on newton raphson iteration for multiplicative inverses modulo prime powers we study algorithms for the fast computation of modular inverses newton raphson iteration over p adic numbers gives a recurrence relation computing modular inverse modulo p m that is logarithmic in m we solve the recurrence to obtain an explicit formula for the inverse then we study different implementation variants of this iteration and show that our explicit formula is interesting for small exponent values but slower or large exponent say of more than 700 bits overall we thus propose a hybrid combination of our explicit formula and the best asymptotic variants this hybrid combination yields then a constant factor improvement also for large exponents

weyl groups of fine gradings on matrix algebras octonions and the albert algebra given a grading a g gag on a nonassociative algebra a by an abelian group g we have two subgroups of aut a the automorphisms that stabilize each component ag as a subspace and the automorphisms that permute the components by the weyl group of we mean the quotient of the latter subgroup by the former in the case of a cartan decomposition of a semisimple complex lie algebra this is the automorphism group of the root system i e the so called extended weyl group a grading is called fine if it cannot be refined we compute the weyl groups of all fine gradings on matrix algebras octonions and the albert algebra over an algebraically closed field of characteristic different from 2 in the case of the albert algebra

graded morita theory for infinite groups abstract having shown that not all graded rings are graded equivalent via classical morita theory to a skew group ring we extend classical morita theory which is based on rings with identity to a generalized graded morita theory for rings with local units this enables us to give necessary and sufficient conditions for graded morita equivalence between two rings graded by a group we show that the strongly graded property is a graded morita invariant and we show that a graded ring is graded morita equivalent to a skew group ring namely r g g if and only if it is strongly graded these results extend the classical cohen montgomery duality theory to rings graded by infinite groups the fundamental tools include rings with local units and the results of boisen

on lie gradings ii abstract to obtain fine gradings of gl n c it is necessary to study maximal abelian subgroups mad subgroups of diagonizable automorphisms in a ut gl n c we described two finite sets h n and k n such that each mad subgroup consisting only from inner automorphisms is conjugated to some element of h n and each mad subgroup containing at least one outer automorphism is conjugated to some element of k n using these results concerning k n we then easily find all mad subgroups in a ut o n c for n 8 and in a ut sp 2n c

the brauer splitting theorem and projective representations of finite groups over rings in the theory of representations of finite groups the projective representations arise naturally when one studies the relations between the representations of the group and representations of certain subgroups the role played by the group rings in representation theory is taken by the twisted groups rings when one considers projective representations for representations over algebraically closed fields the theory of schur multipliers provides a very satisfactory tool that may be used to lift projective representations of a finite group to usual representations of a finite central extension of that group furthermore there exists a satisfactory theory of projective characters block theory for projective representations etc for which we may refer to karpilovsky s recent book 3 since for finite groups the twisted group rings defined over commutative rings are azumaya algebras when a mild condition holds it is natural to ask for an equivalent of the brauer splitting theorem for group rings in this paper we derive this splitting theorem for twisted group rings over commutative rings without nontrivial idempotents this is obtained by introducing a suitable extension of the schur multiplier theorem in such a way that this result may be applied to a single representation and over arbitrary fields or rings of the forementioned type along the way we get involved in some cocycle manipulations that will lead to the determination of the center of a twisted group ring in terms of ray classes the results derived in this paper should be of some use in the study of properties and

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

equivalence relations on separated nets arising from linear toral flows in 1998 burago kleiner and mcmullen independently proved the existence of separated nets in r which are not bi lipschitz equivalent bl to a lattice a finer equivalence relation than bl is bounded displacement bd separated nets arise naturally as return times to a section for minimal r actions we analyze the separated nets which arise via these constructions focusing particularly on nets arising from linear r actions on tori we show that generically these nets are bl to a lattice and for some choices of dimensions and sections they are generically bd to a lattice we also show the existence of such nets which are not bd to a lattice

