Showing papers on "Product (mathematics) published in 1990"

Order entry and inventory control method

Abstract: A method of order entry, product selection and inventory control for building products, building product accessories and building product components sold by a dealer to a customer. The method comprises the steps of providing a computer having a display screen and a plurality of files. A first file lists descriptions, dimensions and styles of available building products by product code along with building product accessories. A second file lists rough opening dimensions and building products fitting the rough opening dimensions and a third file lists building product components. A customer can enter into the computer a product code, a rough opening dimension or plural product codes for a desired building product. Based on the entry, the computer will select a building product. The computer then determines and displays the desired building product along with a price. The customer can then order the desired building product from an inventory.

On the singular values of a product of operators

Abstract: For compact Hilbert space operators A and B, the singular values of $A^ * B$ are shown to be dominated by those of $\frac{1}{2}(AA^* + BB^* )$

Are product and Process Innovations Independent of Each Other

Abstract: This paper studies the relationship between product- and process-innovation. After discussing the determinants of product- and process-innovation, a simultaneous equation model is estimated. A positive impact of product-innovation on process-innovation can be proved, but no evidence for a reverse effect to the respect to the inclusion of product-innovation.

Closed-form solution of pure proportional navigation

Abstract: The closed-form solution of the equations of motion of an ideal missile pursuing a nonmaneuvering target according to the pure proportional navigation law is obtained as a function of the polar coordinates for all real navigation constants N>or=2. The solution is given in the form of a uniformly convergent infinite product which reduces to a product of a finite number of factors if the navigation constant is a rational number. The solution is discussed, and necessary and sufficient conditions are stated for vanishing, bounded, and unbounded missile acceleration in the final phase of pursuit. >

Tuples, Projections and Cartesian Products

Abstract: The purpose of this article is to define projections of ordered pairs, and to introduce triples and quadruples, and their projections. The theorems in this paper may be roughly divided into two groups: theorems describing basic properties of introduced concepts and theorems related to the regularity, analogous to those proved for ordered pairs by Cz. Bylinski [1]. Cartesian products of subsets are redefined as subsets of Cartesian products.

Konig's Theorem

Abstract: Summary. In the article the sum and product of any number of cardinals are introduced and their relationships to addition, multipli

Analysis of the asymmetric shortest queue problem

Abstract: In this paper we study a system consisting of two parallel servers with different service rates. Jobs arrive according to a Poisson stream and generate an exponentially distributed workload. On arrival a job joins the shortest queue and in case both queues have equal lengths, he joins the first queue with probability q and the second one with probability 1- q, where q is an arbitrary number between 0 and 1. In a previous paper we showed for the symmetric problem, that is for equal service rates and q = ½, that the equilibrium distribution of the lengths of the two queues can be exactly represented by an infinite sum of product form solutions by using an elementary compensation procedure. The main purpose of the present paper is to prove a similar product form result for the asymmetric problem by using a generalization of the compensation procedure. Furthermore, it is shown that the product form representation leads to a numerically efficient algorithm. Essentially, the method exploits the convergence properties of the series of product forms. Because of the fast convergence an efficient method is obtained with upper and lower bounds for the exact solution. For states further away from the origin the convergence is faster. This aspect is also exploited in the paper. Keywords: bounds, difference equation, product form, queues in parallel, stationary queue length distribution, shortest queue problem.

A Generalization of $D$- and $D_1$-Optimal Designs in Polynomial Regression

Abstract: In the class of polynomial regression models up to degree $n$ we determine the design on $\lbrack -1, 1\rbrack$ that maximizes a product of $n + 1$ determinants of information matrices weighted with a prior $\beta$, where the $l$-th information matrix corresponds to a polynomial regression model of degree $l$, for $l = 0, 1, \cdots, n$. The designs are calculated using canonical moments. We identify a special class of priors $\beta(z)$ depending on one real parameter $z$ so that analogous results are obtained as in the classical $D$- and $D_1$-optimal design problems. The interior support of the optimal design with respect to the prior $\beta(z)$ is given by the zeros of a Jacobi polynomial and all the interior support points have the same masses. The masses at the boundary points $-1$ and $1$ are $(z + 1)/2$ times bigger than the masses of the interior points. The results found in one dimension are generalized to the problem of determining optimal product designs in the case of multivariate polynomial regression on the $q$-cube $\lbrack -1,1\rbrack^q$. Explicit solutions are obtained for the $D$- and $D_1$-optimal product designs in the polynomial model of degree $n$ for all $n \in \mathbb{N}$ and $q \in \mathbb{N}$.

