Showing papers on "Product (mathematics) published in 1986"
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TL;DR: Algorithms for optimal lot sizing of products with a complex product structure are developed that convert the classical formulation of the general structure problem into a simple but expanded assembly structure with additional constraints, and solve the transformed problem by a branch-and-bound based procedure.
Abstract: Lot sizing of products that have complex bills of materials plays an important role in the efficient operations of modern manufacturing and assembly processes. In this paper we develop algorithms for optimal lot sizing of products with a complex product structure. We convert the classical formulation of the general structure problem into a simple but expanded assembly structure with additional constraints, and solve the transformed problem by a branch-and-bound based procedure. The algorithm uses a Lagrangean relaxation and subgradient optimization procedure to generate tight lower bounds on the optimal solutions. In computational experiments, a code based on this method was able to solve single end product problems with up to 40 stages in the product structure. The model is extended to handle problems with multi-end items in the product structure, but with less favorable computational results.
177 citations
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TL;DR: This article deals with the boundedness properties of singular integrals that generalize the double Hilbert transform on product domains.
Abstract: This article deals with the boundedness properties of singular integrals that generalize the double Hilbert transform on product domains. The action of these operators on Hardy spaces and on L(log L) is discussed.
103 citations
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TL;DR: In this article, the singular value decomposition of a product of two general matrices without explicitly forming the product is computed using plane rotations applied to the two matrices separately, based on an earlier Jacobi-like method due to Kogbetliantz.
Abstract: An algorithm is developed for computing the singular value decomposition of a product of two general matrices without explicitly forming the product. The algorithm is based on an earlier Jacobi-like method due to Kogbetliantz and uses plane rotations applied to the two matrices separately. A triangular variant of the basic algorithm is developed that reduces the amount of work required.
100 citations
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TL;DR: Describing circuits for computation of a large class of algebraic functions on polynomials, power series, and integers, for which, it has been a long standing open problem to compute in depth less than $\Omega (\log n)^2 $.
Abstract: This paper describes circuits for computation of a large class of algebraic functions on polynomials, power series, and integers, for which, it has been a long standing open problem to compute in depth less than $\Omega (\log n)^2 $.Algebraic circuits assume unit cost for elemental addition and multiplication. This paper describes $O(\log n)$ depth algebraic circuits which given as input the coefficients of n degree polynomials (over an appropriate ring), compute the product of $n^{O(1)} $ polynomials, the symmetric functions, as well as division and interpolation of real polynomials. Also described are $O(\log n)$ depth algebraic circuits which are given as input the first n coefficients of a power series (over an appropriate ring) compute the product of $n^{O(1)} $ power series, as well as division, reciprocal and reversion of real power series.Furthermore this paper describes boolean circuits of depth $O(\log n(\log \log n))$ which, given n-bit binary numbers, compute the product of n numbers and integ...
70 citations
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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
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TL;DR: In this article, the ITEP group has analyzed the structure of the operator product expansion in the 1/N expansion of some two-dimensional models and showed that, despite their claim, their analysis does not invalidate the conclusions of previous studies by the present author on the role of renormalon singularities in the operation product expansion.
60 citations
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55 citations
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TL;DR: In this paper, a theory of perceptual organization is presented that accords with the Description Postulate presented in Paper I; i.e., the theory claims that perceptual organisation is defined by the input group of a machine which has a decomposition sequence prescribed by an algebraic stability ordering.
48 citations
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TL;DR: In this paper, it was shown that a tensor product of cyclic algebras is not a cross product, and that the splitting fields of the cyclic product can be constructed by the Brauer group.
