Topological Properties of a Class of Higher-dimensional Self-affine Tiles

2019 ◽  
Vol 62 (4) ◽  
pp. 727-740
Author(s):  
Guotai Deng ◽  
Chuntai Liu ◽  
Sze-Man Ngai
Keyword(s):  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Topological Properties ◽  
Higher Dimensional ◽  
Necessary And Sufficient

AbstractWe construct a family of self-affine tiles in $\mathbb{R}^{d}$ ($d\geqslant 2$) with noncollinear digit sets, which naturally generalizes a class studied originally by Q.-R. Deng and K.-S. Lau in $\mathbb{R}^{2}$, and its extension to $\mathbb{R}^{3}$ by the authors. We obtain necessary and sufficient conditions for the tiles to be connected and for their interiors to be contractible.

Asymptotic stability of heteroclinic cycles in systems with symmetry. II

2004 ◽  
Vol 134 (6) ◽  
pp. 1177-1197 ◽  
Author(s):  
Martin Krupa ◽  
Ian Melbourne
Keyword(s):  
Asymptotic Stability ◽  
Transition Matrix ◽  
Sufficient Conditions ◽  
Systematic Investigation ◽  
Necessary And Sufficient Conditions ◽  
Sufficient Condition ◽  
Heteroclinic Cycles ◽  
Higher Dimensional ◽  
Necessary And Sufficient

Systems possessing symmetries often admit robust heteroclinic cycles that persist under perturbations that respect the symmetry. In previous work, we began a systematic investigation into the asymptotic stability of such cycles. In particular, we found a sufficient condition for asymptotic stability, and we gave algebraic criteria for deciding when this condition is also necessary. These criteria are satisfied for cycles in R3.Field and Swift, and Hofbauer, considered examples in R4 for which our sufficient condition for stability is not optimal. They obtained necessary and sufficient conditions for asymptotic stability using a transition-matrix technique.In this paper, we combine our previous methods with the transition-matrix technique and obtain necessary and sufficient conditions for asymptotic stability for a larger class of heteroclinic cycles. In particular, we obtain a complete theory for ‘simple’ heteroclinic cycles in R4 (thereby proving and extending results for homoclinic cycles that were stated without proof by Chossat, Krupa, Melbourne and Scheel). A partial classification of simple heteroclinic cycles in R4 is also given. Finally, our stability results generalize naturally to higher dimensions and many of the higher-dimensional examples in the literature are covered by this theory.

SOME TOPOLOGICAL RESULTS ON LEVEL SETS AND MULTIFRACTAL SETS

Fractals ◽  
2019 ◽  
Vol 28 (01) ◽  
pp. 2050009
Author(s):  
CHUNTAI LIU
Keyword(s):  
Level Sets ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Topological Properties ◽  
Symbolic Space ◽  
Necessary And Sufficient

In this paper, we study topological properties of some level sets and some multifractal sets induced by Rademacher’s series and Takagi’s series, respectively. By using symbolic space, we obtain necessary and sufficient conditions for them to be residual.

On Janowski harmonic functions

2015 ◽  
Vol 21 (2) ◽  
Author(s):  
Jacek Dziok
Keyword(s):  
Harmonic Functions ◽  
Sufficient Conditions ◽  
Extreme Points ◽  
Necessary And Sufficient Conditions ◽  
Topological Properties ◽  
Janowski Functions ◽  
Distortion Theorems ◽  
Necessary And Sufficient

AbstractIn this paper we define classes of harmonic functions related to the Janowski functions and we give some necessary and sufficient conditions for these classes. Some topological properties and extreme points of the classes are also considered. By using extreme points theory we obtain coefficients estimates, distortion theorems, integral mean inequalities for the classes of functions.

scholarly journals On the generalized Riesz B-difference sequence spaces

Filomat ◽  
2010 ◽  
Vol 24 (4) ◽  
pp. 35-52 ◽  
Author(s):  
Metin Başarir
Keyword(s):  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Sequence Spaces ◽  
Matrix Transformations ◽  
Difference Sequence ◽  
Topological Properties ◽  
Linear Spaces ◽  
The Matrix ◽  
Necessary And Sufficient ◽  
Difference Sequence Spaces

In this paper, we define the new generalized Riesz B-difference sequence spaces rq? (p, B), rqc (p, B), rq0 (p, B) and rq (p, B) which consist of the sequences whose Rq B-transforms are in the linear spaces l?(p), c (p), c0(p) and l(p), respectively, introduced by I.J. Maddox[8],[9]. We give some topological properties and compute the ?-, ?- and ?-duals of these spaces. Also we determine the necessary and sufficient conditions on the matrix transformations from these spaces into l? and c.

scholarly journals Lattice separation, coseparation and regular measures

1996 ◽  
Vol 19 (4) ◽  
pp. 773-779
Author(s):  
Maurice C. Figueres
Keyword(s):  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Topological Properties ◽  
Finitely Additive Measures ◽  
Necessary And Sufficient ◽  
Additive Measures

LetXbe an arbitrary non-empty set, and letℒ,ℒ1,ℒ2be lattices of subsets ofXcontainingϕandX.𝒜(ℒ)designates the algebra generated byℒandM(ℒ), these finite, non-trivial, non-negative finitely additive measures on𝒜(ℒ).I(ℒ)denotes those elements ofM(ℒ)which assume only the values zero and one. In terms of aμ∈M(ℒ)orI(ℒ), various outer measures are introduced. Their properties are investigated. The interplay of measurability, smoothness ofμ, regularity ofμand lattice topological properties on these outer measures is also investigated.Finally, applications of these outer measures to separation type properties between pairs of latticesℒ1,ℒ2whereℒ1⊂ℒ2are developed. In terms of measures fromI(ℒ), necessary and sufficient conditions are established forℒ1to semi-separateℒ2, forℒ1to separateℒ2, and finally forℒ1to coseparateℒ2.

