an a posteriori parameter choice for tikhonov regularization in hilbert scales leading to optimal convergence rates an a posteriori parameter choice for tikhonov regularization in hilbert scales is proposed that leads to optimal convergence rates the advantage of this method is that knowledge of the smoothness parameters of both the operator and the exact solution is not needed the finite dimensional realization of this method is also discussed numerical examples confirm the theoretical results

on the distribution of a random variable occurring in 1d disordered systems the authors consider the random variable z 1 x1 x1x2 x1x2x3 where the xi are independent identically distributed variables they derive some asymptotic properties of the distribution of z which are related e g to the low temperature behaviour of the random field ising chain for a special class of distributions of the xi exact solutions are presented they also study the cases where the distribution function of z exhibits a power law fall off modulated by a periodic critical amplitude

geometrically reductive hopf algebras and their invariants abstract by analogy with the mumford definition of geometrically reductive algebraic group we introduce the concept of geometrically reductive hopf algebra over a field then we prove that if h is a geometrically reductive hopf algebra and a is a commutative finitely generated and locally finite h module algebra then the algebra of invariants a h is finitely generated we also prove that in characteristic 0 a hopf algebra h is geometrically reductive if and only if every finite dimensional h module is semisimple and that in positive characteristic every finite dimensional hopf algebra is geometrically reductive finally we prove that in positive characteristic the quantum enveloping hopf algebras u q sl n n 2 are geometrically reductive for any parameter q 1

locally nilpotent derivations over a ufd and an application to rank two locally nilpotent derivations ofk x1 xn abstract given a ufd r containing the rational numbers we study locally nilpotent r derivations of the polynomial ring r x y in particular we give a generalization of rentschler s theorem and a criterion for the existence of a slice these results are then applied to describe rank two locally nilpotent derivations of k x 1 x n where k is a field of characteristic zero we also give an example of a nontriangulable locally nilpotent derivation of k x y z whose set of fixed points is a line

optimal discretization of inverse problems in hilbert scales regularization and self regularization of projection methods we study the efficiency of the approximate solution of ill posed problems based on discretized noisy observations which we assume to be given beforehand a basic purpose of the paper is the consideration of stochastic noise but deterministic noise is also briefly discussed we restrict ourselves to problems which can be formulated in hilbert scales within this framework we shall quantify the degree of ill posedness provide general conditions on projection schemes to achieve the best possible order of accuracy we pay particular attention on the problem of self regularization vs tikhonov regularization moreover we study the information complexity asymptotically any method which achieves the best possible order of accuracy must use at least such amount of noisy observations

the separating variety for the basic representations of the additive group abstract for a group g acting on an affine variety x the separating variety is the closed subvariety of x x encoding which points of x are separated by invariants we concentrate on the indecomposable rational linear representations v n of dimension n 1 of the additive group of a field of characteristic zero and decompose the separating variety into the union of irreducible components we show that if n is odd divisible by four or equal to two the closure of the graph of the action which has dimension n 2 is the only component of the separating variety in the remaining cases there is a second irreducible component of dimension n 1 we conclude that in these cases there are no polynomial separating algebras

cancellation over commutative rings of dimension one and two all rings in this paper are commutative and noetherian and all modules are finitely generated we will study the question if a c z b c is a z b here a b and c are modules over a ring r in 1962 chase 4 gave an affirmative answer for r k x y the polynomial ring in two variables over an algebraically closed field k provided a and b are torsionfree and char k trank a in the first section of this paper we extend chase s result to include all two dimensional regular affine k domains most of the paper deals with one dimensional rings in section 2 we prove some general results which we hope will eventally lead to some sort of structure theory for torsionfree modules over one dimensional reduced rings with finite normalization in this context we show that one can cancel from projectives that is the answer is yes when a and b are projective if and only if pit r pit z where g is the normalization when a and b are assumed to be merely torsionfree the problem is more subtle but we have some partial results the last three sections deal with three naturally occurring examples of one dimensional rings coordinate rings of curves quadratic orders and integral group rings of finite abelian groups in each of these cases we are able to give a fairly complete answer to the cancellation problem for torsionfree modules many of the ideas in this paper are the direct or indirect results of innumerable conversations with raymond heitmann and lawrence levy in particular levy discovered a serious error in the proof of the main theorem of an earlier version and heitmann showed by example that the statement itself was incorrect i am extremely grateful to both of them for their interest and insight

