Homeomorphic Embedding Research Articles

On Trajectories Entirely Situated Near a Hyperbolic Set

Let I 1 be a set of points such that their trajectories under a diffeomorphism f 1 are entirely close enough to a hyperbolic set F 1 of this diffeomorphism. Then it is proved that the structure of I 1 and restriction \( {f}_1\left|{}_{I_1}\right. \) (“motion in I 1”) are essentially defined (up to an equivariant homeomorphism) by “internal dynamics” in F 1 , i.e., by the restriction \( {f}_1\left|{}_{{}_{F_1}}\right. \). (In more detail: the equivariant homeomorphism g 1 of the set F 1 on the hyperbolic set F 2 of the second diffeomorphism f 2 (probably, acting on another manifold M 2) is extendable to an equivariant homeomorphic embedding I 1 → M 2 . The image of the imbedding contains all the trajectories f 2 close enough to F 2 .)

A Comparison of Well-Quasi Orders on Trees

Well-quasi orders such as homeomorphic embedding are commonly used to ensure termination of program analysis and program transformation, in particular supercompilation. We compare eight well-quasi orders on how discriminative they are and their computational complexity. The studied well-quasi orders comprise two very simple examples, two examples from literature on supercompilation and four new proposed by the author. We also discuss combining several well-quasi orders to get well-quasi orders of higher discriminative power. This adds 19 more well-quasi orders to the list.

On Shuffe Ideals of General Algebras

We extend a word language concept called shuffle ideal to general algebras. For this purpose, we introduce the relation SH and show that there exists a natural connection between this relation and the homeomorphic embedding order on trees. We establish connections between shuffle ideals, monotonically ordered algebras and automata, and piecewise testable tree languages.

On the structure of perfect sets in various topologies associated with tree forcings

AbstractWe prove that the Ellentuck, Hechler and dual Ellentuck topologies are perfect isomorphic to one another. This shows that the structure of perfect sets in all these spaces is the same. We prove this by finding homeomorphic embeddings of one space into a perfect subset of another. We prove also that the space corresponding to eventually different forcing cannot contain a perfect subset homeomorphic to any of the spaces above.

Type-based homeomorphic embedding for online termination

Online termination techniques dynamically guarantee termination of computations by supervising them in such a way that computations whose termination can no longer be guaranteed are stopped. Homeomorphic Embedding ( HEm) has proven to be very useful for online termination provided that the computations supervised are performed over a finite signature, i.e., the number of constants and function symbols involved is finite. However, there are many situations, for example numeric computations, which involve an infinite signature and thus HEm does not guarantee termination. Some extensions to HEm for the case of infinite signatures have been proposed which guarantee termination. However, the existing techniques either do not provide systematic means for generating such extensions or the extensions are too simplistic and do not produce the expected results in practice. We propose Type-based Homeomorphic Embedding ( TbHEm) as an extension of the standard, untyped, HEm. By taking static information about the behavior of the computation into account, expressed as types, TbHEm allows obtaining more precise results than those of the previous extensions to HEm for the case of infinite signatures. We show that the existing extensions to HEm which are currently used in state-of-the-art specialization tools can be reconstructed as instances of TbHEm. We illustrate the applicability of our proposal in a realistic case study: partial evaluation of an interpreter. We argue that the results obtained provide empirical evidence of the interest of our proposal.

Attractor and spatial chaos for the Brusselator in [formula omitted

Brusselator is an important dynamical system which can be described by a reaction–diffusion system. In this paper it is proved that this reaction–diffusion system possesses a global attractor A in the corresponding phase space. The upper and lower bounds of Kolmogorov ε -entropy of A are obtained. Moreover we give a more detailed study of spatial chaos of the attractor A for the Brusselator in R N . We interpret a group of spatial shifts as a dynamical system which acts on the attractor A . By using the technique of unstable manifolds, it is proved that this dynamical system is chaotic. In order to clarify the nature of this chaos, we construct the Lipschitz-continuous homeomorphic embedding of a typical model dynamical system whose chaotic behavior is evident, into the spatial shifts on the attractor A . This typical dynamical system generalizes the symbolic system. It was first introduced by Zelik.

