generalized macdonald ruijsenaars systems abstract we consider the polynomial representation of double affine hecke algebras dahas and construct its submodules as ideals of functions vanishing on the special collections of affine planes this generalizes certain results of kasatani in types a n c n c n we obtain commutative algebras of difference operators given by the action of invariant combinations of cherednik dunkl operators in the corresponding quotient modules of the polynomial representation this gives known and new generalized macdonald ruijsenaars systems thus in the cases of dahas of types a n and c n c n we derive chalykh sergeev veselov operators and a generalization of the koornwinder operator respectively together with complete sets of quantum integrals in the explicit form

differential operators and entire functions with simple real zeros abstract let and f be functions in the laguerre polya class write z e z 2 1 z and f z e z 2 f 1 z where 1 and f 1 have genus 0 or 1 and 0 if 1 4 and has infinitely many zeros then d f z has only simple real zeros where d denotes differentiation

on cohen macaulayness of algebras generated by generalized power sums generalized power sums are linear combinations of ith powers of coordinates we consider subalgebras of the polynomial algebra generated by generalized power sums and study when such algebras are cohen macaulay it turns out that the cohen macaulay property of such algebras is rare and tends to be related to quantum integrability and representation theory of cherednik algebras using representation theoretic results and deformation theory we establish cohen macaulayness of the algebra of q t deformed power sums defined by sergeev and veselov and of some generalizations of this algebra proving a conjecture of brookner corwin etingof and sam we also apply representation theoretic techniques to studying m quasi invariants of deformed calogero moser systems in an appendix to this paper m feigin uses representation theory of cherednik algebras to compute hilbert series for such quasi invariants and show that in the case of one light particle the ring of quasi invariants is gorenstein

the inverse eigenvalue problem for symmetric anti bidiagonal matrices abstract the inverse eigenvalue problem for real symmetric matrices of the form 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 is solved the solution is shown to be unique the problem is also shown to be equivalent to the inverse eigenvalue problem for a certain subclass of jacobi matrices

hermite biehler routh hurwitz and total positivity simple proofs of the hermite biehler and routh hurwitz theorems are presented the total nonnegativity of the hurwitz matrix of a stable real polynomial follows as an immediate corollary

deformed quantum calogero moser problems and lie superalgebras the deformed quantum calogero moser sutherland problems related to the root systems of the contragredient lie superalgebras are introduced the construction is based on the notion of the generalized root systems suggested by v serganova for the classical series a recurrent formula for the quantum integrals is found which implies the integrability of these problems the corresponding algebras of the quantum integrals are investigated the explicit formulas for their poincare series for generic values of the deformation parameter are presented

mechanisms for local search abstract local search methods for combinatorial optimization make a series of steps at each stage improving the current solution by moving to a neighbouring solution this is usually done by considering the neighbouring solutions one at a time and moving to the first one which gives an improvement a first improving method in this paper we consider whether there are circumstances in which some other strategy will have better performance in exploring this question we begin by giving a theoretical treatment of a simple model with random objective values at each solution point we carry out some experiments on the travelling salesman problem and the quadratic assignment problem using varying values of a spread parameter k the value of k corresponds to the number of improving solutions looked at before making a move we also make some conjectures relating the overall performance of the local search method to the average number of solutions which are evaluated before a local minimum is reached

new results on the old k opt algorithm for the traveling salesman problem local search with k change neighborhoods is perhaps the oldest and most widely used heuristic method for the traveling salesman problem yet almost no theoretical performance guarantees for it were previously known this paper develops several results some worst case and some probabilistic on the performance of 2 and k opt local search for the traveling salesman problem with respect to both the quality of the solution and the speed with which it is obtained

