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 A New Approach to Mannheim Curve in Euclidean 3-Space

2021 ◽  
Vol 54 ◽  
Author(s):  
Ali UÇUM ◽  
Çetin Camcı ◽  
Kazım İlarslan
Keyword(s):  
Euclidean Space ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Dimensional Euclidean Space ◽  
New Approach ◽  
3 Dimensional ◽  
Mannheim Curve ◽  
Necessary And Sufficient

In this article, a new approach is given for Mannheim curves in 3-dimensional Euclidean space. Thanks to this approach, the necessary and sufficient conditions including the known results have been obtained for a curve to be Mannheim curve in E³. In addition, related examples and graphs are given by showing that there can be Mannheim curves in Salkowski or anti-Salkowski curves as well as giving Mannheim mate curves, which are not in literature. Finally, the Mannheim partner curves are characterized in E³.

Matrix Resolving Function in the Nonstationary Linear Group Pursuit Problem Concepting Multiple Capture

2021 ◽  
pp. 2150016
Author(s):  
N. N. Petrov
Keyword(s):  
Euclidean Space ◽  
Linear Group ◽  
Sufficient Conditions ◽  
Dimensional Euclidean Space ◽  
Pursuit Problem ◽  
Mathematical Basis ◽  
Group Pursuit ◽  
Finite Dimensional

In finite-dimensional Euclidean space, an analysis is made of the problem of pursuit of a single evader by a group of pursuers, which is described by a system of the form [Formula: see text] The goal of the group of pursuers is the capture of the evader by no less than [Formula: see text] different pursuers (the instants of capture may or may not coincide). Matrix resolving functions, which are a generalization of scalar resolving functions, are used as a mathematical basis of this study. Sufficient conditions are obtained for multiple capture of a single evader in the class of quasi-strategies. Examples illustrating the results obtained are given.

scholarly journals Finite dimensional H-invariant spaces

1997 ◽  
Vol 56 (3) ◽  
pp. 353-361
Author(s):  
K.E. Hare ◽  
J.A. Ward
Keyword(s):  
Compact Group ◽  
Closed Subspace ◽  
Sufficient Conditions ◽  
Trigonometric Polynomials ◽  
Left Translation ◽  
Almost Periodic ◽  
Locally Compact ◽  
Finite Dimensional ◽  
Necessary And Sufficient ◽  
Invariant Spaces

A subset V of M(G) is left H-invariant if it is invariant under left translation by the elements of H, a subset of a locally compact group G. We establish necessary and sufficient conditions on H which ensure that finite dimensional subspaces of M(G) when G is compact, or of L∞(G) when G is locally compact Abelian, which are invariant in this weaker sense, contain only trigonometric polynomials. This generalises known results for finite dimensional G-invariant subspaces. We show that if H is a subgroup of finite index in a compact group G, and the span of the H-translates of μ is a weak*-closed subspace of L∞(G) or M(G) (or is closed in Lp(G)for 1 ≤ p < ∞), then μ is a trigonometric polynomial.We also obtain some results concerning functions that possess the analogous weaker almost periodic condition relative to H.

Products of Normal Operators

1988 ◽  
Vol 40 (6) ◽  
pp. 1322-1330 ◽  
Author(s):  
Pei Yuan Wu
Keyword(s):  
Hilbert Space ◽  
Linear Operator ◽  
Separable Hilbert Space ◽  
Polar Decomposition ◽  
Sufficient Conditions ◽  
Bounded Linear ◽  
Underlying Space ◽  
Finite Dimensional ◽  
Normal Operators ◽  
Necessary And Sufficient

Which bounded linear operator on a complex, separable Hilbert space can be expressed as the product of finitely many normal operators? What is the answer if “normal” is replaced by “Hermitian”, “nonnegative” or “positive”? Recall that an operator T is nonnegative (resp. positive) if (Tx, x) ≧ 0 (resp. (Tx, x) ≥ 0) for any x ≠ 0 in the underlying space. The purpose of this paper is to provide complete answers to these questions.If the space is finite-dimensional, then necessary and sufficient conditions for operators expressible as such are already known. For normal operators, this is easy. By the polar decomposition, every operator is the product of two normal operators. An operator is the product of Hermitian operators if and only if its determinant is real; moreover, in this case, 4 Hermitian operators suffice and 4 is the smallest such number (cf. [10]).

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.

