on the optimal shape parameter for gaussian radial basis function finite difference approximation of the poisson equation we investigate the influence of the shape parameter in the meshless gaussian radial basis function finite difference rbf fd method with irregular centres on the quality of the approximation of the dirichlet problem for the poisson equation with smooth solution numerical experiments show that the optimal shape parameter strongly depends on the problem but insignificantly on the density of the centres therefore we suggest a multilevel algorithm that effectively finds a near optimal shape parameter which helps to significantly reduce the error comparison to the finite element method and to the generalised finite differences obtained in the flat limits of the gaussian rbf is provided

an h adaptive method in the generalized finite differences abstract this paper describes an h adaptive method in generalized finite difference gfd to solve second order partial differential equations these equations representing the behaviour of many physical processes the explicit difference formulae obtained make it possible to propose an a posteriori error indicator which serves as starting point for an h adaptive method to improve the solution by selectively adding nodes to the domain this paper also analyses the influence of key parameters as the number of nodes to add in each step or the minimum distance between nodes through the analysis of the obtained solutions for different types of differential equations

two stage approximation methods with extended b splines we develop and analyze a framework for two stage methods with eb splines applicable to continuous and discrete approximation problems in particular we propose a weighted discrete least squares fit that yields optimal convergence rates for sufficiently dense data on lipschitz domains in rd

minimal positive stencils in meshfree finite difference methods for the poisson equation meshfree finite difference methods for the poisson equation approximate the laplace operator on a point cloud desirable are positive stencils i e all neighbor entries are of the same sign classical least squares approaches yield large stencils that are in general not positive we present an approach that yields stencils of minimal size which are positive we provide conditions on the point cloud geometry so that positive stencils always exist the new discretization method is compared to least squares approaches in terms of accuracy and computational performance

error estimates for scattered data interpolation on spheres we study sobolev type estimates for the approximation order resulting from using strictly positive definite kernels to do interpolation on the n sphere the interpolation knots are scattered our approach partly follows the general theory of golomb and weinberger and related estimates these error estimates are then based on series expansions of smooth functions in terms of spherical harmonics the markov inequality for spherical harmonics is essential to our analysis and is used in order to find lower bounds for certain sampling operators on spaces of spherical harmonics

adaptive rbf fd method for elliptic problems with point singularities in 2d abstract we describe and test numerically an adaptive meshless generalized finite difference method based on radial basis functions that competes well with the finite element method on standard benchmark problems with reentrant corners of the boundary sharp peaks and rapid oscillations in the neighborhood of an isolated point this is achieved thanks to significant improvements introduced into the earlier algorithms of davydov and oanh 2011 including a new error indicator of zienkiewicz zhu type

divisibility and laws in finite simple groups we provide new bounds for the divisibility function of the free group mathbf f 2 and construct short laws for the symmetric groups mathrm sym n the construction is random and relies on the classification of the finite simple groups we also give bounds on the length of laws for finite simple groups of lie type

on the orders of primitive groups abstract almost all primitive permutation groups of degree n have order at most n i 0 log 2 n 1 n 2 i 1 log 2 n or have socle isomorphic to a direct power of some alternating group the mathieu groups m11 m12 m23 and m24 are the four exceptions as a corollary the sharp version of a theorem of praeger and saxl is established where m12 turns out to be the largest primitive group for an application a bound on the orders of permutation groups without large alternating composition factors is given this sharpens a lemma of babai cameron palfy and generalizes a theorem of dixon

effective bounds for the maximal order of an element in the symmetric group cet article donne des bornes effectives pour g n l ordre maximal des permutations d un groupe symetrique a n elements et p g n le plus grand diviseur premier de g n des preuves theoriques sont dans un article anterieur les outils utilises ici sont les techniques des nombres superieurs hautement composites dans le sens de ramanujan et les bornes de rosser et schoenfeld sur la fonction de chebyshev

set measure directed multi source information fusion our concern here is with the multi source fusion problem two important aspects of this problem are the representation of information provided by the sources and the formulation of the instructions on how to fuse the information provided which we refer to as the fusion imperative we investigate the use of a monotonic set measure as a means of representing the fusion imperative we look at the fusion of various different types of information precise data uncertain information such as probabilistic and possibilistic we also consider the case of imprecise uncertain information such as that represented by a dempster shafer belief structure

intersections of matrix algebras and permutation representations of psl n q abstract if g is a group h a subgroup of g and a transitive g set we ask under what conditions one can guarantee that h has a regular orbit of size h on here we prove that if psl n q g pgl n q and h is cyclic then h has a regular orbit in every non trivial g set with few exceptions this result is obtained via a mixture of group theoretical and ring theoretical methods let r be the ring of all n n matrices over the finite field f and let z be the subring of scalar matrices we show that if a and m are proper subrings of r containing z and if a is commutative and semisimple then there exists an element x sl n f such that xax 1 m z or n 2 f

