the burnside ring of fusion systems abstract given a saturated fusion system f on a finite p group s we define a ring a f modeled on the burnside ring a g of finite groups we show that these rings have several properties in common when f is the fusion system of g we describe the relationship between these rings

rational points on generalized flag varieties and unipotent conjugacy in finite groups of lie type let g be a connected reductive algebraic group defined over the finite field fq where q is a power of a good prime for g we write f for the frobenius morphism of g corresponding to the fq structure so that gf is a finite group of lie type let p be an f stable parabolic subgroup of g and u the unipotent radical of p in this paper we prove that the number of uf conjugacy classes in gf is given by a polynomial in q under the assumption that the centre of g is connected this answers a question of j alperin in 1 in order to prove the result mentioned above we consider for unipotent u gf the variety p0u of g conjugates of p whose unipotent radical contains u we prove that the number of fq rational points of p0u is given by a polynomial in q with integer coefficients moreover in case g is split over fq and u is split in the sense of 22 5 the coefficients of this polynomial are given by the betti numbers of p0u we also prove the analogous results for the variety pu consisting of conjugates of p that contain u

component groups of unipotent centralizers in good characteristic abstract let g be a connected reductive group over an algebraically closed field of good characteristic for u g unipotent we describe the conjugacy classes in the component group a u of the centralizer of u our results extend work of the second author done for simple adjoint g over the complex numbers when g is simple and adjoint the previous work of the second author makes our description combinatorial and explicit moreover it turns out that knowledge of the conjugacy classes suffices to determine the group structure of a u thus we obtain the result previously known through case checking that the structure of the component group a u is independent of good characteristic

character sheaves v this paper is part of a series s 131 devoted to the study of a class g of irreducible perverse sheaves called character sheaves on a connected reductive algebraic group g the numbering of chapters sections and references will continue that of s 131 this paper is a step towards the classification of character sheaves on g one of the main results is the following one under certain assumptions there is a natural surjective map with finite fibers from g to the set of all pairs 9 c up to conjugacy by the weyl group where 9 is a tame local system on the maximal torus and c is a two sided cell in the stabilizer w of 9 in the weyl group the assumptions made on g are

irreducible bimodules over alternative algebras and superalgebras the irreducible alternative superbimodules are studied the complete classification is obtained for even bimodules of arbitrary dimension and for finite dimensional irreducible superbimodules over an algebraically closed field

resolutions of mesh algebras periodicity and calabi yau dimensions a triangulated category mathcal t sigma is said to be calabi yau of dimension d if d is a serre functor we determine which stable module categories of self injective algebras of finite type are calabi yau and compute their calabi yau dimensions correcting errors in previous work we first show that the calabi yau property of mod can be detected in the minimal projective resolution of the stable auslander algebra of over its enveloping algebra we then describe the beginning of such a minimal resolution for any mesh algebra of a stable translation quiver and apply covering theory to relate these minimal resolutions to those of the generalized preprojective algebras of dynkin graphs for representation finite self injective algebras of torsion order t 1 we obtain a complete description of their stable calabi yau properties but only partial results for those algebras of torsion order t 2 we also obtain some new information about the periods of the representation finite self injective algebras of torsion order t 1 finally we describe how these questions can also be approached by realizing the stable categories of representation finite self injective algebras as orbit categories of the bounded derived categories of hereditary algebras and illustrate this technique with several explicit computations that our previous methods left unsettled

localization and ideal theory in noetherian strongly group graded rings abstract we study primality hypercentrality simplicity and localization and the second layer condition in skew groups rings and group graded rings we give necessary and sufficient conditions for the skew group ring of a torsion free nilpotent group to be a simple ring and if the coefficient ring is commutative we give necessary and sufficient conditions for the skew group ring of an abelian group to be simple our method involves showing certain group graded rings are hypercentral our main results show that if g is a polycyclic by finite group and r is an artinian ring or a commutative noetherian ring then a strongly g graded ring with base ring r satisfies the second layer condition we discuss consequences of this for localization in such rings

numerical experiments on the accuracy of the chebyshev frobenius companion matrix method for finding the zeros of a truncated series of chebyshev polynomials for a function f x that is smooth on the interval email protected a b but otherwise arbitrary the real valued roots on the interval can always be found by the following two part procedure first expand f x as a chebyshev polynomial series on the interval and truncate for sufficiently large n second find the zeros of the truncated chebyshev series the roots of an arbitrary polynomial of degree n when written in the form of a truncated chebyshev series are the eigenvalues of an nxn matrix whose elements are simple explicit functions of the coefficients of the chebyshev series this matrix is a generalization of the frobenius companion matrix we show by experimenting with random polynomials wilkinson s notoriously ill conditioned polynomial and polynomials with high order roots that the chebyshev companion matrix method is remarkably accurate for finding zeros on the target interval yielding roots close to full machine precision we also show that it is easy and cheap to apply newton s iteration directly to the chebyshev series so as to refine the roots to full machine precision using the companion matrix eigenvalues as the starting point lastly we derive a couple of theorems the first shows that simple roots are stable under small perturbations of magnitude e to a chebyshev coefficient the shift in the root x is bounded by e df dx x o e 2 for sufficiently small e second we show that polynomials with definite parity only even or only odd powers of x can be solved by a companion matrix whose size is one less than the number of nonzero coefficients a vast cost saving

