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This paper investigates the stability of complex-valued neural networks (CVNNs) with semi-Markovian jump (sMJ) and generally uncertain transition rates. Each transition rate may be totally unknown or its estimate is determined. Firstly, two improved reciprocally convex inequalities (RCIs) and three less conservative integral inequalities are generalized to the complex-valued domain. Secondly, the existence and uniqueness of the addressed networks are proposed by the complex-valued homeomorphism theorem. Thirdly, by constructing a Lyapunov–Krasovskii functional (LKF), delay-dependent robust stability criteria of the CVNNs are obtained by utilizing the improved complex-valued RCIs and integral inequalities. Finally, simulations are presented to demonstrate the effectivenes and practicality of the established method. [ABSTRACT FROM AUTHOR]

The existence of core solutions for general cooperative games (Game Econ Behav 4:145–160, 1992) is extended to the case with discontinuous and non-quasiconcave payoffs. The analysis shows that the core of a discontinuous general cooperative game is homeomorphic to a KKMS-related region of a simplex. Each discontinuous general cooperative game has a refined stable set consisting of some efficient core solutions. [ABSTRACT FROM AUTHOR]

Through this paper, via the operators (.)* and ψ, we presented notion of *-Locally set in an ideal topological space Zi as a new stronger form of locally closed set, and considered relations with various existing weak form of locally closed set. Preservations of direct images as well as inverse images of (.)*, ψ, *-perfect and various weak forms of locally closed set including *-Locally closed set are important investigating part. Besides, we pointed out that consideration of 'bijectivity' in Lemma 3.1 of [24] is sufficient, and the Lemma 3.3 of [24] is wrong. We demonstrated two modifications of the last one. [ABSTRACT FROM AUTHOR]

Let f : X → X be a hereditarily locally connected continuum homeomorphism and denote respectively by P(f), AP(f) and Ω (f) , the sets of periodic points, almost periodic points and the non-wandering points of f. We show that any ω -limit set (resp. α -limit set) is minimal. Moreover, we show that Ω (f) = A P (f) . We also prove that if P (f) = ∅ , then there exists a unique minimal set. On the other hand, if P (f) ≠ ∅ then we prove that any infinite minimal set has the adding machine structure and the absence of Li-Yorke pairs. Consequently, we partially solve the positive entropy conjecture which remains open even in the case of hereditarily locally connected continuum. [ABSTRACT FROM AUTHOR]

The operator Ψ has been introduced as an associated set-valued set function. Although it has importance for the study of minimal open sets as well as minimal -open sets. As a result of this study, we introduce minimal I*-has been introduced as an associated set-valued set function. Although it has importance for the study of minimal open sets as well as minimal -open sets. As a result of this study, we introduce minimal I* -open sets are also investigated. This study also discusses the role of minimal I*-open sets in the * -locally finite spaces. In an aspect of topological invariant, the homeomorphic images of minimal I* -open set has been discussed here. [ABSTRACT FROM AUTHOR]

We prove that there are no 6-cycles of the form xn+3 = xif(xj, xk), with i, j, k ∈ {n, n+1, n+2} pairwise distinct, whenever f : (0,∞) x (0,∞) → (0,∞) is a continuous symmetric function, that is, f(x, y) = f(y, x), for all x, y > 0. Moreover, we obtain all the 6-cycles of potential form and present some open questions relative to the search of p-cycles whenever symmetry does not hold. [ABSTRACT FROM AUTHOR]

