Strong convergence theorems for a new iterative method of -strictly pseudo-contractive mappings in Hilbert spaces

In this paper, we introduce a new iterative method of a k-strictly pseudo-contractive mapping for some 0≤k<1 and prove that the sequence {x_n} converges strongly to a fixed point of T, which solves a variational inequality related to the linear operator A. Our results have extended and improved the corresponding results of Y.J. Cho, S.M. Kang and X. Qin [Some results on k-strictly pseudo-contractive mappings in Hilbert spaces, Nonlinear Anal. 70 (2008) 1956–1964], and many others.     

Suboptimal robust synthesis for MIMO plant under coprime factor perturbations

Suboptimal robust synthesis for MIMO nominal system under coprime factor perturbations is considered in classical and non-classical statements. In the classical statement, weights of perturbations and upper bound on magnitude bounded exogenous disturbance are assumed to be known to controller designer. Suboptimal synthesis within ε tolerance is reduced to the solution of log₂(1/ε) standard mixed sensitivity problems of ℓ₁ optimization. In the non-classical statement, the upper bounds on perturbations and exogenous disturbance are to be estimated from measurement data and suboptimal synthesis is reduced to the solution of 1/ε mixed sensitivity problems.     

Improved upper bounds on the L(2,1) -labeling of the skew and converse skew product graphs

An L(2,1)-labeling of a graph G is a function f from the vertex set V(G) to the set of all nonnegative integers such that |f(x)−f(y)|≥2 if d(x,y)=1 and |f(x)−f(y)|≥1 if d(x,y)=2, where d(x,y) denotes the distance between x and y in G. The L(2,1)-labeling number λ(G) of G is the smallest number k such that G has an L(2,1)-labeling with max{f(v):vV(G)}=k. Griggs and Yeh conjecture that λ(G)≤Δ² for any simple graph with maximum degree Δ≥2. This paper considers the graph formed by the skew product and the converse skew product of two graphs with a new approach on the analysis of adjacency matrices of the graphs as in [W.C. Shiu, Z. Shao, K.K. Poon, D. Zhang, A new approach to the L(2,1)-labeling of some products of graphs, IEEE Trans. Circuits Syst. II: Express Briefs (to appear)] and improves the previous upper bounds significantly.     

Nonlinear relaxed cocoercive variational inclusions involving (A, η)-accretive mappings in banach spaces

In this paper, we introduce a new concept of (A, η)-accretive mappings, which generalizes the existing monotone or accretive operators. We study some properties of (A, η)-accretive mappings and define resolvent operators associated with (A, η)-accretive mappings. By using the new resolvent operator technique, we also construct a new perturbed iterative algorithm with mixed errors for a class of nonlinear relaxed Cocoercive variational inclusions involving (A, η)-accretive mappings and study applications of (A, η)-accretive mappings to the approximation-solvability of this class of nonlinear relaxed Cocoercive variational inclusions in q-uniformly smooth Banach spaces. Our results improve and generalize the corresponding results of recent works.     

Stabilization of some fractional delay systems of neutral type

In this note, we give new stability tests which enable one to fully characterize the H_∞-stability of systems with transfer function , where h>0 and p,q,r are real polynomials in the variable s^μ for 0<μ<1.As an application of this, in the case r(s)=1 and degp=degq=1, families of H_∞-stabilizing controllers are given and a complete parametrization of all H_∞-stabilizing controllers is obtained when .     

The -basis and implicitization of a rational parametric surface

The concept of a μ-basis was introduced in the case of parametrized curves in 1998 and generalized to the case of rational ruled surfaces in 2001. The μ-basis can be used to recover the parametric equation as well as to derive the implicit equation of a rational curve or surface. Furthermore, it can be used for surface reparametrization and computation of singular points. In this paper, we generalize the notion of a μ-basis to an arbitrary rational parametric surface. We show that: (1) the μ-basis of a rational surface always exists, the geometric significance of which is that any rational surface can be expressed as the intersection of three moving planes without extraneous factors; (2) the μ-basis is in fact a basis of the moving plane module of the rational surface; and (3) the μ-basis is a basis of the corresponding moving surface ideal of the rational surface when the base points are local complete intersections. As a by-product, a new algorithm is presented for computing the implicit equation of a rational surface from the μ-basis. Examples provide evidence that the new algorithm is superior than the traditional algorithm based on direct computation of a Gröbner basis. Problems for further research are also discussed.     