parameter shifts d4 symmetry and joint eigenfunctions for commuting askey wilson type difference operators in previous papers we studied a generalized hypergeometric function that is a joint eigenfunction of four hyperbolic askey wilson difference operators using parameter shifts related to the d4 weight lattice we show that this function is the even linear combination of two elementary joint eigenfunctions for a dense subset of the parameter space more specifically the latter eigenfunctions are equal to a plane wave multiplied by hyperbolic functions

differential equations in the spectral parameter we determine all the potentialsv x for the schrodinger equation 2 v x k2 such that some family of eigenfunctions satisfies a differential equation in the spectral parameterk of the formb k k o x o for each suchv x we determine the algebra of all possible operatorsb and the corresponding functions x

complex zero strip decreasing operators abstract let z be a function in the laguerre polya class write z e z 2 1 z where 0 and where 1 z is a real entire function of genus 0 or 1 let f z be any real entire function of the form f z e z 2 f 1 z where 0 and f 1 z is a real entire function of genus 0 or 1 having all of its zeros in the strip s r z c r im z r where r 0 if 1 4 the linear differential operator d f z where d denotes differentiation is known to converge to a real entire function whose zeros also belong the strip s r we describe several necessary and sufficient conditions on z such that all zeros of d f z belong to a smaller strip s r 1 z c r 1 im z r 1 where 0 r 1 r and r 1 depends on z but is independent of f z we call a linear operator having this property a complex zero strip decreasing operator or czsdo we examine several relevant examples in certain cases we give explicit upper and lower bounds for r and we state several conjectures and open problems regarding complex zero strip decreasing operators

nonsymmetric koornwinder polynomials and duality in the fundamental work of lusztig l on affine hecke algebras a special role is played by the root system of type cn the affine hecke algebra is a deformation of the group algebra of an affine weyl group which usually depends on as many parameters as there are distinct root lengths i e one or two for an irreducible root system however in the cn case the hecke algebra h has three parameters corresponding to the fact that there is a simple coroot which is divisible by 2 recently cherednik c1 c3 has introduced the notion of a double affine hecke algebra and has used it to prove several conjectures on macdonald polynomials these polynomials and cherednik s double affine hecke algebra involve two or three parameters i e one more than the number of root lengths

residue construction of hecke algebras abstract hecke algebras are usually defined algebraically via generators and relations we give a new algebra geometric construction of affine and double affine hecke algebras the former is known as the iwahori hecke algebra and the latter was introduced by cherednik in terms of residues more generally to any generalized cartan matrixaand a pointqin a one dimensional complex algebraic group c we associate an associate algebrahq ifais of finite type and c c the algebrahqis theaffinehecke algebra of the corresponding finite root system ifais of affine type and c c thenhqis essentially the cherednik algebra the case c c corresponds to degenerate counterparts of the above objects considered by drinfeld and lusztig finally taking c to be an elliptic curve one gets some new elliptic analogues of the affine hecke algebra

a probabilistic analysis of the switching algorithm for the euclidean tsp the well known switching algorithm proposed by lin and kernighan for the euclidean travelling salesman problem has proved to be a simple efficient algorithm for medium size problems though it often gets trapped in local optima although its complexity status is still open it has been observed to be polynomially bounded in practice when applied to uniformly distributed points in the unit square in this paper this polynomial behaviour is derived theoretically however we will come up with a bound of o n18 with probability 1 c n whereas in practice the algorithm works slightly better

an asymptotic formula for the koornwinder polynomials a formula for the large degree asymptotics of koornwinder s multivariate askey wilson polynomials is presented in the special case of a single variable this asymptotic formula agrees with the known leading asymptotics of the askey wilson polynomials determined by ismail and wilson

deformed macdonald ruijsenaars operators and super macdonald polynomials it is shown that the deformed macdonald ruijsenaars operators can be described as the restrictions on certain affine subvarieties of the usual macdonald ruijsenaars operator in infinite number of variables the ideals of these varieties are shown to be generated by the macdonald polynomials related to young diagrams with special geometry the super macdonald polynomials and their shifted version are introduced the combinatorial formulas for them are given

