Semigroup generation and stabilization by A-bounded feedback perturbations

Let A be a generator of a strongly continuous semigroup of operators, and assume that C and H are operators such that A + CH generates a strongly continuous semigroup S_H(t) on X. Let λ₀ be a real number in the resolvent set of A, and let ε [−1, 1]. Then there are some fairly unrestrictive conditions under which A+(λ₀ − A)CH(λ₀ − A)⁻ also generates a strongly continuous semigroup S_K(t) on X which has the same exponential growth rate as S_H(t). Given an input operator B, we can use this to identify a class of feedback perturbations K such that A + BK generates a strongly continuous semigroup. We can also use this result to identify classes of feedbacks which can and cannot uniformly stabilize a system. For example, we show that if the control on a cantilever beam in the state space H₀²[0, 1] × L²[0, 1] is a moment force on the free end, then we cannot stabilize the beam with an A^−1/2-bounded feedback, but we can find an A^−1/4-bounded feedback, for any > 0, which does stabilize the beam.     

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.     

Maximum flow and critical cutset as descriptors of multi-state systems with randomly capacitated components

Let G = (V, E, s, t) denote a directed network with node set V, arc set E = {1,…, n}, source node s and sink node t. Let Γ denote the set of all minimal s−t cutsets and b₁(τ), …, B_n(τ), the random arc capacities at time τ with known joint probability distribution function. Let Λ(τ) denote the maximum s−t flow at time τ and D(τ), the corresponding critical minimal s−t cutset. Let Ω denote a set of minimal s−t cutsets. This paper describes a comprehensive Monte Carlo sampling plan for efficiently estimating the probability that D(τ)εΩ-Γ and x<λ(τ)y at time τ and the probability that D(τ) Ω given that x < Λ(τ) y at time τ. The proposed method makes use of a readily obtainable upper bound on the probability that Λ(τ) > x to gain its computational advantage. Techniques are described for computing confidence intervals and credibility measures for assessing that specified accuracies have been achieved. The paper includes an algorithm for performing the Monte Carlo sampling experiment, an example to illustrate the technique and a listing of all steps needed for implementation.     

Spectral properties of infinite-dimensional closed-loop systems

We consider feedback systems obtained from infinite-dimensional well-posed linear systems by output feedback. Thus, our framework allows for unbounded control and observation operators. Our aim is to investigate the relationship between the open-loop system, the feedback operator K and the spectrum (in particular, the eigenvalues and eigenvectors) of the closed-loop generator A^K. We give a useful characterization of that part of the spectrum σ(A^K) which lies in the resolvent set of A, the open-loop generator, via the “characteristic equation” involving the open-loop transfer function. We show that certain points of σ(A) cannot be eigenvalues of A^K if the input and output are scalar (so that K is a number) and K≠0. We devote special attention to the case when the output feedback operator K is compact. It is relatively easy to prove that in this case, σ_e(A), the essential spectrum of A, is invariant, that is, it is equal to σ_e(A^K). A related but much harder problem is to determine the largest subset of σ(A) which remains invariant under compact output feedback. This set, which we call the immovable spectrum of A, strictly contains σ_e(A). We give an explicit characterization of the immovable spectrum and we investigate the consequences of our results for a certain class of well-posed systems obtained by interconnecting an infinite chain of identical systems.     

Recursive Data Types in Algebraically ω-Complete Categories

A category (of data types) is called algebraically ω-complete provided that for each endofunctor T the data-type equation T(X) X has a solution constructed as a colimit of the ω-chain → T() → T²()..., where is the initial data-type. Examples include the categories of (1) countable sets and (total, partial, or nondeterministic) functions, (2) countably dimensional vector spaces and linear functions, (3) countable well-ordered sets and join-preserving functions. In the case of categories enriched over CPO (the category of complete partial orders and strict, continuous functions) a stronger property holds for all locally continuous functors T: the data-type equation is both a limit and a colimit of the finite iterations of T over the initial data-type.     

On the connection between the exponential stability of C0-semigroups and the admissibility of a certain Sobolev space

Let T be a strongly continuous semigroup on a Banach space X and A its infinitesimal generator. We will prove that T is exponentially stable, if and only if, there exist p[1,∞) such that the space is admissible to the system Σ(A,I,I), defined below (i.e for all f belonging to the Sobolev space the convolution T*f lies in .     