We establish existence results for differential equations of the form: (φ(u′))′=f(t,u,u′),$$ {\left(\varphi \left({u}^{\prime}\right)\right)}^{\prime }=f\left(t,u,{u}^{\prime}\right), $$where f$$ f $$ is continuous and φ$$ \varphi $$ is a homeomorphism, with nonlinear boundary conditions: g(u(0),G(u′))=h(u′(1),H(u))=0;g(u′(0),G(u))=h(u(1),H(u′))=0;g(u(0),G(u))=h(u′(1),H(u′))=0org(u′(0),G(u′))=h(u(1),H(u))=0,$$ {\displaystyle \begin{array}{cc}\hfill g\left(u(0),G\left({u}^{\prime}\right)\right)=h\left({u}^{\prime }(1),H(u)\right)=& 0;\kern6.05pt g\left({u}^{\prime }(0),G(u)\right)=h\left(u(1),H\left({u}^{\prime}\right)\right)=0;\hfill \\ {}\hfill g\left(u(0),G(u)\right)=h\left({u}^{\prime }(1),H\left({u}^{\prime}\right)\right)=& 0\kern6.05pt \mathrm{or}\kern6.05pt g\left({u}^{\prime }(0),G\left({u}^{\prime}\right)\right)=h\left(u(1),H(u)\right)=0,\hfill \end{array}} $$where g,h$$ g,h $$ are continuous and G,H$$ G,H $$ are continuous functionals. Our methods of proofs are based on the extension of Mawhin's continuation theorem for quasi‐linear operators. [ABSTRACT FROM AUTHOR]

The Hyers-Ulam stability of the iterative equation f n = F for continuous functions F was studied under the assumptions that F is a homeomorphism on its range, and the equation has stability on its range. It is important to study the stability of the equation for homeomorphisms on intervals. In this paper, theorems on stability are obtained using the properties of monotonic approximate solutions. The method is based on the stability of two derived iterative equations. [ABSTRACT FROM AUTHOR]

In this work, we define ZG a topology on the vertex set of a graph G which preserves the connectivity of the graph, called Z-graphic topology. We prove that two isomorphic graphs have homeomorphic and symmetric Z-graphic topologies. We show that ZG is an Alexandroff topology and we give a necessary and sufficient condition for a topology to be Z-graphic. [ABSTRACT FROM AUTHOR]

We consider the generalized ordered spaces and such that the tightness coincides with the tightness but and have different cardinalities. Some sufficient conditions are found under which such spaces and are not -equivalent. [ABSTRACT FROM AUTHOR]

Topological spaces can be induced by various algebraic ordering relations such as, linear, partial and the inclusion-ordering of open sets forming chains and chain complexes. In general, the classifications of covering spaces are made by using fundamental groups and lifting. However, the Riesz ordered n-spaces and Urysohn interpretations of real-valued continuous functions as ordered chains provide new perspectives. This paper proposes the formulation of covering spaces of n-space charts of a foliated n-manifold containing linearly ordered chains, where the chains do not form topologically separated components within a covering section. The chained subspaces within covering spaces are subjected to algebraic split–join operations under a bijective function within chain-subspaces to form simply directed chains and twisted chains. The resulting sets of chains form simply directed chain-paths and oriented chain-paths under the homotopy path-products involving the bijective function. It is shown that the resulting embedding of any chain in a leaf of foliated n-manifold is homogeneous and unique. The finite measures of topological subspaces containing homotopies of chain-paths in covering spaces generate multiplicative and cyclic group varieties of different orders depending upon the types of measures. As a distinction, the proposed homotopies of chain-paths in covering spaces and the homogeneous chain embedding in a foliated n-manifold do not consider the formation of circular nerves and the Nachbin topological preordering, thereby avoiding symmetry/asymmetry conditions. [ABSTRACT FROM AUTHOR]

In this article, we derive sufficient conditions for the existence of countably infinitely many positive solutions for an iterative system of higher order singular Rimean- Liouville(R-L) fractional order boundary value problems with Riemann–Stieltjes(R-S) integral boundary conditions involving increasing homeomorphism and positive homomorphism operator(IHPHO) by applying Hölder's inequality and Krasnoselskii's cone fixed point theorem in a Banach space. Also, we derive sufficient conditions for the uniqueness of the solution for the problem by fixed point theorem on a complete metric space. [ABSTRACT FROM AUTHOR]

