Homeomorphism Theorem Research Articles

PL homeomorphisms of surfaces and codimension 2 PL foliations

The Haefliger–Thurston conjecture predicts that Haefliger's classifying space for $C^r$ -foliations of codimension $n$ whose normal bundles are trivial is $2n$ -connected. In this paper, we confirm this conjecture for piecewise linear (PL) foliations of codimension $2$ . Using this, we use a version of the Mather–Thurston theorem for PL homeomorphisms due to the author to derive new homological properties for PL surface homeomorphisms. In particular, we answer the question of Epstein in dimension $2$ and prove the simplicity of the identity component of PL surface homeomorphisms.

A homeomorphism theorem for sums of translates

For a fixed positive integer n consider continuous functions K1,⋯\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$K_1,\\dots $$\\end{document}, Kn:[-1,1]→R∪{-∞}\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ K_n:[-1,1]\\rightarrow {{\\mathbb {R}}}\\cup \\{-\\infty \\}$$\\end{document} that are concave and real valued on [-1,0)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$[-1,0)$$\\end{document} and on (0, 1], and satisfy Kj(0)=-∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$K_j(0)=-\\infty $$\\end{document}. Moreover, let J:[0,1]→R∪{-∞}\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$J:[0,1]\\rightarrow {{\\mathbb {R}}}\\cup \\{-\\infty \\}$$\\end{document} be upper bounded and such that [0,1]\\J-1({-∞})\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$[0,1]\\setminus J^{-1}(\\{-\\infty \\})$$\\end{document} has at least n+1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$n+1$$\\end{document} elements, but it is arbitrary otherwise. For x0:=0<x1<⋯<xn≤xn+1:=1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$x_0:=0<x_1<\\dots < x_n \\le x_{n+1}:=1$$\\end{document}, so called nodes, and for t∈[0,1]\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$t\\in [0,1]$$\\end{document} consider the sum of translates function F(x1,…,xn,t):=J(t)+∑j=1nKj(t-xj)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$F(x_1,\\ldots ,x_n,t):=J(t)+\\sum _{j=1}^n K_j(t-x_j)$$\\end{document}, and the vector of interval maximum values mj:=mj(x1,…,xn):=maxt∈[xj,xj+1]F(x1,…,xn,t)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$m_j:=m_j(x_1,\\ldots ,x_n):=\\max _{t\\in [x_j,x_{j+1}]}F(x_1,\\ldots ,x_n,t)$$\\end{document} (j=0,1,…,n\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$j=0,1,\\ldots ,n$$\\end{document}). We describe the structure of the arising interval maxima as the nodes run over the n-dimensional simplex. Applications presented here range from abstract moving node Hermite–Fejér interpolation for generalized algebraic and trigonometric polynomials via Bojanov’s problem to more abstract results of interpolation theoretic flavour.

Asymptotic stability for quaternion-valued fuzzy BAM neural networks via integral inequality approach

The existence-uniqueness (EU) and globally asymptotic stability (GAS) of equilibrium solutions (ESS) for a class of quaternion-valued fuzzy BAM neural networks (QVFBAMNNS) with delays are concerned. Firstly, by using Homeomorphism theorem (HT) and by using the properties of unary high degree inequality, a sufficient condition ensuring the EU of ESS of the concerned QVFBAMNNS is achieved. Then, without utilizing V′(t)≤0, by applying integral inequality approach (IIA), a condition assuring the GAS of ESS for above networks is achieved. Utilizing the properties of high degree inequality studies the EU of ESS and utilizing IIA studies the GAS for neural networks (NNS) are new research approaches.

Thurston’s fragmentation and c-principles

Abstract In this paper, we generalize the original idea of Thurston for the so-called Mather-Thurston’s theorem for foliated bundles to prove new variants of this theorem for PL homeomorphisms and contactormorphisms. These versions answer questions posed by Gelfand-Fuks ([GF73, Section 5]) and Greenberg ([Gre92]) on PL foliations and Rybicki ([Ryb10, Section 11]) on contactomorphisms. The interesting point about the original Thurston’s technique compared to the better-known Segal-McDuff’s proof of the Mather-Thurston theorem is that it gives a compactly supported c-principle theorem without knowing the relevant local statement on open balls. In the appendix, we show that Thurston’s fragmentation implies the non-abelian Poincare duality theorem and its generalization using blob complexes ([MW12, Theorem 7.3.1]). To the memory of John Mather.

