induced hopf galois structures for a nite galois extension k k and an intermediate eld f such that gal k f has a normal complement in gal k k we construct and characterize hopf galois structures on k k which are induced by a pair of hopf galois structures on k f and f k

estimation of generalized mixtures and its application in image segmentation we introduce the notion of a generalized mixture and propose some methods for estimating it along with applications to unsupervised statistical image segmentation a distribution mixture is said to be generalized when the exact nature of the components is not known but each belongs to a finite known set of families of distributions for instance we can consider a mixture of three distributions each being exponential or gaussian the problem of estimating such a mixture contains thus a new difficulty we have to label each of three components there are eight possibilities we show that the classical mixture estimation algorithms expectation maximization em stochastic em sem and iterative conditional estimation ice can be adapted to such situations once as we dispose of a method of recognition of each component separately that is when we know that a sample proceeds from one family of the set considered we have a decision rule for what family it belongs to considering the pearson system which is a set of eight families the decision rule above is defined by the use of skewness and kurtosis the different algorithms so obtained are then applied to the problem of unsupervised bayesian image segmentation we propose the adaptive versions of sem em and ice in the case of blind i e pixel by pixel segmentation global segmentation methods require modeling by hidden random markov fields and we propose adaptations of two traditional parameter estimation algorithms gibbsian em gem and ice allowing the estimation of generalized mixtures corresponding to pearson s system the efficiency of different methods is compared via numerical studies and the results of unsupervised segmentation of three real radar images by different methods are presented

galois objects with normal bases for free hopf algebras of prime degree suppose r is a commutative ring and h a commutative cocommutative hopf algebra which is a free r module of prime rank the set of isomorphism classes of r algebras which are galois h objects forms an abelian group see chase and sweedler a in this paper we show that when h is an algebra over a special subring of the p adic integers containing a full set of p 1 st roots of unity then the subring consisting of those galois objects which are isomorphic to h as h module h hom h r is isomorphic to ue r u r where the notation u r means units of r congruent to 1 modx and 5 is a p 1 st root of an element b of r which will be seen to characterize the hopf algebra h the paper is organised as follows we briefly review notation related to hopf algebras integrals and galois objects in section 0 in section 1 we present the description of hopf algebras of order p given by tate and oort in their paper group schemes of prime order s such a hopf algebra h is associated with a pair of constants a b in fact hz r x x ax and h x r x xp bx where ab wi some unit of r in section 2 we describe the h module structure of h and the h module structure of a galois h object s and we state what it means for s to have a normal basis s z h as h module in section 3 we assume r contains a p l st root of 6 we show rz pz c h from which it follows that s contains a kummer order c modp s with s r and sis si i

noether s theorem for hopf orders in group algebras let k be a local field with valuation ring r of residue character istic p containing a primitive pth root of unity fp we state an analog to noether s theorem for modules over i hopf algebras and use induction tech niques to deduce a criterion for this analog to hold we then construct a family of noncommutative hopf algebras which satisfy the criterion if l k is a galois extension of number fields with group g and rings of integers s r then in a natural way l is a a g module and s is an rg module the normal basis theorem asserts that l kg as kg modu es the analogous result does not generally hold for s and rg even after localizing r in 1931 e noether n showed that locally for all primes p of r s rg as left 5g modules if and only if the extension l k is tame that is for every prime p of r the ramification index of any prime p of s lying over p is relatively prime to the characteristic of the residue field r p in fact the following are equivalent 1 s rg locally as 5g modules 2 l k is tame 3 the trace map s r is surjective 4 5 is a projective 5g module in 1986 l childs and s hurley c h generalized the notions of tameness and local normal basis to modules where h is an 5 hopf algebra their definitions specialized to those above when it rg and they obtained in par ticular the result that when h is commutative s h locally as modules if and only if s is tame waterhouse w notes that the childs hurley notions of tameness do not imply the existence of local normal basis when h is noncommutative even for group rings over fields the counterexample is not entirely satisfactory in that any natural number theoretic example would have normal basis over the quotient field in accordance with a result of kreimer and cook k c note that the childs hurley results assume commutativity of h yet noether s theorem holds for arbitrary finite groups accordingly there seems reason to believe that analogous results should hold for at least some non commutative h other than group algebras we say that noether s theorem

unsupervised texture segmentation in a deterministic annealing framework we present a novel optimization framework for unsupervised texture segmentation that relies on statistical tests as a measure of homogeneity texture segmentation is formulated as a data clustering problem based on sparse proximity data dissimilarities of pairs of textured regions are computed from a multiscale gabor filter image representation we discuss and compare a class of clustering objective functions which is systematically derived from invariance principles as a general optimization framework we propose deterministic annealing based on a mean field approximation the canonical way to derive clustering algorithms within this framework as well as an efficient implementation of mean field annealing and the closely related gibbs sampler are presented we apply both annealing variants to brodatz like microtexture mixtures and real word images

on the modularity of elliptic curves over q wild 3 adic exercises in this paper building on work of wiles wi and of wiles and one of us r t tw we will prove the following two theorems see 2 2 theorem a if e q is an elliptic curve then e is modular theorem b if gal q q gl2 f5 is an irreducible continuous representation with cyclotomic determinant then is modular we will first remind the reader of the content of these results and then briefly outline the method of proof if n is a positive integer then we let 1 n denote the subgroup of sl2 z consisting of matrices that modulo n are of the form 1 0 1

