Showing papers on "Product (mathematics) published in 1978"
TL;DR: Product formulas constitute one of several bridges between numerical and functional analysis as discussed by the authors, and they represent algorithms intended to approximate some evolution equation and, in functional analysis, they are used to prove estimates, existence and representation theorems.
Abstract: Product formulas constitute one of several bridges between numerical and functional analysis. In numerical analysis, they represent algorithms intended to approximate some evolution equation and, in functional analysis, they are used to prove estimates, existence and representation theorems. Our aim is to survey the setting for product formulas and to discuss some recent results. Needless to say, we do not attempt to accommodate all the complex variations which occur in practical algorithms, nor the sharpest possible theoretical results. Nevertheless, we hope that our middle ground approach and some of the examples will be of interest to both groups. Because of its survey nature, we have not hesitated to include some well-known examples which are important for understanding the ideas. The general idea of product formulas is the following. Suppose one is interested in an initial value problem
338 citations
TL;DR: In this paper, the special point formulas generated from product Gauss-Chebyshev integration formulas for fcc and bcc lattices are compared with those resulting from the special treatment suggested by Monkhorst and Pack.
Abstract: The special-point formulas generated from product Gauss-Chebyshev integration formulas for fcc and bcc lattices are compared with those resulting from the special treatment suggested by Monkhorst and Pack. It is found that the Monkhorst-Pack special-point formulas are more efficient for formulas with few points and less efficient for formulas with many points.
100 citations
Book•
01 Jan 1978
TL;DR: The authors constructed and tested models incorporating this dependence and tested them on a set of product classes (e.g., magazines, stereo albums, liquor in a home bar) and found that the assumption that product choices are made independently of each other is implausible.
Abstract: By focusing on consumer preference for single items, researchers have implicitly assumed that product choices are made independently of each other. For many product classes (e.g., magazines, stereo albums, liquor in a home bar) that assumption is implausible. “Balancing” or “rounding out” a group of items implies dependence among selections. Models incorporating this dependence are constructed and tested.
99 citations
TL;DR: An instrumental variable modification to the product moment technique which overcomes problems when significant amounts of extraneous noise are present with a negligible amount of extra computation.
98 citations
91 citations
TL;DR: In this paper, it was shown that the Jacobian of the Nth Fermat curve is isogenous over C. The purpose of this paper is to determine when two factors in this product are isogenous over C, and whether they are absolutely simple.
Abstract: Let denote the Nth Fermat curve. The period lattice of F(N) is contained with finite index in the product of certain lattices Lr,s (see [6]), and to this inclusion of lattices there corresponds an isogeny of the Jacobian of F(N) onto a product of abelian varieties. The purpose of this paper is to determine when two factors in this product are isogenous over C, and whether they are absolutely simple.
85 citations
TL;DR: The author empirically illustrates how to identify penetrable market segments in terms of image-direct measurement influences and discusses the implications of this type of information in forming a realistic product positioning strategy.
Abstract: Product positioning studies for both new and established products generally involve consumers’ perception of the current product space in terms of the salient product attributes. Perceptions per br...
84 citations
TL;DR: For every field F and every pair (n,k) of positive integers, an n×n matrix S over F is a product of k idempotent matrices over F iff rank(I − S)⩽k· nullity S.
77 citations
TL;DR: In this article, a product of topological amplitudes with specific genus and boundary structure is defined, also specified by a genus and a boundary structure, which is independent of the particular graphical representation of the members of the product.
76 citations
Patent•
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TL;DR: A printed promotional coupon comprises a multiple-product base coupon and a plurality of product stamps, each of which bears an indicia defining a coupon redemption value for a particular product purchase.
Abstract: A printed promotional coupon comprises a multiple-product base coupon and a plurality of product stamps. The product stamps bear indicia defining a coupon redemption value for a particular product purchase. The coupon acquires value and becomes effective only by the affixing of one or more stamps onto the base coupon, each at a location which bears indicia corresponding to the associated stamp. The coupon holder can determine the overall redemption value of the coupon by selecting the number of stamps affixed, corresponding to the number of different products purchased at any retail store carrying the products. In one embodiment, the base coupon and product stamp form a single sheet, with the product stamps being detachable from the base coupon. In another embodiment, the product stamps are on a card or sheet separate from but distributed with the base coupon.
65 citations
TL;DR: The inner product computer greatly reduces computational costs for the solution of a large class of problems, including computerized tomography, image restoration, weather forecasting, and economic modeling.
