parasupercoherent states and color supergroups parabosonic coherent states are illustrated from the displacement operator point of view such a construction requires the use of the green cusson ansatze a specific representation of parastatistics leading to color supergroups a comparison with a previous result is shown through the annihilation operator and the minimum uncertainty contexts the extension to the parasupersymmetric case is also considered

transformation groups for soliton equations iv a new hierarchy of soliton equations of kp type abstract a new approach to soliton equations based on functions or hirota s dependent variables vertex operators and the clifford algebra of free fermions is applied to study a new hierarchy of kadomtsev petviashvili type equations the bkp hierarchy the infinite dimensional orthogonal group acts on the space of bkp functions the sawada kotera equation is obtained as a reduction of bkp its infinitesimal transformations constitute the euclidean lie algebra a2 2

lie structures in parasupersymmetric quantum mechanics i the standard supersymmetrization procedure parasupersymmetric quantum mechanics leads to the construction of new lie structures that are called lie parasuperalgebras they are characterized by double commutation relations between even and odd generators and include as particular substructures the usual lie superalgebras oscillatorlike systems in one spatial dimension are especially exploited in order to put such structures in evidence but n dimensional extensions are easily determined these developments in the standard supersymmetrization procedure will be considered first and the spin orbit coupling one will be considered in a subsequent article j beckers and n debergh j math phys 1991

fermionic approach to soliton equations in this paper we exploit the algebraic structure of the soliton equations and find solutions in terms of fermion particles we show how determinants arise naturally in the fermionic approach to soliton equations we write the function for charged free fermions in terms of determinants examples of how to get soliton rational and dromion solutions from functions for the various soliton equations are given

darboux transformations for the two dimensional toda system binary darboux transformations are constructed for the two dimensional a toda lattice and for a generalized non abelian toda lattice invariance of these transformations under certain reductions produces the b and c toda lattices as well as various periodic reductions corresponding to kac moody algebras the reduction to the d lattice from the non abelian lattice is studied in some detail

a note on parasupersymmetric quantum mechanics of arbitrary order we revisit parasupersymmetric quantum mechanics of arbitrary order and present a set of nontrivial relations which characterizes the most general multilinear part of the associated parasupersymmetric algebra we then show that the formulation of multilinear relations leads immediately to a polynomial of parasupersymmetric hamiltonian in terms of the corresponding parasupercharges the deduction of higher derivative supersymmetric quantum mechanics directly via this parasupersymmetric formulation is discussed the complete degenerate structure of the energy spectrum for parasupersymmetric quantum mechanics of order p is systematically analyzed finally the notion of cyclic symmetry is introduced and the algebra of cyclic charge operators of arbitrary order is developed based on the parasupersymmetric formalism

a note on recent lie parasuperstructures the authors discuss two recently published triple products seen as generalizations in parasupersymmetric quantum mechanics of the usual anticommutator between odd operators in the supersymmetric context they lead to lie parasuperalgebras which are not isomorphic in general

pfaffian formulas and schur q function identities abstract we establish pfaffian analogues of the cauchy binet formula and the ishikawa wakayama minor summation formula each of these pfaffian analogues expresses a sum of products of subpfaffians of two skew symmetric matrices in terms of a single pfaffian by using these pfaffian formulas we give transparent proofs to several identities for schur q functions

symmetric function products and plethysms and the boson fermion correspondence we use the boson fermion correspondence for s and q functions to establish some interesting properties concerning outer products and plethysms of s functions or q functions by power sum symmetric functions the techniques which are developed are also applied to computing the inverse kostka foulkes matrix which is the transition matrix between hall littlewood symmetric functions and s functions in some simple cases