Conformal and related changes of metric on the product of two almost contact metric manifolds

Abstract: This paper studies conformal and related changes of the product metric on the product of two almost contact metric manifolds. It is shown that if one factor is Sasakian, the other is not, but that locally the second factor is of the type studied by Kenmotsu. The results are more general and given in terms of trans-Sasakian, a-Sasakian and s-Kenmotsu structures.

Efficient computation of two-electron - repulsion integrals and their nth-order derivatives using contracted Gaussian basis sets

Abstract: We present an general algorithm for the evaluation of the nth derivatives (with respect to the nuclear Cartesian coordinates) of two-electron-repulsion integrals (ERIs) over Gaussian basis functions. The algorithm is a generalization of our recent synthesis of the McMurchie/Davidson and Head-Gordon/Pople methodologies for ERI generation. Any ERI nth derivative may be viewed as an inner product between a function (which we term a bra) of electron 1 and a function (which we term a key) of electron 2. After defining bras and kets appropriately, we derive five recurrence relations that enable any bra to be constructed recursively from very simple bras which we call p-bras

Optimal design of multipurpose batch plants. 1. Problem formulation

Abstract: The design of multipurpose plants is posed as a mixed integer nonlinear program in which the binary variables are the structural choice variables. The proposed model is able to accommodate equipment used in and out of phase, units available in two or more sizes within a processing stage, multiple choices of equipment types for each product task, and allocation of products to campaigns and to determine the campaign lengths and sizing of the processing equipment. The proposed formulation is presented and its properties are discussed

Processes for the preparation of photogenerating pigments

Abstract: A process which comprises adding a pigment to a solution of trihaloacetic acid and toluene; adding the solution to a nonsolvent for the pigment; and separating the product from the solution.

Integrable equation of state for noisy cosmic string.

Abstract: It is argued that, independently of the detailed (thermal or more general) noise spectrum of the microscopic extrinsic excitations that can be expected on an ordinary cosmic string, their effect can be taken into account at a macroscopic level by replacing the standard isotropic Goto-Nambu-type string model by the nondegenerate string model characterized by an equation of state of the nondispersive "fixed determinant" type, with the effective surface stress-energy tensor satisfying ${(T^{\ensuremath{ u}}_{\ensuremath{ u}})}^{2}\ensuremath{-}T^{\ensuremath{\mu}}_{\ensuremath{ u}}T^{\ensuremath{ u}}_{\ensuremath{\mu}}=2{T}_{0}^{2}$, where ${T}_{0}$ is a constant representing the null-state limit of the string tension $T$, whose product with the energy density $U$ of the string is thereby held fixed: $TU={T}_{0}^{2}$. It is shown that this equation of state has the special property of giving rise (in a flat background) to explicitly integrable dynamical equations.

Lucas' theorem for prime powers

Abstract: Lucas' theorem on binomial coefficients states that ( A B ) ≡ ( a r b r ) ⋯ ( a 1 b 1 ) ( a 0 b 0 ) (mod p) where p is a prime and A = arpr + ⋯ + a0p + a0, B = brpr + ⋯ + b1p + b0 + are the p-adic expansions of A and B. If s ⩾ 2, it is shown that a similar formula holds modulo ps where the product involves a slightly modified binomial coefficient evaluated on blocks of s digits.w

Asymptotic ordering of distribution functions and convolution semigroups

Abstract: It is well-known that the setH of distribution functions on [0,∞] supplied with the convolution product * is a semigroup. We introduce a preorder on (H,*) with attractive algebraic properties. The corresponding equivalence relation ≈W extends the concept of tail-equivalence of distribution functions. We show that the idempotents of the factorsemigroupH W form a subsemigroup ofH W.

