the gauss map of a complete non flat minimal surface cannot omit 7 points of the sphere a well known theorem of osserman 21 3 4 states that the gauss map of a complete minimal surface m2 c r3 cannot omit a set of positive logarithmic capacity unless the surface is a plane in this paper we improve osserman s theorem by showirg that the gauss map of m2 omits at most 6 points provided m2 is not flat it should be pointed out however that no example is known where the omitted set has 5 points therefore the problem of determining the exact size of the omitted set remains unsolved we refer to 2 and 31 for a historical account as well as for general facts pertaining to minimal surfaces we take the opportunity to thank l petrola for sharing his insights with us and professor j shah for some very enlightening conversations on related matters

gauss map of minimal surfaces with ramification we prove that for any complete minimal surface m immersed in r n if in cp n 1 there are q n n 1 2 hyperplanes h j in general position such that the gauss map of m is ramified over h j with multiplicity at least e j for each j and formule then m must be flat

coordinate algebras of extended affine lie algebras of type a1 abstract the cores of extended affine lie algebras of reduced types were classified except for type a1 in this paper we determine the coordinate algebra of extended affine lie algebras of type a1 it turns out that such an algebra is a unital z n graded jordan algebra of a certain type called a jordan torus we classify jordan tori and get five types of jordan tori

embedding of the lie superalgebra d 2 1 α into the lie superalgebra of pseudodifferential symbols on s1 2 we obtain an embedding of the one parameter family of exceptional simple lie superalgebras d 2 1 into the lie superalgebra of pseudodifferential symbols on the supercircle s1 2 correspondingly there is an embedding of d 2 1 into a nontrivial central extension of the derived contact superconformal algebra k 4 realized in terms of 4 4 matrices over a weyl algebra

group gradings on associative algebras abstract let r g g rg be a g graded ring we describe all types of gradings on r if g is torsion free and r is artinian semisimple if r is a matrix algebra over an algebraically closed field f then we give a description of all g gradings on r provided that g is an abelian group in the case of an abelian group g we also classify all finite dimensional graded simple algebras and finite dimensional graded division algebras over an algebraically closed field of characteristic zero

non integrated defect relations for the gauss maps of complete minimal surfaces with finite total curvature abstract in this article we give the non integrated defect relations in the sense of fujimoto for the gauss maps of complete minimal surfaces with finite total curvature in r 3 r 4 these are some strict improvements of previous results for the ramifications and modified defect relations of the gauss maps of complete minimal surfaces with finite total curvature

lie algebras graded by finite root systems and intersection matrix algebras this paper classifies the lie algebras graded by doubly laced finite root systems and applies this classification to identify the intersection matrix algebras arising from multiply affinized cartan matrices of types b c f and g this completes the determination of the lie algebras graded by finite root systems initiated by berman and moody who studied the simply laced finite root systems of rank 2

on p p structural matrix rings abstract a ring is called a left p p ring if every principal left ideal is projective the objective here is to completely determine the left p p structural matrix rings over a von neumann regular ring

classification of quantum groups and lie bialgebra structures on sl n f relations with brauer group given an arbitrary field f of characteristic 0 we study lie bialgebra structures on sl n f based on the description of the corresponding classical double for any lie bialgebra structure 5 the classical double d sl n f delta is isomorphic to sl n f circle times f a where a is either f epsilon with epsilon 2 0 or f circle plus f or a quadratic field extension of f in the first case the classification leads to quasi frobenius lie subalgebras of sl n f in the second and third cases a belavin drinfeld cohomology can be introduced which enables one to classify lie bialgebras on sl n f up to gauge equivalence the belavin drinfeld untwisted and twisted cohomology sets associated to an r matrix are computed for the cremmer gervais r matrix in sl 3 we also construct a natural map of sets between the total belavin drinfeld twisted cohomology set and the brauer group of the field f

