Unit Ball Research Articles

Octonionic monogenic and slice monogenic Hardy and Bergman spaces

Abstract In this paper we discuss some basic properties of octonionic Bergman and Hardy spaces. In the first part we review some fundamental concepts of the general theory of octonionic Hardy and Bergman spaces together with related reproducing kernel functions in the monogenic setting. We explain how some of the fundamental problems in well-defining a reproducing kernel can be overcome in the non-associative setting by looking at the real part of an appropriately defined para-linear octonion-valued inner product. The presence of a weight factor of norm 1 in the definition of the inner product is an intrinsic new ingredient in the octonionic setting. Then we look at the slice monogenic octonionic setting using the classical complex book structure. We present explicit formulas for the slice monogenic reproducing kernels for the unit ball, the right octonionic half-space and strip domains bounded in the real direction. In the setting of the unit ball we present an explicit sequential characterization which can be obtained by applying the special Taylor series representation of the slice monogenic setting together with particular octonionic calculation rules that reflect the property of octonionic para-linearity.

Essential norm of Riemann–Stieltjes operator on weighted Bergman spaces with doubling weights

Abstract Let ω be a doubling weight and 0 < p ≤ q < ∞ {0<p\leq q<\infty} . The essential norm of Riemann–Stieltjes operator T g {T_{g}} from the weighted Bergman space A ω p {A^{p}_{\omega}} to A ω q {A^{q}_{\omega}} was investigated in the unit ball of ℂ n {\mathbb{C}^{n}} .

Periodic solutions for second-order even and noneven Hamiltonian systems

AbstractIn this paper, we consider the second-order Hamiltonian system $$ \ddot{x}+V^{\prime}(x)=0,\quad x\in \mathbb{R}^{N}. $$ x ¨ + V ′ ( x ) = 0 , x ∈ R N . We use the monotonicity assumption introduced by Bartsch and Mederski (Arch. Ration. Mech. Anal. 215:283–306, 2015). When V is even, we can release the strict convexity hypothesis, which is used by Bartsch and Mederski combined with the monotonicity assumption. When V is noneven, we weaken the strict convexity assumption and introduce another hypothesis (see (V10)). Then in both cases, we can build the homomorphism between the Nehari manifold and the unit sphere of some suitable space. Using the Nehari manifold method introduced by Szulkin (J. Funct. Anal. 257:3802–3822 2009), we prove the existence of T-periodic solutions with minimal period T.

The core of the unit sphere of a Banach space

<abstract><p>A geometric invariant or preserver is essentially a geometric property of the unit sphere of a real Banach space that remains invariant under the action of a surjective isometry onto the unit sphere of another real Banach space. A new geometric invariant of the unit ball of a real Banach space was introduced and analyzed in this manuscript: the core of the unit sphere. This geometric invariant consists of all points in the unit sphere of a real Banach space, which are contained in a unique maximal face. It is, in a geometrical sense, the opposite of fractal-like sets such as starlike sets. Classical geometric properties, such as smoothness and strict convexity, were employed to characterize the core of the unit sphere. Also, the core was related to a recently introduced new index: the index of strong rotundity. A characterization of the core in terms of the index of strong rotundity was provided. Finally, applications to longstanding open problems, such as Tingley's problem, were provided by presenting a new notion: Mazur-Ulam classes of Banach spaces.</p></abstract>

Difference of composition operators on the Békollè weighted Bergman spaces in the unit ball

Difference of composition operators on the Békollè weighted Bergman spaces in the unit ball

Non-simple blow-up for singular Liouville equations in unit ball

Non-simple blow-up for singular Liouville equations in unit ball

Investigating the characteristics of Clifford hypersurfaces and the unit sphere via a minimal immersion in $ S^{n+1} $

<p>In this article, we find the different sufficient conditions for a compact minimal hypersurface $ M $ of the unit sphere $ S^{n+1}, n\in \mathbb{Z}^{+} $ to be the Clifford hypersurface $ S^{\ell }(\sqrt{\frac{\ell }{n}})\times S^{m}(\sqrt{\frac{m}{n}}), $ where $ \ell, m\in \mathbb{Z}^{+}, \; \ell +m = n $ or the sphere $ S^{n} $. This classification is achieved by applying constraints to the tangent and normal components of the immersion.</p>

Unit balls of polyhedral Banach spaces with many extreme points

Unit balls of polyhedral Banach spaces with many extreme points

On subspaces of Orlicz spaces spanned by independent copies of a mean zero function

We study the subspaces of the Orlicz spaces $L_M$ spanned by independent copies $f_k$, $k=1,2,…$, of a function $f\in L_M$, $\int_0^1 f(t) dt=0$. Any such a subspace $H$ is isomorphic to some Orlicz sequence space $\ell_\psi$. In terms of dilations of the function $f$, a description of strongly embedded subspaces of this type is obtained, and conditions guaranteeing that the unit ball of such a subspace consists of functions with equicontinuous norms in $L_M$ are found. In particular, we prove that there is a wide class of Orlicz spaces $L_M$ (containing the $L^p$-spaces, $1\le p< 2$), for which each of the above properties of $H$ holds if and only if the Matuszewska-Orlicz indices of the functions $M$ and $\psi$ satisfy $\alpha_\psi^0>\beta_M^\infty$.

