Unit Ball Research Articles

Linear Preserves of BP-quasi invertible elements in JB*-algebras

In this note, we study one of the main outcomes of the Russo-Dye Theorem of JB*-algebra: a linear operator that preserves Brown-Pedersen-quasi invertible elements between two JB*-algebras is characterized by a Jordan ∗-homomorphism. Earlier, in C*-setting of algebras, Russo and Dye gave a characterization of any linear operator that maps unitary elements into unitary elements; namely a Jordan ∗-homomorphism. Special sorts of linear preservers between C*-algebras and between JB*-triples were introduced by Burgos et al. As a result, if G is a linear operator between two JB*-algebras having non-empty sets of extreme points of the closed unit sphere that preserves extreme points, then there exists a Jordan ∗-homomorphism Φ which also preserves extreme points and characterizes the linear operator G. We also explore the connection between linear operators that strongly preserve Brown-Pedersen-quasi nvertible elements between two JB*-triples and the λ-property of both JB*-triples. Other geometric properties, such as extremally richness and the Bade property of two JB*-algebras or triples under linear preservers, are to be elaborated on in forthcoming research.

First eigenvalues of free boundary hypersurfaces in the unit ball along the inverse mean curvature flow

In this note, we consider the first nonzero eigenvalue λp,1 of the p-Laplacian on free boundary proper hypersurfaces in the unit ball evolving along the inverse mean curvature flow. We show that λp,1 is monotone decreasing along the flow. Using the convergence of free boundary disks in the unit ball, we give a lower bound of λp,1 of a free boundary disk type hypersurface in the unit ball.

Symmetry breaking for ground states of biharmonic NLS via Fourier extension estimates

We consider ground state solutions u ∈ H2(ℝN) of biharmonic (fourth-order) nonlinear Schrödinger equations of the form Δ2u+2aΔu+bu−∣u∣p−2u=0inℝN\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\Delta ^2}u + 2a\\Delta u + bu - |u{|^{p - 2}}u = 0\\,\\,\\,\\,{\\rm{in}}\\,\\,{\\mathbb{R}^N}$$\\end{document} with positive constants a, b > 0 and exponents 2 < p < 2*, where 2∗=2NN−4\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${2^ * } = {{2N} \\over {N - 4}}$$\\end{document} if N > 4 and 2* = ∞ if N ≤ 4. By exploiting a connection to the adjoint Stein–Tomas inequality on the unit sphere and by using trial functions due to Knapp, we prove a general symmetry breaking result by showing that all ground states u ∈ H2(ℝN) in dimension N ≥ 2 fail to be radially symmetric for all exponents 2<p<2N+2N−1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$2 < p < {{2N + 2} \\over {N - 1}}$$\\end{document} in a suitable regime of a, b > 0.As applications of our main result, we also prove symmetry breaking for a minimization problem with constrained L2-mass and for a related problem on the unit ball in ℝN subject to Dirichlet boundary conditions.

PLO3S: Protein LOcal Surficial Similarity Screening

The study of protein molecular surfaces enables to better understand and predict protein interactions. Different methods have been developed in computer vision to compare surfaces that can be applied to protein molecular surfaces. The present work proposes a method using the Wave Kernel Signature: Protein LOcal Surficial Similarity Screening (PLO3S). The descriptor of the PLO3S method is a local surface shape descriptor projected on a unit sphere mapped onto a 2D plane and called Surface Wave Interpolated Maps (SWIM). PLO3S allows to rapidly compare protein surface shapes through local comparisons to filter large protein surfaces datasets in protein structures virtual screening protocols.

The Slice Approximating Property and Figiel-Type Problem on Unit Spheres

The Slice Approximating Property and Figiel-Type Problem on Unit Spheres

Some Optimal Inequalities for Anti-invariant Submanifolds of the Unit Sphere

Some Optimal Inequalities for Anti-invariant Submanifolds of the Unit Sphere

On the approximation of the function on the unite sphere by the spherical harmonics

In this paper we discuss convergence and summability of the Fourier series of distributions in the domains where it coincides with smooth functions in eigenfunction expansions of the Laplace operator on the unite sphere. We consider representation of the distributions defined on the unit sphere by its Fourier-Laplace series by the spherical harmonics in different topologies. Mainly we study the Chesaro method of summation such a series