the lie algebra of homeomorphisms of the circle we define and study an infinite dimensional lie algebrahomeo which is shown to be naturally associated to thetopologicallie grouphomeo of all orientation preserving homeomorphisms of the circle roughly we rely on the universal decorated teichmuller theory developed before as motivation to provide frechet coordinates on the homogeneous space given byhomeo modulo the group of real fractional linear transformations whose corresponding vector fields on the circle we then extend by the usual lie algebrasl2of real traceless two by two matrices in order to definehomeo surprisingly homeo turns out to be equal to the algebra of all vector fields on the circle which are piecewisesl2 in the obvious sense it is evidently important to consider the relationship between our new frechet coordinates and the usual trigonometric functions on the circle and we undertake here both natural infinitesimal calculations we finally apply some further previous work in order to give sufficient conditions on the fourier coefficients of a certain class of homeomorphisms of the circle which arises naturally in topology and number theory

better bootstrap confidence intervals abstract we consider the problem of setting approximate confidence intervals for a single parameter in a multiparameter family the standard approximate intervals based on maximum likelihood theory can be quite misleading in practice tricks based on transformations bias corrections and so forth are often used to improve their accuracy the bootstrap confidence intervals discussed in this article automatically incorporate such tricks without requiring the statistician to think them through for each new application at the price of a considerable increase in computational effort the new intervals incorporate an improvement over previously suggested methods which results in second order correctness in a wide variety of problems in addition to parametric families bootstrap intervals are also developed for nonparametric situations

semiprime prime and simple jordan superpairs covered by grids in this paper we continue the investigation of jordan superpairs covered by a grid which have recently been described by the second author e neher j algebra 269 2003 28 73 for such a jordan superpair we characterize semiprimeness primeness and simplicity in terms of its associated supercoordinate system

on the graded identities for elementary gradings in matrix algebras over infinite fields we find a basis for the g graded identities of the n n matrix algebra mn k over an infinite field k of characteristic p 0 with an elementary grading such that the neutral component corresponds to the diagonal of mn k

exceptional n 6 superconformal algebra we realize the exceptional n 6 superconformal algebra spanned by 32 fields inside the lie superalgebra of pseudodifferential symbols on the supercircle s1 3 correspondingly we obtain two nonisomorphic one parameter families of irreducible representations of this superconformal algebra in a superspace spanned by 8 fields

statistical significance for hierarchical clustering cluster analysis has proved to be an invaluable tool for the exploratory and unsupervised analysis of high dimensional datasets among methods for clustering hierarchical approaches have enjoyed substantial popularity in genomics and other fields for their ability to simultaneously uncover multiple layers of clustering structure a critical and challenging question in cluster analysis is whether the identified clusters represent important underlying structure or are artifacts of natural sampling variation few approaches have been proposed for addressing this problem in the context of hierarchical clustering for which the problem is further complicated by the natural tree structure of the partition and the multiplicity of tests required to parse the layers of nested clusters in this article we propose a monte carlo based approach for testing statistical significance in hierarchical clustering which addresses these issues the approach is implemented as a sequential testing procedure guaranteeing control of the family wise error rate theoretical justification is provided for our approach and its power to detect true clustering structure is illustrated through several simulation studies and applications to two cancer gene expression datasets

bootstrap confidence intervals for a class of parametric problems summary we consider the following class of problems having observed a multivariate normal data vector y with unknown mean vector q covariance matrix the identity find an approximate confidence interval for 0 t q a real valued function of q a simple geometric construction is given which leads to highly accurate solutions this construction shows that the standard approximation based on maximum likelihood theory 04 cz00 can be quite misleading when 0 is nonlinear in q we discuss bootstrap based confidence intervals which remove most of the error in the standard approximation at the expense of considerably more calculation the bootstrap intervals are invariant under transformation of both y and i and so they automatically produce accurate solutions in problems which can be transformed to multivariate normality without requiring knowledge of the normalizing transformation

covering algebras ii isomorphism of loop algebras this paper studies the loop algebras that arise from pairs consisting of a symmetrizable kac moody lie algebra g and a finite order automorphism of g we obtain necessary and sufficient conditions for two such loop algebras to be isomorphic