regulator constants and the parity conjecture the p parity conjecture for twists of elliptic curves relates multiplicities of artin representations in p selmer groups to root numbers in this paper we prove this conjecture for a class of such twists for example if e is semistable at 2 and 3 k is abelian and k is its maximal pro p extension then the p parity conjecture holds for twists of e by all orthogonal artin representations of mathop mathrm gal k infty mathbb q we also give analogous results when k is non abelian the base field is not and e is replaced by an abelian variety the heart of the paper is a study of relations between permutation representations of finite groups their regulator constants and compatibility between local root numbers and local tamagawa numbers of abelian varieties in such relations

homology of the zero set of a nilpotent vector field on a flag manifold 0 1 let x be a linear transformation of a finite dimensional vector space v the configuration of flags in v which are fixed by x has rather remarkable properties when x is unipotent though this case is especially interesting the proper generality in which to study such configurations is in the theory of reductive algebraic groups where their definition can be reformulated in the language of borel subalgebras as follows let g be a connected reductive group over c with lie algebra g and let n e g be a nilpotent element let q be the variety of all borel subalgebras of g and let

transfer and characteristic idempotents for saturated fusion systems abstract we construct a well behaved transfer map from the p local burnside ring of the underlying p group s to the p local burnside ring of a saturated fusion system f using this transfer map we give new results on the characteristic idempotent of f the unique idempotent in the p local double burnside ring of s satisfying properties of linckelmann and webb we describe this idempotent explicitly both in terms of fixed points and as a linear combination of transitive bisets additionally using fixed points we determine the map for burnside rings given by multiplication with the characteristic idempotent and show that this is the transfer map previously constructed applying these results we show that for every saturated fusion system the ring generated by all not necessarily idempotent characteristic elements in the p local double burnside ring is isomorphic as rings to the p local single burnside ring of the fusion system and we disprove a conjecture by park ragnarsson stancu on the composition product of fusion systems

a test based on conversion to the bernstein polynomial basis for an interval to be free of zeros applicable to polynomials in chebyshev form and to transcendental functions approximated by chebyshev series abstract polynomials of high degree are much less vulnerable to roundoff error when expressed as truncated chebyshev series rather than the usual power series form recent articles have developed subdivision methods in which all real roots on the canonical chebyshev interval x 1 1 are found by subdividing the interval and finding the roots of separate chebyshev series of moderate degree on each subdomain this strategy can be applied either to polynomials or to transcendental functions if the latter are analytic on the search interval and thus have rapidly convergent chebyshev polynomial approximations the last step is to compute the eigenvalues of the chebyshev frobenius companion matrix for each local polynomial here we propose a simple strategy for flagging some subdomains as zero free so that eigensolving can be omitted the test requires conversion of the polynomial from chebyshev form to a bernstein polynomial basis the interval is zero free if all coefficients in the bernstein basis are of the same sign we give the conversion matrices for various small and moderate n and quote rababah s formulas for general n we show how to exploit parity so as to halve the cost of the conversion for large n to about n 2 floating point operations we show that the conversion matrices have condition numbers that are approximately 5 8 2 n

ramanujan s inversion formulas for the lemniscate and allied functions abstract in the unorganized portions of his second notebook ramanujan states without proofs 10 inversion formulas for the lemniscate integral and two similar integrals these 10 formulas are proven here

computing the zeros of a fourier series or a chebyshev series or general orthogonal polynomial series with parity symmetries in recent years good algorithms have been developed for finding the zeros of trigonometric polynomials and of ordinary polynomials when written in the form of a truncated chebyshev polynomial or legendre polynomial series in each case the roots can be found from the eigenvalues of a generalized frobenius companion matrix whose elements are trivial functions of the fourier coefficients or chebyshev coefficients however the qr method for computing the companion matrix eigenvalues has a cost that grows proportionally to n 3 where n is the polynomial degree by exploiting the special structure of the companion matrices the cost can be reduced to o n 2 but only for large n here we show that if the polynomial has definite parity such as a trigonometric polynomial composed only of cosines or a polynomial that is a sum only of chebyshev polynomials of odd degree one can exploit these symmetries to halve the size of the problem this reduces costs in the companion matrix method by a factor ranging between four and eight for trigonometric polynomials we give transformations that dramatically reduce costs even if the roots are found by an algorithm other than the companion matrix procedure we further give reductions for trigonometric polynomials with double parity symmetries which save a factor of sixteen to a factor of sixty four in the companion matrix algorithm special functions such as spherical harmonics mathieu functions prolate spheroidal wavefunctions and hough functions all represented by truncated fourier series with double parity are a rich source of applications

