equivariant stable homotopy and segal s burnside ring conjecture analogous results were proved later for ko in the generality of compact lie groups by atiyah and segal 8 and for kfq the algebraic k theory spectrum associated to the finite field fq by rector 29 using quillen s 26 computation of it kfq in each case the answer involves an appropriate completed representation ring of g and the cohomology theory in question is constructed from the permutative category of finite dimensional vector spaces over a field see 31 if one considers cohomology theories constructed from other permutative categories one expects to find analogous computations in terms of a completed representation ring of g in the given category appropriately defined in particular stable cohomotopy ts is constructed from the category of finite sets 31 and for this category the analogue of the representation ring is a well known object the burnside ring a g 13 a g is a commutative ring with augmentation so one may speak of a g the completed burnside ring moreover there is

levenberg marquardt methods for constrained nonlinear equations with strong local convergence properties we consider the problem of finding a solution of a constrained and not necessarily square system of equations i e we consider systems of nonlinear equations and want to find a solution that belongs to a certain feasible set to this end we present two levenberg marquardt type algorithms that differ in the way they compute their search directions the first method solves a strictly convex minimization problem at each iteration whereas the second one solves only one system of linear equations in each step both methods are shown to converge locally quadratically under an error bound assumption that is much weaker than the standard nonsingularity condition both methods can be globalized in an easy way some numerical results for the second method indicate that the algorithm works quite well in practice

polynomials associated with equilibrium positions in calogero moser systems in a previous paper corrigan sasaki many remarkable properties of classical calogero and sutherland systems at equilibrium are reported for example the minimum energies frequencies of small oscillations and the eigenvalues of lax pair matrices at equilibrium are all integer valued the equilibrium positions of calogero and sutherland systems for the classical root systems ar br cr and dr correspond to the zeros of hermite laguerre jacobi and chebyshev polynomials here we define and derive the corresponding polynomials for the exceptional e6 e7 e8 f4 and g2 and non crystallographic i2 m h3 and h4 root systems they do not have orthogonality but share many other properties with the above mentioned classical polynomials

products and duality in waldhausen categories the natural transformatione from l theory to the tate cohomol ogy of z 2 acting on k theory commutes with external products corollary the tate cohomology of z 2 acting on the k theory of any ring with inrvolu tion is a generalized eilenberg mac lane spectrum and it is 4 periodic categories with cofibrations and weak equivalences waldhausen categories for short were introduced in wald generalizing earlier constructions due to quillen and others waldhausen defined the k theory spectrum k c of a waldhausen category c it was realized early on vo that suitable notions of spanier whitehead duality in c give rise to involutions on k c we formalized this idea in wwd by introducing the notion of a spanier whitehead product sw product for short in c to some extent this will be recalled below when c is equipped with an sw product d and c 0 satisfies certain conditions 2 of wwd then the following are defined

equilibrium positions shape invariance and askey wilson polynomials we show that the equilibrium positions of the ruijsenaars schneider van diejen systems with the trigonometric potential are given by the zeros of the askey wilson polynomials with five parameters the corresponding single particle quantum version which is a typical example of discrete quantum mechanical systems with a q shift type kinetic term is shape invariant and the eigenfunctions are the askey wilson polynomials this is an extension of our previous study 1 2 which established the discrete analogue of the well known fact the equilibrium positions of the calogero systems are described by the hermite and laguerre polynomials whereas the corresponding single particle quantum versions are shape invariant and the eigenfunctions are the hermite and laguerre polynomials

solution dependence on initial conditions in differential variational inequalities in the first part of this paper we establish several sensitivity results of the solution x t to the ordinary differential equation ode initial value problem ivp dx dt f x x 0 as a function of the initial value for a nondifferentiable f x specifically we show that for xi t equiv x t xi 0 0 leq t leq t a if f is b differentiable on xi t then so is the solution operator x t at 0 b if f is semismooth on xi t then so is x t at 0 c if f has a linear newton approximation on xi t then so does x t at 0 moreover the linear newton approximation of the solution operator can be obtained from the solution of a linear differential inclusion in the second part of the paper we apply these ode sensitivity results to a differential variational inequality dvi and discuss a the existence uniqueness and lipschitz dependence of solutions to subclasses of the dvi subject to boundary conditions via an implicit function theorem for semismooth equations and b the convergence of a nonsmooth shooting method for numerically computing such boundary value solutions