graded identities of matrix algebras and the universal graded algebra in the last decade group gradings and graded identities of finite dimensional central simple algebras have been an active area of research we refer the reader to bahturin et al 6 and 7 there are two basic kinds of group grading elementary and fine it was proved by bahturin and zaicev 7 that any group grading of mn c is given by a certain composition of an elementary grading and a fine grading in this paper we are concerned with fine gradings on mn c and their corresponding graded identities let r be a simple algebra finite dimensional over its center k and g a finite group we say that r is fine graded by g if r g grg is a grading and dimk rg 1 thus any component is either 0 or isomorphic to k as a k vector space it is easy to show that supp r the subset of elements of g for which rg is not 0 is a subgroup of g moreover

representability and specht problem for g graded algebras abstract let w be an associative pi algebra over a field f of characteristic zero graded by a finite group g let id g w denote the t ideal of g graded identities of w we prove 1 g graded pi equivalence there exists a field extension k of f and a finite dimensional z 2 z g graded algebra a over k such that id g w id g a where a is the grassmann envelope of a 2 g graded specht problem the t ideal id g w is finitely generated as a t ideal 3 g graded pi equivalence for affine algebras let w be a g graded affine algebra over f then there exists a field extension k of f and a finite dimensional algebra a over k such that id g w id g a

discrete and continuous graded contractions of lie algebras and superalgebras grading preserving contractions of lie algebras and superalgebras of any type over the complex number field are defined and studied such contractions fall naturally into two classes the wigner inonu like continuous contractions and new discrete contractions a general method is described for any abelian grading semigroup and any lie algebra or superalgebra admitting such a grading all contractions preserving z2 z3 and z2 z2 gradings are found examples of these gradings and contractions for the simple lie algebra a2 affine kac moody algebra a 1 and the simple superalgebra osp 2 1 are shown

fast iterative division of p adic numbers a fast iterative scheme based on the newton method is described for finding the reciprocal of a finite segment p adic numbers hensel code the rate of generation of the reciprocal digits per step can be made quadratic or higher order by a proper choice of the starting value and the iterating function the extension of this method to find the inverse transform of the hensel code of a rational polynomial over a finite field is also indicated

graded artin algebras a graded artin algebra is a graded ring which neglecting the grading is an artin algebra although this definition has the merit of brevity it does not expose some of the basic properties of graded artin algebras for instance that there is a bound on absolute values of degrees of homogeneous elements indeed a different definition is given in section 1 and the discrepancy is rectified by theory 1 4 we admit that our chief interest in graded artin algebras lies in applications to the representation theory of artin algebras see 5 1 we feel nonetheless that graded artin algebras are interesting objects of study in their own right and some of our techniques especially in sections 5 and 6 are equally applicable to arbitrary graded rings thus this paper unlike its sequel 51 is designed to be read by a general ring theoretical audience it is in addition self contained from the standpoint of knowledge concerning graded rings required the major concern of the paper is the existence of gradings on given modules over a graded artin algebra we use the term gradable when a gradation exists our transcendent result along these lines is that direct summands of finitely generated gradable modules are gradable it follows immediately see section 3 that simple modules and projective modules over a graded artin algebra are gradable also inasmuch as we do not restrict ourselves to positively graded rings endomorphism rings of graded finitely generated modules over graded artin algebras are graded artin algebras in a similar vein duality and horn

global anomalies on orbifolds we consider global anomalies for heterotic string theory formulated on orbifolds the vanishing of certain characteristic classes in group cohomology provides sufficient conditions for the absence of global anomalies for abelian orbifolds level matching implies these cohomology conditions so suffices for the absence of anomalies for nonabelian orbifolds level matching does not suffice and there are additional constraints we give some examples to illustrate these new constraints

entire majorants via euler maclaurin summation it is the aim of this article to give extremal majorants of type 2 for the class of functions f n x sgn x x n where n n as applications we obtain positive definite extensions to r of it m defined on r 1 1 where m n optimal bounds in hilbert type inequalities for the class of functions it m and majorants of type 2 for functions whose graphs are trapezoids

