On-line graph coloring of {\mathbb{P}_5}-free graphs

Kierstead et al. (SIAM J Discret Math 8:485–498, 1995) have shown 1 that the competitive function of on-line coloring for -free graphs (i.e., graphs without induced path on 5 vertices) is bounded from above by the exponential function . No nontrivial lower bound was known. In this paper we show the quadratic lower bound . More precisely, we prove that is the exact competitive function for ()-free graphs. In this paper we also prove that 2 - 1 is the competitive function of the best clique covering on-line algorithm for ()-free graphs.     

Simple a posteriori error estimators for the <Emphasis Type="Italic">h</Emphasis>-version of the boundary element method

The h-h/2-strategy is one well-known technique for the a posteriori error estimation for Galerkin discretizations of energy minimization problems. One considers to estimate the error , where is a Galerkin solution with respect to a mesh and is a Galerkin solution with respect to the mesh obtained from a uniform refinement of . This error estimator is always efficient and observed to be also reliable in practice. However, for boundary element methods, the energy norm is non-local and thus the error estimator η does not provide information for a local mesh-refinement. We consider Symm’s integral equation of the first kind, where the energy space is the negative-order Sobolev space . Recent localization techniques allow to replace the energy norm in this case by some weighted L ²-norm. Then, this very basic error estimation strategy is also applicable to steer an h-adaptive algorithm. Numerical experiments in 2D and 3D show that the proposed method works well in practice. A short conclusion is concerned with other integral equations, e.g., the hypersingular case with energy space and , respectively, or a transmission problem. Dedicated to Professor Ernst P. Stephan on the occasion of his 60th birthday.     

A Novel Class of Symmetric and Nonsymmetric Periodizing Variable Transformations for Numerical Integration

Variable transformations for numerical integration have been used for improving the accuracy of the trapezoidal rule. Specifically, one first transforms the integral via a variable transformation that maps [0,1] to itself, and then approximates the resulting transformed integral by the trapezoidal rule. In this work, we propose a new class of symmetric and nonsymmetric variable transformations which we denote , where r and s are positive scalars assigned by the user. A simple representative of this class is . We show that, in case , or but has algebraic (endpoint) singularities at x = 0 and/or x = 1, the trapezoidal rule on the transformed integral produces exceptionally high accuracies for special values of r and s. In particular, when and we employ , the error in the approximation is (i) O(h ^r ) for arbitrary r and (ii) O(h ^2r ) if r is a positive odd integer at least 3, h being the integration step. We illustrate the use of these transformations and the accompanying theory with numerical examples.      

Uniform stabilization for linear systems with persistency of excitation: the neutrally stable and the double integrator cases

Consider the controlled system dx/dt = Ax + α(t)Bu where the pair (A, B) is stabilizable and α(t) takes values in [0, 1] and is persistently exciting, i.e., there exist two positive constants μ, T such that, for every t ≥ 0, ${\int_t^{t+T}\alpha(s){\rm d}s \geq \mu}Consider the controlled system dx/dt = Ax + α(t)Bu where the pair (A, B) is stabilizable and α(t) takes values in [0, 1] and is persistently exciting, i.e., there exist two positive constants μ, T such that, for every t ≥ 0, . In particular, when α(t) becomes zero the system dynamics switches to an uncontrollable system. In this paper, we address the following question: is it possible to find a linear time-invariant state-feedback u = Kx, with K only depending on (A, B) and possibly on μ, T, which globally asymptotically stabilizes the system? We give a positive answer to this question for two cases: when A is neutrally stable and when the system is the double integrator. Notation  A continuous function is of class , if it is strictly increasing and is of class if it is continuous, non-increasing and tends to zero as its argument tends to infinity. A function is said to be a class -function if, for any t ≥ 0, and for any s ≥ 0. We use |·| for the Euclidean norm of vectors and the induced L ₂-norm of matrices.     

