N-dimensional Unit Sphere Research Articles

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.

Wavelet Based Solutions to the Poisson and the Helmholtz Equations on the n-Dimensional Unit Sphere

We present a method of solving partial differential equations on the n-dimensional unit sphere using methods based on the continuous wavelet transform derived from approximate identities. We give an explicit analytical solution to the Poisson equation and to the Helmholtz equations. For the first one and for some special values of the parameter in the latter one, we derive a closed formula for the generalized Green function.

A sharp regularity estimate for the Schrödinger propagator on the sphere

Let ΔSn denote the Laplace-Beltrami operator on the n-dimensional unit sphere Sn, n≥2. In this paper we show that‖eitΔSnf‖L4([0,2π)×Sn)≤C‖f‖Wα,4(Sn) holds if α>(n−2)/4. The range of α is sharp in the sense that the estimate fails for α<(n−2)/4. As a consequence, we obtain space-time Lp-estimates for eitΔSn for 2≤p≤∞. We also prove that the maximal operator f→sup0≤t<2π⁡|eitΔSnf| is bounded from Wα,2(Sn) to L6n/(3n−2)(Sn) for α>1/3 whenever f are zonal functions on Sn.

On the rotational variance of the differential evolution algorithm

In this study we examine the rotational (in)variance of the differential evolution (DE) algorithm. We show that the classic DE/rand/1/bin algorithm, which uses constant mutation and standard crossover, is rotationally variant. We then study a previously proposed rotationally invariant formulation in which the crossover operation takes place in an orthogonal base constructed using Gramm-Schmidt orthogonalization.We propose two new formulations by firstly considering a very simple rotationally invariant formulation using constant mutation and whole arithmetic crossover. This rudimentary formulation performs badly, due to lack of diversity. We introduce diversity into the formulation using two distinctly different strategies. The first adjusts the crossover step by perturbing the direction of the linear combination between the target vector and the mutant vector. This formulation is invariant in a stochastic sense only. The other formulation adds a self-scaling random vector with a standard normal distribution, sampled uniformly from the surface of an n-dimensional unit sphere to the unaltered whole arithmetic crossover vector. This formulation is strictly invariant, if in a stochastic sense only.We compare the four invariant formulations in terms of numerical efficiency for a modest set of test problems; the intention not being to propose yet another competitive and/or superior DE variant, but rather to present formulations that are both diverse and invariant, in the hope that this will stimulate additional future contributions, since rotational invariance in general is a desirable, salient feature for an optimization algorithm.

Weakly biharmonic maps from the ball to the sphere

The aim of this paper is to investigate the existence of proper, weakly biharmonic maps within a family of rotationally symmetric maps $$u_a : B^n \rightarrow {\mathbb {S}}^n$$, where $$B^n$$ and $${\mathbb {S}}^n$$ denote the Euclidean n-dimensional unit ball and sphere respectively. We prove that there exists a proper, weakly biharmonic map $$u_a$$ of this type if and only if $$n=5$$ or $$n=6$$. We shall also prove that these critical points are unstable.

Extremes of spherical fractional Brownian motion

Let $\{B_\beta (x), x \in \mathbb{S}^N\}$ be a fractional Brownian motion on the $N$-dimensional unit sphere $\mathbb{S}^N$ with Hurst index $\beta$. We study the excursion probability $\mathbb{P}\{\sup_{x\in T} B_\beta(x) > u \}$ and obtain the asymptotics as $u\to \infty$, where $T$ can be the entire sphere $\mathbb{S}^N$ or a geodesic disc on $\mathbb{S}^N$.

Reconstruction of functions on the sphere from their integrals over hyperplane sections

We obtain new inversion formulas for the Funk type transforms of two kinds associated to spherical sections by hyperplanes passing through a common point $A$ which lies inside the n-dimensional unit sphere or on the sphere itself. Transforms of the first kind are defined by integration over complete subspheres and can be reduced to the classical Funk transform. Transforms of the second kind perform integration over truncated subspheres, like spherical caps or bowls, and can be reduced to the hyperplane Radon transform. The main tools are analytic families of cosine transforms, Semyanisyi's integrals, and modified stereorgraphic projection with the pole at $A$. Assumptions for functions are close to minimal.

Polar Transform of Conformally Flat Metrics

In the theory of convex subsets in a Euclidean space, an important role is played by Minkowski duality (the polar transform of a convex set, or the Legendre transform of a convex set). We consider conformally flat Riemannian metrics on the n-dimensional unit sphere and their embeddings into the isotropic cone of the Lorentz space. For a given class of metrics, we define and carry out a detailed study of the Legendre transform.

