on some minimal supervarieties of exponential growth abstract in the present paper we investigate minimal supervarieties of given superexponent over fields of characteristic zero we show that any minimal supervariety of finite basic rank is generated by one of the minimal superalgebras introduced by giambruno and zaicev in 2003 furthermore it is proved that any minimal superalgebra whose graded simple components of the semisimple part are simple generates a minimal supervariety finally we state that the same conclusion holds when the semisimple part of a minimal superalgebra has exactly two arbitrary graded simple components

exponential codimension growth of pi algebras an exact estimate abstract letabe an associative pi algebra over a fieldfof characteristic zero by studying the exponential behavior of the sequence of codimensions cn a ofa we prove thatinv a limn c n a always exists and is an integer we also give an explicit way for computing such integer letbbe a finite dimensionalz2 graded algebra whose grassmann envelopeg b satisfies the same identities ofa theninv a inv g b dim c 0 dim c 1 wherec 0 c 1 is a suitablez2 graded semisimple subalgebra ofb

minimal superalgebras generating minimal supervarieties it has been shown that in characteristic zero the generators of the minimal supervarieties of finite basic rank belong to the class of minimal superalgebras introduced by giambruno and zaicev trans am math soc 355 5091 5117 2003 in the present paper the complete list of minimal supervarieties generated by minimal superalgebras whose maximal semisimple homogeneous subalgebra is the sum of three graded simple algebras is provided as a consequence we negatively answer the question of whether any minimal superalgebra generates a minimal supervariety

codimension growth and minimal superalgebras a celebrated theorem of kemer 1978 states that any algebra satisfying a polynomial identity over a field of characteristic zero is pi equivalent to the grassmann envelope g a of a finite dimensional superalgebra a in this paper by exploiting the basic properties of the exponent of a pi algebra proved by giambruno and zaicev 1999 we define and classify the minimal superalgebras of a given exponent over a field of characteristic zero in particular we prove that these algebras can be realized as block triangular matrix algebras over the base field the importance of such algebras is readily proved a is a minimal superalgebra if and only if the ideal of identities of g a is a product of verbally prime t ideals also such superalgebras allow us to classify all minimal varieties of a given exponent i e varieties v such that exp v d 2 and exp u d for all proper subvarieties u of v this proves in the positive a conjecture of drensky 1988 as a corollary we obtain that there is only a finite number of minimal varieties for any given exponent a classification of minimal varieties of finite basic rank was proved by the authors 2003 as an application we give an effective way for computing the exponent of a t ideal given by generators and we discuss the problem of what functions can appear as growth functions of varieties of algebras

cubic unipotent arthur parameters and multiplicities of square integrable automorphic forms l g f g a ld g f g a lcont g f g a into the direct sum of its discrete spectrum and its continuous spectrum the theory of eisenstein series reduces the description of lcont g f g a to that of the discrete spectrum of certain reductive subgroups of g and thus the basic question is the understanding of the discrete spectrum ld g f g a the discrete spectrum has a further orthogonal decomposition

polynomial identities on superalgebras and exponential growth abstract let a be a finitely generated superalgebra over a field f of characteristic 0 to the graded polynomial identities of a one associates a numerical sequence cnsup a n 1 called the sequence of graded codimensions of a in case a satisfies an ordinary polynomial identity such sequence is exponentially bounded and we capture its exponential growth by proving that for any such algebra lim n c n sup a n exists and is a non negative integer we denote such integer by supexp a and we give an effective way for computing it as an application we construct eight superalgebras ai i 1 8 characterizing the identities of any finitely generated superalgebra a with supexp a 2 in the following way supexp a 2 if and only if idsup a idsup ai for some i 1 8 where idsup b is the ideal of graded identities of the algebra b we also compare the superexponent and the exponent see a giambruno m zaicev adv math 140 1998 145 155 of any finitely generated superalgebra

exact solitary wave solutions of nonlinear evolution and wave equations using a direct algebraic method the authors present a systematic and formal approach toward finding solitary wave solutions of nonlinear evolution and wave equations from the real exponential solutions of the underlying linear equations the physical concept is one of the mixing of these elementary solutions through the nonlinearities in the system the emphasis is however on the mathematical aspects i e the formal procedure necessary to find single solitary wave solutions by means of examples it is shown how various cases of pulse type and kink type solutions are to be obtained by this method an exhaustive list of equations so treated is presented in tabular form together with the particular intermediate relations necessary for deriving their solutions the extension of the technique to construct n soliton solutions and indicate connections with other existing methods is outlined