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

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

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

deformations of complete minimal surfaces a notion of deformation is defined and studied for complete minimal surfaces in r3 and r3 c c a group of translations the catenoid enneper s surface and the surface of meeks jorge modelled on a 3 punctured sphere are shown to be isolated minimal surfaces of total curvature 4sr in r3 z and r3 z2 are studied it is proved that the helicoid and scherk s surface are isolated under periodic perturbations let m be a submanifold of a riemannian manifold n and let te m be a tubular neighborhood of m in n of radius e a c1 e variation of m in n is a submanifold m1 c te m which is a graph over m and is e c1 close to m this means m1 is pointwise e close to m in each fibre of te m and the tangent planes as well we are interested in complete minimal submanifolds m of n abbreviated c m s and c1 e variations which are also c m s s and henceforth we always assume m and m1 are c m s s we say m is isolated if for some e 0 the only e c1 variations of m differ from m by an ambient isometry of n in this paper we shall study this question when n is r3 or a translation space r3 modulo a group of translations a flat plane in r3is isolated this follows immediately from bernstein s theorem a function of two variables on r2 whose graph is a c m s is linear we shall prove enneper s surface the catenoid and a certain 3 punctured sphere discovered by meeks and jorge are isolated in r3 we prove the helicoid and scherk s surface are isolated under periodic deformations i e the helicoid of total curvature 4sr in r3 modulo one translation is isolated in this translation space and scherk s surface of total curvature 4sr is isolated in r3 modulo two translations this scherk surface is conformally a 4 punctured sphere and the helicoid a 2 punctured sphere we shall study minimal submanifolds of translation spaces many of the tech niques developed by r osserman can be adapted to this context and yield information about periodic minimal surfaces in r3 for example we classify c m s s vf total curvature 4sr in translation space in r3 osserman has proved the catenoid and enneper s surface are the only c m s s of total curvature 4sr we obtain an analogous classification in r3 g in vi we describe these surfaces for g z we study c m s s m of finite total curvature in translation spaces we prove a deformation i e an e c1 variation m1 of a finite total curvature c m s in a translation space is also of finite total curvature and is conformally a compact

graphs with relations coverings and group graded algebras the paper studies the interrelationship between coverings of finite directed graphs and gradings of the path algebras associated to the directed graphs to include gradings of all basic finite dimensional algebras over an algebraically closed field a theory of coverings of graphs with relations is introduced the object of this paper is to relate group gradings on algebras to coverings of a graph which is associated to the algebra the linking of the theories allows one to relate purely algebraic questions to questions in algebraic topology group theory or combinatorics in the representation theory of artin algebras the association to each algebra of a finite directed graph called the quiver of the algebra has been a useful tool the reason that the quiver of such an algebra is of interest is that there is a natural definition of representations of the quiver so that the category of representations of the quiver satisfying certain relations is equivalent to the category of finitely generated modules over the algebra 1 gives a slight extension of these concepts to finitely generated algebras the main emphasis of the paper is to show that the theory of coverings of graphs with relations introduced by c riedtmann 9 and expanded by p gabriel 2 and the theory of group graded algebras are essentially the same although the original connection between coverings and gradings was inspired by the similarity of results for z graded artin algebras 3 4 and p gabriel s announced results 2 the context of this paper is more general and deals with all finitely generated algebras over a field we associate to each such algebra a finite directed graph which we still call a quiver of the algebra we show that for each regular covering t of a quiver t0 of an algebra a with certain prescribed restrictions we get a g grading of the algebra a where g is the automorphism group of the covering y over ro conversely given a certain type of g grading of a where g is a group we construct a regular covering y of the quiver t0 of the algebra such that g is isomorphic to the automorphism group of t over ro furthermore if y is a regular covering of t0 with automorphism group g we show that the category of representations of y satisfying a certain set of relations is equivalent to the category of finite dimensional graded g modules we

