On convergence of formal sums of series over two-dimensional complete fields

Series over two-dimensional complete fields are studied; sufficient conditions for convergence of a series at all points of the maximal ideal and sufficient conditions for convergence of formal sums and infinite superpositions of formal sums of series are obtained. Bibliography: 5 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 227, 1995, pp. 89–92.     

Gap, Excess and Bornological Convergence

Let be the nonempty subsets of a metric space 〈X, d〉. Some classical convergences in - such as convergence in Hausdorff distance, Attouch-Wets convergence and Wijsman convergence - have been shown to be compatible with the weak topology on induced by all gap and excess functionals with fixed left argument ranging in some bornology. Here we consider an arbitrary ideal of subsets of X and compare the gap and excess topology so generated with the corresponding convergence defined in terms of truncations by elements of the ideal. Dedicated to the memory of Flora Daniel.     

Quadratic Newton Iteration for Systems with Multiplicity

Newton's iterator is one of the most popular components of polynomial equation system solvers, either from the numeric or symbolic point of view. This iterator usually handles smooth situations only (when the Jacobian matrix associated to the system is invertible). This is often a restrictive factor. Generalizing Newton's iterator is still an open problem: How to design an efficient iterator with a quadratic convergence even in degenerate cases? We propose an answer for an m -adic topology when the ideal m can be chosen generic enough: compared to a smooth case we prove quadratic convergence with a small overhead that grows with the square of the multiplicity of the root.     

On Convergence to Infinity

 By a metric mode of convergence to infinity in a regular Hausdorff space X, we mean a sequence of closed subsets of X with and , and a sequence (or net) in X is convergent to infinity with respect to provided for each contains eventually. Modulo a natural equivalence relation, these correspond to one-point extensions of the space with a countable base at the ideal point, and in the metrizable setting, they correspond to metric boundedness structures for the space. In this article, we study the interplay between these objects and certain continuous functions that may determine the metric mode of convergence to infinity, called forcing functions. Falling out of our results is a simple proof that each noncompact metrizable space admits uncountably many distinct metric uniformities. (Received 2 March 1999)     

On Convergence to Infinity

 By a metric mode of convergence to infinity in a regular Hausdorff space X, we mean a sequence of closed subsets of X with and , and a sequence (or net) in X is convergent to infinity with respect to provided for each contains eventually. Modulo a natural equivalence relation, these correspond to one-point extensions of the space with a countable base at the ideal point, and in the metrizable setting, they correspond to metric boundedness structures for the space. In this article, we study the interplay between these objects and certain continuous functions that may determine the metric mode of convergence to infinity, called forcing functions. Falling out of our results is a simple proof that each noncompact metrizable space admits uncountably many distinct metric uniformities.     

On the convergence of quasi-newton methods for nonsmooth problems

We develop a theory of quasi-New ton and least-change update methods for solving systems of nonlinear equations F(x) = 0. In this theory, no differentiability conditions are necessary. Instead, we assume that Fcan be approximated, in a weak sense, by an affine function in a neighborhood of a solution. Using this assumption, we prove local and ideal convergence. Our theory can be applied to B-differentiable functions and to partially differentiable functions.     

Pointwise versus equal (quasi-normal) convergence via ideals

We prove a characterization showing when the ideal pointwise convergence does not imply the ideal equal (aka quasi-normal) convergence. The characterization is expressed in terms of a cardinal coefficient related to the bounding number b$\mathfrak{b}$. We also prove a characterization showing when the ideal equal limit is unique.     

On ideal convergence of double sequences in probabilistic normed spaces

The notion of ideal convergence is a generalization of statistical convergence which has been intensively investigated in last few years.For an admissible ideal ∮N× N,the aim of the present paper is to introduce the concepts of ∮-convergence and ∮*-convergence for double sequences on probabilistic normed spaces(PN spaces for short).We give some relations related to these notions and find condition on the ideal ∮ for which both the notions coincide.We also define ∮-Cauchy and ∮*-Cauchy double sequences on PN spaces and show that ∮-convergent double sequences are ∮-Cauchy on these spaces.We establish example which shows that our method of convergence for double sequences on PN spaces is more general.     

