complete branching rules generating function for so 7 contains implies su 2 3 and polynomial basis states a branching rules generating function is given for so 7 contains implies su 2 3 along with its interpretation in terms of basis states and instructions for calculating generator matrix elements the generator matrix elements for the degenerate 0 0 c representations are calculated as an example of this procedure

a general approach to the systematic derivation of so 3 shift operator relations i theory a new technique is established for the construction of relations between products consisting of two three or more so 3 shift operators within the framework of the lie algebras of so 3 tensor operators constructed out of an underlying single boson structure

time space trade offs for reversible computation a reversible turing machine is one whose transition function is 1 1 so that no instantaneous description id has more than one predecessor using a pebbling argument this paper shows that for any varepsilon 0 ordinary multitape turing machines using time t and space s can be simulated by reversible ones using time o t 1 varepsilon and space o s log t or in linear time and space o st varepsilon the former result implies in particular that reversible machines can simulate ordinary ones in quadratic space these results refer to reversible machines that save their input thereby insuring a global 1 1 relation between initial and final ids even when the function being computed is many to one reversible machines that instead erase their input can of course compute only 1 1 partial recursive functions and indeed provide a godel numbering of such functions the time space cost of computing a 1 1 function on such a machine is equal within a small polynomial to the cost of co

the schrodinger propagator for scattering metrics let g be a scattering metric on a compact manifold x with boundary i e a smooth metric giving the interior x the structure of a complete riemannian manifold with asymptotically conic ends an example is any compactly supported perturbation of the standard metric on rn consider the operator h 1 2 v where is the positive laplacian with respect to g and v is a smooth real valued function on x vanishing to second order at iox assuming that g is nontrapping we construct a global parametrix u z w t for the kernel of the schri odinger propagator u t e ith where z w io x and t 0 the parametrix is such that the difference between u and u is smooth and rapidly decreasing both as t iae 0 and as z iae iox uniformly for w on compact subsets of x let r x 1 where x is a boundary defining function for x be an asymptotic radial variable and let w t be the kernel e ir2 2tu t using the parametrix we show that w t belongs to a class of i legendre distributionsi on x i x i r 0 previously considered by hassell vasy when the metric is trapping then the parametrix construction goes through microlocally in the nontrapping part of the phase space we apply this result to determine the singularities of u t f for any tempered distribution f and for any fixed t 0 in terms of the oscillation of f near iox if the metric is nontrapping then we precisely determine the wavefront set of u t f and hence also precisely determine its singular support more generally we are able to determine the wavefront set of u t f for t 0 resp t 0 on the non backward trapped resp non forward trapped subset of the phase space this generalizes results of craig kappeler strauss and wunsch

efficient and exact quantum compression we present a divide and conquer based algorithm for optimal quantum compression decompression using o n log 4n loglogn elementary quantum operations our result provides the first quasi linear time algorithm for asymptotically optimal in size and fidelity quantum compression and decompression we also outline the quantum gate array model to bring about this compression in a quantum computer our method uses various classical algorithmic tools to significantly improve the bound from the previous best known bound of o n 3 for this operation

shift operator techniques for the classification of multipole phonon states xi properties of mixed type quadratic product operators in r 7 expressions connecting r 3 scalar and nonscalar product operators of the type p jl kokl and okl jp jl are constructed within the group r 7

a pair of commuting missing label operators for g su 2 n a general procedure which can lead to a pair of commuting subgroup scalars in the enveloping algebra of a lie group g decomposed in su 2 n is discussed the technique is illustrated by means of three explicit examples

shift operator techniques for the classification of multipole phonon states xii o0l eigenstate and eigenvalue determination in r 7 on account of previously derived relations between quadratic shift operator products in the group g2 part of the eigenvalue spectrum of the scalar shift operator p0l is derived in closed form the corresponding eigenstates which are closely related to the octupole phonon states are defined in terms of angular momentum lowering shift operator actions upon the maximum angular momentum state

the shift operator technique for so 7 in an su 2 3 basis ii applications reduced matrix elements of the tensor operator which forms part of so 7 generator basis are calculated for the three respective classes of so 7 representations v 0 0 v n0 v 0 1 v n0 0 0 v v n0 the calculation is based on the relations between quadratic products of shift operators established in a previous paper by the same authors