Stationary Harry-Dym's equation and its relation with geodesics on ellipsoid

Abstract: Harry-Dym's equation (HD) $$r_\tau = (r^{ - \tfrac{1}{2}} )_{xxx} $$ is well-known for its cusp soliton solutions. In this paper, relations are revealed between HD and a completely integrable Hamiltonian system in Liouville sense given by $$H = \frac{1}{2}\left\langle {p,p} \right\rangle - \left\langle {q,q} \right\rangle ^{ - 1} $$ in the symplectic manifold (ℝ2N ,dp/∩). Here 〈ξ,η〉 is the standard inner product in ℝ N .

Normally ordering some multimode exponential operators by virtue of the IWOP technique

Abstract: The author shows that the technique of integration within an ordered product (IWOP) provides one with a very simple approach to deriving the normal product form of some multimode exponential operators, which greatly simplifies the calculations of normalising some state vectors in Hilbert space.

On construction of Martin boundaries for second order elliptic equations

Abstract: Introduction §1. Preliminaries 1.1. Comparison Theorem 1.2. Existence of Positive Solutions 1.3. Criticality, Subcriticality, and Minimal Growth 1.4. Martin Boundary §2. Imbedding of a Boundary Part into the Minimal Martin Boundary § 3. Tensor Product Decomposition §4. Proof of Theorem 3.5 §5. Schrodinger Equations in a Cone §6. Equations in the Product of a Cone and a Bounded Domain §7. Direct Sum Decomposition Appendix. A Uniqueness Theorem References

A generalized inner product model for the analysis of asymmetry

Abstract: A least squares procedure called GIPSCAL (a Generalized Inner Product multidimensional SCALing) is proposed which extends Chino’s ASYMSCAL into higher dimensions than three. GIPSCAL fits the inner product of two vectors and the area of the parallelogram spanned by these vectors, respectively, for the symmetric and skew-symmetric parts of observed similarity judgements. It is shown that GIPSCAL has a very desirable property that the geometrical interpretation of asymmetric parts in similarity judgements is reducible to that of the area of the parallelogram spanned by vectors in two dimensions. It is also shown that GIPSCAL permits a social psychological justification for the cause of asymmetry. Relation to distance model is discussed. Examples of application are given to demonstrate the feasibility of the model.

Linear circuits over GF(2)

Abstract: For $n=2^k $, let S be an $n \times n$ matrix whose rows and columns are indexed by $\operatorname{GF}(2)^k $ and, for $i, j \in \operatorname{GF}(2)^k , S_{i.j}=\langle i, j \rangle $, the standard inner product. Size-depth trade-oils are investigated for computing $S{\bf x}$ with circuits using only linear operations. In particular, linear size circuits with depth bounded by the inverse of an Ackerman function are constructed, and it is shown that depth two circuits require $\Omega (n \log n)$ size. The lower bound applies to any Hadamard matrix.

Programmable logic device with global and local product terms

Abstract: A programmable logic device having a programmable AND (product term) array formed with input terms on both global and local busses and with both global and local product term lines. Each macrocell of the device, whether as input/output macrocell connected to an I/O pin or a buried macrocell providing only feedback, connects to and receives an inputs both global and local product terms. In one embodiment, global product terms are connectable to the global bus and to a local bus corresponding to a particular group or quadrant of macrocells. Local product terms are only connectable to that local bus, and thus only a fraction of the terms available to the global product terms. In an alternate embodiment, global product terms are connectable to the global bus and to a set of local busses which is a prope subset of all of the local busses. Local product terms are connectable only to the particular local bus assigned to a particular group or quadrant of macrocells and to a fraction of the terms on the global bus.

Thermal properties of directed polymers in a random medium

Abstract: We calculate, by performing a product of random matrices, the specific heat and its derivative for the problem of directed polymers in a random medium. Our results are consistent with the existence of a phase transition both in 3+1 and 2+1 dimensions and with the absence of a phase transition in 1+1 dimensions. Our finite-size scaling analysis leads to \ensuremath{\alpha}=-0.1\ifmmode\pm\else\textpm\fi{}0.1 in 3+1 and \ensuremath{\alpha}=-1.0\ifmmode\pm\else\textpm\fi{}0.2 in 2+1 dimensions.