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Showing papers on "Product (mathematics) published in 1973"


Journal ArticleDOI
TL;DR: In this article, the authors investigate right modules over a B*algebra B which posses a B-valued "inner product" respecting the module action, and show that such self-dual modules have important properties in common with both Hilbert spaces and W*-algebras.
Abstract: This paper is an investigation of right modules over a B*algebra B which posses a B-valued "inner product" respecting the module action. Elementary properties of these objects, including their normability and a characterization of the bounded module maps between two such, are established at the beginning of the exposition. The case in which B is a W*-algebra is of especial interest, since in this setting one finds an abundance of inner product modules which satisfy an analog of the self-duality property of Hilbert space. It is shown that such self-dual modules have important properties in common with both Hilbert spaces and W*-algebras. The extension of an inner product module over B by a B*-algebra A containing B as a *-subalgebra is treated briefly. An application of some of the theory described above to the representation and analysis of completely positive maps is given.

686 citations



Journal ArticleDOI
TL;DR: In this article, the binding energy of an excitonic molecule is calculated as a function of electron-to-hole mass ratio and of anisotropy in the hole bands, which is a product of a Hylleraas and Ore correlated excition function and an overlap function of the hole-hole separation.
Abstract: The binding energy of the excitonic molecule is calculated as a function of electron-to-hole mass ratio and of anisotropy in the hole bands. A variational wave function is used which is a product of a Hylleraas and Ore correlated excition function and an overlap function of the hole-hole separation. The results are tabulated for CuCl, CuBr, ${\mathrm{Cu}}_{2}$O, and for a number of II-VI compounds.

150 citations


Journal ArticleDOI
TL;DR: In this article, it was shown that certain rank-one and rank-two corrections to symmetric positive definite matrices can be expressed in the form of a product, which gives control over the positive definiteness, determinant value and conditioning of the corrected matrix.
Abstract: It is shown that certain rank-one and rank-two corrections to symmetric positive definite matrices may be expressed in the form of a product. This product form gives control over the positive definiteness, determinant value and conditioning of the corrected matrix. An application to updating formulae of quasi-Newton methods for unconstrained minimization is discussed.

63 citations


Journal ArticleDOI
TL;DR: In this paper, the Boltzmann-Enskog memory function can be written in terms of a product of phase-space correlation functions, and the velocity-autocorrelation function is calculated, in particular, over the complete range of times with no adjustable parameters.
Abstract: In a previous paper a general formalism was developed for treating the time-dependent correlation functions that arise in the theory of self-diffusion. In this paper this formalism is used in conjunction with the approximation where the two-particle source is approximated by a sum of one-particle sources. This approximation follows from physical arguments and from an analysis of the exact equation for the two-particle source. The resulting expression for the memory function is similar to that found previously by Pomeau and is related to the ring terms studied by Kawasaki and Oppenheim. It is further shown that this correction to the Boltzmann-Enskog memory function can be written in terms of a product of phase-space correlation functions. This theory, to the extent that the hydrodynamical projection onto these correlation functions is dominant, provides a microscopic basis for the various mode-mode coupling theories. The associated long-time behavior of the velocity-autocorrelation function is shown to go as ${t}^{\frac{\ensuremath{-}3}{2}}$ and the coefficient agrees with that found by Dorfman and Cohen for low densities. For higher densities there are differences. It is further demonstrated how one can remove the wave-number cutoffs used in other theories, and the velocity-autocorrelation function is calculated, in a particular approximation, over the complete range of times with no adjustable parameters.

50 citations


Journal ArticleDOI
TL;DR: In this article, the authors considered a problem where a product is packaged simultaneously into more than one type container after its manufacture, and set-up costs are associated with the making of the product as well as the packaging of the various items.
Abstract: We consider a problem typical of the batch processing industry. This problem arises when a product is packaged simultaneously into more than one type container after its manufacture, and when set-up costs are associated with the making of the product as well as the packaging of the various items. We formulate the production inventory problem so as to minimize the total holding and set-up costs and give a graphical solution for the two-item case.