robust fixed size confidence procedures for a restricted parameter space on obtient des procedures robustes basees sur la resolution d un probleme de decision minimax robuste associe

sparse recovery by the standard tikhonov method it is a common belief that the tikhonov scheme with the cdot l 2 penalty fails to reconstruct a sparse structure with respect to a given system i however in this paper we present a procedure for the sparse recovery which is totally based on the standard tikhonov method this procedure consists of two steps at first the tikhonov scheme is used as a sieve to find the coefficients near i which are suspected to be non zero within this step the performance of the standard tikhonov method is controlled in some sparsity promoting space rather than in the original hilbert one in the second step of the proposed procedure the coefficients with indices selected in the previous step are estimated by means of the data functional strategy the choice of the regularization parameter is a crucial issue for both steps we show that a recently developed parameter choice rule called the balancing principle can be effectively used here we also present the results of computational experiments giving the evidence of the reliability of our approach

kernel of locally nilpotent r derivations of r x y in this paper we study the kernel of a non zero locally nilpotent r derivation of the polynomial ring r x y over a noetherian integral domain r containing a field of characteristic zero we show that if r is normal then the kernel has a graded r algebra structure isomorphic to the symbolic rees algebra of an unmixed ideal of height one in r and conversely the symbolic rees algebra of any unmixed height one ideal in r can be embedded in r x y as the kernel of a locally nilpotent r derivation of r x y we also give a necessary and sufficient criterion for the kernel to be a polynomial ring in general

derivations of r x y z with a slice the question under consideration is whether every locally nilpotent r derivation of r x y z with a slice has kernel a generated by two elements over r where r is a polynomial ring over a field of characteristic zero theorem 1 1 gives a fundamental property of such kernels namely that a is an a2 fibration over r while it is an open question whether every a2 fibration over r is trivial the property of a asserted in the theorem is necessary to the condition that a is a polynomial ring in two variables over r the last section of the paper presents a family of examples n n 1 which are quite simple to define but whose status relative to the kernel question is not known

random fibonacci sequences solutions to the random fibonacci recurrence xn 1 xn xn 1 decrease increase exponentially xn exp n for sufficiently small large in the limits 0 and we expand the lyapunov exponent in powers of and 1 respectively for the classical case of 1 we obtain exact non perturbative results in particular an invariant measure associated with ricatti variable rn xn 1 xn is shown to exhibit plateaux around all rational r

on the rigidity of small domains abstract let k be an algebraically closed field of arbitrary characteristic let a be an affine domain over k with transcendence degree 1 which is not isomorphic to k x and let b be a domain over k we show that the ak invariant distributes over the tensor product of a by b as a consequence we obtain a generalization of the cancellation theorem of s abhyankar p eakin and w heinzer

singularities in spectra of disordered systems abstract this paper starts with a review of work on singularities in the spectral density of random harmonic chains detailed descriptions of singular behavior near the maximal frequency near special frequencies and near island frequencies are discussed next the site disordered anderson model is considered here the discussion is limited to the singularity in the density of states near the low energy band edge for systems with arbitrary disorder in dimensions d 1 2 3 in a specific example it is shown that renormalization improves the prediction for the band tail quantitatively

regularization of some linear ill posed problems with discretized random noisy data for linear statistical ill posed problems in hilbert spaces we introduce an adaptive procedure to recover the unknown so lution from indirect discrete and noisy data this procedure is shown to be order optimal for a large class of problems smooth ness of the solution is measured in terms of general source condi tions the concept of operator monotone functions turns out to be an important tool for the analysis

singular behaviour of certain infinite products of random 2 2 matrices the authors consider an infinite product of random matrices which appears in several physical problems in particular the ising chain in a random field the random matrices depend analytically on a parameter epsilon in such a way that for 0 they all commute under certain conditions they find that the lyapunov index of this product behaves approximately as c 2 for to 0 their approach is based on the decomposition of an exact integral equation for this problem into two reduced equations they give an expression for the exponent alpha in terms of the probability distribution of the matrices and for the proportionality constant c in terms of the solutions of the reduced integral equations in cases where exact results are available agreement is obtained the physical consequences for disordered one dimensional systems are pointed out

