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In this paper we consider convex subsets of locally-convex topological vector spaces. Given a fixed point in such a convex subset, we show that there exists a curve completely contained in the convex subset and leaving the point in a given…

In real-world systems, the relationships and connections between components are highly complex. Real systems are often described as networks, where nodes represent objects in the system and edges represent relationships or connections…

The method of alternating projections involves orthogonally projecting an element of a Hilbert space onto a collection of closed subspaces. It is known that the resulting sequence always converges in norm if the projections are taken…

We develop a homotopy theory of directed graphs based on cubical homotopy groups, also referred to as A-groups or reduced GLMY homotopy groups. Localizing the category of directed graphs at morphisms that induce isomorphisms on these groups…

Let \Fc be a holomorphic foliation by curves defined in a neighborhood of 0 in \C^2 having 0 as a hyperbolic singularity. Let T be a harmonic current directed by \Fc which does not give mass to any of the two separatrices. Then we show that…

We explore the well-known Jaeger's directed cycle double cover conjecture which is equivalent to the assertion that every cubic bridgeless graph has an embedding on a closed orientable surface with no dual loop. We associate each cubic…

Given a graph, does there exist an orientation of the edges such that the resulting directed graph is strongly connected? Robbins' theorem [Robbins, Am. Math. Monthly, 1939] states that such an orientation exists if and only if the graph is…

This paper shows that finitely additive measures occur naturally in very general Divergence Theorems. The main results are two such theorems. The first proves the existence of pure normal measures for sets of finite perime- ter, which yield…

We prove a counterpart of the log-convex density conjecture in the hyperbolic plane.

Answering a question of Diestel, we develop a topological notion of gammoids in infinite graphs which, unlike traditional infinite gammoids, always define a matroid. As our main tool, we prove for any infinite graph $G$ with vertex sets $A$…

Let $X$ be an infinite Riemann surface equipped with its conformal hyperbolic metric such that the action of the covering group $\pi_1(X)$ on $\tilde{X}$ is of the first kind-i.e., the surface $X$ is equal to its convex core. We first prove…

We consider the systems of diffusion-orthogonal polynomials, defined in the work [1] of D. Bakry, S. Orevkov and M. Zani and (particularly) explain why these systems with boundary of maximal possible degree should always come from the…

In this article we have studied bicomplex valued measurable functions on an arbitrary measurable space. We have established the bicomplex version of Lebesgue's dominated convergence theorem and some other results related to this theorem.…

A directed hypergraph is a hypergraph in which the vertex set of each hyperedge is partitioned into two disjoint parts, a head and a tail. Keszegh and P\'alv\"olgyi posed the following conjecture. Let $H$ be a directed hypergraph such that…

We consider homogenization of Dirichlet problems for semilinear elliptic systems with non-smooth data. We suppose that the diffusion tensors H-converge if the homogenization parameter tends to zero. Our result is of implicit function…

We formulate a very general conjecture relating the analytical invariants of a normal surface singularity to the Seiberg-Witten invariants of its link provided that the link is a rational homology sphere. As supporting evidence, we…

We study monotone simultaneous embeddings of upward planar digraphs, which are simultaneous embeddings where the drawing of each digraph is upward planar, and the directions of the upwardness of different graphs can differ. We first…

We study the existence of directed Hamilton cycles in random digraphs with $m$ edges where we condition on minimum in- and out-degree $\d \ge k+1$, where $k \ge 1$. Denote such a random graph by $D_{n,m}^{(\delta\geq k+1)}$. Let $m=cn$ and…

In this paper, we introduce a parameterized discrete curvature ($\alpha$-curvature) for piecewise linear metrics on polyhedral surfaces, which is a generalization of the classical discrete curvature. A discrete uniformization theorem is…

We propose a "decomposition method" to prove non-asymptotic bound for the convergence of empirical measures in various dual norms. The main point is to show that if one measures convergence in duality with sufficiently regular observables,…