Using dual techniques to derive componentwise and mixed condition numbers for a linear function of a linear least squares solution

We prove duality results for adjoint operators and product norms in the framework of Euclidean spaces. We show how these results can be used to derive condition numbers especially when perturbations on data are measured componentwise relatively to the original data. We apply this technique to obtain formulas for componentwise and mixed condition numbers for a linear function of a linear least squares solution. These expressions are closed when perturbations of the solution are measured using a componentwise norm or the infinity norm and we get an upper bound for the Euclidean norm.      

Radicals of 0-regular algebras

Summary We consider a generalisation of the Kurosh--Amitsur radical theory for rings (and more generally multi-operator groups) which applies to 0-regular varieties in which all operations preserve 0. We obtain results for subvarieties, quasivarieties and element-wise equationally defined classes. A number of examples of radical and semisimple classes in particular varieties are given, including hoops, loops and similar structures. In the first section, we introduce 0-normal varieties (0-regular varieties in which all operations preserve 0), and show that a key isomorphism theorem holds in a 0-normal variety if it is subtractive, a property more general than congruence permutability. We then define our notion of a radical class in the second section. A number of basic results and characterisations of radical and semisimple classes are then obtained, largely based on the more general categorical framework of L. M\'arki, R. Mlitz and R. Wiegandt as in [13]. We consider the subtractive case separately. In the third section, we obtain results concerning subvarieties and quasivarieties based on the results of the previous section, and also generalise to subtractive varieties some results for multi-operator group radicals defined by simple equational rules. Several examples of radical and semisimple classes are given for a range of fairly natural 0-normal varieties of algebras, most of which are subtractive.     

Stability and bifurcations for the chronic state in Marchuk's model of an immune system

In the paper we present known and new results concerning stability and the Hopf bifurcation for the positive steady state describing a chronic disease in Marchuk's model of an immune system. We describe conditions guaranteeing local stability or instability of this state in a general case and for very strong immune system. We compare these results with the results known in the literature. We show that the positive steady state can be stable only for very specific parameter values. Stability analysis is illustrated by Mikhailov's hodographs and numerical simulations. Conditions for the Hopf bifurcation and stability of arising periodic orbit are also studied. These conditions are checked for arbitrary chosen realistic parameter values. Numerical examples of arising due to the Hopf bifurcation periodic solutions are presented.     

Wandering vector multipliers for unitary groups

A wandering vector multiplier is a unitary operator which maps the set of wandering vectors for a unitary system into itself. A special case of unitary system is a discrete unitary group. We prove that for many (and perhaps all) discrete unitary groups, the set of wandering vector multipliers is itself a group. We completely characterize the wandering vector multipliers for abelian and ICC unitary groups. Some characterizations of special wandering vector multipliers are obtained for other cases. In particular, there are simple characterizations for diagonal and permutation wandering vector multipliers. Similar results remain valid for irrational rotation unitary systems. We also obtain some results concerning the wandering vector multipliers for those unitary systems which are the ordered products of two unitary groups. There are applications to wavelet systems.

     

Homogenization of the Maxwell Equations: Case II. Nonlinear Conductivity

The Maxwell equations with uniformly monotone nonlinear electric conductivity in a heterogeneous medium, which may be non-periodic, are homogenized by two-scale convergence. We introduce a new set of function spaces appropriate for the nonlinear Maxwell system. New compactness results, of two-scale type, are proved for these function spaces. We prove existence of a unique solution for the heterogeneous system as well as for the homogenized system. We also prove that the solutions of the heterogeneous system converge weakly to the solution of the homogenized system. Furthermore, we prove corrector results, important for numerical implementations.     

