Showing papers on "Product (mathematics) published in 1981"

Maximum entropy and conditional probability

Abstract: It is well-known that maximum entropy distributions, subject to appropriate moment constraints, arise in physics and mathematics. In an attempt to find a physical reason for the appearance of maximum entropy distributions, the following theorem is offered. The conditional distribution of X_{l} given the empirical observation (1/n)\sum^{n}_{i}=_{l}h(X_{i})=\alpha , where X_{1},X_{2}, \cdots are independent identically distributed random variables with common density g converges to f_{\lambda}(x)=e^{\lambda^{t}h(X)}g(x) (Suitably normalized), where \lambda is chosen to satisfy \int f_{lambda}(x)h(x)dx= \alpha . Thus the conditional distribution of a given random variable X is the (normalized) product of the maximum entropy distribution and the initial distribution. This distribution is the maximum entropy distribution when g is uniform. The proof of this and related results relies heavily on the work of Zabell and Lanford.

Overlapping Clustering: A New Method for Product Positioning:

Abstract: Most clustering techniques used in product positioning and market segmentation studies render mutually exclusive equivalence classes of the relevant products or subjects space. Such classificatory ...

Aggregation of Product Data for Hierarchical Production Planning

Abstract: When applying an aggregate view to production planning, the production system is given a hierarchical structure. The problem description at the upper level which is less detailed concerns product groups and machine groups rather than individual products and machines. This means that product structures and capacity requirements also must be expressed in terms of product groups and machine groups. In this paper it is shown under what conditions a perfect aggregation of product data can be obtained. If a perfect aggregation is not possible, our problem is to find a good approximate solution. A mathematical formulation of this approximation problem and its general solution is also given.

Joint continuity of separately continuous functions

Abstract: ABsmRAcr. It is shown that a separately continuous function f: X x Y -+ Z from the product of a certain type of Hausdorff space X and a compact Hausdorff space Y into a metrizable space Z is jointly continuous on a set of the type A x Y, where A is a dense G8 set in X. The class of Hausdorff spaces X in question is defined by a gametheoretic condition. The result improves (and simplifies the proof of) a recent result of Namioka. Many "deep" theorems in functional analysis and automatic continuity theory are easy corollaries.

Cost and performance of VLSI computing structures

Abstract: Using VLSI technology, it will soon be possible to implement entire computing systems on one monolithic silicon chip. Conducting paths are required for communicating information throughout any integrated system. The length and organization of these communication paths place a lower bound on the area and time required for system operations. Optimal designs can be achieved in only a few of the many alternative structures. Two illustrative systems are analyzed in detail: a RAM-based system and an associative system. It is shown that in each case an optimum design is possible using the area-time product as a cost function.

Singular integrals on product domains

Abstract: Introduction. In their well-known theory of singular integrals on R, Calderón and Zygmund [1] obtained the boundedness of certain convolution operators on R which generalize the Hubert transform on R. Thus, we know that if Tf = ƒ * K and K(x) is defined on R and satisfies the analogous estimates that 1/x satisfies on/? , namely (i) \\K(x)\\2\\H\\ \\ ( )~ K(x)\\dx < C for all h * 0, then T is a bounded operator on L(R) for 1 < p < <*>. (See Stein [2].) Now if we take the space R x R along with the two parameter family of dilations (x, y) —* (dxx9 82y), xeR n 9yGR , Ô. > 0, instead of the usual one parameter dilations, we are led to consider operators which generalize the double Hubert transform on R, H f = ƒ * 1/xy. The boundedness properties of H are usually very easy to obtain by an argument which iterates the one-dimensional theory of the Hubert transform. But if we consider, more generally, operators Tf—f*K where K satisfies analogous estimates to those satisfied by 1/xy but cannot be written in the form Kx(x) • K2(y) then the argument which deals with H fails. We wish to announce here that for various classes of kernels K which \"look like\" l/xy on R, but are not products of two functions on the x and y variables respectively, the convolution operators are bounded on L for 1 < p < °° and take L log*L(R x R) boundedly to weak L. In particular this involves the problem of formulating the right two parameter versions of the assumptions on the kernel K. We wish to take this opportunity to thank E. M. Stein for his help in the course of this work. The formulation of several of our theorems follows his suggestions, and Theorem 3 in its L form is due to him. We also wish to thank the Institute for Advanced Study for its hospitality and the National Science Foundation for its financial support. Statement of results. We shall state three results dealing with the action of convolution operators. These deal with the action of these operators on L, LP for 1 < p < «>, and L logZ, respectively.

Relative Importance of Perceptual Factors to Consumer Acceptance: Linear vs Quadratic Analysis

Abstract: This paper introduces a new approach to evaluating the relative importance of sensory characteristics to acceptance, using quadratic relations. The analysis illustrates that the relative importance of any specific characteristic cannot be defined as a single number, but depends upon the level of that characteristic as perceived by the consumer, and upon the other sensory factors of the product.

The Rogers-Ramanujan identities: Lie theoretic interpretation and proof.

