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).     

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.     

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.     

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.     

On Weighted Mean Convergence of Lagrange Interpolation for General Arrays

For n1, let {x_jn}_j=1ⁿ be n distinct points and let L_n[·] denote the corresponding Lagrange Interpolation operator. Let W : →[0,∞). What conditions on the array {x_jn}_{1jn, n1} ensure the existence of p>0 such

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for every continuous f : → with suitably restricted growth, and some “weighting factor” φ^b? We obtain a necessary and sufficient condition for such a p to exist. The result is the weighted analogue of our earlier work for interpolation arrays contained in a compact set.     

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.     

On a Conjecture of Ditzian and Runovskii

Let M_θ be the mean operator on the unit sphere in ⁿ, n3, which is an analogue of the Steklov operator for functions of single variable. Denote by D the Laplace–Beltrami operator on the sphere which is an analogue of second derivative for functions of single variable. Ditzian and Runovskii have a conjecture on the norm of the operator θ²D(M_θ)^m, m2 from X=L^p (1p∞) to itself which can be expressed as

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. We give a proof of this conjecture.     

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 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.     

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).     

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.     

Finite automata presentable abelian groups

We give new examples of FA presentable torsion-free abelian groups. Namely, for every n2, we construct a rank n indecomposable torsion-free abelian group which has an FA presentation. We also construct an FA presentation of the group in which every nontrivial cyclic subgroup is not FA recognizable.     

Universal Polynomial Majorants on Convex Bodies

Let K be a convex body in ^d (d2), and denote by B_n(K) the set of all polynomials p_n in ^d of total degree n such that |p_n|1 on K. In this paper we consider the following question: does there exist a p*_nB_n(K) which majorates every element of B_n(K) outside of K? In other words can we find a minimal γ1 and p*_nB_n(K) so that |p_n(x)|γ |p*_n(x)| for every p_nB_n(K) and x ^d\K? We discuss the magnitude of γ and construct the universal majorants p*_n for evenn. It is shown that γ can be 1 only on ellipsoids. Moreover, γ=O(1) on polytopes and has at most polynomial growth with respect to n, in general, for every convex body K.     

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.     

Eigenvalue estimates for non-normal matrices and the zeros of random orthogonal polynomials on the unit circle

We prove that for any n×n matrix, A, and z with |z|A, we have that . We apply this result to the study of random orthogonal polynomials on the unit circle.     

On triconnected and cubic plane graphs on given point sets

Let S be a set of n4 points in general position in the plane, and let h<n be the number of extreme points of S. We show how to construct a 3-connected plane graph with vertex set S, having max{3n/2,n+h−1} edges, and we prove that there is no 3-connected plane graph on top of S with a smaller number of edges. In particular, this implies that S admits a 3-connected cubic plane graph if and only if n4 is even and hn/2+1. The same bounds also hold when 3-edge-connectivity is considered. We also give a partial characterization of the point sets in the plane that can be the vertex set of a cubic plane graph.     

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.

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.     

Uniqueness and stability of regional blow-up in a porous-medium equation

We study the blow-up phenomenon for the porous-medium equation in R^N, N1, u_t=Δu^m+u^m, m>1, for nonnegative, compactly supported initial data. A solution u(x,t) to this problem blows-up at a finite time . Our main result asserts that there is a finite number of points x₁,…,x_kR^N, with |x_i−x_j|2R^* for i≠j, such that Here w_*(|x|) is the unique nontrivial, nonnegative compactly supported, radially symmetric solution of the equation in R^N and R^* is the radius of its support. Moreover u(x,t) remains uniformly bounded up to its blow-up time on compact subsets of . The question becomes reduced to that of proving that the ω-limit set in the problem consists of a single point when its initial condition is nonnegative and compactly supported.     

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.