COMPLEX WIENER-IT? CHAOS DECOMPOSITION REVISITED

In this article, some properties of complex Wiener-It? multiple integrals and complex Ornstein-Uhlenbeck operators and semigroups are obtained. Those include Stroock's formula, Hu-Meyer formula, Clark-Ocone formula, and the hypercontractivity of complex Ornstein-Uhlenbeck semigroups. As an application, several expansions of the fourth moments of complex Wiener-It? multiple integrals are given.     

Green's equivalences in finite semigroups of binary relations

This paper contains an algorithm which, given a set of generators of a semigroupS of binary relations on a finite set, computes the structure ofS in terms of Green's equivalences. The algorithm is a generalization to semigroups of binary relations of an algorithm obtained by Lallement and McFadden for semigroups of transformations. Part of this research was supported by a Mary Washington College Faculty Development Grant.     

Sandwich semigroups of binary relations

This paper introduces new semigroups of binary relations that arose naturally from investigating the transfer of information between automata and semigroups associated with automata. In particular we introduce a new multiplication on binary relations by means of an arbitrary but fixed “sandwich” relation. R.J. Plemmons and M. West have characterized Green's relations in the usual semigroup of binary relations, and we use these to investigate Green's relations in our semigroups. We give algorithms for constructing idempotents and regular elements in these new semigroups.     

Absolute flatness of regular semigroups with a finite height function

In this paper we give a sufficient condition for regular semigroups with a finite height function to be left absolutely flat. As a consequence, we can show that the semigroup Λ(S) of all right translations of a primitive regular semigroupS with only finitely manyR-classes, with composition being from left to right, is absolutely flat and give a generalization of a Bulman-Fleming and McDowell result concerning absolutely flat semigroups from primitive regular semigroups to regular semigroups with a finite height function. These results give examples of semigroups which are amalgamation bases in the class of semigroups. The author thanks the referee for finding errors in the original version of this paper.     

Euler’s Exponential Formula for Semigroups

The aim of this paper is to show that Eulers exponential formula $\lim_{n\rightarrow\infty}\linebreak[4] (I-tA/n)^{-n}x = e^{tA}x$, well known for $C_0$ semigroups in a Banach space $X\ni x$, can be used for semigroups not of class $C_0$, the sense of the convergence being related to the regularity of the semigroup for $t>0$. Although the strong convergence does not hold in general for not strongly continuous semigroups, an integrated version is stated for once integrated semigroups. Furthermore by replacing the initial topology on $X$ by some (coarser) locally convex topology $\tau$, the strong $\tau$-convergence takes place provided the semigroup is strongly $\tau$-continuous; in particular this applies to the class of bi-continuous semigroups. More generally if a $k$-times integrated semigroup $S(t)$ in a Banach space $X$ is strongly $k$-times $\tau$-differentiable, then Eulers formula holds in this topology with limit $S^{(k)}(t)$. On the other hand, for bounded holomorphic semigroups not necessarily of class $C_0$, Eulers formula is shown to hold in operator norm, with the error bound estimate ${\cal O}(\ln n/n)$, uniformly in $t>0$. All these results also concern degenerate semigroups.     

Decision functions for chain classifiers based on Bayesian networks for multi-label classification

Multi-label classification problems require each instance to be assigned a subset of a defined set of labels. This problem is equivalent to finding a multi-valued decision function that predicts a vector of binary classes. In this paper we study the decision boundaries of two widely used approaches for building multi-label classifiers, when Bayesian network-augmented naive Bayes classifiers are used as base models: Binary relevance method and chain classifiers. In particular extending previous single-label results to multi-label chain classifiers, we find polynomial expressions for the multi-valued decision functions associated with these methods. We prove upper boundings on the expressive power of both methods and we prove that chain classifiers provide a more expressive model than the binary relevance method.     