a study of exponential neighborhoods for the travelling salesman problem and for the quadratic assignment problem this paper deals with exponential neighborhoods for combinatorial optimization problems exponential neighborhoods are large sets of feasible solutions whose size grows exponentially with the input length we are especially interested in exponential neighborhoods over which the tsp respectively the qap can be solved in polynomial time and we investigate combinatorial and algorithmical questions related to such neighborhoods first we perform a careful study of exponential neighborhoods for the tsp we investigate neighborhoods that can be defined in a simple way via assignments matchings in bipartite graphs partial orders trees and other combinatorial structures we identify several properties of these combinatorial structures that lead to polynomial time optimization algorithms and we also provide variants that slightly violate these properties and lead to np complete optimization problems whereas it is relatively easy to find exponential neighborhoods over which the tsp can be solved in polynomial time the corresponding situation for the qap looks pretty hopeless every exponential neighborhood that is considered in this paper provably leads to an np complete optimization problem for the qap

complete integrability of relativistic calogero moser systems and elliptic function identities poincare invariant generalizations of the galilei invariant calogero moser a particle systems are studied a quantization of the classical integrals s1 sn is presented such that the operators si9 sn mutually commute as a corollary it follows that si9 9sn poisson commute these results hinge on functional equations satisfied by the weierstrass and 0 functions a generalized cauchy identity involving the function leads to an n x n matrix l whose symmetric functions are proportional to sl5 9sn

lectures on affine hecke algebras and macdonald s conjectures this paper gives a review of cherednik s results on the representation theoretic approach to macdonald polynomials and related special functions macdonald polynomials are a remarkable 2 parameter family of polynomials which can be associated to every root system as special cases they include the schur functions the q jacobi polynomials and certain spherical functions on real and p adic symmetric spaces they have a number of elegant combinatorial properties which however are extremely difficult to prove in this paper we show that a natural setup for studying these polynomials is provided by the representation theory of hecke algebras and show how this can be used to prove some of the combinatorial identities for macdonald polynomials

poisson lie interpretation of trigonometric ruijsenaars duality a geometric interpretation of the duality between two real forms of the complex trigonometric ruijsenaars schneider system is presented the phase spaces of the systems in duality are viewed as two different models of the same reduced phase space arising from a suitable symplectic reduction of the standard heisenberg double of u n the collections of commuting hamiltonians of the systems in duality are shown to descend from two families of free hamiltonians on the double which are dual to each other in a poisson lie sense our results give rise to a major simplification of ruijsenaars proof of the crucial symplectomorphism property of the duality map

self interlacing polynomials abstract we describe a new subclass of the class of real polynomials with real simple roots called self interlacing polynomials this subclass is isomorphic to the class of real hurwitz stable polynomials all roots in the open left half plane in the work we present basic properties of self interlacing polynomials and their relations with hurwitz and hankel matrices as well as with stiltjes type of continued fractions we also establish self interlacing analogues of the well known hurwitz and lienard chipart criterions for stable polynomials a criterion of hurwitz stability of polynomials in terms of minors of certain hankel matrices is established

a survey for the quadratic assignment problem the quadratic assignment problem qap one of the most difficult problems in the np hard class models many real life problems in several areas such as facilities location parallel and distributed computing and combinatorial data analysis combinatorial optimization problems such as the traveling salesman problem maximal clique and graph partitioning can be formulated as a qap in this paper we present some of the most important qap formulations and classify them according to their mathematical sources we also present a discussion on the theoretical resources used to define lower bounds for exact and heuristic algorithms we then give a detailed discussion of the progress made in both exact and heuristic solution methods including those formulated according to metaheuristic strategies finally we analyze the contributions brought about by the study of different approaches

