Weak type estimates for maximal operators with a cylindric distance function

We establish sharp weak-type estimates for the maximal operators T^λ_* associated with cylindric Riesz means for functions on H^p(ℝ³) when 4/5 <p<1 and λ=3/p−5/2, and when p=4/5 and λ>3/p−5/2. The first author was supported by the Korean Research Foundation Grant funded by the Korean Government (MOEHRD) No. R04-2002-000-20028-0. The third author was supported by a Korea University Grant.     

The Hadamard Theorem and Borel–Carathéodory Theorem on Riemann Surfaces

In this paper, we first define a kind of pseudo–distance function and annulus domain on Riemann surfaces, then prove the Hadamard Theorem and the Borel–Carathéodory Theorem on any Riemann surfaces. Supported by NSFC 10501052     

Novel evolutionary models and applications to sequence alignment problems

In this paper, we present a novel graph-theoretical approach for representing a wide variety of sequence analysis problems within a single model. The model allows incorporation of the operations “insertion”, “deletion”, and “substitution”, and various parameters such as relative distances and weights. Conceptually, we refer the problem as the minimum weight common mutated sequence (MWCMS) problem. The MWCMS model has many applications including multiple sequence alignment problem, the phylogenetic analysis, the DNA sequencing problem, and sequence comparison problem, which encompass a core set of very difficult problems in computational biology. Thus the model presented in this paper lays out a mathematical modeling framework that allows one to investigate theoretical and computational issues, and to forge new advances for these distinct, but related problems. Through the introduction of supernodes, and the multi-layer supergraph, we proved that MWCMS is -complete. Furthermore, it was shown that a conflict graph derived from the multi-layer supergraph has the property that a solution to the associated node-packing problem of the conflict graph corresponds to a solution of the MWCMS problem. In this case, we proved that when the number of input sequences is a constant, MWCMS is polynomial-time solvable. We also demonstrated that some well-known combinatorial problems can be viewed as special cases of the MWCMS problem. In particular, we presented theoretical results implied by the MWCMS theory for the minimum weight supersequence problem, the minimum weight superstring problem, and the longest common subsequence problem. Two integer programming formulations were presented and a simple yet elegant decomposition heuristic was introduced. The integer programming instances have proven to be computationally intensive. Consequently, research involving simultaneous column and row generation and parallel computing will be explored. The heuristic algorithm, introduced herein for multiple sequence alignment, overcomes the order-dependent drawbacks of many of the existing algorithms, and is capable of returning good sequence alignments within reasonable computational time. It is able to return the optimal alignment for multiple sequences of length less than 1500 base pairs within 30 minutes. Its algorithmic decomposition nature lends itself naturally for parallel distributed computing, and we continue to explore its flexibility and scalability in a massive parallel environment.     

Estimates in the invariance principle in terms of truncated power moments

We obtain estimates for the distributions of errors which arise in approximation of a random polygonal line by a Wiener process on the same probability space. The polygonal line is constructed on the whole axis for sums of independent nonidentically distributed random variables and the distance between it and the Wiener process is taken to be the uniform distance with an increasing weight. All estimates depend explicitly on truncated power moments of the random variables which is an advantage over the earlier estimates of Komlos, Major, and Tusnady where this dependence was implicit.     

On the complexity of suboptimal decoding for list and decision feedback schemes

We consider bounded distance list decoding, such that the decoder calculates the list of all codewords within a sphere around the received vector. We analyze the performance and the complexity of this suboptimum list decoding scheme for the binary symmetric channel. The reliability function of the list decoding scheme is equivalent to the sphere-packing bound, where the decoding complexity is asymptotically bounded by 2^nR(1-R). Furthermore, we investigate a decision feedback strategy that is based on bounded distance list decoding. Here, any output with zero or many codewords will call for a repeated transmission. In this case the decoding complexity will be of the order 2^nR(1-C), where C denotes the channel capacity. The reliability function is close to Forney's feedback exponent.     

Improved estimation of duality gap in binary quadratic programming using a weighted distance measure

We present in this paper an improved estimation of duality gap between binary quadratic program and its Lagrangian dual. More specifically, we obtain this improved estimation using a weighted distance measure between the binary set and certain affine subspace. We show that the optimal weights can be computed by solving a semidefinite programming problem. We further establish a necessary and sufficient condition under which the weighted distance measure gives a strictly tighter estimation of the duality gap than the existing estimations.     

Uniqueness and asymptotic behaviour for solutions of semilinear problems with boundary blow-up

In this paper we prove uniqueness of positive solutions to logistic singular problems , , 1$">, 0$"> in , where the main feature is the fact that . More importantly, we provide exact asymptotic estimates describing, in the form of a two-term expansion, the blow-up rate for the solutions near . This expansion involves both the distance function and the mean curvature of .

     

Maximum Matchings in the n-Dimensional Cube

The problem of efficient computation of maximum matchings in the n-dimensional cube, which is applied in coding theory, is solved. For an odd n, such a matching can be found by the method given in our Theorem 2. This method is based on the explicit construction (Theorem 1) of the maps of the vertex set that induce largest matchings in any bipartite subgraph of the n-dimensional cube for any n.