the center of the generic division ring and twisted multiplicative group actions abstract let f be a field and let p be a prime the problem we study is whether the center cp of the division ring of p p generic matrices is stably rational over f given a finite group g and a z g lattice we let f m be the quotient field of the group algebra of the abelian group m procesi and formanek linear multilinear algebra 7 1979 203 212 have shown that for all n there is a z s n lattice gn such that cn is stably isomorphic to the fixed field under the action of sn of f gn let h be a p sylow subgroup of sp let a be the root lattice and let l f z s p h we show that there exists an action of sp on l z s p z h a twisted by an element ext 1 s p z s p z h a l such that l z s p z h a s p is stably isomorphic to cp the extension corresponds to an element of the relative brauer group of l over lh since z s p z h a and z s p h are quasi permutations l z s p z h a s p is stably rational over f however it is not known whether l z s p z h a s p is stably rational over f thus the result represents a reduction on the problem since z s p z h a is quasi permutation however the twist introduces a new level of complexity

the zak transform and sampling theorems for wavelet subspaces the zak transform is used for generalizing a sampling theorem of g waiter see ieee trans informat theory vol 38 p 881 884 1992 for wavelet subspaces cardinal series based on signal samples f a n n in z with a possibly unequal to 0 waiter s case are considered the condition number of the sampling operator and worst case aliasing errors are expressed in terms of zak transforms of scaling function and wavelet this shows that the stability of the resulting interpolation formula depends critically on a

a sufficient condition for the convergence of the inexact uzawa algorithm for saddle point problems in this paper the convergence property of the inexact uzawa algorithm for solving symmetric indefinite linear systems is studied a simple sufficient condition for the convergence of the inexact uzawa algorithm is obtained two examples and numerical experiments illustrating the conclusion are provided

uzawa type algorithms for nonsymmetric saddle point problems in this paper we consider iterative algorithms of uzawa type for solving linear nonsymmetric saddle point problems specifically we consider systems written as usual in block form where the upper left block is an invertible linear operator with positive definite symmetric part such saddle point problems arise for example in certain finite element and finite difference discretizations of navier stokes equations oseen equations and mixed finite element discretization of second order convection diffusion problems we consider two algorithms each of which utilizes a preconditioner for the operator in the upper left block convergence results for the algorithms are established in appropriate norms the convergence of one of the algorithms is shown assuming only that the preconditioner is spectrally equivalent to the inverse of the symmetric part of the operator the other algorithm is shown to converge provided that the preconditioner is a sufficiently accurate approximation of the inverse of the upper left block applications to the solution of steady state navier stokes equations are discussed and finally the results of numerical experiments involving the algorithms are presented

a sampling theorem for wavelet subspaces the classical shannon sampling theorem is extended to the subspaces used in the multiresolution analysis in wavelet theory under weak hypotheses these subspaces are first shown to have a riesz basis formed from the reproducing kernels these in turn are used to construct the sampling sequences examples are given

an optimal preconditioner for a class of saddle point problems with a penalty term iterative methods are considered for a class of saddle point problems with a penalty term arising from finite element discretizations of certain elliptic problems an optimal preconditioner which is independent of the discretization and the penalty parameter is constructed this approach is then used to design an iterative method with a convergence rate bounded independently of the lame parameters occurring in the equations of linear elasticity

lattice invariants and the center of the generic division ring let g be a finite group let m be a zg lattice and let f be a field of characteristic zero containing primitive p th roots of 1 let f m be the quotient field of the group algebra of the abelian group m it is well known that if m is quasi permutation and g faithful then f m g is stably equivalent to f zg g let c n be the center of the division ring of n n generic matrices over f let s n be the symmetric group on n symbols let p be a prime we show that there exist a split group extension g of sp by a p elementary group a g faithful quasi permutation zg lattice m and a one cocycle in ext 1 g m f such that cp is stably isomorphic to f m g this represents a reduction of the problem since we have a quasi permutation action however the twist introduces a new level of complexity the second result which is a consequence of the first is that if f is algebraically closed there is a group extension e of sp by an abelian p group such that cp is stably equivalent to the invariants of the noether setting f e

irregular sampling for spline wavelet subspaces spline wavelets spl psi sub m t are important in time frequency localization due to i spl psi sub m can be arbitrarily close to the optimal case as m is sufficiently large ii spl psi sub m has compact support and simple analytic expression which lead to effective computation although the spline wavelet subspaces are so simple walter s well known sampling theorem does not hold if the order of spline m is even moreover when irregular sampling is considered in these spaces it is hard to determine the sampling density which is a serious problem in applications in this correspondence a general sampling theorem is obtained for m spl ges 3 in the sense of iterative construction and the sampling density spl delta sub m is estimated

stable rationality of certain invariant fields abstract let f be a field for a finite group g let f g be the purely transcendental extension of f with transcendency basis x g g g let f g g denote the fixed field of f g under the action of g let w be a primitive p 1 st root of 1 and let i be the ideal p w a in z w where a is a primitive p 1 st root of 1mod p we show that if g be the semi direct product of a cyclic group of order p by a cyclic group of order prime to p if i is principal and if f contains a primitive g th root of 1 then f g g is stably rational over f it is not known whether the set of primes p for which i is principal is finite or infinite we also show that if p is an odd prime and g is a non abelian group of order p 3 then f g g is stably rational over f provided that f contains a primitive g th root of 1

