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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Patent•
26 Sep 1938
TL;DR: In this paper, an improved condensation product suitable for use in resins, lacquers, paints, varnishes, enamels, plastic masses, and for other similar purposes is described.
Abstract: My invention relates to methods of producing an improved condensation product suitable for use in resins, lacquers, paints, varnishes, enamels, plastic masses, and for other similar purposes, and to the new product so produced. This application is a continuation in part of my copending applications,...
67 citations
TL;DR: In this paper, the authors give a geometric characterization of finite dimensional normed spaces with a 1-unconditional basis, such that their volumetric product is minimal, and they show that this is the case for any normed space.
Abstract: We give a geometric characterization of finite dimensional normed spacesE, with a 1-unconditional basis, such that their volumetric product is minimal.
67 citations
TL;DR: In this article, a systematic framework was proposed to classify (2+1)-dimensional (2 + 1D) fermionic topological orders without symmetry and 2+1D fermion/bosonic topological order with symmetry.
Abstract: We propose a systematic framework to classify (2+1)-dimensional (2+1D) fermionic topological orders without symmetry and 2+1D fermionic/bosonic topological orders with symmetry $G$. The key is to use the so-called symmetric fusion category $\mathcal{E}$ to describe the symmetry. Here, $\mathcal{E}=\mathrm{sRep}({\mathbb{Z}}_{2}^{f})$ describing particles in a fermionic product state without symmetry, or $\mathcal{E}=\mathrm{sRep}({G}^{f}) [\mathcal{E}=\mathrm{Rep}(G)]$ describing particles in a fermionic (bosonic) product state with symmetry $G$. Then, topological orders with symmetry $\mathcal{E}$ are classified by nondegenerate unitary braided fusion categories over $\mathcal{E}$, plus their modular extensions and total chiral central charges. This allows us to obtain a list that contains all 2+1D fermionic topological orders without symmetry. For example, we find that, up to $p+\phantom{\rule{1.0pt}{0ex}}\mathrm{i}\phantom{\rule{1.0pt}{0ex}}p$ fermionic topological orders, there are only four fermionic topological orders with one nontrivial topological excitation: (1) the $K=\left(\begin{array}{cc}\hfill \ensuremath{-}1a \hfill 0\\ \hfill 0a \hfill 2\end{array}\right)$ fractional quantum Hall state, (2) a Fibonacci bosonic topological order stacking with a fermionic product state, (3) the time-reversal conjugate of the previous one, and (4) a fermionic topological order with chiral central charge $c=\frac{1}{4}$, whose only topological excitation has non-Abelian statistics with spin $s=\frac{1}{4}$ and quantum dimension $d=1+\sqrt{2}$.
67 citations
Patent•
30 Apr 1993
TL;DR: A system for detecting product loss from a fluid product storage and dispensing means including at least one storage tank comprising tank volume monitoring devices, fluid dispenser monitoring devices and computer means to analyze the data from such devices at various times as determined by the status of the various elements to determine if there are unexplained shortages of product as discussed by the authors.
Abstract: A system for detecting product loss from a fluid product storage and dispensing means including at least one storage tank comprises tank volume monitoring devices, fluid dispenser monitoring devices and computer means to analyze the data from such devices at various times as determined by the status of the various elements to determine if there are unexplained shortages of product.
67 citations