scholarly journals Characterizations of an MW-topological rough set structure

Filomat ◽  
2020 ◽  
Vol 34 (7) ◽  
pp. 2357-2366
Author(s):  
Sang-Eon Han
Keyword(s):  
Euclidean Space ◽  
Compact Subset ◽  
Rough Set ◽  
Dimensional Euclidean Space ◽  
Finite Covering ◽  
Locally Finite ◽  
Target Set ◽  
The Universe

Regarding the study of digital topological rough set structures, the present paper explores some mathematical and systemical structures of the Marcus-Wyse (MW-, for brevity) topological rough set structures induced by the locally finite covering approximation (LFC-, for brevity) space (R2,C) (see Proposition 3.4 in this paper), where R2 is the 2-dimensional Euclidean space. More precisely, given the LFC-space (R2,C), based on the set of adhesions of points in R2 inducing certain LFC-rough concept approximations, we systematically investigate various properties of the MW-topological rough concept approximations (D -M, D+M) derived from this LFC-space (R2,C). These approaches can facilitate the study of an estimation of roughness in terms of an MW-topological rough set. In the present paper each of a universe U and a target set X(? U) need not be finite and further, a covering C is locally finite. In addition, when regarding both an M-rough set and an MW-topological rough set in Sections 3, 4, and 5, the universe U(? R2) is assumed to be the set R2 or a compact subset of R2 or a certain set containing the union of all adhesions of x ? X (see Remark 3.6).

scholarly journals On targeting an integral funnel of control system at a target set in the phase space

2020 ◽  
Vol 56 ◽  
pp. 79-101
Author(s):  
V.N. Ushakov ◽  
A.V. Ushakov
Keyword(s):  
Exact Solution ◽  
Control System ◽  
Phase Space ◽  
Euclidean Space ◽  
Time Moment ◽  
Dimensional Euclidean Space ◽  
Time Interval ◽  
Finite Dimensional ◽  
Approximate Construction ◽  
Target Set

A control system in finite-dimensional Euclidean space is considered. On a given time interval, we investigate the problem of constructing an integral funnel for which a section corresponding to the last time moment of interval is equal to a target set in a phase space. Since the exact solution of such a funnel is possible only in rare cases, the question of the approximate construction of an integral funnel is being studied.

Unisolvence on Multidimensional Spaces

1968 ◽  
Vol 11 (3) ◽  
pp. 469-474 ◽  
Author(s):  
Charles B. Dunham
Keyword(s):  
Continuous Function ◽  
Euclidean Space ◽  
Compact Subset ◽  
Compact Space ◽  
General Condition ◽  
Chebyshev Approximation ◽  
Continuous Functions ◽  
Dimensional Euclidean Space ◽  
Variable Degree ◽  
Convexity Conditions

In this note we consider the possibility of unisolvence of a family of real continuous functions on a compact subset X of m-dimensional Euclidean space. Such a study is of interest for two reasons. First, an elegant theory of Chebyshev approximation has been constructed for the case where the approximating family is unisolvent of degree n on an interval [α, β]. We study what sort of theory results from unisolvence of degree n on a more general space. Secondly, uniqueness of best Chebyshev approximation on a general compact space to any continuous function on X can be shown if the approximating family is unisolvent of degree n and satisfies certain convexity conditions. It is therefore of importance to Chebyshev approximation to consider the domains X on which unisolvence can occur. We will also study a more general condition on involving a variable degree.

Smooth Finite Dimensional Embeddings

1999 ◽  
Vol 51 (3) ◽  
pp. 585-615 ◽  
Author(s):  
R. Mansfield ◽  
H. Movahedi-Lankarani ◽  
R. Wells
Keyword(s):  
Hilbert Space ◽  
Euclidean Space ◽  
Compact Subset ◽  
Structure Theorem ◽  
Sufficient Conditions ◽  
Dimensional Euclidean Space ◽  
Locally Compact ◽  
Finite Dimensional ◽  
Necessary And Sufficient ◽  
Compact Subsets

AbstractWe give necessary and sufficient conditions for a norm-compact subset of a Hilbert space to admit a C1 embedding into a finite dimensional Euclidean space. Using quasibundles, we prove a structure theorem saying that the stratum of n-dimensional points is contained in an n-dimensional C1 submanifold of the ambient Hilbert space. This work sharpens and extends earlier results of G. Glaeser on paratingents. As byproducts we obtain smoothing theorems for compact subsets of Hilbert space and disjunction theorems for locally compact subsets of Euclidean space.

scholarly journals Euclidean null controllability of infinite neutral differential systems

2006 ◽  
Vol 48 (2) ◽  
pp. 285-293 ◽  
Author(s):  
Davies Iyai
Keyword(s):  
Euclidean Space ◽  
Compact Subset ◽  
Differential System ◽  
Null Controllability ◽  
Dimensional Euclidean Space ◽  
Controlled System ◽  
Differential Systems ◽  
Free System ◽  
Measurable Functions ◽  
The Stability

AbstractThis paper is aimed at establishing sufficient computable criteria for the Euclidean null controllability of an infinite neutral differential system, when the controls are essentially bounded measurable functions on finite intervals, with values in a compact subset U of an m-dimensional Euclidean space with zero in its interior. Our results are obtained by exploiting the stability of the free system and the rank criterion for properness of the controlled system. An example is also given.

Full Development of Tarski's Geometry of Solids

2008 ◽  
Vol 14 (4) ◽  
pp. 481-540 ◽  
Author(s):  
Rafaŀ Gruszczyński ◽  
Andrzej Pietruszczak
Keyword(s):  
Euclidean Space ◽  
Three Dimensional ◽  
Final Part ◽  
Dimensional Euclidean Space ◽  
Short Paper ◽  
Logical Status ◽  
Full Development ◽  
Definition Of ◽  
The Universe ◽  
Universe Of Discourse

AbstractIn this paper we give probably an exhaustive analysis of the geometry of solids which was sketched by Tarski in his short paper [20, 21]. We show that in order to prove theorems stated in [20, 21] one must enrich Tarski's theory with a new postulate asserting that the universe of discourse of the geometry of solids coincides with arbitrary mereological sums of balls, i.e., with solids. We show that once having adopted such a solution Tarski's Postulate 4 can be omitted, together with its versions 4′ and 4″. We also prove that the equivalence of postulates 4, 4′ and 4″ is not provable in any theory whose domain contains objects other than solids. Moreover, we show that the concentricity relation as defined by Tarski must be transitive in the largest class of structures satisfying Tarski's axioms.We build a model (in three-dimensional Euclidean space) of the theory of so called T*-structures and present the proof of the fact that this is the only (up to isomorphism) model of this theory.Moreover, we propose different categorical axiomatizations of the geometry of solids. In the final part of the paper we answer the question concerning the logical status (within the theory of T*-structures) of the definition of the concentricity relation given by Tarski.

On Vitali's Theorem for Groups of Motions of Euclidean Space

1999 ◽  
Vol 6 (4) ◽  
pp. 323-334
Author(s):  
A. Kharazishvili
Keyword(s):  
Euclidean Space ◽  
Dimensional Euclidean Space ◽  
Finite Dimensional

Abstract We give a characterization of all those groups of isometric transformations of a finite-dimensional Euclidean space, for which an analogue of the classical Vitali theorem [Sul problema della misura dei gruppi di punti di una retta, 1905] holds true. This characterization is formulated in purely geometrical terms.

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