Abstract: The two Rogers-Ramanujan identities, which equate certain infinite products with infinite sums, are among the most intriguing of the classical formal power series identitites It has been found by Lepowsky and Milne that the product side of each of them differs by a certain factor from the principally specialized character of a certain standard module for the Euclidean Kac-Moody Lie algebra A1(1) On the other hand, the present authors have introduced an infinite-dimensional Heisenberg subalgebra [unk] of A1(1) which leads to a construction of A1(1) in terms of differential operators given by the homogeneous components of an “exponential generating function” In the present announcement, we use [unk] to formulate a natural “abstract Rogers-Ramanujan identity” for an arbitrary standard A1(1)-module which turns out to coincide with the classical identities in the cases of the two corresponding standard modules The abstract identity equates two expressions, one a product and the other a sum, for the principally specialized character of the space Ω of highest weight vectors or “vacuum states” for [unk] in the module The construction of A1(1) leads to a concrete realization of Ω as the span of certain spaces of symmetric polynomials occurring as the homogeneous components of exponential generating functions The summands in the Rogers-Ramanujan identities turn out to “count” the dimensions of these spaces For general standard A1(1)-modules, we conjecture that the abstract identities agree with generalizations of the Rogers-Ramanujan identities due to Gordon, Andrews, and Bressoud

Generalization of the Trotter-Lie formula

Abstract: Our generalization of the Trotter-Lie formula is based upon the following principle: starting with a "general mean" of semigroups, the corresponding formula must converge to the semigroup generated by an extension of the arithmetic mean of the generators. Accordingly, the usual product formula is associated with a "geometric mean". We establish "average formulae" for nonlinear semigroups generated by the subdifferentials of convex functionals. We then explore in detail the case of self-adjoint semigroups. A great advantage of these average formulae is that they naturally hold for an arbitrary number of operators whereas the traditional product formulae do not usually hold for more than two operators. This fact is important for numerical and theoretical applications. For instance, it enables us to handle with ease the Schrodinger operator withmagnetic vector potential and with arbitrarily singular scalar potential.

Microscopic Calculation of the Low-Temperature Properties of 3 He- 4 He Mixtures

Abstract: The effective interaction between two $^{3}\mathrm{He}$ atoms in $^{4}\mathrm{He}$ is obtained by assuming the ground-state wave function to be a product of pair correlations. It is used to evaluate some of the low-temperature properties of mixtures.

Article display package and blank therefor

Abstract: This package is particularly adapted for carrying an article to display the latter at the point of sale. The package is formed from a unitary paperboard blank and includes product retaining openings along with integral product encircling portions to retain the product in the openings. The preferred embodiment of the package is adapted for use in carrying flashlights along with batteries carried separately by the package.

Transfer function factorization of SISO 2-D systems using state feedback

Abstract: The problem of factorizing the transfer function of single-input single-output 2-D systems to a product of two 1-D transfer functions using state feedback is considered. It is shown that if the numerator of the open-loop transfer function can, by itself, be written as a product of two 1-D polynomials, then the closed-loop transfer function can, under certain conditions, be factorized as a product of two 1-D transfer functions. In cases where the numerator is not by itself factorizable, then the present results restrict themselves to factoring the denominator of the closed-loop system to a product of two 1-D polynomials.

Dual layer record element and method

Abstract: A method of making a dual layer record element for photographically and magnetically recording information respectively on opposite faces thereof and product.

$K$-theory of Azumaya algebras

Abstract: Quillen has defined a K-theory for symmetric monoidal categories. We show that Quillen's groups agree with the groups KO, K,, and K2 defined by Bass. Finally, we compute the K-theory of the Azumaya algebras over a commutative ring. The purpose of this paper is to advertise the K-theory of symmetric monoidal categories, and to compute the K-theory of the category of Azumaya R-algebras. The point is that Quillen's theory (introduced in [61) is a natural generalization of the "classical" theory for Ko, K1, K2 defined by Bass in [2], [3], [4]. On the other hand, it provides a wealth of examples of infinite loop spaces (see [1], [9], [10], [12] and [13]). A symmetric monoidal category is a category S with a unit 0: * S and a product [Z: S x S -> S which is commutative and associative up to coherent natural isomorphism; the precise definition may be found in [7]. We shall be especially interested in the following examples (from [2]): (1) P, the fin. gen. projective modules over a ring R. The product [1 is direct sum, and we consider only isomorphisms. (2) FP, the fin. gen. faithful projective modules over a commutative ring R. The product [1 is the tensor product, and the arrows are isomorphisms. (3) Pic, the full subcategory of FP of rank one projective modules. (4) Az, the Azumaya algebras over a commutative ring R. The arrows are R-algebra isomorphisms, and the product is the tensor product. If R is a field an Azumaya algebra is just a central simple algebra. In the language of [3, Chapter VII], a symmetric monoidal category is a "category with product [1", with the additional condition that there be a special object 0 and natural isomorphisms 0 LI s s -s [1 0 satisfying the coherence conditions on page 159 of [7]. Groups Kdet(S) (i = 0, 1, 2) were defined and studied in [2], [3] and [4], using only the objects, isomorphisms and product of the category S. We will restrict our attention to the category SMCat of small symmetric monoidal categories and relaxed morphisms. We require in addition that every symmetric monoidal category S in SMCat satisfies (i) every arrow is an isomorphism, and (ii) every translation s[1]: Aut(t) -> Aut(s LI t) is an injection. The categories P, FP, Pic, Az all belong to SMCat, as do the categories: Received by the editors November 29, 1979. AMS (MOS) subject classifications (1970). Primary 18F25; Secondary 18D10, 16A16, 55P47.

