Existence of HSOLSSOMs of type

In this paper we investigate the existence of holey self-orthogonal Latin squares with a symmetric orthogonal mate of type 2ⁿu¹ (HSOLSSOM(2ⁿu¹)). For u2, necessary conditions for existence of such an HSOLSSOM are that u must be even and n3u/2+1. Xu Yunqing and Hu Yuwang have shown that these HSOLSSOMs exist whenever either (1) n9 and n3u/2+1 or (2) n263 and n2(u-2). In this paper we show that in (1) the condition n9 can be extended to n30 and that in (2), the condition n263 can be improved to n4, except possibly for 19 pairs (n,u), the largest of which is (53,28).     

Steiner centers and Steiner medians of graphs

Let G be a connected graph and S a set of vertices of G. The Steiner distance of S is the smallest number of edges in a connected subgraph of G that contains S and is denoted by d_G(S) or d(S). The Steiner n-eccentricity e_n(v) and Steiner n-distance d_n(v) of a vertex v in G are defined as e_n(v)=max{d(S)| SV(G), |S|=n and vS} and d_n(v)=∑{d(S)| SV(G), |S|=n and vS}, respectively. The Steiner n-center C_n(G) of G is the subgraph induced by the vertices of minimum n-eccentricity. The Steiner n-median M_n(G) of G is the subgraph induced by those vertices with minimum Steiner n-distance. Let T be a tree. Oellermann and Tian [O.R. Oellermann, S. Tian, Steiner centers in graphs, J. Graph Theory 14 (1990) 585–597] showed that C_n(T) is contained in C_n+1(T) for all n2. Beineke et al. [L.W. Beineke, O.R. Oellermann, R.E. Pippert, On the Steiner median of a tree, Discrete Appl. Math. 68 (1996) 249–258] showed that M_n(T) is contained in M_n+1(T) for all n2. Then, Oellermann [O.R. Oellermann, On Steiner centers and Steiner medians of graphs, Networks 34 (1999) 258–263] asked whether these containment relationships hold for general graphs. In this note we show that for every n2 there is an infinite family of block graphs G for which C_n(G)C_n+1(G). We also show that for each n2 there is a distance–hereditary graph G such that M_n(G)M_n+1(G). Despite these negative examples, we prove that if G is a block graph then M_n(G) is contained in M_n+1(G) for all n2. Further, a linear time algorithm for finding the Steiner n-median of a block graph is presented and an efficient algorithm for finding the Steiner n-distances of all vertices in a block graph is described.     

Uniform Kazhdan constant for some families of linear groups

Let R be a ring generated by l elements with stable range r. Assume that the group EL_d(R) has Kazhdan constant ₀>0 for some dr+1. We prove that there exist (₀,l)>0 and , s.t. for every nd, EL_n(R) has a generating set of order k and a Kazhdan constant larger than . As a consequence, we obtain for where n3, a Kazhdan constant which is independent of n w.r.t. generating set of a fixed size.     

A fourth degree integration formula for the n-dimensional simplex

A fourth degree integration formula is given for the n-dimensional simplex for all n2, which is invariant under the group G of all affine transformations of T_n onto itself. The formula contains (n²+5n+6)/2 nodes.     

Perfect codes in direct products of cycles—a complete characterization

Let be a direct product of cycles. It is known that for any r1, and any n2, each connected component of G contains a so-called canonical r-perfect code provided that each ℓ_i is a multiple of rⁿ+(r+1)ⁿ. Here we prove that up to a reasonably defined equivalence, these are the only perfect codes that exist.     

Rational Approximation with Varying Weights III

Approximation by weighted rationals of the form wⁿr_n, where r_n=p_n/q_n, p_n and q_n are polynomials of degree at most [αn] and [βn], respectively, and w is an admissible weight, is investigated on compact subsets of the real line for a general class of weights and given α0, β0, with α+β>0. Conditions that characterize the largest sets on which such approximation is possible are given. We apply the general theorems to Laguerre and Freud weights.     

On the code generated by the incidence matrix of points and k-spaces in and its dual

In this paper, we study the p-ary linear code C_k(n,q), q=p^h, p prime, h1, generated by the incidence matrix of points and k-dimensional spaces in PG(n,q). For kn/2, we link codewords of C_k(n,q)C_k(n,q) of weight smaller than 2q^k to k-blocking sets. We first prove that such a k-blocking set is uniquely reducible to a minimal k-blocking set, and exclude all codewords arising from small linear k-blocking sets. For k<n/2, we present counterexamples to lemmas valid for kn/2. Next, we study the dual code of C_k(n,q) and present a lower bound on the weight of the codewords, hence extending the results of Sachar [H. Sachar, The F_p span of the incidence matrix of a finite projective plane, Geom. Dedicata 8 (1979) 407–415] to general dimension.     

Cycles through a prescribed vertex set in N-connected graphs

A well-known result of Dirac (Math. Nachr. 22 (1960) 61) says that given n vertices in an n-connected G, G has a cycle through all of them. In this paper, we generalize Dirac's result as follows:Given at most vertices in an n-connected graph G when n3 and , then G has a cycle through exactly n vertices of them.This improves the previous known bound given by Kaneko and Saito (J. Graph Theory 15(6) (1991) 655).     

Continued fractions with multiple limits

For integers m2, we study divergent continued fractions whose numerators and denominators in each of the m arithmetic progressions modulo m converge. Special cases give, among other things, an infinite sequence of divergence theorems, the first of which is the classical Stern–Stolz theorem.We give a theorem on a class of Poincaré-type recurrences which shows that they tend to limits when the limits are taken in residue classes and the roots of their characteristic polynomials are distinct roots of unity.We also generalize a curious q-continued fraction of Ramanujan's with three limits to a continued fraction with k distinct limit points, k2. The k limits are evaluated in terms of ratios of certain q-series.Finally, we show how to use Daniel Bernoulli's continued fraction in an elementary way to create analytic continued fractions with m limit points, for any positive integer m2.     

