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We discuss a dynamical systems perspective on discrete optimization. Departing from the fact that many combinatorial optimization problems can be reformulated as finding low energy spin configurations in corresponding Ising models, we…
This paper can be seen as an attempt of rethinking the {\em Extra-Gradient Philosophy} for solving Variational Inequality Problems. We show that the properly defined {\em Reduced Gradients} can be used instead for finding approximate…
Probabilistic graphical models that encode indistinguishable objects and relations among them use first-order logic constructs to compress a propositional factorised model for more efficient (lifted) inference. To obtain a lifted…
A class of two-dimensional systems of second-order ordinary differential equations is identified in which a system requires fewer Lie point symmetries than required to solve it. The procedure distinguishes among those which are…
Numerical solving differential equations with fractional derivatives requires elimination of the singularity which is inherent in the standard definition of fractional derivatives. The method of integration by parts to eliminate this…
The issue of inheriting periodicity of an exact solution of a dynamic system by a difference scheme is considered. It is shown that some difference schemes (midpoint scheme, Kahan scheme) in some special cases provide approximate solutions…
Background: Many attempts have been made to resolve in time the folding of model proteins in computer simulations. Different computational approaches have emerged. Some of these approaches suffer from the insensitivity to the geometrical…
We present algorithms to solve coupled systems of linear differential equations, arising in the calculation of massive Feynman diagrams with local operator insertions at 3-loop order, which do {\it not} request special choices of bases.…
We obtain closed-form solutions of several inhomogeneous Lienard equations by the factorization method. The two factorization conditions involved in the method are turned into a system of first-order differential equations containing the…
The implicit compact finite-difference scheme was developed for evolutionary partial differential parabolic and Schr\"odinger-type equations and systems with a weak nonlinearity. To make a temporal step of the compact implicit scheme we…
This paper investigates model-order reduction methods for geometrically nonlinear structures. The parametrisation method of invariant manifolds is used and adapted to the case of mechanical systems expressed in the physical basis, so that…
Lie symmetries of systems of second-order linear ordinary differential equations with constant coefficients are exhaustively described over both the complex and real fields. The exact lower and upper bounds for the dimensions of the maximal…
This paper introduces a novel approach for the construction of bulk--surface splitting schemes for semi-linear parabolic partial differential equations with dynamic boundary conditions. The proposed construction is based on a reformulation…
The mathematics of K-conserving functional differentiation, with K being the integral of some invertible function of the functional variable, is clarified. The most general form for constrained functional derivatives is derived from the…
We describe a compatible finite element discretisation for the shallow water equations on the rotating sphere, concentrating on integrating consistent upwind stabilisation into the framework. Although the prognostic variables are velocity…
Fractional Differential Equations (FDEs) are essential tools for modelling complex systems in science and engineering. They extend the traditional concepts of differentiation and integration to non-integer orders, enabling a more precise…
Topologically ordered systems are characterized by topological invariants that are often calculated from the momentum space integration of a certain function that represents the curvature of the many-body state. The curvature function may…
In this work, in order to obtain higher-order schemes for solving forward backward stochastic differential equations, we adopt the high-order multi-step method in [W. Zhao, Y. Fu and T. Zhou, SIAM J. Sci. Comput., 36(4) (2014),…
The vast majority of real world classification problems are imbalanced, meaning there are far fewer data from the class of interest (the positive class) than from other classes. We propose two machine learning algorithms to handle highly…
Based on an original classification of differential equations by types of regular Lie group actions, we offer a systematic procedure for describing partial differential equations with prescribed symmetry groups. Using a new powerful…