Showing papers on "Product (mathematics) published in 1984"

Calculus on arithmetic surfaces

Abstract: In [A2] and [A3], Arakelov introduces an intersection calculus for arithmetic surfaces, that is, for stable models of curves over a number field. In this paper we intend to show that his intersection product has a lot of useful properties. More precisely, we show that the following properties from the theory of algebraic surfaces have an analogue in our situation:

Automatic synthesis of systolic arrays from uniform recurrent equations

Abstract: We describe a systematic method for the design of systolic arrays. This method may be used for algorithms that can be expressed as a set of uniform recurrent equations over a convex set D of Cartesian coordinates. Most of the algorithms already considered for systolic implementation may be represented in this way. The methods consists of two steps: finding a timing-function for the computations that is compatible with the dependences introduced by the equations, then mapping the domain D onto another finite set of coordinates, each representing a processor of the systolic array, in such a way that concurrent computations are mapped onto different processors. The scheduling and mapping functions meet conditions that allow the full automation of the method. The method is exemplified on the convolution product and the matrix product.

Decomposition through formalization in a product space

Abstract: When an optimization problem is posed in a product space it is classical to decompose this problem. The goal of this paper is to show how such an approach can be used when the problem to be solved is not naturally posed in a product space. By associating systematically to this problem an equivalent one posed in ann-fold cartesian product space, we obtain by decomposition of the latter both a splitting of operators and a desintegration of constraints for the former. Applications to three rather classical mathematical programming problems are given.

Generalized Vector Products, Duality and Octonionic Identities in $D=8$ Geometry

Abstract: In an explicit, unified, and covariant formulation, we study and generalize the exceptional vector products in R8 of Zvengrowski, Gray, and Kleinfeld. We derive the associated general quadratic, cubic, and quartic G2 invariant algebraic identities and uncover an octonionic counterpart to the d=4 quaternionic duality. When restricted to seven dimensions the latter is an algebraic statement of absolute parallelism on S7. We further link up with the Ogievetski–Tzeitlin vector product and obtain explicit tensor forms of the SO(7) and G2 structure constants. SO(8) transformations related by Triality are characterized by means of invariant tensors. Possible physical applications are discussed.

Experimental Investigation of Resonances in Reactive Scattering: The F + H 2 Reaction

Abstract: The reaction $\mathrm{F}+p\ensuremath{-}{\mathrm{H}}_{2}\ensuremath{\rightarrow}\mathrm{HF}+\mathrm{H}$ was studied in a high-resolution crossed-molecular-beams experiment at a collision energy of 184 kcal/mole Center-of-mass translational energy and angular distributions were determined for each product vibrational state The $v=3$ product showed intense forward scattering while the $v=2$ product was backward peaked These results suggest that dynamical resonances play an important role in the reaction dynamics of this system

Computation of matrix chain products. Part II

Abstract: This paper considers the computation of matrix chain products of the form $M_1 \times M_2 \times \cdots \times M_{n - 1} $. If the matrices are of different dimensions, the order in which the product is computed affects the number of operations. An optimum order is an order which minimizes the total number of operations. We present some theorems about an optimum order of computing the matrices. Based on these theorems, an $O(n\log n)$ algorithm for finding an optimum order will be presented in Part II.

Arithmetic Functions and Integer Products

Abstract: Duality and the Differences of Additive Functions.- First Motive.- 1 Variants of Well-Known Arithmetic Inequalities.- Multiplicative Functions.- Generalized Turan-Kubilius Inequalities.- Selberg's Sieve Method.- Kloosterman Sums.- 2 A Diophantine Equation.- 3 A First Upper Bound.- The First Inductive Proof.- The Second Inductive Proof.- Concluding Remarks.- 4 Intermezzo: The Group Q*/?.- 5 Some Duality.- Duality in Finite Spaces.- Self-adjoint Maps.- Duality in Hilbert Space.- Duality in General.- Second Motive.- 6 Lemmas Involving Prime Numbers.- The Large Sieve and Prime Number Sums.- The Method of Vinogradov in Vaughan's Form.- Dirichlet L-Series.- 7 Additive Functions on Arithmetic Progressions with Large Moduli.- Additive Functions on Arithmetic Progressions.- Algebraicanalytic Inequalities.- 8 The Loop.- Third Motive.- 9 The Approximate Functional Equation.- 10 Additive Arithmetic Functions on Differences.- The Basic Inequality.- The Decomposition of the Mean.- Concluding Remarks.- 11 Some Historical Remarks.- 12 From L2 to L?.- 13 A Problem of Katai.- 14 Inequalities in L?.- 15 Integers as Products.- More Duality Additive Functions as Characters.- Divisible Groups and Modules.- Sets of Uniqueness.- Algorithms.- 16 The Second Intermezzo.- 17 Product Representations by Values of Rational Functions.- A Ring of Operators.- Practical Measures.- 18 Simultaneous Product Representations by Values of Rational Functions.- Linear Recurrences in Modules.- Elliptic Power Sums.- Concluding Remarks.- 19 Simultaneous Product Representations with aix + bi.- 20 Information and Arithmetic.- Transition to Arithmetic.- Information as an Algebraic Object.- 21 Central Limit Theorem for Differences.- 22 Density Theorems.- Groups of Bounded Order.- Measures on Dual Groups.- Arithmic Groups.- Concluding Remarks.- 23 Problems.- Exercises.- Unsolved Problems.- Supplement Progress in Probabilistic Number Theory.- Analogues of the Turan-Kubilius Inequality.- References.

