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.     

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.     

Catalan paths and quasi-symmetric functions

We investigate the quotient ring of the ring of formal power series over the closure of the ideal generated by non-constant quasi-symmetric functions. We show that a Hilbert basis of the quotient is naturally indexed by Catalan paths (infinite Dyck paths). We also give a filtration of ideals related to Catalan paths from and above the line . We investigate as well the quotient ring of polynomial ring in variables over the ideal generated by non-constant quasi-symmetric polynomials. We show that the dimension of is bounded above by the th Catalan number.

     

Binary shuffle bases for quasi-symmetric functions

We construct bases of quasi-symmetric functions whose product rule is given by the shuffle of binary words, as for multiple zeta values in their integral representations, and then extend the construction to the algebra of free quasi-symmetric functions colored by positive integers. As a consequence, we show that the fractions introduced in Guo and Xie (Ramanujan J 25:307–317, 2011) provide a realization of this algebra by rational moulds extending that of free quasi-symmetric functions given in Chapoton et al. (Int Math Res Not IMRN 2008, no. 9, Art. ID rnn018, 2008).     

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.     

Some characterizations of quasi-symmetric designs with a spread

The designPG ₂ (4,q) of the points and planes ofPG (4,q) forms a quasi-symmetric 2-design with block intersection numbersx=1 andy=q+1. We give some characterizations of quasi-symmetric designs withx=1 which have a spread through a fixed point. For instance, it is proved that if such a designD is also smooth, thenDPG ₂ (4,q).     

Non-existence of quasi-symmetric designs with restricted block graphs

Designs, Codes and Cryptography - Quasi-symmetric designs (QSDs) with particular block graphs are investigated. We rule out the possibility of a QSD with block graph that has the same parameters as...     

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].     

On quasi-symmetric designs with intersection difference three

In a recent paper, Pawale (Des Codes Cryptogr, 2010) investigated quasi-symmetric 2-(v, k, λ) designs with intersection numbers x > 0 and y = x + 2 with λ > 1 and showed that under these conditions either λ = x + 1 or λ = x + 2, or D{mathcal{D}} is a design with parameters given in the form of an explicit table, or the complement of one of these designs. In this paper, quasi-symmetric designs with y − x = 3 are investigated. It is shown that such a design or its complement has parameter set which is one of finitely many which are listed explicitly or λ ≤ x + 4 or 0 ≤ x ≤ 1 or the pair (λ, x) is one of (7, 2), (8, 2), (9, 2), (10, 2), (8, 3), (9, 3), (9, 4) and (10, 5). It is also shown that there are no triangle-free quasi-symmetric designs with positive intersection numbers x and y with y = x + 3.     

New classes of self-complementary codes and quasi-symmetric designs

We present a new technique for constructing binary error correcting codes and give some examples of codes that can be constructed via this method. Among the examples is an infinite family of self-complementary codes with parameters (2u ²−u, 8u ², u ²−u) that can be constructed whenever there exists a u × u Hadamard Matrix. These codes meet the Grey–Rankin bound and imply the existence of quasi-symmetric designs on 2u ²−u points.     

Bidendriform bialgebras, trees, and free quasi-symmetric functions

We introduce bidendriform bialgebras, which are bialgebras such that both product and coproduct can be split into two parts satisfying good compatibilities. For example, the Malvenuto-Reutenauer Hopf algebra and the non-commutative Connes-Kreimer Hopf algebras of planar decorated rooted trees are bidendriform bialgebras. We prove that all connected bidendriform bialgebras are generated by their primitive elements as a dendriform algebra (bidendriform Milnor-Moore theorem) and then is isomorphic to a Connes-Kreimer Hopf algebra. As a corollary, the Hopf algebra of Malvenuto-Reutenauer is isomorphic to the Connes-Kreimer Hopf algebra of planar rooted trees decorated by a certain set. We deduce that the Lie algebra of its primitive elements is free in characteristic zero (G. Duchamp, F. Hivert and J.-Y. Thibon conjecture).     