a counterexample to the fourteenth problem of hilbert in dimension four abstract this paper presents a family of new counterexamples to hilbert s fourteenth problem they are realized as subfields of the rational function field of four variables over a field of characteristic zero of transcendence degree three it was previously not known whether any counterexample could be found as a subfield of a rational function field of four variables or whether counterexamples with minimal transcendence degree three could exist

linear counterexamples to the fourteenth problem of hilbert abstract following ideas of a campo neuen indag math n s 5 1994 253 257 we construct linear counterexamples to the fourteenth problem of hilbert from non linear counterexamples d daigle g freudenburg j algebra 221 1999 528 535 g freudenburg transform groups 5 2000 61 71

ml invariant of the symmetric algebra of a projective module over a one dimensional affine domain abstract let k be an algebraically closed field and let a be an affine domain of dimension one over k let p be a finitely generated projective a module of rank d and let r sym a p be the symmetric algebra of p over a assume that a is not a polynomial algebra over k in this article we show that under these assumptions the makar limanov invariant ml r a

dynamical surface structures in multiparticle correlated surface growths we investigate the scaling properties of the interface fluctuation width for the q mer and q particle correlated deposition evaporation models these models are constrained with a global conservation law that the particle number at each height is conserved modulo q in equilibrium the stationary roughness is anomalous but universal with the roughness exponent alpha 1 3 while the early time evolution shows nonuniversal behavior with the growth exponent beta varying with models and q nonequilibrium surfaces display diverse growing and stationary behaviors the q mer model shows a faceted structure while the q particle correlated model shows a macroscopically grooved structure

fixed point free locally nilpotent derivations of a2 fibrations abstract let r be a regular ring containing q and let a be an a 2 fibration over r we prove in this paper that every fixed point free locally nilpotent r derivation of a has a slice as a consequence an a 2 fibration over a factorial regular ring r is trivial if and only if it has a fixed point free locally nilpotent r derivation our results answer a question raised by freudenburg 2009 11 question 2 which actually was the main motivation of this work

estimating a bounded normal mean the problem of estimating a normal mean has received much attention in recent years if one assumes however that the true mean lies in a bounded interval the problem changes drastically in this paper we show that if the interval is small approximately two standard deviations wide then the bayes rule against a two point prior is the unique minimax estimator under squared error loss for somewhat wider intervals we also derive sufficient conditions for minimaxity of the bayes rule against a three point prior

on the finite generation of additive group invariants in positive characteristic abstract roberts freudenburg and daigle and freudenburg have given the smallest counterexamples to hilbert s fourteenth problem as rings of invariants of algebraic groups each is of an action of the additive group on a finite dimensional vector space over a field of characteristic zero and thus each is the kernel of a locally nilpotent derivation in positive characteristic additive group actions correspond to locally finite iterative higher derivations we set up characteristic free analogs of the three examples and show that contrary to characteristic zero in every positive characteristic the invariants are finitely generated

on the remarkable spectrum of a non hermitian random matrix model a non hermitian random matrix model proposed a few years ago has a remarkably intricate spectrum various attempts have been made to understand the spectrum but even its dimension is not known using the dyson schmidt equation we show that the spectrum consists of a non denumerable set of lines in the complex plane each line is the support of the spectrum of a periodic hamiltonian obtained by the infinite repetition of any finite sequence of the disorder variables our approach is based on the theory of words we make a complete study of all four letter words the spectrum is complicated because our matrix contains everything that will ever be written in the history of the universe including this particular paper

on algebras which are locally mathbb a 1 in codimension one let r be a noetherian normal domain call an r algebra a locally a1 in codimension one if rp r a is a polynomial ring in one variable over rp for every height one prime ideal p in r we shall describe a general structure for any faithfully flat r algebra a which is locally a1 in codimensionone and deduce results giving sufficient conditions for such an r algebra to be a locally polynomial algebra we also give a recipe for constructing r algebras which are locally a1 in codimension one when r is a normal affine spot i e a normal local domain obtained by a localisation of an affine domain we give criteria for a faithfully flat r algebra a which is locally a1 in codimensionone to be krull and a further condition for a to be noetherian the results are used to construct intricate examples of faithfully flat r algebras locally a1 in codimension one which are noetherian normal but not finitely generated