counting hopf galois structures on cyclic field extensions of squarefree degree abstract we investigate hopf galois structures on a cyclic field extension l k of squarefree degree n by a result of greither and pareigis each such hopf galois structure corresponds to a group of order n whose isomorphism class we call the type of the hopf galois structure we show that every group of order n can occur and we determine the number of hopf galois structures of each type we then express the total number of hopf galois structures on l k as a sum over factorisations of n into three parts as examples we give closed expressions for the number of hopf galois structures on a cyclic extension whose degree is a product of three distinct primes there are several cases depending on congruence conditions between the primes we also consider one case where the degree is a product of four primes

hopf galois structures on extensions of degree p2q and skew braces of order p2q the cyclic sylow p subgroup case abstract let p q be distinct primes with p 2 we classify the hopf galois structures on galois extensions of degree p 2 q such that the sylow p subgroups of the galois group are cyclic this we do according to greither and pareigis and byott by classifying the regular subgroups of the holomorphs of the groups g of order p 2 q in the case when the sylow p subgroups of g are cyclic this is equivalent to classifying the skew braces g furthermore we prove that if g and are groups of order p 2 q with non isomorphic sylow p subgroups then there are no regular subgroups of the holomorph of g which are isomorphic to equivalently a galois extension with galois group has no hopf galois structures of type g our method relies on the alternate brace operation on g which we use mainly indirectly that is in terms of the functions g aut g defined by g x x g g 1 these functions are in one to one correspondence with the regular subgroups of the holomorph of g and are characterised by the functional equation g h h g h for g h g we develop methods to deal with these functions with the aim of making their enumeration easier and more conceptual

maximal tame extensions over hopf orders in rings of integers of p adic number fields abstract let k k be a cyclic totally ramified kummer extension of degree pn with galois group g let o and o be the rings of integers in k and k respectively for n 1 f bertrandias and m j ferton determined the ring end o g o of o g endomorphisms of o and l n childs constructed the maximal order s in o whose ring of o g endomorphisms is a hopf order a hopf order whose linear dual is a larson order in kg is called a dual larson order in this paper the generators of a dual larson order are given we determine end o g o and obtain a maximal dual larson order contained in end o g o we construct a hopf galois extension s in o whose ring h s of o g endomorphisms is a dual larson order we affirm an order s in o with a dual larson order h s is a tame h s extension and show that s is maximal in such orders s

mean field annealing a formalism for constructing gnc like algorithms optimization problems are approached using mean field annealing mfa which is a deterministic approximation using mean field theory and based on peierls s inequality to simulated annealing the mfa mathematics are applied to three different objective function examples in each case mfa produces a minimization algorithm that is a type of graduated nonconvexity when applied to the weak membrane objective mfa results in an algorithm qualitatively identical to the published gnc algorithm one of the examples mfa applied to a piecewise constant objective function is then compared experimentally with the corresponding gnc weak membrane algorithm the mathematics of mfa are shown to provide a powerful and general tool for deriving optimization algorithms

skew braces and hopf galois structures of heisenberg type abstract we classify all skew braces of heisenberg type for a prime number p 3 furthermore we determine the automorphism group of each one of these skew braces as well as their socle and annihilator hence by utilising a link between skew braces and hopf galois theory we can determine all hopf galois structures of heisenberg type on galois field extensions of fields of degree p 3

hopf galois kummer theory of formal groups let f g be an isogeny between finite n dimensional formal groups defined over r the valuation ring of some field extension k of qp let h be the hopf algebra which arises from this isogeny for such h we classify gal h the group of 7 galois objects let ai be the maximal ideal of r and let p f k denote the n tuples of m under the group operation induced by f our main result is the construction of an isomorphism from the cokernel of p f to gal where p f is the induced map from p f k to p g k in geometric language gal describes the group of isomorphism classes of principal homogeneous spaces for spec over spec geometric methods have been used by mazur to establish the above isomorphism but the proof is nonconstructive a geometric approach has also provided a formula for the cardinality of gal we give an alternative derivation of this result using formal group techniques introduction given a field k and a finite abelian group g a question in classical field theory is to determine gal k g the isomorphism classes of galois field extensions of k with group g when k has characteristic zero and contains a primitive nth root of unity classical kummer theory asserts that gal k cn u k u k n where u k denotes the units of k and cn is the cyclic group of order n furthermore there is an explicit isomorphism from u k u k n to gal cn given by sending u g u k mod u k n to k x xn u in their development of hopf galois theory chase and sweedier 1 provide criteria for an action or co action of a cocommutative hopf algebra on an exten sion of rings to be galois an algebra s which admits a galois co action by an hopf algebra h is called an galois object given a ring r and a cocommu tative hopf algebra a basic question in this theory is to determine gal the group of isomorphism classes of galois objects when k is a field and h is the dual of the group ring kg g a finite group gal becomes gal k g determining gal is also of great interest because in geometric language this describes the group of isomorphism classes of principal homogeneous spaces for spec over spec we make the conventions that all rings are commutative all formal groups are commutative and all hopf algebras are abelian commutative and cocommu tative an hopf algebra is called finite if it is finitely generated and projective as an module

the p parity conjecture for elliptic curves with a p isogeny for an elliptic curve e over a number field k one consequence of the birch and swinnerton dyer conjecture is the parity conjecture the global root number matches the parity of the mordell weil rank assuming finiteness of xpe kqrp8s for a prime p this is equivalent to the p parity conjecture the global root number matches the parity of the zp corank of the p8 selmer group we complete the proof of the p parity conjecture for elliptic curves that have a p isogeny for p 3 the cases p 3 were known t and v dokchitser have showed this in the case when e has semistable reduction at all places above p by establishing respective cases of a conjectural formula for the local root number we remove the restrictions on reduction types by proving their formula in the remaining cases we apply our result to show that the p parity conjecture holds for every e with complex multiplication defined over k consequently if for such an elliptic curve xpe kqrp8s is infinite it must contain pqp zpq