Екстремальні значення найкращих наближень лінійних комбінацій гармонічних функцій

Extreme problems and their practical applications have been under the scrutiny of mathematicians since ancient times. An important step in the development of extreme problems was made by P. L. Chebyshev, who in the 50s of the 19th century laid the foundations of a section of destructive function theory – the theory of approximation. A significant role of the formation of the theory of approximation of functions was played by Carl Weierstrass’s theorem on the convergence to zero of best approximations by polynomials of a continuous function. As is well known, Weierstrass’s theorem is not constructive – it does not contained estimates of the approach speed. Thanks to the work of D. Jackson, S. N. Bernstein, Vallee-Poussin and others, such estimates began to appear in works on approximation theory. At the same time, at the first stages of the development of the theory of approximation, approximations of individual functions were studied. That beginning of a new period, a dipper study of the deviation values of functions from their approximating polynomials, dates back to the 30s and 40s of the 20th century and is associated with the names of A. M. Kolmogorov, S. M. Nikolsky, J. Favard, N. I. Achieser, M. G. Crane and B. Nagy. Thanks to their works, the main emphasis in the theory of approximations is shifted to the study of the best approximations or other approximation characteristics of functions that have certain differential-difference or smoothness properties. In particular, in 1936, J. Favard calculated the exact values of the best uniform approximations by trigonometric polynomials of order no higher than n – 1 on classes of differentiable 2π-periodic functions, whose r-th (r – natural) derivatives are in a unit sphere of the space of essentially bounded functions. The problem of obtaining exact values of the best approximations in uniform and integral metrics for various functional compacts was in sight of many prominent mathematicians of the XX century. General issues related to the study of the best approximation functional: the existence of a polynomial of the best approximation, its characteristic properties, are destribed in detail in many monographs, in particular, for example, in the book by M. P. Korneichuk [1]. In the 80s and 90s of the XX century, O. I. Stepanets (see, [2, section III]) developed a new approach to the classification of periodic functions, which allowed for a fairly fine classification of extremely wide sets of periodic functions. At the same time, the results obtained for these classes are, on the one hand, general, and on the other hand, they give a number of new, hitherto unknown results that were impossible to obtain on previously known classes. Following the approaches to the requirements of function classification, we can consider a linear combination of function classes of a more complex nature. And then the problem of finding the exact values of the upper bounds of the best joint approximations will be reduced to the problem of the best approximation of this composite class corresponding to convolutions with the composite kernel.

Quantitative Steinitz theorem: A polynomial bound

AbstractThe classical Steinitz theorem states that if the origin belongs to the interior of the convex hull of a set , then there are at most points of whose convex hull contains the origin in the interior. Bárány, Katchalski, and Pach proved the following quantitative version of Steinitz's theorem. Let be a convex polytope in containing the standard Euclidean unit ball . Then there exist at most vertices of whose convex hull satisfies with . They conjectured that holds with a universal constant . We prove , the first polynomial lower bound on . Furthermore, we show that is not greater than .

The Group of Transformation Which Preserving Distance on Cuboctahedron and Truncated Octahedron Space

Minkowski geometry is a non-Euclidean geometry in a finite number of dimensions. In a Minkowski geometry the unit ball is a symmetric, convex closed set instead of the usual sphere in Euclidean space. In [14], it is shown that there are some geometries which unit spheres are cuboctahedron and truncated octahedron-which are Archimedean solids-, they are also Minkowski geometries. In geometry determining the group of isometries of a space with a metric is a fundamental problem. In this article we show that the group of isometries of the 3−dimensional spaces covered by CO − metric and TO − metric are the semi-direct product of Oh and T(3), where octahedral group Oh is the (Euclidean) symmetry group of the octahedron and T(3) is the group of all translations of the 3−dimensional space.

Sign-changing solutions for a weighted Schrödinger–Kirchhoff equation with double exponential nonlinearities growth

This work is devoted to the existence of least energy nodal solutions for a nonlocal weighted Schrödinger–Kirchhoff problem, under boundary Dirichlet condition in the unit ball [Formula: see text] of [Formula: see text]. The nonlinearity of the equation is assumed to have double exponential growth in view of Trudinger–Moser type inequalities. By using the constrained minimization in Nehari set, the quantitative deformation Lemma and degree theory results, we prove our existence result.

