Unit Ball Research Articles

On tiling spherical triangles into quadratic subpatches

Various interpolation and approximation methods arising in several practical applications in geometric modeling deal, at a particular step, with the problem of computing suitable rational patches (of low degree) on the unit sphere. Therefore, we are concerned with the construction of a system of spherical triangular patches with prescribed vertices that globally meet along common boundaries. In particular, we investigate various possibilities for tiling a given spherical triangular patch into quadratically parametrizable subpatches. We revisit the condition that the existence of a quadratic parameterization of a spherical triangle is equivalent to the sum of the interior angles of the triangle being π, and then circumvent this limitation by studying alternative scenarios and present constructions of spherical macro-elements of the lowest possible degree. Applications of our method include algorithms relying on the construction of (interpolation) surfaces from prescribed rational normal vector fields.

Surface areas of equifacetal polytopes inscribed in the unit sphere 𝕊2

Surface areas of equifacetal polytopes inscribed in the unit sphere 𝕊2

Shortest closed curve to contain a sphere in its convex hull

AbstractWe show that in Euclidean 3‐space any closed curve which contains the unit sphere within its convex hull has length , and characterize the case of equality. This result generalizes the authors' recent solution to a conjecture of Zalgaller. Furthermore, for the analogous problem in dimensions, we include the estimate by Nazarov, which is sharp up to the constant .

On conformally flat minimal Legendrian submanifolds in the unit sphere

This paper is concerned with the study on an open problem of classifying conformally flat minimal Legendrian submanifolds in the $(2n+1)$ -dimensional unit sphere $\mathbb {S}^{2n+1}$ admitting a Sasakian structure $(\varphi,\,\xi,\,\eta,\,g)$ for $n\ge 3$ , motivated by the classification of minimal Legendrian submanifolds with constant sectional curvature. First of all, we completely classify such Legendrian submanifolds by assuming that the tensor $K:=-\varphi h$ is semi-parallel, which is introduced as a natural extension of $C$ -parallel second fundamental form $h$ . Secondly, such submanifolds have also been determined under the condition that the Ricci tensor is semi-parallel, generalizing the Einstein condition. Finally, as direct consequences, new characterizations of the Calabi torus are presented.

Linearization of holomorphic Lipschitz functions

AbstractLet and be complex Banach spaces with denoting the open unit ball of . This paper studies various aspects of the holomorphic Lipschitz space , endowed with the Lipschitz norm. This space consists of the functions in the intersection of the sets of Lipschitz mappings and of bounded holomorphic mappings, from to . Thanks to the Dixmier–Ng theorem, is indeed a dual space, whose predual shares linearization properties with both the Lipschitz‐free space and Dineen–Mujica predual of . We explore the similarities and differences between these spaces, and combine techniques to study the properties of the space of holomorphic Lipschitz functions. In particular, we get that contains a 1‐complemented subspace isometric to and that has the (metric) approximation property whenever has it. We also analyze when is a subspace of , and we obtain an analog of Godefroy's characterization of functionals with a unique norm preserving extension in the holomorphic Lipschitz context.

Structure of strongly proximinal subspaces

In this paper we extended some results of G. Godefroy, originally proved for c0 using its special geometry, to the class of complex Banach spaces X whose dual is isometric to L1(μ). We exhibit classes of complex Banach spaces in which every strongly proximinal finite codimensional subspace is a finite dimensional perturbation of a M-ideal. We also show for this class, when the unit ball has an extreme point, if every proximinal subspace of codimension one is strongly proximinal, then the space is isometric to the k-dimensional space with the maximum norm, ℓ∞(k).

Von Neumann algebras of analytic functions on the unit ball

This paper is an attempt to answer the challenge posed by Ma and Zhu in [3]. We provide examples of von Neumann algebras of holomorphic functions over the unit disc as well as over the unit ball. In the end, we introduce a novel multiplication operation on the set of all holomorphic functions over the unit disc, ensuring that this collection forms a ⁎-algebra of analytic functions over D.

Target Tracking Systems on a Sphere With Topographic Information.

We consider the target tracking problem on a sphere with topographic structure. For a given moving target on the unit sphere, we suggest a double-integrator autonomous system of multiple agents that track the given target under the topographic influence. Through this dynamic system, we can obtain a control design for target tracking on the sphere and the adapted topographic data provides an efficient agent trajectory. The topographic information, described as a form of friction in the double-integrator system, affects the velocity and acceleration of the target and agents. The target information required by the tracking agents consists of position, velocity, and acceleration. We can obtain practical rendezvous results when agents utilize only target position and velocity information. If the acceleration data of the target is accessible, we can get the complete rendezvous result using an additional control term in the form of the Coriolis force. We provide mathematically rigorous proofs for these results and present numerical experiments that can be visually confirmed.