50 citations



Journal ArticleDOI
TL;DR: In this article, it was shown that the various types of normal terms arising from the normal ordering of a disordered product of fermion operators can be enumerated with the aid of a well-known combinatorial problem, called the problem of the rooks.
Abstract: It is shown that the various types of normal terms arising from the normal ordering of a disordered product of fermion operators can be enumerated with the aid of a well-known combinatorial problem, called the problem of the rooks

45 citations


Journal ArticleDOI
TL;DR: A probabilistic algorithm is presented to calculate the Boolean product of two n × n Boolean matrices using an expected number of elementary operations of 0(n2).
Abstract: A probabilistic algorithm is presented to calculate the Boolean product of two n × n Boolean matrices using an expected number of elementary operations of 0(n2). Asymptotically in n, almost all pairs of matrices may be multiplied using this algorithm in 0(n2+e) elementary operations for any e > 0.

38 citations


Journal ArticleDOI
R.S Hales1
TL;DR: In this paper, the independence number, cliquecovering number, and Rosenfeld number of graphs are studied in relation to themselves and to the strong product of two graphs, and applications are made to limiting values, such as capacity.

36 citations


Journal ArticleDOI
TL;DR: In this paper, it was shown that under natural conditions, the Haar measure and normal distribution of the rotations group jointly converged weakly to the product distribution for a certain closed subgroup of the rotation group.
Abstract: Let $g_1, g_2,\cdots$ be random elements of the Euclidean group of motions of $d$-dimensional Euclidean space $R^d (d \geqq 1)$, that are independent and identically distributed. The product $g_1\cdots g_n$ is represented in the form $t(n)r(n)$, where $t(n)$ is a translation and $r(n)$ is a rotation. In this paper it is shown that under natural conditions $r(n)$ and $n^{-\frac{1}{2}}t(n)$ jointly converge weakly as $n \rightarrow \infty$ to the product distribution of the Haar measure on a certain closed subgroup of the rotations group, and a normal distribution on $R^d$, with mean zero and covariance matrix $\sigma^2\mathbf{I}$ ($\mathbf{I}$ is the identity matrix), and the value of $\sigma^2$ is identified.


Journal ArticleDOI
C. Becchi1
TL;DR: The absence of strong interaction corrections to the axial anomaly in the σ model is proved in a cut-off independent way using Zimmermann's normal product algorithm.
Abstract: The absence of strong interaction corrections to the axial anomaly in the σ model is proved in a cut-off independent way using Zimmermann's normal product algorithm.

Journal ArticleDOI
TL;DR: In this article, a new conjugacy class was introduced which has the property that every permutation of S is a product of two permutations from that class, and make use of this and a result from [l] in considering a more general question which now arises naturally: given a permutation P of S with infinite support, is there an integer m(P) such that any permutation in Sym(S) can be represented as a product m permutations, each conjugate to P?

Journal ArticleDOI
TL;DR: In this article, it was shown that a regular semigroup is always a regular subsemigroup of S and investigated relationships between it and S, where = S is of particular interest.
Abstract: Suppose S is a regular semigroup and E is its set of idempotents. If E is subsemigroup of S, then S has been called orthodox and studied recently by Hall [3], Meakin [6], and Yamada [8]. In this paper we assume that E is not (necessarily) a subsemigroup of S and consider the subsemigroup generated by E, denoted . If E denotes the set of all elements of S which can be written E, denoted . If E denotes the set of all elements of S which can be written as the product of n (not necessarily distinct) idempotents of S, then . We show that is always a regular subsemigroup of S and investigate relationships between it and S. The case where = S is of particular interest to us; such semigroups will be referred to as idempotent-generated regular semi- groups.