Asymptotic stability for quaternion-valued BAM neural networks via a contradictory method and two Lyapunov functionals

&lt;abstract&gt;&lt;p&gt;We explore the existence and asymptotic stability of equilibrium point for a class of quaternion-valued BAM neural networks with time-varying delays. Firstly, by employing Homeomorphism theorem and a contradictory method with novel analysis skills, a criterion ensuring the existence of equilibrium point of the considered quaternion-valued BAM neural networks is acquired. Secondly, by constructing two Lyapunov functionals, a criterion assuring the global asymptotic stability of equilibrium point for above discussed quaternion-valued BAM is presented. Applying a contradictory method to study the equilibrium point and applying two Lyapunov functionals to study stability of equilibrium point are completely new methods.&lt;/p&gt;&lt;/abstract&gt;

Existence, Uniqueness, and Exponential Stability of Uncertain Delayed Neural Networks with Inertial Term: Nonreduced Order Case

In this work, we mainly focus on uncertain delayed neural network system with inertial term. Here, the existence, uniqueness, and exponential stability of inertial neural networks are derived without shifting the second order differential system into first order through substituting variables. Initially, we construct a proper Lyapunov–Krasovskii functional to investigate the stability of novel uncertain delayed inertial neural networks, which is different from the classical Lyapunov functional approach. By utilizing the Kirchhoff’s matrix tree theorem, Cauchy–Schwartz inequality, homeomorphism theorem, and some inequality techniques, the necessary and sufficient conditions are derived for the designed framework. Subsequently, to exhibit the strength of this outcome, we framed a quantitative example.

Stochastic stability of fractional-order Markovian jumping complex-valued neural networks with time-varying delays

This paper is concerned with the problem of stochastic stability analysis for fractional-order Markovian jumping complex-valued neural networks (MJCVNNs) with time-varying delays. The novelty of this study is emphasized in two phases. In first phase, MJCVNNs is considered in the form of fractional-order systems. Secondly, complex-valued Wirtinger based integral inequality is newly constructed. The existence and uniqueness conditions of the proposed systems are derived based on the homeomorphism theorem in the complex domain. Then, the stochastic stability condition is derived in terms of linear matrix inequalities (LMIs) for the MJCVNNs by employing Lyapunov indirect method. The feasibility of the derived conditions is verified by the numerical examples and their simulation results are demonstrated to show the effectiveness of the fractional-order derivatives and proposed approach.

Anthropic Principle’s Predicting Symmetric Distribution Matter Strata, Their Physics Laws, and Verifications

This paper shows anthropic principle’s predicting symmetric distribution matter strata, their physics laws, and verifications, concretely deduces characteristic time, energy, and temperature expressions at different scales, discovers four interesting invariant quantities, shows homeomorphic theorem of space map, and naturally presents a supersymmetric scale energy. We further discover that any infinitesimal space has the same proportional structure space; namely, they have renormalization group invariance. Consequently, this paper shows that the region of any nth level Plank-scope is from the nth level Planck scale to the (n+1)​th level Planck scale, where the different matters of the nthlevel Planck scale build up the (n+1)th level Planck scale matter. The branches of physics science for this region include the nth level Planck scale matter dynamics and the nth level Planck scale matter group dynamics. The nth level Planck scale matter group dynamics describe how the nth level Planck scale matter constructs the (n+1)th level Planck scale matter and how the different matters of the nth level Planck scale evolve in the group system. This paper discovers that the different matters below Planck scale can exist with our matter world at the same time and same place and may be some candidates for dark matter; furthermore, this paper shows a relative theorem of matter scale: for the world of any nth level, the matters’ sizes are relative, not absolute. Evidently, the discoveries of both the symmetrical distribution scales and the relations among the corresponding different physics laws from infinitesimal to infinitely large scales give a scientific solid development platform for formation of new scientific branches and deeper development of old scientific branches, because we can precisely construct many kinds of scientific theories relevant to all the corresponding matter strata. All the branch sciences of different matter strata up to now naturally need to be included in the framework of the new scientific system of physics.

