scholarly journals Realizing arbitrary $d$-dimensional dynamics by renormalization of $C^d$-perturbations of identity

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Bassam Fayad ◽  
Maria Saprykina
Keyword(s):  
Unit Ball ◽  
Dimensional Unit ◽  
Return Dynamics ◽  
Perturbation Of Identity

<p style='text-indent:20px;'>Any <inline-formula><tex-math id="M3">\begin{document}$ C^d $\end{document}</tex-math></inline-formula> conservative map <inline-formula><tex-math id="M4">\begin{document}$ f $\end{document}</tex-math></inline-formula> of the <inline-formula><tex-math id="M5">\begin{document}$ d $\end{document}</tex-math></inline-formula>-dimensional unit ball <inline-formula><tex-math id="M6">\begin{document}$ {\mathbb B}^d $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M7">\begin{document}$ d\geq 2 $\end{document}</tex-math></inline-formula>, can be realized by renormalized iteration of a <inline-formula><tex-math id="M8">\begin{document}$ C^d $\end{document}</tex-math></inline-formula> perturbation of identity: there exists a conservative diffeomorphism of <inline-formula><tex-math id="M9">\begin{document}$ {\mathbb B}^d $\end{document}</tex-math></inline-formula>, arbitrarily close to identity in the <inline-formula><tex-math id="M10">\begin{document}$ C^d $\end{document}</tex-math></inline-formula> topology, that has a periodic disc on which the return dynamics after a <inline-formula><tex-math id="M11">\begin{document}$ C^d $\end{document}</tex-math></inline-formula> change of coordinates is exactly <inline-formula><tex-math id="M12">\begin{document}$ f $\end{document}</tex-math></inline-formula>.</p>

scholarly journals Algebras of Toeplitz Operators on the n-Dimensional Unit Ball

2018 ◽  
Vol 13 (2) ◽  
pp. 493-524 ◽  
Author(s):  
Wolfram Bauer ◽  
Raffael Hagger ◽  
Nikolai Vasilevski
Keyword(s):  
Unit Ball ◽  
Toeplitz Operators ◽  
Algebras Of Toeplitz Operators ◽  
Dimensional Unit

A KNOTTED MINIMAL TREE

1999 ◽  
Vol 01 (01) ◽  
pp. 71-86
Author(s):  
KRYSTYNA KUPERBERG
Keyword(s):  
Unit Ball ◽  
Three Dimensional ◽  
Minimal Tree ◽  
Dimensional Unit ◽  
Finite Set ◽  
Set Of Points

There is a finite set of points on the boundary of the three-dimensional unit ball whose minimal tree is knotted. This example answers a problem posed by Michael Freedman.

scholarly journals HOW LARGE IS THE SHADOW OF A SYMPLECTIC BALL?

2013 ◽  
Vol 05 (01) ◽  
pp. 87-119 ◽  
Author(s):  
ALBERTO ABBONDANDOLO ◽  
ROSTISLAV MATVEYEV
Keyword(s):  
Vector Space ◽  
Unit Ball ◽  
Orthogonal Projection ◽  
Real Dimension ◽  
Symplectic Embedding ◽  
Dimensional Unit ◽  
Dimensional Ball ◽  
Symplectic Vector Space

Consider the image of the 2n-dimensional unit ball by a symplectic embedding into the standard symplectic vector space of dimension 2n. Its 2k-dimensional shadow is its orthogonal projection onto a complex subspace of real dimension 2k. Is it true that the volume of this 2k-dimensional shadow is at least the volume of the unit 2k-dimensional ball? This statement is trivially true when k = n, and when k = 1 it is a reformulation of Gromov's non-squeezing theorem. Therefore, this question can be considered as a middle-dimensional generalization of the non-squeezing theorem. We investigate the validity of this statement in the linear, nonlinear and perturbative setting.

scholarly journals Toeplitz Operators on the Weighted Bergman Space over the Two-Dimensional Unit Ball

2015 ◽  
Vol 2015 ◽  
pp. 1-10 ◽  
Author(s):  
Alma García ◽  
Nikolai Vasilevski
Keyword(s):  
Unit Ball ◽  
Toeplitz Operators ◽  
Operator Algebras ◽  
Banach Algebras ◽  
Weighted Bergman Space ◽  
Two Dimensional ◽  
Spherical Coordinates ◽  
Extra Condition ◽  
Dimensional Unit ◽  
Commutative Banach Algebras

We extend the known results on commutative Banach algebras generated by Toeplitz operators with radial quasi-homogeneous symbols on the two-dimensional unit ball. Spherical coordinates previously used hid a possibility to detect an essentially wider class of symbols that can generate commutative Banach Toeplitz operator algebras. We characterize these new algebras describing their properties and, under a certain extra condition, construct the corresponding Gelfand theory.

