Product (mathematics)

About: Product (mathematics) is a research topic. Over the lifetime, 44382 publications have been published within this topic receiving 377809 citations.

Modules over coproducts of rings

Abstract: Let Ro be a skew field, or more generally, a finite product of full matrix rings over skew fields. Let (RX)XEA be a family of faithful Rorings (associative unitary rings containing RO) and let R denote the coproduct ("free product") of the RX as R0-rings. An easy way to obtain an R-module M is to choose for each X E A U {0} an Rx-module Mx, and put M = MA OR; R. Such an M will be called a "standard" R-module. (Note that these include the free R-modules.) We obtain results on the structure of standard R-modules and homomorphisms between them, and hence on the homological properties of R. In particular: (1) Every submodule of a standard module is isomorphic to a standard

On asymptotic dimension of groups

Abstract: We prove a version of the countable union theorem for asymp- totic dimension and we apply it to groups acting on asymptotically nite dimensional metric spaces. As a consequence we obtain the following nite dimensionality theorems. A) An amalgamated product of asymptotically nite dimensional groups has nite asymptotic dimension: asdimAC B< 1 . B) Suppose that G 0 is an HNN extension of a group G with asdimG < 1. Then asdimG 0 <1. C) Suppose that is Davis' group constructed from a group with asdim <1. Then asdim <1. AMS Classication 20H15; 20E34, 20F69

Uniform Budgets and the Envy-Free Pricing Problem

Abstract: We consider the unit-demand min-buying pricing problem, in which we want to compute revenue maximizing prices for a set of products $\mathcal{P}$ assuming that each consumer from a set of consumer samples $\mathcal{C}$ will purchase her cheapest affordable product once prices are fixed. We focus on the special uniform-budget case, in which every consumer has only a single non-zero budget for some set of products. This constitutes a special case also of the unit-demand envy-free pricing problem. We show that, assuming specific hardness of the balanced bipartite independent set problem in constant degree graphs or hardness of refuting random 3CNF formulas, the unit-demand min-buying pricing problem with uniform budgets cannot be approximated in polynomial time within $\mathcal{O}(\log ^{\varepsilon} |\mathcal{C}|)$ for some ?> 0. This is the first result giving evidence that unit-demand envy-free pricing, as well, might be hard to approximate essentially better than within the known logarithmic ratio. We then introduce a slightly more general problem definition in which consumers are given as an explicit probability distribution and show that in this case the envy-free pricing problem can be shown to be inapproximable within $\mathcal{O}(|\mathcal{P}|^{\varepsilon})$ assuming NP $ subseteq \bigcap _{\delta >0}$ BPTIME($2^{\mathcal{O}(n^{\delta})}$). Finally, we briefly argue that all the results apply to the important setting of pricing with single-minded consumers as well.

Large $N$ topologically twisted index: necklace quivers, dualities, and Sasaki-Einstein spaces

Abstract: In this paper, we calculate the topological free energy for a number of $$ \mathcal{N} $$ ≥ 2 Yang-Mills-Chern-Simons-matter theories at large N and fixed Chern-Simons levels. The topological free energy is defined as the logarithm of the partition function of the theory on S 2 × S 1 with a topological A-twist along S 2 and can be reduced to a matrix integral by exploiting the localization technique. The theories of our interest are dual to a variety of Calabi-Yau four-fold singularities, including a product of two asymptotically locally Euclidean singularities and the cone over various well-known homogeneous Sasaki-Einstein seven-manifolds, N 0,1,0, V 5,2, and Q 1,1,1. We check that the large N topological free energy can be matched for theories which are related by dualities, including mirror symmetry and $$ \mathrm{S}\mathrm{L}\left(2,\mathbb{Z}\right) $$ duality.

Basic Hoops: an Algebraic Study of Continuous t-norms

Abstract: A continuous t-norm is a continuous map * from [0, 1]2 into [0, 1] such that $$\langle [0, 1], *, 1 \rangle$$ is a commutative totally ordered monoid. Since the natural ordering on [0, 1] is a complete lattice ordering, each continuous t-norm induces naturally a residuation → and $$\langle [0, 1], *,\rightarrow, 1\rangle$$ becomes a commutative naturally ordered residuated monoid, also called a hoop. The variety of basic hoops is precisely the variety generated by all algebras $$\langle [0, 1], *,\rightarrow, 1\rangle$$ , where * is a continuous t-norm. In this paper we investigate the structure of the variety of basic hoops and some of its subvarieties. In particular we provide a complete description of the finite subdirectly irreducible basic hoops, and we show that the variety of basic hoops is generated as a quasivariety by its finite algebras. We extend these results to Hajek’s BL-algebras, and we give an alternative proof of the fact that the variety of BL-algebras is generated by all algebras arising from continuous t-norms on [0, 1] and their residua. The last part of the paper is devoted to the investigation of the subreducts of BL-algebras, of Godel algebras and of product algebras.