Input–output approach to stability and -gain analysis of systems with time-varying delays

Stability and L₂ (l₂)-gain of linear (continuous-time and discrete-time) systems with uncertain bounded time-varying delays are analyzed under the assumption that the nominal delay values are not equal to zero. The delay derivatives (in the continuous-time) are not assumed to be less than q<1. An input–output approach is applied by introducing a new input–output model, which leads to effective frequency domain and time domain criteria. The new method significantly improves the existing results for delays with derivatives not greater than 1, which were treated in the past as fast-varying delays (without any constraints on the delay derivatives). New bounded real lemmas (BRLs) are derived for systems with state and objective vector delays and norm-bounded uncertainties. Numerical examples illustrate the efficiency of the new method.     

Algorithmic proofs of two theorems of Stafford

Two classical results of Stafford say that every (left) ideal of the n-th Weyl algebra A_n can be generated by two elements, and every holonomic A_n-module is cyclic, i.e. generated by one element. We modify Stafford’s original proofs to make the algorithmic computation of these generators possible.     

Delay-dependent robust stability and control for jump linear systems with delays

This paper considers robust stochastic stability, stabilization and H_∞ control problems for a class of jump linear systems with time delays. By using some zero equations, neither model transformation nor bounding for cross terms is required to obtain the delay-dependent results, which are given in terms of linear matrix inequalities (LMIs). Maximum sizes of time delays are also studied for system stability. Furthermore, solvability conditions and corresponding H_∞ control laws are given which provide robust stabilization with a prescribed H_∞ disturbance attenuation level. Numerical examples show that the proposed methods are much less conservative than existing results.     

Powers in a class of -strict standard episturmian words

This paper concerns a specific class of strict standard episturmian words whose directive words resemble those of characteristic Sturmian words. In particular, we explicitly determine all integer powers occurring in such infinite words, extending recent results of Damanik and Lenz [D. Damanik, D. Lenz, Powers in Sturmian sequences, European J. Combin. 24 (2003) 377–390, doi:10.1016/S0195-6698(03)00026-X], who studied powers in Sturmian words. The key tools in our analysis are canonical decompositions and a generalization of singular words, which were originally defined for the ubiquitous Fibonacci word. Our main results are demonstrated via some examples, including the k-bonacci word, a generalization of the Fibonacci word to a k-letter alphabet (k≥2).     

Approximation by Bézier type of Meyer–König and Zeller operators

In this paper, we give direct, inverse and equivalence approximation theorems for the Bézier type of Meyer–König and Zeller operator with unified Ditzian–Totik modulus ω_φ^λ(f,t) (0≤λ≤1).     

The complete inclusion structure of leaf power classes

Let k≥2 be an integer and G=(V,E) be a finite simple graph. A tree T is a k-leaf root of G, if V is the set of leaves of T and, for any two distinct x,yV, the distance between x and y in T is at most k if and only if xyE. We say that G is a k-leaf power if there is a k-leaf root of G. The main result of this paper is that, for all 2≤k<k^′, the classes of k- and k^′-leaf powers are inclusion-incomparable, if and only if k^′≤2k−3 and k^′−k is an odd number. With this result, an open problem from the literature about the inclusion structure of these graph classes is solved completely. In addition, the intersection of the smallest pair of inclusion-incomparable classes is studied.     

Existence of positive solutions for th-order singular superlinear boundary value problems

This paper investigates the existence of positive solutions for 2nth-order (n>1) singular superlinear boundary value problems. A necessary and sufficient condition for the existence of C²ⁿ⁻²[0,1] as well as C²ⁿ⁻¹[0,1] positive solutions is given by constructing a special cone and with the e-Norm.     

Finding an Euclidean anti--centrum location of a set of points

An obnoxious facility is to be located inside a polygonal region of the plane, maximizing the sum of the k smallest weighted Euclidean distances to n given points, each protected by some polygonal forbidden region. For the unweighted case and k fixed an O(n²logn) time algorithm is presented. For the weighted case a thorough study of the relevant structure of the multiplicatively weighted order-k-Voronoi diagram leads to the design of an O(kn³+n³logn) time algorithm for finding an optimal solution to the anti-t-centrum problem for every t=1,…,k, simultaneously.     

