Showing papers on "Product (mathematics) published in 1985"

Optimal Policy for a Two-Stage Assembly System under Random Demand

Abstract: This paper considers an inventory system in which an end product is assembled from two components, each of which is ordered from an external supplier. Only the end product has final demand, which is assumed to be random. Using the functional equation approach of dynamic programming, we characterize the forms of both the optimal order policy for the components and the optimal assembly policy of the end product for a multiperiod problem of arbitrary, but finite length. We believe that this work represents the only exact analysis of an MRP-type assembly system when demand is stochastic.

On the chromatic number of the product of graphs

Abstract: This paper concerns an intriguing conjecture involving vertex colorings of products of graphs. First we specify the product. Given (irreflexive, symmetric) graphs G and H, the product G x H has vertex set V(G) X V ( H ) and edges all pairs {(g, h), (g‘, h’)} such that gg‘ and hh’ are edges of G and H, respectively. We avoid modifying the label “product,” in part because of the variety of its names in the literature (e.g., Kronecker product, weak direct product, categorical product, conjunction, cardinal product). Given a coloring cp of G, there is a natural induced coloring p’ of G X H, namely cp’(g, h) = cp(g). Letting x, as usual, denote the chromatic number, we see that

Classifying immersions into IR[4] over stable maps of 3-manifolds into IR[2]

Abstract: The singularities of stable maps of three manifolds into the plane.- Canonical coordinatized product neighborhoods (C2PN).- Lifting stable maps of three manifolds into the plane to immersions in ?4.

Field effects on Rydberg product state distribution from dielectronic recombination

Abstract: The effects of state mixing by extrinsic fields in the collision region have been investigated for the dielectronic recombination process ${\mathrm{Mg}}^{+}$(3s)+e\ensuremath{\rightarrow}Mg(3p,nl)\ensuremath{\rightarrow}Mg(3s ,nl)+h\ensuremath{ u}. By field ionization of the Rydberg atoms produced, cross sections \ensuremath{\sigma}(${n}_{f}$) have been measured. The observed large changes of \ensuremath{\sigma}(${n}_{f}$) with alteration of the extrinsic field provide the first incontrovertible experimental evidence that dielectronic recombination can be changed by external fields.

Element Preconditioning Using Splitting Techniques

Abstract: The finite element method is a good way to turn elliptic boundary value problems into large symmetric systems of equations. These large matrices, ${\bf A}$, are usually assembled from small ones. It is simple to omit the assembly process and use the code to accumulate the product ${\bf {Av}}$ for any v. Consequently the conjugate gradient algorithm (CG) can be used to solve ${\bf {Ax}} = {\bf b}$ without ever forming ${\bf A}$.It is well known, however, that CG performs best when applied to preconditioned systems. In this paper we show one way to precondition without forming any large matrices.The trade-off between time and storage is examined for the 1-D model problem and for the analysis of several realistic structures.

Indecomposable representations with invariant inner product: A theory of the Gupta-Bleuler triplet

Abstract: AbstractConsequences of the existence of an invariant (necessarily indefinite) non-degenerate inner product for an indecomposable representation π of a groupG on a space $$\\mathfrak{H}$$ are studied. If π has an irreducible subrepresentation π1 on a subspace $$\\mathfrak{H}_1 $$ , it is shown that there exists an invariant subspace $$\\mathfrak{H}_2 $$ of $$\\mathfrak{H}$$ containing $$\\mathfrak{H}_1 $$ and satisfying the following conditions: (1) the representation π1#=π mod $$\\mathfrak{H}_2 $$ on $$\\mathfrak{H}$$ mod $$\\mathfrak{H}_2 $$ is conjugate to the representation (π1, $$\\mathfrak{H}_1 $$ ), (2) $$\\mathfrak{H}_1 $$ is a null space for the inner product, and (3) the induced inner product on $$\\mathfrak{H}_2 $$ mod $$\\mathfrak{H}_1 $$ is non-degenerate and invariant for the representation $$\\pi _2 = (\\pi _2 |_{\\mathfrak{H}_2 } )\\bmod \\mathfrak{H}_1 ,$$ a special example being the Gupta-Bleuler triplet for the one-particle space of the free classical electromagnetic field with $$\\mathfrak{H}_1 $$ =space of longitudinal photons and $$\\mathfrak{H}_2 $$ =the space defined by the subsidiary condition.