Topology is the main branch of mathematics and it lays the new line foundation for the future study in analysis geometry and algebraic topology new line The term topology can be formally defined as the qualitative properties of new line certain objects called sets mappings and topological spaces The objects that new line are invariant under certain kind of transformations are called continuous new line maps and those under a certain kind of equivalence are called new line homeomorphism The concept of topological space grew out of the study of new line the real line and Euclidean spaces and the study of continuous functions on new line these spaces The present research is done under the basis of the earlier and new line traditional studies It is a brief monograph which explores the new class of new line generalized be closed sets which investigate the intersection of two new line generalized be closed sets and the union of two generalized be closed sets It new line also discusses its other properties new line The study of the separation axioms continuous function closed new line maps open maps and homeomorphism are also presented as a part of the new line present thesis The main class of generalized be homeomorphism which new line . [ABSTRACT FROM AUTHOR]

Topological indices are numbers related to molecular graphs that catch symmetry of molecular structures and give it a scientific dialect to foresee properties, such as boiling points, viscosity, the radius of gyrations and so on. In this paper, we aim to compute the general formula for first and second K.Banhatti indices, first and second k hyper-Banhatti indices for graph gluing of web graph and also subdivision and semi-total point graph for the same. [ABSTRACT FROM AUTHOR]

The author of this paper and coauthors in 2022 studied a family of self-affine tiles in R d with noncollinear digit sets, and gave a sufficient and necessary condition for such tiles to be tame balls. We in this paper mainly present a simpler proof of such equivalent condition. We replace quadric surfaces by some zigzag planes, and redefine the quasi-invariant plane which plays a key role in the construction of the desired homeomorphism. This adjustment greatly simplifies the proof. [ABSTRACT FROM AUTHOR]

Human beings exist in a biological and social system from a micro to a macro level, by means of "collectivity", a dynamic collaboration that they have established together with the elements in that system in a way to complement each other and realize a common goal. Many neuroscientific concepts used today to explain neuronal processes from which mental functions originate are far from searching answers to traditional philosophical questions. However, the brain — as the generator of highly abstract concepts — is so complex that it cannot be explained by minimalistic approaches. The concept of connectome used in recent years to describe neuronal connections from which brain functions originate exemplifies this minimalistic approach, because it only describes structural and functional connections but does not look at brain functions in a holistic view. For this reason, we propose the concept of collectome — to replace the concept of connectome — that describes a homeomorphic and homotopic neuronal framework that has a bicontinuous style of work from micro to macroscale which is based on fractal rules. [ABSTRACT FROM AUTHOR]

We prove that every finite-expansive homeomorphism with the shadowing property has a kind of stability. This stability will be good enough to imply both the shadowing property and the denseness of periodic points in the chain recurrent set. Next we analyze the N-shadowing property which is really stronger than the multishadowing property in Cherkashin and Kryzhevich (Topol Methods Nonlinear Anal 50(1): 125–150, 2017). We show that an equicontinuous homeomorphism has the N-shadowing property for some positive integer N if and only if it has the shadowing property. [ABSTRACT FROM AUTHOR]

This paper seeks to demonstrate the relation between homology group and homotopy group. The result in this paper is a construction of the homology of a complex torus. [ABSTRACT FROM AUTHOR]

Is there a topology on the set of integers whose proper subtopology is homeomorphic to the Khalimsky line topology? To address this open problem, the present paper focuses on the topologies T S k and T S k ′ on the set of integers, k ∈ Z. The former is generated by the set S k : = { S k , t | S k , t : = { 2 t , 2 t + 1 , 2 t + 2 k + 1 } , t ∈ Z } and the latter is generated by the set S k ′ : = { S k , t ′ | S k , t ′ : = { 2 t , 2 t + 1 , 2 t + 2 k } , t ∈ Z } as a subbase. Then, we first investigate some properties of semi-open and semi-closed subsets of these two topological spaces (Z , T S k ) and (Z , T S k ′ ). Next, we prove that for k ∈ Z ∖ { 0 } the topological space (Z , T S k ) (resp. (Z , T S k ′ )) has | k | numbers of proper subspaces P i : = (P i , (T S k ) P i ) (resp. C i : = (C i , (T S k ′ ) C i )), i ∈ [ 1 , k ] Z , such that each of them is homeomorphic to the Khalimsky line topological space. Besides, for i ∈ [ 1 , k ] Z , each of all P i and C i is proved to be homeomorphic to the other. Finally, we prove that for k ∈ N each of the product topological spaces (Z n , (T S k ) n) and (Z n , (T S k ′ ) n) has a proper subspace which is homeomorphic to the n -dimensional Khalimsky topological space. [ABSTRACT FROM AUTHOR]