quadratic jordan superpairs covered by grids abstract in this paper we are proposing a theory of jordan superpairs defined over super commutative superrings our framework has two novelties we allow scalars of even and odd parity and we do not assume that 1 2 lies in our base superring to demonstrate that it is possible to work in this generality we classify jordan superpairs covered by a grid

the classical theory of minimal surfaces we present here a survey of recent spectacular successes in classical minimal surface theory we highlight this article with the theorem that the plane the helicoid the catenoid and the one parameter family frtgt2 0 1 of riemann minimal examples are the only complete properly embedded minimal planar domains in r 3 the proof of this result depends primarily on work of colding

hyperbolic complete minimal surfaces with arbitrary topology we show a method to construct orientable minimal surfaces in et3 with arbitrary topology this procedure giares complete examples of two different kinds surfaces whose gauss map omits four points of the sphere and surfaces with a bounded coordinate function we also apply these ideas to construct stable minimal surfaces with high topology which are incomplete or complete with boundary

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

flexible lie admissible algebras abstract this paper investigates finite dimensional flexible lie admissible algebras a over fields of characteristic 0 under these hypotheses the vector space a with the lie product x y xy yx is a lie algebra denoted by a the main result of this work gives a characterization of those flexible lie admissible algebras for which the solvable radical of a is a direct summand of a included in this class of algebras are all flexible lie admissible a for which a is a reductive lie algebra our technique is to view a as a module for a certain semisimple lie algebra of derivations of a and to see what restrictions the module structure imposes on the multiplication of a a subsequent investigation will show that this module approach can also be used to determine the flexible lie admissible algebras a for which the radical of a is abelian

the alternative torus and the structure of elliptic quasi simple lie algebras of type we present the complete classification of the tame irreducible elliptic quasi simple lie algebras of type a2 and in particular specialize on the case where the coordinates are not associative here the coordinates are cayley dickson algebras over laurent polynomial rings in v 3 variables which we call alternative tori in giving our classification we need to present much information on these alternative tori and the lie algebras coordinatized by them

multiloop realization of extended affine lie algebras and lie tori an important theorem in the theory of infinite dimensional lie algebras states that any affine kac moody algebra can be realized that is to say constructed explicitly using loop algebras in this paper we consider the corresponding problem for a class of lie algebras called extended affine lie algebras ealas that generalize affine algebras ealas occur in families that are constructed from centreless lie tori so the realization problem for ealas reduces to the realization problem for centreless lie tori we show that all but one family of centreless lie tori can be realized using multiloop algebras in place of loop algebras we also obtain necessary and sufficient conditions for two centreless lie tori realized in this way to be isotopic a relation that corresponds to isomorphism of the corresponding families of ealas

automorphism groups of generalized triangular matrix rings abstract we call a ring strongly indecomposable if it cannot be represented as a non trivial i e m 0 generalized triangular matrix ring r m 0 s for some rings r and s and some r s bimodule r m s examples of such rings include rings with only the trivial idempotents 0 and 1 as well as endomorphism rings of vector spaces or more generally semiprime indecomposable rings we show that if r and s are strongly indecomposable rings then the triangulation of the non trivial generalized triangular matrix ring r m 0 s is unique up to isomorphism to be more precise if r m 0 s r m 0 s is an isomorphism then there are isomorphisms r r and s s such that m m m is an r s bimodule isomorphism relative to and in particular this result describes the automorphism groups of such upper triangular matrix rings r m 0 s

triangular matrix representations in this paper we develop the theory of generalized triangular matrix representation in an abstract setting this is accomplished by introducing the concept of a set of left triangulating idempotents these idempotents determine a generalized triangular matrix representation for an algebra the existence of a set of left triangulating idempotents does not depend on any specific conditions on the algebras however if the algebra satisfies a mild finiteness condition then such a set can be refined to a complete set of left triangulating idempotents in which each diagonal subalgebra has no nontrivial generalized triangular matrix representation we then apply our theory to obtain new results on generalized triangular matrix representations including extensions of several well known results

