the as cohen macaulay property for quantum flag manifolds of minuscule weight abstract it is shown that quantum homogeneous coordinate rings of generalised flag manifolds corresponding to minuscule weights their schubert varieties big cells and determinantal varieties are as cohen macaulay the main ingredient in the proof is the notion of a quantum graded algebra with a straightening law introduced by t h lenagan and l rigal t h lenagan l rigal quantum graded algebras with a straightening law and the as cohen macaulay property for quantum determinantal rings and quantum grassmannians j algebra 301 2006 670 702 using stanley s theorem it is moreover shown that quantum generalised flag manifolds of minuscule weight and their big cells are as gorenstein

algebraic structure of multiparameter quantum groups let g be a connected semi simple complex lie group we define and study the multi parameter quantum group cq p g in the case where q is a complex parameter that is not a root of unity using a method of twisting bigraded hopf algebras by a cocycle 2 we develop a unified approach to the construction of cq p g and of the multi parameter drinfeld double dq p using a general method of deforming bigraded pairs of hopf algebras we construct a hopf pairing between these algebras from which we deduce a peterweyl type theorem for cq p g we then describe the prime and primitive spectra of cq p g generalizing a result of joseph in the one parameter case this description was conjectured and established in the sl n case by the first and second authors in 15 16 it was proved in the general case by joseph in 18 19 in particular the orbits in prim cq p g under the natural action of the maximal torus h are indexed as in the one parameter case by the elements of the double weyl group w w unlike the one parameter case there is not in general a bijection between symp g and prim cq p g however in the case when

study of q commutations in uq n algebra abstract let g be a semi simple lie algebra with maximal nilpotent subalgebra n let q g and u q n be the quantized enveloping algebras we describe the set of normal elements of u q n we prove that each ideal of u q n can be generated by a normal sequence

morphological reversible contour representation in this paper a novel morphological reversible contour representation of discrete binary images is proposed a binary image is represented by a set of nonoverlapping multilevel contours and a residual image in this proposed representation the total number of pixels representing an image is far less than the total number of pixels obtained by the seed based morphological contour skeleton lossless representation the proposed contour representation is simple unique and general without restrictions on the binary image to be represented the resulting multicontour image component is robust to noise an efficient differential chain contour coding scheme is employed to further compress the represented image the proposed method yields very low bit rates compared to the existing morphological techniques an automatic filling procedure which properly fills a proper multicontour image according to its topological structure without need of seed points is proposed

connected graded gorenstein algebras with enough normal elements we generalize 12 1 1 and 1 2 to the following situation theorem 1 let a be a connected graded noetherian algebra of injective dimension d such that every nonsimple graded prime factor ring of a contains a homogeneous normal element of positive degree then 1 a is auslander gorenstein and cohen macaulay 2 a has a quasi frobenius quotient ring 3 every minimal prime ideal p is graded andgkdim a p d 4 if moreover a has finite global dimension then a is a domain and a maximal order in its quotient division ring to prove the above we need the following result which is a generalization of 3 2 46 ii theorem 2 let a be a connected graded noetherian as gorenstein algebra of injective dimension d then 1 the last term of the minimal injective resolution of aais isomorphic to a shift of a 2 for every noetherian graded a module m extd m a is finite dimensional over k

the epsilon expansion from conformal field theory conformal multiplets of and 3 recombine at the wilson fisher xed point as a consequence of the equations of motion using this fact and other constraints from conformal symmetry we reproduce the lowest nontrivial order results for the anomalous dimensions of operators without any input from perturbation theory

on critical exponents without feynman diagrams in order to achieve a better analytic handle on the modern conformal bootstrap program we re examine and extend the pioneering 1974 work of polyakov s which was based on consistency between the operator product expansion and unitarity as in the bootstrap approach this method does not depend on evaluating feynman diagrams we show how this approach can be used to compute the anomalous dimensions of certain operators in the o n model at the wilson fisher fixed point in 4 epsilon dimensions up to o epsilon 2

rigid dualizing complex for quantum enveloping algebras and algebras of generalized differential operators abstract in the first part of this article we compute the rigid dualizing complex of a quantum enveloping algebra we consider the generic case and the case of a specialization at a non root of unity this answers a question of yekutieli j pure appl algebra 150 2000 85 in bull soc math france 122 1994 371 and math z 232 1999 367 we generalized d module theory to lie algebroids using these results we compute explicitly the rigid dualizing complex of the algebra of differential operators defined by an affine lie algebroid this generalizes results of yekutieli j pure appl algebra 150 2000 85