magri morosi gel fand dorfman s bi hamiltonian constructions in the action angle variables a constructive method of transforming a completely integrable in liouville s sense hamiltonian system into magri morosi gel fand dorfman s mmgd bi hamiltonian form is presented the approach is carried out by making use of the action angle coordinates the classical kepler problem is shown to be a mmgd bi hamiltonian system explicit plethoras of higher order conserved quantities for the kepler problem are derived by employing oevel s method based on the existence of the mmgd bi hamiltonian representation

cayley s theorem and hopf galois structures for semidirect products of cyclic groups abstract for g any finite group the left and right regular representations respectively of g into perm g map g into inhol g g inn g we determine regular embeddings of g into inhol g modulo equivalence by conjugation in hol g by automorphisms of g for groups g that are semidirect products g z h z k of cyclic groups and have trivial centers if h is the power of an odd prime p then the number of equivalence classes of regular embeddings of g into inhol g is equal to twice the number of fixed point free endomorphisms of g and we determine that number each equivalence class of regular embeddings determines a hopf galois structure on a galois extension of fields l k with galois group g we show that if h 1 is the hopf algebra that gives the standard non classical hopf galois structure on l k then h 1 gives a different hopf galois structure on l k for each fixed point free endomorphism of g

relative galois module structure of rings of integers and elliptic functions iii in this paper we consider the problem of describing the galois module structure of rings of integers of abelian extensions of a quadratic imaginary number field k in certain cases by use of elliptic functions we are able to exhibit specific generators of such rings of integers as galois modules the general problem of finding galois integral generators by means of evaluation of transcendental functions has been considered by a number of mathematicians our results represent the first major advance outside the cyclotomic case we recall that the principal cyclotomic results are the hilbert speiser theorem and more generally the following result of leopoldt cf le theorem 1 let e denote an abelian extension of q let f gal e q

theory of tensor invariants of integrable hamiltonian systems i incompatible poisson structures this paper develops a new theory of tensor invariants of a completely integrable non degenerate hamiltonian system on a smooth manifoldmn the central objects in this theory are supplementary invariant poisson structurespc which are incompatable with the original poisson structurep1 for this hamiltonian system a complete classification of invariant poisson structures is derived in a neighbourhood of an invariant toroidal domain this classification resolves the well known inverse problem that was brought into prominence by magri s 1978 paper deveoted to the theory of compatible poisson structures applications connected with the kam theory with the kepler problem with the basic integrable problem of celestial mechanics and with the harmonic oscillator are pointed out a cohomology is defined for dynamical systems on smooth manifolds the physically motivated concepts of dynamical compatibility and strong dynamical compatibility of pairs of poisson structures are introduced to study the diversity of pairs of poisson structures incompatible in magri s sense it is proved that if a dynamical systemv preserves two strongly dynamically compatible poisson structuresp1 andp2 in a general position then this system is completely integrable such a systemv generates a hierarchy of integrable dynamical systems which in general are not hamiltonian neither with respect top1 nor with respect top2 necessary conditions for dynamical compatibility and for strong dynamical compatibility are derived which connect these global properties with new local invariants of an arbitrary pair of incompatible poisson structures

hopf galois structures on separable field extensions of odd prime power degree abstract we study hopf galois structures on separable field extensions l k of degree p n p an odd prime and denote by g the galois group of the normal closure over k we prove that if l k has a hopf galois structure of cyclic type then it has no structure of noncyclic type for n 3 we characterize the transitive groups g such that l k has cyclic hopf galois structures we prove that if l k has a nonabelian hopf galois structure of type n then it has an abelian structure whose type has the same exponent as n we obtain that for p 3 the two abelian noncyclic hopf galois structures do not occur on the same extension finally we list all possible sets of hopf galois structure types on a separable extension of degree p 3 for p 3 a prime

a covering theory for non involutive set theoretic solutions to the yang baxter equation abstract an algebraic theory of non involutive set theoretic solutions to the yang baxter equation by means of a non commutative version of braces is established the structure group and the permutation group of a solution is put into a context of discrete lie theory this leads to a natural concept of covering which shares the good properties of coverings in geometry and topology in the special case of quandle coverings in knot theory known anomalies are removed

monogenic hopf algebras over discrete valuation rings with low ramification abstract let k be an extension of q p with absolute ramification index 1 e p 1 let r o k and let k be the residue field of r we show that any monogenic finite abelian local hopf algebra with local dual lifts to r and we construct all such lifts for h an r hopf algebra arising from one of these lifts we realize spech as the kernel of an isogeny of one dimensional formal groups we then obtain a complete list of fields l for which l k is an h k galois object

hopf galois structures on field extensions of degree twice an odd prime square and their associated skew left braces abstract we determine the hopf galois structures on a galois field extension of degree twice an odd prime square and classify the corresponding skew left braces besides we determine the separable field extensions of degree twice an odd prime square allowing a cyclic hopf galois structure and the number of these structures

on bi hamiltonian formulation of the perturbed kepler problem the perturbed kepler problem is shown to be a bi hamiltonian system in spite of the fact that the graph of the hamilton function is not a hypersurface of translation which goes against a necessary condition for the existence of the bi hamiltonian structure according to the fernandes theorem in fact the initial and perturbed kepler systems are both isochronous systems and therefore the fernandes theorem cannot be applied to them

