Showing papers on "Product (mathematics) published in 1979"

Optimal Open-Loop Maneuver Profiles for Flexible Spacecraft

Abstract: Using the Calculus of Variations, optimal slewing profiles minimizing a structural e xcitation criterion are e stablished for a dynamically s imple spacecraft maneuvering between two quiescent states. Two problem types are considered. In the free end point problem, the structural deformation and its time derivative are unconstrained at maneuver's end. For the constrained end point problem, these variables a re r equired to vanish, which necessarily degrades the excitation criterion. Several figures are presented that illustrate both the n ature and the limitations inherent in maneuvering the spacecraft from one attitude state to another. For a given maneuver amplitude, en, the key parameter influencing structural e xcitation is the product of the maneuver time, Ta, and the lowest significant structural frequency, w. It is shown that when wTa 10, however, this penalty is fairly minor, and some reasonable control of terminal conditions is then practical. Thus it is generally d esirable that all maneuver times meet this criterion. When this is the case, it is possible to derive a normalized suboptimal slewing profile, F(x), applicable to a1 1 maneuvers. Given 0" and Ta, the commanded maneuver rate becomes e( t) = eO F(t/Ta)/Ta. Only a minor computational and memory burden is therefore necessary to perform almost optimal re-orientations.

Form-fill-seal packaging method and apparatus

Abstract: A form-fill-seal packaging method and apparatus in which a web of flexible packaging material is pulled from a supply and fed over a device for forming it into tubing, product to be packaged is provided in the tubing, and the tubing is sealed to form packages, and in which the web is pulled forward from the supply as one operation and the tubing is drawn forward and thereby taken up as an accompanying but separate operation with the take-up such that the web is pulled over the forming device under tension.

ZIPMAP — A Zero-One Integer Programming Model for Market Segmentation and Product Positioning

Abstract: An optimization model is proposed to aid marketing managers to search for and develop new product brand ideas. The model, which is founded on individual consumer behaviour constructs, determines a position for a new brand, in the consumers' perceptual space of product attributes, that maximizes company sales. The resulting mathematical model is stated as a large-scale, mixed, zero-one, integer, non-linear mathematical programming problem whose solution is sought through a two-stage optimization approach.

Apparatus for dispensing a striped product and method of producing the striped product

Abstract: A product to be dispensed is packaged in an axially collapsible product bag in distinct transversely extending axial layers. When the product is dispensed, it enters into a discharge nozzle generally from the center of the bag with the result that there is an almost immediate combining of the product layers to be simultaneously dispensed in a striped stream.

A Generalized Queueing Discipline for Product Form Network Solutions

Abstract: Certain queuemg discaphnes, such as processor sharing, the preemptive last-come-first-served discipline, and the infinite server queue, are known to result in network eqmltbnum state probabihties that have a convenient product form A generalization of the above dlsciphnes is introduced The general class is presented in the form of a parametenzed disciphne, called the last-batch-processor-sharing (LBPS) discipline The eqmhbrlum state probabilities for disciplines of the LBPS class are shown, and, by use of the concept of local balance, at ~s shown that arbitrary networks of LBPS queues have product form eqmlibrium state probabilities It as also shown that within the class of symmetric dlsciphnes, the LBPS form is necessary ff the product form solution ~s to be obtained for general service time dtstrlbutmns A dasctphne ~s symmetric ff the processor assignments to the customers in the queue depend on total queue occupancy and queue poslUon (relatwe arrival time) only Generahzations of the LBPS rule beyond the symmetric dlsciphnes are discussed A multiple customer-class form of the LBPS disclphne is also demonstrated, and it is shown to have the local balance property

Selected Families of Location Problems

Abstract: This exposition is concerned with two important families of static, deterministic, single criterion, one product location problems: Center problems and median problems. A similar treatise covering additional families is at present under preparation and will be published elsewhere. Reference is made to the introduction for an overview of the families of problems considered. Each part, which is introduced by its own abstract and concluded by its own bibliography, can be read independently of the others as cross references are virtually non-existent.