Random circles in the d-dimensional unit ball

1989 ◽  
Vol 26 (2) ◽  
pp. 408-412 ◽  
Author(s):  
Fernando Affentranger
Keyword(s):  
Euclidean Space ◽  
Unit Ball ◽  
Dimensional Unit

This note gives the solution of the following problem concerning geometric probabilities. What is the probability p(Bd; 2) that the circumference determined by three points P, P1 and P2 chosen independently and uniformly at random in the interior of a d-dimensional unit ball Bd in Euclidean space Ed (d ≧ 2) is entirely contained in Bd? From our result we conclude that p(Bd; 2) →π /(3√3) as d →∞.

scholarly journals Two-weighted inequalities for the derivatives of holomorphic functions and Carleson measures on the unit ball

2000 ◽  
Vol 158 ◽  
pp. 107-131 ◽  
Author(s):  
Hyeonbae Kang ◽  
Hyungwoon Koo
Keyword(s):  
Unit Ball ◽  
Carleson Measure ◽  
Positive Measure ◽  
Holomorphic Functions ◽  
Carleson Measures ◽  
Weighted Inequalities ◽  
Higher Dimensional ◽  
Dimensional Unit ◽  
Derivatives Of

AbstractWe characterize those positive measure µ’s on the higher dimensional unit ball such that “two-weighted inequalities” hold for holomorphic functions and their derivatives. Characterizations are given in terms of the Carleson measure conditions. The results of this paper also distinguish between the fractional and the tangential derivatives.

Global structure of solutions for equations of Brezis–Nirenberg type on the unit ball

2001 ◽  
Vol 131 (3) ◽  
pp. 647-665 ◽  
Author(s):  
Yoshitsugu Kabeya ◽  
Eiji Yanagida ◽  
Shoji Yotsutani
Keyword(s):  
Boundary Condition ◽  
Unit Ball ◽  
Existence And Uniqueness ◽  
Three Dimensional ◽  
Critical Value ◽  
Global Structure ◽  
Radial Solutions ◽  
Structure Of Solutions ◽  
Dimensional Unit ◽  
Robin Condition

The Brezis–Nirenberg equation and the scalar field equation on the three-dimensional unit ball are studied. Under the Robin condition, we show the existence and uniqueness of radial solutions in a unified way. In particular, it is shown that the global structure of solutions changes qualitatively when a parameter in the boundary condition exceeds a certain critical value.

scholarly journals On the approximation of a ball by random polytopes

1994 ◽  
Vol 26 (4) ◽  
pp. 876-892 ◽  
Author(s):  
K.-H. Küfer
Keyword(s):  
Convex Hull ◽  
Unit Ball ◽  
Surface Area ◽  
Limiting Distribution ◽  
Spherically Symmetric ◽  
Random Vectors ◽  
Symmetric Distributions ◽  
Random Polytopes ◽  
Random Polytope ◽  
Dimensional Unit

Let be a sequence of independent and identically distributed random vectors drawn from the d-dimensional unit ball Bd and let Xn be the random polytope generated as the convex hull of a1,· ··, an. Furthermore, let Δ(Xn): = Vol (BdXn) be the volume of the part of the ball lying outside the random polytope. For uniformly distributed ai and 2 we prove that the limiting distribution of Δ(Xn)/Ε (Δ (Xn)) for n → ∞ (satisfies a 0–1 law. In particular, we show that Var for n → ∞. We provide analogous results for spherically symmetric distributions in Bd with regularly varying tail. In addition, we indicate similar results for the surface area and the number of facets of Xn.

Random circles in the d-dimensional unit ball

1989 ◽  
Vol 26 (02) ◽  
pp. 408-412 ◽  
Author(s):  
Fernando Affentranger
Keyword(s):  
Euclidean Space ◽  
Unit Ball ◽  
Dimensional Unit

This note gives the solution of the following problem concerning geometric probabilities. What is the probabilityp(Bd; 2) that the circumference determined by three pointsP, P1andP2chosen independently and uniformly at random in the interior of ad-dimensional unit ballBdin Euclidean spaceEd(d≧ 2) is entirely contained inBd? From our result we conclude thatp(Bd; 2) →π /(3√3) asd→∞.

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