information theoretic determination of minimax rates of convergence we present some general results determining minimax bounds on statistical risk for density estimation based on certain information theoretic considerations these bounds depend only on metric entropy conditions and are used to identify the minimax rates of convergence

carl s inequality for quasi banach spaces we prove that for any two quasi banach spaces x and y and any 0 there exists a constant 0 such that sup1 k n k ek t sup1 k n k ck t holds for all linear and bounded operators t x y here ek t is the k th entropy number of t and ck t is the k th gelfand number of t for banach spaces x and y this inequality is widely used and well known as carl s inequality for general quasi banach spaces it is a new result

eigenvalue distribution of some fractal semi elliptic differential operators we consider differential operators of type au x u x 1 t 1 frac partial 2t 1 u x partial x 1 2t 1 1 t 2 frac partial 2t 2 u x partial x 2 2t 2 x x 1 x 2 in r 2 and sierpinski carpets g subset r 2 the aim of the paper is to investigate spectral properties of the fractal differential operator a 1 circ tr gamma acting in the anisotropic sobolev space w t 1 t 2 2 bbb r 2 where tr gamma is closely related to the trace operator tr gamma

tuning to resonances with iterative time reversal this paper considers a distributed wave based sensing system that probes a scene consisting of multiple interacting idealized targets each sensor is a collocated transmit receive pair that is capable of transmitting arbitrary wideband waveforms we address the problem of finding the space time transmit waveform that provides the best target detection performance in the sense of maximizing the energy scattered back into the receivers our approach is based on earlier theoretical work that showed for an idealized infinite half space geometry that the solution could be constructed by an iterative time reversal tr process in this paper we give a more realistic example involving a two sensor time domain system we show that for this system the iterative tr process can be used to tune automatically to all the target resonances that are within the bandwidth of the interrogating radar system we show that although obtaining eigenvalues and hence resonances of the scattering operator is in general unstable using the iterative tr process to obtain the resonances is a stable process moreover we show that these resonance frequencies are connected to the poles of the singularity expansion method

some inverse problems with spectral limitations on data we consider the recovery of the impedance a y for a one dimensional seismic problem v n av y y a where y is travel time with impulse input v y 0 delta y and readout v 0 t delta t g t g t 2 pi int 0 infty hat g lambda operatorname cos lambda t d lambda hat g being the fourier cosine transform first it follows that for delta denoting change delta a y leqq tilde c y delta g infty on 0 2y the problem of estimating delta g infty on 0 2y is ill posed if only estimates delta g 2 leqq varepsilon can be obtained using regularization and summability we construct a sequence of approximants g n which converges to g pointwise on 0 2y uniformly thus g n to g in l infty 0 2y and the corresponding a n y converse to a y pointwise this paper also serves as an introduction to the connection of regularization and summability ideas with transmutation formulas applications to

the essential spectrum of neumann laplacians on some bounded singular domains in the present paper we consider neumann laplacians on singular domains of the type rooms and passages or combs and we show that in typical situations the essential spectrum can be determined from the geometric data moreover given an arbitrary closed subset s of the non negative reals we construct domains s such that the essential spectrum of the neumann laplacian on is just this set s

centred bimodules over prime rings closed submodules and applications to ring extensions let m be a centred bimodule over a prime ring r in this paper we define and study a very useful class of sub bimodules of m the class of closed sub bimodules there is a canonical torsion free extension of m to a q bimodule m which is always free over q where q is the complete ring of right quotients of r we prove that closed sub bimodules of m are in one to one correspondence with closed sub bimodules of m the results are applied to study the torsion free rank of a sub bimodule of m and to study non singular and strongly closed sub bimodules also the results are applied to study prime ideals in centred extensions and intermediate extensions in particular we complete and extend the results obtained in m ferrero j algebra 148 1992 1 16

entropy numbers s numbers and eigenvalue problems abstract we establish inequalities between entropy numbers and approximation numbers for operators acting between banach spaces furthermore we derive inequalities between eigenvalues and entropy numbers for operators acting on a banach space the results are compared with the classical inequalities of bernstein and jackson

almost optimal estimates for entropy numbers of b2 21 2 and its consequences triebel s conjecture on the entropy numbers for the limiting embedding case b p p 1 p e e being suitable orlicz spaces is almost proved for p 2 almost in the sense that a log factor occurs in the upper estimate furthermore it is shown that the breaking point where the radii of the balls in the covering of the b p q 1 p unit ball stop to be constant essentially depends on the second parameter q the methods used in the paper also give improvements of the results till now known in the case p 2 explains a result of kashin and temlyakov on entropy numbers of continuous functions of low smoothness and indicates how the e scale can be refined