On a conjecture of jungnickel and tonchev for quasi-symmetric designs

Jungnickel and Tonchev conjectured in [4] that if a quasi-symmetric design D is an s-fold quasi-multiple of a symmetric (v,k,λ) design with (k,(s ? 1)λ) = 1, then D is a multiple. We prove this conjecture under any one of the conditions: s ≤ 7, k ? 1 is prime, or the design D is a 3-design. It is shown that for any fixed s, the conjecture is true with at most finitely many exceptions. The unique quasi-symmetric 3-(22,7,4) design is characterized as the only quasi-symmetric 3-design, which as a 2-design is an s-fold quasi-multiple with s ≡ 1 (mod k). © 1994 John Wiley & Sons, Inc.     

Polarities,quasi-symmetric designs,and Hamada’s conjecture

We prove that every polarity of PG(2k − 1,q), where k≥ 2, gives rise to a design with the same parameters and the same intersection numbers as, but not isomorphic to, PG _k (2k,q). In particular, the case k = 2 yields a new family of quasi-symmetric designs. We also show that our construction provides an infinite family of counterexamples to Hamada’s conjecture, for any field of prime order p. Previously, only a handful of counterexamples were known.      

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.     

The application of invariant theory to the existence of quasi-symmetric designs

Gleason and Mallows and Sloane characterized the weight enumerators of maximal self-orthogonal codes with all weights divisible by 4. We apply these results to obtain a new necessary condition for the existence of 2 − (v, k, λ) designs where the intersection numbers s₁…,s_n satisfy s₁ ≡ s₂ ≡ … ≡ s_n (mod 2). Non-existence of quasi-symmetric 2−(21, 18, 14), 2−(21, 9, 12), and 2−(35, 7, 3) designs follows directly from the theorem. We also eliminate quasi-symmetric 2−(33, 9, 6) designs. We prove that the blocks of quasi-symmetric 2−(19, 9, 16), 2−(20, 10, 18), 2-(20,8, 14), and 2−(22, 8, 12) designs are obtained from octads and dodecads in the [24, 12] Golay code. Finally we eliminate quasi-symmetric 2−(19,9, 16) and 2-(22, 8, 12) designs.     

Exponential number of quasi-symmetric SDP designs and codes meeting the Grey-Rankin bound

It is shown that quasi-symmetric designs which are derived or residual designs of nonisomorphic symmetric designs with the symmetric difference property are also nonisomorphic. Combined with a result by W. Kantor, this implies that the number of nonisomorphic quasi-symmetric designs with the symmetric difference property grows exponentially. The column spaces of the incidence matrices of these designs provide an exponential number of inequivalent codes meeting the Grey-Rankin bound. A transformation of quasi-symmetric designs by means of maximal arcs is described. In particular, a residual quasi-symmetric design with the symmetric difference property is transformed into a quasi-symmetric design with the same block graph but higher rank over GF(2).Dedicated to Professor Hanfried Lenz on the occasion of his 75th birthday.This paper was written while the author was at the University of Giessen as a Research Fellow of the Alexander von Humboldt Foundation, on leave from the University of Sofia, Bulgaria.     

Vibration analysis of cracked post-buckled beams

Vibration analysis of cracked post-buckled beam is investigated in this study. Crack, assumed to be open, is modeled by a massless rotational spring. The beam is divided into two segments and the governing nonlinear equations of motion for the post-buckled state are derived. The solution consists of static and dynamic parts, both leading to nonlinear differential equations. The differential quadrature has been used to solve the problem. First, it is applied to the equilibrium equations, leading to a nonlinear algebraic system of equations that will be solved utilizing an arc length strategy. Next, the differential quadrature is applied to the linearized dynamic differential equations of motion and their corresponding boundary and continuity conditions. Upon solution of the resulting eigenvalue problem, the natural frequencies and mode shapes of the cracked beam are extracted. Several experimental as well as numerical case studies are performed to demonstrate the effectiveness of the proposed method. The investigation also includes an examination of several parameters influencing the dynamic behavior of the problem. The results show that the position and size of the crack as well as the geometric imperfection and applied load largely affect the modal shapes and natural frequencies of the beam.     

A description via second degree character of a family of quasi-symmetric forms

Periodica Mathematica Hungarica - The purpose of this paper is to give, through the second degree character, new characterizations of a part of the family of quasi-symmetric forms. In fact, thanks...