on the ramanujan conjecture and finiteness of poles for certain l functions in this paper we prove a number of general results about automorphic l functions which appear in the constant terms of eisenstein series for an arbitrary quasi split connected reductive algebraic group over a number field these l functions were first introduced by langlands for chevalley groups 20 and through them he was led to make some of his deep conjectures 21 one significance of these l functions is that all the automorphic l functions studied so far are among them there are three general results first we obtain a uniform line of absolute convergence for all of these l functions theorem 5 1 a uniform estimate for the hecke eigenvalues of generic cusp forms on many absolutely simple quasi split connected reductive algebraic groups over number fields corollary 5 4 and an improvement on the best available estimate for the fourier coefficients of maass wave forms corollary 5 5 also follow next we establish a meromorphic continuation and functional equation for each of these l functions theorem 6 1 finally in theorem 6 2 we prove finiteness of poles on the whole complex plane for an important class of these l functions corollaries 6 6 through 6 10 more precisely let g be a quasi split connected reductive algebraic group over a number field f set g g a f where a f is the ring of adeles of f fix a borel subgroup b of g over f and let u be its unipotent radical let p be a maximal f parabolic subgroup of g with p d b in the context of the problems studied here nothing new will be obtained if one drops the maximality condition on p write p mn a levi decomposition n c u and let b u p m and n be the corresponding groups of adelic points for every place v of f let gv g fj similarly we have bv us pv me and nv let x xv be a generic character of u f u cf section 3 then each xv is generic let r ov 7tv be a cusp form on m we shall say 7t is

modulational instability and exact solutions of the nonlinear schrodinger equation coupled with the nonlinear klein gordon equation the modulational instability mi of the coupled nonlinear schrodinger and nonlinear klein gordon equations is investigated it is found that there are a number of possibilities for the mi regions due to the generalized dispersion relation which relates the frequency and wavenumber of the modulating perturbations some exact travelling wave solutions are constructed via the solutions of a 4 model through a simple mapping relation furthermore we present five different types of solutions representing possible final states of modulationally unstable perturbations the profiles of solitary wave structures are displayed for some fixed parameters

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 unitary dual of p adic g_2 in this paper is completly solved the unitarizability problem for the p adic reductive group of type g 2 the unitary dual of p adic g 2 is described in terms of supercuspidal representations of its levi subgroups

a proof of langland s conjecture on plancherel measures complementary series of p adic groups analysis of p adic reductive groups our first result theorem 7 9 proves a conjecture of langlands on normalization of intertwining operators by means of local langlands root numbers and l functions at least when the group is quasi split and the inducing representation is generic assuming two natural conjectures in harmonic analysis of p adic groups we also prove the validity of the conjecture in general theorem 9 5 as our second result we obtain all the complementary series and special representations of quasi split p adic groups coming from rank one parabolic subgroups and generic supercuspidal represen

graded polynomial identities and codimensions computing the exponential growth abstract let g be a finite abelian group and a a g graded algebra over a field of characteristic zero this paper is devoted to a quantitative study of the graded polynomial identities satisfied by a we study the asymptotic behavior of c n g a n 1 2 the sequence of graded codimensions of a and we prove that if a satisfies an ordinary polynomial identity lim n c n g a n exists and is an integer we give an explicit way of computing such integer by proving that it equals the dimension of a suitable finite dimension semisimple g z 2 graded algebra related to a

on regular square integrable representations of p adic groups we study the problem of construction of noncuspidal irreducible square integrable representations of classical p adic groups and questions related to it starting from cuspidal reducibilities in the generalized rank one case we give a construction of regular irreducible square integrable representations of p adic gsp n sp n and so 2 n 1 under an assumption expected to hold in general we show that all such representations should come from this construction further we obtain a number of general results about places where square integrable representations can appear in parabolically induced representation among others we show that in irreducible cuspidal representations of general linear groups only the self contragredient ones play a role in the construction of square integrable representations of the above groups at the end we use our regular square integrable representations to study some basic properties of whittaker models we also show that there exists nondegenerate standard modules of classical groups which contain degenerate irreducible subrepresentations

a characterization of minimal varieties of zp graded pi algebras abstract let f be a field of characteristic zero and p a prime in the present paper it is proved that a variety of z p graded associative pi f algebras of finite basic rank is minimal of fixed z p exponent d if and only if it is generated by an upper block triangular matrix algebra u t z p a 1 a m equipped with a suitable elementary z p grading whose diagonal blocks are isomorphic to z p graded simple algebras a 1 a m satisfying dim f a 1 a m d

on the minimality of algebras with involution abstract in di vincenzo and la scala 2007 1 given a n tuple a 1 a n of finite dimensional simple algebras over a field of characteristic zero a block triangular matrix algebra with involution denoted by r ut a 1 a n was introduced and it was proved that any finite dimensional algebra with involution which is minimal with respect to its exponent is pi equivalent to r for a suitable choice of the algebras a i motivated by a conjecture stated in the same paper here we show that r is minimal when either it is symmetric or n 2

reduction to real infinitesimal character in affine hecke algebras the main result of this paper is to show that the problem of the determination of the unramified unitary dual of a split p adic group is equivalent to the problem of determining the unitary dual of the corresponding graded hecke algebra in bm the authors established this equivalence in the case of iwahori spherical representations under a certain restriction namely it was essential for the infinitesimal character to be real in the terminology of bmj in terms of the langlands deligne lusztig parameters s u p kul the restriction is that s e lg be a purely hyperbolic element the technique used in bm was to combine the notion of the signature of a k character in v with some facts which follow from kl namely that the 2w characters of tempered representations are linearly independent this is essentially true precisely when the infinitesimal character is real for if not s has an elliptic part se such that

