ternary 2n point lagrange subdivision schemes ternary 2n point lagrange subdivision schemes with a tension parameter are analyzed it is shown that the resulting curves are c 3 c 4 and c 5 continuous for the values of n 2 3 and 4 respectively the role of the tension parameter in subdivision scheme is demonstrated by a few examples

a moufang loop the exceptional jordan algebra and a cubic form in 27 variables abstract let i be the exceptional 27 dimensional jordan algebra over c its automorphism group is the lie group f4 c and this group is known to have a finite subgroup al where a is a self centralizing elementary abelian of order 27 l sl 3 3 and l normalizes a as an a module i decomposes into a direct sum of 1 dimensional spaces i x which afford the 27 distinct linear characters x a hom a c x these spaces satisfy i x i y i xy let e 2 i 3 there are a basis of i of the form ex for x a and a function g a a f 3 such that exey 2 e x y g x y exy where c x y 0 if x and y are linearly dependent and c x y 1 otherwise identifying a with f 33 we write x x1 x2 x3 and y y1 y2 y3 a function g which has the above properties is g x y x1 x2 x3 x3 y1 y2 x2 x3 y1 x1 y2 y3 the elements ex x a generate the infinite commutative loop l 2 m nex m z n z 3 x a under jordan multiplication the loop s is not moufang but has as quotient a moufang loop m of order 81 and exponent 3 conversely the loop m may be constructed from scratch using g and used to define the jordan algebra i using the formula this gives a new existence proof for a simple 27 dimensional jordan algebra over fields of characteristic not 2 or 3 with a primitive cube root of unity in characteristic 3 we get the group algebra of a we discuss some finite groups associated to m and the lie groups f4 c and 3e6 c and compare the analogous situation with the loop o 16 the cayley numbers and lie groups g2 c and d4 c we also get a new construction of the cubic form in 27 variables whose group is 3e6 c and an easy and natural construction of the exotic 3 local subgroup 31 3 3 sl 3 3

generation of fractal curves and surfaces using ternary 4 point interpolatory subdivision scheme in this paper the generation of fractal curves and surfaces along with their properties using ternary 4 point interpolatory subdivision scheme with one parameter are analyzed the relationship between the tension parameter and the fractal behavior of the limiting curve is demonstrated through different examples the specific range of the tension parameter has also been depicted which provides a clear way to generate fractal curves since the fractal scheme introduces in the paper have more number of control points therefore it gives more degree of freedom to control the shape of the fractal curve

flots et series de taylor stochastiques we study the expansion of the solution of a stochastic differential equation as an infinite sum of iterated stochastic stratonovitch integrals this enables us to give a universal and explicit formula for any invariant diffusion on a lie group in terms of lie brackets as well as a universal and explicit formula for the brownian motion on a riemannian manifold in terms of derivatives of the curvature tensor the first of these formulae contains and extends to the non nilpotent case the results of doss 6 sussmann 17 yamato 18 fliess and normand cyrot 7 krener and lobry 19 and kunita 11 on the representation of solutions of stochastic differential equations

short presentations for three dimensional unitary groups abstract we give a presentation of length o log2 g for the groups g psu3 q this result has applications in recent algorithms to compute the structure of permutation groups and matrix groups

twisted loop groups and their affine flag varieties we develop a theory of affine flag varieties and of their schubert varieties for reductive groups over a laurent power series local field k t with k a perfect field this can be viewed as a generalization of the theory of affine flag varieties for loop groups to a twisted case a consequence of our results is that our construction also includes the flag varieties for kac moody lie algebras of affine type we also give a coherence conjecture on the dimensions of the spaces of global sections of the natural ample line bundles on the partial flag varieties attached to a fixed group over k t and some applications to local models of shimura varieties

