Isolation concepts for efficiently enumerating dense subgraphs

In an undirected graph G=(V,E), a set of k vertices is called c-isolated if it has less than ck outgoing edges. Ito and Iwama [H. Ito, K. Iwama, Enumeration of isolated cliques and pseudo-cliques, ACM Transactions on Algorithms (2008) (in press)] gave an algorithm to enumerate all c-isolated maximal cliques in O(4^cc⁴|E|) time. We extend this to enumerating all maximal c-isolated cliques (which are a superset) and improve the running time bound to O(2.89^cc²|E|), using modifications which also facilitate parallelizing the enumeration. Moreover, we introduce a more restricted and a more general isolation concept and show that both lead to faster enumeration algorithms. Finally, we extend our considerations to s-plexes (a relaxation of the clique notion), providing a W[1]-hardness result when the size of the s-plex is the parameter and a fixed-parameter algorithm for enumerating isolated s-plexes when the parameter describes the degree of isolation.     

Limitations of nonlinear discrete periodic control for disturbance attenuation and robust stabilization

This paper presents a performance analysis of nonlinear periodically time-varying discrete controllers acting upon a linear time-invariant discrete plant. Time-invariant controllers are distinguished from strictly periodically time-varying controllers. For a given nonlinear periodic controller, a time-invariant controller is constructed. Necessary and sufficient conditions are given under which the time-invariant controller gives strictly better control performance than the time-invariant controller from which it was obtained, for the attenuation of l_p exogenous disturbances and the robust stabilization of l_p unstructured perturbations, for all p[1,∞].     

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.     

-admissibility of the right-shift semigroup on

It is shown that the right-shift semigroup on does not satisfy the weighted Weiss conjecture for α(0,1). In other words, α-admissibility of scalar valued observation operators cannot always be characterised by a simple resolvent growth condition. This result is in contrast to the unweighted case, where 0-admissibility can be characterised by a simple growth bound. The result is proved by providing a link between discrete and continuous α-admissibility and then translating a counterexample for the unilateral shift on to continuous time systems.     

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.     

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 .     

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.     

Robust filtering for 2-D discrete-time linear systems with convex-bounded parameter uncertainty

This paper is concerned with the problems of robust H_∞ and H₂ filtering for 2-dimensional (2-D) discrete-time linear systems described by a Fornasini-Marchesini second model with matrices that depend affinely on convex-bounded uncertain parameters. By a suitable transformation, the system is represented by an equivalent difference-algebraic representation. A parameter-dependent Lyapunov function approach is then proposed for the design of 2-D stationary discrete-time linear filters that ensure either a prescribed H_∞ performance or H₂ performance for all admissible uncertain parameters. The filter designs are given in terms of linear matrix inequalities. Numerical examples illustrate the effectiveness of the proposed filter design methods.     

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.     

Filter design for LPV systems using quadratically parameter-dependent Lyapunov functions

This paper considers design problems of robust gain-scheduled H_∞ and H₂ filters for linear parameter-varying (LPV) systems whose state-space matrices are represented as parametrically affine matrices, using quadratically parameter-dependent Lyapunov functions, and proposes methods of filter design via parametrically affine linear matrix inequalities (LMIs). For robust filters, our design methods theoretically encompass those that use constant Lyapunov functions. Several numerical examples are included that demonstrate the effectiveness of gain-scheduled and robust filters using our proposed methods compared with robust filters using existing methods.     

filtering for uncertain stochastic time-delay systems with sector-bounded nonlinearities

In this paper, we deal with the robust H_∞ filtering problem for a class of uncertain nonlinear time-delay stochastic systems. The system under consideration contains parameter uncertainties, Itô-type stochastic disturbances, time-varying delays, as well as sector-bounded nonlinearities. We aim at designing a full-order filter such that, for all admissible uncertainties, nonlinearities and time delays, the dynamics of the filtering error is guaranteed to be robustly asymptotically stable in the mean square, while achieving the prescribed H_∞ disturbance rejection attenuation level. By using the Lyapunov stability theory and Itô’s differential rule, sufficient conditions are first established to ensure the existence of the desired filters, which are expressed in the form of a linear matrix inequality (LMI). Then, the explicit expression of the desired filter gains is also characterized. Finally, a numerical example is exploited to show the usefulness of the results derived.     

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.     

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.     

An LMI approach to boundary control of semilinear parabolic and hyperbolic systems

Exponential stability analysis and L₂-gain analysis are developed for scalar uncertain distributed parameter systems, governed by semilinear partial differential equations of parabolic and hyperbolic types. Sufficient exponential stability conditions with a given decay rate are derived in the form of Linear Matrix Inequalities (LMIs) for both systems. These conditions are then utilized to synthesize  static output feedback boundary controllers of the systems in question.     

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.     

Reduction of Permutation-Invariant Polynomials: A Noncommutative Case Study

Let R be a commutative ring with 1, let RX₁,…,X_n/I be the polynomial algebra in the n≥4 noncommuting variables X₁,…,X_n over R modulo the set of commutator relations I={(X₁+···+X_n)*X_i=X_i*(X₁+···+X_n)|1≤i≤n}. Furthermore, let G be an arbitrary group of permutations operating on the indeterminates X₁,…,X_n, and let RX₁,…,X_n/I^G be the R-algebra of G-invariant polynomials in RX₁,…,X_n/I. The first part of this paper is about an algorithm, which computes a representation for any fRX₁,…,X_n/I^G as a polynomial in multilinear G-invariant polynomials, i.e., the maximal variable degree of the generators of RX₁,…,X_n/I^G is at most 1. The algorithm works for any ring R and for any permutation group G. In addition, we present a bound for the number of necessary generators for the representation of all G-invariant polynomials in RX₁,…,X_n/I^G with a total degree of at most d. The second part contains a first but promising analysis of G-invariant polynomials of solvable polynomial rings.     

Computing the nearest polynomial with a zero in a given domain by using piecewise rational functions

For a real univariate polynomial f and a closed complex domain D whose boundary C is a simple curve parameterized by a univariate piecewise rational function, a rigorous method is given for finding a real univariate polynomial such that has a zero in D and is minimal. First, it is proved that the minimum distance between f and polynomials having a zero at α∈C is a piecewise rational function of the real and imaginary parts of α. Thus, on C, the minimum distance is a piecewise rational function of a parameter obtained through the parameterization of C. Therefore, can be constructed by using the property that has a zero on C and computing the minimum distance on C. We analyze the asymptotic bit complexity of the method and show that it is of polynomial order in the size of the input.     

Full-rank representation of generalized inverse AT,S and its application

In this paper, we introduce a full-rank representation of the generalized inverse of a given complex matrix A, which is based on an arbitrary full-rank decomposition of G, where G is a matrix such that R(G)=T and N(G)=S. Using this representation, we introduce the minor of the generalized inverse ; as a special case of the minor, a determinantal representation of the generalized inverse is obtained. As an application, we use an example to demonstrate that this representation is correct.     

Robust observers for neutral jumping systems with uncertain information

In this paper, the problem of designing observer for a class of uncertain neutral systems. The uncertainties are parametric and norm-bounded. Both robust observation and robust H_∞ observation methods are developed by using linear state-delayed observers. In case of robust observation, sufficient conditions are established for asymptotic stability of the system, which is independent of time delay. The results are then extended to robust H_∞ observation which renders the augmented system asymptotically stable independent of delay with a guaranteed performance measure. Furthermore, a memoryless state-estimate feedback is designed to stabilize the closed-loop neutral system. In all cases, the gain matrices are determined by linear matrix inequality approach. Two numerical examples are presented to illustrate the validity of the theoretical results.