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

tensor operators and projection techniques in infinite dimensional representations of semi simple lie algebras polynomial identities satisfied by the infinitesimal generators of a semi simple lie group are employed to construct projection operators which project a tensor product representation v lambda x v v lambda finite dimensional and irreducible v an infinite dimensional representation admitting an infinitesimal character onto a primary subrepresentation such projection operators project out the generalised shift components of a tensor operator acting on an infinite dimensional representation v in direct analogy with the finite dimensional analysis of bracken and green 1971 applications of these methods to evaluating the matrix elements of the generators of the locally compact groups u p q and o p q in unitary discrete irreducible representations are discussed

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

on the structure of the observable algebra of qcd on the lattice we prove that the matter field subalgebra of the observable algebra for qed on a finite lattice is isomorphic to the enveloping algebra of the lie algebra sl 2n c factorized by a certain ideal using this result we give a new proof of the decomposition of the physical hilbert space into charge superselection sectors

representations of supergroups an explicit construction of representations of supergroups is given in terms of direct products of covariant and contravariant fundamental representations the rules of supersymmetrization are characterized by extended young supertableaux this constructive approach leads to explicit transformation properties of higher representations as well as to closed explicit formulas for characters from which other invariants such as dimensions and eigenvalues of all casimir operators can be calculated we have applied this approach so far to the supergroups su n m osp n 2m p n for which we have obtained all the representations constructible as direct products of the fundamental defining representations an argument is presented toward the irreducibility of all these representations

an asymptotic theory for a weakly nonlinear beam equation with a quadratic perturbation in this paper an initial boundary value problem for a weakly nonlinear beam equation with a quadratic nonlinearity will be studied the initial boundary value problem can be regarded as a simple model describing free oscillations of flexible structures like suspension bridges using a two timescales perturbation method an approximation for the solution of this initial boundary problem will be constructed for a class of initial boundary value problems the existence and uniqueness of solutions and the asymptotic validity of approximations on a large timescale are shown it will be shown that for specific values of the beam parameters complicated internal resonances occur

classification of star and grade star representations of c n 1 the two types of and grade representations of the lie superalgebra c n 1 are classified a type 1 irreducible representation is characterized by the single condition s 0 and s being the highest weight and the odd simple root respectively while the type 2 representations are duals of type 1 representations for n 1 only the identity and vector representations of c n 1 are shown to be grade the superalgebra c 2 proves to be an exception and admits several classes of nontrivial irreducible grade representations

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

efficient classical simulation of clifford circuits with nonstabilizer input states we investigate the problem of evaluating the output probabilities of clifford circuits with nonstabilizer product input states first we consider the case when the input state is mixed and give an efficient classical algorithm to approximate the output probabilities with respect to the l 1 norm of a large fraction of clifford circuits the running time of our algorithm decreases as the inputs become more mixed second we consider the case when the input state is a pure nonstabilizer product state and show that a similar efficient algorithm exists to approximate the output probabilities when a suitable restriction is placed on the number of qubits measured this restriction depends on a magic monotone that we call the pauli rank we apply our results to give an efficient output probability approximation algorithm for some restricted quantum computation models such as clifford circuits with solely magic state inputs pauli based computation and instantaneous quantum polynomial time circuits finally we discuss the relationship between pauli rank and stabilizer rank

galloping of tall prismatic structures a two dimensional analysis abstract the galloping of tall prismatic structure with square cross section due to smooth wind flow is considered wind tunnel data for lift and drag coefficients are used to generate polynomials to describe the self excited wind forces the problem is treated as a two degree of freedom one in both the along wind and the across wind directions the two coupled non linear differential equations are solved by the method of multiple scales with first order and second order perturbation analysis comparison of the results with those obtained by using numerical integration indicates the accuracy of the perturbation analysis moreover it is shown that steady state along wind response is small as compared with the across wind response and one can estimate galloping amplitudes from one dimensional analysis