the matrix representation of so7 in an su2 3 basis vector coherent state vcs theory is applied to so7 in an su2 3 basis some reduced matrix elements of the tensor operator that forms part of the so7 algebra are derived by using the k matrix technique

semilinear pseudodifferential equations in spaces of tempered ultradistributions abstract we study a class of semilinear elliptic equations on spaces of tempered ultradistributions of beurling and roumieu type assuming that the linear part of the equation is a pseudodifferential operator of infinite order satisfying a suitable ellipticity condition we prove a regularity result in the functional setting above for weak sobolev type solutions

regularity and decay of solutions of nonlinear harmonic oscillators abstract we prove sharp analytic regularity and decay at infinity of solutions of variable coefficients nonlinear harmonic oscillators namely we show holomorphic extension to a sector in the complex domain with a corresponding gaussian decay according to the basic properties of the hermite functions in r d our results apply in particular to nonlinear eigenvalue problems for the harmonic oscillator associated to a real analytic scattering or asymptotically conic metric in r d as well as to certain perturbations of the classical harmonic oscillator

on the spectra of so 3 scalars in the enveloping algebra of su 3 formulas are given that make it possible to calculate the eigenvalues of the two independent so 3 scalars o0l and q0l in the su 3 enveloping algebra

shift operators and the u n multiplicity problem a computationally effective method for decomposing r fold tensor products of irreducible representations of u n in a basis independent fashion is given the multiplicity arising from the tensor decomposition is resolved with the eigenvalues of invariant operators chosen from the universal enveloping algebra generated by the infinitesimal operators of the dual or complementary representation shift operators which commute with the u n invariant operators but not the dual invariant operators are introduced to compute the eigenvectors and eigenvalues of the dual invariant operators algebraically a three fold tensor product of irreducible representations of su 4 is decomposed to illustrate the power and generality of the method

scattering metrics and geodesic flow at infinity any compact manifold with boundary admits a riemann metric on its interior taking the form x 4 dx 2 x 2 h near the boundary where x is a boundary defining function and h is a smooth symmetric 2 cotensor restricting to be positive definite and hence a metric h on the boundary the scattering theory associated to the laplacian for such a scattering metric was discussed by the first author and here it is shown as conjectured that the scattering matrix is a fourier integral operator which quantizes the geodesic flow on the boundary for the metric h at time to prove this the poisson operator of the associated generalized boundary problem is constructed as a fourier integral operator associated to a singular legendre manifold

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

quantum computational networks the theory of quantum computational networks is the quantum generalization of the theory of logic circuits used in classical computing machines quantum gates are the generalization of classical logic gates a single type of gate the universal quantum gate together with quantum unit wires is adequate for constructing networks with any possible quantum computational property

a general approach to the systematic derivation of so 3 shift operator relations ii applications the technique reported on in the preceding paper is applied to construct shift operator relations in the su 3 and so 5 lie algebras comments are made concerning the integrity basis for so 3 scalar operators appearing in their respective enveloping algebras

on the radius of spatial analyticity for semilinear symmetric hyperbolic systems abstract we study the problem of propagation of analytic regularity for semi linear symmetric hyperbolic systems we adopt a global perspective and we prove that if the initial datum extends to a holomorphic function in a strip of radius width e 0 the same happens for the solution u t for a certain radius e t as long as the solution exists our focus is on precise lower bounds on the spatial radius of analyticity e t as t grows we also get similar results for the schrodinger equation with a real analytic electromagnetic potential

pointwise decay and smoothness for semilinear elliptic equations and travelling waves abstract we derive sharp decay estimates and prove holomorphic extensions for the solutions of a class of semilinear nonlocal elliptic equations with linear part given by a sum of fourier multipliers with finitely smooth symbols at the origin applications concern the decay and smoothness of travelling waves for nonlinear evolution equations in fluid dynamics and plasma physics

