nonlocal discrete regularization on weighted graphs a framework for image and manifold processing we introduce a nonlocal discrete regularization framework on weighted graphs of the arbitrary topologies for image and manifold processing the approach considers the problem as a variational one which consists of minimizing a weighted sum of two energy terms a regularization one that uses a discrete weighted dirichlet energy and an approximation one this is the discrete analogue of recent continuous euclidean nonlocal regularization functionals the proposed formulation leads to a family of simple and fast nonlinear processing methods based on the weighted laplace operator parameterized by the degree of regularity the graph structure and the graph weight function these discrete processing methods provide a graph based version of recently proposed semi local or nonlocal processing methods used in image and mesh processing such as the bilateral filter the tv digital filter or the nonlocal means filter it works with equal ease on regular 2 d and 3 d images manifolds or any data we illustrate the abilities of the approach by applying it to various types of images meshes manifolds and data represented as graphs

ladders and simplicity of derived module categories abstract recollements of derived module categories are investigated using a new technique ladders of recollements which are maximal mutation sequences the position in the ladder is shown to control whether a recollement restricts from unbounded to another level of derived category ladders also turn out to control derived simplicity on all levels an algebra is derived simple if its derived category cannot be deconstructed that is if it is not the middle term of a non trivial recollement whose outer terms are again derived categories of algebras derived simplicity on each level is characterised in terms of heights of ladders these results are complemented by providing new classes of examples of derived simple rings in particular indecomposable commutative rings as well as by a finite dimensional counterexample to the jordan holder problem for derived module categories moreover recollements are used to compute homological and k theoretic invariants

irreductible representations of the simple jordan superalgebra of grassmann poisson bracket abstract we obtain classification of the irreducible bimodules over the jordan superalgebra kan n the kantor double of the grassmann poisson superalgebra g n on n odd generators for all n 2 and an algebraically closed field of characteristic 2 this generalizes the corresponding result of c martinez and e zelmanov announced in 12 for n 4 and a field of characteristic zero

the capelli eigenvalue problem for lie superalgebras for a finite dimensional unital complex simple jordan superalgebra j the tits kantor koecher construction yields a 3 graded lie superalgebra mathfrak g flat cong mathfrak g flat 1 oplus mathfrak g flat 0 oplus mathfrak g flat 1 such that mathfrak g flat 1 cong j set v mathfrak g flat 1 and mathfrak g mathfrak g flat 0 in most cases the space mathscr p v of superpolynomials on v is a completely reducible and multiplicity free representation of mathfrak g and there exists a direct sum decomposition mathscr p v bigoplus lambda in omega v lambda where left v lambda right lambda in omega is a family of irreducible mathfrak g modules parametrized by a set of partitions omega in these cases one can define a natural basis left d lambda right lambda in omega of capelli operators for the algebra mathscr pd v mathfrak g of mathfrak g invariant superpolynomial differential operators on v in this paper we complete the solution to the capelli eigenvalue problem which asks for the determination of the scalar c mu lambda by which d mu acts on v lambda we associate a restricted root system varsigma to the symmetric pair mathfrak g mathfrak k that corresponds to j which is either a deformed root system of type mathsf a m n or a root system of type mathsf q n we prove a necessary and sufficient condition on the structure of varsigma for mathscr p v to be completely reducible and multiplicity free when varsigma satisfies the latter condition we obtain an explicit formula for the eigenvalue c mu lambda in terms of sergeev veselov s shifted super jack polynomials when varsigma is of type mathsf a m n and okounkov ivanov s factorial schur q polynomials when varsigma is of type mathsf q n along the way we prove that the natural map from the centre of the enveloping algebra of mathfrak g into mathscr pd v mathfrak g is surjective in all cases except when j cong f where f is the 10 dimensional exceptional jordan superalgebra

infinite dimensional primitive linearly compact lie superalgebras we classify open maximal subalgebras of all infinite dimensional linearly compact simple lie superalgebras this is applied to the classification of infinite dimensional lie superalgebras of vector fields acting transitively and primitively in a formal neighborhood of a point of a finite dimensional supermanifold

representation theory of jordan superalgebras i we complete the classification of irreducible bimodules over simple finite dimensional jordan superalgebras