Comments on "A Computer Algorithm for Calculating the Product AB Modulo M"

Abstract: The modular multiplication algorithm (MMA) was presented as a method of calculating " the smallest nonnegative integer R congruent modulo M to the product AB of two nonegative integers without dividing by M."1 The claim that division is avoided is technically correct, but misleading. A minor modification calculates both R and Q such that AB = MQ + R. A simplified version of the new algorithm is given and an alternate derivation is shown to illustrate the key ideas behind the method.

Product formulae for nielsen numbers of fibre maps

Abstract: This work simplifies proofs of a recent publication by You and gives simple sufficient conditions for Brown's product formula for the Nielsen number of a fibre map, as well as new product formulae in this context. Product formulae are also given relating absolute and relative Nielsen numbers, together with corresponding results for Reidemeister numbers. Introduction. Let p: E -> B be a fibration in which E, B and all fibres are compact connected ANR's, and let /:£->£ be a fibre preserving map inducing self maps/on B and/fo on the fibre Fb over some fixed point b in the base. Since Brown (1) introduced his multiplicative formula N(f) = N(fb)N(f) for the Nielsen number N(f) of/, various attempts have been made both to improve his results (cf. (4), (5)) and to generalize his formula (cf. (6), (14), (18)). In a recent paper which super- cedes most of what precedes it in both of the aspects mentioned above, You (20) gives, among other things, necessary and sufficient conditions for Brown's formula together with a new result relating the Nielsen numbers of/, /and a relative Nielsen number Nκ(fb) oίfh. Here K is the kernel of the inclusion induced homomorphism Π^ -> TlλE. In this work we consider the second of these two results and use it (1) as a focus to give what we feel are more eccessible proofs of the results in (20); (2) as a springboard to give new product theorems for fibre maps. Our results here include conditions under which N(f) = Nk(fb)N(f) and also conditions under which Nκ(fb) = N(fb). By combining these we thus obtain new sufficient conditions for Brown's formula. These conditions are simpler to verify than You's. We investigate the hypotheses of You's theorems giving conditions under which they hold. In the process we develop product formulae relating relative and absolute Nielsen numbers, together with corresponding results for Reidemeister numbers. The unifying tool in this work is a certain exact sequence associated with a self morphism of a short exact sequence of groups. This result is a kind of non-abelian snake lemma and is a special case of a theorem (cf. (9)) originally proved in connection with localization of orbit sets. All the product theorems mentioned above ultimately derive from this sequence.

Products of idempotent linear transformations

Abstract: In 1966, J. M. Howie characterised the transformations of an arbitrary set that can be written as a product (under composition) of idempotent transformations of the same set. In 1967, J. A. Erdos considered the analogous problem for linear transformations of a finite-dimensional vector space and in 1983, R. J. Dawlings investigated the corresponding idea for bounded operators on a separable Hilbert space. In this paper we study the case of arbitrary vector spaces.

The chromatic number of the product of two ℵ1-chromatic graphs can be countable

Abstract: We prove (in ZFC) that for every infinite cardinal ϰ there are two graphsG 0,G 1 with χ(G 0)=χ(G 1)=ϰ+ and χ(G 0×G 1)=ϰ. We also prove a result from the other direction. If χ(G 0)≧≧ℵ0 and χ(G 1)=k<ω, then χ(G 0×G 1)=k.

Alternative estimates of the real product of India, 1900-46:

Abstract: The second volume of the recently published Cambridge Economic History of India (Kumar and Desai, 1983) includes a lengthy article by Alan Heston which surveys the literature on measurement of Indian economic growth, and provides new annual estimates of net domestic product (NDP) and net national income (NNI) for fiscal years 1884-1946, and for three earlier years beginning 1868 for undivided India. This note is concerned only with the period 1900-46. It compares Heston’s estimates with those of Sivasubramonian (1965) and Maddison (1971). In fact both Heston’s and Maddison’s estimates were based mainly on Sivasubramonian’s unpublished series, with two main modifications which give higher growth estimates for crops, and lower estimates for growth in services. In the case of Heston the impact of the adjustments in offsetting and his NDP growth is virtually the same of that of Sivasubramonian, whereas my estimate showed substantially lower growth than both of these. Heston criticises the level of my estimates and on reworking them, I discovered that I had made an error for agriculture. Correction of this lowers my estimate of the NDP level but does not significantly affect the estimate of growth. I still