The conventional triangulation of 2-spheres and subdivision of tetrahedrons in R3\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\mathbb {R}^3$$\end{document} are difficult to generalize to higher dimensions. The challenge lies in finding a systematic way to characterize each of subcells after the division. This work discusses the dissection of high-dimensional hypercubes and presents a way where all subsequent subcells and their symmetries can be systematically enumerated. Of particular interest is a generic coordinate system that is employed to construct all cells through suitable homeomorphisms. By repeatedly applying this generic coordinate to all cells, multitasking in parallel is possible. On the other hand, the Borsuk-Ulam theorem asserts that every continuous function from an n-sphere into the Euclidian n-space maps at least one pair of antipodal points on the sphere with the same function value. The exquisiteness lies in that only the continuity is assumed in the theorem with yet such profound applications. As an application, this enumeration scheme can be employed to find the Borsuk-Ulam antipodal pair guaranteed without evoking any derivative information for the task. Numerical experiments manifest the effectiveness and potential of this enumeration scheme. [ABSTRACT FROM AUTHOR]

In the present paper, a characterization of the intuitionistic fuzzy sets, the interval-valued intuitionistic fuzzy sets and their set-operations are given. By making use of these characterizations, the relationships between the interval-valued intuitionistic fuzzy topology and four fuzzy topologies associated to it are studied. For this reason, some subclasses of the family of interval-valued intuitionistic fuzzy topologies on a set which we call pre-suitable and suitable are introduced. Furthermore, the concepts of homeomorphism functions and compactness in the framework of interval-valued intuitionistic fuzzy topological spaces are introduced and studied. [ABSTRACT FROM AUTHOR]

In this paper, we consider diffeomorphic mappings of a nondegenerate parallelepiped onto another such parallelepiped. We prove that the image of any face of the first parallelepiped under such a mapping is a certain face of the second parallelepiped. Moreover, we also obtain several auxiliary results on the transformation of boundary points of a nondegenerate parallelepiped under the action of a diffeomorphism. [ABSTRACT FROM AUTHOR]

We will use continuum-theory to obtain an upper bound for the Kolmogorov-Sinai metric entropy. This bound (throughout called continuum-wise entropy) though different satisfies some properties resembling the metric entropy. Moreover, we prove that every ergodic measure with positive continuum-wise entropy is continuum-wise expansive. [ABSTRACT FROM AUTHOR]

The aim of this paper is to generalize some ideas related to the concept of causality in General Relativity and to carry them over to the theory of control systems. Suppose that M is a smooth manifold, and let (Σ) q. = fu(q), u in u, be a control system on M satisfying certain natural conditions. It is proved that if F: M → M is a bijection such that F and F-1 map the tra jectories of (\Sigma) onto tra jectories of (Σ), then F is a homeomorphism. [ABSTRACT FROM AUTHOR]

The K-Banhatti indices was introduced by Kulli in 2016, defined as B1(G) = X ue [dG(u) + dG(e)] and B2(G) = X ue dG(u).dG(e), where ue means that the vertex u and edge e are incident and dG(e) denotes the degree of the edge e in G. In this paper, we formulate general formula for certain graphs. [ABSTRACT FROM AUTHOR]

In this article, we establish the existence of countably infinitely many positive solutions for singular fractional order Boundary value problem involving Riemann-Stieltjes integral boundary condition with increasing homeomorphism and positive homomorphism operator by using Hölder's inequality and Krasnoselskii's fixed-point theorem (in cone) in a Banach spaces. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to define a new topology called omega topology over a new structure called omega algebra and discuss some of its topological properties. Four different examples of omega topology are introduced. Furthermore, we define a new topology over a semiring in conventional algebra and we study the relationship between omega topology and weaker kinds of normality. [ABSTRACT FROM AUTHOR]

