Unit Ball Research Articles

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.

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 $$\ell _q(\ell _p)$$

Almost Ricci–Yamabe soliton on contact metric manifolds

PurposeThis paper aims to study almost Ricci–Yamabe soliton in the context of certain contact metric manifolds.Design/methodology/approachThe paper is designed as follows: In Section 3, a complete contact metric manifold with the Reeb vector field ξ as an eigenvector of the Ricci operator admitting almost Ricci–Yamabe soliton is considered. In Section 4, a complete K-contact manifold admits gradient Ricci–Yamabe soliton is studied. Then in Section 5, gradient almost Ricci–Yamabe soliton in non-Sasakian (k, μ)-contact metric manifold is assumed. Moreover, the obtained result is verified by constructing an example.FindingsWe prove that if the metric g admits an almost (α, β)-Ricci–Yamabe soliton with α ≠ 0 and potential vector field collinear with the Reeb vector field ξ on a complete contact metric manifold with the Reeb vector field ξ as an eigenvector of the Ricci operator, then the manifold is compact Einstein Sasakian and the potential vector field is a constant multiple of the Reeb vector field ξ. For the case of complete K-contact, we found that it is isometric to unit sphere S2n+1 and in the case of (k, μ)-contact metric manifold, it is flat in three-dimension and locally isometric to En+1 × Sn(4) in higher dimension.Originality/valueAll results are novel and generalizations of previously obtained results.

Sparse recovery from quadratic measurements with external field

Motivated by recent results in the statistical physics of spin glasses, we study the recovery of a sparse vector x0∈Sn−1, where Sn−1 denotes the n-dimensional unit sphere, ‖x0‖ℓ0=k<n, from m quadratic measurements of the form (1−λ)〈Ai,x0x0⊺〉+λ〈ci,x0〉 where Ai,ci have i.i.d Gaussian entries. This can be related to a constrained version of the 2-spin Hamiltonian with external field for which it was shown (in the absence of any structural constraint and in the asymptotic regime) in [1] that the geometry of the energy landscape becomes trivial above a certain threshold λ>λc∈(0,1). Building on this idea, we characterize the recovery of x0 as a function of λ∈[0,1]. We show that recovery of the vector x0 can be guaranteed as soon as m≳k2(1−λ)2/λ2∨k, λ>1/2 provided that this vector satisfies a sufficiently strong incoherence condition, thus retrieving the linear regime for an external field (1−λ)/λ≲k−1/2. A similar result (with a slightly deteriorating sample complexity) can be shown for weaker fields. Our proof relies on an interpolation between the linear and quadratic settings, as well as on standard convex geometry arguments.

Investigation of the solvability of boundary value problems for the nonlocal Poisson equation with periodic conditions in circular domains

In this paper, boundary value problems with transformed arguments are studied in the unit ball. The transformation of the arguments is specified using the involution type mapping. These mappings participate both in the equation and in the boundary conditions. The equation under consideration is a nonlocal analog of the Poisson equation. Boundary conditions are specified as a relationship between the value of the desired function in the upper hemisphere and the value of the lower hemisphere. These conditions generalize the known periodic conditions for spherical regions. When studying boundary value problems, the properties of involutive mappings are used. The problems under consideration are solved by reducing them to analogues of boundary value problems with periodic conditions for the classical Poisson equation. Using well-known statements for periodic problems for the problems under consideration, theorems on the existence and uniqueness of solutions are proved. Exact conditions for the solvability of the problems under study are found. Spectral questions related to periodic problems are also studied. Eigenfunctions and eigenvalues of these problems are found.

On the classification of ancient solutions to curvature flows on the sphere

We consider the evolution of hypersurfaces on the unit sphere $\mathbb{S}^{n+1}$ by smooth functions of the Weingarten map. We introduce the notion of `quasi-ancient' solutions for flows that do not admit non-trivial, convex, ancient solutions. Such solutions are somewhat analogous to ancient solutions for flows such as the mean curvature flow, or 1-homogeneous flows. The techniques presented here allow us to prove that any convex, quasi-ancient solution of a curvature flow which satisfies a backwards in time uniform bound on mean curvature must be stationary or a family of shrinking geodesic spheres. The main tools are geometric, employing the maximum principle, a rigidity result in the sphere and an Alexandrov reflection argument. We emphasize that no homogeneity or convexity/concavity restrictions are placed on the speed, though we do also offer a short classification proof for several such restricted cases.

