branching rules for the supergroup su n m from those of su n m the decomposition of representations of supergroups into representations of subgroups is needed in practical applications in this paper we set up and exploit a fruitful one to one correspondence between the lie group branching su n m su n su m u 1 and the supergroup branchings su n m su n su m u 1 and su n1 n2 m1 m2 su n1 m1 su n2 m2 u 1 a simple and useful prescription is discovered for obtaining the su n m branching rules from those of su n m for any representation a large class of examples sufficient for many physical applications we can foresee are explicitly worked out and tabulated

a conjectured character formula for atypical irreducible modules of the lie superalgebra sl m n a formula is used for the character of each finite dimensional irreducible representation of the lie superalgebra sl m n this new formula appears to cover not only typical but also both singly and multiply atypical representations a comparison is made with previous conjectures and the problem of generalising the formula to cover all of the basic classical lie superalgebras is discussed

branching rules for a class of typical and atypical representations of gl m n the irreducible representations of the lie superalgebra gl m n with highest weights of the form 1 2 m are investigated using a recently introduced induced module construction for atypical modules the gl m n gl m n 1 branching rules are obtained and a suitable gel fand tsetlin basis is introduced the class of representations considered includes some multiply atypical irreducible representations of gl m n and all irreducible representations of gl m 1

general eigenvalue formula for casimir invariants of type i quantum superalgebras induced invariant forms and the multiplicity labeling problem are investigated for typical summands of the tensor product of an arbitrary finite dimensional irreducible representation and a typical one for the type i quantum superalgebras the results are applied to obtain a general eigenvalue formula for casimir invariants corresponding to an arbitrary finite dimensional irreducible reference representation

representations of osp 2 1 and the metaplectic representation shift operator techniques are used to treat the irreducible representations of the superalgebra osp 2 1 apart from obtaining the well known gradestar dispin representations which arise when the even part is the compact su 2 algebra the case when the star conditions on the even part are those satisfied by the noncompact su 1 1 algebra is also treated in this case no gradestar representations arise and the star representations are found to consist of the direct sum of two discrete series representations of su 1 1 one of these representations can be realized in terms of functions of a single complex variable and turns out to be a simple example of a metaplectic representation

branching rules for typical and atypical representations of gl n 1 the gl n 1 down arrow gl n gl 1 branching rules are determined for all finite dimensional irreducible typical and atypical representations of gl n mod 1 using a recently introduced induced module construction for atypical modules and confirm those found recently by palev 1988 by another method the lowest weights and characters of the irreducible representations are found and the validity for gl n mod 1 of the character formula conjectured by hughes and king 1987 is confirmed some generalisations and extensions of the method are discussed

temperley lieb algebra and a new integrable electronic model a new correlated electron model is presented which is derived using the quantum inverse scattering method and is thus integrable in one dimension on a periodic lattice the r matrix is constructed from a representation of the temperley lieb algebra the associated hamiltonian describes correlated hopping electron pair hopping and a generalized spin interaction

classification of all star and grade star irreps of gl n 1 a method for inducing nondegenerate forms on irreducible gl m n modules that implies some general results on star and grade star representations is investigated these results are applied to obtain a complete classification in terms of highest weights of the irreducible star and grade star representations of gl n 1 it is demonstrated that while gl n 1 admits a large class of star representations the irreducible grade star representations are comparatively rare moreover for n 2 all grade star irreducible representations are also star representations and for n 2 are atypical the superalgebra gl 2 1 proves to be a special case and admits a two parameter family of four dimensional typical grade star irreducible representations that are not star representations in particular typical grade star irreducible representations of gl n 1 exist only for n 1 2

matrix elements for type 1 unitary irreducible representations of the lie superalgebra gl m n using our recent results on eigenvalues of invariants associated to the lie superalgebra gl m n we use characteristic identities to derive explicit matrix element formulae for all gl m n generators particularly non elementary generators on finite dimensional type 1 unitary irreducible representations we compare our results with existing works that deal with only subsets of the class of type 1 unitary representations all of which only present explicit matrix elements for elementary generators our work therefore provides an important extension to existing methods and thus highlights the strength of our techniques which exploit the characteristic identities

representations of orthosymplectic superalgebras ii young diagrams and weight space techniques for pt i see ibid vol 16 p 473 1983 finite dimensional graded tensor representations of osp m n are enumerated via standard young diagrams and their corresponding highest weight and kac dynkin labels are given a uniform set of conditions on the diagram shape necessary and sufficient for atypicality are presented weight space techniques are used to provide a complete analysis of an atypical representations for the low rank cases osp 2 2 osp 3 2 and osp 4 2

criticality and the λ property for elliptic equations abstract let l denote a nonnegative second order elliptic expression defined on c 0 and let l be an associated self adjoint operator l is said to have the property in iff l for some fixed positive function and any c 0 l is termed subcritical iff a generalized green s function may be defined for l in various applications of these and closely related concepts may be found in the literature it is the main purpose of this article to show that l is subcritical iff l has the property several corollaries and an example are given as consequences of this equivalence

representations of n 2 extended supergravity and unitarity conditions in osp n 4 the structure of unitary irreducible representations of the lie superalgebra osp 2 4 the algebra of n 2 extended supergravity is investigated in detail in particular four new classes of shortened multiplets are found and the complete unitarity conditions are given the shortened multiplets are shown to correspond to atypical infinite dimensional representations finally unitarity conditions for osp n 4 3 n 8 are constructed

