scholarly journals On topology of manifolds admitting gradient-like calscades with surface dynamics and on growth of the number of non-compact heteroclinic curves

2021 ◽  
Vol 23 (4) ◽  
pp. 379-393
Author(s):  
Vyacheslav Z. Grines ◽  
Elena Ya. Gurevich ◽  
Evgenii Iv. Yakovlev
Keyword(s):  
Sufficient Conditions ◽  
Periodic Points ◽  
Orientable Surface ◽  
Fixed Number ◽  
Surface Dynamics ◽  
Oriented Manifold ◽  
Mapping Torus ◽  
Mapping Tori ◽  
Minimum Number ◽  
Topology Of Manifolds

We consider a class GSD(M3) of gradient-like diffeomorphisms with surface dynamics given on closed oriented manifold M3 of dimension three. Earlier it was proved that manifolds admitting such diffeomorphisms are mapping tori under closed orientable surface of genus g, and the number of non-compact heteroclinic curves of such diffeomorphisms is not less than 12g. In this paper, we determine a class of diffeomorphisms GSDR(M3)⊂GSD(M3) that have the minimum number of heteroclinic curves for a given number of periodic points, and prove that the supporting manifold of such diffeomorphisms is a Seifert manifold. The separatrices of periodic points of diffeomorphisms from the class GSDR(M3) have regular asymptotic behavior, in particular, their closures are locally flat. We provide sufficient conditions (independent on dynamics) for mapping torus to be Seifert. At the same time, the paper establishes that for any fixed g geq1, fixed number of periodic points, and any integer n≥12g, there exists a manifold M3 and a diffeomorphism f∈GSD(M3) having exactly n non-compact heteroclinic curves.

scholarly journals Ranks of mapping tori via the curve complex

2019 ◽  
Vol 2019 (748) ◽  
pp. 153-172 ◽  
Author(s):  
Ian Biringer ◽  
Juan Souto
Keyword(s):  
Fundamental Group ◽  
Orientable Surface ◽  
Curve Complex ◽  
Mapping Torus ◽  
Closed Orientable Surface ◽  
Mapping Tori

Abstract We show that if ϕ is a homeomorphism of a closed, orientable surface of genus g, and ϕ has large translation distance in the curve complex, then the fundamental group of the mapping torus {M_{\phi}} has rank {2g+1} .

scholarly journals On Embedding of the Morse-Smale Diffeomorphisms in a Topological Flow

2020 ◽  
Vol 66 (2) ◽  
pp. 160-181
Author(s):  
V. Z. Grines ◽  
E. Ya. Gurevich ◽  
O. V. Pochinka
Keyword(s):  
Invariant Manifolds ◽  
Three Dimensional ◽  
Sufficient Conditions ◽  
Periodic Points ◽  
Topological Classification ◽  
Higher Dimension ◽  
Topological Information ◽  
Topology Of Manifolds ◽  
Necessary And Sufficient

This review presents the results of recent years on solving of the Palis problem on finding necessary and sufficient conditions for the embedding of Morse-Smale cascades in topological flows. To date, the problem has been solved by Palis for Morse-Smale diffeomorphisms given on manifolds of dimension two. The result for the circle is a trivial exercise. In dimensions three and higher new effects arise related to the possibility of wild embeddings of closures of invariant manifolds of saddle periodic points that leads to additional obstacles for Morse-Smale diffeomorphisms to embed in topological flows. The progress achieved in solving of Paliss problem in dimension three is associated with the recently obtained complete topological classification of Morse-Smale diffeomorphisms on three-dimensional manifolds and the introduction of new invariants describing the embedding of separatrices of saddle periodic points in a supporting manifold. The transition to a higher dimension requires the latest results from the topology of manifolds. The necessary topological information, which plays key roles in the proofs, is also presented in the survey.

scholarly journals Genus Ranges of Chord Diagrams

2015 ◽  
Vol 24 (04) ◽  
pp. 1550022 ◽  
Author(s):  
Jonathan Burns ◽  
Nataša Jonoska ◽  
Masahico Saito
Keyword(s):  
Outer Boundary ◽  
Orientable Surface ◽  
Fixed Number ◽  
Chord Diagram ◽  
Boundary Circle ◽  
Line Segments ◽  
Chord Diagrams ◽  
Computer Calculations

A chord diagram consists of a circle, called the backbone, with line segments, called chords, whose endpoints are attached to distinct points on the circle. The genus of a chord diagram is the genus of the orientable surface obtained by thickening the backbone to an annulus and attaching bands to the inner boundary circle at the ends of each chord. Variations of this construction are considered here, where bands are possibly attached to the outer boundary circle of the annulus. The genus range of a chord diagram is the genus values over all such variations of surfaces thus obtained from a given chord diagram. Genus ranges of chord diagrams for a fixed number of chords are studied. Integer intervals that can be, and those that cannot be, realized as genus ranges are investigated. Computer calculations are presented, and play a key role in discovering and proving the properties of genus ranges.

scholarly journals On number of moduli for gradient surface height function flows

