VIABILITY SOLUTIONS TO STRUCTURED HAMILTON-JACOBI EQUATIONS UNDER CONSTRAINTS
2011-01-01
International audience; Structured Hamilton-Jacobi partial differential equations are Hamilton-Jacobi equations where the time variable is replaced by a vector-valued variable "structuring" the system. It could be the time-age pair (Hamilton-Jacobi-McKendrick equations) or candidates for initial or terminal conditions (Hamilton-Jacobi-Cournot equations) among a manifold of examples. Here, we define the concept of "viability solution" which always exists and can be computed by viability algori...
Quantum Potential Via General Hamilton - Jacobi Equation
Mollai, Maedeh; Jami, Safa; Ahanj, Ali
2011-01-01
In this paper, we sketch and emphasize the automatic emergence of a quantum potential (QP) in general Hamilton-Jacobi equation via commuting relations, quantum canonical transformations and without the straight effect of wave function. The interpretation of QP in terms of independent entity is discussed along with the introduction of quantum kinetic energy. The method has been extended to relativistic regime, and same results have been concluded.
Hamilton Jacobi method for solving ordinary differential equations
Mei, Feng-Xiang; Wu, Hui-Bin; Zhang, Yong-Fa
2006-08-01
The Hamilton-Jacobi method for solving ordinary differential equations is presented in this paper. A system of ordinary differential equations of first order or second order can be expressed as a Hamilton system under certain conditions. Then the Hamilton-Jacobi method is used in the integration of the Hamilton system and the solution of the original ordinary differential equations can be found. Finally, an example is given to illustrate the application of the result.
Numerical Solution of Hamilton-Jacobi Equations in High Dimension
2012-11-23
high dimension FA9550-10-1-0029 Maurizio Falcone Dipartimento di Matematica SAPIENZA-Universita di Roma P. Aldo Moro, 2 00185 ROMA AH930...solution of Hamilton-Jacobi equations in high dimension AFOSR contract n. FA9550-10-1-0029 Maurizio Falcone Dipartimento di Matematica SAPIENZA
Wave-Particle Duality and the Hamilton-Jacobi Equation
Sivashinsky, Gregory I
2009-01-01
The Hamilton-Jacobi equation of relativistic quantum mechanics is revisited. The equation is shown to permit solutions in the form of breathers (oscillating/spinning solitons), displaying simultaneous particle-like and wave-like behavior. The de Broglie wave thus acquires a clear deterministic meaning of a wave-like excitation of the classical action function. The problem of quantization in terms of the breathing action function and the double-slit experiment are discussed.
Solution Hamilton-Jacobi equation for oscillator Caldirola-Kanai
LEONARDO PASTRANA ARTEAGA
2016-12-01
Full Text Available The method allows Hamilton-Jacobi explicitly determine the generating function from which is possible to derive a transformation that makes soluble Hamilton's equations. Using the separation of variables the partial differential equation of the first order called Hamilton-Jacobi equation is solved; as a particular case consider the oscillator Caldirola-Kanai (CK, which is characterized in that the mass presents a temporal evolution exponentially . We demonstrate that the oscillator CK position presents an exponential decay in time similar to that obtained in the damped sub-critical oscillator, which reflects the dissipation of total mechanical energy. We found that in the limit that the damping factor is small, the behavior is the same as an oscillator with simple harmonic motion, where the effects of energy dissipation is negligible.
Studying on Opinion Evolution by Hamilton-Jacobi Equation
Feng, Chen-Jie; Huo, Jie; Hao, Rui; Wang, Xu-Ming
2016-01-01
A physical description of an opinion evolution is conducted based on the Hamilton-Jacobi equation derived from a generalized potential and the corresponding Langevin equation. The investigation mainly focuses on the heterogeneities such as age, connection circle and overall quality of the participants involved in the opinion exchange process. The evolutionary patterns of opinion can be described by solution of the Hamilton-Jacobi equation, information entropy. The results show that the overall qualities of the participants play critical roles in forming an opinion. The higher the overall quality is, the easier the consensus can reach. The solution also demonstrates that the age and the connection circle of the agents play equally important roles in forming an opinion. The essence of the age, overall quality, and connection circle corresponds to the maturity of thought (opinion inertia), reason and intelligence, influence strength of the environment, respectively. So the information entropy distributes in the ...
Beam dynamics with the Hamilton-Jacobi equation
Gabella, W.E.; Ruth, R.D.; Warnock, R.L.
1989-03-01
We describe a non-perturbative method to solve the Hamilton-Jacobi equation for invariant surfaces in phase space. The problem is formulated in action-angle variables with a general nonlinear perturbation. The solution of the Hamilton-Jacobi equation is regarded as the fixed point of a map on the Fourier coefficients of the generating function. Periodicity of the generator in the independent variable is enforced with a shooting method. We present two methods for finding the fixed point and hence the invariant surface. A solution by plain iteration is economical but has a restricted domain of convergence. The Newton iteration is costly but yields solutions up to the dynamic aperture. Examples of lattices with sextupoles for chromatic correction are discussed. 10 refs., 5 figs., 1 tab.
THE RELAXING SCHEMES FOR HAMILTON-JACOBI EQUATIONS
Hua-zhong Tang; Hua-mu Wu
2001-01-01
Hamilton-Jacobiequation appears frequently in applications, e.g., in differential games and control theory, and is closely related to hyperbolic conservation laws[3, 4, 12]. This is helpful in the design of difference approximations for Hamilton-Jacobi equation and hyperbolic conservation laws. In this paper we present the relaxing system for HamiltonJacobiequations in arbitrary space dimensions, and high resolution relaxing schemes for Hamilton-Jacobi equation, based on using the local relaxation approximation. The schemes are numerically tested on a variety of 1D and 2D problems, including a problem related to optimal control problem. High-order accuracy in smooth regions, good resolution of discontinuities, and convergence to viscosity solutions are observed.
DISCONTINUOUS SOLUTIONS IN L∞ FOR HAMILTON-JACOBI EQUATIONS
无
2000-01-01
An approach is introduced to construct global discontinuous solutions in L∞ for Hamilton Jacobi equations. This approach allows the initial data only in L∞ and applies to the equations with nonconvex Hamiltonians. The profit functions are introduced to formulate the notion of discontinuous solutions in L. The existence of global discontinuous solutions in L∞ is established. These solutions in L∞ coincide with the viscosity solutions and the minimax solutions, provided that the initial data are continuous. A prototypical equation is analyzed to examine the L∞ stability of our L∞ solutions. The analysis also shows that global discontinuous solutions are determined by the topology in which the initial data are approximated.
Fast methods for static Hamilton-Jacobi Partial Differential Equations
Vladimirsky, Alexander Boris
2001-05-01
The authors develop a family of fast methods approximating the solution to a wide class of static Hamilton-Jacobi partial differential equations. These partial differential equations are considered in the context of control-theoretic and front-propagation problems. In general, to produce a numerical solution to such a problem, one has to solve a large system of coupled non-linear discretized equations. The techniques use partial information about the characteristic directions to de-couple the system. Previously known fast methods, available for isotropic problems, are discussed in detail. They introduce a family of new Ordered Upwinding Methods (OUM) for general (anisotropic) problems and prove convergence to the viscosity solution of the corresponding Hamilton-Jacobi partial differential equation. The hybrid methods introduced here are based on the analysis of the role played by anisotropy in the context of front propagation and optimal trajectory problems. The performance of the methods is analyzed and compared to that of several other numerical approaches to these problems. Computational experiments are performed using test problems from control theory, computational geometry and seismology.
Fast methods for static Hamilton-Jacobi Partial Differential Equations
Vladimirsky, Alexander Boris [Univ. of California, Berkeley, CA (United States)
2001-01-01
The authors develop a family of fast methods approximating the solution to a wide class of static Hamilton-Jacobi partial differential equations. These partial differential equations are considered in the context of control-theoretic and front-propagation problems. In general, to produce a numerical solution to such a problem, one has to solve a large system of coupled non-linear discretized equations. The techniques use partial information about the characteristic directions to de-couple the system. Previously known fast methods, available for isotropic problems, are discussed in detail. They introduce a family of new Ordered Upwinding Methods (OUM) for general (anisotropic) problems and prove convergence to the viscosity solution of the corresponding Hamilton-Jacobi partial differential equation. The hybrid methods introduced here are based on the analysis of the role played by anisotropy in the context of front propagation and optimal trajectory problems. The performance of the methods is analyzed and compared to that of several other numerical approaches to these problems. Computational experiments are performed using test problems from control theory, computational geometry and seismology.
Unconditionally stable methods for Hamilton-Jacobi equations
Karlsen, Kenneth Hvistendal; Risebro, Nils Henrik
2000-05-01
We present new numerical methods for constructing approximate solutions to the Cauchy problem for Hamilton-Jacobi equations of the form u{sub t} + H(D{sub x}u) = 0. The methods are based on dimensional splitting and front tracking for solving the associated (non-strictly hyperbolic) system of conservation laws p{sub t} + D{sub x}H(p) = 0, where p = D{sub x}u. In particular, our methods depend heavily on a front tracking method for one-dimensional scalar conservation laws with discontinuous coefficients. The proposed methods are unconditionally stable in the sense that the time step is not limited by the space discretization and they can be viewed as ''large time step'' Godunov type (or front tracking) methods. We present several numerical examples illustrating the main features of the proposed methods. We also compare our methods with several methods from the literature. (author)
Quantitative Compactness Estimates for Hamilton-Jacobi Equations
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.
Singular Hamilton-Jacobi equation for the tail problem
Mirrahimi, Sepideh; Perthame, Benoit; Souganidis, Panagiotis E
2010-01-01
In this paper we study the long time-long range behavior of reaction diffusion equations with negative square root -type reaction terms. In particular we investigate the exponential behavior of the solutions after a standard hyperbolic scaling. This leads to a Hamilton-Jacobi variational inequality with an obstacle that depends on the solution itself and defines the open set where the limiting solution does not vanish. Counter-examples show a nontrivial lack of uniqueness for the variational inequality depending on the conditions imposed on the boundary of this open set. Both Dirichlet and state constraints boundary conditions play a role. When the competition term does not change sign, we can identify the limit, while, in general, we find lower and upper bounds for the limit. Although models of this type are rather old and extinction phenomena are as important as blow-up, our motivation comes from the so-called "tail problem" in population biology. One way to avoid meaningless exponential tails, is to impose...
Value Functions for Certain Class of Hamilton Jacobi Equations
Anup Biswas; Rajib Dutta; Prosenjit Roy
2011-08-01
We consider a class of Hamilton Jacobi equations (in short, HJE) of type $$u_t+\\frac{1}{2}\\left(|u_{x_n}|^2+\\cdots+|u_{x_{n-1}}|^2\\right)+\\frac{e^u}{m}|u_{x_n}|^m=0,$$ in $\\mathbb{R}^n×\\mathbb{R}_+$ and >1, with bounded, Lipschitz continuous initial data. We give a Hopf-Lax type representation for the value function and also characterize the set of minimizing paths. It is shown that the minimizing paths in the representation of value function need not be straight lines. Then we consider HJE with Hamiltonian decreasing in of type $$u_t+H_1(u_{x_1},\\ldots,u_{x_i})+e^{-u}H_2(u_{x_{i+1}},\\ldots,u_{x_n})=0$$ where $H_1,H_2$ are convex, homogeneous of degree $n,m>1$ respectively and the initial data is bounded, Lipschitz continuous. We prove that there exists a unique viscosity solution for this HJE in Lipschitz continuous class. We also give a representation formula for the value function.
Computational method for the quantum Hamilton-Jacobi equation: one-dimensional scattering problems.
Chou, Chia-Chun; Wyatt, Robert E
2006-12-01
One-dimensional scattering problems are investigated in the framework of the quantum Hamilton-Jacobi formalism. First, the pole structure of the quantum momentum function for scattering wave functions is analyzed. The significant differences of the pole structure of this function between scattering wave functions and bound state wave functions are pointed out. An accurate computational method for the quantum Hamilton-Jacobi equation for general one-dimensional scattering problems is presented to obtain the scattering wave function and the reflection and transmission coefficients. The computational approach is demonstrated by analysis of scattering from a one-dimensional potential barrier. We not only present an alternative approach to the numerical solution of the wave function and the reflection and transmission coefficients but also provide a computational aspect within the quantum Hamilton-Jacobi formalism. The method proposed here should be useful for general one-dimensional scattering problems.
On the convergence rate of operator splitting for Hamilton-Jacobi equations with source terms
Jakobsen, Espen R.; Karlsen, Kenneth H.; Risebro, Nils Henrik
2000-02-01
We establish a rate of convergence for a semi-discrete operator splitting method applied to Hamilton-Jacobi equations with source terms. The method is based on sequentially solving a Hamilton-Jacobi equation and an ordinary differential equation. The Hamilton-Jacobi equation is solved exactly while the ordinary differential equation is solved exactly or by an explicit Euler method. We prove that the L{sup {infinity}} error associated with the operator splitting method is bounded by O({delta}t), where {delta}t is the splitting (or time) step. This error bound is an improvement over the existing O((sqroot)({delta}t)) bound due to Souganidis [40]. In the one dimensional case, we present a fully discrete splitting method based on an unconditionally stable front tracking method for homogenuous Hamilton-Jacobi equations. It is proved that this fully discrete splitting method possesses a linear convergence rate. Moreover, numerical results are presented to illustrate the theoreticle convergence results. (author)
An optimal L1-minimization algorithm for stationary Hamilton-Jacobi equations
Guermond, Jean-Luc
2009-01-01
We describe an algorithm for solving steady one-dimensional convex-like Hamilton-Jacobi equations using a L1-minimization technique on piecewise linear approximations. For a large class of convex Hamiltonians, the algorithm is proven to be convergent and of optimal complexity whenever the viscosity solution is q-semiconcave. Numerical results are presented to illustrate the performance of the method.
Hamilton-Jacobi equation and the breaking of the WKB approximation
Canfora, F. [Istituto Nazionale di Fisica Nucleare, GC di Salerno (Italy) and Dipartimento di Fisica E.R. Caianiello, Universita di Salerno, Via S. Allende, 84081 Baronissi (Salerno) (Italy)]. E-mail: canfora@sa.infn.it
2005-03-17
A simple method to deal with four-dimensional Hamilton-Jacobi equation for null hypersurfaces is introduced. This method allows to find simple geometrical conditions which give rise to the failure of the WKB approximation on curved spacetimes. The relation between such failure, extreme blackholes and the Cosmic Censor hypothesis is briefly discussed.
Yang, Shu-Zheng; Feng, Zhong-Wen; Li, Hui-Ling
2017-02-01
We derive the Hamilton-Jacobi equation from the Dirac equation, then, with the help of the Hamilton-Jacobi equation, the the tunneling radiation behavior of the non-stationary spherical symmetry de Sitter black hole is discussed, at last, we obtained the tunneling rate and Hawking temperature. Our results showed that the Hamilton-Jacobi equation is a fundamental dynamic equation, it can widely be derived from the dynamic equations which describe the particles with any spin. Therefore, people can easy calculate the tunneling behavior from the black holes.
Superluminal Neutrinos and a Curious Phenomenon in the Relativistic Quantum Hamilton-Jacobi Equation
Matone, Marco
2011-01-01
OPERA's results, if confirmed, pose the question of superluminal neutrinos. We investigate the kinematics defined by the quantum version of the relativistic Hamilton-Jacobi equation, i.e. E^2=p^2c^2+m^2c^4+2mQc^2, with Q the quantum potential of the free particle. The key point is that the quantum version of the Hamilton-Jacobi equation is a third-order differential equation, so that it has integration constants which are missing in the Schr\\"odinger and Klein-Gordon equations. In particular, a non-vanishing imaginary part of an integration constant leads to a quantum correction to the expression of the velocity which is curiously in agreement with OPERA's results.
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.
A Large Deviation, Hamilton-Jacobi Equation Approach to a Statistical Theory for Turbulence
2012-09-03
and its associated compressible Euler equations, Comptes Rendus Mathematique , (09 2011): 973. doi: 10.1016/j.crma.2011.08.013 2012/09/03 14:17:15 6...Hamilton-Jacobi PDE is shown to be well-posed. (joint work with T Nguyen, Journal de Mathematique Pures et Appliquees). Future works focusing on large time behavior for such equations is currently under its way. Technology Transfer
On the Hamilton-Jacobi-Bellman Equation by the Homotopy Perturbation Method
Abdon Atangana
2014-01-01
Full Text Available Our concern in this paper is to use the homotopy decomposition method to solve the Hamilton-Jacobi-Bellman equation (HJB. The approach is obviously extremely well organized and is an influential procedure in obtaining the solutions of the equations. We portrayed particular compensations that this technique has over the prevailing approaches. We presented that the complexity of the homotopy decomposition method is of order O(n. Furthermore, three explanatory examples established good outcomes and comparisons with exact solution.
Computational method for the quantum Hamilton-Jacobi equation: bound states in one dimension.
Chou, Chia-Chun; Wyatt, Robert E
2006-11-07
An accurate computational method for the one-dimensional quantum Hamilton-Jacobi equation is presented. The Mobius propagation scheme, which can accurately pass through singularities, is used to numerically integrate the quantum Hamilton-Jacobi equation for the quantum momentum function. Bound state wave functions are then synthesized from the phase integral using the antithetic cancellation technique. Through this procedure, not only the quantum momentum functions but also the wave functions are accurately obtained. This computational approach is demonstrated through two solvable examples: the harmonic oscillator and the Morse potential. The excellent agreement between the computational and the exact analytical results shows that the method proposed here may be useful for solving similar quantum mechanical problems.
Relations between low-lying quantum wave functions and solutions of the Hamilton-Jacobi equation
Friedberg, R; Zhao Wei Qin
1999-01-01
We discuss a new relation between the low lying Schroedinger wave function of a particle in a one-dimentional potential V and the solution of the corresponding Hamilton-Jacobi equation with -V as its potential. The function V is $\\geq 0$, and can have several minina (V=0). We assume the problem to be characterized by a small anhamornicity parameter $g^{-1}$ and a much smaller quantum tunneling parameter $\\epsilon$ between these different minima. Expanding either the wave function or its energy as a formal double power series in $g^{-1}$ and $\\epsilon$, we show how the coefficients of $g^{-m}\\epsilon^n$ in such an expansion can be expressed in terms of definite integrals, with leading order term determined by the classical solution of the Hamilton-Jacobi equation. A detailed analysis is given for the particular example of quartic potential $V={1/2}g^2(x^2-a^2)^2$.
Large time behavior of weakly coupled systems of first-order Hamilton-Jacobi equations
Camilli, Fabio; Loreti, Paola; Nguyen, Vinh Duc
2011-01-01
We show a large time behavior result for class of weakly coupled systems of first-order Hamilton-Jacobi equations in the periodic setting. We use a PDE approach to extend the convergence result proved by Namah and Roquejoffre (1999) in the scalar case. Our proof is based on new comparison, existence and regularity results for systems. An interpretation of the solution of the system in terms of an optimal control problem with switching is given.
A Discontinuous Galerkin Finite Element Method for Hamilton-Jacobi Equations
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.
程晓良; 徐渊辑; 孟炳泉
2005-01-01
An algorithm for numerical solution of discrete Hamilton-Jacobi-Bellman equations is proposed.The method begins with a suitable initial guess value of the solution,then finds a suitable matrix to linearize the system and constructs an iteration algorithm to generate the monotone sequence.The convergence of the algorithm for nonlinear discrete Hamilton-Jacobi-Bellman equations is proved.Some numerical examples are presented to confirm the effciency of this algorithm.
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…
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…
Viscosity solutions of two classes of coupled Hamilton-Jacobi-Bellman equations
Başar Tamer
2001-01-01
Full Text Available This paper studies viscosity solutions of two sets of linearly coupled Hamilton-Jacobi-Bellman (HJB equations (one for finite horizon and the other one for infinite horizon which arise in the optimal control of nonlinear piecewise deterministic systems where the controls could be unbounded. The controls enter through the system dynamics as well as the transitions for the underlying Markov chain process, and are allowed to depend on both the continuous state and the current state of the Markov chain. The paper establishes the existence and uniqueness of viscosity solutions for these two sets of HJB equations, whose Hamiltonian structures are different from the standard ones.
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.
Hamilton-Jacobi equation, heteroclinic chains and Arnol'd diffusion in three time scales systems
Gallavotti, G; Mastropietro, V; Gallavotti, Giovanni; Gentile, Guido; Mastropietro, Vieri
1998-01-01
Interacting systems consisting of two rotators and a point mass near a hyperbolic fixed point are considered, in a case in which the uncoupled systems have three very different characteristic time scales. The abundance of quasi periodic motions in phase space is studied via the Hamilton-Jacobi equation. The main result, a high density theorem of invariant tori, is derived by the classical canonical transformation method extending previous results. As an application the existence of long heteroclinic chains (and of Arnol'd diffusion) is proved for systems interacting through a trigonometric polynomial in the angle variables.
Dey, Bijoy K; Janicki, Marek R; Ayers, Paul W
2004-10-08
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.
HERMITE WENO SCHEMES WITH LAX-WENDROFF TYPE TIME DISCRETIZATIONS FOR HAMILTON-JACOBI EQUATIONS
Jianxian Qiu
2007-01-01
In this paper, we use Hermite weighted essentially non-oscillatory (HWENO) schemes with a Lax-Wendroff time discretization procedure, termed HWENO-LW schemes, to solve Hamilton-Jacobi equations. The idea of the reconstruction in the HWENO schemes comes from the original WENO schemes, however both the function and its first derivative values are evolved in time and are used in the reconstruction. One major advantage of HWENO schemes is its compactness in the reconstruction. We explore the possibility in avoiding the nonlinear weights for part of the procedure, hence reducing the cost but still maintaining non-oscillatory properties for problems with strong discontinuous derivative. As a result,comparing with HWENO with Runge-Kutta time discretizations schemes (HWENO-RK) of Qiu and Shu [19] for Hamilton-Jacobi equations, the major advantages of HWENO-LW schemes are their saving of computational cost and their compactness in the reconstruction.Extensive numerical experiments are performed to illustrate the capability of the method.
Gabella, W.E.; Ruth, R.D.; Warnock, R.L.
1988-05-01
Periodic solutions of the Hamilton-Jacobi equation determine invariant tori in phase space. The Fourier spectrum of a torus with respect to angular coordinates gives useful information about nonlinear resonances and their potential for causing instabilities. We describe a method to solve the Hamilton-Jacobi equation for an arbitrary accelerator lattice. The method works with Fourier modes of the generating functions, and imposes periodicity in the machine azimuth by a shooting method. We give examples leading to three-dimensional plots in a surface of section. It is expected that the technique will be useful in lattice optimization. 14 refs., 6 figs., 1 tab.
Leclerc, M
2012-01-01
We introduce a symmetric Poisson bracket that allows us to describe anticommuting fields on a classical level in the same way as commuting fields, without the use of Grassmann variables. By means of a simple example, we show how the Dirac bracket for the elimination of the second class constraints can be introduced, how the classical Hamiltonian equations can be derived and how quantization can be achieved through a direct correspondence principle. Finally, we show that the semiclassical limit of the corresponding Schroedinger equation leads back to the Hamilton-Jacobi equation of the classical theory. Summarizing, it is shown that the relations between classical and quantum theory are valid for fermionic fields in exactly the same way as in the bosonic case, and that there is no need to introduce anticommuting variables on a classical level.
Solutions to estimation problems for scalar hamilton-jacobi equations using linear programming
Claudel, Christian G.
2014-01-01
This brief presents new convex formulations for solving estimation problems in systems modeled by scalar Hamilton-Jacobi (HJ) equations. Using a semi-analytic formula, we show that the constraints resulting from a HJ equation are convex, and can be written as a set of linear inequalities. We use this fact to pose various (and seemingly unrelated) estimation problems related to traffic flow-engineering as a set of linear programs. In particular, we solve data assimilation and data reconciliation problems for estimating the state of a system when the model and measurement constraints are incompatible. We also solve traffic estimation problems, such as travel time estimation or density estimation. For all these problems, a numerical implementation is performed using experimental data from the Mobile Century experiment. In the context of reproducible research, the code and data used to compute the results presented in this brief have been posted online and are accessible to regenerate the results. © 2013 IEEE.
Equation of motion of canonical tensor model and Hamilton-Jacobi equation of general relativity
Chen, Hua; Sato, Yuki
2016-01-01
The canonical tensor model (CTM) is a rank-three tensor model formulated as a totally constrained system in the canonical formalism. The constraint algebra of CTM has a similar structure as that of the ADM formalism of general relativity, and is studied as a discretized model for quantum gravity. In this paper, we analyze the classical equation of motion (EOM) of CTM in a formal continuum limit through a derivative expansion of the tensor up to the forth order, and show that it is the same as the EOM of a coupled system of gravity and a scalar field derived from the Hamilton-Jacobi equation with an appropriate choice of an action. The action contains a scalar field potential of an exponential form, and the system classically respects a dilatational symmetry. We find that the system has a critical dimension, given by six, over which it becomes unstable due to the wrong sign of the scalar kinetic term. In six dimensions, de Sitter spacetime becomes a solution to the EOM, signaling the emergence of a conformal s...
Hybrid massively parallel fast sweeping method for static Hamilton-Jacobi equations
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.
Probabilistic formulation of estimation problems for a class of Hamilton-Jacobi equations
Hofleitner, Aude
2012-12-01
This article presents a method for deriving the probability distribution of the solution to a Hamilton-Jacobi partial differential equation for which the value conditions are random. The derivations lead to analytical or semi-analytical expressions of the probability distribution function at any point in the domain in which the solution is defined. The characterization of the distribution of the solution at any point is a first step towards the estimation of the parameters defining the random value conditions. This work has important applications for estimation in flow networks in which value conditions are noisy. In particular, we illustrate our derivations on a road segment with random capacity reductions. © 2012 IEEE.
Sakamoto, Noboru; Schaft, Arjan J. van der
2007-01-01
In this paper, an analytical approximation approach for the stabilizing solution of the Hamilton-Jacobi equation using stable manifold theory is proposed. The proposed method gives approximated flows on the stable manifold of the associated Hamiltonian system and provides approximations of the
Sakamoto, Noboru; Schaft, Arjan J. van der
2007-01-01
In this paper, an analytical approximation approach for the stabilizing solution of the Hamilton-Jacobi equation using stable manifold theory is proposed. The proposed method gives approximated flows on the stable manifold of the associated Hamiltonian system and provides approximations of the stabl
Cardaliaguet, Pierre
2008-01-01
We investigate the regularity of solutions of first order Hamilton-Jacobi equation with super linear growth in the gradient variable. We show that the solutions are locally H\\"older continuous with H\\"older exponent depending only on the growth of the Hamiltonian. The proof relies on a reverse H\\"older inequality.
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. © 2011 American Institute of Physics
High-Order Semi-Discrete Central-Upwind Schemes for Multi-Dimensional Hamilton-Jacobi Equations
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.
2016-05-01
Algorithm for Overcoming the Curse of Dimensionality for Certain Non-convex Hamilton-Jacobi Equations, Projections and Differential Games Yat Tin...complexity of the resulting algorithm is polynomial in the problem dimension; hence, it overcomes the curse of dimensionality [1, 2]. We extend previous work...compute the evolution of geometric objects [25], which was first used for reachability problems in [21, 22] to our knowledge . Numerical solutions to HJ PDE
Cagnetti, Filippo
2013-11-01
We consider a numerical scheme for the one dimensional time dependent Hamilton-Jacobi equation in the periodic setting. This scheme consists in a semi-discretization using monotone approximations of the Hamiltonian in the spacial variable. From classical viscosity solution theory, these schemes are known to converge. In this paper we present a new approach to the study of the rate of convergence of the approximations based on the nonlinear adjoint method recently introduced by L.C. Evans. We estimate the rate of convergence for convex Hamiltonians and recover the O(h) convergence rate in terms of the L∞ norm and O(h) in terms of the L1 norm, where h is the size of the spacial grid. We discuss also possible generalizations to higher dimensional problems and present several other additional estimates. The special case of quadratic Hamiltonians is considered in detail in the end of the paper. © 2013 IMACS.
Osetrin, Konstantin; Osetrin, Evgeny
2015-01-01
The characteristics of dust matter in space-time 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.
On the Hamilton-Jacobi Treatment of Constrained Systems
Sami.I.Muslih; Hosam A.El-Zalan
2003-01-01
Systems with singular-higher order Lagrangians are investigated by two methods: Dirac method and Hamilton-Jacobi method. An example is studied and it is shown that the Hamilton-Jacobi method gives the correct canonical generalized equations of motion, contrary to Dirac method, where Dirac conjecture is invalid.
Invariant surfaces and tracking by the Hamilton-Jacobi method
Warnock, R.L.; Ruth, R.D.
1986-09-01
The Hamilton-Jacobi method is described for a model of betatron motion in one degree of freedom, namely, a harmonic oscillator perturbed by a lattice of sextupoles. The Hamilton-Jacobi equation is given in terms of Fourier amplitudes. Invariant surfaces have been obtained in phase space, and finite time symplectic maps were obtained for tracking of single particles. (LEW)
Canepa, Edward S.
2017-06-19
Nowadays, traffic management has become a challenge for urban areas, which are covering larger geographic spaces and facing the generation of different kinds of traffic data. This article presents a robust traffic estimation framework for highways modeled by a system of Lighthill Whitham Richards equations that is able to assimilate different sensor data available. We first present an equivalent formulation of the problem using a Hamilton–Jacobi equation. Then, using a semi-analytic formula, we show that the model constraints resulting from the Hamilton–Jacobi equation are linear ones. We then pose the problem of estimating the traffic density given incomplete and inaccurate traffic data as a Mixed Integer Program. We then extend the density estimation framework to highway networks with any available data constraint and modeling junctions. Finally, we present a travel estimation application for a small network using real traffic measurements obtained obtained during Mobile Century traffic experiment, and comparing the results with ground truth data.
Extinction for viscous Hamilton-Jacobi equations%粘性Hamilton-Jacobi方程的熄灭
田娅; 穆春来
2009-01-01
Extinction in finite time for non-negative classical solutions p∈(0,1). The occurrence of these phenomena is shown to concerned with the domain Ω. For the Cauchy problem,extinction in finite time occurs to the classical solution of the above equation depends on the behavior of the initial data u0(x) as |x|→∞.%作者研究了初边值问题ut-Δu-u+|Δu|p=0,p∈(0,1) 的非负古典解的有限时间熄灭,证明了熄灭现象的发生与区域Ω有关.对于Cauchy问题,上述方程的古典解在有限时间熄灭与否依赖于初值u0(x)在|x|→∞时的性态.
Utility maximization with partial information: Hamilton-Jacobi-Bellman equation approach
BAI Lihua; GUO Junyi
2007-01-01
This paper deals with the problem of maximizing the expected utility of the terminal wealth when the stock price satisfies a stochastic differential equation with instantaneous rates of return modelled as an Ornstein-Uhlenbeck process.Here, only the stock price and interest rate can be observable for an investor.It is reduced to a partially observed stochastic control problem.Combining the filtering theory with the dynamic programming approach, explicit representations of the optimal value functions and corresponding optimal strategies are derived. Moreover, closed-form solutions are provided in two cases of exponential utility and logarithmic utility.In particular, logarithmic utility is considered under the restriction of shortselling and borrowing.
Geometric Hamilton-Jacobi theory on Nambu-Poisson manifolds
de León, M.; Sardón, C.
2017-03-01
The Hamilton-Jacobi theory is a formulation of classical mechanics equivalent to other formulations as Newtonian, Lagrangian, or Hamiltonian mechanics. The primordial observation of a geometric Hamilton-Jacobi theory is that if a Hamiltonian vector field XH can be projected into the configuration manifold by means of a 1-form dW, then the integral curves of the projected vector field XHd Wcan be transformed into integral curves of XH provided that W is a solution of the Hamilton-Jacobi equation. Our aim is to derive a geometric Hamilton-Jacobi theory for physical systems that are compatible with a Nambu-Poisson structure. For it, we study Lagrangian submanifolds of a Nambu-Poisson manifold and obtain explicitly an expression for a Hamilton-Jacobi equation on such a manifold. We apply our results to two interesting examples in the physics literature: the third-order Kummer-Schwarz equations and a system of n copies of a first-order differential Riccati equation. From the first example, we retrieve the original Nambu bracket in three dimensions and from the second example, we retrieve Takhtajan's generalization of the Nambu bracket to n dimensions.
Quantum Hamilton-Jacobi Cosmology and Classical-Quantum Correlation
Fathi, M.; Jalalzadeh, S.
2017-07-01
How the time evolution which is typical for classical cosmology emerges from quantum cosmology? The answer is not trivial because the Wheeler-DeWitt equation is time independent. A framework associating the quantum Hamilton-Jacobi to the minisuperspace cosmological models has been introduced in Fathi et al. (Eur. Phys. J. C 76, 527 2016). In this paper we show that time dependence and quantum-classical correspondence both arise naturally in the quantum Hamilton-Jacobi formalism of quantum mechanics, applied to quantum cosmology. We study the quantum Hamilton-Jacobi cosmology of spatially flat homogeneous and isotropic early universe whose matter content is a perfect fluid. The classical cosmology emerge around one Planck time where its linear size is around a few millimeter, without needing any classical inflationary phase afterwards to make it grow to its present size.
Geometric Hamilton-Jacobi theory for higher-order autonomous systems
Colombo, Leonardo; de León, Manuel; Prieto-Martínez, Pedro Daniel; Román-Roy, Narciso
2014-06-01
The geometric framework for the Hamilton-Jacobi theory is used to study this theory in the background of higher-order mechanical systems, in both the Lagrangian and Hamiltonian formalisms. Thus, we state the corresponding Hamilton-Jacobi equations in these formalisms and apply our results to analyze some particular physical examples.
Dynamical and geometric aspects of Hamilton-Jacobi and linearized Monge-Ampère equations VIASM 2016
Tran, Hung
2017-01-01
Consisting of two parts, the first part of this volume is an essentially self-contained exposition of the geometric aspects of local and global regularity theory for the Monge–Ampère and linearized Monge–Ampère equations. As an application, we solve the second boundary value problem of the prescribed affine mean curvature equation, which can be viewed as a coupling of the latter two equations. Of interest in its own right, the linearized Monge–Ampère equation also has deep connections and applications in analysis, fluid mechanics and geometry, including the semi-geostrophic equations in atmospheric flows, the affine maximal surface equation in affine geometry and the problem of finding Kahler metrics of constant scalar curvature in complex geometry. Among other topics, the second part provides a thorough exposition of the large time behavior and discounted approximation of Hamilton–Jacobi equations, which have received much attention in the last two decades, and a new approach to the subject, the n...
