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.

scholarly journals On the construction of resolving control in the problem of getting close at a fixed time moment

2021 ◽  
Vol 58 ◽  
pp. 73-93
Author(s):  
V.N. Ushakov ◽  
A.V. Ushakov ◽  
O.A. Kuvshinov
Keyword(s):  
Euclidean Space ◽  
Compact Space ◽  
Time Moment ◽  
Fixed Time ◽  
Dimensional Euclidean Space ◽  
Controlled System ◽  
Finite Dimensional ◽  
Solvability Set

The problem of getting close of a controlled system with a compact space in a finite-dimensional Euclidean space at a fixed time is studied. A method of constructing a solution to the problem is proposed which is based on the ideology of the maximum shift of the motion of the controlled system by the solvability set of the getting close problem.

scholarly journals Cores of Potential Operators for Processes With Stationary Independent Increments

1972 ◽  
Vol 48 ◽  
pp. 129-145
Author(s):  
Ken-Iti Sato
Keyword(s):  
Banach Space ◽  
Stochastic Process ◽  
Euclidean Space ◽  
Continuous Functions ◽  
Dimensional Euclidean Space ◽  
Semigroup Of Operators ◽  
Transition Semigroup ◽  
Potential Operators ◽  
Independent Increments

Let Xt(ω)) be a stochastic process with stationary independent increments on the N-dimensional Euclidean space RN, right continuous in t ≧ 0 and starting at the origin. Let C0(RN) be the Banach space of real-valued continuous functions on RN vanishing at infinity with norm . The process induces a transition semigroup of operators Tt on C0(RN) :Ttf(x) = Ef(x + Xt).

scholarly journals ON A PROBLEM OF TALAGRAND CONCERNING SEPARATELY CONTINUOUS FUNCTIONS

2020 ◽  
pp. 1-10
Author(s):  
Volodymyr Mykhaylyuk ◽  
Roman Pol
Keyword(s):  
Continuous Function ◽  
Compact Space ◽  
Continuous Functions ◽  
Baire Space ◽  
Negative Solution ◽  
Borel Set ◽  
Joint Continuity

We construct a separately continuous function $e:E\times K\rightarrow \{0,1\}$ on the product of a Baire space $E$ and a compact space $K$ such that no restriction of $e$ to any non-meagre Borel set in $E\times K$ is continuous. The function $e$ has no points of joint continuity, and, hence, it provides a negative solution of Talagrand’s problem in Talagrand [Espaces de Baire et espaces de Namioka, Math. Ann.270 (1985), 159–164].

scholarly journals A note on convex functions

1999 ◽  
Vol 22 (3) ◽  
pp. 525-534 ◽  
Author(s):  
Yu-Ru Syau
Keyword(s):  
Continuous Function ◽  
Euclidean Space ◽  
Convex Function ◽  
Convex Functions ◽  
Dimensional Euclidean Space ◽  
Strictly Convex ◽  
Lower Semi

In this paper, we give two weak conditions for a lower semi-continuous function on then-dimensional Euclidean spaceRnto be a convex function. We also present some results for convex functions, strictly convex functions, and quasi-convex functions.

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.

AN ACCESS THEOREM FOR CONTINUOUS FUNCTIONS

2001 ◽  
Vol 64 (1) ◽  
pp. 81-92
Author(s):  
ALEXANDER BORICHEV ◽  
IGOR KLESCHEVICH
Keyword(s):  
Continuous Function ◽  
Compact Subset ◽  
Open Subset ◽  
Higher Dimensions ◽  
Continuous Functions ◽  
Continuous Map

Let f be a continuous function on an open subset Ω of ℝ2 such that for every x ∈ Ω there exists a continuous map γ : [−1, 1] → Ω with γ(0) = x and f ∘ γ increasing on [−1, 1]. Then for every γ ∈ Ω there exists a continuous map γ : [0, 1) → Ω such that γ(0) = y, f ∘ γ is increasing on [0; 1), and for every compact subset K of Ω, max{t : γ(t) ∈ K} < 1. This result gives an answer to a question posed by M. Ortel. Furthermore, an example shows that this result is not valid in higher dimensions.

On the optimal search for a target whose motion is a Markov process

1973 ◽  
Vol 10 (4) ◽  
pp. 847-856 ◽  
Author(s):  
Lauri Saretsalo
Keyword(s):  
Markov Process ◽  
Euclidean Space ◽  
Markov Processes ◽  
General Condition ◽  
Necessary Condition ◽  
Dimensional Euclidean Space ◽  
Optimal Search ◽  
Continuous Transition ◽  
Transition Functions ◽  
Multiplicative Functionals

We will consider the optimal search for a target whose motion is a Markov process. The classical detection law leads to the use of multiplicative functionals and the search is equivalent to the termination of the Markov process with a termination density. A general condition for the optimality is derived and for Markov processes in n-dimensional Euclidean space with continuous transition functions we derive a simple necessary condition which generalizes the result of Hellman (1972).

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.

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).

On the optimal search for a target whose motion is a Markov process

1973 ◽  
Vol 10 (04) ◽  
pp. 847-856 ◽  
Author(s):  
Lauri Saretsalo
Keyword(s):  
Markov Process ◽  
Euclidean Space ◽  
Markov Processes ◽  
General Condition ◽  
Necessary Condition ◽  
Dimensional Euclidean Space ◽  
Optimal Search ◽  
Continuous Transition ◽  
Transition Functions ◽  
Multiplicative Functionals

We will consider the optimal search for a target whose motion is a Markov process. The classical detection law leads to the use of multiplicative functionals and the search is equivalent to the termination of the Markov process with a termination density. A general condition for the optimality is derived and for Markov processes in n-dimensional Euclidean space with continuous transition functions we derive a simple necessary condition which generalizes the result of Hellman (1972).

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