2018 ◽  
Vol 20 (4) ◽  
pp. 419-428
Author(s):  
V. E. Kruglov
Keyword(s):  
Finite Number ◽  
Sufficient Conditions ◽  
Height Function ◽  
Orientable Surface ◽  
Topological Conjugacy ◽  
Topological Equivalence ◽  
Gradient Flows ◽  
A Chain ◽  
Gradient Surface ◽  
Equivalence Invariant

In 1978 J. Palis invented continuum topologically non-conjugate systems in a neighbourhood of a system with a heteroclinic contact; in other words, he invented so-called moduli. W. de Melo and С. van Strien in 1987 described a diffeomorphism class with a finite number of moduli. They discovered that a chain of saddles taking part in the heteroclinic contact of such diffeomorphism includes not more than three saddles. Surprisingly, such effect does not happen in flows. Here we consider gradient flows of the height function for an orientable surface of genus g>0. Such flows have a chain of 2g saddles. We found that the number of moduli for such flows is 2g−1 which is the straight consequence of the sufficient topological conjugacy conditions for such systems given in our paper. A complete topological equivalence invariant for such systems is four-colour graph carrying the information about its cells relative position. Equipping the graph's edges with the analytical parameters --- moduli, connected with the saddle connections, gives the sufficient conditions of the flows topological conjugacy.

scholarly journals Classification of rough transformations of a circle from a modern point of view

2018 ◽  
Vol 20 (4) ◽  
pp. 408-418
Author(s):  
A. E. Kolobyanina ◽  
E. V. Nozdrinova ◽  
O. V. Pochinka
Keyword(s):  
Invariant Manifolds ◽  
Sufficient Conditions ◽  
Periodic Points ◽  
Point Of View ◽  
Topological Classification ◽  
Topological Invariants ◽  
Modern Theory ◽  
Ambient Manifold ◽  
Periodic Data

In this paper the authors use modern methods and approaches to present a solution to the problem of the topological classification of circle’s rough transformations in canonical formulation. In the modern theory of dynamical systems such problems are understood as the complete topological classification: finding topological invariants, proving the completeness of the set of invariants found and constructing a standard representative from a given set of topological invariants. Namely, in the first theorem of this paper the type of periodic data of circle’s rough transformations is established. In the second theorem necessary and sufficient conditions of their conjugacy are proved. These conditions mean coincidence of periodic data and rotation numbers. In the third theorem the admissible set of parameters is implemented by a rough transformation of a circle. While proving the theorems, we assume that the results on the local topological classification of hyperbolic periodic points, as well as the results on the global representation of the ambient manifold as a union of invariant manifolds of periodic points, are known.

Some Applications of Artamonov-Quillen-Suslin Theorems to Metabelian Inner Rank and Primitivity

1994 ◽  
Vol 46 (2) ◽  
pp. 298-307 ◽  
Author(s):  
C. K. Gupta ◽  
N. D. Gupta ◽  
G. A. Noskov
Keyword(s):  
Metabelian Group ◽  
Sufficient Conditions ◽  
Surface Group ◽  
Maximal Rank ◽  
Orientable Surface ◽  
Surface Groups ◽  
Modular Element ◽  
Free Metabelian Group ◽  
One Relator Groups ◽  
Necessary And Sufficient

AbstractFor any variety of groups, the relative inner rank of a given groupG is defined to be the maximal rank of the -free homomorphic images of G. In this paper we explore metabelian inner ranks of certain one-relator groups. Using the well-known Quillen-Suslin Theorem, in conjunction with an elegant result of Artamonov, we prove that if r is any "Δ-modular" element of the free metabelian group Mn of rank n > 2 then the metabelian inner rank of the quotient group Mn/(r) is at most [n/2]. As a corollary we deduce that the metabelian inner rank of the (orientable) surface group of genus k is precisely k. This extends the corresponding result of Zieschang about the absolute inner ranks of these surface groups. In continuation of some further applications of the Quillen-Suslin Theorem we give necessary and sufficient conditions for a system g = (g1,..., gk) of k elements of a free metabelian group Mn, k ≤ n, to be a part of a basis of Mn. This extends results of Bachmuth and Timoshenko who considered the cases k = n and k < n — 3 respectively.

Secure Publishing using Schema-level Role-based Access Control Policies for Fragments of XML Documents

2009 ◽  
Author(s):  
Tomasz Müldner ◽  
Robin McNeill ◽  
Jan Krzysztof Miziołek
Keyword(s):  
Access Control ◽  
Fixed Number ◽  
Original Structure ◽  
Role Based Access Control ◽  
Control Policies ◽  
Xml Documents ◽  
Access Control Policies ◽  
Minimum Number ◽  
Role Based ◽  
Implementation Tool