extremal functions in de branges and euclidean spaces ii this paper presents the gaussian subordination framework to generate optimal one sided approximations to multidimensional real valued functions by functions of prescribed exponential type such extremal problems date back to the works of beurling and selberg and provide a variety of applications in analysis and analytic number theory here we majorize and minorize on bbb r n the gaussian bf x mapsto e pi lambda bf x 2 where lambda0 is a free parameter by functions with distributional fourier transforms supported on euclidean balls optimizing weighted l 1 errors by integrating the parameter lambda against suitable measures we solve the analogous problem for a wide class of radial functions applications to inequalities and periodic analogues are discussed the constructions presented here rely on the theory of de branges spaces of entire functions and on new interpolation tools derived from the theory of laplace transforms of laguerre p o lya functions

interpolation formulas with derivatives in de branges spaces ii abstract we investigate necessary and sufficient conditions under which entire functions in de branges spaces can be recovered from function values and values of derivatives our main focus is on spaces with a structure function whose logarithmic derivative is bounded in the upper half plane

reduction techniques for strongly graded rings and finite representation type abstract we present reduction techniques for studying the category of lattices over strongly graded orders in particular we apply these techniques in order to reduce the problem of classifying those strongly graded orders with finite representation type to the case where the coefficient ring is a maximal order in a division ring

group graded rings and duality we give an alternative construction of the duality between finite group actions and group gradings on rings which was shown by cohen and montgomery in 1 this duality is then used to extend known results on skew group rings to corresponding results for large classes of group graded rings finally we modify the construction slightly to handle infinite groups introduction in the first section we give an alternate construction of the duality between finite group actions and group gradings on rings which was shown by cohen and montgomery in 1 this duality is then used to extend results on skew group rings and their modules to corresponding results for large classes of groupgraded rings in the second section we modify the construction slightly to handle infinite groups and apply it to a theorem on prime and semiprime infinite crossed products i would like to thank my thesis advisor d s passman for his guidance and encouragement throughout the writing of this paper 1 finite group graded rings and the smash product let g be a multiplicative group an associative ring with identity is said to be g graded if 1 r e r x xec is a direct sum of additive subgroups r x with r x r y c r xy it follows that necessarily 1r e r 1 so that each r x is a unitary r l bimodule if r e r we write r x for the component of r in r x so that r exgcr x r is said to be strongly g graded if r x r y r xy for all x y e g throughout this section r is assumed to be g graded where g is a finite group with igi n let mc r denote the set of n x n matrices over r with the rows and columns indexed by the elements of g if a c mg r we write ax y for the entry in the x y position of a then if a 1 c mc r the matrix product a 8 is given by 2 x x y e xx zaz y z e c if u c g is any subset of g let r u ex eur x in particular r r g now suppose h c g is a subgroup of g we define r h c mc r by r h a e ms r ax y e r xhy l that is r h ex ygcr xhy l e x y where e x y is the matrix unit with 1r in the x y position and zeroes elsewhere note that r g mc r received by the editors september 27 1984 1980 mathematics subject classification primary 16a03 c 1985 american mathematical society 0002 9947 85 1 00 25 per page

the pauli matrices in n dimensions and finest gradings of simple lie algebras of type a n 1 properties of the lie algebra gl n c are described for a basis which is a generalization of the 2 2 pauli matrices the 3 3 case is described in detail the remarkable properties of that basis are the grading of the lie algebra it offers each grading subspace is one dimensional and the matrix group it generates it is a finite group with the center of sl n c as its commutator group

on the structure of graded commutative algebras abstract consider a a commutative algebra graded by means of an abelian group g we show that under the hypothesis a 0 g a g a g where g g a g 0 the algebra a is of the form a j i j with any i j a well described graded ideal of a satisfying i j i k 0 if j k under certain conditions the simplicity of a is characterized and it is shown that a is the direct sum of the family of its minimal graded ideals each one being a simple graded commutative algebra