Polar SAT and related graphs

We introduce polar SAT and show that a general SAT can be reduced to it in polynomial time. A set of clauses C is called polar if there exists a partition C^pCⁿ=C, called a polar partition, such that each clause in C^p involves only positive (i.e., non-complemented) variables, while each clause in Cⁿ contains only negative (i.e., complemented) variables. A polar set of clauses C=(C^p,Cⁿ) is called (p,n)-polar, where p1 and n1, if each clause in C^p (respectively, in Cⁿ) contains exactly p (respectively, exactly n) literals. We classify all (p,n)-polar SAT Problems according to their complexity. Specifically, a (p,n)-Polar SAT problem is NP-complete if either p>n2 or n>p2. Otherwise it can be solved in polynomial time. We introduce two new hereditary classes of graphs, namely polar satgraphs and polar (3,2)-satgraphs, and we characterize them in terms of forbidden induced subgraphs. Both characterization involve an infinite number of minimal forbidden induced subgraphs. As are result, we obtain two narrow hereditary subclasses of weakly chordal graphs where Independent Domination is an NP-complete problem.     

L'algèbre de Lie des transpositions

For any n3 we obtain the decomposition in simple factors of the Lie subalgebra of the group algebra of the symmetric group on n letters generated by the transpositions. This enables us to determine the algebraic hull of the braid group B_n and of several of its subgroups inside the representations of the Iwahori–Hecke algebra of type A.     

Algorithms for graphs of bounded treewidth via orthogonal range searching

We show that, for any fixed constant k3, the sum of the distances between all pairs of vertices of an abstract graph with n vertices and treewidth at most k can be computed in O(nlog^k−1n) time.We also show that, for any fixed constant k2, the dilation of a geometric graph (i.e., a graph drawn in the plane with straight-line segments) with n vertices and treewidth at most k can be computed in O(nlog^k+1n) expected time. The dilation (or stretch-factor) of a geometric graph is defined as the largest ratio, taken over all pairs of vertices, between the distance measured along the graph and the Euclidean distance.The algorithms for both problems are based on the same principle: data structures for orthogonal range searching in bounded dimension provide a compact representation of distances in abstract graphs of bounded treewidth.     

Hankel determinants and orthogonal polynomials

We develop a general context for the computation of the determinant of a Hankel matrix H_n = (α_i+j)0i,jn, assuming some suitable conditions for the exponential (or ordinary) generating function of the sequence (α_n)_n0. Several well-known particular cases are thus derived in a unified way.     

On Mean Convergence of Lagrange Interpolation for General Arrays

For n1, let {x_jn}ⁿ_j=1 be n distinct points in a compact set K and letL_n[·] denote the corresponding Lagrange interpolation operator. Let v be a suitably restricted function on K. What conditions on the array {x_jn}_1jn, n1 ensure the existence of p>0 such that lim_n→∞ (f−L_n[f]) v_Lp(K)=0 for very continuous f: K→ ? We show that it is necessary and sufficient that there exists r>0 with sup_n1 π_nv_Lr(K) ∑ⁿ_j=1 (1/|π′_n| (x_jn))<∞. Here for n1, π_n is a polynomial of degree n having {x_jn}ⁿ_j=1 as zeros. The necessity of this condition is due to Ying Guang Shi.     

A Characterization of Graphs with No Cube Minor

In this paper it is shown that any 4-connected graph that does not contain a minor isomorphic to the cube is a minor of the line graph of V_n for some n6 or a minor of one of five graphs. Moreover, there exists a unique 5-connected graph on at least 8 vertices with no cube minor and a unique 4-connected graph with a vertex of degree at least 8 with no cube minor. Further, it is shown that any graph with no cube minor is obtained from 4-connected such graphs by 0-, 1-, and 2-summing, and 3-summing over a specified triangles.     

An iterative approximation procedure for the distribution of the maximum of a random walk

Let I(F) be the distribution function (d.f.) of the maximum of a random walk whose i.i.d. increments have the common d.f. F and a negative mean. We derive a recursive sequence of embedded random walks whose underlying d.f.'s F_k converge to the d.f. of the first ladder variable and satisfy FF₁F₂ on [0,∞) and I(F)=I(F₁)=I(F₂)=. Using these random walks we obtain improved upper bounds for the difference of I(F) and the d.f. of the maximum of the random walk after finitely many steps.     

A supplement to the Davis–Gut law

Let {X,X_n;n1} be a sequence of i.i.d. real-valued random variables and set , n1. Let h() be a positive nondecreasing function such that . Define Lt=log_emax{e,t} for t0. In this note we prove that

if and only if E(X)=0 and E(X²)=1, where , t1. When h(t)≡1, this result yields what is called the Davis–Gut law. Specializing our result to h(t)=(Lt)^r, 0<r1, we obtain an analog of the Davis–Gut law.     

disjoint cycles containing specified independent vertices

Let k1 be an integer and G be a graph of order n3k satisfying the condition that σ₂(G)n+k-1. Let v₁,…,v_k be k independent vertices of G, and suppose that G has k vertex-disjoint triangles C₁,…,C_k with v_iV(C_i) for all 1ik.Then G has k vertex-disjoint cycles such that

(i) for all 1ik.

(ii) , and

(iii) At least k-1 of the k cycles are triangles.

The condition of degree sum σ₂(G)n+k-1 is sharp.