Fuzzy logic models in a category of fuzzy relations

We investigate interpretations of formulas ψ in a first order fuzzy logic in models which are based on objects of a category SetR(Ω) which consists of Ω-sets, i.e. sets with similarity relations with values in a complete MV-algebra Ω and with morphisms defined as special fuzzy relations between Ω-sets. The interpretations are then morphisms in a category SetR(Ω) from some Ω-set to the object . We define homomorphisms between models in a category SetR(Ω) and we prove that if is a (special) homomorphism of models in a category SetR(Ω) then there is a relation between interpretations of a formula ψ in models . Supported by MSM6198898701, grant 201/07/0191 of GAČR and grant 1M0572.     

Strong non-standard completeness for fuzzy logics

In this paper we are going to introduce the notion of strong non-standard completeness (SNSC) for fuzzy logics. This notion naturally arises from the well known construction by ultraproduct. Roughly speaking, to say that a logic is strong non-standard complete means that, for any countable theory Γ over and any formula φ such that , there exists an evaluation e of -formulas into a -algebra such that the universe of is a non-Archimedean extension of the real unit interval [0,1], e is a model for Γ, but e(φ) < 1. Then we will apply SNSC to prove that various modal fuzzy logics allowing to deal with simple and conditional probability of infinite-valued events are complete with respect to classes of models defined starting from non-standard measures, that is measures taking value in .     

On the Relation between States and Maps in Infinite Dimensions

Relations between states and maps, which are known for quantum systems in finitedimensional Hilbert spaces, are formulated rigorously in geometrical terms with no use of coordinate (matrix) interpretation. In a tensor product realization they are represented simply by a permutation of factors. This leads to natural generalizations for infinite-dimensional Hilbert spaces and a simple proof of a generalized Choi Theorem. The natural framework is based on spaces of Hilbert-Schmidt operators and the corresponding tensor products of Hilbert spaces. It is proved that the corresponding isomorphisms cannot be naturally extended to compact (or bounded) operators, nor reduced to the trace-class operators. On the other hand, it is proven that there is a natural continuous map from trace-class operators on (with the nuclear norm) into compact operators mapping the space of all bounded operators on into trace class operators on (with the operator-norm). Also in the infinite-dimensional context, the Schmidt measure of entanglement and multipartite generalizations of state-maps relations are considered in the paper.     

Population Variance under Interval Uncertainty: A New Algorithm

In statistical analysis of measurement results, it is often beneficial to compute the range V of the population variance when we only know the intervals of possible values of the x_i. In general, this problem is NP-hard; a polynomialtime algorithm is known for the case when the measurements are sufficiently accurate, i.e., when for all In this paper, we show that we can efficiently compute V under a weaker (and more general) condition .     

\mathcal{C}oo\mathcal{PS} – Towards a method for coordinating personalized services

This paper presents oo which is a method for oordinating ersonalized Services. These services are primarily offered to mobile users. The concept of services is the object of intense investigations from both academia and industry. However, very little has been accomplished so far regarding first, personalizing services for the benefit of mobile users, and second, providing the appropriate methodological support for those (i.e., designers) who will be specifying the operations of personalization. Various obstacles still exist such as lack of techniques for modeling and specifying the integration of personalization into services, and existing approaches for service composition typically facilitate orchestration only, while neglecting contexts of users and services. ooconsists of several steps ranging from service definition and personalization to service deployment. Each step has some representation techniques, which aim at facilitating the specification and validation of the operations of coordinating personalized services. Zakaria Maamar is an associate professor in computer sciences at Zayed University, Dubai, United Arab Emirates. His research interests include Web services, software agents, and context-aware computing. Maamar has a PhD in computer sciences from Laval University. Djamal Benslimane is a full professor in computer sciences at Claude Bernard Lyon 1 University and a member of the Laboratoire d'InfoRmatique en Images et Systèmes d'information- Centre National De la Recherche Scientifique (LIRIS-CNRS), both in Lyon, France. His research interests include interoperability, Web services, and ontologies. Benslimane has a PhD in computer sciences from Blaise Pascal University. Michael Mrissa is a Ph.D. candidate in computer sciences at Claude Bernard Lyon 1 University and a member of the Laboratoire d'InfoRmatique en Images et Systèmes d'information - Centre National De la Recherche Scientifique (LIRIS-CNRS), both in Lyon, France. His research interests include semantic Web services, interoperability and peer-to-peer networks. Chirine Ghedira is an associate professor in computer sciences at Claude Bernard Lyon 1 University and a member of the Laboratoire d'InfoRmatique en Images et Systèmes d'information- Centre National De la Recherche Scientifique (LIRIS-CNRS), both in Lyon, France. Her research interests include Web services and context-aware computing. Ghedira has a PhD in computer sciences from the National Institute for Applied Sciences (INSA).     