Quantum ChemicalStudyof a Dihydroazulene‐SpiropyranMulti‐StateMulti‐FunctionalMolecularSwitch

Quantum chemistry methods have been enacted to characterize the second‐order nonlinear optical (NLO) properties of a multi‐state molecular switch combining dihydroazulene and spiropyran units. They have demonstrated that the first hyperpolarizabilities constitute a novel output signal from the viewpoint of molecular logic, owing to the contrasts of NLO responses between the different forms of the dyad. On the one hand, the DFT calculations demonstrate that switching either unit has only a negligible impact on the geometrical parameters of the other one, as well as on the thermodynamics of their transformations. Similarly, as determined by time‐dependent DFT calculations, the UV/vis absorption spectra are in good approximation a superposition of those from the parent units. On the other hand, this additivity trend is not consistently observed for the NLO responses. However, an interpretation of these NLO responses has been proposed by using the unit sphere representation of the first hyperpolarizability tensors to address their orientational aspects as well as by resorting to the few‐state approximation and the missing state analysis to highlight the localized or charge‐transfer nature of the dominant excited states.

Tensor Tomography on Negatively Curved Manifolds of Low Regularity

We prove solenoidal injectivity for the geodesic X-ray transform of tensor fields on simple Riemannian manifolds with C1,1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C^{1,1}$$\\end{document} metrics and non-positive sectional curvature. The proof of the result rests on Pestov energy estimates for a transport equation on the non-smooth unit sphere bundle of the manifold. Our low regularity setting requires keeping track of regularity and making use of many functions on the sphere bundle having more vertical than horizontal regularity. Some of the methods, such as boundary determination up to gauge and regularity estimates for the integral function, have to be changed substantially from the smooth proof. The natural differential operators such as covariant derivatives are not smooth.

The Constants to Measure the Differences Between Isosceles and α-β Orthogonalities

In this paper, by combining the isosceles orthogonality and α−β orthogonality of Banach spaces, we first introduce a new geometric constant. We demonstrate some basic properties about it, such as calculating its value in the common norm spaces. Moreover, the necessary and sufficient conditions for the new constant to characterize Hilbert spaces are given. Finally, only consider the points on the unit sphere, we introduce another new geometric constant and some basic properties are also obtained.

Unconditional bases and Daugavet renormings

We prove that the space ℓ2 – and more generally every infinite dimensional Banach space with an unconditional Schauder basis – can be renormed to have a Daugavet point. As a starting point, we introduce a new diametral notion for points of the unit sphere of Banach spaces, that complements the notion of Δ-points, but is weaker than the notion of Daugavet points. We show that this notion can be used to provide another geometric characterization of the Daugavet property, as well as to recover – and even to provide new – results about Daugavet points in various contexts such as absolute sums of Banach spaces or projective tensor products. Then, we observe that the study of this notion naturally leads to new renorming ideas in the context, and combine these with previous techniques from the literature to show that Daugavet points may exist even in super-reflexive spaces.

Learning Representations on the Unit Sphere: Investigating Angular Gaussian and Von Mises-Fisher Distributions for Online Continual Learning

We use the maximum a posteriori estimation principle for learning representations distributed on the unit sphere. We propose to use the angular Gaussian distribution, which corresponds to a Gaussian projected on the unit-sphere and derive the associated loss function. We also consider the von Mises-Fisher distribution, which is the conditional of a Gaussian in the unit-sphere. The learned representations are pushed toward fixed directions, which are the prior means of the Gaussians; allowing for a learning strategy that is resilient to data drift. This makes it suitable for online continual learning, which is the problem of training neural networks on a continuous data stream, where multiple classification tasks are presented sequentially so that data from past tasks are no longer accessible, and data from the current task can be seen only once. To address this challenging scenario, we propose a memory-based representation learning technique equipped with our new loss functions. Our approach does not require negative data or knowledge of task boundaries and performs well with smaller batch sizes while being computationally efficient. We demonstrate with extensive experiments that the proposed method outperforms the current state-of-the-art methods on both standard evaluation scenarios and realistic scenarios with blurry task boundaries. For reproducibility, we use the same training pipeline for every compared method and share the code at https://github.com/Nicolas1203/ocl-fd.