the hesselink stratification of nullcones and base change let g be a connected reductive algebraic group over an algebraically closed field of characteristic p 0 we give a case free proof of lusztig s conjectures lusztig in transform groups 10 449 487 2005 on so called unipotent pieces this presents a uniform picture of the unipotent elements of g which can be viewed as an extension of the dynkin kostant theory but is valid without restriction on p we also obtain analogous results for the adjoint action of g on its lie algebra mathfrak g and the coadjoint action of g on mathfrak g

computing zeros on a real interval through chebyshev expansion and polynomial rootfinding robust polynomial rootfinders can be exploited to compute the roots on a real interval of a nonpolynomial function f x by the following i expand f as a chebyshev polynomial series ii convert to a polynomial in ordinary series of powers form and iii apply the polynomial rootfinder complex valued roots and real roots outside the target interval are discarded the expansion is most efficiently done by adaptive chebyshev interpolation with n equal to a power of two where n is the degree of the truncated chebyshev series all previous evaluations of f can then be reused when n is increased adaption stops when n is sufficiently large so that further increases produce no significant change in the interpolant we describe two conversion strategies the convert to powers method uses multiplication by mildly ill conditioned matrices to create a polynomial of degree n the degree doubling strategy defines a polynomial of larger degree 2n but is very well conditioned the convert to powers method although faster restricts n to moderate values this can always be accomplished by subdividing the target interval both these strategies allow simultaneous approximation of many roots on an interval whether simple or multiple for nonpolynomial f x

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

theta functions gaussian series and spatially periodic solutions of the korteweg de vries equation it has been shown by novikov funct anal appl 8 236 1974 dubrovin et al russian math surveys 31 59 1976 lax commun pure appl math 28 141 1975 mckean and van moerbeke inv math 30 217 1975 and others that the nonlinear evolution equations which admit solitary waves also have spatially periodic exact solutions polycnoidal waves which can be expressed in terms of multidimensional riemann theta functions here it is shown that via poisson summation the fourier series that define the theta functions can be transformed into an infinite series of gaussian functions because the lowest terms of the gaussian series generate the usual solitary waves it is possible to intimately explore the relationship between solitary waves and these spatially periodic polycnoidal waves also by using the gaussian series one can perturbatively calculate phase velocities and wave structure for the polycnoidal wave even in the strongly nonlinear regime for which the soliton or mult

fixed rings and integrality abstract in this note we offer some variants of a theorem of pare and schelter on integrality in matrix rings we use these to deduce that a ring r is integral over the fixed subring r g under the action of a finite abelian group g provided g r r we also obtain an integrality version of the bergman isaacs theorem

chebyshev expansion on intervals with branch points with application to the root of kepler s equation a chebyshev hermite pade method when two or more branches of a function merge the chebyshev series of u l will converge very poorly with coefficients a n of t n l falling as o 1 n a for some small positive exponent a however as shown in j p boyd chebyshev polynomial expansions for simultaneous approximation of two branches of a function with application to the one dimensional bratu equation appl math comput 143 2002 189 200 it is possible to obtain approximations that converge exponentially fast in n if the roots that merge are denoted as u 1 l and u 2 l then both branches can be written without approximation as the roots of u u 1 l u u 2 l u email protected l email protected l by expanding the nonsingular coefficients of the quadratic b l and c l as chebyshev series and then applying the usual roots of a quadratic formula we can approximate both branches simultaneously with error that decreases proportional to exp email protected for some constant s 0 where n is the truncation of the chebyshev series this is dubbed the chebyshev shafer or chebyshev hermite pade method because it substitutes chebyshev series for power series in the generalized pade approximants known variously as shafer or hermite pade approximants here we extend these ideas first we explore square roots with branches that are both real valued and complex valued in the domain of interest illustrated by meteorological baroclinic instability second we illustrate triply branched functions via roots of the kepler equation f u l e email protected u email protected 0 only one of the merging roots is real valued and the root depends on two parameters l e rather than one nonetheless the chebyshev hermite pade scheme is successful over the whole two dimensional parameter plane we also discuss how to cope with poles and logarithmic singularities that arise in our examples at the extremes of the expansion domain

