Related papers: Analytic representation of discrete and continuous…
In this article, we generalize the theory of discrete Lagrangian mechanics and variational integrators in two principal directions. First, we show that Lagrangian submanifolds of symplectic groupoids give rise to discrete dynamical systems,…
A general exact theory of autoresonance (self-sustained resonance) in both dissipative and Hamiltonian nonautonomous systems is presented. The equations that together govern the autoresonance solutions and excitations are derived with the…
We review some results on the logarithmic convexity for evolution equations, a well-known method in inverse and ill-posed problems. We start with the classical case of self-adjoint operators. Then, we analyze the case of analytic…
A general theorem on conservation laws for arbitrary difference equations is proved. The theorem is based on an introduction of an adjoint system related with a given difference system, and it does not require the existence of a difference…
Derivative-based algorithms are ubiquitous in statistics, machine learning, and applied mathematics. Automatic differentiation offers an algorithmic way to efficiently evaluate these derivatives from computer programs that execute relevant…
The Lorentz transformations are represented by Einstein velocity addition on the ball of relativistically admissible velocities. This representation is by projective maps. The Lie algebra of this representation defines the relativistic…
This paper is a continuation of [13], where new variational principles were introduced based on the concept of anti-selfdual (ASD) Lagrangians. We continue here the program of using these Lagrangians to provide variational formulations and…
In this paper, we will give a rigorous construction of the exact discrete Lagrangian formulation associated to a continuous Lagrangian problem. Moreover, we work in the setting of Lie groupoids and Lie algebroids which is enough general to…
We show that, under certain smoothness conditions, a Brownian martingale, when evaluated at a fixed time, can be represented via an exponential formula at a later time. The time-dependent generator of this exponential operator only depends…
We consider the general class of time-homogeneous stochastic dynamical systems, both discrete and continuous, and study the problem of learning a representation of the state that faithfully captures its dynamics. This is instrumental to…
We show that any second order linear ordinary diffrential equation with constant coefficients (including the damped and undumped harmonic oscillator equation) admits an exact discretization, i.e., there exists a difference equation whose…
The inverse problem of the calculus of variations consists in determining if the solutions of a given system of second order differential equations correspond with the solutions of the Euler-Lagrange equations for some regular Lagrangian.…
Differential equations based on physical principals are used to represent complex dynamic systems in all fields of science and engineering. Through repeated use in both academics and industry, these equations have been shown to represent…
We study the known coherent states of a quantum harmonic oscillator from the standpoint of the original developed noncommutative integration method for linear partial differential equations. The application of the method is based on the…
There is developed a differential-algebraic approach to studying the representations of commuting differentiations in functional differential rings under nonlinear differential constraints. An example of the differential ideal with the only…
In this paper, we propose a distributed algorithm for solving large-scale separable convex problems using Lagrangian dual decomposition and the interior-point framework. By adding self-concordant barrier terms to the ordinary Lagrangian, we…
This work develops a rigorous numerical framework for solving time-dependent Optimal Control Problems (OCPs) governed by partial differential equations, with a particular focus on biomedical applications. The approach deals with…
A Lagrangian is introduced which includes the coupling between magnetic moments $\mathbf{m}$ and the degrees of freedom $\boldsymbol{\sigma}$ of a reservoir. In case the system-reservoir coupling breaks the time reversal symmetry the…
We address the problem of constructing numerical integrators for nonholonomic Lagrangian systems that enjoy appropriate discrete versions of the geometric properties of the continuous flow, including the preservation of energy. Building on…
The existence and analyticity of solutions to linear systems of moment differential equations with analytic coefficients is studied. The relation of solutions of such systems with respect to linear moment differential equations is…