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.     

A new algebra of Toeplitz-plus-Hankel matrices and applications

We introduce a new simultaneously diagonalizable real algebra of symmetrical centrosymmetrical matrices having a Toeplitz-plus-Hankel structure. We give the corresponding orthonormal basis of eigenvectors which are alternately symmetrical and skewsymmetrical vectors. An application is the construction of a symmetrical Toeplitz-plus-centrosymmetrical Hankel matrix of equal row sums having a prescribed real spectrum. This matrix can be used as the starting matrix for symmetrical centrosymmetrical isospectral flows. In particular, for the isospectral flow corresponding to the construction of a regular Toeplitz matrix having prescribed eigenvalues. Moreover, if A is a noise representation of an unknown matrix in of rank k then we give a procedure to approximate A by a matrix in of rank k.     

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.     

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.     

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 algorithms for finding disjoint subsequences with extremal densities

We derive fast algorithms for the following problem: given a set of n points on the real line and two parameters s and p, find s disjoint intervals of maximum total length that contain at most p of the given points. Our main contribution consists of algorithms whose time bounds improve upon a straightforward dynamic programming algorithm, in the relevant case that input size n is much bigger than parameters s and p. These results are achieved by selecting a few candidate intervals that are provably sufficient for building an optimal solution via dynamic programming. As a byproduct of this idea we improve an algorithm for a similar subsequence problem of Chen et al. [Disjoint segments with maximum density, in: International Workshop on Bioinformatics Research and Applications IWBRA 2005, (within ICCS 2005), Lecture Notes in Computer Science, vol. 3515, Springer, Berlin, pp. 845–850]. The problems are motivated by the search for significant patterns in biological data. Finally, we propose several heuristics that further reduce the time complexity in typical instances. One of them leads to an apparently open subsequence sum problem of independent interest.     

Multi-degree reduction of triangular Bézier surfaces with boundary constraints

Given a triangular Bézier surface of degree n, the problem of multi-degree reduction by a triangular Bézier surface of degree m with boundary constraints is investigated. This paper considers the continuity of triangular Bézier surfaces at the three corners, so that the boundary curves preserve endpoints continuity of any order . The l₂- and L₂-norm combined with the constrained least-squares method are used to get the matrix representations for the control points of the degree reduced surfaces. Both methods can be applied to piecewise continuous triangular patches or to only a triangular patch with the combination of surface subdivision. And the resulting piecewise approximating patches are globally C⁰ continuous. Finally, error estimation is given and numerical examples demonstrate the effectiveness of our methods.     

Robust control of continuous-time Markov jump linear systems

This paper is concerned with the problem of designing robust H₂ state-feedback controllers for continuous-time Markov jump linear systems subject to polytopic-type parameter uncertainty. Based on the parameter-dependent Lyapunov function approach, a new method for designing robust H₂ controllers is presented in terms of solutions to a set of linear matrix inequalities. A numerical example is given to illustrate the effectiveness of the proposed design method.     

The Lambert W function and the spectrum of some multidimensional time-delay systems

In this note we find an explicit expression for the eigenvalues of a retarded time-delay system with one delay, for the special case that the system matrices are simultaneously triangularizable, which includes the case where they commute. Using matrix function definitions we define a matrix version of the Lambert W function, from which we form the expression. We prove by counter-example that some expressions in other publications on Lambert W for time-delay systems do not always hold.     

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.     

Squaring the circle: An algorithm for generating polyhedral invariant sets from ellipsoidal ones

This paper presents a new (geometrical) approach to the computation of polyhedral (robustly) positively invariant (PI) sets for general (possibly discontinuous) nonlinear discrete-time systems possibly affected by disturbances. Given a β-contractive ellipsoidal set , the key idea is to construct a polyhedral set that lies between the ellipsoidal sets and . A proof that the resulting polyhedral set is contractive and thus, PI, is given, and a new algorithm is developed to construct the desired polyhedral set. The problem of computing polyhedral invariant sets is formulated as a number of quadratic programming (QP) problems. The number of QP problems is guaranteed to be finite and therefore, the algorithm has finite termination. An important application of the proposed algorithm is the computation of polyhedral terminal constraint sets for model predictive control based on quadratic costs.     