elementary abelian hopf galois structures and polynomial formal groups abstract we find a relationship between regular embeddings of g an elementary abelian p group of order p n into unipotent upper triangular matrices with entries in f p and commutative dimension n degree 2 polynomial formal groups with nilpotent upper triangular structure matrices we classify the latter when n 3 up to linear isomorphism and use that classification to determine the number of hopf galois structures on a galois extension l k of fields with galois group g elementary abelian of order p 3 we also obtain a lower bound on the number of hopf galois structures on a galois extension l k when the galois group is elementary abelian of order p n

stochastic relaxation gibbs distributions and the bayesian restoration of images we make an analogy between images and statistical mechanics systems pixel gray levels and the presence and orientation of edges are viewed as states of atoms or molecules in a lattice like physical system the assignment of an energy function in the physical system determines its gibbs distribution because of the gibbs distribution markov random field mrf equivalence this assignment also determines an mrf image model the energy function is a more convenient and natural mechanism for embodying picture attributes than are the local characteristics of the mrf for a range of degradation mechanisms including blurring nonlinear deformations and multiplicative or additive noise the posterior distribution is an mrf with a structure akin to the image model by the analogy the posterior distribution defines another imaginary physical system gradual temperature reduction in the physical system isolates low energy states annealing or what is the same thing the most probable states under the gibbs distribution the analogous operation under the posterior distribution yields the maximum a posteriori map estimate of the image given the degraded observations the result is a highly parallel relaxation algorithm for map estimation we establish convergence properties of the algorithm and we experiment with some simple pictures for which good restorations are obtained at low signal to noise ratios

galois correspondences for hopf bigalois extensions abstract lethbe a hopf algebra with bijective antipode over a commutative ringk a righth galois extension ofkis a righth comodule algebraasuch thatk aco hand a certain canonical mapa a a his a bijection we investigate galois connections for hopf galois extensions that can be formulated with the help of an additional hopf algebralover whichais also a leftl galois extension ofk and anl h bicomodule such an additional hopf algebra always exists and is unique up to isomorphism the galois connection between quotient coalgebras and left modules oflandh subcomodule algebras ofainduces a bijection between those quotients over whichlis faithfully coflat and those subalgebras over whichais faithfully flat as a consequence the lattices of hopf subalgebras oflandhover which these are faithfully flat are isomorphic

forms of coalgebras and hopf algebras abstract we study forms of coalgebras and hopf algebras i e coalgebras and hopf algebras which are isomorphic after a suitable extension of the base field we classify all forms of grouplike coalgebras according to the structure of their simple subcoalgebras for hopf algebras given a w galois field extension k l for w a finite dimensional semisimple hopf algebra and a k hopf algebra h we show that all l forms of h are invariant rings l h w under appropriate actions of w on l h we apply this result to enveloping algebras duals of finite dimensional hopf algebras and adjoint actions of finite dimensional semisimple cocommutative hopf algebras

monogenic hopf algebras and local galois module theory let k be an unramified extension of qp and denote the ring of integers of k by r ok let h be an r hopf algebra with monogenic dual h we realize h as the kernel of an isogeny of one dimensional formal groups this allows us to give a complete list of fields l for which l k is h k hopf galois and s ol is a free h module

multiple holomorphs of finite p groups of class two abstract let g be a group and s g be the group of permutations on the set g the abstract holomorph of g is the natural semidirect product aut g g we will write hol g for the normalizer of the image in s g of the right regular representation of g hol g n s g g aut g g aut g g and also refer to it as the holomorph of g more generally if n is any regular subgroup of s g then n s g n is isomorphic to the holomorph of n g a miller has shown that the group t g n s g hol g hol g acts regularly on the set of the regular subgroups n of s g which are isomorphic to g and have the same holomorph as g in the sense that n s g n hol g if g is non abelian inversion on g yields an involution in t g other non abelian regular subgroups n of s g having the same holomorph as g yield other involutions in t g in the cases studied in the literature t g turns out to be a finite 2 group which is often elementary abelian in this paper we exhibit an example of a finite p group g p of class 2 for p 2 a prime which is the smallest p group such that t g p is non abelian and not a 2 group moreover t g p is not generated by involutions when p 3 more generally we develop some aspects of a theory of t g for g a finite p group of class 2 for p 2 in particular we show that for such a group g there is an element of order p 1 in t g and exhibit examples where t g p 1 and others where t g contains a large elementary abelian p subgroup

a markovian model for contour grouping abstract classical methods of contour extraction do not always allow the detection of all the contours for instance the contours obtained by a canny deriche filter have some gaps especially at corners or at t junctions in this paper we present an algorithm which restores incomplete contours we model the image by a markov random field in order to complete the contours several criteria are defined and introduced in an energy function which has to be optimized the deterministic icm iterated conditional mode relaxation algorithm is implemented to minimize this energy function the result is a contour image consisting of closed contours this method has been tested on different images which present different types of difficulties indoors outdoors satellite spot industrial and medical images

homological integral of hopf algebras the left and right homological integrals are introduced for a large class of infinite dimensional hopf algebras using the homological integrals we prove a version of maschke s theorem for infinite dimensional hopf algebras the generalization of maschke s theorem and homological integrals are the keys to studying noetherian regular hopf algebras of gelfand kirillov dimension one

groups that have the same holomorph as a finite perfect group abstract we describe the groups that have the same holomorph as a finite perfect group our results are complete for centerless groups when the center is non trivial some questions remain open the peculiarities of the general case are illustrated by a couple of examples that might be of independent interest