gelfand numbers of operators with values in a hilbert space in the paper we prove two inequalities involving gelfand numbers of operators with values in a hilbert space the first inequality is a rademacher version of the main result in pa to 1 which relates the gelfand numbers of an operator from a banach spacex intol 2 n with a certain rademacher average for the dual operator the second inequality states that the gelfand numbers of an operatoru froml 1 n into a hilbert space satisfy the inequality k 1 2 c k u leqq c parallel u parallel log 1 n k 1 2 wherec is a universal constant several applications of these inequalities in the geometry of banach spaces are given

extrapolation estimates for entropy numbers we establish abstract extrapolation results for entropy numbers of operators in banach spaces the results apply to extrapolation in the source spaces and also in the target spaces as an illustration of the abstract results applications to limiting sobolev embeddings are given

the singular function expansion in time dependent scattering a class of linear initial boundary value problems relevant to time dependent scattering is solved exactly in terms of the complete orthogonal set of singular functions of the appropriate integral operator the method is primarily presented as a mathematically defensible alternative to the eigenmode expansion method the results extend the singularity expansion method to include the steady state response at source frequencies and provide a connection between resonance behavior in the steady state and low loss natural modes the method furthermore leads to an efficient automatic algorithm for the determination of natural modes and natural frequencies by observation of the singular values the conclusions are supported by numerical calculations

hp theory on a smooth domain in rn and elliptic boundary value problems in this paper we answer the following two questions question 1 let be an appropriate domain inrn what are the possible natural notions of hp that generalize the usual hardy spaces hp rn we shall see that several versions are possible we concentrate on the two most relevant here which may be considered to be the largest version hpr and the smallest version hpz for the second question consider the dirichlet and neumann problems for the former consists in finding the u g which solves u in and u 0 the latter consists in finding the u g which solves u in and u n 0 question 2 in the context of the relevant hp can one obtain the boundedness of 2g xi xj and g xi xj

perturbation of resonances abstract resonances are often treated under the assumption that they are simple poles of the resolvent kernel green s function there are no physical or mathematical reasons which exclude the existence of multiple poles of green s functions even in the case when the potential is real valued in this case the poles in the physical sheet are simple since the schrodinger operator is selfadjoint but the poles on the unphysical sheet are not necessarily simple we prove that sufficiently small perturbations of the potential or boundary in diffraction problems do not increase the multiplicity of the resonances on any fixed compact domain of the complex plane we also observe that a knowledge of the set of the green s function poles does not determine the potential uniquely

on a characterization of operators from lq into a banach space of type p with some applications to eigenvalue problems abstract let s be an operator admitting a factorization where d is a diagonal operator and t an arbitrary operator with the image in a banach space of type p we shall characterize these operators by entropy numbers using a well known inequality between eigenvalues and entropy numbers we establish some results about distributions of eigenvalues

equisummability of eigenfunction expansions under analytic multipliers in this paper we present some abstract criteria for equisummability of expansions in eienfunctions of certain pairs of elliptic operators on general domains of w the criteria to be developed originate in some cases from general banach space arguments and in others from the more specific spatial nature of the differential operators involved the importance of our analysis of l equisummability of two operators is that the question of convergence of the two summability means is reduced essentially to showing that the difference of the modified resolvent operators is uniformly bounded we apply these criteria to give a simple proof of an equisummability result of gurarie and kon 4 for a certain class of elliptic operators whose leading terms are positive and lower order terms have coefficients which are singular on a nowhere dense set the key technique is to analyze kernels of the resolvents and use radial bounds of the resolvents developed in 3 the prototypical case for equisummability is found in the study of equiconvergence for differential operators classically haar 51 walsh lo birkhoff 2 and tamarkin lo have shown that the difference between expansions with respect to eigenfunctions of a sturm liouville operator or birkhoff series and the ordinary fourier series tends to zero uniformly in every finite interval in this case the multiplier or summator function is the characteristic function in some one dimensional cases where such equiconvergence fails stone 9 levitan and sargsjan 6 and benzinger 1 have showed equisummability for riesz typical means of eigenfunction expansions of differential operators where the summator function is

application of the singular function expansion to an integral equation for scattering the dirichlet scattering problem is solved exactly for a surface of arbitrary smooth shape the solution is given in terms of the complete orthonormal set of functions defined on the surface and arising as singular functions of the integral operator since the singular values do not have a point of accumulation at zero the method is of much greater practical value than the eigenmode expansion method eem since the terms in the infinite series are naturally ordered by size of singular value the dominance of one or several terms in the series may be established without explicit evaluation of the coupling coefficients extentions of the method to other boundary value problems and applications to the singularity expansion method sem are described

