The two-equal-disjoint path cover problem of Matching Composition Network

Embedding of paths have attracted much attention in the parallel processing. Many-to-many communication is one of the most central issues in various interconnection networks. A graph G is globally two-equal-disjoint path coverable if for any two distinct pairs of vertices (u,v) and (w,x) of G, there exist two disjoint paths P and Q satisfied that (1) P (Q, respectively) joins u and v (w and x, respectively), (2) |P|=|Q|, and (3) V(PQ)=V(G). The Matching Composition Network (MCN) is a family of networks which two components are connected by a perfect matching. In this paper, we consider the globally two-equal-disjoint path cover property of MCN. Applying our result, the Crossed cube CQ_n, the Twisted cube TQ_n, and the Möbius cube MQ_n can all be proven to be globally two-equal-disjoint path coverable for n5.     

Hermite interpolation with simple planar PH curves by speed reparametrization

We introduce a new method of solving C¹ Hermite interpolation problems, which makes it possible to use a wider range of PH curves with potentially better shapes. By characterizing PH curves by roots of their hodographs in the complex representation, we introduce PH curves of type K(t−c)²ⁿ⁺¹+d. Next, we introduce a speed reparametrization. Finally, we show that, for C¹ Hermite data, we can use PH curves of type K(t−c)²ⁿ⁺¹+d or strongly regular PH quintics satisfying the G¹ reduction of C¹ data, and use these curves to solve the original C¹ Hermite interpolation problem.     

Fast algorithm for multicast and data gathering in wireless networks

Given a wireless network G=(V,E), we consider a maximum critical energy problem [J. Park, S. Sahni, Maximum lifetime broadcasting in wireless networks, IEEE Transactions on Computers 54 (9) (2005) 1081–1090] that has an objective of increasing the chances of doing a sequence of broadcasts. We present an optimal generalized solution algorithm running in improved optimal O(|V|+|E|) time, where V stands for a set of nodes and E stands for a set of links in the network. Our approach is applicable in an omnidirectional antenna model and can be used to solve the problem of multicasting traffic so as to maximize the lifetime of the network [A. Orda, B.-A. Yassour, Maximum-lifetime routing algorithms for networks with omnidirectional and directional antennas, in: Proc. ACM MOBIHOC, 2005] and a data gathering problem [K. Kalpakis, K. Dasgupta, P. Namjoshi, Maximum lifetime data gathering and aggregation in wireless sensor networks, Computer Networks 42 (2003) 697–716; Y. Xue, Y. Cui, K. Nahrstedt, Maximizing lifetime for data aggregation in wireless sensor networks, ACM Moble Networks and Applications 10 (6) (2005) 853–864] with an improved running time.     

A novel lossless data compression scheme based on the error correcting Hamming codes

This paper introduces a novel lossless binary data compression scheme that is based on the error correcting Hamming codes, namely the HCDC scheme. In this scheme, the binary sequence to be compressed is divided into blocks of n bits length. To utilize the Hamming codes, the block is considered as a Hamming codeword that consists of p parity bits and d data bits (n=d+p). Then each block is tested to find if it is a valid or a non-valid Hamming codeword. For a valid block, only the d data bits preceded by 1 are written to the compressed file, while for a non-valid block all n bits preceded by 0 are written to the compressed file. These additional 1 and 0 bits are used to distinguish the valid and the non-valid blocks during the decompression process. An analytical formula is derived for computing the compression ratio as a function of block size, and fraction of valid data blocks in the sequence. The performance of the HCDC scheme is analyzed, and the results obtained are presented in tables and graphs. Finally, conclusions and recommendations for future works are pointed out.     

Fault-tolerant and progressive transmission of images

A fault-tolerant progressive image transmission method is proposed. The advantages include the following: (1) Unlike most progressive methods, the image is divided into n parts with equal importance to avoid worrying about which part is lost or transmitted first. (2) If the image is a secret image, then the transmission can use n distinct channels (one shared result per channel), and intercepting up to r₁-1 channels by the enemy (r₁r_kn are all pre-set constants) will not reveal any secret. Meanwhile, the disconnection up to n-r_k channels will not affect the lossless recovery of the secret image.     

Farthest line segment Voronoi diagrams

The farthest line segment Voronoi diagram shows properties different from both the closest-segment Voronoi diagram and the farthest-point Voronoi diagram. Surprisingly, this structure did not receive attention in the computational geometry literature. We analyze its combinatorial and topological properties and outline an O(nlogn) time construction algorithm that is easy to implement. No restrictions are placed upon the n input line segments; they are allowed to touch or cross.     