on two dimensional markov spectral estimation we give systematic methods for two dimensional 2 d spectral estimation from raw data using gaussian markov random field mrf models the mrf models considered in this paper characterize the statistical dependence of the observation y s on its neighbors in all directions due to the modeling assumption the 2 d spectrum is an explicit function of the parameters of the mrf model thus the spectral estimation problem reduces to that of estimating the appropriate structure and the parameters of the mrf model a tractable algorithm for maximum likelihood ml estimation of parameters is obtained by using doubly periodic boundary conditions for the given data asymptotically consistent decision rules are used to choose the appropriate neighbor set of the mrf model the mrf spectral estimate obtained using the doubly periodic boundary conditions has the property that its inverse discrete fourier transforms are in perfect agreement with the periodic sample correlation values of the given observations over a symmetric window identical to the structure of the mrf model using this result and the fact that the 2 d maximum entropy spectral mes estimate has a structure similar to the mrf power spectrum we show that the mrf spectral estimate developed in this paper is a good approximation to the mes estimate this result provides a reasonable basis i e maximum likelihood on which to justify the use of the mes estimate with correlation estimates obtained from a finite piece of data samples

opposite skew left braces and applications abstract given a skew left brace b we introduce the notion of an opposite skew left brace b which is closely related to the concept of the opposite of a group and provide several applications skew left braces are closely linked with both solutions to the yang baxter equation and hopf galois structures on galois field extensions we show that the set theoretic solution to the ybe given by b is the inverse to the solution given by b every hopf galois structure on a galois field extension l k gives rise to a skew left brace b if the underlying hopf algebra is not commutative then one can construct an additional opposite hopf galois structure see 1 which relates the hopf galois module structures of each and refers to the structures as commuting the corresponding skew left brace to this second structure is precisely b we show how left ideals and a newly introduced family of quasi ideals of b allow us to identify the intermediate fields of l k which occur as fixed fields of sub hopf algebras under this correspondence and to identify which of these are galois or hopf galois over k finally we use the opposite to connect the inverse solution to the ybe and the structure of the hopf algebra h acting on l k this allows us to identify the group like elements of h

skew braces and the galois correspondence for hopf galois structures abstract let l k be a galois extension of fields with galois group and suppose l k is also an h hopf galois extension using the recently uncovered connection between hopf galois structures and skew left braces we introduce a method to quantify the failure of surjectivity of the galois correspondence from subhopf algebras of h to intermediate subfields of l k given by the fundamental theorem of hopf galois theory suppose l k h l n where n g then there exists a skew left brace g where g we show that there is a bijective correspondence between the set of intermediate fields e between k and l that correspond to k subhopf algebras of h and a set of sub skew left braces of g that we call the stable subgroups of g counting these subgroups and comparing that number with the number of subgroups of g describes how far the galois correspondence for the h hopf galois structure is from being surjective the method is illustrated by a variety of examples

statistical mechanical approach to image processing the basic frameworks and techniques of the bayesian approach to image restoration are reviewed from the statistical mechanical point of view first a few basic notions in digital image processing are explained to convince the reader that statistical mechanics has a close formal similarity to this problem second the basic formulation of the statistical estimation from the observed degraded image by using the bayes formula is demonstrated the relationship between bayesian statistics and statistical mechanics is also listed particularly it is explained that some correlation inequalities on the nishimori line of the random spin model also play an important role in bayesian image restoration third the framework of bayesian image restoration for binary images by means of the ising model is reviewed some practical algorithms for binary image restoration are given by employing the mean field and the bethe approximations finally bayesian image restoration for a grey level image using the gaussian model is reviewed and the gaussian model is extended to a more practical probabilistic model by introducing the line state to treat the effects of edges the line state is also extended to quantized values

compound gauss markov random fields for image estimation algorithms for obtaining approximations to statistically optimal estimates for images modeled as compound gauss markov random fields are discussed the authors consider both the maximum a posteriori probability map estimate and the minimum mean squared error mmse estimate for image estimation and image restoration compound image models consist of several submodels having different characteristics along with an underlying structure model which govern transitions between these image submodels two different compound random field models are employed the doubly stochastic gaussian dsg random field and a compound gauss markov cgm random field the authors present map estimators for dsg and cgm random fields using simulated annealing a fast converging algorithm called deterministic relaxation which however converges to only a locally optimal map estimate is also presented as an alternative for reducing computational loading on sequential machines for comparison purposes the authors include results on the fixed lag smoothing mmse estimator for the dsg field and its suboptimal m algorithm approximation

finitary galois extensions over noncommutative bases we study galois extensions m co h m for h co module algebras m if h is a frobenius hopf algebroid the relation between the action and coaction pictures is analogous to that found in hopf galois theory for finite dimensional hopf algebras over fields so we obtain generalizations of various classical theorems of kreimer takeuchi doi takeuchi and cohen fischman montgomery we find that the galois extensions n m over some frobenius hopf algebroid are precisely the balanced depth 2 frobenius extensions we prove that the yetter drinfeld categories over h are always braided and their braided commutative algebras play the role of noncommutative scalar extensions by a slightly generalized brzezinski militaru theorem contravariant fiber functors are used to prove an analogue of ulbrich s theorem and to get a monoidal embedding of the module category me of the endomorphism hopf algebroid e endmnn into mnopn