On the Complexity of 2-Monotone Restarting Automata

The -automaton is the weakest form of the nondeterministic version of the restarting automaton that was introduced by Jančar et al. to model the so-called analysis by reduction. Here it is shown that the class ℒ(R) of languages that are accepted by -automata is incomparable under set inclusion to the class of Church-Rosser languages and to the class of growing context-sensitive languages. In fact this already holds for the class of languages that are accepted by 2-monotone -automata. In addition, we prove that already the latter class contains -complete languages, showing that already the 2-monotone -automaton has a surprisingly large expressive power. The results of this paper have been announced at DLT 2004 in Auckland, New Zealand. This work was mainly carried out while T. Jurdziński was visiting the University of Kassel, supported by a grant from the Deutsche Forschungsgemeinschaft (DFG). F. Mráz and M. Plátek were partially supported by the Grant Agency of the Czech Republic under Grant-No. 201/04/2102 and by the program ‘Information Society’ under project 1ET100300517. F. Mráz was also supported by the Grant Agency of Charles University in Prague under Grant-No. 358/2006/A-INF/MFF.     

Quantum Error Correction of Time-correlated Errors

The complexity of the error correction circuitry forces us to design quantum error correction codes capable of correcting a single error per error correction cycle. Yet, time-correlated error are common for physical implementations of quantum systems; an error corrected during the previous cycle may reoccur later due to physical processes specific for each physical implementation of the qubits. In this paper, we study quantum error correction for a restricted class of time-correlated errors in a spin-boson model. The algorithm we propose allows the correction of two errors per error correction cycle, provided that one of them is time-correlated. The algorithm can be applied to any stabilizer code when the two logical qubits and are entangled states of 2 ⁿ basis states in .      

Strong Completeness and Limited Canonicity for PDL

Propositional dynamic logic () is complete but not compact. As a consequence, strong completeness (the property ) requires an infinitary proof system. In this paper, we present a short proof for strong completeness of relative to an infinitary proof system containing the rule from [α; β ⁿ ]φ for all , conclude . The proof uses a universal canonical model, and it is generalized to other modal logics with infinitary proof rules, such as epistemic knowledge with common knowledge. Also, we show that the universal canonical model of lacks the property of modal harmony, the analogue of the Truth lemma for modal operators.     

Almost direct sum decomposition and implementation of 2D behaviors

This paper deals with multidimensional systems, for example, systems described by linear, constant coefficient partial differential/difference equations. In the behavioral approach, the notion of interconnection is the basis of control. In this setting, feedback interconnection of systems is based on the still more fundamental concept of regular interconnection, which has been introduced by J.C. Willems. The dual problem of regular interconnection is the one of direct sum decomposition. The following two problems are addressed: given a behavior and one of its sub-behaviors , under what conditions does there exist another sub-behavior such that has finite dimension and has finite codimension with respect to i.e. we treat the direct sum decomposition of up to finite-dimensional behaviors, which, in this context, are considered negligible. The second related problem concerns regular interconnections and reads as follows: given a plant behavior together with a desired behavior, find, if possible, another behavior (a controller) such that the interconnection is regular and has finite codimension with respect to the given desired behavior. A constructive solution to the problems is provided for two-dimensional behaviors.     

