Ordinary differential equations (ODEs) are the primary means to modelling dynamical systems in many natural and engineering sciences. The number of equations required to describe a system with high heterogeneity limits our capability of effectively performing analyses. This has motivated a large body of research, across many disciplines, into abstraction techniques that provide smaller ODE syst...

We study accelerated mirror descent dynamics in continuous and discrete time. Combining the original continuous-time motivation of mirror descent with a recent ODE interpretation of Nesterov’s accelerated method, we propose a family of continuous-time descent dynamics for convex functions with Lipschitz gradients, such that the solution trajectories converge to the optimum at a O(1/t2) rate. We...

This paper discusses the discrete analogue of the gradient of a function and shows how discrete gradients can be used in the numerical integration of ordinary diieren-tial equations (ODE's). Given an ODE and one or more rst integrals (i.e., constants of the motion) and/or Lyapunov functions, it is shown that the ODE can be rewritten as a `linear-gradient system.' Discrete gradients are used to ...

A systematic algorithm for building integrating factors of the form μ(x, y), μ(x, y) or μ(y, y) for second order ODEs is presented. The algorithm can determine the existence and explicit form of the integrating factors themselves without solving any differential equations, except for a linear ODE in one subcase of the μ(x, y) problem. Examples of ODEs not having point symmetries are shown to be...

For the complicated input-output systems with nonlinearity and stochasticity, Deep State Space Models (SSMs) are effective for identifying in latent state space, which of great significance representation, forecasting, planning online scenarios. However, most SSMs designed discrete-time sequences inapplicable when observations irregular time. To solve problem, we propose a novel continuous-time...

We consider a linear 2 × 2 matrix ODE with two coalescing regular singularities. This coalescence is restricted with an isomonodromy condition with respect to the distance between the merging singularities in a way consistent with the ODE. In particular, a zero-distance limit for the ODE exists. The monodromy group of the limiting ODE is calculated in terms of the original one. This coalescing ...

We give a necessary and sufficient condition for an algebraic ODE to have a rational type general solution. For a first order autonomous ODE F = 0, we give an exact degree bound for its rational solutions, based on the connection between rational solutions of F = 0 and rational parameterizations of the plane algebraic curve defined by F = 0. For a first order autonomous ODE, we further give a p...

In this paper we review many aspects of the well-posedness theory for the Cauchy problem for the continuity and transport equations and for the ordinary differential equation (ODE). In this framework, we deal with velocity fields that are not smooth, but enjoy suitable ‘weak differentiability’ assumptions. We first explore the connection between the partial differential equation (PDE) and the O...

The analysis of large-scale software is difficult in the absence of supporting tools, due to the sheer size and complexity of today’s systems. We present CODECITY, a language-independent interactive 3D visualization tool for the analysis of large object-oriented software systems. Using a city metaphor, it depicts classes as buildings and packages as districts of a “software city”. We start pres...

We study a number of embedded DSLs for autonomous ordinary differential equations (autonomous ODEs) in Haskell. A naive implementation based on the lazy tower of derivatives is straightforward but has serious time and space leaks due to the loss of sharing when handling cyclic and infinite data structures. In seeking a solution to fix this problem, we explore a number of DSLs ranging from shall...