dpa a deterministic approach to the map problem deterministic pseudo annealing dpa is a new deterministic optimization method for finding the maximum a posteriori map labeling in a markov random field in which the probability of a tentative labeling is extended to a merit function on continuous labelings this function is made convex by changing its definition domain this unambiguous maximization problem is solved and the solution is followed down to the original domain yielding a good if suboptimal solution to the original labeling assignment problem the performance of dpa is analyzed on randomly weighted graphs

the valuative condition and r hopf algebra orders in kc p 3 let k be a finite extension of qp endowed with the p adic valuation and let r be its ring of integers in this paper we give a complete classification of r hopf algebra orders in the group ring kcp3 for an arbitrary hopf order h we show that either the group scheme sph or the group scheme sph with h the linear dual can be viewed as the middle term of the baer product of a distinguished extension with a generically trivial extension using this characterization we then compute algebra generators for either h or h

groups of order 4p twisted wreath products and hopf galois theory abstract the work of greither and pareigis details the enumeration of the hopf galois structures if any on a given separable field extension we consider the cases where l k is already classically galois with gal l k where 4 p for p 3 a prime the goal is to determine those regular transitive and fixed point free subgroups n of perm that are normalized by the left regular representation of a key fact that aids in this search is the observation that any such regular subgroup necessarily of order 4p has a unique subgroup of order p this allows us to show that all such n are contained in a twisted wreath product a subgroup of high index in perm which has a very computationally convenient description that allows us to perform the aforementioned enumeration

hopf galois structures of squarefree degree abstract let n be a squarefree natural number and let g be two groups of order n we determine the number of hopf galois structures of type g admitted by a galois extension of fields with galois group isomorphic to we give some examples including a full treatment of the case where n is the product of three primes

non isomorphic hopf galois structures with isomorphic underlying hopf algebras abstract we give a degree 8 non normal separable extension having two non isomorphic hopf galois structures with isomorphic underlying hopf algebras

the convergence of mean field procedures for mrfs image processing convergence of the mean field equations is analyzed for the ising model with a nonhomogeneous external field when spl beta is small the equations are a contraction mapping and thereby convergent when spl beta is large however they can be noncontracting and in this case contraction is restored by a slight modification of the equations

hopf galois structures on almost cyclic field extensions of 2 power degree let l k be a finite separable field extension and let e be the normal closure of l k let g gal e k and g gal e l we call l k almost cyclic if g has a normal cyclic complement in g this includes the case that l k is a cyclic galois extension or a radical extension we give a method for counting hopf galois structures on an almost cyclic extension l k we then count the hopf galois structures on an almost cyclic extension of degree 2n n 3 and determine how many of them are almost classical this is analogous to a result of t kohl t kohl classification of the hopf galois structures on prime power radical extensions j algebra 207 1998 525 546 which counts the hopf galois structures on a radical extension of odd prime power degree in contrast to the odd prime power degree case however we find that an almost cyclic extension l k of 2 power degree has hopf galois structures for which the hopf algebra acting on l is not commutative

simulated annealing in compound gaussian random fields image processing recently a stochastic relaxation technique called simulated annealing has been developed to search for a globally optimal solution in image estimation and restoration problems the convergence of simulated annealing has been proved only for random fields with a compact range space because of this images were modeled as random fields with bounded discrete or continuous values however in most image processing problems it is more natural to model the image as a random field with values in a noncompact space e g conditional gaussian models the proof of convergence of the stochastic relaxation method is extended to a class of compound gauss markov random fields simulation results are provided to show the power of these methods

monogenic hopf orders and associated orders of valuation rings abstract let r be a complete discrete valuation ring of mixed characteristic with perfect residue field and let h be a finite local commutative r hopf algebra we consider when there exists a finite extension of the field of fractions of r whose valuation ring is a galois h object if this occurs then h is monogenic conversely if h is also cocommutative and h is monogenic then there exists a valuation ring which is a galois h object to prove this result we represent h as the kernel of an isogeny of a special type between formal groups over r we deduce that if a is a finite abelian r hopf algebra such that both a and its dual are local then a is the associated order of a valuation ring if and only if the dual of a is monogenic

commutative hopf galois module structure of tame extensions abstract we prove three theorems concerning the hopf galois module structure of fractional ideals in a finite tamely ramified extension of p adic fields or number fields which is h galois for a commutative hopf algebra h firstly we show that if l k is a tame galois extension of p adic fields then each fractional ideal of l is free over its associated order in h we also show that this conclusion remains valid if l k is merely almost classically galois finally we show that if l k is an abelian extension of number fields then every ambiguous fractional ideal of l is locally free over its associated order in h

unsupervised image restoration and edge location using compound gauss markov random fields and the mdl principle discontinuity preserving bayesian image restoration typically involves two markov random fields one representing the image intensities gray levels to be recovered and another one signaling discontinuities edges to be preserved the usual strategy is to perform joint maximum a posterori map estimation of the image and its edges which requires the specification of priors for both fields instead of taking an edge prior we interpret discontinuities in fact their locations as deterministic unknown parameters of the compound gauss markov random field cgmrf which is assumed to model the intensities this strategy should allow inferring the discontinuity locations directly from the image with no further assumptions however an additional problem emerges the number of parameters edges is unknown to deal with it we invoke the minimum description length mdl principle according to mdl the best edge configuration is the one that allows the shortest description of the image and its edges taking the other model parameters noise and cgmrf variances also as unknown we propose a new unsupervised discontinuity preserving image restoration criterion implementation is carried out by a continuation type iterative algorithm which provides estimates of the number of discontinuities their locations the noise variance the original image variance and the original image itself restored image experimental results with real and synthetic images are reported