A modified hybrid projection method for solving generalized mixed equilibrium problems and fixed point problems in Banach spaces

In this paper, we introduce a modified new hybrid projection method for finding the set of solutions of the generalized mixed equilibrium problems and the convex feasibility problems for an infinite family of closed and uniformly quasi-?-asymptotically nonexpansive mappings. Strong convergence theorems are established in a uniformly smooth and strictly convex Banach space which also enjoys the Kadec-Klee property. Our results improve and extend the corresponding results announced by Qin et al. (2010) and many authors.     

Stability, robust stabilization and control of singular-impulsive systems via switching control

In this paper, stability, robust stabilization and H_∞ control of singular-impulsive systems are studied. Some new fundamental properties are derived for switched singular systems subject to impulse effects. Applying the Lyapunov function theory, several sufficient conditions are established for exponential stability, robust stabilization and H_∞ control of the corresponding singular-impulsive closed-loop systems. Some numerical examples are given to demonstrate the effectiveness of the proposed control and stabilization methods.     

Minimality considerations for graph energy over a class of graphs

Let G be a graph on n vertices, and let CHP(G;λ) be the characteristic polynomial of its adjacency matrix A(G). All n roots of CHP(G;λ), denoted by , are called to be its eigenvalues. The energy E(G) of a graph G, is the sum of absolute values of all eigenvalues, namely, . Let be the set of n-vertex unicyclic graphs, the graphs with n vertices and n edges. A fully loaded unicyclic graph is a unicyclic graph taken from with the property that there exists no vertex with degree less than 3 in its unique cycle. Let be the set of fully loaded unicyclic graphs. In this article, the graphs in with minimal and second-minimal energies are uniquely determined, respectively.     

Molecular surfaces on proteins via beta shapes

A protein consists of linearly combined amino acids via peptide bonds, and an amino acid consists of atoms. It is known that the geometric structure of a protein is the primary factor which determines the functions of the protein.Given the atomic complex of a protein, one of the most important geometric structures of a protein is its molecular surface since this distinguishes between the interior and exterior of the protein and plays an important role in protein folding, docking, interactions between proteins, and other functions.This paper presents an algorithm for the precise and efficient computation of the molecular surface of a protein, using a recently proposed geometric construct called the β-shape based on the Voronoi diagram of atoms in a protein. Given a Voronoi diagram of atoms, based on the Euclidean distance from the atom surfaces, the proposed algorithm first computes the β-shape with an appropriate sized probe. Then, the molecular surface is computed by employing a blending operation on the atomic complex of the protein. In this paper, it is also shown that for a given Voronoi diagram of atoms, the multiple molecular surfaces can be computed by using various sized probes.     

Reduced-order filtering for singular systems

This paper solves the problem of reduced-order H_∞ filtering for singular systems. The purpose is to design linear filters with a specified order lower than the given system such that the filtering error dynamic system is regular, impulse-free (or causal), stable, and satisfies a prescribed H_∞ performance level. One major contribution of the present work is that necessary and sufficient conditions for the solvability of this problem are obtained for both continuous and discrete singular systems. These conditions are characterized in terms of linear matrix inequalities (LMIs) and a coupling non-convex rank constraint. Moreover, an explicit parametrization of all desired reduced-order filters is presented when these inequalities are feasible. In particular, when a static or zeroth-order H_∞ filter is desired, it is shown that the H_∞ filtering problem reduces to a convex LMI problem. All these results are expressed in terms of the original system matrices without decomposition, which makes the design procedure simple and directly. Last but not least, the results have generalized previous works on H_∞ filtering for state-space systems. An illustrative example is given to demonstrate the effectiveness of the proposed approach.     

control for fast sampling discrete-time singularly perturbed systems

This paper is concerned with the H_∞ control problem via state feedback for fast sampling discrete-time singularly perturbed systems. A new H_∞ controller design method is given in terms of solutions to linear matrix inequalities (LMIs), which eliminates the regularity restrictions attached to the Riccati-based solution. A method for evaluating the upper bound of singular perturbation parameter with meeting a prescribed H_∞ performance bound requirement is also given. Furthermore, the results are extended to robust controller design for fast sampling discrete-time singularly perturbed systems with polytopic uncertainties. Numerical examples are given to illustrate the validity of the proposed methods.