Robust stabilization of infinite dimensional systems by finite dimensional controllers

The problem of robustly stabilizing an infinite dimensional system with transfer function G, subject to an additive perturbation Δ is considered. It is assumed that: G ε ₀(σ) of systems introduced by Callier and Desoer [3]; the perturbation satisfies |W₁ΔW₂| < ε, where W₁ and W₂ are stable and minimum phase; and G and G + Δ have the same number of poles in ⁺. Now write W₁GW₂=G₁ + G₁, where G₁ is rational and totally unstable and G₂ is stable. Generalizing the finite dimensional results of Glover [12] this family of perturbed systems is shown to be stabilizable if and only if ε σ_min (G^*₁)( = the smallest Hankel singular value of G^*₁). A finite dimensional stabilizing controller is then given by where 2 is a rational approximation of G₂ such that

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) and K₁ robustly stabilizes G₁ to margin ε. The feedback system (G, K) will then be stable if |W₁ΔW₂| _∞< ε − Δ.     

Simultaneously controllable fuzzy matrices

Let F be a set of n × n fuzzy matrices. F is called simultaneously controllable if there exists a permutation matrix P such that for each A ε F, C = [c_ij] = P A P^T satisfies c_ij ≥ c_ji for i > j, where is the max-min composition. In this paper, the necessary and sufficient conditions for a set of n × n fuzzy matrices to be simultaneously controllable will be established. A constructive algorithm which can determine a simultaneously controllable set of n × n fuzzy matrices is presented as well.     

Toward a theory of Pollard's rho method

Pollard's “rho” method for integer factorization iterates a simple polynomial map and produces a nontrivial divisor of n when two such iterates agree modulo this divisor. Experience and heuristic arguments suggest that a prime divisor p should be detected in steps, but this has never been proved. Indeed, nothing seems to be have been rigorously proved about the probability of success that would improve the obvious lower bound of 1/p. This paper shows that for fixed k, this probability is at least (₂^k)/p + O(p^−3/2) as p → ∞. This leads to an Ω(log²p)/p estimate of the success probability.     

Strategy for a class of games with dynamic ties

Let 0 < 2b ≤ a < c be integers. Two players play alternately with a pile of stones. Each player at his turn selects one move from the following two: (i) Remove k stones from the pile subject to 1 ≤ k ≤ a or c + 1 ≤ k ≤ c + a. (ii) If the number m of stones in the pile satisfies m ≡ 2b (mod 2a), add a stones to the pile. The player making the last move wins. If there is no last move, the game is a (dynamic) tie. The Generalized Sprague—Grundy function G is determined, thus giving the strategy of play for the game and its disjunctive compound. An algorithm requiring O(a²) steps for computing G is given. It turns out that G = G (a, b, c) is of a rather complicated form. The main interest of the paper is in presenting a complete strategy for a class of games with dynamic ties.     

Embedding meshes in Boolean cubes by graph decomposition

This paper explores the embeddings of multidimensional meshes into minimal Boolean cubes by graph decomposition. The dilation and the congestion of the product graph (G₁ × G₂) → (H₁ × H₂) is the maximum of the dilation and congestion for the two embeddings G₁ → H₁ and G₂ → H₂. The graph decomposition technique can be used to improve the average dilation and average congestion. The graph decomposition technique combined with some particular two-dimensional embeddings allows for minimal-expansion, dilation-two, congestion-two embeddings of about 87% of all two-dimensional meshes, with a significantly lower average dilation and congestion than by modified line compression. For three-dimensional meshes we show that the graph decomposition technique, together with two three-dimensional mesh embeddings presented in this paper and modified line compression, yields dilation-two embeddings of more than 96% of all three-dimensional meshes contained in a 512 × 512 × 512 mesh. The graph decomposition technique is also used to generalize the embeddings to meshes with wrap-around. The dilation increases by at most one compared to a mesh without wraparound. The expansion is preserved for the majority of meshes, if a wraparound feature is added to the mesh.     

Inclusion dynamics hybrid automata

Hybrid systems are dynamical systems with the ability to describe mixed discrete-continuous evolution of a wide range of systems. Consequently, at first glance, hybrid systems appear powerful but recalcitrant, neither yielding to analysis and reasoning through a purely continuous-time modeling as with systems of differential equations, nor open to inferential processes commonly used for discrete state-transition systems such as finite state automata. A convenient and popular model, called hybrid automata, was introduced to model them and has spurred much interest on its tractability as a tool for inference and model checking in a general setting. Intuitively, a hybrid automaton is simply a “finite-state” automaton with each state augmented by continuous variables, which evolve according to a set of well-defined continuous laws, each specified separately for each state. This article investigates both the notion of hybrid automaton and the model checking problem over such a structure. In particular, it relates first-order theories and analysis results on multivalued maps and reduces the bounded reachability problem for hybrid automata whose continuous laws are expressed by inclusions (x′f(x,t)) to a decidability problem for first-order formulæ over the reals. Furthermore, the paper introduces a class of hybrid automata for which the reachability problem can be decided and shows that the problem of deciding whether a hybrid automaton belongs to this class can be again decided using first-order formulæ over the reals. Despite the fact that the bisimulation quotient for this class of hybrid automata can be infinite, we show that our techniques permit effective model checking for a nontrivial fragment of CTL.     

