Binary bases of spaces of continuous functions

This paper addresses the question of density of binomials in C[a,b]. This seemingly innocuous question turns out to be equivalent in certain instances to the moment problem.     

Crystal graphs for general linear Lie superalgebras and quasi-symmetric functions

We give a new representation theoretic interpretation of the ring of quasi-symmetric functions. This is obtained by showing that the super analogue of Gessel's fundamental quasi-symmetric function can be realized as the character of a connected crystal for the Lie superalgebra associated to its non-standard Borel subalgebra with a maximal number of odd isotropic simple roots. We also present an algebraic characterization of these super quasi-symmetric functions.     

Free quasi-symmetric functions and descent algebras for wreath products, and noncommutative multi-symmetric functions

We introduce analogs of the Hopf algebra of Free quasi-symmetric functions with bases labeled by colored permutations. When the color set is a semigroup, an internal product can be introduced. This leads to the construction of generalized descent algebras associated with wreath products Γ?S_n and to the corresponding generalizations of quasi-symmetric functions. The associated Hopf algebras appear as natural analogs of McMahon’s multisymmetric functions. As a consequence, we obtain an internal product on ordinary multi-symmetric functions. We extend these constructions to Hopf algebras of colored parking functions, colored non-crossing partitions and parking functions of type B.     

Inversion of some series of free quasi-symmetric functions

We give a combinatorial formula for the inverses of the alternating sums of free quasi-symmetric functions of the form where I runs over compositions with parts in a prescribed set C. This proves in particular three special cases (no restriction, even parts, and all parts equal to 2) which were conjectured by B.C.V. Ung in [B.C.V. Ung, Combinatorial identities for series of quasi-symmetric functions, in: Proc. FPSAC’08, Toronto, 2008].     

Quasi-residual quasi-symmetric designs

It is shown that for each λ ? 3, there are only finitely many quasi-residual quasi-symmetric (QRQS) designs and that for each pair of intersection numbers (x, y) not equal to (0, 1) or (1, 2), there are only finitely many QRQS designs.A design is shown to be affine if and only if it is QRQS with x = 0. A projective design is defined as a symmetric design which has an affine residual. For a projective design, the block-derived design and the dual of the point-derivate of the residual are multiples of symmetric designs.     

On quasi-symmetric designs

In this paper we show that Lander's coding-theoretic proof of (parts of) the Bruck-Ryser-Chowla Theorem can be suitably modified to obtain analogous number theoretic restrictions on the parameters of quasi-symmetric designs. These results may be thought of as extensions to odd primes of the recent binary nonexistence results due to Calderbank et al. The results in this paper kill infinitely many feasible parameters for quasi-symmetric designs.     

Inequalities and bounds for quasi-symmetric 3-designs

Quasi-symmetric 3-designs with block intersection numbers x and y(0x<y<k) are studied, several inequalities satisfied by the parameters of a quasi-symmetric 3-designs are obtained. Let D be a quasi-symmetric 3-design with the block size k and intersection numbers x, y; y>x1 and suppose D′ denote the complement of D with the block size k′ and intersection numbers x′ and y′. If k −1 x + y then it is proved that x′ + y′ k′. Using this it is shown that the quasi-symmetric 3-designs corresponding to y = x + 1, x + 2 are either extensions of symmetric designs or designs corresponding to the Witt-design (or trivial design, i.e., v = k + 2) or the complement of above designs.     

On a class of bases for boolean functions

We prove that up to congruence there exist exactly fourty-four primitive bases of Boolean functions. We also apply our results in order to improve an algorithm of finding the maximal strong depth of a Boolean function. Bibliography: 4 titles. Dedicated to Yuri Matiyasevich on the occasion of his 60th birthday Published in Zapiski Nauchnykh Seminarov POMI, Vol. 358, 2008, pp. 271–281.     

Randomization time for the overhand shuffle

This paper analyzes repeated shuffling of a deck ofN cards. The measure studied is a model for the popularoverhand shuffle introduced by Aldous and Diaconis. It is shown that convergence to the uniform distribution requires at least orderN ² shuffles, and that orderN ² log(N) shuffles suffice. For a 52-card deck, more than 1000 shuffles are needed.     

