Extraneous multipliers of cyclic difference sets

Mann (Canad. J. Math. (1952), 222–226) has proved that 2 is a multiplier for a cyclic difference set only if 2 | n = k − λ, and 3 is a multiplier for a simple cyclic difference set only if 3 | n = k − λ = k − 1, provided that these cyclic difference sets are not trivial. In this paper, we prove that 3 is a multiplier for a nontrivial cyclic difference set only if 3 | n = k − λ for arbitrary λ, and 5 is a multiplier for a non-trivial simple cyclic difference set only if 5 | n = k − λ = k − 1, as well as a necessary and sufficient condition for a prime to be an extraneous multiplier—multipliers not dividing n = k − λ.     

Embedding up to homotopy type— The first obstruction

For a 2n?m connected map from an n-dimensional complex to a m-dimensional manifold, an obstruction to embedding up to homotopy type is defined. The vanishing of this obstruction is a necessary and sufficient condition (in the 2n?m connected case, 2n?m ? 2, m?n ?3) to obtain an embedding up to homotopy type. In case the target manifold is Euclidean space, it is shown that the obstruction vanishes if and only if certain Thom operations are trivial. A classification theorem is given in the 2n?m+1 connected case.     

Further gossip problems

n people have distinct bits of information. They can communicate via k-party conference calls. How many such calls are needed to inform everyone of everyone else's information? Let f(n,k) be this minimum number. Then we give a simple proof that f(n,k)= [(n?k)(k?1)]+[nk] for 1?n?k², f(n,k)=2[(n?k)(k?1)] for n>k².In the 2-party case we consider the case in which certain of the calls may permit information flow in only one direction. We show that any 2n-4 call scheme that conveys everone's information to all must contain a 4-cycle, each of whose calls is “two way”, along with some other results. The method follows that of Bumby who first proved the 4-cycle conjecture.     

Bounding the number of zeros of certain Abelian integrals

In this paper we prove a criterion that provides an easy sufficient condition in order for any nontrivial linear combination of n Abelian integrals to have at most n+k−1 zeros counted with multiplicities. This condition involves the functions in the integrand of the Abelian integrals and it can be checked, in many cases, in a purely algebraic way.     

Robustness and surgery of frames

We characterize frames in Rⁿ that are robust to k erasures. The characterization is given in terms of the support of the null space of the synthesis operator of the frame. A necessary and sufficient condition is given for when an (r, k)-surgery on unit-norm tight frames in R² are possible. Also a generalization of a known characterization of the existence of tight frames with prescribed norms is given.     

Cohomology rings and formality properties of nilpotent groups

We analyze k-stage formality and relate resonance with this type of formality properties. For instance, we show that, for a finitely generated nilpotent group that is k-stage formal, the resonance varieties are trivial up to degree k. We also show that the cohomology ring, truncated up to degree k+1, of a finitely generated nilpotent, k-stage formal group is generated in degree 1; this criterion is necessary and sufficient for a finitely generated, 2-step nilpotent group to be k-stage formal. We compute resonance varieties for Heisenberg-type groups and deduce the degree of partial formality for this class of groups.     

(k)-Characters and the triviality of symmetry classes

The author introduces the notion of (k)-character and its application to symmetry classes of tensors. A necessary and sufficient condition for the symmetry class V_λ(G) to be trivial in the case degree of G ? 2dim V has also been found.     

Latin squares and superqueens

Let L be a Latin square of order n with entries from {0, 1,…, n ? 1}. In addition, L is said to have the (n, k) property if, in each right or left wrap around diagonal, the number of cells with entries smaller than k is exactly k. It is established that a necessary and sufficient condition for the existence of Latin squares having the (n, k) property is that of (2|n ? 2| k) and (3|n ? 3| k). Also, these Latin squares are related to a problem of placing nonattacking queens on a toroidal chessboard.     

Unbordered factors and Lyndon words

A primitive word w is a Lyndon word if w is minimal among all its conjugates with respect to some lexicographic order. A word w is bordered if there is a nonempty word u such that w=uvu for some word v. A right extension of a word w of length n is a word wu where all factors longer than n are bordered. A right extension wu of w is called trivial if there exists a positive integer k such that w^k=uv for some word v.We prove that Lyndon words have only trivial right extensions. Moreover, we give a conjecture which characterizes a property of every word w which has a nontrivial right extension of length 2|w|-2.     

Conjugate connections and Radon's theorem in affine differential geometry

For a given nondegenerate hypersurfaceM ⁿ in affine space ? ⁿ⁺¹ there exist an affine connection ?, called the induced connection, and a nondegenerate metrich, called the affine metric, which are uniquely determined. The cubic formC=?h is totally symmetric and satisfies the so-called apolarity condition relative toh. A natural question is, conversely, given an affine connection ? and a nondegenerate metrich on a differentiable manifoldM ⁿ such that ?h is totally symmetric and satisfies the apolarity condition relative toh, canM ⁿ be locally immersed in ? ⁿ⁺¹ in such a way that (?,h) is realized as the induced structure? In 1918J. Radon gave a necessary and sufficient condition (somewhat complicated) for the problem in the casen=2. The purpose of the present paper is to give a necessary and sufficient condition for the problem in casesn=2 andn≥3 in terms of the curvature tensorR of the connection ?. We also provide another formulation valid for all dimensionsn: A necessary and sufficient condition for the realizability of (?,h) is that the conjugate connection of ? relative toh is projectively flat.     

