On the k-error linear complexity over \mathbb{F}_p of Legendre and Sidelnikov sequences

We determine exact values for the k-error linear complexity L _k over the finite field of the Legendre sequence of period p and the Sidelnikov sequence of period p ^m  − 1. The results are

The Maximum Size Of Graphs With A Unique<Emphasis Type="Italic">k</Emphasis>- Factor

For suitable positive integers n and k let m(n, k) denote the maximum number of edges in a graph of order n which has a unique k-factor. In 1964, Hetyei and in 1984, Hendry proved for even n and , respectively. Recently, Johann confirmed the following conjectures of Hendry: for and kn even and for n = 2kq, where q is a positive integer. In this paper we prove for and kn even, and we determine m(n, 3).     

Minors in Expanding Graphs

We propose a unifying framework for studying extremal problems related to graph minors. This framework relates the existence of a large minor in a given graph to its expansion properties. We then apply the developed framework to several extremal problems and prove in particular that: (a) Every -free graph G with average degree r ( are constants) contains a minor with average degree , for some constant ; (b) Every C_2k-free graph G with average degree r (k ≥ 2 is a constant) contains a minor with average degree , for some constant c = c(k) > 0. We also derive explicit lower bounds on the minor density in random, pseudo-random and expanding graphs. Received: March 2008, Accepted: May 2008     

Spherical Monogenics: An Algebraic Approach

The space of spherical monogenics in can be regarded as a model for the irreducible representation of Spin(m) with weight . In this paper we construct an orthonormal basis for . To describe the symmetry behind this procedure, we define certain Spin(m − 2)-invariant representations of the Lie algebra (2) on . Received: October, 2007. Accepted: February, 2008.     

Some Limit Theorems for a Particle System of Single Point Catalytic Branching Random Walks

We study the scaling limit for a catalytic branching particle system whose particles perform random walks on Z and can branch at 0 only. Varying the initial （finite） number of particles, we get for this system different limiting distributions. To be more specific, suppose that initially there are n^β particles and consider the scaled process Zt^n（·） = Znt（√n·）, where Zt is the measure-valued process 1 and to a representing the original particle system. We prove that Ztn converges to 0 when β 〈1/4 and to a nondegenerate discrete distribution when β=1/4.In addition,if 1/4〈β〈1/2 then n-^（2β-1/2）Zt^n converges to a random limit,while if β 〉21then n^-βZtn converges to a deterministic limit.     

Möbius parametrizations of curves in {\mathbb{R}}^n

We use Ahlfors’ definition of Schwarzian derivative for curves in euclidean spaces to present new results about M?bius or projective parametrizations. The class of such parametrizations is invariant under compositions with M?bius transformations, and the resulting curves are simple. The analysis is based on the oscillatory behavior of the associated linear equation , where k = k(s) is the curvature as a function of arclength. Received: 24 November 2008     

On Confining Potentials and Essential Self-Adjointness for Schrödinger Operators on Bounded Domains in {\mathbb{R}}^n

Let Ω be a bounded domain in with C²-smooth boundary, , of co-dimension 1, and let be a Schr?dinger operator on Ω with potential . We seek the weakest conditions we can find on the rate of growth of the potential V close to the boundary which guarantee essential self-adjointness of H on . As a special case of an abstract condition, we add optimal logarithmic type corrections to the known condition where . More precisely, we show that if, as x approaches ,

Hamiltonicity in balancedk-partite graphs

One of the earliest results about hamiltonian graphs was given by Dirac. He showed that if a graphG has orderp and minimum degree at least thenG is hamiltonian. Moon and Moser showed that a balanced bipartite graph (the two partite sets have the same order)G has orderp and minimum degree more than thenG is hamiltonian. In this paper, their idea is generalized tok-partite graphs and the following result is obtained: LetG be a balancedk-partite graph with orderp = kn. If the minimum degree

On zero-sum sequences of prescribed length

Summary. Let k ≥ 1 be any integer. Let G be a finite abelian group of exponent n. Let s_k(G) be the smallest positive integer t such that every sequence S in G of length at least t has a zero-sum subsequence of length kn. We study this constant for groups when d = 3 or 4. In particular, we prove, as a main result, that for every k ≥ 4, and for every prime p ≥ 5.     