connection coefficients for basic harish chandra series basic harish chandra series are asymptotically free meromorphic solutions of the system of basic hypergeometric difference equations associated to root systems the associated connection coefficients are explicitly computed in terms of jacobi theta functions we interpret the connection coefficients as the transition functions for asymptotically free meromorphic solutions of cherednik s root system analogs of the quantum knizhnik zamolodchikov equations they thus give rise to explicit elliptic solutions of root system analogs of dynamical yang baxter and reflection equations applications to quantum c functions basic hypergeometric functions reflectionless difference operators and multivariable baker akhiezer functions are discussed

multidimensional baker akhiezer functions and huygens principle a notion of the rational baker akhiezer ba function related to a configuration of hyperplanes in c n is introduced it is proved that the ba function exists only for very special configurations locus configurations which satisfy a certain overdetermined algebraic system the ba functions satisfy some algebraically integrable schrodinger equations so any locus configuration determines such an equation some results towards the classification of all locus configurations are presented this theory is applied to the famous hadamard problem of description of all hyperbolic equations satisfying huygens principle we show that in a certain class all such equations are related to locus configurations and the corresponding fundamental solutions can be constructed explicitly from the ba functions

orthogonality relations and cherednik identities for multivariable baker akhiezer functions abstract we establish orthogonality relations for the baker akhiezer ba eigenfunctions of the macdonald difference operators we also obtain a version of cherednik macdonald mehta integral for these functions as a corollary we give a simple derivation of the norm identity and cherednik macdonald mehta integral for macdonald polynomials in the appendix written by the first author we prove a summation formula for ba functions we also consider more general identities of cherednik type which we use to introduce and construct more general twisted ba functions this leads to a construction of new quantum integrable models of macdonald ruijsenaars type

on zero sector reducing operators abstract we prove a jensen disc type theorem for polynomials p r z having all their zeros in a sector of the complex plane this result is then used to prove the existence of a collection of linear operators t r z r z which map polynomials with their zeros in a closed convex sector arg z 2 to polynomials with zeros in a smaller sector arg z we therefore provide the first example of a zero sector reducing operator

a thermodynamically motivated simulation procedure for combinatorial optimization problems abstract exchange algorithms are an important class of heuristics for hard combinatorial optimization problems as e g salesman problems or quadratic assignment problems in kirkpatrick s and cerny s exchange algorithms for the travelling salesman problem and placement problem they propose to perform an exchange not only if the objective function value decreases by this exchange but also in certain cases if the objective function value increases an exchange increasing the objective function value is performed stochastically depending on the size of the increment computational tests with quadratic assignment problems revealed an excellent behaviour in such an approach suboptimal solutions differing 1 2 from the best known solutions are obtained by a simple program in short time by starting this program several times with different starting values all known minimal objective function values were reached thus this approach is well suited also for smaller computers and leads in short time to acceptable solutions

the fox wright functions and laguerre multiplier sequences abstract linear operators which 1 preserve the reality of zeros of polynomials having only real zeros and 2 map stable polynomials into stable polynomials are investigated using recently established results concerning the zeros of certain fox wright functions and generalized mittag leffler functions the paper includes several open problems and questions

structured matrices continued fractions and root localization of polynomials we give a detailed account of various connections between several classes of objects hankel hurwitz toeplitz vandermonde and other structured matrices stietjes and jacobi type continued fractions cauchy indices moment problems total positivity and root localization of univariate polynomials along with a survey of many classical facts we provide a number of new results

polya schur master theorems for circular domains and their boundaries we characterize all linear operators on finite or infinite dimensional polynomial spaces that preserve the property of having the zero set inside a prescribed region omega subset of c for arbitrary

macdonald polynomials and algebraic integrability we construct explicitly nonpolynomial eigenfunctions of the difference operators by macdonald in the case t qk k z this leads to a new more elementary proof of several macdonald conjectures proved first by cherednik we also establish the algebraic integrability of macdonald operators at t qk k z generalizing the result of etingof and styrkas our approach works uniformly for all root systems including the bcn case and related koornwinder polynomials moreover we apply it for a certain deformation of the an root system where the previously known methods do not work