a phase cell cluster expansion for a hierarchical o 3 4 model the formalism developed in a previous paper is applied to yield a phase cell cluster expansion for a hierarchical b model the field is expanded into modes with specific renormatization group scaling properties the present cluster expansion for a vacuum expectation value is formally the natural factorization of each term in the perturbation expansion into the contribution of modes connected to the variables in the expectation via interactions and that of the complementary set the expectation value is thus realized as a sum of contributions due to finite subsets of the modes we emphasize the following additional features 1 partitions of unity are not used 2 there are essentially no cut offs 3 the expansion is developed directly without an initial need to prove an ultraviolet stability bound the most difficult part of the traditional approach our main interest in the present phase cell cluster expansion is founded in the belief that it may be the right vehicle for proving the existence of a non trivial four dimensional field theory

the q deformation of symmetric functions and the symmetric group the q deformation of symmetric functions is introduced leading to q analogues of many well known relationships in the theory of symmetric functions q deformed scalar products are developed and used to define q dependent symmetric functions the symmetric functions commonly associated with the names hall littlewood schur and jack are all special cases of the q deformation of macdonald s 1988 new symmetric functions plambda s t a q analogue of the spin and ordinary characters of sn is given and illustrated by the explicit calculation of examples of q deformed characters the methods used are closely parallel to those of quantum groups

on colour superalgebras in parasupersymmetric quantum mechanics the simplest lie parasuperalgebra psqm 2 is identified as an example with a c 2 s colour superalgebra through simple connections between trilinear and bilinear structure relations but by using green cusson ansatze developed in parastatistics some general properties and conclusions are then obtained and illustrated

the spectra of q state vertex models and related antiferromagnetic quantum spin chains some exactly solvable q state vertex models are investigated the author employs inversion relations to determine directly the spectra of the transfer matrices in the thermodynamic limit by avoiding the more cumbersome bethe ansatz the results are applied to related quantum spin chains of which two families are su q and so q invariant respectively various quantities are calculated e g energy momentum excitations and the correlation length for q 3 the su q invariant chain is the pure biquadratic spin 1 hamiltonian which turns out to be non critical the ground state energy the gap and the correlation length are given

schur s q functions and the bkp hierarchy in this letter we define q shift operators q lambda of hirota derivatives that leads to a simple technique of constructing the hierarchy of bkp soliton equations using schur s q functions

the crum transformation for a third order scattering problem it is well known see for example calogero and degasperis 1982 that the classical darboux theorem darboux 1882 relating solutions of a pair of schrodinger equations may be exploited in the context of inverse scattering theory expressed most simply it is a transformation between a pair of schrodinger equations where the discrete spectrum of the second contains the discrete eigenvalues of the first together with an extra one

wronskian determinants the kp hierarchy and supersymmetric polynomials by using the wronskian representation of the solutions of the bilinear kp hierarchy a connection between hirota derivatives and supersymmetric polynomials is brought to light this correspondence is used in order to give an alternative construction of the hierarchy

spin swapping operator as an entanglement witness for quantum heisenberg spin s systems using the su n representation of the group theory we derive the general form of the spin swapping operator for the quantum heisenberg spin s systems we further prove that such a spin swapping operator is equal to the spin singlet pairing operator under the partial transposition for su 2 invariant states it is shown that the expectation value of the spin swapping operator and its generalizations the permutations can be used as an entanglement witness especially for the formulation of observable conditions of entanglement

hierarchy of the parasupersymmetric hamiltonians in quantum mechanics the simplest parasupersymmetric model found in one dimensional quantum mechanics is considered it contains one bosonic and one parafermionic degree of freedom by successive restriction of the superpotentials the author shows how the algebra of symmetry is simplified some discussion of the general parasupersymmetry algebras is presented

travelling wave solutions of two nonlinear evolution equations by using the g g expansion method abstract by using the g g expansion method proposed recently we give the exact travelling wave solutions of two different types of nonlinear evolution equations in this paper the travelling wave solutions are expressed by three types of functions the computation for the method appears to be easier and faster by general mathematical software

hall littlewood symmetric functions and the bkp equation the connection between solutions of the bkp equation and hall littlewood symmetric functions is utilised in a unified approach to soliton and polynomial solutions this is analogous to the wronskian formulation of the solution of the kp equation as a by product two novel expressions for certain hall littlewood functions in terms of pfaffians are derived

