Worst inputs and a bound on the highest peak statistics of a class of non-linear systems

A method is developed by which the input leading to the highest possible response in an interval of time can be determined for a class of non-linear systems. The input, if deterministic, is constrained to have a known finite energy (or norm) in the interval under consideration. In the case of random inputs, the energy is constrained to have a known probability distribution function. The approach has applications when a system has to be put to maximum advantage by getting the largest possible output or when a system has to be designed to the highest maximum response with only the input energy or the energy distribution known. The method is also useful in arriving at a bound on the highest peak distribution of the response, when the excitation is a known random process.As an illustration the Duffing oscillator has been analysed and some numerical results have also been presented.

Optimal infinite-horizon feedback laws for a general class of constrained discrete-time systems: stability and moving-horizon approximations

Stability results are given for a class of feedback systems arising from the regulation of time-varying discrete-time systems using optimal infinite-horizon and moving-horizon feedback laws. The class is characterized by joint constraints on the state and the control, a general nonlinear cost function and nonlinear equations of motion possessing two special properties. It is shown that weak conditions on the cost function and the constraints are sufficient to guarantee uniform asymptotic stability of both the optimal infinite-horizon and movinghorizon feedback systems. The infinite-horizon cost associated with the moving-horizon feedback law approaches the optimal infinite-horizon cost as the moving horizon is extended.

Properties of Toeplitz operators on analytic function spaces : from function symbols to distributions

Toeplitz operators are among the most important classes of concrete operators with applications to several branches of pure and applied mathematics. This doctoral thesis deals with Toeplitz operators on analytic Bergman, Bloch and Fock spaces. Usually, a Toeplitz operator is a composition of multiplication by a function and a suitable projection. The present work deals with generalizing the notion to the case where the function is replaced by a distributional symbol. Fredholm theory for Toeplitz operators with matrix-valued symbols is also considered. The subject of this thesis belongs to the areas of complex analysis, functional analysis and operator theory. This work contains five research articles. The articles one, three and four deal with finding suitable distributional classes in Bergman, Fock and Bloch spaces, respectively. In each case the symbol class to be considered turns out to be a certain weighted Sobolev-type space of distributions. The Bergman space setting is the most straightforward. When dealing with Fock spaces, some difficulties arise due to unboundedness of the complex plane and the properties of the Gaussian measure in the definition. In the Bloch-type spaces an additional logarithmic weight must be introduced. Sufficient conditions for boundedness and compactness are derived. The article two contains a portion showing that under additional assumptions, the condition for Bergman spaces is also necessary. The fifth article deals with Fredholm theory for Toeplitz operators having matrix-valued symbols. The essential spectra and index theorems are obtained with the help of Hardy space factorization and the Berezin transform, for instance. The article two also has a part dealing with matrix-valued symbols in a non-reflexive Bergman space, in which case a condition on the oscillation of the symbol (a logarithmic VMO-condition) must be added.

A structural principle for a class of adaptive systems

A feature common to many adaptive systems for identification and control is the adjustment.of gain parameters in a manner ensuring the stability of the overall system. This paper puts forward a principle which assures such a result for arbitrary systems which are linear and time invariant except for the adjustable parameters. The principle only demands that a transfer function be positive real. This transfer function dependent on the structure of the system with respect to the parameters. Several examples from adaptive identification, control and observer schemes are given as illustrations of the conceptual simplification provided by the structural principle.

Residence time distribution for a class of Gaussian Markov processes

We study the distribution of residence time or equivalently that of "mean magnetization" for a family of Gaussian Markov processes indexed by a positive parameter alpha. The persistence exponent for these processes is simply given by theta=alpha but the residence time distribution is nontrivial. The shape of this distribution undergoes a qualitative change as theta increases, indicating a sharp change in the ergodic properties of the process. We develop two alternate methods to calculate exactly but recursively the moments of the distribution for arbitrary alpha. For some special values of alpha, we obtain closed form expressions of the distribution function. [S1063-651X(99)03306-1].

Perfluoroalkyl bile esters: a new class of efficient gelators of organic and aqueous-organic media

A new class of fluorinated gelators derived from bile acids is reported. Perfluoroalkyl chains were attached to the bile acids through two different ester linkages and were synthesized following simple transformations. The gelation property of these derivatives is a function of the bile acid moiety, the spacer and the fluoroalkyl chain length. By varying these parameters, gels were obtained in aromatic hydrocarbons, DMSO and DMSO/DMF-H(2)O mixtures of different proportions. Several derivatives of deoxycholic and lithocholic acids were found to be efficient organogelators, while the reported bile-acid based organogelators are mostly derived from the cholic acid moiety. The efficient gelators among these compounds formed gels well below 1.0% (w/v) and hence they can be termed as supergelators. The mechanical properties of these gels could be modulated by changing either the bile acid moiety or by varying the length of the fluoroalkyl segment. The presence of CO(2)-philic perfluoroalkyl groups is also expected to enhance their solubility in supercritical CO(2) and hence these compounds are promising candidates for making aerogels.