Kinetic derivation of a Hamilton-Jacobi traffic flow model
Borsche, Raul; Kimathi, Mark
2012-01-01
Kinetic models for vehicular traffic are reviewed and considered from the point of view of deriving macroscopic equations. A derivation of the associated macroscopic traffic flow equations leads to different types of equations: in certain situations modified Aw-Rascle equations are obtained. On the other hand, for several choices of kinetic parameters new Hamilton-Jacobi type traffic equations are found. Associated microscopic models are discussed and numerical experiments are presented discussing several situations for highway traffic and comparing the different models.
Hamilton-Jacobi formalism of tachyon inflation and cosmological perturbations
LIU Daojun
2014-08-01
Full Text Available We study the cosmological inflation models driven by the rolling tachyon field which has a Born-Infeld-type action.We drive the Hamilton-Jacobi equation for the cosmological dynamics of tachyon inflation and the mode equations for the scalar and tensor perturbations of tachyon field and spacetime, then a solution under the slow-roll condition is given. In the end,a realistic model from string theory is discussed.
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.
A Hamilton-Jacobi Formalism for Thermodynamics
Rajeev, S G
2007-01-01
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 \\m{n} thermodynamic degrees of freedom it is \\m{2n+1}-dimensional. The equations of state of a substance pick out an \\m{n}-dimensional submanifold. A family of substances whose equations of state depend on \\m{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...
Hamilton-Jacobi method for a simple resonance
Rudnev, Mischa
2003-01-01
It is well known that a generic small perturbation of a Liouville-integrable Hamiltonian system causes breakup of resonant and near-resonant invariant tori. A general approach to the simple resonance case in the convex real-analytic setting is developed, based on a new technique for solving the Hamilton-Jacobi equation. It is shown that a generic perturbation creates in the core of a resonance a partially hyperbolic lower-dimensional invariant torus, whose Lagrangian stable and unstable manif...
Hamilton-Jacobi theorems for regular reducible Hamiltonian systems on a cotangent bundle
Wang, Hong
2017-09-01
In this paper, some of formulations of Hamilton-Jacobi equations for Hamiltonian system and regular reduced Hamiltonian systems are given. At first, an important lemma is proved, and it is a modification for the corresponding result of Abraham and Marsden (1978), such that we can prove two types of geometric Hamilton-Jacobi theorem for a Hamiltonian system on the cotangent bundle of a configuration manifold, by using the symplectic form and dynamical vector field. Then these results are generalized to the regular reducible Hamiltonian system with symmetry and momentum map, by using the reduced symplectic form and the reduced dynamical vector field. The Hamilton-Jacobi theorems are proved and two types of Hamilton-Jacobi equations, for the regular point reduced Hamiltonian system and the regular orbit reduced Hamiltonian system, are obtained. As an application of the theoretical results, the regular point reducible Hamiltonian system on a Lie group is considered, and two types of Lie-Poisson Hamilton-Jacobi equation for the regular point reduced system are given. In particular, the Type I and Type II of Lie-Poisson Hamilton-Jacobi equations for the regular point reduced rigid body and heavy top systems are shown, respectively.
Conformal invariance and Hamilton Jacobi theory for dissipative systems
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 renormalization for Lifshitz spacetime
Baggio, M.; de Boer, J.; Holsheimer, K.
2012-01-01
Just like AdS spacetimes, Lifshitz spacetimes require counterterms in order to make the on-shell value of the bulk action finite. We study these counterterms using the Hamilton-Jacobi method. Rather than imposing boundary conditions from the start, we will derive suitable boundary conditions by
Hamilton--Jacobi meet M\\"obius
Faraggi, Alon E
2015-01-01
Adaptation of the Hamilton--Jacobi formalism to quantum mechanics leads to a cocycle condition, which is invariant under $D$--dimensional M\\"obius transformations with Euclidean or Minkowski metrics. In this paper we aim to provide a pedagogical presentation of the proof of the M\\"obius symmetry underlying the cocycle condition. The M\\"obius 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 M\\"obius 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 M\\"obius symmetry can only be implemented consistently if spatial space is compact, and correspondingly if there exist a finite ultraviolet length scale. Evidence for non--trivial space topology may exist in the cosmic microwave background radiation.
Hamilton-Jacobi method for curved domain walls and cosmologies
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.
On the Hamilton-Jacobi method in classical and quantum nonconservative systems
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 Method and Gravitation
Di Criscienzo, R; Zerbini, S
2010-01-01
Studying the behaviour of a quantum field in a classical, curved, spacetime is an extraordinary task which nobody is able to take on at present time. Independently by the fact that such problem is not likely to be solved soon, still we possess the instruments to perform exact predictions in special, highly symmetric, conditions. Aim of the present contribution is to show how it is possible to extract quantitative information about a variety of physical phenomena in very general situations by virtue of the so-called Hamilton-Jacobi method. In particular, we shall prove the agreement of such semi-classical method with exact results of quantum field theoretic calculations.
Hamilton-Jacobi Method and Gravitation
di Criscienzo, R.; Vanzo, L.; Zerbini, S.
Studying the behaviour of a quantum field in a classical, curved, spacetime is an extraordinary task which nobody is able to take on at present time. Independently by the fact that such problem is not likely to be solved soon, still we possess the instruments to perform exact predictions in special, highly symmetric, conditions. Aim of the present contribution is to show how it is possible to extract quantitative information about a variety of physical phenomena in very general situations by virtue of the so-called Hamilton-Jacobi method. In particular, we shall prove the agreement of such semi-classical method with exact results of quantum field theoretic calculations.
Deformed Hamilton-Jacobi Method in Covariant Quantum Gravity Effective Models
Benrong, Mu; Yang, Haitang
2014-01-01
We first briefly revisit the original Hamilton-Jacobi method and show that the Hamilton-Jacobi equation for the action $I$ of tunnelings of a fermionic particle from a charged black hole can be written in the same form as that of a scalar particle. For the low energy quantum gravity effective models which respect covariance of the curved spacetime, we derive the deformed model-independent KG/Dirac and Hamilton-Jacobi equations using the methods of effective field theory. We then find that, to all orders of the effective theories, the deformed Hamilton-Jacobi equations can be obtained from the original ones by simply replacing the mass of emitted particles $m$ with a parameter $m_{eff}$ that includes all the quantum gravity corrections. Therefore, in this scenario, there will be no corrections to the Hawking temperature of a black hole from the quantum gravity effects if its original Hawking temperature is independent of the mass of emitted particles. As a consequence, our results show that breaking covariance...
Viscous warm inflation: Hamilton-Jacobi formalism
Akhtari, L.; Mohammadi, A.; Sayar, K.; Saaidi, Kh.
2017-04-01
Using Hamilton-Jacobi formalism, the scenario of warm inflation with viscous pressure is considered. The formalism gives a way of computing the slow-rolling parameter without extra approximation, and it is well-known as a powerful method in cold inflation. The model is studied in detail for three different cases of the dissipation and bulk viscous pressure coefficients. In the first case where both coefficients are taken as constant, it is shown that the case could not portray warm inflationary scenario compatible with observational data even it is possible to restrict the model parameters. For other cases, the results shows that the model could properly predicts the perturbation parameters in which they stay in perfect agreement with Planck data. As a further argument, r -ns and αs -ns are drown that show the acquired result could stand in acceptable area expressing a compatibility with observational data.
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.
曹玉松
2013-01-01
从保险公司的角度出发,在投资基金价格服从带漂移的几何布朗运动的假定下,基于Hamilton-Jacobi-Bellman理论,给出了使得盈余终值的期望指数效用最大化的比例再保险函数的最优比例,及其各个风险市场的最优投资比例.%From the insurer's point of view,on the assumption that investment fund follows the Geometric Brownian motion,based on the theory of Hamilton-Jacobi-Bellman equation,the paper gives the optimal proportion of the proportional reinsurance and the capital amount of each risky investment markets,which can make the expected exponential utility of terminal wealth maximum.
Viscosity Solutions of Hamilton-Jacobi Equations.
1981-08-01
the result follove. Remark 4.2. W* could replace u5 e c 2 ( ) above by uC e W 2 ,p(Q), p > N , via Bony’s boc maximum principle (5]. Remark 4.3. If...semigroup property S(t)S() S(t+T) for tr N 0 as usual. We remark that (6.7) also follows from (6.8) and the translation invariance of this model...Symposium, L. Cesari, J. Hale, J. LaSalle ode., Academic Press, New York, (1976), 131-165. a. Crandall, M. G. and P. L. Lions, Condition d’unicite pour
A Practical Approach to the Hamilton-Jacobi Formulation of Holographic Renormalization
Elvang, Henriette
2016-01-01
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 rel...
Adaptive dynamics via Hamilton-Jacobi approach and entropy methods for a juvenile-adult model.
Carrillo, José Antonio; Cuadrado, Sílvia; Perthame, Benoît
2007-01-01
We consider a nonlinear system describing a juvenile-adult population undergoing small mutations. We analyze two aspects: from a mathematical point of view, we use an entropy method to prove that the population neither goes extinct nor blows-up; from an adaptive evolution point of view, we consider small mutations on a long time scale and study how a monomorphic or a dimorphic initial population evolves towards an Evolutionarily Stable State. Our method relies on an asymptotic analysis based on a constrained Hamilton-Jacobi equation. It allows to recover earlier predictions in Calsina and Cuadrado [A. Calsina, S. Cuadrado, Small mutation rate and evolutionarily stable strategies in infinite dimensional adaptive dynamics, J. Math. Biol. 48 (2004) 135; A. Calsina, S. Cuadrado, Stationary solutions of a selection mutation model: the pure mutation case, Math. Mod. Meth. Appl. Sci. 15(7) (2005) 1091.] that we also assert by direct numerical simulation. One of the interests here is to show that the Hamilton-Jacobi approach initiated in Diekmann et al. [O. Diekmann, P.-E. Jabin, S. Mischler, B. Perthame, The dynamics of adaptation: an illuminating example and a Hamilton-Jacobi approach, Theor. Popul. Biol. 67(4) (2005) 257.] extends to populations described by systems.
On the Connection between the Hamilton-Jacobi-Bellman and the Fokker-Planck Control Frameworks
Annunziato, Mario
2014-09-01
In the framework of stochastic processes, the connection between the dynamic programming scheme given by the Hamilton-Jacobi-Bellman equation and a recently proposed control approach based on the Fokker-Planck equation is discussed. Under appropriate assumptions it is shown that the two strategies are equivalent in the case of expected cost functionals, while the FokkerPlanck formalism allows considering a larger class of objectives. To illustrate the connection between the two control strategies, the cases of an Itō stochastic process and of a piecewise-deterministic process are considered.
Scalar particles emission from black holes with topological defects using Hamilton-Jacobi method
Jusufi, Kimet
2015-01-01
We study quantum tunneling of charged and uncharged scalar particles from the event horizon of Schwarzschild-de Sitter and Reissner-Nordstr\\"{o}m-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.
Scalar particles emission from black holes with topological defects using Hamilton-Jacobi method
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.
Park, Chandeok
This dissertation presents a general methodology for solving the optimal feedback control problem in the context of Hamiltonian system theory. It is first formulated as a two point boundary value problem for a standard Hamiltonian system, and the associated phase flow is viewed as a canonical transformation. Then relying on the Hamilton-Jacobi theory, we employ generating functions to develop a unified methodology for solving a variety of optimal feedback control formulations with general types of boundary conditions. The major accomplishment is to establish a theoretical connection between the optimal cost function and a special kind of generating function. Guided by this recognition, we are ultimately led to a new flexible representation of the optimal feedback control law for a given system, which is adjustable to various types of boundary conditions by algebraic conversions and partial differentiations. This adaptive property provides a substantial advantage over the classical dynamic programming method in the sense that we do not need to solve the Hamilton-Jacobi-Bellman equation repetitively for varying types of boundary conditions. Furthermore for a special type of boundary condition, it also enables us to work around an inherent singularity of the Hamilton-Jacobi-Bellman equation by a special algebraic transformation. Taking full advantage of these theoretical insights, we develop a systematic algorithm for solving a class of optimal feedback control problems represented by smooth analytic Hamiltonians, and apply it to problems with different characteristics. Then, broadening the practical utility of generating functions for problems where the relevant Hamiltonian is non-smooth, we construct a pair of Cauchy problems from the associated Hamilton-Jacobi equations. This alternative formulation is justified by solving problems with control constraints which usually feature non-smoothness in the control logic. The main result of this research establishes that
A practical approach to the Hamilton-Jacobi formulation of holographic renormalization
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.
A practical approach to the Hamilton-Jacobi formulation of holographic renormalization
Elvang, Henriette; Hadjiantonis, Marios [Department of Physics and Michigan Center for Theoretical Physics, University of Michigan,450 Church Str., Ann Arbor MI 48109 (United States)
2016-06-08
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.
The Hamilton-Jacobi method and Hamiltonian maps
Abdullaev, S.S. [Institut fuer Plasmaphysik, Forschungszentrum Juelich GmbH, EURATOM Association, Trilateral Euregio Cluster, Juelich (Germany)
2002-03-29
A method for constructing time-step-based symplectic maps for a generic Hamiltonian system subjected to perturbation is developed. Using the Hamilton-Jacobi method and Jacobi's theorem in finite periodic time intervals, the general form of the symplectic maps is established. The generating function of the map is found by the perturbation method in the finite time intervals. The accuracy of the maps is studied for fully integrable and partially chaotic Hamiltonian systems and compared to that of the symplectic integration method. (author)
The Hamilton-Jacobi method and Hamiltonian maps
Abdullaev, S. S.
2002-03-01
A method for constructing time-step-based symplectic maps for a generic Hamiltonian system subjected to perturbation is developed. Using the Hamilton-Jacobi method and Jacobi's theorem in finite periodic time intervals, the general form of the symplectic maps is established. The generating function of the map is found by the perturbation method in the finite time intervals. The accuracy of the maps is studied for fully integrable and partially chaotic Hamiltonian systems and compared to that of the symplectic integration method.
Hamilton-Jacobi method for molecular distribution function in a chemical oscillator.
Nakanishi, Hiizu; Sakaue, Takahiro; Wakou, Jun'ichi
2013-12-07
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.
Initialization of the Shooting Method via the Hamilton-Jacobi-Bellman Approach
Cristiani, Emiliano
2009-01-01
The aim of this paper is to investigate from the numerical point of view the possibility of coupling the Hamilton-Jacobi-Bellman (HJB) equation and Pontryagin's Minimum Principle (PMP) to solve some control problems. A rough approximation of the value function computed by the HJB method is used to obtain an initial guess for the PMP method. The advantage of our approach over other initialization techniques (such as continuation or direct methods) is to provide an initial guess close to the global minimum. Numerical tests involving multiple minima, discontinuous control, singular arcs and state constraints are considered. The CPU time for the proposed method is less than four minutes up to dimension four, without code parallelization.
Hamilton-Jacobi formalism for inflation with non-minimal derivative coupling
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.
Holographic renormalization and Ward identities with the Hamilton-Jacobi method
Martelli, Dario E-mail: d.martelli@qmul.ac.uk; Mueck, Wolfgang E-mail: mueck@na.infn.it
2003-03-24
A systematic procedure for performing holographic renormalization, which makes use of the Hamilton-Jacobi method, is proposed and applied to a bulk theory of gravity interacting with a scalar field and a U(1) gauge field in the Stueckelberg formalism. We describe how the power divergences are obtained as solutions of a set of 'descent equations' stemming from the radial Hamiltonian constraint of the theory. In addition, we isolate the logarithmic divergences, which are closely related to anomalies. The method allows to determine also the exact one-point functions of the dual field theory. Using the other Hamiltonian constraints of the bulk theory, we derive the Ward identities for diffeomorphisms and gauge invariance. In particular, we demonstrate the breaking of U(1){sub R} current conservation, recovering the holographic chiral anomaly recently discussed in and.
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.
The classical limit of minimal length uncertainty relation: revisit with the Hamilton-Jacobi method
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.
Hamilton-Jacobi formalism for inflation with non-minimal derivative coupling
Sheikhahmadi, Haidar; Aghamohammadi, Ali; Saaidi, Khaled
2016-01-01
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.
Hamilton-Jacobi method for classical mechanics in Grassmann algebra (in English)
Tabunshchyk, K. V.
We present the Hamilton--Jacobi method for the classical mechanics with the constrains in Grassmann algebra. Within the framework of this method the solution for the classical system characterized by the SUSY Lagrangian is obtained.
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.”
Hamilton-Jacobi Many-Worlds Theory and the Heisenberg Uncertainty Principle
Tipler, Frank J
2010-01-01
I show that the classical Hamilton-Jacobi (H-J) equation can be used as a technique to study quantum mechanical problems. I first show that the the Schr\\"odinger equation is just the classical H-J equation, constrained by a condition that forces the solutions of the H-J equation to be everywhere $C^2$. That is, quantum mechanics is just classical mechanics constrained to ensure that ``God does not play dice with the universe.'' I show that this condition, which imposes global determinism, strongly suggests that $\\psi^*\\psi$ measures the density of universes in a multiverse. I show that this interpretation implies the Born Interpretation, and that the function space for $\\psi$ is larger than a Hilbert space, with plane waves automatically included. Finally, I use H-J theory to derive the momentum-position uncertainty relation, thus proving that in quantum mechanics, uncertainty arises from the interference of the other universes of the multiverse, not from some intrinsic indeterminism in nature.
Study of invariant surfaces and their break-up by the Hamilton-Jacobi method
Warnock, R.L.; Ruth, R.D.
1986-08-01
A method is described to compute invariant tori in phase space for calssical non-integrable Hamiltonian systems. Our procedure is to solve the Hamilton-Jacobi equation stated as a system of equations for Fourier coefficients of the generating function. The system is truncated to a finite number of Fourier modes and solved numerically by Newton's method. The resulting canonical transformation serves to reduce greatly the non-integrable part of the Hamiltonian. In examples studied to date the convergence properties of the method are excellent, even near chaotic regions and on the separatrices of isolated broad resonances. We propose a criterion for breakup of invariant surfaces, namely the vanishing of the Jacobian of the canonical transformation to new angle variables. By comparison with results from tracking, we find in an example with two nearly overlapping resonances that this criterion can be implemented with sufficient accuracy to determine critical parameters for the breakup ('transition to chaos') to an accuracy of 5 to 10%.
Hawking radiation of Kerr-de Sitter black holes using Hamilton-Jacobi method
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.
McGann, M; Dewar, R L; von Nessi, G
2010-01-01
The vanishing of the divergence of the total stress tensor (magnetic plus kinetic) in a neighborhood of an equilibrium plasma containing a toroidal surface of discontinuity gives boundary and jump conditions that strongly constrain allowable continuations of the magnetic field across the surface. The boundary conditions allow the magnetic fields on either side of the discontinuity surface to be described by surface magnetic potentials, reducing the continuation problem to that of solving a Hamilton--Jacobi equation. The characteristics of this equation obey Hamiltonian equations of motion, and a necessary condition for the existence of a continued field across a general toroidal surface is that there exist invariant tori in the phase space of this Hamiltonian system. It is argued from the Birkhoff theorem that existence of such an invariant torus is also, in general, sufficient for continuation to be possible. An important corollary is that the rotational transform of the continued field on a surface of disco...
Hamilton-Jacobi method and effective actions of D-brane and M-brane in supergravity
Sato, Matsuo E-mail: machan@het.phys.sci.osaka-u.ac.jp; Tsuchiya, Asato E-mail: tsuchiya@het.phys.sci.osaka-u.ac.jp
2003-11-03
We show that the effective actions of D-brane and M-brane are solutions to the Hamilton-Jacobi (H-J) equations in supergravities. This fact means that these effective actions are on-shell actions in supergravities. These solutions to the H-J equations reproduce the supergravity solutions that represent D-branes in a B{sub 2} field, M2 branes and the M2-M5 bound states. The effective actions in these solutions are those of a probe D-brane and a probe M-brane. Our findings can be applied to the study of the gauge/gravity correspondence, especially the holographic renormalization group, and a search for new solutions of supergravity.
Hamilton-Jacobi method and effective actions of D-brane and M-brane in supergravity
Sato, Matsuo; Tsuchiya, Asato
2003-11-01
We show that the effective actions of D-brane and M-brane are solutions to the Hamilton-Jacobi (H-J) equations in supergravities. This fact means that these effective actions are on-shell actions in supergravities. These solutions to the H-J equations reproduce the supergravity solutions that represent D-branes in a B2 field, M2 branes and the M2-M5 bound states. The effective actions in these solutions are those of a probe D-brane and a probe M-brane. Our findings can be applied to the study of the gauge/gravity correspondence, especially the holographic renormalization group, and a search for new solutions of supergravity.
Rahman, M Atiqur
2013-01-01
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\\rq s opinion and gives a correction to the Hawking radiation of SAdS black hole.
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.
Salisbury, Donald; Sundermeyer, Kurt
2015-01-01
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 spacetime curvature based coordinates grant equal status to all geometric backgrounds. They play an essential role as a starting point for inequivalent semi-classical quantizations. The scheme calls into question Wheeler's geometrodynamical approach and the associated Wheeler-DeWitt equation in which three-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...
Bound States and Band Structure - a Unified Treatment through the Quantum Hamilton - Jacobi Approach
Ranjani, S S; Panigrahi, P K
2005-01-01
We analyze the Scarf potential, which exhibits both discrete energy bound states and energy bands, through the quantum Hamilton-Jacobi approach. The singularity structure and the boundary conditions in the above approach, naturally isolate the bound and periodic states, once the problem is mapped to the zero energy sector of another quasi-exactly solvable quantum problem. The energy eigenvalues are obtained without having to solve for the corresponding eigenfunctions explicitly. We also demonstrate how to find the eigenfunctions through this method.
Coordinates Used in Derivation of Hawking Radiation via Hamilton-Jacobi Method
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.
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.
The Classical Limit of Minimal Length Uncertainty Relation:Revisit with the Hamilton-Jacobi Method
Guo, Xiaobo; Yang, Haitang
2015-01-01
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 use the Hamilton-Jacobi method to study motions of particles in the context of deformed Newtonian mechanics and general relativity. Specifically, the precession of planetary orbits, deflection of light, and time delay in radar propagation are considered in this paper. 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.
Hawking radiation from a Vaidya black hole by Hamilton-Jacobi method
Ding, Han; Liu, Wen-Biao
2011-03-01
Using the Hamilton-Jacobi method, Hawking radiation from the apparent horizon of a dynamical Vaidya black hole is calculated. The black hole thermodynamics can be built successfully on the apparent horizon. If a relativistic perturbation is given to the apparent horizon, a similar calculation can also lead to a purely thermal spectrum, which corresponds to a modified temperature from the former. The first law of thermodynamics can also be constructed successfully at a new supersurface which has a small deviation from the apparent horizon. When the event horizon is thought as such a deviation from the apparent horizon, the expressions of the characteristic position and temperature are consistent with the previous result that asserts that thermodynamics should be built on the event horizon. It is concluded that the thermodynamics should be constructed on the apparent horizon exactly while the event horizon thermodynamics is just one of the perturbations near the apparent horizon.
Hawking radiation of Reissner-Nordstrom-de Sitter black hole by Hamilton-Jacobi method
Hossain, M Ilias
2013-01-01
In Refs. (M. Atiqur Rahman, M. Ilias Hossain (2012) Phys. Lett. B {\\bf 712} 1), we have developed Hamilton-Jacobi method for dynamical spacetime and discussed Hawking radiation of Schwarzschild-de Sitter black hole by massive particle tunneling method. In this letter, we have investigated the hawking purely thermal and nonthermal radiations of Reissner-Nordstr\\"{o}m-de Sitter (RNdS) black hole. We have considered energy and angular momentum as conserved and shown that the tunneling rate is related to the change of Bekenstein-Hawking entropy and the derived emission spectrum deviates from the pure thermal spectrum. The results we have obtained for RNdS black hole is also in accordance with Parikh and Wilczek\\rq s opinion and recovered the new result for Hawking radiation of RNdS black hole.
Back-Reaction of Black Hole Radiation from Hamilton-Jacobi Method
Ding, Chikun
2013-10-01
In the frame of Hamilton-Jacobi method, the back-reactions of the radiating particles together with the total entropy change of the whole system are investigated. The emission probability from this process is found to be equivalent to the null geodesic method. However its physical picture is more clear: the negative energy one of a virtual particle pair is absorbed by the black hole, resulting in the temperature, electric potential and angular velocity increase; then the black hole amount of heat, electric charge and angular momentum can spontaneously transfer to the positive energy particle; when obtaining enough energy, it can escape away to infinity, visible to distant observers. And this method can be applied to any sort of horizons and particles without a specific choice of (regular-across-the-horizon) coordinates.
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
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.
Videla, Nelson [FCFM, Universidad de Chile, Departamento de Fisica, Santiago (Chile)
2017-03-15
In the present work we study the consequences of considering an inflationary universe model in which the Hubble rate has a quasi-exponential dependence in the inflaton field, given by H(φ) = H{sub inf} exp[((φ)/(m{sub p}))/(p(1+(φ)/(m{sub p})))]. We analyze the inflation dynamics under the Hamilton-Jacobi approach, which allows us to consider H(φ), rather than V(φ), as the fundamental quantity to be specified. By comparing the theoretical predictions of the model together with the allowed contour plots in the n{sub s} - r plane and the amplitude of primordial scalar perturbations from the latest Planck data, the parameters charactering this model are constrained. The model predicts values for the tensor-to-scalar ratio r and for the running of the scalar spectral index dn{sub s}/d ln k consistent with the current bounds imposed by Planck, and we conclude that the model is viable. (orig.)
Hossain, M Ilias
2013-01-01
Incorporating Parikh and Wilczek's opinion to the Kerr de-Sitter (KdS) black hole Hawking non-thermal and purely thermal radiations have been investigated using Hamilton-Jacobi method. We have taken the background spacetime of KdS black hole as dynamical, involving the self-gravitation effect of the emitted particles, energy and angular momentum has been taken as conserved 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. The explored results gives a correction to the Hawking radiation of KdS black hole.
Some Inverse Problems in Periodic Homogenization of Hamilton-Jacobi Equations
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.
1985-10-01
many more resu -s of this kind, including existence results in cases where nonuniqueness is possible and the existence of minimal solutions. We also...in these works. -4- . .. . . . . . . . . * .. ..-.. .. ... . .. . .. . .. . .. . .. . .. . . H CONTENTS I. Lipschitz Hamiltonians and the stationary...condition~s at infinity VII. Further remarks on the Cauchy problem V -5-r 1. LIPSCHITZ HAMILTONIANS AN-D THE STATIONARY PROBLEM. wilIn this section we
Gallavotti, G
1997-01-01
Local integrability of hyperbolic oscillators is discussed to provide an introductory example of the Arnold's diffusion phenomenon in a forced pendulum. This is a text prepared for the the ISI summer school of June 1997 and deals with developments of the topics treated in the lectures.
Lindstedt series and Hamilton-Jacobi equation for hyperbolic tori in three time scales problems
Gallavotti, G; Mastropietro, V; Gallavotti, Giovannni; Gentile, Guido; Astropietro, Vieri M
1998-01-01
Interacting systems consisting of two rotators and a pendulum are considered, in a case in which the uncoupled systems have three very different characteristic time scales. The abundance of unstable quasi periodic motions in phase space is studied via Lindstedt series. The result is a strong improvement, compared to our previous results, on the domain of validity of bounds that imply existence of invariant tori, large homoclinic angles, long heteroclinic chains and drift--diffusion in phase space.
Shuhuan Wen
2012-01-01
Full Text Available This paper works on hybrid force/position control in robotic manipulation and proposes an improved radial basis functional (RBF neural network, which is a robust relying on the Hamilton Jacobi Issacs principle of the force control loop. The method compensates uncertainties in a robot system by using the property of RBF neural network. The error approximation of neural network is regarded as an external interference of the system, and it is eliminated by the robust control method. Since the conventionally fixed structure of RBF network is not optimal, resource allocating network (RAN is proposed in this paper to adjust the network structure in time and avoid the underfit. Finally the advantage of system stability and transient performance is demonstrated by the numerical simulations.
Sakalli, I., E-mail: izzet.sakalli@emu.edu.tr; Mirekhtiary, S. F., E-mail: fatemeh.mirekhtiary@emu.edu.tr [Eastern Mediterranean University G. Magosa, Department of Physics (Turkey)
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.
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
2013-01-01
Hawking radiation of a non-asymptotically flat (NAF) 4-dimensional spherically symmetric and static dilatonic black hole (BH) via the Hamilton-Jacobi (HJ) method has been studied. In addition to the naive coordinates, we have used four more different coordinate systems which are well-behaved at the horizon. Except the isotropic coordinates, direct computation of the HJ method leads us the standard Hawking temperature for all coordinate systems. The isotropic coordinates render possible to get the index of refraction extracting from the Fermat metric. It is explicitly shown that the index of refraction determines the value of the tunneling rate and its natural consequence, Hawking temperature. The isotropic coordinates within the conventional HJ method produce wrong result for the temperature of the LDBH. Here, we explain how this discrepancy can be resolved by regularizing the integral possessing a pole at the horizon.
Dirac equation of spin particles and tunneling radiation from a Kinnersly black hole
Li, Guo-Ping; Zu, Xiao-Tao [University of Electronic Science and Technology of China, School of Physical Electronics, Chengdu (China); Feng, Zhong-Wen [University of Electronic Science and Technology of China, School of Physical Electronics, Chengdu (China); China West Normal University, College of Physics and Space Science, Nanchong (China); Li, Hui-Ling [University of Electronic Science and Technology of China, School of Physical Electronics, Chengdu (China); Shenyang Normal University, College of Physics Science and Technology, Shenyang (China)
2017-04-15
In curved space-time, the Hamilton-Jacobi equation is a semi-classical particle equation of motion, which plays an important role in the research of black hole physics. In this paper, starting from the Dirac equation of spin 1/2 fermions and the Rarita-Schwinger equation of spin 3/2 fermions, respectively, we derive a Hamilton-Jacobi equation for the non-stationary spherically symmetric gravitational field background. Furthermore, the quantum tunneling of a charged spherically symmetric Kinnersly black hole is investigated by using the Hamilton-Jacobi equation. The result shows that the Hamilton-Jacobi equation is helpful to understand the thermodynamic properties and the radiation characteristics of a black hole. (orig.)
Transient behavior in absorptive optical bistability by the Hamilton-Jacobi method
Sarkar, S.; Satchell, J. S.
1986-04-01
One-, two-, and five-dimensional Fokker-Planck equations for absorptive bistability are solved with use of small-noise asymptotic expansions, which are different from Gaussian linearized analysis. The cases studied are the bifurcation point for the start of hysteresis, where there is critical slowing down and the fluctuations are large, and the evolution of a steady-state distribution when the input field has a step change. The time evolution of the probability distribution is calculated.
Propagation of singularities for Schr\\"odinger equations with modestly long range type potentials
2013-01-01
In a previous paper by the second author, we discussed a characterization of the microlocal singularities for solutions to Schr\\"odinger equations with long range type perturbations, using solutions to a Hamilton-Jacobi equation. In this paper we show that we may use Dollard type approximate solutions to the Hamilton-Jacobi equation if the perturbation satisfies somewhat stronger conditions. As applications, we describe the propagation of microlocal singularities for $e^{itH_0}e^{-itH}$ when ...
Guibout, Vincent M.
This dissertation has been motivated by the need for new methods to address complex problems that arise in spacecraft formation design. As a direct result of this motivation, a general methodology for solving two-point boundary value problems for Hamiltonian systems has been found. Using the Hamilton-Jacobi theory in conjunction with the canonical transformation induced by the phase flow, it is shown that generating functions solve two-point boundary value problems. Traditional techniques for addressing these problems are iterative and require an initial guess. The method presented in this dissertation solves boundary value problems at the cost of a single function evaluation, although it requires knowledge of at least one generating function. Properties of this method are presented. Specifically, we show that it includes perturbation theory and generalizes it to nonlinear systems. Most importantly, it predicts the existence of multiple solutions and allows one to recover all of these solutions. To demonstrate the efficiency of this approach, an algorithm for computing the generating functions is proposed and its convergence properties are studied. As the method developed in this work is based on the Hamiltonian structure of the problem, particular attention must be paid to the numerics of the algorithm. To address this, a general framework for studying the discretization of certain dynamical systems is developed. This framework generalizes earlier work on discretization of Lagrangian and Hamiltonian systems on tangent and cotangent bundles respectively. In addition, it provides new insights into some symplectic integrators and leads to a new discrete Hamilton-Jacobi theory. Most importantly, it allows one to discretize optimal control problems. In particular, a discrete maximum principle is presented. This dissertation also investigates applications of the proposed method to solve two-point boundary value problems. In particular, new techniques for designing
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.
Almost Periodic Viscosity Solutions of Nonlinear Parabolic Equations
Zhang Shilin
2009-01-01
Full Text Available We generalize the comparison result 2007 on Hamilton-Jacobi equations to nonlinear parabolic equations, then by using Perron's method to study the existence and uniqueness of time almost periodic viscosity solutions of nonlinear parabolic equations under usual hypotheses.
Da Lio, Francesca; 10.1137/S0363012904440897
2010-01-01
In this paper, we prove a comparison result between semicontinuous viscosity sub and supersolutions growing at most quadratically of second-order degenerate parabolic Hamilton-Jacobi-Bellman and Isaacs equations. As an application, we characterize the value function of a finite horizon stochastic control problem with unbounded controls as the unique viscosity solution of the corresponding dynamic programming equation.
Time-Inconsistent Optimal Control Problems and the Equilibrium HJB Equation
Yong, Jiongmin
2012-01-01
A general time-inconsistent optimal control problem is considered for stochastic differential equations with deterministic coefficients. Under suitable conditions, a Hamilton-Jacobi-Bellman type equation is derived for the equilibrium value function of the problem. Well-posedness and some properties of such an equation is studied, and time-consistent equilibrium strategies are constructed. As special cases, the linear-quadratic problem and a generalized Merton's portfolio problem are investigated.
GUSTAVO ARAÚJO DE ANDRADE
2014-01-01
Neste trabalho o principal objetivo é apresentar o desenvolvimento de algoritmos de aprendizagem para execução online para a solução da equação algébrica de Hamilton-Jacobi-Bellman. Os conceitos abordados se concentram no desenvolvimento da metodologia para sistemas de controle, por meio de técnicas que tem como objetivo o projeto online de controladores adaptativos são projetados para rejeitar ruídos de sensores, variações paramétricas e erros de modelagem. Conceitos de programação neurodinâ...