Min-phase-isometries on the unit sphere of Lp-type spaces

Min-phase-isometries on the unit sphere of Lp-type spaces

Quantum chemistry study on the hyper-Rayleigh scattering optical activity of R-carvone and (1R,5R)-α-pinene.

In this work, the HRS-OA circular differential scattering ratios of medium-size chiral molecules, R-carvone and (1R,5R)-α-pinene, has been computed by means of TD-DFT calculations. The results have been discussed as a function of the basis set size. For R-carvone a conformational analysis has been taken into account. In particular, the three most stable conformations of R-carvone present chiral terms that contribute to the HRS-OA circular differential scattering ratio, with opposite signs. Finally, the modified unit sphere representations (mUSR) have been produced for the molecules under study.

Efficient and accurate numerical-projection of electromagnetic multipoles for scattering objects

In this paper, we develop an efficient and accurate procedure of electromagnetic multipole decomposition by using the Lebedev and Gaussian quadrature methods to perform the numerical integration. Firstly, we briefly review the principles of multipole decomposition, highlighting two numerical projection methods including surface and volume integration. Secondly, we discuss the Lebedev and Gaussian quadrature methods, provide a detailed recipe to select the quadrature points and the corresponding weighting factor, and illustrate the integration accuracy and numerical efficiency (that is, with very few sampling points) using a unit sphere surface and regular tetrahedron. In the demonstrations of an isotropic dielectric nanosphere, a symmetric scatterer, and an anisotropic nanosphere, we perform multipole decomposition and validate our numerical projection procedure. The obtained results from our procedure are all consistent with those from Mie theory, symmetry constraints, and finite element simulations.Graphical

Adaptive mesh methods on compact manifolds via Optimal Transport and Optimal Information Transport

Moving mesh methods were devised to redistribute a mesh in a smooth way, while keeping the number of vertices of the mesh and their connectivity unchanged. A fruitful theoretical point-of-view is to take such moving mesh methods and think of them as an application of the diffeomorphic density matching problem. Given two probability measures μ0 and μ1, the diffeomorphic density matching problem consists of finding a diffeomorphic pushforward map T such that T#μ0=μ1. Moving mesh methods are seen to be an instance of the diffeomorphic density matching problem by treating the probability density as the local density of nodes in the mesh. It is preferable that the restructuring of the mesh be done in a smooth way that avoids tangling the connections between nodes, which would lead to numerical instability when the mesh is used in computational applications. This then suggests that a diffeomorphic map T is desirable to avoid tangling. The first tool employed to solve the moving mesh problem between source and target probability densities on the sphere was Optimal Transport (OT). Recently Optimal Information Transport (OIT) was rigorously derived and developed allowing for the computation of a diffeomorphic mapping by simply solving a Poisson equation. Not only is the equation simpler to solve numerically in OIT, but with Optimal Transport there is no guarantee that the mapping between probability density functions defines a diffeomorphism for general 2D compact manifolds.In this manuscript, we perform a side-by-side comparison of using Optimal Transport and Optimal Information Transport on the sphere for adaptive mesh problems. We choose to perform this comparison with recently developed provably convergent solvers, but these are, of course, not the only numerical methods that may be used. We believe that Optimal Information Transport is preferable in computations due to the fact that the partial differential equation (PDE) solve step is simply a Poisson equation. For more general surfaces M, we show how the Optimal Transport and Optimal Information Transport problems can be reduced to solving on the sphere, provided that there exists a diffeomorphic mapping Φ:M→S2. This implies that the Optimal Transport problem on M with a special cost function can be solved with regularity guarantees, while computations for the problem are performed on the unit sphere.

Complexity for a class of elliptic ordinary integro-differential equations

Consider the variational form of the ordinary integro-differential equation (OIDE)−u″+u+∫01q(⋅,y)u(y)dy=f on the unit interval I, subject to homogeneous Neumann boundary conditions. Here, f and q respectively belong to the unit ball of Hr(I) and the ball of radius M1 of Hs(I2), where M1∈[0,1). For ε>0, we want to compute ε-approximations for this problem, measuring error in the H1(I) sense in the worst case setting. Assuming that standard information is admissible, we find that the nth minimal error is Θ(n−min⁡{r,s/2}), so that the information ε-complexity is Θ(ε−1/min⁡{r,s/2}); moreover, finite element methods of degree max⁡{r,s} are minimal-error algorithms. We use a Picard method to approximate the solution of the resulting linear systems, since Gaussian elimination will be too expensive. We find that the total ε-complexity of the problem is at least Ω(ε−1/min⁡{r,s/2}) and at most O(ε−1/min⁡{r,s/2}ln⁡ε−1), the upper bound being attained by using O(ln⁡ε−1) Picard iterations.