four dimensional yetter drinfeld module algebras over h4 abstract in this paper we classify four dimensional yetter drinfeld module algebras over sweedler s 4 dimensional hopf algebra h 4 all h 4 azumaya algebras of dimension 4 have been given in particular all yetter drinfeld h 4 module algebra structures on the matrix algebra m 2 k are classified

elliptic lie algebras and tubular algebras abstract this article is to study relations between tubular algebras of ringel and elliptic lie algebras in the sense of saito yoshii using the explicit structure of the derived categories of tubular algebras given by happel ringel we prove that the elliptic lie algebra of type d 4 1 1 e 6 1 1 e 7 1 1 or e 8 1 1 is isomorphic to the ringel hall lie algebra of the root category of the tubular algebra with the same type as a by product of our proof we obtain a chevalley basis of the elliptic lie algebra following indecomposable objects of the root category of the corresponding tubular algebra this can be viewed as an analogue of the frenkel malkin vybornov theorem in which they described a chevalley basis for each untwisted affine kac moody lie algebra by using indecomposable representations of the corresponding affine quiver

the problem of regions in the problem of regions we wish to know which one of a discrete set of possibilities applies to a continuous parameter vector this problem arises in the following way we compute a descriptive statistic from a set of data notice an interesting feature and wish to assign a confidence level to that feature for example we compute a density estimate and notice that the estimate is bimodal what confidence can we assign to bimodality a natural way to measure confidence is via the bootstrap we compute our descriptive statistic on a large number of bootstrap data sets and record the proportion of times that the feature appears this seems like a plausible measure of confidence for the feature the paper studies the construction of such confidence values and examines to what extent they approximate frequentist p values and bayesian a posteriori probabilities we derive more accurate confidence levels using both frequentist and objective bayesian approaches the methods are illustrated with a number of examples including polynomial model selection and estimating the number of modes of a density

epsilon strongly graded rings separability and semisimplicity abstract we introduce the class of epsilon strongly graded rings and show that it properly contains both the class of strongly graded rings and the class of unital partial crossed products we determine precisely when an epsilon strongly graded ring is separable over its principal component thereby we simultaneously generalize a result for strongly group graded rings by n st sescu van den bergh and van oystaeyen and a result for unital partial crossed products by bagio lazzarin and paques we also show that the class of unital partial crossed products appears in the class of epsilon strongly graded rings in a fashion similar to how the classical crossed products present themselves in the class of strongly graded rings thereby we obtain in the special case of unital partial crossed products a short proof of a general result by dokuchaev exel and simon concerning when graded rings can be presented as partial crossed products we also provide some interesting classes of examples of separable epsilon strongly graded rings with finite as well as infinite grading groups in particular we obtain an answer to a question raised by le bruyn van den bergh and van oystaeyen in 1988

the minimum number of idempotent generators of a complete blocked triangular matrix algebra abstract let r be a complete blocked triangular matrix algebra over an infinite field f assume that r is not an upper triangular matrix algebra or a full matrix algebra we prove that the minimum number r such that r can be generated as an f algebra by idempotents is given by formula where m1 is the number of 1 1 diagonal blocks of r we also show that r can be generated as an f algebra by two elements and if m1 0 r can be generated by an idempotent and a nilpotent element

