from two equation-solving exercises are reported. Data analysis shows that there is a significant relationship between…
Local Observed-Score Kernel Equating
ERIC Educational Resources Information Center
Wiberg, Marie; van der Linden, Wim J.; von Davier, Alina A.
2014-01-01
Three local observed-score kernel equating methods that integrate methods from the local equating and kernel equating frameworks are proposed. The new methods were compared with their earlier counterparts with respect to such measures as bias--as defined by Lord's criterion of equity--and percent relative error. The local kernel item response…
The Effect of Repeaters on Equating
ERIC Educational Resources Information Center
Kim, HeeKyoung; Kolen, Michael J.
2010-01-01
Test equating might be affected by including in the equating analyses examinees who have taken the test previously. This study evaluated the effect of including such repeaters on Medical College Admission Test (MCAT) equating using a population invariance approach. Three-parameter logistic (3-PL) item response theory (IRT) true score and…
The Effects of Repeaters on Test Equating.
ERIC Educational Resources Information Center
Andrulis, Richard S.; And Others
The purpose of this investigation was to establish the effects of repeaters on test equating. Since consideration was not given to repeaters in test equating, such as in the derivation of equations by Angoff (1971), the hypothetical effect needed to be established. A case study was examined which showed results on a test as expected; overall mean…
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.
From invasion to extinction in heterogeneous neural fields.
Bressloff, Paul C
2012-03-26
In this paper, we analyze the invasion and extinction of activity in heterogeneous neural fields. We first consider the effects of spatial heterogeneities on the propagation of an invasive activity front. In contrast to previous studies of front propagation in neural media, we assume that the front propagates into an unstable rather than a metastable zero-activity state. For sufficiently localized initial conditions, the asymptotic velocity of the resulting pulled front is given by the linear spreading velocity, which is determined by linearizing about the unstable state within the leading edge of the front. One of the characteristic features of these so-called pulled fronts is their sensitivity to perturbations inside the leading edge. This means that standard perturbation methods for studying the effects of spatial heterogeneities or external noise fluctuations break down. We show how to extend a partial differential equation method for analyzing pulled fronts in slowly modulated environments to the case of neural fields with slowly modulated synaptic weights. The basic idea is to rescale space and time so that the front becomes a sharp interface whose location can be determined by solving a corresponding local Hamilton-Jacobi equation. We use steepest descents to derive the Hamilton-Jacobi equation from the original nonlocal neural field equation. In the case of weak synaptic heterogenities, we then use perturbation theory to solve the corresponding Hamilton equations and thus determine the time-dependent wave speed. In the second part of the paper, we investigate how time-dependent heterogenities in the form of extrinsic multiplicative noise can induce rare noise-driven transitions to the zero-activity state, which now acts as an absorbing state signaling the extinction of all activity. In this case, the most probable path to extinction can be obtained by solving the classical equations of motion that dominate a path integral representation of the stochastic
From invasion to extinction in heterogeneous neural fields.
Bressloff, Paul C
2012-01-01
In this paper, we analyze the invasion and extinction of activity in heterogeneous neural fields. We first consider the effects of spatial heterogeneities on the propagation of an invasive activity front. In contrast to previous studies of front propagation in neural media, we assume that the front propagates into an unstable rather than a metastable zero-activity state. For sufficiently localized initial conditions, the asymptotic velocity of the resulting pulled front is given by the linear spreading velocity, which is determined by linearizing about the unstable state within the leading edge of the front. One of the characteristic features of these so-called pulled fronts is their sensitivity to perturbations inside the leading edge. This means that standard perturbation methods for studying the effects of spatial heterogeneities or external noise fluctuations break down. We show how to extend a partial differential equation method for analyzing pulled fronts in slowly modulated environments to the case of neural fields with slowly modulated synaptic weights. The basic idea is to rescale space and time so that the front becomes a sharp interface whose location can be determined by solving a corresponding local Hamilton-Jacobi equation. We use steepest descents to derive the Hamilton-Jacobi equation from the original nonlocal neural field equation. In the case of weak synaptic heterogenities, we then use perturbation theory to solve the corresponding Hamilton equations and thus determine the time-dependent wave speed. In the second part of the paper, we investigate how time-dependent heterogenities in the form of extrinsic multiplicative noise can induce rare noise-driven transitions to the zero-activity state, which now acts as an absorbing state signaling the extinction of all activity. In this case, the most probable path to extinction can be obtained by solving the classical equations of motion that dominate a path integral representation of the stochastic
The Riesz-Bessel Fractional Diffusion Equation
Anh, V.V. McVinish, R.
2004-05-15
This paper examines the properties of a fractional diffusion equation defined by the composition of the inverses of the Riesz potential and the Bessel potential. The first part determines the conditions under which the Green function of this equation is the transition probability density function of a Levy motion. This Levy motion is obtained by the subordination of Brownian motion, and the Levy representation of the subordinator is determined. The second part studies the semigroup formed by the Green function of the fractional diffusion equation. Applications of these results to certain evolution equations is considered. Some results on the numerical solution of the fractional diffusion equation are also provided.
Binomial moment equations for stochastic reaction systems.
Barzel, Baruch; Biham, Ofer
2011-04-15
A highly efficient formulation of moment equations for stochastic reaction networks is introduced. It is based on a set of binomial moments that capture the combinatorics of the reaction processes. The resulting set of equations can be easily truncated to include moments up to any desired order. The number of equations is dramatically reduced compared to the master equation. This formulation enables the simulation of complex reaction networks, involving a large number of reactive species much beyond the feasibility limit of any existing method. It provides an equation-based paradigm to the analysis of stochastic networks, complementing the commonly used Monte Carlo simulations. PMID:21568538
Bogomol'nyi equations of classical solutions
NASA Astrophysics Data System (ADS)
Atmaja, Ardian N.; Ramadhan, Handhika S.