generalised burnside rings g categories and module categories this note describes an application of the theory of generalised burnside rings to algebraic representation theory tables of marks are given explicitly for the groups s4 and s5 which are of particular interest in the context of reductive algebraic groups as an application the base sets for the nilpotent element f4 a3 are computed

all p local finite groups of rank two for odd prime p in this paper we give a classification of the rank two p local finite groups for odd p this study requires the analysis of the possible saturated fusion systems in terms of the outer automorphism group of the possible f radical subgroups also for each case in the classification either we give a finite group with the corresponding fusion system or we check that it corresponds to an exotic p local finite group getting some new examples of these for p 3

rootfinding through global newton iteration and chebyshev polynomials for the amplitude of an electronic oscillator the amplitude z and oscillator gain k of a balanced oscillator are connected by the transcendental equation f z k 0 where f z k 2 z 0 1tanh zsin email protected sin email protected dt 1 k where the email protected 1 and email protected 0 it is trivial to find k z through numerical quadrature but the inverse function z k is a non trivial exercise in rootfinding first we derive a power series for small k and an asymptotic expansion for large k by blending these we obtain an analytic approximation to initialize newton s iteration it is found that the iteration converges to full machine precision within at most four iterations over the entire parameter range however the newton method is rather slow because each iteration requires two numerical quadratures because the integrands are periodic these can be done by the rectangle rule with an accuracy that increases exponentially with the number of quadrature points n but n must be large when z is large a faster method is to apply the newton quadrature algorithm to generate z k at a few selected points and apply chebyshev interpolation twice each chebyshev expansion is to choose the mimic the corresponding power series or asymptotic series so that singularities are avoided just 15 terms of each series plus an asymptotic approximation for very large k are sufficient to approximate z k to within about eight decimal places over the entire parameter range

superspecies and their representations abstract superspecies are introduced to provide a construction of all finite dimensional superalgebras all acyclic superspecies are classified according to their graded representation types

conjugacy classes in unitriangular matrices let gn gn q be the group of the upper unitriangular matrices of size n n over fq the finite field with q pt elements higman has conjectured that for each n the number of conjugacy classes of elements of gn is a polynomial expression in q in this paper we develop the algorithm given in j algebra 177 1995 899 introducing new ideas and theoretical properties which lead us to get the conjugacy vector of gn for n 13 these vectors for n 5 resp for n 6 7 8 were given in j algebra 152 1 1992 1 resp j algebra 177 1995 899 in particular we conclude that for these values the number of conjugacy classes of gn is a polynomial in q with integral coefficients and degree n n 6 12 thus higman s conjecture holds for n 13 on the other hand for each positive integer n we find explicitly a set of q n n 6 12 different conjugacy classes

numerical perturbative and chebyshev inversion of the incomplete elliptic integral of the second kind abstract the incomplete elliptic integral of the second kind e sin t m 0 t 1 m sin 2 t dt where m 0 1 is the elliptic modulus can be inverted with respect to angle t by solving the transcendental equation e sin t m z 0 we show that newton s iteration t n 1 t n e sin t m z 1 m sin 2 t always converges to t z m e 1 z m within a relative error of less than 10 10 in three iterations or less from the first guess t 0 z m 2 r 2 where defining 1 z e 1 m r 1 m 2 2 and atan 1 m we briefly discuss three alternative initialization strategies homotopy initialization t 0 z m 1 m z 0 mt z m m 1 perturbation series in powers of m and inversion of the chebyshev interpolant of the incomplete elliptic integral although all work well and are general strategies applicable to a very wide range of problems none of these three alternatives is as efficient as the empirical initialization which is completely problem specific this illustrates tai tsun wu s maxim usefulness is often inversely proportional to generality

actions of divided power hopf algebras abstract we study prime ideals in enveloping algebra smash products and use a duality construction to obtain results on prime ideals in rings on which divided power hopf algebras act these actions correspond to higher derivations first we consider chains of prime ideals in an enveloping algebra smash product over an arbitrary ring where the lie algebra is assumed to be finite dimensional abelian over a field of positive characteristic we give a bound on the length of such a chain where the ideals all have the same intersection with the coefficient ring then using an explicit construction of a duality theorem of blattner and montgomery in this context we are able to apply results on enveloping algebra smash products to study the invariant ideals of prime ideals in a ring under a locally nilpotent divided power hopf algebra action

