Cyclotomic factors of the descent set polynomial

We introduce the notion of the descent set polynomial as an alternative way of encoding the sizes of descent classes of permutations. Descent set polynomials exhibit interesting factorization patterns. We explore the question of when particular cyclotomic factors divide these polynomials. As an instance we deduce that the proportion of odd entries in the descent set statistics in the symmetric group Sn only depends on the number on 1's in the binary expansion of n. We observe similar properties for the signed descent set statistics.     

On continuous descent functions for polynomial equations

In his recent book, Henrici (1974) gave an axiomatic treatment of the method of descent applied to the solution of polynomial equations, dealing in particular with the non-existence of continuous descent functions defined on the whole complex plane. This note presents an alternative account of this question, in which a somewhat stronger theorem is proved. At the same time, a certain problematical step, to which Henrici himself drew attention, is avoided.

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Zusammenfassung Henrici (1974) gibt in einem kürzlich erschienenen Buch eine axiomatische Behandlung der Absteigungsmethode zur Lösung von Polynomialgleichungen. Dort wird insbesondere die Nichtexistenz von stetigen Absteigungsfunktionen, die auf der ganzen komplexen Ebene definiert sind, behandelt. In dieser Arbeit wird das gleiche Problem von einem anderen Standpunkt aus betrachtet, und es wird ein etwas stärkerer Satz bewiesen. Dabei wird eine kleine Schwierigkeit vermieden, auf die Henrici selber aufmerksam gemacht hat.

     

A point set puzzle revisited

Two discrete point sets in Rⁿ are said to be homometric if their difference sets coincide. Homometric point sets were first studied in the 1930s in connection with the interpretation of x-ray diffraction patterns; today they appear in many contexts. Open questions still abound, even for point sets on the line. Under what conditions does a difference set S−S characterize S uniquely? If it does not, how can we find all the sets S_i,i=1,…, that give rise to it, and how are these sets related?     

Cycle type and descent set in wreath products

We express the number of elements of the hyperoctahedral group B_n, which have descent set K and such that their inverses have descent set J, as a scalar product of two representations of B_n. We also give the number of elements of B_n, which have a prescribed descent set and which are in a given conjugacy class of B_n by another scalar product of representations of B_n. For this, we first establish corresponding results for certain wreath products. We finally give, by generating series of symmetric functions, some analogs of the classical formulas which express the exponential generating series of alternating elements in the B_n's.     

Trees, set compositions and the twisted descent algebra

We first show that increasing trees are in bijection with set compositions, extending simultaneously a recent result on trees due to Tonks and a classical result on increasing binary trees. We then consider algebraic structures on the linear span of set compositions (the twisted descent algebra). Among others, a number of enveloping algebra structures are introduced and studied in detail. For example, it is shown that the linear span of trees carries an enveloping algebra structure and embeds as such in an enveloping algebra of increasing trees. All our constructions arise naturally from the general theory of twisted Hopf algebras.     

The polynomial hull of a set of finite linear measure in Cn

Journal d'Analyse Mathématique -     

The linking set problem: a polynomial special case of the multiple-choice knapsack problem

We consider the linking set problem, which can be seen as a particular case of the multiple-choice knapsack problem. This problem occurs as a subproblem in a decomposition procedure for solving large-scale p-median problems such as the optimal diversity management problem. We show that if a non-increasing diference property of the costs in the linking set problem holds, then the problem can be solved by a greedy algorithm and the corresponding linear gap is null.     

A nonlinear descent method for a variational inequality on a nonconvex set

In this paper, we consider a generalized variational inequality problem which involves the integrable cost mapping and a nonsmooth mapping with convex components. We propose a new gradient-type method which determines a stepsize by using the smooth part of the cost function. Thus, the method does not utilize analogs of derivatives of nonsmooth functions. We show that its convergence does not require additional assumptions.     

The componentwise descent method

The problem of minimizing a function of several variables on a parallelepiped is considered. A method of componentwise descent, not requiring the computation of derivatives of the function being minimized, is suggested for its solution. Its convergence for smooth convex functions is proved.Translated from Matematicheskie Zametki, Vol. 22, No. 1, pp. 137–142, July, 1977.The author thanks F. P. Vasil'ev for assistance with the article.