calculating resonances natural frequencies and extracting them from transient fields mathematical formulation and analysis of numerical methods for calculating the natural frequencies resonances are given stability of these methods towards roundoff errors and small perturbations of the obstacles is established some formulas for the variations of the natural frequencies due to small perturbations of the surface of the obstacle are given a simple new method for extraction of resonances from transient fields is given

a discrete transform and decompositions of distribution spaces abstract we study a representation formula of the form q q q for a distribution on r n this formula is obtained by discretizing and localizing a standard littlewood paley decomposition the map taking to the sequence q q with q running over the dyadic cubes in r n is called the transform the functions q and q have a particularly simple form moreover most of the familiar distribution spaces lp spaces 1 r n we also consider pointwise multipliers for the characteristic function of a domain we obtain boundedness results for a general class of domains which properly includes lipschitz domains several interpolation methods are easily analyzed via the sequence spaces for real interpolation we obtain among other things an extension to the case p 0 this in turn gives a new approach to the traditional atomic decomposition of hardy spaces

non smooth atomic decomposition of 2 microlocal spaces and application to pointwise multipliers abstract we provide non smooth atomic characterizations for 2 microlocal besov and triebel lizorkin spaces with variable integrability b p q w r n and f p q w r n these spaces cover the classical besov and triebel lizorkin spaces as well as spaces of variable smoothness and integrability as an application we state a pointwise multipliers result for these spaces

integrality for pi rings abstract in this paper we study schelter integral extensions of pi rings we prove in particular lying over going up and incomparability for prime ideals a major result is transitivity of integrality if r s b are pi rings such that b is integral over s and s is integral over r then b is integral over r next we obtain a powerful criterion for integrality if s is a prime pi ring such that its center is integral over a noetherian subring r of s then s is integral over r this allows interesting applications to the theory of finite group actions further topics concern eakin nagata type results and embeddings of quotient rings for integral extensions finally we analyze the relationship between module finite extensions and finitely generated integral extensions obtaining positive results for affine noetherian pi algebras and algebras satisfying certain restrictions on pi degrees e g algebras of low pi degree

thomas bayes walk on manifolds convergence of the bayes posterior measure is considered in canonical statistical settings where observations sit on a geometrical object such as a compact manifold or more generally on a compact metric space verifying some conditions a natural geometric prior based on randomly rescaled solutions of the heat equation is considered upper and lower bound posterior contraction rates are derived

s numbers of integral operators with holder continuous kernels over metric compacta abstract we determine the asymptotic behaviour of kolmogorov and approximation numbers of integral operators which act in c x where x is a compact metric space and which are defined by means of holder continuous kernels the results show the connection of the decay of these s numbers with the entropy and in this way with the dimension of x

closed and prime ideals in free centred extensions prime ideals in ring extensions r g s have extensively been studied in the last years for example when the extension s is finite and generated by a set of r centralizing elements s is called a liberal extension 15 161 a normalizing extension is again a finite extension which is generated by a set of r normalizing elements e g 8 121 also prime ideals in more general types of extensions not necessarily finite have been considered e g l 2 13 141 on the other hand the author in 4 studied prime ideals in polynomial rings r x actually a more general class of ideals called the principal closed ideals are defined and considered in 4 and the results on prime ideals are obtained as applications of the general results a main tool in that paper is the representation of closed ideals by using manic polynomials in c x where c is the extended centroid of the prime ring r closed ideals have also been used to study prime ideals in ore extensions c3 61 the purpose of this paper is to extend the results in 4 to free centred extensions the ring s is said to be a free centred extension of r if s is free as an r module and there exists a basis e ei rer of r centralizing elements such that the common identity 1 of r and s is in e there are several usual examples of free centred extensions namely a group or even a semigroup ring rg in particular a polynomial ring in any set of either commuting or non commuting indeterminates also a matrix ring even a ring of infinite matrices provided that every matrix in the ring has a finite number of non zero entries and the identity is adjoined finally a tensor product s r gl k where l is a field and r and k are l algebras

nonparametric estimation with nonlinear budget sets choice models with nonlinear budget sets provide a precise way of accounting for the nonlinear tax structures present in many applications in this paper we propose a nonparametric approach to estimation of these models the basic idea is to think of the choice in our case hours of labor supply as being a function of the entire budget set then we can do nonparametric regression where the variable in the regression is the budget set we reduce the dimensionality of this problem by exploiting structure implied by utility maximization with piecewise linear convex budget sets this structure leads to estimators where the number of segments can differ across observations and does not affect accuracy we give consistency and asymptotic normality results for these estimators the usefulness of the estimator is demonstrated in an empirical example where we find it has a large impact on estimated effects of the swedish tax reform