Computable metrization

AbstractEvery second‐countable regular topological space X is metrizable. For a given “computable” topological space satisfying an axiom of computable regularity M. Schröder [10] has constructed a computable metric. In this article we study whether this metric space (X, d) can be considered computationally as a subspace of some computable metric space [15]. While Schröder's construction is “pointless”, i. e., only sets of a countable base but no concrete points are known, for a computable metric space a concrete dense set of computable points is needed. But there may be no computable points in X. By converging sequences of basis sets instead of Cauchy sequences of points we construct a metric completion ($ \tilde X $, $ \tilde d $) of a space (X, d) together with a canonical representation $ \tilde X $, $ \tilde \delta $. We show that there is a computable embedding of (X, d) in ($ \tilde X $, $ \tilde d $) with computable inverse. Finally, we construct a notation of a dense set of points in ($ \tilde X $, $ \tilde d $) with computable mutual distances and prove that the Cauchy representation of the resulting computable metric space is equivalent to $ \tilde \delta $. Therefore, every computably regular space has a computable homeomorphic embedding in a computable metric space, which topologically is its completion. By the way we prove a computable Urysohn lemma. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Spatial and Dynamical Chaos Generated by Reaction–Diffusion Systems in Unbounded Domains

We consider in this article a nonlinear reaction–diffusion system with a transport term (L,∇ x )u, where L is a given vector field, in an unbounded domain Ω. We prove that, under natural assumptions, this system possesses a locally compact attractor $$\mathcal{A}$$ in the corresponding phase space. Since the dimension of this attractor is usually infinite, we study its Kolmogorov’s ɛ-entropy and obtain upper and lower bounds of this entropy. Moreover, we give a more detailed study of the spatio-temporal chaos generated by the spatially homogeneous RDS in $$\Omega = \mathbb{R}^{n}$$ . In order to describe this chaos, we introduce an extended (n + 1)-parametrical semigroup, generated on the attractor by 1-parametrical temporal dynamics and by n-parametrical group of spatial shifts ( = spatial dynamics). We prove that this extended semigroup has finite topological entropy, in contrast to the case of purely temporal or purely spatial dynamics, where the topological entropy is infinite. We also modify the concept of topological entropy in such a way that the modified one is finite and strictly positive, in particular for purely temporal and for purely spatial dynamics on the attractor. In order to clarify the nature of the spatial and temporal chaos on the attractor, we use (following Zelik, 2003, Comm. Pure. Appl. Math. 56(5), 584–637) another model dynamical system, which is an adaptation of Bernoulli shifts to the case of infinite entropy and construct homeomorphic embeddings of it into the spatial and temporal dynamics on $$\mathcal{A}$$ . As a corollary of the obtained embeddings, we finally prove that every finite dimensional dynamics can be realized (up to a homeomorphism) by restricting the temporal dynamics to the appropriate invariant subset of $$\mathcal{A}$$ .

Attractors of reaction‐diffusion systems in unbounded domains and their spatial complexity

AbstractThe nonlinear reaction‐diffusion system in an unbounded domain is studied. It is proven that, under some natural assumptions on the nonlinear term and on the diffusion matrix, this system possesses a global attractor 𝒜 in the corresponding phase space. Since the dimension of the attractor happens to be infinite, we study its Kolmogorov's ϵ‐entropy. Upper and lower bounds of this entropy are obtained.Moreover, we give a more detailed study of the attractor for the spatially homogeneous RDE in ℝn. In this case, a group of spatial shifts acts on the attractor. In order to study the spatial complexity of the attractor, we interpret this group as a dynamical system (with multidimensional “time” if n > 1) acting on a phase space 𝒜. It is proven that the dynamical system thus obtained is chaotic and has infinite topological entropy.In order to clarify the nature of this chaos, we suggest a new model dynamical system that generalizes the symbolic dynamics to the case of the infinite entropy and construct the homeomorphic (and even Lipschitz‐continuous) embedding of this system into the spatial shifts on the attractor.Finally, we consider also the temporal evolution of the spatially chaotic structures in the attractor and prove that the spatial chaos is preserved under this evolution. © 2003 Wiley Periodicals, Inc.