Interpolation on the unit hypersphere using the n-dimensional generalized Kuramoto model

Computer graphics, robotics, and physics are one of the many domains where interpolation on the unit sphere S n (often called a unit hypersphere or unit n-sphere) plays a crucial role. In this paper, we introduce a novel approach for achieving smooth and precise interpolation on the unit sphere S n−1 using the n-dimensional generalized Kuramoto model. The proposed algorithm finds the shortest and most direct path between two points on that non-Euclidean manifold. Our simulation results demonstrate that it achieves performance comparable to that of a Spherical Linear Interpolation algorithm. Also, the paper proposes the application of our algorithm in the interpolation of rotations that are presented in the form of four-dimensional data.

Turing Patterns in a $$p$$-Adic FitzHugh-Nagumo System on the Unit Ball

Turing Patterns in a $$p$$-Adic FitzHugh-Nagumo System on the Unit Ball

On the Convergence Properties of a Heterogeneous Multiagent System on the Unit Sphere

On the Convergence Properties of a Heterogeneous Multiagent System on the Unit Sphere

The Gleason’s problem on normal weight general function spaces in the unit ball of ℂn

The Gleason’s problem on normal weight general function spaces in the unit ball of ℂn

Quantum quantities near the Grothendieck bound in a single quantum system

The Grothendieck theorem considers a ‘classical’ quadratic form that uses complex scalars in the unit disc, and the corresponding ‘quantum’ quadratic form that replaces the scalars with vectors in the unit ball of a Hilbert space. It shows that when then might take values greater than 1, up to the complex Grothendieck constant kG . Previous work in a quantum context, used Grothendieck’s theorem with multipartite entangled systems, in contrast to the present work which uses it for a single quantum system. The emphasis in the paper is in examples with , which is a classically forbidden region in the sense that cannot take values in it.

Averaging data on the unit hypersphere

Averaging data on the unit sphere S d (also called a unit hypersphere) is a common problem in computer vision, robotics and other fields, with applications ranging from motion planning to DNA modelling. In this paper, we introduce a new method for averaging data represented as points on the unit sphere S d−1 using the d-dimensional generalized Kuramoto model. Our method is verified on a range of benchmark data sets and compared with common data averaging algorithms. Also, we showcase the applicability of this method for solving rotation averaging problem.

Approximative Solar Properties of Sets and Local Geometry of the Unit Sphere

Approximative Solar Properties of Sets and Local Geometry of the Unit Sphere

The equivalence between CPCP and strong regularity under Krein-Milman property

We obtain a result in the spirit of the well-known W. Schachermayer and H. P. Rosenthal research about the equivalence between Radon-Nikodym and Krein-Milman properties, by showing that, for closed, bounded and convex subsets C of a separable Banach space, under Krein-Milman property for C, one has the equivalence between convex point of continuity property and strong regularity both defined for every locally convex topology on C, containing the weak topology on C. Then, under Krein-Milman property, not only the classical convex point of continuity property and strong regularity are equivalent, but also when they are defined for an arbitrary locally convex topology containing the weak topology. We also show that while the unit ball B of c0 fails convex point of continuity property and strong regularity (both defined for the weak topology), there is a locally convex topology τ on B, containing the weak topology on B, such that B still fails convex point of continuity property for τ, but B surprisingly enjoys strong regularity for τ-open sets. Moreover, B satisfies the diameter two property for the topology τ, that is, every nonempty τ-open subset of B has diameter two even though every τ-open subset of B contains convex combinations of relative τ-open subsets with arbitrarily small diameter, that is, B fails the strong diameter two property for the topology τ. This stresses the known extreme differences up to now between those diameter two properties from a topological point of view.

Inclusion Properties of the Triangular Ratio Metric Balls

Inclusion properties are studied for balls of the triangular ratio metric, the hyperbolic metric, the j∗\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$j^*$$\\end{document}-metric, and the distance ratio metric defined in the unit ball domain. Several sharp results are proven and a conjecture about the relation between triangular ratio metric balls and hyperbolic balls is given. An algorithm is also built for drawing triangular ratio circles or three-dimensional spheres.