rough paths analysis of general banach space valued wiener processes abstract in this article we carry out a rough paths analysis for banach space valued wiener processes we show that most of the features of the classical wiener process pertain to its rough path analog to be more precise the enhanced process has the same scaling properties and it satisfies a fernique type theorem a support theorem and a large deviation principle in the same holder topologies as the classical wiener process does moreover the canonical rough paths of finite dimensional approximating wiener processes converge to the enhanced wiener process finally a new criterion for the existence of the enhanced wiener process is provided which is based on compact embeddings this criterion is particularly handy when analyzing kunita flows by means of rough paths analysis which is the topic of a forthcoming article

prime ideals in normalizing extensions abstract in this paper we offer a new approach to the study of prime ideals in finite normalizing extensions of rings we introduce a primitivity machine which translates certain problems on prime ideals to analogous questions on primitive ideals we then use the well known clifford theory of irreducible modules to study these primitive ideals

inference in nonparametric instrumental variables with partial identification this paper develops methods for hypothesis testing in a nonparametric instrumental variables setting within a partial identification framework we construct and derive the asymptotic distribution of a test statistic for the hypothesis that at least one element of the identified set satisfies a conjectured restriction the same test statistic can be employed under identification in which case the hypothesis is whether the true model satisfies the posited property an almost sure consistent bootstrap procedure is provided for obtaining critical values possible applications include testing for semiparametric specifications as well as building confidence regions for certain functionals on the identified set as an illustration we obtain confidence intervals for the level and slope of brazilian fuel engel curves a monte carlo study examines finite sample performance

spectral theory of some degenerate elliptic operators with local singularities abstract this paper is based on our previous results haroske and skrzypczak 2008 23 haroske and skrzypczak in press 25 on compact embeddings of muckenhoupt weighted function spaces of besov and triebel lizorkin type with example weights of polynomial growth near infinity and near some local singularity our approach also extends haroske and triebel 1994 21 in various ways we obtain eigenvalue estimates of degenerate pseudodifferential operators of type b 2 p x d b 1 where b i l r i r n w i w i a i 1 2 and p x d 1 0 0 finally we deal with the negative spectrum of some operator h a v for where the potential v may have singularities in terms of muckenhoupt weights and a is a positive elliptic pseudodifferential operator of order 0 self adjoint in l 2 r n this part essentially relies on the birman schwinger principle we conclude this paper with a number of examples also comparing our results with preceding ones

calderon reproducing formulas and new besov spaces associated with operators abstract let l be the generator of an analytic semigroup whose heat kernel satisfies an upper bound of poisson type acting on l 2 x where x is a possibly non doubling space of polynomial upper bound on volume growth the aim of this paper is to introduce a new class of besov spaces associated with the operator l so that when l is the laplace operator or its square root acting on the euclidean space r n the new besov spaces are equivalent to the classical besov spaces depending on the choice of l the new besov spaces are natural settings for generic estimates for certain singular integral operators such as the fractional powers l since our approach does not require the doubling volume property of the underlying space it is applicable to any subset of euclidean spaces without any smoothness requirement on the boundary we will also develop a number of calderon reproducing formulas which play an important role in the theory of function spaces and are of independent interest as an application we study besov spaces associated with schrodinger operators with non negative potentials satisfying reverse holder estimates

an isomorphic version of the slicing problem abstract here we show that any centrally symmetric convex body k r n has a perturbation t r n which is convex and centrally symmetric such that the isotropic constant of t is universally bounded t is close to k in the sense that the banach mazur distance between t and k is o log n if k is a body of a non trivial type then the distance is universally bounded the distance is also universally bounded if the perturbation t is allowed to be non convex our technique involves the use of mixed volumes and alexandrov fenchel inequalities some additional applications of this technique are presented here

hardy spaces associated with magnetic schrodinger operators on strongly lipschitz domains abstract let p 0 1 be a strongly lipschitz domain in r n and a i a i a v a magnetic schrodinger operator on l 2 satisfying the dirichlet boundary condition where a a 1 a n l loc 2 r n and 0 v l loc 1 in this paper the authors introduce the hardy space h a p by the lusin area function associated with a and establish its equivalent characterization via the non tangential maximal function associated with e t a t 0 as applications the authors obtain the boundedness of the riesz transforms l k a 1 2 k 1 n from h a p to l p for p 0 1 and the fractional integral a from h a p to h a q for 0 p q 1 and n 2 1 p 1 q where l k is the closure of x k i a k in l 2