On inequalities of the Tchebychev type

1963 ◽  
Vol 59 (1) ◽  
pp. 135-146 ◽  
Author(s):  
J. F. C. Kingman
Keyword(s):  
Probability Theory ◽  
Euclidean Space ◽  
Random Variable ◽  
Dimensional Euclidean Space ◽  
Real Random Variable ◽  
Finite Dimensional

1. A type of problem which frequently occurs in probability theory and statistics can be formulated in the following way. We are given real-valued functions f(x), gi(x) (i = 1, 2, …, k) on a space (typically finite-dimensional Euclidean space). Then the problem is to set bounds for Ef(X), where X is a random variable taking values in , about which all we know is the values of Egi(X). For example, we might wish to set bounds for P(X > a), where X is a real random variable with some of its moments given.

scholarly journals Biquasitriangularity and spectral continuity

1985 ◽  
Vol 26 (2) ◽  
pp. 177-180 ◽  
Author(s):  
Ridgley Lange
Keyword(s):  
Hilbert Space ◽  
Sufficient Conditions ◽  
Invariant Subspaces ◽  
Extension Property ◽  
Point Of View ◽  
Single Valued Extension Property ◽  
Continuity Theorem ◽  
Spectral Continuity ◽  
Necessary And Sufficient ◽  
Continuity Of The Spectrum

In [6] Conway and Morrell characterized those operators on Hilbert space that are points of continuity of the spectrum. They also gave necessary and sufficient conditions that a biquasitriangular operator be a point of spectral continuity. Our point of view in this note is slightly different. Given a point T of spectral continuity, we ask what can then be inferred. Several of our results deal with invariant subspaces. We also give some conditions characterizing a biquasitriangular point of spectral continuity (Theorem 3). One of these is that the operator and its adjoint both have the single-valued extension property.

scholarly journals Topological entropy in totally disconnected locally compact groups

2016 ◽  
Vol 37 (7) ◽  
pp. 2163-2186 ◽  
Author(s):  
ANNA GIORDANO BRUNO ◽  
SIMONE VIRILI
Keyword(s):  
Topological Entropy ◽  
Sufficient Conditions ◽  
Invariant Subgroup ◽  
Locally Compact Groups ◽  
Compact Groups ◽  
Locally Compact ◽  
Image Position ◽  
Necessary And Sufficient ◽  
Totally Disconnected ◽  
Dynamical Interpretation

Let $G$ be a topological group, let $\unicode[STIX]{x1D719}$ be a continuous endomorphism of $G$ and let $H$ be a closed $\unicode[STIX]{x1D719}$-invariant subgroup of $G$. We study whether the topological entropy is an additive invariant, that is, $$\begin{eqnarray}h_{\text{top}}(\unicode[STIX]{x1D719})=h_{\text{top}}(\unicode[STIX]{x1D719}\restriction _{H})+h_{\text{top}}(\bar{\unicode[STIX]{x1D719}}),\end{eqnarray}$$ where $\bar{\unicode[STIX]{x1D719}}:G/H\rightarrow G/H$ is the map induced by $\unicode[STIX]{x1D719}$. We concentrate on the case when $G$ is totally disconnected locally compact and $H$ is either compact or normal. Under these hypotheses, we show that the above additivity property holds true whenever $\unicode[STIX]{x1D719}H=H$ and $\ker (\unicode[STIX]{x1D719})\leq H$. As an application, we give a dynamical interpretation of the scale $s(\unicode[STIX]{x1D719})$ by showing that $\log s(\unicode[STIX]{x1D719})$ is the topological entropy of a suitable map induced by $\unicode[STIX]{x1D719}$. Finally, we give necessary and sufficient conditions for the equality $\log s(\unicode[STIX]{x1D719})=h_{\text{top}}(\unicode[STIX]{x1D719})$ to hold.

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.

On the Integral Extensions of Quadratic Forms Over Local Fields

1970 ◽  
Vol 22 (2) ◽  
pp. 297-307 ◽  
Author(s):  
Melvin Band
Keyword(s):  
Prime Ideal ◽  
Local Field ◽  
Quadratic Forms ◽  
Sufficient Conditions ◽  
Local Fields ◽  
Ring Of Integers ◽  
Finite Dimensional ◽  
Integral Analogue ◽  
Necessary And Sufficient ◽  
Special Case

Let F be a local field with ring of integers and unique prime ideal (p). Suppose that V a finite-dimensional regular quadratic space over F, W and W′ are two isometric subspaces of V (i.e. τ: W → W′ is an isometry from W to W′). By the well-known Witt's Theorem, τ can always be extended to an isometry σ ∈ O(V).The integral analogue of this theorem has been solved over non-dyadic local fields by James and Rosenzweig [2], over the 2-adic fields by Trojan [4], and partially over the dyadics by Hsia [1], all for the special case that W is a line. In this paper we give necessary and sufficient conditions that two arbitrary dimensional subspaces W and W′ are integrally equivalent over non-dyadic local fields.

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