on the congruence subgroup problem for anisotropic groups of inner type a_n an let g be an absolutely almost simple simply connected algebraic group of inner form of type a n defined and anisotropic over a number field k let v denote a complete set of inequivalent valuations of k and v f resp infty the subset of non archimedean resp archimedean valuations in v this work deals with the congruence subgroup problem for groups g of inner form of type a n with respect to a particular class of valuations s subset v which we term as d amenable fix a realization g subset mathrm gl n as a k subgroup and set lambda s g k cap gl n mathcal o s where mathcal o s is the ring of s integers in k then we show that if s is d amenable and gamma is any subgroup of finite index in lambda s there exists a finite set s gamma and positive integers r v mid v in s gamma such that gamma contains gamma in lambda s mid gamma in 1 mathfrak p v r v text for v in s gamma the d amenability property holds for any s with v smallsetminus s finite we also show that the set of primes in v in an arithmetic progression is d amenable for any d so that we recover the result of prasad and rapinchuk proc steklov inst math 292 1 216 246 2016

presentations on standard generators for classical groups abstract for each family of finite classical groups and their associated simple quotients we provide an explicit presentation on a specific generating set of size at most 8 since there exist efficient algorithms to construct this generating set in any copy of the group our presentations can be used to verify claimed isomorphisms between representations of the classical group the presentations are available in magma

interpolation through an iterative scheme abstract a new method of interpolation is generated through an iterative scheme on dyadic rationals the process is linear local and produces almost twice differentiable functions a fundamental interpolating function is analyzed error formulas are derived the method is applied for curve fitting

depth zero supercuspidal l packets and their stability in this paper we verify the local langlands correspondence for pure inner forms of unramied p adic groups and tame langlands parameters in general position for each such parameter we explicitly construct in a natural way a nite set l packet of depth zero supercuspidal representations of the appropriate p adic group and we verify some expected properties of this l packet in particular we prove with some conditions on the base eld that the appropriate sum of characters of the representations in our l packet is stable no proper subset of our l packets can form a stable combination our l packets are also consistent with the conjectures of b gross and d prasad on restriction from so2n 1 to so2n 24 these l packets are in general quite large for example sp2n has an l packet containing 2 n representations of which exactly two are generic in fact on a quasi split form eachl packet contains exactly one generic representation for every rational orbit of hyperspecial vertices in the reduced bruhattits building when the group has connected center every depth zero generic supercuspidal representation appears in one of these l packets we emphasize that there is nothing new about the representations we construct they are induced from deligne lusztig representations on subgroups of nite

on the automorphism group of extraspecial 2 groups fix a nonsingular quadratic space u q of dimension 2n over 2 n 1 then either q has witt index n q q or index n 1 2 n q q let e e q and o q be the associated extraspecial 2 group 2 n 1 2 n and full orthogonal group respectively usually one writes e 2 if 1 2 n q q and e 2 otherwise the orthogonal group is written o q 2 n o 2 it is well known that the group of outer automorphisms out e 2 n o q acting on e z e u in the natural way by griess 3 the group extension

a combined ternary 4 point subdivision scheme combined binary 6 point interpolating and approximating subdivision scheme with tension parameters is analyzed it is shown that the resulting curves are c1 c2 interpolating continuous and c1 c2 c3 approximating continuous for different values of tension parameters the role of the tension parameters in subdivision scheme are illustrated using a few examples

subgroup growth and congruence subgroups letk be a global field o its ring of integers g an almost simple simply connected connected algebraic subgroups ofgl m defined overk and g o which is assumed to be infinite let n resp n be the number of all resp congruence subgroups of index at mostn in we show a if char k 0 then i c 1 frac log 2 n log log n leqq log gamma n gamma leqq c 2 frac log 2 n log log n for suitable constantsc1 andc2 under some mild assumptions we also have ii has the congruence subgroup property if and only if log n o log2n iii if is boundedly generated the has the congruence subgroup property this confirms a conjecture of rapinchuk r1 which was also proved by platonov and rapinchuk pr3 b if char k 0 and under somewhat stronger conditions ong then for suitable constantsc3 andc4 c3 log2n log n c4log3n