set membership identification: A survey

Robustness had become in past years a central issue in system and control theory, focusing the attention of researchers from the study of a single model to the investigation of a set of models, described by a set of perturbations of a “nominal” model. Such a set, often indicated as an uncertainty model set or model set for short, has to be suitably constructed to describe the inherent uncertainty about the system under consideration and to be used for analysis and design purposes. H_∞ identification methods deliver uncertainty model sets in a suitable form to be used by well-established robust design techniques, based on H_∞ or μ optimization methods. The literature on H_∞ identification is now very extensive. In this paper, some of the most relevant contributions related to assumption validation, evaluation of bounds on unmodeled dynamics, convergence analysis and optimality properties of linear, two-stage and interpolatory algorithms are surveyed from a deterministic point of view.     

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.     

A parallel method for large sparse generalized eigenvalue problems using a GridRPC system

In this paper we present a master–worker type parallel method for finding several eigenvalues and eigenvectors of a generalized eigenvalue problem , where A and B are large sparse matrices. A moment-based method that finds all of the eigenvalues that lie inside a given domain is used. In this method, a small matrix pencil that has only the desired eigenvalues is derived by solving large sparse systems of linear equations constructed from A and B. Since these equations can be solved independently, we solve them on remote servers in parallel. This approach is suitable for master–worker programming models. We have implemented and tested the proposed method in a grid environment using a grid RPC (remote procedure call) system called OmniRPC. The performance of the method on PC clusters that were used over a wide-area network was evaluated.     

LTL over integer periodicity constraints

Periodicity constraints are used in many logical formalisms, in fragments of Presburger LTL, in calendar logics, and in logics for access control, to quote a few examples. In the paper, we introduce the logic PLTL^mod, an extension of Linear-Time Temporal Logic LTL with past-time operators whose atomic formulae are defined from a first-order constraint language dealing with periodicity. Although the underlying constraint language is a fragment of Presburger arithmetic shown to admit a PSPACE-complete satisfiability problem, we establish that PLTL^mod model-checking and satisfiability problems remain in PSPACE as plain LTL (full Presburger LTL is known to be highly undecidable). This is particularly interesting for dealing with periodicity constraints since the language of PLTL^mod has a language more concise than existing languages and the temporalization of our first-order language of periodicity constraints has the same worst case complexity as the underlying constraint language. Finally, we show examples of introduction the quantification in the logical language that provide to PLTL^mod, EXPSPACE-complete problems. As another application, we establish that the equivalence problem for extended single-string automata, known to express the equality of time granularities, is PSPACE-complete by designing a reduction from QBF and by using our results for PLTL^mod.     

Stability for first order delay dynamic equations on time scales

In this paper, we establish some sufficient conditions for the uniform stability and the uniformly asymptotical stability of the first order delay dynamic equation

where is a time scale, p(.) is rd-continuous and positive, the delay function . Our results unify the corresponding ones for differential and difference equations. To the best of our knowledge, this is the first time to discuss the asymptotical behavior of delay dynamic equations on time scales.     

output feedback control design for uncertain fuzzy singularly perturbed systems: an LMI approach

This paper examines the problem of designing a robust H_∞ output feedback controller for a class of singularly perturbed systems described by a Takagi–Sugeno fuzzy model. Based on a linear matrix inequality (LMI) approach, LMI-based sufficient conditions for the uncertain singularly perturbed nonlinear systems to have an H_∞ performance are derived. To eliminate the ill-conditioning caused by the interaction of slow and fast dynamic modes, solutions to the problem are presented in terms of LMIs which are independent of the singular perturbation . The proposed approach does not involve the separation of states into slow and fast ones and it can be applied not only to standard, but also to nonstandard singularly perturbed nonlinear systems. A numerical example is provided to illustrate the design developed in this paper.     

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).     

Robust information filter for decentralized estimation

This paper presents a robust information filter which inherits the structural simplicity of the information filter and the robustness property of the H_∞ filter with respect to noise statistics. In this filter, an assurance level γ on the noise bound is reflected in the newly defined covariance matrix. It provides robustness against uncertainty in the noise model by sacrificing performance in the minimum variance sense. All these are achieved while retaining the simple structure of the standard information filter. Thus, it is able to realize robust decentralized estimation with less communication and computational load while without the need to model the system noise accurately.