Perfluoroalkyl bile esters: a new class of efficient gelators of organic and aqueous-organic media

A new class of fluorinated gelators derived from bile acids is reported. Perfluoroalkyl chains were attached to the bile acids through two different ester linkages and were synthesized following simple transformations. The gelation property of these derivatives is a function of the bile acid moiety, the spacer and the fluoroalkyl chain length. By varying these parameters, gels were obtained in aromatic hydrocarbons, DMSO and DMSO/DMF-H(2)O mixtures of different proportions. Several derivatives of deoxycholic and lithocholic acids were found to be efficient organogelators, while the reported bile-acid based organogelators are mostly derived from the cholic acid moiety. The efficient gelators among these compounds formed gels well below 1.0% (w/v) and hence they can be termed as supergelators. The mechanical properties of these gels could be modulated by changing either the bile acid moiety or by varying the length of the fluoroalkyl segment. The presence of CO(2)-philic perfluoroalkyl groups is also expected to enhance their solubility in supercritical CO(2) and hence these compounds are promising candidates for making aerogels.

On Convergence of Differential Evolution Over a Class of Continuous Functions With Unique Global Optimum

Differential evolution (DE) is arguably one of the most powerful stochastic real-parameter optimization algorithms of current interest. Since its inception in the mid 1990s, DE has been finding many successful applications in real-world optimization problems from diverse domains of science and engineering. This paper takes a first significant step toward the convergence analysis of a canonical DE (DE/rand/1/bin) algorithm. It first deduces a time-recursive relationship for the probability density function (PDF) of the trial solutions, taking into consideration the DE-type mutation, crossover, and selection mechanisms. Then, by applying the concepts of Lyapunov stability theorems, it shows that as time approaches infinity, the PDF of the trial solutions concentrates narrowly around the global optimum of the objective function, assuming the shape of a Dirac delta distribution. Asymptotic convergence behavior of the population PDF is established by constructing a Lyapunov functional based on the PDF and showing that it monotonically decreases with time. The analysis is applicable to a class of continuous and real-valued objective functions that possesses a unique global optimum (but may have multiple local optima). Theoretical results have been substantiated with relevant computer simulations.

Abstract Characteristic Function

The characteristic function for a contraction is a classical complete unitary invariant devised by Sz.-Nagy and Foias. Just as a contraction is related to the Szego kernel k(S)(z, w) = ( 1 - z(w)over bar)- 1 for |z|, |w| < 1, by means of (1/k(S))( T, T *) = 0, we consider an arbitrary open connected domain Omega in C(n), a kernel k on Omega so that 1/k is a polynomial and a tuple T = (T(1), T(2), ... , T(n)) of commuting bounded operators on a complex separable Hilbert spaceHsuch that (1/k)( T, T *) >= 0. Under some standard assumptions on k, it turns out that whether a characteristic function can be associated with T or not depends not only on T, but also on the kernel k. We give a necessary and sufficient condition. When this condition is satisfied, a functional model can be constructed. Moreover, the characteristic function then is a complete unitary invariant for a suitable class of tuples T.

Perturbative expansion of the QCD Adler function improved by renormalization-group summation and analytic continuation in the Borel plane

We examine the large-order behavior of a recently proposed renormalization-group-improved expansion of the Adler function in perturbative QCD, which sums in an analytically closed form the leading logarithms accessible from renormalization-group invariance. The expansion is first written as an effective series in powers of the one-loop coupling, and its leading singularities in the Borel plane are shown to be identical to those of the standard ``contour-improved'' expansion. Applying the technique of conformal mappings for the analytic continuation in the Borel plane, we define a class of improved expansions, which implement both the renormalization-group invariance and the knowledge about the large-order behavior of the series. Detailed numerical studies of specific models for the Adler function indicate that the new expansions have remarkable convergence properties up to high orders. Using these expansions for the determination of the strong coupling from the hadronic width of the tau lepton we obtain, with a conservative estimate of the uncertainty due to the nonperturbative corrections, alpha(s)(M-tau(2)) = 0.3189(-0.0151)(+0.0173), which translates to alpha(s)(M-Z(2)) = 0.1184(-0.0018)(+0.0021). DOI: 10.1103/PhysRevD.87.014008

A nested linear codes approach to distributed function computation over subspaces