Coproducts of Kleene algebras

Abstract: The coproduct of a family of Kleene algebras is determined firstly by describing the maximal homomorphic image of a De Morgan algebra in the subvariety of Kleene algebras and, secondly, by characterizing the categorical product in the dual category of Kleene spaces.

Minimization of Modulo-2 Sum of Products

Abstract: This correspondence attempts to solve the problem of expressing an arbitrary switching function in a modulo-2 sum-of-product terms form, having a minimum number of product terms. Each product term may contain variables in complemented or uncomplemented form.

High-speed programmable logic array adders

Abstract: Programmable Logic Array (PLA) adders are described which perform an addition in one cycle with a single pass through a PLA and require a reasonable number of product terms for an 8-, 16-, or even a 32-bit adder. The PLA features two-bit input decoders feeding an AND array followed by an OR array whose outputs are pairwise Exclusive-ORed. Carry-look-ahead adder equations, adapted to the PLA to require relatively few product terms, are adjusted for maximum sharing of product terms. The number of unique product terms is a relative measure of one of the physical dimensions of the PLA. Equations for contiguous sum bits are grouped into strings, each using a common input carry. A procedure optimally assigns sum bits to strings to further minimize the total number of unique product terms. The methods are extended to PLAs with decoders of increased inputs and substantially reduced product terms. They can serve as dedicated macro functions on a chip, using special decoders relevant to adders. As a result, the other PLA dimension comprising the number of outputs from all input decoders increases only moderately, and can even decrease, with larger decoders. Finally, the PLA adder can be further substantially compressed by splitting the OR array into two parts such that a row of the AND array is shared between two product terms, and an OR array column is shared between two sums of product terms.

Energy distribution among reaction products. XII. F + HBr → HF(v′⩽ 4) + Br*(2P32, 2P12)

Abstract: As part of a study of product energy distributions in the family of reactions F + HX → HF(v′, J′) + X (X Cl, Br, I) we have studied the second reaction in this family, F + HBr → HF(v′,J′) + Br(2P 3 2 , 2P 1 2 ) (2) using the arrested relaxation variant of the infrared chemiluminescence approach to obtain the detailed rate constant k(V′, R′, T′) (V′, R′, T′ are product vibrational, rotational and translational excitation), as well as the fraction of reaction yielding excited electronic state atomic product (reaction (2b), → Br*(2P 1 2 )). The product vibrational excitation is indicative of quite different reaction dynamics for reaction (2) as compared with reaction (1) [(1) was F + HCl, studied previously by the same technique]. Reaction (1) favoured the formation of vibrationally excited product; reaction (2) gives rise predominantly to product with low vibrational excitation. With respect to this product-attribute the reaction F + HBr (2) gives a markedly more-nearly statistical outcome than does reaction (1). If, however, the observed product rotational distribution is used as an index of behaviour, the outcome of the reaction F + HBr (2) is markedly less statistical than is the rotational distribution in the product of reaction (1). The mean fractions of the available energy entering vibration, rotation and translation for reaction (2) were found to be (f′V) = 0.24, (f′R) = 0.09 and (f′T) = 0.67. The fraction of reaction (2) which proceeds on the upper energy-surface to yield Br* (with energy, 10.5 kcal mole−1 in excess of Br) is k*/k = 7 ± 3‰

Preparing centerfilled food product

Abstract: A tubular centerfilled food product having a rigid, friable, thermoplastic baked outer shell and a core of edible filling material is produced by a continuous, straight-through process. A semi-liquid batter having a relatively high content of mono- and/or disaccharides is carried between a pair of spaced-apart moving heated surfaces and baked to form a continuous, elongated flat thermoplastic sheet having a controlled thickness. The continuous thermoplastic sheet while warm and pliable is rolled around its longitudinal axis to form a continuous tube having a closed longitudinal, non-overlapping seam. As the baked sheet is rolled around its longitudinal axis, an edible filling material is injected into the core of the continuous tube as it is formed. The filled tube is cooled until the outer shell becomes rigid and is then cut into pieces of a desired length.

