Unit Ball Research Articles

SLERP-TVDRK (STVDRK) Methods for Ordinary Differential Equations on Spheres

We mimic the conventional explicit Total Variation Diminishing Runge–Kutta (TVDRK) schemes and propose a class of numerical integrators to solve differential equations on a unit sphere. Our approach utilizes the exponential map inherent to the sphere and employs spherical linear interpolation (SLERP). These modified schemes, named SLERP-TVDRK methods or STVDRK, offer improved accuracy compared to typical projective RK methods. Furthermore, they eliminate the need for any projection and provide a straightforward implementation. While we have successfully constructed STVDRK schemes only up to third-order accuracy, we explain the challenges in deriving STVDRK-r for r≥4\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$r\\ge 4$$\\end{document}. To showcase the effectiveness of our approach, we will demonstrate its application in solving the eikonal equation on the unit sphere and simulating p-harmonic flows using our proposed method.

Nodal solutions for a weighted (N,p)-Kirchhoff-type problem with double exponential growth non-linearity

This paper investigates the existence of sign-changing solutions for a logarithmic weighted Kirchhoff [Formula: see text]-Laplacian problem in the unit ball [Formula: see text] of [Formula: see text] where [Formula: see text]. The non-linearity of the equation is assumed to have double exponential growth, which is motivated by Trudinger–Moser-type inequalities. To prove the existence of these solutions, we employ techniques such as constrained minimization within the Nehari set, a quantitative deformation lemma and results from degree theory.

A class of Hausdorff–Berezin operators on the unit ball

Abstract The article introduces and studies Hausdorff–Berezin operators on the unit ball in a complex space. These operators are a natural generalization of the Berezin transform. In addition, the class of such operators contains, for example, the invariant Green potential, and some other operators of complex analysis. Sufficient and necessary conditions for boundedness in the space of p – integrable functions with Haar measure (invariant with respect to involutive automorphisms of the unit ball) are given. We also provide results on compactness of Hausdorff–Berezin operators in Lebesgue spaces on the unit ball. Such operators have previously been introduced and studied in the context of the unit disc in the complex plane. Present work is a natural continuation of these studies.

On the classification of function algebras on subvarieties of noncommutative operator balls

We study algebras of bounded noncommutative (nc) functions on unit balls of operator spaces (nc operator balls) and on their subvarieties. Considering the example of the nc unit polydisk we show that these algebras, while having a natural operator algebra structure, might not be the multiplier algebra of any reasonable nc reproducing kernel Hilbert space (RKHS). After examining additional subtleties of the nc RKHS approach, we turn to study the structure and representation theory of these algebras using function theoretic and operator algebraic tools. We show that the underlying nc variety is a complete invariant for the algebra of uniformly continuous nc functions on a homogeneous subvariety, in the sense that two such algebras are completely isometrically isomorphic if and only if the subvarieties are nc biholomorphic. We obtain extension and rigidity results for nc maps between subvarieties of nc operator balls corresponding to injective spaces that imply that a biholomorphism between homogeneous varieties extends to a biholomorphism between the ambient balls, which can be modified to a linear isomorphism. Thus, the algebra of uniformly continuous nc functions on nc operator balls, and even its restriction to certain subvarieties, completely determine the operator space up to completely isometric isomorphism.

Submanifolds with constant scalar curvature in space forms

Let [Formula: see text] be an [Formula: see text]-dimensional oriented compact submanifold with constant normalized scalar curvature [Formula: see text] in the space form [Formula: see text]. Denote by [Formula: see text] and [Formula: see text] the mean curvature and the Ricci curvature of [Formula: see text] respectively. By applying Cheng-Yau’s self-adjoint operator, we first prove that if [Formula: see text] is a hypersurface in a unit sphere, and [Formula: see text], then [Formula: see text] is totally umbilical. Furthermore, we investigate the submanifolds in [Formula: see text] with flat normal bundle satisfying [Formula: see text], and obtain a complete classification theorem.

A proof of Guo-Wang’s conjecture on the uniqueness of positive harmonic functions in the unit ball

A proof of Guo-Wang’s conjecture on the uniqueness of positive harmonic functions in the unit ball

On dual Kadec norms

Let (X,‖⋅‖) be a Banach space such that all w⁎-convergent sequences in the dual unit sphere SX⁎ are also norm convergent. Then the weak⁎ and norm topologies agree on SX⁎. By known results it implies that X has a renorming whose dual is locally uniformly rotund, hence also C1-Fréchet smooth. In particular, X is an Asplund space. Our results also lend an alternative proof of the celebrated Josefson-Nissenzweig theorem.

