N-dimensional Unit Ball Research Articles

New classes of p-adic evolution equations and their applications

In this article, we study new classes of evolution equations in the p-adic context. We establish rigorously that the fundamental solutions of the homogeneous Cauchy problem, naturally associated to these equations, are transition density functions of some strong Markov processes {mathfrak {X}} with state space the n-dimensional p-adic unit ball ({mathbb {Z}}_{p}^{n}). We introduce a family of operators {T_{t}}_{tge 0} (obtained explicitly) that determine a Feller semigroup on C_{0}({mathbb {Z}}_{p}^{n}). Also, we study the asymptotic behavior of the survival probability of a strong Markov processes {mathfrak {X}} on a ball B_{-m}^{n}subset {mathbb {Z}}_{p}^{n}, min {mathbb {N}}. Moreover, we study the inhomogeneous Cauchy problem and we will show that its mild solution is associated with the mentioned above Feller semigroup.

Weighted Hardy Spaces of Quasiconformal Mappings

We study the integral characterizations of weighted Hardy spaces of quasiconformal mappings on the n-dimensional unit ball using the weight \((1-r)^{n-2 + \alpha }\). We extend the known results for univalent functions on the unit disk. Some of our results are new even in the unweighted setting for quasiconformal mappings.

Blow up of solutions for a Parabolic-Elliptic chemotaxis system with gradient dependent chemotactic coefficient

We consider a Parabolic-Elliptic system of PDE’s with a chemotactic term in a N-dimensional unit ball describing the behavior of the density of a biological species “u” and a chemical stimulus “v.” The system includes a nonlinear chemotactic coefficient depending of “” i.e. the chemotactic term is given in the form for a positive constant χ when v satisfies the poisson equation We study the radially symmetric solutions under the assumption in the initial mass For χ large enough, we present conditions in the initial data, such that any regular solution of the problem blows up at finite time.

On extremals for the Trudinger-Moser inequality with vanishing weight in the N-dimensional unit ball

On extremals for the Trudinger-Moser inequality with vanishing weight in the N-dimensional unit ball

Weakly biharmonic maps from the ball to the sphere

The aim of this paper is to investigate the existence of proper, weakly biharmonic maps within a family of rotationally symmetric maps $$u_a : B^n \rightarrow {\mathbb {S}}^n$$, where $$B^n$$ and $${\mathbb {S}}^n$$ denote the Euclidean n-dimensional unit ball and sphere respectively. We prove that there exists a proper, weakly biharmonic map $$u_a$$ of this type if and only if $$n=5$$ or $$n=6$$. We shall also prove that these critical points are unstable.

English

The purpose of this short note is to illustrate the utility of (semi-) dendroidal objects in describing certain 'up-to-homotopy' operads. Specifically, we exhibit a semi-dendroidal space satisfying the Segal condition, whose evaluation at a k-corolla is the space of ordered configurations of k points in the n-dimensional unit ball.

Algebras of Toeplitz Operators on the n-Dimensional Unit Ball

We study $C^*$-algebras generated by Toeplitz operators acting on the standard weighted Bergman space $\mathcal{A}_{\lambda}^2(\mathbb{B}^n)$ over the unit ball $\mathbb{B}^n$ in $\mathbb{C}^n$. The symbols $f_{ac}$ of generating operators are assumed to be of a certain product type. By choosing $a$ and $c$ in different function algebras $\mathcal{S}_a$ and $\mathcal{S}_c$ over lower dimensional unit balls $\mathbb{B}^{\ell}$ and $\mathbb{B}^{n-\ell}$, respectively, and by assuming the invariance of $a\in \mathcal{S}_a$ under some torus action we obtain $C^*$-algebras $\boldsymbol{\mathcal{T}}_{\lambda}(\mathcal{S}_a, \mathcal{S}_c)$ whose structural properties can be described. In the case of $k$-quasi-radial functions $\mathcal{S}_a$ and bounded uniformly continuous or vanishing oscillation symbols $\mathcal{S}_c$ we describe the structure of elements from the algebra $\boldsymbol{\mathcal{T}}_{\lambda}(\mathcal{S}_a, \mathcal{S}_c)$, derive a list of irreducible representations of $\boldsymbol{\mathcal{T}}_{\lambda}(\mathcal{S}_a, \mathcal{S}_c)$, and prove completeness of this list in some cases. Some of these representations originate from a `quantization effect', induced by the representation of $\mathcal{A}_{\lambda}^2(\mathbb{B}^n)$ as the direct sum of Bergman spaces over a lower dimensional unit ball with growing weight parameter. As an application we derive the essential spectrum and index formulas for matrix-valued operators.