On ideal equal convergence

We consider ideal equal convergence of a sequence of functions. This is a generalization of equal convergence introduced by Császár and Laczkovich [Császár Á., Laczkovich M., Discrete and equal convergence, Studia Sci. Math. Hungar., 1975, 10(3–4), 463–472]. Our definition of ideal equal convergence encompasses two different kinds of ideal equal convergence introduced in [Das P., Dutta S., Pal S.K., On and *-equal convergence and an Egoroff-type theorem, Mat. Vesnik, 2014, 66(2), 165–177]_and [Filipów R., Szuca P., Three kinds of convergence and the associated I-Baire classes, J. Math. Anal. Appl., 2012, 391(1), 1–9]. We also solve a few problems posed in the paper by Das, Dutta and Pal.     

On <Emphasis Type="Italic">I</Emphasis>-convergence in random 2-normed spaces

Recently the concepts of statistical convergence and ideal convergence have been studied in 2-normed and 2-Banach spaces by various authors. In this paper we define and study the notion of ideal convergence in random 2-normed space and construct some interesting examples.     

Korovkin-type Theorems for Abstract Modular Convergence

We give some Korovkin-type theorems on convergence and estimates of rates of approximations of nets of functions, satisfying suitable axioms, whose particular cases are filter/ideal convergence, almost convergence and triangular A-statistical convergence, where A is a non-negative summability method. Furthermore, we give some applications to Mellin-type convolution and bivariate Kantorovich-type discrete operators.     

More about the kernel convergence and the ideal convergence

Let I 2N be an ideal and let XI = span{χI ： I ∈ I}, and let pI be the quotient norm of l∞/XI. In this paper, we show first that for each proper ideal I 2N, the ideal convergence deduced by I is equivalent to pI-kernel convergence. In addition, let K = {x＊oχ（·） ： x＊∈ p（e）}, where p（x） = lim supn→∞1/n（∑k=1n｜x（k）｜, and let Iμ = {A N ： μ（A） = 0} for all μ = x＊oχ（·） ∈ K. Then Iμ is a proper ideal. We also show that the ideal convergence deduced by the proper ideal Iμ, the p-kernel convergence and the statistical convergence are also equivalent.     

Bivariate Lagrange interpolation at the Padua points: the ideal theory approach

The Padua points are a family of points on the square [−1, 1]² given by explicit formulas that admits unique Lagrange interpolation by bivariate polynomials. Interpolation polynomials and cubature formulas based on the Padua points are studied from an ideal theoretic point of view, which leads to the discovery of a compact formula for the interpolation polynomials. The L ^p convergence of the interpolation polynomials is also studied. S. De Marchi and M. Vianello were supported by the “ex-60%” funds of the University of Padua and by the INdAM GNCS (Italian National Group for Scientific Computing). Y. Xu was partially supported by NSF Grant DMS-0604056.     

Two‐level Fourier analysis of multigrid for higher‐order finite‐element discretizations of the Laplacian

In this paper, we employ local Fourier analysis (LFA) to analyze the convergence properties of multigrid methods for higher‐order finite‐element approximations to the Laplacian problem. We find that the classical LFA smoothing factor, where the coarse‐grid correction is assumed to be an ideal operator that annihilates the low‐frequency error components and leaves the high‐frequency components unchanged, fails to accurately predict the observed multigrid performance and, consequently, cannot be a reliable analysis tool to give good performance estimates of the two‐grid convergence factor. While two‐grid LFA still offers a reliable prediction, it leads to more complex symbols that are cumbersome to use to optimize parameters of the relaxation scheme, as is often needed for complex problems. For the purposes of this analytical optimization as well as to have simple predictive analysis, we propose a modification that is “between” two‐grid LFA and smoothing analysis, which yields reasonable predictions to help choose correct damping parameters for relaxation. This exploration may help us better understand multigrid performance for higher‐order finite element discretizations, including for Q₂‐Q₁ (Taylor‐Hood) elements for the Stokes equations. Finally, we present two‐grid and multigrid experiments, where the corrected parameter choice is shown to yield significant improvements in the resulting two‐grid and multigrid convergence factors.     

