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In multi-phase fluid flow, fluid-structure interaction, and other applications, partial differential equations (PDEs) often arise with discontinuous coefficients and singular sources (e.g., Dirac delta functions). These complexities arise…
We study the classical evaluation problem for regular path queries: Given an edge-labeled graph and a regular path query, compute the set of pairs of vertices that are connected by paths that match the query. The Product Graph (PG) is the…
First-order logic (FO) can express many algorithmic problems on graphs, such as the independent set and dominating set problem, parameterized by solution size. On the other hand, FO cannot express the very simple algorithmic question of…
In the first part of the paper, we classify linear integrable (multi-dimensionally consistent) quad-equations on bipartite isoradial quad-graphs in $\mathbb C$, enjoying natural symmetries and the property that the restriction of their…
In this paper, we continue the study of the Davey-Stewartson system which is one of the most important$(2+1)$ dimensional integrable models. As we showed in the previous paper, the dDS (dispersionless Davey-Stewartson) system arises from…
We construct linear Hamilton systems without usual dichotomy property. The Ljapunov spectra of these systems are unfamiliar and conflicting, the behaviour of trajectories is very complicated. The paper's subject refers to some problems of…
Transformations of differential equations to other equivalent equations play a central role in many routines for solving intricate equations. A class of differential equations that are particularly amenable to solution techniques based on…
The purpose of this paper is to present an example of an Ordinary Differential Equation $x'=F(x)$ in the infinite-dimensional Hilbert space $\ell^2$ with $F$ being of class $\mathcal{C}^1$ in the Fr\'{e}chet sense, such that the origin is…
Vector fields and line fields, their counterparts without orientations on tangent lines, are familiar objects in the theory of dynamical systems. Among the techniques used in their study, the Morse--Smale decomposition of a (generic) field…
We consider the problem of complex interpolation of certain Hardy-type subspaces of K\"othe function spaces. For example, suppose $X_0$ and $X_1$ are K\"othe function spaces on the unit circle $\bold T,$ and let $H_{X_0}$ and $H_{X_1}$ be…
This paper is devoted to Hardy type inequalities with remainders for compactly supported smooth functions on open sets in the Euclidean space. We establish new inequalities with weight functions depending on the distance function to the…
A theorem of Ding, Oporowski, Oxley, and Vertigan implies that any sufficiently large twin-free graph contains a large matching, a co-matching, or a half-graph as a semi-induced subgraph. The sizes of these unavoidable patterns are measured…
We consider a natural generalisation of the Painlev\'e property and use it to identify the known integrable cases of the Lane-Emden equation with a real positive index. We classify certain first-order ordinary differential equations with…
We study orbit-finite systems of linear equations, in the setting of sets with atoms. Our principal contribution is a decision procedure for solvability of such systems. The procedure works for every field (and even commutative ring) under…
The separation of two sets (or more specific of two cones) plays an important role in different fields of mathematics such as variational analysis, convex analysis, convex geometry, optimization. In the paper, we derive some new results for…
In this paper, we survey constructions of and nonexistence results on combinatorial/geometric structures which arise from unions of cyclotomic classes of finite fields. In particular, we survey both classical and recent results on…
The enumeration of independent sets of regular graphs is of interest in statistical mechanics, as it corresponds to the solution of hard-particle models. In 2004, it was conjectured by Fendleyet al. that for some rectangular grids, with…
We study first order equations of continuity and transport type on metric spaces of martingale dimension one, including finite metric graphs, p.c.f. self-similar sets and classical Sierpi\'nski carpets. On such spaces solutions of the…
We consider a relatively new hybrid generalized F-contraction involving a pair of mappings and utilize the same to prove a common fixed point theorem for a hybrid pair of occasionally coincidentally idempotent mappings satisfying…
We extend the potential theory on almost minimzers from Part 1. We introduce so-called Hardy structures to study many classical operators using the tools from part 1. Furthermore, we show that for a naturally defined operator L, minimal…