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).     

Algorithms for extracting motifs from biological weighted sequences

In this paper we present three algorithms for the Motif Identification Problem in Biological Weighted Sequences. The first algorithm extracts repeated motifs from a biological weighted sequence. The motifs correspond to repetitive words which are approximately equal, under a Hamming distance, with probability of occurrence 1/k, where k is a small constant. The second algorithm extracts common motifs from a set of N2 weighted sequences. In this case, the motifs consists of words that must occur with probability 1/k, in 1qN distinct sequences of the set. The third algorithm extracts maximal pairs from a biological weighted sequence. A pair in a sequence is the occurrence of the same word twice. In addition, the algorithms presented in this paper improve previous work on these problems.     

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.     

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.

Arithmetic progressions consisting of unlike powers

In this paper we present some new results about unlike powers in arithmetic progression. We prove among other things that for given k 4 and L 3 there are only finitely many arithmetic progressions of the form with x_i , gcd(x₀, x_l) = 1 and 2 l_i L for i = 0, 1, …, k − 1. Furthermore, we show that, for L = 3, the progression (1, 1,…, 1) is the only such progression up to sign. Our proofs involve some well-known theorems of Faltings [9], Darmon and Granville [6] as well as Chabauty's method applied to superelliptic curves.     

Hilbert's Tenth Problem for function fields of varieties over number fields and p-adic fields

Let k be a subfield of a p-adic field of odd residue characteristic, and let be the function field of a variety of dimension n1 over k. Then Hilbert's Tenth Problem for is undecidable. In particular, Hilbert's Tenth Problem for function fields of varieties over number fields of dimension 1 is undecidable.     

Stable extendibility of vector bundles over lens spaces mod 3 and the stable splitting problem

Let Lⁿ(3) denote the (2n+1)-dimensional standard lens space mod 3. In this paper, we study the conditions for a given real vector bundle over Lⁿ(3) to be stably extendible to L^m(3) for every mn, and establish the formula on the power ζ^k=ζζ (k-fold) of a real vector bundle ζ over Lⁿ(3). Moreover, we answer the stable splitting problem for real vector bundles over Lⁿ(3) by means of arithmetic conditions.     

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.