Showing papers on "Product (mathematics) published in 1987"

Harmonic analysis on product spaces

Abstract: Rn x R' and commute with the dilations p33 '8(x2, x2) = (61x1, 62x2) for x1 E Rn, x2 E Rm, and 61, 82 > 0, then the corresponding operators are related to many problems in the theory of multiple Fourier integrals (see Zygmund [21]). The maximal operator which commutes with the p31382 is the Jessen, Marcinkiewicz, and Zygmund [14] strong maximal operator Ms defined by 1 Ms(f)(x1, x2) = sup RI A|(Y1, Y2)1 dy1 dy2 (xl, X2)GR JR1

Electronic product information display system

Abstract: An electronic product information display system includes a plurality of self-contained display modules (10) mounted proximate to respective products. A signal (22) is formed that comprises an address part and a data part, the adderess part corresponding to a selected module address and the data part encoding product information (18, 20) to be displayed by the addressed module (10). The signal a broadcast from a transmitter (24) remote from a plurality of modules (10). Modules (10) each receive the signal, decode the address part into a received address, and decode the data part into product information (18, 20). The product information is stored and displayed responsive to a match between the module address and the received address. The signal (22) is preferably infrared.

A Heuristic Approach to Product Design

Abstract: A dynamic-programming heuristic is described to find approximate solutions to the problem of identifying a new, multi-attribute product profile associated with the highest share-of-choices in a competitive market. The input data consist of idiosyncratic multi-attribute preference functions estimated using conjoint or hybrid-conjoint analysis. An individual is assumed to choose a new product profile if he/she associates a higher utility with it than with a status-quo alternative. Importance weights are assigned to individuals to account for differences in their purchase and/or usage rates and the performance of a new product profile is evaluated after taking into account its cannibalization of a seller's existing brands. In a simulation with real-sized problems, the proposed heuristic strictly dominates an alternative lagrangian-relaxation heuristic in terms of both computational time and approximation of the optimal solution. Across 192 simulated problems, the dynamic-programming heuristic identifies product profiles whose share-of-choices, on average, are 98.2% of the share-of-choices of the optimal product profile, suggesting that it closely approximates the optimal solution.

On the expectation of the product of four matrix-valued Gaussian random variables

Abstract: The formula for the expectation of the product of four scalar real Gaussian random variables is generalized to matrix-valued (real or complex) Gaussian random variables. As an application of the extended formula, a simple derivation is presented of the covariance matrix of instrumental variable estimates of parameters in multivariable regression models. >

L-Functions and the Oscillator Representation

Abstract: Notation and preliminaries.- Special Eisenstein series on orthogonal groups.- Siegel formula revisited.- Inner product formulae.- Siegel formula - Compact case.- Local l-factors.- Global theory.

Scalar curvature and warped products of Riemann manifolds

Abstract: We establish the relationship between the scalar curvature of a warped product M Xf N of Riemann manifolds and those ones of M and N. Then we search for weights f to obtain constant scalar curvature on M Xf N when M is compact.

Product term sharing/allocation in an EPROM array

Abstract: An erasable programmable logic device which includes a programmable AND memory array and a macrocell processing the output of the AND array allows product term sharing/allocation by adjacent macrocells. Two groups of four product terms each are coupled to each macrocell, wherein the OR'ing of each group of four product terms is each coupled to a multiplexor. One group is also coupled to a previously adjacent macrocell and the second group is coupled to a subsequently adjacent macrocell. A third and fourth multiplexor accept four product terms from each of the adjacent macrocells and the output of the four multiplexors are coupled to an OR gate. When a multiplexor is activated, it couples each grouping of four product terms to the OR gate and the output of the OR gate is coupled to an I/O circuit which emulates combinatory and sequential logic circuits. By selecting appropriate multiplexors each eight product term macrocell is capable of processing 0, 4, 8, 12 or 16 product terms. An alternative embodiment has three groupings of product terms wherein only two of the groupings are shared by adjacent macrocells.

Methods and apparatus for optical product inspection

Abstract: Products having optically detectable straight line segments are inspected for acceptability by forming one or more one-dimensional images of the product in which properly aligned straight line segments are respectively focused to points in the image. Such parameters as the location and image intensity of these one-dimensional image points are used to determine whether or not the product is acceptable. An optical Hough transform underlies these product inspection techniques.

The parallel complexity of Abelian permutation group problems

Abstract: We classify Abelian permutation group problems with respect to their parallel complexity. For such groups specified by generating permutations we show that testing membership, computing order and testing isomorphism are $NC^1 $-equivalent to (and therefore have essentially the same parallel complexity as) determining solvability of a system of linear equations modulo a product of small prime powers; we show that intersecting two such groups is $NC^1 $-equivalent to computing setwise stabilizers; we show that each of these problems is $NC^1 $-reducible to the problem of computing a generator-relator presentation. Then we prove that the aforementioned problems belong to $NC^3 $, thus identifying several natural set recognition problems in $NC$ which may lie outside $NC^2 $. Finally we prove that $NC^4 $ contains the problem of computing the cyclic decomposition of an Abelian permutation group. Background results include an $NC^1 $ solution to the problem of computing the product of n integers modulo a $\lce...