Hamilton-Jacobi Method and Gravitation
Di Criscienzo, R.; Vanzo, L.; Zerbini, S.
2010-01-01
Studying the behaviour of a quantum field in a classical, curved, spacetime is an extraordinary task which nobody is able to take on at present time. Independently by the fact that such problem is not likely to be solved soon, still we possess the instruments to perform exact predictions in special, highly symmetric, conditions. Aim of the present contribution is to show how it is possible to extract quantitative information about a variety of physical phenomena in very general situations by ...
Cagnetti, Filippo
2015-01-01
We investigate large-time asymptotics for viscous Hamilton-Jacobi equations with possibly degenerate diffusion terms. We establish new results on the convergence, which are the first general ones concerning equations which are neither uniformly parabolic nor first order. Our method is based on the nonlinear adjoint method and the derivation of new estimates on long time averaging effects. It also extends to the case of weakly coupled systems.
Partial differential equations
Evans, Lawrence C
2010-01-01
This text gives a comprehensive survey of modern techniques in the theoretical study of partial differential equations (PDEs) with particular emphasis on nonlinear equations. The exposition is divided into three parts: representation formulas for solutions; theory for linear partial differential equations; and theory for nonlinear partial differential equations. Included are complete treatments of the method of characteristics; energy methods within Sobolev spaces; regularity for second-order elliptic, parabolic, and hyperbolic equations; maximum principles; the multidimensional calculus of variations; viscosity solutions of Hamilton-Jacobi equations; shock waves and entropy criteria for conservation laws; and, much more.The author summarizes the relevant mathematics required to understand current research in PDEs, especially nonlinear PDEs. While he has reworked and simplified much of the classical theory (particularly the method of characteristics), he primarily emphasizes the modern interplay between funct...
The geometry of ordinary variational equations
Krupková, Olga
1997-01-01
The book provides a comprehensive theory of ODE which come as Euler-Lagrange equations from generally higher-order Lagrangians. Emphasis is laid on applying methods from differential geometry (fibered manifolds and their jet-prolongations) and global analysis (distributions and exterior differential systems). Lagrangian and Hamiltonian dynamics, Hamilton-Jacobi theory, etc., for any Lagrangian system of any order are presented. The key idea - to build up these theories as related with the class of equivalent Lagrangians - distinguishes this book from other texts on higher-order mechanics. The reader should be familiar with elements of differential geometry, global analysis and the calculus of variations.
Espen R. Jakobsen
2002-05-01
Full Text Available Using the maximum principle for semicontinuous functions [3,4], we prove a general ``continuous dependence on the nonlinearities'' estimate for bounded Holder continuous viscosity solutions of fully nonlinear degenerate elliptic equations. Furthermore, we provide existence, uniqueness, and Holder continuity results for bounded viscosity solutions of such equations. Our results are general enough to encompass Hamilton-Jacobi-Bellman-Isaacs's equations of zero-sum, two-player stochastic differential games. An immediate consequence of the results obtained herein is a rate of convergence for the vanishing viscosity method for fully nonlinear degenerate elliptic equations.
Luo, Songting; Payne, Nicholas
2017-07-01
We present an effective asymptotic method for approximating the density of particles for kinetic equations with a Bhatnagar-Gross-Krook (BGK) relaxation operator in the large scale hyperbolic limit. The density of particles is transformed via a Hopf-Cole transformation, where the phase function is expanded as a power series with respect to the Knudsen number. The expansion terms can be determined by solving a sequence of equations. In particular, it has been proved in [3] that the leading order term is the viscosity solution of an effective Hamilton-Jacobi equation, and we show that the higher order terms can be formally determined by solving a sequence of transport equations. Both the effective Hamilton-Jacobi equation and the transport equations are independent of the Knudsen number, and are formulated in the physical space, where the effective Hamiltonian is obtained as the solution of a nonlinear equation that is given as an integral in the velocity variable, and the coefficients of the transport equations are given as integrals in the velocity variable. With appropriate Gauss quadrature rules for evaluating these integrals effectively, the effective Hamilton-Jacobi equation and the transport equations can be solved efficiently to obtain the expansion terms for approximating the density function. In this work, the zeroth, first and second order terms in the expansion are used to obtain second order accuracy with respect to the Knudsen number. The proposed method balances efficiency and accuracy, and has the potential to deal with kinetic equations with more general BGK models. Numerical experiments verify the effectiveness of the proposed method.
Paul, Wolfgang; Koeppe, Jeanette [Institut fuer Physik, Martin Luther Universitaet, 06099 Halle (Germany); Grecksch, Wilfried [Institut fuer Mathematik, Martin Luther Universitaet, 06099 Halle (Germany)
2016-07-01
The standard approach to solve a non-relativistic quantum problem is through analytical or numerical solution of the Schroedinger equation. We show a way to go around it. This way is based on the derivation of the Schroedinger equation from conservative diffusion processes and the establishment of (several) stochastic variational principles leading to the Schroedinger equation under the assumption of a kinematics described by Nelson's diffusion processes. Mathematically, the variational principle can be considered as a stochastic optimal control problem linked to the forward-backward stochastic differential equations of Nelson's stochastic mechanics. The Hamilton-Jacobi-Bellmann equation of this control problem is the Schroedinger equation. We present the mathematical background and how to turn it into a numerical scheme for analyzing a quantum system without using the Schroedinger equation and exemplify the approach for a simple 1d problem.
First-order partial differential equations in classical dynamics
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.
Vedenyapin, V. V.; Negmatov, M. A.; Fimin, N. N.
2017-06-01
We give a derivation of the Vlasov-Maxwell and Vlasov-Poisson-Poisson equations from the Lagrangians of classical electrodynamics. The equations of electromagnetic hydrodynamics (EMHD) and electrostatics with gravitation are derived from them by means of a `hydrodynamical' substitution. We obtain and compare the Lagrange identities for various types of Vlasov equations and EMHD equations. We discuss the advantages of writing the EMHD equations in Godunov's double divergence form. We analyze stationary solutions of the Vlasov-Poisson-Poisson equation, which give rise to non-linear elliptic equations with various properties and various kinds of behaviour of the trajectories of particles as the mass passes through a critical value. We show that the classical equations can be derived from the Liouville equation by the Hamilton-Jacobi method and give an analogue of this procedure for the Vlasov equation as well as in the non-Hamiltonian case.
Non-Hamiltonian systems separable by Hamilton Jacobi method
Marciniak, Krzysztof; Błaszak, Maciej
2008-05-01
We show that with every separable classical Stäckel system of Benenti type on a Riemannian space one can associate, by a proper deformation of the metric tensor, a multi-parameter family of non-Hamiltonian systems on the same space, sharing the same trajectories and related to the seed system by appropriate reciprocal transformations. These systems are known as bi-cofactor systems and are integrable in quadratures as the seed Hamiltonian system is. We show that with each class of bi-cofactor systems a pair of separation curves can be related. We also investigate the conditions under which a given flat bi-cofactor system can be deformed to a family of geodesically equivalent flat bi-cofactor systems.
Dirac Mass Dynamics in Multidimensional Nonlocal Parabolic Equations
Lorz, Alexander
2011-01-17
Nonlocal Lotka-Volterra models have the property that solutions concentrate as Dirac masses in the limit of small diffusion. Is it possible to describe the dynamics of the limiting concentration points and of the weights of the Dirac masses? What is the long time asymptotics of these Dirac masses? Can several Dirac masses coexist? We will explain how these questions relate to the so-called "constrained Hamilton-Jacobi equation" and how a form of canonical equation can be established. This equation has been established assuming smoothness. Here we build a framework where smooth solutions exist and thus the full theory can be developed rigorously. We also show that our form of canonical equation comes with a kind of Lyapunov functional. Numerical simulations show that the trajectories can exhibit unexpected dynamics well explained by this equation. Our motivation comes from population adaptive evolution a branch of mathematical ecology which models Darwinian evolution. © Taylor & Francis Group, LLC.
Dirac mass dynamics in a multidimensional nonlocal parabolic equation
Lorz, Alexander; Perthame, Benoit
2010-01-01
Nonlocal Lotka-Volterra models have the property that solutions concentrate as Dirac masses in the limit of small diffusion. Is it possible to describe the dynamics of the limiting concentration points and of the weights of the Dirac masses? What is the long time asymptotics of these Dirac masses? Can several Dirac masses co-exist? We will explain how these questions relate to the so-called "constrained Hamilton-Jacobi equation" and how a form of canonical equation can be established. This equation has been established assuming smoothness. Here we build a framework where smooth solutions exist and thus the full theory can be developed rigorously. We also show that our form of canonical equation comes with a structure of gradient flow. Numerical simulations show that the trajectories can exhibit unexpected dynamics well explained by this equation. Our motivation comes from population adaptive evolution a branch of mathematical ecology which models darwinian evolution.
Philip Broadbridge
2012-11-01
Full Text Available Olver and Rosenau studied group-invariant solutions of (generally nonlinear partial differential equations through the imposition of a side condition. We apply a similar idea to the special case of finite-dimensional Hamiltonian systems, namely Hamilton-Jacobi, Helmholtz and time-independent Schrödinger equations with potential on N-dimensional Riemannian and pseudo-Riemannian manifolds, but with a linear side condition, where more structure is available. We show that the requirement of N−1 commuting second-order symmetry operators, modulo a second-order linear side condition corresponds to nonregular separation of variables in an orthogonal coordinate system, characterized by a generalized Stäckel matrix. The coordinates and solutions obtainable through true nonregular separation are distinct from those arising through regular separation of variables. We develop the theory for these systems and provide examples.
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.
Addendum to "Phenomenological Derivation of the Schrödinger Equation"
Ogiba F.
2014-04-01
Full Text Available This addendum to the article [1] is crucial for understandin g how the complex effective action, despite its derivation based on classical concepts , prevents quantal particles to move along extreme action trajectories. The reason relates to homogenous, isotropic and unpredictable impulses received from the environment. These random impulses al- lied to natural obedience to the dynamical principle imply t hat such particles are perma- nently and randomly passing from an extreme action trajecto ry to another; all of them belonging to the ensemble given by the stochastic Hamilton- Jacobi-Bohm equation. Also, to correct a wrong interpretation concerning energy c onservation, it is shown that the remaining energy due to these permanent particle-mediu m interactions (absorption- emission phenomena is the so-called quantum potential.
On Gaussian Beams Described by Jacobi's Equation
Smith, Steven Thomas
2013-01-01
Gaussian beams describe the amplitude and phase of rays and are widely used to model acoustic propagation. This paper describes four new results in the theory of Gaussian beams. (1) It is shown that the \\v{C}erven\\'y equations for the amplitude and phase are equivalent to the classical Jacobi Equation of differential geometry. The \\v{C}erven\\'y equations describe Gaussian beams using Hamilton-Jacobi theory, whereas the Jacobi Equation expresses how Gaussian and Riemannian curvature determine geodesic flow on a Riemannian manifold. Thus the paper makes a fundamental connection between Gaussian beams and an acoustic channel's so-called intrinsic Gaussian curvature from differential geometry. (2) A new formula $\\pi(c/c")^{1/2}$ for the distance between convergence zones is derived and applied to several well-known profiles. (3) A class of "model spaces" are introduced that connect the acoustics of ducting/divergence zones with the channel's Gaussian curvature $K=cc"-(c')^2$. The "model" SSPs yield constant Gauss...
Efficient Traveltime Solutions of the TI Acoustic Eikonal Equation
Waheed, Umair bin
2014-10-22
Numerical solutions of the eikonal (Hamilton-Jacobi) equation for transversely isotropic (TI) media are essential for integral imaging and traveltime tomography applications. Such solutions, however, suffer from the inherent higher-order nonlinearity of the TI eikonal equation, which requires solving a quartic polynomial at each computational step. Using perturbation theory, we approximate the first-order discretized form of the TI eikonal equation with a series of simpler equations for the coefficients of a polynomial expansion of the eikonal solution in terms of the anellipticity anisotropy parameter. Such perturbation, applied to the discretized form of the eikonal equation, does not impose any restrictions on the complexity of the perturbed parameter field. Therefore, it provides accurate traveltime solutions even for the anisotropic Marmousi model, with complex distribution of velocity and anellipticity anisotropy parameter. The formulation allows tremendous cost reduction compared to using the exact TI eikonal solver. Furthermore, comparative tests with previously developed approximations illustrate remarkable gain in accuracy of the proposed approximation, without any addition to the computational cost.
Efficient traveltime solutions of the acoustic TI eikonal equation
Waheed, Umair bin
2015-02-01
Numerical solutions of the eikonal (Hamilton-Jacobi) equation for transversely isotropic (TI) media are essential for imaging and traveltime tomography applications. Such solutions, however, suffer from the inherent higher-order nonlinearity of the TI eikonal equation, which requires solving a quartic polynomial for every grid point. Analytical solutions of the quartic polynomial yield numerically unstable formulations. Thus, it requires a numerical root finding algorithm, adding significantly to the computational load. Using perturbation theory we approximate, in a first order discretized form, the TI eikonal equation with a series of simpler equations for the coefficients of a polynomial expansion of the eikonal solution, in terms of the anellipticity anisotropy parameter. Such perturbation, applied to the discretized form of the eikonal equation, does not impose any restrictions on the complexity of the perturbed parameter field. Therefore, it provides accurate traveltime solutions even for models with complex distribution of velocity and anisotropic anellipticity parameter, such as that for the complicated Marmousi model. The formulation allows for large cost reduction compared to using the direct TI eikonal solver. Furthermore, comparative tests with previously developed approximations illustrate remarkable gain in accuracy in the proposed algorithm, without any addition to the computational cost.
Mild solutions of semilinear elliptic equations in Hilbert spaces
Federico, Salvatore; Gozzi, Fausto
2017-03-01
This paper extends the theory of regular solutions (C1 in a suitable sense) for a class of semilinear elliptic equations in Hilbert spaces. The notion of regularity is based on the concept of G-derivative, which is introduced and discussed. A result of existence and uniqueness of solutions is stated and proved under the assumption that the transition semigroup associated to the linear part of the equation has a smoothing property, that is, it maps continuous functions into G-differentiable ones. The validity of this smoothing assumption is fully discussed for the case of the Ornstein-Uhlenbeck transition semigroup and for the case of invertible diffusion coefficient covering cases not previously addressed by the literature. It is shown that the results apply to Hamilton-Jacobi-Bellman (HJB) equations associated to infinite horizon optimal stochastic control problems in infinite dimension and that, in particular, they cover examples of optimal boundary control of the heat equation that were not treatable with the approaches developed in the literature up to now.
Properties-preserving high order numerical methods for a kinetic eikonal equation
Luo, Songting; Payne, Nicholas
2017-02-01
For the BGK (Bhatnagar-Gross-Krook) equation in the large scale hyperbolic limit, the density of particles can be transformed as the Hopf-Cole transformation, where the phase function converges uniformly to the viscosity solution of an effective Hamilton-Jacobi equation, referred to as the kinetic eikonal equation. In this work, we present efficient high order finite difference methods for numerically solving the kinetic eikonal equation. The methods are based on monotone schemes such as the Godunov scheme. High order weighted essentially non-oscillatory techniques and Runge-Kutta procedures are used to obtain high order accuracy in both space and time. The effective Hamiltonian is determined implicitly by a nonlinear equation given as integrals with respect to the velocity variable. Newton's method is applied to solve the nonlinear equation, where integrals with respect to the velocity variable are evaluated either by a Gauss quadrature formula or as expansions with respect to moments of the Maxwellian. The methods are designed such that several key properties such as the positivity of the viscosity solution and the positivity of the effective Hamiltonian are preserved. Numerical experiments are presented to demonstrate the effectiveness of the methods.
First-arrival Tomography Using the Double-square-root Equation Solver Stepping in Subsurface Offset
Serdyukov, A.S.
2013-01-01
Double-square-root (DSR) equation can be viewed as a Hamilton-Jacobi equation describing kinematics of downward data continuation in depth. It describes simultaneous propagation of source and receiver rays assuming that they are nowhere horizontal. Thus it is not suitable for describing diving waves. This equation can be rewritten in a new form when stepping is made in subsurface offset instead of depth. In this form it can be used for describing traveltimes of diving waves in prestack seismic data. This equation can be solved using WENO-RK numerical scheme. Prestack traveltimes (for multiple sources) can be computed in one run thus speeding up solution of the forward problem. We derive linearized version of this new DSR equation that can be used for tomographic inversion of first-arrival traveltimes. Here we used a ray-based tomographic inversion consisting of the following steps: get numerical solution of the offset DSR equation in the background velocity model, back trace DSR rays connecting receivers to sources, update velocity model using truncated SVD pseudoinverse. This approach was tested on a synthetic model generating diving waves.
Band theory in the context of the Hamilton-Jacobi formulation
Bouda, A
2006-01-01
In the one-dimensional periodic potential case, we formulate the condition of Bloch periodicity for the reduced action by using the relation between the wave function and the reduced action established in the context of the equivalence postulate of quantum mechanics. Then, without appealing to the wave function properties, we reproduce the well-known dispersion relations which predict the band structure for the energy spectrum in the Kr\\"onig-Penney model.
Hawking radiation of Schwarzschild-de Sitter black hole by Hamilton-Jacobi method
Rahman, M. Atiqur, E-mail: atirubd@yahoo.com [Department of Applied Mathematics, Rajshahi University (Bangladesh); Hossain, M. Ilias, E-mail: ilias_math@yahoo.com [Department of Mathematics, Rajshahi University, Rajshahi, 6205 (Bangladesh)
2012-05-30
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.
Hawking radiation of Schwarzschild-de Sitter black hole by Hamilton-Jacobi method
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.
Hawking Radiation of Schwarzschild-de Sitter Black Hole by Hamilton-Jacobi method
Rahman, M Atiqur; 10.1016/j.physletb.2012.04.049
2012-01-01
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\\rq s opinion and gives a correction to the Hawking radiation of SdS black hole.
Stochastic Differential Games with Reflection and Related Obstacle Problems for Isaacs Equations
Rainer BUCKDAHN; Juan LI
2011-01-01
In this paper we first investigate zero-sum two-player stochastic differential games with reflection,with the help of theory of Reflected Backward Stochastic Differential Equations (RBSDEs).We will establish the dynamic programming principle for the upper and the lower value functions of this kind of stochastic differential games with reflection in a straightforward way.Then the upper and the lower value functions are proved to be the unique viscosity solutions to the associated upper and the lower Hamilton-Jacobi-Bellman-Isaacs equations with obstacles,respectively.The method differs significantly from those used for control problems with reflection,with new techniques developed of interest on its own.Further,we also prove a new estimate for RBSDEs being sharper than that in the paper of El Karoui,Kapoudjian,Pardoux,Peng and Quenez (1997),which turns out to be very useful because it allows us to estimate the Lp-distance of the solutions of two different RBSDEs by the p-th power of the distance of the initial values of the driving forward equations.We also show that the unique viscosity solution to the approximating Isaacs equation constructed by the penalization method converges to the viscosity solution of the Isaacs equation with obstacle.
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.
Ling Huang; Chi-Wang Shu; Mengping Zhang
2008-01-01
High order fast sweeping methods have been developed recently in the literature to solve static Hamilton-Jacobi equations efficiently. Comparing with the first order fast sweeping methods, the high order fast sweeping methods are more accurate, but they often require additional numerical boundary treatment for several grid points near the boundary because of the wider numerical stencil. It is particularly important to treat the points near the inflow boundary accurately, as the information would flow into the computational domain and would affect global accuracy. In the literature, the numerical solution at these boundary points are either fixed with the exact solution, which is not always feasible, or computed with a first order discretization, which could reduce the global accuracy. In this paper, we discuss two strategies to handle the inflow boundary conditions. One is based on the numerical solutions of a first order fast sweeping method with several different mesh sizes near the boundary and a Richardson extrapolation, the other is based on a Lax-Wendroff type procedure to repeatedly utilizing the PDE to write the normal spatial derivatives to the inflow boundary in terms of the tangential derivatives, thereby obtaining high order solution values at the grid points near the inflow boundary. We explore these two approaches using the fast sweeping high order WENO scheme in [18] for solving the static Eikonal equation as a representative example. Numerical examples are given to demonstrate the performance of these two approaches.
Shaojun Xia, Lingen Chen, Fengrui Sun
2012-01-01
Full Text Available A multistage endoreversible Carnot heat engine system operating with a finite thermal capacity high-temperature black photon fluid reservoir and the heat transfer law is investigated in this paper. Optimal control theory is applied to derive the continuous Hamilton-Jacobi-Bellman (HJB equations, which determine the optimal fluid temperature configurations for maximum power output under the conditions of fixed initial time and fixed initial temperature of the driving fluid. Based on the general optimization results, the analytical solution for the case with pseudo-Newtonian heat transfer law is further obtained. Since there are no analytical solutions for the radiative heat transfer law, the continuous HJB equations are discretized and the dynamic programming (DP algorithm is adopted to obtain the complete numerical solutions, and the relationships among the maximum power output of the system, the process period and the fluid temperatures are discussed in detail. The optimization results obtained for the radiative heat transfer law are also compared with those obtained for pseudo-Newtonian heat transfer law and stage-by-stage optimization strategy. The obtained results can provide some theoretical guidelines for the optimal designs and operations of solar energy conversion and transfer systems.
Xinzhi Liu
1998-01-01
Full Text Available This paper studies a class of high order delay partial differential equations. Employing high order delay differential inequalities, several oscillation criteria are established for such equations subject to two different boundary conditions. Two examples are also given.
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.
Optimal portfolio strategies under a shortfall constraint
Hamilton-Jacobi-Bellman equation to insert the constraint into the resolution ... A numerical method is applied to obtain an approximate solution to the ... risk measure, has emerged as an industry standard with regulatory authorities, such as.
A Student's Guide to Lagrangians and Hamiltonians
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.
Han, Song; Zhang, Wei; Zhang, Jie
2017-09-01
A fast sweeping method (FSM) determines the first arrival traveltimes of seismic waves by sweeping the velocity model in different directions meanwhile applying a local solver. It is an efficient way to numerically solve Hamilton-Jacobi equations for traveltime calculations. In this study, we develop an improved FSM to calculate the first arrival traveltimes of quasi-P (qP) waves in 2-D tilted transversely isotropic (TTI) media. A local solver utilizes the coupled slowness surface of qP and quasi-SV (qSV) waves to form a quartic equation, and solve it numerically to obtain possible traveltimes of qP-wave. The proposed quartic solver utilizes Fermat's principle to limit the range of the possible solution, then uses the bisection procedure to efficiently determine the real roots. With causality enforced during sweepings, our FSM converges fast in a few iterations, and the exact number depending on the complexity of the velocity model. To improve the accuracy, we employ high-order finite difference schemes and derive the second-order formulae. There is no weak anisotropy assumption, and no approximation is made to the complex slowness surface of qP-wave. In comparison to the traveltimes calculated by a horizontal slowness shooting method, the validity and accuracy of our FSM is demonstrated.
Davies, Ian M.; Truman, Aubrey; Zhao, Huaizhong
2005-04-01
We study the inviscid limit, μ →0, of the stochastic viscous Burgers equation, for the velocity field vμ(x,t), t >0, x εRd, (∂vμ/∂t)+(vμ.∇)vμ=-∇c(x,t)-ε∇k(x,t)Ẇt+(μ2/2)Δvμ, for small ε, with vμ(x,0)≡∇S0(x) for some given S0, Ẇt representing white noise. Here we use the Hopf-Cole transformation, vμ=-μ2∇lnuμ, where uμ satisfies the stochastic heat equation of Stratonovich-type and the Feynmac-Kac Truman-Zhao formula for uμ, where dutμ(x )=[(μ2/2)Δutμ(x)+μ-2c(x,t)utμ(x)]dt+εμ-2k(x,t)utμ(x)∘dWt, with u0μ(x)=T0(x)exp(-S0(x)/μ2), S0 as before and T0 a smooth positive function. In an earlier paper, Davies, Truman, and Zhao [J. Math. Phys. 43, 3293 (2002)], an exact solution of the stochastic viscous Burgers equation was used to show how the formal "blow-up" of the Burgers velocity field occurs on random shockwaves for the vμ =0 solution of Burgers equation coinciding with the caustics of a corresponding Hamiltonian system with classical flow map Φ. Moreover, the uμ =0 solution of the stochastic heat equation has its wavefront determined by the behavior of the Hamilton principal function of the corresponding stochastic mechanics. This led in particular to the level surface of the minimizing Hamilton-Jacobi function developing cusps at points corresponding to points of intersection of the corresponding prelevel surface with the precaustic, "pre" denoting the preimage under Φ determined algebraically. These results were primarily of a geometrical nature. In this paper we consider small ε and derive the shape of the random shockwave for the inviscid limit of the stochastic Burgers velocity field and also give the equation determining the random wavefront for the stochastic heat equation both correct to first order in ε. In the case c (x,t)=1/2xTΩ2x, ∇k(x,t)=-a(t), we obtain the exact random shockwave and prove that its shape is unchanged by the addition of noise, it merely being displaced by a random Brownian vector
Cosymplectic and contact structures for time-dependent and dissipative Hamiltonian systems
de León, M.; Sardón, C.
2017-06-01
In this paper, we apply the geometric Hamilton-Jacobi theory to obtain solutions of classical hamiltonian systems that are either compatible with a cosymplectic or a contact structure. As it is well known, the first structure plays a central role in the theory of time-dependent hamiltonians, whilst the second is here used to treat classical hamiltonians including dissipation terms. The interest of a geometric Hamilton-Jacobi equation is the primordial observation that if a hamiltonian vector field X H can be projected into a configuration manifold by means of a 1-form dW , then the integral curves of the projected vector field X_HdW can be transformed into integral curves of X H provided that W is a solution of the Hamilton-Jacobi equation. In this way, we use the geometric Hamilton-Jacobi theory to derive solutions of physical systems with a time-dependent hamiltonian formulation or including dissipative terms. Explicit, new expressions for a geometric Hamilton-Jacobi equation are obtained on a cosymplectic and a contact manifold. These equations are later used to solve physical examples containing explicit time dependence, as it is the case of a unidimensional trigonometric system, and two dimensional nonlinear oscillators as Winternitz-Smorodinsky oscillators and for explicit dissipative behavior, we solve the example of a unidimensional damped oscillator.
Faraggi, Alon E
2013-01-01
The equivalence postulate of quantum mechanics offers an axiomatic approach to quantum field theories and quantum gravity. The equivalence hypothesis can be viewed as adaptation of the classical Hamilton-Jacobi formalism to quantum mechanics. The construction reveals two key identities that underly the formalism in Euclidean or Minkowski spaces. The first is a cocycle condition, which is invariant under $D$--dimensional Mobius transformations with Euclidean or Minkowski metrics. The second is a quadratic identity which is a representation of the D-dimensional quantum Hamilton--Jacobi equation. In this approach, the solutions of the associated Schrodinger equation are used to solve the nonlinear quantum Hamilton-Jacobi equation. A basic property of the construction is that the two solutions of the corresponding Schrodinger equation must be retained. The quantum potential, which arises in the formalism, can be interpreted as a curvature term. I propose that the quantum potential, which is always non-trivial and...
Optimal Consumption in a Stochastic Ramsey Model with Cobb-Douglas Production Function
Md. Azizul Baten
2013-01-01
Full Text Available A stochastic Ramsey model is studied with the Cobb-Douglas production function maximizing the expected discounted utility of consumption. We transformed the Hamilton-Jacobi-Bellman (HJB equation associated with the stochastic Ramsey model so as to transform the dimension of the state space by changing the variables. By the viscosity solution method, we established the existence of viscosity solution of the transformed Hamilton-Jacobi-Bellman equation associated with this model. Finally, the optimal consumption policy is derived from the optimality conditions in the HJB equation.
ShuZheng Yang
2016-01-01
Full Text Available Based on semiclassical tunneling method, we focus on charged fermions tunneling from higher-dimensional Reissner-Nordström black hole. We first simplify the Dirac equation by semiclassical approximation, and then a semiclassical Hamilton-Jacobi equation is obtained. Using the Hamilton-Jacobi equation, we study the Hawking temperature and fermions tunneling rate at the event horizon of the higher-dimensional Reissner-Nordström black hole space-time. Finally, the correct entropy is calculation by the method beyond semiclassical approximation.
Minimal Length Effects on Tunnelling from Spherically Symmetric Black Holes
Benrong Mu
2015-01-01
Full Text Available We investigate effects of the minimal length on quantum tunnelling from spherically symmetric black holes using the Hamilton-Jacobi method incorporating the minimal length. We first derive the deformed Hamilton-Jacobi equations for scalars and fermions, both of which have the same expressions. The minimal length correction to the Hawking temperature is found to depend on the black hole’s mass and the mass and angular momentum of emitted particles. Finally, we calculate a Schwarzschild black hole's luminosity and find the black hole evaporates to zero mass in infinite time.
刘道军
2014-01-01
讨论带有Born-Infeld作用量的快子场所驱动的暴涨模型.首先给出了快子暴涨方程的Hamilton-Jacobi形式并且考虑如何去解Hamilton-Jacobi方程;然后讨论快子场的宇宙学扰动给出扰动场的模方程;最后,讨论了一个真实的由弦理论得来的滚动快子模型.
1989-07-01
sample paths Track D Room B239 Diffusion and Partial Differential Equations 2:00 - 2:20 p.m. G. Galeazzi Uniqueness of viscosity solutions of Ham...V.S.) OF HAMILTON-JACOBI--BELLMAN (IB) EQUATIONS FOR CONTROLLED DIFFUSIONS ON FINITE-DIMENSIONAL RIEMANNIAN MANIFOLDS WITH BOUNDARY Giuliano Galeazzi
蔚喜军
2001-01-01
In this paper, a numerical method is developed for solvingone-dimensional hy perbolic system of conservation laws by the Taylor-Galerkin finite element method. The scheme is obtained by solving conservation equations associated HamiltonJacobi equations. The scheme has the TVD-like property under the uniform meshes. Numerical examples are given.
Motion of Massive and Massless Test particles in Dyadosphere Geometry
Raychaudhuri, B; Kalam, M; Ghosh, A
2008-01-01
Motion of massive and massless test particle in equilibrium and non-equilibrium case is discussed in a dyadosphere geometry through Hamilton-Jacobi method. Geodesics of particles are discussed through Lagrangian method too. Scalar wave equation for massless particle is analyzed to show the absence of superradiance.
Busseman functions for the N-body problem
Percino, Boris; Morgado, Héctor Sánchez
2013-01-01
We prove that the Busemann function of the parabolic homotetic motion for a minimal central coniguration of the N-body problem is a viscosity solution of the Hamilton-Jacobi equation and that its calibrating curves are asymptotic to the homotetic motion.
SCIENTIFIC PRINCIPLES AND MATHEMATICAL MODELS OF PROCESSES OF MINING
Kriuchkov, Anatolii Ivanovych
2016-01-01
The connection between mathematical models of the mining industry with the basic scientific principles. The method of simulation of random non-stationary processes in the form of a set of Hamilton-Jacobi equations and Fokker-Planck-Kolmogorov using the principle of duality movement of mass in space
Biological evolution model with conditional mutation rates
Saakian, David B.; Ghazaryan, Makar; Bratus, Alexander; Hu, Chin-Kun
2017-05-01
We consider an evolution model, in which the mutation rates depend on the structure of population: the mutation rates from lower populated sequences to higher populated sequences are reduced. We have applied the Hamilton-Jacobi equation method to solve the model and calculate the mean fitness. We have found that the modulated mutation rates, directed to increase the mean fitness.
Hamiltonian approach to GR - Part 1: covariant theory of classical gravity
Cremaschini, Claudio
2016-01-01
A challenging issue in General Relativity concerns the determination of the manifestly-covariant continuum Hamiltonian structure underlying the Einstein field equations and the related formulation of the corresponding covariant Hamilton-Jacobi theory. The task is achieved by adopting a synchronous variational principle requiring distinction between the prescribed deterministic metric tensor $\\hat{g}(r)\\equiv \\left\\{ \\hat{g}_{\\mu \
Anisotropic cosmology in S\\'aez-Ballester theory: classical and quantum solutions
Socorro, J; G., M A Sánchez; Palos, M G Frías
2010-01-01
We use the S\\'aez-Ballester theory on anisotropic Bianchi I cosmological model, with barotropic fluid and cosmological constant. We obtain the classical solution by using the Hamilton-Jacobi approach. Also the quantum regime is constructed and exact solutions to the Wheeler-DeWitt equation are found.
High-Order Methods for Computational Physics
1999-03-01
307-344, 1982. 14. Z. Chen, B. Cockburn, C. Gardner, and J. Jerome. Quantum hydrodynamic simulation of hysteresis in the resonant tunneling diode. J...Hamilton-Jacobi equations, SIAM Journal on Numerical Analysis, v28 (1991), pp.907-922. 80. C.S. Peskin , Numerical analysis of blood flow in the heart
Gomes, Diogo A.
2014-01-06
In this talk we will report on new results concerning the existence of smooth solutions for time dependent mean-field games. This new result is established through a combination of various tools including several a-priori estimates for time-dependent mean-field games combined with new techniques for the regularity of Hamilton-Jacobi equations.
Greenwood, Donald T
1997-01-01
Graduate-level text for science and technology students provides strong background in the more abstract and intellectually satisfying areas of dynamical theory. Topics include d'Alembert's principle and the idea of virtual work, Hamilton's equations, Hamilton-Jacobi theory, canonical transformations, more. Problems and references at chapter ends.
Action Quantization, Energy Quantization, and Time Parametrization
Floyd, Edward R.
2017-03-01
The additional information within a Hamilton-Jacobi representation of quantum mechanics is extra, in general, to the Schrödinger representation. This additional information specifies the microstate of ψ that is incorporated into the quantum reduced action, W. Non-physical solutions of the quantum stationary Hamilton-Jacobi equation for energies that are not Hamiltonian eigenvalues are examined to establish Lipschitz continuity of the quantum reduced action and conjugate momentum. Milne quantization renders the eigenvalue J. Eigenvalues J and E mutually imply each other. Jacobi's theorem generates a microstate-dependent time parametrization t-τ =partial _E W even where energy, E, and action variable, J, are quantized eigenvalues. Substantiating examples are examined in a Hamilton-Jacobi representation including the linear harmonic oscillator numerically and the square well in closed form. Two byproducts are developed. First, the monotonic behavior of W is shown to ease numerical and analytic computations. Second, a Hamilton-Jacobi representation, quantum trajectories, is shown to develop the standard energy quantization formulas of wave mechanics.