Pointwise Estimates for Linear Combinations of Gamma Operators

For linear combinations of Gamma operators G_n,r(f,x), we give an equivalent theorem with (f,t), where (f,t) are the Ditzian-Totik moduli of smoothness ( ).AMS Subject Classification (1991): 41A25, 41A36.Supported by NSF of Hebei ProvinceSupported by NSF of Hebei Province     

Upper bounds for prime <Emphasis Type="Italic">k</Emphasis>-tuples of size log <Emphasis Type="Italic">N</Emphasis> and oscillations

We prove the estimate for the number E_k(N) of k-tuples (n + a₁,..., n + a_k) of primes not exceeding N, for k of size c₁ log N and N sufficiently large. A bound of this strength was previously known in the special case < only, (Vaughan, 1973). For general a_i this is an improvement upon the work of Hofmann and Wolke (1996). The number of prime tuples of this size has considerable oscillations, when varying the prime pattern. Received: 20 December 2002     

Cubic elliptic functions

The function

Eigenvalues and homology of flag complexes and vector representations of graphs

The flag complex of a graph G = (V, E) is the simplicial complex X(G) on the vertex set V whose simplices are subsets of V which span complete subgraphs of G. We study relations between the first eigenvalues of successive higher Laplacians of X(G). One consequence is the following:Theorem: Let λ₂(G) denote the second smallest eigenvalue of the Laplacian of G. If \,\frac{k}{k+1}|V|$$" align="middle" border="0"> then Applications include a lower bound on the homological connectivity of the independent sets complex I(G), in terms of a new graph domination parameter Γ(G) defined via certain vector representations of G. This in turns implies Hall type theorems for systems of disjoint representatives in hypergraphs.Received: January 2004 Revised: August 2004 Accepted: August 2004     

Hyperbolic Function Theory in the Clifford Algebra {\mathcal {C}}\ell_{n+1, 0}

The aim of this paper is to give the basic principles of hyperbolic function theory on the Clifford algebra . The structure of the theory is quite similar to the case of Clifford algebras with negative generators, but the proofs are not obvious. The (real) Clifford algebra is generated by unit vectors with positive squares e²_i = + 1. The hyperbolic Dirac operator is of the form where Q₀f is represented by the composition . If is a solution of H_kf = 0, then f is called k-hypergenic in Ω, where is an open set. We introduce some basic results of hyperbolic function theory and give some representation theorems on . Received: October, 2007. Accepted: February, 2008.     

A Vector-valued H p Corona Theorem on the Polydisk

For the corona problem on the bidisk, we find analytic solutions belonging to the Orlicz-type space In addition, for 1 ≤ p < ∞, an corona theorem is established. Similar techniques can be used for the polydisk.     

On the minimum length of some linear codes

We determine the minimum length n _q (k, d) for some linear codes with k ≥ 5 and q ≥ 3. We prove that n _q (k, d) = g _q (k, d) + 1 for when k is odd, for when k is even, and for . This work was supported by the Korea Research Foundation Grant funded by the Korean Government(MOEHRD). (KRF-2005-214-C00175). This research has been partially supported by Grant-in-Aid for Scientific Research of Japan Society for the Promotion of Science under Contract Number 17540129.     

Semilinear representations of PGL

Let L be the function field of a projective space over an algebraically closed field k of characteristic zero, and H be the group of projective transformations. An H-sheaf on is a collection of isomorphisms for each g ∈ H satisfying the chain rule. We construct, for any n > 1, a fully faithful functor from the category of finite-dimensional L-semilinear representations of H extendable to the semigroup End(L/k) to the category of coherent H-sheaves on The paper is motivated by a study of admissible representations of the automorphism group G of an algebraically closed extension of k of countable transcendence degree undertaken in [4]. The semigroup End(L/k) is considered as a subquotient of G, hence the condition on extendability. In the appendix it is shown that, if is either H, or a bigger subgroup in the Cremona group (generated by H and a certain pair of involutions), then any semilinear of degree one is an integral L-tensor power of It is also shown that this bigger subgroup has no non-trivial representations of finite degree if n > 1.