Relaxing support vectors for classification

We introduce a novel modification to standard support vector machine (SVM) formulations based on a limited amount of penalty-free slack to reduce the influence of misclassified samples or outliers. We show that free slack relaxes support vectors and pushes them towards their respective classes, hence we use the name relaxed support vector machines (RSVM) for our method. We present theoretical properties of the RSVM formulation and develop its dual formulation for nonlinear classification via kernels. We show the connection between the dual RSVM and the dual of the standard SVM formulations. We provide error bounds for RSVM and show it to be stable, universally consistent and tighter than error bounds for standard SVM. We also introduce a linear programming version of RSVM, which we call RSVMLP. We apply RSVM and RSVMLP to synthetic data and benchmark binary classification problems, and compare our results with standard SVM classification results. We show that relaxed influential support vectors may lead to better classification results. We develop a two-phase method called RSVM² for multiple instance classification (MIC) problems, where RSVM formulations are used as classifiers. We extend the two-phase method to the linear programming case and develop RSVMLP². We demonstrate the classification characteristics of RSVM² and RSVMLP², and report our classification results compared to results obtained by other SVM-based MIC methods on public benchmark datasets. We show that both RSVM² and RSVMLP² are faster and produce more accurate classification results.     

Invariance results for linear,time-invariant,discrete-time systems

We establish a general invariance result for linear, time-invariant, discrete-time systems. This result, which constitutes necessary and sufficient conditions for asymptotic stability in the large is very general and yields several important variants. The results presented herein are in the same spirit as the authors’ earlier results for linear, time-invariant continuous-time systems. We demonstrate the applicability of the present results by means of a specific example.     

trackr: A Framework for Enhancing Discoverability and Reproducibility of Data Visualizations and Other Artifacts in R

Research is an incremental, iterative process, with new results relying and building upon previous ones. Scientists need to find, retrieve, understand, and verify results to confidently extend them, even when the results are their own. We present the trackr framework for organizing, automatically annotating, discovering, and retrieving results. We identify sources of automatically extractable metadata for computational results, and we define an extensible system for organizing, annotating, and searching for results based on these and other metadata. We present an open-source implementation of these concepts for plots, computational artifacts, and woven dynamic reports generated in the R statistical computing language. Supplementary materials for this article are available online.     

Second-order asymptotic analysis for noncoercive convex optimization

We use second-order asymptotic analysis to deal with the minimization problem of a noncoercive convex function in a reflexive Banach space. To that end, we first introduce the definition of a second-order asymptotic cone, and its respective function, based on previous results for the finite dimensional case. We provide necessary and sufficient conditions for the existence of solutions for noncoercive convex minimization problems. Examples for which our assumptions are easier to verify than other well-known results are also provided.     

APPROXIMATE DUALITY OF g-FRAMES IN HILBERT SPACES

In this article, we introduce and characterize approximate duality for g-frames. We get some important properties and applications of approximate duals. We also obtain some new results in approximate duality of frames, and generalize some of the known results in approximate duality of frames to g-frames. We also get some results for fusion frames, and perturbation of approximately dual g-frames. We show that approximate duals are stable under small perturbations and they are useful for erasures and reconstruction.     

Effective bandwidths: Call admission,traffic policing and filtering for ATM networks

In this paper we review and extend the effective bandwidth results of Kelly [28], and Kesidis, Walrand and Chang [29, 6]. These results provide a framework for call admission schemes which are sensitive to constraints on the mean delay or the tail distribution of the workload in buffered queues. We present results which are valid for a wide variety of traffic streams and discuss their applicability for traffic management in ATM networks. We discuss the impact of traffic policing schemes, such as thresholding and filtering, on the effective bandwidth of sources. Finally we discuss effective bandwidth results for Brownian traffic models for which explicit results reveal the interaction arising in finite buffers.     

Solitons and Other Solutions for the Generalized KdV–mKdV Equation with Higher-order Nonlinear Terms

The generalized sub-ODEmethod, the rational (G' ⁄ G)-expansionmethod, the exp-function method and the sine-cosine method are applied for constructing many traveling wave solutions of nonlinear partial differential equations (PDEs). Some illustrative equations are investigated by these methods and many hyperbolic, trigonometric and rational function solutions are found. We apply these methods to obtain the exact solutions for the generalized KdV-mKdV (GKdV-mKdV) equation with higher-order nonlinear terms. The obtained results confirm that the proposed methods are efficient techniques for analytic treatment of a wide variety of nonlinear partial differential equations in mathematical physics. We compare between the results yielding from these methods. Also, a comparison between our new results in this paper and the well-known results are given.     

