Spheres In Euclidean Space Research Articles

Elliptic Quaternion and Elliptic linear Interpolation

Spherical spline quaternion interpolation has been done on sphere in Euclidean space using quaternions. In this paper, we have been done elliptic quaternion linear interpolation on ellipsoid using elliptic quaternions. This interpolation curve is called Elerp elliptic linear interpolation. In addition, ESquad (spline elliptic quaternion interpolation) is defined by using the group structure feature of elliptic quaternion on ellipsoid.

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.

Asymptotic interplay of states and adaptive coupling gains in the Lohe Hermitian sphere model

<p style='text-indent:20px;'>We study emergent dynamics of the Lohe Hermitian sphere (LHS) model with the same free flows under the dynamic interplay between state evolution and adaptive couplings. The LHS model is a complex counterpart of the Lohe sphere (LS) model on the unit sphere in Euclidean space, and when particles lie in the Euclidean unit sphere embedded in <inline-formula><tex-math id="M1">\begin{document}$ \mathbb C^{d+1} $\end{document}</tex-math></inline-formula>, it reduces to the Lohe sphere model. In the absence of interactions between states and coupling gains, emergent dynamics have been addressed in [<xref ref-type="bibr" rid="b23">23</xref>]. In this paper, we further extend earlier results in the aforementioned work to the setting in which the state and coupling gains are dynamically interrelated via two types of coupling laws, namely anti-Hebbian and Hebbian coupling laws. In each case, we present two sufficient frameworks leading to complete aggregation depending on the coupling laws, when the corresponding free flow is the same for all particles.</p>

Quantum jamming: Critical properties of a quantum mechanical perceptron

In this Letter, we analyze the quantum dynamics of the perceptron model: a particle is constrained on a $N$-dimensional sphere, with $N\to \infty$, and subjected to a set of randomly placed hard-wall potentials. This model has several applications, ranging from learning protocols to the effective description of the dynamics of an ensemble of infinite-dimensional hard spheres in Euclidean space. We find that the jamming transition with quantum dynamics shows critical exponents different from the classical case. We also find that the quantum jamming transition, unlike the typical quantum critical points, is not confined to the zero-temperature axis, and the classical results are recovered only at $T=\infty$. Our findings have implications for the theory of glasses at ultra-low temperatures and for the study of quantum machine-learning algorithms.

Smoothness of solutions of a convolution equation of restricted type on the sphere

Abstract Let $\mathbb {S}^{d-1}$ denote the unit sphere in Euclidean space $\mathbb {R}^d$ , $d\geq 2$ , equipped with surface measure $\sigma _{d-1}$ . An instance of our main result concerns the regularity of solutions of the convolution equation $$\begin{align*}a\cdot(f\sigma_{d-1})^{\ast {(q-1)}}\big\vert_{\mathbb{S}^{d-1}}=f,\text{ a.e. on }\mathbb{S}^{d-1}, \end{align*}$$ where $a\in C^\infty (\mathbb {S}^{d-1})$ , $q\geq 2(d+1)/(d-1)$ is an integer, and the only a priori assumption is $f\in L^2(\mathbb {S}^{d-1})$ . We prove that any such solution belongs to the class $C^\infty (\mathbb {S}^{d-1})$ . In particular, we show that all critical points associated with the sharp form of the corresponding adjoint Fourier restriction inequality on $\mathbb {S}^{d-1}$ are $C^\infty $ -smooth. This extends previous work of Christ and Shao [4] to arbitrary dimensions and general even exponents and plays a key role in the companion paper [24].