bm3d frames and variational image deblurring a family of the block matching 3 d bm3d algorithms for various imaging problems has been recently proposed within the framework of nonlocal patchwise image modeling in this paper we construct analysis and synthesis frames formalizing bm3d image modeling and use these frames to develop novel iterative deblurring algorithms we consider two different formulations of the deblurring problem i e one given by the minimization of the single objective function and another based on the generalized nash equilibrium gne balance of two objective functions the latter results in the algorithm where deblurring and denoising operations are decoupled the convergence of the developed algorithms is proved simulation experiments show that the decoupled algorithm derived from the gne formulation demonstrates the best numerical and visual results and shows superiority with respect to the state of the art in the field confirming a valuable potential of bm3d frames as an advanced image modeling tool

derived tame local and two point algebras we determine derived representation type of complete finitely generated local and two point algebras over an algebraically closed field one of the main problems in the representation theory of algebras is a classification of indecomposable finitely generated modules the dichotomy theorem 14 divides all finite dimensional algebras according to their rep resentation type into tame and wild in the case of tame algebras a clas sification of indecomposable modules is relatively easy for each dimension d they admit a parametrization of d dimensional indecomposable modules by a finite number of 1 parameter families the situation is much more complicated in the case of wild algebras this singles out the problem of establishing the representation type of a given algebra the answer is fully known for complete local algebras those algebras whose quiver contains a single vertex 6 7 13 21 23 28 31 32 and for finite dimensional two point algebras 4 8 18 19 20 25 27 in the case of infinite dimensional two point algebras the problem is still open except the pure noetherian algebras 15 during the last years there has been an active study of derived categories in particular a notion of derived representation type was introduced for finite dimensional algebras 22 the tame wild dichotomy for derived categories over finite dimensional algebras was established in 3 the structure of the derived category is known for a few classes of finite dimensional algebras e g 9 10 24 26

cartoon texture image decomposition using blockwise low rank texture characterization using a novel characterization of texture we propose an image decomposition technique that can effectively decomposes an image into its cartoon and texture components the characterization rests on our observation that the texture component enjoys a blockwise low rank nature with possible overlap and shear because texture in general is globally dissimilar but locally well patterned more specifically one can observe that any local block of the texture component consists of only a few individual patterns based on this premise we first introduce a new convex prior named the block nuclear norm bnn leading to a suitable characterization of the texture component we then formulate a cartoon texture decomposition model as a convex optimization problem where the simultaneous estimation of the cartoon and texture components from a given image or degraded observation is executed by minimizing the total variation and bnn in addition patterns of texture extending in different directions are extracted separately which is a special feature of the proposed model and of benefit to texture analysis and other applications furthermore the model can handle various types of degradation occurring in image processing including blur missing pixels with several types of noise by rewriting the problem via variable splitting the so called alternating direction method of multipliers becomes applicable resulting in an efficient algorithmic solution to the problem numerical examples illustrate that the proposed model is very selective to patterns of texture which makes it produce better results than state of the art decomposition models

is tame open abstract is tame open no answer so far one may pose the tame open conjecture tame is open but how to support it no effective way to date in this note the rank of a wild algebra is introduced the wild rank conjecture which implies the tame open conjecture is formulated the wild rank conjecture is improved to the basic wild rank conjecture a covering criterion on the rank of a basic wild algebra is given which can be effectively applied to verify the basic wild rank conjecture for concrete algebras it makes all conjectures much reliable

edge direction preserving image zooming a mathematical and numerical analysis we focus in this paper on some reconstruction restoration methods whose aim is to improve the resolution of digital images the main point here is to study the ability of such methods to preserve one dimensional 1d structures indeed such structures are important since they are often carried by the image edges first we focus on linear methods give a general framework to design them and show that the preservation of 1d structures pleads in favor of the cancellation of the periodization of the image spectrum more precisely we show that preserving 1d structures implies the linear methods to be written as a convolution of the sinc interpolation as a consequence we cannot cope linearly with gibbs effects sharpness of the results and the preservation of the 1d structure second we study variational nonlinear methods and in particular the one based on total variation we show that this latter permits us to avoid these shortcomings we also prove the existence and consistency of an approximate solution to this variational problem at last this theoretical study is highlighted by experiments both on synthetic and natural images which show the effects of the described methods on images as well as on their spectrum