actions of picard groups on graded rings throughout this paper g denotes a group with identity e if r is a g graded ring we write r g grg and we refer to re as the coefficient ring we denote the category of graded left r modules i e those left r modules m with a g grading m g gmg such that rhmg mhg by r gr the aim of this paper is to present a method to reduce the study of one strongly graded ring r to the study of another strongly graded ring r that is more tractable this reduction process has two aspects reducing the coefficient ring or reducing the grading since we apply this method in our study of module theoretic properties such as semisimplicity for graded rings we require that this reduction process preserve the category of modules r mod and the category of graded modules r gr this leads us to the notion of graded equivalence in this paper a morita context is the usual tuple a b p q where we assume and are isomorphisms a g graded context is a morita context r r p q where r and r are g graded rings rpr and r qr are graded bimodules i e they have a grading which makes them into both graded r modules and graded r modules on the corresponding side and and are graded bimodule homomorphisms see 14 for details about graded modules and homomorphisms given a functor f r gr r gr we say that f is a graded equivalence of categories if any of the following equivalent conditions hold see for example 8

graded embeddings of finite dimensional simple graded algebras abstract let a b be finite dimensional g graded algebras over an algebraically closed field k with char k 0 where g is an abelian group and let id g a be the set of graded identities of a resp id g b we show that if a b are g simple then there is a graded embedding a b if and only if id g b id g a we also give a weaker generalization for the case where a is g semisimple and b is arbitrary

identities and isomorphisms of graded simple algebras abstract let g be an arbitrary abelian group and let a and b be two finite dimensional g graded simple algebras over an algebraically closed field f such that the orders of all finite subgroups of g are invertible in f we prove that a and b are isomorphic if and only if they satisfy the same g graded identities we also describe all isomorphism classes of finite dimensional g graded simple algebras

the construction of solvable lie algebras from equidimensional nilpotent algebras abstract the study of gradings of solvable lie algebras l of finite dimensionover a field f of zero characteristic led the authors to the discovery of equidimensional nilpotent algebras l uniquely determined by l up to f isomorphy conversely for any finite dimensional lie algebra l over f an algorithm is developed which yields in parametric form all solvable lie algebras l determining l as the corresponding nilpotent algebra the exposition is independent of lie grading theory it organizes in a novel way the classification of solvable lie algebras of given dimension around the same task for nilpotent algebras every isomorphy class of solvable f algebras is obtained in this way

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

hopf cocycle deformations and invariant theory for a given finite dimensional hopf algebra h we describe the set of all equivalence classes of cocycle deformations of h as an affine variety using methods of geometric invariant theory we show how our results specialize to the universal coefficients theorem in the case of a group algebra and we also give examples from other families of hopf algebras including dual group algebras and bosonizations of nichols algebras in particular we use the methods developed here to classify the cocycle deformations of a dual pointed hopf algebra associated to the symmetric group on three letters we also give an example of a cocycle deformation over a dual group algebra which has only rational invariants but which is not definable over the rational field this differs from the case of group algebras in which every 2 cocycle is equivalent to one which is definable by its invariants

solvable lie algebras of dimension 4 over perfect fields abstract the solvable lie algebras of dimension not greater than four over a perfect field of reference are described in terms of their nilpotent frame counted over finite fields of q elements and classified over the real and complex number fields the number dq n of nonisomorphic solvable lie algebras of dimension n over f q roughly speaking grows as a polynomial in n with qn 2 as highest term but there are smaller additional terms depending on the residue class of q modulo n 1 if n 4

