Homeomorphic Research Articles

Conjugacy between piecewise monotone functions decreasing on characteristic interval

It is known that topological conjugacy is important in the study of functional equations and dynamical systems, since all functions share topological dynamical properties if they are topologically conjugate. In this paper, the necessary and sufficient conditions of topological conjugacy between two piecewise monotone functions of non-monotonicity height 1, which are decreasing on their characteristic intervals, are given.

Solid–Vapor Equilibria under Desolvation of Solid Solutions. Topological Isomorphism with Phase Diagrams of Polymorphic Solid Solutions

Solid–Vapor Equilibria under Desolvation of Solid Solutions. Topological Isomorphism with Phase Diagrams of Polymorphic Solid Solutions

Distinguishing between topological isomorphism and topological equivalence of power electronic converters

In the process of deducing the topology of power electronic converters, scholars often use topological equivalence or topological isomorphism to identify topologies with different structures but the same performance to avoid repeated research. However, the connotations of topological equivalence and topological isomorphism are not the same. This paper will clarify the method for accurately identifying topologies with the same performance by distinguishing the differences and connections between the two. First, it is derived that the necessary condition for topological isomorphism is that the determinants of their adjacency matrices are equal; then, it is derived that the necessary and sufficient conditions for topological equivalence are that their component composition is the same and their simple circuits correspond one to one; finally, the above two conditions are analyzed from the perspective of topological subgraphs, and it is found that topological isomorphism is a sufficient but not necessary condition for topological equivalence, and topological equivalence is a necessary and sufficient condition for the same topological performance. Therefore, in practice, equivalence rather than isomorphism should be used to identify topologies with the same performance. This paper verifies the correctness of the theoretical analysis through a case analysis. In addition, this paper also introduces a method for automatically determining equivalent topologies based on the depth-first search algorithm to help quickly and accurately identify converter topologies with the same performance.

An Establishment and Transfer Strategy for Formation Configurations Based on Ω–u Torus Topological Equivalence

An Establishment and Transfer Strategy for Formation Configurations Based on Ω–u Torus Topological Equivalence

Multiplication cubes and multiplication automata

Abstract We extend previously known two-dimensional multiplication tiling systems that simulate multiplication by two natural numbers p and q in base $pq$ to higher dimensional multiplication tessellation systems. We develop the theory of these systems and link different multiplication tessellation systems with each other via macrotile operations that glue cubes in one tessellation system into larger cubes of another tessellation system. The macrotile operations yield topological conjugacies and factor maps between cellular automata performing multiplication by positive numbers in various bases.

Topological equivalence and phase transition rate in holographic thermodynamics of regularized Maxwell theory

Utilizing the holographic dictionary from the proposal that treats Newton’s constant as a thermodynamic variable, we establish a thermodynamic topological equivalence between the AdS black holes in the bulk and the thermal states in the dual CFT. The findings further reveal that the thermodynamic topological characteristics of the RegMax AdS black holes are strongly influenced by the characteristic parameter of the regularized Maxwell theory. Additionally, we investigate the phase transition between low and high entropy thermal states within a canonical ensemble in the dual CFT. Our observations indicate that the phase transition behavior of the thermal states mirrors that of the black holes. By modeling the phase transition process as a stochastic process, we are able to calculate the rates of phase transition between the thermal states. This result enhances our understanding of the dominant processes involved in the phase transition of the thermal states in the dual CFT.

Homological aspects of topological gauge-gravity equivalence

In the works of Achúcarro and Townsend and also by Witten, a duality between three-dimensional Chern–Simons gauge theories and gravity was established. In all cases, the results made use of the field equations. In a previous work, we were capable of generalizing Witten’s work to the off-shell cases, as well as to the four-dimensional Yang–Mills theory with de Sitter gauge symmetry. The price we paid is that curvature and torsion must obey some constraints under the action of the interior derivative. These constraints implied on the partial breaking of diffeomorphism invariance. In this work, we first formalize our early results in terms of fiber bundle theory by establishing the formal aspects of the map between a principal bundle (gauge theory) and a coframe bundle (gravity) with partial breaking of diffeomorphism invariance. Then, we study the effect of the constraints on the homology defined by the interior derivative. The main result is the emergence of a non-trivial homology in Riemann–Cartan manifolds.

