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Plasticity in metric spaces
Abstract
In this paper we examine the properties of EC-plastic metric spaces, spaces which have the property that any noncontractive bijection from the space onto itself must be an isometry.
... The only general result concerning plasticity of metric space states that every totally bounded metric space is plastic, see Naimpally et al. [4] for details. In fact, in the study mentioned in the reference [4], a more general result was obtained, i. e., so-called strong plasticity of totally bounded metric spaces was shown. ...
... The only general result concerning plasticity of metric space states that every totally bounded metric space is plastic, see Naimpally et al. [4] for details. In fact, in the study mentioned in the reference [4], a more general result was obtained, i. e., so-called strong plasticity of totally bounded metric spaces was shown. Definition 2.2. ...
... On the other hand, it is easy to show that the set of integers Z with the same usual metric is plastic in spite of its unboundedness and the set R \ Z. The proof of the plasticity of both mentioned spaces may be found in the study mentioned in the reference [4]. In the proof of plasticity of the set R \ Z, one of the possible cases was missed; nevertheless, the statement is still correct. ...
The study deals with plastic and non-plastic sub-spaces A of the real-line ℝ with the usual Euclidean metric d . It investigates non-expansive bijections, proves properties of such maps, and demonstrates their relevance by hands of examples. Finally, it is shown that the plasticity property of a sub-space A contains at least two complementary questions, a purely geometric and a topological one. Both contribute essential aspects to the plasticity property and get more critical in higher dimensions and more abstract metric spaces.
... This property was introduced and studied in depth in [5]. The initial point of that study was the fact that every totally bounded metric space is EC-plastic [3]. ...
... The initial point of that study was the fact that every totally bounded metric space is EC-plastic [3]. In reality, totally bounded spaces possess a stronger property [5,Theorem 1.1] or [3,Satz IV], which for our convenience we formalize in the following definition. Definition 1.2. ...
... Remark, that previously known for the case of X = Y approach from [3,6,7] and [5] used the dynamical system generated by f : ...
... This property was introduced and studied in depth in [5]. The initial point of that study was the fact, that every totally bounded metric space is EC-plastic [3]. ...
... The initial point of that study was the fact, that every totally bounded metric space is EC-plastic [3]. In reality, totally bounded spaces possess a stronger property [5,Theorem 1.1] or [3,Satz IV], which for our convenience we formalize in the following definition. Definition 1.2. ...
... Remark, that previously known for the case of X = Y approach from [3,6,7] and [5] used the dynamical system generated by f : ...
We address pairs of metric spaces with the following property: for every mapping the existence of points with implies the existence of for which . We give sufficient conditions for this property and for its uniform version in terms of finite -nets and finite -separated subsets.
... Following [9], a metric space (M, ρ) is said to be plastic if every non-expansive bijection F : M → M is in fact an isometry ("non-expansive" means "1-Lipschitz"). For example, it is a well known classical fact that any compact metric space is plastic; more generally, any totally bounded metric space is plastic (see [9] for a proof and for historical references, and see [1] for an extension to the setting of uniform spaces). ...
... Following [9], a metric space (M, ρ) is said to be plastic if every non-expansive bijection F : M → M is in fact an isometry ("non-expansive" means "1-Lipschitz"). For example, it is a well known classical fact that any compact metric space is plastic; more generally, any totally bounded metric space is plastic (see [9] for a proof and for historical references, and see [1] for an extension to the setting of uniform spaces). On the other hand, every non-trivial normed space is non-plastic, as shown by F (x) = 1 2 x. ...
... On the other hand, every non-trivial normed space is non-plastic, as shown by F (x) = 1 2 x. Several interesting examples can be found in [9]. ...
We show that if K is a compact metrizable space with finitely many accumulation points, then the closed unit ball of C(K) is a plastic metric space, which means that any non-expansive bijection from onto itself is in fact an isometry. We also show that if K is a zero-dimensional compact Hausdorff space with a dense set of isolated points, then any non-expansive homeomorphism of is an isometry.
... We call a metric space X Expand-Contract plastic (or simply plastic) if every non-expansive bijection from X onto itself is an isometry. The last notion was introduced by S. A. Naimpally, Z. Piotrowski, and E. J. Wingler in [8]. It is known that every totally bounded metric space is plastic, see [3,Satz IV] or [8,Theorem 1.1]. ...