The sufficient conditions are derived for the existence of denumerably many positive solutions for fractional order boundary value problem... whereDε0+, Dη0+ denote fractional derivatives of Riemann-Liouville type with 0 < ε < 1,2 < η ≤3, Iδ0+ (δ ∈ R)denotes Riemann-Liouvillefractional integral, φ: R → Ris an increasing homeomorphism and positive homomorphism operator (IHPHO), by applying Hölder's inequalityand Krasnoselskii's cone fixed point theorem in a Banach space. [ABSTRACT FROM AUTHOR]

Given a space X and topological group H, C(X;H) denotes the set of all continuous functions from a space X to a topological group H. For a subset A of a space X, X=A denotes the quotient space with quotient topology obtained from X by identifying A to a point. We study some homeomorphisms between C(X;H) and C(Y;H) for different X and Y under the point-open topology, open-point topology and bi-point-open topology. [ABSTRACT FROM AUTHOR]

The orbits of slowly perturbed Hamilton systems and the associated ergodic deformations of Lagrange manifolds are studied. The main results are based on the Mather approach [18, 19] to the construction of the homologies of invariant probabilistic measures, which minimize some Lagrange functionals, and on the elliptic Gromov–Salamon–Zehnder–Floer theory [7, 9, 12, 20, 26] of the construction of invariant manifolds. We have constructed the invariant submanifolds, which are the supports of invariant ergodic measures and have a structure of locally homeomorphic metric spaces. We analyze the problem of construction of efficient criteria of their global homeomorphism, which was posed by Professor A. M. Samoilenko during the study of ergodic deformations of nonlinear Hamilton systems and their adiabatic invariants. It is established that the mapping f : X → Y from a linearly connected Hausdorff space X onto a simply connected (in particular, contractible) space Y is a homeomorphism iff f is local and homeomorphic, and the preimage f−1(y) of every point y ∈ Y is a nonempty compact subset in X. [ABSTRACT FROM AUTHOR]

We present a novel C 0 -characterization of symplectic embeddings and diffeomorphisms in terms of Lagrangian embeddings. Our approach is based on the shape invariant, which was discovered by Sikorav and Eliashberg, intersection theory and the displacement energy of Lagrangian submanifolds, and the fact that non-Lagrangian submanifolds can be displaced immediately. This characterization gives rise to a new proof of C 0 -rigidity of symplectic embeddings and diffeomorphisms. The various manifestations of Lagrangian rigidity that are used in our arguments come from J -holomorphic curve methods. An advantage of our techniques is that they can be adapted to a C 0 -characterization of contact embeddings and diffeomorphisms in terms of coisotropic (or pre-Lagrangian) embeddings, which in turn leads to a proof of C 0 -rigidity of contact embeddings and diffeomorphisms. We give a detailed treatment of the shape invariants of symplectic and contact manifolds, and demonstrate that shape is often a natural language in symplectic and contact topology. We consider homeomorphisms that preserve shape, and propose a hierarchy of notions of Lagrangian topological submanifold. Moreover, we discuss shape-related necessary and sufficient conditions for symplectic and contact embeddings, and define a symplectic capacity from the shape. [ABSTRACT FROM AUTHOR]

The n-deck of a topological space X is the set Dn(X) = {[X - {x1x2; . . .; xn}]: x1; x2; . . .; xn ∊ X}; where [Z] denotes the homeomorphism class of Z: A space X is called topologically n-reconstructible if whenever Dn(X) = Dn(Y); then X is homeomorphic to Y: The multi n-deck of a space X is the set that contains all n-cards of X (that is, it includes all homeomorphism types of n-cards). The space X is n-reconstructible from its multi n-deck, we will say it is weakly n-reconstructible. It is shown that the topological properties Ti are n-reconstructible for i = 0; 1; 2; 3; and 2-reconstructible for i = 31/2; 5; 6: It is also proved that the weight of a space and the number of isolated vertices in a T1 space are 2-reconstructible. Also we prove all Hausdorff spaces with a compact subspace obtained by deleting n points, all compact Hausdorff spaces whose reconstructions are compact, all T1 spaces with a finite number of isolated points, the discrete space, the space of rationals and the space endowed with the countable complement topology are n-reconstructible. Finally, we show the property that a bounded ordered space attaining its bounds to be path connected is weakly 1-reconstructible, and that a space to be hyper-connected and an Alexandroff space to be ultra-connected are weakly 2-reconstructible. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to study the fuzzy structure space of a semiring as well as of a Γ-semiring. We study separation axioms, compactness etc. in the fuzzy structure space of a semiring. Similar study has also been accomplished in the setting of a Γ-semiring S by using the nice interplay between S and its left operator semiring L. [ABSTRACT FROM AUTHOR]