Journal ArticleDOI
TL;DR: In this paper, a random process developed by Silverman can be used to describe the wave fluctuations in localized smoothly varying turbulence, and it is shown that for a plane wave case, if the distance L is within (L 0 √ 2 ) 2/λ 2, then the usual formula given by Tatarski is valid.
Abstract: In the past, smoothly varying turbulence has been studied by changing the structure constant to the function C_{n}^{2}(\bar{r} ). The purpose of this paper is to show that this approach is insufficient, and that a random process developed by Silverman can be used to describe the wave fluctuations in localized smoothly varying turbulence. The localized turbulence is characterized by a correlation function which is a product of a function of the average coordinate and a function of the difference coordinate. The corresponding spectrum is also given by a product of a function of the difference wavenumber and a function of the average wavenumber. They are related to each other through two Fourier transform pairs. Making use of the preceding representations, the fluctuations of a wave propagating through such a turbulence can be given either by the integrals with respect to the two wavenumbers or by a convolution integral of the structure constant C_{n}^{2}(\bar{r} ) and a function involving the outer scale of the turbulence L_{0} . It is shown that for a plane wave case, if the distance L is within ( L_{0}^{2}/\lambda ), then the usual formula given by Tatarski is valid. But if the distance is between L_{0}^{2}/\lambda and (bL_{0})/\lambda where b is the total transverse size of the turbulence, the variance of the wave is nearly constant, and if L \gg (bL_{0})/\lambda , the variance decays as L^{-2} . Similar conclusions are shown for a spherical wave case. Some examples are shown illustrating the effectiveness of this method.

Journal ArticleDOI
TL;DR: In this article, it was proved that for any finitely based variety of groups there exists a finitely-based left factor such that their product is infinitely based (the left factor is a Burnside variety).
Abstract: It is proved that for any finitely based variety of groups there exists a finitely based left factor such that their product is infinitely based (the left factor is a Burnside variety).


Journal ArticleDOI
01 May 1973
TL;DR: In this article, it was shown that the existence of a measurable cardinal would imply that a separately continuous real function on a product of two topological finite complete measure spaces need not be product-measurable.
Abstract: It is known that a real-valued function f of two real variables which is continuous in each variable separately need not be continuous in (x, y), but must be in the first Baire class (1). Moreover if f is continuous in x for each y and merely measurable in y for each x then it must be Lebesgue-measurable (7), and this result can be extended to more general product spaces (2). However, the continuum hypothesis implies that this result fails if continuity is replaced by approximate continuity, as can be seen from the proof of Theorem 2 of (2). This makes Misik's question (5) very natural: is a function which is separately approximately continuous in both variables necessarily Lebesgue-measurable? Our main aim is to establish an affirmative answer. It will be shown that such a function must in fact be in the second Baire class, although not necessarily in the first Baire class (unlike approximately continuous functions of one variable (3)). Finally, we show that the existence of a measurable cardinal would imply that a separately continuous real function on a product of two topological finite complete measure spaces need not be product-measurable.

Patent
02 Mar 1973
TL;DR: A ply-bonded, embossed product and apparatus, and method wherein at least two nested-embossed webs are advanced between a pair of wheels and anvil surfaces to subject the webs to plybonding compressive forces.
Abstract: A ply-bonded, embossed product and apparatus, and method wherein at least two nested-embossed webs are advanced between a pair of wheels and anvil surfaces to subject the webs to ply-bonding compressive forces.

Patent
10 Sep 1973
TL;DR: In this paper, an improved apparatus is provided for economically producing bacon analog products, where at least one slurry comprising oil, water and coagulable protein is continuously mixed and layered.
Abstract: An improved apparatus is provided for economically producing meat analog products. According to the invention, at least one slurry comprising oil, water and coagulable protein is continuously mixed and layered. The layered slurry is heat set under moist heat to provide a cohesive mass without surface product drying. The heat set product is then sliced for marketing as is, or after subjecting it to varying degrees of frying. The preferred product is a partially fried bacon analog product.

