The generalized nonlinear initial-boundary Riemann problem for linearly degenerate quasilinear hyperbolic systems of conservation laws

This work is a continuation of our previous work, in the present paper we study the generalized nonlinear initial-boundary Riemann problem with small BV data for linearly degenerate quasilinear hyperbolic systems of conservation laws with nonlinear boundary conditions in a half space . We prove the global existence and uniqueness of piecewise C¹ solution containing only contact discontinuities to a class of the generalized nonlinear initial-boundary Riemann problem, which can be regarded as a small BV perturbation of the corresponding nonlinear initial-boundary Riemann problem, for general n×n linearly degenerate quasilinear hyperbolic system of conservation laws; moreover, this solution has a global structure similar to the one of the self-similar solution to the corresponding nonlinear initial-boundary Riemann problem. Some applications to quasilinear hyperbolic systems of conservation laws arising in the string theory and high energy physics are also given.     

Convergence rates in the SLLN for some classes of dependent random fields

Let be a random field i.e. a family of random variables indexed by Nr, r?2. We discuss complete convergence and convergence rates under assumption on dependence structure of random fields in the case of nonidentical distributions. Results are obtained for negatively associated random fields, ρ^?-mixing random fields (having maximal coefficient of correlation strictly smaller then 1) and martingale random fields.     

Regions of meromorphy and value distribution of geometrically converging rational functions

Let D be a region, {rn}_n∈N a sequence of rational functions of degree at most n and let each rn have at most m poles in D, for m∈N fixed. We prove that if {rn}_n∈N converges geometrically to a function f on some continuum S⊂D and if the number of zeros of rn in any compact subset of D is of growth o(n) as n→∞, then the sequence {rn}_n∈N converges m₁-almost uniformly to a meromorphic function in D. This result about meromorphic continuation is used to obtain Picard-type theorems for the value distribution of m₁-maximally convergent rational functions, especially in Padé approximation and Chebyshev rational approximation.     

Characterization of equality in a generalized Dunkl-Williams inequality

We establish a generalization of the Dunkl-Williams inequality and its inverse in the framework of Hilbert C^?-modules and characterize the equality case. As applications, we get some new results and some known results due to J.E. Pe?ari? and R. Raji? [The Dunkl-Williams equality in pre-Hilbert C^?-modules, Linear Algebra Appl. 425 (1) (2007) 16-25].     

On L-Tychonoff spaces II

This paper is a sequel to the 1995 paper On L-Tychonoff spaces. The embedding theorem for L-topological spaces is shown to hold true for L an arbitrary complete lattice without imposing any order reversing involution (·)^′ on L. Some results on completely L-regular spaces and on L-Tychonoff spaces, which have previously been known to hold true for (L,^′) a frame, are exhibited as ones holding for (L,^′) a meet-continuous lattice. For such a lattice an insertion theorem for completely L-regular spaces is given. Some weak forms of separating families of maps are discussed. We also clarify the dependence between the sub-T₀ separation axiom of Liu and the L-T₀ separation axiom of Rodabaugh.     

Identifying codes and locating-dominating sets on paths and cycles

Let G=(V,E) be a graph and let r≥1 be an integer. For a set D⊆V, define N_r[x]={y∈V:d(x,y)≤r} and D_r(x)=N_r[x]∩D, where d(x,y) denotes the number of edges in any shortest path between x and y. D is known as an r-identifying code (r-locating-dominating set, respectively), if for all vertices x∈V (x∈V?D, respectively), D_r(x) are all nonempty and different. Roberts and Roberts [D.L. Roberts, F.S. Roberts, Locating sensors in paths and cycles: the case of 2-identifying codes, European Journal of Combinatorics 29 (2008) 72-82] provided complete results for the paths and cycles when r=2. In this paper, we provide results for a remaining open case in cycles and complete results in paths for r-identifying codes; we also give complete results for 2-locating-dominating sets in cycles, which completes the results of Bertrand et al. [N. Bertrand, I. Charon, O. Hudry, A. Lobstein, Identifying and locating-dominating codes on chains and cycles, European Journal of Combinatorics 25 (2004) 969-987].     

A note on the generalized min-sum set cover problem

In this paper, we consider the generalized min-sum set cover problem, introduced by Azar, Gamzu, and Yin (STOC 2009). Bansal, Gupta, and Krishnaswamy (SODA 2010) give a 485-approximation algorithm for the problem. We are able to alter their algorithm and analysis to obtain a 28-approximation algorithm, improving the performance guarantee by an order of magnitude. We use concepts from α-point scheduling to obtain our improvements.