Symmetry breaking and instability for semilinear elliptic equations in spherical sectors and cones

We consider semilinear elliptic equations with mixed boundary conditions in spherical sectors inside a cone. The aim of the paper is to show that a radial symmetry result of Gidas-Ni-Nirenberg type for positive solutions does not hold in general nonconvex cones. This symmetry breaking result is achieved by studying the Morse index of radial positive solutions and analyzing how it depends on the domain D on the unit sphere which spans the cone. In particular it is proved that the Neumann eigenvalues of the Laplace Beltrami operator on D play a role in computing the Morse index. A similar breaking of symmetry result is obtained for the positive solutions of the critical Neumann problem in the whole unbounded cone. In this case it is proved that the standard bubbles, which are the only radial solutions, become unstable for a class of nonconvex cones.

Analytic representations of backward shifts on weighted Bergman and Dirichlet spaces

The backward shifts (the adjoints of multiplication by z) on weighted Bergman spaces of the unit disk are important in operator theory. For example, Agler [1] shows their restrictions to invariant subspaces of vector-valued weighted Bergman spaces are model operators for hypercontractions. We present analytic representations of backward shifts on weighted Bergman spaces and Dirichlet type spaces. Since these analytic representations are not space specific, it is interesting to study these operators on Hilbert spaces of holomorprhic functions on the unit disk. We initiate this study by developing several properties of these operators. In particular, we obtain simple proofs and extend the invariant subspace theorem of shift plus Volterra operator on the Hardy space by Čučković and Paudyal [17]. Furthermore, we describe invariant subspaces of shift plus Volterra operator on the Bergman space by relating them to the invariant subspaces of the Dirichlet shift. Analytic representations of the tuple of backward shifts on weighted Bergman spaces of the unit ball are also discussed.

FIRST EIGENVALUE CHARACTERISATION OF CLIFFORD HYPERSURFACES AND VERONESE SURFACES

Abstract We give a sharp estimate for the first eigenvalue of the Schrödinger operator $L:=-\Delta -\sigma $ which is defined on the closed minimal submanifold $M^{n}$ in the unit sphere $\mathbb {S}^{n+m}$ , where $\sigma $ is the square norm of the second fundamental form.

Banach spaces with small weakly open subsets of the unit ball and massive sets of Daugavet and Δ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Delta $$\\end{document}-points

We prove that there exists an equivalent norm ·\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\left| \\left| \\left| \\cdot \\right| \\right| \\right| $$\\end{document} on L∞[0,1]\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L_\\infty [0,1]$$\\end{document} with the following properties: The unit ball of (L∞[0,1],·)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(L_\\infty [0,1],\\left| \\left| \\left| \\cdot \\right| \\right| \\right| )$$\\end{document} contains non-empty relatively weakly open subsets of arbitrarily small diameter;The set of Daugavet points of the unit ball of (L∞[0,1],·)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(L_\\infty [0,1],\\left| \\left| \\left| \\cdot \\right| \\right| \\right| )$$\\end{document} is weakly dense;The set of ccw Δ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Delta $$\\end{document}-points of the unit ball of (L∞[0,1],·)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(L_\\infty [0,1],\\left| \\left| \\left| \\cdot \\right| \\right| \\right| )$$\\end{document} is norming. We also show that there are points of the unit ball of (L∞[0,1],·)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(L_\\infty [0,1],\\left| \\left| \\left| \\cdot \\right| \\right| \\right| )$$\\end{document} which are not Δ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Delta $$\\end{document}-points, meaning that the space (L∞[0,1],·)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(L_\\infty [0,1],\\left| \\left| \\left| \\cdot \\right| \\right| \\right| )$$\\end{document} fails the diametral local diameter 2 property. Finally, we observe that the space (L∞[0,1],·)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(L_\\infty [0,1],\\left| \\left| \\left| \\cdot \\right| \\right| \\right| )$$\\end{document} provides both alternative and new examples that illustrate the differences between the various diametral notions for points of the unit ball of Banach spaces.