shift operator techniques for the classification of multipole phonon states viii r 5 su 2 su 2 reduction for obtaining quadrupole o0l eigenvalues with the aid of previously derived expressions for the reduced matrix elements of the r 5 generators which are obtained by considering r 5 su 2 su 2 reduction a method is set up to evaluate analytical expressions for the eigenvalues of the quadrupole scalar shift operator o0l explicit expressions are listed for states with angular momenta l 2v k k 0 up to 13

the so 7 polynomial basis for symmetric representations a polynomial basis is derived for the symmetric irreducible representations of the group so 7 the reduction of so 7 into su 2 3 is considered the so 7 generators not belonging to su 2 3 are grouped into a bispinor vector of which matrix elements are calculated an explicit expression for the state vector is given

the theory of angular momenta and higher su n symmetries the authors discuss interrelations between generating invariants gi for the reduction pi mi 1 d pi to d on 1 of the m fold tensor product of su n irreducible representations irreps d pi and polynominal bases of the su m irreps d on 1j om n 1 ok 0 0 0 k times a realisation of the su m irrep d oj om 3 bases is given in terms of gi for su 2 coupling wigner coefficients as a byproduct an expression is obtained for su 2 6 j symbols in terms of only two wigner coefficients the authors also discuss some possibilities of the analysis involved in solving the wigner biedenharn problem construction of orthonormal sets of the wigner coefficients for su n groups n or 3

nonscalar extension of shift operator techniques for su 3 in an o 3 basis iii shift operators of second degree in the tensor components shift operators qkl 2 k 2 of second degree in the tensor components q 2 2 are constructed relations connecting quadratic shift operator products of the type ojl k qkl or qkl j ojl and of the type qjl k qkl are derived the usefulness of these relations is demonstrated by the example of the ool and qol eigenvalue calculation for various irreducible respresentations p q of su 3

fractional derivative estimates in gevrey spaces global regularity and decay for solutions to semilinear equations in rn abstract we propose a unified functional analytic approach to study the uniform analytic gevrey regularity and the decay of solutions to semilinear elliptic equations on r n first we develop a fractional calculus for nonlinear maps in banach spaces of l p based gevrey functions 1 h p s r n regularity with s scr 0 depending on the nonlinearity next we investigate the type of decay polynomial or exponential of the derivatives of solutions to semilinear elliptic equations provided they decay a priori slowly as o x x for some small 0 the restrictions involved in our results are optimal in particular given a hyperplane l we construct 2d 2 strongly singular solutions locally in h p s r n for s

regularity in time of solutions to nonlinear schrodinger equations abstract in this paper we consider the regularity of solutions to nonlinear schrodinger equations nls i t u 1 2 u f u u t x r r n u 0 x r n where f is a polynomial of degree p with complex coefficients we prove that if the initial function is in some gevrey class then there exists a positive constant t such that the solution u of nls is in the gevrey class of the same order as in the initial data in time variable t t t 0 in particilar we show that if the initial function has an analytic continuation on the complex domain a 1 a 2 z c n z j x j iy j x j a 2 tan x j y j a 2 tan x j j 1 2 n a 2 0 see fig 1 where 0 1 a 1 a 1 t and such that the solution u of nls is analytic in time variable t t t 0 and has an analytic continuation on z 0 t i arg z 0 t t where sin a 1 1 2 a 1 2 a 2 3 a 2 formula 1 r when x r

the shift operator technique for so 7 in an su 2 3 basis i theory shift operators for the chain of lie groups so 7 su 2 3 are constructed and several properties are examined quite a number of relations connecting quadratic products of shift operators are built up finally the connection between matrix elements of the shift operators and reduced matrix elements of the tensor operator which forms part of the so 7 generator basis is explained

su 2 su 2 shift operators and representations of so 5 su 2 su 2 shift operators analogous to the su 2 shift operators developed and used by the author for the classification and analysis of representations of lie algebras in an su 2 or so 3 basis are obtained for the su 2 su 2 lie algebra in the case where one has an additional set of operators forming an irreducible four dimensional tensor representation of su 2 su 2 the shift operators obtained are used to treat the representations of so 5 in an su 2 su 2 basis