Sharp Error Bound on Norm Convergence of Exponential Product Formula and Approximation to Kernels of Schrödinger Semigroups

Abstract

We consider semigroups generated by Schrödinger operators with smooth potentials, and show that the exponential product formula approximates the Schrödinger semigroups with a remarkably sharp error bound in the norm of operators on the Sobolev spaces. As an application, we discuss the approximation to the integral kernels of semigroups by the product formula. The analysis is based on the pseudodifferential calculus.     

Asymptotic expansions and convergence rates for Euler’s approximations of semigroups

We provide optimal bounds for errors in Euler’s approximations of semigroups in Banach algebras and of semigroups of operators in Banach spaces. Furthermore, we construct asymptotic expansions for such approximations with optimal bounds for remainder terms. The sizes of errors are controlled by smoothness properties of semigroups. In this paper we use Fourier–Laplace transforms and a reduction of the problem to the convergence rates and asymptotic expansions in the Law of Large Numbers. The research was partially supported by the Lithuanian State Science and Studies Foundation, grant No. T-70/09. This paper was written in 2004. In the interim, several related articles were published; let us mention [14, 13, 15].     

Binary quadratic and cubic forms,and unipositive matrices

The classical method of reducing a positive binary quadratic form to a semi-reduced form applies translations alternately left and right to minimize the absolute value of the middle coefficient — and may therefore be called absolute reduction. There is an alternative method which keeps the sign of the middle coefficient constant before the end: we call this method positive reduction. Positive reduction seems to make possible an algorithm for finding the representations of 1 by a binary cubic form with real linear factors, and has various properties somewhat simpler than those of absolute reduction. Some of these properties involve unipositive matrices (with nonnegative integer elements and determinant 1). Certain semigroups of unipositive matrices with unique factorization into primes are described. Two of these semigroups give a neat approach to the reduction of indefinite binary quadratic forms—which may generalize. Some remarks on unimodular automorphs occur in Section 6.     

Unital affine semigroups

Affine semigroups are convex sets on which there exists an associative binary operation which is affine separately in either variable. They were introduced by Cohen and Collins in 1959. We look at examples of affine semigroups which are of interest to matrix and operator theory and we prove some new results on the extreme points and the absorbing elements of certain types of affine semigroups. Most notably we improve a result of Wendel that every invertible element in a compact affine semigroup is extreme by extending this result to linearly bounded affine semigroups.     

A product formula for semigroups of Lipschitz operators associated with abstract quasilinear evolution equations

The notion of semigroups of Lipschitz operators associated with abstract quasilinear evolution equations is introduced and a product formula for such semigroups is established. The product formula obtained in the paper is applied to the solvability of the Cauchy problem for a first order quasilinear system through a finite difference scheme of the Lax‐Friedrichs type.     

Enhancing evolutionary fuzzy systems for multi-class problems: Distance-based relative competence weighting with truncated confidences (DRCW-TC)

Classification problems with multiple classes suppose a challenge in Data Mining tasks. There is a difficulty inherent to the learning process when trying to find the most adequate discrimination functions among the different concepts within the dataset. Using Fuzzy Rule Based Classification Systems in general, and Evolutionary Fuzzy Systems in particular, provide the advantage of describing smoother borderline areas, thanks to the linguistic label-based representation.In multi-classification, the pairwise learning approach (One-vs-One) has gained a notorious attention. However, there is certain dependence between the goodness of the confidence degrees or scores of binary classifiers, and the final performance shown by the global model. Regarding this fact, the problem of non-competent classifiers is of special relevance. It occurs when a binary classifier outputs a positive score for a couple of classes unrelated with the input example, which may degrade the final accuracy. Precisely, the previously exposed properties of fuzzy classifiers make them more prone to the former condition.In this paper, we propose an extension of the distance-based combination strategy to overcome this non-competence problem. It is based on the truncation of the confidence degrees of the classes prior to the distance-based tuning. This allows taking advantage of the good classification abilities of Evolutionary Fuzzy Systems, while diminishing the adverse effect of the aforementioned non-competence. Experimental results, using FARC-HD with overlap functions as the fuzzy learning algorithm, show that this new adaptation of the Distance-based Relative Competence Weighting model outperforms both the OVO and standard distance-based approaches, and it is competitive with robust classifiers such as Support Vector Machines.     