irreducible representations of the superalgebras type ii the representation of the orthosymplectic algebras and the other members of their class are built explicitly with simple techniques

large time behavior of the lp norm of schrodinger semigroups abstract letv 0 v c0 rv with v 3 be such that h 1 2 v 0 but for any e 0 1 2 1 e v is not positive we determine the exact rate of divergence of the norm of e th as a map from l to l a number of related problems are discussed

on positive solutions of second order elliptic equations stability results and classification soit p un operateur uniformement elliptique d ordre 2 defini dans un domaine cr n soit p le cone convexe de toutes les solutions positives de l equation pu 0 dans on etudie les proprietes generales de p pour p et generaux

irreducible representations of the exceptional lie superalgebras d 2 1 α the shift operator technique is used to give a complete analysis of all finite and infinite dimensional irreducible representations of the exceptional lie superalgebrasd 2 1 for all cases the star or grade star conditions for the algebra are investigated among the finite dimensional representations there are no star and only a few grade star representations but an infinite class of infinite dimensional star representations is found explicit expressions are given for the doublet representation of d 2 1 the one missing label problem d 2 1 su 2 su 2 su 2 is discussed in detail and solved explicitly

young supertableaux of the basic lie superalgebras among the simple lie superalgebras lsa the basic classical ones are well known together with many of their finite dimensional representations this is particularly true for the typical representations which are known to be completely reducible it is well known that reducible but indecomposable representations of the basic lsa are not completely characterised by their highest weight hw or the eigenvalues of their casimir operators in fact there may be several different finite representations characterised by the same hw such representations are called non typical or atypical the aim of this paper is to establish a complete characterisation of the typical and non typical representations by means of properly defined young supertableaux yst therefore the definition we are going to use for our yst differs slightly from the standard definitions the examples given throughout this paper help the illustration of this new interpretation for the yst which makes possible the complete classification of the indecom posable representations characterised by a generalised notion of hw

diagram and superfield techniques in the classical superalgebras introduces the concept of graded permutation group in the analysis of tensor operators in the classical superalgebras for u m n and su m n irreducible tensor representations correspond to classes of young tableaux with definite graded symmetry type diagram techniques are given for kronecker products dimensions and branching rules the tensor techniques are complemented by the introduction of a superfluid formalism in which u m n and su m n act on polynomial functions over the appropriate superspace such superfluids may admit constraints a general superfield interpolates between the classes of young tableaux which correspond to particular types of constraint the tensor and superfield techniques are illustrated with case studies of su 2 1 and su m 1

structure of positive solutions to δ v u 0 in rn on etudie la structure de toutes les solutions positives d une equation de schrodinger stationnaire ou le potentiel v est suppose etre une fonction a valeur reelle appartenant a l p loc r n avec p n 2 pour n 2 et p 1 pour n 1 on considere une formule de representation explicite pour des solutions positives leurs asymptotes a l infini et le probleme de dirichlet avec une condition aux limites a l infini

classification of finite dimensional unitary irreps for uq gl m n all finite dimensional irreducible unitary representations of the quantum supergroup uq gl m n are classified in terms of their highest weights for generic q 0

adaptive control of a class of non linearly parametrized systems using convexification in this paper we deal with the problem of adaptive control of a class of non linearly parametrized plants the main contribution is to show that through a well chosen reparametrization it is possible to transform the non convex nonlinear parametrization into a convex one a property that is used to design a globally stable adaptive controller

matrix elements and duality for type 2 unitary representations of the lie superalgebra gl m n the characteristic identity formalism discussed in our recent articles is further utilized to derive matrix elements of type 2 unitary irreducible gl m n modules in particular we give matrix element formulae for all gl m n generators including the non elementary generators together with their phases on finite dimensional type 2 unitary irreducible representations which include the contravariant tensor representations and an additional class of essentially typical representations remarkably we find that the type 2 unitary matrix element equations coincide with the type 1 unitary matrix element equations for non vanishing matrix elements up to a phase

a q superdimension formula for irreps of type i quantum superalgebras we derive a closed formula for the q superdimensions of a wide class of irreps including all unitary irreps of the type i quantum superalgebras

classification of irreducible super unitary representations of su p q n in this paper we classify all the irreducible super unitary representations ofsu p q n which can be integrated up to a unitary representation ofs u p q u n a lie group corresponding to the even part ofsu p q n note that a real form of the lie superalgebrasl m n which has non trivial superunitary representations is of the formsu p q n p q m orsu m r s r s n moreover we give an explicit realization for each irreducible super unitary representation using the oscillator representation of an orthosymplectic lie superalgebra

finite and infinite dimensional representations of the orthosymplectic superalgebra osp 3 2 the shift operator technique is used to give a complete analysis of all finite and infinite dimensional irreducible representations of the orthosymplectic superalgebra osp 3 2 for all cases the star or grade star conditions for the algebra are investigated only two finite dimensional representations are grade star representations if the representation space is required to be a graded hilbert space when the even part is so 3 sp 2 r su 2 su 1 1 an infinite class of infinite dimensional star representations is found one of them can be realized in terms of two valued functions of a complex variable this representation reduces to the sum of two metaplectic representations of sp 2 we show that it is precisely this metaplectic representation for osp 3 2 which gives the spin energy eigenstates for the one dimensional harmonic oscillator with spin 1 2 states