... The last notion was introduced by S. A. Naimpally, Z. Piotrowski, and E. J. Wingler in [8]. It is known that every totally bounded metric space is plastic, see [3,Satz IV] or [8,Theorem 1.1]. On the other hand, a plastic metric space need not be totally bounded nor bounded -e.g., the set of integers with the usual metric is plastic [8,Theorem 3.1]. ...
... It is known that every totally bounded metric space is plastic, see [3,Satz IV] or [8,Theorem 1.1]. On the other hand, a plastic metric space need not be totally bounded nor bounded -e.g., the set of integers with the usual metric is plastic [8,Theorem 3.1]. There are also examples of bounded metric spaces that are not plastic, one of our favorite examples here is a solid ellipsoid in Hilbert space ℓ 2 (Z) with infinitely many semi-axes equal to 1 and infinitely many semi-axes equal to 2, see [2,Example 2.7]. ...
We prove that Banach spaces ℓ1 ⊕2 R and X ⊕∞ Y , with strictly convex X and Y , have plastic unit balls (we call a metric space plastic if every non-expansive bijection from this space onto itself is an isometry).
... We call a metric space X plastic if every non-expansive bijection from X onto itself is an isometry. The last notion was introduced by S. A. Naimpally, Z. Piotrowski, and E. J. Wingler in [8]. It is known that every totally bounded metric space is plastic, see [3,Satz IV] or [8,Theorem 1.1]. ...
... The last notion was introduced by S. A. Naimpally, Z. Piotrowski, and E. J. Wingler in [8]. It is known that every totally bounded metric space is plastic, see [3,Satz IV] or [8,Theorem 1.1]. On the other hand, a plastic metric space need not be totally bounded nor bounded -e.g., the set of integers with the usual metric is plastic [8,Theorem 3.1]. ...
... It is known that every totally bounded metric space is plastic, see [3,Satz IV] or [8,Theorem 1.1]. On the other hand, a plastic metric space need not be totally bounded nor bounded -e.g., the set of integers with the usual metric is plastic [8,Theorem 3.1]. There are also examples of bounded metric spaces that are not plastic, one of our favorite examples here is a solid ellipsoid in Hilbert space ℓ 2 (Z) with infinitely many semi-axes equal to 1 and infinitely many semi-axes equal to 2, see [1,Example 2.7]. ...
We prove that Banach spaces and , with strictly convex X and Y, have plastic unit balls (we call a metric space plastic if every non-expansive bijection from this space onto itself is an isometry).
... On the other hand, if there is a counterexample, it is not an easy task to find it, because of known partial positive results. Namely, in finite-dimensional case the Expand-Contract plasticity of B X follows from compactness argument: it is known [5] that every totally bounded metric space is Expand-Contract plastic. For infinite-dimensional case, the main result of [2] ensures Expand-Contract plasticity of the unit ball of every strictly convex Banach space, in particular of Hilbert spaces and of all L p with 1 < p < ∞. ...
... Observe, that U n and V n are isometric to the unit ball of n-dimensional 1 , so they can be considered as two copies of the same compact metric space. Hence Expand-Contract plasticity of totally bounded metric spaces [5] implies that every bijective non-expansive map from U n onto V n is an isometry. In particular, F maps U n onto V n isometrically. ...
Extending recent results by Cascales, Kadets, Orihuela and Wingler (2016), Kadets and Zavarzina (2017), and Zavarzina (2017) we demonstrate that for every Banach space X and every collection of strictly convex Banach spaces every non-expansive bijection from the unit ball of X to the unit ball of sum of by is an isometry.
... On the other hand, if there is a counterexample, it is not an easy task to find it, because of known partial positive results. Namely, in finite-dimensional case the Expand-Contract plasticity of B X follows from compactness argument: it is known [5] that every totally bounded metric space is Expand-Contract plastic. For infinite-dimensional case, the main result of [2] ensures Expand-Contract plasticity of the unit ball of every strictly convex Banach space, in particular of Hilbert spaces and of all L p with 1 < p < ∞. ...