Approximate duality of g-frames in Hilbert spaces

In this article, we introduce and characterize approximate duality for g-frames. We get some important properties and applications of approximate duals. We also obtain some new results in approximate duality of frames, and generalize some of the known results in approximate duality of frames to g-frames. We also get some results for fusion frames, and perturbation of approximately dual g-frames. We show that approximate duals are stable under small perturbations and they are useful for erasures and reconstruction.     

On vertex algebras and their modules associated with even lattices

We study vertex algebras and their modules associated with possibly degenerate even lattices, using an approach somewhat different from others. Several known results are recovered and a number of new results are obtained. We also study modules for Heisenberg algebras and we classify irreducible modules satisfying certain conditions and obtain a complete reducibility theorem.     

Liberating the dimension for function approximation: Standard information

This is a follow-up paper of “Liberating the dimension for function approximation”, where we studied approximation of infinitely variate functions by algorithms that use linear information consisting of finitely many linear functionals. In this paper, we study similar approximation problems, however, now the algorithms can only use standard information consisting of finitely many function values. We assume that the cost of one function value depends on the number of active variables. We focus on polynomial tractability, and occasionally also study weak tractability. We present non-constructive and constructive results. Non-constructive results are based on known relations between linear and standard information for finitely variate functions, whereas constructive results are based on Smolyak’s construction generalized to the case of infinitely variate functions. Surprisingly, for many cases, the results for standard information are roughly the same as for linear information.     

High frequency asymptotics for wavelet-based tests for Gaussianity and isotropy on the torus

We prove a multivariate CLT for skewness and kurtosis of the wavelets coefficients of a stationary field on the torus. The results are in the framework of the fixed-domain asymptotics, i.e. we refer to observations of a single field which is sampled at higher and higher frequencies. We consider also studentized statistics for the case of an unknown correlation structure. The results are motivated by the analysis of high-frequency financial data or cosmological data sets, with a particular interest towards testing for Gaussianity and isotropy.     

Long term dynamics of a reaction-diffusion system

We study a reaction-diffusion system of two parabolic differential equations describing the behavior of a nuclear reactor. We provide existence results for nontrivial periodic solutions, nonexistence results for stationary solutions and we prove that, depending on the value of the parameters, either the system admits a compact global attractor, or the solutions are unbounded.     

Momentum Operators in the Unit Square

We investigate the skew-adjoint extensions of a partial derivative operator acting in the direction of one of the sides a unit square. We investigate the unitary equivalence of such extensions and the spectra of such extensions. It follows from our results, that such extensions need not have discete spectrum. We apply our techniques to the problem of finding commuting skew-adjoint extensions of the partial derivative operators acting in the directions of the sides of the unit square. While our results are most easily stated for the unit square, they are established for a larger class of domains, including certain fractal domains.     

Isochronous systems and their quantization

We review recent results about classical isochronous systems characterized by the presence of an open (hence fully dimensional) region in their phase space in which all their solutions are completely periodic (i.e., periodic in all degrees of freedom) with the same fixed period (independent of the initial data provided they are inside the isochronicity region). We report a technique for generating such systems, whose wide applicability justifies the statement that isochronous systems are not rare. We also present an analogous technique applicable to a vast class of Hamiltonian systems and generating isochronous Hamiltonian systems. We also report some results concerning the quantized versions of such systems.     

Error bounds for monotone approximation schemes for parabolic Hamilton-Jacobi-Bellman equations

We obtain nonsymmetric upper and lower bounds on the rate of convergence of general monotone approximation/numerical schemes for parabolic Hamilton-Jacobi-Bellman equations by introducing a new notion of consistency. Our results are robust and general - they improve and extend earlier results by Krylov, Barles, and Jakobsen. We apply our general results to various schemes including Crank-Nicholson type finite difference schemes, splitting methods, and the classical approximation by piecewise constant controls. In the first two cases our results are new, and in the last two cases the results are obtained by a new method which we develop here.