a generalized newton method for absolute value equations associated with second order cones in this paper we introduce the absolute value equations associated with second order cones socave in short which is a generalization of the absolute value equations discussed recently in the literature it is proved that the socave is equivalent to a class of second order cone linear complementarity problems soclcp in short in particular we propose a generalized newton method for solving the socave and show that the proposed method is globally linearly and locally quadratically convergent under suitable assumptions we also report some preliminary numerical results of the proposed method for solving the socave and the soclcp which show the efficiency of the proposed method

on the computation of an element of clarke generalized jacobian for a vector valued max function abstract in this paper we present an algorithm for calculating an element of clarke generalized jacobian for a vector valued max type function the algorithm reduces the computational cost of an existing algorithm

the homotopy fixed point theorem and the quillen lichtenbaum conjecture in hermitian k theory abstract we settle two conjectures for computing higher grothendieck witt groups also known as hermitian k groups of noetherian schemes x under some mild conditions it is shown that the comparison map from the hermitian k theory of x to the homotopy fixed points of k theory under the natural z 2 action is a 2 adic equivalence we also prove that the mod 2 comparison map between the hermitian k theory of x and its etale version is an isomorphism on homotopy groups in the same range as for the quillen lichtenbaum conjecture in k theory applications compute higher grothendieck witt groups of complex algebraic varieties and rings of 2 integers in number fields and hence values of dedekind zeta functions

semismoothness and decomposition of maximal normal operators abstract a function is semismooth if and only if its subdifferential is semicontinuous all the semicontinuous operators form a big linear subspace of the maximal normal operator space by big we mean that this subspace includes all the continuous operators and all the maximal monotone operators all the decomposable maximal normal operators form another big subspace

quantum versus classical integrability in calogero moser systems calogero moser systems are classical and quantum integrable multiparticle dynamics defined for any root system the quantum calogero systems having 1 q2 potential and a confining q2 potential and the sutherland systems with 1 sin2q potentials have integer energy spectra characterized by the root system various quantities of the corresponding classical systems e g minimum energy frequencies of small oscillations the eigenvalues of the classical lax pair matrices etc at the equilibrium point of the potential are investigated analytically as well as numerically for all root systems to our surprise most of these classical data are also integers or they appear to be quantized to be more precise these quantities are polynomials of the coupling constant s with integer coefficients the close relationship between quantum and classical integrability in calogero moser systems deserves fuller analytical treatment which would lead to better understanding of these systems and of integrable systems in general

polynomials associated with equilibria of affine toda sutherland systems an affine toda sutherland system is aquasi exactly solvable multi particle dynamics based on an affine simple root system it is a cross between two well known integrable multi particle dynamics an affine toda moleculeand a sutherland system polynomials describing the equilibrium positions of affine toda sutherland systems are determined for all affine simple root systems

finite termination of a newton type algorithm for a class of affine variational inequality problems abstract many optimization problems can be reformulated as a system of equations one may use the generalized newton method or the smoothing newton method to solve the reformulated equations so that a solution of the original problem can be found such methods have been powerful tools to solve many optimization problems in the literature in this paper we propose a newton type algorithm for solving a class of monotone affine variational inequality problems avips for short in the proposed algorithm the techniques based on both the generalized newton method and the smoothing newton method are used in particular we show that the algorithm can find an exact solution of the avip in a finite number of iterations under an assumption that the solution set of the avip is nonempty preliminary numerical results are reported

coassembly and the k theory of finite groups abstract we study the k theory and swan theory of the group ring r g when g is a finite group and r is any ring or ring spectrum in this setting the well known assembly map for k r g has a companion called the coassembly map we prove that their composite is the equivariant norm of k r this gives a splitting of both assembly and coassembly after k n localization a new map between whitehead torsion and tate cohomology and a partial computation of k theory of representations in the category of spectra

second order necessary conditions in constrained semismooth optimization first and second order conditions are given which must necessarily be satisfied by local minimizers for certain finite dimensional nonsmooth nonlinear programming problems the problems considered are of standard form having a finite number of equality and inequality constraints the principal result does not require a constraint qualification but does require that the functions be semismooth at the minimizer the necessary conditions are stated in terms of the generalized gradients of nonsmooth analysis and certain second order directional derivatives

quasi newton trust region algorithm for non smooth least squares problems1this work was supported by cnpq of brazil and the national natural science foundation of china 1 in this paper we present a quasi newton trust region algorithm for non smooth least squares problems which possess the smooth plus non smooth decomposition this method uses a smooth subproblem with second order information to approximate the locally lipschitzian function which is important to increase the convergence rate of the algorithm and deal with the large residual non smooth least squares problems the global and superlinear convergence of this algorithm are established