sampling theorem and irregular sampling theorem for multiwavelet subspaces we generalize the multiwavelet sampling theorem by reproducing a kernel that is easy to use multiscaling functions that have orthogonality regularity short compact support symmetry and high approximation order satisfy the conditions in the sampling theorem which is not possible in the case of the scalar wavelet we consider the general cases of the uniform noninteger and irregular sampling a reconstruction from more general sets of points is necessary since the measurements may not be made at uniform points finally we establish a general irregular sampling theorem for multiwavelet subspaces and derive an estimate for the perturbations of uniform noninteger sampling in shift invariant spaces

the convergence factor of preconditioned algorithms of the arrow hurwicz type the convergence properties of the arrow hurwicz algorithm for solving mixed finite element problems are studied using a purely algebraic technique the convergence factor of the algorithm will be estimated the iteration parameters depending on the smallest and greatest eigenvalues of some generalized eigenvalue problems can be chosen optimally for special cases the convergence factor of the preconditioned arrow hurwicz algorithm is independent of the discretization parameter finally a quite general method that allows us to construct preconditioning operators of the arrow hurwicz algorithm for a wide class of problems is presented

on the ring of 13 13 generic matrices abstract let f be a field of characteristic 0 we investigate the rationality of the center c n of the division algebra of two n n generic matrices over f for n 13 if n is a prime power then the only cases for which c n is known to be stably rational over f are n 2 3 4 5 and 7 with rationality proven for 2 3 and 4 there is a certain z s n lattice denoted a which has played an essential role in the proofs of these results formanek proved the case n 4 by showing that c n is stably isomorphic to f a s n the invariants of f a under the action of s n here a is a z and z is the sign representation of s n lebruyn and bessenrodt proved the cases n 5 and 7 by showing that c n is stably isomorphic to f a s n we show that for n 13 there exists a z s n lattice m which is stably permutation when restricted to the alternating group such that f a m s n and c n are stably isomorphic we then show that a field extension of degree 2 of c n is stably isomorphic to a field extension of degree 2 of a rational extension of f

induction theorems on the stable rationality of the center of the ring of generic matrices following procesi and formanek the center of the division ring of rn x n generic matrices over the complex numbers c is stably equivalent to the fixed field under the action of sn of the function field of the group algebra of a zsn lattice denoted by gn we study the question of the stable rationality of the center cn over the complex numbers when rn is a prime in this module theoretic setting let n be the normalizer of an rn sylow subgroup of sn let m be a zsn lattice we show that under certain conditions on m inductionrestriction from n to sn does not affect the stable type of the corresponding field in particular c gn and c zg0zn gn are stably isomorphic and the isomorphism preserves the sn action we further reduce the problem to the study of the localization of gn at the prime n all other primes behave well we also present new simple proofs for the stable rationality of cn over c in the cases r 5 and rn 7

on the iterative algorithm for saddle point problems abstract in this paper we point out that the generalized iteration method presented in a recent paper by ling and hu x f ling x z hu on the iterative algorithm for large sparse saddle point problems appl math comput 178 2006 372 379 belongs to the inexact uzawa algorithm and an improper statement in a theorem in that paper is corrected

analysis of the inexact uzawa algorithm for saddle point problems in this paper we consider the so called inexact uzawa algorithm for iteratively solving linear block saddle point problems such saddle point problems arise for example in finite element and finite difference discretizations of stokes equations the equations of elasticity and mixed finite element discretization of second order problems we consider both the linear and nonlinear variants of the inexact uzawa iteration we show that the linear method always converges as long as the preconditioners defining the algorithm are properly scaled bounds for the rate of convergence are provided in terms of the rate of convergence for the preconditioned uzawa algorithm and the reduction factor corresponding to the preconditioner for the upper left hand block in the case of nonlinear iteration the inexact uzawa algorithm is shown to converge provided that the nonlinear process approximating the inverse of the upper left hand block is of sufficient accuracy bounds for the nonlinear iteration are given in terms of this accuracy parameter and the rate of convergence of the preconditioned linear uzawa algorithm applications to the stokes equations and mixed finite element discretization of second order elliptic problems are discussed and finally the results of numerical experiments involving the algorithms are presented