asymptotically exact nonparametric hypothesis testing in sup norm and at a fixed point for the signal in gaussian white noise model we consider the problem of testing the hypothesis h 0 f 0 the signal f is zero against the nonparametric alternative h 1 f where is a set of functions on r 1 of the form f f f c here is a holder or sobolev class of functions f is either the sup norm of f or the value of f at a fixed point c 0 is a constant is the minimax rate of testing and 0 is the asymptotic parameter of the model we find exact separation constants c 0 such that a test with the given summarized asymptotic errors of first and second type is possible for c c and is not possible for c c we propose asymptotically minimax test statistics

even visiting random walks exact and asymptotic results in one dimension we reconsider the problem of even visiting random walks in one dimension this problem is mapped onto a non hermitian anderson model with binary disorder we develop very efficient numerical tools to enumerate and characterize even visiting walks the number of closed walks is obtained as an exact integer up to 1828 steps i e some 10535 walks on the analytical side the concepts and techniques of one dimensional disordered systems allow one to obtain explicit asymptotic estimates for the number of closed walks of 4k steps up to an absolute prefactor of the order of unity which is determined numerically all the cumulants of the maximum height reached by such walks are shown to grow as k1 3 with exactly known prefactors these results illustrate the tight relationship between even visiting walks trapping models and the lifshitz tails of disordered electron or phonon spectra

statistical estimation and optimal recovery new formulas are given for the minimax linear risk in estimating a linear functional of an unknown object from indirect data contaminated with random gaussian noise the formulas cover a variety of loss functions and do not require the symmetry of the convex a priori class it is shown that affine minimax rules are within a few percent of minimax even among nonlinear rules for a variety of loss functions it is also shown that difficulty of estimation is measured by the modulus of continuity of the functional to be estimated the method of proof exposes a correspondence between minimax affine estimates in the statistical estimation problem and optimal algorithms in the theory of optimal recovery

on finite generation of kernels of locally nilpotent r derivations of r x y z let r be a noetherian domain containing the field of rationals we show that if r is dedekind then the kernel of any locally nilpotent r derivation of r x y z is a finitely generated r algebra conversely we show that if r is neither a field nor a dedekind domain then there exists a locally nilpotent r derivation of r x y z whose kernel is not finitely generated over r

deconvolution when classifying noisy data involving transformations in the present study we consider the problem of classifying spatial data distorted by a linear transformation or convolution and contaminated by additive random noise in this setting we show that classifier performance can be improved if we carefully invert the data before the classifier is applied however the inverse transformation is not constructed so as to recover the original signal and in fact we show that taking the latter approach is generally inadvisable we introduce a fully data driven procedure based on cross validation and use several classifiers to illustrate numerical properties of our approach theoretical arguments are given in support of our claims our procedure is applied to data generated by light detection and ranging lidar technology where we improve on earlier approaches to classifying aerosols this article has supplementary materials online

multi parameter regularization and its numerical realization in this paper we propose and analyse a choice of parameters in the multi parameter regularization of tikhonov type a modified discrepancy principle is presented within the multi parameter regularization framework an order optimal error bound is obtained under the standard smoothness assumptions we also propose a numerical realization of the multi parameter discrepancy principle based on the model function approximation numerical experiments on a series of test problems support theoretical results finally we show how the proposed approach can be successfully implemented in laplacian regularized least squares for learning from labeled and unlabeled examples

triangular derivations of k x1 x2 x3 x4 abstract it is shown that if k is a field of characteristic zero then the kernel of any triangular k derivation of k x 1 x 2 x 3 x 4 is finitely generated as a k algebra this is obtained as a corollary of a more general result concerning triangular r derivations of r x y z for certain rings r

lifshitz singularities in the total and the wavenumber dependent spectral density of random harmonic chains abstract lifshitz has shown that the density of states of random systems becomes exponentially small near the band edge we perform a detailed computation in a one dimensional system and calculate the periodic prefactor of this essential singularity the analysis is extended to the wavenumber dependent spectral density the periodic amplitude is derived for the exactly soluble cases where the distribution of masses is exponential for binary distributions the leading behavior in the concentration of light masses is also derived