The notion of porouscontinuity was introduced by Borsík and Holos. We find classes of functions whose compositions with porouscontinuous functions are still porouscontinuous. Next, we study compositions of porouscontinuous functions with homeomorphisms. We investigate connections between 𝓘𝓒(𝓐) ∩ 𝓗𝓞𝓜 and 𝓘𝓒(𝓑) ∩ 𝓗𝓞𝓜 for different classes 𝓐,𝓑 ∈ 𝓟𝓒, where 𝓟𝓒 is the set of all classes of porouscontinuous functions and 𝓗𝓞𝓜 is the set of all homeomorphisms f : ℝ → ℝ. [ABSTRACT FROM AUTHOR]

Abstract We first prove that the set of shadowable measures of a homeomorphism of a compact metric space X is an F σ δ set of the space M (X) of Borel probability measures of X equipped with the weak*-topology. Next that the set of shadowable measures is dense in M (X) if and only if the set of shadowable points is dense in X. Therefore, if X has no isolated points and every non-atomic Borel probability measure has the shadowing property, then the shadowable points are dense in X (this is false when the space has isolated points). Afterwards, we consider the almost shadowable measures (measures for which the shadowable point set has full measure) and prove that all of them are weak* approximated by shadowable ones. In addition the set of almost shadowable measures is a G δ set of M (X). Furthermore, the closure of the shadowable points is the union of the supports of the almost shadowable measures. Finally, we prove that every almost shadowable measure can be weak* approximated by ones with support equals to the closure of the shadowable points. [ABSTRACT FROM AUTHOR]

Abstract We introduce a new class of planar mappings that allows for cavitations and fractures. The class is the set of strict limits of planar BV homeomorphisms. Each mapping from this class has a proper pointwise representative which is a multifunction, we show that it maps disjoint sets to essentially disjoint sets and that they have an inverse as a proper multifunction. We also characterize and study cavities and fractures of these mappings. [ABSTRACT FROM AUTHOR]

Highlights • A topology-based rigid spherical quadrilateral mesh method is proposed in this paper. • The nondifferentiable problem of free-form surface is solved based on topological transformation. • The model of feature extraction for free-form surface based on topological transformation is built. • The tangent bundle of smooth free-form surface manifold is analyzed. Abstract Free-form surface reconstruction is the key technology of mathematical modeling of engineering. It is important for accurate reconstruction of the entity model that conforms to the design. In order to improve the measurement accuracy of the workpiece, a method for free-form surface feature recognition based on topological transformation is proposed in this paper. Firstly, the information of scanning points is expressed as the form of topological space. Based on this, a method of surface constructing that selects the natural-quadratic-surface as the manifold of free-form surface is proposed. Then, the free-form surface that is constructed from the original point cloud is transformed into a smooth free-form surface which is homeomorphic to the original free-form surface based on the theory of topological embedding. The features of the transformed free-form surface can be easily analyzed and recognized. Finally, the qualitative description of an arbitrary point on the surface is obtained by analyzing the tangent bundle of smooth free-form surface manifolds. The feature recognition of free-form surface for workpiece is realized. The experimental results on workpiece model are reported in this paper, which validate the effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]

We prove that the set of expansive measures of a homeomorphism of a compact metric space is a G δ σ subset of the space of Borel probability measures equipped with the weak* topology. Next that every expansive measure of a homeomorphism of a compact metric space can be weak* approximated by expansive measures with invariant support. In addition, if the expansive measures of a homeomorphism of a compact metric space are dense in the space of Borel probability measures, then there is an expansive measure whose support is both invariant and close to the whole space with respect to the Hausdorff metric. Henceforth, if the expansive measures are dense in the space of Borel probability measures, the set of heteroclinic points has no interior and the space has no isolated points. [ABSTRACT FROM AUTHOR]