a series of exact solutions for coupled higgs field equation and coupled schrodinger boussinesq equation abstract in this paper we consider complex coupled higgs field equation and coupled schrodinger boussinesq equation an algebraic method is applied to construct solitary wave solutions jacobi periodic wave solutions and a range of other solutions of physical interest it is shown that the jacobi elliptic periodic wave solutions exactly degenerate to the soliton solutions at a certain limit condition

minimal varieties of algebras of exponential growth abstract the exponent of a variety of algebras over a field of characteristic zero has been recently proved to be an integer through this scale we can now classify all minimal varieties of given exponent and of finite basic rank as a consequence we describe the corresponding t ideals of the free algebra and we compute the asymptotics of the related codimension sequences verifying in this setting some known conjectures we also show that the number of these minimal varieties is finite for any given exponent we finally point out some relations between the exponent of a variety and the gelfand kirillov dimension of the corresponding relatively free algebras of finite rank

graded polynomial identities on upper block triangular matrix algebras let g be an arbitrary abelian group in the present paper we study the question of whether g graded upper block triangular matrix algebras over a field of characteristic zero are determined up to g graded isomorphism by their g graded polynomial identities we obtain some results for an elementary grading and as a consequence for any grading over an algebraically closed field

new exact solutions for the konopelchenko dubrovsky equation using an extended riccati equation rational expansion method and symbolic computation in this paper based on computerized symbolic computation and a new general ansatz an extended riccati equation rational expansion method is presented to construct multiple exact solutions for nonlinear evolution equations and implemented in a computer algebraic system the validity and reliability of the method are tested by its application to the konopelchenko dubrovsky equation the method can be applied to other nonlinear evolution equations in mathematical physics

on the siegel weil formula for quaternionic unitary groups we extend the siegel weil formula to all quaternion dual pairs applications include the classification problem of skew hermitian forms over a quaternion algebra over a number field and a product formula for the weighted average of the representation numbers of a skew hermitian form by another skew hermitian form

varieties of superalgebras of almost polynomial growth abstract let v gr be a variety of superalgebras and let c n gr v gr n 1 2 be its sequence of graded codimensions such a sequence is polynomially bounded if and only if v gr does not contain a list of five superalgebras consisting of a commutative superalgebra the infinite dimensional grassmann algebra and the algebra of 2 2 upper triangular matrices with trivial and natural z 2 gradings in this paper we completely classify all subvarieties of the varieties generated by these five superalgebras by giving a complete list of finite dimensional generating superalgebras

n fold darboux transformation and solitonic interactions of a variable coefficient generalized boussinesq system in shallow water abstract under consideration in this paper is a variable coefficient generalized boussinesq system for the long weakly nonlinear and weakly dispersive surface waves in shallow water with the aid of symbolic computation n fold darboux transformation n dt is constructed for that system analytic solutions of the system are obtained via the n dt elastic interactions of three bell shaped and periodic bell shaped solitons are obtained fusion interactions and periodic fusion fission interactions of the solitary waves are graphically analyzed which are inelastic

block triangular matrix algebras and factorable ideals of graded polynomial identities abstract we study the graded polynomial identities of block triangular matrix algebras with respect to the grading defined by an abelian group g in particular we describe conditions for the t g ideal of a such algebra to be factorable as a product of t g ideals corresponding to the algebras defining the diagonal blocks moreover for the factorable t 2 ideal of a superalgebra we give a formula for computing its sequence of graded cocharacters once given the sequences of cocharacters of the t 2 ideals that factorize it we finally apply these results to a specific example of block triangular matrix superalgebra

new exact solutions for some coupled nonlinear partial differential equations using extended coupled sub equations expansion method abstract an extended coupled sub equations expansion method is proposed to seek new traveling wave solutions of the coupled nonlinear partial differential equations the generalized schrodinger boussinesq and coupled nonlinear klein gordon schrodinger equations are used to illustrate the validity and the advantages of this method

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

on minimal representations of chevalley groups of type dn en and g2 let g be a simply connected chevalley group of type dn en or g2 in this paper we show that the minimal representation of g is unique for types dn and en and it does not exist for the type g2

on codimension growth of finitely generated associative algebras abstract letabe a pi algebra over a fieldf we study the asymptotic behavior of the sequence of codimensionscn a ofa we show that ifais finitely generated overftheninv a limn c n a always exists and is an integer we also obtain the following characterization of simple algebras ais finite dimensional central simple overfif and only ifinv a dim a

minimal algebras with respect to their exponent abstract given an m tuple a 1 a m of finite dimensional simple algebras we introduce a block triangular matrix algebra with involution denoted as ut a 1 a m where each a i can be embedded as algebra we describe the t ideal of r ut a 1 a m in terms of the ideals t a i and prove that any algebra with involution which is minimal with respect to its exponent is pi equivalent to r for a suitable choice of a 1 a m moreover we show that if m 1 or a i f for all i then r itself is a minimal algebra the assumption for the base field f is characteristic zero