adaptive estimation of linear functionals in hilbert scales from indirect white noise observations we consider adaptive estimating the value of a linear functional from indirect white noise observations for a flexible approach the problem is embedded in an abstract hilbert scale we develop an adaptive estimator that is rate optimal within a logarithmic factor simultaneously over a wide collection of balls in the hilbert scale it is shown that the proposed estimator has the best possible adaptive properties for a wide range of linear functionals the case of discretized indirect white noise observations is studied and the adaptive estimator in this setting is developed

constants of weitzenbock derivations and invariants of unipotent transformations acting on relatively free algebras abstract in commutative algebra a weitzenbock derivation is a nonzero triangular linear derivation of the polynomial algebra k x 1 x m in several variables over a field k of characteristic 0 the classical theorem of weitzenbock states that the algebra of constants is finitely generated this algebra coincides with the algebra of invariants of a single unipotent transformation in this paper we study the problem of finite generation of the algebras of constants of triangular linear derivations of finitely generated not necessarily commutative or associative algebras over k assuming that the algebras are free in some sense in most of the cases relatively free algebras in varieties of associative or lie algebras in this case the algebra of constants also coincides with the algebra of invariants of some unipotent transformation the main results are the following 1 we show that the subalgebra of constants of a factor algebra can be lifted to the subalgebra of constants 2 for all varieties of associative algebras which are not nilpotent in lie sense the subalgebras of constants of the relatively free algebras of rank 2 are not finitely generated 3 we describe the generators of the subalgebra of constants for all factor algebras k x y i modulo a gl 2 k invariant ideal i 4 applying known results from commutative algebra we construct classes of automorphisms of the algebra generated by two generic 2 2 matrices we obtain also some partial results on relatively free lie algebras

enumeration of simple random walks and tridiagonal matrices we present some old and new results in the enumeration of random walks in one dimension mostly developed in work on enumerative combinatorics the relation between the trace of the nth power of a tridiagonal matrix and the enumeration of weighted paths of n steps allows an easier combinatorial enumeration of paths it also seems promising for the theory of tridiagonal random matrices

non hermitian tridiagonal random matrices and returns to the origin of a random walk we study a class of tridiagonal matrix models the q roots of unity models which includes the sign q 2 and the clock q models by feinberg and zee we find that the eigenvalue densities are bounded by and have the symmetries of the regular polygon with 2q sides in the complex plane furthermore the averaged traces of mk are integers that count closed random walks on the line such that each site is visited a number of times multiple of q we obtain an explicit evaluation for them

fields defined by locally nilpotent derivations and monomials abstract we prove a necessary and sufficient condition for certain fields defined by locally nilpotent derivations and monomials to be algebraically closed in a rational function field this implies that a counterexample to the fourteenth problem of hilbert in dimension four which was recently given by the author is obtained as the kernel of a derivation it was previously unknown only in dimension four whether there exists such a counterexample

hilbert s fourteenth problem and algebraic extensions let k x be the polynomial ring in n variables over a field k for some n n and k x its field of fractions assume that l is a subfield of k x containing k over which k x is algebraic in spite of being an important issue in hilbert s fourteenth problem relations between finite generation of the k subalgebra l k x of k x and the extension degree k x l of k x over l have not been investigated in the present paper we give an example of l with k x l d such that l k x is not finitely generated for each d n with d 3 for n 3

hilbert s fourteenth problem and field modifications abstract let k x k x 1 x n be the rational function field and k l k x an intermediate field hilbert s fourteenth problem asks whether the k algebra l k x 1 x n is finitely generated various counterexamples to this problem were already given but there are still many open cases in this paper we study the problem in terms of the field theoretic properties of l our result implies that for any s k x 1 x n 1 with k s k x 1 x n 1 and tr deg k k s 2 there exists aut k k x such that k s x n is a counterexample to hilbert s fourteenth problem this implies the existence of several interesting new counterexamples such as one with k x l 2