on groups of central type a finite group is said to be of central type if it possesses an irreducible complex character which takes the value zero on all noncentral elements equivalently the degree of this character is the square root of the index of the center in 1964 iwahori and matsumoto 15 conjectured that a group of central type must be solvable the paper of liebler and yellen 16 aims to prove this but there is a gap in their proof as explained below nevertheless they do correctly obtain substantial information about a minimal counterexample to the conjecture the aim of this paper is to fill the gap in the liebler yellen proof using as they do the classification of finite simple groups and to provide further information about which solvable groups can have central type we show for instance that every normal subgroup of the central factor group of a solvable group of central type has the property that its system normalizers have square index it is interesting to contrast this restrictive condition with the result of gagola that every solvable group is embeddable in the central factor group of a group of central type see 6 theorem 1 2 the arguments in 16 show that a minimal counterexample g to the iwahori matsumoto conjecture must have the following structure if k z is any minimal normal subgroup of g z where z z g the center then k is abelian k z is a 2 group s cg k is the maximal solvable normal subgroup of g and any minimal normal subgroup his of g s is the direct product of a number l of copies of gl 3 2 indeed the liebler yellen methods can be used to show that the sylow 2 subgroup of k considered as an h s module is a product o f copies of a gl 3 2 module of dimension 4 this module having a unique minimal submodule which is trivial and an irreducible factor module of dimension 3 the minimal submodules are amalgamated in the product

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

verbally prime t ideals and graded division algebras abstract let f be an algebraically closed field of characteristic zero and let g be a finite group we consider graded verbally prime t ideals in the free g graded algebra it turns out that equivalent definitions in the ordinary case i e ungraded extend to nonequivalent definitions in the graded case namely verbally prime g graded t ideals and strongly verbally prime t ideals at first following kemer s ideas we classify g graded verbally prime t ideals the main bulk of the paper is devoted to the stronger notion we classify g graded strongly verbally prime t ideals which are t ideal of affine g graded algebras or equivalently g graded t ideals that contain a capelli polynomial it turns out that these are precisely the t ideal of g graded identities of finite dimensional g graded central over f i e z a e f which admit a g graded division algebra twisted form over a field k which contains f or equivalently over a field k which contains enough roots of unity e g a primitive n root of unity where n o r d g

bounding s t and s_1 t on the riemann hypothesis let pi s t denote the argument of the riemann zeta function zeta s at the point s frac 1 2 it assuming the riemann hypothesis we present two proofs of the bound begin aligned s t le left frac 1 4 o 1 right frac log t log log t end aligned for large t this improves a result of goldston and gonek by a factor of 2 the first method consists of bounding the auxiliary function s 1 t int 0 t s u text d u using extremal functions constructed by carneiro littmann and vaaler we then relate the size of s t to the size of the functions s 1 t pm h s 1 t when h asymp 1 log log t the alternative approach bounds s t directly relying on the solution of the beurling selberg extremal problem for the odd function f x arctan left frac 1 x right frac x 1 x 2 this draws upon recent work by carneiro and littmann

hilbert spaces and the pair correlation of zeros of the riemann zeta function montgomery s pair correlation conjecture predicts the asymptotic behavior of the function n t defined to be the number of pairs and of ordinates of nontrivial zeros of the riemann zetafunction satisfying 0 0 using montgomery s formula and some extremal functions of exponential type these functions are optimal in the sense that they majorize and minorize the characteristic function of the interval in a way to minimize the l1 r 1 sin x x 2 dx error we give a complete solution for this extremal problem using the framework of reproducing kernel hilbert spaces of entire functions this extends previous work of p x gallagher 18 in 1985 where the case 1 2 n was considered using non extremal majorants and minorants

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

on calculating multiplicative inverses modulo 2 m this paper presents a procedure for calculating multiplicative inverses modulo 2m based on a novel mathematical approach the procedure is suitable for software implementation on a general purpose processor when counting the total number of word level processor multiplications the computational effort involved in calculating a multiplicative inverse is 2 3 that of a single multiplication of m bit values in addition to a few word level multiplications for standard processor word sizes the number of these additional multiplications does not exceed 12 this introduces a clear advantage of the proposed method when compared to other known methods presented in the literature

more on squaring and multiplying large integers methods of squaring and multiplying large integers are discussed the obvious o n sup 2 methods turn out to be best for small numbers existing o n sup log sup 3 log sup 2 spl ap o n sup 1 585 methods become better as the numbers get bigger new methods that are o sup log5 log sup 3 spl ap 0 n sup 1 465 o n sup log sup 7 log sup 4 spl ap o n sup 1 404 and o n sup log sup 9 log sup 5 spl ap o n sup 1 365 presented in actual experiments all of these methods turn out to be faster than fft multipliers for numbers that can be quite large 37 000 000 bits squaring seems to be fundamentally faster than multiplying but it is shown that t sub multiplyspl les 2t sub square o n