efficient morphological shape representation mathematical morphology is well suited to capturing geometric information hence morphology based approaches have been popular for object shape representation the two primary morphology based approaches the morphological skeleton and the morphological shape decomposition msd each represent an object as a collection of disjoint sets a practical shape representation scheme though should give a representation that is computationally efficient to use unfortunately little work has been done for the morphological skeleton and the msd to address efficiency we propose a flexible search based shape representation scheme that typically gives more efficient representations than the morphological skeleton and msd our method decomposes an object into a number of simple components based on homothetics of a set of structuring elements to form the representation the components are combined using set union and set difference operations we use three constituent component types and a thorough cost based search strategy to find efficient representations we also consider allowing object representation error which may yield even more efficient representations

dualizing complexes and tilting complexes over simple rings simple rings like fields are literally simple in many ways hence quite a few invariants of rings become trivial for simple rings we show that this principle applies to the derived picard group which classifies dualizing complexes over a ring in this paper all rings are algebras over a base field k ring homomorphisms are all over k and bimodules are all k central the symbol denotes k for a ring b b denotes the opposite ring we shall write moda for the category of left a modules and db moda will stand for the bounded derived category a brief review of key definitions such as dualizing complexes two sided tilting complexes and the derived picard group dpic a is included in the body of the paper

boundary constrained morphological skeleton minimization and skeleton reconstruction a new algorithm for minimizing a morphological skeleton entitled boundary constrained skeleton minimization bcsm as well as a new algorithm for reconstructing an original image from its minimized skeletal structure termed boundary constrained skeleton reconstruction bcsr are proposed the new algorithms are shown to reduce data storage requirements from n 1 binary images represented as separate skeleton subsets with their corresponding indices to 2 binary images composed of a binary morphological skeleton and its corresponding morphological boundary structure in addition to a reduction in memory storage bcsm and bcsr result in substantial savings in computational complexity the proposed algorithms are evaluated in the context of image analysis and coding and their performance is compared to previous algorithms proposed by serra 1982 1988 and by maragos and schafer 1986 sample evaluations indicate a greater than 22 fold savings in computational requirements and 11 fold reduction in memory requirements

homological identities for noncommutative rings abstract the auslander buchsbaum formula and bass s theorem are proved for a class of nonlocal noncommutative algebras including certain affine pi hopf algebras we also show that each filtered noetherian affine pi hopf algebra has finite injective dimension which partially answers a question of brown

noncommutative projective schemes an analogue of the concept of projective scheme is defined for noncommutative n graded algebras using the quotient category c of graded right a modules modulo its full subcategory of torsion modules we define proj a c a s where a is the object corresponding to the module aa and s is the autoequivalence defined by the shift of degrees the triples equivalent to proj a for a right noetherian graded algebra a are characterized in terms of a condition on extensions

rings with auslander dualizing complexes abstract a ring with an auslander dualizing complex is a generalization of an auslander gorenstein ring we show that many results which hold for auslander gorenstein rings also hold in the more general setting on the other hand we give criteria for existence of auslander dualizing complexes which show these occur quite frequently the most powerful tool we use is the local duality theorem for connected graded algebras over a field filtrations allow the transfer of results to nongraded algebras we also prove some results of a categorical nature most notably the functoriality of rigid dualizing complexes

threshold superposition in morphological image analysis systems it is shown that four composite morphological systems namely morphological edge detection peak valley extraction skeletonization and shape size distributions obey a weak linear superposition called threshold linear superposition the output image signal or measurement from each system is shown to be the sum of outputs due to input binary images that result from thresholding the input gray level image at all levels these results are generalized to a vector space formulation e g to any finite linear combination of simple morphological systems thus many such systems processing gray level images are reduced to corresponding binary image processing systems which are easier to analyze and implement

existence theorems for dualizing complexes over non commutative graded and filtered rings abstract in this note we prove existence theorems for dualizing complexes over graded and filtered rings thereby generalizing some results by zhang yekutieli and jorgensen

