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.     

Defining and demonstrating an equivalence way of thinking in enumerative combinatorics

Counting problems offer opportunities for rich mathematical thinking, and yet there is evidence that students struggle to solve counting problems correctly. There is a need to identify useful approaches and thought processes that can help students be successful in their combinatorial activity. In this paper, we propose a characterization of an equivalence way of thinking, we discuss examples of how it arises mathematically in a variety of combinatorial concepts, and we offer episodes from a paired teaching experiment with undergraduate students that demonstrate useful ways in which students developed and leverage this way of thinking. Ultimately, we argue that this way of thinking can apply to a variety of combinatorial situations, and we make the case that it is a valuable way of thinking that should be prioritized for students learning combinatorics.     

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.     

Galois hulls of special Goppa codes and related codes with application to EAQECCs

Galois hulls of MDS codes can be applied to construst MDS entanglement-assisted quantum error-correcting codes (EAQECCs). Goppa codes and expurgated Goppa codes (resp., extended Goppa codes) over ${\mathbb{F}}_{q^m}$ are GRS codes (resp., extended GRS codes) when $m=1$. In this paper, we investigate the Galois dual codes of a special kind of Goppa codes and related codes and provide a necessary and sufficient condition for the Galois dual codes of such codes to be Goppa codes and related codes. Then we determine the Galois hulls of the above codes. In particular, we completely characterize Galois LCD, Galois self-orthogonal, Galois dual-containing and Galois self-dual codes among such family of codes. Moreover, we apply the above results to EAQECCs.     

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.     

Loss of regularity for the solutions to hyperbolic equations with non‐regular coefficients—an application to Kirchhoff equation

We consider the Cauchy problem for second‐order strictly hyperbolic equations with time‐depending non‐regular coefficients. There is a possibility that singular coefficients make a regularity loss for the solution. The main purpose of this paper is to derive an optimal singularity for the coefficient that the Cauchy problem is C^∞ well‐posed. Moreover, we will apply such a result to the estimate of the existence time of the solution for Kirchhoff equation. Copyright © 2003 John Wiley & Sons, Ltd.