A comparative survey of selected results and open problems concerning topological groups, fields and vector spaces

It is natural to expect that the behaviour of some topological properties tends to improve in the presence of an additional algebraic structure interacting with the topology (for example, in topological groups, topological fields, or topological vector spaces). The purpose of this survey is to compare topological groups, topological vector spaces and topological fields as to how far each of these classes of spaces is from the class of Tychonoff spaces. In other words, we want to compare the degree of how much of an additional strain an algebraic structure of a group, vector space or field which agrees with the topology of the space imposes on the topology of that space. We cover selected results and open problems related to normality-type properties, covering properties, Cartesian products, homeomorphic embeddings and dimension theory.

A categorical version of the Lefschets-Nöbeling-Pontryagin theorem on embedding compacta in R n

For a category K we use Ob(K) to denote the class of all objects of K; if X, Y ϵ Ob( K), then Mor K ( X, Y) is the set of all K-morphisms from X into Y. Let A and B be subcategories of the category of all topological spaces and their continuous maps. We say that a covariant functor F: A → B is an embedding functor if there exists a class { i x : X ϵ Ob( A)} satisfying the following conditions: 1. (i) i x : X → F( X) is a homeomorphic embedding for every X ϵ Ob( A), and 2. (ii) if X, Y ϵ Ob( A) and f ϵ Mor K ( X, Y), then F( f) o i x = i y o f. For a natural number n let C( n) denote the category of all n-dimensional compact metric spaces and their continuous maps. Let G(< ∞) be the category of all Hausdorff finite-dimensional topological groups and their continuous group homomorphisms. We prove that there is no embedding covariant functor F: C(1) → G(< ∞), but there exists a covariant embedding functor F: C(0) → G(0), where G(0) is the category consisting of the single (zero-dimensional) compact metric group Z 2 ω and all its continuous group homomorphisms into itself, i.e., Ob( G(0)) = { Z 2 ω } and Mor G(0)(Z 2 ω, Z 2 ω) is the set of all continuous group homomorphisms from Z 2 ω into Z 2 ω.

Affine and homeomorphic embeddings into $\ell ^2$

Affine and homeomorphic embeddings into $\ell ^2$

Generating polynomial orderings

There are various methods of proving termination of term rewriting systems (TRS). Most of these are based on reduction orderings, which are well-founded, compatible with the structure of terms and stable with respect to substitutions. The notion of reduction ordering allows the following characterization of termination of TRS: A TRS 9? terminates iff there exists a reduction ordering + such that I t r for each rule I + r of 9. Polynomial orderings are special reduction orderings, mapping terms into a well-founded set by attaching monotonic polynomials (so-called interpretations) to operators. They have been studied by Manna and Ness [12], Lankford [lo], Dershowitz [3], Huet and Oppen [8] and BenCherifa and Lescanne [l]. In contrast to the others, Dershowitz uses an arbitrary set of functions (polynomials) by requiring them to possess the subterm property, i.e. his ordering is a simplification ordering including the homeomorphic embedding relation. One of the main problems concerning polynomial orderings is the choice of adequate interpretations for a given TRS. The object of this paper is to describe an algorithm (being powerful in practice, see Section 5) based on the Simplex method for finding polynomial interpretations for a given TRS.

Embeddings of countable closed sets and reverse mathematics

If there is a homeomorphic embedding of one set into another, the sets are said to be topologically comparable. Friedman and Hirst have shown that the topological comparability of countable closed subsets of the reals is equivalent to the subsystem of second order arithmetic denoted byATR0. Here, this result is extended to countable closed locally compact subsets of arbitrary complete separable metric spaces. The extension uses an analogue of the one point compactification of ℝ.

