Binomial Factors Research Articles

The constrained multinomial logit: A semi-compensatory choice model

The traditional formulation of logit models applied to transport demand assumes a compensatory (indirect) utility function in which the consumers’ strategy assumes a trade-off between attributes. Several authors have criticized this approach because it fails to recognize attribute thresholds in consumer behavior, or a more generic domain where such a compensatory strategy is contained. In this paper, a mixed strategy is proposed, which combines the compensatory strategy valid in the interior of the choice domain with cutoff factors that restrain choices to the domain edge. The proposed model combines the multinomial logit model with a binomial logit factor that represents soft cutoffs. This approach extends previous contributions in several ways and allows multiple dimensions for cutoff factors. In addition to considering individual behavior, it introduces system constraints such as capacity and inter-agent interactions (choice externalities). This extension yields a non-linear problem, which is solved by analyzing the fixed point problem. Additionally, a set of evaluation tools, a social utility of the constrained problem, and a measure of the shadow price of each constraint, are proposed.

A multiple boundary interpolation problem for contracting matrix-valued functions in the unit circle

The method of solving the interpolation problem of analysis, presented by Vo P. Potapov [i, 2], the foundation of which consists in the adequacy of the interpolation problem for the assignment of definite objects of the j-theory of analytic matrix-valued functions, allows us to connect also this problem with objects of j-theory, such as the multiple j-elementary matrix-valued function with pole ~0 on the boundary of the unit circle (!~r01 = i) and the Blaschke--Potapov product of binomial j-elementary factors.

Moments and characteristic polynomials of bipartite H�ckel graphs

Moments (u k ) and coefficients of characteristic polynomials (a k ) have been evaluated in terms of molecular fragments up tok = 12 for bipartite Huckel graphs. Based on combinatorial analysis, each coefficient can be derived as a combination of binomial factors mapping to the corresponding multi-component graphs. The general formula becomes lengthy as k increases, but can be considerably simplified for a homologous series. This has been illustrated by dealing with the cata-condensed benzenoid hydrocarbons as a corollary where a rather compact set ofa k has been deduced. On combining the present result with Coulson's formula, one gains insight into the relative stability of isomers in relationship to the energy contribution of fragments classified as stabilized and destabilized species.

A graphical evaluation of characteristic polynomials of huckel trees

AbstractCharacteristic polynomials of acyclic carbon chains (Huckel trees) are treated in a systematic way. Formulas of coefficients (ak) of the polynomial are obtained in terms of connectivities that were introduced for dealing with moments in a previous paper. Based on the meaning of ak, a graph‐theoretical analysis is given such that ak can be expressed as a linear combination of binomial factors specified by a set of graphs containing ½k edges. The numerical relationship is disclosed between each binomial factor and its specified graph. This stimulates the proposal of a novel approach for evaluating ak by simply collecting the graph set of defnite edges. The approach is equally applicable for the evaluation of matching polynomials of cyclic systems and extendable to the investigation of general trees.

Combinatorial formulae for one-dimensional generalised random walks

The calculation of ensemble averages in one-dimensional walks with weights given by exp(-g Sigma (ni)alpha ), where ni is the number of times a site is visited, is simplified by reducing the problem to one of calculating binomial factors for partitions and their permutations. The combinatorial factors are generalised to the case of Cayley trees and all lattices without closed loops. In one dimension the combinatorial formulae are shown to reduce the known cases when alpha =0 and alpha =1. The authors also use these formulae to extend the exact enumeration results, for the one-dimensional generalised random walk, from 21 to 29 terms; and the series are analysed using Pade approximant techniques.

Vibrational product distributions from moments methods: A systematic investigation for a simplistic model system

The application of moments methods for the determination of vibrational distributions in reactive collisions is investigated for a simple model system, the isoergic rotated harmonic potential. An information-theoretic technique minimizing the entropy deficiency and a method minimizing the least-squares deviation relative to a reference distribution are tested as well as direct analytical representations. For different reagent conditions and potential parameters, the dependence of the results on the choice of the reference distribution and on the number of constraining moments is studied, using both quantum and classical moments as input. The reference distributions included the generalized Poisson distribution used by Hofacker and Levine, the generalized binomial distributions of Basilevsky and Ryaboy, the newly introduced binomial shape factor, and others. For c = 0 good agreement with quantum distributions is obtained using only the first two moments for almost all reference distributions including the uniform distribution. With increasing v, the choice of the reference distribution becomes more important. Universally useful results are obtained only with specific parameterizations of the binomial shape factor in conjunction with the information-theoretic method. At the same time more than two moments are usually required.