Free boundary minimal hypersurfaces outside of the ball

In this paper, we obtain two classification theorems for free boundary minimal hypersurfaces outside of the unit ball (exterior FBMH, for short) in Euclidean space. The first result states that the only exterior stable FBMH with parallel embedded regular ends are the catenoidal hypersurfaces. To achieve this, we prove a Bôcher-type result for positive Jacobi functions on regular minimal ends in \mathbb{R}^{n+1} that, after some calculations, implies the first theorem. The second theorem states that any exterior FBMH \Sigma with one regular end is a catenoidal hypersurface. Its proof is based on a symmetrization procedure similar to that of R. Schoen. We also give a complete description of the catenoidal hypersurfaces, including the calculation of their indices.

Confining Co1.11Te2 nanoparticles within mesoporous hollow carbon combination sphere for fast and ultralong sodium storage

Co1.11Te2 nanoparticles are in-situ uniformly grown within mesoporous hollow carbon combination sphere (MHCCS@Co1.11Te2) using a hard-template and spray drying process, solution impregnation and pyrolysis tellurization. Material characterizations reveal that Co1.11Te2, with a diameter of ∼ 20 nm, is attached to the internal walls of the unit spheres or embedded in the mesopore shells of the unit spheres, presenting a distinctive “ships-in-combination-bottles” nanoencapsulation structure. In sodium-ion half-cells, MHCCS@Co1.11Te2 exhibits excellent cycling stability, achieving reversible capacities of 257 mAh/g at 0.5 A/g after 250 cycles, 235 mAh/g at 1.0 A/g after 300 cycles and 161 mAh/g at 10.0 A/g after 1900 cycles. Electrochemical kinetic analyses and ex-situ characterizations reveal rapid electron/Na+ transport kinetics, prominent surface pseudocapacitive behavior, robust nanocomposite structure, and multi-step conversion reactions of sodium polytellurides. In sodium-ion full-cells, MHCCS@Co1.11Te2 still demonstrates stable cycling performance at 1.0 and 5.0 A/g and excellent rate capability. The superior electrochemical performance is associated with the nanoencapsulation structure based on mesoporous hollow carbon combination spheres, which promotes electron conduction and Na+ transport. The space-confined effect maintains the high electrochemical activity and cycling stability of Co1.11Te2.

The Inverse Weber Problem on the Plane and the Sphere

Weber’s inverse problem in the plane is to modify the positive weights associated with n fixed points in the plane at minimum cost, ensuring that a given point a priori becomes the Euclidean weighted geometric median. In this paper, we investigate Weber’s inverse problem in the plane and generalize it to the surface of the sphere. Our study uses a subspace orthogonal to a subspace generated by two vectors X and Y associated with the given points and weights. The main achievement of our work lies in determining a vector perpendicular to the vectors X and Y, in Rn; which is used to determinate a solution of Weber’s inverse problem. In addition, lower bounds are obtained for the minimum of the Weber function, and an upper bound for the difference of the minimal of Weber’s direct and inverse problems. Examples of application at the plane and unit sphere are given.

Multiplication Operators on Weighted Zygmund Spaces of the First Cartan Domain

Inspired by some recent studies of the multiplication operators on holomorphic function spaces of the classical domains such as the open unit disk, the unit ball and the unit polydisk, the purpose of the present paper is to study just the operators that are defined on weighted Zygmund spaces of the first Cartan domain. We obtain some necessary conditions and sufficient conditions for the operators to be bounded and compact.

Representation of the Green’s Function of the Dirichlet Problem for the Polyharmonic Equation in the Ball

We define the elementary solution of the polyharmonic equation, with the help of which an explicit representation of the Green’s function of the Dirichlet problem for the polyharmonic equation in the unit ball is given for all space dimensions except for some finite set. On the basis of the obtained Green’s function, the solution of the homogeneous Dirichlet problem in the unit ball is constructed. As an example, an explicit form of the solution of the homogeneous Dirichlet problem for the inhomogeneous polyharmonic equation with the simplest polynomial right-hand side is found.

Sphere bundle over the set of inner products in a Hilbert space

Let (H,〈,〉) be a complex Hilbert space and B(H) the space of bounded linear operators in H. Any other equivalent inner product in H is of the form 〈f,g〉A=〈Af,g〉 (f,g∈H) for some positive invertible operator A∈B(H). In this paper we study the bundle M which consist of the unit sphere {f∈H:〈f,f〉A=1} over each (equivalent) inner product 〈,〉A, which due to the observation above can be definedM={(A,f)∈B(H)×H:A is positive and invertible and 〈Af,f〉=1}. We prove that M is a complemented submanifold of the Banach space B(H)×H and a homogeneous space of the Banach-Lie group G(H)⊂B(H) of invertible operators. We introduce a reductive structure in M, and study properties of the geodesics of the linear connection induced by this reductive structure. We consider certain submanifolds of M, for instance, the one obtained when the positive elements A describing the inner products lie in a prescribed C⁎-algebra A⊂B(H).