The Essential Spectrum of Toeplitz Operators on the Unit Ball

In this paper we study the Fredholm properties of Toeplitz operators acting on weighted Bergman spaces $$A^p_{\nu }(\mathbb {B}^n)$$ , where $$p \in (1,\infty )$$ and $$\mathbb {B}^n \subset \mathbb {C}^n$$ denotes the n-dimensional open unit ball. Let f be a continuous function on the Euclidean closure of $$\mathbb {B}^n$$ . It is well-known that then the corresponding Toeplitz operator $$T_f$$ is Fredholm if and only if f has no zeros on the boundary $$\partial \mathbb {B}^n$$ . As a consequence, the essential spectrum of $$T_f$$ is given by the boundary values of f. We extend this result to all operators in the algebra generated by Toeplitz operators with bounded symbol (in a sense to be made precise down below). The main ideas are based on the work of Suarez et al. (Integral Equ Oper Theory 75:197–233, 2013, Indiana Univ Math J 56(5):2185–2232, 2007) and limit operator techniques coming from similar problems on the sequence space $$\ell ^p(\mathbb {Z})$$ (Hagger et al. in J Math Anal Appl 437(1):255–291, 2016; Lindner and Seidel in J Funct Anal 267(3):901–917, 2014; Rabinovich et al. Integral Equ Oper Theory 30(4): 452–495, 1998 and references therein).

Area Bounds for Minimal Surfaces that Pass Through a Prescribed Point in a Ball

Let $${\Sigma}$$ be a k-dimensional minimal submanifold in the n-dimensional unit ball B n which passes through a point $${y \in B^{n}}$$ and satisfies $${\partial \Sigma \subset \partial B^{n}}$$ . We show that the k-dimensional area of $${\Sigma}$$ is bounded from below by $${{|B^{k}| (1-|y|^{2})}^{\frac{k}{2}}}$$ . This settles a question left open by the work of Alexander and Osserman in 1973.

Series associated to some expressions involving the volume of the unit ball and applications

The aim of this paper is to construct asymptotic series associated to some expressions involving the volume of the n-dimensional unit ball. New refinements and sharpenings of some old and recent inequalities on the volume of the n-dimensional unit ball are presented.

An Equivalence Between the Dirichlet and the Neumann Problem for the Laplace Operator

We give a representation of the solution of the Neumann problem for the Laplace operator on the n-dimensional unit ball in terms of the solution of an associated Dirichlet problem. The representation is extended to other operators besides the Laplacian, to smooth simply connected planar domains, and to the infinite-dimensional Laplacian on the unit ball of an abstract Wiener space, providing in particular an explicit solution for the Neumann problem in this case. As an application, we derive an explicit formula for the Dirichlet-to-Neumann operator, which may be of independent interest.

Approximation and convergence of the intrinsic volume

We introduce a modification of the classic notion of intrinsic volume using persistence moments of height functions. Evaluating the modified first intrinsic volume on digital approximations of a compact body with smoothly embedded boundary in Rn, we prove convergence to the first intrinsic volume of the body as the resolution of the approximation improves. We have weaker results for the other modified intrinsic volumes, proving they converge to the corresponding intrinsic volumes of the n-dimensional unit ball.

Maps Close to Identity and Universal Maps in the Newhouse Domain

Given an n-dimensional C r-diffeomorphism g, its renormalized iteration is an iteration of g, restricted to a certain n-dimensional ball and taken in some C r-coordinates in which the ball acquires radius 1. We show that for any r ≥ 1 the renormalized iterations of C r-close to identity maps of an n-dimensional unit ball B n (n ≥ 2) form a residual set among all orientation-preserving C r-diffeomorphisms B n→ R n. In other words, any generic n-dimensional dynamical phenomenon can be obtained by iterations of C r-close to identity maps, with the same dimension of the phase space. As an application, we show that any C r-generic two-dimensional map that belongs to the Newhouse domain (i.e., it has a so-called wild hyperbolic set, so it is not uniformly-hyperbolic, nor uniformly partially-hyperbolic) and that neither contracts, nor expands areas, is C r-universal in the sense that its iterations, after an appropriate coordinate transformation, C r-approximate every orientation-preserving two-dimensional diffeomorphism arbitrarily well. In particular, every such universal map has an infinite set of coexisting hyperbolic attractors and repellers.