Computing Population Variance and Entropy under Interval Uncertainty: Linear-Time Algorithms

In statistical analysis of measurement results it is often necessary to compute the range of the population variance when we only know the intervals of possible values of the x _i . While can be computed efficiently, the problem of computing is, in general, NP-hard. In our previous paper “Population Variance under Interval Uncertainty: A New Algorithm” (Reliable Computing 12 (4) (2006), pp. 273–280) we showed that in a practically important case we can use constraints techniques to compute in time O(n · log(n)). In this paper we provide new algorithms that compute (in all cases) and (for the above case) in linear time O(n). Similar linear-time algorithms are described for computing the range of the entropy when we only know the intervals of possible values of probabilities p _i . In general, a statistical characteristic ƒ can be more complex so that even computing ƒ can take much longer than linear time. For such ƒ, the question is how to compute the range in as few calls to ƒ as possible. We show that for convex symmetric functions ƒ, we can compute in n calls to ƒ.     

Ordered fields and {{{\rm \L}\Pi\frac{1}{2}}} -algebras

In this work we further explore the connection between -algebras and ordered fields. We show that any two -chains generate the same variety if and only if they are related to ordered fields that have the same universal theory. This will yield that any -chain generates the whole variety if and only if it contains a subalgebra isomorphic to the -chain of real algebraic numbers, that consequently is the smallest -chain generating the whole variety. We also show that any two different subalgebras of the -chain over the real algebraic numbers generate different varieties. This will be exploited in order to prove that the lattice of subvarieties of -algebras has the cardinality of the continuum. Finally, we will also briefly deal with some model-theoretic properties of -chains related to real closed fields, proving quantifier-elimination and related results.     

On the computability power and the robustness of set agreement-oriented failure detector classes

Solving agreement problems deterministically, such as consensus and k-set agreement, in asynchronous distributed systems prone to an unbounded number of process failures has been shown to be impossible. To circumvent this impossibility, unreliable failure detectors for the crash failure model have been widely studied. These are oracles that provide information on failures. The exact nature of such information is defined by a set of abstract properties that a particular class of failure detectors satisfy. The weakest class of such failure detectors that allow to solve consensus is Ω. This paper considers failure detector classes from the literature that solve k-set agreement in the crash failure model, and studies their relative power. It shows that the family of failure detector classes (1 ≤ x ≤ n), and (0 ≤ y ≤ n), can be “added” to provide a failure detector of the class Ω ^z (1 ≤ z ≤ n, a generalization of Ω). It also characterizes the power of such an “addition”, namely, , can construct Ω ^z iff y + z > t, and can construct Ω ^z iff x + z > t + 1, where t is the maximum number of processes that can crash in a run. As an example, the paper shows that, while allows solving 2-set agreement (but not consensus) and allows solving t-set agreement (but not (t − 1)-set agreement), a system with failure detectors of both classes can solve consensus for any value of t. More generally, the paper studies the failure detector classes , and Ω ^z , and shows which reductions among these classes are possible and which are not. The paper also presents a message-passing Ω ^k -based k-set agreement protocol and shows that Ω ^k is not enough to solve (k − 1)-set agreement. In that sense, it can be seen as a step toward the characterization of the weakest failure detector class that allows solving the k-set agreement problem. An extended abstract of this paper has appeared in the proceedings of PODC 2006 [20]. This work has been supported partially by a grant from LAFMI (Franco-Mexican Lab in Computer Science), the European Network of Excellence ReSIST and PAPIIT-UNAM.     

Sufficient conditions for robustness of $$\mathcal{K}\mathcal{L}$$ -stability for difference inclusions

Difference inclusions arise naturally in the study of discrete-time or sampled-data systems. We develop two novel sufficient conditions for robustness of a stability property referred to as -stability with respect to an arbitrary measure; i.e., where a continuous positive definite function of the solutions satisfies a class- estimate of time and the continuous positive definite function of the initial condition. Christopher M. Kellett was supported by the Australian Research Council under Discovery Project Grant DP0771131. Andrew R. Teel was supported by NSF grants ECS-0324679, ECS-0622253, and AFOSR grants F49620-03-1-0203 and FA9550-06-1-0134.