Showing papers on "Product (mathematics) published in 1975"

Representations of a semi-direct product by quantization

Abstract: 1.Introduction. The purpose of this note is to apply the Kostant-Souriau quantization theory (2, 3, 4, 5, 7) to construct representations of a semi-direct product.

On commutative power-associative nilalgebras of low dimension

Abstract: Commutative power-associative nilalgebras of dimension 4 and characteristic *2 are shown to be nilpotent and all their isomorphism classes are determined. The long-standing conjecture, originally due to A. A. Albert, that a commutative power-associative nilalgebra of finite dimension over a field is nilpotent, has recently been disproved by Suttles [41, who gave a counterexample of dimension 5. This dimension is generally the best possible, for we show here that every commutative power-associative nilalgebra of dimension 4 over a field of characteristic 4 2 is nilpotent, and we determine the isomorphism classes of all such algebras. The proof is elementary and cannot be significantly simplified by the use of the general results of [1] and [2], which require, moreover, additional assumptions about the characteristic. Throughout, A will denote a commutative power-associative nilalgebra of dimension 4 over a field F of characteristic 4 2. The subspace of A generated by elements u, v, w, ... will be denoted by (u, v, w, ). Since every x E A is nilpotent, the powers of x are linearly independent, so we must have xn = 0 for n > 5; the least n such that xn = 0 for anl x E A is called the nilindex. Now the product of any two elements of A can be written as a linear combination of squares, for xy = ,'4[(x + y)2 _ x 2 _ y 2 ]. Therefore, if the nilindex is 2, then every product vanishes. If it is 5, then A = (x, x2, x3, x4) for any x with x4 A 0. These are trivial cases, in each of which there is, up to isomorphism, a unique, associative, algebra. Only the cases of nilindex 3 and 4 are of interest. 1. Nilindex 3. Linearizing the identity (x2)x 0 yields 2[(xy)z + (yz)x + (zx)y] = 0 for all x, y, z E A. Therefore, denoting right Received by the editors January 2, 1974. AMS (MOS) subject classifications (1970). Primary 17A10.

The Complexity of the Quaternion Product

Abstract: Let X and Y be two quaternions over an arbitrary ring. Eight multiplications are necessary and sufficient for computing the product XY. If the ring is assumed to be commutative, at least seven multiplications are still necessary and eight are sufficient.

A notion of limit for enriched categories

Abstract: For a V-category B, where V is a symmetric monoidal closed category, various limit-like notions have been recognized: ordinary limits (in the underlying category B0 ) preserved by the V-valued representable functors; cotensor products; ends; pointwise Kan extensions. It has further been recognized that, to be called complete, B should admit all of these; for which it suffices to demand the first two. Hitherto, however, there has been no single limit-notion of which all these are special cases, and particular instances of which may exist even when B is not complete or even cotensored. In consequence it has not been possible even to state, say, the representability criterion for a V-functor T: B → V, or even to define, say, pointwise Kan extensions into B, except for cotensored B. (It is somewhat as if, for ordinary categories, we had the notions of product and equalizer, but lacked that of general limit, and could not discuss pullbacks in the absence of products.) In this paper we provide such a general limit-notion for V-categories.

Fundamental properties of fractional derivatives via pochhammer integrals

Abstract: In this paper, various representations of fractional differentiation are explored, and a definition using Pochhammer contour integrals emerges as deserving special emphasis. The analyticity of Dαzpf(z) and Dαzpln z f(z) is investigated with reference to the three variables z, α, and p. The validity of the operation DβDα=Dα+α is studied. An improvement in the Leibniz rule for the fractional derivative of the product of two functions published previously is given.

Sui fibrati con struttura quaternionale generalizzata

Abstract: Quaternion generalized fiber bundles\(\tilde E_n^\mathbb{Q} \to X\) are studied, both isomorphic to global tensorial product\(E_n^\mathbb{Q} \otimes _\mathbb{Q} E_1^\mathbb{Q} (E_n^\mathbb{Q} , E_1^\mathbb{Q}\) ordinary quaternion fiber bundles right and left respectively) and quite general ones A cohomology class\(\varepsilon (\tilde E_n^\mathbb{Q} ) \in H^2 (X;\mathbb{Z}_2 )\) is considered which represents the obstruction in order the fiber bundle be a tensorial product Several properties and a splitting principle are proved for bundles\(\tilde E_n^\mathbb{Q}\) On this ground and founding on a convenient bundle BE → X associated to jaz (that we call Bonan's bundle and for which ɛ(\(\tilde E_n^\mathbb{Q}\)=ɛ(BE)) relations are stated among Stiefel-Whitney classes of\(\tilde E_n^\mathbb{Q}\), BE and the class ɛ

Toward a Product Theory for Orthocompactness

Abstract: Publisher Summary This chapter discusses the behavior of orthocompactness in finite Cartesian products—products with a metric factor and products of ordinals. The results show that the product theory for orthocompactness exhibits striking parallels with that of normality, with metacompactness playing the role of paracompactness. The chapter presents the target theorem to discuss the product theory for orthocompactness. It discusses the preservation of orthocompactness in products with arbitrary metric spaces. One justification for studying the normality of products of ordinals is the well-known fact that every linearly ordered topological space (LOTS) is hereditarily collection-wise normal. Every LOTS is also hereditarily orthocompact and in fact a finite product of ordinals is orthocompact if it is normal.