Recursive matrices and umbral calculus

Two major objectives can be seen to guide much recent work in enumeration: (1) to single out a limited variety of recurrences for numerical sequences which will encompass counting problems of wide-enough type; (2) to recover from empirical data an underlying set-theoretic structure which would reveal the source of the given recursion. We are here concerned with the first of these objectives, though the eventual understanding of the second is tacitly present, if only as a goal. We noticed the coincidence of several computations which, similar as they are in retrospect, had failed to realize their kinship. On leafing through the unique assembly of recursively solvable combinatorial problems in Comtet’s and Sloane’s invaluable collections, one is struck by the repeated occurrence of one and the same kind of double recursion. More strikingly, the same recursion is seen to occur in the polynomial sequences of the Umbra1 Calculus of Roman and Rota (see [25]). Everywhere, the Lagrange inversion formula for power series plays a pivotal role. Much work is nowadays going into the unraveling of the everdeeper layers of combinatorial significance of this formula, both in the ordinary case and in its as yet partially worked out noncommutative and qanalogs (Andrews, Foata, Garsia, Gessel, Joni, Raney, Reiner, Schtitzenberger, to name but a few). Whatever their origins, the identities abutting Lagrange inversion are expressed by integers alone. This suggests not only a hidden set-theoretic layer, but a characteristic-free generalization as well: this generalization is the central theme of our work. We define a monoid of infinite matrices-“recursive matrices” for short. The entries of these matrices give the sought-out recursion, for example, that for coefficients of binomial and Sheffer polynomials and factor sequences, as well as that of the special sequences recently introduced by Roman in [26]. 546 002 l-8693/82/040546-28%02.00/‘0

Multiplication and Division by Binomial Factors

Multiplication and Division by Binomial Factors

High Common Factor

As the colleges now are requiring highest common factor by factoring methods only, any plan whereby the number of factors to be tried can be lessened is certainly worth while. For in general we must regard as possible any binomial factor whose first-degree term is an integral divisor of the highest term of the given expression and whose independent term is an integral divisor of the independent term of the expression. Thus in such a problem as, “Find the H.C.F. of 5x3 − 21x2 + 5x − 4 and 5x3 − 19x2 + 5x + 4” (McCurdy’s Exercise Book, page 40, example 4) the possible factors that a student might try and must try, unless he were very lucky in those he chose to try first, are x ± 1, x ± 2, x ± 4, 5x ± 1, 5x ± 2, 5x ± 4. And further if the given expressions were, say, cubics having no common binomial factor at all but with a quadratic one instead (e. g., 2x3 + 5x2 + x − 3 and 2x3 − x2 − 5x + 3) I doubt if the ordinary first-year student would get any result unless it were unity.

XXIV. On the calculus of symbols.—Third memoir

In my second Memoir “On the Calculus of Symbols,” I worked out the general case of multiplication according to one of the two systems of combination of non-commutative symbols previously given. In the present paper I propose to investigate the general case of multiplication according to the other system. I commence with the Binomial Theorem, to which the second system gives rise. In my previous researches I obtained the general term of the binomial theorem when the symbols combine according to the first system by equating symbolical coefficients; here, on the other hand, I consider the nature of the combinations which arise from the symbolical multiplication, and obtain the general term by summation. I next proceed to the multiplication of binomial factors. Here the general term is obtained by considering the alteration of weight undergone by certain symbols in the process of multiplication. The multinomial theorem according to the second system is next considered and its general term calculated. I conclude the memoir with some applications of the calculus of symbols to successive differentiation. This paper completes the investigation of symbolical multiplication and division according to the two systems of combination, the general case of division having been worked out by Mr. Spottiswoode in a very beautiful memoir recently published in the Transactions of this Society.

VII. On the calculus of symbols.—Third memoir

In my second Memoir “On the Calculus of Symbols,” I worked out the general case of multiplication according to one of the two systems of combination of non-commutative symbols previously given. In the present paper I propose to investigate the general case of multiplication according to the other system. I commence with the Binomial Theorem, to which the second system gives rise. In my previous researches I obtained the general term of the binomial theorem when the symbols combine according to the first system by equating symbolical coefficients; here, on the other hand, I consider the nature of the combinations which arise from the symbolical multiplication, and obtain the general term by summation. I next proceed to the multiplication of binomial factors.

XIII. On the calculus of symbols. — Second memoir

This Memoir is a continuation of one on the Calculus of Symbols which I had the honour to lay before the Society in December 1860, and which has since been published in the Philosophical Transactions. I commence this paper with some extensions of the method given in the former memoir for resolving functions of non-commutative symbols into binomial factors. I then explain a method, analogous to the process for extracting the square root in ordinary algebra, for resolving such functions into equal factors. I next investigate a process for finding the highest common internal divisor of two functions of non-commutative symbols, or, in other words, of finding if two linear differential equations admit of a common solution. After this, I give a rule for multiplying linear factors of non-commutative symbols, analogous to the ordinary algebraical rule for linear algebraical factors. I then resume the consideration of the binomial theorem explained in the former memoir. Two new forms of this binomial theorem are here given; and the method by which these forms are proved identical will, I hope, be considered an interesting portion of symbolical algebra, and as exhibiting in a remarkable manner its peculiar nature. In the next place, I proceed to calculate the general values of the coefficients which occur in the form of the binomial theorem given in the first memoir; I then obtain an expression for the symbolical coefficient of the general term of the multinomial theorem as previously explained; and also a theorem for the multiplication of symbolical factors emanating from each other after a given law; lastly, I investigate a binomial theorem reciprocal to the binomial theorem already considered.

I. On the calculus of symbols.—Second memoir

This memoir is the continuation of one on the calculus of symbols which I had the honour to lay before the Society in December 1860, and which has since been published in the ‘Philosophical Transactions.’ I commence this paper with some extensions of the method given in the former memoir for resolving functions of non-commutative symbols into binomial factors. I then explain a method, analogous to the process for extracting the square root in ordinary algebra, for resolving such functions into equal factors.