Finite Projective Geometries and Classification of the Weight Hierarchies of Codes （Ⅰ）

The weight hierarchy of a binary linear [n,κ] code C is the sequence (d1,d2,...,dκ), where dr is the smallest support of an r-dimensional subcode of C. The codes of dimension 4 are collected in classes and the possible weight hierarchies in each class is determined by finite projective geometries.The possible weight hierarchies in class A, B, C, D are determined in Part (Ⅰ). The possible weight hierarchies in class E, F, G, H, I are determined in Part (Ⅱ).     

An Assmus–Mattson-Type Approach for Identifying 3-Designs from Linear Codes over Z 4

The complete weight enumerator of the Delsarte–Goethals code over Z ₄ is derived and an Assmus–Mattson-type approach at identifying t-designs in linear codes over Z ₄ is presented. The Assmus–Mattson-type approach, in conjunction with the complete weight enumerator are together used to show that the codewords of constant Hamming weight in both the Goethals code over Z ₄ as well as the Delsarte–Goethals code over Z ₄ yield 3-designs, possibly with repeated blocks.     

A Lower Bound on the Greedy Weights of Product Codes

A greedy 1-subcode is a one-dimensional subcode of minimum (support) weight. A greedy r-subcode is an r-dimensional subcode with minimum support weight under the constraint that it contain a greedy (r - 1)-subcode. The r-th greedy weight e _r is the support weight of a greedy r-subcode. The greedy weights are related to the weight hierarchy. We use recent results on the weight hierarchy of product codes to develop a lower bound on the greedy weights of product codes.     

The Polynomial Behavior of Weight Multiplicities for the Affine Kac–Moody Algebras A(1)r

We prove that the multiplicity of an arbitrary dominant weight for an irreducible highest weight representation of the affine Kac–Moody algebra A ⁽¹⁾ _r is a polynomial in the rank r. In the process we show that the degree of this polynomial is less than or equal to the depth of the weight with respect to the highest weight. These results allow weight multiplicity information for small ranks to be transferred to arbitrary ranks.     

Hyperspaces of Normed Linear Spaces with the Attouch–Wets Topology

Let Cld _AW (X) be the hyperspace of nonempty closed subsets of a normed linear space X with the Attouch–Wets topology. It is shown that the space Cld _AW (X) and its various subspaces are AR's. Moreover, if X is an infinite-dimensional Banach space with weight w(X) then Cld _AW (X) is homeomorphic to a Hilbert space with weight 2 ^w(X).     

The Six-Vertex Model at Roots of Unity and some Highest Weight Representations of the sl 2 Loop Algebra

We discuss irreducible highest weight representations of the sl₂ loop algebra and reducible indecomposable ones in association with the sl₂ loop algebra symmetry of the six-vertex model at roots of unity. We formulate an elementary proof that every highest weight representation with distinct evaluation parameters is irreducible. We present a general criteria for a highest weight representation to be irreducible. We also give an example of a reducible indecomposable highest weight representation and discuss its dimensionality. Communicated by Vincent Rivasseau Dedicated to Daniel Arnaudon Submitted: March 3, 2006; Accepted: March 13, 2006     

Function Spaces with Exponential Weights I

In this paper we define weighted function spaces of type B^s_pq(u) and F^s_pq(u) on the Euclidean space IRⁿ, where u is a weight function of at most exponential growth. In particular, u(x) = exp(±|χ|) is an admissible weight. We prove some basic properties of these spaces, such as completeness and density of the smooth functions.     

An elliptic boundary problem in a half–space

The spectral theory for general non–selfadjoint elliptic boundary problems involving a discontinuous weight function has been well developed under certain restrictions concerning the weight function. In the course of extending the results so far established to a more general weight function, there arises the problem of establishing, in an L_p Sobolev space setting, the existence of and a priori estimates for solutions for a boundary problem for the half–space ?ⁿ₊ involving a weight function which vanishes at the boundary x_n = 0. In this paper we resolve this problem.     

