gaudin models with irregular singularities abstract we introduce a class of quantum integrable systems generalizing the gaudin model the corresponding algebras of quantum hamiltonians are obtained as quotients of the center of the enveloping algebra of an affine kac moody algebra at the critical level extending the construction of higher gaudin hamiltonians from b feigin et al 1994 17 to the case of non highest weight representations of affine algebras we show that these algebras are isomorphic to algebras of functions on the spaces of opers on p 1 with regular as well as irregular singularities at finitely many points we construct eigenvectors of these hamiltonians using wakimoto modules of critical level and show that their spectra on finite dimensional representations are given by opers with trivial monodromy we also comment on the connection between the generalized gaudin models and the geometric langlands correspondence with ramification

yang s system of particles and hecke algebras the graded hecke algebra has a simple realization as a certain algebra of operators acting on a space of smooth functions this operator algebra arises from the study of the root system analogue of yang s system of n particles on the real line with delta function potential it turns out that the spectral problem for this generalization of yang s system is related to the problem of finding the spherical tempered representations of the graded hecke algebra this observation turns out to be very useful for both these problems application of our technique to affine hecke algebras yields a simple formula for the formal degree of the generic iwahori spherical discrete series representations

periodic integrable systems with delta potentials in this paper we study root system generalizations of the quantum bose gas on the circle with pair wise delta function interactions the underlying symmetry structures are shown to be governed by the associated graded algebra of cherednik s suitably filtered degenerate double affine hecke algebra acting by dunkl type differential reflection operators we use gutkin s generalization of the equivalence between the impenetrable bose gas and the free fermi gas to derive the bethe ansatz equations and the bethe ansatz eigenfunctions

projective modules and involutions let g be a finite group and let t g t2 1 then is a g set under conjugation let k be an algebraically closed field of characteristic 2 it is shown that each projective indecomposable summand of the g permutation module k is irreducible and self dual whence it belongs to a real 2 block of defect zero this together with the fact that each irreducible kg module that belongs to a real 2 block of defect zero occurs with multiplicity 1 as a direct summand of k establishes a bijection between the projective components of k and the real 2 blocks of g of defect zero

connection between yangian symmetry and the quantum inverse scattering method the quantum nonlinear schrodinger model with two component fermions exhibits a yangian symmetry when considered on an infinite interval we construct the generators of the yangian using dunkl operators we show an equivalence between the monodromy matrix constructed from the dunkl operators and that of the quantum inverse scattering method under the yangian algebra the space of states with a fixed particle number forms a tensor product representation of fundamental representations

the q deformed knizhnik zamolodchikov bernard heat equation we introduce a q deformation of the genus one sl 2 knizhnik zamolodchikov bernard heat equation we show that this equation for the dependence on the moduli of elliptic curves is compatible with the qkzb equations which give the dependence on the marked points

sofic systems and encoding data techniques of symbolic dynamics are applied to prove the existence of codes suitable for certain input restricted channels this generalizes the earlier work of adler coppersmith and hassner on the same problem

state splitting for variable length graphs corresp the state splitting algorithm of adler coppersmith and hassner for graphs with edges of fixed length is extended to graphs with edges of variable lengths this has the potential to improve modulation code construction techniques although the ideas of state splitting come from dynamical systems completely graph theoretic terms are used

on the harishchandra homomorphism for infinite dimensional lie algebras a natural generalization of the finite dimensional semisimple lie algebras to the infinite dimensional case is the kac moody lie algebra g a or more generally the contragredient lie algebra associated to an arbitrary square matrix a 9a 111 the category a introduced in la b to study the representations of finite dimensional semisimple lie algebras was extended 9b to the case of kac moody lie algebras in lo kac and kazhdan describe the irreducible subquotients of verma modules using this a decomposition of the category b was obtained in 3 in both 3 and lo the center 3 of the universal enveloping algebra plays no role since there is no analog to the theory of central characters in the infinite dimensional setup let g g a be the kac moody lie algebra associated to a symmetrizable generalized cartan matrix a and let h h a be the abelian diagonalizable subalgebra of g a in this article we introduce alalgebra i u g a of operators on the category p and a surjectio2 i of onto u h the weyl group w acts on u h and the restriction of 3 to the cznter 3 of li maps into the invariants u eqw in section 3 we prove that 3 ij is injective and that 3 injects into 3 i 3 4 3 if g a is infinite dimensional then the existence of the casimir operator 5 9b implies that i j f 3 and hence it follows that the harishchandra homomorphism is never surjective in section 4 we deal with the case of affme kac moody lie algebras using a theorem of hochschild and mostow 6 we deduce that u h w is the polynomial algebra on two generators this disproves the conjecture in 3 that the image of the harishchandra homomorphism separ es the weyl group orbits further we show that the restriction of p to 3 is an isomorphism onto u h w