Reverse mathematics and homeomorphic embeddings

Extrapolating from the work of Mahlo (1911), one can prove that given any pair of countable closed totally bounded subsets of complete separable metric spaces, one subset can be homeomorphically embedded in the other. This sort of topological comparability is reminiscent of the statements concerning comparability of well orderings which Friedman has shown to be equivalent to ATR 0 over the weak base system RCA 0. The main result of this paper states that topological comparability is also equivalent to ATR 0. In Section 1, the pertinent subsystems of second-order arithmetic and results on well orderings are reviewed. Sections 2 and 3 overview the encoding of metric spaces and homeomorphisms in second-order arithmetic. Section 4 contains a proof of the topological comparability result in ATR 0. Section 5 contains the reversal, a derivation of ATR 0 from the topological comparability result. In Section 6, additional information about the structure of the embeddings is obtained, culminating in an application to closed subsets of the real numbers.

On J. Nagata's universal spaces

The main result of this paper is the following extension of an embedding theorem by Nagata: given a sequence of zero-dimensional sets X 1, X 2,… in a metrizable space X of weight τ⩾ℵ 0, the set of homeomorphic embeddings h of X into S(τ) ℵ 0 , satisfying h( X n )⊂ K n-1 ( τ) for n= 1,2,…, is dense in the function space of all continuous mappings of X into S(τ) ℵ 0 , where K n (τ) is the n-dimensional universal Nagata's space in the countable product of the star-space S(τ) of weight τ. This seems to be a new result even in the separable case τ=χ 0 and provides in particular an answer to a question asked by Kuratowski (see Remark 2.6 for the details).

Homeomorphism connecting the hydrogenic system and the free-particle system

The topological equivalence between the hydrogenic system and the free-particle system is established by a homeomorphic embedding of ${\mathrm{R}}_{3}$ into ${\mathrm{R}}_{4}$. The oscillator representation of the generators of the dynamical group SO(4,2) is realized for both systems in a unified way. It is shown that the free-particle wave functions form the basis of the unitary irreducible representation D(0,0) of the homogeneous Lorentz group instead of $\mathrm{D}{(0,(\ensuremath{-}\frac{\frac{1}{2}Z}{E})}^{\frac{1}{2}})$, $E&gt;0$, in the case of the hydrogen atom.

On Homeomorphic Embeddings of Km,n in the Cube

Homeomorphic embeddings of Knin the m-cube were investigated in [6]. In particular, it was proved that any homeomorph of Kn+1embedded in the m-cube has at least n2edges. Furthermore, homeomorphic embeddings of Kn+1having exactly n2edges are unique up to isomorphism. In this paper a similar problem for the complete bipartite graph is considered.We adopt the notation and terminology of [5].All graphs considered are without loops and multiple edges.Let x = uv be an edge of a graph G; x will be called subdivided if it is replaced by a vertex ω and by edges uω and ωv. A graph G’ is called a subdivision of G if it is obtained from G by a subdivision of an edge of G. A refinement Ĝ of G, is a graph isomorphic to a graph obtained from G by a finite sequence of subdivisions.

The homeomorphic embedding of Kn in the m-cube

In this paper we find a homeomorphic embedding of Kn in Qm. Then it is proved that any homeomorphic of Kn embedded in Qm has at least (n 1)2 edges. Moreover, if the homeomorphic embedding has exactly (n 1)2 edges, then it is unique up to isomorphism.

Approximation by homeomorphisms and solution of P. Blass problem on pseudo-isotopy

For every map of f : S 1 → S 1 = { z ∈ C : | z | = 1 } f:{S^1} \to {S^1} = \{ z \in C:|z| = 1\} of degree 1, existence of apseudo-isotopy h : S 1 × I → R = { z ∈ C : | z | ≧ 1 } h:{S^1} \times I \to R = \{ z \in C:|z| \geqq 1\} such that h ( z , 0 ) = z h(z,0) = z and h ( z , 1 ) = f ( z ) h(z,1) = f(z) is established. On the other hand (i) maps of I n {I^n} into I n × 0 ⊂ E n + 1 {I^n} \times 0 \subset {E^{n + 1}} cannot be, in general, uniformly approximated by homeomorphic embeddings of I n {I^n} in E n + 1 {E^{n + 1}} for n &gt; 1 n &gt; 1 , and (ii) maps of S n {S^n} into S n ⊂ E n {S^n} \subset {E^n} of degree 1 cannot be, in general, extended to a pseudo-isotopy of S n {S^n} into E n + 1 {E^{n + 1}} .