Showing papers on "Product (mathematics) published in 1970"

Limit of Cross Sections at Infinite Energy

Abstract: At infinite energy, we predict: (1) ${\ensuremath{\sigma}}_{\mathrm{tot}}$ approaches infinity; (2) the ratio of the real part to the imaginary part of the forward elastic amplitude approaches zero; (3) $\frac{{\ensuremath{\sigma}}_{\mathrm{el}}}{{\ensuremath{\sigma}}_{\mathrm{tot}}}$ approaches \textonehalf{}; (4) the width of diffraction peak approaches zero; its product with ${\ensuremath{\sigma}}_{\mathrm{tot}}$ is a constant. We give theoretical evidence based on massive quantum electrodynamics as well as experimental evidence in support of these predictions, and a physical picture for high-energy scattering.

Operator-product expansions and anomalous dimensions in the thirring model.

Abstract: An example of an operator product expansion is worked out for the . Thirring model D The Thirring model involves a two-dimensional zero mass Dirac field $ interacting via the Fermi interaction. The model is scale invariant but the dimensions of local fields in the model vary with the coupling constant A p It is shown that ti has dimension l/2 + h2/4*2 (1h2/41r2)-l, while the composite fields F$ and $ y5ti, appropriately defined, have the dimension (1A/2 n) (l+ h/27r)-l. (Submitted to Phys. Rev.) ,,*Work supported by the U. S. Atomic Energy Commission. Permanent address (after September, 1970).

Eigenvectors of real and complex matrices byLR andQR triangularizations

Abstract: In a recent paper [4] the triangularization of complex Hessenberg matrices using the LR algorithm was described. Denoting the Hessenberg matrix by H and the final triangular matrix by T we have $${P^{ - 1}}HP = T$$ (1) , where P is the product of all the transformation matrices used in the execution of the LR algorithm. In practice H will almost invariably have been derived from a general complex matrix A using the procedure comhes [3] and hence for some nonsingular S we have $${P^{ - 1}}{S^{ - 1}}ASP = T$$ (2) .

Kronecker Matrices, Computer Implementation, and Generalized Spectra

Abstract: Closed product form and simplified algorithm for efficient operation of generalized Kronecker matrices, considering role in spectral analysis

Container loader for compressible product

Abstract: A method and apparatus for packing a stack of compressible material, such as disposable diapers, into a carton wherein the apparatus includes a rotatable turret which is serially indexed to a number of stations. The compressible stack is loaded between a pair of platens which rotate with the turret and the stack is compressed to the desired height, with the rotational movement of the turret causing the platens to move relative to each other for compression. After the stack has been compressed to the desired level, a carton is placed over the platens and the stack at one station; and, at a further station, the diapers are pushed out from between the platens so that the carton is also thereby moved and the stack is contained in the carton. Provision is also made for placing a coupon or sheet containing advertising, instructions or the like, into the carton in such a fashion that the coupon is not damaged and for pushing the carton over the platens with a straight line pushing motion to insure proper container alignment and to allow motion with a minimum applied force.

The Interaction of Two Electronic Systems at Short and Medium Range

Abstract: In the quantum theory of intermolecular forces, emphasis usually falls on long-range effects such as the dispersion interactions. At shorter range the electron distributions begin to overlap appreciably, serious non-orthogonality problems arise, and the nature of the interactions is less well understood. This paper is devoted to a general analysis of the short-range interactions between any pair of electronic systems, A and B, in states of arbitrary spin multiplicity and described by wavefunctions of arbitrarily high accuracy. The interaction energy is developed as a series of terms associated with interchange of 0, 1, 2,... electrons (arising from the use of a fully antisymmetric wave function): the final expression is written as a power series in the expectation value of the spin scalar product S$^{\text{A}}$$\cdot $S$^{\text{B}}$, with numerical coefficients which are determined by the electron density, spin density, etc., of the separate systems. Numerical results are presented for two neon atoms (the closed-shell case, with spins S$\_{\text{A}}$ = S$\_{\text{B}}$ = 0) and for two nitrogen atoms (open shells with S$\_{\text{A}}$ = S$\_{\text{B}}$ = $\frac{3}{2}$. The results, obtained from the use of one-determinant wavefunctions for the separate systems, are in good agreement with the best results so far available.

