Related papers: On the Nagumo uniqueness theorem
We provide a sufficient geometric condition for $\mathbb{R}^n$ to be countably $(\mu,m)$ rectifiable of class $\mathscr{C}^{1,\alpha}$ (using the terminology of Federer), where $\mu$ is a Radon measure having positive lower density and…
We present a rather unknown version of the change of variables formula for non-autonomous functions. We will show that this formula is equivalent to Green's Theorem for regions of the plane bounded by the graphs of two continuously…
We obtain a uniform stability of recovering entire functions of a special form from their zeros. To this form, one can reduce the characteristic determinants of strongly regular differential operators and pencils of the first and the second…
In this paper, we give an algorithm for finding general rational solutions of a given first-order ODE with parametric coefficients that occur rationally. We present an analysis, complete modulo Hilbert's irreducibility problem, of the…
We study the connections between ordinary differential equations and optimization algorithms in a non-Euclidean setting. We propose a novel accelerated algorithm for minimising convex functions over a convex constrained set. This algorithm…
Differential equations are frequently used in engineering domains, such as modeling and control of industrial systems, where safety and performance guarantees are of paramount importance. Traditional physics-based modeling approaches…
A novel static algorithm is proposed for numerical reparametrization of periodic planar curves. The method identifies a monitor function of the arclength variable with the true curvature of an open planar curve and considers a simple…
We develop a numerical method to reconstruct systems of ordinary differential equations (ODEs) from time series data without {\it a priori} knowledge of the underlying ODEs using sparse basis learning and sparse function reconstruction. We…
In their 1968 paper Fujita and Watanabe considered the issue of uniqueness of the trivial solution of semilinear parabolic equations with respect to the class of bounded, non-negative solutions. In particular they showed that if the…
It is hereby established that, in Euclidean spaces of finite dimension, bounded self-contracted curves have finite length. This extends the main result of Daniilidis, Ley, and Sabourau (J. Math. Pures Appl. 2010) concerning continuous…
We develop spectral methods for ODEs and operator eigenvalue problems that are based on a least-squares formulation of the problem. The key tool is a method for rectangular generalized eigenvalue problems, which we extend to quasimatrices…
A spatio-temporal version of the well-known Omori-Utsu law of aftershock sequences is proposed. This 'diffusive Omori-Utsu law' satisfies a nonlinear partial differential equation (PDE). A similarity reduction is obtained that reduces the…
It is shown that $N$ points on a real algebraic curve of degree $n$ in $\mathbb{R}^d$ always determine $\gtrsim_{n,d}N^{1+\frac{1}{4}}$ distinct distances, unless the curve is a straight line or the closed geodesic of a flat torus. In the…
We establish elliptic regularity for nonlinear inhomogeneous Cauchy-Riemann equations under minimal assumptions, and give a counterexample in a borderline case. In some cases where the inhomogeneous term has a separable factorization, the…
Solutions of Rough Differential Equations (RDE) may be defined as paths whose increments are close to an approximation of the associated flow. They are constructed through a discrete scheme using a non-linear sewing lemma. In this article,…
Ordinary differential equations (ODEs), via their induced flow maps, provide a powerful framework to parameterize invertible transformations for the purpose of representing complex probability distributions. While such models have achieved…
For the ordinary differential equation (ODE) $\dot{x}(t) = f(t,x)$, $x(0) = x_0$, $t\geq 0$, $x\in R^d$, assume $f$ to be at least continuous in $t$ and locally Lipshitz in $x$, and if necessary, several times continuously differentiable in…
This papers studies the expressive and computational power of discrete Ordinary Differential Equations (ODEs). It presents a new framework using discrete ODEs as a central tool for computation and provides several implicit characterizations…
We give a definition of integration by quadratures of first-order ordinary differential equations, and recover a little known result by Maximovic which states that a first-order ordinary differential equation can be integrated by…
This paper is devoted to order-one explicit approximations of random periodic solutions to multiplicative noise driven stochastic differential equations (SDEs) with non-globally Lipschitz coefficients. The existence of the random periodic…