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We develop a unified representation theory for the categories of finite subsets and relation-preserving maps of highly homogeneous relational structures classified by Cameron. For any commutative coefficient ring $k$, we extend the…

Let $Q$ be a quiver of type $\mathbb{A}_n$ with linear orientation and $\operatorname{rep}(Q,\mathbb{F}_1)$ the category of representations of $Q$ over the virtual field $\mathbb{F}_1$.It is proved that $\operatorname{rep}(Q,\mathbb{F}_1)$…

Degeneracy loci polynomials for quiver representations generalize several important polynomials in algebraic combinatorics. In this paper we give a nonconventional generating sequence description of these polynomials, when the quiver is of…

Suppose that Q is a family of seminorms on a locally convex space E which determines the topology of E. We study the existence of Q-nonexpansive retractions for families of Q-nonexpansive mappings and prove that a separated and sequentially…

We develop a general theory for irreducible homogeneous spaces $M= G/H$, in relation to the nullity $\nu$ of their curvature tensor. We construct natural invariant (different and increasing) distributions associated with the nullity, that…

This article is a summary of the author's unpublished Ph.D thesis. Its purpose is to generalise a construction by H. Cassens and P. Slodowy of the semiuniversal deformations of the simple singularities of type $A_r$, $D_r$, $E_6$, $E_7$ and…

For an arbitrary noninvertible evolution family on the half-line and for $\rho \colon [0, \infty)\to [0, \infty)$ in a large class of rate functions, we consider the notion of a $\rho$-dichotomy with respect to a family of norms and…

Given a bigraded exact couple of modules over some ring, we determine the meaning of the $E^{\infty}$-terms of its associated spectral sequence: Let $L^{\ast}$ and $L_{\ast}$ denote the limit and colimit abutting objects of the exact…

In this paper we introduce a special kind of relative (co)resolutions associated to a pair of classes of objects in an abelian category $\mathcal{C}.$ We will see that, by studying these relative (co)resolutions, we get a possible…

Given finite metric spaces $(X, d_X)$ and $(Y, d_Y)$, we investigate the persistent homology $PH_*(X \times Y)$ of the Cartesian product $X \times Y$ equipped with the sum metric $d_X + d_Y$. Interpreting persistent homology as a module…

We show that the algebraic K-theory of semi-valuation rings with stably coherent regular semi-fraction ring satisfies homotopy invariance. Moreover, we show that these rings are regular if their valuation is non-trivial. Thus they yield…

In this article, we define the tensor product $V\otimes W$ of a representation $V$ of a quiver $Q$ with a representation $W$ of an another quiver $Q'$, and show that the representation $V\otimes W$ is semistable if $V$ and $W$ are…

For any non-zero finite module M of finite projective dimension over a noetherian local ring R with maximal ideal m and residue field k, it is proved that the natural map Ext_R(k,M)-->Ext_R(k,M/mM) is non-zero when R is regular and is zero…

The purpose of this article is to study the space of stability conditions $\stab(P_n)$ on the bounded derived category $\D^b(P_n)$ of finite dimensional representations of a quiver $P_n$ with two vertices and $n$ parallel arrows. There is a…

In this third paper in a series on type I Howe duality for finite fields, we give a complete description of the restriction of the oscillator representation over a finite field to products of dual pairs of symplectic and orthogonal groups…

We define and study the category $\RepQ$ of representations of a quiver in $\VFun$ - the category of vector spaces "over $\Fun$". $\RepQ$ is an $\Fun$-linear category possessing kernels, co-kernels, and direct sums. Moreover, $\RepQ$…

We study rational double points over algebraically closed fields in arbitrary characteristics and completely classify the indecomposable objects in their singularity categories, which correspond to the vertices in their Auslander-Reiten…

We study the representation category of thread quivers and their quotients. A thread quiver is a quiver in which some arrows have been replaced by totally ordered sets. Pointwise finite-dimensional (pwf) representations of such a thread…

In this paper we continue our study of property (RD) for Hecke pairs initiated in (V. Shirbisheh, Property (RD) for Hecke pairs. Mathematical Physics, Analysis and Geometry, vol. 15, no. 2, (2012), 173--192). We study the behavior of…

In this paper, we study the semi-stable subcategories of the category of representations of a Euclidean quiver, and the possible intersections of these subcategories. Contrary to the Dynkin case, we find out that the intersection of…