new series for the cosine lemniscate function and the polynomialization of the lemniscate integral we discuss the numerical computation of the cosine lemniscate function and its inverse the lemniscate integral these were previously studied by bernoulli euler gauss abel jacobi and ramanujan we review general elliptic formulas for this special case and provide some novelties we show that a fourier series by ramanujan converges twice as fast as the standard elliptic cosine fourier series specialized to this case the so called imbricate series however converges geometrically fast over the entire complex plane we derive two new expansions the rational plus fourier series converges much faster than ramanujan s for real z each term is asymptotically 12 400 times smaller than its immediate predecessor coslem z 4b q 1 q cos bz 1 q 2 4qcos 2 bz n 1 q n 1 2 1 1 q 2 n 1 1 cos 2n 1 bz where q exp p is the elliptic nome k 1 85 is the complete elliptic integral of the first kind for a modulus m 1 2 and b p k2 the rational imbricate series is uniformly valid over the complex plane but converges twice as fast as the sech imbricate series coslem z 4bq 1 q j q 1 q cos bz 1 q 2 4qcos 2 b z jpi where p 4 2 k is the period in both the real and imaginary directions we devise a new approximation for the lemniscate integral for real argument as the arccosine of a chebyshev series and show that 17 terms yield about 15 digits of accuracy for complex argument we show that the lemniscate integral can be found to near machine precision assumed as sixteen decimal digits by computing the roots of a polynomial of degree thirteen alternatively newton s iteration converges in three iterations with an initialization accurate to four decimal places that is the chosen root of a cubic equation

group graded algebras and almost polynomial growth let f be a field of characteristic 0 g a finite abelian group and a a g graded algebra we prove that a generates a variety of g graded algebras of almost polynomial growth if and only if a has the same graded identities as one of the following algebras 1 fcp the group algebra of a cyclic group of order p where p is a prime number and p g 2 ut2g f the algebra of 2 2 upper triangular matrices over f endowed with an elementary g grading 3 e the infinite dimensional grassmann algebra with trivial g grading 4 in case 2 g ez2 the grassmann algebra with canonical z2 grading

generalized burnside rings and group cohomology we define the cohomological burnside ring b n g m of a finite group g with coefficients in a zgmodule m as the grothendieck ring of the isomorphism classes of pairs x u where x is a g set and u is a cohomology class in a cohomology group h n x g m the cohomology groups h x g m are defined in such a way that h x g m i h hi m when x is the disjoint union of transitive g sets g hi i fa is an abelian group with trivial action then b 1 g a is the same as the monomial burnside ring over a and when m is taken as a g monoid then b 0 g m is equal to the crossed burnside ring b c g m we discuss the generalizations of the ghost ring and the mark homomorphism and prove the fundamental theorem for cohomological burnside rings we also give an interpretation of b 2 g m in terms of twisted

hopf algebra actions abstract finite groups acting on rings by automorphisms and group graded rings are instances of hopf algebras h acting on h module algebras a we study such actions let a h a a h a e h a all h h the ring of h invariants and form the smash product a h we study the ring extensions ah a a h we prove a maschke type theorem for a h modules we form an associated morita context ah a a a h and use these to get connections between the various rings

constants of lie algebra actions let r be an associative algebra over a field k and let l be a lie algebra over k acting on r as derivations we will be interested in studying the relationship between r and the ring of constants rl our situation is a special case of the study of rings of invariants of hopf algebra actions and is also analogous to another special case of hopf algebra actions namely the study of fixed rings of finite group actions many of the results in this paper are analogs of results on group actions and give rise to more general questions regarding hopf algebra actions in section 1 we study actions on non nilpotent algebras we first show that any algebraic derivation has non zero constants and then prove that any finite dimensional nilpotent lie algebra of algebraic derivations must act with non zero constants it then follows as a corollary that any finite dimensional nilpotent restricted lie algebra must act with non zero constants these are analogs of results on groups proved in 12 5 in section 2 we prove a necessary condition for the existence of nonzero invariants of certain hopf algebra actions we then use this result to prove the converse of our theorem on nilpotent restricted lie algebras we conclude the section with several questions on general hopf algebra actions

representations of super yang mills algebras we study in this article the representation theory of a family of super algebras called the super yang mills algebras by exploiting the kirillov orbit method a la dixmier for nilpotent super lie algebras these super algebras are an extension of the so called yang mills algebras introduced by a connes and m dubois violette in lett math phys 61 2 149 158 2002 and in fact they appear as a background independent formulation of supersymmetric gauge theory considered in physics in a similar way as yang mills algebras do the same for the usual gauge theory our main result states that under certain hypotheses all clifford weyl super algebras rm cliff q k otimes a p k for p 3 or p 2 and q 2 appear as a quotient of all super yang mills algebras for n 3 and s 1 this provides thus a family of representations of the super yang mills algebras