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

extremal functions in de branges and euclidean spaces abstract in this work we obtain optimal majorants and minorants of exponential type for a wide class of radial functions on r n these extremal functions minimize the l 1 r n x 2 2 n d x distance to the original function where 1 is a free parameter to achieve this result we develop new interpolation tools to solve an associated extremal problem for the exponential function f x e x where 0 in the general framework of de branges spaces of entire functions we then specialize the construction to a particular family of homogeneous de branges spaces to approach the multidimensional euclidean case finally we extend the result from the exponential function to a class of subordinated radial functions via integration on the parameter 0 against suitable measures applications of the results presented here include multidimensional versions of hilbert type inequalities extremal one sided approximations by trigonometric polynomials for a class of even periodic functions and extremal one sided approximations by polynomials for a class of functions on the sphere s n 1 with an axis of symmetry

the picard group of a structural matrix algebra abstract we investigate the picard group of a structural matrix or incidence algebra a of a finite preordered set p over a field and we consider five interrelated problems our main technique involves establishing a connection between the group of outer automorphisms out a of a and the group of outer automorphisms of the basic algebra a which is an incidence algebra of the associated partially ordered set p of p we discuss necessary and sufficient conditions for out a to be a natural invariant for the morita equivalence class of a and necessary and sufficient conditions for m n k to be strongly graded by a group g and coefficient ring a containing n primitive orthogonal idempotents

representation theory of graded artin algebras having initiated the study of graded artin algebras a in 91 here we initiate the study of their representation theory our point of view is to study graded i modules which we believe to be somewhat more tractable than ungraded ones in order to obtain information about all i modules this point of view leads to the introduction in section 1 of the full subcategory mod of mod i consisting of the gradable objects of mod i that is the finitely generated i modules which support a gradation in these terms one of the major results of the paper asserts that mod has finite representation type if and only if mod i has finite representation type thus in section 3 we introduce a number g g 4 designed to measure the size of mod i if g co we show that i has infinite representation type if g co we show that there is a graded artin algebra q of a certain specified form such that i has infinite representation type precisely when 0 has infinite representation type the artin algebra fl has in particular desirable diagrammatic properties and these we will exploit elsewhere we speculate that when g is finite every finitely generated i module is gradable now in case n has finite representation type it is obvious that g is finite we show indeed that for every gradation of a given artin algebra of finite representation type every module is gradable this result and the others cited are chiefly consequences of a result proved in section 4 if a component of the auslander reiten graph of a graded artin algebra contains a gradable module then the component

gaussian subordination for the beurling selberg extremal problem we determine extremal entire functions for the problem of ma jorizing minorizing and approximating the gaussian function e x 2 by en tire functions of exponential type the combination of the gaussian and a general distribution approach provides the solution of the extremal problem for a wide class of even functions that includes most of the previously known examples for instance 3 4 10 and 17 plus a variety of new interesting functions such asjxj for 1 log x 2 2 x 2 2 for 0 log x2 2 andx2n logx2 forn2 n further applications to number theory include optimal approximations of theta functions by trigonometric polynomi als and optimal bounds for certain hilbert type inequalities related to the discrete hardy littlewood sobolev inequality in dimension one

