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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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90 citations
TL;DR: Five special forms for a von Neumann-Morgenstern utility function u on a product set A×X are examined, each expresses u as a combination of functions defined on the separate factors A and X.
Abstract: Five special forms for a von Neumann-Morgenstern utility function u on a product set A×X are examined. Each expresses u as a combination of functions defined on the separate factors A and X. The five forms include the additive and multiplicative forms, u=u1+u2 and u=f1f2, respectively, and the additive form with independent product expressed by u=u1+u2+f1f2. Necessary and sufficient preference conditions are identified for each form, along with admissible transformations on the single-argument functions.
90 citations
90 citations
TL;DR: In this article, the causality condition was reformulated as follows: at any time the total probability of finding the particle outside the scattering center shall not be greater than 1, for every form of the incident wave packet.
Abstract: The application of the "causality condition" to the $S$ matrix for nonrelativistic particles encounters several difficulties: (a) there is no maximum velocity; (b) the interference of ingoing and outgoing waves has to be taken into account; (c) wave packets with a sharp front do not exist. The condition is therefore reformulated as follows: At any time the total probability of finding the particle outside the scattering center shall not be greater than 1, for every form of the incident wave packet. From this follows for spherical waves that $S$, as a function of the momentum $p$, is analytic and holomorphic in the first quadrant and that ${e}^{2iap}S(p)$ (where $a$ is the radius of the scattering center) has an imaginary part \ensuremath{\leqslant} 1. That suffices to give an explicit integral representation and a product expansion for $S$, but these permit a more general form for $S$ than is usually envisaged. If, however, the usual symmetry relation $S(\ensuremath{-}p)=S{(p)}^{*}$ is assumed in addition to the causality condition, more specific equations can be derived, which are direct generalizations of those in Part I. In particular, integral relations between the real and imaginary parts of $S$, and the properties that Wigner found for the $R$ matrix can be deduced.
90 citations
TL;DR: In this paper, it was shown that every element in a non-abelian finite simple group can be written as a product of two rth powers for any prime power r (in particular, a product-of-two squares).
Abstract: We prove several results on products of conjugacy classes in finite simple groups. The first result is that there always exists a uniform generating triple. This result and other ideas are used to solve a 1966 conjecture of Peter Neumann about the existence of elements in an irreducible linear group with small fixed space. We also show that there always exist two conjugacy classes in a finite non-abelian simple group whose product contains every nontrivial element of the group. We use this to show that every element in a non-abelian finite simple group can be written as a product of two rth powers for any prime power r (in particular, a product of two squares).
90 citations