counting hopf galois structures on non abelian galois field extensions let l be a field which is a galois extension of the field k with galois group g greither and pareigis gp87 showed that for many g there exist k hopf algebras h other than the group ring kg which make l into an h hopf galois extension of k or a galois h object in the sense of chase and sweedler cs69 using galois descent they translated the problem of determining the hopf galois structures on l k into one which depends only on the galois group g by this translation they showed for example that any galois extension with non abelian g admits at least one non classical hopf galois structure byott by96 further translated the problem to a more amenable group theoretic problem and showed that a galois extension l k of fields with group g has a unique hopf galois structure namely that by kg iff n the order of g is a burnside number that is is coprime to n euler s phi function of n this implies that g is cyclic of square free order

estimation of markov random field prior parameters using markov chain monte carlo maximum likelihood developments in statistics now allow maximum likelihood estimators for the parameters of markov random fields mrfs to be constructed we detail the theory required and present an algorithm that is easily implemented and practical in terms of computation time we demonstrate this algorithm on three mrf models the standard potts model an inhomogeneous variation of the potts model and a long range interaction model better adapted to modeling real world images we estimate the parameters from a synthetic and a real image and then resynthesize the models to demonstrate which features of the image have been captured by the model segmentations are computed based on the estimated parameters and conclusions drawn

mean field annealing using compound gauss markov random fields for edge detection and image estimation the authors consider the problem of edge detection and image estimation in nonstationary images corrupted by additive gaussian noise the noise free image is represented using the compound gauss markov random field developed by f c jeng and j w woods 1990 and the problem of image estimation and edge detection is posed as a maximum a posteriori estimation problem since the a posteriori probability function is nonconvex computationally intensive stochastic relaxation algorithms are normally required a deterministic relaxation method based on mean field annealing with a compound gauss markov random cgmrf field model is proposed the authors present a set of iterative equations for the mean values of the intensity and both horizontal and vertical line processes with or without taking into account some interaction between them the relationship between this technique and two other methods is considered edge detection and image estimation results on several noisy images are included

classification of the hopf galois structures on prime power radical extensions abstract let p be an odd prime and n a positive integer and let k be a field of characteristic zero let k k w with w p n a k where a is such that k k p n and let r denote the largest integer between 0 and n such that k k p r k p r where p r denotes a primitive p r th root of unity the extension k k is separable but not necessarily normal and by greither and pareigis is h galois with h a k hopf algebra form of a group ring kn where k is the normal closure of k k h is said to be almost classical if n k k the result is that if r n then there are p r hopf galois structures on k k for which the associated group n is cyclic of order p n of these p min r n r are almost classical and the rest are non almost classical when r n there are p n 1 h galois structures for which n c p n of which only one is almost classical finally we show that these are the only structures possible that is for this class of extensions n must be cyclic

integral hopf galois structures on degree p2 extensions of p adic fields abstract let l k be a totally ramified normal extension of p adic fields of degree p2 we investigate the behavior of the valuation ring o l in the various hopf galois structures on l k specifically we determine when o l is hopf galois with respect to a hopf order in the corresponding hopf algebra when this occurs o l is necessarily a free module over this hopf order we also determine which hopf orders can arise in this way for cyclic extensions l k of degree p2 l n childs has shown under certain restrictions on the ramification numbers that if o l is hopf galois with respect to a hopf order in one of the hopf galois structures on l k then the same is true in all p hopf galois structures on l k we show that this no longer holds if the ramification conditions are relaxed or if elementary abelian extensions of degree of p2 are considered we illustrate our results with a special family of kummer extensions and with certain extensions arising from lubin tate formal groups

tame realisable classes over hopf orders abstract letobe the ring of algebraic integers in a number fieldkand letgbe a finite abelian group mcculloh has characterised the classes in the locally free classgroup cl og which are realisable by rings of integers of tame normal extensions ofkwith galois groupg we extend this to certain extensions which in general need be neither tame nor normal by replacingogwith a hopf order a and introducing the strongly a tame orders as an analogue of tame rings of integers we also describe the classes realised by principal homogeneous spaces over the dual of a

realizable hopf orders in kcp3 abstract let k be a finite extension of the p adic rationals q p with ring of integers r let c p 3 denote the cyclic group of order p 3 in this paper we construct a new collection of hopf orders in the group ring kc p 3 we determine which of these hopf orders are realizable as galois groups

a model based approach for estimation of two dimensional maximum entropy power spectra a stochastic model based approach is presented for estimation of the two dimensional maximum entropy power spectrum meps from given finite uniform array data the method consists of fitting an appropriate two dimensional noncausal gaussian markov random field gmrf model to the given data using the maximum likelihood ml technique for parameter estimation the nonlinear criterion function used for ml estimation is similar in structure to the function arising in the deterministic approach of lang and mcclellan the model based approach provides new insights into the two dimensional meps estimation problem for example using the asymptotic normality of ml estimates we derive simultaneous confidence bands for the estimated meps it turns out that when the true correlations are generated by a noncausal gmrf model the two dimensional meps can be obtained by solving linear equations this approach also suggests techniques for realizing two dimensional gmrf models from the given correlation data several numerical examples are given to illustrate the usefulness of the approach

hopf galois structures on a galois s_n extension abstract in this paper we shall determine the exact number of hopf galois structures on a galois s n extension where s n denotes the symmetric group on n letters

