is denoising dead image denoising has been a well studied problem in the field of image processing yet researchers continue to focus attention on it to better the current state of the art recently proposed methods take different approaches to the problem and yet their denoising performances are comparable a pertinent question then to ask is whether there is a theoretical limit to denoising performance and more importantly are we there yet as camera manufacturers continue to pack increasing numbers of pixels per unit area an increase in noise sensitivity manifests itself in the form of a noisier image we study the performance bounds for the image denoising problem our work in this paper estimates a lower bound on the mean squared error of the denoised result and compares the performance of current state of the art denoising methods with this bound we show that despite the phenomenal recent progress in the quality of denoising algorithms some room for improvement still remains for a wide class of general images and at certain signal to noise levels therefore image denoising is not dead yet

projective techniques and functional integration for gauge theories a general framework for integration over certain infinite dimensional spaces is first developed using projective limits of a projective family of compact hausdorff spaces the procedure is then applied to gauge theories to carry out integration over the non linear infinite dimensional spaces of connections modulo gauge transformations this method of evaluating functional integrals can be used either in the euclidean path integral approach or the lorentzian canonical approach a number of measures discussed are diffeomorphism invariant and therefore of interest to the connection dynamics version of quantum general relativity the account is pedagogical in particular prior knowledge of projective techniques is not assumed

refined algebraic quantization in the oscillator representation of sl 2 r we investigate refined algebraic quantization raq with group averaging in a constrained hamiltonian system with unreduced phase space t r4 and gauge group sl 2 r the reduced phase space m is connected and contains four mutually disconnected regular sectors with topology r s1 but these sectors are connected to each other through an exceptional set where m is not a manifold and where m has non hausdorff topology the raq physical hilbert space hphys decomposes as hphys hi where the four subspaces hi naturally correspond to the four regular sectors of m the raq observable algebra aobs represented on hphys contains natural subalgebras represented on each hi the group averaging takes place in the oscillator representation of sl 2 r on l2 r2 2 and ensuring convergence requires a subtle choice for the test state space the classical analog of this choice is to excise from m the exceptional set while nevertheless retaining information about the connections between the regular sectors a quantum

f purity and rational singularity we investigate singularities which are f pure respectively f pure type a ring r of characteristic p is f pure if for every r module m 0 m r m r is exact where 1r denotes the r algebra structure induced on r via the frobenius map if r e r and s e 1r then r s rps in 1r f pure type is defined in characteristic 0 by reducing to characteristicp it is proven that when r s i is the quotient of a regular local ring s r is f pure at the prime ideal q if and only if i pi i q p here j p denotes the ideal ap i a e j several theorems result from this criterion if f is a quasihomogeneous hypersurface having weights r1 r and an isolated singularity at the origin 1 17 1ri i implies k x1 x f has f pure type at m x1 x 2 1n i ri i implies k x1 xn f does not have f pure type at m 3 1 1ri i remains unsolved but does connect with a problem that number theorists have studied for many years this theorem parallels known results about rational singularities it is also proven that classifying f pure singularities for complete intersection ideals can be reduced to classifying such singularities for hypersurfaces and that the f pure locus in the maximal spectrum of k x1 xn i where k is a perfect field of characteristic p is zariski open an important conjecture is that r fr is f pure type should imply r is f pure type whenever r is a cohen macauley normal local ring it is proven that ext1 1r r 0 is a sufficient though not necessary condition a local ring r m of characteristic p is f injective if the frobenius map induces an injection on the local cohomology modules h r h 1r an example is constructed which is f injective but not f pure from this a counterexample to the conjecture that r fr is f pure implies r is f pure is constructed however it is not a domain much less normal moreover it does not lead to a counterexample to the characteristic 0 version of the conjecture 0 introduction let r be a ring of characteristic p and let r denote the ring r viewed as an r module via the frobenius map f r rp r is f pure if for every r module m 0 r m 4 r m is exact a notion of f pure type is then defined in characteristic 0 by reduction to characteristicp f pure rings are connected with invariant theory and appear in the proof that the ring of invariants of a linearly reductive affine linear group acting on a regular ring is cohen macaulay 3 it has also been demonstrated that f purity measures good singularities in the sense that it implies a great deal of simplification in the computation of local cohomology 1 received by the editors march 30 1982 and in revised form august 3 1982 1980 mathematics subject classification primary 13h99 1983 american mathematical society 0002 9947 82 0000 0829 05 25

a generalization of tight closure and multiplier ideals we introduce a new variant of tight closure associated to any fixed ideal a which we call tight closure and study various properties thereof in our theory the annihilator ideal r a of all tight closure relations which is a generalization of the test ideal in the usual tight closure theory plays a particularly important role we prove the correspondence of the ideal a and the multiplier ideal associated to a or the adjoint of a in lipman s sense in normal q gorenstein rings reduced from characteristic zero to characteristic p 0 also in fixed prime characteristic we establish some properties of r a similar to those of multiplier ideals e g a briancon skoda type theorem subadditivity etc with considerably simple proofs and study the relationship between the ideal and the f rationality of rees algebras

