Hyperbolic cross approximation is a special type of multivariate approximation. Recently, driven by applications in engineering, biology, medicine and other areas of science new challenging problems have appeared. The common feature of these problems is high dimensions. We present here a survey on classical methods developed in multivariate approximation theory, which are known to work very well for moderate dimensions and which have potential for applications in really high dimensions. The theory of hyperbolic cross approximation and related theory of functions with mixed smoothness are under detailed study for more than 50 years. It is now well understood that this theory is important both for theoretical study and for practical applications. It is also understood that both theoretical analysis and construction of practical algorithms are very difficult problems. This explains why many fundamental problems in this area are still unsolved. Only a few survey papers and monographs on the topic are published. This and recently discovered deep connections between the hyperbolic cross approximation (and related sparse grids) and other areas of mathematics such as probability, discrepancy, and numerical integration motivated us to write this survey. We try to put emphases on the development of ideas and methods rather than list all the known results in the area. We formulate many problems, which, to our knowledge, are open problems. We also include some very recent results on the topic, which sometimes highlight new interesting directions of research. We hope that this survey will stimulate further active research in this fascinating and challenging area of approximation theory and numerical analysis.

Hyperbolic Cross Approximation

General topology I , edited by A. V. Arkhangel’skii and L. S. Pontryagin. Pp 230. DM128. 1990. ISBN 3-540-18178-4 (Springer)

This paper is essentially a survey on grand and small Lebesgue spaces, which are rearrangement-invariant Banach function spaces of interest not only from the point of view of Function Spaces theory, but also from the point of view of their applications: the corresponding Sobolev spaces are of interest, for instance, in the theory of PDEs. We discuss results of existence, uniqueness and regularity of certain Dirichlet problems, where the knowledge of these spaces plays a central role. The novelty of this paper relies in an unified treatment containing a number of equivalent quasinorms, all written making explicit the dependence of |Ω| , in the discussion of the sharpness of Hölder’s inequality, and in the connection of the results in PDEs with some existing literature. 1. Grand and small Lebesgue spaces: a short overview 1.1. The original motivation Let Ω ⊂ Rn be a bounded domain and f : Ω →Rn , f = ( f 1, ..., f n) be a mapping of Sobolev class W 1,n loc (Ω,R n) . Let us denote by Df (x) : Rn → Rn the differential and by J(x, f ) = detD f (x) the Jacobian of f . After the elementary remark that by Hölder’s inequality the Jacobian J(x, f ) is in Lloc(Ω) , the first fundamental result on the integrability of the Jacobian was due to Müller ([135]): f ∈W (Ω,R), J(x, f ) 0 a.e. ⇒ J(x, f ) ∈ L logLloc(Ω). Mathematics subject classification (2010): 46E30, 35J65.

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On grand and small Lebesgue and Sobolev spaces and some applications to PDE's

Optimal approximation of multivariate periodic Sobolev functions in the sup-norm

Rough set theory is a non-statistical approach to handle uncertainty and uncertain knowledge. It is characterized by two methods called classification (lower and upper approximations) and accuracy measure. The closeness of notions and results in topology and rough set theory motivates researchers to explore the topological aspects and their applications in rough set theory. To contribute to this area, this paper applies a topological concept called “somewhere dense sets” to improve the approximations and accuracy measure in rough set theory. We firstly discuss further topological properties of somewhere dense and cs-dense sets and give explicitly formulations to calculate S-interior and S-closure operators. Then, we utilize these two sets to define new concepts in rough set context such as SD-lower and SD-upper approximations, SD-boundary region, and SD-accuracy measure of a subset. We establish the fundamental properties of these concepts as well as show their relationships with the previous ones. In the end, we compare the current method of approximations with the previous ones and provide two examples to elucidate that the current method is more accurate.

Improvement of the approximations and accuracy measure of a rough set using somewhere dense sets

One of the considerable subjects in mathematics is the study of topology. Deducing topology from arbitrary binary relations has enticed the attention of many researchers. So, we devote this article...

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, Various topologies generated from Ej-neighbourhoods via ideals

In this paper, we generalize rough set theory by introducing concepts of  δβ-I lower and δβ-I -upper approximation for any ideal  I on X which depends on the concept δβ-I -open sets Some of their basic properties with the help of examples are investigated and the interrelation between them are obtained Also, the connections between the rough approximations de_ned in [2] and our new approximations are studied

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δβ-I APPROXIMATION SPACES

The idea of neighborhood systems is induced from the geometric idea of “near,” and it is primitive in the topological structures. Now, the idea of neighborhood systems has been extensively applied in rough set theory. The master contribution of this manuscript is to generate various topologies by means of the concepts of - adhesion neighborhoods and ideals. Then, we define a new rough set model derived from these topologies and discussed main features. We show that these topologies are finer than those given in the previous ones under arbitrary binary relations. In addition, we elucidate that these topologies are finer than those topologies initiated based on different neighborhoods and ideals under reflexive relations. Several examples are provided to validate that our model is better than the previous ones.

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Some Topological Approaches for Generalized Rough Sets via Ideals

Many scientists have studied and improved the soft set theory, which is initiated by Molodtsov (33) and easily applied to many problems having uncertainties from social life. The main purpose of our paper, is to introduce new soft separation axioms based on the b-open soft sets which are more general than of the open soft sets. We show that, the properties of soft b-Ti-spaces (i = 1, 2) are soft topological properties under the bijection and irr esolute open soft mapping. Also, the property of being soft b-regular and soft b-normal are soft topological properties under bijection, irresolute soft and irresolute open soft functions. Furthe r, we show that the properties of being soft b-Ti-spaces (i = 1, 2, 3, 4) are hereditary properties.

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Rodyna A. Hosny

Papers

, Various topologies generated from Ej-neighbourhoods via ideals

δβ-I APPROXIMATION SPACES

Some Topological Approaches for Generalized Rough Sets via Ideals

Characterizations of b-Soft Separation Axioms in Soft Topological Spaces

Θβ-Ideal Approximation Spaces and Their Applications