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.     

Renaming in synchronous message passing systems with Byzantine failures

We study the renaming problem in a fully connected synchronous network with Byzantine failures. We show that when the original namespace of the processors is unbounded, this problem cannot be solved in an a priori bounded number of rounds for , where n is the size of the network and t is the number of failures. On the other hand, for n > 3t, we present a Byzantine renaming algorithm that runs in O(lg n) rounds. In addition, we present a fast, efficient strong renaming algorithm for n > t, which runs in rounds, where N ₀ is the value of the highest identifier among all the correct processors.     

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.     

Asymptotic Fourier Coefficients for a C∞ Bell (Smoothed-“Top-Hat”) & the Fourier Extension Problem

In constructing local Fourier bases and in solving differential equations with nonperiodic solutions through Fourier spectral algorithms, it is necessary to solve the Fourier Extension Problem. This is the task of extending a nonperiodic function, defined on an interval , to a function which is periodic on the larger interval . We derive the asymptotic Fourier coefficients for an infinitely differentiable function which is one on an interval , identically zero for , and varies smoothly in between. Such smoothed “top-hat” functions are “bells” in wavelet theory. Our bell is (for x ≥ 0) where where . By applying steepest descents to approximate the coefficient integrals in the limit of large degree j, we show that when the width L is fixed, the Fourier cosine coefficients a _j of on are proportional to where Λ(j) is an oscillatory factor of degree given in the text. We also show that to minimize error in a Fourier series truncated after the Nth term, the width should be chosen to increase with N as . We derive similar asymptotics for the function f(x)=x as extended by a more sophisticated scheme with overlapping bells; this gives an even faster rate of Fourier convergence     

A new bound for the D0L sequence equivalence problem

The D0L sequence equivalence problem consists of deciding, given two morphisms , and a word , whether or not g ⁱ (w) = h ⁱ (w) for all i ≥ 0. We show that in case of smooth and loop-free morphisms, to decide the D0L sequence equivalence problem, it suffices to consider the terms of the sequences with , where n is the cardinality of X.     

Classifications of Dense Languages

Let X be a finite alphabet containing more than one letter. A dense language over X is a language containing a disjunctive language. A language L is an n-dense language if for any distinct n words there exist two words such that In this paper we classify dense languages into strict n-dense languages and study some of their algebraic properties. We show that for each n ≥ 0, the n-dense language exists. For an n-dense language L, n ≠ 1, the language L ∩ Q is a dense language, where Q is the set of all primitive words over X. Moreover, for a given n ≥ 1, the language L is such that , then for some m, n ≤ m ≤ 2n + 1. Characterizations on 0-dense languages and n-dense languages are obtained. It is true that for any dense language L, there exist such that for some. We show that everyn-dense language, n ≥ 0, can be split into disjoint union of infinitely many n-dense languages.     

Efficient Solution of A, x ( k )?= b ( k ) Using A ?1

In this work, we consider the problem of solving , , where b ^(k+1) = f(x ^(k)). We show that when A is a full matrix and , where depends on the specific software and hardware setup, it is faster to solve for by explicitly evaluating the inverse matrix A ⁻¹ rather than through the LU decomposition of A. We also show that the forward error is comparable in both methods, regardless of the condition number of A.     

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.     

Generalized fuzzy filters of <Emphasis Type="Italic">R</Emphasis><Subscript>0</Subscript>-algebras

In this paper, we introduce the notions of interval valued -fuzzy filters and interval valued -fuzzy Boolean (implicative) filters in R ₀-algebras and investigate some of their related properties. Some characterization theorems of these generalized fuzzy filters are derived. In particular, we prove that an interval valued fuzzy set F in R ₀-algebras is an interval valued -fuzzy Boolean filter if and only if it is an interval valued -fuzzy implicative filter.     

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.     

Improved Bounds on Quantum Learning Algorithms

In this article we give several new results on the complexity of algorithms that learn Boolean functions from quantum queries and quantum examples.

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.      

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 .     