Digital products inspection system

Abstract: A Digital Product Inspection System using a digital pseudorandom generator in combination with a charactertistic of the product being inspected to produce a unique set of data combinations which when compared with previously taken data from the test of a ''''known good unit'''' will provide an output indicating whether or not the characteristic of the unit under test is within acceptable limits.

Process for forming a multi-colored food product

Abstract: A PROCESS FOR MAKING A MULTI-COLORED COOD PRODUCT BY INTRODUCING AT LEAST TWO DIFFERENT COLORED DYES INTO A CONTINUOUS COOKER ON AN ALTERNATE OR SEQUENTIAL BASIS. A PRODUCT IS CONTINUOUSLY DISCHARGED FROM THE COOKER WHICH VARIES IN COLOR FROM EACH OF THE INDIVIDUAL COLORS INTRODUCED, TO A MIXTURE OF THE TWO COLORS.

Extensions of the Weierstrass Product Inequalities

Abstract: (1970). Extensions of the Weierstrass Product Inequalities. Mathematics Magazine: Vol. 43, No. 3, pp. 137-141.

On the Divisibility Properties of Sequences of Integers

Abstract: Let a, < n2 < . . . be a sequence of integers denoted by A. Put A(x) = xCniCs 1. If no ai divides any other then A is called a p&m&e sequence. It is well known and easy to see that, for a primitive sequence, max A(z) = [3(x+ l)]. B esicovitch (1) constructed a primitive sequence of positive upper density and Behrend and Erd6s (1) proved that every primitive sequence has lower density 0. Davenport and Erdijs (1) proved that if A has positive upper logarithmic density then there is an infinite subsequence (ni,)j,r,. of A such that CZ,~ j aij+l. Erdijs (2) proved that’, if we assume that no ai divides the product of two others, then$

The Construction of Uniformly Minimum Variance Unbiased Estimators for Exponential Distributions

Abstract: Consider a sample $(x_1, x_2, \cdots, x_N)$ from a population with a distribution function $F_\theta(x), (\theta \epsilon \mathbf{\Omega})$ for which a complete sufficient statistic, $s(x)$, exists. Then any parametric function $g(\theta)$ possesses a unique minimum variance unbiased estimator U.M.V.U.E., which may be obtained by the Rao-Blackwell theorem provided an unbiased estimator of $g(\theta)$ with finite variance for each $\theta \epsilon \mathbf{\Omega}$ is available. In this paper we will consider the Koopman-Darmois class of exponential densities and develop a method for obtaining the U.M.V.U.E., $t_g$, of $g(\theta)$ without explicit knowledge of any unbiased estimator of $g(\theta)$. The U.M.V.U.E. $t_g$ is given as the limit in the mean (l.i.m.) of a series and a convergent series is also given for the variance. For any arbitrary but fixed $\theta_0 \epsilon \mathbf{\Omega}$, it can be verified that the complete sufficient statistic $s(x)$ has moments of all orders and that these moments determine its distribution function. Hence the set of polynomials in $s(x)$ is dense in the Hilbert space, $V$ (with the usual inner product), of Borel measurable functions of $s(x)$. Since $t_g$ is an element of $V$, we may obtain a generalized Fourier series for it by constructing a complete orthonormal set $\{\varphi_n\}$ for $V$. Such a set $\{\varphi_n\}$ may be obtained from the density function and its derivatives with respect to $\theta$. For a subclass of the exponential family, Seth [18] has obtained $\{\varphi_n\}$ in a form which is convenient for our purposes. We will study this case in Section 3 and use Seth's results to give an explicit construction of $t_g$. Criteria for the pointwise convergence of the series will also be given. In Section 4 examples illustrating the use of the method are given and some related results are discussed. The general theory for the representation of minimum variance unbiased estimates, both local and uniform, has been developed in depth, for example in [5], [18], [19], [16], [3], and [4]. The present remarks, though founded in the general theory (in particular [3] and [4]), are tailored specifically to the exponential family.