Some observations concerning multidimensional quasiconformal mappings

For several important objects and quantities in the theory of quasiconformal space mappings, we discuss their dependence on the dimension of the space. In particular, in connection with the global homeomorphism theorem and the theorem on the injectivity radius of quasiconformal immersions, we consider the asymptotic behaviour of the moduli of Grötzsch and Teichmüller rings with respect to the dimension. Bibliography: 23 titles.

Poincaré Theory for Decomposable Cofrontiers

We extend Poincar\'e's theory of orientation-preserving homeomorphisms from the circle to circloids with decomposable boundary. As special cases, this includes both decomposable cofrontiers and decomposable cobasin boundaries. More precisely, we show that if the rotation number on an invariant circloid $A$ of a surface homeomorphism is irrational and the boundary of $A$ is decomposable, then the dynamics are monotonically semiconjugate to the respective irrational rotation. This complements classical results by Barge and Gillette on the equivalence between rational rotation numbers and the existence of periodic orbits and yields a direct analogue to the Poincar\'e Classification Theorem for circle homeomorphisms. Moreover, we show that the semiconjugacy can be obtained as the composition of a monotone circle map with a `universal factor map', only depending on the topological structure of the circloid. This implies, in particular, that the monotone semiconjugacy is unique up to post-composition with a rotation. If, in addition, $A$ is a minimal set, then the semiconjugacy is almost one-to-one if and only if there exists a biaccessible point. In this case, the dynamics on $A$ are almost automorphic. Conversely, we use the Anosov-Katok method to build a $C^\infty$-example where all fibres of the semiconjugacy are non-trivial.

Stability Analysis of Fuzzy Genetic Regulatory Networks with Various Time Delays

The synthetic genetic regulatory networks (GRNs) prove to be a powerful tool in studying gene regulation processes in living organisms. In order to take vagueness into consideration, fuzzy theory is incorporated in the GRNs and we found a novel system, namely fuzzy genetic regulatory networks (FGRNs). By applying the Homeomorphism theorem, we demonstrated an existence, uniqueness, and asymptotic stability analysis of the equilibrium point of FGRNs with time delay in the leakage term and unbounded distributed delays. By utilizing Lyapunov functional method and the linear matrix inequality (LMI) techniques, some new and useful criteria of the FGRNs with respect to the equilibrium point are derived. The derived criteria are of the form of LMI, and hence they can be verified easily. Finally, illustrative example with simulations are demonstrated to prove the effectiveness of the proposed results.

On a generalization of the Cartwright–Littlewood fixed point theorem for planar homeomorphisms

We prove a generalization of the fixed point theorem of Cartwright and Littlewood. Namely, suppose that $h:\mathbb{R}^{2}\rightarrow \mathbb{R}^{2}$ is an orientation preserving planar homeomorphism, and let $C$ be a continuum such that $h^{-1}(C)\cup C$ is acyclic. If there is a $c\in C$ such that $\{h^{-i}(c):i\in \mathbb{N}\}\subseteq C$, or $\{h^{i}(c):i\in \mathbb{N}\}\subseteq C$, then $C$ also contains a fixed point of $h$. Our approach is based on Brown’s short proof of the result of Cartwright and Littlewood. In addition, making use of a linked periodic orbits theorem of Bonino, we also prove a counterpart of the aforementioned result for orientation reversing homeomorphisms, that guarantees a $2$-periodic orbit in $C$ if it contains a $k$-periodic orbit ($k&gt;1$).