on the support of the ashtekar lewandowski measure we show that the ashtekar isham extension overline a g of the configuration space of yang mills theories a g is topologically and measure theoretically the projective limit of a family of finite dimensional spaces associated with arbitrary finite lattices

extension of weakly and strongly f regular rings by flat maps this is a preprint of an article published in the journal of algebra vol 241 2001 no 2 799 807

spin networks for noncompact groups spin networks are a natural generalization of wilson loop functionals they have been extensively studied in the case where the gauge group is compact and it has been shown that they naturally form a basis of gauge invariant observables physically the restriction to compact gauge groups is enough for the study of yang mills theories however it is well known that noncompact groups naturally arise as internal gauge groups for lorentzian gravity models in this context a proper construction of gauge invariant observables is needed the purpose of the present work is to define the notion of spin network states for noncompact groups we first build by a careful gauge fixing procedure a natural measure and a hilbert space structure on the space of gauge invariant graph connections spin networks are then defined as generalized eigenvectors of a complete set of hermitic commuting operators we show how the delicate issue of taking the quotient of a space by noncompact groups can be address in term of algebraic geometry we finally construct the full hilbert space containing all spin network states having in mind applications to gravity we illustrate our results for the groups sl 2 r and sl 2 c

quantization of diffeomorphism invariant theories of connections with local degrees of freedom quantization of diffeomorphism invariant theories of connections is studied and the quantum diffeomorphism constraint is solved the space of solutions is equipped with an inner product that is shown to satisfy the physical reality conditions this provides in particular a quantization of the husain kucha model the main results also pave the way to quantization of other diffeomorphism invariant theories such as general relativity in the riemannian case i e signature the approach appears to contain all the necessary ingredients already in the lorentzian case it will have to be combined in an appropriate fashion with a coherent state transform to incorporate complex connections

graded annihilators and tight closure test ideals abstract let r be a commutative noetherian local ring of prime characteristic p with maximal ideal m the main purposes of this paper are to show that if the injective envelope e of r m has a structure as an x torsion free left module over the frobenius skew polynomial ring over r in the indeterminate x then r has a tight closure test element for modules and is f pure and to relate the test ideal of r to the smallest e special ideal of r of positive height a byproduct is an analogue of a result of janet cowden vassilev she showed in the case where r is an f pure homomorphic image of an f finite regular local ring that there exists a strictly ascending chain 0 0 1 t r of radical ideals of r such that for each i 0 t 1 the reduced local ring r i is f pure and its test ideal has positive height and is exactly i 1 i this paper presents an analogous result in the case where r is complete but not necessarily f finite and e has a structure as an x torsion free left module over the frobenius skew polynomial ring whereas cowden vassilev s results were based on r fedder s criterion for f purity the arguments in this paper are based on the author s work on graded annihilators of left modules over the frobenius skew polynomial ring

discreteness and rationality of f jumping numbers on singular varieties we prove that the f jumping numbers of the test ideal tau x delta mathfrak a t are discrete and rational under the assumptions that x is a normal and f finite scheme over a field of positive characteristic p kx is mathbb q cartier of index not divisible p and either x is essentially of finite type over a field or the sheaf of ideals mathfrak a is locally principal this is the largest generality for which discreteness and rationality are known for the jumping numbers of multiplier ideals in characteristic zero

adjoint ideals along closed subvarieties of higher codimension in this paper we introduce a notion of adjoint ideal sheaves along closed subvarieties of higher codimension and study its local properties using char acteristic p methods when x is a normal gorenstein closed subvariety of a smooth complex variety a we formulate a restriction property of the adjoint ideal sheaf adj x a of a along x involving the l c i ideal sheaf dx of x the proof relies on a modification of generalized test ideals of hara and yoshida 11

the inverse loop transform the loop transform in quantum gauge field theory can be recognized as the fourier transform or characteristic functional of a measure on the space of generalized connections modulo gauge transformations since this space is a compact hausdorff space conversely we know from the riesz markov theorem that every positive linear functional on the space of continuous functions thereon qualifies as the loop transform of a regular borel measure on the moduli space in the present article we show how one can compute the finite joint distributions of a given characteristic functional that is we derive the inverse loop transform

projective loop quantum gravity i state space instead of formulating the state space of a quantum field theory over one big hilbert space it has been proposed by kijowski to describe quantum states as projective families of density matrices over a collection of smaller simpler hilbert spaces beside the physical motivations for this approach it could help designing a quantum state space holding the states we need in a latter work by okolow the description of a theory of abelian connections within this framework was developed an important insight being to use building blocks labeled by combinations of edges and surfaces the present work generalizes this construction to an arbitrary gauge group g in particular g is neither assumed to be abelian nor compact this involves refining the definition of the label set as well as deriving explicit formulas to relate the hilbert spaces attached to different labels if the gauge group happens to be compact we also have at our disposal the well established ashtekar lewandowski hilbert space which is