Emergent behaviors of Lohe Hermitian sphere particles under time-delayed interactions

<p style='text-indent:20px;'>We study emergent behaviors of the Lohe Hermitian sphere(LHS) model with a time-delay for a homogeneous and heterogeneous ensemble. The LHS model is a complex counterpart of the Lohe sphere(LS) aggregation model on the unit sphere in Euclidean space, and it describes the aggregation of particles on the unit Hermitian sphere in <inline-formula><tex-math id="M1">\begin{document}$ \mathbb C^d $\end{document}</tex-math></inline-formula> with <inline-formula><tex-math id="M2">\begin{document}$ d \geq 2 $\end{document}</tex-math></inline-formula>. Recently it has been introduced by two authors of this work as a special case of the Lohe tensor model. When the coupling gain pair satisfies a specific linear relation, namely the Stuart-Landau(SL) coupling gain pair, it can be embedded into the LS model on <inline-formula><tex-math id="M3">\begin{document}$ \mathbb R^{2d} $\end{document}</tex-math></inline-formula>. In this work, we show that if the coupling gain pair is close to the SL coupling pair case, the dynamics of the LHS model exhibits an emergent aggregate phenomenon via the interplay between time-delayed interactions and nonlinear coupling between states. For this, we present several frameworks for complete aggregation and practical aggregation in terms of initial data and system parameters using the Lyapunov functional approach.</p>

On Some Properties of Distance in TO-Space

The aim of this work is to investigate some properties of the truncated octahedron metric introduced in the space in further studies on metric geometry. With this metric, the 3-dimensional analytical space is a Minkowski geometry which is a non-Euclidean geometry in a finite number of dimensions. In a Minkowski geometry, the unit ball is a certain symmetric closed convex set instead of the usual sphere in Euclidean space. The unit ball of the truncated octahedron geometry is a truncated octahedron which is an Archimedean solid. In this study, first, metric properties of truncated octahedron distance, d_TO, in R^2 has been examined by metric approach. Then, by using synthetic approach some distance formulae in R_TO^3, 3-dimensional analytical space furnished with the truncated octahedron metric has been found.

Monte Carlo simulation for a gas of hard spheres in d-dimensional space: Equilibrium structure and state equations

In this work, we emphasize that it is possible using a personal computer to perform a MonteCarlo simulations in a reasonable computing time, and find the equilibrium structure of a hardsphere gas for a Euclidean multi-dimensional spaces. We study the properties of equilibrium and determine the equation of state of gas of hard spheres in Euclidean spaces from two to seven dimensions. The results show that the pressure is in agreement with different theoretical models based on virial expansion in spaces from two to five dimensions, also our results are extended for seven dimensions. As expected, it was found that the system of hard spheres loses its structure and the pressure of the system decreases when the dimension of the space increases.

Schoenberg’s Theorem for Positive Definite Functions on Products: A Unifying Framework

The main contribution in the present paper is a characterization for positive definiteness and strict positive definiteness of a kernel on the product $$X \times S^d$$ , in which X is a nonempty set and $$S^d$$ is the usual d-dimensional unit sphere in Euclidean space, through Fourier-like expansions. The setting presupposes continuity and isotropy on the $$S^d$$ side and no algebraic structure or topology on X. The result may be interpreted as another extension of a classical result of I. J. Schoenberg on positive definite functions on spheres. We take a closer look at our results in the case in which X is a locally compact group, paying special attention to usual Euclidean spaces and high dimensional tori.

QUANTITATIVE OSCILLATION ESTIMATES FOR ALMOST-UMBILICAL CLOSED HYPERSURFACES IN EUCLIDEAN SPACE

We prove${\it\epsilon}$-closeness of hypersurfaces to a sphere in Euclidean space under the assumption that the traceless second fundamental form is${\it\delta}$-small compared to the mean curvature. We give the explicit dependence of${\it\delta}$on${\it\epsilon}$within the class of uniformly convex hypersurfaces with bounded volume.

Spline Split Quaternion Interpolation in Minkowski Space

Spherical spline quaternion interpolation has been done on sphere in Euclidean space using quaternions. In this paper, we have done the spline split quaternion interpolation on hyperbolic sphere in Minkowski space using split quaternions and metric Lorentz. This interpolation curve is called spherical spline split quaternion interpolation in Minkowski space (MSquad).