besov spaces on domains in rd we study besov spaces 5 lp i2 0 p q a oo on do mains i2 in rd we show that there is an extension operator w which is a bounded mapping from b lp u onto b lp rd this is then used to derive various properties of the besov spaces such as interpolation theorems for a pair of b lp il atomic decompositions for the elements of 5 lp f2 and a description of the besov spaces by means of spline approximation

hardy spaces and divergence operators on strongly lipschitz domains of rn abstract let be a strongly lipschitz domain of r n consider an elliptic second order divergence operator l including a boundary condition on and define a hardy space by imposing the non tangential maximal function of the extension of a function f via the poisson semigroup for l to be in l1 under suitable assumptions on l we identify this maximal hardy space with h 1 r n if r n with h r 1 under the dirichlet boundary condition and with h z 1 under the neumann boundary condition

non smooth atomic decompositions traces on lipschitz domains and pointwise multipliers in function spaces abstract we provide non smooth atomic decompositions for besov spaces b p q s r n s 0 0 p q defined via differences the results are used to compute the trace of besov spaces on the boundary of bounded lipschitz domains with smoothness s restricted to 0 s 1 and no further restrictions on the parameters p q we conclude with some more applications in terms of pointwise multipliers

weyl numbers in sequence spaces and sections of unit balls using techniques of a geometrical nature the following result is proved denoting by iqp m lpm lqm the natural embedding and by xk the kth weyl number if 0 p q 2 there are positive constants c1 and c2 such that for all k m n with k m2 c1k1q 1p xk iqp m c2k1q 1p the upper estimate being even valid for all k 1 m as a consequence of the approach used some results about sections of unit balls are also derived namely volh h bpm volk bpk for 0 p 2 where bpm bpk are the closed unit balls centred at zero of the spaces lpm and lpk respectively h is a k dimensional subspace of lpm and volk volh denote lebesgue measures in rk and h respectively

radial bounds for perturbations of elliptic operators abstract elliptic operators a m b x d a multi index with leading term positive and constant coefficient and with lower order coefficients b x l r l with n r defined on r n or a quotient space r n r n u u r n are considered it is shown that the lp spectrum of a is contained in a parabolic region of the complex plane enclosing the positive real axis uniformly in p outside the kernel of the resolvent of a is shown to be uniformly bounded by an l1 radial convolution kernel some consequences are a can be closed in all lp 1 p and is essentially self adjoint in l2 if it is symmetric a generates an analytic semigroup e ta in the right half plane strongly lp and pointwise continuous at t 0 a priori estimates relating the leading term and remainder are obtained and summability ea e 0 0 with analytic is proved for l p with convergence in lp and on the lebesgue set of more comprehensive summability results are obtained when a has constant coefficients

volume ratio and other s numbers of operators related to local properties of banach spaces abstract volume ratio numbers are studied and inequalities relating them to other s numbers of operators are established characterizations of spaces of weak cotype 2 in terms of s numbers and of operator ideal norms are obtained

mean separations in banach spaces under abstract interpolation and extrapolation abstract we introduce the mean separation for bounded sequences in banach spaces and the related seminorm for bounded linear operators the introduced quantities are closely related to the geometric characterizations of the banach saks property and the alternate signs banach saks property we investigate the behavior of the mean separations for a class of operators between vector valued banach sequence spaces e x providing that a banach sequence lattice e has the banach saks property we estimate the mean separations for operators under abstract interpolation and extrapolation methods in particular we obtain quantitative and qualitative results on the heredity of the banach saks properties under these methods

variations on a theme of jost and pais abstract we explore the extent to which a variant of a celebrated formula due to jost and pais which reduces the fredholm perturbation determinant associated with the schrodinger operator on a half line to a simple wronski determinant of appropriate distributional solutions of the underlying schrodinger equation generalizes to higher dimensions in this multi dimensional extension the half line is replaced by an open set r n n n n 2 where has a compact nonempty boundary satisfying certain regularity conditions our variant involves ratios of perturbation determinants corresponding to dirichlet and neumann boundary conditions on and invokes the corresponding dirichlet to neumann map as a result we succeed in reducing a certain ratio of modified fredholm perturbation determinants associated with operators in l 2 d n x n n to modified fredholm determinants associated with operators in l 2 d n 1 n 2 applications involving the birman schwinger principle and eigenvalue counting functions are discussed