Hamiltonian approach to GR - Part 2: covariant theory of quantum gravity
Cremaschini, Claudio
2016-01-01
A non-perturbative quantum field theory of General Relativity is presented which leads to a new realization of the theory of Covariant Quantum-Gravity (CQG-theory). The treatment is founded on the recently-identified Hamiltonian structure associated with the classical space-time, i.e., the corresponding manifestly-covariant Hamilton equations and the related Hamilton-Jacobi theory. As shown here the connection with CQG-theory is achieved via the classical GR Hamilton-Jacobi equation, leading to the realization of the CQG-wave equation in 4-scalar form for the corresponding CQG-state and wave-function. The new quantum wave equation exhibits well-known formal properties characteristic of quantum mechanics, including the validity of quantum hydrodynamic equations and suitably-generalized Heisenberg inequalities. In addition, it recovers the classical GR equations in the semiclassical limit, while admitting the possibility of developing further perturbative approximation schemes. Applications of the theory are po...
The CEV Model and Its Application in a Study of Optimal Investment Strategy
Aiyin Wang
2014-01-01
Full Text Available The constant elasticity of variance (CEV model is used to describe the price of the risky asset. Maximizing the expected utility relating to the Hamilton-Jacobi-Bellman (HJB equation which describes the optimal investment strategies, we obtain a partial differential equation. Applying the Legendre transform, we transform the equation into a dual problem and obtain an approximation solution and an optimal investment strategies for the exponential utility function.
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
Quantum mechanics from an equivalence principle
Faraggi, A.E. [Univ. of Florida, Gainesville, FL (United States). Inst. for Fundamental Theory; Matone, M. [Univ. of Padova (Italy)
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.
Conformal invariance and particle aspects in general relativity
Salehi, H; Darabi, F
2000-01-01
We study the breakdown of conformal symmetry in a conformally invariantgravitational model. The symmetry breaking is introduced by defining apreferred conformal frame in terms of the large scale characteristics of theuniverse. In this context we show that a local change of the preferredconformal frame results in a Hamilton-Jacobi equation describing a particlewith adjustable mass. In this equation the dynamical characteristics of theparticle substantially depends on the applied conformal factor and localgeometry. Relevant interpretations of the results are also discussed.
Stochastic control of Itô-Lévy processes with applications to finance
Øksendal, Bernt; Sulem, Agnès
2014-01-01
We give a short introduction to the stochastic calculus for Itô-Lévy processes and review briefly the two main methods of optimal control of systems described by such processes: (i) Dynamic programming and the Hamilton-Jacobi-Bellman (HJB) equation (ii) The stochastic maximum principle and its associated backward stochastic differential equation (BSDE). The two methods are illustrated by application to the classical portfolio optimization problem in finance. A second application is t...
Canepa, Edward S.
2012-09-01
This article presents a new mixed integer programming formulation of the traffic density estimation problem in highways modeled by the Lighthill Whitham Richards equation. We first present an equivalent formulation of the problem using an Hamilton-Jacobi equation. Then, using a semi-analytic formula, we show that the model constraints resulting from the Hamilton-Jacobi equation result in linear constraints, albeit with unknown integers. We then pose the problem of estimating the density at the initial time given incomplete and inaccurate traffic data as a Mixed Integer Program. We then present a numerical implementation of the method using experimental flow and probe data obtained during Mobile Century experiment. © 2012 IEEE.
Reconstruction of boundary conditions from internal conditions using viability theory
Hofleitner, Aude
2012-06-01
This article presents a method for reconstructing downstream boundary conditions to a HamiltonJacobi partial differential equation for which initial and upstream boundary conditions are prescribed as piecewise affine functions and an internal condition is prescribed as an affine function. Based on viability theory, we reconstruct the downstream boundary condition such that the solution of the Hamilton-Jacobi equation with the prescribed initial and upstream conditions and reconstructed downstream boundary condition satisfies the internal value condition. This work has important applications for estimation in flow networks with unknown capacity reductions. It is applied to urban traffic, to reconstruct signal timings and temporary capacity reductions at intersections, using Lagrangian sensing such as GPS devices onboard vehicles.
CONTROL ÓPTIMO INVERSO PARA SISTEMAS NO LINEALES EN TIEMPO CONTINUO
Carlos Vega-Pérez
2014-01-01
Full Text Available Background: Optimization theory applied to automatic control allows governing actions reaching desired conditions but minimizing a given performance index. Such optimization tasks imply to solve complicated mathematical expressions. The inverse optimal control appears as alternative to find the optimal control law without the explicit solution for the Hamilton-Jacobi-Bellman equation. Objective: To show the potential of the inverse optimal control for solving complex optimization problems in control theory. Methods: A general description of the optimal control problem is performed, followed by the justification of an inverse optimal approach. Examples for illustration are properly selected. Results: Mathematical formulations given are applied to solve analytically the cases of a linear optimal quadratic regulator (LQR and a nonlinear inverse optimal Lyapunov-based control problem (CLF. Conclusion: It is possible to solve optimal control problems for nonlinear systems, without explicitly facing the Hamilton-Jacobi-Bellman equation, by means of the inverse optimal control approach.
Computation of invariant tori in 2 1/2 degrees of freedom
Gabella, W.E. (Colorado Univ., Boulder, CO (USA). Dept. of Physics); Ruth, R.D.; Warnock, R.L. (Stanford Linear Accelerator Center, Menlo Park, CA (USA))
1991-01-01
Approximate invariant tori in phase space are found using a non-perturbative, numerical solution of the Hamilton-Jacobi equation for a nonlinear, time-periodic Hamiltonian. The Hamiltonian is written in the action-angle variables of its solvable part. The solution of the Hamilton-Jacobi equation is represented as a Fourier series in the angle variables but not in the time' variable. The Fourier coefficients of the solution are regarded as the fixed point of a nonlinear map. The fixed point is found using a simple iteration or a Newton-Broyden iteration. The Newton-Broyden method is slower than the simple iteration, but it yields solutions at amplitudes that are significant compared to the dynamic aperture.' Invariant tori are found for the dynamics of a charged particle in a storage ring with sextupole magnets. 10 refs., 3 figs., 3 tabs.
Fedkiw, R P
2003-01-01
In this paper we review the algorithm development and applications in high resolution shock capturing methods, level set methods, and PDE based methods in computer vision and image processing. The emphasis is on Stanley Osher's contribution in these areas and the impact of his work. We will start with shock capturing methods and will review the Engquist-Osher scheme, TVD schemes, entropy conditions, ENO and WENO schemes, and numerical schemes for Hamilton-Jacobi type equations. Among level set methods we will review level set calculus, numerical techniques, fluids and materials, variational approach, high codimension motion, geometric optics, and the computation of discontinuous solutions to Hamilton-Jacobi equations. Among computer vision and image processing we will review the total variation model for image denoising, images on implicit surfaces, and the level set method in image processing and computer vision.
Chang, Shuhua; Wang, Xinyu; Wang, Zheng
2015-01-01
Transboundary industrial pollution requires international actions to control its formation and effects. In this paper, we present a stochastic differential game to model the transboundary industrial pollution problems with emission permits trading. More generally, the process of emission permits price is assumed to be stochastic and to follow a geometric Brownian motion (GBM). We make use of stochastic optimal control theory to derive the system of Hamilton-Jacobi-Bellman (HJB) equations sati...
$\\psi$ = W e$^{\\pm\\phi}$ quantum cosmological solutions for Class A Bianchi models
Obregón, O
1995-01-01
We find solutions for quantum Class A Bianchi models of the form \\rm \\Psi=W\\, e^{\\pm \\Phi} generalizing the results obtained by Moncrief and Ryan in standard quantum cosmology. For the II and IX Bianchi models there are other solutions \\rm \\tilde\\Phi_2, \\rm \\tilde\\Phi_9 to the Hamilton-Jacobi equation for which \\rm \\Psi is necessarely zero, in contrast with solutions found in supersymmetric quantum cosmology.
Feedback Control Method Using Haar Wavelet Operational Matrices for Solving Optimal Control Problems
Waleeda Swaidan; Amran Hussin
2013-01-01
Most of the direct methods solve optimal control problems with nonlinear programming solver. In this paper we propose a novel feedback control method for solving for solving affine control system, with quadratic cost functional, which makes use of only linear systems. This method is a numerical technique, which is based on the combination of Haar wavelet collocation method and successive Generalized Hamilton-Jacobi-Bellman equation. We formulate some new Haar wavelet oper...
Chubing Zhang
2013-01-01
Full Text Available We study the optimal investment strategies of DC pension, with the stochastic interest rate (including the CIR model and the Vasicek model and stochastic salary. In our model, the plan member is allowed to invest in a risk-free asset, a zero-coupon bond, and a single risky asset. By applying the Hamilton-Jacobi-Bellman equation, Legendre transform, and dual theory, we find the explicit solutions for the CRRA and CARA utility functions, respectively.
Optimal Control of a PEM Fuel Cell for the Inputs Minimization
José de Jesús Rubio
2014-01-01
Full Text Available The trajectory tracking problem of a proton exchange membrane (PEM fuel cell is considered. To solve this problem, an optimal controller is proposed. The optimal technique has the objective that the system states should reach the desired trajectories while the inputs are minimized. The proposed controller uses the Hamilton-Jacobi-Bellman method where its Riccati equation is considered as an adaptive function. The effectiveness of the proposed technique is verified by two simulations.
High-Accurate, Physics-Based Wake Simulation Techniques
2015-01-27
the development of this method the Gauss-Lobatto rule was applied since the Gauss rule does not include the boundary points thus a Jacobi - Lobato...Taylor and Francis, second ed., 2004. [24] Yan, J., and Osher, S., ”A local discontinuous Galerkin method for directly solving Hamilton- Jacobi ...code was developed that utilizes the discontinuous Galerkin method to solve the Euler equations while utilizing a modal artificial viscosity sensor
Formation of the remnant close to Planck scale and the Schwarzschild black hole with global monopole
Li, Hui-Ling; Chen, Shuai-Ru
2017-10-01
In this paper, we use the generalized uncertainty principle (GUP) and quantum tunneling method to research the formation of the remnant from a Schwarzschild black hole with global monopole. Based on the corrected Hamilton-Jacobi equation, the corrections to the Hawking temperature, heat capacity and entropy are calculated. We not only find the remnant close to Planck scale by employing GUP, but also research the thermodynamic stability of the black hole remnant according to the phase transition and heat capacity.
Massive vector particles tunneling from Kerr and Kerr-Newman black holes
Li, Xiang-Qian
2015-01-01
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.
Near horizon extremal Myers-Perry black holes and integrability of associated conformal mechanics
Hakobyan, Tigran; Nersessian, Armen; Sheikh-Jabbari, M. M.
2017-09-01
We investigate dynamics of probe particles moving in the near-horizon limit of (2 N + 1)-dimensional extremal Myers-Perry black hole with arbitrary rotation parameters. We observe that in the most general case with non-equal non-vanishing rotational parameters the system admits separation of variables in N-dimensional ellipsoidal coordinates. This allows us to find solution of the corresponding Hamilton-Jacobi equation and write down the explicit expressions of Liouville constants of motion.
Separation of variables on a non-hyperelliptic curve
Marikhin, V. G.; Sokolov, V. V.
2004-01-01
In the paper we consider several dynamical systems that admit a separation of variables on the algebraic curve of genus 4. The main result of the paper is an explicit formula for the action of these systems. We find it directly from the Hamilton-Jacobi equation. We find the action and a separation of variables for the Kowalewski hyrostat, the Clebsch and the so(4) Schottky-Manakov spinning tops.
Hyperbolicity and Exponential Convergence of the Lax Oleinik Semigroup for Time Periodic Lagrangians
Morgado, Héctor Sánchez
2012-01-01
We prove exponential convergence to time-periodic states of the solutions of time-periodic Hamilton-Jacobi equations on the torus, assuming that the Aubry set is the union of a finite number of hyperbolic periodic orbits of the the Euler Lagrange flow. The period of limiting solutions is the least common multiple of the periods of the orbits in the Aubry set. This extends a result that we obtained in the autonomous case.
The speed of reaction-diffusion wavefronts in nonsteady media
Mendez, Vicenc [Departament de Medicina, Facultat de Ciencies de la Salut, Universitat Internacional de Catalunya. c/Gomera s/n, 08190-Sant Cugat del Valles (Barcelona) (Spain); Fort, Joaquim [Departament de Fisica, Universitat de Girona, Campus Montilivi, 17071 Girona, Catalonia (Spain); Pujol, Toni [Departament de Fisica, Universitat de Girona, Campus Montilivi, 17071 Girona, Catalonia (Spain)
2003-04-11
The evolution of the speed of wavefronts for reaction-diffusion equations with time-varying parameters is analysed. We make use of singular perturbative analysis to study the temporal evolution of the speed for pushed fronts. The analogy with Hamilton-Jacobi dynamics allows us to consider the problem for pulled fronts, which is described by Kolmogorov-Petrovskii-Piskunov (KPP) reaction kinetics. Both analytical studies are in good agreement with the results of numerical solutions.
Hawking Radiation of Spin-1 Particles From Three Dimensional Rotating Hairy Black Hole
Sakalli, I
2015-01-01
In the present article, we study the Hawking radiation (HR) of spin-1 particles -- so-called vector particles -- from a three dimensional (3D) rotating black hole with scalar hair (RBHWSH) using Hamilton-Jacobi (HJ) ansatz. Putting the Proca equation amalgamated with the WKB approximation in process, the tunneling spectrum of vector particles is obtained. We recover the standard Hawking temperature corresponding to the emission of these particles from RBHWSH.
The speed of reaction-diffusion wavefronts in nonsteady media
Méndez, V; Pujol, T
2003-01-01
The evolution of the speed of wavefronts for reaction-diffusion equations with time-varying parameters is analysed. We make use of singular perturbative analysis to study the temporal evolution of the speed for pushed fronts. The analogy with Hamilton-Jacobi dynamics allows us to consider the problem for pulled fronts, which is described by Kolmogorov-Petrovskii-Piskunov (KPP) reaction kinetics. Both analytical studies are in good agreement with the results of numerical solutions.
Symplectic maps for accelerator lattices
Warnock, R.L.; Ruth, R.; Gabella, W.
1988-05-01
We describe a method for numerical construction of a symplectic map for particle propagation in a general accelerator lattice. The generating function of the map is obtained by integrating the Hamilton-Jacobi equation as an initial-value problem on a finite time interval. Given the generating function, the map is put in explicit form by means of a Fourier inversion technique. We give an example which suggests that the method has promise. 9 refs., 9 figs.
Mean field games systems of first order
Cardaliaguet, Pierre; Graber, Philip Jameson
2014-01-01
International audience; We consider a system of mean field games with local coupling in the deterministic limit. Under general structure conditions on the Hamiltonian and coupling, we prove existence and uniqueness of the weak solution, characterizing this solution as the minimizer of some optimal control of Hamilton-Jacobi and continuity equations. We also prove that this solution converges in the long time average to the solution of the associated ergodic problem.
On an optimal control design for Roessler system
Rafikov, Marat [Universidade Regional do Noroeste do Estado do Rio Grande do Sul, 98700-000 Ijui, RS (Brazil)]. E-mail: rafikov@admijui.unijui.tche.br; Balthazar, Jose Manoel [Universidade Estadual Paulista, C.P. 178, 13500-230 Rio Claro, SP (Brazil)
2004-12-06
In this Letter, an optimal control strategy that directs the chaotic motion of the Roessler system to any desired fixed point is proposed. The chaos control problem is then formulated as being an infinite horizon optimal control nonlinear problem that was reduced to a solution of the associated Hamilton-Jacobi-Bellman equation. We obtained its solution among the correspondent Lyapunov functions of the considered dynamical system.
郭淑妹; 郭杰; 张宁
2013-01-01
In this paper, the authors consider the optimal investment of the ruin probability with perturbed diffusion and dividends. The insurance company can invest the surplus in risky asset and no risky asset. We consider the optimization problem of minimizing the probability of ruin. By solving the corresponding Hamilton-Jacobi-Bellman equations, the optimal strategies and an upper bound of Lundberg for the minimal ruin probability are obtained.%研究了带干扰和支付红利的经典风险模型,在保险公司对外投资风险资产和无风险资产时,通过求解相应的Hamilton-Jacobi-Bellman方程,得到破产概率最小的最优投资比例以及最小破产概率的Lundberg上界.
Hawking Radiation of Mass Generating Particles From Dyonic Reissner Nordstrom Black Hole
Sakalli, I
2016-01-01
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 analyze 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-Nordstr\\"{o}m black hole as a test background. Comparing the tunneling probability obtained with the Boltzmann formula, we succeed to read the standard Hawking temperature of the dyonic Reissner-Nordstr\\"{o}m black hole.
Hawking Radiation of Mass Generating Particles from Dyonic Reissner–Nordström Black Hole
I. Sakalli; A Ovgun
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.
Magnus, Wilhelm
2004-01-01
The hundreds of applications of Hill's equation in engineering and physics range from mechanics and astronomy to electric circuits, electric conductivity of metals, and the theory of the cyclotron. New applications are continually being discovered and theoretical advances made since Liapounoff established the equation's fundamental importance for stability problems in 1907. Brief but thorough, this volume offers engineers and mathematicians a complete orientation to the subject.""Hill's equation"" connotes the class of homogeneous, linear, second order differential equations with real, period
Moiseiwitsch, B L
2005-01-01
Two distinct but related approaches hold the solutions to many mathematical problems--the forms of expression known as differential and integral equations. The method employed by the integral equation approach specifically includes the boundary conditions, which confers a valuable advantage. In addition, the integral equation approach leads naturally to the solution of the problem--under suitable conditions--in the form of an infinite series.Geared toward upper-level undergraduate students, this text focuses chiefly upon linear integral equations. It begins with a straightforward account, acco
Lloyd K. Williams
1987-01-01
Full Text Available In this paper we find closed form solutions of some Riccati equations. Attention is restricted to the scalar as opposed to the matrix case. However, the ones considered have important applications to mathematics and the sciences, mostly in the form of the linear second-order ordinary differential equations which are solved herewith.
Prentis, Jeffrey J.
1996-05-01
One of the most challenging goals of a physics teacher is to help students see that the equations of physics are connected to each other, and that they logically unfold from a small number of basic ideas. Derivations contain the vital information on this connective structure. In a traditional physics course, there are many problem-solving exercises, but few, if any, derivation exercises. Creating an equation poem is an exercise to help students see the unity of the equations of physics, rather than their diversity. An equation poem is a highly refined and eloquent set of symbolic statements that captures the essence of the derivation of an equation. Such a poetic derivation is uncluttered by the extraneous details that tend to distract a student from understanding the essential physics of the long, formal derivation.
Young, C.W. [Applied Research Associates, Inc., Albuquerque, NM (United States)
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.
Tricomi, FG
2013-01-01
Based on his extensive experience as an educator, F. G. Tricomi wrote this practical and concise teaching text to offer a clear idea of the problems and methods of the theory of differential equations. The treatment is geared toward advanced undergraduates and graduate students and addresses only questions that can be resolved with rigor and simplicity.Starting with a consideration of the existence and uniqueness theorem, the text advances to the behavior of the characteristics of a first-order equation, boundary problems for second-order linear equations, asymptotic methods, and diff
2015-01-01
Matlab for simplicity. 6.1 Hamilton-Jacobi Ray Tracing approximation The Hamilton-Jacobi ray tracing or geometric optics approximation is valid...SCATTERING SIMULATIONS: RAY TRACING ..................................... 56 6.0 6.1 Hamilton-Jacobi Ray Tracing approximation...58 6.3 Integration of Ray Tracing Algorithm with Vortex Solutions ............................................ 60 6.4 Fokker-Planck
Barbu, Viorel
2016-01-01
This textbook is a comprehensive treatment of ordinary differential equations, concisely presenting basic and essential results in a rigorous manner. Including various examples from physics, mechanics, natural sciences, engineering and automatic theory, Differential Equations is a bridge between the abstract theory of differential equations and applied systems theory. Particular attention is given to the existence and uniqueness of the Cauchy problem, linear differential systems, stability theory and applications to first-order partial differential equations. Upper undergraduate students and researchers in applied mathematics and systems theory with a background in advanced calculus will find this book particularly useful. Supplementary topics are covered in an appendix enabling the book to be completely self-contained.
The Discontinuous Galerkin Method for the Multiscale Modeling of Dynamics of Crystalline Solids
2007-08-26
can be used to provide the equation of state, and therefore close the continuum equations . 3 Numerical methodology 3.1 Macroscale solver: the DG method...Hamilton-Jacobi equations . In the standard RKDG method, we seek the solution in the finite dimen- sional polynomial space V̄ 7,k h = { v = (v1, v2, v3...Comput. Methods. Appl. Mech. En- grg., 193(17–20):1645–1669, 2004. [49] J. Yan and C.-W. Shu. A local discontinuous Galerkin method for KdV type equations
Lorentz Invariance Violation and Modified Hawking Fermions Tunneling Radiation
Shu-Zheng Yang
2016-01-01
Full Text Available Recently the modified Dirac equation with Lorentz invariance violation has been proposed, which would be helpful to resolve some issues in quantum gravity theory and high energy physics. In this paper, the modified Dirac equation has been generalized in curved spacetime, and then fermion tunneling of black holes is researched under this correctional Dirac field theory. We also use semiclassical approximation method to get correctional Hamilton-Jacobi equation, so that the correctional Hawking temperature and correctional black hole’s entropy are derived.
Li, Juan
2012-01-01
In this paper we study the optimal stochastic control problem for stochastic differential systems reflected in a domain. The cost functional is a recursive one, which is defined via generalized backward stochastic differential equations developed by Pardoux and Zhang [17]. The value function is shown to be the viscosity solution to the associated Hamilton-Jacobi-Bellman equation, which is a fully nonlinear parabolic partial differential equation with a nonlinear Neumann boundary condition. The method of stochastic "backward semigroups" introduced by Peng [18] is adapted to our context.
Dynamic optimization and its relation to classical and quantum constrained systems
Contreras, Mauricio; Pellicer, Rely; Villena, Marcelo
2017-08-01
We study the structure of a simple dynamic optimization problem consisting of one state and one control variable, from a physicist's point of view. By using an analogy to a physical model, we study this system in the classical and quantum frameworks. Classically, the dynamic optimization problem is equivalent to a classical mechanics constrained system, so we must use the Dirac method to analyze it in a correct way. We find that there are two second-class constraints in the model: one fix the momenta associated with the control variables, and the other is a reminder of the optimal control law. The dynamic evolution of this constrained system is given by the Dirac's bracket of the canonical variables with the Hamiltonian. This dynamic results to be identical to the unconstrained one given by the Pontryagin equations, which are the correct classical equations of motion for our physical optimization problem. In the same Pontryagin scheme, by imposing a closed-loop λ-strategy, the optimality condition for the action gives a consistency relation, which is associated to the Hamilton-Jacobi-Bellman equation of the dynamic programming method. A similar result is achieved by quantizing the classical model. By setting the wave function Ψ(x , t) =e iS(x , t) in the quantum Schrödinger equation, a non-linear partial equation is obtained for the S function. For the right-hand side quantization, this is the Hamilton-Jacobi-Bellman equation, when S(x , t) is identified with the optimal value function. Thus, the Hamilton-Jacobi-Bellman equation in Bellman's maximum principle, can be interpreted as the quantum approach of the optimization problem.
Covariance, Curved Space, Motion and Quantization
Apostol M.
2008-01-01
Full Text Available Weak external forces and non-inertial motion are equivalent with thefree motion in a curved space. The Hamilton-Jacobi equation is derivedfor such motion and the effects of the curvature upon the quantizationare analyzed, starting from a generalization of the Klein-Gordon equation in curved spaces. It is shown that the quantization is actually destroyed, in general, by a non-inertial motion in the presence of external forces, in the sense that such a motion may produce quantum transitions. Examples are given for a massive scalar field and for photons.
OPTIMAL PORTFOLIO ON TRACKING THE EXPECTED WEALTH PROCESS WITH LIQUIDITY CONSTRAINTS
Luo Kui; Wang Guangming; Hu Yijun
2011-01-01
In this article, the authors consider the optimal portfolio on tracking the expected wealth process with liquidity constraints. The constrained optimal portfolio is first formulated as minimizing the cumulate variance between the wealth process and the expected wealth process. Then, the dynamic programming methodology is applied to reduce the whole problem to solving the Hamilton-Jacobi-Bellman equation coupled with the liquidity constraint, and the method of Lagrange multiplier is applied to handle the constraint. Finally, a numerical method is proposed to solve the constrained HJB equation and the constrained optimal strategy. Especially, the explicit solution to this optimal problem is derived when there is no liquidity constraint.
Jusufi, Kimet
2016-01-01
In this paper we study the quantum tunneling of charged and magnetized particles (magnetic monopoles) from the global monopole black hole by incorporating the quantum gravity effects. Starting from the modified Maxwell's equations and Generalized Uncertainty Relation (GUP), we recover the GUP corrected temperate for the global monopole black hole by solving the modified Dirac equation via Hamilton-Jacobi method. Furthermore, we also include the quantum corrections beyond the semiclassical approximation, in particular, first we find the logarithmic corrections of GUP corrected entropy and finally we calculate the GUP corrected specific heat capacity.
An approximate atmospheric guidance law for aeroassisted plane change maneuvers
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.
Hawking Radiation of Scalar and Vector Particles From 5D Myers-Perry Black Holes
Jusufi, Kimet
2016-01-01
In the present paper we explore the Hawking radiation as a quantum tunneling effect from a rotating 5 dimensional Myers-Perry (5D-MPBH) black hole with two independent angular momentum componentes. First, we investigate the Hawking temperature by considering the tunneling of massive scalar particles and spin-1 vector particles from the 5D-MPBH in the Painlev\\'e coordinates and then in the corotating frames. More specifically, we solve the Klein-Gordon and Proca equation by applying the WKB method and Hamilton-Jacobi equation in both cases. Finally, we recover the Hawking temperature and show that coordinates systems do not affect the Hawking temperature.
Tunneling of Massive Vector Particles From Rotating Charged Black Strings
Jusufi, Kimet
2015-01-01
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.
Hawking Radiation of Scalar and Vector Particles from 5D Myers-Perry Black Holes
Jusufi, Kimet; Övgün, Ali
2017-06-01
In the present paper we explore the Hawking radiation as a quantum tunneling effect from a rotating 5 dimensional Myers-Perry black hole (5D-MPBH) with two independent angular momentum components. First, we investigate the Hawking temperature by considering the tunneling of massive scalar particles and spin-1 vector particles from the 5D-MPBH in the Painlevé coordinates and then in the corotating frames. More specifically, we solve the Klein-Gordon and Proca equations by applying the WKB method and Hamilton-Jacobi equation in both cases. Finally, we recover the Hawking temperature and show that coordinates systems do not affect the Hawking temperature.
Hawking Radiation of Scalar and Vector Particles from 5D Myers-Perry Black Holes
Jusufi, Kimet; Övgün, Ali
2017-02-01
In the present paper we explore the Hawking radiation as a quantum tunneling effect from a rotating 5 dimensional Myers-Perry black hole (5D-MPBH) with two independent angular momentum components. First, we investigate the Hawking temperature by considering the tunneling of massive scalar particles and spin-1 vector particles from the 5D-MPBH in the Painlevé coordinates and then in the corotating frames. More specifically, we solve the Klein-Gordon and Proca equations by applying the WKB method and Hamilton-Jacobi equation in both cases. Finally, we recover the Hawking temperature and show that coordinates systems do not affect the Hawking temperature.
Time-Dependent Mean-Field Games with Logarithmic Nonlinearities
Gomes, Diogo A.
2015-10-06
In this paper, we prove the existence of classical solutions for time-dependent mean-field games with a logarithmic nonlinearity and subquadratic Hamiltonians. Because the logarithm is unbounded from below, this nonlinearity poses substantial mathematical challenges that have not been addressed in the literature. Our result is proven by recurring to a delicate argument which combines Lipschitz regularity for the Hamilton-Jacobi equation with estimates for the nonlinearity in suitable Lebesgue spaces. Lipschitz estimates follow from an application of the nonlinear adjoint method. These are then combined with a priori bounds for solutions of the Fokker-Planck equation and a concavity argument for the nonlinearity.
Jusufi, Kimet; Apostolovska, Gordana
2016-12-01
In this paper we study the quantum tunneling of Dirac magnetic monopoles from the global monopole black hole under quantum gravity effects. We start from the modified Maxwell's equations and the Generalized Uncertainty Relation (GUP), to recover the GUP corrected temperature for the global monopole black hole by solving the modified Dirac equation via Hamilton-Jacobi method. Furthermore, we also include the quantum corrections beyond the semiclassical approximation, in particular, first we find the logarithmic corrections of GUP corrected entropy and finally we calculate the GUP corrected specific heat capacity. It is argued that the GUP effects may prevent a black hole from complete evaporation and leave remnants.
Tunnelling of scalar and Dirac particles from squashed charged rotating Kaluza-Klein black holes
Stetsko, M.M. [Ivan Franko National University of Lviv, Department of Theoretical Physics, Lviv (Ukraine)
2016-02-15
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. (orig.)
Tunnelling of scalar and Dirac particles from squashed charged rotating Kaluza-Klein black holes
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.
How fast is the wave function collapse?
Ignatiev, A Yu
2012-01-01
Using complex quantum Hamilton-Jacobi formulation, a new kind of non-linear equations is proposed that have almost classical structure and extend the Schroedinger equation to describe the collapse of the wave function as a finite-time process. Experimental bounds on the collapse time are reported (of order 0.1 ms to 0.1 ps) and its convenient dimensionless measure is introduced. This parameter helps to identify the areas where sensitive probes of the possible collapse dynamics can be done. Examples are experiments with Bose-Einstein condensates, ultracold neutrons or ultrafast optics.
Time-Dependent Mean-Field Games in the Subquadratic Case
Gomes, Diogo A.
2014-10-14
In this paper we consider time-dependent mean-field games with subquadratic Hamiltonians and power-like local dependence on the measure. We establish existence of classical solutions under a certain set of conditions depending on both the growth of the Hamiltonian and the dimension. This is done by combining regularity estimates for the Hamilton-Jacobi equation based on the Gagliardo-Nirenberg interpolation inequality with polynomial estimates for the Fokker-Planck equation. This technique improves substantially the previous results on the regularity of time-dependent mean-field games.
Expected Utility Optimization - Calculus of Variations Approach
Tran, Khoa
2007-01-01
In this paper, I'll derive the Hamilton-Jacobi (HJ) equation for Merton's problem in Utility Optimization Theory using a Calculus of Variations (CoV) Approach. For stochastic control problems, Dynamic Programming (DP) has been used as a standard method. To the best of my knowledge, no one has used CoV for this problem. In addition, while the DP approach cannot guarantee that the optimum satisfies the HJ equation, the CoV approach does. Be aware that this is the first draft of this paper and many flaws might be introduced.
Tricomi, Francesco Giacomo
1957-01-01
This classic text on integral equations by the late Professor F. G. Tricomi, of the Mathematics Faculty of the University of Turin, Italy, presents an authoritative, well-written treatment of the subject at the graduate or advanced undergraduate level. To render the book accessible to as wide an audience as possible, the author has kept the mathematical knowledge required on the part of the reader to a minimum; a solid foundation in differential and integral calculus, together with some knowledge of the theory of functions is sufficient. The book is divided into four chapters, with two useful
Stochastic partial differential equations
Chow, Pao-Liu
2014-01-01
Preliminaries Introduction Some Examples Brownian Motions and Martingales Stochastic Integrals Stochastic Differential Equations of Itô Type Lévy Processes and Stochastic IntegralsStochastic Differential Equations of Lévy Type Comments Scalar Equations of First Order Introduction Generalized Itô's Formula Linear Stochastic Equations Quasilinear Equations General Remarks Stochastic Parabolic Equations Introduction Preliminaries Solution of Stochastic Heat EquationLinear Equations with Additive Noise Some Regularity Properties Stochastic Reaction-Diffusion Equations Parabolic Equations with Grad
Hamiltonian approach to GR. Pt. 1. Covariant theory of classical gravity
Cremaschini, Claudio [Silesian University in Opava, Faculty of Philosophy and Science, Institute of Physics and Research Center for Theoretical Physics and Astrophysics, Opava (Czech Republic); Tessarotto, Massimo [University of Trieste, Department of Mathematics and Geosciences, Trieste (Italy); Silesian University in Opava, Faculty of Philosophy and Science, Institute of Physics, Opava (Czech Republic)
2017-05-15
A challenging issue in General Relativity concerns the determination of the manifestly covariant continuum Hamiltonian structure underlying the Einstein field equations and the related formulation of the corresponding covariant Hamilton-Jacobi theory. The task is achieved by adopting a synchronous variational principle requiring distinction between the prescribed deterministic metric tensor g(r) ≡ {g_μ_ν(r)} solution of the Einstein field equations which determines the geometry of the background space-time and suitable variational fields x ≡ {g,π} obeying an appropriate set of continuum Hamilton equations, referred to here as GR-Hamilton equations. It is shown that a prerequisite for reaching such a goal is that of casting the same equations in evolutionary form by means of a Lagrangian parametrization for a suitably reduced canonical state. As a result, the corresponding Hamilton-Jacobi theory is established in manifestly covariant form. Physical implications of the theory are discussed. These include the investigation of the structural stability of the GR-Hamilton equations with respect to vacuum solutions of the Einstein equations, assuming that wave-like perturbations are governed by the canonical evolution equations. (orig.)
Hamiltonian approach to GR. Pt. 2. Covariant theory of quantum gravity
Cremaschini, Claudio [Faculty of Philosophy and Science, Silesian University in Opava, Institute of Physics and Research Center for Theoretical Physics and Astrophysics, Opava (Czech Republic); Tessarotto, Massimo [University of Trieste, Department of Mathematics and Geosciences, Trieste (Italy); Faculty of Philosophy and Science, Silesian University in Opava, Institute of Physics, Opava (Czech Republic)
2017-05-15
A non-perturbative quantum field theory of General Relativity is presented which leads to a new realization of the theory of covariant quantum gravity (CQG-theory). The treatment is founded on the recently identified Hamiltonian structure associated with the classical space-time, i.e., the corresponding manifestly covariant Hamilton equations and the related Hamilton-Jacobi theory. The quantum Hamiltonian operator and the CQG-wave equation for the corresponding CQG-state and wave function are realized in 4-scalar form. The new quantum wave equation is shown to be equivalent to a set of quantum hydrodynamic equations which warrant the consistency with the classical GR Hamilton-Jacobi equation in the semiclassical limit. A perturbative approximation scheme is developed, which permits the adoption of the harmonic oscillator approximation for the treatment of the Hamiltonian potential. As an application of the theory, the stationary vacuum CQG-wave equation is studied, yielding a stationary equation for the CQG-state in terms of the 4-scalar invariant-energy eigenvalue associated with the corresponding approximate quantum Hamiltonian operator. The conditions for the existence of a discrete invariant-energy spectrum are pointed out. This yields a possible estimate for the graviton mass together with a new interpretation about the quantum origin of the cosmological constant. (orig.)
