In this paper we extend the Eilenberg-Steenrod axiomatic description of a homology theory from the category of topological spaces to an arbitrary category and, in particular, to a topos. Implicit in this extension is an extension of the notions of homotopy and excision. A general discussion of such homotopy and excision structures on a category is given along with several examples including the...

It is pointed out that one-component φ theory in four dimensions has a non-perturbative sector that can be studied by means of an exact duality transformation of its Ising limit. This duality maps it to a membrane model. As a consequence, the φ-theory turns out to have, in the broken symmetry phase, a remarkably rich spectrum of physical states corresponding to membrane excitations. A numerical...

Using a construction similar to an iterated function system, but with functions changing at each step of iteration, we provide a natural example of a continuous one-parameter family of holomorphic functions of infinitely many variables. This family is parametrized by the compact space of positive integer sequences of prescribed growth and hence it can also be viewed as a parametric description ...

‎Triangular matrix rings are examples of trivial extensions‎. ‎In this article we determine the structure of derivations and biderivations of the trivial extensions‎, ‎and thereby we describe the derivations and biderivations of the upper triangular matrix rings‎. ‎Some related results are also obtained‎.

Black holes with hair represented by generic fields surrounding the central source of vacuum Schwarzschild metric are examined under minimal set requirements consisting i) existence a well defined event horizon and ii) strong or dominant energy condition for outside horizon. We develop our analysis means gravitational decoupling approach. find that trivial deformations seed preserve conditions ...

In this paper, we study the Landis-type conjecture, i.e., unique continuation property from infinity, of fractional Schr\"{o}dinger equation with drift and potential terms. We show that if any solution decays at a certain exponential rate, then it must be trivial. The main ingredients our proof are Caffarelli-Silvestre extension Armitage's Liouville-type theorem.

Several kinds of quotient triangulated categories arising naturally in representations of algebras are studied; their relations with the stable categories of Frobenius exact categories are investigated; the derived categories of Gorenstein algebras are explicitly computed inside the stable categories of the graded module categories of the corresponding trivial extension algebras; new descriptio...

In this paper we introduce and investigate a class of those rings in which every projective ideal is free. We establish the transfer of this notion to the trivial ring extension and pullbacks and then generate new and original families of rings satisfying this property.

We show that if H is a Fitting pseudovariety of groups and V is a local pseudovariety of monoids, then LH ©m V is local if either V contains the six element Brandt monoid, or H is a non-trivial pseudovariety of groups closed under extension.

Given any representation of an arbitrary Lie algebra g over a field K of characteristic 0, we construct representations of a central extension of g on bosonic and fermionic Fock space. The method gives an explicit formula for a (sometimes trivial) 2-cocycle in H(g;K). We illustrate these techniques with several concrete examples. MSC: 17B10, 17B56, 17B65, 17B68.