Unbounded functional calculus for bounded groups with applications

In this paper, we develop the unbounded extension of the Hille–Phillips functional calculus for generators of bounded groups. Mathematical applications include the generalised Lévy–Khintchine formula for subordinate semigroups, the analyticity of semigroups generated by fractional powers of group generators, where the power is not an odd integer, and a shifted abstract Grünwald formula. We also give an application of the theory to subsurface hydrology, modeling solute transport on a regional scale using fractional dispersion along flow lines. M. Kovács is partially supported by postdoctoral grant No. 623-2005-5078 of the Swedish Research Council (VR).     

关于C_0半群族的等度指数稳定性

本文给出了Hilbert空间中C0半群的一个表示公式，利用它讨论C0半群算子族的等度指数稳定性．     

Sharp Extensions for Convoluted Solutions of Abstract Cauchy Problems

In this paper we give sharp extension results for convoluted solutions of abstract Cauchy problems in Banach spaces. The main technique is the use of the algebraic structure (for the usual convolution product *) of these solutions which are defined by a version of the Duhamel formula. We define algebra homomorphisms from a new class of test-functions and apply our results to concrete operators. Finally, we introduce the notion of k-distribution semigroups to extend previous concepts of distribution semigroups.     

Growth Estimates for Semigroups Generated by 2Sx2 Operator Matrices

In this paper we give an explicit formula for semigroups generated by 2×2 operator matrices. Using this formula we can derive an estimate for the growth bound of the associated matrix generator. Finally these results are applied to a complete second order Cauchy problem yielding an explicit formula and a growth estimate for the solution of this problem.     

Generalization of the Trotter-Lie formula

Our generalization of the Trotter-Lie formula is based upon the following principle: starting with a "general mean" of semigroups, the corresponding formula must converge to the semigroup generated by an extension of the arithmetic mean of the generators. Accordingly, the usual product formula is associated with a "geometric mean". We establish "average formulæ" for nonlinear semigroups generated by the subdifferentials of convex functionals. We then explore in detail the case of self-adjoint semigroups. A great advantage of these average formulæ is that they naturally hold for an arbitrary number of operators whereas the traditional product formulæ do not usually hold for more than two operators. This fact is important for numerical and theoretical applications. For instance, it enables us to handle with ease the Schrödinger operator withmagnetic vector potential and with arbitrarily singular scalar potential.This work is a portion of the author's "Doctorat d'Université" at the Université Paris VI, which was supervised by M. Haïm Brezis. It was financially supported by a Research Allocation DGRST (Paris VI, contract n° 78170) and a Georges Lurcy Fellowship (Berkeley, academic year 1979–80). The final version of this paper was written while the author was an Assistant Professor at the University of Southern California.     

The complex inversion formula in UMD spaces for families of bounded operators

In this article, we prove that in UMD Banach spaces the complex inversion formula of the Laplace transform is valid, in the strong sense, for wide classes of families of bounded linear operators. Our approach allows us to recover (in a unified way) known results about C ₀-semigroups, cosine functions and resolvent families as well as to prove new results for k-convoluted semigroups and integrated semigroups, among others.     

Constructing 2 × 2 Bricks from Unitary Numerical Semigroups

This paper investigates numerical semigroups that yield 2 × 2 bricks. We demonstrate the existence of an infinite family of 2 × 2 bricks that includes all of the perfect 2 × 2 bricks. We provide a formula for the Frobenius numbers of these semigroups as well as a necessary and sufficient condition for the semigroups to be symmetric.