optimal inference in a class of regression models we consider the problem of constructing confidence intervals cis for a linear functional of a regression function such as its value at a point the regression discontinuity parameter or a regression coefficient in a linear or partly linear regression our main assumption is that the regression function is known to lie in a convex function class which covers most smoothness and or shape assumptions used in econometrics we derive finite sample optimal cis and sharp efficiency bounds under normal errors with known variance we show that these results translate to uniform over the function class asymptotic results when the error distribution is not known when the function class is centrosymmetric these efficiency bounds imply thatminimax cis are close to efficient at smooth regression functions this implies in particular that it is impossible to form cis that are tighter using data dependent tuning parameters and maintain coverage over the whole function class we specialize our results to inference in a linear regression and inference on the regression discontinuity parameter and illustrate them in an empirical application

a triviality criterion for a2 fibrations over a ring containing q let r be an arbitrary commutative ring with unity containing q and let a be a stably trivial a2 fibration over r in this paper we prove that a is trivial if and only if it has a fixed point free locally nilpotent r derivation

an algebraic proof of a cancellation theorem for surfaces let k be an algebraically closed field of arbitrary characteristic we give a self contained algebraic proof of the following statement if v is an affine surface over k such that v k k3 then v k2 this fact which is due to fujita miyanishi sugie and russell solves the zariski cancellation problem for surfaces to achieve our proof we first show that if a is a finitely generated domain with ak a a then ak a x a

minimax estimation of the mean of a normal distribution when the parameter space is restricted if x is a n theta 1 random variable let rho m be the minimax risk for estimation with quadratic loss subject to theta leq m then rho m 1 pi 2 m 2 o m 2 we exhibit estimates which are asymptotically minimax to this order as well as approximations to the least favorable prior distributions the approximate least favorable distributions correct to order m 2 have density m 1 cos 2 big frac pi 2m s big s leq m rather than the naively expected uniform density on lbrack m m rbrack we also show how our results extend to estimation of a vector mean and give some explicit solutions

do linear problems have linear optimal algorithms a selective introduction to the field of information based complexity is presented in the context of the question raised in the title after introducing some of the basic ideas of this relatively new area that interfaces mathematics and theoretical computer science the article surveys results on the existence of linear optimal algorithms for linear problems several illustrative examples are given along with an indication of the variety of disciplines in which techniques of information based complexity are being applied known under aliases such as the general theory of optimal algorithms and the information centered approach to algorithms information based complexity seeks to create a general theory about problems with partial or approximate information and to apply the results to specific problems some of the disciplines where problems with such incomplete information have arisen include computer science economics control theory signal processing and geophysics the goal of this article is to present a selective introduction to the field of information based complexity and to describe the seesaw nature of results in the last decade on the extent to which linear problems have linear optimal algorithms the approach we take will certainly not do justice to the breadth of ideas techniques and applications that are coming into play in the general theory instead we will take a direct route to the results on linear optimal algorithms pausing occasionally to survey the pleasing way in which ideas from linear algebra and functional analysis are used in the final section we mention some other theoretical and applied topics currently under investigation in information based complexity a more general expository overview can be found in traub and wozniakowski 15 and more technical surveys appear in packel and wozniakowski 10 and wozniakowski 21 2 two preliminary examples we set the stage for an exposition of the general linear theory by imposing partial information on two problems which are known to virtually every student of mathematics the problem of integrating a continuous function needs no introduction the fourier series approach to approximating func tions by linear combinations of sines and cosines is almost as familiar we do not develop these ideas as a testimony to the power of the general theory more about that later but rather to have a comfortable base to refer to when the generalizations and abstractions are introduced integration example let f be a function from the space w 0 1 of real valued continuous functions on the interval 0 1 we would like to obtain a value for the

limits to estimation in stochastic ill conditioned inverse problems using information theoretic methods we develop simple results quantifying a lower bound for minimax estimation a kind of infinite dimensional cramer rao lower bound for signal estimation in possibly nonlinear ill conditioned inverse problems our results reduce calculation to a geometric computation based on a modulus of continuity and make explicit connections with results in the literature on deterministic ill conditioned inverse problems several applications are discussed

on the cancellation problem for the affine space mathbb a 3 in characteristic p we show that the cancellation conjecture does not hold for the affine space mathbb a 3 k over any field k of positive characteristic we prove that an example of t asanuma provides a three dimensional k algebra a for which a is not isomorphic to k x1 x2 x3 although a t is isomorphic to k x1 x2 x3 x4