a second quantized kolmogorov chentsov theorem via the operator product expansion we establish a direct connection between two fundamental topics one in probability theory and one in quantum field theory the first topic is the problem of pointwise multiplication of random schwartz distributions which has been the object of recent progress thanks to hairer s theory of regularity structures and the theory of paracontrolled distributions introduced by gubinelli imkeller and perkowski the second topic is wilson s operator product expansion which is a general property of models of quantum field theory and a cornerstone of the bootstrap approach to conformal field theory our main result is a general theorem for the almost sure construction of products of random distributions by mollification and suitable additive as well as multiplicative renormalizations the hypothesis for this theorem is the operator product expansion with precise bounds for pointwise correlations we conjecture these bounds to be universal features of quantum field theories with gapped dimension spectrum our theorem can accommodate logarithmic corrections anomalous scaling dimensions and even lack of translation invariance however it only applies to fields with short distance singularities that are milder than white noise as an application we provide a detailed treatment of a scalar conformal field theory of mean field type i e the fractional massless free field also known as the fractional gaussian field

dualizing complexes over noncommutative graded algebras the motivation for the work presented in this paper was two questions posed by m artin let a be a d dimensional regular graded algebra over the field k with augmentation ideal m see def 4 13 denote by hi a resp hi a the dth left resp right local cohomology of a at the ideal m i e hi a lim extd a m a and similarly for hi a as left modules and as right modules hi a and h a are both isomorphic to a hom a k up to a twist in degrees all operations are taking place in the graded category

relations between short range and long range ising models we perform a numerical study of the long range lr ferromagnetic ising model with power law decaying interactions j ensuremath propto r ensuremath d ensuremath ensuremath sigma on both a one dimensional chain d 1 and a square lattice d 2 we use advanced cluster algorithms to avoid the critical slowing down we first check the validity of the relation connecting the critical behavior of the lr model with parameters d ensuremath sigma to that of a short range sr model in an equivalent dimension d we then study the critical behavior of the d 2 lr model close to the lower critical ensuremath sigma uncovering that the spatial correlation function decays with two different power laws the effect of the subdominant power law is much stronger than finite size effects and actually makes the estimate of critical exponents very subtle by including this subdominant power law the numerical data are consistent with the standard renormalization group rg prediction by sak phys rev b 8 281 1973 thus making not necessary and unlikely according to occam s razor the recent proposal by picco arxiv 1207 1018 of having a new set of rg fixed points in addition to the mean field one and the sr one

bounds on renormalized feynman graphs we prove two bounds on the value of renormalized euclidean feynman graphs one is a relatively crude but widely applicable bound the other a finer bound applicable to 4 4 like models

infinite chiral symmetry in four dimensions we describe a new correspondence between four dimensional conformal field theories with extended supersymmetry and two dimensional chiral algebras the meromorphic correlators of the chiral algebra compute correlators in a protected sector of the four dimensional theory infinite chiral symmetry has far reaching consequences for the spectral data correlation functions and central charges of any four dimensional theory with mathcal n 2 superconformal symmetry

hall algebras hereditary algebras and quantum groups let r be an associative hereditary algebra over a finite field k and let r fin be the full subcategory of r mod whose objects are those left r modules x which are finite as sets ix oc assume also that r isfinitary in c ringel s sense i e that iext l s s c for all simple s s in r fin this condition is met for example if r is finitely generated as k algebra 4 pp 435 436 let be the set of all isomorphism classes in r fin if 2 e then u will denote an r module in class 2 the class of all zero left r modules is denoted 0 let i c be the set of all isomorphism classes of simple modules in r fin thus u i e i is a complete set of simple finite left r modules we identify the grothendieck group k0 r fin with the free abelian group e1 v i v e 7z having i as free basis so that if x e r fin then the corresponding element of k0 r fin is the dimension vector dimx y i vii where for each i e vi is the multiplicity of the simple module u in any composition series of x clearly d imx lies in the subsemigroup