orthogonality of macdonald polynomials with unitary parameters for any admissible pair of irreducible reduced crystallographic root systems we present discrete orthogonality relations for a finite dimensional system of macdonald polynomials with parameters on the unit circle subject to a truncation relation

the n city travelling salesman problem statistical mechanics and the metropolis algorithm in any n city travelling salesman problem there are n 1 mathord left vphantom 2 right kern nulldelimiterspace 2 possible tours we use the metropolis algorithm to generate a sequence of such tours this sequence may be viewed as the random evolution of a physical system in contact with a heat bath as the temperature is lowered the tours generated approach the optimal tour it appears that for large n one arrives within a few percent of the optimal solution in better than quadratic time

bispectral commuting difference operators for multivariable askey wilson polynomials we construct a commutative algebra a z generated by d algebraically independent q difference operators acting on variables z 1 z 2 z d which is diagonalized by the multivariable askey wilson polynomials p n z considered by gasper and rahman 2005 iterating sears 4 3 transformation formula we show that the polynomials p n z possess a certain duality between z and n analytic continuation allows us to obtain another commutative algebra a n generated by d algebraically independent difference operators acting on the discrete variables n 1 n 2 n d which is also diagonalized by p n z this leads to a multivariable q askey scheme of bispectral orthogonal polynomials which parallels the theory of symmetric functions

double affine hecke algebras and macdonald s conjectures in this paper we prove the main results about the structure of double affine hecke algebras announced in c1 c2 the technique is based on the realization of these algebras in terms of demazure lusztig operators bgg d l2 ls c3 and rather standard facts from the theory of affine weyl groups in particular it completes the proof partially published in c2 of the macdonald scalar product conjecture see ml 12 6 including the famous macdonald constant term conjecture the q t case we mainly follow the opdam paper 0 where the macdonald mehta conjectures in the degenerate differential case were deduced from certain properties of the heckman opdam operators ho and the existence of the shift operators heckman s interpretation of these operators via the so called dunkl operators see he and also c5 was important to our approach we note that the ho operators are closely related to the so called quantum many body problem calogero sutherland moser olshanetsky perelomov the conformal field theory knizhnik zamolodchikov equations the harmonic analysis on symmetric spaces harish chandra helgason etc and last but not the least the classic theory of the hypergeometric functions establishing the connection between the difference counterparts of heckman opdam operators introduced in c4 and the macdonald theory ml m2 including the construction of the difference shift operators is the main thrust of this paper once the connection is established it is not very difficult to calculate the scalar squares of the macdonald polynomials and to prove the constant term conjecture from his fundamental paper m3 to simplify the exposition we discuss the reduced root systems only and impose the relation q tk for k e z to avoid infinite products in the definition of macdonald s pairing the purpose of this work is to present a concrete application of the new technique arbitrary q t can be handled in a

scattering theory of discrete pseudo laplacians on a weyl chamber to a crystallographic root system we associate a system of multivariate orthogonal polynomials diagonalizing an integrable system of discrete pseudo laplacians on the weyl chamber we develop the time dependent scattering theory for these discrete pseudo laplacians and determine the corresponding wave operators and scattering operators in closed form as an application we describe the scattering behavior of certain hyperbolic ruijsenaars schneider type lattice calogero moser models associated with the macdonald polynomials

bispectrality for the quantum ruijsenaars model and its integrable deformation an elementary construction of the eigenfunctions for the quantum rational ruijsenaars model with integer coupling parameter is presented as a by product we establish the bispectral duality between this model and the trigonometric calogero moser model in particular this gives a new way for calculating jack polynomials we propose also a certain one parameter deformation of the ruijsenaars model proving its integrability and bispectrality the generalizations related to other root systems and difference operators by macdonald are considered