Strictly and non-strictly positive definite functions on spheres

Isotropic positive definite functions on spheres play important roles in spatial statistics, where they occur as the correlation functions of homogeneous random fields and star-shaped random particles. In approximation theory, strictly positive definite functions serve as radial basis functions for interpolating scattered data on spherical domains. We review characterizations of positive definite functions on spheres in terms of Gegenbauer expansions and apply them to dimension walks, where monotonicity properties of the Gegenbauer coefficients guarantee positive definiteness in higher dimensions. Subject to a natural support condition, isotropic positive definite functions on the Euclidean space $\mathbb{R}^3$, such as Askey's and Wendland's functions, allow for the direct substitution of the Euclidean distance by the great circle distance on a one-, two- or three-dimensional sphere, as opposed to the traditional approach, where the distances are transformed into each other. Completely monotone functions are positive definite on spheres of any dimension and provide rich parametric classes of such functions, including members of the powered exponential, Mat\'{e}rn, generalized Cauchy and Dagum families. The sine power family permits a continuous parameterization of the roughness of the sample paths of a Gaussian process. A collection of research problems provides challenges for future work in mathematical analysis, probability theory and spatial statistics.Comment: Published in at http://dx.doi.org/10.3150/12-BEJSP06 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm

Bounds on Effective Dynamic Properties of Elastic Composites

We present general, computable, improvable, and rigorous bounds for the total energy of a finite heterogeneous volume element or a periodically distributed unit cell of an elastic composite of any known distribution of inhomogeneities of any geometry and elasticity, undergoing a harmonic motion at a fixed frequency or supporting a single-frequency Bloch-form elastic wave of a given wave-vector. These bounds are rigorously valid for \emph{any consistent boundary conditions} that produce in the finite sample or in the unit cell, either a common average strain or a common average momentum. No other restrictions are imposed. We do not assume statistical homogeneity or isotropy. Our approach is based on the Hashin-Shtrikman (1962) bounds in elastostatics, which have been shown to provide strict bounds for the overall elastic moduli commonly defined (or actually measured) using uniform boundary tractions and/or linear boundary displacements; i.e., boundary data corresponding to the overall uniform stress and/or uniform strain conditions. Here we present strict bounds for the dynamic frequency-dependent constitutive parameters of the composite and give explicit expressions for a direct calculation of these bounds

Generalized Convolution Roots of Positive Definite Kernels on Complex Spheres

Convolution is an important tool in the construction of positive definite kernels on a manifold. This contribution provides conditions on an $L^2$-positive definite and zonal kernel on the unit sphere of $\mathbb{C}^q$ in order that the kernel can be recovered as a generalized convolution root of an equally positive definite and zonal kernel

M-tensors and The Positive Definiteness of a Multivariate Form

We study M-tensors and various properties of M-tensors are given. Specially, we show that the smallest real eigenvalue of M-tensor is positive corresponding to a nonnegative eigenvector. We propose an algorithm to find the smallest positive eigenvalue and then apply the property to study the positive definiteness of a multivariate form

Scalar tachyons in the de Sitter universe

We provide a construction of a class of local and de Sitter covariant tachyonic quantum fields which exist for discrete negative values of the squared mass parameter and which have no Minkowskian counterpart. These quantum fields satisfy an anomalous non-homogeneous Klein-Gordon equation. The anomaly is a covariant field which can be used to select the physical subspace (of finite codimension) where the homogeneous tachyonic field equation holds in the usual form. We show that the model is local and de Sitter invariant on the physical space. Our construction also sheds new light on the massless minimally coupled field, which is a special instance of it.Comment: 9 page