Cohomology of step functions under irrational rotations

Abstract: This paper studies the solvability of the functional equationg(x+θ)=φ(x)g(x), given an irrationalθ and a step functionf mappingR/Z (with Lebesgue measure) to the unit circle. Results are applied to find parameterized families of representations of non-regular semi-direct product groups and to display irregularities in the uniform distribution of the sequenceZθ.

The Reaction of 4‐Oxo‐4H‐[1]benzopyran‐3‐carbaldehyde with 1,2‐Benzenediamine. X‐Ray Structure Analysis of the Reaction Product

Abstract: The reaction product of 4-oxo-4H-[1]benzopyran-3-carbaldehyde and 1,2-benzenediamine is 1,8-dihydro-6,13-bis(2-hydroxybenzoyl)dibenzo[b,i]-1,4,8,11-tetraazacyclotetradeca-4,6,11,13-tetraene (1), and its dehydrogenation product is 3-(2-benzimidazolyl)-4H-[1]benzopyran-4-one (2) as shown by X-ray structure analysis confirming our earlier report.

Eigenvalues of the difference and product of projections

Abstract: (1985). Eigenvalues of the difference and product of projections. Linear and Multilinear Algebra: Vol. 17, No. 3-4, pp. 295-299.

Product supply system for accumulation packaging machine

Abstract: Disclosed herein is a product supply system for an accumulation packaging machine which may permit a high speed operation of a product discharge section and smoothly supply a product to the accumulation packaging device with a row arrangement of the product retained. The product supply system comprises a push conveyance line provided between the product discharge section and the accumulation packaging machine through an acceleration feeding device, and a product separating device and a product retaining device each provided in the push conveyance line and adapted to be interlocked with the accumulation packaging machine, wherein a product feed velocity of the acceleration feeding device is higher than that of the product discharge section.

On product and change of base for toposes

Abstract: Le produit de deux S-topos born6s coincide avec leur produit tensoriel comme categories cocomplbtes S-index6es. De plus, la cat6gorie index6e cocomplete sous-jacente au produit fibr6 d’un S-topos born6 le long d’un morphisme g6om6trique est obtenue en consid6rant le topos comme une cat6gorie index6e cocomplbte, puis en appliquant un certain foncteur adjoint associe. au morphisme g6om6trique. Un corollaire montre la stabilite des morphismes essentiels et localement connexes par changement de base. 0. INTRODUCTION. In this paper we prove some results about Grothendieck toposes and geometric morphisms that derive from regarding a Grothendieck topos (resp., the inverse image part of a geometric morphism) as a cocomplete category (resp., cocontinuous functor) with additional properties. Specifically, if S is an elementary topos with natural number object, then the product of two Grothendieck S -toposes coincides with their \"tensor product\" as cocomplete S -indexed categories. Moreover if p : E->S is a geometric morphism between elementary toposes (with natural number objects), then the pullback of a Grothendieck S topos F along p coincides with the cocomplete S -indexed category obtained from F (regarded as a cocomplete S -indexed category) by \"changing base along p\". An immediate corollary of the latter fact is a result about essential geometric morphisms which generalises a theorem of Tierney on the pullback of locally connected geometric morphisms (unpublished, but see [12J, V). These results are seen as favourable evidence for the speculation that one may be able to describe Grothendieck toposes in terms of cocomplete categories in a way analogous to that in which Joyal and Tierney have recently described the theory of locales as part of the \"commutative algebra\" of complete lattices and arbitrary sup preserving maps : see [8]. Thus replacing locales by Grothendieck toposes and sups by colimits, Theorems 2.3 and 3.6 below are the analogues of the results in III.2 and VI.1 of [8]. However the proofs are not analogous : this is because in the one case they are immediate corollaries of the characterization of a locale as a certain kind of commutative monoid in the monoidal category of complete lattices and sup-preserving maps equipped with its tensor product (the monoid multplication giving binary meet in the locale and the unit giving the