... Observe, that U n and V n are isometric to the unit ball of n-dimensional 1 , so they can be considered as two copies of the same compact metric space. Hence Expand-Contract plasticity of totally bounded metric spaces [5] implies that every bijective non-expansive map from U n onto V n is an isometry. In particular, F maps U n onto V n isometrically. ...
Extending recent results by Cascales, Kadets, Orihuela and Wingler (2016), Kadets and Zavarzina (2017), and Zavarzina (2017) we demonstrate that for every Banach space X and every collection of strictly convex Banach spaces every non-expansive bijection from the unit ball of X to the unit ball of sum of by is an isometry.
... Satz IV of [3] or Theorem 1.1 of [6] imply that every totally bounded metric space is an EC-space, but there are also examples of EC-spaces that are not totally bounded. According to [2,Theorem 2.6], the unit ball of every strictly convex Banach space is an EC-space, so in particular the closed unit ball of an infinite-dimensional Hilbert space is an example of not totally bounded EC-space. ...
... Remark, that U N is isometric to V N and, by finite dimensionality, U N and V N are compacts. So, U N and V N can be considered as two copies of one the same compact metric space, and Theorem 1.1 of [6] (which we mentioned in the beginning of the Introduction) implies that every bijective non-expansive map from U N onto V N is an isometry. In particular, F maps U N onto V N isometrically. ...
In the recent paper by Cascales, Kadets, Orihuela and Wingler it is shown that for every strictly convex Banach space X every non-expansive bijection is an isometry. We extend this result to the space , which is not strictly convex.
http://vestnik-math.univer.kharkov.ua/Vestnik-KhNU-83-2016-kadets.pdf
... This short note is motivated by [3], where the following property of metric spaces was studied in depth. Definition 1.1. ...
... Satz IV of [1] or Theorem 1.1 of [3] imply that every totally bounded metric space is an EC-space, but there are examples of EC-spaces that are not totally bounded (and even unbounded). It is an open question whether there exists a simple characterization of these spaces. ...
https://rdcu.be/6Cru
Let X be a strictly convex Banach space, and let be its unit ball. Then every non-expansive bijection is an isometry.
... Observe that by (3.1) U N is isometric to V N and, by finite dimensionality, U N and V N are compacts. So, U N and V N can be considered as two copies of one the same compact metric space, and Theorem 1.1 of [7] implies that every bijective non-expansive map from U N onto V N is an isometry. In particular, F maps U N onto V N isometrically. ...
It is known that if M is a finite-dimensional Banach space, or a strictly convex space, or the space , then every non-expansive bijection is an isometry. We extend these results to non-expansive bijections between unit balls of two different Banach spaces. Namely, if E is an arbitrary Banach space and M is finite-dimensional or strictly convex, or the space then every non-expansive bijection is an isometry.
... The corresponding metric space (X, d s ) is precompact because for any sequence in (X, d s ) there is a Cauchy subsequence due to item (c) of [7, Theorems 3] from any sequence in (X, d) one may extract a left K-Cauchy subsequence and then extract a right K-Cauchy subsequence from this left K-Cauchy subsequence using the same item of the same theorem but for X, d −1 . Our hypothesis together with [8,Corollary 1.2] imply that F is a d s -isometry, i.e. d s F(x), F(y) = d s (x, y) for all x, y ∈ X. In particular, d s F(p), F(q) = d s (p, q). ...
It is known that if any function acting from precompact metric space to itself increases the distance between some pair of points then it must decrease distance between some other pair of points. We show that this is not the case for quasi-metric spaces. After that, we present some sufficient conditions under which the previous property holds true for hereditarily precompact quasi-metric spaces.
... There is a number of relatively recent publications devoted to plasticity of the unit balls of Banach spaces (see [1,3,4,6,12]). Here we give only one theorem which is a simple consequence of Theorem 1 in [7] or Theorem 3.8 in [12]. Theorem 1. ...