nonlocally centralized sparse representation for image restoration sparse representation models code an image patch as a linear combination of a few atoms chosen out from an over complete dictionary and they have shown promising results in various image restoration applications however due to the degradation of the observed image e g noisy blurred and or down sampled the sparse representations by conventional models may not be accurate enough for a faithful reconstruction of the original image to improve the performance of sparse representation based image restoration in this paper the concept of sparse coding noise is introduced and the goal of image restoration turns to how to suppress the sparse coding noise to this end we exploit the image nonlocal self similarity to obtain good estimates of the sparse coding coefficients of the original image and then centralize the sparse coding coefficients of the observed image to those estimates the so called nonlocally centralized sparse representation ncsr model is as simple as the standard sparse representation model while our extensive experiments on various types of image restoration problems including denoising deblurring and super resolution validate the generality and state of the art performance of the proposed ncsr algorithm

a q analogue of the riordan group abstract the riordan group consisting of riordan matrices shows up naturally in a variety of combinatorial settings in this paper we define a q riordan matrix to be a q analogue of the exponential riordan matrix by using the eulerian generating functions of the form n 0 f n z n n q we first prove that the set of q riordan matrices forms a loop a quasigroup with an identity element and find its loop structures next it is shown that q riordan matrices associated to the counting functions may be applied to the enumeration problem on set partitions by block inversions this notion leads us to find q analogues of the composition formula and the exponential formula respectively

unital bimodules over the simple jordan superalgebra d t we classify indecomposable finite dimensional bimodules over jordan superalgebras d t t 1 0 1

representations of simple noncommutative jordan superalgebras i abstract in this article we begin the study of representations of simple finite dimensional noncommutative jordan superalgebras in the case of degree 3 we show that any finite dimensional representation is completely reducible and depending on the superalgebra quasiassociative or jordan then we study representations of superalgebras d t and k 3 and prove the kronecker factorization theorem for superalgebras d t in the last section we use a new approach to study noncommutative jordan representations of simple jordan superalgebras

solving inverse problems with piecewise linear estimators from gaussian mixture models to structured sparsity a general framework for solving image inverse problems with piecewise linear estimations is introduced in this paper the approach is based on gaussian mixture models which are estimated via a maximum a posteriori expectation maximization algorithm a dual mathematical interpretation of the proposed framework with a structured sparse estimation is described which shows that the resulting piecewise linear estimate stabilizes the estimation when compared with traditional sparse inverse problem techniques we demonstrate that in a number of image inverse problems including interpolation zooming and deblurring of narrow kernels the same simple and computationally efficient algorithm yields results in the same ballpark as that of the state of the art

new q laguerre polynomials having factorized permutation interpretations abstract in this paper we generalize the laguerre polynomials in terms of q analogue for riordan matrices to be more specific for n 0 we introduce new q laguerre polynomials l n x q by defining the eulerian generating function for l n x q as j 0 1 1 q j 1 z e q x z 1 z interestingly it turns out that l n x q have combinatorial descriptions in the aspect of the inversions of factorized permutations and q rook numbers we locate their zeros and develop their algebraic properties as well

image zoom completion we consider the problem of recovering a high resolution image from a pair consisting of a complete low resolution image and a high resolution but incomplete one we refer to this task as the image zoom completion problem after discussing possible contexts in which this setting may arise we introduce a nonlocal regularization strategy giving full details concerning the numerical optimization of the corresponding energy and discussing its benefits and shortcomings we also derive two total variation based algorithms and evaluate the performance of the proposed methods on a set of natural and textured images we compare the results and get with those obtained with two recent state of the art single image super resolution algorithms

a new n 6 superconformal algebra in this paper we construct a newn 6 superconformal algebra which extends the virasoro algebra by theso 6 current algebra by 6 odd primary fields of conformal weight 3 2 and by 10 odd primary fields of conformal weight 1 2 the commutation relations of this algebra which we will refer to asck 6 are represented by short distance operator product expansions ope we constructck 6 as a subalgebra of theso 6 superconformal algebra k6 thus giving it a natural representation as first order differential operators on the circle withn 6 extended symmetry we show thatck 6 has no nontrivial central extensions

stirling probability and q bosons normal ordering of powers of the bosonic number operator can be used to define a discrete probability distribution associated with the number of elements of a random set we represent these random sets by vectors in a hilbert space and obtain q bosons

some combinatorial applications of the q riordan matrix abstract recently the authors developed a q analogue for riordan matrices by means of eulerian generating functions of the form g z n 0 g n z n n q where n q is the q factorial we apply this concept to give q analogues of some familiar objects from the set partitions with double restrictions on blocks namely the r s bessel numbers of both types by setting r 0 and letting s these numbers may be reduced to the q stirling numbers of both kinds several algebraic formulas for the q analogues are also derived using combinatorial methods together with the concept of q riordan matrices in particular q analogues of the classical bessel numbers of both kinds and their combinatorial interpretations are obtained