lattices of minimum covolume in chevalley groups over local fields of positive characteristic in this article we show that if g is a simply connected chevalley group of either classical type of rank bigger than 1 or type e6 and q 9 is a power of a prime number p 5 then g g fq t 1 up to an automorphism has a unique lattice of minimum covolume which is g fq t 1

stationary measures and invariant subsets of homogeneous spaces ii let g be a real lie group be a lattice in g and be a compactly generated closed subgroup of g if the zariski closure of the group ad is semisimple with no compact factor we prove that every orbit closure in g is a nite volume homogeneous space we also establish related equidistribution properties

supercuspidal l packets of positive depth and twisted coxeter elements the local langlands correspondence is a conjectural connection between representations of groups g k for connected reductive groups g over a padic field k and certain homomorphisms langlands parameters from the galois or weil deligne group of k into a complex lie group g which is dual in a certain sense to g and which encodes the splitting structure of g over k more introductory remarks on the local langlands correspondence can be found in 21 wheng gl1 this correspondence should reduce to local abelian class field theory for g gln the langlands correspondence is uniquely determined by local factors 24 and was shown to exist in 23 and 25 so far this correspondence is not completely explicit but much progress has been made in this direction see 9 10 for example for groups other than gln or pgln the theory is much less advanced new phenomena appear arising on the arithmetic side from the difference between conjugacy and stable conjugacy and on the dual side from nontrivial monodromy of langlands parameters this means that a single langlands parameter should determine not just one but a finite set of representations these are the l packets of the title however since local factors have not been defined in general there is no precise characterization of an l packet for general groups one can at present only hope to define finite sets of representations attached to langlands parameters and show that they have properties expected or perhaps unexpected of l packets see 14 chap 3 for some of these properties one is thereby proposing a definition of local factors for the representations in the sets cf 4 chap 3 this paper is a sequel to 14 the aim of both papers is to verify in an explicit and natural way the local langlands correspondence for the simplest kinds of non abelian extensions of k and the simplest kinds of

generators relations and coverings of algebraic groups ii theorem 1 1 let g be the group of rational points of a simply connected simple afine algebraic group defined and split over a fmite j eld of q elements then the schur multiplier m of g is trivial with the following exceptions a zf g is a 4 a 2 a 3 c 2 f 2 or g 4 then h4 is z cyclic of order 2 if g is a 9 b 3 or g 3 it is iz b zf g is a 4 resp d 2 then m is the direct product of z resp z with itself

presentations of finite simple groups a quantitative approach there is a constant such that all nonabelian finite simple groups of rank over with the possible exception of the ree groups have presentations with at most generators and relations and total length at most as a corollary we deduce a conjecture of holt there is a constant such that for every finite simple group every prime and every irreducible module

similarity classes of integral p adic matrices and representation zeta functions of groups of type a2 we compute explicitly dirichlet generating functions enumerating finite dimensional irreducible complex representations of various p adic analytic and adelic profinite groups of type a2 this has consequences for the representation zeta functions of arithmetic groups h k where k is a number field and h is a k form of sl3 assuming that possesses the strong congruence subgroup property we obtain precise uniform estimates for the representation growth of our results are based on explicit uniform formulae for the representation zeta functions of the p adic analytic groups sl3 o and su3 o where o is a compact discrete valuation ring of characteristic 0 these formulae build on our classification of similarity classes of integral p adic 3 3 matrices in gl3 o and gu3 o where o is a compact discrete valuation ring of arbitrary characteristic organising the similarity classes by invariants which we call their shadows allows us to combine the kirillov orbit method with clifford theory to obtain explicit formulae for representation zeta functions in a different direction we introduce and compute certain similarity class zeta functions our methods also yield formulae for representation zeta functions of various finite subquotients of groups of the form sl3 o su3 o gl3 o and gu3 o arising from the respective congruence filtrations these formulae are valid in case that the characteristic of o is either 0 or sufficiently large analysis of some of these formulae leads us to observe p adic analogues of ennola duality

