2,436 research outputs found
On power series expansions of the S-resolvent operator and the Taylor formula
The -functional calculus is based on the theory of slice hyperholomorphic
functions and it defines functions of -tuples of not necessarily commuting
operators or of quaternionic operators. This calculus relays on the notion of
-spectrum and of -resolvent operator. Since most of the properties that
hold for the Riesz-Dunford functional calculus extend to the S-functional
calculus it can be considered its non commutative version. In this paper we
show that the Taylor formula of the Riesz-Dunford functional calculus can be
generalized to the S-functional calculus, the proof is not a trivial extension
of the classical case because there are several obstructions due to the non
commutativity of the setting in which we work that have to be overcome. To
prove the Taylor formula we need to introduce a new series expansion of the
-resolvent operators associated to the sum of two -tuples of operators.
This result is a crucial step in the proof of our main results,but it is also
of independent interest because it gives a new series expansion for the
-resolvent operators. This paper is devoted to researchers working in
operators theory and hypercomplex analysis
Testing for Bivariate Spherical Symmetry
An omnibus test for spherical symmetry in R2 is proposed, employing localized empirical likelihood. The thus obtained test statistic is distri- bution-free under the null hypothesis. The asymptotic null distribution is established and critical values for typical sample sizes, as well as the asymptotic ones, are presented. In a simulation study, the good perfor- mance of the test is demonstrated. Furthermore, a real data example is presented.Asymptotic distribution;distribution-free;empirical like- lihood;hypothesis test;spherical symmetry.
The Half-Half Plot
The Half-Half (HH) plot is a new graphical method to investigate qualitatively the shape of a regression curve. The empirical HH-plot counts observations in the lower and upper quarter of a strip that moves horizontally over the scatter plot. The plot displays jumps clearly and reveals further features of the regression curve. We prove a functional central limit theorem for the empirical HH-plot, with rate of convergence 1=p n. In a simulation study the good performance of the plot is demonstrated. The method is also applied to two case studies.Data analysis;functional central limit theorem;graphical methods;jump detection;nonparametric regression
Reliable and efficient a posteriori error estimation for adaptive IGA boundary element methods for weakly-singular integral equations
We consider the Galerkin boundary element method (BEM) for weakly-singular
integral equations of the first-kind in 2D. We analyze some residual-type a
posteriori error estimator which provides a lower as well as an upper bound for
the unknown Galerkin BEM error. The required assumptions are weak and allow for
piecewise smooth parametrizations of the boundary, local mesh-refinement, and
related standard piecewise polynomials as well as NURBS. In particular, our
analysis gives a first contribution to adaptive BEM in the frame of
isogeometric analysis (IGABEM), for which we formulate an adaptive algorithm
which steers the local mesh-refinement and the multiplicity of the knots.
Numerical experiments underline the theoretical findings and show that the
proposed adaptive strategy leads to optimal convergence
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