2 groups 2 characters and burnside rings abstract we study 2 representations i e actions of 2 groups on 2 vector spaces our main focus is character theory for 2 representations to this end we employ the technique of extended burnside rings our main theorem is that the ganter kapranov 2 character is a particular mark homomorphism of the burnside ring as an application we give a new proof of osorno s formula for the ganter kapranov 2 character of a finite group

two clifford s theorems for strongly group graded rings abstract let a be a strongly group graded ring in this paper we obtain the main results as following 1 in a a similar result to clifford s theorem of the decomposition of induced modules from normal subgroups it corrects the result of 4 2 frobenius reciprocity for a 3 we extend the theory of stable modules in two ways one of which is the localization theory of rings and modules in some special cases we can obtain the corresponding results of 1 2 3

solving the helmholtz equation for membranes of arbitrary shape numerical results i calculate the modes of vibration of membranes of arbitrary shape using a collocation approach based on little sinc functions the matrix representation of the pde obtained using this method is explicit and does not require the calculation of integrals to illustrate the virtues of this approach i have considered a large number of examples part of them are taken from the literature and part of them new when possible i have tested the accuracy of these results by comparing them with the exact results when available or with results from the literature in particular in the case of the l shaped membrane the first example discussed in the paper i show that it is possible to extrapolate the results obtained with different grid sizes to obtain highly precise results finally i also show that the present collocation technique can be easily combined with conformal mapping to provide numerical approximations to the energies which quite rapidly converge to the exact results

cnoidal waves as exact sums of repeated solitary waves new series for elliptic functions via poisson summation of the corresponding fourier series it is shown that the cnoidal wave of the korteweg de vries equation can be written as an infinite sum of periodically repeated solitary waves a result first proved by toda through a different method similar series of hyperbolic secant or tangent functions are also derived for the elliptic dn sn and cn functions which make it possible to express the cnoidal wave of almost any evolution equation in terms of a series of the corresponding solitary waves as a by product new series are also obtained for the complete elliptic integral k and modulus k

poisson maps and integrable deformations of the kowalevski top we construct a poisson map between manifolds with linear poisson brackets corresponding to the lie algebras e 3 and so 4 using this map we establish a connection between the deformed kowalevski top on e 3 proposed by sokolov and the kowalevski top on so 4 the connection between these systems leads to the separation of variables for the deformed system on e 3 and yields the natural 5 5 lax pair for the kowalevski top on so 4

hopf actions on vertex operator algebras ii smash product abstract the smash product v h of a hopf algebra h and an h module vertex operator algebra v are investigated a n v theory and contragredient module theory are founded for v h if h is the group algebra of a finite subgroup of aut v all the irreducible inequivalent admissible v h modules are classified we also prove the maschke theorem in vertex operator algebra version finally we prove weak duality theorems for actions and coactions in vertex operator algebras version

solitons from sine waves analytical and numerical methods for non integrable solitary and cnoidal waves abstract the fkdv equation u t uu x u xxxxx 0 is used as a testbed for a variety of analytical and numerical methods that can be applied to solitary waves and cnoidal waves of non integrable differential equations that is to say to equations which cannot be solved by the inverse scattering transform the basic tools are i pade approximants formed from power series in the amplitude ii a newton kantorovich pseudospectral fourier continuation numerical method iii singular perturbation theory for two interacting solitons of almost identical phase speed iv bifurcation and branch switching methods v the imbricate soliton series a number of new results for the fkdv equation are obtained including extensive numerical calculations of the spatially periodic solutions with one peak cnoidal wave and two peaks bicnoidal wave per period an analytical expression for the double peaked soliton bion calculation of both the limit and bifurcation points for the bicnoidal wave and finally the computation of accurate analytical approximations to the cnoidal wave for all amplitudes more important all of these analytical and numerical tools are highly effective for this equation in spite of the fact that it cannot be solved by the inverse scattering transform work now in progress will apply these methods to non integrable equations in two space dimensions

an extension of matlab to continuous functions and operators an object oriented matlab system is described for performing numerical linear algebra on continuous functions and operators rather than the usual discrete vectors and matrices about eighty matlab functions from plot and sum to svd and cond have been overloaded so that one can work with our chebfun objects using almost exactly the usual matlab syntax all functions live on 1 1 and are represented by values at sufficiently many chebyshev points for the polynomial interpolant to be accurate to close to machine precision each of our overloaded operations raises questions about the proper generalization of familiar notions to the continuous context and about appropriate methods of interpolation differentiation integration zerofinding or transforms applications in approximation theory and numerical analysis are explored and possible extensions for more substantial problems of scientific computing are mentioned

