Schubert varieties, linear codes and enumerative combinatorics

We consider linear error correcting codes associated to higher-dimensional projective varieties defined over a finite field. The problem of determining the basic parameters of such codes often leads to some interesting and difficult questions in combinatorics and algebraic geometry. This is illustrated by codes associated to Schubert varieties in Grassmannians, called Schubert codes, which have recently been studied. The basic parameters such as the length, dimension and minimum distance of these codes are known only in special cases. An upper bound for the minimum distance is known and it is conjectured that this bound is achieved. We give explicit formulae for the length and dimension of arbitrary Schubert codes and prove the minimum distance conjecture in the affirmative for codes associated to Schubert divisors.     

Nondecomposable solutions to group equations and an application to polyhedral combinatorics

This paper is based on the study of the set of nondecomposable integer solutions in a Gomory corner polyhedron, which was recently used in a reformulation method for integer linear programs. In this paper, we present an algorithm for efficiently computing this set. We precompute a database of nondecomposable solutions for cyclic groups up to order 52. As a second application of this database, we introduce an algorithm for computing nontrivial simultaneous lifting coefficients. The lifting coefficients are exact for a discrete relaxation of the integer program that consists of a group relaxation plus bound constraints. Received: November 2004 / Revised version: June 2005 AMS classification: 90C10, 90C57     

Existence varieties with lattices of regular subsemigroups

A class of regular semigroups is called an existence variety, ore-variety, if it is closed under taking homomorphic images, regular subsemigroups, and direct products. For a regular semigroupS, the set of all regular subsemigroups ofS forms a partially ordered set under set inclusion. We determine for whiche-varietiesV the set of regular subsemigroups of members ofV forms a lattice. This includes the known result that the regular subsemigroups of an orthodox semigroup form a lattice.Presented by R. Freese.     

Comma-free codes: An extension of certain enumerative techniques to recursively defined sequences

The number X_n of increasing subsequences of the n-long random permutation is studied. Asymptotics of the moments of X_n are found; In X_n is shown to grow, in probability, as $\text{a}\text{n}$, 2 ln 2 ? a ? 2.     

On congruence lattices of regular semigroups with Q-inverse transversals

Communicated by F. Pastijn     

A remark on partial linear spaces of girth 5 with an application to strongly regular graphs

We derive a lower bound on the number of points of a partial linear space of girth 5. As an application, certain strongly regular graphs with=2 are ruled out by observing that the first subconstituents are partial linear spaces.     

From extended regular variation to regular variation with application in extreme value statistics

This paper aims to provide a study of a variety of concepts involving power behavior of eventually positive functions which, falling under the umbrella of the Theory of Regular Variation and its second order refinements, are prone to application in Extreme Value Theory. To this extent, some well-known properties shall be resumed, others will be designed with the ultimate purpose of establishing a relation between regular variation and extended regular variation of second order. As a by-product, new ways of looking at some common estimators for the extreme value index, in particular the maximum likelihood estimator, will be unveiled.     

An application of probability to combinatorics: a proof of Vandermonde identity

In this paper, we give possible suggestions for a classroom lesson about an application of probability using basic mathematical notions. We will approach to some combinatoric results without using ‘induction’, ‘polynomial identities’ nor ‘generating functions’, and will give a proof of the ‘Vandermonde Identity’ using elementary notions of probability.     

On a conjecture of Martin on the parameters of completely regular codes and the classification of the completely regular codes in the biggs-smith graph

In this paper we study a conjecture of Martin [7] on the paramaters of completely regular codes in distance-regular graphs. We show that this conjecture is not true in general, but for most classical graphs it is true. we also show that there is a counterexample in the Biggs-Smith graph for a weakened version of Martin's conjecture. Furthermore we classify the completely regular codes in the Biggs-Smith graph.     

A class of negacyclic BCH codes and its application to quantum codes

In this paper, we study negacyclic BCH codes over \(\mathbb {F}_{q}\) of length \(n=(q^{2m}-1)/(q-1)\), where q is an odd prime power and m is a positive integer. In particular, the dimension, the minimum distance and the weight distribution of some negacyclic BCH codes over \(\mathbb {F}_{q}\) of length \(n=(q^{2m}-1)/(q-1)\) are determined. Two classes of negacyclic BCH codes meeting the Griesmer bound are obtained. As an application, we construct quantum codes with good parameters from this class of negacyclic BCH codes.     

Duality between prime factors and an application to the prime number theorem for arithmetic progressions

We study in this paper a new duality identity between large and small prime factors of integers and its relationship with the prime number theorem for arithmetic progressions. The asymptotic behavior of large prime factors of integers leads to interesting relations involving the Möbius function.     

Approximate extension of partial ε-characters of abelian groups to characters with application to integral point lattices

Let G be an abelian group, S ⊆ G be a finite set, and T denote the multiplicative group of complex unitswith the invariant arc metric | arg(a/b)|. We will show that for a mapping ? : S → T to be ε-close on S to a character φ : G → T it is enough that ? be extendable to a mapping ¯f : (S U {1} US⁻¹)ⁿ → T, where n is big enough and ¯f violates the homomorphy condition at most up to an arbitrary σ < min(ε, π/2). Moreover, n can be chosen uniformly, independently of G and both ? and ¯f, depending just on σ, ε and the number of elements of S. The proof is non-constructive, using the ultraproduct construction and Pontryagin duality, hence yielding no estimate on the actual size of n. As one of the applications we show that, for a vector u ∈ R ^q to be ε-close to some vector from the dual lattice H^★ of a full rank integral point lattice H ⊆ ?^q, it is enough for the scalar product ux to be δ-close (with δ < 1/3) to an integer for all vectors xH satisfying Σ_i|x_i | < n, where n depends on δ, ε and q only.     

Densities of lattices corresponding to spaces of positive, negative, and variational dimension, and their application to time series

We prove a general theorem concerning a distribution of Bose-Einstein type. Using this theorem, we apply the notions of lattice dimension and lattice density to oscillatory time series.