Incomplete Methods for Solving $A^T Ax = b$

Abstract: Incomplete methods are considered for the solution of $A^T Ax = b$. One possibility is to construct the product $A^T A$ and proceed with an ICCG solution. However, it is also possible to utilize incomplete orthogonal decompositions of A obtained by modifying either the Gram–Schmidt or the Givens rotation method. These three methods are applied to four small structural analyses and the numerical efficiencies are compared.

The Product Form for Sojourn Time Distributions in Cyclic Exponential Queues

Abstract: Consider a closed cyclic queuing system consisting of M exponential queues. The Laplace-Stieltjes transform of the joint distribution of the consecutive sojourn times of a customer at the M queues is determined and shown to have a product form. The proof is based on a reversibility argument.

Removing Perceptual Distortions in Product Space Analysis

Abstract: Removing distortion in perceptual judgments has been the subject of several recent research studies. The authors examine the issue of identifying and removing perceptual distortions in product spac...

Convergence of product formulas for the numerical evaluation of certain two-dimensional Cauchy principal value integrals

Abstract: We consider product rules of interpolatory type for the numerical approximation of certain two-dimensional Cauchy principal value integrals We present convergence results which generalize those known in the one-dimensional case

Determination of economic production quantities for a two-product single machine system

Abstract: In this paper, a search procedure is given for determining production quantities for two products manufactured on a single machine. The search procedure permits unequal batch quantities for the more frequently manufactured product. An example has been solved to illustrate the method and the results have been compared with the solution obtained on applying the existing methods.

A characterization of a/0-spaces

Abstract: The following result is obtained. Theorem. A Tyspace has a o-closure-preserving base consisting of clopen sets iff the space can be embedded in the product of countably many o-discrete stratifiable spaces.

Method and apparatus for slicing a product in accordance with its anticipated weight distribution

Abstract: A slicing machine for slicing a product, comprising a blade 1 and a feeding mechanism 8, 9, 11, 12, to feed the product towards the blade 1 also includes a programmed computer 18 to control the feed rate of the feeding mechanism 8, 9, 11, 12 and programmed with a function corresponding to the typical weight distribution of at least one type of product. The computer 18 is programmed to respond to the input of information representing the weight and length of a particular product to modify the typical weight distribution function in accordance with the input values to provide an anticipated weight distribution for that particular product. The computer is also programmed to control the operation of the feeding mechanism 8, 9, 11, 12, so that the product is fed towards the blade 1 at a rate which varies with the anticipated weight distribution.

A method of defect evaluation for ceramic products

Abstract: A ceramic product is exposed to a light from a tungsten lamp, so that a specified region of the product is heated. In this heating process, acoustic emissions from the ceramic product are detected by an AE sensor which is acoustically coupled to the ceramic product by means of a waveguide passage. A temperature gradient directed from the surface toward the inner part of the ceramic product is formed by heating, and thermal stress is applied to the product so that tensile stress is produced in the inner part of the product. In practical use, a crack constituting the core of fracture is caused to grow by the thermal stress, and an acoustic emission is released. The existence of a defect can be evaluated by detecting the acoustic emission. Also, the location of the defect in the product can be detected by heating part of the product. The equivalent crack dimension ae of the defect is obtained by measuring the rate of production of acoustic emission dN/dt, and the product can be checked for acceptability on the basis of the dimension ae.

Signatures Through Approximate Representations by Quadratic Forms

Abstract: We propose a signature scheme where the private key is a random (n, n)-matrix T with coefficients in ℤm=ℤ/mℤ, m a product of two large primes. The corresponding public key is A,m with A = T⊤T. A signature y of a message z ∈ ℤm is any y∈(ℤm)n such that y⊤ Ay approximates z, e.g. \(\left| z-{{y}^{T}}Ay \right|<4{{m}^{{{2}^{-n}}}}\). Messages z can be efficiently signed using the private key T and by approximating z as a sum of squares. Even tighter approximations | z− y⊤Ay| can be achieved by tight signature procedures. Heuristical arguments show that forging signatures is not easier than factoring m. The prime decomposition of m is not needed for signing messages, however knowledge of this prime decomposition enables forging signatures. Distinct participants of the system may share the same modulus m provided that its prime decomposition is unknown. Our signature scheme is faster than the RSA-scheme.

Maximum antichains in the product of chains

Abstract: Let P be the poset k1 × ... × k n , which is a product of chains, where n≥1 and k1≥ ... ≥k n ≥2. Let \(M = k_1 - \sum olimits_{i = 2}^n {(k_i - 1)} \). P is known to have the Sperner property, which means that its maximum ranks are maximum antichains. Here we prove that its maximum ranks are its only maximum antichains if and only if either n=1 or M≤1. This is a generalization of a classical result, Sperner's Theorem, which is the case k1= ... =k n =2. We also determine the number and location of the maximum ranks of P.

Package for food products

Abstract: A package for a rigid or semi-rigid food product has a wrapper sheet which encloses the product and which can be torn along a transverse rupture line, and a take-up member which at least partly surrounds the portion of the wrapper sheet between one end of the product and the rupture line and by means of which the product may be held for consumption.

Supplement to G. V. Kuz'mina's paper: “On the problem of the maximum of the product of the conformal radii of nonoverlapping domains” (Zap. Nauchn. Sem. Lomi, 100 , 131–145, 1980)

Abstract: One gives a generalization of a result of G. V. Kuz'mina.

Arithmetic circuit for obtaining the vector product of two vectors

Abstract: In an arithmetic circuit, a first input of m bits representing the vector expression of a first set of elements of a finite field GF(2m) is converted into the components of a matrix, and each component of the matrix is multiplied by second input of m bits representing the vector expression of a second set of elements of the finite field. A vector product of the first and second inputs is thereby obtained.