Properties of abelian categories via recollements

A recollement is a decomposition of a given category (abelian or triangulated) into two subcategories with functorial data that enables the glueing of structural information. This paper is dedicated to investigating the behaviour under glueing of some basic properties of abelian categories (well-poweredness, Grothendieck's axioms AB3, AB4 and AB5, existence of a generator) in the presence of a recollement. In particular, we observe that in a recollement of a Grothendieck abelian category the other two categories involved are also Grothendieck abelian and, more significantly, we provide an example where the converse does not hold and explore multiple sufficient conditions for it to hold.     

On the Hilbert Evolution Algebras of a Graph

Evolution algebras are a special class of nonassociative algebras exhibiting connections with various fields of mathematics. Hilbert evolution algebras generalize the concept in the framework of Hilbert spaces. This allows us to deal with a wide class of infinite-dimensional spaces. We study Hilbert evolution algebras associated to a graph. Inspired by the definitions of evolution algebras we define the Hilbert evolution algebra that is associated to a given graph and the Hilbert evolution algebra that is associated to the symmetric random walk on a graph. For a given graph, we provide the conditions for these structures to be or not to be isomorphic. Our definitions and results extend to the graphs with infinitely many vertices. We also develop a similar theory for the evolution algebras associated to finite graphs.

     

Topological systems and Artin glueing

Using methods of categorical fuzzy topology, the paper shows a relation between topological systems of S. Vickers and Artin glueing of M. Artin. Inspired by the problem of interrelations between algebra and topology, we show the necessary and sufficient conditions for the category, obtained by Artin glueing along an adjoint functor, to be (co)algebraic and (co)monadic, incorporating the respective result of G. Wraith. As a result, we confirm the algebraic nature of the category of topological systems, showing that it is monadic.     

Cluster expansion formulas and perfect matchings

We study cluster algebras with principal coefficient systems that are associated to unpunctured surfaces. We give a direct formula for the Laurent polynomial expansion of cluster variables in these cluster algebras in terms of perfect matchings of a certain graph G _T,γ that is constructed from the surface by recursive glueing of elementary pieces that we call tiles. We also give a second formula for these Laurent polynomial expansions in terms of subgraphs of the graph G _T,γ .     

The glueing construction and double categories

We introduce Artin–Wraith glueing and locally closed inclusions in double categories. Examples include locales, toposes, topological spaces, categories, and posets. With appropriate assumptions, we show that locally closed inclusions are exponentiable, and the exponentials are constructed via Artin–Wraith glueing. Thus, we obtain a single theorem establishing the exponentiability of locally closed inclusions in these five cases.     

Stochastic averaging with a flattened Hamiltonian: A Markov process on a stratified space (a whiskered sphere)

We consider a random perturbation of a 2-dimensional Hamiltonian ODE. Under an appropriate change of time, we identify a reduced model, which in some aspects is similar to a stochastically averaged model. The novelty of our problem is that the set of critical points of the Hamiltonian has an interior. Thus we can stochastically average outside this set of critical points, but inside we can make no model reduction. The result is a Markov process on a stratified space which looks like a whiskered sphere (i.e, a 2-dimensional sphere with a line attached). At the junction of the sphere and the line, glueing conditions identify the behavior of the Markov process.

     

Decomposable Near Polygons

We give a common construction for the product and the glued near polygons by generalizing the glueing construction given in [5]. We call the near polygons arising from this generalized glueing construction decomposable or (again) glued. We will study the geodetically closed sub near polygons of decomposable near polygons. Each decomposable near hexagon has a nice pair of partitions in geodetically closed near polygons. We will give a characterization of the decomposable near polygons using this property.     

Eulerian detachments with local edge-connectivity

For a graph G, a detachment operation at a vertex transforms the graph into a new graph by splitting the vertex into several vertices in such a way that the original graph can be obtained by contracting all the split vertices into a single vertex. A graph obtained from a given graph G by applying detachment operations at several vertices is called a detachment of graph G. While detachment operations may decrease the connectivity of graphs, there are several works on conditions for preserving the connectivity. In this paper, we present necessary and sufficient conditions for a given graph/digraph to have an Eulerian detachment that satisfies a given local edge-connectivity requirement. We also discuss conditions for the detachment to be loopless.     

Fox calculus, symplectic forms, and moduli spaces

An ``open pre-symplectic form' on surfaces with boundary and glueing formulae are provided to symplectically integrate the symplectic form on the deformation space of representations of the fundamental group of a Riemann surface into a reductive Lie group .

     

Möbius strips with flat metrics

This paper is focused on flat Möbius strips. The first section presents some results related to asymptotic parameterization of such surfaces. The second section contains studies of their bendability. It is proved that any Möbius strip isometric to a surface obtained by suitably glueing a flat rectangle admits a non-trivial bending which is a slide along itself. A concrete numerical computation of such bending is given.     

Glueing continuous functions constructively

The glueing of (sequentially, pointwise, or uniformly) continuous functions that coincide on the intersection of their closed domains is examined in the light of Bishop-style constructive analysis. This requires us to pay attention to the way that the two domains intersect.     