Topological Conjugacy of Gradient-Like Flows on Surfaces and Efficient Algorithms for its Distinction

Topological Conjugacy of Gradient-Like Flows on Surfaces and Efficient Algorithms for its Distinction

Topological equivalence of nonhyperbolic nonautonomous systems

Backes, Dragičević and Palmer proved in [1, J. Differential Equations 2021] a C0 linearization result for nonautonomous coupled systems, whose linear part is nonhyperbolic but the second subsystem in their equations remains unchanged. In this paper, we establish a topological equivalence sending the solutions of the semilinear evolution equationx′=A(t)x+f(t,x,y),y′=r(t,x,y), onto those of its nonhyperbolic linearized partx′=A(t)x,y′=0. We require that one of the linear subsystem x′=A(t)x admits a nonuniform strong exponential dichotomy, but the other is a critical case, i.e., its linear part is a null operator. Then the difficulty arises: how to implement the linearization of the second subsystem? We will overcome the difficulty by introducing an implicit function, which enable us to construct equivalent functions to achieve the topological conjugacy between nonhyperbolic systems.

The isomorphism problem for oligomorphic groups with weak elimination of imaginaries

AbstractIn Kechris et al. [J. Symb. Log. 83 (2018), no. 3, 1190–1203], it was asked if equality on the reals is sharp as a lower bound for the complexity of topological isomorphism between oligomorphic groups. We prove that under the assumption of weak elimination of imaginaries, this is indeed the case. Our methods are model theoretic and they also have applications on the classical problem of reconstruction of isomorphisms of permutation groups from (topological) isomorphisms of automorphisms groups. As a concrete application, we give an explicit description of for any vector space of dimension over a finite field, in affinity with the classical description for finite‐dimensional spaces due to Schreier and van der Waerden.

Extending finite free actions of surfaces

AbstractWe prove the existence of finite groups acting freely as orientation‐preserving homeomorphisms on closed orientable surfaces that extend as a group of homeomorphisms of some compact orientable 3‐manifold but that cannot extend to a handlebody. This solves a basic problem in low‐dimensional equivariant topology going back to the work of Reni and Zimmermann in the mid‐1990s.

Topological Deformations of Manifolds by Algebraic Compositions in Polynomial Rings

The interactions between topology and algebraic geometry expose various interesting properties. This paper proposes the deformations of topological n-manifolds over the automorphic polynomial ring maps and associated isomorphic imbedding of locally flat submanifolds within the n-manifolds. The manifold deformations include topologically homeomorphic bending of submanifolds at multiple directions under algebraic operations. This paper introduces the concept of a topological equivalence class of manifolds and the associated equivalent class of polynomials in a real ring. The concepts of algebraic compositions in a real polynomial ring and the resulting topological properties (homeomorphism, isomorphism and deformation) of manifolds under algebraic compositions are introduced. It is shown that a set of ideals in a polynomial ring generates manifolds retaining topological isomorphism under algebraic compositions. The numerical simulations are presented in this paper to illustrate the interplay of topological properties and the respective real algebraic sets generated by polynomials in a ring within affine 3-spaces. It is shown that the coefficients of polynomials generated by a periodic smooth function can induce mirror symmetry in manifolds. The proposed formulations do not consider the simplectic class of manifolds and associated quantizable deformations. However, the proposed formulations preserve the properties of Nash representations of real algebraic manifolds including Nash isomorphism.

Quantum analogues of white noise delta functions

In this paper, we construct a quantum analogue of a white noise delta function. For our main purpose, we first discuss several white noise delta functions, specially, an infinite-dimensional analogue of Donsker’s delta function. Then as a general setting, we construct white noise delta functions on white noise test functionals and we derive a Cauchy problem whose solution is given by the white noise delta function. Also, we prove that the white noise delta function is an eigenvector of the exotic Laplacian. By applying the canonical topological isomorphisms between the spaces of white noise operators and the spaces of white noise functionals, we formulate a quantum analogue of the white noise delta function which satisfies a quantum white noise differential equation associated with the quantum Volterra Laplacian. Finally, we prove that quantum analogue of the white noise delta function is an eigenoperator of the quantum exotic Laplacian.

