Topic
Product (mathematics)
About: Product (mathematics) is a research topic. Over the lifetime, 44382 publications have been published within this topic receiving 377809 citations.
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TL;DR: This paper first introduces the by-now standard setting of weighted Hilbert spaces of functions with square-integrable mixed first derivatives, and then indicates alternative settings, such as non-Hilbert spaces, that can sometimes be more suitable.
Abstract: This paper is a contemporary review of quasi-Monte Carlo (QMC) methods, that is, equal-weight rules for the approximate evaluation of high-dimensional integrals over the unit cube \([0,1]^s\). It first introduces the by-now standard setting of weighted Hilbert spaces of functions with square-integrable mixed first derivatives, and then indicates alternative settings, such as non-Hilbert spaces, that can sometimes be more suitable. Original contributions include the extension of the fast component-by-component (CBC) construction of lattice rules that achieve the optimal convergence order (a rate of almost \(1/N\), where \(N\) is the number of points, independently of dimension) to so-called “product and order dependent” (POD) weights, as seen in some recent applications. Although the paper has a strong focus on lattice rules, the function space settings are applicable to all QMC methods. Furthermore, the error analysis and construction of lattice rules can be adapted to polynomial lattice rules from the family of digital nets.
doi:10.1017/S1446181112000077
88 citations
TL;DR: In this article, the Hilbert-Schmidt norms of products of composition and differentiation operators on the Bergman space and Hardy space were derived in terms of the generalized Nevanlinna counting function.
Abstract: Motivated by a recent paper by S. Ohno we calculate Hilbert-Schmidt norms of products of composition and differentiation operators on the Bergman space $A^2_\alpha,$ $\alpha>-1$ and the Hardy space $H^2$ on the unit disk. When the convergence of sequences $(\varphi_n)$ of symbols to a given symbol $\varphi$ implies the convergence of product operators $C_{\varphi_n}D^k$ is also studied. Finally, the boundedness and compactness of the operator $C_{\varphi}D^k: A^2_\alpha\to A^2_\alpha$ are characterized in terms of the generalized Nevanlinna counting function.
88 citations
Patent•
21 Sep 2012TL;DR: A server for registration of products may include a processor and computer storage as mentioned in this paper, which can be configured to receive, via a network, a communication from a product application executed on a computing device, the communication including a product identification (ID associated with a product and an indication of registration of the product by a user of the application.
Abstract: A server for registration of products may include a processor and computer storage. The processor may be configured to receive, via a network, a communication from a product application executed on a computing device, the communication including a product identification (ID) associated with a product and an indication of registration of the product by a user of the product application. The processor may further be configured to register the product at least by storing, in the computer storage, the product ID and a relationship between the product and an identification of the user of the product application. The processor may further be configured to send a subsequent communication to the product application on the computing device based on the registration of the product.
87 citations