Showing papers on "Product (mathematics) published in 1983"

Group graded rings

Abstract: Let R be a group graded ring . The map ( , ): R × R →1 defined by: (x,y) = (xy)1 , is an inner product on R. In this paper we investigate aspects of nondegeneracy of the product, which is a generalization of the notion of strongly G —graded rings,introduced by Dade. We show that various chain conditions are satisfied by R if and only if they are satisfied by R1 , and that when R1 is simple artinian, then R is a crossed product R1 * G. We give conditions for simple R-modules to be completely reducible R1 -modules . Finally, we prove an incomparability theorem,when G is finite abelian.

On the history of the kronecker product

Abstract: History reveals that what is today called the Kronecker product should be called the Zehfuss product.

A Characterization of Product-Form Queuing Networks

Abstract: Sunple criterta are given for determining whether a queuing network model has a product-form solution and is thus computationally tractable Categories and SubJect Descriptors. D.4.8 [Operating Systems]: Performance--queuing theory, stochasuc analysts General Terms Performance, Theory Additional

An Approach to the Optimal Positioning of a New Product

Abstract: This study examines the problem faced by a firm which wishes to position a new choice object in an existing product class. It is assumed that both the consumer and the firm are involved in a two-stage decision process. The consumer first decides on his budget for the product class. He then evaluates, within the product class, that subset of competing objects which have prices approximately equal to his budget constraint. This evaluation is performed through a weighted multi-attribute utility model. The product classes considered here are ones for which the consumer has a finite ideal level on each attribute. The consumer is hypothesized to choose, without error, that object which is closest to his ideal. Different individuals are assumed to be heterogeneous in both attribute weights and ideal levels. The firm is assumed to first identify, in the attribute space which contains ideal points and competing objects, promising product positions which would attract a large number of consumers. It then evaluates these positions in terms of costs and resulting profits. The problem of identifying the promising positions, given information on a sample of individuals, is formulated as a Mixed Integer Nonlinear Program. Due to the inability of such a program to solve even small sample problems, the spatial properties of the problem are examined. An exact solution algorithm which is computationally feasible for small samples is developed. It is based on an examination of intersections of indifference hyperellipsoids. For larger sample problems an efficient heuristic which is an extension of the random point search used in nonlinear programming is provided. It involves random line search procedures for our noncontinuous-type problem. The positioning approach and the heuristic are illustrated in a simulated positioning problem in the small car market.

Unbounded fan-in circuits and associative functions

Abstract: We consider the computation of finite semigroups using unbounded fan-in circuits. There are constant-depth, polynomial size circuits for semigroup product iff the semigroup does not contain a nontrivial group as a subset. In the case that the semigroup in fact does not contain a group, then for any primitive recursive function f, circuits of size O(nf−1(n)) and constant depth exist for the semigroup product of n elements. The depth depends upon the choice of the primitive recursive function f. The circuits not only compute the semigroup product, but every prefix of the semigroup product. A consequence is that the same bounds apply for circuits computing the sum of two n-bit numbers.

Products of idempotent matrices

Abstract: A result of J. Erdos [2] states that if A is a singularn×n matrix with entries in a field F then A can be written as the product of idempotents over F. C. S. Ballantinc III quantified this result by relating the minimum number of idempotents required to the rank of A and in particular proved that A can be written as the product of n idempotents over F. In this paper we consider the case where F is replaced by a ring. We show thai if R is a division ring or a Euclidean ring, then every singular n×n matrix with entries in R can be expressed as a product of idempotents over R.

Networks of queues in discrete time

Abstract: We study a new class of networks of queues whose nodes operate in round-robin fashion and other ways of interest to computer science. We compute a stationary law of product form for the Markov process describing the state of the network. Moreover, we obtain the conditional expected travel time of a job given the job's requested processing times at particular nodes along its route.

The KS-transformation in hypercomplex form

Abstract: In this note the KS-transformation introduced by Kustaanheimo and Stiefel into Celestial Mechanics is formulated in terms of hypercomplex numbers as the product of a quaternion and its antiinvolute. Therefore it represents a particular morphism of the real algebra of quaternions-having for image a three-dimensional real linear subspace-and also anatural generalization of the Levi-Civita transformation. The quaternion matrix of the product leads to the KS-matrix; the bilinear relation and the two identities which play a central role in the KS-theory are easily derived. A suitable quaternion gauge-transformation is given which leads to the well-known fibration of the four-dimensional space. In addition several geometrical interpretations are brought out.

