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...

With the development of biotechnology, high-throughput studies on protein-protein, protein-gene, and gene-gene interactions become possible and attract remarkable attention. To explore the interactions in dynamic gene regulatory networks, we propose a single-index ordinary differential equation (ODE) model and develop a variable selection procedure. We employ the smoothly clipped absolute devia...

Video generation models often operate under the assumption of fixed frame rates, which leads to suboptimal performance when it comes handling flexible rates (e.g., increasing rate more dynamic portion video as well missing frames). To resolve restricted nature existing models' ability handle arbitrary timesteps, we propose continuous-time by combining neural ODE (Vid-ODE) with pixel-level proce...