lie superalgebras graded by p n and q n abstract in this article we study lie superalgebras graded by the root systems p n and q n

classification of linearly compact simple jordan and generalized poisson superalgebras abstract we classify all linearly compact simple jordan superalgebras over an algebraically closed field of characteristic zero as a corollary we deduce the classification of all linearly compact unital simple generalized poisson superalgebras

mertens equimodular matrices of redheffer type abstract asymptotic behavior of the mertens function m n k 0 n k which is equal to the determinant of the n n redheffer matrix is known to be closely related to the riemann hypothesis an infinite matrix whose nth leading principal minor is equal to m n for all n 1 is called a mertens equimodular matrix we use riordan matrices to find a large class of mertens equimodular matrices each element of which is called by us a riordan redheffer matrix briefly an r r matrix we also give the generating function for the characteristic polynomials of r r matrices as a result we introduce several examples of r r matrices that reveal interesting spectral properties further we pose two conjectures on the eigenvalues of those r r matrices finally we find a sufficient condition for the riemann hypothesis using the smallest singular value of a r r matrix

analysis operator learning and its application to image reconstruction exploiting a priori known structural information lies at the core of many image reconstruction methods that can be stated as inverse problems the synthesis model which assumes that images can be decomposed into a linear combination of very few atoms of some dictionary is now a well established tool for the design of image reconstruction algorithms an interesting alternative is the analysis model where the signal is multiplied by an analysis operator and the outcome is assumed to be sparse this approach has only recently gained increasing interest the quality of reconstruction methods based on an analysis model severely depends on the right choice of the suitable operator in this paper we present an algorithm for learning an analysis operator from training images our method is based on lp norm minimization on the set of full rank matrices with normalized columns we carefully introduce the employed conjugate gradient method on manifolds and explain the underlying geometry of the constraints moreover we compare our approach to state of the art methods for image denoising inpainting and single image super resolution our numerical results show competitive performance of our general approach in all presented applications compared to the specialized state of the art techniques

restricted growth functions rank row matchings of partition lattices and q stirling numbers abstract in this paper we use restricted growth functions to give a direct algebraic description of the incidence relations of the lattice of partitions of an n set that is independent of the partitions themselves this description of the incidence relations not only gives new information about rank row matchings of partition lattices but is of independent interest since every finite lattice is a sublattice of some partition lattice as a further application of restricted growth functions we show how the polynomials in q known as q stirling numbers of the second kind may be viewed as generating functions

on the tits kantor koecher construction of unital jordan bimodules abstract in this paper we explore relationship between representations of a jordan algebra j and the lie algebra g obtained from j by the tits kantor koecher construction more precisely we construct two adjoint functors l i e j mod 1 g mod 1 and j o r g mod 1 j mod 1 where j mod 1 is the category of unital j bimodules and g mod 1 is the category of g modules admitting a short grading using these functors we classify j such that the semisimple part is of clifford type and the category j mod 1 is tame

dual norms and image decomposition models following a recent work by y meyer decomposition models into a geometrical component and a textured component have recently been proposed in image processing in such approaches negative sobolev norms have seemed to be useful to modelize oscillating patterns in this paper we compare the properties of various norms that are dual of sobolev or besov norms we then propose a decomposition model which splits an image into three components a first one containing the structure of the image a second one the texture of the image and a third one the noise our decomposition model relies on the use of three different semi norms the total variation for the geometrical component a negative sobolev norm for the texture and a negative besov norm for the noise we illustrate our study with numerical examples

controlled wild algebras the controlled wild algebra is introduced a covering criterion for a finite dimensional algebra to be controlled wild is given and this criterion is applied to the algebras with radical square zero algebras with zero relations local algebras and finite p group algebras 2000 mathematical subject classification 16g20 16g60

q riordan representation abstract the aim of this study is to establish further theory of q riordan representation we emphasize that any q matrices can be obtained by q riordan representation then we analyse the structure of the entries in the matrix product of general q riordan matrices

structurable superalgebras of cartan type abstract we describe the simple lie superalgebras arising from the unital structurable superalgebras of characteristic 0 and construct four series of the unital simple structurable superalgebras of cartan type we give a classification of simple structurable superalgebras of cartan type over an algebraically closed field f of characteristic 0 together with the faulkner theorem on the classification of classical such superalgebras it gives a classification of the simple structurable superalgebras over f