a chebyshev polynomial interval searching method lanczos economization for solving a nonlinear equation with application to the nonlinear eigenvalue problem to search a given real interval for roots our algorithm is to replace f by fn its n term chebyshev expansion on the search interval min max and compute the roots of this proxy this strategy is efficient if and only if f itself is expensive to evaluation such as when f is the determinant of a large matrix whose elements depend nonlinearly on for such expensive functions it is much cheaper to search for zeros of fn which can be evaluated in o n operations than to iterate or look for sign changes on a fine grid with f itself it is possible to systematically increase n until the chebyshev series converges acceptably fast without wasting previously computed values of f by imitating the clenshaw curtis quadrature our strategy of replacing f by fn is similar to lanczos economization of power series which also replaces an expensive function by a chebyshev approximation that is more rapidly evaluated the errors induced by the chebyshev approximation can be eliminated by a final step of one or two iterations with f itself using the zeros of fn as initial guesses we show through numerical examples that the algorithm works well the only sour note is that it is sometimes necessary to split the search interval into subintervals each with a separate chebyshev expansion when f varies by many orders of magnitude on the search interval

nilpotent orbits in good characteristic and the kempf rousseau theory let g be a connected reductive algebraic group over an algebraically closed field k of characteristic p 0 image and suppose that p is a good prime for the root system of g in this paper we give a fairly short conceptual proof of pommerening s theorem pommerening j algebra 49 1977 525 536 j algebra 65 1980 373 398 which states that any nilpotent element in image is richardson in a distinguished parabolic subalgebra of the lie algebra of a levi subgroup of g as a by product we obtain a short noncomputational proof of the existence theorem for good transverse slices to the nilpotent g orbits in image for earlier proofs of this theorem see kawanaka invent math 84 1986 575 616 premet trans amer math soc 347 1995 2961 2988 spaltenstein j fac sci univ tokyo sect ia math 31 1984 283 286 we extend recent results of sommers internal math res notices 11 1998 539 562 to reductive lie algebras of good characteristic thus providing a satisfactory approach to computing the component groups of the centralisers of nilpotent elements in image and unipotent elements in g earlier computations of these groups in positive characteristics relied mostly on work of mizuno j fac sci univ tokyo sect ia math 24 1977 525 563 tokyo j math 3 1980 391 459 our approach is based on the theory of optimal parabolic subgroups for g unstable vectors also known as the kempf rousseau theory which provides a good substitute for the image theory prominent in the characteristic zero case

on a trajectory isomorphism of the kowalevski gyrostat and the clebsch problem for the kowalevski gyrostat a change of variables similar to that for the kowalevski top is done we establish one to one correspondence between solutions of the kowalevski gyrostat and the clebsch system and demonstrate that kowalevski variables for the gyrostat practically coincide with elliptic coordinates on a sphere for the clebsch case

generalized gelfand graev representations of exceptional simple algebraic groups over a finite field i introduction let ffi be an adjoint simple algebraic group of type e n 6 7 8 f 4 or g2 over an algebraically closed field k of characteristic p 0 we assume that 5 has a fixed fq rational structure for a finite subfield fq of k the purpose of this paper is to determine explicitly the values of the irreducible characters of g ffi fq at the unipotent elements when p is good 38 i 4 3 in this part i we shall do this only for large p and the full result will be proved in the forthcoming part ii after submitting the first version of the present paper the author was informed by lusztig that he solved analogous problems for classical groups using his theory of character sheaves and that he can also recover the result of this paper mentioned above our method depends upon 1 a parametrization of the irreducible characters of g and the determination of their multiplicities in deligne lusztig virtual representations lusztig 25 2 the computation of the green functions of g in large characteristic springer 36 7 1 shoji 33 beynon and spaltenstein 3 4 3 the determination of the values of generalized gelfand graev characters 20 of g in good characteristic see section 3 by 1 and 2 the uniform parts of the irreducible characters are already known explicitly in large characteristic this implies that in large characteristic our problem for groups of type e 6 o r e 7 has already been solved it should be mentioned here that the whole character table of the g2 group is also known by chang and ree 8 hence in the g2 case this paper gives just another approach to a part of the result of chang and ree if g is of type g2 f4 or e 8 the restriction of an irreducible character r of g to the set of unipotent elements can be written as a sum of its uniform part with a function of the form

a lax pair for kowasevski s top abstract we show that on each level surface of the invariants the equations of the kowalevski top are equivalent to a neumann system describing the motion of a mass point on the sphere s2 p 1 under the influence of force qp this allows us to write a global lax pair for the kowalevski system and to show that kowalevski s original reduction of the integration of the equations of motion to jacobi s inversion theorem is identical to the introduction of elliptical spherical coordinates for integrating c neumann s system

on group gradings on pi algebras abstract we show that there exists a constant k such that for any pi algebra w and any nondegenerate g grading on w where g is any group possibly infinite there exists an abelian subgroup u of g with g u exp w k a g grading w g g w g is said to be nondegenerate if w g 1 w g 2 w g r 0 for any r 1 and any r tuple g 1 g 2 g r in g r