Abstract: New examples of noncrossed product division algebras are obtained, using methods different from all previous noncrossed product constructions The examples are division algebras over intersections of />-Henselian valued fields, and they have Schur index /?"' and exponent p" for any prime number p and any integers m>n>2(n>3if^ = 2) The basic tools used in the construction are valuation theory and Galois cohomology; no generic methods are applied and there is no pi theory Along the way, local-global principles are proved for central simple algebras over intersections of p-Henselian valued fields The first examples of central simple division algebras which are not crossed products were obtained by Amitsur in (Am) in 1972 Amitsur thereby settled a question that had been one of the outstanding open problems in the theory of algebras for at least thirty years His examples were the generic division algebras UD(Q, n) of index n over Q (the rational numbers) for any natural number n such that p2
, p an odd prime, or 8|« All subsequent constructions of noncrossed products in (SS, 83,-883, R'» R°> an^ Ti) have heretofore been based on Amitsur's specialization argument, and they are all generic division algebras or extensions of generic division algebras The centers of these noncrossed product algebras are not known, nor are the Brauer groups, nor the absolute Galois groups of the centers We present here a new method of constructing noncrossed product algebras In our approach, the noncrossed product is realized as the underlying division algebra D of a tensor product of suitably chosen cyclic algebras over a field F = L, n L2, where each L, is a pth root Henselian valued field We prove local global principles relating the splitting fields of D to those of D ®F Lt, i = 1, 2 It is shown that the />-part of the Brauer group of F is completely determined by that of Ll and L2 Computations for central simple F-algebras thus become very tractable For exam- ple, we not only show that D is not a crossed product, but also calculate exactly how large r must be so that the matrix ring Mr{D) is a crossed product, and how large s must be so that MS(D) is a tensor product of cyclic algebras Indeed, the structure of the Brauer group of F is so nice that we were somewhat surprised that noncrossed products could possibly exist over F
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22 May 1986
TL;DR: In this article, a flexible package made of plastic and capable of standing by itself, comprising a generally cylindrically-shaped body, comprised of a flexible plastic film; a first cover part at one end of the cylindrical shape, wherein the cover parts are bonded to the plastic film by heat-sealing.
Abstract: Disclosed is a flexible package made of plastic and capable of standing by itself, comprising a generally cylindrically-shaped body, comprised of a flexible plastic film; a first cover part at one end of the cylindrically-shaped body; and a second cover part at the opposite end of the cylindrically-shaped body, wherein the cover parts are bonded to the plastic film by heat-sealing. Also disclosed is a packaged liquid product, in particular a product under gas pressure, and a method for producing the packaged product.
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TL;DR: The mixed (or synchronous) product is used to construct particular asynchronous automata and it is shown that each asynchronous automaton is the image by a strictly alphabetic morphism of a mixed product of automata of a free partially commutative monoids.
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TL;DR: The product operator description of magnetic resonance experiments is developed in the spherical-tensor basis, and a computer simulation scheme, employing this description, is reported in this article, where the utility of simulations in the design of selective phase cycles and in the study of the effects of spectrometer imperfections is demonstrated.
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TL;DR: A group G is said to be factorizable if G = AB with A,B proper subgroups of G. The general problem is to obtain information on structure of a factorizabl... as mentioned in this paper.
Abstract: All the groups in this article are finite. A group G is said to be factorizable if G = AB with A,B proper subgroups of G . The general problem is to obtain information on structure of a factorizabl...
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01 Dec 1986TL;DR: An algorithm is presented in this paper for computing state space balancing transformations directly from a state space realization using the singular value decomposition of a certain product of matrices without explicitly forming the product.
Abstract: An algorithm is presented in this paper for computing state space balancing transformations directly from a state space realization. The algorithm requires no "squaring up." Various algorithmic aspects are discussed in detail. Applications to numerous other closely-related problems are also mentioned. The key idea throughout involves determining a contragredient transformation through computing the singular value decomposition of a certain product of matrices without explicitly forming the product.
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01 Mar 1986TL;DR: In this article, a necessary and sufficient condition for the validity of the product theorem for the Hilbert transform of a product of two signals x(t) and y(t), is derived.
Abstract: The usual product theorem for Hilbert transforms states that under certain conditions x(t)y(t), the Hilbert transform of a product of two signals x(t) and y(t), is equal to x(t)y^(t), a result having repeated use in simplifying calculations involving modulated waveforms. Several sufficient conditions for the theorem to hold are well-known and appear in the literature. Here, using a time-domain approach, we derive a necessary and sufficient condition for the validity of the theorem and show as an example two signals which have the desired property but which do not satisfy any of the earlier sufficient conditions.
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TL;DR: It is shown that any Boolean function f which can be expressed in a sum-of-products form using m product terms can contain as many as 2m− 1 implicants but no more.
Abstract: We show that any Boolean function f which can be expressed in a sum-of-products form using m product terms can contain as many as 2m− 1 implicants but no more.
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TL;DR: It is proved that an orthogonal product is generally more informative than its multipliers, namely, if cost is not considered a constraining factor, then there is a nonnegativevalue to obtaining a second opinion.
Abstract: The paper discusses the value of information when a numberof independent sources provide information related to acommon set of states of nature.The starting point is the Information Economic model ofInformation Structures. The model is augmented to representindependence of informational sources by means oforthogonality of the information structures.A new mathematical operator, orthogonal product, is definedand its properties are probed. It is shown that thisoperator maintains some mathematical properties such asclosure, association, unity element, null element, etc. Itis demonstrated how the orthogonal product represents thenotion of multi-source information.The paper proves that an orthogonal product is generallymore informative than its multipliers, namely, if cost isnot considered a constraining factor, then there is a nonnegativevalue to obtaining a second opinion.The paper concludes with a numerical example and adiscussion on the applicability of the model oforthogonality.