tameness and local normal bases for objects of finite hopf algebras let r be a commutative ring s an r algebra h a hopf ralgebra both finitely generated and projective as r modules and suppose s is an h object so that h homr h r acts on s via a measuring let i be the space of left integrals of h we say s has normal basis if s h as h modules and s has local normal bases if sp hp as hp modules for all prime ideals p of r when r is a perfect field h is commutative and cocommutative and certain obvious necessary conditions on s hold then s has normal basis if and only if is r sh if r is a domain with quotient field k h is cocommutative and l s or k has normal basis as h 0 k module then s has local normal bases if and only if is r sh suppose k is a number field with ring of integers r l is a finite galois extension of k with galois group g and s is the integral closure of r in l then g acts as a group of automorphisms of s relative galois module theory seeks to understand s as an rg module via this action the most basic question is to inquire whether s is locally isomorphic to rg that is for all primes p of r sp has a local normal basis as a free rp module this question was answered by emmy noether who showed that s has a local normal basis at every prime p of r if and only if l k is tamely ramified here tamely ramified means that at each prime ideal p of r the ramification index of any prime p of s lying over p is relatively prime to the residue field characteristic it is well known that this latter condition is equivalent to the surjectivity of the trace map tr s r tr s ezacg s the purpose of this paper is to formulate and prove an extension of noether s theorem to objects of finite cocommutative hopf algebras assume now only that r is a commutative ring h is a hopf r algebra which is a finitely generated projective r module and h homr h r the dual hopf algebra a commutative r algebra s is an h object if s is a right h comodule via a map a s s 0 h which is an r algebra homomorphism if s is an h object then a induces a measuring a h s s in the sense of sweedler 21 galois module theory in this setting is the study of s as an h module via the measuring a the trivial example is s h itself with a a the comultiplication on h so we say that the h object has normal basis if s h as h modules and s has local normal bases if for all primes p of r sp hp as h modules since rgf rg as rg modules for g any finite group this notion of normal basis when specialized to h rg is equivalent to the classical notion received by the editors may 7 1985 and in revised form january 7 1986 1980 mathematics subject classification 1985 revision primary 13b05 secondary 16a24 1986 american mathematical society 0002 9947 86 1 00 25 per page

duality for hopf orders in this paper we use duality to construct new classes of hopf orders in the group algebra kc p3 where k is a finite extension of q p and c p3 denotes the cyclic group of order p 3 included in this collection is a subcollection of hopf orders which are realizable as galois groups

nilpotent and abelian hopf galois structures on field extensions abstract let l k be a finite galois extension of fields with group when is nilpotent we show that the problem of enumerating all nilpotent hopf galois structures on l k can be reduced to the corresponding problem for the sylow subgroups of we use this to enumerate all nilpotent resp abelian hopf galois structures on a cyclic extension of arbitrary finite degree when is abelian we give conditions under which every abelian hopf galois structure on l k has type we also give a criterion on n such that every hopf galois structure on a cyclic extension of degree n has cyclic type

unsupervised parallel image classification using markovian models abstract this paper deals with the problem of unsupervised classification of images modeled by markov random fields mrf if the model parameters are known then we have various methods to solve the segmentation problem simulated annealing sa iterated conditional modes icm etc however when the parameters are unknown the problem becomes more difficult one has to estimate the hidden label field parameters only from the observed image herein we are interested in parameter estimation methods related to monogrid and hierarchical mrf models the basic idea is similar to the expectation maximization em algorithm we recursively look at the maximum a posteriori map estimate of the label field given the estimated parameters then we look at the maximum likelihood ml estimate of the parameters given a tentative labeling obtained at the previous step the only parameter supposed to be known is the number of classes all the other parameters are estimated the proposed algorithms have been implemented on a connection machine cm200 comparative experiments have been performed on both noisy synthetic data and real images

on a conjecture of conrad diamond and taylor we prove a conjecture of conrad diamond and taylor on the size of certain deformation rings parametrizing potentially barsotti tate ga lois representations to achieve this we extend results of breuil and m ezard classifying galois lattices in semistable representations in terms of strongly divisible modules to the potentially crystalline case in hodge tate weights 0 1 we then use these strongly divisible modules to compute the desired deformation rings as a corollary we obtain new results on the modularity of potentially barsotti tate representations

enumerating dihedral hopf galois structures acting on dihedral extensions abstract the work of greither and pareigis details the enumeration of the hopf galois structures if any on a given separable field extension for an extension l k which is classically galois with g g a l l k the hopf algebras in question are of the form l n g where n b p e r m g is a regular subgroup that is normalized by the left regular representation g b we consider the case where both g and n are isomorphic to a dihedral group d n for any n 3 using the normal block systems inherent to the left regular representation of each d n and every other regular permutation group isomorphic to d n we explicitly enumerate all possible such n which arise

isogenies of polynomial formal groups leth hv i j be a greither order inkcp2and let spec hv i j spec b spec cbe a resolution of spec hv i j in the flat topology which appears as p2 gm gm gm gmover the fieldk under certain conditions the formal completions ofbandcgive rise to two dimensional polynomial formal groups f and g respectively the given resolution will then translate to an isogeny of polynomial formal groups f g with kernel spec hv i j

stereo matching with nonlinear diffusion one of the central problems in stereo matching and other image registration tasks is the selection of optimal window sizes for comparing image regions this paper addresses this problem with some novel algorithms based on iteratively diffusing support at different disparity hypotheses and locally controlling the amount of diffusion based on the current quality of the disparity estimate it also develops a novel bayesian estimation technique which significantly outperforms techniques based on area based matching ssd and regular diffusion we provide experimental results on both synthetic and real stereo image pairs