This work is aimed to describe linearly expand-contract plastic ellipsoids given via quadratic form of a bounded positively defined self-adjoint operator in terms of its spectrum.Let Y be a metric space and be a map. F is called non-expansive if it does not increase distance between points of the space Y. We say that a subset M of a normed space X is linearly expand-contract plastic (briefly an LEC-plastic) if every linear operator whose restriction on M is a non-expansive bijection from M onto M is an isometry on M.In the paper, we consider a fixed separable infinite-dimensional Hilbert space H. We define an ellipsoid in H as a set of the following form where A is a self-adjoint operator for which the following holds: and .We provide an example which demonstrates that if the spectrum of the generating operator A has a non empty continuous part, then such ellipsoid is not linearly expand-contract plastic.In this work, we also proof that an ellipsoid is linearly expand-contract plastic if and only if the spectrum of the generating operator A has empty continuous part and every subset of eigenvalues of the operator A that consists of more than one element either has a maximum of finite multiplicity or has a minimum of finite multiplicity.
... The corresponding metric space (X, d s ) is precompact because for any sequence in (X, d s ) there is a Cauchy subsequence (due to item (c) of Theorems 3 of [5] from any sequence in (X, d) one may extract a left K-Cauchy subsequence and then extract a right K-Cauchy subsequence from this left K-Cauchy subsequence using the same item of the same theorem but for (X, d −1 )). Corollary 1.2 from [6] together with our hypothesis imply that F is a d s -isometry, i.e. d s (F (x), F (y)) = d s (x, y) for all x, y ∈ X. In particular, d s (F (p), F (q)) = d s (p, q). ...
It is known that if any function acting from precompact metric space to itself increases the distance between some pair of points then it must decrease distance between some other pair of points. We show that this is not the case for quasi-metric spaces. After that, we present some sufficient conditions under which the previous property holds true for hereditarily precompact quasi-metric spaces.
... There is a number of relatively recent publications devoted to plasticity of the unit balls of Banach spaces (see [1,3,4,6,12]). Here we give only one theorem which is a simple consequence of Theorem 1 in [7] or Theorem 3.8 in [12]. Theorem 1.1. ...
This work is aimed to describe linear expand-contract plastic ellipsoids given via quadratic form of a bounded positively defined self-adjoint operator in terms of its spectrum.
... It can also be shown that a bounded space need not be plastic. These results were obtained in [1]. ...
We prove the plasticity of the unit ball of c. That is, we show that every non-expansive bijection from the unit ball of c onto itself is an isometry. We also demonstrate a slightly weaker property for the unit ball of c0 – we prove that a non-expansive bijection is an isometry, provided that it has a continuous inverse.
... It can also be shown that a bounded space need not be plastic. These results were obtained in [1]. ...
We prove the plasticity of the unit ball of c. That is, we show that every non-expansive bijection from the unit ball of c onto itself is an isometry. We also demonstrate a slightly weaker property for the unit ball of -- we prove that a non-expansive bijection is an isometry, provided that it has a continuous inverse.
... ∈ M such that ρ(F (x 1 ), F (x 2 )) < ρ(x 1 , x 2 ), then there are other points y 1 , y 2 ∈ M such that ρ(F (y 1 ), F (y 2 )) > ρ(y 1 , y 2 ). It is well known that every compact metric space is EC-plastic, moreover every precompact space (in particular every bounded metric subset of a finite-dimensional normed space, equipped with the induced metric) has the same property ([3, Satz IV], see also [7,Theorem 1.1]). Using bases of uniformities one can define non-expansive maps in uniform spaces and extend the above result to compact or totally bounded uniform spaces [1, Theorem 2.3 and Corollary 2.5]. ...
... This property of M can be reformulated in the following way: for every bijection F : M → M , if there are points x 1 , x 2 ∈ M such that ρ(F (x 1 ), F (x 2 )) < ρ(x 1 , x 2 ), then there are other points y 1 , y 2 ∈ M such that ρ(F (y 1 ), F (y 2 )) > ρ(y 1 , y 2 ). It is well known that every compact metric space is EC-plastic, moreover every precompact space (in particular every bounded metric subset of a finite-dimensional normed space, equipped with the induced metric) has the same property ([3, Satz IV], see also [7,Theorem 1.1]). Using bases of uniformities one can define non-expansive maps in uniform spaces and extend the above result to compact or totally bounded uniform spaces [1, Theorem 2.3 and Corollary 2.5]. ...