a duality theorem for crossed products of hopf algebras several variations have appeared recently see l 5 8 101 among the best are 2 2 1 3 2 2 but the former is restricted to the case of a trivial cocycle 0 and the latter to the case of finite dimensional h the main purpose of this paper is to prove a common generalization of these two theorems in section 2 we first recall preliminaries about crossed products then we introduce some new algebra constructions and we prove related useful results in particular we define an algebra a h op u that will replace a h u in the main theorem and we show a relationship between the two algebras in the new construction u is allowed to be more general for example we may take u h section 3 introduces a subalgebra h of h with a comodule structure co h h h they are our tools to formulate and handle certain awkward assumptions in the main theorem these can be regarded as finiteness conditions and they correspond to the u local finiteness and rl condition of 2 section 4 contains the duality theorem and its proof in section 5 we draw some corollaries in particular we show that 2 2 1 3 2 2 do follow from our theorem 153 0021 8693 92 3 00

calculating conjugacy classes in sylow p subgroups of finite chevalley groups abstract in s m goodwin on the conjugacy classes in maximal unipotent subgroups of simple algebraic groups transform groups 11 1 2006 51 76 8 the first author outlined an algorithm for calculating a parametrization of the conjugacy classes in a sylow p subgroup u q of a finite chevalley group g q valid when q is a power of a good prime for g q in this paper we develop this algorithm and discuss an implementation in the computer algebra language gap using the resulting computer program we are able to calculate the parametrization of the conjugacy classes in u q when g q is of rank at most 6 in these cases we observe that the number of conjugacy classes of u q is given by a polynomial in q with integer coefficients

the homotopy theory of fusion systems the main goal of this paper is to identify and study a certain class of spaces which in many ways behave like p completed classifying spaces of finite groups these spaces occur as the classifying spaces of certain algebraic objects which we call p local finite groups a p local finite group consists roughly speaking of a finite p group s and fusion data on subgroups of s encoded in a way explained below our starting point is our earlier paper blo on p completed classifying spaces of finite groups together with the axiomatic treatment by lluis puig pu pu2 of systems of fusion among subgroups of a given p group the p completion of a space x is a space xp which isolates the properties of x at the prime p and more precisely the properties which determine its mod p cohomology for example a map of spaces x f y induces a homotopy equivalence xp y p if and only if f induces an isomorphism in mod p cohomology and h xp fp h x fp in favorable cases if x is p good when g is a finite group the p completion bgp of its classifying space encodes many of the properties of g at p for example not only the mod p cohomology of bg but also the sylow p subgroup of g together with all fusion among its subgroups are determined up to isomorphism by the homotopy type of bgp our goal here is to give a direct link between p local structures and homotopy types which arise from them this theory tries to make explicit the essence of what it means to be the p completed classifying space of a finite group and at the same time yields new spaces which are not of this type but which still enjoy most of the properties a space of the form bgp would have we hope that the ideas presented here will have further applications and generalizations in algebraic topology but this theory also fits well with certain aspects of modular representation theory in particular it may give a way of constructing classifying spaces for blocks in the group ring of a finite group over an algebraically closed field of characteristic p a saturated fusion system f over a p group s consists of a set homf p q of monomorphisms for each pair of subgroups p q s which form a category under composition include all monomorphisms induced by conjugation in s and satisfy certain other axioms formulated by puig definitions 1 1 and 1 2 below in particular these axioms are satisfied by the conjugacy homomorphisms in a finite group we refer to pu and pu2 for more details of puig s work on saturated

kowalewski s top on the lie algebras o 4 e 3 and o 3 1 the authors extend the lax pair for kowalewski s top kt obtained by haine and horozov 1987 to the lie algebras o 4 and o 3 1 the results are expressed in terms of a solution of the neumann system they derive formulae for the action variables by analogy with the e 3 kt case and obtain a useful representation for the equations of motion that demonstrate explicitly the separation of variables

computing real roots of a polynomial in chebyshev series form through subdivision with linear testing and cubic solves abstract an arbitrary polynomial of degree n f n x can always be represented as a truncated chebyshev polynomial series chebyshev form this representation is much better conditioned than the usual power form of a polynomial we describe a new method for finding the real roots of f n x in chebyshev form the canonical interval x 1 1 is subdivided into n s subintervals each is tested for zeros using the error bound for linear interpolation on zero possible intervals f n is approximated by a cubic polynomial whose roots are then found by the usual sixteenth century formulas

the classification of p local finite groups over the extraspecial group of order p 3 and exponent p in this paper we classify the p local finite groups over p1 2 the extraspecial group of order p3 and exponent p for odd p this study reduces to the classification of the saturated fusion systems over p1 2 which will be characterized by the outer automorphism group the number of open image in new window radical subgroups and the automorphism group of each nontrivial open image in new window radical subgroup as part of this classification we obtain three new exotic 7 local finite groups