Alternating time versus deterministic time: A separation

Paulet al. [12] proved that deterministic Turing machines can be speeded up by a factor of log*t (n) using four alternations; that is, DTIME(t(n) log*t(n)) σ₄(t(n)). Balcázaret al. [1] noted that two alternations are sufficient to achieve this speed-up of deterministic Turing machines; that is, DTIME(t(n) log*t(n)) Σ₂(t (n)). We supply a proof of this speed-up and using that show that for each time-constructible functiont(n), DTIME(t(n)) ⊂ Σ₂(t(n)); that is, two alternations are strictly more powerful than deterministic time. An important corollary is that at least one (or both) of the immediate generalizations of the result DTIME(n) ⊂ NTIME(n) [12] must be true: NTIME(n) ≠ co-NTIME(n) or, for each time-constructible functiont(n), DTIME(t (n)) ⊂ NTIME(t (n)). This work was supported in part by NSF Grant CCR-8909071. The preliminary version of the work was done when the author was a graduate student at The Ohio State University.     

Topology on the set of R 0 semantics for R 0 algebras

This paper introduces the concepts of R ₀ valuation, R ₀ semantic, countable R ₀ category , R ₀ fuzzy topological category , etc. It is established in a natural way that the fuzzy topology δ and its cut topology on the set Ω _M consisting of all R ₀ valuations of an R ₀ algebra M, and some properties of fuzzy topology δ and its cut topology are investigated carefully. Moreover, the representation theorem for R ₀ algebras by means of fuzzy topology is given, that is to say the category is equivalent to the category . By studying the relation between valuations and filters, the Loomis–Sikorski theorem for R ₀ algebras is obtained. As an application, K-compactness of the R ₀ logic is discussed.     

Construction and enumeration of Boolean functions with maximum algebraic immunity

Algebraic immunity is a new cryptographic criterion proposed against algebraic attacks. In order to resist algebraic attacks, Boolean functions used in many stream ciphers should possess high algebraic immunity. This paper presents two main results to find balanced Boolean functions with maximum algebraic immunity. Through swapping the values of two bits, and then generalizing the result to swap some pairs of bits of the symmetric Boolean function constructed by Dalai, a new class of Boolean functions with maximum algebraic immunity are constructed. Enumeration of such functions is also given. For a given function p(x) with deg(p(x)) < , we give a method to construct functions in the form p(x)+q(x) which achieve the maximum algebraic immunity, where every term with nonzero coefficient in the ANF of q(x) has degree no less than . Supported by the National Natural Science Foundation of China (Grant No. 60673068), and the Natural Science Foundation of Shandong Province (Grant Nos. Y2007G16, Y2008G01)     

Fast Range Searching with Delaunay Triangulations

This paper studies the idea of answering range searching queries using simple data structures. The only data structure we need is the Delaunay Triangulation of the input points. The idea is to first locate a vertex of the (arbitrary) query polygon and walk along the boundary of the polygon in the Delaunay Triangulation and report all the points enclosed by the query polygon. For a set of uniformly distributed random points in 2-D and a query polygon the expected query time of this algorithm is O(n ^1/3 + Q + E K + L _r n ^1/2), where Q is the size of the query polygon , {\bf E}K = O(n\bcdot area is the expected number of output points, L _r is a parameter related to the shape of the query polygon and n, and L _r is always bounded by the sum of the edge lengths of . Theoretically, when L _r = O(1/n^1/6) the expected query time is O(n^1/3 + Q + E K), which improves the best known average query time for general range searching. Besides the theoretical meaning, the good property of this algorithm is that once the Delaunay Triangulation is given, no additional preprocessing is needed. In order to obtain empirical results, we design a new algorithm for generating random simple polygons within a given domain. Our empirical results show that the constant coefficient of the algorithm is small, at least for the special (practical) cases when the query polygon is either a triangle (simplex range searching) or an axis-parallel box (orthogonal range searching) and for the general case when the query polygons are generated by our new polygon-generating algorithms and their sizes are relatively small.