the basis for the symmetric irreducible representations of so 9 an explicit basis is constructed for the symmetric irreducible representation irrep of su 9 so 9 so 5 su1 2 su2 2 it is also indicated how good angular momentum states can be constructed the techniques used are based on the well known tensor algebra for the infinitesimal generators of simple lie groups

nonscalar extension of shift operator techniques for su 3 in an o 3 basis ii applications in a preceding paper relations have been derived which connect nonscalar quadratic shift operator products here the extreme usefulness of these relations is demonstrated by the example of the o0l eigenvalue calculation for the cases l p i i 0 1 2 3 and 4 where p q is any su 3 representation for the first time a case of threefold l degeneracy is completely solved in a pure analytical way

on a class of spinor representations of so 7 the reduction of the irreducible representations v 0 0 1 of so 2n 1 with respect to su 2 su 2 so 2n 3 is considered for the n 3 case all the reduced matrix elements of the so 7 generators in the su 2 2 so 3 basis are calculated with the use of recursion relations

wave packet analysis of schrodinger equations in analytic function spaces abstract we consider a class of linear schrodinger equations in r d with analytic symbols we prove a global in time integral representation for the corresponding propagator as a generalized gabor multiplier with a window analytic and decaying exponentially at infinity which is transported by the hamiltonian flow we then provide three applications of the above result the exponential sparsity in phase space of the corresponding propagator with respect to gabor wave packets a wave packet characterization of fourier integral operators with analytic phases and symbols and the propagation of analytic singularities

so 2n 1 in an so 2n 3 su 2 su 2 basis ii detailed study of the symmetric representations of the so 7 group for pt i see ibid vol 15 p 2665 76 1982 the so 7 down arrow su 2 3 reduction is analysed in detail explicit forms for the different su 2 generators and the remaining ones are derived using the results of the previous paper basis states describing the symmetric irreducible unitary representations are introduced expressions for the reduced matrix elements of the occurring generators with respect to these basis states are determined

a pair of commuting scalars for g 2 su 2 su 2 g 2 su 2 su 2 is a two missing labels problem and therefore in order to give a complete and orthogonal specification of states of irreducible representations of g 2 in an su 2 su 2 basis one needs to find a pair of commuting hermitian operators which are scalar with respect to the su 2 su 2 subalgebra a theorem due to peccia and sharp states that there are apart from the lie algebra invariants twice as many functionally independent scalars as missing labels here two commuting su 2 su 2 scalars are obtained both of sixth order in the g 2 basis elements they are in fact combinations of five scalars of different tensorial types indicating that the functionally independent ones are in general insufficient to provide the lowest order commuting scalars an expression for the sixth order invariant of g 2 is also obtained

some group theoretical aspects of the so0 1 4 invariant theory some properties of unitary representations of the de sitter group so0 1 4 are considered in the case of spin zero and positivity of the mass operator these representations involve a decomposition which ensures the fulfilment of the cluster separability condition and allows for the conventional interpretation of the de sitter analogues of momentum and boost operators the decomposition of the tensor product of representations belonging to the principal series contains the representations of the principal series with all masses under natural assumptions concerning the interaction operators there are no bound states in the theory which is due to the universal de sitter antigravity

so 2n 1 in an so 2n 3 su 2 su 2 basis i reduction of the symmetric representations the branching rule for the reduction of symmetric irreducible unitary representations iur of the simple lie group so 2n 1 into iur of its maximal subgroup so 2n 3 x su 2 x su 2 is established for all n or 3 after the particular case n 3 is analysed in detail a general proof is presented which is valid for all n or 3 all branching rules n 3 4 can be summed up in one formula also a dimension verification is carried out the generators of so 2n 1 not belonging to the semi simple subgroup can be combined into a mixed tensor spinor representation with respect to the simple groups which occur in the direct product the precise nature of that representation is indicated and discussed

finiteness of physics and its possible consequences the modular analog of representations of the so 1 4 algebra for a system of two spinless particles is considered in the framework of approach proposed by the author earlier in which physical systems are described by the elements of a linear space over a finite field and operators of physical quantities by linear operators in this space the eigenvalues of the free mass operator and corresponding eigenvectors are found it is shown that if the finite field under consideration satisfies some properties then the particles necessarily interact with each other the corresponding interaction is universal since it is fully defined by the masses of particles and characteristics of the finite field the formula for the kernel of the interaction operator is derived but since the kernel is defined by the sum over the galois field and this sum cannot be calculated explicitly then it cannot be determined how far the above interaction manifests itself in the usual conditions