On the chromatic numbers of spheres in Euclidean spaces

On the chromatic numbers of spheres in Euclidean spaces

Inscribed and circumscribed polyhedra for a convex body and continuous functions on a sphere in Euclidean space

Inscribed and circumscribed polyhedra for a convex body and continuous functions on a sphere in Euclidean space

Perturbed kernel approximation on homogeneous manifolds

Current methods for interpolation and approximation within a native space rely heavily on the strict positive-definiteness of the underlying kernels. If the domains of approximation are the unit spheres in euclidean spaces, then zonal kernels (kernels that are invariant under the orthogonal group action) are strongly favored. In the implementation of these methods to handle real world problems, however, some or all of the symmetries and positive-definiteness may be lost in digitalization due to small random errors that occur unpredictably during various stages of the execution. Perturbation analysis is therefore needed to address the stability problem encountered. In this paper we study two kinds of perturbations of positive-definite kernels: small random perturbations and perturbations by Dunkl's intertwining operators [C. Dunkl, Y. Xu, Orthogonal polynomials of several variables, Encyclopedia of Mathematics and Its Applications, vol. 81, Cambridge University Press, Cambridge, 2001]. We show that with some reasonable assumptions, a small random perturbation of a strictly positive-definite kernel can still provide vehicles for interpolation and enjoy the same error estimates. We examine the actions of the Dunkl intertwining operators on zonal (strictly) positive-definite kernels on spheres. We show that the resulted kernels are (strictly) positive-definite on spheres of lower dimensions.

Distribution of the points of a design on the sphere

We determine the size of spherical caps with centres at the points of a design that cover the whole sphere in Euclidean space with a given multiplicity. By projecting -designs on one-dimensional subspaces, we obtain the nodes of a Chebyshev-type quadrature formula of the same precision . For large values of we establish that the points of a minimal -design are uniformly distributed on the sphere. We construct a weighted cubature formula on the sphere with the minimum number of nodes.

Almost sure weak convergence for the circular ensembles of Dyson

The circular ensembles of Dyson satisfy isoperimetric inequalities and concentration of measure phenomena for large particle numbers analogous to the isoperimetric inequality for surface measure on the sphere in Euclidean space of high dimension. This leads to a geometrical proof of a result of Johansson [Bull. Sci. Math. (2) 112, (1988), 257–304] that the empirical distribution of energy levels under such ensembles converge weakly almost surely to normalized arclength on the unit circle as

Approximation with interpolatory constraints

Given a triangular array of points on [ − 1 , 1 ] [-1,1] satisfying certain minimal separation conditions, a classical theorem of Szabados asserts the existence of polynomial operators that provide interpolation at these points as well as a near-optimal degree of approximation for arbitrary continuous functions on the interval. This paper provides a simple, functional-analytic proof of this fact. This abstract technique also leads to similar results in general situations where an analogue of the classical Jackson-type theorem holds. In particular, it allows one to obtain simultaneous interpolation and a near-optimal degree of approximation by neural networks on a cube, radial-basis functions on a torus, and Gaussian networks on Euclidean space. These ideas are illustrated by a discussion of simultaneous approximation and interpolation by polynomials and also by zonal-function networks on the unit sphere in Euclidean space.

On geometric properties of stochastic flows related to the Lyapunov spectrum

We study the geometric properties of two stochastic flows on spheres in Euclidean space. The underlying one-point motion in both cases is Brownian. Both flows arise from the action of a Lie group valued Brownian motion on a quotient. For both flows the curvature of a curve moving under the flow is shown to be a diffusion, null recurrent in one case and transient in the other.

Continuous causal representation of space–time via sphere orderings

A space–time can be regarded as a partially ordered set, or poset, where the partial ordering is causal precedence. The question of when this poset can be represented by spheres in Euclidean space is investigated, and it is shown that spaces which can be so represented are rare, if the position and radius of the spheres are required to be continuous functions on the space–time.

On geometric properties of stochastic flows related to the Lyapunov spectrum

We study the geometric properties of two stochastic flows on spheres in Euclidean space. The underlying one-point motion in both cases is Brownian. Both flows arise from the action of a Lie group valued Brownian motion on a quotient. For both flows the curvature of a curve moving under the flow is shown to be a diffusion, null recurrent in one case and transient in the other.