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.     

Viscosity approximation to common fixed points of a nonexpansive semigroup with a generalized contraction mapping

The purpose of this paper is to introduce and construct the implicit and explicit viscosity iterative processes by a generalized contraction mapping f and a nonexpansive semigroup {T(t):t0}, and to prove that under suitable conditions these iterative processes converge strongly to a unique common fixed point of {T(t):t0} in reflexive Banach spaces which admits a weakly sequentially continuous duality mapping.     

Ellipticity criteria for ballistic behavior of random walks in random environment

We introduce ellipticity criteria for random walks in i.i.d. random environments under which we can extend the ballisticity conditions of Sznitman and the polynomial effective criteria of Berger, Drewitz and Ramírez originally defined for uniformly elliptic random walks. We prove under them the equivalence of Sznitman’s \((T')\) condition with the polynomial effective criterion \((P)_M\) , for \(M\) large enough. We furthermore give ellipticity criteria under which a random walk satisfying the polynomial effective criterion, is ballistic, satisfies the annealed central limit theorem or the quenched central limit theorem.     

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.     

On the structure of semi-invariant polynomials in Ore extensions

Let D be an X-outer S-derivation of a prime ring R, where S is an automorphism of R. The following is proved among other things: The degree of the minimal semi-invariant polynomial of the Ore extension R[X;S,D] is ν if charR=0, and is p^kν for some k0 if charR=p2, where ν is the least integer ν1 such that S^νDS^−ν−D is X-inner. A similar result holds for cv-polynomials. These are done by introducing the new notion of k-basic polynomials for each integer k0, which enable us to analyze semi-invariant polynomials inductively.     

Pushing up point stabilizers, II

This paper is a continuation of the paper [G. Parmeggiani, Pushing up point stabilizers, I, J. Algebra 319 (9) (2008) 3854–3884]. Under the same hypotheses, we determine those amalgams which involve SL_n(q), n3, as factors.     

On the number of independent chorded cycles in a graph

Hajnal and Corrádi proved that any simple graph on at least 3k vertices with minimal degree at least 2k contains k independent cycles. We prove the analogous result for chorded cycles. Let G be a simple graph with |V(G)|4k and minimal degree δ(G)3k. Then G contains k independent chorded cycles. This result is sharp.     

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.     

Resource augmentation for online bounded space bin packing

We study online bounded space bin packing in the resource augmentation model of competitive analysis. In this model, the online bounded space packing algorithm has to pack a list L of items in (0,1] into a small number of bins of size b1. Its performance is measured by comparing the produced packing against the optimal offline packing of the list L into bins of size 1.We present a complete solution to this problem: For every bin size b1, we design online bounded space bin packing algorithms whose worst case ratio in this model comes arbitrarily close to a certain bound ρ(b). Moreover, we prove that no online bounded space algorithm can perform better than ρ(b) in the worst case.     

Orthogonal graphs of odd characteristic and their automorphisms

The (isotropic) orthogonal graph O(2ν+δ,q) over of odd characteristic, where ν1 and δ=0,1 or 2 is introduced. When ν=1, O(21+δ,q) is a complete graph. When ν2, O(2ν+δ,q) is strongly regular and its parameters are computed, as well as its chromatic number. The automorphism groups of orthogonal graphs are also determined.     

Representable orientations of the free spikes

All orientations of binary and ternary matroids are representable [R.G. Bland, M. Las Vergnas, Orientability of matroids, J. Combinatorial Theory Ser. B 24 (1) (1978) 94–123; J. Lee, M. Scobee, A characterization of the orientations of ternary matroids, J. Combin. Theory Ser. B 77 (2) (1999) 263–291]. In this paper we show that this is not the case for matroids that are representable over GF(p^k) where k2. Specifically, we show that there are orientations of the rank-k free spike that are not representable for all k4. The proof uses threshold functions to obtain an upper bound on the number of representable orientations of the free spikes.     

A note on transformations of edge colorings of bipartite graphs

The author and A. Mirumian proved the following theorem: Let G be a bipartite graph with maximum degree Δ and let t,n be integers, tnΔ. Then it is possible to obtain, from one proper edge t-coloring of G, any proper edge n-coloring of G using only transformations of 2-colored and 3-colored subgraphs such that the intermediate colorings are also proper. In this note we show that if t>Δ then we can transform f to g using only transformations of 2-colored subgraphs. We also correct the algorithm suggested in [A.S. Asratian, Short solution of Kotzig's problem for bipartite graphs, J. Combin. Theory Ser. B 74 (1998) 160–168] for transformation of f to g in the case when t=n=Δ and G is regular.     

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.     

Existence of nonstationary bubbles in higher dimensions

We are interested in the existence of travelling-waves for the nonlinear Schrödinger equation in R^N with “ψ³−ψ⁵”-type nonlinearity. First, we prove an abstract result in critical point theory (a local variant of the classical saddle-point theorem). Using this result, we get the existence of travelling-waves moving with sufficiently small velocity in space dimension N4.     

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.     

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.     

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).     

On the mean value of

Let χ be the Dirichlet character modulo q3 and L(s,χ) denote the corresponding Dirichlet L-function. The mean value of is studied and a few asymptotic formulae are given. Hybrid mean value of , general Kloosterman sums and general quadratic Gauss sums are considered.