Redundancy of constraints in the classical and quantum theories of gravitation.
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.
Editorial Special issue on approximate dynamic programming and reinforcement learning
Silvia Ferrari; Jagannathan Sarangapani; Frank L. Lewis
2011-01-01
We are extremely pleased to present this special issue of the Journal of Control Theory and Applications.Approximate dynamic programming (ADP) is a general and effective approach for solving optimal control and estimation problems by adapting to uncertain environments over time.ADP optimizes the sensing objectives accrued over a future time interval with respect to an adaptive control law,conditioned on prior knowledge of the system,its state,and uncertainties.A numerical search over the present value of the control minimizes a Hamilton-Jacobi-Bellman (HJB) equation providing a basis for real-time,approximate optimal control.
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.
Mechanics classical and quantum
Taylor, T T
2015-01-01
Mechanics: Classical and Quantum explains the principles of quantum mechanics via the medium of analytical mechanics. The book describes Schrodinger's formulation, the Hamilton-Jacobi equation, and the Lagrangian formulation. The author discusses the Harmonic Oscillator, the generalized coordinates, velocities, as well as the application of the Lagrangian formulation to systems that are partially or entirely electromagnetic in character under certain conditions. The book examines waves on a string under tension, the isothermal cavity radiation, and the Rayleigh-Jeans result pertaining to the e
Cosmic censorship of rotating Anti-de Sitter black hole
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.
Nonsingularity in the no-boundary Universe
吴忠超
1997-01-01
In the no-boundary Universe of Hartle and Hawking, 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.
A Michelson Interferometer in the Field of a Plane Gravitational Wave
Poplawski, N J
2006-01-01
In this paper we treat the problem of a Michelson interferometer in the field of a weak, monochromatic, plane gravitational wave in the framework of the general theory of relativity. The arms of the interferometer are regarded as world lines, whose motion is determined by the equations of geodesics in the Hamilton-Jacobi formalism. We find that interference appears in the second approximation. Moreover, the measurement of the light beam delay between both arms can be used for determining the wavelength of such a wave.
Discrete-time inverse optimal control for nonlinear systems
Sanchez, Edgar N
2013-01-01
Discrete-Time Inverse Optimal Control for Nonlinear Systems proposes a novel inverse optimal control scheme for stabilization and trajectory tracking of discrete-time nonlinear systems. This avoids the need to solve the associated Hamilton-Jacobi-Bellman equation and minimizes a cost functional, resulting in a more efficient controller. Design More Efficient Controllers for Stabilization and Trajectory Tracking of Discrete-Time Nonlinear Systems The book presents two approaches for controller synthesis: the first based on passivity theory and the second on a control Lyapunov function (CLF). Th
Tunneling of Vector Particles from Lorentzian Wormholes in 3+1 Dimensions
Sakalli, I
2015-01-01
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)ans\\"{a}tz 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.
Analytical methods of optimization
Lawden, D F
2006-01-01
Suitable for advanced undergraduates and graduate students, this text surveys the classical theory of the calculus of variations. It takes the approach most appropriate for applications to problems of optimizing the behavior of engineering systems. Two of these problem areas have strongly influenced this presentation: the design of the control systems and the choice of rocket trajectories to be followed by terrestrial and extraterrestrial vehicles.Topics include static systems, control systems, additional constraints, the Hamilton-Jacobi equation, and the accessory optimization problem. Prereq
Eigenvalues of Killing Tensors and Separable Webs on Riemannian and Pseudo-Riemannian Manifolds
Giovanni Rastelli
2007-02-01
Full Text Available Given a $n$-dimensional Riemannian manifold of arbitrary signature, we illustrate an algebraic method for constructing the coordinate webs separating the geodesic Hamilton-Jacobi equation by means of the eigenvalues of $m leq n$ Killing two-tensors. Moreover, from the analysis of the eigenvalues, information about the possible symmetries of the web foliations arises. Three cases are examined: the orthogonal separation, the general separation, including non-orthogonal and isotropic coordinates, and the conformal separation, where Killing tensors are replaced by conformal Killing tensors. The method is illustrated by several examples and an application to the L-systems is provided.
Dirac particles tunneling from black holes with topological defects
Jusufi, Kimet
2015-01-01
We study Hawking radiation of Dirac particles with spin-$1/2$ as a tunneling process from Schwarzschild-de Sitter and Reissner-Nordstr\\"{o}m-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 black holes remains unchanged in presence of topological defects in both cases.
Numerical solution of continuous-time mean-variance portfolio selection with nonlinear constraints
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.
Jun Long
2016-01-01
Full Text Available We analyze a continuous-time model for corporate international investment problem (CIIP with mean-variance criterion. Based on Nash subgame perfect equilibrium theory, we define an infinitesimal operator and directly derive an extended Hamilton-Jacobi-Bellman (HJB equation. Besides, we also obtain the equilibrium time-consistent strategy for CIIP. In addition, we discuss two cases of risk aversion coefficient; one is constant and the other is state dependent. Finally, the simulation results are given to illustrate our conclusions and the influence of some parameters on the optimal solution.
方建印; 慕小武
2005-01-01
In this paper, the nonlinear singular stabilization, H∞ control problem of systems with ordinary homogeneous properties is considered. At first, we discuss the stabilization problems of nonlinear systems with homogeneous. Secondly, by vitue of Hamilton-JacobiIsaacs equations or inequalities, we solve regular H∞ of nonlinear systems with homogeneous properties. To overcome the H∞ problem of singular nonlinear system, we try to transform inputs of the singular nonlinear system into two parts: regular part input and singular part input. Following the previous results, we solve the singular nonlinear system H∞ control,we give the Lyapunov function and the state feedback controller of the singular nonlinear systems with homogeneous properties.
Ergodic optimization in the expanding case concepts, tools and applications
Garibaldi, Eduardo
2017-01-01
This book focuses on the interpretation of ergodic optimal problems as questions of variational dynamics, employing a comparable approach to that of the Aubry-Mather theory for Lagrangian systems. Ergodic optimization is primarily concerned with the study of optimizing probability measures. This work presents and discusses the fundamental concepts of the theory, including the use and relevance of Sub-actions as analogues to subsolutions of the Hamilton-Jacobi equation. Further, it provides evidence for the impressively broad applicability of the tools inspired by the weak KAM theory.
Hawking Radiation of Massive Vector Particles From Warped AdS$_{\\text{3}}$ Black Hole
Gursel, H
2015-01-01
Hawking radiation (HR) of massive vector particles from a rotating Warped Anti-de Sitter black hole in 2+1 dimensions (WAdS$_{\\text{3}}$BH) is studied in detail. The quantum tunneling approach with the Hamilton-Jacobi method (HJM) is applied in the Proca equation (PE), and we show that the radial function yields the tunneling rate of the outgoing particles. Comparing the result obtained with the Boltzmann factor, we satisfactorly reproduce the Hawking temperature (HT) of the WAdS$_{\\text{3}}$BH.
A level set approach for damage identification of continuum structures based on dynamic responses
Zhang, Weisheng; Du, Zongliang; Sun, Guo; Guo, Xu
2017-01-01
The present paper aims to propose a novel approach for damage identification of continuum structures based on their dynamic performances. The main idea is resorting to the level set model, which is used to describe the shape and topology of the damage regions implicitly. The original natural frequency-based inverse problem is thus transferred into a generalized shape optimization problem which can be tackled by solving an evolution type Hamilton-Jacobi equation. Compared to traditional approaches, the distinctive advantage of the proposed approach is that it can deal with damage regions of complex shapes in a convenient way. Numerical examples demonstrate the effectiveness of the proposed approach.
Dividend Maximization when Cash Reserves Follow a Jump-diffusion Process
LI LI-LI; FENG JIN-GHAI; SONG LI-XIN
2009-01-01
This paper deals with the dividend optimization problem for an insur-ance company, whose surplus follows a jump-diffusion process. The objective of the company is to maximize the expected total discounted dividends paid out until the time of ruin. Under concavity assumption on the optimal value function, the paper states some general properties and, in particular, smoothness results on the optimal value function, whose analysis mainly relies on viscosity solutions of the associated Hamilton-Jacobi-Bellman (HJB) equations. Based on these properties, the explicit expression of the optimal value function is obtained. And some numerical calculations are presented as the application of the results.
Finite population size effects in quasispecies models with single-peak fitness landscape
Saakian, David B.; Deem, Michael W.; Hu, Chin-Kun
2012-04-01
We consider finite population size effects for Crow-Kimura and Eigen quasispecies models with single-peak fitness landscape. We formulate accurately the iteration procedure for the finite population models, then derive the Hamilton-Jacobi equation (HJE) to describe the dynamic of the probability distribution. The steady-state solution of HJE gives the variance of the mean fitness. Our results are useful for understanding the population sizes of viruses in which the infinite population models can give reliable results for biological evolution problems.
有限时区非线性系统的最优切换控制%Optimal Switching Control for Nonlinear Systems in A Finite Duration
慕小武; 刘海军
2006-01-01
This paper proposes a optimal control problem for a general nonlinear systems with finitely many admissible control settings and with costs assigned to switching of controls. With dynamic programming and viscosity solution theory we show that the switching lower-value function is a viscosity solution of the appropriate systems of quasi-variational inequalities(the appropriate generalization of the Hamilton-Jacobi equation in this context)and that the minimal such switching-storage function is equal to the continuous switching lower-value for the game. With the lower value function a optimal switching control is designed for minimizing the cost of running the systems.
Optimal control linear quadratic methods
Anderson, Brian D O
2007-01-01
This augmented edition of a respected text teaches the reader how to use linear quadratic Gaussian methods effectively for the design of control systems. It explores linear optimal control theory from an engineering viewpoint, with step-by-step explanations that show clearly how to make practical use of the material.The three-part treatment begins with the basic theory of the linear regulator/tracker for time-invariant and time-varying systems. The Hamilton-Jacobi equation is introduced using the Principle of Optimality, and the infinite-time problem is considered. The second part outlines the
EPR & Klein Paradoxes in Complex Hamiltonian Dynamics and Krein Space Quantization
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
Stochastic Optimal Control Models for Online Stores
Bradonjić, Milan
2011-01-01
We present a model for the optimal design of an online auction/store by a seller. The framework we use is a stochastic optimal control problem. In our setting, the seller wishes to maximize her average wealth level, where she can control her price per unit via her reputation level. The corresponding Hamilton-Jacobi-Bellmann equation is analyzed for an introductory case. We then turn to an empirically justified model, and present introductory analysis. In both cases, {\\em pulsing} advertising strategies are recovered for resource allocation. Further numerical and functional analysis will appear shortly.
Equatorial gravitational lensing by accelerating and rotating black hole with NUT parameter
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.
Time dependent optimal switching controls in online selling models
Bradonjic, Milan [Los Alamos National Laboratory; Cohen, Albert [MICHIGAN STATE UNIV
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.
Polymer quantization and the saddle point approximation of partition functions
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.
Partial Differential Equations
1988-01-01
The volume contains a selection of papers presented at the 7th Symposium on differential geometry and differential equations (DD7) held at the Nankai Institute of Mathematics, Tianjin, China, in 1986. Most of the contributions are original research papers on topics including elliptic equations, hyperbolic equations, evolution equations, non-linear equations from differential geometry and mechanics, micro-local analysis.
Towards Quantum Cybernetics:. Optimal Feedback Control in Quantum Bio Informatics
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.
Path integrals and symmetry breaking for optimal control theory
Kappen, H J
2005-01-01
This paper considers linear-quadratic control of a non-linear dynamical system subject to arbitrary cost. I show that for this class of stochastic control problems the non-linear Hamilton-Jacobi-Bellman equation can be transformed into a linear equation. The transformation is similar to the transformation used to relate the classical Hamilton-Jacobi equation to the Schr\\"odinger equation. As a result of the linearity, the usual backward computation can be replaced by a forward diffusion process, that can be computed by stochastic integration or by the evaluation of a path integral. It is shown, how in the deterministic limit the PMP formalism is recovered. The significance of the path integral approach is that it forms the basis for a number of efficient computational methods, such as MC sampling, the Laplace approximation and the variational approximation. We show the effectiveness of the first two methods in number of examples. Examples are given that show the qualitative difference between stochastic and d...
Duits, Remco; Ghosh, Arpan; Haije, Tom Dela
2011-01-01
In this article we study both left-invariant (convection-)diffusions and left-invariant Hamilton-Jacobi equations on the space SE(3)/({0} \\times SO(2)) of 3D-positions and orientations naturally embedded in the group SE(3) of 3D-rigid body movements. The general motivation for these (convection-)diffusions and erosions is to obtain crossing-preserving fiber enhancement on probability densities defined on the space of positions and orientations. The linear left-invariant (convection-)diffusions are forward Kolmogorov equations of Brownian motions on SE(3)/({0}\\timesSO(2)) and can be solved by convolution with the corresponding Green's functions or by a finite difference scheme. The left-invariant Hamilton-Jacobi equations are Bellman equations of cost processes on SE(3)/({0}\\timesSO(2)) and they are solved by a morphological convolution with the corresponding Green's functions. Furthermore, we consider pseudo-linear scale spaces on the space of positions and orientations that combines dilation and diffusion in...
Discrete-time dynamic graphical games:model-free reinforcement learning solution
Mohammed I ABOUHEAF; Frank L LEWIS; Magdi S MAHMOUD; Dariusz G MIKULSKI
2015-01-01
This paper introduces a model-free reinforcement learning technique that is used to solve a class of dynamic games known as dynamic graphical games. The graphical game results from multi-agent dynamical systems, where pinning control is used to make all the agents synchronize to the state of a command generator or a leader agent. Novel coupled Bellman equations and Hamiltonian functions are developed for the dynamic graphical games. The Hamiltonian mechanics are used to derive the necessary conditions for optimality. The solution for the dynamic graphical game is given in terms of the solution to a set of coupled Hamilton-Jacobi-Bellman equations developed herein. Nash equilibrium solution for the graphical game is given in terms of the solution to the underlying coupled Hamilton-Jacobi-Bellman equations. An online model-free policy iteration algorithm is developed to learn the Nash solution for the dynamic graphical game. This algorithm does not require any knowledge of the agents’ dynamics. A proof of convergence for this multi-agent learning algorithm is given under mild assumption about the inter-connectivity properties of the graph. A gradient descent technique with critic network structures is used to implement the policy iteration algorithm to solve the graphical game online in real-time.
Conceptual design of compliant mechanisms using level set method
Shi-kui CHEN; Michael Yu WANG
2006-01-01
We propose a level set method-based framework for the conceptual design of compliant mechanisms.In this method,the compliant mechanism design problem is recast as an infinite dimensional optimization problem,where the design variable is the geometric shape of the compliant mechanism and the goal is to find a suitable shape in the admissible design space so that the objective functional can reach a minimum.The geometric shape of the compliant mechanism is represented as the zero level set of a one-higher dimensional level set function,and the dynamic variations of the shape are governed by the Hamilton-Jacobi partial differential equation.The application of level set methods endows the optimization process with the particular quality that topological changes of the boundary,such as merging or splitting,can be handled in a natural fashion.By making a connection between the velocity field in the Hamilton-Jacobi partial differential equation with the shape gradient of the objective functional,we go further to transform the optimization problem into that of finding a steady-state solution of the partial differential equation.Besides the above-mentioned methodological issues,some numerical examples together with prototypes are presented to validate the performance of the method.
On Functional and Holographic Renormalization Group Methods in Stochastic Theory of Turbulence
Ogarkov, S L
2016-01-01
A nonlocal quantum-field model is constructed for the system of hydrodynamic equations for incompressible viscous fluid (the stochastic Navier--Stokes (NS) equation and the continuity equation). This model is studied by the following two mutually parallel methods: the Wilson--Polchinski functional renormalization group method (FRG), which is based on the exact functional equation for the generating functional of amputated connected Green's functions (ACGF), and the Heemskerk--Polchinski holographic renormalization group method (HRG), which is based on the functional Hamilton--Jacobi (HJ) equation for the holographic boundary action. Both functional equations are equivalent to infinite hierarchies of integro-differential equations (coupled in the FRG case) for the corresponding families of Green's functions (GF). The RG-flow equations can be derived explicitly for two-particle functions. Because the HRG-flow equation is closed (contains only a two-particle GF), the explicit analytic solutions are obtained for ...
Kinetic energy equations for the average-passage equation system
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.
Kinetic energy equations for the average-passage equation system
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.
Li, Yanning
2013-10-01
This article presents a new robust control framework for transportation problems in which the state is modeled by a first order scalar conservation law. Using an equivalent formulation based on a Hamilton-Jacobi equation, we pose the problem of controlling the state of the system on a network link, using boundary flow control, as a Linear Program. Unlike many previously investigated transportation control schemes, this method yields a globally optimal solution and is capable of handling shocks (i.e. discontinuities in the state of the system). We also demonstrate that the same framework can handle robust control problems, in which the uncontrollable components of the initial and boundary conditions are encoded in intervals on the right hand side of inequalities in the linear program. The lower bound of the interval which defines the smallest feasible solution set is used to solve the robust LP (or MILP if the objective function depends on boolean variables). Since this framework leverages the intrinsic properties of the Hamilton-Jacobi equation used to model the state of the system, it is extremely fast. Several examples are given to demonstrate the performance of the robust control solution and the trade-off between the robustness and the optimality. © 2013 IEEE.
Efficient robust control of first order scalar conservation laws using semi-analytical solutions
Li, Yanning
2014-01-01
This article presents a new robust control framework for transportation problems in which the state is modeled by a first order scalar conservation law. Using an equivalent formulation based on a Hamilton-Jacobi equation, we pose the problem of controlling the state of the system on a network link, using initial density control and boundary flow control, as a Linear Program. We then show that this framework can be extended to arbitrary control problems involving the control of subsets of the initial and boundary conditions. Unlike many previously investigated transportation control schemes, this method yields a globally optimal solution and is capable of handling shocks (i.e. discontinuities in the state of the system). We also demonstrate that the same framework can handle robust control problems, in which the uncontrollable components of the initial and boundary conditions are encoded in intervals on the right hand side of inequalities in the linear program. The lower bound of the interval which defines the smallest feasible solution set is used to solve the robust LP/MILP. Since this framework leverages the intrinsic properties of the Hamilton-Jacobi equation used to model the state of the system, it is extremely fast. Several examples are given to demonstrate the performance of the robust control solution and the trade-off between the robustness and the optimality.
Solving Nonlinear Wave Equations by Elliptic Equation
FU Zun-Tao; LIU Shi-Da; LIU Shi-Kuo
2003-01-01
The elliptic equation is taken as a transformation and applied to solve nonlinear wave equations. It is shown that this method is more powerful to give more kinds of solutions, such as rational solutions, solitary wave solutions,periodic wave solutions and so on, so it can be taken as a generalized method.
Introduction to differential equations
Taylor, Michael E
2011-01-01
The mathematical formulations of problems in physics, economics, biology, and other sciences are usually embodied in differential equations. The analysis of the resulting equations then provides new insight into the original problems. This book describes the tools for performing that analysis. The first chapter treats single differential equations, emphasizing linear and nonlinear first order equations, linear second order equations, and a class of nonlinear second order equations arising from Newton's laws. The first order linear theory starts with a self-contained presentation of the exponen
The Modified Magnetohydrodynamical Equations
EvangelosChaliasos
2003-01-01
After finding the really self-consistent electromagnetic equations for a plasma, we proceed in a similar fashion to find how the magnetohydrodynamical equations have to be modified accordingly. Substantially this is done by replacing the "Lorentz" force equation by the correct (in our case) force equation. Formally we have to use the vector potential instead of the magnetic field intensity. The appearance of the formulae presented is the one of classical vector analysis. We thus find a set of eight equations in eight unknowns, as previously known concerning the traditional MHD equations.
Geometrical Unification of Gravitation and Dark Energy: The Universe as a Relativistic Particle
Hojman, Sergio A; Rubio, Carlos A
2014-01-01
The Lagrangian, the Hamilton--Jacobi equation and the Schr\\"{o}dinger, Dirac and Klein--Gordon equations for the Friedmann--Robertson--Walker--Quintessence (FRWQ) system are presented and solved exactly for different interesting scenarios. The classical Lagrangian reproduces the usual two (second order) dynamical equations for the radius of the Universe and for the scalar field as well as the (first order) constraint equation. The approach naturally unifies gravity and dark energy, which may be related to the tlaplon (scalar torsion potential). The Lagrangian and the equations of motion are those of a relativistic particle moving on a two dimensional spacetime where the conformal metric factor is related to the dark energy scalar field potential. This allows us to quantize the system, obtaining a Klein-Gordon equation when the Universe is considered as a spinless particle, and a Dirac equation when the Universe is thought as a relativistic spin particle.
George F R Ellis
2007-07-01
The Raychaudhuri equation is central to the understanding of gravitational attraction in astrophysics and cosmology, and in particular underlies the famous singularity theorems of general relativity theory. This paper reviews the derivation of the equation, and its significance in cosmology.
Beginning partial differential equations
O'Neil, Peter V
2014-01-01
A broad introduction to PDEs with an emphasis on specialized topics and applications occurring in a variety of fields Featuring a thoroughly revised presentation of topics, Beginning Partial Differential Equations, Third Edition provides a challenging, yet accessible,combination of techniques, applications, and introductory theory on the subjectof partial differential equations. The new edition offers nonstandard coverageon material including Burger's equation, the telegraph equation, damped wavemotion, and the use of characteristics to solve nonhomogeneous problems. The Third Edition is or
Renormalizing Partial Differential Equations
Bricmont, J.; Kupiainen, A.
1994-01-01
In this review paper, we explain how to apply Renormalization Group ideas to the analysis of the long-time asymptotics of solutions of partial differential equations. We illustrate the method on several examples of nonlinear parabolic equations. We discuss many applications, including the stability of profiles and fronts in the Ginzburg-Landau equation, anomalous scaling laws in reaction-diffusion equations, and the shape of a solution near a blow-up point.
Ordinary differential equations
Greenberg, Michael D
2014-01-01
Features a balance between theory, proofs, and examples and provides applications across diverse fields of study Ordinary Differential Equations presents a thorough discussion of first-order differential equations and progresses to equations of higher order. The book transitions smoothly from first-order to higher-order equations, allowing readers to develop a complete understanding of the related theory. Featuring diverse and interesting applications from engineering, bioengineering, ecology, and biology, the book anticipates potential difficulties in understanding the various solution steps
林祥; 李娜
2011-01-01
In this paper,we consider an insurance company whose surplus (reserve) is modeled by a compound Poisson-Geometric risk process perturbed by a diffusion. The insurance company can invest part of the surplus in a risky asset and purchase proportional reinsurance for claims. We consider the optimization problem of minimizing the probability of ruin. By solving the corresponding Hamilton-Jacobi-Bellman equations,explicit expressions for the minimal ruin probability and the corresponding optimal strategies are obtained.%本文对索赔次数为复合Poisson-Geometric过程的风险模型,在保险公司的盈余可以投资于风险资产,以及索赔购买比例再保险的策略下,研究使得破产概率最小的最优投资和再保险策略.通过求解相应的Hamilton-Jacobi-Bellman方程,得到使得破产概率最小的最优投资和比例再保险策略,以及最小破产概率的显示表达式.
Optimal Investment-consumption Policies Selection for Heston Model%Heston模型下最优投资-消费策略选择
杨鹏
2016-01-01
The optimal investment-consumption policies selection problems were studied with stochastic financial market. In stochastic financial market, assets are composed of risk-free and risky asset, and the volatility of the risky asset was described by a Heston model. Optimal investment-consumption policies which maximize the expected discounted utility of terminal wealth and accumulative consumption was found. By solving the corresponding Hamilton-Jacobi-Bellman ( HJB) equation, closed-form solutions for the value function as well as the investment-consumption policies in the power utility function case are ob-tained.%在随机金融市场模型中，研究了最优投资-消费策略选择问题。随机金融市场由无风险资产和风险资产构成，在风险资产的方差满足Heston模型下，求得最优投资-消费策略最大化终端财富和累积消费的期望折现效用。在幂效用函数情形下，通过求解值函数满足的Hamilton-Jacobi-Bellman ( HJB)方程，得到了最优投资-消费策略以及值函数的显式解。
The Modified Magnetohydrodynamical Equations
Evangelos Chaliasos
2003-01-01
After finding the really self-consistent electromagnetic equations for a plasma, we proceed in a similarfashion to find how the magnetohydrodynamical equations have to be modified accordingly. Substantially this is doneby replacing the "Lorentz" force equation by the correct (in our case) force equation. Formally we have to use the vectorpotential instead of the magnetic field intensity. The appearance of the formulae presented is the one of classical vectoranalysis. We thus find a set of eight equations in eight unknowns, as previously known concerning the traditional MHDequations.
Singular stochastic differential equations
Cherny, Alexander S
2005-01-01
The authors introduce, in this research monograph on stochastic differential equations, a class of points termed isolated singular points. Stochastic differential equations possessing such points (called singular stochastic differential equations here) arise often in theory and in applications. However, known conditions for the existence and uniqueness of a solution typically fail for such equations. The book concentrates on the study of the existence, the uniqueness, and, what is most important, on the qualitative behaviour of solutions of singular stochastic differential equations. This is done by providing a qualitative classification of isolated singular points, into 48 possible types.
Fractional Differential Equations
Jianping Zhao
2012-01-01
Full Text Available An extended fractional subequation method is proposed for solving fractional differential equations by introducing a new general ansätz and Bäcklund transformation of the fractional Riccati equation with known solutions. Being concise and straightforward, this method is applied to the space-time fractional coupled Burgers’ equations and coupled MKdV equations. As a result, many exact solutions are obtained. It is shown that the considered method provides a very effective, convenient, and powerful mathematical tool for solving fractional differential equations.
Geometrical vs wave optics under gravitational waves
Angélil, Raymond
2015-01-01
We present some new derivations of the effect of a plane gravitational wave on a light ray. A simple interpretation of the results is that a gravitational wave causes a phase modulation of electromagnetic waves. We arrive at this picture from two contrasting directions, namely null geodesics and Maxwell's equations, or, geometric and wave optics. Under geometric optics, we express the geodesic equations in Hamiltonian form and solve perturbatively for the effect of gravitational waves. We find that the well-known time-delay formula for light generalizes trivially to massive particles. We also recover, by way of a Hamilton-Jacobi equation, the phase modulation obtained under wave optics. Turning then to wave optics, rather than solving Maxwell's equations directly for the fields, as in most previous approaches, we derive a perturbed wave equation (perturbed by the gravitational wave) for the electromagnetic four-potential. From this wave equation it follows that the four-potential and the electric and magnetic...
Differential equations for dummies
Holzner, Steven
2008-01-01
The fun and easy way to understand and solve complex equations Many of the fundamental laws of physics, chemistry, biology, and economics can be formulated as differential equations. This plain-English guide explores the many applications of this mathematical tool and shows how differential equations can help us understand the world around us. Differential Equations For Dummies is the perfect companion for a college differential equations course and is an ideal supplemental resource for other calculus classes as well as science and engineering courses. It offers step-by-step techniques, practical tips, numerous exercises, and clear, concise examples to help readers improve their differential equation-solving skills and boost their test scores.
Wei Khim Ng
2009-02-01
Full Text Available We construct nonlinear extensions of Dirac's relativistic electron equation that preserve its other desirable properties such as locality, separability, conservation of probability and Poincaré invariance. We determine the constraints that the nonlinear term must obey and classify the resultant non-polynomial nonlinearities in a double expansion in the degree of nonlinearity and number of derivatives. We give explicit examples of such nonlinear equations, studying their discrete symmetries and other properties. Motivated by some previously suggested applications we then consider nonlinear terms that simultaneously violate Lorentz covariance and again study various explicit examples. We contrast our equations and construction procedure with others in the literature and also show that our equations are not gauge equivalent to the linear Dirac equation. Finally we outline various physical applications for these equations.
Fractional Chemotaxis Diffusion Equations
Langlands, T A M
2010-01-01
We introduce mesoscopic and macroscopic model equations of chemotaxis with anomalous subdiffusion for modelling chemically directed transport of biological organisms in changing chemical environments with diffusion hindered by traps or macro-molecular crowding. The mesoscopic models are formulated using Continuous Time Random Walk master equations and the macroscopic models are formulated with fractional order differential equations. Different models are proposed depending on the timing of the chemotactic forcing. Generalizations of the models to include linear reaction dynamics are also derived. Finally a Monte Carlo method for simulating anomalous subdiffusion with chemotaxis is introduced and simulation results are compared with numerical solutions of the model equations. The model equations developed here could be used to replace Keller-Segel type equations in biological systems with transport hindered by traps, macro-molecular crowding or other obstacles.
3D shape reconstruction of medical images using a perspective shape-from-shading method
Yang, Lei; Han, Jiu-qiang
2008-06-01
A 3D shape reconstruction approach for medical images using a shape-from-shading (SFS) method was proposed in this paper. A new reflectance map equation of medical images was analyzed with the assumption that the Lambertian reflectance surface was irradiated by a point light source located at the light center and the image was formed under perspective projection. The corresponding static Hamilton-Jacobi (H-J) equation of the reflectance map equation was established. So the shape-from-shading problem turned into solving the viscosity solution of the static H-J equation. Then with the conception of a viscosity vanishing approximation, the Lax-Friedrichs fast sweeping numerical method was used to compute the viscosity solution of the H-J equation and a new iterative SFS algorithm was gained. Finally, experiments on both synthetic images and real medical images were performed to illustrate the efficiency of the proposed SFS method.
Mean-field games with logistic population dynamics
Gomes, Diogo A.
2013-12-01
In its standard form, a mean-field game can be defined by coupled system of equations, a Hamilton-Jacobi equation for the value function of agents and a Fokker-Planck equation for the density of agents. Traditionally, the latter equation is adjoint to the linearization of the former. Since the Fokker-Planck equation models a population dynamic, we introduce natural features such as seeding and birth, and nonlinear death rates. In this paper we analyze a stationary meanfield game in one dimension, illustrating various techniques to obtain regularity of solutions in this class of systems. In particular we consider a logistic-type model for birth and death of the agents which is natural in problems where crowding affects the death rate of the agents. The introduction of these new terms requires a number of new ideas to obtain wellposedness. In a forthcoming publication we will address higher dimensional models. ©2013 IEEE.
Duchkov, Anton A.
2013-09-22
The double-square-root (DSR) equation can be viewed as a Hamilton-Jacobi equation describing kinematics of downward data continuation in depth. It describes simultaneous propagation of source and receiver rays which allows computing reflection wave prestack traveltimes (for multiple sources) in a one run thus speeding up solution of the forward problem. Here we give and overview of different alternative forms of the DSR equation which allows stepping in two-way time and subsurface offset instead of depth. Different forms of the DSR equation are suitable for computing different types of waves including reflected, head and diving waves. We develop a WENO-RK numerical scheme for solving all mentioned forms of the DSR equation. Finally the extended exploding reflector concept can be used for computing prestack traveltimes while initiating the numerical solver as if a reflector was exploding in extended imaging space.
K. Banoo
1998-01-01
equation in the discrete momentum space. This is shown to be similar to the conventional drift-diffusion equation except that it is a more rigorous solution to the Boltzmann equation because the current and carrier densities are resolved into M×1 vectors, where M is the number of modes in the discrete momentum space. The mobility and diffusion coefficient become M×M matrices which connect the M momentum space modes. This approach is demonstrated by simulating electron transport in bulk silicon.
Developmental Partial Differential Equations
Duteil, Nastassia Pouradier; Rossi, Francesco; Boscain, Ugo; Piccoli, Benedetto
2015-01-01
In this paper, we introduce the concept of Developmental Partial Differential Equation (DPDE), which consists of a Partial Differential Equation (PDE) on a time-varying manifold with complete coupling between the PDE and the manifold's evolution. In other words, the manifold's evolution depends on the solution to the PDE, and vice versa the differential operator of the PDE depends on the manifold's geometry. DPDE is used to study a diffusion equation with source on a growing surface whose gro...
Differential equations I essentials
REA, Editors of
2012-01-01
REA's Essentials provide quick and easy access to critical information in a variety of different fields, ranging from the most basic to the most advanced. As its name implies, these concise, comprehensive study guides summarize the essentials of the field covered. Essentials are helpful when preparing for exams, doing homework and will remain a lasting reference source for students, teachers, and professionals. Differential Equations I covers first- and second-order equations, series solutions, higher-order linear equations, and the Laplace transform.
Ordinary differential equations
Pontryagin, Lev Semenovich
1962-01-01
Ordinary Differential Equations presents the study of the system of ordinary differential equations and its applications to engineering. The book is designed to serve as a first course in differential equations. Importance is given to the linear equation with constant coefficients; stability theory; use of matrices and linear algebra; and the introduction to the Lyapunov theory. Engineering problems such as the Watt regulator for a steam engine and the vacuum-tube circuit are also presented. Engineers, mathematicians, and engineering students will find the book invaluable.
Hazewinkel, M.
1995-01-01
Dedication: I dedicate this paper to Prof. P.C. Baayen, at the occasion of his retirement on 20 December 1994. The beautiful equation which forms the subject matter of this paper was invented by Wouthuysen after he retired. The four complex variable Wouthuysen equation arises from an original space-
Shabat, A. B.
2016-12-01
We consider the class of entire functions of exponential type in relation to the scattering theory for the Schrödinger equation with a finite potential that is a finite Borel measure. These functions have a special self-similarity and satisfy q-difference functional equations. We study their asymptotic behavior and the distribution of zeros.
Dissipative Boussinesq equations
Dutykh, D; Dias, Fr\\'{e}d\\'{e}ric; Dutykh, Denys
2007-01-01
The classical theory of water waves is based on the theory of inviscid flows. However it is important to include viscous effects in some applications. Two models are proposed to add dissipative effects in the context of the Boussinesq equations, which include the effects of weak dispersion and nonlinearity in a shallow water framework. The dissipative Boussinesq equations are then integrated numerically.