New representations for the spherical tensor gradient and the spherical delta function

Abstract: In this article, we analyze representations for the product Ym1l1(∇)Fm2l2(r) with Yml(∇) specifying a solid harmonic whose argument is the nabla operator ∂/∂r instead of the vector r. Since both Ym1l1(∇) and Fm2l2(r) are irreducible spherical tensors, we can use angular momentum algebra for evaluating the product. Accordingly, the problem of finding a representation for the product is reduced to the determination of the radial functions generated by the product. Analytical expressions for these radial functions are derived by direct differentiation and with the help of Fourier transforms. Closely related to the spherical tensor gradient Yml(∇) is the spherical delta function δml(r). We derive new representations for δml by considering convolution integrals involving B functions. These functions are closely related to the modified Bessel functions and also to the Yukawa potential e−αr/r. We show that the definition of the B functions can be extended to include a large class of derivatives of the delta func...

Interlaced programmable logic array having shared elements

Abstract: A programmable PLA circuit in which an interlaced AND/OR array is provided which has both common input and common output lines. Separate AND and OR functions are generated during two different timing intervals such that both of the logical arrays can physically share input and output circuit elements. A binary adder is described in which pairs of array output lines are applied to the same Exclusive-NOR circuits during the two time intervals to provide the Exclusive-NOR of product terms during the AND array time interval and to provide the Exclusive-NOR of sum of product terms or the sum of the Exclusive-NOR of product terms during the second time interval.

The recognition of deterministic CFLs in small time and space

Abstract: Let S(n) be a nice space bound such that log2 n ⩽ S(n) ⩽ n. Then every DCFL is recognized by a multitape Turing machine simultaneously in time O(n2/S(n)) and space O(S(n)), and this time bound is optimal. If the machine is allowed a random access input, then the time bound can be improved so that the time-space product is O(n1 + ɛ).

Product quality level monitoring and control for logic chips and modules

Abstract: In the manufacture of chips and modules, it is important to minimize defects in order to maximize quality levels and product reliability at each level of product assembly (chips, module, card, system). These objectives are best achieved by controlling defects through manufacturing process controls and testing at the lowest possible level of assembly. Defective product remaining after test and inspection must be repaired or discarded. The ability to detect and reduce or eliminate these defects is crucial to ensuring maximum product quality. The amount of such defective product is typically described quantitatively in terms of statistical sampling plans. The problem with such approaches is that the absolute defect level is imprecisely defined. This paper defines an absolute number for such product defects, which we will call "product quality level" (PQL). PQL categories found in logic chips and modules after completion of electrical testing are described and a methodology for the monitoring and control of the PQL in chips is presented. The impact of chip defects on module, card, and system performances is discussed with the aid of examples. By using the described comprehensive design, process control, testing, and user-feedback approach at each assembly level, final product can be manufactured with the lowest possible level of defects that must then be repaired at the machine level.

A note on two measures of dependence and mixing sequences

Abstract: In this note we establish an inequality between the maximal coefficient of correlation and the φ -mixing coefficient which is symmetric in its arguments. Motivated by this inequality, we introduce a mixing coefficient which is the product of two φ -mixing coefficients. We also study an invariance principle under conditions imposed on this new mixing coefficient. As a consequence of this result it follows that the invariance principle holds when either the direct-time process or its time-reversed process is φ -mixing; when both processes are φ-mixing the invariance principle holds for sequences of L 2-integrable random variables under a mixing rate weaker than that used by Ibragimov.

Improvements in or relating to in-mill marking of metal products

Abstract: The invention provides a method of marking a metal product comprising the steps of producing a laser capable of visibly thermally affecting the surface of the product concerned, directing the laser at the product and controlling the operation and direction of the laser to apply thereby a required marking to the surface of the product.

Coupling of Quantum Logics

Abstract: A quantum logic is a couple (L, M), whereL is a logic andM is a quite full set of states onL. A tensor product in the category of quantum logics is defined and a comparison with the definition of free orthodistributive product of orthomodular σ lattices is given. Several physically important cases are treated.

Analytical angular integration of a product of hydrogenic bound-free first Born matrix elements

Abstract: An explicit expression is given for the average over the angular part of the continuum wavefunction of the product of two first Born matrix elements linking the 1s state and the continuum of a hydrogen-like atom

The Singular Value Decomposition in Product Form

Abstract: A gain of about $50\% $ in the CP-time required for the calculation of the singular value decomposition of a general matrix can be achieved by not forming the orthogonal factors explicitly, but storing the Householder reflections and Jacobi rotations that compose them. An efficient method for storing the Jacobi rotations is given. The storage required for the resulting decomposition is, for general matrices, about $1\frac{1} {2}$ times what is usual, but it is not larger than usual for matrices arising in ill-posed problems. The storage required for the decomposition of a general matrix can be reduced to what is usual by giving up part of the gain in efficiency and applying an “ultimate shift” strategy.