α pfaffian pfaffian point process and shifted schur measure abstract for any complex number and any even size skew symmetric matrix b we define a generalization pf b of the pfaffian pf b which we call the pfaffian the pfaffian is a pfaffian analogue of the determinant studied in t shirai and y takahashi j funct anal 205 2003 414 463 and d vere jones linear algebra appl 111 1988 119 124 it gives the pfaffian at 1 we give some formulas for pfaffians and study the positivity further we define point processes determined by the pfaffian also we provide a linear algebraic proof of the explicit pfaffian expression obtained in s matsumoto correlation functions of the shifted schur measure math co 0312373 for the correlation function of the shifted schur measure

ondelettes and phase cell cluster expansions a vindication y meyer has recently developed a particularly useful o n basis forl2 r d expansions using this basis i e expansions into ondelettes or wavelets have yielded important new results in soft and hard analysis the expansion into ondelettes of a boson scalar field naturally leads to phase cell cluster expansions a formalism already developed by the authors using other related bases adoption of ondelettes expansions into the phase cell program gives improvements of some extant results and excises an early error

phase space localization theorem for ondelettes it is proven that an orthonormal basis of ondelettes can never have exponential localization in both position space and momentum space

a block spin construction of ondelettes part i lemarie functions using block spin assignments we construct anl2 orthonormal basis consisting of dyadic scalings and translates of just a finite number of functions these functions have exponential localization and for even values of a construction parameterm one can make them classcm 1 with vanishing moments up to orderm inclusive such a basis has an important application to phase cell cluster expansions in quantum field theory

k theoretic analogues of factorial schur p and q functions abstract we introduce two families of symmetric functions generalizing the factorial schur p and q functions due to ivanov we call them k theoretic analogues of factorial schur p and q functions we prove various combinatorial expressions for these functions e g as a ratio of pfaffians a sum over set valued shifted tableaux and a sum over excited young diagrams as a geometric application we show that these functions represent the schubert classes in the k theory of torus equivariant coherent sheaves on the maximal isotropic grassmannians of symplectic and orthogonal types this generalizes a corresponding result for the equivariant cohomology given by the authors we also discuss a remarkable property enjoyed by these functions which we call the k theoretic q cancellation property we prove that the k theoretic p functions form a formal basis of the ring of functions with the k theoretic q cancellation property

dimensional reductions of bkp and ckp hierarchies a discussion of dimensional reductions which are not classical symmetry reductions is made for the bkp and ckp hierarchies of integrable evolution equations a novel direct method for testing pfaffian solutions to bilinear identities is presented and applied to these reductions

state space structure and entanglement of rotationally invariant spin systems we investigate the structure of so 3 invariant quantum systems which are composed of two particles with spins j1 and j2 the states of the composite spin system are represented by means of two complete sets of rotationally invariant operators namely by the projections pj onto the eigenspaces of the total angular momentum j and by certain invariant operators qk which are built out of spherical tensor operators of rank k it is shown that these representations are connected by an orthogonal matrix whose elements are expressible in terms of wigner s 6 j symbols the operation of the partial time reversal of the combined spin system is demonstrated to be diagonal in the qk representation these results are employed to obtain a complete characterization of spin systems with j1 1 and arbitrary j2 1 we prove that the peres horodecki criterion of positive partial transposition ppt is necessary and sufficient for separability if j2 is an integer while for half integer spins j2 there always exist entangled ppt states bound entanglement we construct an optimal entanglement witness for the case of half integer spins and design a protocol for the detection of entangled ppt states through measurements of the total angular momentum

symmetric functions and the kp and bkp hierarchies we study the kp hierarchy through its relationship with s functions using results from the classical theory of symmetric functions the plucker equations for the hierarchy are derived from the tau function bilinear identity and are given in terms of composite s functions their connection to the hirota bilinear form of the hierarchy is clarified a novel combinatorial proof is given of the fact that schur polynomials solve the kp hierarchy we show how the analysis can be carried through for the bkp hierarchy in a completely parallel fashion with the s functions replaced by schur q functions