2014-11-01
We review the Bogomol'nyi equations and investigate an alternative route in obtaining it. It can be shown that the known Bogomol'nyi-Prasad-Sommerfield equations can be derived directly from the corresponding Euler-Lagrange equations via the separation of variables, without having to appeal to the Hamiltonian. We apply this technique to the Dirac-Born-Infeld solitons and obtain the corresponding equations and the potentials. This method is suitable for obtaining the first-order equations and determining the allowed potentials for noncanonical defects.
Sparse dynamics for partial differential equations
Schaeffer, Hayden; Caflisch, Russel; Hauck, Cory D.; Osher, Stanley
2013-01-01
We investigate the approximate dynamics of several differential equations when the solutions are restricted to a sparse subset of a given basis. The restriction is enforced at every time step by simply applying soft thresholding to the coefficients of the basis approximation. By reducing or compressing the information needed to represent the solution at every step, only the essential dynamics are represented. In many cases, there are natural bases derived from the differential equations, which promote sparsity. We find that our method successfully reduces the dynamics of convection equations, diffusion equations, weak shocks, and vorticity equations with high-frequency source terms. PMID:23533273
Spectrum Analysis of Some Kinetic Equations
NASA Astrophysics Data System (ADS)
Yang, Tong; Yu, Hongjun
2016-11-01
We analyze the spectrum structure of some kinetic equations qualitatively by using semigroup theory and linear operator perturbation theory. The models include the classical Boltzmann equation for hard potentials with or without angular cutoff and the Landau equation with {γ≥q-2}. As an application, we show that the solutions to these two fundamental equations are asymptotically equivalent (mod time decay rate {t^{-5/4}}) as {tto∞} to that of the compressible Navier-Stokes equations for initial data around an equilibrium state.
The equations of medieval cosmology
NASA Astrophysics Data System (ADS)
Buonanno, Roberto; Quercellini, Claudia
2009-04-01
In Dantean cosmography the Universe is described as a series of concentric spheres with all the known planets embedded in their rotation motion, the Earth located at the centre and Lucifer at the centre of the Earth. Beyond these "celestial spheres", Dante represents the "angelic choirs" as other nine spheres surrounding God. The rotation velocity increases with decreasing distance from God, that is with increasing Power (Virtù). We show that, adding Power as an additional fourth dimension to space, the modern equations governing the expansion of a closed Universe (i.e. with the density parameter Ω0 > 1) in the space-time, can be applied to the medieval Universe as imaged by Dante in his Divine Comedy. In this representation, the Cosmos acquires a unique description and Lucifer is not located at the centre of the hyperspheres.
Evolution equation for quantum coherence
NASA Astrophysics Data System (ADS)
Hu, Ming-Liang; Fan, Heng
2016-07-01
The estimation of the decoherence process of an open quantum system is of both theoretical significance and experimental appealing. Practically, the decoherence can be easily estimated if the coherence evolution satisfies some simple relations. We introduce a framework for studying evolution equation of coherence. Based on this framework, we prove a simple factorization relation (FR) for the l1 norm of coherence, and identified the sets of quantum channels for which this FR holds. By using this FR, we further determine condition on the transformation matrix of the quantum channel which can support permanently freezing of the l1 norm of coherence. We finally reveal the universality of this FR by showing that it holds for many other related coherence and quantum correlation measures.
Evolution equation for quantum coherence
Hu, Ming-Liang; Fan, Heng
2016-01-01
The estimation of the decoherence process of an open quantum system is of both theoretical significance and experimental appealing. Practically, the decoherence can be easily estimated if the coherence evolution satisfies some simple relations. We introduce a framework for studying evolution equation of coherence. Based on this framework, we prove a simple factorization relation (FR) for the l1 norm of coherence, and identified the sets of quantum channels for which this FR holds. By using this FR, we further determine condition on the transformation matrix of the quantum channel which can support permanently freezing of the l1 norm of coherence. We finally reveal the universality of this FR by showing that it holds for many other related coherence and quantum correlation measures. PMID:27382933
Evolution equation for quantum coherence.
Hu, Ming-Liang; Fan, Heng
2016-01-01
The estimation of the decoherence process of an open quantum system is of both theoretical significance and experimental appealing. Practically, the decoherence can be easily estimated if the coherence evolution satisfies some simple relations. We introduce a framework for studying evolution equation of coherence. Based on this framework, we prove a simple factorization relation (FR) for the l1 norm of coherence, and identified the sets of quantum channels for which this FR holds. By using this FR, we further determine condition on the transformation matrix of the quantum channel which can support permanently freezing of the l1 norm of coherence. We finally reveal the universality of this FR by showing that it holds for many other related coherence and quantum correlation measures. PMID:27382933
Entropic corrections to Friedmann equations
Sheykhi, Ahmad
2010-05-15
Recently, Verlinde discussed that gravity can be understood as an entropic force caused by changes in the information associated with the positions of material bodies. In Verlinde's argument, the area law of the black hole entropy plays a crucial role. However, the entropy-area relation can be modified from the inclusion of quantum effects, motivated from the loop quantum gravity. In this note, by employing this modified entropy-area relation, we derive corrections to Newton's law of gravitation as well as modified Friedmann equations by adopting the viewpoint that gravity can be emerged as an entropic force. Our study further supports the universality of the log correction and provides a strong consistency check on Verlinde's model.