Abstract: The inner product computer is a special-purpose computational unit intended to be used as an adjunct to a general-purpose digital computer to perform numerical processing tasks which previously exceeded the capacity of the general-purpose computer. The algorithmic structure of the inner product is briefly reviewed in the first section of this paper. Methods are described for computing the inner product of complex vectors with a series of four real inner products. Several hardware implementations of the inner product computer are described and then compared in terms of speed and complexity; a figure of merit is developed to simplify the comparison. The utility of this computational unit is demonstrated via the examination of a large-scale numerical problem, computerized three-dimensional x-ray reconstruction (computerized tomography) arising in the biomedical sciences. Finally, a comparison is given of the size of a general-purpose computer required to execute a large-scale processing task with that of an inner product computer to execute the same task. The inner product computer greatly reduces computational costs for the solution of a large class of problems, including computerized tomography, image restoration, weather forecasting, and economic modeling.
Patent•
08 Dec 1978
TL;DR: In this paper, a dispensing nozzle for an aerosol package of the barrier type is presented, where a liquid product per se is directed out of the container through the usual valve.
Abstract: A dispensing nozzle for an aerosol package of the barrier type wherein a liquid product per se is directed out of the container through the usual valve. The nozzle is constructed to mix with the liquid product being dispensed air which is drawn in by the product so as to atomize the product and permit the dispensingthereof as a spray.
TL;DR: In this article, the operation of Walsh product matrix on a coefficient vector equals the product of a coefficient matrix and a Walsh vector, and this unique property of Walsh function is used to determine the unknown parameters of a general bilinear system from the input-output data.
Abstract: Walsh product matrix is formed by the multiplication of Walsh vector and its transpose. The operation of Walsh product matrix on a coefficient vector equals the product of a coefficient matrix and a Walsh vector. This unique property of Walsh function is used to determine the unknown parameters of a general bilinear system from the input-output data. An example with satisfactory result is given.
TL;DR: A separable metric space X and a paracompact space Y are constructed such that dim X + dim Y = 0 < 1 = dim(X x Y).
Abstract: The covering dimension of a product space can exceed the sum of the dimensions of its factors. A separable metric space X and a paracompact space Y are constructed such that dim X + dim Y = 0 < 1 = dim(X x Y). Related results are also discussed.
Journal Article•
01 Jan 1978
TL;DR: In this paper, it was shown that if X is a homogeneous Hausdorff space, then |X"| < 2"(;f); no power of ß (w) u, or of ÞQ Q or of ǫ R R is homogeneous.
Abstract: We prove a general nonhomogeneity result which implies among others (1) if X is a homogeneous Hausdorff space, then |X"| < 2"(;f); (2) no power of ß (w) u, or of ßQ Q or of ßR R is homogeneous.
Patent•
26 Apr 1978
TL;DR: In this paper, a method and apparatus for storing, on a product card, information relating to the description and the unit price of a product to be weighed, and for providing a label printer with the stored information as well as the total price of the product in a simple and reliable operation.
Abstract: Described herein is a method and apparatus for storing, on a product card, information relating to the description and the unit price of a product to be weighed, and for providing a label printer with the stored information as well as the total price of the product in a simple and reliable operation. Preferably, the invention makes use of a card reader for reading the information stored on the product card, a computer for storing the information and for calculating the total price of the weighed product. The information read from the product card and the total product price are transmitted to a label printer for printing thereof on a label. In its preferred form, the invention provides for the modification of the unit price information on the product card to reflect a manually set unit price.
TL;DR: In this article, product integration is defined for a class of bounded-operator-valued functions on a Banach space X and properties of the product integral in the new setting are derived, including material on improper product integration.
TL;DR: In this article, the exoergic reactive scattering of F + H 2 → FH + H + H was calculated in the framework of distorted wave approximation, and the Vibration-rotation distribution of the product molecule obtained in the present calculation was discussed in terms of available experimental results.
TL;DR: In this paper, a relation to the permutation-inversion group (Longuet-Higgins group) for semi-rigid molecules is established. But the relation is not necessarily homomorphic to the isometric group.
Abstract: After a short recapitulation of the basic assumptions underlying the isometric group concept for semi-rigid molecules a relation to the permutation-inversion group (Longuet-Higgins group) is established. Conditions will be put forward under which the permutation-inversion group is homomorphic (not isomorphic) to the isometric group. It will be shown that the familiar symmetry concept of quasi-rigid molecules is identical with the isometric group concept. Finally the latter will be generalized to non-rigid molecules; the isometric group of such systems will be shown to be isomorphic with the isometric group of the associated semi-rigid model. For semi-rigid models with proper covering group the isometric group is shown to be a semi-direct product of the covering group and the internal isometric group.