Reproducing property of bounded linear operators and kernel regularized least square regressions

We consider bounded linear operators from the view of functional reproducing property. For some bounded linear operators associated with orthogonal polynomials we define an inner product space associated with a kernel constructed with orthogonal polynomials, show that it is a functional reproducing kernel Hilbert space (FRKHS) associated with these bounded linear operators and give decay rate for the best FRKHS approximation with a [Formula: see text]-functional associated with the FRKHS. On this basis, we provide a learning rate for kernel regularized regression whose hypothesis space is the defined FRKHS. As applications, we define some concrete FRKHSs associated with polynomial operators such as the Bernstein–Durrmeyer operators, the de la Vallée Poussin operators on both the unit sphere [Formula: see text] and the unit ball [Formula: see text]. We show that these polynomial operators have reproducing property with respect to the corresponding concrete FRKHSs and show the learning rate for the kernel regularized regression. In short, we provide a way of constructing FRKHS operators with Fourier multipliers and show a learning framework from the view of operator approximation.

Lipschitz continuity of the solutions to the Dirichlet problems for the invariant Laplacians

This short note is motivated by an attempt to understand the distinction between the Laplace operator and the hyperbolic Laplacian on the unit ball of Rn, regarding the Lipschitz continuity of the solutions to the corresponding Dirichlet problems.We investigate the Dirichlet problem{Δϑu=0, in Bn,u=ϕ, on Sn−1, whereΔϑ:=(1−|x|2){1−|x|24Δ+ϑ∑j=1nxj∂∂xj+ϑ(n2−1−ϑ)I}.We show that: (i) when ϑ>0, the Lipschitz continuity of boundary data always implies the Lipschitz continuity of the solutions; (ii) when ϑ=0, there exists a Lipschitz continuous function ϕ:Sn−1→R such that the solution is not Lipschitz continuous; (iii) when ϑ<0, the solution is definitely not Lipschitz continuous unless ϕ≡0.

A Reliable Iteration Algorithm for One-Bit Compressive Sensing on the Unit Sphere

A Reliable Iteration Algorithm for One-Bit Compressive Sensing on the Unit Sphere

Tent Carleson measures for Hardy spaces

We completely characterize those positive Borel measures μ on the unit ball Bn such that the Carleson embedding from Hardy spaces Hp into the tent-type spaces Tsq(μ) is bounded, for all possible values of 0<p,q,s<∞.

A note on extreme points of the unit ball of Hardy-Lorentz spaces

We show that inner functions are extreme points of the unit ball of the Hardy-Lorentz space H ( Λ ( φ ) ) H(\Lambda (\varphi )) , for Λ ( φ ) \Lambda (\varphi ) a Lorentz space with φ \varphi strictly increasing and strictly concave.

On extreme points and representer theorems for the Lipschitz unit ball on finite metric spaces

AbstractIn this note, we provide a characterization for the set of extreme points of the Lipschitz unit ball in a specific vectorial setting. While the analysis of the case of real-valued functions is covered extensively in the literature, no information about the vectorial case has been provided up to date. Here, we aim at partially filling this gap by considering functions mapping from a finite metric space to a strictly convex Banach space that satisfy the Lipschitz condition. As a consequence, we present a representer theorem for such functions. In this setting, the number of extreme points needed to express any point inside the ball is independent of the dimension, improving the classical result from Carathéodory.

PAL-SLAM2: Visual and visual–inertial monocular SLAM for panoramic annular lens

This paper presents PAL-SLAM2, a visual and visual–inertial monocular simultaneous localization and mapping (SLAM) system for a panoramic annular lens (PAL) with an ultra-hemispherical field of view (FoV), overcoming the limitations of traditional frameworks in handling fast turns, nighttime conditions and rapid lighting changes. The system incorporates modules for initialization, tracking, local mapping, loop and map merging. To fully exploit information from the negative space (z<0), keypoints are projected onto the unit sphere for visual initialization and tightly-coupled visual–inertial optimization. Leveraging a multimap strategy, the feature-based PAL-SLAM2 is capable of detecting unidirectional and bidirectional public areas using PAL images, thereby enhancing the performance of loop correction and map merging. Testing indicates that it achieves an average accuracy of 7.1 cm on the PALVIO indoor dataset, surpassing similar state-of-the-art frameworks. Furthermore, we have collected a large-scale dataset comprising over 120,000 images with corresponding inertial measurement unit (IMU) data, demonstrating PAL-SLAM2’s robustness under challenging outdoor conditions. Relying solely on visual and inertial inputs, our system is particularly suited for environments where Global Navigation Satellite System (GNSS) signals are frequently obstructed, such as indoor spaces or dense urban areas. The dataset can be accessed at: https://github.com/wwendy233/Plaza-Dataset.

On Extension of Norm-Additive Maps Between the Positive Unit Spheres of $$\ell _q(\ell _p)$$

On Extension of Norm-Additive Maps Between the Positive Unit Spheres of $$\ell _q(\ell _p)$$