the symmetric irreducible representations of so7 in su2 3 basis the symmetric irreducible representations of so7 in su2 3 basis are constructed some reduced matrix elements and closed expressions of simple isoscalar factors for so7 contains implies su2 3 are obtained finally an example is given to show how to obtain physical states from these mathematical ones

nonscalar extension of shift operator techniques for su 3 in an o 3 basis i theory a set of relations is set up which connect quadratic products of the shift operators okl k 0 1 2 which are nonscalar with respect to the o 3 subgroup of su 3 the usefulness of these relations is illustrated by the calculation of the eigenvalues of the scalar shift operator o0l for various irreducible representations p q of su 3

totally symmetric irreducible representations of the group so 6 in the principal so 3 subgroup basis explicit matrix elements are found for the generators of the group so 6 in an arbitrary totally symmetric irreducible representation using the physical principal so 3 subgroup in the chain so 6 so 5 so 3 the internal one missing label problem is solved through the definition of intrinsic states associated to the su 2 su 2 subgroup in the chain so 6 so 5 su 2 su 2 and out of which is projected a complete set of states in the physical basis by integrations over the physical rotation group manifold the matrix elements of the so 6 generators in the su 2 su 2 basis are themselves obtained by the intermediate use of an su 2 su 2 u 1 basis the latter group being a subgroup of so 6 but not of so 5

modular representations as a possible basis of finite physics the approach in which physical systems are described by the elements of a linear space over a finite field and operators of physical quantities by linear operators in this space is discussed the mathematical formulation of the correspondence between such a description and the conventional one is given for a case when the characteristic of the finite field is sufficiently large the above correspondence is considered in the examples of finite analogs of representations of so 2 3 so 1 4 and osp 1 4 algebras finite analogs of representations of infinite dimensional algebras are briefly discussed

super exponential decay and holomorphic extensions for semilinear equations with polynomial coefficients abstract we show that all eigenfunctions of linear partial differential operators in r n with polynomial coefficients of shubin type are extended to entire functions in c n of finite exponential type 2 and decay like exp z 2 for z in conic neighbourhoods of the form im z re z we also show that under semilinear polynomial perturbations all nonzero homoclinics keep the super exponential decay of the above type whereas a loss of the holomorphicity occurs namely we show holomorphic extension into a strip z c n im z t for some t 0 the proofs are based on geometrical and perturbative methods in gelfand shilov spaces the results apply in particular to semilinear schrodinger equations of the form u x 2 u u f x u u our estimates on homoclinics are sharp in fact we exhibit examples of solutions of with super exponential decay which are meromorphic functions the key point of our argument being the celebrated great picard theorem in complex analysis

on the spectrum of a third order so 3 scalar in the enveloping algebra of so 6 with the aid of previously derived expressions for the so 3 reduced matrix elements of the so 6 generators which were obtained by considering an intermediate so 6 su 2 su 2 reduction a method is set up to evaluate analytic expressions for the eigenvalues of a third order scalar operator belonging to the integrity basis of so 3 scalar operators in the enveloping algebra of so 6

irreducible representations of the central extension of sl 2 λt2 using shift operator techniques a classification is given of the irreducible star representations of the central extension algebra c sl 2 t2 it is found to possess two generic series of such representations together with an isolated representation which is just the metaplectic representation of sl 2 this is the only representation it possesses in common with the superalgebra osp 2 1

holomorphic extension of solutions of semilinear elliptic equations abstract we prove sharp analytic estimates and holomorphic extensions in sectors of c d for the solutions of a wide class of semilinear elliptic differential and pseudodifferential equations in r d improving earlier results where the extension was shown for a strip moreover we derive exponential decay estimates for such extended solutions the results presented apply in particular to solitary wave solutions of many classical nonlinear evolution equations as kdv type long wave type and schrodinger equations