Product spaces in locales

Abstract: The embedding of sober spaces in locales preserves products X X Y where X is quasi-locally compact. A completely regular space X has this preservation property, for all Y, if and only if X is a complemented sublocale of a compact space. Equivalently, every closed subset of X is locally compact somewhere. This paper assumes such familiarity with locales as can be gotten from the first nine pages of [4]. There will be a couple of references to later results from [4], but the first nine pages (through 1.5) establish the conceptual apparatus. The referee suggested that "a brief description of the product locale of two spaces . . . would be helpful". Yes, indeed. The existence of these products, i.e. of coproducts in the dual category, was an unsolved problem for some time until Benabou proved existence [1]. But it happens that, after the first version of this paper was written, D. Wigner sent me his beautiful brief description (to be published). Recall, the dual object of a locale X is the complete Heyting algebra T(X) of open parts of X. If A1 is a (sober) space, T(X) is just its topology. Wigner gets the product X X Y from Galois connections between T(X) and T( Y). Note, this can be skipped; Wigner's theorem will not be used below. But a Galois connection between two partially ordered sets P, Q, consists of two functions (order-reversing) /: P —» Q, g: Q -> P, such that y < f(x) if and only if x < g(y)These connections are partially ordered, (/, g) < (/', g') if / < /' pointwise. (Equivalently, g < g'.) It was known [5] that the Galois connections between two complete Heyting algebras A, B, so ordered, form a complete Heyting algebra A <8> B. Wigner showed that T(X) ® T(Y) is T(X X Y). Turning toward proofs, we need to name the spaces X whose topology is a continuous lattice. Various names have been used (e.g. in [4]), but A. J. Ward's quasi-locally compact seems to be prevailing. The defining (lattice-theoretic) property: each open set U is the union of those open sets B such that every open cover of U has a finite subcollection covering B, i.e. B is bounded in U. Actually, to justify identifying topological spaces X with the corresponding locales, we need to assume all spaces are sober. (It is easy to check that the obvious interpretations of the theorems below for nonsober spaces remain true—vacuously for Hausdorff spaces, which are sober.) Then one should note another nice, rather new theorem, which will not be used: sober quasi-locally compact spaces are locally quasi-compact [3]. Received by the editors September 27, 1979 and, in revised form, November 19, 1979. 1980 Mathematics Subject Classification. Primary 54H99; Secondary 54B10.

The axiom of determinacy and the prewellordering property

Abstract: Let ω = (0,1,2,...) be the set of natural numbers and R = ω^ω the set of all functions from ω into ω, or for simplicity reals. A product space is of the form X = X_1 x X_2 x ... x X_k, where X_1 = ω or R. Subsets of these product spaces are called pointsets. A boldface pointclass is a class of pointsets closed under continuous preimages and containing all clopen pointsets (in all product spaces).

Slicing method and apparatus

Abstract: An apparatus and method are disclosed for slicing elongated products, such as sausages. A pair of product engaging members are driven simultaneously in opposite directions so that one member can feed the product to a slicer while the other member is returning to the start position. A magazine may be provided for storing a supply of products and automatically discharging the product in timed relationship with the product engaging members. In addition, the product engaging members may have tines for gripping the product during feeding to the slicer and for releasing the waste end of the product during the return stroke.

Method and apparatus for manufacturing metal sections

Abstract: In making metal sections, starting material having a quadrangular cross-section is hot-forged into a blank resembling the desired product in cross-section, and the length of the piece is divided into sections of a given length. Then, the blank is hot-rolled into the product having the desired cross-section.

Products of conjugate permutations.

Abstract: If s, pβSv, \s\ S \p\ and \p\ is infinite, then s is a product of 4 elements each conjugate to p. Furthermore, 4 is minimal with this property. The latter follows by examining s = (123) and any permutation p containing only transpositions (without fixed points) in its disjoint cycle decomposition, cf. G. Moran [6, p. 76] and [4, p. 288, 289]. If p is odd and s is even (with finite supports), then obviously s $ (pή\ and similar examples with finite | p | show

On States on the Product of Logics

Abstract: We take up the question of when a state (= σ-additive measure) on the product of logics (=σ-orthomodular posets) depends on at most countably many coordinates. We show that it is always so provided there are no real-measurable cardinals. The manner of dependence is a kind of convex combination. We derive some consequences of the latter statement.