error bounds for kernel based numerical differentiation the literature on meshless methods shows that kernel based numerical differentiation formulae are robust and provide high accuracy at low cost this paper analyzes the error of such formulas using the new technique of growth functions it allows to bypass certain technical assumptions that were needed to prove the standard error bounds on interpolants and their derivatives since differentiation formulas based on polynomials also have error bounds in terms of growth functions we have a convenient way to compare kernel based and polynomial based formulas it follows that kernel based formulas are comparable in accuracy to the best possible polynomial based formulas a variety of examples is provided

bivariate polynomial interpolation on the square at new nodal sets abstract as known the problem of choosing good nodes is a central one in polynomial interpolation while the problem is essentially solved in one dimension all good nodal sequences are asymptotically equidistributed with respect to the arc cosine metric in several variables it still represents a substantially open question in this work we consider new nodal sets for bivariate polynomial interpolation on the square first we consider fast leja points for tensor product interpolation on the other hand for interpolation in p n 2 on the square we experiment four families of points which are asymptotically equidistributed with respect to the dubiner metric which extends to higher dimension the arc cosine metric one of them nicknamed padua points gives numerically a lebesgue constant growing like log square of the degree

on the minimal degree of a primitive permutation group abstract we improve a result of liebeck and saxl concerning the minimal degree of a primitive permutation group and use it to strengthen a result of guralnick and neubauer on generic covers of riemann surfaces

on the maximum orders of elements of finite almost simple groups and primitive permutation groups we determine upper bounds for the maximum order of an element of a finite almost simple group with socle t in terms of the minimum index m t of a maximal subgroup of t for t not an alternating group we prove that with finitely many excep tions the maximum element order is at most m t moreover apart from an explicit list of groups the bound can be reduced to m t 4 these results are applied to determine all primitive permutation groups on a set of size n that contain permutations of order greater than or equal to n 4 we note again that this result gives upper bounds for meo aut t in terms of m t and for meo g in terms of m g since m t m g moreover equality in the up per bound meo aut t m t holds when t psld q for all but two pairs d q see table 3 and theorem 2 16 theorem 2 16 and table 3 provide good estimates for

finite primitive permutation groups and regular cycles of their elements we conjecture that if g is a finite primitive group and if g is an element of g then either the element g has a cycle of length equal to its order or for some r m and k the group gsym m wrsym r preserving a prod uct structure of r direct copies of the natural action of sym m or alt m on k sets in this paper we reduce this conjecture to the case that g is an almost simple group with socle a classical group

on the fusion of imprecise uncertainty measures using belief structures our interest is in the fusion of information from multiple sources when the information provided by the individual sources is expressed in terms of an imprecise uncertainty measure we observe that the dempster shafer belief structure provides a framework for the representation of a wide class of imprecise uncertainty measures we then discuss the fusion of multiple dempster shafer belief structures using the dempster rule and note the problems that can arise when using this fusion method because of the required normalization in the face of conflicting focal elements we then suggest some alternative approaches fusing multiple belief structures that avoid the need for normalization

regular orbits of cyclic subgroups in permutation representations of certain simple groups if h acts on then an h orbit is regular if its cardinality is h the alternating groups already in their natural representation do not have the property of the theorem hence the exception the other known simple groups with a doubly transitive permutation representation are psl n q sp 2n 2 two representations u3 q b2 q g2 q and a short list of sporadic examples which are reproduced in section 5 if one assumes the completeness of the classification of finite simple groups then these are all doubly transitive representations of finite simple groups and the word known can be omitted in the theorem in our paper 8 the theorem 1 1 was proved for psl n q here we consider the remaining doubly transitive groups the same method can in

minimal numerical differentiation formulas we investigate numerical differentiation formulas on irregular centers in two or more variables that are exact for polynomials of a given order and minimize an absolute seminorm of the weight vector error bounds are given in terms of a growth function that carries the information about the geometry of the centers specific forms of weighted ell 1 and weighted least squares minimization are proposed that produce numerical differentiation formulas with particularly good performance in numerical experiments the results are of interest in particular for meshless generalized finite difference methods as they provide a consistency error analysis for such methods

a classification of the maximal subgroups of the finite alternating and symmetric groups following the classification of finite simple groups one of the major problems in finite group theory today is the determination of the maximal subgroups of the almost simple groups that is of groups x such that x0 u x aut x for some finite non abelian simple group x0 the problem has been solved for most sporadic groups and groups of lie type of low rank see 19 sects 3 43 for discussion and references this paper is a contribution to the case where x0 is an alternating group a so that for n 6 x is a or s the maximal subgroups of a and s are known for several classes of degrees n

adaptive meshless centres and rbf stencils for poisson equation we consider adaptive meshless discretisation of the dirichlet problem for poisson equation based on numerical differentiation stencils obtained with the help of radial basis functions new meshless stencil selection and adaptive refinement algorithms are proposed in 2d numerical experiments show that the accuracy of the solution is comparable with and often better than that achieved by the mesh based adaptive finite element method