Numerical evaluation of orientation averages and its application to molecular physics.

In molecular physics, it is often necessary to average over the orientation of molecules when calculating observables, in particular when modeling experiments in the liquid or gas phase. Evaluated in terms of Euler angles, this is closely related to integration over two- or three-dimensional unit spheres, a common problem discussed in numerical analysis. The computational cost of the integration depends significantly on the quadrature method, making the selection of an appropriate method crucial for the feasibility of simulations. After reviewing several classes of spherical quadrature methods in terms of their efficiency and error distribution, we derive guidelines for choosing the best quadrature method for orientation averages and illustrate these with three examples from chiral molecule physics. While Gauss quadratures allow for achieving numerically exact integration for a wide range of applications, other methods offer advantages in specific circumstances. Our guidelines can also be applied to higher-dimensional spherical domains and other geometries. We also present a Python package providing a flexible interface to a variety of quadrature methods.

Area of Minimal Hypersurfaces in the Unit Sphere (II)

Area of Minimal Hypersurfaces in the Unit Sphere (II)

Equivariant insight into hyperspaces of convex bodies

Let cc(Rn) denote the hyperspace of all nonempty compact convex subsets of the Euclidean space Rn endowed with the Hausdorff metric, and let cc(Bn)={A∈cc(Rn)|A⊂Bn}, where Bn is the closed unit ball of Rn. Let cb(Rn) be the subset of cc(Rn) consisting of all convex bodies, i.e., of sets A∈cc(Rn) with non-empty interior. In this survey article we reveal fundamental properties of the natural action Aff(n)↷cb(Rn) of the affine group that leads to structure results for cb(Rn) and for several important subsets of it. Orbit spaces of cb(Rn) and its subsets by some appropriate subgroups G<Aff(n) are homeomorphic to such important objects like the Hilbert cube or the Banach-Mazur compacta. Among other hyperspaces studied here are cw(Rn) - the hyperspace of convex bodies of constant width, Bp - the hyperspace of convex bodies with smooth boundary of positive Gaussian curvature, L(n) - the hyperspace of convex bodies for which the Euclidean unit ball is the Löwner ellipsoid, cˇ(n) - the hyperspace of convex sets for which the Euclidean unit ball is the Chebyshev ball, etc. The paper contains also several new results concerning equivariant extension properties of the hyperspaces in question, as well as the homeomorphism type of the hyperspace cb(Bn)=cb(Rn)∩cc(Bn) and its orbit spaces. Equivariant extension properties of these hyperspaces are applied to provide a very short proof of the classical Grünbaum Conjecture from Convex Geometry. Along the paper seventeen open problems are raised.

Rigid Formation Control on a Sphere: A Heterogeneous System Approach.

We consider a heterogeneous multiagent system for tracking multiple targets with a rigid formation on a unit sphere, where the targets and chasing agents are governed by single- and double-integrator models, respectively. To make asymptotic rendezvous between agents and their corresponding targets, we use an autonomous system consisting of attraction forces and velocity alignments. If the target's position, velocity, and acceleration information are available, we derive a multiagent system for complete rendezvous and obtain its exponential convergence result. If we have only the location and velocity information of the targets, we provide an autonomous system for practical rendezvous and the corresponding mathematical analysis. To prove the main results, we employ frame-rotation-structure decomposition for the double-integrator model and the geometric properties of a rigid formation on a sphere. We also provide numerical simulations to confirm our mathematical results and apply them to multiagent dynamics with a rigid formation that patrols the boundary line for a certain area on the sphere.

Reverse Carleson Measures for Hardy Spaces in the Unit Ball

Reverse Carleson Measures for Hardy Spaces in the Unit Ball

Toeplitz operators with symmetric, alternating and anti-symmetric separately radial symbols on the unit ball

Toeplitz operators with symmetric, alternating and anti-symmetric separately radial symbols on the unit ball

Density Waves of Vortex Fluids on a Sphere

Abstract We aim to find one highly nontrivial example of the solutions to the vortex fluid dynamical equation on the unit sphere (S^2) and compare it with the numerical simulation. Since the rigid rotating steady solution for vortex fluids on S^2 is&amp;#xD;already known to us, we consider the perturbations above it. After decomposing the perturbation of the vortex number density and vortex charge density into spherical harmonics, we find that the perturbations are propagating waves. To&amp;#xD;be precise, the velocities for different single-mode vortex number density waves are all the same, while the velocities for single-mode vortex charge density waves depend on the degree of the spherical harmonics l, which is a signal of the existence of dispersion. Meanwhile, we find that there is a beat phenomenon for the positive (or negative) vortex density wave. Numerical simulation based on the canonical equations for point vortex model agrees perfectly with our theoretical calculations.