Dehydrated food product

Abstract: A dehydrated food product in the form of grains which dissolve instantly in water and which have a porous, continuous structure, a smooth surface and an apparent density of from 30 to 600 g/l the product is prepared by which comprises extruding a thermoplastic starting material in powder or paste form into a chamber where a sub-atmospheric pressure prevails, and cutting the extruded product into fragments.

Scattering theory for quantum electrodynamics. I. Infrared renormalization and asymptotic fields

Abstract: The present article lays the theoretical foundation for a scattering theory of quantum electrodynamics, which is completed into a practical calculational scheme in the accompanying article. In order to circumvent infrared divergences, an infrared renormalization procedure is instituted whereby a Lorentz-invariant, but indefinite, inner product is defined for a class of photon test functions defined on the future light cone ${k}^{\ensuremath{\mu}}=\ensuremath{\omega}(1, \stackrel{^}{k})$, $\ensuremath{\omega}\ensuremath{\ge}0$. This class includes test functions whose low-frequency behavior is given by ${\ensuremath{\varphi}}^{\ensuremath{\mu}}(k)\ensuremath{\sim}\frac{e{p}^{\ensuremath{\mu}}}{p}\ifmmode\cdot\else\textperiodcentered\fi{}k$, for which the usual inner product $\ensuremath{\int}{d}^{3}k{(2\ensuremath{\omega})}^{\ensuremath{-}1}{{\ensuremath{\varphi}}_{\ensuremath{\mu}}}^{*}(k)(\ensuremath{-}{g}^{\ensuremath{\mu}\ensuremath{ u}}){\ensuremath{\varphi}}_{\ensuremath{ u}}(k)$ is infrared-divergent. The Fock space of such test functions provides a representation space for the asymptotic fields of quantum electrodynamics. It contains subspaces in which the indefinite metric is non-negative which, when completed in the norm, yield physical Hilbert spaces. This Fock space of test functions thereby replaces the nonphysical Hilbert space of the usual Gupta-Bleuler method and its positive-definite but noncovariant metric. As an application the $S$ matrix and finite transition probabilities are found for the bremsstrahlung emitted by the classical external current of a scattered charged particle. A final result is a simple weak asymptotic limit of the charged field $\ensuremath{\psi}$. It is used as a starting point in the accompanying article, for the derivation of reduction formulas for the quantum electrodynamical $S$ matrix.

The Power of Negative Thinking in Multiplying Boolean Matrices

Abstract: We show that $n^3 $ distinct and-gate inputs appear in any circuit constructed from and-gates and or-gates that computes the product of two $n \times n$ Boolean matrices. Using not-gates as well, it is possible to realize a circuit for this problem using only $O(n^{\log _2 7} \log ^2 n)$ gates, whence we infer a much larger complexity gap between and-or and and-or-not circuits than was previously known.

Multistep Methods for Solving Linear Volterra Integral Equations of the First Kind

Abstract: This paper is concerned with deriving sufficient conditions for multistep methods to be convergent. The conditions require that the eigenvalues of a product of companion matrices lie inside the unit circle. A specific high order convergent method is determined and numerical results are presented.

Countable products and sums of lines and circles: their closed subgroups, quotients and duality properties

Abstract: Introduction. I t is well-known ((2), Theorem 9-11) that any closed subgroup of R is isomorphic (topologically and algebraically) to R x Z, where a, b are suitable non-negative integers. For an infinite product of copies of R, it is also known that any locally compact (hence closed) subgroup is a product of copies R and Z, and that any connected subgroup is a product of copies of R (see (7), (3), respectively). Some information is also given in (3) on closed subgroups of products of copies of R and T, where T = R/Z is the circle group. In this paper, we study the class @>n consisting of all Hausdorff Abelian groups topologically isomorphic to a product of a compact group with a countable product of copies of JR and Z. In §4, we prove:

Tensor Products and Approximation Problems of C*-Algebras

Abstract: For C*-algebras C and D and their C*-subalgebras A and B, it is shown with other related results that if all irreducible representations of A are finite dimensional with bounded degree then the family of product functionals in the minimal C*-tensor product C®D determines completely the subalgebra A®B. Relations to Effros' recent work are also discussed.

Sum and product separabilities of multivariable functions and applications

Abstract: Necessary and sufficient conditions for reducibility of certain classes of multivariable polynomials are considered. The special case when a polynomial in that class can be factored into a finite product of single variable polynomials is investigated. A set of necessary and sufficient conditions for decomposing a multivariable p.r.f. into a sum of single variable p.r.f.'s is given. Examples to illustrate applications of sum and product separabilities of functions are given.

Conveyor for staggering adjacent lanes of product

Abstract: Apparatus for staggering product delivered to multiple product lanes. A merging conveyor receives the staggered product from the product lanes and merges the product into a single lane. A split conveyor is disclosed in which alternate groups of product positioning supports and product conveying supports position and convey the product in staggered relation to a merging conveyor for merging into a single lane without product accumulation.

On mechanization of vector multiplication

Abstract: This letter discusses an organization of an approach to form the inner product of a pair of data vectors. The criterion is the minimization of the number of adders, the independent variable is the number of bits to be formed in the product. The cost ratio of addition-to-storage provided the motivation for this investigation. The most efficient mechanization appears to be the combining of the vectors in sets of 3 or 4 elements at a time, then summing the results.