Product (mathematics)

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

Fitting the Semantic Differential to the Marketing Problem

Abstract: How can we QUANTIFY abstract qualitative data that deal with consumers’ reactions to the image of a brand, product, or company? This article suggests new uses for the semantic differential, and pre...

New Proofs of Pl\"unnecke-type Estimates for Product Sets in Groups

Abstract: We present a new method to bound the cardinality of triple product sets in groups and give three applications. A new and unexpectedly short proof of the Plunnecke-Ruzsa sumset inequalities for Abelian groups. A new proof of a theorem of Tao on triple products, which generalises these inequalities when no assumption on commutativity is made. A further generalisation of the Plunnecke-Ruzsa inequalities in general groups.

Characterizations of bounded mean oscillation on the polydisk in terms of Hankel operators and Carleson measures

Abstract: A scale of BMO spaces appears naturally in product spaces corresponding to the different, yet equivalent, characterizations of the class of functions of bounded mean oscillation in one variable. S.-Y. Chang and R. Fefferman characterized product BMO, the dual of the (real) Hardy spaceH Re 1 on product domains, in terms of Carleson measures. Here we describe two other BMO spaces, one contained in and the other containing product BMO, in terms of Carleson measures and Hankel operators. Both of these spaces play a significant role in harmonic analysis of the polydisk and in multivariable operator theory.

Method of identifying game controllers in multi-player game

Abstract: In one embodiment, an operating system may assign a product ID to the game controller that matches the product ID provided with or for the game controller.

Stringy K-theory and the Chern character

Abstract: We construct two new G-equivariant rings: $\mathcal{K}(X,G)$ , called the stringy K-theory of the G-variety X, and $\mathcal{H}(X,G)$ , called the stringy cohomology of the G-variety X, for any smooth, projective variety X with an action of a finite group G. For a smooth Deligne–Mumford stack $\mathcal{X}$ , we also construct a new ring $\mathsf{K}_{\mathrm{orb}}(\mathcal{X})$ called the full orbifold K-theory of $\mathcal{X}$ . We show that for a global quotient $\mathcal{X} = [X/G]$ , the ring of G-invariants $K_{\mathrm{orb}}(\mathcal{X})$ of $\mathcal{K}(X,G)$ is a subalgebra of $\mathsf{K}_{\mathrm{orb}}([X/G])$ and is linearly isomorphic to the “orbifold K-theory” of Adem-Ruan [AR] (and hence Atiyah-Segal), but carries a different “quantum” product which respects the natural group grading. We prove that there is a ring isomorphism $\mathcal{C}\mathbf{h}:\mathcal{K}(X,G)\to\mathcal{H}(X,G)$ , which we call the stringy Chern character. We also show that there is a ring homomorphism $\mathfrak{C}\mathfrak{h}_\mathrm{orb}:\mathsf{K}_{\mathrm{orb}}(\mathcal{X}) \rightarrow H^\bullet_{\mathrm{orb}}(\mathcal{X})$ , which we call the orbifold Chern character, which induces an isomorphism $Ch_{\mathrm{orb}}:K_{\mathrm{orb}}(\mathcal{X})\rightarrow H^\bullet_{\mathrm{orb}}(\mathcal{X})$ when restricted to the sub-algebra $K_{\mathrm{orb}}(\mathcal{X})$ . Here $H_{\mathrm{orb}}^\bullet(\mathcal{X})$ is the Chen–Ruan orbifold cohomology. We further show that $\mathcal{C}\mathbf{h}$ and $\mathfrak{C}\mathfrak{h}_\mathrm{orb}$ preserve many properties of these algebras and satisfy the Grothendieck–Riemann–Roch theorem with respect to etale maps. All of these results hold both in the algebro-geometric category and in the topological category for equivariant almost complex manifolds. We further prove that $\mathcal{H}(X,G)$ is isomorphic to Fantechi and Gottsche’s construction [FG, JKK]. Since our constructions do not use complex curves, stable maps, admissible covers, or moduli spaces, our results greatly simplify the definitions of the Fantechi–Gottsche ring, Chen–Ruan orbifold cohomology, and the Abramovich–Graber–Vistoli orbifold Chow ring. We conclude by showing that a K-theoretic version of Ruan’s Hyper-Kahler Resolution Conjecture holds for the symmetric product of a complex projective surface with trivial first Chern class.