A canonical subspace of modular forms of half-integral weight

We characterise the space of newforms of weight k + 1/2 on Γ₀(4N), N odd and square-free (studied by the second and third authors with Vasudevan) under the Atkin-Lehner W(4) operator. As an application, we show that the (±1)-eigensubspaces of the W(4) operator on the space of modular forms of weight k + 1/2 on Γ₀(4N) is mapped to modular forms of weight 2k on Γ₀(N), under a class of Shimura maps. The existence of such subspaces having this mapping property was conjectured by Zagier in a private communication. One of the special features of the (±1)-eigensubspaces is that the (2k + 1)-th power of the classical theta series of weight 1/2 belongs to the +  eigensubspace and hence this gives interesting congruences for r _2k+1(p ²).     

A P‐Completeness Result for Visibility Graphs of Simple Polygons

For each vertex of a simple polygon P an integer valued weight is given. We consider the path p₁, p₂, ..., p_k in P which is created according to the following strategy: p₁ is a designated start vertex s and p_i+1 is obtained by choosing the vertex with smallest weight among all vertices visible from p_i and different from p₁, p₂, ..., p_i. If there is no such vertex the path is finished. This path is called geometric lexicographic dead end path. We shall prove the problem of determining whether a distinguished vertex t of P is on the geometric lexicographic dead end path or not to be P‐complete.     

Optimal doubly constant weight codes

A doubly constant weight code is a binary code of length n₁ + n₂, with constant weight w₁ + w₂, such that the weight of a codeword in the first n₁ coordinates is w₁. Such codes have applications in obtaining bounds on the sizes of constant weight codes with given minimum distance. Lower and upper bounds on the sizes of such codes are derived. In particular, we show tight connections between optimal codes and some known designs such as Howell designs, Kirkman squares, orthogonal arrays, Steiner systems, and large sets of Steiner systems. These optimal codes are natural generalization of Steiner systems and they are also called doubly Steiner systems. © 2007 Wiley Periodicals, Inc. J Combin Designs 16: 137–151, 2008     

Critical values of L-functions on GSp 2 × Gl 2

This paper deals with Deligne's conjecture on the critical values of L-functions. Let Z _{G ⊗ h} (s) denote the tensor product L-function attached to a Siegel modular form G of weight k and an elliptic cusp form h of weight l. We assume that the first Fourier-Jacobi coefficient of G is not identically zero. Then Deligne's conjecture is fully proven for Z _{G ⊗ h} (s), when l≤2k−2 and partly for the remaining case. Supported by the von Neumann Fund at the Institute for Advanced Study, Princeton     

Representations of algebraic groups of type <Emphasis Type="Italic">C</Emphasis><Subscript><Emphasis Type="Italic">n</Emphasis></Subscript> with small weight multiplicities

Lower estimates for the maximal weight multiplicities in irreducible representations of the algebraic groups of type C _n in characteristic p ≤ 7 are found. If n ≥ 8 and p ≠ 2 , then for an irreducible representation either such a multiplicity is at least n− 4 − [n]₄,where [n]₄ is the residue of n modulo 4, or all the weight multiplicities are equal to 1.For p = 2, the situation is more complicated, and for every n and l there exists a class of representations with the maximal weight multiplicity equal to 2 ^l . For symplectic groups in characteristic p > 7 and spinor groups similar results were obtained earlier. Bibliography: 15 titles.     

A basis for the symplectic group branching algebra

The symplectic group branching algebra, B\mathcal {B}, is a graded algebra whose components encode the multiplicities of irreducible representations of Sp_2n−2(ℂ) in each finite-dimensional irreducible representation of Sp_2n (ℂ). By describing on B\mathcal {B} an ASL structure, we construct an explicit standard monomial basis of B\mathcal {B} consisting of Sp_2n−2(ℂ) highest weight vectors. Moreover, B\mathcal {B} is known to carry a canonical action of the n-fold product SL₂×⋯×SL₂, and we show that the standard monomial basis is the unique (up to scalar) weight basis associated to this representation. Finally, using the theory of Hibi algebras we describe a deformation of Spec(B)\mathrm{Spec}(\mathcal {B}) into an explicitly described toric variety.     