Hannelore Breckner
2000-01-01
Full Text Available We consider a stochastic equation of Navier-Stokes type containing a noise part given by a stochastic integral with respect to a Wiener process. The purpose of this paper is to approximate the solution of this nonlinear equation by the Galerkin method. We prove the convergence in mean square.
Differential Equation of Equilibrium
user
than the classical method in the solution of the aforementioned differential equation. Keywords: ... present a successful approximation of shell ... displacement function. .... only applicable to cylindrical shell subject to ..... (cos. 4. 4. 4. 3 β. + β. + β. -. = β. - β x x e ex. AL. xA w. Substituting equations (29); (30) and (31) into.
Applied partial differential equations
Logan, J David
2004-01-01
This primer on elementary partial differential equations presents the standard material usually covered in a one-semester, undergraduate course on boundary value problems and PDEs. What makes this book unique is that it is a brief treatment, yet it covers all the major ideas: the wave equation, the diffusion equation, the Laplace equation, and the advection equation on bounded and unbounded domains. Methods include eigenfunction expansions, integral transforms, and characteristics. Mathematical ideas are motivated from physical problems, and the exposition is presented in a concise style accessible to science and engineering students; emphasis is on motivation, concepts, methods, and interpretation, rather than formal theory. This second edition contains new and additional exercises, and it includes a new chapter on the applications of PDEs to biology: age structured models, pattern formation; epidemic wave fronts, and advection-diffusion processes. The student who reads through this book and solves many of t...
William Margulies
2004-11-01
Full Text Available In this paper, we study a specific stochastic differential equation depending on a parameter and obtain a representation of its probability density function in terms of Jacobi Functions. The equation arose in a control problem with a quadratic performance criteria. The quadratic performance is used to eliminate the control in the standard Hamilton-Jacobi variational technique. The resulting stochastic differential equation has a noise amplitude which complicates the solution. We then solve Kolmogorov's partial differential equation for the probability density function by using Jacobi Functions. A particular value of the parameter makes the solution a Martingale and in this case we prove that the solution goes to zero almost surely as time tends to infinity.
Shaolin Ji
2013-01-01
Full Text Available This paper is devoted to a stochastic differential game (SDG of decoupled functional forward-backward stochastic differential equation (FBSDE. For our SDG, the associated upper and lower value functions of the SDG are defined through the solution of controlled functional backward stochastic differential equations (BSDEs. Applying the Girsanov transformation method introduced by Buckdahn and Li (2008, the upper and the lower value functions are shown to be deterministic. We also generalize the Hamilton-Jacobi-Bellman-Isaacs (HJBI equations to the path-dependent ones. By establishing the dynamic programming principal (DPP, we derive that the upper and the lower value functions are the viscosity solutions of the corresponding upper and the lower path-dependent HJBI equations, respectively.
Quantum potential and symmetries in extended phase space
Nasiri, S
2005-01-01
Here, we study the concept of the quantum potential using an extended phase space technique. It seems that, for a given potential, there exist an extended canonical transformation that removes the expression for quantum potential in dynamical equation. The situation, mathematically, is similar to the appearance of centrifugal potential in going from Cartesian to spherical coordinates that changes the physical potential to an effective one. As Examples, the cases of harmonic oscillator, particle in a box and hydrogen atom are worked out, where the quantum potential disappears from the Wigner equation as a possible representation of quantum mechanics in the phase space. This representation that keeps the Hamilton-Jacobi equation form invariant could be obtained by a particular extended canonical transformation on Sobouti-Nasiri equation in extended phase space.
Tunnelling of Massive/Massless Bosons from the Apparent Horizon of FRW Universe
Kimet Jusufi
2017-01-01
Full Text Available We investigate the Hawking radiation of vector particles from the apparent horizon of a Friedmann-Robertson-Walker (FRW universe in the framework of quantum tunnelling method. Furthermore we use Proca equation, a relativistic wave equation for a massive/massless spin-1 particle (massless γ photons, weak massive W± and Z0 bosons, strong massless gluons, and ρ and ω mesons together with a Painlevé space-time metric for the FRW universe. We solve the Proca equation via Hamilton-Jacobi (HJ equation and the WKB approximation method. We recover the same result for the Hawking temperature associated with vector particles as in the case of scalar and Dirac particles tunnelled from outside to the inside of the apparent horizon in a FRW universe.
Moussa Kounta
2016-01-01
Full Text Available We consider the so-called mean-variance portfolio selection problem in continuous time under the constraint that the short-selling of stocks is prohibited where all the market coefficients are random processes. In this situation the Hamilton-Jacobi-Bellman (HJB equation of the value function of the auxiliary problem becomes a coupled system of backward stochastic partial differential equation. In fact, the value function V often does not have the smoothness properties needed to interpret it as a solution to the dynamic programming partial differential equation in the usual (classical sense; however, in such cases V can be interpreted as a viscosity solution. Here we show the unicity of the viscosity solution and we see that the optimal and the value functions are piecewise linear functions based on some Riccati differential equations. In particular we solve the open problem posed by Li and Zhou and Zhou and Yin.
Risk-sensitive mean-field games
Tembine, Hamidou
2014-04-01
In this paper, we study a class of risk-sensitive mean-field stochastic differential games. We show that under appropriate regularity conditions, the mean-field value of the stochastic differential game with exponentiated integral cost functional coincides with the value function satisfying a Hamilton -Jacobi- Bellman (HJB) equation with an additional quadratic term. We provide an explicit solution of the mean-field best response when the instantaneous cost functions are log-quadratic and the state dynamics are affine in the control. An equivalent mean-field risk-neutral problem is formulated and the corresponding mean-field equilibria are characterized in terms of backward-forward macroscopic McKean-Vlasov equations, Fokker-Planck-Kolmogorov equations, and HJB equations. We provide numerical examples on the mean field behavior to illustrate both linear and McKean-Vlasov dynamics. © 1963-2012 IEEE.
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.
Differential equations problem solver
Arterburn, David R
2012-01-01
REA's Problem Solvers is a series of useful, practical, and informative study guides. Each title in the series is complete step-by-step solution guide. The Differential Equations Problem Solver enables students to solve difficult problems by showing them step-by-step solutions to Differential Equations problems. The Problem Solvers cover material ranging from the elementary to the advanced and make excellent review books and textbook companions. They're perfect for undergraduate and graduate studies.The Differential Equations Problem Solver is the perfect resource for any class, any exam, and
Ordinary differential equations
Miller, Richard K
1982-01-01
Ordinary Differential Equations is an outgrowth of courses taught for a number of years at Iowa State University in the mathematics and the electrical engineering departments. It is intended as a text for a first graduate course in differential equations for students in mathematics, engineering, and the sciences. Although differential equations is an old, traditional, and well-established subject, the diverse backgrounds and interests of the students in a typical modern-day course cause problems in the selection and method of presentation of material. In order to compensate for this diversity,
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.
Beginning partial differential equations
O'Neil, Peter V
2011-01-01
A rigorous, yet accessible, introduction to partial differential equations-updated in a valuable new edition Beginning Partial Differential Equations, Second Edition provides a comprehensive introduction to partial differential equations (PDEs) with a special focus on the significance of characteristics, solutions by Fourier series, integrals and transforms, properties and physical interpretations of solutions, and a transition to the modern function space approach to PDEs. With its breadth of coverage, this new edition continues to present a broad introduction to the field, while also addres
Hyperbolic partial differential equations
Witten, Matthew
1986-01-01
Hyperbolic Partial Differential Equations III is a refereed journal issue that explores the applications, theory, and/or applied methods related to hyperbolic partial differential equations, or problems arising out of hyperbolic partial differential equations, in any area of research. This journal issue is interested in all types of articles in terms of review, mini-monograph, standard study, or short communication. Some studies presented in this journal include discretization of ideal fluid dynamics in the Eulerian representation; a Riemann problem in gas dynamics with bifurcation; periodic M
Wu Zhuo Qun; Li Hui Lai; Zhao Jun Ning
2001-01-01
Nonlinear diffusion equations, an important class of parabolic equations, come from a variety of diffusion phenomena which appear widely in nature. They are suggested as mathematical models of physical problems in many fields, such as filtration, phase transition, biochemistry and dynamics of biological groups. In many cases, the equations possess degeneracy or singularity. The appearance of degeneracy or singularity makes the study more involved and challenging. Many new ideas and methods have been developed to overcome the special difficulties caused by the degeneracy and singularity, which
Partial differential equations
Friedman, Avner
2008-01-01
This three-part treatment of partial differential equations focuses on elliptic and evolution equations. Largely self-contained, it concludes with a series of independent topics directly related to the methods and results of the preceding sections that helps introduce readers to advanced topics for further study. Geared toward graduate and postgraduate students of mathematics, this volume also constitutes a valuable reference for mathematicians and mathematical theorists.Starting with the theory of elliptic equations and the solution of the Dirichlet problem, the text develops the theory of we
Introduction to functional equations
Sahoo, Prasanna K
2011-01-01
Introduction to Functional Equations grew out of a set of class notes from an introductory graduate level course at the University of Louisville. This introductory text communicates an elementary exposition of valued functional equations where the unknown functions take on real or complex values. In order to make the presentation as manageable as possible for students from a variety of disciplines, the book chooses not to focus on functional equations where the unknown functions take on values on algebraic structures such as groups, rings, or fields. However, each chapter includes sections hig
Uncertain differential equations
Yao, Kai
2016-01-01
This book introduces readers to the basic concepts of and latest findings in the area of differential equations with uncertain factors. It covers the analytic method and numerical method for solving uncertain differential equations, as well as their applications in the field of finance. Furthermore, the book provides a number of new potential research directions for uncertain differential equation. It will be of interest to researchers, engineers and students in the fields of mathematics, information science, operations research, industrial engineering, computer science, artificial intelligence, automation, economics, and management science.
A Comparison of IRT Equating and Beta 4 Equating.
Kim, Dong-In; Brennan, Robert; Kolen, Michael
Four equating methods were compared using four equating criteria: first-order equity (FOE), second-order equity (SOE), conditional mean squared error (CMSE) difference, and the equipercentile equating property. The four methods were: (1) three parameter logistic (3PL) model true score equating; (2) 3PL observed score equating; (3) beta 4 true…
Applied partial differential equations
Logan, J David
2015-01-01
This text presents the standard material usually covered in a one-semester, undergraduate course on boundary value problems and PDEs. Emphasis is placed on motivation, concepts, methods, and interpretation, rather than on formal theory. The concise treatment of the subject is maintained in this third edition covering all the major ideas: the wave equation, the diffusion equation, the Laplace equation, and the advection equation on bounded and unbounded domains. Methods include eigenfunction expansions, integral transforms, and characteristics. In this third edition, text remains intimately tied to applications in heat transfer, wave motion, biological systems, and a variety other topics in pure and applied science. The text offers flexibility to instructors who, for example, may wish to insert topics from biology or numerical methods at any time in the course. The exposition is presented in a friendly, easy-to-read, style, with mathematical ideas motivated from physical problems. Many exercises and worked e...
Frédéric, Pierret
2014-01-01
The equations of celestial mechanics that govern the variation of the orbital elements are completely derived for stochastic perturbation which generalized the classic perturbation equations which are used since Gauss, starting from Newton's equation and it's solution. The six most understandable orbital element, the semi-major axis, the eccentricity, the inclination, the longitude of the ascending node, the pericenter angle and the mean motion are express in term of the angular momentum vector $\\textbf{H}$ per unit of mass and the energy $E$ per unit of mass. We differentiate those expressions using It\\^o's theory of differential equations due to the stochastic nature of the perturbing force. The result is applied to the two-body problem perturbed by a stochastic dust cloud and also perturbed by a stochastic dynamical oblateness of the central body.
Kinetic equations: computation
Pareschi, Lorenzo
2013-01-01
Kinetic equations bridge the gap between a microscopic description and a macroscopic description of the physical reality. Due to the high dimensionality the construction of numerical methods represents a challenge and requires a careful balance between accuracy and computational complexity.
Saaty, Thomas L
1981-01-01
Covers major types of classical equations: operator, functional, difference, integro-differential, and more. Suitable for graduate students as well as scientists, technologists, and mathematicians. "A welcome contribution." - Math Reviews. 1964 edition.
Geometry of differential equations
Khovanskiĭ, A; Vassiliev, V
1998-01-01
This volume contains articles written by V. I. Arnold's colleagues on the occasion of his 60th birthday. The articles are mostly devoted to various aspects of geometry of differential equations and relations to global analysis and Hamiltonian mechanics.
Regularized Structural Equation Modeling.
Jacobucci, Ross; Grimm, Kevin J; McArdle, John J
A new method is proposed that extends the use of regularization in both lasso and ridge regression to structural equation models. The method is termed regularized structural equation modeling (RegSEM). RegSEM penalizes specific parameters in structural equation models, with the goal of creating easier to understand and simpler models. Although regularization has gained wide adoption in regression, very little has transferred to models with latent variables. By adding penalties to specific parameters in a structural equation model, researchers have a high level of flexibility in reducing model complexity, overcoming poor fitting models, and the creation of models that are more likely to generalize to new samples. The proposed method was evaluated through a simulation study, two illustrative examples involving a measurement model, and one empirical example involving the structural part of the model to demonstrate RegSEM's utility.
A.I.Arbab
2013-01-01
A unified complex model of Maxwell's equations is presented.The wave nature of the electromagnetic field vector is related to the temporal and spatial distributions and the circulation of charge and current densities.A new vacuum solution is obtained,and a new transformation under which Maxwell's equations are invariant is proposed.This transformation extends ordinary gauge transformation to include charge-current as well as scalar-vector potential.An electric dipole moment is found to be related to the magnetic charges,and Dirac's quantization is found to determine an uncertainty relation expressing the indeterminacy of electric and magnetic charges.We generalize Maxwell's equations to include longitudinal waves.A formal analogy between this formulation and Dirac's equation is also discussed.
Applied partial differential equations
DuChateau, Paul
2012-01-01
Book focuses mainly on boundary-value and initial-boundary-value problems on spatially bounded and on unbounded domains; integral transforms; uniqueness and continuous dependence on data, first-order equations, and more. Numerous exercises included.
Singular Renormalization Group Equations
Minoru, HIRAYAMA; Department of Physics, Toyama University
1984-01-01
The possible behaviour of the effective charge is discussed in Oehme and Zimmermann's scheme of the renormalization group equation. The effective charge in an example considered oscillates so violently in the ultraviolet limit that the bare charge becomes indefinable.
Problems in differential equations
Brenner, J L
2013-01-01
More than 900 problems and answers explore applications of differential equations to vibrations, electrical engineering, mechanics, and physics. Problem types include both routine and nonroutine, and stars indicate advanced problems. 1963 edition.
Relativistic Guiding Center Equations
White, R. B. [PPPL; Gobbin, M. [Euratom-ENEA Association
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.
Asymptotics for dissipative nonlinear equations
Hayashi, Nakao; Kaikina, Elena I; Shishmarev, Ilya A
2006-01-01
Many of problems of the natural sciences lead to nonlinear partial differential equations. However, only a few of them have succeeded in being solved explicitly. Therefore different methods of qualitative analysis such as the asymptotic methods play a very important role. This is the first book in the world literature giving a systematic development of a general asymptotic theory for nonlinear partial differential equations with dissipation. Many typical well-known equations are considered as examples, such as: nonlinear heat equation, KdVB equation, nonlinear damped wave equation, Landau-Ginzburg equation, Sobolev type equations, systems of equations of Boussinesq, Navier-Stokes and others.
Functional Equations and Fourier Analysis
2010-01-01
By exploring the relations among functional equations, harmonic analysis and representation theory, we give a unified and very accessible approach to solve three important functional equations -- the d'Alembert equation, the Wilson equation, and the d'Alembert long equation, on compact groups.
Bianchini, Stefano; Mielke, Alexander; Villani, Cédric
2011-01-01
This volume collects the notes of the CIME course "Nonlinear PDE’s and applications" held in Cetraro (Italy) on June 23–28, 2008. It consists of four series of lectures, delivered by Stefano Bianchini (SISSA, Trieste), Eric A. Carlen (Rutgers University), Alexander Mielke (WIAS, Berlin), and Cédric Villani (Ecole Normale Superieure de Lyon). They presented a broad overview of far-reaching findings and exciting new developments concerning, in particular, optimal transport theory, nonlinear evolution equations, functional inequalities, and differential geometry. A sampling of the main topics considered here includes optimal transport, Hamilton-Jacobi equations, Riemannian geometry, and their links with sharp geometric/functional inequalities, variational methods for studying nonlinear evolution equations and their scaling properties, and the metric/energetic theory of gradient flows and of rate-independent evolution problems. The book explores the fundamental connections between all of these topics and po...
A minimax optimal control strategy for uncertain quasi-Hamiltonian systems
Yong WANC; Zu-guang YING; Wei-qiu ZHU
2008-01-01
A minimax optimal control strategy for quasi-Hamiltonian systems with bounded parametric and/or external disturbances is proposed based on the stochastic averaging method and stochastic differential game. To conduct the system energy control, the partially averaged It6 stochastic differential equations for the energy processes are first derived by using the stochastic averaging method for quasi-Hamiltonian systems. Combining the above equations with an appropriate performance index, the proposed strategy is searching for an optimal worst-case controller by solving a stochastic differential game problem. The worst-case disturbances and the optimal controls are obtained by solving a Hamilton-Jacobi-Isaacs (HJI) equation. Numerical results for a controlled and stochastically excited Duffing oscillator with uncertain disturbances exhibit the efficacy of the proposed control strategy.
Jusufi, Kimet
2016-01-01
In the present paper we extend the study of Hawking radiation as a quantum tunneling effect of spin-$1$ particles to the case of a five-dimensional, spherically symmetric, Einstein-Yang-Mills-Gauss-Bonnet (EYMGB) black hole. We solve the Proca equation (PE) by applying the WKB approximation and separation of variables via Hamilton-Jacobi (HJ) equation which results with a set of five differential equations, and reproduce in this way, the Hawking temperature. In the second part of this paper, we extend our results beyond the semiclassical approximation. In particular, we derive the logarithmic correction to the entropy of the EYMGB black hole and show that the quantum corrected specific heat indicates the possible existence of a remnant.
Modified Semi-Classical Methods for Nonlinear Quantum Oscillations Problems
Moncrief, Vincent; Maitra, Rachel
2012-01-01
We develop a modified semi-classical approach to the approximate solution of Schrodinger's equation for certain nonlinear quantum oscillations problems. At lowest order, the Hamilton-Jacobi equation of the conventional semi-classical formalism is replaced by an inverted-potential-vanishing-energy variant thereof. Under smoothness, convexity and coercivity hypotheses on its potential energy function, we prove, using the calculus of variations together with the Banach space implicit function theorem, the existence of a global, smooth `fundamental solution'. Higher order quantum corrections, for ground and excited states, are computed through the integration of associated systems of linear transport equations, and formal expansions for the corresponding energy eigenvalues obtained by imposing smoothness on the quantum corrections to the eigenfunctions. For linear oscillators our expansions naturally truncate, reproducing the well-known solutions for the energy eigenfunctions and eigenvalues. As an application, w...
Electromagnetic and Gravitational Waves: the Third Dimension
Marsh, Gerald E
2011-01-01
Plane electromagnetic and gravitational waves interact with particles in such a way as to cause them to oscillate not only in the transverse direction but also along the direction of propagation. The electromagnetic case is usually shown by use of the Hamilton-Jacobi equation and the gravitational by a transformation to a local inertial frame. Here, the covariant Lorentz force equation and the second order equation of geodesic deviation followed by the introduction of a local inertial frame are respectively used. It is often said that there is an analogy between the motion of charged particles in the field of an electromagnetic wave and the motion of test particles in the field of a gravitational wave. This analogy is examined and found to be rather limited. It is also shown that a simple special relativistic relation leads to an integral of the motion, characteristic of plane waves, that is satisfied in both cases.
Properties of diffusive systems near a saddle point: application to a quartic double well
Battezzati, M
2003-01-01
This paper aims at the analysis of diffusive properties of unidimensional mechanical systems in the environment of maxima and minima of the potential. It begins with a study of the properties of the singular solutions of the Hamilton-Jacobi-Yasue equation in the above-mentioned environment, in both strong or very small frictional forces. For the quartic symmetrical double-well potential, approximate solutions are found for local validity and the diffusion operator is then calculated in the limits of deep wells and small temperature, the regime being supposed to be aperiodic, with high or moderate values of frictional coefficient. This equation is proved to be nonunique. This operator is then reduced to second order by imposing suitable boundary conditions. Thus an appropriate eigenvalue equation is obtained to describe stationary states in the environment of extremal points of the potential energy function. The main interest of this work relies upon the fact that transition times between wells mainly depend u...
Black holes and Rindler superspace: classical singularity and quantum unitarity
Chou, C H; Yu, H L; Chou, Chung-Hsien; Soo, Chopin; Yu, Hoi-Lai
2007-01-01
Canonical quantization of spherically symmetric initial data which is appropriate to classical interior black hole solutions in four dimensions is carried out and solved exactly without gauge fixing. The resultant mini-superspace manifold and arena for quantum geometrodynamics is two-dimensional, of signature (+, -), non-singular, and can in fact be identified precisely with the first Rindler wedge. The associated Wheeler-DeWitt equation with evolution in intrinsic superspace time can be formulated as a free massive Klein-Gordon equation; and the Hamilton-Jacobi semiclassical limit of plane wave solutions can be matched precisely to the interiors of Schwarzschild black holes. Furthermore, classical black hole horizons and singularities correspond to the boundaries of the Rindler wedge. Exact wavefunctions of the first-order-in-superspace intrinsic time Dirac equation are also considered. Precise correspondence between Schwarzschild black holes and free particle mechanics in superspace is noted. Another intrig...
Jusufi, K.
2016-12-01
In the present paper we study the Hawking radiation as a quantum tunneling effect of spin-1 particles from a five-dimensional, spherically symmetric, Einstein-Yang-Mills-Gauss-Bonnet (5D EYMGB) black hole. We solve the Proca equation (PE) by applying the WKB approximation and separation of variables via Hamilton-Jacobi (HJ) equation which results in a set of five differential equations, and reproduces, in this way, the Hawking temperature. In the second part of this paper, we extend our results beyond the semiclassical approximation. In particular, we derive the logarithmic correction to the entropy of the EYMGB black hole and show that the quantum corrected specific heat indicates the possible existence of a remnant.
Equilibrium policies when preferences are time inconsistent
Ekeland, Ivar
2008-01-01
This paper characterizes differentiable and subgame Markov perfect equilibria in a continuous time intertemporal decision problem with non-constant discounting. Capturing the idea of non commitment by letting the commitment period being infinitesimally small, we characterize the equilibrium strategies by a value function, which must satisfy a certain equation. The equilibrium equation is reminiscent of the classical Hamilton-Jacobi-Bellman equation of optimal control, but with a non-local term leading to differences in qualitative behavior. As an application, we formulate an overlapping generations Ramsey model where the government maximizes a utilitarian welfare function defined as the discounted sum of successive generations' lifetime utilities. When the social discount rate is different from the private discount rate, the optimal command allocation is time inconsistent and we retain subgame perfection as a principle of intergenerational equity. Existence of multiple subgame perfect equilibria is establishe...
A Dynamic Programming Approach to Finite-horizon Coherent Quantum LQG Control
Vladimirov, Igor G
2011-01-01
The paper considers the coherent quantum Linear Quadratic Gaussian (CQLQG) control problem for time-varying quantum plants governed by linear quantum stochastic differential equations over a bounded time interval. A controller is sought among quantum linear systems satisfying physical realizability (PR) conditions. The latter describe the dynamic equivalence of the system to an open quantum harmonic oscillator and relate its state-space matrices to the free Hamiltonian, coupling and scattering operators of the oscillator. Using the Hamiltonian parameterization of PR controllers, the CQLQG problem is recast into an optimal control problem for a deterministic system governed by a differential Lyapunov equation. The state of this subsidiary system is the symmetric part of the quantum covariance matrix of the plant-controller state vector. The resulting covariance control problem is treated using dynamic programming and Pontryagin's minimum principle. The associated Hamilton-Jacobi-Bellman equation for the minimu...
Feedback Optimal Control of Low-thrust Orbit Transfer in Central Gravity Field
Ashraf H. Owis
2013-05-01
Full Text Available Low-thrust trajectories with variable radial thrust is studied in this paper. The problem is tackled by solving the Hamilton- Jacobi-Bellman equation via State Dependent Riccati Equation( STDE technique devised for nonlinear systems. Instead of solving the two-point boundary value problem in which the classical optimal control is stated, this technique allows us to derive closed-loop solutions. The idea of the work consists in factorizing the original nonlinear dynamical system into a quasi-linear state dependent system of ordinary differential equations. The generating function technique is then applied to this new dynamical system, the feedback optimal control is solved. We circumvent in this way the problem of expanding the vector field and truncating higher-order terms because no remainders are lost in the undertaken approach. This technique can be applied to any planet-to-planet transfer; it has been applied here to the Earth-Mars low-thrust transfer
Coarse-graining two-dimensional turbulence via dynamical optimization
Turkington, Bruce; Thalabard, Simon
2015-01-01
A model reduction technique based on an optimization principle is employed to coarse-grain inviscid, incompressible fluid dynamics in two dimensions. In this reduction the spectrally-truncated vorticity equation defines the microdynamics, while the macroscopic state space consists of quasi-equilibrium trial probability densities on the microscopic phase space, which are parameterized by the means and variances of the low modes of the vorticity. A macroscopic path therefore represents a coarse-grained approximation to the evolution of a nonequilibrium ensemble of microscopic solutions. Closure in terms of the vector of resolved variables, namely, the means and variances of the low modes, is achieved by minimizing over all feasible paths the time integral of their mean-squared residual with respect to the Liouville equation. The equations governing the optimal path are deduced from Hamilton-Jacobi theory. The coarse-grained dynamics derived by this optimization technique contains a scale-dependent eddy viscosit...
Scaling Equation for Invariant Measure
LIU Shi-Kuo; FU Zun-Tao; LIU Shi-Da; REN Kui
2003-01-01
An iterated function system (IFS) is constructed. It is shown that the invariant measure of IFS satisfies the same equation as scaling equation for wavelet transform (WT). Obviously, IFS and scaling equation of WT both have contraction mapping principle.
Florian Ion Tiberiu Petrescu
2015-09-01
Full Text Available This paper presents the dynamic, original, machine motion equations. The equation of motion of the machine that generates angular speed of the shaft (which varies with position and rotation speed is deduced by conservation kinetic energy of the machine. An additional variation of angular speed is added by multiplying by the coefficient dynamic D (generated by the forces out of mechanism and or by the forces generated by the elasticity of the system. Kinetic energy conservation shows angular speed variation (from the shaft with inertial masses, while the dynamic coefficient introduces the variation of w with forces acting in the mechanism. Deriving the first equation of motion of the machine one can obtain the second equation of motion dynamic. From the second equation of motion of the machine it determines the angular acceleration of the shaft. It shows the distribution of the forces on the mechanism to the internal combustion heat engines. Dynamic, the velocities can be distributed in the same way as forces. Practically, in the dynamic regimes, the velocities have the same timing as the forces. Calculations should be made for an engine with a single cylinder. Originally exemplification is done for a classic distribution mechanism, and then even the module B distribution mechanism of an Otto engine type.
Introduction to partial differential equations
Greenspan, Donald
2000-01-01
Designed for use in a one-semester course by seniors and beginning graduate students, this rigorous presentation explores practical methods of solving differential equations, plus the unifying theory underlying the mathematical superstructure. Topics include basic concepts, Fourier series, second-order partial differential equations, wave equation, potential equation, heat equation, approximate solution of partial differential equations, and more. Exercises appear at the ends of most chapters. 1961 edition.
Generalization of Hopf Functional Equation
无
2002-01-01
This paper generalizes the Hopf functional equation in order to apply it to a wider class of not necessarily incompressible fluid flows. We start by defining characteristic functionals of the velocity field, the density field and the temperature field of a compressible field. Using the continuity equation, the Navier-Stokes equations and the equation of energy we derive a functional equation governing the motion of an ideal gas flow and a van der Waals gas flow, and then give some general methods of deriving a functional equation governing the motion of any compressible fluid flow. These functional equations can be considered as the generalization of the Hopf functional equation.
Stochastic porous media equations
Barbu, Viorel; Röckner, Michael
2016-01-01
Focusing on stochastic porous media equations, this book places an emphasis on existence theorems, asymptotic behavior and ergodic properties of the associated transition semigroup. Stochastic perturbations of the porous media equation have reviously been considered by physicists, but rigorous mathematical existence results have only recently been found. The porous media equation models a number of different physical phenomena, including the flow of an ideal gas and the diffusion of a compressible fluid through porous media, and also thermal propagation in plasma and plasma radiation. Another important application is to a model of the standard self-organized criticality process, called the "sand-pile model" or the "Bak-Tang-Wiesenfeld model". The book will be of interest to PhD students and researchers in mathematics, physics and biology.
Quasirelativistic Langevin equation.
Plyukhin, A V
2013-11-01
We address the problem of a microscopic derivation of the Langevin equation for a weakly relativistic Brownian particle. A noncovariant Hamiltonian model is adopted, in which the free motion of particles is described relativistically while their interaction is treated classically, i.e., by means of action-to-a-distance interaction potentials. Relativistic corrections to the classical Langevin equation emerge as nonlinear dissipation terms and originate from the nonlinear dependence of the relativistic velocity on momentum. On the other hand, similar nonlinear dissipation forces also appear as classical (nonrelativistic) corrections to the weak-coupling approximation. It is shown that these classical corrections, which are usually ignored in phenomenological models, may be of the same order of magnitude, if not larger than, relativistic ones. The interplay of relativistic corrections and classical beyond-the-weak-coupling contributions determines the sign of the leading nonlinear dissipation term in the Langevin equation and thus is qualitatively important.
Boussinesq evolution equations
Bredmose, Henrik; Schaffer, H.; Madsen, Per A.
2004-01-01
This paper deals with the possibility of using methods and ideas from time domain Boussinesq formulations in the corresponding frequency domain formulations. We term such frequency domain models "evolution equations". First, we demonstrate that the numerical efficiency of the deterministic...... Boussinesq evolution equations of Madsen and Sorensen [Madsen, P.A., Sorensen, O.R., 1993. Bound waves and triad interactions in shallow water. Ocean Eng. 20 359-388] can be improved by using Fast Fourier Transforms to evaluate the nonlinear terms. For a practical example of irregular waves propagating over...... a submerged bar, it is demonstrated that evolution equations utilising FFT can be solved around 100 times faster than the corresponding time domain model. Use of FFT provides an efficient bridge between the frequency domain and the time domain. We utilise this by adapting the surface roller model for wave...
Nonlocal electrical diffusion equation
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 0numerical 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.
Equations of mathematical physics
Tikhonov, A N
2011-01-01
Mathematical physics plays an important role in the study of many physical processes - hydrodynamics, elasticity, and electrodynamics, to name just a few. Because of the enormous range and variety of problems dealt with by mathematical physics, this thorough advanced-undergraduate or graduate-level text considers only those problems leading to partial differential equations. The authors - two well-known Russian mathematicians - have focused on typical physical processes and the principal types of equations deailing with them. Special attention is paid throughout to mathematical formulation, ri
Gas Dynamics Equations: Computation
Chen, Gui-Qiang G
2012-01-01
Shock waves, vorticity waves, and entropy waves are fundamental discontinuity waves in nature and arise in supersonic or transonic gas flow, or from a very sudden release (explosion) of chemical, nuclear, electrical, radiation, or mechanical energy in a limited space. Tracking these discontinuities and their interactions, especially when and where new waves arise and interact in the motion of gases, is one of the main motivations for numerical computation for the gas dynamics equations. In this paper, we discuss some historic and recent developments, as well as mathematical challenges, in designing and formulating efficient numerical methods and algorithms to compute weak entropy solutions for the Euler equations for gas dynamics.
Theory of differential equations
Gel'fand, I M
1967-01-01
Generalized Functions, Volume 3: Theory of Differential Equations focuses on the application of generalized functions to problems of the theory of partial differential equations.This book discusses the problems of determining uniqueness and correctness classes for solutions of the Cauchy problem for systems with constant coefficients and eigenfunction expansions for self-adjoint differential operators. The topics covered include the bounded operators in spaces of type W, Cauchy problem in a topological vector space, and theorem of the Phragmén-Lindelöf type. The correctness classes for the Cau
Systematic Equation Formulation
Lindberg, Erik
2007-01-01
A tutorial giving a very simple introduction to the set-up of the equations used as a model for an electrical/electronic circuit. The aim is to find a method which is as simple and general as possible with respect to implementation in a computer program. The “Modified Nodal Approach”, MNA, and th......, and the “Controlled Source Approach”, CSA, for systematic equation formulation are investigated. It is suggested that the kernel of the P Spice program based on MNA is reprogrammed....
Generalized estimating equations
Hardin, James W
2002-01-01
Although powerful and flexible, the method of generalized linear models (GLM) is limited in its ability to accurately deal with longitudinal and clustered data. Developed specifically to accommodate these data types, the method of Generalized Estimating Equations (GEE) extends the GLM algorithm to accommodate the correlated data encountered in health research, social science, biology, and other related fields.Generalized Estimating Equations provides the first complete treatment of GEE methodology in all of its variations. After introducing the subject and reviewing GLM, the authors examine th
Ding Yi
2009-01-01
In this article, the author derives a functional equation η(s)=［(π/4)s-1/2√2/πг(1-s)sin(πs/2)]η(1-s) of the analytic function η(s) which is defined by η(s)=1-s-3-s-5-s+7-s…for complex variable s with Re s>1, and is defined by analytic continuation for other values of s. The author proves (1) by Ramanujan identity (see [1], [3]). Her method provides a new derivation of the functional equation of Riemann zeta function by using Poisson summation formula.
Comparison of Kernel Equating and Item Response Theory Equating Methods
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…
Test equating methods and practices
Kolen, Michael J
1995-01-01
In recent years, many researchers in the psychology and statistical communities have paid increasing attention to test equating as issues of using multiple test forms have arisen and in response to criticisms of traditional testing techniques This book provides a practically oriented introduction to test equating which both discusses the most frequently used equating methodologies and covers many of the practical issues involved The main themes are - the purpose of equating - distinguishing between equating and related methodologies - the importance of test equating to test development and quality control - the differences between equating properties, equating designs, and equating methods - equating error, and the underlying statistical assumptions for equating The authors are acknowledged experts in the field, and the book is based on numerous courses and seminars they have presented As a result, educators, psychometricians, professionals in measurement, statisticians, and students coming to the subject for...