Ordered sum and tensor product of linear utility structures

Abstract: Measurement-theoretic behavior of the category of social utility models is studied in terms of ordered sums and tensor products of mixture spaces Necessary and sufficient conditions are given for the existence of individual utility functions and a social utility function (being a weighted average, and in another case a product, of the individual utilities) in terms of individual and social preference rankings

Computer Aided Progressive Die Design

Abstract: The MEL system which offers advantages in improving the design process is described. Our system consists of ten steps. Each step can be used independently of the others. This method enables us to design a new die, and, in addition, changes and corrections of part of an existing die can be done easily. In our system, a product shape is input hierarchically using elements such as points, lines, and circles, closed figures consisting of these elements, patterns consisting of these closed figures, and the product consisting of these patterns. We introduce a concept of the basic pattern which corresponds to the shape of a standard punch. There are several of these. These patterns can be input efficiently using the macro commands for the pattern in the hierarchy, hence input is simple and the programming can be done quickly. The great benifits accrue from these patterns, i.e., the most effective blank and strip layout including information of the tool configuration can be done quickly.

On the information theoretic synthesis of three dimensional vibrotational reaction probabilities from collinear results

Abstract: A modified version of the Bernstein—Levine collinear to three dimensional (1D→3D) transformation is described in detail and tested. As input data, the procedure requires 1D reaction probabilities and a preassigned value of the average fraction of rotational product energy. We obtain these quantities from accurate quantum and classical trajectory calculations respectively and compare the 1D→3D vibrotational reaction probabilities with those from the trajectory calculations. We consider first the H + F 2 → HF + F reaction, which previous work suggests is a favourable case. The procedure is then applied to the F + H 2 → HF + H reaction at various fixed energies. It is found that the 1D→3D transformation is useful at low energies but becomes unreliable at higher energies. This may be because bent configurations play a more important role as the energy increases. The significance of almost linear surprisal plots is also discussed.

Direct synthesis of 2,2-diaryl-3-methyl-2,3-dihydrobenzothiazoles from 3-methyl-2,3-dihydrobenzothiazole-2-thione and some mechanistic aspects.

Abstract: Reactions of 3-methyl-2,3-dihydrobenzothiazole-2-thione (1) and 3-methyl-2-(methylthio)benzothiazoliumiodide with Grignard reagents gave 2,2-disubstituted 3-methyl-2,3-dihydrobenzothiazole (2) as a major product as well as 2,2′-disubstituted 3,3′-dimethyl-2,2′,3,3′-tetrahydrobi(2-benzothiazolyl) (3). Reactions of 1 with organolithiums gave 2 as a major product. The results demonstrate that the formation of 3 is due to the presence of transition metals as an impurity in magnesium.

On the viscosity and thermal conductivity of dense gases

Abstract: The modified Enskog theory (MET) of transport in dense gases can be used to predict the density dependence of transport coefficients provided the function y , which is a product of a dimensionless density and a pair correlation function, is known as a function of density. In this communication we consider the inverse problem, extracting information on the y function from measured transport property data, both to resolve some peculiarities found by others, and to provide the basis for improved MET predictions.

Tournaments and Vandermond's determinant

Abstract: We prove that det |xii–1|n × n = Π1≤i

Optimal product attributes in single choice models

Abstract: Product positioning is concerned with the determination of new product attributes such that certain objectives can be met. Recently, Zufryden proposed a model for optimal product positioning with respect to sales. It relies on the single choice assumption, i.e. consumers choose that product closest to their ideal product perceptions. To improve solvability, Zufryden suggests to solve his model via an approximate procedure, called ZIPMAP. This procedure is compared with an exactly optimising algorithm for the same problem, called PROPOSAS. Comparisons are drawn with respect to the solution values of the objective functions and the CPU-time requirements. The results indicate a superiority of PROPOSAS for a vast number of problem types.