impossibility of classically simulating one clean qubit model with multiplicative error the one clean qubit model or the deterministic quantum computation with one quantum bit model is a restricted model of quantum computing where all but a single input qubits are maximally mixed it is known that the probability distribution of measurement results on three output qubits of the one clean qubit model cannot be classically efficiently sampled within a constant multiplicative error unless the polynomial time hierarchy collapses to the third level t morimae k fujii and j f fitzsimons phys rev lett 112 130502 2014 prltao0031 900710 1103 physrevlett 112 130502 it was open whether we can keep the no go result while reducing the number of output qubits from three to one here we solve the open problem affirmatively we also show that the third level collapse of the polynomial time hierarchy can be strengthened to the second level one the strengthening of the collapse level from the third to the second also holds for other subuniversal models such as the instantaneous quantum polynomial model m bremner r jozsa and d j shepherd proc r soc a 467 459 2011 prlaaz1364 502110 1098 rspa 2010 0301 and the boson sampling model s aaronson and a arkhipov stoc 2011 p 333 we additionally study the classical simulatability of the one clean qubit model with further restrictions on the circuit depth or the gate types

structure of atypical representations of the lie superalgebras sl m n in contrast to lie algebras lie superalgebras contain typical and atypical irreducible representations character formulae for typical representations of classical lie superalgebras have been known for a long time but the structure of atypical representations remained unsolved the author constructs the character formula for atypical representations of the special linear lie superalgebras sl m n

induced module construction for highest weight representations of uq gl n at roots of unity a method is investigated for inducing highest weight representations for the quantum group uq gl n from the canonical subalgebra uq gl n 1 when q is a root of unity we classify the irreps into two types typical and atypical where the former is a generalization of the class of irreps with maximal dimensionality the structures of both the typical and atypical irreps are studied in particular a sufficiency condition is given for an irrep to be typical as examples we consider flat representations induced from a one dimensional representation of the canonical subalgebra and representations induced from vector representation

a class of quadratic deformations of lie superalgebras we study certain graded finite dimensional polynomial algebras of degree 2 which are a special class of deformations of lie superalgebras which we call quadratic lie superalgebras starting from the formal definition we discuss the generalized jacobi relations in the context of the koszul property and give a proof of the poincare birkhoff witt basis theorem we give several concrete examples of quadratic lie superalgebras for low dimensional cases and discuss aspects of their structure constants for the type i class we derive the equivalent of the kac module construction for typical and atypical modules and a related direct construction of irreducible modules due to gould we investigate in detail one specific case the quadratic generalization gl2 n 1 of the lie superalgebra sl n 1 we formulate the general atypicality conditions at level 1 and present an analysis of zero and one step atypical modules for a certain family of kac modules

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

atypical representations for type i lie superalgebras a new modified induced module construction is presented for all finite dimensional irreducible typical and atypical modules of a type i basic classical lie superalgebra the method is illustrated with some low dimensional representations of gl m mod n and all representations of gl 2 mod 1

invariants and reduced matrix elements associated with the lie superalgebra gl m n we construct explicit formulae for the eigenvalues of certain invariants of the lie superalgebra gl m n using characteristic identities we discuss how such eigenvalues are related to reduced wigner coefficients and the reduced matrix elements of generators and thus provide a first step to a new algebraic derivation of matrix element formulae for all generators of the algebra

classical simulation of commuting quantum computations implies collapse of the polynomial hierarchy we consider quantum computations comprising only commuting gates known as iqp computations and provide compelling evidence that the task of sampling their output probability distributions is unli

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

analysis of galloping responses in cantilever beams abstract the method of multiple scales is used to determine a fourth order expansion of the galloping amplitude of cantilever beams the lift wind force is assumed to be given by a general fifth order polynomial in the effective flow incidence the results show that in addition to determining the amplitude of oscillation the odd power terms modify its frequency moreover the even power terms affect both the amplitude and frequency of the response

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

asymptotic approach for non linear periodical vibrations of continuous structures abstract an asymptotic approach for determining periodic solutions of non linear vibration problems of continuous structures such as rods beams plates etc is proposed starting with the well known perturbation technique the independent displacement and frequency is expanded in a power series of a natural small parameter it leads to infinite systems of interconnected non linear algebraic equations governing the relationships between modes amplitudes and frequencies a non trivial asymptotic technique based on the introduction of an artificial small parameter is used to solve the equations an advantage of the procedure is the possibility to take into account a number of vibration modes as examples free longitudinal vibrations of a rod and lateral vibrations of a beam under cubically non linear restoring force are considered resonance interactions between different modes are investigated and asymptotic formulae for corresponding backbone curves are derived

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