reconstruction algorithms in irregular sampling a constructive solution of the irregular sampling problem for band limited functions is given we show how a band limited function can be com pletely reconstructed from any random sampling set whose density is higher than the nyquist rate and give precise estimates for the speed of convergence of this iteration method variations of this algorithm allow for irregular sampling with derivatives reconstruction of band limited functions from local averages and irregular sampling of multivariate band limited functions in the irregular sampling problem one is asked whether and how a band limited function can be completely reconstructed from its irregularly sam pled values f xi this has many applications in signal and image processing seismology meteorology medical imaging etc finding constructive solutions of this problem has received considerable attention among mathematicians and engineers the mathematical literature provides several uniqueness results 1 2 17 18 19 it is now part of the folklore that for stable sampling the sampling rate must be at least the nyquist rate 18 these results as deep as they are have had little impact for the applied sciences because they were not constructive if the sampling set is just a perturbation of the regular oversampling then a reconstruction method has been obtained in a seminal paper by duffin and schaeffer 6 see also 29 if for some l 0 a 0 and o 0 the sampling points xk k e z satisfy a xk ok a k i then the norm equivalence a ir f x 2dx with w n o this norm equivalence implies that it is possible to reconstruct through an iterative procedure the so called frame method most of the later work on constructive methods consists of variations of this method 3 21 22 26 the above conditions on the sampling set exclude random irregular sampling sets e g sets with regions of higher sampling density a partial but undesirable remedy to handle highly irregular sampling sets would be to force the above conditions by throwing away information on part of the points and accept a very slow convergence of the iteration

a preconditioning technique for indefinite systems resulting from mixed approximations of elliptic problems this paper provides a preconditioned iterative technique for the solution of saddle point problems these problems typically arise in the numerical approximation of partial differential equations by lagrange multiplier techniques and or mixed methods the saddle point problem is reformulated as a symmetric positive definite system which is then solved by conjugate gradient iteration applications to the equations of elasticity and stokes are discussed and the results of numerical experiments are given

monomial actions of the symmetric group abstract let f be a field let g be a finite group and let m be a g faithful zg lattice we investigate the stable rationality of f m g over f when g is s p the symmetric group on p letters and p is a prime it follows from work of endo miyata lenstra and swan that g faithful zg lattices are in the same flasque class if and only if they have g isomorphic corresponding fields thus the study of flasque classes of zs p lattices plays a fundamental role in this investigation let n be the normalizer of a p sylow subgroup of s p we show that there are classes of zs p lattices for which induction restriction from n to s p does not affect the flasque class in particular we study the flasque class of a specific lattice g p which has the property that f g p s p is stably isomorphic to the center of the division ring of generic matrices formanek linear and multilinear algebra 7 1979 203 212 for a finite group g lattices in the same genus are not in general in the same flasque class however they are for g s n we show that there is a class of zs p lattices containing the genus of g p whose elements are in the flasque of g p

stable rationality of certain pgln quotients stable rationality of the field of matrix invariants m n m n pgl n is proved forn 5 andn 7 in combination with existing results this shows that v pgl n is stably rational wheneverv is an almost free representation ofpgl n andn divides 420 22 3 5 7

wavelets of multiplicity r a multiresolution approximation km m z of l2 r is of multi plicity r 0 if there are r functions x r whose translates form a riesz basis for v in the general theory we derive necessary and sufficient conditions for the translates of x r y x y r to form a riesz basis for v the resulting reconstruction and decomposition sequences lead to the construction of dual bases for v0 and its orthogonal complement w0 in vx the general theory is applied in the construction of spline wavelets with mul tiple knots algorithms for the construction of these wavelets for some special cases are given

lattices over cyclic groups and noether settings abstract for each positive integer m let m be a primitive mth root of 1 given a prime p let a be an integer which is a primitive root of 1 mod p let i p m be the ideal in z m generated by p and m a let g be a cyclic group of order n such that the flasque class of any maximal projective ideal in z g contains a maximal ideal u of index p for some prime p such that n divides p 1 and such that i p n is principal it is not known whether the number of groups satisfying this property is finite or infinite we show that if f is a field of characteristic zero containing all nth roots of 1 then for any z g lattice m the fixed subfield of f m is stably rational over f if f also contains primitive mth roots of 1 for all natural numbers m then for any finite g module t the fixed subfield of the noether setting of the group g t g is stably rational over f

global and superlinear convergence of inexact uzawa methods for saddle point problems with nondifferentiable mappings this paper investigates inexact uzawa methods for nonlinear saddle point problems we prove that the inexact uzawa method converges globally and superlinearly even if the derivative of the nonlinear mapping does not exist we show that the newton type decomposition method for saddle point problems is a special case of a newton uzawa method we discuss applications of inexact uzawa methods to separable convex programming problems and coupling of finite elements boundary elements for nonlinear interface problems

weighted sampling and reconstruction in weighted reproducing kernel spaces in this paper we discuss sampling and reconstruction of signals in the weighted reproducing kernel space associated with an idempotent integral operator we show that any signal in that space can be stably reconstructed from its weighted samples taken on a relatively separated set with sufficiently small gap we also develop an iterative reconstruction algorithm for the reconstruction of a signal from its weighted samples taken on a relatively separated set with sufficiently small gap

on preconditioned uzawa methods and sor methods for saddle point problems this paper studies convergence analysis of a preconditioned inexact uzawa method for nondifferentiable saddle point problems the sor newton method and the sor bfgs method are special cases of this method we relax the bramble pasciak vassilev condition on preconditioners for convergence of the inexact uzawa method for linear saddle point problems the relaxed condition is used to determine the relaxation parameters in the sor newton method and the sor bfgs method furthermore we study global convergence of the multistep inexact uzawa method for nondifferentiable saddle point problems