Hamiltonian-connected graphs

For a simple graph G, let . In this paper, we prove that if NCD(G)≥|V(G)|, then either G is Hamiltonian-connected, or G belongs to a well-characterized class of graphs. The former results by Dirac, Ore and Faudree et al. are extended.     

Triple positive solutions for some second-order boundary value problem on a measure chain

In this paper, we study the existence of three positive solutions for the second-order two-point boundary value problem on a measure chain,

where f:[t₁,σ(t₂)]×[0,∞)×R→[0,∞) is continuous and p:[t₁,σ(t₂)]→[0,∞) a nonnegative function that is allowed to vanish on some subintervals of [t₁,σ(t₂)] of the measure chain. The method involves applications of a new fixed-point theorem due to Bai and Ge [Z.B. Bai, W.G. Ge, Existence of three positive solutions for some second order boundary-value problems, Comput. Math. Appl. 48 (2004) 699–707]. The emphasis is put on the nonlinear term f involved with the first order delta derivative x^Δ(t).     

Vector extrapolation methods with applications to solution of large systems of equations and to PageRank computations

An important problem that arises in different areas of science and engineering is that of computing the limits of sequences of vectors {x_n}, where with N very large. Such sequences arise, for example, in the solution of systems of linear or nonlinear equations by fixed-point iterative methods, and lim_n→∞x_n are simply the required solutions. In most cases of interest, however, these sequences converge to their limits extremely slowly. One practical way to make the sequences {x_n} converge more quickly is to apply to them vector extrapolation methods. In this work, we review two polynomial-type vector extrapolation methods that have proved to be very efficient convergence accelerators; namely, the minimal polynomial extrapolation (MPE) and the reduced rank extrapolation (RRE). We discuss the derivation of these methods, describe the most accurate and stable algorithms for their implementation along with the effective modes of usage in solving systems of equations, nonlinear as well as linear, and present their convergence and stability theory. We also discuss their close connection with the method of Arnoldi and with GMRES, two well-known Krylov subspace methods for linear systems. We show that they can be used very effectively to obtain the dominant eigenvectors of large sparse matrices when the corresponding eigenvalues are known, and provide the relevant theory as well. One such problem is that of computing the PageRank of the Google matrix, which we discuss in detail. In addition, we show that a recent extrapolation method of Kamvar et al. that was proposed for computing the PageRank is very closely related to MPE. We present a generalization of the method of Kamvar et al. along with a very economical algorithm for this generalization. We also provide the missing convergence theory for it.     

Uniqueness theorems on meromorphic functions sharing one value

In this paper, we study with a weighted sharing method the uniqueness problem of [fⁿ(z)]^(k) and [gⁿ(z)]^(k) sharing one value and obtain some results which extend the theorems given by M. Fang, S. Bhoosnurmath and S. Dyavanal et al.     

Improved construction for universality of determinant and permanent

Valiant [L. Valiant, Completeness classes in algebra, in: Proc. 11th Annual ACM Symposium on the Theory of Computing, Atlanta, GA, 1979, pp. 249–261] proved that every polynomial of formula size e is a projection of the (e+2)×(e+2) determinant polynomial. We improve “e+2” to “e+1”, also for a definition of formula size that does not count multiplications by constants as gates. Our proof imitates the “2e+2” proof of von zur Gathen [J. von zur Gathen, Feasible arithmetic computations: Valiant's hypothesis, Journal of Symbolic Computation 4 (1987) 137–172], but uses different invariants and a tighter set of base cases.     

Scheduling in polling systems

We present a simple mean value analysis (MVA) framework for analyzing the effect of scheduling within queues in classical asymmetric polling systems with gated or exhaustive service. Scheduling in polling systems finds many applications in computer and communication systems. Our framework leads not only to unification but also to extension of the literature studying scheduling in polling systems. It illustrates that a large class of scheduling policies behaves similarly in the exhaustive polling model and the standard M/GI/1 model, whereas scheduling policies in the gated polling model behave very differently than in an M/GI/1.     

Fibonacci mean and golden section mean

In this paper we show that there is a mapping D:M→DM on means such that if M is a Fibonacci mean so is DM, that if M is the harmonic mean, then DM is the arithmetic mean, and if M is a Fibonacci mean, then lim_n→∞DⁿM is the golden section mean.     

An efficient algorithm for computing moments on a block representation of a grey-scale image

Computing lower order moments is important in image processing. Suppose the input grey image with size N×N has been compressed into the block representation where the number of blocks is K, commonly K<N² due to the compression effect. This correspondence presents an efficient algorithm for computing lower order moments on the block representation directly. Our proposed algorithm takes O(K) time which is proportional to the number of blocks. Experimental results reveal the computational advantage of our proposed algorithm. In addition, the results of this paper can be viewed as a generalization of the previous result by Spiliotis and Mertzios for computing lower order moments from the binary image domain to the grey image domain.     