A Homeomorphism Theorem for the Universal Type Space with the Uniform Weak Topology

Kolmogorov’s extension theorem provides a natural mapping from the space of coherent hierarchies of an agent’s first-order, second-order, etc. beliefs to the space of probability measures over the exogenous parameters and the other agents' belief hierarchies. Mertens and Zamir (1985) showed that, if the spaces of belief hierarchies are endowed with the product topology, then this mapping is a homeomorphism. This paper shows that this mapping is also a homeomorphism if the spaces of belief hierarchies are endowed with the uniform weak topology of Chen et al. (2010) or the universal strategic topology of Dekel et al. (2006), both of which ensure that strategic behaviour exhibits desirable continuity properties.

New global exponential stability conditions for inertial Cohen–Grossberg neural networks with time delays

In this paper, global exponential stability of inertial Cohen–Grossberg neural networks with time delays is investigated. By using Homeomorphism theorem and inequality technique, a LMI-based global exponential stability condition and inequality form global exponential stability condition are obtained for the above neural networks. In our result, the assumptions for the differentiability and monotonicity on the behaved functions in Ke and Miao (2013) [23] are removed. Thus our results are less conservative than those obtained in Ke and Miao (2013) [23]. Hence, we obtain new global exponential stability for this neural network.

Mappings of Finite Inner Distortion: Global Homeomorphism Theorem

We show that if \(f\in W_{loc}^{1,n-1}(\mathbb {R}^n,\mathbb {R}^n)\), \(n\ge 3\), is a local homeomorphism of \(K\)-bounded 1-mean inner distortion, then \(f\) is a homeomorphism onto \(\mathbb {R}^n\). We also establish the \(K_I\)-weighted Vaisala’s inequality for mappings of finite inner distortion under minimal assumptions.

Quasisymmetric homeomorphisms on reducible Carnot groups

We show that quasisymmetric homeomorphisms between (most) reducible Carnot groups are biLipschitz. This implies rigidity for quasi-isometries between certain negatively curved homogeneous manifolds. The proof uses Pansu differentiability theorem for quasisymmetric homeomorphisms between Carnot groups.

On the homeomorphism of continuous mappings

Denote by C(X) the partially ordered (PO) set of all continuous epimorphisms of a space X under the natural identification of homeomorphic epimorphisms. The following homeomorphism theorem for bicompacta is implicitly contained in Magill’s 1968 paper: two bicompacta X and Y are homeomorphic if and only if the PO sets C(X) and C(Y) are isomorphic. In the present paper, Magill’s theorem is extended to the category of mappings in which the role of bicompacta is played by perfect mappings. The results are obtained in two versions, namely, in the category TOPZ (of triangular commutative diagrams) and in the category MAP (of quadrangular commutative diagrams).

Carathéodory and Smirnov type theorems for harmonic mappings of the unit disk onto surfaces

We prove representation theorems, the versions of Smirnov's theorem and Caratheo- dory type theorem for harmonic homeomorphisms of the unit disk onto Jordan surfaces with rectifi- able boundaries. Further we establish the classical isoperimetric inequality and the Riesz-Zygmund inequality for Jordan harmonic surfaces without any smoothness assumptions on the boundary.

Stability analysis of inertial Cohen–Grossberg-type neural networks with time delays

In this paper, the global exponential stability of inertial Cohen–Grossberg-type neural networks with time delays is investigated. First, by properly chosen variable substitution the system is transformed to the first order differential equation. Second, some sufficient conditions which can ensure the global exponential stability of the system are obtained using homeomorphism and differential mean value theorem, applying the analysis method and inequality technique. Finally, two examples are given to illustrate the effectiveness of the results.

Handel’s fixed point theorem revisited

AbstractMichael Handel proved in [A fixed-point theorem for planar homeomorphisms. Topology38 (1999), 235–264] the existence of a fixed point for an orientation-preserving homeomorphism of the open unit disk that can be extended to the closed disk, provided that it has points whose orbits form an oriented cycle of links at infinity. Later, Patrice Le Calvez gave a different proof of this theorem based only on Brouwer theory and plane topology arguments in [Une nouvelle preuve du théorème de point fixe de Handel. Geom. Topol.10(2006), 2299–2349]. These methods improved the result by proving the existence of a simple closed curve of index 1. We give a new, simpler proof of this improved version of the theorem and generalize it to non-oriented cycles of links at infinity.