Elliptic problems on the ball endowed with Funk-type metrics

We study Sobolev spaces on the n-dimensional unit ball Bn(1) endowed with a parameter-depending Finsler metric Fa, a∈[0,1], which interpolates between the Klein metric (a=0) and Funk metric (a=1), respectively. We show that the standard Sobolev space defined on the Finsler manifold (Bn(1),Fa) is a vector space if and only if a∈[0,1). Furthermore, by exploiting variational arguments, we provide non-existence and existence results for sublinear elliptic problems on (Bn(1),Fa) involving the Finsler–Laplace operator whenever a∈[0,1).

Integration and polar coordinates

A simple proof of the formula for the integration of radial functions on ℝN, N ≥ 2, is given. As an application, the volume of the N-dimensional unit ball is computed.

On the structure of commutative Banach algebras generated by Toeplitz operators on the unit ball. Quasi-elliptic case. I: Generating subalgebras

Extending recent results in [3] to the higher dimensional setting n⩾3 we provide a further step in the structural analysis of a class of commutative Banach algebras generated by Toeplitz operators on the standard weighted Bergman space over the n-dimensional complex unit ball. The algebras Bk(h) under study are subordinated to the quasi-elliptic group of automorphisms of Bn and in terms of their generators they were described in [23]. We show that Bk(h) is generated in fact by an essentially smaller set of operators, i.e., the Toeplitz operators with k-quasi-radial symbols and a finite set of Toeplitz operators with “elementary” k-quasi-homogeneous symbols. Then we analyze the structure of the commutative subalgebras corresponding to these two types of generating symbols. In particular, we describe spectra, joint spectra, maximal ideal spaces and the Gelfand transform.

Geometric Characterization of Weyl’s Discrepancy Norm in Terms of Its n-Dimensional Unit Balls

Weyl’s discrepancy measure induces a norm on ℝn which shows a monotonicity and a Lipschitz property when applied to differences of index-shifted sequences. It turns out that its n-dimensional unit ball is a zonotope that results from a multiple sheared projection from the (n+1)-dimensional hypercube which can be interpreted as a discrete differentiation. This characterization reveals that this norm is the canonical metric between sequences of differences of values from the unit interval in the sense that the n-dimensional unit ball of the discrepancy norm equals the space of such sequences.

Partitions by congruent sets and optimal positions

AbstractLet X be a metrizable space with a continuous group or semi-group action G. Let D be a non-empty subset of X. Our problem is how to choose a fixed number of sets in {g−1D; g∈G}, say σ−1D with σ∈τ, to maximize the cardinality of the partition ℙ({σ−1D; σ∈τ}) generated by them. Let An infinite subset Σ of G is called an optimal position of the triple (X,G,D) if holds for any k=1,2,… and τ⊂Σ with #τ=k. In this paper, we discuss examples of the triple (X,G,D) admitting or not admitting an optimal position. Let X=G=ℝn (n≥1) , where the action g∈G to x∈X is the translation x−g. If D is the n-dimensional unit ball, then holds and the triple (X,G,D) admits an optimal position. In fact, if n≥2 and Σ is an infinite subset of G such that for some δ with 0<δ<1 , Σ⊂{x∈ℝn ; ‖x‖=δ}, and that any subset of Σ with cardinality n+1 is not on a hyperplane, then Σ is an optimal position of the triple (X,G,D) . We have determined the primitive factor of the uniform sets coming from these optimal positions. We also show that in the above setting with n=2 and the unit square D′ in place of the unit disk D, the maximal pattern complexity is unchanged and p*X,G,D′(k)=k2 −k+2 , but there is no optimal position.

A geometric interpretation of Ungar's addition and of gyration in the hyperbolic plane

We present a geometric interpretation of the operation a ⊕ b and the gyration on the unit-disc as defined by A.A. Ungar. Using this geometric interpretation we show that the two known generalizations to the n-dimensional unit ball are identical. The interpretation in the plane leads us to the notion of outer-median of a triangle and we discuss some possible properties of this median.

A Note on Noncommutative Interpolation

AbstractIn this paper we formulate and solve Nevanlinna-Pick and Carathéodory type problems for tensor algebras with data given on the N-dimensional operator unit ball of a Hilbert space. We develop an approach based on the displacement structure theory.