approximately unbiased tests of regions using multistep multiscale bootstrap resampling approximately unbiased tests based on bootstrap probabilities are considered for the exponential family of distributions with unknown expectation parameter vector where the null hypothesis is represented as an arbitraryshaped region with smooth boundaries this problem has been discussed previously in efron and tibshirani ann statist 26 1998 1687 1718 and a corrected p value with second order asymptotic accuracy is calculated by the two level bootstrap of efron halloran and holmes proc natl acad sci u s a 93 1996 13429 13434 based on the abc bias correction of efron j amer statist assoc 82 1987 171 185 our argument is an extension of their asymptotic theory where the geometry such as the signed distance and the curvature of the boundary plays an important role we give another calculation of the corrected p value without finding the nearest point on the boundary to the observation which is required in the two level bootstrap and is an implementational burden in complicated problems the key idea is to alter the sample size of the replicated dataset from that of the observed dataset the frequency of the replicates falling in the region is counted for several sample sizes and then the p value is calculated by looking at the change in the frequencies along the changing sample sizes this is the multiscale bootstrap of shimodaira systematic biology 51 2002 492 508 which is third order accurate for the multivariate normal model here we introduce a newly devised multistep multiscale bootstrap calculating a third order accurate p value for the exponential family of distributions in fact our p value is asymptotically equivalent to those obtained by the double bootstrap of hall the bootstrap and edgeworth expansion 1992 springer new york and the modified signed likelihood ratio of barndorff nielsen biometrika 73 1986 307 322 ignoring o n 3 2 terms yet the computation is less demanding and free from model specification the algorithm is remarkably simple despite complexity of the theory behind it the differences of the pvalues are illustrated in simple examples and the accuracies of the bootstrap methods are shown in a systematic way

derivations of the cheng kac jordan superalgebras abstract the derivations of the cheng kac jordan superalgebras are studied it is shown that assuming 1 is a square in the ground field the lie superalgebra of derivations of a cheng kac jordan superalgebra is isomorphic to the lie superalgebra obtained from a simpler jordan superalgebra a kantor double superalgebra of vector type by means of the tits kantor koecher construction this is done by exploiting the s 4 symmetry of the cheng kac jordan superalgebra

more accurate confidence intervals in exponential families summary fisher s theory of maximum likelihood estimation routinely provides approximate confidence intervals for a parameter of interest 0 the standard intervals 0 za5 s where 0 is the maximum likelihood estimator 5 is an estimate of standard error based on differentiation of the log likelihood function and za is a normal percentile point recent work has produced systems of better approximate confidence intervals which look more like exact intervals when exact intervals exist and in general have coverage probabilities an order of magnitude more accurate than the standard intervals this paper develops an efficient and dependable algorithm for calculating highly accurate approximate intervals on a routine basis for parameters 0 defined in the framework of a multiparameter exponential family the better intervals require only a few times as much computational effort as the standard intervals a variety of numerical and theoretical arguments are used to show that the algorithm works well and that the improvement over the standard intervals can be striking in realistic situations

matrix rings satisfying column sum conditions versus structural matrix rings abstract the internal characterization of a structural matrix ring in terms of a set of matrix units associated with a quasiorder relation is used to obtain isomorphisms between seemingly different classes of subrings of a complete matrix ring

vertex operators arising from the homogeneous realization for we use the underlying fock space for the homogeneous vertex operator representation of the affine lie algebra to construct a family of vertex operators as an application an irreducible module for an extended affine lie algebra of type a n 1 coordinatized by a quantum torus q of 2 variables or 3 variables is obtained moreover this module turns out to be a highest weight module which is an analog of the basic module for affine lie algebras

exceptional zero formulae and a conjecture of perrin riou let a mathbf q be an elliptic curve with split multiplicative reduction at a prime p we prove an analogue of a conjecture of perrin riou relating p adic beilinson kato elements to heegner points in a mathbf q and a large part of the rank one case of the mazur tate teitelbaum exceptional zero conjecture for the cyclotomic p adic l function of a more generally let f be the weight two newform associated with a let f infty be the hida family of f and let l p f infty k s be the mazur kitagawa two variable p adic l function attached to f infty we prove a p adic gross zagier formula expressing the quadratic term of the taylor expansion of l p f infty k s at k s 2 1 as a non zero rational multiple of the extended height weight of a heegner point in a mathbf q

artinian rings with restricted primeness conditions abstract complete blocked triangular matrix rings over division rings are characterized as left artinian rings which satisfy suitably restricted primeness conditions a matrix representation is also provided for the class of nonsingular left artinian c prime rings