variable length state splitting with applications to average runlength constrained arc codes a new class of constrained systems average runlength constraints arcs is defined by requiring that the sum of n consecutive run lengths be bounded above by a linear function of n in particular the running average runlength of every sequence in the system is bounded above by a constant a general result is given on the capacity of arc systems the state splitting algorithm is then improved for variable length graphs this is then applied to obtain high fixed rate codes from the free binary source to arc systems as an example a rate 1 2 d k

runlength codes from source codes a class of binary runlength codes also known as d k codes is analyzed these codes are developed by constructing a lossless source code that maps runlengths into unconstrained binary sequences the source code is constructed for the maxentropic distribution on runlengths the inverse of the source code which outputs runlengths guided toward the ideal maxentropic distribution is the d k code four types of source codes are investigated for this purpose huffman enumerative variable length to block and elias or arithmetic the rates of the codes are each proven to converge to the capacity with increasing complexity the codes are not state dependent and are variable rate except for the fixed rate enumerative code a combined source d k code is presented that is based on the arithmetic code

extended trigonometric cherednik algebras and nonstationary schrodinger equations with delta potentials we realize an extended version of the trigonometric cherednik algebra as affine dunkl operators involving heaviside functions we use the quadratic casimir element of the extended trigonometric cherednik algebra to define an explicit nonstationary schrodinger equation with delta potential we use coordinate bethe ansatz methods to construct solutions of the nonstationary schrodinger equation in terms of generalized bethe wave functions it is shown that the generalized bethe wave functions satisfy affine difference knizhnik zamolodchikov equations as functions of the momenta the relation to the vector valued root system analogs of the quantum bose gas on the circle with delta function interactions is indicated

quantum knizhnik zamolodchikov equations and affine root systems quantum difference knizhnik zamolodchikov equations s1 fr are generalized for ther matrices from ch1 with the arguments in arbitrary root systems and their formal counterparts in particular qkz equations with certain boundary conditions are introducted the self consistency of the equations from fr and the cross derivative integrability conditions for ther matrix kz equations from ch2 are obtained as corollaries a difference counterpart of the quantum many body problem connected with macdonald s operators is defined as an application

projective indecomposable modules scott modules and the frobenius schur indicator abstract let be a principal indecomposable character of a finite group g in characteristic 2 the frobenius schur indicator of is shown to equal the rank of a bilinear form defined on the span of the involutions in g moreover if the principal indecomposable module corresponding to affords a quadratic geometry then 0 this result is used to prove a more precise form of a theorem of benson and carlson on the existence of scott components in the endomorphism ring of an indecomposable g module in case the module affords a g invariant symmetric form

higgs bundles gauge theories and quantum groups the appearance of the bethe ansatz equation for the nonlinear schrodinger equation in the equivariant integration over the moduli space of higgs bundles is revisited we argue that the wave functions of the corresponding two dimensional topological u n gauge theory reproduce quantum wave functions of the nonlinear schrodinger equation in the n particle sector this implies the full equivalence between the above gauge theory and the n particle sub sector of the quantum theory of the nonlinear schrodinger equation this also implies the explicit correspondence between the gauge theory and the representation theory of the degenerate double affine hecke algebra we propose a similar construction based on the g g gauged wzw model leading to the representation theory of the double affine hecke algebra

the method of poles a coding method for constrained channels a method for solving coding problems involving channels restricted by finite state constraints is presented the method was introduced by p a franaszek 1969 and is based on the computation of a set of principal states it is proved that state independent decoding is guaranteed the main result is that the transducer constructed using the method is a local automaton in its output the method is also related to recent work in symbolic dynamics by r l adler 1987 b marcus 1985 and others typical applications are coding methods for magnetic storage

involutions cohomology and metabolic spaces abstract we prove that over a field of characteristic two the involution module of a finite group allows a natural splitting into a direct sum of cohomology modules

generalized korteweg de vries equation induced from position dependent effective mass quantum models and mass deformed soliton solution through inverse scattering transform we consider one dimensional stationary position dependent effective mass quantum model and derive a generalized korteweg de vries kdv equation in 1 1 dimension through lax pair formulation one being the effective mass schrodinger operator and the other being the time evolution of wave functions we obtain an infinite number of conserved quantities for the generated nonlinear equation and explicitly show that the new generalized kdv equation is an integrable system inverse scattering transform method is applied to obtain general solution of the nonlinear equation and then n soliton solution is derived for reflectionless potentials finally a special choice has been made for the variable mass function to get mass deformed soliton solution the influence of position and time dependence of mass and also of the different representations of kinetic energy operator on the nature of such solitons is investigated in detail the remarkable features of such solitons are demonstrated in several interesting figures and are contrasted with the conventional kdv soliton associated with constant mass quantum model

