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.     

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.     

An involute spiral that matches Hermite data in the plane

A construction is given for a planar rational Pythagorean hodograph spiral, which interpolates any two-point G² Hermite data that a spiral can match. When the curvature at one of the points is zero, the construction gives the unique interpolant that is an involute of a rational Pythagorean hodograph curve of the form cubic over linear. Otherwise, the spiral comprises an involute of a Tschirnhausen cubic together with at most two circular arcs. The construction is by explicit formulas in the first case, and requires the solution of a quadratic equation in the second case.     

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).     

Existence and uniqueness theorems for fourth-order singular boundary value problems

By constructing a special cone and using cone compression and expansion fixed point theorem, the existence and uniqueness are established for the following singular fourth-order boundary value problems:

where f(t,x,y) may be singular at t=0,1; x=0 and y=0.     

On admissible pairs and equivalent feedback—Youla parameterization in iterative learning control

This paper revisits a well-known synthesis problem in iterative learning control, where the objective is to optimize a performance criterion over a class of causal iterations. The approach taken here adopts an infinite-time setting and looks at limit behavior.The first part of the paper considers iterations without current-cycle-feedback (CCF) term. A notion of admissibility is introduced to distinguish between pairs of operators that define a robustly converging iteration and pairs that do not. The set of admissible pairs is partitioned into disjoint equivalence classes. Different members of an equivalence class are shown to correspond to different realizations of a (stabilizing) feedback controller. Conversely, every stabilizing controller is shown to allow for a (non-unique) factorization in terms of admissible pairs. Class representatives are introduced to remove redundancy. The smaller set of representative pairs is shown to have a trivial parameterization that coincides with the Youla parameterization of all stabilizing controllers (stable plant case).The second part of the paper considers the general family of CCF-iterations. Results derived in the non-CCF case carry over, with the exception that the set of equivalent controllers now forms but a subset of all stabilizing controllers. Necessary and sufficient conditions for full generalization are given.     

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.     

New applications of clique separator decomposition for the Maximum Weight Stable Set problem

Graph decompositions such as decomposition by clique separators and modular decomposition are of crucial importance for designing efficient graph algorithms. Clique separators in graphs were used by Tarjan as a divide-and-conquer approach for solving various problems such as the Maximum Weight Stable Set (MWS) problem, Colouring and Minimum Fill-in. The basic tool is a decomposition tree of the graph whose leaves have no clique separator (so-called atoms), and the problem can be solved efficiently on the graph if it is efficiently solvable on its atoms. We give new examples where the clique separator decomposition works well for the MWS problem; our results improve and extend various recently published results. In particular, we describe the atom structure for some new classes of graphs whose atoms are P₅-free (the P₅ is the induced path with five vertices) and obtain new polynomial time results for the MWS problem. The complexity of this problem on the class of P₅-free graphs is still unknown.     

On standard control problems for systems with infinitely many unstable poles

In this paper, H^∞ control for a class of linear time invariant systems with infinitely many unstable poles is studied. An example of such a plant is a high gain system with delayed feedback. We formulate the problem via a generalized plant which consists of a rational transfer matrix and the inverse of a scalar (possibly irrational) inner function. It is shown that the problem can be decomposed into a finite-dimensional H^∞ control problem and an additional rank condition.     

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.     

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.     

Constrained curve drawing using trigonometric splines having shape parameters

The problem of drawing a smooth obstacle avoiding curve has attracted the attention of many people working in the area of CAD/CAM and its applications. In the present paper we propose a method of constrained curve drawing using certain C¹-quadratic trigonometric splines having shape parameters, which have been recently introduced in [Han X. Quadratic trigonometric polynomial curves with a shape parameter. Computer Aided Geometric Design 2002;19:503–12]. Besides this, we have also presented a simpler approach for studying the approximation properties of the trigonometric spline curves.     

VCPSS: A two-in-one two-decoding-options image sharing method combining visual cryptography (VC) and polynomial-style sharing (PSS) approaches

This paper presents a novel method to combine two major branches of image sharing: VC and PSS. n transparencies are created for a given gray-valued secret image. If the decoding computer is temporarily not available at (or, not connected to) the decoding scene, we can still physically stack any t received transparencies (tn is a threshold value) to get a vague black-and-white view of the secret image immediately. On the other hand, when the decoding computer is finally available, then we can get a much finer gray-valued view of the secret image using the information hidden in the transparencies. In summary, each transparency is a two-in-one carrier of the information, and the decoding has two options.     

Solvability of nonlocal boundary value problems for ordinary differential equation of higher order with a -Laplacian

Nonlocal boundary value problems at resonance for a higher order nonlinear differential equation with a p-Laplacian are considered in this paper. By using a new continuation theorem, some existence results are obtained for such boundary value problems. An explicit example is also given in this paper to illustrate the main results.     

A class of iterative methods for solving nonsymmetric algebraic Riccati equations arising in transport theory

In this paper, we consider the nonsymmetric algebraic Riccati equation arising in transport theory. An important feature of this equation is that its minimal positive solution can be obtained via computing the minimal positive solution of a vector equation. We propose a class of iterative methods to solve the vector equation. The convergence analysis shows that the sequence of vectors generated by iterative methods with two kinds of specific iterative matrices is monotonically increasing and converges to the minimal positive solution of the vector equation. Numerical experiments show that the new methods outperform the modified simple iterative method and Newton’s method.     

On the decoding of binary cyclic codes with the Newton identities

We revisit in this paper the concept of decoding binary cyclic codes with Gröbner bases. These ideas were first introduced by Cooper, then Chen, Reed, Helleseth and Truong, and eventually by Orsini and Sala. We discuss here another way of putting the decoding problem into equations: the Newton identities. Although these identities have been extensively used for decoding, the work was done manually, to provide formulas for the coefficients of the locator polynomial. This was achieved by Reed, Chen, Truong and others in a long series of papers, for decoding quadratic residue codes, on a case-by-case basis. It is tempting to automate these computations, using elimination theory and Gröbner bases.Thus, we study in this paper the properties of the system defined by the Newton identities, for decoding binary cyclic codes. This is done in two steps, first we prove some facts about the variety associated with this system, then we prove that the ideal itself contains relevant equations for decoding, which lead to formulas.Then we consider the so-called online Gröbner basis decoding, where the work of computing a Gröbner basis is done for each received word. It is much more efficient for practical purposes than preprocessing and substituting into the formulas. Finally, we conclude with some computational results, for codes of interesting length (about one hundred).     

Loop subdivision surfaces interpolating -spline curves

This paper presents a novel method for defining a Loop subdivision surface interpolating a set of popularly-used cubic B-spline curves. Although any curve on a Loop surface corresponding to a regular edge path is usually a piecewise quartic polynomial curve, it is found that the curve can be reduced to a single cubic B-spline curve under certain constraints of the local control vertices. Given a set of cubic B-spline curves, it is therefore possible to define a Loop surface interpolating the input curves by enforcing the interpolation constraints. In order to produce a surface of local or global fair effect, an energy-based optimization scheme is used to update the control vertices of the Loop surface subjecting to curve interpolation constraints, and the resulting surface will exactly interpolate the given curves. In addition to curve interpolation, other linear constraints can also be conveniently incorporated. Because both Loop subdivision surfaces and cubic B-spline curves are popularly used in engineering applications, the curve interpolation method proposed in this paper offers an attractive and essential modeling tool for computer-aided design.     

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 .