Harmonic Measures on the Sphere via Curvature-Dimension

We show that the family of probability measures on the n-dimensional unit sphere, having density proportional to:

On rotationally invariant continuous-parameter genetic algorithms

We examine the rotational (in)variance of the continuous-parameter genetic algorithm (CPGA). We show that a standard CPGA, using blend crossover and standard mutation, is rotationally variant.To construct a rotationally invariant CPGA it is possible to modify the crossover operation to be rotationally invariant. This however results in a loss of diversity. Hence we introduce diversity in two ways: firstly using a modified mutation scheme, and secondly by adding a self-scaling random vector with a standard normal distribution, sampled uniformly from the surface of a n-dimensional unit sphere to the offspring vector. This formulation is strictly invariant, albeit in a stochastic sense only.We compare the three formulations in terms of numerical efficiency for a modest set of test problems; the intention not being the contribution of yet another competitive and/or superior CPGA variant, but rather to present formulations that are both diverse and invariant, in the hope that this will stimulate additional future contributions, since rotational invariance in general is a desirable, salient feature for an optimization algorithm.

The 1-harmonic flow with values in a hyperoctant of theN-sphere

We prove the existence of solutions to the 1-harmonic flow — that is, the formal gradient flow of the total variation of a vector field with respect to the L2-distance — from a domain of ℝm into a hyperoctant of the N-dimensional unit sphere, S+N−1, under homogeneous Neumann boundary conditions. In particular, we characterize the lower-order term appearing in the Euler–Lagrange formulation in terms of the “geodesic representative” of a BV-director field on its jump set. Such characterization relies on a lower semicontinuity argument which leads to a nontrivial and nonconvex minimization problem: to find a shortest path between two points on S+N−1 with respect to a metric which penalizes the closeness to their geodesic midpoint.

Determinants of the Laplacians on the n-dimensional unit sphere S n

During the last three decades, the problem of evaluation of the determinants of the Laplacians on Riemann manifolds has received considerable attention from many authors. The functional determinant for the k-dimensional unit sphere with the standard metric has been computed in several ways. Here we aim at computing the determinants of the Laplacians on ( ) by mainly using certain closed-form evaluations of the series involving zeta function. MSC:11M35, 11M36, 11M06, 33B15.

Using QR Decomposition to Obtain a New Instance of Mesh Adaptive Direct Search with Uniformly Distributed Polling Directions

The purpose of this paper is to introduce a new instance of the Mesh Adaptive Direct Search (Mads) class of algorithms, which utilizes a more uniform distribution of poll directions than do other common instances, such as OrthoMads and LtMads. Our new implementation, called QrMads, bases its poll directions on an equal area partitioning of the n-dimensional unit sphere and the QR decomposition to obtain an orthogonal set of directions. While each instance produces directions which are dense in the limit, QrMads directions are more uniformly distributed in the unit sphere. This uniformity is the key to enhanced performance in higher dimensions and for constrained problems. The trade-off is that QrMads is no longer deterministic and at each iteration the set of polling directions is no longer orthogonal. Instead, at each iteration, the poll directions are only ‘nearly orthogonal,’ becoming increasingly closer to orthogonal as the mesh size decreases. Finally, we present a variety of test results on smooth, nonsmooth, unconstrained, and constrained problems and compare them to OrthoMads on the same set of problems.

A superspace formulation of Yang–Mills theory on sphere

A superspace approach to the Becchi–Rouet–Stora–Tyutin (BRST) formalism for the Yang–Mills theory on an n-dimensional unit sphere S1n is developed in a manifestly covariant manner based on the rotational supersymmetry characterized by the supergroup OSp(n+1∣2). This is done by employing an (n+2)-dimensional unit supersphere S1n∣2 parametrized by n commutative and two anticommutative coordinate variables so that it includes S1n as a subspace and realizes the OSp(n+1∣2) supersymmetry. In this superspace formulation, referred to as the supersphere formulation, the so-called horizontality condition is concisely expressed in terms of the rank-3 field strength tensor of a Yang–Mills superfield on S1n∣2. The supersphere formulation completely covers the BRST gauge-fixing procedure for the Yang–Mills theory on S1n provided by us [R. Banerjee and S. Deguchi, Phys. Lett. B 632, 579 (2006); arXiv:hep-th/0509161]. Furthermore, this formulation admits the (massive) Curci–Ferrari model defined on S1n, describing the gauge-fixing and mass terms on S1n together as a mass term on S1n∣2.