a combined approximating and interpolating ternary 4 point subdivision scheme abstract in this paper a new combined approximating and interpolating ternary 4 point subdivision scheme with multiple parameters is proposed a set of nice properties such as support continuity and polynomial generation are briefly discussed the new combined scheme not only contains a lot of classical ternary schemes as special cases but also generates brand new ternary schemes compared to other approximating subdivision schemes limit curves generated by the new scheme are more consistent with the corresponding control polygons and keep detail features better examples are given to show the effectiveness of the scheme furthermore fractal property is analyzed and fractal curves are also given

deligne s topological central extension is universal abstract we give a purely local proof of the fact that the topological central extension of g k g an absolutely almost simple algebraic group defined and isotropic over a nonarchimedean local field k by the finite group k of roots of unity in k constructed by pierre deligne is a universal topological central extension of g k

a representation of solution of stochastic differential equations we prove that the logarithm of the formal power series obtained from a stochastic differential equation is an element in the closure of the lie algebra generated by vector fields being coefficients of equations by using this result we obtain a representation of the solution of stochastic differential equations in terms of lie brackets and iterated stratonovich integrals in the algebra of formal power series

first eigenvalue of laplace operator on locally symmetric space this paper is concerned with the lower bound for the first positive eigenvalue lambda 1 gamma of the laplace operator bigtriangleup on the space of square integrable functions on some locally symmetric space gamma backslash g k where g so 2n 1 mathbb c k is a maximal compact subgroup of g and gamma is a lattice in g which arises from some imaginary quadratic field it is well known that the casimir operator fancyscript c acts by a scalar lambda pi on every irreducible admissible representation pi of g by diximir s schur lemma in this paper we prove that when n ge 5 there exists a spectral gap between lambda 1 gamma and the infimum lambda 1 g of lambda pi when pi passes through all non trivial irreducible spherical unitary representations of g

on the congruence subgroup problem determination of the metaplectic kernel let f be a global field i e either a number field or a function field in one variable over a finite field and let oo be the set of its archimedean places let g be a finite non empty set of places of f containing oo let o o denote the ring of g integers of f let a denote the ring of g adeles i e the restricted direct product of the completions f v for r eg let f be an absolutely simple simply connected subgroup of sl defined over f recall that a subgroup f of c f is an g arithmetic subgroup if fnsl n o has finite index in f as well as in o fy f c sl n o an g arithmetic subgroup f is a congruence subgroup if there exists a non zero ideal a in o o g such that

a ternary 4 point approximating subdivision scheme in the implementation of subdivision scheme three of the most important issues are smoothness size of support and approximation order our objective is to introduce an improved ternary 4 point approximating subdivision scheme derived from cubic polynomial interpolation which has smaller support and higher smoothness comparing to binary 4 point and 6 point schemes ternary 3 point and 4 point schemes see table 2 the method is easily generalized to ternary 2n 2 point approximating subdivision schemes we choose a ternary scheme because a way to get smaller support is to raise arity and we use polynomial reproduction to get higher approximation order easily 2007 elsevier inc all rights reserved

short presentations for finite groups we conjecture that every finite groupghas a short presentation in terms of generators and relations in the sense that the totallengthof the relations is log g o 1 we show that it suffices to prove this conjecture for simple groups motivated by applications in computational complexity theory we conjecture that for finite simple groups such a short presentation is computable in polynomial time from the standard name ofg assuming in the case of lie type simple groups overgf pm that an irreducible polynomialfof degreemovergf p and a primitive root ofgf pm are given we verify this stronger conjecture for all finite simple groups except for the three families of rank 1 twisted groups we do not handle the unitary groupspsu 3 q 2a2 q the suzuki groupssz q 2b2 q and the ree groupsr q 2g2 q in particular all finite groups g without composition factors of these types have presentations of length o log g 3 for groups of lie type normal or twisted of rank 2 we use a reduced version of the curtis steinberg tits presentation