congruence properties of q analogs abstract using group actions and generating functions we derive various arithmetic properties of the q binomial coefficients and q stirling numbers these include recurrence relations and computation of residues modulo a cyclotomic polynomial we also obtain periodicity and non periodicity results

a q analogue of mahler expansions i abstract we examine a q analogue of mahler expansions for continuous functions in p adic analysis replacing binomial coefficient polynomials x n with a q analogue x n q for a p adic variable q with q 1 p q 1 and we consider the p adic q gamma function p q of koblitz relative to its q mahler expansion

wild two point algebras abstract the representation type of two point algebras is determined it is also shown that a two point algebra is wild if and only if it is controlled wild moreover it turns out that a minimal wild two point algebra is strongly minimal wild if and only if it is not strictly wild

cranks and t cores new statistics on partitions calledcranks are defined which combinatorially prove ramanujan s congruences for the partition function modulo 5 7 11 and 25 explicit bijections are given for the equinumerous crank classes the cranks are closely related to thet core of a partition usingq series some explicit formulas are given for the number of partitions which aret cores some related questions for self conjugate and distinct partitions are discussed

modular representations of the jordan superalgebras d t and k3 abstract up to the irreducible representations of the simple three dimensional lie algebra s l 2 we classify the unital finite dimensional irreducible jordan representations of the simple superalgebra d t in the case of an algebraically closed field of characteristic p 2 as a corollary we obtain a classification of the finite dimensional irreducible representations of the kaplansky superalgebra k 3 in the case of characteristic p 2

jordan bimodules over the superalgebras p n and q n we extend the jacobson s coordinatization theorem to jordan superalgebras using it we classify jordan bimodules over superalgebras of types q n and jp n n 3 then we use the tits kantor koecher construction and representation theory of lie superalgebras to treat the remaining case q 2

the delta conjecture we conjecture two combinatorial interpretations for the symmetric function eken where f is an eigenoperator for the modified macdonald polynomials defined by bergeron garsia haiman and tesler both interpretations can be seen as generalizations of the shuffle conjecture a statement originally conjectured by haglund haiman remmel loehr and ulyanov and recently proved by carlsson and mellit we show how previous work of the second and third authors on tesler matrices and ordered set partitions can be used to verify several cases of our conjectures furthermore we use a reciprocity identity and llt polynomials to prove another case finally we show how our conjectures inspire 4 variable generalizations of the catalan numbers extending work of garsia haiman and the first author

image super resolution via sparse representation this paper presents a new approach to single image superresolution based upon sparse signal representation research on image statistics suggests that image patches can be well represented as a sparse linear combination of elements from an appropriately chosen over complete dictionary inspired by this observation we seek a sparse representation for each patch of the low resolution input and then use the coefficients of this representation to generate the high resolution output theoretical results from compressed sensing suggest that under mild conditions the sparse representation can be correctly recovered from the downsampled signals by jointly training two dictionaries for the low and high resolution image patches we can enforce the similarity of sparse representations between the low resolution and high resolution image patch pair with respect to their own dictionaries therefore the sparse representation of a low resolution image patch can be applied with the high resolution image patch dictionary to generate a high resolution image patch the learned dictionary pair is a more compact representation of the patch pairs compared to previous approaches which simply sample a large amount of image patch pairs reducing the computational cost substantially the effectiveness of such a sparsity prior is demonstrated for both general image super resolution sr and the special case of face hallucination in both cases our algorithm generates high resolution images that are competitive or even superior in quality to images produced by other similar sr methods in addition the local sparse modeling of our approach is naturally robust to noise and therefore the proposed algorithm can handle sr with noisy inputs in a more unified framework

representations of simple jordan superalgebras abstract this paper completes description of categories of representations of finite dimensional simple unital jordan superalgebras over algebraically closed field of characteristic zero

wedderburn principal theorem for jordan superalgebras i abstract we consider finite dimensional jordan superalgebras j over an algebraically closed field of characteristic 0 with solvable radical n such that n 2 0 and j n is a simple jordan superalgebra of one of the following types kac k 10 kaplansky k 3 superform or d t we prove that an analogue of the wedderburn principal theorem wpt holds if certain restrictions on the types of irreducible subbimodules of n are imposed where n is considered as a j n bimodule using counterexamples it is shown that the imposed restrictions are essential