Estimates for eigenvalues of the poly-Laplacian with any order in a unit sphere

In this paper we study eigenvalues of the poly-Laplacian with any order on a domain in an n-dimensional unit sphere and obtain estimates for eigenvalues. In particular, the optimal result of Cheng and Yang (Math Ann 331:445–460, 2005) is included in our ones. In order to prove our results, we introduce 2(l + 1) functions ai and bi, for i = 0, 1, . . . , l and two operators μ and η. First of all, we study properties of functions ai and bi and the operators μ and η. By making use of these properties and introducing k free constants, we obtain estimates for eigenvalues.

Closed forms of the Green's function and the generalized Green's function for the Helmholtz operator on the N-dimensional unit sphere

The Green's function for the Helmholtz differential operator ∇2 + λ(λ + N − 1) on the N-dimensional (with N ⩾ 1) hyperspherical surface of unit radius is investigated. Its closed form is shown to be where SN is the area of , C(α)λ(x) is the Gegenbauer function of the first kind, while n and n′ are radius vectors, with respect to the centre of , of the observation and source points, respectively. The Green's function G(N)(λ; n, n′) fails to exist whenever λ is such that it holds that λ(λ + N − 1) = L(L + N − 1), with . For these exceptional cases, the generalized (known also as ‘modified’ or ‘reduced’) Green's function is considered. It is shown that may be expressed compactly in terms of the Gegenbauer polynomial C((N−1)/2)L(n ⋅ n′) and the derivative [∂C((N−1)/2)λ(−n ⋅ n′)/∂λ]λ=L. Explicit expressions for the derivatives [∂C(n)λ(x)/∂λ]λ=L and [∂C(n+1/2)λ(x)/∂λ]λ=L, with , are found and used to transform the functions and to potentially more useful forms.

Decomposition of Besov and Triebel–Lizorkin spaces on the sphere

A discrete system of almost exponentially localized elements (needlets) on the n-dimensional unit sphere S n is constructed. It shown that the needlet system can be used for decomposition of Besov and Triebel–Lizorkin spaces on the sphere. As an application of Besov spaces on S n , a Jackson estimate for nonlinear m-term approximation from the needlet system is obtained.

Turbulence Phenomena in Real Analysis

The purpose of this paper is first to show that if X is any locally compact but not compact perfect Polish space and Open image in new window stands for the one-point compactification of X, while EX is the equivalence relation which is defined on the Polish group C(X,R+*) by Open image in new window where f, g are in C(X,R+*), then EX is induced by a turbulent Polish group action. Second we show that given any Open image in new window if we identify the n-dimensional unit sphere Sn with the one-point compactification of Rn via the stereographic projection, while En,r is the equivalence relation which is defined on the Polish group Cr(Rn,R+*) by Open image in new window where f, g are in Cr(Rn,R+*), then En,r is also induced by a turbulent Polish group action.

Estimates on Eigenvalues of Laplacian

In this paper, we study eigenvalues of Laplacian on either a bounded connected domain in an n-dimensional unit sphere Sn(1), or a compact homogeneous Riemannian manifold, or an n-dimensional compact minimal submanifold in an N-dimensional unit sphere SN(1). We estimate the k+1-th eigenvalue by the first k eigenvalues. As a corollary, we obtain an estimate of difference between consecutive eigenvlaues. Our results are sharper than ones of P. C. Yang and Yau [25], Leung [19], Li [20] and Harrel II and Stubbe [12], respectively. From Weyl’s asymptotical formula, we know that our estimates are optimal in the sense of the order of k for eigenvalues of Laplacian on a bounded connected domain in an n-dimensional unit sphere Sn(1).

Series involving the Zeta function and multiple Gamma functions

The theory of multiple Gamma functions, which was recently revived in the study of the determinants of the Laplacians, was applied in several earlier works in order to evaluate some families of series involving the Riemann Zeta function as well as to compute the determinants of the Laplacians. Here, in the present paper, the authors address the converse problem and apply various (known or new) formulas for series associated with the Zeta and related functions with a view to developing the corresponding theory of multiple Gamma functions and then using these series to compute the determinants of the Laplacians on the n-dimensional unit sphere S n ( n=5,6,7) explicitly.