Efficient Spatial Modeling Using the SPDE Approach With Bivariate Splines

Gaussian fields (GFs) are frequently used in spatial statistics for their versatility. The associated computational cost can be a bottleneck, especially in realistic applications. It has been shown that computational efficiency can be gained by doing the computations using Gaussian Markov random fields (GMRFs) as the GFs can be seen as weak solutions to corresponding stochastic partial differential equations (SPDEs) using piecewise linear finite elements. We introduce a new class of representations of GFs with bivariate splines instead of finite elements. This allows an easier implementation of piecewise polynomial representations of various degrees. It leads to GMRFs that can be inferred efficiently and can be easily extended to nonstationary fields. The solutions approximated with higher order bivariate splines converge faster, hence the computational cost can be alleviated. Numerical simulations using both real and simulated data also demonstrate that our framework increases the flexibility and efficiency. Supplementary materials are available online.     

Uncertainty principles associated with quaternionic linear canonical transforms

In the present paper, we generalize the linear canonical transform (LCT) to quaternion‐valued signals, known as the quaternionic LCT (QLCT). Using the properties of the LCT, we establish an uncertainty principle for the two‐sided QLCT. This uncertainty principle prescribes a lower bound on the product of the effective widths of quaternion‐valued signals in the spatial and frequency domains. It is shown that only a Gaussian quaternionic signal minimizes the uncertainty. Copyright © 2016 John Wiley & Sons, Ltd.     

Some subgradient extragradient type algorithms for solving split feasibility and fixed point problems

In this paper, we consider the split feasibility problem (SFP) in infinite‐dimensional Hilbert spaces and propose some subgradient extragradient‐type algorithms for finding a common element of the fixed‐point set of a strict pseudocontraction mapping and the solution set of a split feasibility problem by adopting Armijo‐like stepsize rule. We derive convergence results under mild assumptions. Our results improve some known results from the literature. Copyright © 2016 John Wiley & Sons, Ltd.     

Noncyclic mappings and best proximity pair results in Hilbert and uniformly convex Banach spaces

A new class of noncyclic mappings, called generalized noncyclic relatively nonexpansive, is introduced and used to study the existence of best proximity pairs in the setting of uniformly convex Banach spaces. We also obtain a weak convergence theorem for noncyclic relatively nonexpansive mappings in the setting of Hilbert spaces.     

Second-order hyperbolic s.p.d.e.'s driven by homogeneous Gaussian noise on a hyperplane

We study a class of hyperbolic stochastic partial differential equations in Euclidean space, that includes the wave equation and the telegraph equation, driven by Gaussian noise concentrated on a hyperplane. The noise is assumed to be white in time but spatially homogeneous within the hyperplane. Two natural notions of solutions are function-valued solutions and random field solutions. For the linear form of the equations, we identify the necessary and sufficient condition on the spectral measure of the spatial covariance for existence of each type of solution, and it turns out that the conditions differ. In spatial dimensions 2 and 3, under the condition for existence of a random field solution to the linear form of the equation, we prove existence and uniqueness of a random field solution to non-linear forms of the equation.

     

On the Equivalence of Multiparameter Gaussian Processes

Our purpose is to characterize the multiparameter Gaussian processes, that is Gaussian sheets, that are equivalent in law to the Brownian sheet and to the fractional Brownian sheet. We survey multiparameter analogues of the Hitsuda, Girsanov and Shepp representations. As an application, we study a special type of stochastic equation with linear noise.      

Leggett-Williams norm-type theorems for coincidences

We study the existence of positive solutions to the operator equation Lx = Nx, where L is a linear Fredholm mapping of index zero and N is a nonlinear operator. Using the properties of cones in Banach spaces and Leray-Schauder degree for completely continuous operators, k-set contractions and condensing mappings, we obtain some refinements of the results established in [3] and [14]. Received: 9 July 2005; revised: 12 January 2006     

Limit theorems for the canonical von Mises statistics with dependent data

We study the limit behavior of the canonical (i.e., degenerate) von Mises statistics based on samples from a sequence of weakly dependent stationary observations satisfying the ψ-mixing condition. The corresponding limit distributions are defined by the multiple stochastic integrals of nonrandom functions with respect to the nonorthogonal Hilbert noises generated by Gaussian processes with nonorthogonal increments.     

LITTLEWOOD-PALEY THEOREM FOR SCHR(O)DINGER OPERATORS

Let H be a Schr(o)dinger operator on Rn. Under a polynomial decay condition for the kernel of its spectral operator, we show that the Besov spaces and Triebel-Lizorkin spaces associated with H are well defined. We further give a Littlewood-Paley characterization of Lp spaces in terms of dyadic functions of H. This generalizes and strengthens the previous result when the heat kernel of H satisfies certain upper Gaussian bound.     

The Galois representations associated to a Drinfeld module in special characteristic—III: Image of the group ring

Let K be a finitely generated field of transcendence degree 1 over a finite field, and set G_K?Gal(K^sep/K). Let φ be a Drinfeld A-module over K in special characteristic. Set E?End_K(φ) and let Z be its center. We show that for almost all primes p of A, the image of the group ring A_p[G_K] in End_A(T_p(φ)) is the commutant of E. Thus, for almost all p it is a full matrix ring over Z⊗_AA_p. In the special case E=A it follows that the representation of G_K on the p-torsion points φ[p] is absolutely irreducible for almost all p.