Considering the spaces Bp[1, α] of all Baire functions x: [1, α] → ℝ on the ordinal segments [1, α] that are endowed with the topology of pointwise convergence, we give a complete topological classification of these spaces. [ABSTRACT FROM AUTHOR]

Abstract We investigate homeomorphisms of different types of Hashimoto topologies based on the Euclidean topology on the real line and classic σ -ideals. [ABSTRACT FROM AUTHOR]

For the metric spaces, the homeomorphisms more general than conformal mappings are studied. It is proved that the families of ine indicated mappings are equicontinuous in their closure under definite conditions imposed on the boundaries of the corresponding domains. [ABSTRACT FROM AUTHOR]

A homeomorphism of the 2-torus with Lefschetz number different from zero has a fixed point. We give a version of this result for nilpotent groups of diffeomorphisms. We prove that a nilpotent group of 2-torus diffeomorphims has finite orbits when the group has some element with Lefschetz number different from zero. [ABSTRACT FROM AUTHOR]

A topological space SP, which is a modification of the Sorgenfrey line S, is considered. It is defined as follows: if x ∈ P ⊂ S, then a base of neighborhoods of x is the family {[x, x + ε), ε > 0} of half-open intervals, and if x ∈ SP, then a base of neighborhoods of x is the family {(x − ε, x], ε > 0}. A necessary and sufficient condition under which the space SP is homeomorphic to S is obtained. Similar questions were considered by V. A. Chatyrko and I. Hattori, who defined the neighborhoods of x ∈ P to be the same as in the natural topology of the real line. [ABSTRACT FROM AUTHOR]

In this paper, we generalize Rees-Shishikura's theorem to the class of geometrically finite rational maps. [ABSTRACT FROM AUTHOR]

The symbol S c ( X ) denotes the hyperspace of all nontrivial convergent sequences in a Hausdorff space X . This hyperspace is endowed with the Vietoris topology. For a given mapping between Hausdorff spaces f : X → Y , define the induced mapping S c ( f ) by S c ( f ) ( A ) = f [ A ] (the image of A under f ) for every A ∈ S c ( X ) . In the current paper, we study under which conditions the fact that f belongs to a given class of mappings M implies that S c ( f ) belongs to M , and vice versa. [ABSTRACT FROM AUTHOR]

In the present paper we introduce two new types of mappings called gspsg-homeomorphism and sggsp-homeomorphism and then shown that one of these mapping has a group structure. Further we investigate some properties of these two homeomorphisms. [ABSTRACT FROM AUTHOR]

Conditions on equivalence of topological or uniform spaces having equivalent compactifications are investigated. For instance: (1) Let X , Y be Čech-complete, pseudoradial spaces complete in their compactifications b X , b Y . If the compactifications are homeomorphic then X , Y are homeomorphic. (2) Let X , Y be products of complete uniform spaces having linearly ordered bases. If their Samuel compactifications are homeomorphic then X and Y are uniformly homeomorphic. [ABSTRACT FROM AUTHOR]

Let G be a subgroup of the group Homeo(E) of homeomorphisms of a Hausdorff topological space E. The class of an orbit O of G is the union of all orbits having the same closure as O. We denote by E/G the space of classes of orbits called quasi-orbit space. A space X is called a quasi-orbital space if it is homeomorphic to E/G where E is a compact Hausdorff space. In this paper, we show that every infinite second countable quasi-compact T0-space is the quotient of a quasiorbital space. [ABSTRACT FROM AUTHOR]

Let H stand for the set of homeomorphisms ɸ: [0, 1] → [0, 1]. We prove the following dichotomy for Borel subsets A ⊂ [0, 1]: • either there exists a homeomorphism ɸ ∈ H such that the image ɸ(A) contains no 3-term arithmetic progressions; • or, for every ɸ ∈ H, the image ɸ(A) contains arithmetic progressions of arbitrary finite length. In fact, we show that the first alternative holds if and only if the set A is meager (a countable union of nowhere dense sets). [ABSTRACT FROM AUTHOR]