Verallgemeinerte topogene Ordnungen

Topogenous orders in the sense of Császár are a common generalization of proximity and topology. ech closures are a generalization of the topological closure operators in the sense of Kuratowski. We show that the topogenous orders as well as the ech closures are special cases of the so called compressed operators. Moreover, the now defined categoryCOM (in germanBAL) of compress spaces and compress faithful maps is a properly fibred topological category in the sense of Herrlich which is weakly cartesian closed, that means the product map of two quotient maps inCOM is a quotient map inCOM. Therefore by results of L. D. Nel it is possible to construct a cartesian closed properly fibred topological category in whichCOM can be nicely embedded. Further it turns out that the compressed operators be in a natural connexion with the uniform convergence structures in the sense of Cook and Fischer and in addition with the limit structures in the sense of Fischer. For principal ideal uniform convergence structures we prove that they are precompact and complete iff the properly constructed compressed operator is compact.     

Maximal Homomorphic Group Image and Convergence of Convolution Sequences on a Semigroup

Let be a probability measure generating a locally compact semigroup S. If the convolution sequence ⁿ is tight, in particular if S is compact, S admits a closed minimal ideal K. The convergence of ⁿ is characterized in terms of convergence of a homomorphic image (~) ⁿ on a factor group of the compact group G in the Rees–Suschkewitsch decomposition of K.     

Semidefinite representations for finite varieties

We consider the problem of minimizing a polynomial over a set defined by polynomial equations and inequalities. When the polynomial equations have a finite set of complex solutions, we can reformulate this problem as a semidefinite programming problem. Our semidefinite representation involves combinatorial moment matrices, which are matrices indexed by a basis of the quotient vector space ℝ[x ₁, . . . ,x _n ]/I, where I is the ideal generated by the polynomial equations in the problem. Moreover, we prove the finite convergence of a hierarchy of semidefinite relaxations introduced by Lasserre. Semidefinite approximations can be constructed by considering truncated combinatorial moment matrices; rank conditions are given (in a grid case) that ensure that the approximation solves the original problem to optimality. Supported by the Netherlands Organisation for Scientific Research grant NWO 639.032.203.     

On conditional weak convergence

In this paper we discuss a number of technical issues associated with conditional weak convergence. The main modes of convergence of conditional probability distributions areuniform, probability, andalmost sure convergence in the conditioning variable. General results regarding conditional convergence are obtained, including details of sufficient conditions for each mode of convergence, and characterization theorems for uniform conditional convergence.     

On success rates for controlled random search

Controlled Random Search (CRS) is a simple population based algorithm which despite its attractiveness for practical use, has never been very popular among researchers on Global Optimization due to the difficulties in analysing the algorithm. In this paper, a framework to study the behaviour of algorithms in general is presented and embedded into the context of our view on questions in Global Optimization. By using as a reference a theoretical ideal algorithm called N-points Pure Adaptive Search (NPAS) some new analytical results provide bounds on speed of convergence and the Success Rate of CRS in the limit once it has settled down into simple behaviour. To relate the performance of the algorithm to characteristics of functions to be optimized, constructed simple test functions, called extreme cases, are used.     

Backfitting neural networks

Summary  Regression and classification problems can be viewed as special cases of the problem of function estimation. It is rather well known that a two-layer perceptron with sigmoidal transformation functions can approximate any continuous function on the compact subsets ofRP if there are sufficient number of hidden nodes. In this paper, we present an algorithm for fitting perceptron models, which is quite different from the usual backpropagation or Levenberg-Marquardt algorithm. This new algorithm based on backfitting ensures a better convergence than backpropagation. We have also used resampling techniques to select an ideal number of hidden nodes automatically using the training data itself. This resampling technique helps to avoid the problem of overfitting that one faces for the usual perceptron learning algorithms without any model selection scheme. Case studies and simulation results are presented to illustrate the performance of this proposed algorithm.