Involution codes: with application to DNA coded languages

For an involution θ : Σ^* → Σ^* over a finite alphabet Σ we consider involution codes: θ-infix, θ-comma-free, θ-k -codes and θ-subword-k-codes. These codes arise from questions on DNA strand design. We investigate conditions under which both X and X⁺ are same type of involution codes. General methods for generating such involution codes are given. The information capacity of these codes show to be optimized in most cases. A specific set of these codes was chosen for experimental testing and the results of these experiments are presented.     

A Cubic Kernel for Feedback Vertex Set and Loop Cutset

The Feedback Vertex Set problem on unweighted, undirected graphs is considered. Improving upon a result by Burrage et al. (Proceedings 2nd International Workshop on Parameterized and Exact Computation, pp. 192–202, 2006), we show that this problem has a kernel with O(k ³) vertices, i.e., there is a polynomial time algorithm, that given a graph G and an integer k, finds a graph G′ with O(k ³) vertices and integer k′≤k, such that G has a feedback vertex set of size at most k, if and only if G′ has a feedback vertex set of size at most k′. Moreover, the algorithm can be made constructive: if the reduced instance G′ has a feedback vertex set of size k′, then we can easily transform a minimum size feedback vertex set of G′ into a minimum size feedback vertex set of G. This kernelization algorithm can be used as the first step of an FPT algorithm for Feedback Vertex Set, but also as a preprocessing heuristic for Feedback Vertex Set.     

Approximating the Minimum Chain Completion problem

A bipartite graph G=(U,V,E) is a chain graph [M. Yannakakis, Computing the minimum fill-in is NP-complete, SIAM J. Algebraic Discrete Methods 2 (1) (1981) 77–79] if there is a bijection such that Γ(π(1))Γ(π(2))Γ(π(|U|)), where Γ is a function that maps a node to its neighbors.We give approximation algorithms for two variants of the Minimum Chain Completion problem, where we are given a bipartite graph G(U,V,E), and the goal is find the minimum set of edges F that need to be added to G such that the bipartite graph G^′=(U,V,E^′) (E^′=EF) is a chain graph.     

Uniform finite generation of compact Lie groups

Consider a compact connected Lie group G and the corresponding Lie algebra . Let {X₁,…,X_m} be a set of generators for the Lie algebra . We prove that G is uniformly finitely generated by {X₁,…,X_m}. This means that every element KG can be expressed as K=e^Xt₁e^Xt₂···e^Xt_l, where the indeterminates X are in the set {X₁,…,X_m}, , and the number l is uniformly bounded. This extends a previous result by F. Lowenthal in that we do not require the connected one dimensional Lie subgroups corresponding to the X_i, i=1,…,m, to be compact. We link the results to the existence of universal logic gates in quantum computing and discuss the impact on bang bang control algorithms for quantum mechanical systems.     

On Unbounded Tolerable Solution Sets

We prove that the nonempty tolerable solution set of the interval linear system Ax = b is unbounded if and only if the interval matrix of the system has linearly dependent degenerate columns. Also, we prove that the tolerable solution set may be represented as a sum of the linear subspace {c ε ℝⁿ ∨ Ac = 0} and a bounded convex polyhedron and propose a way for suitable estimation of unbounded tolerable solution sets.     

Cluster points and stability

The relationship is examined between the stability properties of the solution of the discrete time system where A(·): R ⁿ →R ⁿ is a continuous non-linear map, and the set of cluster points L(x*) of the sequence {x_i } _i=0 ^∞. We show that the sequence is bounded if and only if the set L(x*) is contained in a compact set. We review also the contributions of other authors in this area.     

Needed reduction and spine strategies for the lambda calculus

A redex R in a lambda-term M is called needed if in every reduction of M to normal form (some residual of) R is contracted. Among others the following results are proved: 1. R is needed in M iff R is contracted in the leftmost reduction path of M. 2. Let : M₀ →M₁ → M₂ → … reduce redexes R_i: M_i → M_i+1, and have the property that i.j≥i.R_j is needed in M_j. Then is normalising, i.e., if M₀ has a normal form, then is finite and terminates at that normal form. 3. Neededness is an undecidable property, but has several efficiently decidable approximations, various versions of the so-called spine redexes.     

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.