the jordan socle and finitary lie algebras in this paper we introduce the notion of jordan socle for nondegenerate lie algebras which extends the definition of socle given in a fernandez lopez et al 3 graded lie algebras with jordan finiteness conditions comm algebra in press for 3 graded lie algebras any nondegenerate lie algebra with essential jordan socle is an essential subdirect product of strongly prime ones having nonzero jordan socle these last algebras are described up to exceptional cases in terms of simple lie algebras of finite rank operators and their algebras of derivations when working with lie algebras which are infinite dimensional over an algebraically closed field of characteristic 0 the exceptions disappear and the algebras of derivations are computed

group gradings on full matrix rings we study g gradings of the matrix ring mn k k a field and give a complete description of the gradings where all the elements ei j are homogeneous called good gradings among these we determine the ones that are strong gradings or crossed products if g is a finite cyclic group and k contains a primitive g th root of 1 we show how all g gradings of mn k can be produced in particular we give a precise description of all c2 gradings of m2 k and show that for algebraically closed k any such grading is isomorphic to one of the two good gradings

gradings on the algebra of upper triangular matrices and their graded identities abstract let k be an infinite field and let ut n k denote the algebra of n n upper triangular matrices over k we describe all elementary gradings on this algebra further we describe the generators of the ideals of graded polynomial identities of ut n k and we produce linear bases of the corresponding relatively free graded algebras we prove that one can distinguish the elementary gradings by their graded identities we describe bases of the graded polynomial identities in several typical cases although in these cases we consider elementary gradings by cyclic groups the same methods serve for elementary gradings by any finite group

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

notions of limiting p values based on data depth and bootstrap abstract in this article we propose some new notions of limiting p values for hypothesis testing the limiting p value lp here not only provides the usual attractive interpretation of a p value as the strength in support of the null hypothesis coming from the observed data but also has several advantages first it allows us to resample directly from the empirical distribution in the bootstrap implementations rather than from the estimated population distribution satisfying the null constraints second it serves as a test statistic and as a p value simultaneously and thus enables us to obtain test results directly without having to construct an explicit test statistic and then establish or approximate its sampling distribution these are the two steps generally required in a standard testing procedure using bootstrap and the concept of data depth we have provided lp s for a broad class of testing problems where the parameters of interest can be either finite or infinite dimensional some compute

graded polynomial identities of matrices we consider g graded polynomial identities of the p p matrix algebra mp k over a field k of characteristic 0 graded by an arbitrary group g we find relations between the g graded identities of the g graded algebra mp k and the g h graded identities of the tensor product of mp k and the h graded algebra mq k with a fine h grading we also find a basis of the g graded identities of mp k with an elementary grading such that the identity component coincides with the diagonal of mp k

infereni on full or partial parameters based on the standardized signed log likelihood ratio on etudie des tests et des limites de confiance bases sur la statistique du rapport de log vraisemblance signee en ajustant cette statistique de telle sorte qu a un ordre superieur d approximation la distribution normale standard soit valable

the graded grothendieck group and the classification of leavitt path algebras this paper is an attempt to show that parallel to elliott s classification of af c algebras by means of k theory the graded k0 group classifies leavitt path algebras completely in this direction we prove this claim at two extremes namely for the class of acyclic graphs graphs with no cycles and multi headed comets or rose graphs graphs in which each head is connected to a cycle or to a collection of loops or a mixture of these graphs i e polycephaly graphs

partial representations of hopf algebras in this work the notion of a partial representation of a hopf algebra is introduced and its relationship with partial actions of hopf algebras is explored given a hopf algebra h one can associate to it a hopf algebroid hpar which has the universal property that each partial representation of h can be factorized by an algebra morphism from hpar we define also the category of partial modules over a hopf algebra h which is the category of modules over its associated hopf algebroid hpar the hopf algebroid structure of hpar enables us to enhance the category of partial h modules with a monoidal structure and such that the algebra objects in this category are the symmetric partial actions some examples of categories of partial h modules are explored in particular we can describe fully the category of partially z2 graded modules