klein gordon propagation of daubechies wavelets scale commensurate decay estimates inside the light cone for the solution of the cauchy problem for the klein gordon equation with a daubechies wavelet as initial data were obtained similar estimates on the solution of the radiation problem with a space time daubechies wavelet as the source were derived

schubert classes in the equivariant cohomology of the lagrangian grassmannian abstract let lg n denote the lagrangian grassmannian parametrizing maximal isotropic lagrangian subspaces of a fixed symplectic vector space of dimension 2n for each strict partition 1 k with 1 n there is a schubert variety x let t denote a maximal torus of the symplectic group acting on lg n consider the t equivariant cohomology of lg n and the t equivariant fundamental class of x the main result of the present paper is an explicit formula for the restriction of the class to any torus fixed point the formula is written in terms of factorial analogue of the schur q function introduced by ivanov as a corollary to the restriction formula we obtain an equivariant version of the giambelli type formula for lg n as another consequence of the main result we obtained a presentation of the ring h t lg n

on quantum deformations of the simplest lie parasuperalgebra quantum deformations of the simplest lie parasuperalgebra psqm 2 are constructed by taking advantage of the well known representations of suq 2 c as a particular result the authors show that the fundamental witten superalgebra sqm 2 cannot be deformed

soliton solutions of the korteweg de vries and the kadomtsev petviashvili equations the wronskian technique the well known soliton solutions of the kadomtsev petviashvili equations are written in terms of determinants of wronskian form by using this compact representation together with the hirota bilinear form of the equations it is demonstrated by elementary algebraic methods that the n soliton solution satisfies the evolution equation and the n and n 1 soliton solutions satisfy the associated backlund transformation the relation of these results to the eigensolutions of the inverse scattering method and to the more usual representation of the n soliton solution is also given

new exact complex traveling wave solutions for 2 1 dimensional bkp equation by using some exact solutions of the auxiliary ordinary differential equation a direct algebraic method is obtained to construct the exact complex solutions for non linear partial differential equations the method is implemented for the 2 1 dimensional bkp equation new exact complex solutions are obtained

lie structures in parasupersymmetric quantum mechanics ii the spin orbit coupling supersymmetrization procedure the spin orbit coupling supersymmetrization procedure is considered in parasupersymmetric quantum mechanics describing oscillatorlike systems the superposition of bosonic and parafermionic degrees of freedom is mainly studied in the cases of two and three spatial dimensions but the n dimensional case is also determined new lie superalgebras and lie parasuperalgebras are pointed out they differ from those already obtained in connection with the standard procedure of supersymmetrization studied in part i of this series j beckers and n debergh j math phys 1991

spin s quantum chains and temperley lieb algebras the authors construct a family of isotropic spin s quantum chains consisting of sums of operators satisfying a temperley lieb algebra exact values for the infinite lattice limit of the ground state energy per site and for the non zero gap to the lowest energy excited state follow from the temperley lieb equivalence with a bethe ansatz soluble xxz model the family of spin chains includes the biquadratic spin 1 model

shifted tableaux and the projective representations of symmetric groups les groupes de representation de s n classes de conjugaison algebres de clifford et la representation spin de base produits induits pour des representations spin une application caracteristique pour des caracteres de spin tableaux decales et q fonctions de schur les caracteres de spin irreductibles de s n et a n une analogue decalee de la regle lr produits tensoriels interieurs

tetrahedra octahedra and cubo octahedra integrable geometry of multi ratios geometric and algebraic aspects of multi ratios m2n are investigated in detail connections with menelaus theorem clifford configurations and maxwell s reciprocal quadrangles are utilized to associate the multi ratios m4 m8 and m8 with tetrahedra octahedra and cubo octahedra respectively integrable maps defined on face centred fcc lattices and irregular lattices composed of the face centres of simple cubic lattices are constructed and related to the discrete kp and bkp equations and the integrable discrete darboux system governing conjugate lattices an interpretation in terms of integrable irregular lattices of slopes on the plane is also given