The Statistical Drake Equation
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
Variation principle of piezothermoelastic bodies, canonical equation and homogeneous equation
LIU Yan-hong; ZHANG Hui-ming
2007-01-01
Combining the symplectic variations theory, the homogeneous control equation and isoparametric element homogeneous formulations for piezothermoelastic hybrid laminates problems were deduced. Firstly, based on the generalized Hamilton variation principle, the non-homogeneous Hamilton canonical equation for piezothermoelastic bodies was derived. Then the symplectic relationship of variations in the thermal equilibrium formulations and gradient equations was considered, and the non-homogeneous canonical equation was transformed to homogeneous control equation for solving independently the coupling problem of piezothermoelastic bodies by the incensement of dimensions of the canonical equation. For the convenience of deriving Hamilton isoparametric element formulations with four nodes, one can consider the temperature gradient equation as constitutive relation and reconstruct new variation principle. The homogeneous equation simplifies greatly the solution programs which are often performed to solve nonhomogeneous equation and second order differential equation on the thermal equilibrium and gradient relationship.
Neuro-optimal control of helicopter UAVs
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.
Helrich, Carl S
2017-01-01
This advanced undergraduate textbook begins with the Lagrangian formulation of Analytical Mechanics and then passes directly to the Hamiltonian formulation and the canonical equations, with constraints incorporated through Lagrange multipliers. Hamilton's Principle and the canonical equations remain the basis of the remainder of the text. Topics considered for applications include small oscillations, motion in electric and magnetic fields, and rigid body dynamics. The Hamilton-Jacobi approach is developed with special attention to the canonical transformation in order to provide a smooth and logical transition into the study of complex and chaotic systems. Finally the text has a careful treatment of relativistic mechanics and the requirement of Lorentz invariance. The text is enriched with an outline of the history of mechanics, which particularly outlines the importance of the work of Euler, Lagrange, Hamilton and Jacobi. Numerous exercises with solutions support the exceptionally clear and concise treatment...
A General Theory of Markovian Time Inconsistent Stochastic Control Problems
Björk, Tomas; Murgochi, Agatha
We develop a theory for stochastic control problems which, in various ways, are time inconsistent in the sense that they do not admit a Bellman optimality principle. We attach these problems by viewing them within a game theoretic framework, and we look for Nash subgame perfect equilibrium points...... examples of time inconsistency in the literature are easily seen to be special cases of the present theory. We also prove that for every time inconsistent problem, there exists an associated time consistent problem such that the optimal control and the optimal value function for the consistent problem....... For a general controlled Markov process and a fairly general objective functional we derive an extension of the standard Hamilton-Jacobi-Bellman equation, in the form of a system of on-linear equations, for the determination for the equilibrium strategy as well as the equilibrium value function. All known...
Continuous-time mean-variance portfolio selection with value-at-risk and no-shorting constraints
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.
Moving the CFT into the bulk with $T\\bar T$
McGough, Lauren; Verlinde, Herman
2016-01-01
Recent work by Zamolodchikov and others has uncovered a solvable irrelevant deformation of general 2D CFTs, defined by turning on the dimension 4 operator $T \\bar T$, the product of the left- and right-moving stress tensor. We propose that in the holographic dual, this deformation represents a geometric cutoff that removes the asymptotic region of AdS and places the QFT on a Dirichlet wall at finite radial distance $r = r_c$ in the bulk. As a quantitative check of the proposed duality, we compute the signal propagation speed, energy spectrum, and thermodynamic relations on both sides. In all cases, we obtain a precise match. We derive an exact RG flow equation for the metric dependence of the effective action of the $T \\bar T$ deformed theory, and find that it coincides with the Hamilton-Jacobi equation that governs the radial evolution of the classical gravity action in AdS.
Event-Triggered Adaptive Dynamic Programming for Continuous-Time Systems With Control Constraints.
Dong, Lu; Zhong, Xiangnan; Sun, Changyin; He, Haibo
2016-08-31
In this paper, an event-triggered near optimal control structure is developed for nonlinear continuous-time systems with control constraints. Due to the saturating actuators, a nonquadratic cost function is introduced and the Hamilton-Jacobi-Bellman (HJB) equation for constrained nonlinear continuous-time systems is formulated. In order to solve the HJB equation, an actor-critic framework is presented. The critic network is used to approximate the cost function and the action network is used to estimate the optimal control law. In addition, in the proposed method, the control signal is transmitted in an aperiodic manner to reduce the computational and the transmission cost. Both the networks are only updated at the trigger instants decided by the event-triggered condition. Detailed Lyapunov analysis is provided to guarantee that the closed-loop event-triggered system is ultimately bounded. Three case studies are used to demonstrate the effectiveness of the proposed method.
A model of unified quantum chromodynamics and Yang-Mills gravity
HSU Jong-Ping
2012-01-01
Based on a generalized Yang-Mills framework,gravitational and strong interactions can be unified in analogy with the unification in the clectroweak theory.By gauging T(4) × [SU(3)]color in fiat space-time,we have a unified model of chromo-gravity with a new tensor gauge field,which couples universally to all gluons,quarks and anti-quarks.The space-time translational gauge symmetry assures that all wave equations of quarks and gluons reduce to a Hamilton-Jacobi equation with the same ‘effective Riemann metric tensors' in the geometric-optics (or classical) limit.The emergence of effective metric tensors in the classical limit is essential for the unified model to agree with experiments.The unified model suggests that all gravitational,strong and electroweak interactions appear to be dictated by gauge symmetries in the generalized Yang-Mills framework.
Causality of spacetimes admitting a parallel null vector and weak KAM theory
Minguzzi, E
2012-01-01
The causal spacetimes admitting a covariantly constant null vector provide a connection between relativistic and non-relativistic physics. We explore this relationship in several directions. We start proving a formula which relates the Lorentzian distance in the full spacetime with the least action of a mechanical system living in a quotient classical space time. The timelike eikonal equation satisfied by the Lorentzian distance is proved to be equivalent to the Hamilton-Jacobi equation for the least action. We also prove that the Legendre transform on the classical base corresponds to the musical isomorphism on the light cone, and the Young-Fenchel inequality is nothing but a well known geometric inequality in Lorentzian geometry. A strategy to simplify the dynamics passing to a reference frame moving with the E.-L. flow is explained. It is then proved that the causality properties can be conveniently expressed in terms of the least action. In particular, strong causality coincides with stable causality and ...
Pan Wei-Zhen; Yang Xue-Jun; Xie Zhi-Kun
2011-01-01
Using a new tortoise coordinate transformation, this paper investigates the Hawking effect from an arbitrarily accelerating charged black hole by the improved Damour-Ruffini method. After the tortoise coordinate transformation,the Klein-Gordon equation can be written as the standard form at the event horizon. Then extending the outgoing wave from outside to inside of the horizon analytically, the surface gravity and Hawking temperature can be obtained automatically. It is found that the Hawking temperatures of different points on the surface are different. The quantum nonthermal radiation characteristics of a black hole near the event horizon is also discussed by studying the Hamilton Jacobi equation in curved spacetime and the maximum overlap of the positive and negative energy levels near the event horizon is given. There is a dimensional problem in the standard tortoise coordinate and the present results may be more reasonable.
Dynamic programming for infinite horizon boundary control problems of PDE's with age structure
Faggian, Silvia
2008-01-01
We develop the dynamic programming approach for a family of infinite horizon boundary control problems with linear state equation and convex cost. We prove that the value function of the problem is the unique regular solution of the associated stationary Hamilton--Jacobi--Bellman equation and use this to prove existence and uniqueness of feedback controls. The idea of studying this kind of problem comes from economic applications, in particular from models of optimal investment with vintage capital. Such family of problems has already been studied in the finite horizon case by Faggian. The infinite horizon case is more difficult to treat and it is more interesting from the point of view of economic applications, where what mainly matters is the behavior of optimal trajectories and controls in the long run. The study of infinite horizon is here performed through a nontrivial limiting procedure from the corresponding finite horizon problem.
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.
Quantum trajectories, real, surreal or an approximation to a deeper process?
Hiley, B J; Maroney, O
2000-01-01
The proposal that the one-parameter solutions of the real part of the Schrodinger equation (quantum Hamilton-Jacobi equation) can be regarded as `quantum particle trajectories' has received considerable attention recently. Opinions as to their significance differ. Some argue that they do play a fundamental role as actual particle trajectories, others regard them as mere metaphysical appendages without any physical significance. Recent work has claimed that in some cases the Bohm approach gives results that disagree with those obtained from standard quantum mechanics and, in consequence, with experiment. Furthermore it is claimed that these trajectories have such unacceptable properties that they can only be considered as `surreal'. We re-examine these questions and show that the specific objections raised by Englert, Scully, Sussmann and Walther cannot be sustained. We also argue that contrary to their negative view, these trajectories can provide a deeper insight into quantum processes.
A mathematical treatment of bank monitoring incentives
Pagès, Henri
2012-01-01
In this paper, we take up the analysis of a principal/agent model with moral hazard introduced in \\cite{pages}, with optimal contracting between competitive investors and an impatient bank monitoring a pool of long-term loans subject to Markovian contagion. We provide here a comprehensive mathematical formulation of the model and show using martingale arguments in the spirit of Sannikov \\cite{san} how the maximization problem with implicit constraints faced by investors can be reduced to a classic stochastic control problem. The approach has the advantage of avoiding the more general techniques based on forward-backward stochastic differential equations described in \\cite{cviz} and leads to a simple recursive system of Hamilton-Jacobi-Bellman equations. We provide a solution to our problem by a verification argument and give an explicit description of both the value function and the optimal contract. Finally, we study the limit case where the bank is no longer impatient.
Nonlinear feedback control of highly manoeuvrable aircraft
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.
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.
Maximal Abelian subgroups of the isometry and conformal groups of Euclidean and Minkowski spaces
Thomova, Z.; Winternitz, P.
1998-02-01
The maximal Abelian subalgebras (MASAs) of the Euclidean 0305-4470/31/7/016/img1 and pseudo-euclidean 0305-4470/31/7/016/img2 Lie algebras are classified into conjugacy classes under the action of the corresponding Lie groups 0305-4470/31/7/016/img3 and 0305-4470/31/7/016/img4, and also under the conformal groups 0305-4470/31/7/016/img5 and 0305-4470/31/7/016/img6, respectively. The results are presented in terms of decomposition theorems. For 0305-4470/31/7/016/img1 orthogonally indecomposable MASAs exist only for p = 1 and p = 2. For 0305-4470/31/7/016/img2, on the other hand, orthogonally indecomposable MASAs exist for all values of p. The results are used to construct new coordinate systems in which wave equations and Hamilton-Jacobi equations allow the separation of variables.
Holographic renormalisation group flows and renormalisation from a Wilsonian perspective
Lizana, J M; Perez-Victoria, M
2015-01-01
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 the...
Calculus & ordinary differential equations
Pearson, David
1995-01-01
Professor Pearson's book starts with an introduction to the area and an explanation of the most commonly used functions. It then moves on through differentiation, special functions, derivatives, integrals and onto full differential equations. As with other books in the series the emphasis is on using worked examples and tutorial-based problem solving to gain the confidence of students.
Standardized Referente Evapotranspiration Equation
M.D. Mundo–Molina
2009-04-01
Full Text Available In this paper is presented a discussion on the necessity to standardize the Penman–Monteith equations in order to estimate ETo. The proposal is to define an accuracy and standarize equation based in Penman–Monteith. The automated weather station named CIANO (27° 22 ' 144 North latitude and 109" 55' west longitude it was selected tomake comparisons. The compared equations we re: a CIANO weat her station, b Penman–Monteith ASCE (PMA, Penman–Monteith FAO 56 (PM FAO 56, Penman–Monteith estandarizado ASCE (PM Std. ASCE. The results were: a There are important differences between PMA and CIANO weather station. The differences are attributed to the nonstandardization of the equation CIANO weather station, b The coefficient of correlation between both methods was of 0,92, with a standard deviation of 1,63 mm, an average quadratic error of 0,60 mm and one efficiency in the estimation of ETo with respect to the method pattern of 87%.
Modified differential equations
Chartier, Philippe; Hairer, Ernst; Vilmart, Gilles
2007-01-01
Motivated by the theory of modified differential equations (backward error analysis) an approach for the construction of high order numerical integrators that preserve geometric properties of the exact flow is developed. This summarises a talk presented in honour of Michel Crouzeix.
Equational binary decision diagrams
Groote, J.F.; Pol, J.C. van de
2000-01-01
We incorporate equations in binary decision diagrams (BDD). The resulting objects are called EQ-BDDs. A straightforward notion of ordered EQ-BDDs (EQ-OBDD) is defined, and it is proved that each EQ-BDD is logically equivalent to an EQ-OBDD. Moreover, on EQ-OBDDs satisfiability and tautology checkin
Garkavenko A. S.
2011-08-01
Full Text Available The rate equations of the exciton laser in the system of interacting excitons have been obtained and the inverted population conditions and generation have been derived. The possibility of creating radically new gamma-ray laser has been shown.
Equational binary decision diagrams
J.F. Groote (Jan Friso); J.C. van de Pol (Jaco)
2000-01-01
textabstractWe incorporate equations in binary decision diagrams (BDD). The resulting objects are called EQ-BDDs. A straightforward notion of ordered EQ-BDDs (EQ-OBDD) is defined, and it is proved that each EQ-BDD is logically equivalent to an EQ-OBDD. Moreover, on EQ-OBDDs satisfiability and
Structural Equation Model Trees
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…
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.
Hatem Mejjaoli
2008-12-01
Full Text Available We introduce and study the Dunkl symmetric systems. We prove the well-posedness results for the Cauchy problem for these systems. Eventually we describe the finite speed of it. Next the semi-linear Dunkl-wave equations are also studied.
Equational binary decision diagrams
J.F. Groote (Jan Friso); J.C. van de Pol (Jaco)
2000-01-01
textabstractWe incorporate equations in binary decision diagrams (BDD). The resulting objects are called EQ-BDDs. A straightforward notion of ordered EQ-BDDs (EQ-OBDD) is defined, and it is proved that each EQ-BDD is logically equivalent to an EQ-OBDD. Moreover, on EQ-OBDDs satisfiability and tauto
Lie Symmetries of Ishimori Equation
SONG Xu-Xia
2013-01-01
The Ishimori equation is one of the most important (2+1)-dimensional integrable models,which is an integrable generalization of (1+1)-dimensional classical continuous Heisenberg ferromagnetic spin equations.Based on importance of Lie symmetries in analysis of differential equations,in this paper,we derive Lie symmetries for the Ishimori equation by Hirota's direct method.
Lectures on partial differential equations
Petrovsky, I G
1992-01-01
Graduate-level exposition by noted Russian mathematician offers rigorous, transparent, highly readable coverage of classification of equations, hyperbolic equations, elliptic equations and parabolic equations. Wealth of commentary and insight invaluable for deepening understanding of problems considered in text. Translated from the Russian by A. Shenitzer.
Elements of partial differential equations
Sneddon, Ian N
2006-01-01
Geared toward students of applied rather than pure mathematics, this volume introduces elements of partial differential equations. Its focus is primarily upon finding solutions to particular equations rather than general theory.Topics include ordinary differential equations in more than two variables, partial differential equations of the first and second orders, Laplace's equation, the wave equation, and the diffusion equation. A helpful Appendix offers information on systems of surfaces, and solutions to the odd-numbered problems appear at the end of the book. Readers pursuing independent st
Methods for Equating Mental Tests.
1984-11-01
1983) compared conventional and IRT methods for equating the Test of English as a Foreign Language ( TOEFL ) after chaining. Three conventional and...three IRT equating methods were examined in this study; two sections of TOEFL were each (separately) equated. The IRT methods included the following: (a...group. A separate base form was established for each of the six equating methods. Instead of equating the base-form TOEFL to itself, the last (eighth
Differential Equations with Linear Algebra
Boelkins, Matthew R; Potter, Merle C
2009-01-01
Linearity plays a critical role in the study of elementary differential equations; linear differential equations, especially systems thereof, demonstrate a fundamental application of linear algebra. In Differential Equations with Linear Algebra, we explore this interplay between linear algebra and differential equations and examine introductory and important ideas in each, usually through the lens of important problems that involve differential equations. Written at a sophomore level, the text is accessible to students who have completed multivariable calculus. With a systems-first approach, t
SPECIFIC SOLUTIONS GROUNDWATER FLOW EQUATION
Syahruddin, Muhammad Hamzah
2014-01-01
Geophysic publication Groundwater flow under surface, its usually slow moving, so that in laminer flow condition can find analisys using the Darcy???s law. The combination between Darcy law and continuity equation can find differential Laplace equation as general equation groundwater flow in sub surface. Based on Differential Laplace Equation is the equation that can be used to describe hydraulic head and velocity flow distribution in porous media as groundwater. In the modeling Laplace e...
Stochastic differential equations and applications
Friedman, Avner
2006-01-01
This text develops the theory of systems of stochastic differential equations, and it presents applications in probability, partial differential equations, and stochastic control problems. Originally published in two volumes, it combines a book of basic theory and selected topics with a book of applications.The first part explores Markov processes and Brownian motion; the stochastic integral and stochastic differential equations; elliptic and parabolic partial differential equations and their relations to stochastic differential equations; the Cameron-Martin-Girsanov theorem; and asymptotic es
Dyre, Jeppe
1995-01-01
energies chosen randomly according to a Gaussian. The random-walk model is here derived from Newton's laws by making a number of simplifying assumptions. In the second part of the paper an approximate low-temperature description of energy fluctuations in the random-walk modelthe energy master equation...... (EME)is arrived at. The EME is one dimensional and involves only energy; it is derived by arguing that percolation dominates the relaxational properties of the random-walk model at low temperatures. The approximate EME description of the random-walk model is expected to be valid at low temperatures...... of the random-walk model. The EME allows a calculation of the energy probability distribution at realistic laboratory time scales for an arbitrarily varying temperature as function of time. The EME is probably the only realistic equation available today with this property that is also explicitly consistent...
Classical Diophantine equations
1993-01-01
The author had initiated a revision and translation of "Classical Diophantine Equations" prior to his death. Given the rapid advances in transcendence theory and diophantine approximation over recent years, one might fear that the present work, originally published in Russian in 1982, is mostly superseded. That is not so. A certain amount of updating had been prepared by the author himself before his untimely death. Some further revision was prepared by close colleagues. The first seven chapters provide a detailed, virtually exhaustive, discussion of the theory of lower bounds for linear forms in the logarithms of algebraic numbers and its applications to obtaining upper bounds for solutions to the eponymous classical diophantine equations. The detail may seem stark--- the author fears that the reader may react much as does the tourist on first seeing the centre Pompidou; notwithstanding that, Sprind zuk maintainsa pleasant and chatty approach, full of wise and interesting remarks. His emphases well warrant, ...
Numerical Solution of Heun Equation Via Linear Stochastic Differential Equation
Hamidreza Rezazadeh
2014-05-01
Full Text Available In this paper, we intend to solve special kind of ordinary differential equations which is called Heun equations, by converting to a corresponding stochastic differential equation(S.D.E.. So, we construct a stochastic linear equation system from this equation which its solution is based on computing fundamental matrix of this system and then, this S.D.E. is solved by numerically methods. Moreover, its asymptotic stability and statistical concepts like expectation and variance of solutions are discussed. Finally, the attained solutions of these S.D.E.s compared with exact solution of corresponding differential equations.
Dissipative Boussinesq equations
2007-01-01
40 pages, 15 figures, published in C. R. Mecanique 335 (2007) Other author's papers can be downloaded at http://www.cmla.ens-cachan.fr/~dutykh; International audience; The classical theory of water waves is based on the theory of inviscid flows. However it is important to include viscous effects in some applications. Two models are proposed to add dissipative effects in the context of the Boussinesq equations, which include the effects of weak dispersion and nonlinearity in a shallow water fr...
Differential equations with Mathematica
Abell, Martha L
2004-01-01
The Third Edition of the Differential Equations with Mathematica integrates new applications from a variety of fields,especially biology, physics, and engineering. The new handbook is also completely compatible with recent versions of Mathematica and is a perfect introduction for Mathematica beginners.* Focuses on the most often used features of Mathematica for the beginning Mathematica user* New applications from a variety of fields, including engineering, biology, and physics* All applications were completed using recent versions of Mathematica
Arithmetic partial differential equations
Buium, Alexandru; Simanca, Santiago R.
2006-01-01
We develop an arithmetic analogue of linear partial differential equations in two independent ``space-time'' variables. The spatial derivative is a Fermat quotient operator, while the time derivative is the usual derivation. This allows us to ``flow'' integers or, more generally, points on algebraic groups with coordinates in rings with arithmetic flavor. In particular, we show that elliptic curves have certain canonical ``flows'' on them that are the arithmetic analogues of the heat and wave...
Stability in Neutral Equations
1976-02-04
Martinez-Amores Division of Applied Mathematics Brown University Providence, Rhode Island 02912 and Universidad de Granada, Seccion de Matematicas , Spain S...XG w)1- 0 ~t)- >~~~ 0 suc ht j~<kIp, Ii 2 ~ o ~~~ X~ G (t) , y’ip X= 0 y 20 since equation (3.16) is satisfied. Since F = col(f,0), only the col
Trzetrzelewski, Maciej
2016-11-01
Starting with a Nambu-Goto action, a Dirac-like equation can be constructed by taking the square-root of the momentum constraint. The eigenvalues of the resulting Hamiltonian are real and correspond to masses of the excited string. In particular there are no tachyons. A special case of radial oscillations of a closed string in Minkowski space-time admits exact solutions in terms of wave functions of the harmonic oscillator.
D. Diederen
2015-06-01
Full Text Available We present a new equation describing the hydrodynamics in infinitely long tidal channels (i.e., no reflection under the influence of oceanic forcing. The proposed equation is a simple relationship between partial derivatives of water level and velocity. It is formally derived for a progressive wave in a frictionless, prismatic, tidal channel with a horizontal bed. Assessment of a large number of numerical simulations, where an open boundary condition is posed at a certain distance landward, suggests that it can also be considered accurate in the more natural case of converging estuaries with nonlinear friction and a bed slope. The equation follows from the open boundary condition and is therefore a part of the problem formulation for an infinite tidal channel. This finding provides a practical tool for evaluating tidal wave dynamics, by reconstructing the temporal variation of the velocity based on local observations of the water level, providing a fully local open boundary condition and allowing for local friction calibration.
Quantum molecular master equations
Brechet, Sylvain D.; Reuse, Francois A.; Maschke, Klaus; Ansermet, Jean-Philippe
2016-10-01
We present the quantum master equations for midsize molecules in the presence of an external magnetic field. The Hamiltonian describing the dynamics of a molecule accounts for the molecular deformation and orientation properties, as well as for the electronic properties. In order to establish the master equations governing the relaxation of free-standing molecules, we have to split the molecule into two weakly interacting parts, a bath and a bathed system. The adequate choice of these systems depends on the specific physical system under consideration. Here we consider a first system consisting of the molecular deformation and orientation properties and the electronic spin properties and a second system composed of the remaining electronic spatial properties. If the characteristic time scale associated with the second system is small with respect to that of the first, the second may be considered as a bath for the first. Assuming that both systems are weakly coupled and initially weakly correlated, we obtain the corresponding master equations. They describe notably the relaxation of magnetic properties of midsize molecules, where the change of the statistical properties of the electronic orbitals is expected to be slow with respect to the evolution time scale of the bathed system.
M. Paul Gough
2008-07-01
Full Text Available LandauerÃ¢Â€Â™s principle is applied to information in the universe. Once stars began forming there was a constant information energy density as the increasing proportion of matter at high stellar temperatures exactly compensated for the expanding universe. The information equation of state was close to the dark energy value, w = -1, for a wide range of redshifts, 10 > z > 0.8, over one half of cosmic time. A reasonable universe information bit content of only 1087 bits is sufficient for information energy to account for all dark energy. A time varying equation of state with a direct link between dark energy and matter, and linked to star formation in particular, is clearly relevant to the cosmic coincidence problem. In answering the Ã¢Â€Â˜Why now?Ã¢Â€Â™ question we wonder Ã¢Â€Â˜What next?Ã¢Â€Â™ as we expect the information equation of state to tend towards w = 0 in the future.c
Reduction operators of Burgers equation.
Pocheketa, Oleksandr A; Popovych, Roman O
2013-02-01
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.
Bitsadze, A V
1963-01-01
Equations of the Mixed Type compiles a series of lectures on certain fundamental questions in the theory of equations of mixed type. This book investigates the series of problems concerning linear partial differential equations of the second order in two variables, and possessing the property that the type of the equation changes either on the boundary of or inside the considered domain. Topics covered include general remarks on linear partial differential equations of mixed type; study of the solutions of second order hyperbolic equations with initial conditions given along the lines of parab
New application to Riccati equation
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.
Estimation and Control of Networked Distributed Parameter Systems: Application to Traffic Flow
Canepa, Edward
2016-11-01
The management of large-scale transportation infrastructure is becoming a very complex task for the urban areas of this century which are covering bigger geographic spaces and facing the inclusion of connected and self-controlled vehicles. This new system paradigm can leverage many forms of sensing and interaction, including a high-scale mobile sensing approach. To obtain a high penetration sensing system on urban areas more practical and scalable platforms are needed, combined with estimation algorithms suitable to the computational capabilities of these platforms. The purpose of this work was to develop a transportation framework that is able to handle different kinds of sensing data (e.g., connected vehicles, loop detectors) and optimize the traffic state on a defined traffic network. The framework estimates the traffic on road networks modeled by a family of Lighthill-Whitham-Richards equations. Based on an equivalent formulation of the problem using a Hamilton-Jacobi equation and using a semi-analytic formula, I will show that the model constraints resulting from the Hamilton-Jacobi equation are linear, albeit with unknown integer variables. This general framework solve exactly a variety of problems arising in transportation networks: traffic estimation, traffic control (including robust control), cybersecurity and sensor fault detection, or privacy analysis of users in probe-based traffic monitoring systems. This framework is very flexible, fast, and yields exact results. The recent advances in sensors (GPS, inertial measurement units) and microprocessors enable the development low-cost dedicated devices for traffic sensing in cities, 5 which are highly scalable, providing a feasible solution to cover large urban areas. However, one of the main problems to address is the privacy of the users of the transportation system, the framework presented here is a viable option to guarantee the privacy of the users by design.
Evaluating Equating Results: Percent Relative Error for Chained Kernel Equating
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…
Auxiliary equation method for solving nonlinear partial differential equations
Sirendaoreji,; Jiong, Sun
2003-03-31
By using the solutions of an auxiliary ordinary differential equation, a direct algebraic method is described to construct several kinds of exact travelling wave solutions for some nonlinear partial differential equations. By this method some physically important nonlinear equations are investigated and new exact travelling wave solutions are explicitly obtained with the aid of symbolic computation.
New Exact Solutions to NLS Equation and Coupled NLS Equations
FU Zun-Tao; LIU Shi-Da; LIU Shi-Kuo
2004-01-01
A transformation is introduced on the basis of the projective Riccati equations, and it is applied as an intermediate in expansion method to solve nonlinear Schrodinger (NLS) equation and coupled NLS equations. Many kinds of envelope travelling wave solutions including envelope solitary wave solution are obtained, in which some are found for the first time.
The compressible adjoint equations in geodynamics: equations and numerical assessment
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.
Elliptic Equation and New Solutions to Nonlinear Wave Equations
FU Zun-Tao; LIU Shi-Kuo; LIU Shi-Da
2004-01-01
The new solutions to elliptic equation are shown, and then the elliptic equation is taken as a transformationand is applied to solve nonlinear wave equations. It is shown that more kinds of solutions are derived, such as periodicsolutions of rational form, solitary wave solutions of rational form, and so on.
Partial differential equations
Sloan, D; Süli, E
2001-01-01
/homepage/sac/cam/na2000/index.html7-Volume Set now available at special set price ! Over the second half of the 20th century the subject area loosely referred to as numerical analysis of partial differential equations (PDEs) has undergone unprecedented development. At its practical end, the vigorous growth and steady diversification of the field were stimulated by the demand for accurate and reliable tools for computational modelling in physical sciences and engineering, and by the rapid development of computer hardware and architecture. At the more theoretical end, the analytical insight in
Hyperbolic partial differential equations
Lax, Peter D
2006-01-01
The theory of hyperbolic equations is a large subject, and its applications are many: fluid dynamics and aerodynamics, the theory of elasticity, optics, electromagnetic waves, direct and inverse scattering, and the general theory of relativity. This book is an introduction to most facets of the theory and is an ideal text for a second-year graduate course on the subject. The first part deals with the basic theory: the relation of hyperbolicity to the finite propagation of signals, the concept and role of characteristic surfaces and rays, energy, and energy inequalities. The structure of soluti
Dimensional Equations of Entropy
Sparavigna, Amelia Carolina
2015-01-01
Entropy is a quantity which is of great importance in physics and chemistry. The concept comes out of thermodynamics, proposed by Rudolf Clausius in his analysis of Carnot cycle and linked by Ludwig Boltzmann to the number of specific ways in which a physical system may be arranged. Any physics classroom, in its task of learning physics, has therefore to face this crucial concept. As we will show in this paper, the lectures can be enriched by discussing dimensional equations linked to the entropy of some physical systems.
Partial differential equations
Levine, Harold
1997-01-01
The subject matter, partial differential equations (PDEs), has a long history (dating from the 18th century) and an active contemporary phase. An early phase (with a separate focus on taut string vibrations and heat flow through solid bodies) stimulated developments of great importance for mathematical analysis, such as a wider concept of functions and integration and the existence of trigonometric or Fourier series representations. The direct relevance of PDEs to all manner of mathematical, physical and technical problems continues. This book presents a reasonably broad introductory account of the subject, with due regard for analytical detail, applications and historical matters.
Ordinary differential equations
Cox, William
1995-01-01
Building on introductory calculus courses, this text provides a sound foundation in the underlying principles of ordinary differential equations. Important concepts, including uniqueness and existence theorems, are worked through in detail and the student is encouraged to develop much of the routine material themselves, thus helping to ensure a solid understanding of the fundamentals required.The wide use of exercises, problems and self-assessment questions helps to promote a deeper understanding of the material and it is developed in such a way that it lays the groundwork for further
Savvidy, G K
1998-01-01
We discuss the basic properties of the gonihedric string and the problem of its formulation in continuum. We propose a generalization of the Dirac equation and of the corresponding gamma matrices in order to describe the gonihedric string. The wave function and the Dirac matrices are infinite-dimensional. The spectrum of the theory consists of particles and antiparticles of increasing half-integer spin lying on quasilinear trajectories of different slope. Explicit formulas for the mass spectrum allow to compute the string tension and thus demonstrate the string character of the theory.
Generalized estimating equations
Hardin, James W
2013-01-01
Generalized Estimating Equations, Second Edition updates the best-selling previous edition, which has been the standard text on the subject since it was published a decade ago. Combining theory and application, the text provides readers with a comprehensive discussion of GEE and related models. Numerous examples are employed throughout the text, along with the software code used to create, run, and evaluate the models being examined. Stata is used as the primary software for running and displaying modeling output; associated R code is also given to allow R users to replicat
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
Differential Equations as Actions
Ronkko, Mauno; Ravn, Anders P.
1997-01-01
We extend a conventional action system with a primitive action consisting of a differential equation and an evolution invariant. The semantics is given by a predicate transformer. The weakest liberal precondition is chosen, because it is not always desirable that steps corresponding to differential...... actions shall terminate. It is shown that the proposed differential action has a semantics which corresponds to a discrete approximation when the discrete step size goes to zero. The extension gives action systems the power to model real-time clocks and continuous evolutions within hybrid systems....
Classical mechanics systems of particles and Hamiltonian dynamics
Greiner, Walter
2010-01-01
This textbook Classical Mechanics provides a complete survey on all aspects of classical mechanics in theoretical physics. An enormous number of worked examples and problems show students how to apply the abstract principles to realistic problems. The textbook covers Newtonian mechanics in rotating coordinate systems, mechanics of systems of point particles, vibrating systems and mechanics of rigid bodies. It thoroughly introduces and explains the Lagrange and Hamilton equations and the Hamilton-Jacobi theory. A large section on nonlinear dynamics and chaotic behavior of systems takes Classical Mechanics to newest development in physics. The new edition is completely revised and updated. New exercises and new sections in canonical transformation and Hamiltonian theory have been added.
Effective action for the Regge processes in gravity
Lipatov, L.N. [Petersburg Nuclear Physics Institute, Gatchina, St. Petersburg (Russian Federation); Hamburg Univ. (Germany). 2. Inst. fuer Theoretische Physik
2011-05-15
It is shown, that the effective action for the reggeized graviton interactions can be formulated in terms of the reggeon fields A{sup ++} and A{sup --} and the metric tensor g{sub {mu}}{sub {nu}} in such a way, that it is local in the rapidity space and has the property of general covariance. The corresponding effective currents j{sup -} and j{sup +} satisfy the Hamilton-Jacobi equation for a massless particle moving in the gravitational field. These currents are calculated explicitly for the shock wave-like fields and a variation principle for them is formulated. As an application, we reproduce the effective lagrangian for the multi-regge processes in gravity together with the graviton Regge trajectory in the leading logarithmic approximation with taking into account supersymmetric contributions. (orig.)
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.
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.
Optimal traffic control in highway transportation networks using linear programming
Li, Yanning
2014-06-01
This article presents a framework for the optimal control of boundary flows on transportation networks. The state of the system is modeled by a first order scalar conservation law (Lighthill-Whitham-Richards PDE). Based on an equivalent formulation of the Hamilton-Jacobi PDE, the problem of controlling the state of the system on a network link in a finite horizon can be posed as a Linear Program. Assuming all intersections in the network are controllable, we show that the optimization approach can be extended to an arbitrary transportation network, preserving linear constraints. Unlike previously investigated transportation network control schemes, this framework leverages the intrinsic properties of the Halmilton-Jacobi equation, and does not require any discretization or boolean variables on the link. Hence this framework is very computational efficient and provides the globally optimal solution. The feasibility of this framework is illustrated by an on-ramp metering control example.