ContributionsFinitary bases and formal generating functions

Let k be a nonzero, commutative ring with 1, and let R be a k-algebra with a countably-infinite ordered free k-basis B = [p_n: n 0]. We characterize and analyze those bases from which one can construct a k-algebra of ‘formal B-series’ of the form f=∑c_n p_n, with c_n ε k, showing inter alia that many classical polynomial bases fail to have this property.     

Finitary bases and formal generating functions

Let k be a nonzero, commutative ring with 1, and let R be a k-algebra with a countably-infinite ordered free k-basis B = [p_n: n 0]. We characterize and analyze those bases from which one can construct a k-algebra of ‘formal B-series’ of the form f=∑c_np_n, with c_n ε k, showing inter alia that many classical polynomial bases fail to have this property.     

Vibration analysis of quasi-symmetric structures

An efficient computational procedure is presented for the free vibration analysis of structures with unsymmetric geometry. The procedure is based on approximating the unsymmetric vibrational response of the structure by a linear combination of a few symmetric and antisymmetric modes (global approximation vectors), each obtained using approximately half the degrees of freedom of the original model. The three key elements of the procedure are: (a) use of mixed finite element models having independent shape functions for the internal forces (stress resultants) and generalized displacements, with the internal forces allowed to be discontinuous at interelement boundaries, (b) operator splitting, or additive decomposition of the different arrays in the governing finite element equations to delineate the contributions to the symmetric and antisymmetric response vectors, and (c) use of a reduction method through successive application of the finite element method and the classical Bubnov-Galerkin technique. The finite element method is first used to generate a few symmetric and antisymmetric global approximation response vectors. Then, the classical Bubnov-Galerkin technique is used to substantially reduce the size of the eigenvalue problem.

An initial set of global approximation vectors is selected to be a few symmetric and antisymmetric eigenvectors, and their various-order derivatives with respect to a tracing parameter identifying all the correction terms to the symmetric (and antisymmetric) eigenvectors. A modified (improved) set of approximation vectors is obtained by using the inverse iteration procedure. The effectiveness of the proposed procedure is demonstrated by means of a numerical example.     

Binary set functions and parity check matrices

We consider the possibility of extending to a family of sets a binary set function defined on a subfamily so that the extension is, in fact, uniquely determined. We place in this context the problem of finding the least integer n(r) such that every linear code of length n with n n(r), dimension n-r and minimum Hamming distance at least 4 has a parity check matrix composed entirely of odd weight columns and answer this problem by showing that n(r) = 5.2^{r − 4} + 1, r4. This result is applied to yield new constructions and bounds for unequal error protection codes with minimum distances 3 and 4.     

Uniform approximation of continuous functions by probability functions of Boolean bases

Random Boolean expressions obtained by random and independent substitution of the constants 1, 0 with probabilities p, 1 ? p, respectively, into random non-iterated formulas over a given basis are considered. The limit of the probability of appearance of expressions with the value 1 under unrestricted growth of the complexity of expressions, which is called the probability function, is considered. It is shown that for an arbitrary continuous function f(p) mapping the segment [0, 1] into itself there exists a sequence of bases whose probability functions uniformly approximate the function f(p) on the segment [0, 1].     

The generalized faro shuffle

The standard faro shuffle, an idealized riffle shuffle, divides the deck into two equal portions, and perfectly interlaces them. The simple cut takes one card from the top to the bottom of the deck. It is known that for decks of even size, the faro shuffle and simple cut generate all possible permutations, while if the deck is of odu size, only a small fraction are available. This paper considers a generalized faro shuffle wherein the deck is divided into n rather than 2, portions and these portions are “interlaced” together. It is shown that the generalized faro shuffle and the simple cut generate either the symmetric group of the deck, the alternating group of the deck, or in one special case, only a small fraction of the possible permutations. Whether the symmetric group or alternating group is generated depends on the parity of the simple cut and the generalized faro shuffle as group operations.     

Binary search tree recursions with harmonic toll functions

We study recursions that are traditional in the context of binary search trees and quicksort with the (non-standard) toll function H_n.