NONEXISTENCE AND EXISTENCE OF POSITIVE SOLUTIONS FOR 2nTH-ORDER SINGULAR SUPERLINEAR PROBLEMS WITH STRUM-LIOUVILLE BOUNDARY CONDITIONS

This paper investigates a class of 2nth-order singular superlinear problems with Strum-Liouville boundary conditions. We obtain a necessary and sufficient condition for the existence of C 2 n- 2 [0, 1] positive solutions, and a sufficient condition, a necessary condition for the existence of C 2 n-1 [0, 1] positive solutions. Relations between the positive solutions and the Green’s functions are depicted. The results are used to judge nonexistence or existence of positive solutions for given boundary value problems.     

A class of facet producing graphs for vertex packing polyhedra

We examine a family of graphs called webs. For integers n ? 2 and $\text{k, 1 ? k ?}\text{1}\text{2}\text{n}$, the web W(n, k) has vertices V_n = {1, …, n} and edges {(i, j): j = i+k, …, i+n ? k, for i?V_n (sums mod n)}. A characterization is given for the vertex packing polyhedron of W(n, k) to contain a facet, none of whose projections is a facet for the lower dimensional vertex packing polyhedra of proper induced subgraphs of W(n, k). Simple necessary and sufficient conditions are given for W(n, k) to contain W(n′, k′) as an induced subgraph; these conditions are used to show that webs satisfy the Strong Perfect Graph Conjecture. Complements of webs are also studied and it is shown that if both a graph and its complement are webs, then the graph is either an odd hole or its complement.     

P 4k−1-factorization of bipartite multigraphs

Let λK _m,n be a bipartite multigraph with two partite sets having m and n vertices, respectively. A P _v-factorization of λK _m,n is a set of edge-disjoint P _v -factors of λK _m,n which partition the set of edges of λK _m,n. When v is an even number, Ushio, Wang and the second author of the paper gave a necessary and sufficient condition for the existence of a P _v -factorization of λK _m,n. When v is an odd number, we proposed a conjecture. However, up to now we only know that the conjecture is true for v = 3. In this paper we will show that the conjecture is true when v = 4k ? 1. That is, we shall prove that a necessary and sufficient condition for the existence of a P _4k?1-factorization of λK _m,n is (1) (2k ? 1)m ? 2kn, (2) (2k ? 1)n ? 2km, (3) m + n ≡ 0 (mod 4k ? 1), (4) λ(4k ? 1)mn/[2(2k ? 1)(m + n)] is an integer.     

Trivial module for ortho-symplectic Lie superalgebras and Littlewood’s formula

We show that the denominator identity for ortho-symplectic Lie superalgebras osp(k|2n) is equivalent to the Littlewood’s formula. Such an equivalence also implies the relation between the trivial module and generalized Verma modules for osp(k|2n). Furthermore, we discuss the harmonic representative elements of the Kostant’s u-cohomology with trivial coefficients.     

Equality of norm groups of subextensions of Sn(n?5) extensions of algebraic number fields

Let E/k be a Galois extension of algebraic number fields with the Galois group isomorphic to the symmetric group S_n on n?5 letters. For any field extensions k⊆K, L⊆E a necessary and a sufficient condition is given for the equality to hold, where is the group of norms from K to k of the elements of the multiplicative group K∗ of K.     

On cyclic edge-connectivity of fullerenes

A graph is said to be cyclically k-edge-connected, if at least k edges must be removed to disconnect it into two components, each containing a cycle. Such a set of k edges is called a cyclic-k-edge cutset and it is called a trivial cyclic-k-edge cutset if at least one of the resulting two components induces a single k-cycle.It is known that fullerenes, that is, 3-connected cubic planar graphs all of whose faces are pentagons and hexagons, are cyclically 5-edge-connected. In this article it is shown that a fullerene F containing a nontrivial cyclic-5-edge cutset admits two antipodal pentacaps, that is, two antipodal pentagonal faces whose neighboring faces are also pentagonal. Moreover, it is shown that F has a Hamilton cycle, and as a consequence at least 15·2^n/20-1/2 perfect matchings, where n is the order of F.     

Interpolation by splines satisfying mixed boundary conditions

We consider interpolation of Hermite data by splines of degreen withk given knots, satisfying boundary conditions which may involve derivatives at both end points (e.g., a periodicity condition). It is shown that, for a certain class of boundary conditions, a necessary and sufficient condition for the existence of a unique solution is that the data points and knots interlace properly and that there does not exist a polynomial solution of degreen?k. The method of proof is to show that any spline interpolating zero data vanishes identically, rather than the usual determinantal approach.     

Reconstructing Matrices from Minors

Which collections of mn minors of an m-by-n matrix uniquely determine the matrix, given some regularity conditions? For m=n=3, the 585 such collections, that are distinct up to symmetry, are determined. For general m, n, a necessary and a sufficient condition for reconstruction are given in terms of matchings in a bipartite graph. Among other particular results, those collections of entries for which there are minors that permit reconstruction one entry at a time are characterized.     

KI,k-FACTORIZATION OF BIPARTITE GRAPHS

in this paper a necesssary condition for a bipartite graph λK,to be K1-factorizable and a sufficient condition for kk to have a k -factonrization whenever k is a prime number are given.     

Matroid representation of clique complexes

In this paper, we approach the quality of a greedy algorithm for the maximum weighted clique problem from the viewpoint of matroid theory. More precisely, we consider the clique complex of a graph (the collection of all cliques of the graph) which is also called a flag complex, and investigate the minimum number k such that the clique complex of a given graph can be represented as the intersection of k matroids. This number k can be regarded as a measure of “how complex a graph is with respect to the maximum weighted clique problem” since a greedy algorithm is a k-approximation algorithm for this problem. For any k>0, we characterize graphs whose clique complexes can be represented as the intersection of k matroids. As a consequence, we can see that the class of clique complexes is the same as the class of the intersections of partition matroids. Moreover, we determine how many matroids are necessary and sufficient for the representation of all graphs with n vertices. This number turns out to be n-1. Other related investigations are also given.