an algebraic extension of dirac quantization examples an extension of the dirac procedure for the quantization of constrained systems is necessary to address certain issues that are left open in dirac s original proposal these issues play an important role especially in the context of nonlinear diffeomorphism invariant theories such as general relativity recently an extension of the required type was proposed using algebraic quantization methods in this paper the key conceptual and technical aspects of the algebraic program are illustrated through a number of finite dimensional examples the choice of examples and some of the analysis is motivated by certain peculiar problems endemic to quantum gravity however prior knowledge of general relativity is not assumed in the main discussion indeed the methods introduced and conclusions arrived at are applicable to any system with first class constraints in particular they resolve certain technical issues which are present also in the reduced phase space approach to quantization of these systems

cartier modules finiteness results on a locally noetherian scheme x over a field of positive characteristic p we study the category of coherent ox modules m equipped with a pe linear map i e an additive map c ox ox satisfying rc m c r p e m for all m a m r a ox the notion of nilpotence meaning that some power of the map c is zero is used to rigidify this cate gory the resulting quotient category called cartier crystals satisfies some strong finiteness conditions the main result in this paper states that if the frobenius morphism on x is a finite map i e if x is f finite then all cartier crystals have finite length we further show how this and related results can be used to recover and generalize other finiteness results of hartshorne speiser 19 lyubeznik 27 sharp 30 enescu hochster 14 and hochster 20 about the structure of modules with a left action of the frobenius for example we show that over any regular f finite scheme x lyubeznik s f finite modules 27 have finite length

refined algebraic quantisation in a system with nonconstant gauge invariant structure functions in a previous work j louko and e martinez pascual constraint rescaling in refined algebraic quantisation momentum constraint j math phys 52 123504 2011 10 1063 1 3664336 refined algebraic quantisation raq within a family of classically equivalent constrained hamiltonian systems that are related to each other by rescaling one momentum type constraint was investigated in the present work the first steps to generalise this analysis to cases where more constraints occur are developed the system under consideration contains two momentum type constraints originally abelian where rescalings of these constraints by a non vanishing function of the coordinates are allowed these rescalings induce structure functions at the level of the gauge algebra providing a specific parametrised family of real valued scaling functions the implementation of the corresponding rescaled quantum momentum type constraints is performed using raq when the gauge algebra i remains abelian and ii undergoes int

an excellent f pure ring of prime characteristic has a big tight closure test element in two recent papers the author has developed a theory of graded annihilators of left modules over the frobenius skew polynomial ring over a commutative noetherian ring r of prime characteristic p and has shown that this theory is relevant to the theory of test elements in tight closure theory one result of that work was that if r is local and the r module structure on the injective envelope e of the simple r module can be extended to a structure as a torsion free left module over the frobenius skew polynomial ring then r is f pure and has a tight closure test element one of the central results of this paper is the converse namely that if r is f pure then e has a structure as a torsion free left module over the frobenius skew polynomial ring a corollary is that every f pure local ring of prime characteristic even if it is not excellent has a tight closure test element these results are then used along with embedding theorems for modules over the frobenius skew polynomial ring to show that every excellent not necessarily local f pure ring of characteristic p must have a so called big test element

f rational rings have rational singularities it is proved that an excellent local ring of prime characteristic in which a single ideal generated by any system of parameters is tightly closed must be pseudorational a key point in the proof is a characterization of f rational local rings as those cohen macaulay local rings r m in which the local cohomology module h d m r where d is the dimension of r have no submodules stable under the natural action of the frobenius map an analog for finitely generated algebras over a field of characteristic zero is developed which yields a reasonably checkable tight closure test for rational singularities of an algebraic variety over inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 01i without reference to a desingularization

big tight closure test elements for some non reduced excellent rings abstract this paper is concerned with existence of big tight closure test elements for a commutative noetherian ring r of prime characteristic p let r denote the complement in r of the union of the minimal prime ideals of r a big test element for r is an element of r which can be used in every tight closure membership test for every r module and not just the finitely generated ones the main results of the paper are that if r is excellent and satisfies condition r 0 and c r is such that r c is gorenstein and weakly f regular then some power of c is a big test element for r if i r is a homomorphic image of an excellent regular ring of characteristic p for which the frobenius homomorphism is intersection flat or ii r is f pure or iii r is local the gamma construction is not used

graded annihilators of modules over the frobenius skew polynomial ring and tight closure this paper is concerned with the tight closure of an ideal a in a commutative noetherian local ring r of prime characteristic p several authors including r fedder k i watanabe k e smith n hara and f enescu have used the natural frobenius action on the top local cohomology module of such an r to good effect in the study of tight closure and this paper uses that device the main part of the paper develops a theory of what are here called special annihilator submodules of a left module over the frobenius skew polynomial ring associated to r this theory is then applied in the later sections of the paper to the top local cohomology module of r and used to show that if r is cohen macaulay then it must have a weak parameter test element even if it is not excellent