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Neural ordinary differential equations (Neural ODEs) propose the idea that a sequence of layers in a neural network is just a discretisation of an ODE, and thus can instead be directly modelled by a parameterised ODE. This idea has had…
The initial-boundary value problems for linear non-autonomous first order evolution equations are examined. Our assumptions provide a unified treatment which is applicable to many situations, where the domains of the operators may change…
In this paper free harmonic analysis tools are used to study parabolic iteration in the complex upper half-plane. The main result here is a complete characterization for the norming constants in the monotonic central limit theorem. This…
The method of self-consistent expansions is a powerful tool for handling strong coupling problems that might otherwise be beyond the reach of perturbation theory, providing surprisingly accurate approximations even at low order. First…
We introduce and study the filtration on the space of automorphic functions (in the everywhere unramified situation for the function field case) obtained by transferring the filtration on the spectral side of the classical Langlands…
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…
We give estimates for the convolution product of an arbitrary number of endlessly continuable functions. This allows us to deal with nonlinear operations for the corresponding resurgent series, e.g. substitution into a convergent power…
We give an introduction to discrete functional analysis techniques for stationary and transient diffusion equations. We show how these techniques are used to establish the convergence of various numerical schemes without assuming…
The boundary value problems for linear and nonlinear singular degenerate differential-operator equations are studied. We prove a well-posedeness of linear problem and optimal regularity result for the nonlinear problem which occur in fluid…
We apply the zero bias transformation to deduce a recursive asymptotic expansion formula for expectation of functions of sum of independent random variables in terms of normal expectations and we discuss the remainder term estimations.
We prove the existence, uniqueness, and sharp bilateral pointwise estimates for positive bounded solutions to the Lane--Emden type problem \[ \begin{cases} L u = \sum\limits_{i=1}^{m}\sigma_{i} u^{q_{i}}+\sigma_0, \quad u\geq0 & \text{in }…
This paper develops a characterisation of when solutions of forced second order linear differential equations converge to the zero solution of the asymptotically stable and unforced second order equation, or when the solution is bounded,…
Large-time asymptotic properties of solutions to a class of semilinear stochastic wave equations with damping in a bounded domain are considered. First an energy inequality and the exponential bound for a linear stochastic equation are…
We apply the asymptotic iteration method (AIM) [J. Phys. A: Math. Gen. 36, 11807 (2003)] to solve new classes of second-order homogeneous linear differential equation. In particular, solutions are found for a general class of eigenvalue…
Automatic algorithms attempt to provide approximate solutions that differ from exact solutions by no more than a user-specified error tolerance. This paper describes an automatic, adaptive algorithm for approximating the solution to a…
We proposed a framework for solving inverse problems in differential equations based on neural networks and automatic differentiation. Neural networks are used to approximate hidden fields. We analyze the source of errors in the framework…
We discuss linear autonomous evolution equations on function spaces which have the property that a positive initial value leads to a solution which initially changes sign, but then becomes - and stays - positive again for sufficiently large…
Notions of convergence and continuity specifically adapted to Riesz ideals I of the space of continuous real-valued functions on a Lindel\"of locally compact Hausdorff space are given, and used to prove Stone-Weierstra{\ss}-type theorems…
This paper studies the dynamics of families of monotone nonautonomous neutral functional differential equations with nonautonomous operator, of great importance for their applications to the study of the long-term behavior of the…
We propose a new approach to the combinatorial interpretations of linearization coefficient problem of orthogonal polynomials. We first establish a difference system and then solve it combinatorially and analytically using the method of…