In this paper, we consider a distributed function computation setting, where there are m distributed but correlated sources X1,...,Xm and a receiver interested in computing an s-dimensional subspace generated by [X1,...,Xm]Γ for some (m × s) matrix Γ of rank s. We construct a scheme based on nested linear codes and characterize the achievable rates obtained using the scheme. The proposed nested-linear-code approach performs at least as well as the Slepian-Wolf scheme in terms of sum-rate performance for all subspaces and source distributions. In addition, for a large class of distributions and subspaces, the scheme improves upon the Slepian-Wolf approach. The nested-linear-code scheme may be viewed as uniting under a common framework, both the Korner-Marton approach of using a common linear encoder as well as the Slepian-Wolf approach of employing different encoders at each source. Along the way, we prove an interesting and fundamental structural result on the nature of subspaces of an m-dimensional vector space V with respect to a normalized measure of entropy. Here, each element in V corresponds to a distinct linear combination of a set {Xi}im=1 of m random variables whose joint probability distribution function is given.

On the zeta function of a periodic-finite-type shift

Periodic-finite-type shifts (PFT's) are sofic shifts which forbid the appearance of finitely many pre-specified words in a periodic manner. The class of PFT's strictly includes the class of shifts of finite type (SFT's). The zeta function of a PET is a generating function for the number of periodic sequences in the shift. For a general sofic shift, there exists a formula, attributed to Manning and Bowen, which computes the zeta function of the shift from certain auxiliary graphs constructed from a presentation of the shift. In this paper, we derive an interesting alternative formula computable from certain ``word-based graphs'' constructed from the periodically-forbidden word description of the PET. The advantages of our formula over the Manning-Bowen formula are discussed.

Expansions of tau hadronic spectral function moments in a nonpower QCD perturbation theory with tamed large order behavior

The moments of the hadronic spectral functions are of interest for the extraction of the strong coupling alpha(s) and other QCD parameters from the hadronic decays of the tau lepton. Motivated by the recent analyses of a large class of moments in the standard fixed-order and contour-improved perturbation theories, we consider the perturbative behavior of these moments in the framework of a QCD nonpower perturbation theory, defined by the technique of series acceleration by conformal mappings, which simultaneously implements renormalization-group summation and has a tame large-order behavior. Two recently proposed models of the Adler function are employed to generate the higher-order coefficients of the perturbation series and to predict the exact values of the moments, required for testing the properties of the perturbative expansions. We show that the contour-improved nonpower perturbation theories and the renormalization-group-summed nonpower perturbation theories have very good convergence properties for a large class of moments of the so-called ``reference model,'' including moments that are poorly described by the standard expansions. The results provide additional support for the plausibility of the description of the Adler function in terms of a small number of dominant renormalons.

The Lovasz θ function, SVMs and finding large dense subgraphs

The Lovasz θ function of a graph, is a fundamental tool in combinatorial optimization and approximation algorithms. Computing θ involves solving a SDP and is extremely expensive even for moderately sized graphs. In this paper we establish that the Lovasz θ function is equivalent to a kernel learning problem related to one class SVM. This interesting connection opens up many opportunities bridging graph theoretic algorithms and machine learning. We show that there exist graphs, which we call SVM−θ graphs, on which the Lovasz θ function can be approximated well by a one-class SVM. This leads to a novel use of SVM techniques to solve algorithmic problems in large graphs e.g. identifying a planted clique of size Θ(n√) in a random graph G(n,12). A classic approach for this problem involves computing the θ function, however it is not scalable due to SDP computation. We show that the random graph with a planted clique is an example of SVM−θ graph, and as a consequence a SVM based approach easily identifies the clique in large graphs and is competitive with the state-of-the-art. Further, we introduce the notion of a ''common orthogonal labeling'' which extends the notion of a ''orthogonal labelling of a single graph (used in defining the θ function) to multiple graphs. The problem of finding the optimal common orthogonal labelling is cast as a Multiple Kernel Learning problem and is used to identify a large common dense region in multiple graphs. The proposed algorithm achieves an order of magnitude scalability compared to the state of the art.

REPRODUCING KERNEL FOR A CLASS OF WEIGHTED BERGMAN SPACES ON THE SYMMETRIZED POLYDISC

A natural class of weighted Bergman spaces on the symmetrized polydisc is isometrically embedded as a subspace in the corresponding weighted Bergman space on the polydisc. We find an orthonormal basis for this subspace. It enables us to compute the kernel function for the weighted Bergman spaces on the symmetrized polydisc using the explicit nature of our embedding. This family of kernel functions includes the Szego and the Bergman kernel on the symmetrized polydisc.