TL;DR: In this paper, the authors studied the isosingular locus of points in a complex analytic space at which a singularity is complex analytically isomorphic to another singularity in the same space.
Abstract: Let X be a (not necessarily reduced) complex analytic space, and let V be a germ of an analytic space. The locus of points q in X at which the germ Xq is complex analytically isomorphic to V is studied. If it is nonempty it is shown to be a locally closed submanifold of X, and X is locally a Cartesian product along this submanifold. This is used to define what amounts to a coarse partial ordering of singularities. This partial ordering is used to show that there is an essentially unique way to completely decompose an arbitrary reduced singularity as a cartesian product of lower dimensional singularities. This generalizes a result previously known only for irreducible singularities. 0. Introduction. Let X be a complex analytic space. For q E X, Xq will denote the germ of X at q. In this paper I will study the isosingular loci defined by DEFINITION 0. 1. For p E X let Iso(X,p) = {q E XlXq ;)X. (~ here and elsewhere will mean complex analytically isomorphic.) It will be shown that: THEOREM 0.2. For any p E X, Iso(X, p) is a (possibly 0-dimensional) complex submanifold of some open subset of X. Moreover, for any q E Iso(X, p) there is an open neighbornood U of q, and an analytic space Y such that U_ Y x (U n Iso(X,p)). (X is the cartesian product in the category of analytic spaces.) This result is used to introduce what is, in effect, a partial ordering of complex analytic singularities in terms of their complexity. This, in turn, is used to study the ways in which a germ of an analytic space may be written as the cartesian product of other germs of analytic spaces. Let V be a germ of an analytic space (V not the reduced point). By a decomposition of V of length Received by the editors April 1, 1977. AMS (MOS) subject classifications (1970). Primary 32C15, 32C40; Secondary 32B 10, 32C25.
TL;DR: The product of two subsets C, D of a group is defined as. The power Ce is defined inductively by C 0 = {1}, Ce = CCe−1 = Ce−1C as mentioned in this paper.
Abstract: The product of two subsets C, D of a group is defined as . The power Ce is defined inductively by C0 = {1}, Ce = CCe−1 = Ce−1C. It is known that in the alternating group An, n > 4, there is a conjugacy class C such that CC covers An. On the other hand, there is a conjugacy class D such that not only DD≠An, but even De≠An for e<[n/2]. It may be conjectured that as n ← ∞, almost all classes C satisfy C3 = An. In this article, it is shown that as n ← ∞, almost all classes C satisfy C4 = An.
TL;DR: In this paper, a product of two-body and three-body correlation functions is used as a variational wave function for liquid $^{4}\mathrm{He} to take into account the backflows produced by two particles recoiling from each other, and the distribution functions, the energy, and its uncertainty are all calculated using the Lennard-Jones-deBoerMichel potential, and diagrammatic hypernetted-chain summation methods.
Abstract: A product of two-body (${f}_{\mathrm{ij}}$) and three-body (${f}_{\mathrm{ijk}}$) correlation functions is used as a variational wave function for liquid $^{4}\mathrm{He}$. The ${f}_{\mathrm{ijk}}$ take into account the backflows produced by two particles recoiling from each other. The distribution functions, the energy, and its uncertainty are all calculated using the Lennard-Jones-deBoer-Michel potential, and diagrammatic hypernetted-chain summation methods. The calculated equilibrium energy of -6.72 (\ifmmode\pm\else\textpm\fi{}0.2)\ifmmode^\circ\else\textdegree\fi{}K, is significantly lower than the -5.9\ifmmode^\circ\else\textdegree\fi{}K obtained with only a product of ${f}_{\mathrm{ij}}$, and agrees with the -6.84\ifmmode^\circ\else\textdegree\fi{}K estimated from a Monte Carlo integration of the many-body Schr\"odinger equation. The proposed wave function is simple enough to be useful in Fermi liquids.
Patent•
23 Aug 1978TL;DR: In this article, a cutting device for use with an extruder rotates to strike the extruded product with an upward motion to thereby force the product in a generally upward direction.
Abstract: At least one knife on a cutting device for use with an extruder rotates to strike the extruded product with an upward motion to thereby force the product in a generally upward direction. The knife is encased in a vapor-permeable shroud having a size sufficient to prevent substantial striking of the shroud by the cut product.