A Symbolic Computation Approach to Parameterizing Controller for Polynomial Hamiltonian Systems
Zhong Cao
2014-01-01
Full Text Available This paper considers controller parameterization method of H∞ control for polynomial Hamiltonian systems (PHSs, which involves internal stability and external disturbance attenuation. The aims of this paper are to design a controller with parameters to insure that the systems are H∞ stable and propose an algorithm for solving parameters of the controller with symbolic computation. The proposed parameterization method avoids solving Hamilton-Jacobi-Isaacs equations, and thus the obtained controllers with parameters are relatively simple in form and easy in operation. Simulation with a numerical example shows that the controller is effective as it can optimize H∞ control by adjusting parameters. All these results are expected to be of use in the study of H∞ control for nonlinear systems with perturbations.
Ambrosio, Luigi; Savaré, Giuseppe
2012-01-01
We provide a quick overview of various calculus tools and of the main results concerning the heat flow on compact metric measure spaces, with applications to spaces with lower Ricci curvature bounds. Topics include the Hopf-Lax semigroup and the Hamilton-Jacobi equation in metric spaces, a new approach to differentiation and to the theory of Sobolev spaces over metric measure spaces, the equivalence of the L^2-gradient flow of a suitably defined "Dirichlet energy" and the Wasserstein gradient flow of the relative entropy functional, a metric version of Brenier's Theorem, and a new (stronger) definition of Ricci curvature bound from below for metric measure spaces. This new notion is stable w.r.t. measured Gromov-Hausdorff convergence and it is strictly connected with the linearity of the heat flow.
Finite-time H∞ filtering for non-linear stochastic systems
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.
Tao CHENG; Frank L.LEWIS
2007-01-01
In this paper,neural networks are used to approximately solve the finite-horizon constrained input H-infiniy state feedback control problem.The method is based on solving a related Hamilton-Jacobi-Isaacs equation of the corresponding finite-horizon zero-sum game.The game value function is approximated by a neural network wlth timevarying weights.It is shown that the neural network approximation converges uniformly to the game-value function and the resulting almost optimal constrained feedback controller provides closed-loop stability and bounded L2 gain.The result is an almost optimal H-infinity feedback controller with time-varying coefficients that is solved a priori off-line.The effectiveness of the method is shown on the Rotational/Translational Actuator benchmark nonlinear control problem.
Wave propagation in electromagnetic media
Davis, Julian L
1990-01-01
This is the second work of a set of two volumes on the phenomena of wave propagation in nonreacting and reacting media. The first, entitled Wave Propagation in Solids and Fluids (published by Springer-Verlag in 1988), deals with wave phenomena in nonreacting media (solids and fluids). This book is concerned with wave propagation in reacting media-specifically, in electro magnetic materials. Since these volumes were designed to be relatively self contained, we have taken the liberty of adapting some of the pertinent material, especially in the theory of hyperbolic partial differential equations (concerned with electromagnetic wave propagation), variational methods, and Hamilton-Jacobi theory, to the phenomena of electromagnetic waves. The purpose of this volume is similar to that of the first, except that here we are dealing with electromagnetic waves. We attempt to present a clear and systematic account of the mathematical methods of wave phenomena in electromagnetic materials that will be readily accessi...
Feedback Control Method Using Haar Wavelet Operational Matrices for Solving Optimal Control Problems
Waleeda Swaidan
2013-01-01
Full Text Available Most of the direct methods solve optimal control problems with nonlinear programming solver. In this paper we propose a novel feedback control method for solving for solving affine control system, with quadratic cost functional, which makes use of only linear systems. This method is a numerical technique, which is based on the combination of Haar wavelet collocation method and successive Generalized Hamilton-Jacobi-Bellman equation. We formulate some new Haar wavelet operational matrices in order to manipulate Haar wavelet series. The proposed method has been applied to solve linear and nonlinear optimal control problems with infinite time horizon. The simulation results indicate that the accuracy of the control and cost can be improved by increasing the wavelet resolution.
De-Lei Sheng
2016-01-01
Full Text Available Unlike traditionally used reserves models, this paper focuses on a reserve process with dynamic income to study the reinsurance-investment problem for an insurer under Vasicek stochastic interest rate model. The insurer’s dynamic income is given by the remainder after a dynamic reward budget being subtracted from the insurer’s net premium which is calculated according to expected premium principle. Applying stochastic control technique, a Hamilton-Jacobi-Bellman equation is established and the explicit solution is obtained under the objective of maximizing the insurer’s power utility of terminal wealth. Some economic interpretations of the obtained results are explained in detail. In addition, numerical analysis and several graphics are given to illustrate our results more meticulous.
Feng, Z.W.; Zu, X.T. [University of Electronic Science and Technology of China, School of Physical Electronics, Chengdu (China); Li, H.L. [University of Electronic Science and Technology of China, School of Physical Electronics, Chengdu (China); Shenyang Normal University, College of Physics Science and Technology, Shenyang (China); Yang, S.Z. [China West Normal University, Physics and Space Science College, Nanchong (China)
2016-04-15
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. (orig.)
The Lax-Oleinik semi-group: a Hamiltonian point of view
Bernard, Patrick
2012-01-01
The Weak KAM theory was developed by Fathi in order to study the dynamics of convex Hamiltonian systems. It somehow makes a bridge between viscosity solutions of the Hamilton-Jacobi equation and Mather invariant sets of Hamiltonian systems, although this was fully understood only a posteriori. These theories converge under the hypothesis of convexity, and the richness of applications mostly comes from this remarkable convergence. In the present course, we provide an elementary exposition of some of the basic concepts of weak KAM theory. In a companion lecture, Albert Fathi exposes the aspects of his theory which are more directly related to viscosity solutions. Here on the contrary, we focus on dynamical applications, even if we also discuss some viscosity aspects to underline the connections with Fathi's lecture. The fundamental reference on Weak KAM theory is the still unpublished book of Albert Fathi \\textit{Weak KAM theorem in Lagrangian dynamics}. Although we do not offer new results, our exposition is o...
Quantum Tunneling of Massive Spin-1 Particles From Non-stationary Metrics
Sakalli, I
2016-01-01
Hawking radiation (HR) is invariant under the coordinate transformation, and it must be independent of the particle type emitting from the considered black hole (BH). From this fact, we focus on the HR of massive vector (spin-1) particles tunneling from Schwarzschild BH expressed in the Kruskal-Szekeres (KS) and dynamic Lemaitre (DL) coordinates. Using the Proca equation (PE) together with Hamilton-Jacobi (HJ) and 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. This is the first example for the HR of the massive vector particles tunneling from a four dimensional BH expressed in non-stationary regular coordinates.
The GUP effect on Hawking radiation of the 2 + 1 dimensional black hole
Gecim, Ganim; Sucu, Yusuf
2017-10-01
We investigate the Generalized Uncertainty Principle (GUP) effect on the Hawking radiation of the 2 + 1 dimensional Martinez-Zanelli black hole by using the Hamilton-Jacobi method. In this connection, we discuss the tunneling probabilities and Hawking temperature of the spin-1/2 and spin-0 particles for the black hole. Therefore, we use the modified Klein-Gordon and Dirac equations based on the GUP. Then, we observe that the Hawking temperature of the scalar and Dirac particles depend on not only the black hole properties, but also the properties of the tunneling particle, such as angular momentum, energy and mass. And, in this situation, we see that the tunneling probability and the Hawking radiation of the Dirac particle is different from that of the scalar particle.
Polymerization, the Problem of Access to the Saddle Point Approximation, and Thermodynamics
Morales-Técotl, Hugo A; Rastgoo, Saeed
2016-01-01
The saddle point approximation to the 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 can not 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 counter-term to the classical action, which is a solution of the corresponding Hamilton-Jacobi equation. In this work however, we seek an alternative solution to this problem via the polymer quantization which is motivated by the loop quantum gravity.
Regularity theory for mean-field game systems
Gomes, Diogo A; Voskanyan, Vardan
2016-01-01
Beginning with a concise introduction to the theory of mean-field games (MFGs), this book presents the key elements of the regularity theory for MFGs. It then introduces a series of techniques for well-posedness in the context of mean-field problems, including stationary and time-dependent MFGs, subquadratic and superquadratic MFG formulations, and distinct classes of mean-field couplings. It also explores stationary and time-dependent MFGs through a series of a-priori estimates for solutions of the Hamilton-Jacobi and Fokker-Planck equation. It shows sophisticated a-priori systems derived using a range of analytical techniques, and builds on previous results to explain classical solutions. The final chapter discusses the potential applications, models and natural extensions of MFGs. As MFGs connect common problems in pure mathematics, engineering, economics and data management, this book is a valuable resource for researchers and graduate students in these fields.
Symplectic tracking through straight three dimensional fields by a method of generating functions
M. Titze
2016-01-01
Full Text Available 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.
Tunneling method for Hawking radiation in the Nariai case
Belgiorno, F; Piazza, F Dalla
2016-01-01
Particle creation associated with the Hawking effect is calculated in the case of the charged Nariai solution, for the case both of the event horizon and of the cosmological horizon. We apply the Hamilton-Jacobi formalism, and, after a revisitation in the framework of particle creation in external fields {\\sl \\`a la} Nikishov, we consider in particular the case of a charged scalar field on the given background. Due to the knowledge of the exact solutions for the Klein-Gordon equations on Nariai manifold, and to their analytic properties on the extended manifold, we can corroborate the tunneling picture also by means of a direct computation of the flux of particles leaving the horizon.
3D level set methods for evolving fronts on tetrahedral meshes with adaptive mesh refinement
Morgan, Nathaniel R.; Waltz, Jacob I.
2017-05-01
The level set method is commonly used to model dynamically evolving fronts and interfaces. In this work, we present new methods for evolving fronts with a specified velocity field or in the surface normal direction on 3D unstructured tetrahedral meshes with adaptive mesh refinement (AMR). The level set field is located at the nodes of the tetrahedral cells and is evolved using new upwind discretizations of Hamilton-Jacobi equations combined with a Runge-Kutta method for temporal integration. The level set field is periodically reinitialized to a signed distance function using an iterative approach with a new upwind gradient. The details of these level set and reinitialization methods are discussed. Results from a range of numerical test problems are presented.
Comment on Hawking radiation and trapping horizons
Baier, Rudolf
2015-01-01
We consider dynamical black hole formation from a collapsing fluid described by a symmetric and flat FRW metric. Using the Hamilton-Jacobi method the local Hawking temperature for the formed trapping/apparent horizon is calculated. The local Hawking temperature depends on the tunneling path, which we take to be along a null direction $(\\Delta s=0)$. We find that the local Hawking temperature depends directly on the equation of state of the collapsing fluid. We argue that Hawking radiation by quantum tunnelling from future inner and future outer trapping horizons is possible. However, only radiation from a space-like dynamical horizon has a chance to be observed by an external observer. Some comparison to existing literature is made.
Intrinsic Optimal Control for Mechanical Systems on Lie Group
Chao Liu
2017-01-01
Full Text Available The intrinsic infinite horizon optimal control problem of mechanical systems on Lie group is investigated. The geometric optimal control problem is built on the intrinsic coordinate-free model, which is provided with Levi-Civita connection. In order to obtain an analytical solution of the optimal problem in the geometric viewpoint, a simplified nominal system on Lie group with an extra feedback loop is presented. With geodesic distance and Riemann metric on Lie group integrated into the cost function, a dynamic programming approach is employed and an analytical solution of the optimal problem on Lie group is obtained via the Hamilton-Jacobi-Bellman equation. For a special case on SO(3, the intrinsic optimal control method is used for a quadrotor rotation control problem and simulation results are provided to show the control performance.
Gravitinos Tunneling From Traversable Lorentzian Wormholes
Sakalli, I
2015-01-01
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-$3/2$) 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) ans\\"{a}tz 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.
Some reference formulas for the generating functions of canonical transformations
Anselmi, Damiano [Universita di Pisa, Dipartimento di Fisica ' ' Enrico Fermi' ' , Pisa (Italy); INFN, Sezione di Pisa, Pisa (Italy)
2016-02-15
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. (orig.)
Zongyuan Huang
2013-01-01
Full Text Available We present the model of corporate optimal investment with consideration of the influence of inflation and the difference between the market opening and market closure. In our model, the investor has three market activities of his or her choice: investment in project A, investment in project B, and consumption. The optimal strategy for the investor is obtained using the Hamilton-Jacobi-Bellman equation which is derived using the dynamic programming principle. Further along, a specific case, the Hyperbolic Absolute Risk Aversion case, is discussed in detail, where the explicit optimal strategy can be obtained using a very simple and direct method. At the very end, we present some simulation results along with a brief analysis of the relationship between the optimal strategy and other factors.
On holographic three point functions for GKP strings from integrability
Kazama, Yoichi
2011-01-01
Adapting the powerful integrability-based formalism invented previously for the calculation of gluon scattering amplitudes at strong coupling, we develop a method for computing the holographic three point functions for the large spin limit of Gubser-Klebanov- Polyakov (GKP) strings. Although many of the ideas from the gluon scattering problem can be transplanted with minor modifications, the fact that the information of the external states is now encoded in the singularities at the vertex insertion points necessitates several new techniques. Notably, we develop a new generalized Riemann bilinear identity, which allows one to express the area integral in terms of appropriate contour integrals in the presence of such singularities. We also give some general discussions on how semiclassical vertex operators for heavy string states should be constructed systematically from the solutions of the Hamilton-Jacobi equation.
Adaptation and migration of a population between patches
Mirrahimi, Sepideh
2012-01-01
A Hamilton-Jacobi formulation has been established previously for phenotypical structured population models where the solution concentrates as Dirac masses in the limit of small diffusion. Is it possible to extend this approach to spatial models? Are the limiting solutions still in the form of sum of Dirac masses? Does the presence of several habitats lead to polymorphic situations? We study the stationary solutions of a structured population model, while the population is structured by continuous phenotypical traits and discrete positions in space. Several habitable zones are possible and the growth term varies from one zone to another, for instance because of a change in the temperature. The individuals can migrate from one zone to another with a constant rate. The mathematical modeling of this problem, considering mutations between phenotypical traits and competitive interaction of individuals within each zone via a single resource, leads to a system of coupled parabolic integro-differential equations. We ...
Stochastic optimization in insurance a dynamic programming approach
Azcue, Pablo
2014-01-01
The main purpose of the book is to show how a viscosity approach can be used to tackle control problems in insurance. The problems covered are the maximization of survival probability as well as the maximization of dividends in the classical collective risk model. The authors consider the possibility of controlling the risk process by reinsurance as well as by investments. They show that optimal value functions are characterized as either the unique or the smallest viscosity solution of the associated Hamilton-Jacobi-Bellman equation; they also study the structure of the optimal strategies and show how to find them. The viscosity approach was widely used in control problems related to mathematical finance but until quite recently it was not used to solve control problems related to actuarial mathematical science. This book is designed to familiarize the reader on how to use this approach. The intended audience is graduate students as well as researchers in this area.
Dynamics-dependent symmetries in Newtonian mechanics
Holland, Peter
2014-01-01
We exhibit two symmetries of one-dimensional Newtonian mechanics whereby a solution is built from the history of another solution via a generally nonlinear and complex potential-dependent transformation of the time. One symmetry intertwines the square roots of the kinetic and potential energies and connects solutions of the same dynamical problem (the potential is an invariant function). The other symmetry connects solutions of different dynamical problems (the potential is a scalar function). The existence of corresponding conserved quantities is examined using Noethers theorem and it is shown that the invariant-potential symmetry is correlated with energy conservation. In the Hamilton-Jacobi picture the invariant-potential transformation provides an example of a field-dependent symmetry in point mechanics. It is shown that this transformation is not a symmetry of the Schroedinger equation.
Zhu, Yuanheng; Zhao, Dongbin; Yang, Xiong; Zhang, Qichao
2017-01-10
Sum of squares (SOS) polynomials have provided a computationally tractable way to deal with inequality constraints appearing in many control problems. It can also act as an approximator in the framework of adaptive dynamic programming. In this paper, an approximate solution to the H∞ optimal control of polynomial nonlinear systems is proposed. Under a given attenuation coefficient, the Hamilton-Jacobi-Isaacs equation is relaxed to an optimization problem with a set of inequalities. After applying the policy iteration technique and constraining inequalities to SOS, the optimization problem is divided into a sequence of feasible semidefinite programming problems. With the converged solution, the attenuation coefficient is further minimized to a lower value. After iterations, approximate solutions to the smallest L₂-gain and the associated H∞ optimal controller are obtained. Four examples are employed to verify the effectiveness of the proposed algorithm.
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.
Haantjes Manifolds and Integrable Systems
Tempesta, Piergiulio
2014-01-01
A general theory of integrable systems is proposed, based on the theory of Haantjes manifolds. We introduce the notion of symplectic-Haantjes manifold (or $\\omega \\mathcal{H}$ manifold), as the natural setting where the notion of integrability can be formulated. We propose an approach to the separation of variables for classical systems, related to the geometry of Haantjes manifolds. A special class of coordinates, called Darboux-Haantjes coordinates, will be constructed from the Haantjes structure associated with an integrable systems. They enable the additive separation of variables of the Hamilton-Jacobi equation. We also present an application of our approach to the study of some finite-dimensional integrable models, as the H\\'enon-Heiles systems and a stationary reduction of the KdV hierarchy.
Holographic Renormalization of general dilaton-axion gravity
Papadimitriou, Ioannis
2011-01-01
We consider a very general dilaton-axion system coupled to Einstein-Hilbert gravity in arbitrary dimension and we carry out holographic renormalization for any dimension up to and including five dimensions. This is achieved by developing a new systematic algorithm for iteratively solving the radial Hamilton-Jacobi equation in a derivative expansion. The boundary term derived is valid not only for asymptotically AdS backgrounds, but also for more general asymptotics, including non-conformal branes and Improved Holographic QCD. In the second half of the paper, we apply the general result to Improved Holographic QCD with arbitrary dilaton potential. In particular, we derive the generalized Fefferman-Graham asymptotic expansions and provide a proof of the holographic Ward identities.
De-Lei Sheng
2014-01-01
Full Text Available Defined contribution and annuity contract are merged into one pension plan to study both accumulation phase and distribution phase, which results in such effects that both phases before and after retirement being “defined”. Under the Heston’s stochastic volatility model, this paper focuses on mean-variance insurers with the return of premiums clauses to study the optimal time-consistent investment strategy for the DC pension merged with an annuity contract. Both accumulation phase before retirement and distribution phase after retirement are studied. In the time-consistent framework, the extended Hamilton-Jacobi-Bellman equations associated with the optimization problem are established. Applying stochastic optimal control technique, the time-consistent explicit solutions of the optimal strategies and the efficient frontiers are obtained. In addition, numerical analysis illustrates our results and also deepens our knowledge or understanding of the research results.
A representation formula for the Freidlin-Wentzell functional on the one dimensional torus
Faggionato, A
2010-01-01
Inspired by some recent results on fluctuation theory for piecewise deterministic Markov processes, we consider a generic diffusion on the 1D torus and give a simple representation formula for the large deviation rate functional of its invariant probability measure, in the limit of vanishing noise. Previously, this rate functional had been characterized by M.I. Freidlin and A.D. Wentzell as solution of a rather complex optimization problem. We discuss this last problem in full generality and show that it leads to our formula. Finally, we discuss some geometric and regularity properties of the rate functional. In particular, we prove a universality result showing that the rate functional is a viscosity solution of the stationary Hamilton--Jacobi equation associated to any Hamiltonian H satisfying weak suitable conditions.
The Geometry of the Semiclassical Wave Front Set for Schrödinger Eigenfunctions on the Torus
Cardin, Franco, E-mail: cardin@math.unipd.it; Zanelli, Lorenzo, E-mail: lzanelli@math.unipd.it [University of Padova, Department of Mathematics “Tullio Levi Civita” (Italy)
2017-06-15
This paper deals with the phase space analysis for a family of Schrödinger eigenfunctions ψ{sub ℏ} on the flat torus #Mathematical Double-Struck Capital T#{sup n} = (ℝ/2πℤ){sup n} by the semiclassical Wave Front Set. We study those ψ{sub ℏ} such that WF{sub ℏ}(ψ{sub ℏ}) is contained in the graph of the gradient of some viscosity solutions of the Hamilton-Jacobi equation. It turns out that the semiclassical Wave Front Set of such Schrödinger eigenfunctions is stable under viscous perturbations of Mean Field Game kind. These results provide a further viewpoint, and in a wider setting, of the link between the smooth invariant tori of Liouville integrable Hamiltonian systems and the semiclassical localization of Schrödinger eigenfunctions on the torus.
Techniques for developing approximate optimal advanced launch system guidance
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.
Finite-Time Anti-Disturbance Inverse Optimal Attitude Tracking Control of Flexible Spacecraft
Chutiphon Pukdeboon
2013-01-01
Full Text Available We propose a new robust optimal control strategy for flexible spacecraft attitude tracking maneuvers in the presence of external disturbances. An inverse optimal control law is designed based on a Sontag-type formula and a control Lyapunov function. An adapted extended state observer is used to compensate for the total disturbances. The proposed controller can be expressed as the sum of an inverse optimal control and an adapted extended state observer. It is shown that the developed controller can minimize a cost functional and ensure the finite-time stability of a closed-loop system without solving the associated Hamilton-Jacobi-Bellman equation directly. For an adapted extended state observer, the finite-time convergence of estimation error dynamics is proven using a strict Lyapunov function. An example of multiaxial attitude tracking maneuvers is presented and simulation results are included to show the performance of the developed controller.
Feng, Z W; Zu, X T
2016-01-01
We investigate the thermodynamics of Schwarzschild-Tangherlini black hole in the context of the generalized uncertainty principle. 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 Schwarzschild-Tangherlini black hole. Specially, the GUP effect becomes susceptible when the black hole evaporates down to the order of Planck scale, it makes the Hawking radiating stop and leads to remnant. It finds the endpoint of evaporation is a Planck-scale remnant with zero heat capacity. Those phenomenons imply that the GUP may give a way to solve the information. Besides, we also analysis the possibilities to find the black hole at LHC, and show that the black hole can not be produced in the recent LHC.
Jingyun Sun
2016-01-01
Full Text Available We consider a portfolio selection problem for a defined contribution (DC pension plan under the mean-variance criteria. We take into account the inflation risk and assume that the salary income process of the pension plan member is stochastic. Furthermore, the financial market consists of a risk-free asset, an inflation-linked bond, and a risky asset with Heston’s stochastic volatility (SV. Under the framework of game theory, we derive two extended Hamilton-Jacobi-Bellman (HJB equations systems and give the corresponding verification theorems in both the periods of accumulation and distribution of the DC pension plan. The explicit expressions of the equilibrium investment strategies, corresponding equilibrium value functions, and the efficient frontiers are also obtained. Finally, some numerical simulations and sensitivity analysis are presented to verify our theoretical results.
Precommitted Investment Strategy versus Time-Consistent Investment Strategy for a Dual Risk Model
Lidong Zhang
2014-01-01
Full Text Available We are concerned with optimal investment strategy for a dual risk model. We assume that the company can invest into a risk-free asset and a risky asset. Short-selling and borrowing money are allowed. Due to lack of iterated-expectation property, the Bellman Optimization Principle does not hold. Thus we investigate the precommitted strategy and time-consistent strategy, respectively. We take three steps to derive the precommitted investment strategy. Furthermore, the time-consistent investment strategy is also obtained by solving the extended Hamilton-Jacobi-Bellman equations. We compare the precommitted strategy with time-consistent strategy and find that these different strategies have different advantages: the former can make value function maximized at the original time t=0 and the latter strategy is time-consistent for the whole time horizon. Finally, numerical analysis is presented for our results.
Dynamic shortfall constraints for optimal portfolios
Bernd Luderer
2010-06-01
Full Text Available We consider a portfolio problem when a Tail Conditional Expectation constraint is imposed. The financial market is composed of n risky assets driven by geometric Brownian motion and one risk-free asset. The Tail Conditional Expectation is calculated for short intervals of time and imposed as risk constraint dynamically. The method of Lagrange multipliers is combined with the Hamilton-Jacobi-Bellman equation to insert the constraint into the resolution framework. A numerical method is applied to obtain an approximate solution to the problem. We find that the imposition of the Tail Conditional Expectation constraint when risky assets evolve following a log-normal distribution, curbs investment in the risky assets and diverts the wealth to consumption.
Optimal portfolio strategies under a shortfall constraint
D Akuma
2009-06-01
Full Text Available We impose dynamically, a shortfall constraint in terms of Tail Conditional Expectation on the portfolio selection problem in continuous time, in order to obtain optimal strategies. The financial market is assumed to comprise n risky assets driven by geometric Brownian motion and one risk-free asset. The method of Lagrange multipliers is combined with the Hamilton-Jacobi-Bellman equation to insert the constraint into the resolution framework. The constraint is re-calculated at short intervals of time throughout the investment horizon. A numerical method is applied to obtain an approximate solution to the problem. It is found that the imposition of the constraint curbs investment in the risky assets.
Optimal Investment and Excess of Loss Reinsurance with Short-selling Constraint
Sheng Liu; Jing-xiao Zhang
2011-01-01
This paper considers the optimal control problem with constraints for an insurance company. The risk process is assumed to be a jump-diffusion process and the risk can be reduced through an excess of loss (XL) reinsurance. In addition, the surplus can be invested in the financial market. In the financial market, the short-selling constraint is one of the main factors which make models more realistic. Our goal is to find the optimal investment-reinsurance policy without short-selling, which maximizes the expected exponential utility of the terminal wealth. By solving the corresponding Hamilton-Jacobi-Bellman equation, the value function and the optimal investment-reinsurance policy are given in a closed form.
Some reference formulas for the generating functions of canonical transformations
Anselmi, Damiano
2015-01-01
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.
Regularity Theory for Mean-Field Game Systems
Gomes, Diogo A.
2016-09-14
Beginning with a concise introduction to the theory of mean-field games (MFGs), this book presents the key elements of the regularity theory for MFGs. It then introduces a series of techniques for well-posedness in the context of mean-field problems, including stationary and time-dependent MFGs, subquadratic and superquadratic MFG formulations, and distinct classes of mean-field couplings. It also explores stationary and time-dependent MFGs through a series of a-priori estimates for solutions of the Hamilton-Jacobi and Fokker-Planck equation. It shows sophisticated a-priori systems derived using a range of analytical techniques, and builds on previous results to explain classical solutions. The final chapter discusses the potential applications, models and natural extensions of MFGs. As MFGs connect common problems in pure mathematics, engineering, economics and data management, this book is a valuable resource for researchers and graduate students in these fields.
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.
Zhong, Jianghong; Tian, Jie; Yang, Xin; Qin, Chenghu
2010-01-01
Only a planar bioluminescence image acquired from an ordinary cooled charge-coupled device (CCD) array every time, how to re-establish the three-dimensional small animal shape and light intensity distribution on the surface has become urgent to be solved as a bottleneck of bioluminescence tomography (BLT) reconstruction. In this paper, a finite element algorithm to solve the Dirichlet type problem for the first order Hamilton-Jacobi equation related to the shape-fromshading model is adopted. The algorithm outputting the globally maximal solution of the above problem avoids cumbersome boundary conditions on the interfaces between light and shadows and the use of additional information on the surface. The results of the optimization method are satisfied. It demonstrates the feasibility and potential of the finite element shape-fromshading (FE-SFS) model for reconstructing the small animal surface that lays one of key foundations for a fast low-cost application of the BLT in the next future.
Elementary symplectic topology and mechanics
Cardin, Franco
2015-01-01
This is a short tract on the essentials of differential and symplectic geometry together with a basic introduction to several applications of this rich framework: analytical mechanics, the calculus of variations, conjugate points & Morse index, and other physical topics. A central feature is the systematic utilization of Lagrangian submanifolds and their Maslov-Hörmander generating functions. Following this line of thought, first introduced by Wlodemierz Tulczyjew, geometric solutions of Hamilton-Jacobi equations, Hamiltonian vector fields and canonical transformations are described by suitable Lagrangian submanifolds belonging to distinct well-defined symplectic structures. This unified point of view has been particularly fruitful in symplectic topology, which is the modern Hamiltonian environment for the calculus of variations, yielding sharp sufficient existence conditions. This line of investigation was initiated by Claude Viterbo in 1992; here, some primary consequences of this theory are exposed in...
Sandro da Silva Fernandes
2008-01-01
Full Text Available A complete first-order analytical solution, which includes the short periodic terms, for the problem of optimal low-thrust limited-power transfers between arbitrary elliptic coplanar orbits in a Newtonian central gravity field is obtained through canonical transformation theory. The optimization problem is formulated as a Mayer problem of optimal control theory with Cartesian elements—position and velocity vectors—as state variables. After applying the Pontryagin maximum principle and determining the maximum Hamiltonian, classical orbital elements are introduced through a Mathieu transformation. The short periodic terms are then eliminated from the maximum Hamiltonian through an infinitesimal canonical transformation built through Hori method. Closed-form analytical solutions are obtained for the average canonical system by solving the Hamilton-Jacobi equation through separation of variables technique. For transfers between close orbits a simplified solution is straightforwardly derived by linearizing the new Hamiltonian and the generating function obtained through Hori method.
Optimal Proportional Reinsurance and Investment under Stochastic Interest Rates%随机利率下的再保险与投资策略
马威; 顾孟迪
2013-01-01
假设保险公司在考虑比例再保险下,并将盈余资金在无风险资产和风险型资产中进行配置,考虑了利率的随机性,建立了求解相应的HJB方程,并针对CARA效用函数求解HJB方程,得出最优的再保险比例和在各类资产中的投资比例.%Assume that the insurer allocates its surplus between risk-free asset and risky asset, we consider the proportional reinsurance policy with a stochastic interest rate and establish Hamilton-Jacobi-Bellman (HJB) equation. We solve it with a CARA utility and obtain optimal reinsurance and investment strategies.
Optimal proportional reinsurance and investment under stochastic interest rates%随机利率下的再保险与投资策略研究
马威; 顾孟迪
2011-01-01
Considering the proportional reinsurance policy, this paper assumed that the insurer would allocate its surplus between risk-free asset and risky asset, with a stochastic interest rate. Hamilton - Jacobi - Bellman (HJB) equation was established and solved with a CARA utility, and optimal reinsurance and investment strategies were obtained.%假设保险公司在考虑比例再保险下,并将盈余资金在无风险资产和风险型资产中进行配置,其中考虑了利率的随机性,建立了求解相应的HJB方程,并针对CARA效用函数求解HJB方程,得出最优的再保险比例和在各类资产中的投资比例.
Inflationary dynamics in the braneworld scenarios
Zhang Kai-Yuan; Wu Pu-Xun; Yu Hong-Wei
2013-01-01
We analyze the attractor behaviour of the inflation field in braneworld scenarios using the Hamilton-Jacobi formalism,where the Friedmann equation has the form of H2 =ρ + ε√2ρ0ρ or H2 =ρ + ερ2/2σ,with ε =+ 1.We find that in all models the linear homogeneous perturbation can decay exponentially as the scalar field rolls down its potential.However,in the case of a-ρ2 correction to the standard cosmology with ρ ＜ σ,the existence of an attractor solution requires (σ-ρ)/φ2 ＞ 1.Our results show that the perturbation decays more quickly in models with positive-energy correction than in the standard cosmology,which is opposite to the case of negative-energy correction.Thus,the positive-energy modification rather than the negative one can assist the inflation and widen the range of initial conditions.
A Mean-variance Problem in the Constant Elasticity of Variance (CEV) Mo del
Hou Ying-li; Liu Guo-xin; Jiang Chun-lan
2015-01-01
In this paper, we focus on a constant elasticity of variance (CEV) model and want to find its optimal strategies for a mean-variance problem under two con-strained controls: reinsurance/new business and investment (no-shorting). First, a Lagrange multiplier is introduced to simplify the mean-variance problem and the corresponding Hamilton-Jacobi-Bellman (HJB) equation is established. Via a power transformation technique and variable change method, the optimal strategies with the Lagrange multiplier are obtained. Final, based on the Lagrange duality theorem, the optimal strategies and optimal value for the original problem (i.e., the eﬃcient strategies and eﬃcient frontier) are derived explicitly.
Anomaly matching condition in two-dimensional systems
Dubinkin, O; Gubankova, E
2016-01-01
Based on Son-Yamamoto relation obtained for transverse part of triangle axial anomaly in ${\\rm QCD}_4$, we derive its analog in two-dimensional system. It connects the transverse part of mixed vector-axial current two-point function with diagonal vector and axial current two-point functions. Being fully non-perturbative, this relation may be regarded as anomaly matching for conductivities or certain transport coefficients depending on the system. We consider the holographic RG flows in holographic Yang-Mills-Chern-Simons theory via the Hamilton-Jacobi equation with respect to the radial coordinate. Within this holographic model it is found that the RG flows for the following relations are diagonal: Son-Yamamoto relation and the left-right polarization operator. Thus the Son-Yamamoto relation holds at wide range of energy scales.
Weak electromagnetic field admitting integrability in Kerr-NUT-(A)dS spacetimes
Kolar, Ivan
2015-01-01
We investigate properties of higher-dimensional generally rotating black-hole spacetimes, so called Kerr-NUT-(A)dS 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-(A)dS 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.
A LEVEL SET METHOD FOR MICROSTRUCTURE DESIGN OF COMPOSITE MATERIALS
MeiYnlin; WangXiaoming
2004-01-01
Based on a level set model and the homogenization theory, an optimization algorithm for finding the optimal configuration of the microstructure with specified properties is proposed, which extends current research on the level set method for structure topology optimization. The method proposed employs a level set model to implicitly describe the material interfaces of the microstructure and a Hamilton-Jacobi equation to continuously evolve the material interfaces until an optimal design is achieved. Meanwhile, the moving velocities of level set are obtained by conducting sensitivity analysis and gradient projection. Besides, how to handle the violated constraints is also discussed in the level set method for topological optimization, and a return-mapping algorithm is constructed. Numerical examples show that the method exhibits outstanding flexibility of handling topological changes and fidelity of material interface representation as compared with other conventional methods in literatures.
Optimal Investment and Consumption Decisions under the Constant Elasticity of Variance Model
Hao Chang
2013-01-01
Full Text Available We consider an investment and consumption problem under the constant elasticity of variance (CEV model, which is an extension of the original Merton’s problem. In the proposed model, stock price dynamics is assumed to follow a CEV model and our goal is to maximize the expected discounted utility of consumption and terminal wealth. Firstly, we apply dynamic programming principle to obtain the Hamilton-Jacobi-Bellman (HJB equation for the value function. Secondly, we choose power utility and logarithm utility for our analysis and apply variable change technique to obtain the closed-form solutions to the optimal investment and consumption strategies. Finally, we provide a numerical example to illustrate the effect of market parameters on the optimal investment and consumption strategies.