bootstrapping conformal field theories with the extremal functional method the existence of a positive linear functional acting on the space of differences between conformal blocks has been shown to rule out regions in the parameter space of conformal field theories cfts we argue that at the boundary of the allowed region the extremal functional contains in principle enough information to determine the dimensions and operator product expansion ope coefficients of an infinite number of operators appearing in the correlator under analysis based on this idea we develop the extremal functional method efm a numerical procedure for deriving the spectrum and ope coefficients of cfts lying on the boundary of solution space we test the efm by using it to rederive the low lying spectrum and ope coefficients of the two dimensional ising model based solely on the dimension of a single scalar quasiprimary char22 no virasoro algebra required our work serves as a benchmark for applications to more interesting less known cfts in the near future

homological aspects of noetherian pi hopf algebras and irreducible modules of maximal dimension abstract we prove that a noetherian hopf algebra of finite global dimension possesses further attractive homological properties at least when it satisfies a polynomial identity this applies in particular to quantized enveloping algebras and to quantized function algebras at a root of unity as well as to classical enveloping algebras in positive characteristic in all three cases we show that these algebras are auslander regular and macaulay we derive representation theoretic consequences concerning the coincidence of the non azumaya and singular loci for each of the above three classes of algebras

construction and borel summability of infrared φ 4 4 by a phase space expansion we construct the thermodynamic limit of the critical massless 4 model in 4 dimensions with an ultraviolet cutoff by means of a partly renormalized phase space expansion this expansion requires in a natural way the introduction of effective or running constants and the infrared asymptotic freedom of the model i e the decay of the running coupling constant plays a crucial role we prove also that the correlation functions of the model are the borel sums of their perturbation expansion

comparison between the morphological skeleton and morphological shape decomposition the morphological skeleton and morphological shape decomposition msd are two popular approaches for morphological shape representation each method represents an object as an algebraic combination of a number of components where each component is given by a locus of points dilated by a specified structuring element homothetic this correspondence develops a theoretical comparison between the two methods combining the theoretical results with several representation cost measures we make a concrete comparison of the efficiency of the two methods the results indicate that for complex objects i e objects requiring a full range of homothetic sizes in the morphological skeleton representation the msd represents objects more efficiently than the morphological skeleton for three of four suggested cost measures

dualising complexes and twisted hochschild co homology for noetherian hopf algebras we show that many noetherian hopf algebras a have a rigid dualising complex r with r a1 d here d is the injective dimension of the algebra and is a certain k algebra automorphism of a unique up to an inner automorphism in honour of the finite dimensional theory which is hereby generalised we call the nakayama automorphism of a we prove that s2 where s is the antipode of a and is the left winding automorphism of a determined by the left integral of a the hochschild homology and cohomology groups with coefficients in a suitably twisted free bimodule are shown to be non zero in the top dimension d when a is an artin schelter regular noetherian hopf algebra of global dimension d twisted poincare duality holds in this setting as is deduced from a theorem of van den bergh calculating for a using also the opposite coalgebra structure we determine a formula for s4 generalising a 1976 formula of radford for a finite dimensional applications of the results to the cases where a is pi an enveloping algebra a quantum group a quantised function algebra and a group algebra are outlined

noetherian pi hopf algebras are gorenstein we prove that every noetherian affine pi hopf algebra has finite injective dimension which answers a question of brown 1998

long range epidemic spreading in a random environment modeling long range epidemic spreading in a random environment we consider a quenched disordered d dimensional contact process with infection rates decaying with distance as 1 r d ensuremath sigma we study the dynamical behavior of the model at and below the epidemic threshold by a variant of the strong disorder renormalization group method and by monte carlo simulations in one and two spatial dimensions starting from a single infected site the average survival probability is found to decay as p t ensuremath sim t ensuremath d z up to multiplicative logarithmic corrections below the epidemic threshold a griffiths phase emerges where the dynamical exponent z varies continuously with the control parameter and tends to z c d ensuremath sigma as the threshold is approached at the threshold the spatial extension of the infected cluster in surviving trials is found to grow as r t ensuremath sim t 1 z c with a multiplicative logarithmic correction and the average number of infected sites in surviving trials is found to increase as n s t ensuremath sim lnt ensuremath chi with ensuremath chi 2 in one dimension

homological transcendence degree let d be a division algebra over a base field k the homological transcendence degree of d denoted by text htr d is defined to be the injective dimension of the algebra d otimes k d circ we show that text htr has several useful properties which the classical transcendence degree has we extend some results of resco rosenberg schofield and stafford and compute text htr for several classes of division algebras the main tool for the computation is van den bergh s rigid dualizing complex