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TL;DR: In this article, the authors formulate design and manufacturing processes in terms of constraint propagation and satisfaction: designer's intention or design requirements are interpreted into the form of constraints description, and each design process receives these constraints, decomposes them into several small problems, and attempts to solve them.
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11 Sep 1986
TL;DR: A unitary fitted product having a dust ruffle secured to a peripheral boundary of the product is defined in this paper as a unitary product with attached dust ruffles, where the ruffle is attached to the product itself.
Abstract: FITTED PRODUCT WITH ATTACHED DUST RUFFLE A unitary fitted product having a dust ruffle secured to a peripheral boundary thereof.
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TL;DR: The use of p-adic numbers is to allow free use of rational commutative algebra in the systematic study of congruences as discussed by the authors, where the product formula resembles the function-theoretic principle that the sum of the residues is zero.
Abstract: Publisher Summary This chapter discusses the instructive development of p-adic model theory. The use of p-adic numbers is to allow free use of rational commutative algebra in the systematic study of congruences. The observation that the product formula resembles the function-theoretic principle that the sum of the residues is zero is the source of some of the most powerful ideas in modern number theory. There is an important topological difference between the number field case and the general function field case. Quantifier elimination was the original method of Tarski. It was later eclipsed by the methods of model-completeness and saturated models, but with the emphasis on complexity of computation, it was again prominent.
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07 Feb 1986
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14 Apr 1986TL;DR: In this paper, the spatial product sum of a plurality of picture element data stored in a frame memory is calculated using load coefficients with N rows and N columns stored in the coefficient memory.
Abstract: The present invention is directed to an arithmetic which calculates the spatial product sum of each of a plurality of picture element data stored in a frame memory, through utilization of load coefficients with N rows and N columns stored in a coefficient memory. The results of calculation for pieces of picture element data of one row of the frame memory and load coefficients of one row are added to the contents of shift registers corresponding to picture elements. This operation is repeated N times for different rows of the frame memory and different load coefficients. Thus, the spatial product sum calculation is performed at high speed using a small number of multipliers.
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01 Feb 1986
TL;DR: In this article, the authors give an example of a weakly infinite-dimensional space X such that the product X X B of X with a certain subspace B of the space of irrationals is strongly infinite.
Abstract: We give an example of a weakly infinite-dimensional space X such that the product X x B of X and a subspace B of the irrationals is strongly infinite-dimensional; under the assumption of the Continuum Hypothesis, B can be the irrationals. This example answers a question of Addis and Gresham [AG]. 1. Terminology and notation. All spaces under discussion are metrizable and separable. Our terminology follows [AP and E]. We denote by I the real interval [0, 1], by C the usual Cantor set in I and by I' the Hilbert cube. We denote the space of the irrational numbers from I by P and the rational numbers from I by Q. A space X is weakly infinite-dimensional [AP, Chapter 10, ??4-7] if for every sequence {(A1, B1), (A2, B2), ... } of pairs of closed disjoint subsets of X there are partitions L, in X between Ai and Bi such that nfo 1L, = 0. Otherwise, X is strongly infinite-dimensional. A space X is called a C-space if for every sequence 1, .. . of open covers of X there exists a sequence q1, q&2,*.. of families of open subsets of X such that, for (i) the members of q&i are pairwise disjoint, (ii) each member of q&, is contained in a member of 9i, (ii) the union U7 10&i covers X. The notion of C-space was introduced by W. Haver in [H] for metric space and by D. Addis and J. Gresham in [AG] for general topological spaces. LEMMA 1 [AG]. Every C-space is weakly infinite-dimensional. 2. Results. The aim of this note is to construct the following examples. EXAMPLE 1. There exists a weakly infinite-dimensional space X such that the product X X B of X with a certain subspace B of the space of irrationals is strongly infinite-dimensional. Moreover, X is a C-space while X X B is not a C-space. EXAMPLE 2. Under the assumption of the Continuum Hypothesis there exists a weakly infinite-dimensional space X such that the product X X P of X with the space of irrationals P is strongly infinite-dimensional. Moreover, X is a C-space while X X P is
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TL;DR: In this paper, a method for establishing the shelf-life for a pharmaceutical product using data from long-term stability studies is presented, which utilizes the work of Fuller and Battese (1973).
Abstract: A method is presented for establishing the shelf-life for a pharmaceutical product using data from long-term stability studies The method utilizes the work of Fuller and Battese (1973) An example is given which illustrates the use of the method with a set of pharmaceutical stability data