On Near Hexagons and Spreads of Generalized Quadrangles

The glueing-construction described in this paper makes use of two generalized quadrangles with a spread in each of them and yields a partial linear space with special properties. We study the conditions under which glueing will give a near hexagon. These near hexagons satisfy the nice property that every two points at distance 2 are contained in a quad. We characterize the class of the glued near hexagons and give examples, some of which are new near hexagons.     

Optimality Conditions for Switching Operator Differential Equations

We consider optimal switching of hybrid abstract evolution equations. The framework includes switching semilinear partial differential equations of parabolic or hyperbolic type, discontinuous state resets, switching costs and allows switching of the principle parts of the equations. We present adjoint-based formulae for the gradient of the cost functional with respect to position and number of switching time points that lead to first order necessary conditions. Moreover, we discuss an alternate-direction approach for implementing descent methods. As an application we consider optimal open/close-switching of valves and on/off-switching control of compressors in a gas network modelled by a graph with simplified euler equations on edges and suitable coupling conditions at nodes. (© 2017 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Coloring by two-way independent sets

The paper stems from an attempt to investigate a somewhat mysterious phenomenon: conditions which suffice for the existence of a “large” set satisfying certain conditions (e.g., a large independent set in a graph) often suffice (or at least are conjectured to suffice) for the existence of a covering of the ground set by few sets satisfying these conditions (in the example of independent sets in a graph this means that the graph has small chromatic number). We consider two conjectures of this type, on coloring by sets which are “two-way independent”, in the sense of belonging to a matroid and at the same time being independent in a graph sharing its ground set with the matroid. We prove these conjectures for matroids of rank 2. We also consider dual conjectures, on packing bases of a matroid, which are independent in a given graph.     

A fast adaptive spectral graph wavelet method for the viscous Burgers' equation on a star-shaped connected graph

Water wave propagation in an open channel network can be described by the viscous Burgers' equation on the corresponding connected graph, possibly with small viscosity. In this paper, we propose a fast adaptive spectral graph wavelet method for the numerical solution of the viscous Burgers' equation on a star-shaped connected graph. The vital feature of spectral graph wavelets is that they can be constructed on any complex network using the graph Laplacian. The essence of the method is that the same operator can be used for the construction of the spectral graph wavelet and the approximation of the differential operator involved in the Burgers' equation. In this paper, two test problems are considered with homogeneous Dirichlet boundary condition. The numerical results show that the method accurately captures the evolution of the localized patterns at all the scales, and the adaptive node arrangement is accordingly obtained. The convergence of the given method is verified, and efficiency is shown using CPU time.     

Visibility Graphs and Oriented Matroids

   Abstract. We describe a set of necessary conditions for a given graph to be the visibility graph of a simple polygon. For every graph satisfying these conditions we show that a uniform rank 3 oriented matroid can be constructed in polynomial time, which if affinely coordinatizable yields a simple polygon whose visibility graph is isomorphic to the given graph.     

Visibility Graphs and Oriented Matroids

Abstract. We describe a set of necessary conditions for a given graph to be the visibility graph of a simple polygon. For every graph satisfying these conditions we show that a uniform rank 3 oriented matroid can be constructed in polynomial time, which if affinely coordinatizable yields a simple polygon whose visibility graph is isomorphic to the given graph.     

On a time nonlocal problem for inhomogeneous evolution equations

Under consideration is some problem for inhomogeneous differential evolution equation in Banach space with an operator that generates a C ₀-continuous semigroup and a nonlocal integral condition in the sense of Stieltjes. In case the operator has continuous inhomogeneity in the graph norm. We give the necessary and sufficient conditions for existence of a generalized solution for the problem of whether the nonlocal data belong to the generator domain. Estimates on solution stability are given, and some conditions are obtained for existence of the classical solution of the nonlocal problem. All results are extended to a Sobolev-type linear equation, the equation in Banach space with a degenerate operator at the derivative. The time nonlocal problem for the partial differential equation, modeling a filtrating liquid free surface, illustrates the general statements.     

Distance signless Laplacian spectral radius and Hamiltonian properties of graphs

In this paper, we establish a sufficient condition on distance signless Laplacian spectral radius for a bipartite graph to be Hamiltonian. We also give two sufficient conditions on distance signless Laplacian spectral radius for a graph to be Hamilton-connected and traceable from every vertex, respectively. Furthermore, we obtain a sufficient condition for a graph to be Hamiltonian in terms of the distance signless Laplacian spectral radius of the complement of a graph G.     

Lower-bounds on the connectivities of a graph

This article presents a study of the connectivities of a graph as a function of other graph parameters such as the number of vertices, the maximum degree, and the diameter. As a result, lower-bounds on the connectivities of a graph as a function of these parameters are computed. These bounds could serve as sufficient conditions for a graph to be h-edge-connected or k-connected. Consequently, the connectivity characteristics of many of the densest known graphs are determined.