scholarly journals Asymptotic Properties of -Expansive Homeomorphisms on a Metric -Space

2012 ◽  
Vol 2012 ◽  
pp. 1-10
Author(s):  
Ruchi Das ◽  
Tarun Das
Keyword(s):  
Linear Space ◽  
Metric Space ◽  
Normed Linear Space ◽  
Asymptotic Properties ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Bounded Subset ◽  
Necessary And Sufficient

We define and study the notions of positively and negatively -asymptotic points for a homeomorphism on a metric -space. We obtain necessary and sufficient conditions for two points to be positively/negatively -asymptotic. Also, we show that the problem of studying -expansive homeomorphisms on a bounded subset of a normed linear -space is equivalent to the problem of studying linear -expansive homeomorphisms on a bounded subset of another normed linear -space.

A general canonical expression

1933 ◽  
Vol 29 (4) ◽  
pp. 465-469 ◽  
Author(s):  
J. Bronowski
Keyword(s):  
Linear Space ◽  
Dimensional Space ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
General Point ◽  
Linear Forms ◽  
Powers Of Linear Forms ◽  
Necessary And Sufficient

1. In a recent paper I established new conditions for a form φ of order n, homogeneous in r + 1 variables, to be expressible as the sum of nth powers of linear forms in these variables; and for this expression, if it exists, to be unique. These conditions, I further showed, may be stated as general theorems regarding the secant spaces of manifolds Mr in higher space, namely:Necessary and sufficient conditions that through a general point of a space N, of h (r + 1) − 1 dimensions, there passes (i) no, (ii) a unique (h − 1)-dimensional space containing h points of a manifold Mr lying in N are that(i) the space projecting a general point of Mr from the join of h − 1 general r-dimensional tangent spaces of Mr meets Mr in a curve, so that Mr cannot be so projected upon a linear space of r dimensions;(ii) the space projecting a general point of Mr from the join of h − 1 general r-dimensional tangent spaces of Mr does not meet Mr again, so that Mr can be so projected, birationally, upon a linear space of r dimensions..

scholarly journals On the solvability of systems of linear equations on commutative semigroup

2010 ◽  
Vol 43 (4) ◽  
Author(s):  
Nguyen Thi Thu Huyen ◽  
Nguyen Minh Tuan
Keyword(s):  
Linear Operator ◽  
Linear Space ◽  
Commutative Semigroup ◽  
Linear Equations ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Operator Equations ◽  
Systems Of Linear Equations ◽  
Necessary And Sufficient ◽  
Linear Operator Equations

AbstractThis paper deals with the solvability of systems of linear operator equations in a linear space. Namely, the paper provides necessary and sufficient conditions for the operators under which certain kinds of systems of operator equations are solvable.

scholarly journals Further results on images of metric spaces

2015 ◽  
Vol 98 (112) ◽  
pp. 179-191
Author(s):  
Van Dung
Keyword(s):  
Metric Space ◽  
Metric Spaces ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Weakly Continuous ◽  
Locally Separable Metric Spaces ◽  
Necessary And Sufficient ◽  
Separable Metric Space

We introduce the notion of an ls-?-Ponomarev-system to give necessary and sufficient conditions for f:(M,M0) ? X to be a strong wc-mapping (wc-mapping, wk-mapping) where M is a locally separable metric space. Then, we systematically get characterizations of weakly continuous strong wc-images (wc-images, wk-images) of locally separable metric spaces by means of certain networks. Also, we give counterexamples to sharpen some results on images of locally separable metric spaces in the literature.

Characterizations of Spherical Neighbourhoods

1970 ◽  
Vol 22 (2) ◽  
pp. 431-435 ◽  
Author(s):  
C. M. Petty ◽  
J. M. Crotty
Keyword(s):  
Metric Space ◽  
Metric Spaces ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Characterization Problem ◽  
Necessary And Sufficient

If Σ is a specified class of metric spaces and M ∈ Σ, then the characterization problem is to find necessary and sufficient conditions which distinguish the spherical neighbourhoods (open spheres) of M among a specified class of subsets of M.In a metric space M the notation pqr means p ≠ q ≠ r and pq + qr = pr.M is said to be uniformly locally externally convex if there exists δ > 0 such that if p, q ∈ M, p ≠ q, and pq < δ, then there exists r ∈ M such that the relation pqr subsists. We will prove the following result.

scholarly journals Fixed point theorems for non-self maps I

1994 ◽  
Vol 17 (4) ◽  
pp. 713-716 ◽  
Author(s):  
Troy L. Hicks ◽  
Linda Marie Saliga
Keyword(s):  
Fixed Point ◽  
Metric Space ◽  
Closed Subset ◽  
Fixed Point Theorems ◽  
Complete Metric Space ◽  
Sufficient Conditions ◽  
General Setting ◽  
Necessary And Sufficient Conditions ◽  
Necessary And Sufficient

Supposef:C→XwhereCis a closed subset ofX. Necessary and sufficient conditions are given forfto have a fixed point. All results hold whenXis complete metric space. Several results hold in a much more general setting.

scholarly journals On Conditions for Unrectifiability of a Metric Space

2014 ◽  
Vol 3 (1) ◽  
Author(s):  
Piotr Hajłasz ◽  
Soheil Malekzadeh
Keyword(s):  
Metric Space ◽  
Implicit Function Theorem ◽  
Metric Spaces ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Implicit Function ◽  
Heisenberg Groups ◽  
Lipschitz Map ◽  
Necessary And Sufficient

Abstract We find necessary and sufficient conditions for a Lipschitz map f : E ⊂ ℝk → X into a metric space to satisfy ℋk(f(E)) = 0. An interesting feature of our approach is that despite the fact that we are dealing with arbitrary metric spaces, we employ a variant of the classical implicit function theorem. Applications include pure unrectifiability of the Heisenberg groups.

scholarly journals Uniqueness of spaces pretangent to metric spaces at infinity

2019 ◽  
Vol 16 (1) ◽  
pp. 57-87
Author(s):  
Oleksiy Dovgoshey ◽  
Victoria Bilet
Keyword(s):  
Metric Space ◽  
Real Line ◽  
Metric Spaces ◽  
Sufficient Conditions ◽  
Logarithmic Spiral ◽  
Necessary And Sufficient Conditions ◽  
Scaling Sequence ◽  
Necessary And Sufficient

We find the necessary and sufficient conditions under which an unbounded metric space \(X\) has, at infinity, a unique pretangent space \(\Omega^{X}_{\infty,\tilde{r}}\) for every scaling sequence \(\tilde{r}\). In particular, it is proved that \(\Omega^{X}_{\infty,\tilde{r}}\) is unique and isometric to the closure of \(X\) for every logarithmic spiral \(X\) and every \(\tilde{r}\). It is also shown that the uniqueness of pretangent spaces to subsets of a real line is closely related to the ''asymptotic asymmetry'' of these subsets.

scholarly journals On characterizations of fixed points

2001 ◽  
Vol 27 (7) ◽  
pp. 391-397 ◽  
Author(s):  
Zeqing Liu ◽  
Lili Zhang ◽  
Shin Min Kang
Keyword(s):  
Fixed Points ◽  
Metric Space ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Equivalent Condition ◽  
Compact Mapping ◽  
Necessary And Sufficient

We give some necessary and sufficient conditions for the existence of fixed points of a family of self mappings of a metric space and we establish an equivalent condition for the existence of fixed points of a continuous compact mapping of a metric space.

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