Brain chains as topological signatures for Alzheimer’s disease

AbstractTopology is providing new insights for neuroscience. For instance, graphs, simplicial complexes, directed graphs, flag complexes, persistent homology and convex covers have been used to study functional brain networks, synaptic connectivity, and hippocampal place cell codes. We propose a topological framework to study the evolution of Alzheimer’s disease, the most common neurodegenerative disease. The modeling of this disease starts with the representation of the brain connectivity as a graph and the seeding of a toxic protein in a specific region represented by a vertex. Over time, the accumulation of toxic proteins at vertices and their propagation along edges are modeled by a dynamical system on this graph. These dynamics provide an order on the edges of the graph according to the damage created by high concentrations of proteins. This sequence of edges defines a filtration of the graph. We consider different filtrations given by different disease seeding locations. To study these filtrations we propose a new combinatorial and topological method. A filtration defines a maximal chain in the partially ordered set of spanning subgraphs ordered by inclusion. To identify similar graphs, and define a topological signature, we quotient this poset by graph homotopy equivalence, which gives maximal chains in a smaller poset. We provide an algorithm to compute this direct quotient without computing all subgraphs and then propose bounds on the total number of graphs up to homotopy equivalence. To compare the maximal chains generated by this method, we extend Kendall’s $$d_K$$ d K metric for permutations to more general graded posets and establish bounds for this metric. We then demonstrate the utility of this framework on actual brain graphs by studying the dynamics of tau proteins on the structural connectome. We show that the proposed topological brain chain equivalence classes distinguish different simulated subtypes of Alzheimer’s disease.

Chaotic Phenomena for Generalised N-centre Problems

We study a class of singular dynamical systems which generalise the classical N-centre problem of Celestial Mechanics to the case in which the configuration space is a Riemannian surface. We investigate the existence of topological conjugation with the archetypal chaotic dynamical system, the Bernoulli shift. After providing infinitely many geometrically distinct and collision-less periodic solutions, we encode them in bi-infinite sequences of symbols. Solutions are obtained as minimisers of the Maupertuis functional in suitable free homotopy classes of the punctured surface, without any collision regularisation. For any sufficiently large value of the energy, we prove that the generalised N-centre problem admits a symbolic dynamics. Moreover, when the Jacobi-Maupertuis metric curvature is negative, we construct chaotic invariant subsets.

Persistence of heterodimensional cycles

A heterodimensional cycle is an invariant set of a dynamical system consisting of two hyperbolic periodic orbits with different dimensions of their unstable manifolds and a pair of orbits that connect them. For systems which are at least C2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$C^{2}$\\end{document}, we show that bifurcations of a coindex-1 heterodimensional cycle within a generic 2-parameter family create robust heterodimensional dynamics, i.e., a pair of non-trivial hyperbolic basic sets with different numbers of positive Lyapunov exponents, such that the unstable manifold of each of the sets intersects the stable manifold of the second set and these intersections persist for an open set of parameter values. We also give a solution to the so-called local stabilization problem of coindex-1 heterodimensional cycles in any regularity class r=2,…,∞,ω\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$r=2,\\ldots ,\\infty ,\\omega $\\end{document}. The results are based on the observation that arithmetic properties of moduli of topological conjugacy of systems with heterodimensional cycles determine the emergence of Bonatti-Díaz blenders.

Bohr chaoticity of principal algebraic actions and Riesz product measures

Abstract For a continuous $\mathbb {N}^d$ or $\mathbb {Z}^d$ action on a compact space, we introduce the notion of Bohr chaoticity, which is an invariant of topological conjugacy and which is proved stronger than having positive entropy. We prove that all principal algebraic $\mathbb {Z}$ actions of positive entropy are Bohr chaotic. The same is proved for principal algebraic actions of $\mathbb {Z}^d$ with positive entropy under the condition of existence of summable homoclinic points.