projective schur algebras of nilpotent type are brauer equivalent to radical algebras let k be a field and g a finite group for any element h g k k denotes the units of k here as a trivial g module we can form the twisted group algebra kg it is isomorphic to the group algebra as a vector space and its multiplication satisfies the rule xu yu xyf u where f g g k is a 2 cocycle representing it is easily checked that the 2 cocycle condition satisfied by f insures the associativity of the multiplication in kg a projective schur algebra over k is by definition a k central simple algebra which is a homomorphic image of a twisted group algebra kg for some finite group g and some h g k equivalently a k central simple algebra a is a projective schur algebra over k if it contains a subgroup of the units of a which is finite modulo k and spans a as k vector space we often use the notation k clearly a projective schur algebra over k determines an element in the brauer group br k of the field k the subgroup in br k generated by and in fact consisting of such elements is called the projective schur group of k and denoted by ps k the notions of projective schur algebra and projective schur group were introduced in 6 by lorenz and opolka this construction can be viewed as the projective analogue of schur algebras and the schur group denoted by s k of the field k see 10 it is natural to restrict attention to special types of groups g our interest here is in k central simple algebras that are homomorphic images of twisted group algebras kg with g nilpotent resp abelian see 6 we call these algebras projective schur algebras of nilpotent type resp abelian type clearly as above these algebras represent classes in br k and we denote by pnil k resp pab k the subgroup they generate in 6 it is shown that if k is a number field and n is the number of roots of unity in k then pab k nbr k where nb denotes the n torsion subgroup of b more generally since a symbol algebra of degree n over k is a homomorphic

a class of extremal functions for the fourier transform we determine a class of real valued integrable functions f x and corresponding functions m x such that f x 1 and the value of mfo is miimi several applications of these functions to number theory and analysis are given

on weak equivalences of gradings abstract when one studies the structure e g graded ideals graded subspaces radicals or graded polynomial identities of graded algebras the grading group itself does not play an important role but can be replaced by any other group that realizes the same grading here we come to the notion of weak equivalence of gradings two gradings are weakly equivalent if there exists an isomorphism between the graded algebras that maps each graded component onto a graded component the following question arises naturally when a group grading on a finite dimensional algebra is weakly equivalent to a grading by a finite group it turns out that this question can be reformulated purely group theoretically in terms of the universal group of the grading namely a grading is weakly equivalent to a grading by a finite group if and only if the universal group of the grading is residually finite with respect to a special subset of the grading group the same is true for all the coarsenings of the grading if and only if the universal group of the grading is hereditarily residually finite with respect to the same subset we show that if n 349 then on the full matrix algebra m n f there exists an elementary group grading that is not weakly equivalent to any grading by a finite semi group and if n 3 then any elementary grading on m n f is weakly equivalent to an elementary grading by a finite group

extremal problems in de branges spaces the case of truncated and odd functions in this paper we find extremal one sided approximations of exponential type for a class of truncated and odd functions with a certain exponential subordination these approximations optimize the l 1 mathbb r e x 2 mathrm d x error where e is an arbitrary hermite biehler entire function of bounded type in the upper half plane this extends the work of holt and vaaler duke math j 83 203 247 1996 for the signum function we also provide periodic analogues of these results finding optimal one sided approximations by trigonometric polynomials of a given degree to a class of periodic functions with exponential subordination these extremal trigonometric polynomials optimize the l 1 mathbb r mathbb z mathrm d vartheta error where vartheta is an arbitrary nontrivial measure on mathbb r mathbb z the periodic results extend the work of li and vaaler indiana univ math j 48 1 183 236 1999 who considered this problem for the sawtooth function with respect to jacobi measures our techniques are based on the theory of reproducing kernel hilbert spaces of entire functions and of polynomials and on the construction of suitable interpolations with nodes at the zeros of laguerre polya functions

the representation theory of fully group graded algebras let g be a finite group k an algebraically closed field of characteristic p and r a fully g graded k algebra of finite dimension the most trivial example of such an algebra is kg this paper attempts to generalize the p modular representation theory of the group algebra kg to the groupgraded algebra r it is shown that most of the theory is the same for r as for kg two areas in which the representation theory of r differs from that of kg are the dimensions of induced modules and the properties of the defect groups of the blocks of r if h is a subgroup of g and v is an r module then the dimension of vr is not necessarily divisible by the dimension of v if b is a block of r and d is a defect group of b then d is a p group as would be expected however it is not necessarily true that d is a sylow intersection nor that d o n d in the first section we define what it means for an algebra to have a g graded structure we show how to give such a structure to the algebra m k of r by r matrices over the field k whenever igl r a number of basic results about fully g graded algebras are then given the second section deals with r modules proofs are given of the mackey decomposition formula the existence of vertices and sources and the green correspondence these three results were already known to hold for r for example they are mentioned in 6 but proofs have not previously been given in the literature they are proved here to make this paper more or less self contained many of the proofs of supporting results for these theorems will not appear as they are easy generalizations of the known proofs for r kg the reader will often be referred to 1 for these proofs the third and last section contains new results the burry carlson puig theorem is generalized and a new definition of the defect groups of a block