The paper is aimed to establish the interdependence between linear expand-contract plasticity of an ellipsoid in a separable Hilbert spaces and properties of the set of its semi-axes.
... The above theorem generalizes this result for uniformities when the space is compact. We can use some ideas of [10] to get the following results for uniformities in totally bounded spaces. ...
We extend the result of B. Cascales et al. about expand-contract plasticity of the unit ball of strictly convex Banach space to those spaces whose unit sphere is the union of all its finite-dimensional polyhedral extreme subsets. We also extend the definition of expand-contract plasticity to uniform spaces and generalize the theorem on expand-contract plasticity of totally bounded metric spaces to this new setting.
... The above theorem generalizes this result for uniformities when the space is compact. We can use some ideas of [10] to get the following results for uniformities in totally bounded spaces. ...
We extend the result of B. Cascales at al. about expand-contract plasticity of the unit ball of strictly convex Banach space to those spaces whose unit ball is the union of all its finite-dimensional polyhedral extreme subsets. We also extend the definition of expand-contract plasticity to uniform spaces and generalize the theorem on expand-contract plasticity of totally bounded metric spaces to this new setting.
... Remark, that by (2) U N is isometric to V N and, by finite dimensionality, U N and V N are compacts. So, U N and V N can be considered as two copies of one the same compact metric space, and Theorem 1.1 of [8] implies that every bijective non-expansive map from U N onto V N is an isometry. In particular, F maps U N onto V N isometrically. ...
It is known that if M is a finite-dimensional Banach space, or a strictly convex space, or the space , then every non-expansive bijection is an isometry. We extend these results to non-expansive bijections between unit balls of two different Banach spaces. Namely, if E is an arbitrary Banach space and M is finite-dimensional or strictly convex, or the space then every non-expansive bijection is an isometry.
... Work on a basis for near sets began in 2002, motivated by image analysis and inspired by a study of the perception of the nearness of physical objects carried out in cooperation with Zdzisław Pawlak in [22]. This initial work on the perception of typical scenes in nature, especially winter scenes, led to the introduction of near sets [27], elaborated in [26,36,6,7,43,42,28,29,37], inspired by pioneering work on proximity spaces by S.A. Naimpally [16,17,19,18] and my collaboration with A. Skowron and J. Stepaniuk on the nearness of objects and information granulation [31,32,33,38]. This chapter considers the of Z. Pawlak's paintings in three different ways. ...
This chapter commemorates the work of Zdzisław Pawlak as a painter with the focus on the subtleties that come to light in considering the symmetries in his paintings. Specifically, this chapter considers how merotopic distance functions can be used as an aid to visual perception in determining the nearness of Zdzisław Pawlak's paintings. Eventually, the study of the resemblance of perceptual fragments found in nature (e.g., collections of falling snow flakes) in the poem How Near? by Z. Pawlak and J.F. Peters in 2002 led to the discovery of descriptively near sets by J.F. Peters in 2007 and a merotopological approach to measuring the nearness of collections of subsets recently introduced by J.F. Peters, S.A. Naimpally and S. Tiwari. The main contribution of this chapter is the introduction of an approach to measuring the nearness or apartness of Z. Pawlak's paintings in terms of the merotopic distances between collections of neighbourhoods in digital picture regions-of-interest. This study includes a consideration of ε-approach nearness spaces as frameworks in the search for patterns in digital pictures. An application of the proposed approach to measuring visual image nearness is reported relative to resemblances between Z. Pawlak's paintings of waterscapes that span more than a half century, starting in 1954. This study offers a partial answer to the question How near are Zdzisław Pawlak's paintings?.
We prove the existence of periodic solutions in autonomous parabolic system of differential equations with weak diffusion on the circle. We study the existence and stability of an arbitrarily large finite number of cycles for a parabolic system with weak diffusion.
Topological invariants of BPS states and Higgs coupling
measurements at the LHC
Tetiana Obikhod
Institute of Nuclear Research NAS of Ukraine
A categorical approach to physics beyond the Standard Model
Obikhod T.V.
(Institute for Nuclear Research, National Academy of Science of Ukraine 47, prosp. Nauki, Kiev,
03028, Ukraine)
E-mail: obikhod@kinr.kiev.ua
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