hamming distance preserving mappings and trellis codes with constrained binary symbols the transformation of a linear convolutional code into a run length constrained or balanced trellis code with the same or larger free distance is investigated the transformation involves a hamming distance preserving mapping of the set of unconstrained binary symbols of the convolutional code onto a set of suitably constrained symbols simple tests to determine if these mappings can exist and a tree search algorithm for finding such mappings are presented

calculation of norms of bethe wave functions a class of two dimensional completely integrable models of statistical mechanics and quantum field theory is considered eigenfunctions of the hamiltonians are known for these models norms of these eigenfunctions in the finite box are calculated in the present paper these models include in particular the quantum nonlinear schrodinger equation and the heisenbergxxz model

differential difference operators associated to reflection groups there is a theory of spherical harmonics for measures invariant under a finite reflection group the measures are products of powers of linear functions whose zero sets are the mirrors of the reflections in the group times the rotation invariant measure on the unit sphere in rn a commutative set of differential difference operators each homogeneous of degree 1 is the analogue of the set of first order partial derivatives in the ordinary theory of spherical harmonics in the case of r2 and dihedral groups there are analogues of the cauchy riemann equations which apply to gegenbauer and jacobi polynomial expansions the analysis of orthogonality structures for polynomials in several variables is a problem of vast dimensions this paper is part of an ongoing program to establish a workable theory for one particular class the underlying structure is based on finite coxeter groups these are finite groups acting on euclidean space generated by reflections in the zero sets of a collection of linear functions the roots the weight functions for the orthogonality are products of powers of these linear functions restricted to the surface of the unit sphere in addition the weight function is required to be invariant under the action of the group the resulting theory has strong similarities to the theory of spherical harmonics this was established in previous papers of the author 3 4 5 most notably a homogeneous polynomial is orthogonal to all polynomials of lower degree if and only if it is annihilated by a certain second order differential difference operator ordinary partial differentiation acts as an endomorphism on ordinary harmonic functions the use of such operators and their adjoints leads to recurrence formulas and orthogonal decompositions for harmonic polynomials in this paper we construct a commutative set of first order differentialdifference operators associated to the second order operator previously mentioned received by the editors june 1 1987 1980 mathematics subject classification 1985 revision primary 33a45 33a65 20h15 secondary 20c30 42c10 51f15

on the involution module of gln 2f for any group g the set of involutions i in g that is the set of group elements that have order two forms a g set under conjugation the corresponding kg permutation module ki is the involution module of g here k is an algebraically closed field of characteristic two in this paper we discuss aspects of the involution module of the general linear group gln 2f we determine almost all components of this module furthermore we present a vertex and the green correspondent of each component

codes and local constraints etude du probleme de l existance et de la construction de codes dont les elements des mots satisfont certaines contraintes concernant les sequences de lettres qu ils peuvent contenir l equivalence entre les notions de maximalite et de completude pour ces codes est etudiee

on the construction of bounded delay encodable codes for constrained systems we present a new technique to construct sliding block modulation codes with a small decoding window our method which involves both state splitting and look ahead encoding crucially depends on a new local construction method for bounded delay codes we apply our method to construct several new codes all with a smaller decoding window than previously known codes for the same constraints at the same rate

elliptic quantum many body problem and double affine knizhnik zamolodchikov equation the elliptic matrix quantum olshanetsky perelomov problem is introduced for arbitrary root systems by means of an elliptic version of the dunkl operators its equivalence with the double affine generalization of the knizhnik zamolodchikov equation in the induced representations is established

time varying maximum transition run constraints maximum transition run mtr constrained systems are used to improve detection performance in storage channels recently there has been a growing interest in time varying mtr tmtr systems after such codes were observed to eliminate certain error events and thus provide high coding gain for enpr4 channels for n 2 3 in this work tmtr constraints parameterized by a vector whose coordinates specify periodically the maximum runlengths of 1 s ending at the positions are investigated a canonical way to classify such constraints and simplify their minimal graph presentations is introduced it is shown that there is a particularly simple presentation for a special class of tmtr constraints and explicit descriptions of their characteristic equations are derived new upper bounds on the capacity of tmtr constraints are established and an explicit linear ordering by capacity of all tight tmtr constraints up to period 4 is given for mtr constrained systems with unconstrained positions it is shown that the set of sequences restricted to the constrained positions yields a natural tmtr constraint using tmtr constraints a new upper bound on the tradeoff function for mtr systems that relates the density of unconstrained positions to the maximum code rates is determined