The Weight Distribution of C 5(1, n)

In [2] the codes C _q (r,n) over were introduced. These linear codes have parameters , can be viewed as analogues of the binary Reed-Muller codes and share several features in common with them. In [2], the weight distribution of C ₃(1,n) is completely determined.In this paper we compute the weight distribution of C ₅(1,n). To this end we transform a sum of a product of two binomial coefficients into an expression involving only exponentials an Lucas numbers. We prove an effective result on the set of Lucas numbers which are powers of two to arrive to the complete determination of the weight distribution of C ₅(1,n). The final result is stated as Theorem 2.     

Global italian domination in graphs

Abstract

An Italian dominating function (IDF) on a graph G = (V, E) is a function f: V → {0, 1, 2} satisfying the condition that for every vertex v ∈ V (G) with f (v) = 0, either v is adjacent to a vertex assigned 2 under f, or v is adjacent to at least two vertices assigned 1. The weight of an IDF f is the value ∑_v∈V(G) f (v). The Italian domination number of a graph G, denoted by γ_I (G), is the minimum weight of an IDF on G. An IDF f on G is called a global Italian dominating function (GIDF) on G if f is also an IDF on the complement ? of G. The global Italian domination number of G, denoted by γ_gI (G), is the minimum weight of a GIDF on G. In this paper, we initiate the study of the global Italian domination number and we present some strict bounds for the global Italian domination number. In particular, we prove that for any tree T of order n ≥ 4, γ_gI (T) ≤ γ_I (T) + 2 and we characterize all trees with γ_gI (T) = γ_I (T) + 2 and γ_gI (T) = γ_I (T) + 1.     

Modular representations of classical groups with small weight multiplicities

Lower estimates for the maximal weight multiplicities in irreducible representations of the algebraic groups of types B _n , C _n , and D _n in positive characteristic p are found under some minor restrictions on p. If G = B _n (K), C _n (K), or D _n (K), n ≥ 8, p > 2 for types B _n and D _n and p > 7 for type C _n , then either the maximal weight multiplicity for an irreducible representation of G is at least n − 7 or all its weight multiplicities are equal to 1. Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 60, Algebra, 2008.     

Weight Distribution of the Bases of a Matroid

Given a weighted graph, let w₁, w₂, . . . ,w_n denote the increasing sequence of all possible distinct spanning tree weights. In 1992, Mayr and Plaxton proved the following conjecture proposed by Kano: every spanning tree of weight w₁ is at most k−1 edge swaps away from some spanning tree of weight w_k. In this paper, we extend this result for matroids. We also prove that all the four conjectures due to Kano hold for matroids provided one partitions the bases of a matroid by the weight distribution of its elements instead of their weight. The author was partially supported by CNPq (Grant No. 302195/02-5) and ProNEx/CNPq (Grant No. 664107/97-4)     

WEIGHT PROPERTY FOR IDEALS OF U q (sl(2))

Let k be an arbitrary field of characteristic zero, and U be the quantized enveloping algebra U _q (sl(2)) over k. The aim of this present paper is to study the ideals of U at q not a root of unity. It turns out that every non-zero ideal of U can be generated by at most two highest weight vectors under the adjoint action, and by a sum of two highest weight vectors. This weight property make it possible to give a complete list of all prime (primitive, maximal) ideals of U according to their generators.     

Branch-and-price-and-cut on the clique partitioning problem with minimum clique size requirement

Given a complete graph K_n=(V,E)with edge weight c_e on each edge, we consider the problem of partitioning the vertices of graph K_n into subcliques that have at least S vertices, so as to minimize the total weight of the edges that have both endpoints in the same subclique. In this paper, we consider using the branch-and-price method to solve the problem. We demonstrate the necessity of cutting planes for this problem and suggest effective ways of adding cutting planes in the branch-and-price framework. The NP hard pricing problem is solved as an integer programming problem. We present computational results on large randomly generated problems.