the center of the generic g crossed product abstract let g be a finite group and let f be a field of characteristic zero in this paper we construct a generic g crossed product over f using generic graded matrices the center of this generic g crossed product denoted by f g is then the invariant field of a suitable g action on a field of rational functions in several indeterminates the main goal of this paper is to study the extensions f g f given that f contains enough roots of unity and determine how close they are to being purely transcendental in particular we show that f g f is a stably rational extension for g c 2 c 2 n where n is odd and for g n 2 m e 1 1 where g c d n 2 m 1 furthermore we prove that if h k are groups of coprime orders then f h k is stably rationally equivalent to the fraction field of f h f k

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

pauli gradings on lie superalgebras and graded codimension growth abstract we introduce grading on certain finite dimensional simple lie superalgebras of type p t by elementary abelian 2 group this grading gives rise to pauli matrices and is a far generalization of z 2 z 2 grading on lie algebra of 2 2 traceless matrices we use this grading for studying numerical invariants of polynomial identities of lie superalgebras in particular we compute graded pi exponent corresponding to pauli grading

some extremal functions in fourier analysis ii we obtain extremal majorants and minorants of exponential type for a class of even functions on which includes log x and x where 1 1 we also give periodic versions of these results in which the majorants and minorants are trigonometric polynomials of bounded degree as applications we obtain optimal estimates for certain hermitian forms which include discrete analogues of the one dimensional hardy littlewood sobolev inequalities a further application provides an erdos turan type inequality that estimates the sup norm of algebraic polynomials on the unit disc in terms of power sums in the roots of the polynomials

algorithms for inversion mod p k p k this article describes and analyzes all existing algorithms for computing x a 1 pmod p k x a 1 mod p k for a prime p p and also introduces a new algorithm based on the exact solution of linear equations using p p adic expansions the algorithm starts with the initial value c a 1 pmod p c a 1 mod p and iteratively computes the digits of the inverse x a 1 pmod p k x a 1 mod p k in base p p the mod 2 version of the algorithm is more efficient than all existing algorithms for small values of k k moreover it stands out as being the only one that works for any p p any k k and digit by digit while the new algorithm is asymptotically worse off it requires the minimal number of arithmetic operations just a single addition per step as compared to all existing algorithms

a note on the zeros of zeta and l functions let pi s t denote the argument of the riemann zeta function at the point s tfrac 1 2 it assuming the riemann hypothesis we give a new and simple proof of the sharpest known bound for s t we discuss a generalization of this bound for a large class of l functions including those which arise from cuspidal automorphic representations of gl m over a number field we also prove a number of related results including bounding the order of vanishing of an l function at the central point and bounding the height of the lowest zero of an l function

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

graded equivalence theory with applications we develop the foundations for graded equivalence theory and apply them to investigate properties such as primeness finite representation type and vertex theory of graded rings the key fact that we prove is that for any two g graded rings r and s such that there is a category equivalence from gr r to gr s that commutes with suspensions then for any subgroup h of g the categories gr h g r and gr h g s of modules graded by the g set of right cosets are also equivalent

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

the four sets of additive quantum numbers of su 3 the four maximal sets of additive quantum numbers and related fine gradings of the lie algebra sl 3 c are described in detail the quantum numbers are determined by a grading of the lie algebra fine gradings providing maximal sets of them two sets of additive quantum numbers are equivalent precisely if the corresponding gradings are equivalent under a transformation from the automorphism group of the lie algebra