Popularity of social networks is growing rapidly and secure publishing is an important implementation tool for these networks. At the same time, recent implementations of access control policies (ACPs) for sharing fragments of XML documents have moved from distributing to users numerous sanitized sub-documents to disseminating a single document multi-encrypted with multiple cryptographic keys, in such a way that the stated ACPs are enforced. Any application that uses this implementation of ACPs will incur a high cost of generating keys separately for each document. However, most such applications, such as secure publishing, use similar documents, i.e. documents based on a selected schema. This paper describes RBAC defined at the schema level, (SRBAC), and generation of the minimum number of keys at the schema level. The main advantage of our approach is that for any application that uses a fixed number of schemas, keys can be generated (or even pre-generated) only once, and then reused in all documents valid for the given schema. While in general, key generation at the schema level has to be pessimistic, our approach tries to minimize the number of generated keys. Incoming XML documents are efficiently encrypted using single-pass SAX parsing in such a way that the original structure of these documents is completely hidden. We also describe distributing to each user only keys needed for decrypting accessible nodes, and for applying the minimal number of encryption operations to an XML document required to satisfy the protection requirements of the policy.

scholarly journals On the connectivity and diameter of small-world networks

2007 ◽  
Vol 39 (4) ◽  
pp. 853-863 ◽  
Author(s):  
Ayalvadi Ganesh ◽  
Feng Xue
Keyword(s):  
Necessary Conditions ◽  
Sufficient Conditions ◽  
Small World ◽  
Fixed Number ◽  
Total Degree ◽  
Small World Networks ◽  
Fixed Distance ◽  
Stochastic Point Process ◽  
Nearest Neighbours ◽  
Small World Graphs

We consider two different models of small-world graphs on nodes whose locations are modelled by a stochastic point process. In the first model each node is connected to a fixed number of its nearest neighbours, while in the second model each node is connected to all nodes located within some fixed distance. In both models, nodes are additionally connected via shortcuts to other nodes chosen uniformly at random. We obtain sufficient conditions for connectivity in the first model, and necessary conditions in the second model. Thereby, we show that connectivity is achieved at a smaller value of total degree (nearest neighbours plus shortcuts) in the first model. We also obtain bounds on the diameter of the graph in this model.

scholarly journals Immediate conditional hyperbolicity in dynamical systems

1983 ◽  
Vol 3 (4) ◽  
pp. 627-647
Author(s):  
Joseph Rosenblatt ◽  
Richard Swanson
Keyword(s):  
Vector Bundle ◽  
Dynamical Systems ◽  
Compact Manifold ◽  
Hyperbolic Systems ◽  
Sufficient Conditions ◽  
Periodic Points ◽  
Almost Periodic ◽  
Uniquely Ergodic

AbstractFor many diffeomorphisms of a compact manifold X, eventual conditional hyperbolicity implies immediate conditional hyperbolicity in some (possibly new) Finsler structures. That is, if A and B are vector bundle isomorphisms over the mapping ƒ of the base X, such that uniformly on X, then there exist new norms for A and B such that uniformly on X, whenever the mapping ƒ satisfies the condition that there exist infinitely many N ≥ 1 such that any ƒ-invariant. For example, this condition on ƒ holds if any one of the following conditions holds: (1) ƒ is periodic; (2) ƒ is periodic on its non-wandering set; (3) ƒ has a finite non-wandering set (for example, ƒ is a Morse-Smale diffeomorphism); (4) ƒ is an almost periodic mapping of a connected base X; (5) ƒ is a mapping of the circle with no periodic points; or (6) ƒ and all its powers are uniquely ergodic. We consider various types of eventually conditionally hyperbolic systems and describe sufficient conditions on ƒ to have immediate conditional hyperbolicity of these systems in some new Finsler structures. Thus, for a sizable class of dynamical systems, we settle, in the affirmative, a question raised by Hirsch, Pugh, and Shub.

scholarly journals On Sectional Curvature of Metric Connection with Vectorial Torsion

2020 ◽  
pp. 124-127
Author(s):  
E.D. Rodionov ◽  
V.V. Slavsky ◽  
O.P. Khromova
Keyword(s):  
Sectional Curvature ◽  
Curvature Tensor ◽  
Sufficient Conditions ◽  
Civita Connection ◽  
Modern Physics ◽  
Metric Connection ◽  
Levi Civita Connection ◽  
The Difference ◽  
Topology Of Manifolds ◽  
Conformal Deformations

Papers of many mathematicians are devoted to the study of semisymmetric connections or metric connections with vector torsion on Riemannian manifolds. This type of connectivity is one of the three main types discovered by E. Cartan and finds its application in modern physics, geometry, and topology of manifolds. Geodesic lines and the curvature tensor of a given connection were studied by I. Agricola, K. Yano, and other mathematicians. In particular, K. Yano proved an important theorem on the connection of conformal deformations and metric connections with vector torsion. Namely: a Riemannian manifold admits a metric connection with vector torsion and the curvature tensor being equal to zero if and only if it is conformally flat. Although the curvature tensor of a hemisymmetric connection has a smaller number of symmetries compared to the Levi-Civita connection, it is still possible to define the concept of sectional curvature in this case. The question naturally arises about the difference between the sectional curvature of a semisymmetric connection and the sectional curvature of a Levi-Civita connection.This paper is devoted to the study of this issue, and the authors find the necessary and sufficient conditions for the sectional curvature of the semisymmetric connection to coincide with the sectional curvature of the Levi-Civita connection. Non-trivial examples of hemisymmetric connections are constructed when possible.

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