Related papers: Discrete-to-Continuous Extensions: Lov\'asz extens…

We give a presentation theorem for continuous first-order logic and Metric Abstract Elementary classes in terms of $L_{\omega_1, \omega}$ and Abstract Elementary Classes, respectively. This presentation is accomplished by analyzing dense…

Following the previous authors works (joint with I.A.Dynnikov) we develop a theory of the discrete analogs of the differential-geometrical (DG) connections in the triangulated manifolds. We study a nonstandard discretization based on the…

The Lagrange problem is established in the discrete field theory subject to constraints with values in a Lie group. For the admissible sections that satisfy a certain regularity condition, we prove that the critical sections of such…

Homotopy connectedness theorems for complex submanifolds of homogeneous spaces (sometimes referred to as theorems of Barth-Lefshetz type) have been established by a number of authors. Morse Theory on the space of paths lead to an elegant…

This article deals with a continuous closed 1-form defined on a CW-complex. In particular, we show Lusternik-Schnirelmann type theory on continuous closed 1-forms which is related to gradient-like flows. M.Farber defined a continuous closed…

In this work we provide a survey of Fuglede's flux extensions of first order partial differential operators, a concept largely forgotten today. A long the way we also survey the classical weak and strong extensions of PDE operators and the…

We study sublevel set and superlevel set persistent homology on discrete functions through the perspective of finite ordered sets of both linearly ordered and cyclically ordered domains. Finite ordered sets also serve as the codomain of our…

Witten-Helffer-Sj\"ostrand theory is an addition to Morse theory and Hodge-de Rham theory for Riemannian manifolds and considerably improves on them by injecting some spectral theory of elliptic operators. It can serve as a general tool to…

Let $\phi^t$ be a continuous flow on a metric space $X$ and $I$ be an isolated invariant set with an index pair $(N,L)$ and a Morse decomposition $\{M_i\}^n_{i=1}$. For every category $\nu$ on $N/L$, we prove that $\nu(N/L)\leq…

We use discrete Morse theory to give a new proof of the Degree Theorem in Auter space A_n. There is a filtration of A_n into subspaces A_{n,k} using the degree of a graph, and the Degree Theorem says that each A_{n,k} is (k-1)-connected.…

In this paper, we introduce a modification of the Szasz-Mirakjan-Kantorovich operators as well as Stancu operators [9] (or a Dunkl generalization of modified Szasz-Mirakjan-Kantrovich operators [5]) which preserve the linear functions.…

We study the uniqueness of optimal solutions to extremal graph theory problems. Lovasz conjectured that every finite feasible set of subgraph density constraints can be extended further by a finite set of density constraints so that the…

This is a survey paper on Morse theory and the existence problem for closed geodesics. The free loop space plays a central role, since closed geodesics are critical points of the energy functional. As such, they can be analyzed through…

This paper addresses the behavior of the Lov\'asz number for dense random circulant graphs. The Lov\'asz number is a well-known semidefinite programming upper bound on the independence number. Circulant graphs, an example of a Cayley graph,…

Kim defined a very general combinatorial abstraction of the diameter of polytopes called subset partition graphs to study how certain combinatorial properties of such graphs may be achieved in lower bound constructions. Using Lov\'asz'…

The goal of this paper is to present a certain generalization of the classical Kontinuit\"atssatz of Behnke for holomorphic/meromorphic functions in terms of the lift to the envelope of holomorphy. We consider two non-equivalent…

We extend discrete Morse-Bott theory to the setting of loop-free (or acyclic) categories. First of all, we state a homological version of Quillen's Theorem A in this context and introduce the notion of cellular categories. Second, we…

We develop a theory of limits for sequences of dense abstract simplicial complexes, where a sequence is considered convergent if its homomorphism densities converge. The limiting objects are represented by stacks of measurable [0,1]-valued…

Applying a general categorical construction for the extension of dualities, we present a new proof of the Fedorchuk duality between the category of compact Hausdorff spaces with their quasi-open mappings and the category of complete normal…

We extend the BMS(4) group by adding logarithmic supertranslations. This is done by relaxing the boundary conditions on the metric and its conjugate momentum at spatial infinity in order to allow logarithmic terms of carefully designed form…