A note on limiting Calderon–Zygmund theory for transformed n$n$‐Laplace systems in divergence form

AbstractWe consider rotated ‐Laplace systems on the unit ball of the form where , , and for some . We prove that with estimates. As a corollary, we obtain that solutions to , where is the Hardy space, have a higher integrability, namely, .

Douglas-Rudin approximation theorem for operator-valued functions on the unit ball of [formula omitted]

Douglas and Rudin proved that any unimodular function on the unit circle T can be uniformly approximated by quotients of inner functions. We extend this result to the operator-valued unimodular functions defined on the boundary of the open unit ball of Cd. Our proof technique combines the spectral theorem for unitary operators with the Douglas-Rudin theorem in the scalar case to bootstrap the result to the operator-valued case. This yields a new proof and a significant generalization of Barclay's result (2009) [4] on the approximation of matrix-valued unimodular functions on T.

The Gromov–Wasserstein Distance Between Spheres

AbstractThe Gromov–Wasserstein distance—a generalization of the usual Wasserstein distance—permits comparing probability measures defined on possibly different metric spaces. Recently, this notion of distance has found several applications in Data Science and in Machine Learning. With the goal of aiding both the interpretability of dissimilarity measures computed through the Gromov–Wasserstein distance and the assessment of the approximation quality of computational techniques designed to estimate the Gromov–Wasserstein distance, we determine the precise value of a certain variant of the Gromov–Wasserstein distance between unit spheres of different dimensions. Indeed, we consider a two-parameter family $$\{d_{{{\text {GW}}}p,q}\}_{p,q=1}^{\infty }$$ { d GW p , q } p , q = 1 ∞ of Gromov–Wasserstein distances between metric measure spaces. By exploiting a suitable interaction between specific values of the parameters p and q and the metric of the underlying spaces, we are able to determine the exact value of the distance $$d_{{{\text {GW}}}4,2}$$ d GW 4 , 2 between all pairs of unit spheres of different dimensions endowed with their Euclidean distance and their uniform measure.

On questions of uniqueness for the vacant set of Wiener sausages and Brownian interlacements

We consider connectivity properties of the vacant set of (random) ensembles of Wiener sausages in Rd\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {R}}^d$$\\end{document} in the transient dimensions d≥3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$d \\ge 3$$\\end{document}. We prove that the vacant set of Brownian interlacements contains at most one infinite connected component almost surely. For finite ensembles of Wiener sausages, we provide sharp polynomial bounds on the probability that their vacant set contains at least 2 connected components in microscopic balls. The main proof ingredient is a sharp polynomial bound on the probability that several Brownian motions visit jointly all hemiballs of the unit ball while avoiding a slightly smaller ball.

On a subclass of alpha-quasi convex mappings

In this paper, we introduce the class of α-quasi convex mappings on the unit ball in a complex Banach space, with the aim of providing a unified approach to the Fekete-Szegö problem for subclasses of normalized starlike mappings and normalized quasi-convex mappings of type B on the open unit ball B of a complex Banach space. In particular, using some restrictive assumptions, we investigate inclusion relations among the classes of alpha-quasi convex mappings, starlike mappings and quasi-convex mappings of type B.

A probabilistic approach to Lorentz balls [formula omitted]

We develop a probabilistic approach to study the volumetric and geometric properties of unit balls Bq,1n of finite-dimensional Lorentz sequence spaces ℓq,1n. More precisely, we show that the empirical distribution of a random vector X(n) uniformly distributed on its volume normalized unit ball converges weakly to a compactly supported symmetric probability distribution with explicitly given density; as a consequence we obtain a weak Poincaré-Maxwell-Borel principle for any fixed number k∈N of coordinates of X(n) as n→∞. Moreover, we prove a central limit theorem for the largest coordinate of X(n), demonstrating a quite different behavior than in the case of the ℓqn balls, where a Gumbel distribution appears in the limit. Finally, we prove a Schechtman-Schmuckenschläger type result for the asymptotic volume of intersections of volume normalized ℓq,1n and ℓpn balls.