An Empirical Model for Multilane Road Traffic

Abstract: Before we can construct models for multilane traffic flow, we must be able to describe single-lane traffic. This paper considers some of these methods of description including the use of headway and counting distributions. Another method of describing point processes is by means of product densities. These are essentially joint probabilities of two or more vehicles passing a point at different specified times. Product densities have been used in many fields including particle physics, ecology, and road traffic. The idea of using product densities is simple and attractive, but they do have two major disadvantages: except in simple cases, headway distributions are very difficult to derive from product densities, and product density estimates for different time lags are correlated, so that tests of hypotheses are not easy to perform. An alternative to product densities is proposed. For want of a better name these are called termination rates. The termination rate after lag τ is the probability of a vehicle p...

Assembly tolerance control spacer

Abstract: AN ASSEMBLY TOLERANCE CONTROL SPACER TO ABSORB OR TAKEUP SPACE DIFFERENCES OCCURRING IN A PRODUCT MADE UP OF A PLURALITY OF COMPONENTS HAVING SIZE OR TOLERANCE CHARACTERISTICS WHICH MAY VARY FROM COMPONENT TO COMPONENT, THE CONTROL SPACER BEING INCLUDED IN THE ASSEMBLED PRODUCT SO AS TO ABSORB TOLERANCE VARIATIONS BY COLLAPSING MORE OR LESS TO RESULT IN A PREDICTABLE ASSEMBLY RESULT FOR THE PRODUCT.

Marblelized product aerosol dispenser

Abstract: A marblelized product aerosol dispenser, adapted to provide separate drawing of separately contained materials therethrough, to be intermixed and co-dispensed in marblelized product form.

Distribution of product statistics from aPareto population

Abstract: Distributions are derived of the product of sample values, the sample geometric mean, the product of two minimum values from sample of unequal size and product ofk minimum values from sample of equal size from aPareto population. The distributions can be conveniently transformed tox2.

The Economic Theory Of Environmental Life Cycle Inventory Models

Abstract: This paper shows that Life Cycle Inventory theory can be derived directly from the theories of production functions and methods of Input-Output analysis of economics. Linear Economic-Environment (EEP) production functions are formed into a Leontief Input-Output matrix A, which shows the transition of physical quantities of materials through the processes and flows along a product's life cycle. The Life Cycle Inventory E * is then calculated as ={[I-A]xY}xE=the total quantity vector of effluents produced, where Y is the functional unit vector, I an identity matrix, and E the amount of effluents emitted per unit quantity of a product produced. This formulation is usable with both one product and multi-product functional units. Construction principles of dynamic LC and material flow models are then discussed. Examples and results of dynamic LC scenarios are given of the material flows of printing papers in Germany between 1993-2000.

A quantum-chemical study of some properties and reactions of H2CN−, H2CN+, H2CNH2+, H2CNH, and HCN, using small and intermediate-sized sets of basis functions

Abstract: It is suggested that given two molecules potentially capable of being involved in the same reaction, one as a product and one as a reactant, it seems reasonable that the calculated difference in en...

Concerning product integrals and exponentials

Abstract: B. W. Helton, J. S. MacNerney, and H. S. Wall have established various relationships between integral equations, sum integrals, and product integrals. This paper establishes a relationship between exponentials, sum integrals, and product integrals which may be used to evaluate certain product integrals or sum integrals. Integrals used are of the subdivision-refinement type and complete definitions of these and other terms and symbols used in this paper may be found in [1 ] or [2 ]. Suppose S is a linearly ordered set [2 ] and N is the set of real numbers. All functions considered will be functions from S X S to N unless otherwise noted. In [1, Theorem 3.4] it is shown that for functions of bounded variation from S X S to N the following two statements are equivalent: (1) f,' G exists and (2) alIb (1 +G) exists. Under the hypothesis that fb C2 0= , we show that the following two statements are equivalent for functions from SXS to N: (1) f" G exists and (2) a,Hb (1 +G) exists and is not zero. It is also noted that neither of the following two statements is a consequence of the other. (1) fb G2=O and (2) G is of bounded variation on [a, b].