On absolute summability factors of infinite series

In this paper a general theorem on |A,δ|_k-summability methods has been proved. This theorem includes, as a special case, a known result in [E. Savas, Factors for |A|_k Summability of infinite series, Comput. Math. Appl. 53 (2007) 1045–1049].     

Unifying C-curves and H-curves by extending the calculation to complex numbers

Recently, we found that the CB-splines that use basis {sint,cost,t,1} and the HB-splines that use basis {sinht,cosht,t,1} could be unified into a complete curve family, named FB-splines (Zhang and Krause, 2005). FB-splines are a scheme of what we call here F-curves. This paper explains that in the domain of complex numbers, the extended C-curves and extended H-curves are the same curves. Therefore, F-curves can be constructed in two identical styles, C and H. The C style is an extension of C-curves that uses sin and cos, and the H style is an extension of H-curves that uses sinh and cosh. Here the representations of F-curves are clearer and simpler. For real applications, the definitions, equations and main properties for the F-curves in different schemes (FB-splines, F-Bézier and F-Ferguson schemes) are introduced in details. F-curves are shape adjustable, and their curvatures on terminals can be any expected value between 0 and ∞. They can represent the circular (or elliptical) arc, the cylinder, the helix, the cycloid, the hyperbola, the catenary, etc. precisely. Therefore, F-curves are more useful than C-curves or H-curves for the surface modeling in engineering.     

A minimal representation of Markov arrival processes and a moments matching method

The paper investigates the problem of minimal representation of Markov arrival processes of order n (MAP(n)). The minimal representation of MAPs is crucial for developing effective fitting methods. It seems that all existing MAP fitting methods are based on the , representation which is known to be redundant. We present the minimal number of parameters to define a MAP(n) and provide a numerical moments-matching method based on a minimal representation.

The discussion starts with a characterization of phase type (PH) distributions and then the analysis of MAPs follows a similar pattern. This characterization contains essential results on the identity of stationary behaviour of MAPs and on the number of parameters required to describe the stationary behaviour.

The proposed moments matching method is also applicable for PH distributions. In this case it is a unique method that fits a general PH distribution of order n based on 2n−1 parameters.     

An improved lower bound on approximation algorithms for the Closest Substring problem

The Closest Substring problem (the CSP problem) is a basic NP-hard problem in the study of computational biology. It is known that the problem has polynomial time approximation schemes. In this paper, we prove that unless the Exponential Time Hypothesis fails, the CSP problem has no polynomial time approximation schemes of running time f(1/ε)n^o(1/ε) for any function f. This essentially excludes the possibility that the CSP problem has a practical polynomial time approximation scheme even for moderate values of the error bound ε. As a consequence, it is unlikely that the study of approximation schemes for the CSP problem in the literature would lead to practical approximation algorithms for the problem for small error bound ε.     

Multimodal logic programming

We give a framework for developing the least model semantics, fixpoint semantics, and SLD-resolution calculi for logic programs in multimodal logics whose frame restrictions consist of the conditions of seriality (i.e. ) and some classical first-order Horn clauses. Our approach is direct and no special restriction on occurrences of □_i and _i is required. We apply our framework for a large class of basic serial multimodal logics, which are parameterized by an arbitrary combination of generalized versions of axioms T, B, 4, 5 (in the form, e.g. 4:□_i→□_j□_k) and I:□_i→□_j. Another part of the work is devoted to programming in multimodal logics intended for reasoning about multidegree belief, for use in distributed systems of belief, or for reasoning about epistemic states of agents in multiagent systems. For that we also use the framework, and although these latter logics belong to the mentioned class of basic serial multimodal logics, the special SLD-resolution calculi proposed for them are more efficient.     

Identifying a control function in parabolic partial differential equations from overspecified boundary data

Determination of an unknown time-dependent function in parabolic partial differential equations, plays a very important role in many branches of science and engineering. In the current investigation, the Adomian decomposition method is used for finding a control parameter p(t) in the quasilinear parabolic equation u_t=u_xx+p(t)u+, in [0,1]×(0,T] with known initial and boundary conditions and subject to an additional condition in the form of which is called the boundary integral overspecification. The main approach is to change this inverse problem to a direct problem and then solve the resulting equation using the well known Adomian decomposition method. The decomposition procedure of Adomian provides the solution in a rapidly convergent series where the series may lead to the solution in a closed form. Furthermore due to the rapid convergence of Adomian’s method, a truncation of the series solution with sufficiently large number of implemented components can be considered as an accurate approximation of the exact solution. This method provides a reliable algorithm that requires less work if compared with the traditional techniques. Some illustrative examples are presented to show the efficiency of the presented method.