SIMPLICITY OF CROSSED PRODUCTS BY TWISTED PARTIAL ACTIONS

2019 ◽  
Vol 108 (2) ◽  
pp. 202-225
Author(s):  
ALEXANDRE BARAVIERA ◽  
WAGNER CORTES ◽  
MARLON SOARES
Keyword(s):  
Hausdorff Space ◽  
Associative Ring ◽  
Sufficient Conditions ◽  
Continuous Functions ◽  
Necessary And Sufficient Conditions ◽  
Crossed Products ◽  
Topological Properties ◽  
Partial Action ◽  
Necessary And Sufficient ◽  
Skew Group Ring

In this article, we consider a twisted partial action $\unicode[STIX]{x1D6FC}$ of a group $G$ on an associative ring $R$ and its associated partial crossed product $R\ast _{\unicode[STIX]{x1D6FC}}^{w}G$. We provide necessary and sufficient conditions for the commutativity of $R\ast _{\unicode[STIX]{x1D6FC}}^{w}G$ when the twisted partial action $\unicode[STIX]{x1D6FC}$ is unital. Moreover, we study necessary and sufficient conditions for the simplicity of $R\ast _{\unicode[STIX]{x1D6FC}}^{w}G$ in the following cases: (i) $G$ is abelian; (ii) $R$ is maximal commutative in $R\ast _{\unicode[STIX]{x1D6FC}}^{w}G$; (iii) $C_{R\ast _{\unicode[STIX]{x1D6FC}}^{w}G}(Z(R))$ is simple; (iv) $G$ is hypercentral. When $R=C_{0}(X)$ is the algebra of continuous functions defined on a locally compact and Hausdorff space $X$, with complex values that vanish at infinity, and $C_{0}(X)\ast _{\unicode[STIX]{x1D6FC}}G$ is the associated partial skew group ring of a partial action $\unicode[STIX]{x1D6FC}$ of a topological group $G$ on $C_{0}(X)$, we study the simplicity of $C_{0}(X)\ast _{\unicode[STIX]{x1D6FC}}G$ by using topological properties of $X$ and the results about the simplicity of $R\ast _{\unicode[STIX]{x1D6FC}}^{w}G$.

HIGHER DIMENSIONAL RIEMANN MAPS

1998 ◽  
Vol 09 (04) ◽  
pp. 421-442 ◽  
Author(s):  
ZOLTAN M. BALOGH ◽  
CHRISTOPH LEUENBERGER
Keyword(s):  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Pseudoconvex Domains ◽  
Riemann Map ◽  
Higher Dimensional ◽  
Necessary And Sufficient ◽  
Invariant Characterization

We consider the notion of Riemann map of Lempert and Semmes. The purpose of this paper is to give an intrinsic and biholomorphically invariant characterization of strictly pseudoconvex domains in Cn which admit a Riemann map. In this sense necessary and sufficient conditions are given for the existence of a Riemann map in terms of Kobayashi discs and the associated Lempert invariants.

scholarly journals On Topological Properties of Metrics Defined via Generalized “Linking Construction”

2017 ◽  
Vol 2017 ◽  
pp. 1-7
Author(s):  
Marcin Borkowski ◽  
Dariusz Bugajewski ◽  
Adam Burchardt
Keyword(s):  
Metric Spaces ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Topological Properties ◽  
Necessary And Sufficient

We analyze topological properties of metric spaces obtained by using Száz’s construction, which we used to call generalized “linking construction.” In particular, we provide necessary and sufficient conditions for completeness of metric spaces obtained in this way. Moreover, we examine the relation between Száz’s construction and the “linking construction.” A particular attention is drawn to the “floor” metric, the analysis of which provides some interesting observations.

scholarly journals On topological properties of the Hausdorff fuzzy metric spaces

Filomat ◽  
2017 ◽  
Vol 31 (5) ◽  
pp. 1167-1173
Author(s):  
Changqing Li ◽  
Kedian Li
Keyword(s):  
Metric Spaces ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Topological Properties ◽  
Fuzzy Metric Spaces ◽  
Fuzzy Metric ◽  
Necessary And Sufficient

In the paper, necessary and sufficient conditions for two Hausdorff fuzzy metric spaces to be homeomorphic are studied. Also, several properties of the Hausdorff fuzzy metric spaces, as F-boundedness, separability and connectedness are explored.

scholarly journals The application domain of difference type matrix D(r,0,s,0,t) on some sequence spaces

2020 ◽  
pp. 32-32
Author(s):  
Avinoy Paul ◽  
Binod Tripathy
Keyword(s):  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Sequence Spaces ◽  
Application Domain ◽  
Topological Properties ◽  
Difference Type ◽  
The Matrix ◽  
Necessary And Sufficient ◽  
Matrix Mappings

In this paper we introduce new sequence spaces with the help of domain of matrix D(r,0,s,0,t), and study some of their topological properties. Further, we determine ? and ? duals of the new sequence spaces and finally, we establish the necessary and sufficient conditions for characterization of the matrix mappings.

Sign in / Sign up

Export Citation Format

Share Document