Topological Equivalence of VSI and CSI Commutation Cells and Its Application to Switching Resonance Analysis and Damper Design

This paper proposes the concept of the topological equivalence between the commutation cells of a voltage source inverter (VSI) and a current source inverter (CSI) using duality and equivalent circuit transformation techniques. A switching resonance of CSI is analyzed and an effective three-phase <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">RC</i> damper is derived from the dc-link <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">RC</i> damper of VSIs based on the topological equivalence of the VSI and CSI commutation cells. Furthermore, a new three-phase <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">RC</i> damper design framework utilizing the topological equivalence is proposed. Experimental results verify that the proposed three-phase <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">RC</i> damper design for CSI significantly reduces the switching resonance and improves the CSI’s output waveform total harmonics distortions (THDs).

Dielectric Depiction of 1-Butyl-3-methylimidazolium Tetrafluoroborate/1-Butyl-3-methylimidazolium 1,4-Bis(2-ethylhexyl) Sulfosuccinate/Benzene Microemulsions: Percolation, Interface Polarization, and Electrical Parameters.

Two dielectric relaxations located at approximately 10 and 300 MHz were analyzed in the microemulsion composed of the polar ionic liquid (PIL) 1-butyl-3-methylimidazolium tetrafluoroborate ([bmim][BF4]), the surface-active ionic liquid (SAIL) 1-butyl-3-methylimidazolium 1,4-bis(2-ethylhexyl) sulfosuccinate ([bmim][AOT]), and benzene. The curve of the dc conductivity vs PIL weight fraction shows two inflection points, namely, 16.69 and 27.90 wt %, which are used to divide the PIL/O, bicontinuous (B.C.), and O/PIL subregions. The critical exponents of percolation u = 0.75, 0.79, and 0.80 are suggested from the scaling dependence of dc conductivity on the PIL weight fraction, together with frequency dependences of both permittivity and loss angle, which all infer that static percolation occurs in the microemulsion. Only one dielectric relaxation at high frequency was observed in the PIL/O subregion, when the concentration increased to the boundary between the PIL/O and B.C. subregions; the second dielectric relaxation appeared at low frequency. The high-frequency relaxation is caused by interfacial polarization. The low-frequency relaxation is attributed to the dipole-oriented polarization of AOT-. When the oil content of the system was reduced, the interface became softer to allow more AOT- to rotate, and the cation shifted from moving around its long axis to rotating along its short axis. The static dielectric constant of [bmim][AOT] was given. The dielectric constants and conductivity of the dispersed and continuous phases in the PIL/O and O/PIL subregions were calculated from the dielectric parameters of high-frequency relaxation.

Kinetic pathway and micromechanics of fusion/fission for polyelectrolyte vesicles.

Despite the wide existence of vesicles in living cells as well as their important applications like drug delivery, the underlying mechanism of vesicle fusion/fission remains under debate. Classical models cannot fully explain recent observations in experiments and simulations. Here, we develop a constrained self-consistent field theory that allows tracking the shape evolution and free energy as a function of center-of-mass separation distance. Fusion and fission are described in a unified framework. Both the kinetic pathway and the mechanical response can be simultaneously captured. By taking vesicles formed by polyelectrolytes as a model system, we predict discontinuous transitions between the three morphologies: parent vesicle with a single cavity, hemifission/hemifusion, and two separated child vesicles, as a result of breaking topological isomorphism. With the increase in inter-vesicle repulsion, we observe a great reduction in the cleavage energy, indicating that vesicle fission can be achieved without hemifission, in good agreement with simulation results. The force-extension relationship elucidates typical plasticity for separating two vesicles. The super extensibility in the mechanical response of vesicle is in stark contrast to soft particles with other morphologies, such as cylinder and sphere. Our work elucidates the fundamental physical chemistry based on intrinsic topological features of vesicle fusion/fission, which provides insights into various phenomena observed in experiments and simulations.