scholarly journals Summability implies Collet–Eckmann almost surely

2013 ◽  
Vol 34 (4) ◽  
pp. 1184-1209 ◽  
Author(s):  
BING GAO ◽  
WEIXIAO SHEN
Keyword(s):  
Parameter Family ◽  
Density Point ◽  
Strengthened Version ◽  
Interval Map ◽  
Summability Condition ◽  
Lebesgue Density

AbstractWe provide a strengthened version of the famous Jakobson's theorem. Consider an interval map $f$ satisfying a summability condition. For a generic one-parameter family ${f}_{t} $ of maps with ${f}_{0} = f$, we prove that $t= 0$ is a Lebesgue density point of the set of parameters for which ${f}_{t} $ satisfies both the Collet–Eckmann condition and a strong polynomial recurrence condition.

scholarly journals A category analogue of the generalization of Lebesgue density topology

2009 ◽  
Vol 42 (1) ◽  
pp. 11-25
Author(s):  
Wojciech Wojdowski
Keyword(s):  
Real Line ◽  
Continuous Functions ◽  
Density Point ◽  
Density Topology ◽  
The Real ◽  
Lebesgue Density ◽  
Real Anal

Abstract . A notion of AI -topology, a generalization of Wilczy´nski’s I-density topology (see [Wilczy´nski, W.: A generalization of the density topology, Real. Anal. Exchange 8 (1982-1983), 16-20] is introduced. The notion is based on his reformulation of the definition od Lebesgue density point. We consider a category version of the topology, which is a category analogue of the notion of an Ad- -density topology on the real line given in [Wojdowski, W.: A generalization ofdensity topology, Real. Anal. Exchange 32 (2006/2007), 1-10]. We also discuss the properties of continuous functions with respect to the topology.

scholarly journals Corrigendum to a Category Analogue of The Generalization of Lebesgue Density Topology

2016 ◽  
Vol 65 (1) ◽  
pp. 161-164
Author(s):  
Wojciech Wojdowski
Keyword(s):  
Density Operator ◽  
Density Point ◽  
Density Topology ◽  
Lebesgue Density ◽  
Real Functions

Abstract The notion of AI -density point introduced in Wojdowski, W. A topology stronger than the Lebesgue density topology, in: Real Functions, Density Topology and Related Topics, Łódź Univ. Press, 2011, pp. 73-80 [WO1]. leads to the operator ΦAI (A) which is not a lower density operator. We present a counterexample giving a corrected definition which should be used in [WO1] to keep all results valid.

scholarly journals A Uniqueness Result for a Simple Superlinear Eigenvalue Problem

2021 ◽  
Vol 31 (2) ◽  
Author(s):  
Michael Herrmann ◽  
Karsten Matthies
Keyword(s):  
Eigenvalue Problem ◽  
Numerical Simulations ◽  
Convolution Operator ◽  
Existence And Uniqueness ◽  
Uniqueness Result ◽  
Constitutive Laws ◽  
Parameter Family ◽  
Shape Constraint ◽  
Special Case ◽  
Nonlinear Eigenfunctions

AbstractWe study the eigenvalue problem for a superlinear convolution operator in the special case of bilinear constitutive laws and establish the existence and uniqueness of a one-parameter family of nonlinear eigenfunctions under a topological shape constraint. Our proof uses a nonlinear change of scalar parameters and applies Krein–Rutman arguments to a linear substitute problem. We also present numerical simulations and discuss the asymptotics of two limiting cases.

scholarly journals Continuous Maps from Spheres Converging to Boundaries of Convex Hulls

2021 ◽  
Vol 9 ◽  
Author(s):  
Joseph Malkoun ◽  
Peter J. Olver
Keyword(s):  
Convex Hull ◽  
Convex Geometry ◽  
Gauss Map ◽  
Parameter Family ◽  
Convex Hulls ◽  
Dimensional Sphere ◽  
Continuous Maps ◽  
Dimensional Boundary

Abstract Given n distinct points $\mathbf {x}_1, \ldots , \mathbf {x}_n$ in $\mathbb {R}^d$ , let K denote their convex hull, which we assume to be d-dimensional, and $B = \partial K $ its $(d-1)$ -dimensional boundary. We construct an explicit, easily computable one-parameter family of continuous maps $\mathbf {f}_{\varepsilon } \colon \mathbb {S}^{d-1} \to K$ which, for $\varepsilon> 0$ , are defined on the $(d-1)$ -dimensional sphere, and whose images $\mathbf {f}_{\varepsilon }({\mathbb {S}^{d-1}})$ are codimension $1$ submanifolds contained in the interior of K. Moreover, as the parameter $\varepsilon $ goes to $0^+$ , the images $\mathbf {f}_{\varepsilon } ({\mathbb {S}^{d-1}})$ converge, as sets, to the boundary B of the convex hull. We prove this theorem using techniques from convex geometry of (spherical) polytopes and set-valued homology. We further establish an interesting relationship with the Gauss map of the polytope B, appropriately defined. Several computer plots illustrating these results are included.

scholarly journals The holographic conformal manifold of 3d $$ \mathcal{N} $$ = 2 S-fold SCFTs

2021 ◽  
Vol 2021 (7) ◽  
Author(s):  
Nikolay Bobev ◽  
Friðrik Freyr Gautason ◽  
Jesse van Muiden
Keyword(s):  
String Theory ◽  
Three Dimensional ◽  
Parameter Family ◽  
Global Symmetry ◽  
Maximal Supergravity ◽  
All Solutions ◽  
Operator Spectrum ◽  
Type Iib String Theory ◽  
Two Parameter

Abstract We employ a non-compact gauging of four-dimensional maximal supergravity to construct a two-parameter family of AdS4 J-fold solutions preserving $$ \mathcal{N} $$ N = 2 supersymmetry. All solutions preserve $$ \mathfrak{u} $$ u (1) × $$ \mathfrak{u} $$ u (1) global symmetry and in special limits we recover the previously known $$ \mathfrak{su} $$ su (2) × $$ \mathfrak{u} $$ u (1) invariant $$ \mathcal{N} $$ N = 2 and $$ \mathfrak{su} $$ su (2) × $$ \mathfrak{su} $$ su (2) invariant $$ \mathcal{N} $$ N = 4 J-fold solutions. This family of AdS4 backgrounds can be uplifted to type IIB string theory and is holographically dual to the conformal manifold of a class of three-dimensional S-fold SCFTs obtained from the $$ \mathcal{N} $$ N = 4 T [U(N)] theory of Gaiotto-Witten. We find the spectrum of supergravity excitations of the AdS4 solutions and use it to study how the operator spectrum of the three-dimensional SCFT depends on the exactly marginal couplings.

scholarly journals Graviton propagator in a 2-parameter family of de Sitter breaking gauges

2019 ◽  
Vol 2019 (10) ◽  
Author(s):  
D. Glavan ◽  
S.P. Miao ◽  
T. Prokopec ◽  
R.P. Woodard
Keyword(s):  
Parameter Family ◽  
De Sitter ◽  
Graviton Propagator

scholarly journals On density with respect to equivalent measures

2010 ◽  
Vol 43 (1) ◽  
Author(s):  
Artur Bartoszewicz ◽  
Małgorzata Filipczak ◽  
Tadeusz Poreda
Keyword(s):  
Metric Space ◽  
Borel Measure ◽  
Borel Subset ◽  
Density Point

AbstractIn the paper there is disscussed a notion of a density point of a Borel subset of a metric space with respect to a Borel measure

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