Product (mathematics)

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

Partition functions of N = (2,2) gauge theories on S 2 and vortices

Abstract: We apply localization techniques to compute the partition function of a two-dimensional \({\mathcal{N}=(2,2)}\) R-symmetric theory of vector and chiral multiplets on S2. The path integral reduces to a sum over topological sectors of a matrix integral over the Cartan subalgebra of the gauge group. For gauge theories which would be completely Higgsed in the presence of a Fayet–Iliopoulos term in flat space, the path integral alternatively reduces to the product of a vortex times an antivortex partition functions, weighted by semiclassical factors and summed over isolated points on the Higgs branch. For applications, we evaluate the partition function for some U(N) gauge theories, showing equality of the path integrals for theories conjectured to be dual by Hori and Tong and deriving new expressions for vortex partition functions.

Computer-assisted shopping and product location

Abstract: In general, techniques are described that enable crowdsourcing of product-related information. Product-related information may include availability and intra-store locations of various products at particular stores. An example system includes an interface, a memory, and one or more programmable processors configured to receive, at the interface an intra-store product location from a first client device, the intra-store product location indicating a location of a product within a store. The programmable processor(s) are further configured to store, to at least one storage device coupled to the system and in association with the product, the received intra-store product location, receive, at the interface, a request associated with the product from a second client device, and send, from the interface, the intra-store product location to the second client device in response to receiving the request associated with the product.

Analytical Approach to Parallel Repetition

Abstract: We propose an analytical framework for studying parallel repetition, a basic product operation for one-round two-player games. In this framework, we consider a relaxation of the value of a game, $\mathrm{val}_+$, and prove that for projection games, it is both multiplicative (under parallel repetition) and a good approximation for the true value. These two properties imply a parallel repetition bound as $$ \mathrm{val}(G^{\otimes k}) \approx \mathrm{val}_+(G^{\otimes k}) = \mathrm{val}_+(G)^{k} \approx \mathrm{val}(G)^{k}. $$ Using this framework, we can also give a short proof for the NP-hardness of Label-Cover$(1,\delta)$ for all $\delta>0$, starting from the basic PCP theorem. We prove the following new results: - A parallel repetition bound for projection games with small soundness. Previously, it was not known whether parallel repetition decreases the value of such games. This result implies stronger inapproximability bounds for Set-Cover and Label-Cover. - An improved bound for few parallel repetitions of projection games, showing that Raz's counterexample is tight even for a small number of repetitions. Our techniques also allow us to bound the value of the direct product of multiple games, namely, a bound on $\mathrm{val}(G_1\otimes ...\otimes G_k)$ for different projection games $G_1,...,G_k$.