Unitary colligations in Krein spaces and their role in the extension theory of isometries and symmetric linear relations in Hilbert spaces

Abstract: Let H be a Hilbert space and let S be a closed linear relation in H, i.e., S is a * subspace of H 2 such that ScS*cH 2 (for the definition of S and other definitions see Section I). Furthermore, let K be a Krein space such that HcK and the Krein space inner product onKcoincides on Hwith the Hilbert space inner product on H; we denote this situation byH~K. In this case H is anorthocomplemented subspace of K (see [4]). The corresponding orthogonal projection from K onto H is denoted by PH" We consider a selfadjoint relation A in K with a nonempty resolvent set p(A), such that ScA, that a selfadjoint extension of S in K. We define P~2)A-~ ={{PHf,PHg}l{f,g} CA}; it is is, A is clear that S cP(2)A. With A we associate the socalled ~traus extension T of S in H, which is by definition T= (T(£))£E~ U{~}' where T(£) is given by

Arithmetic unit for carrying out both multiplication and addition in an interval for the multiplication

Abstract: In an arithmetic unit comprising a partial product circuit for calculating a plurality of partial products for two numbers and a Wallace tree responsive to the partial products for producing a plurality of tree outputs which gives a total product of the two numbers when summed up, an addend is supplied to the Wallace tree as an additional partial product. The arithmetic unit produces a resultant sum of the total product plus the addend. The addend may be supplied to the Wallace tree from one or more registers therefor. Alternatively, a result register is used for the total product with the total product supplied to the Wallace tree as the addend. As a further alternative, an additional register is used for a third number which is used with bit shifts as the addend. In this last event, the arithmetic unit preferably produces a sum selected from the resultant sum.

Measurement of Discrimination Ability in Taste Tests: An Empirical Investigation

Abstract: For purposes of product testing, several tasks have been used to measure subjects’ discrimination ability (i.e., their ability to distinguish between two slightly different product formulations). T...

Inner Product Structures: Theory and Applications

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A Collection of Problems

Abstract: In this very short chapter we present some problems connected with the inner product structures. This collection of problems is by no means complete and our purpose is to direct the attention of interested readers to problems that are mentioned by different authors. We give references to where the problem is first stated as well as references to where the latest results (to the best knowledge of the present author) can be found.

Blaschke product interpolation and its application to the design of digital filters

Abstract: A new proof is given thatn distinct points on the unit circle can be mapped inton arbitrary points on the unit circle of the complex plane by a finite Blaschke product A result of this proof is that the mapping can be done with at mostn−1 factors in the product The problem is studied in the context of its application to frequency transformations used to design digital filters

The Support of Extremal Probability Measures with Given Marginals

Abstract: The purpose of this paper is to characterize the support of a probability measure P on a product XxY which is extremal in the set of measures having the same marginals as P. The paper continues the investigation started by G. Letac (1966) and by J. Denny (1980) dealing with a non-discrete version of the problem.

Multiple value coupon system

Abstract: A multiple retail coupon and a means for using it to identify and persuade a user of a competitive product to convert to a sponsoring manufacturer's product is disclosed. In particular, an upper sheet is adhered to a lower sheet, the upper sheet having a center section and division lines defining at least one pair of rectangular sections. End sections of each of the rectangular sections correspond to the center section. Cooperating printed matter is provided on an inner face of the lower sheet so that a consumer selecting one of the rectangles will ascertain a value of the sponsor's product which is redeemable. A consumer of a competitor's product will receive a higher value than a consumer of a sponsor's product to persuade the competitive consumer to convert to the desired product.

A new simplicial variable dimension algorithm to find equilibria on the product space of unit simplices

Abstract: A new simplicial variable dimension restart algorithm is introduced to solve the nonlinear complementarity problem on the product spaceS of unit simplices. The triangulation which underlies the algorithm differs from the triangulations ofS used thus far. Moreover, the number of rays along which the algorithm can leave the arbitrarily chosen starting point is much larger. More precisely, there is a ray leading from the starting point to each vertex ofS. In caseS is the product ofn one-dimensional unit simplices the alogrithm is similar to the octahedral algorithm onRn having 2n rays. Also, the accuracy of an approximate solution in the terminal simplex of the algorithm is in general better than for the other algorithms onS. Computational results will show that the number of iterations for the new algorithm is much less. The examples concern the computation of equilibria in noncooperative games, exchange economies and trade models.

Co-extruded semi-moist food product

Abstract: A co-extruded semi-moist food product is provided which is especially suited for use as a dog treat. The product comprises an inner core which is at least partially covered by at least one distinguishable layer. A method of manufacturing such as food product is also provided.

Analysis of models reducible to a class of diffusion processes in the positive quarter plane

Abstract: This paper is concerned with the two-dimensional generalization of the single server queue “diffusion approximation”: this model consists of a stochastic process in $\mathbb{R}^ + \times \mathbb{R}^ + $ being a diffusion in the open quarter plane with absorption on the boundaries $\{ 0 \} \times \mathbb{R}^ + \cup \mathbb{R}^ + \times \{ 0 \}$ and jumps back into this open domain. On the boundary this process is a linear diffusion with absorption on $( {0,0} )$ and jumps back into the x or y axis. This model describes within the same framework such phenomena as coupling of processors or simultaneous arrivals of jobs which do not preserve the classical “product form” solutions. First we exhibit a pathwise construction of the process which leads to the existence, the uniqueness and the Markov property. From this, we derive a functional equation for the Laplace transform of the steady state distribution. The solution of this functional equation is obtained by reduction to a boundary value problem on the righ...