on a stable family of four point nonlinear subdivision schemes eliminating the gibbs phenomenon abstract the chaikin s scheme presents the interesting property of not producing gibbs phenomenon the problem arises when we need a scheme that provides a higher order of approximation in this case linear schemes are not a good option as they produce gibbs phenomenon close to discontinuities in this article a new four point nonlinear family of subdivision schemes that eliminate the gibbs phenomenon is presented it is based on the linear family of four point subdivision schemes depending on a tension parameter introduced in dyn et al 2005 a simple algebraic transformation leads to an easy way of introducing nonlinearity in the original family of schemes the non interpolatory characteristic of the nonlinear scheme can be modulated just varying the value of the tension parameter results about the stability convergence and the elimination of the gibbs phenomenon are presented some numerical comparisons of the results obtained in the generation of curves are also shown leading to the conclusion that the high order nonlinear schemes are more suitable for this purpose

superrigidity theorems in positive characteristic abstract we generalize the superrigidity theorem of g a margulis 1991 discrete subgroups of semisimple lie groups vii 5 19 springer verlag new york berlin to the case of the more general non reductive target group f we also provide a proof of the cocycle superrigidity theorem in the case of algebraic groups defined over a local field of positive characteristic and the generalization of this proof to the case of the more general target group analogous to our generalization of the superrigidity theorem

asymptotic expansion of stochastic flows we study the asymptotic expansion in small time of the solution of a stochastic differential equation we obtain a universal and explicit formula in terms of lie brackets and iterated stochastic stratonovich integrals this formula contains the results of doss 6 sussmann 15 fliess and normand cyrot 7 krener and lobry 10 yamato 17 and kunita 11 in the nilpotent case and extends to general diffusions the representation given by ben arous 3 for invariant diffusions on a lie group the main tool is an asymptotic expansion for deterministic ordinary differential equations given by strichartz 14

tame component of the schur multipliers of finite groups of lie type let f be a finite field of characteristic p let be an almost f simple simply connected algebraic group defined over f for any f subgroup a of y we shall denote by r f the finite group of f rational points of z and shall consider q z as an r f module with trivial x f action let g f f let j be the p primary component of q z and j be the sum of all primeto p primary components of q z then q z jo j and the schur multiplier h2 g q z is the direct sum of h g j and h g j the subgroup h2 g j is the p primary component of the schur multiplier of g we shall call it the wild component of the schur multiplier and h2 g j which is the sum of prime to p primary components of the schur multiplier the tame component it is a classical result of schur 41 that the tame component of the schur multiplier of sl 2 5 is trivial for any finite field 5 griess has proved 3 1 that the schur multiplier of su 3 5 is trivial we shall use these two results and the tits building of y f to give a new proof of the following theorem of steinberg 6 71 and griess 31

black box exceptional groups of lie type if a black box group is known to be isomorphic to an exceptional simple group of lie type of twisted rank 1 other than any f4 q over a field of known size a las vegas algorithm is given to produce a constructive isomorphism in view of its timing this algorithm yields an upgrade of all known nearly linear time monte carlo permutation group algorithms to las vegas algorithms when the input group has no composition factor isomorphic to any group f4 q or g2 q

unrefined minimal k types for p adic groups via their restriction to compact open subgroups was begun by mautner shalika and tanaka for groups of type ai in contrast to real reductive groups where the representation theory of a maximal compact subgroup k is given in terms of slight modifications to cartan s theory of the highest weight the representation theory of a compact parahoric subgroup is quite complicated there is still no comprehensive theory for classifying the irreducible representa tions in the case of