a condition for finite generation of the kernel of a derivation abstract we consider a homogeneous derivation on a finitely generated graded normal domain over a field and give a sufficient condition for finite generation of its kernel as its consequence we show some sufficient conditions for finite generation of the kernel of a derivation on a polynomial ring in particular we prove that the kernel of a locally nilpotent monomial derivation on a polynomial ring in four variables is finitely generated

stability theorems for overrings of polynomial rings let r be a polynomial ring over a field projective modules over rings of the type r ix y r x y where r is a non zero element of r have been considered by murthy swan weibel see mu and w these rings lie between r x and r x x 1 and even when they are regular projective modules over them need not be free mu example 6 2 this leads us to study stability properties of projective modules over rings a which lie between r x and r x x 1 when r is any commutative noetherian ring we prove stability theorems for projective modules over such rings a in w these results theorems 4 2 and 4 3 have been proved for a r x by plumstead p and for a r x x 1 by mandal ma in w we prove similar results for the allied class of rings d of the type r x y xy stability theorems for gl a and gl d are proved in w when a r x or r x x 1 our theorems 6 2 i and 6 4 i had already been proved by suslin s finally in w 7 we study pic r x y r xy when r is a pid theorem 7 1 extends a result of murthy mu corollary 5 3 it would be interesting to know whether theorems 4 1 and 4 2 can be extended to the polynomial ring a t 1 tm where a lies between r x and r x x 1 more precisely for a projective a t 1 t module p of rank dim a we would like to know if the following statements are true i p has a unimodular element ii p has the cancellation property we have been able to show that when a r x the statement i is true b r theorem 3 1

geometrizing rates of convergence ii consider estimating a functional t f of an unknown distribution f in mathbf f from data x 1 cdots x n i i d f let omega varepsilon denote the modulus of continuity of the functional t over mathbf f computed with respect to hellinger distance for well behaved loss functions l t we show that inf t n sup mathbf f e fl t n t f is equivalent to l omega n 1 2 to within constants whenever t is linear and mathbf f is convex the same conclusion holds in three nonlinear cases estimating the rate of decay of a density estimating the mode and robust nonparametric regression we study the difficulty of testing between the composite infinite dimensional hypotheses h 0 t f leq t and h 1 t f geq t delta our results hold in the cases studied because the difficulty of the full infinite dimensional composite testing problem is comparable to the difficulty of the hardest simple two point testing subproblem

a tikhonov based projection iteration for nonlinear ill posed problems with sparsity constraints in this paper we consider nonlinear inverse problems where the solution is assumed to have a sparse expansion with respect to a preassigned basis or frame we develop a scheme which allows to minimize a tikhonov functional where the usual quadratic regularization term is replaced by a one homogeneous typically weighted l p penalty on the coefficients or isometrically transformed coefficients of such expansions for p 2 the regularized solution will have a sparser expansion with respect to the basis or frame under consideration the computation of the regularized solution amounts in our setting to a landweber fixed point iteration with a projection applied in each fixed point iteration step the performance of the resulting numerical scheme is demonstrated by solving the nonlinear inverse single photon emission computerized tomography spect problem

a counterexample to hilbert s fourteenth problem in dimension 5 abstract the kernel of a certain triangular derivation of the polynomial ring k x 1 x 2 x 3 x 4 x 5 is shown to be non finitely generated over k a field of characteristic zero thus giving a new counterexample to hilbert s fourteenth problem in the lowest dimension to date

finite time and asymptotic behaviour of the maximal excursion of a random walk we evaluate the limit distribution of the maximal excursion of a random walk in any dimension for homogeneous environments and for self similar supports under the assumption of spherical symmetry this distribution is obtained in closed form and is an approximation of the exact distribution comparable to that obtained by real space renormalization methods we then focus on the early time behaviour of this quantity the instantaneous diffusion exponent n exhibits a systematic overshooting of the long time exponent exact results are obtained in one dimension up to third order in n 1 2 in two dimensions on a regular lattice and on the sierpinski gasket we find numerically that the analytic scaling n an holds