Journal ArticleDOI
TL;DR: In this article, a necessary and sufficient condition is given for a regular class M to be hereditary, and if the classM satisfies an even stronger condition, then theUM-semisimple objects are subdirectly embeddable in a (direct) product of M-objects.
Abstract: Starting from a regular classM, one can construct the upper radicalUM of the classM in a category which is like that of associative, alternative or not necessarily associative rings, or that of Lie rings. It turns out that in quite a few cases the upper radical is hereditary. (cf.Sulinski [7], Rjabuhin [6], Armendariz [2], Szasz—Wiegandt [8]).W. G. Leavitt has suggested the problem: Give a necessary and sufficient condition to be satisfied by the regular classM so that the upper radical classUM ofM is hereditary. In the present paper we shall give such a necessary and sufficient condition. If the classM satisfies an even stronger condition, then theUM-semisimple objects are subdirectly embeddable in a (direct) product ofM-objects. Also a necessary and sufficient condition is given which assures that eachUM-semisimple object can be subdirectly embedded in a (direct) product ofM-objects.



Journal ArticleDOI
TL;DR: In this paper, a finite-dimensional, complete, metric AR (or ANR) is defined, and X×R∞ is R∞ (or an open subset of R ∞).
Abstract: If X is a finite-dimensional, complete, metric AR (or ANR), then X×R∞ is R∞ (or an open subset of R∞).

Patent
20 Jun 1973
TL;DR: In this article, the elastic or creped wrapping band is wound around the end of a cylindrical product in several layers with one edge of the band extending over the product while stretching the band so that said edge contracts tightly against the end side of the product, the wrapping band thus forming a hooplike protection at the edge of a product.
Abstract: A preferably cylindrical product, such as a paper roll, is wrapped using an elastic wrapping band, particularly in order to protect the edges thereof from damage caused by water and shocks during transport and storing. The elastic or creped wrapping band is wound around the end of the product in several layers with one edge of the band extending over the end of the product while stretching the band so that said edge thereof contracts tightly against the end side of the product, the wrapping band thus forming a hooplike protection at the edge of the product. Alternatively there may be used only one layer of a relatively thick elastic material, such as rubber. Before applying onto the product that part of the wrapping band coming against the mantle surface of the product may be prestretched in order to give the band a suitable form corresponding to the product end to be protected. In a paper roll the mantle surface may be protected by a thinner wrapping or the outest paper layers themselves may form the desired protection in this area.

Journal ArticleDOI
TL;DR: The problem of embedding the inverse semigroup congruences into a certain product lattice is also considered in this paper, where the class of θ-modular bands of groups is defined by means of a type of modularity condition on Λ(S ).
Abstract: It is implicit in a result of Kapp and Schneider [3] that, if S isa completely simple semigroup, then the lattice Λ( S ) of congruences on S can be embedded in the product of certain sublattices. In this paper we consider the problem of embedding Λ( S ) in a product of sublattices, when S is an arbitrary band of groups. The principal tool is the θ-relation of Reilly and Scheiblich [7]. The class of θ-modular bands of groups is definedby means of a type of modularity condition on Λ( S ). It is shown that the θ-modular bands of groups are precisely those for which a certain function is an embedding of Λ( S ) into a product of sublattices. The problem of embedding the inverse semigroup congruences into a certain product lattice is also considered.

Journal ArticleDOI
TL;DR: In this paper, the eigenvalues of many retrocirculants were determined and a necessary and sufficient condition for a permutation matrix to commute with a retrocircular matrix was presented.


Journal ArticleDOI
01 Jan 1973
TL;DR: In this article, the authors give an example of a separable, pseudo-complete metric space X which is totally non-meagre and yet whose square Xx X is not totally nonmeggre.
Abstract: In this note we give an example of a separable, pseudo-complete metric space X which is totally nonmeagre (= every closed subspace of X is a Baire space) and yet whose square Xx X is not totally nonmeagre.