matrices differential operators and polynomials it is noted that the matrix z of order n defined in terms of the n arbitrary numbers xj by the formula zjk jk j l 1 xj xl 1 1 jk xj xk 1 may be considered in an appropriate framework to correspond to the differential operator d dx there follow prescriptions to construct explicit matrices of arbitrary order n in terms of n or more arbitrary numbers or of the n roots of given polynomials matrices whose eigenvalues and eigenvectors are given fully or in part by very simple formulas novel representations of the classical polynomials hermite laguerre lagrange gegenbauer jacobi are also obtained such as the formula for hermite polynomials hn x 2n det xi h where i is the unit matrix of order n and the matrix h of order n is defined by hjk 2n 1 2 jk n 1 exp 2i j 1 2 exp 2i j 1 jk exp 2i j 2i sin j k 1 exp i j k with j j n arbitrary

affine toda sutherland systems a cross between two well known integrable multi particle dynamics an affine toda molecule and a sutherland system is introduced for any affine root system though it is not completely integrable but partially integrable or quasi exactly solvable it inherits many remarkable properties from the parents the equilibrium position is algebraic i e proportional to the weyl vector the frequencies of small oscillations near equilibrium are proportional to the affine toda masses which are essential ingredients of the exact factorizable s matrices of affine toda field theories some lower lying frequencies are integer times a coupling constant for which the corresponding exact quantum eigenvalues and eigenfunctions are obtained an affine toda calogero system with a corresponding rational potential is also discussed

the 1d interacting bose gas in a hard wall box we consider the integrable one dimensional function interacting bose gas in a hard wall box which is exactly solved via the coordinate bethe ansatz the ground state energy including the surface energy is derived from the lieb liniger type integral equations the leading and correction terms are obtained in the weak and strong coupling regimes from both the discrete bethe equations and the integral equations this allows the investigation of both finite size and boundary effects in the integrable model we also study the luttinger liquid behaviour by calculating luttinger parameters and correlations the hard wall boundary conditions are seen to have a strong effect on the ground state energy and phase correlations in the weak coupling regime enhancement of the local two body correlations is shown by application of the hellmann feynman theorem

convergence analysis of some algorithms for solving nonsmooth equations this paper presents convergence analysis of some algorithms for solving systems of nonlinear equations defined by locally lipschitzian functions for the directional derivative based and the generalized jacobian based newton methods both the iterates and the corresponding function values are locally superlinearly convergent globally a limiting point of the iterate sequence generated by the damped directional derivative based newton method is a zero of the system if and only if the iterate sequence converges to this point and the stepsize eventually becomes one provided that the system is strongly bd regular and semismooth at this point in this case the convergence is superlinear a general attraction theorem is presented which can be applied to two algorithms proposed by han pang and rangaraj a hybrid method which is both globally convergent in the sense of finding a stationary point of the norm function of the system and locally quadratically convergent is also presented

quantum calogero moser models integrability for all root systems the issues related to the integrability of quantum calogero moser models based on any root systems are addressed for the models with degenerate potentials i e the rational with without the harmonic confining force the hyperbolic and the trigonometric we demonstrate the following for all the root systems i construction of a complete set of quantum conserved quantities in terms of a total sum of the lax matrix l i e sigma mu nu is an element ofr l n mu nu in which r is a set of r r vectors invariant under the action of the coxeter group they form a single coxeter orbit ii proof of liouville integrability iii triangularity of the quantum hamiltonian and the entire discrete spectrum generalized jack polynomials are defined for all root systems as unique eigenfunctions of the hamiltonian iv equivalence of the lax operator and the dunkl operator v algebraic construction of all excited states in terms of creation operators these are mainly generalizations of the results known for the models based on the a series i e su n type root systems

weak directional closedness and generalized subdifferentials abstract spingarn introduced the notion of a submonotone operator and showed that the clarke subdifferential is submonotone if and only if it is semismooth in the sense of mifflin and regular in the sense of clarke in this article a property of operators referred to as weak directional closedness wdc is introduced the wdc property is used to extend spingarn s result to a broad class of generalized subdifferentials for locally lipschitz functions two members of this class of subdifferentials are the clarke subdifferential which is always wdc and the michel penot subdifferential which may or may not be wdc a subdifferential that is wdc and is contained in the clarke subdifferential is constructed it is shown that this subdifferential coincides with the michel penot subdifferential whenever the michel penot subdifferential is wdc and submonotone

computational limits to nonparametric estimation for ergodic processes a new negative result for nonparametric distribution estimation of binary ergodic processes is shown the problem of estimation of distribution with any degree of accuracy is studied then it is shown that for any countable class of estimators there is a zero entropy binary ergodic process that is inconsistent with the class of estimators our result is different from other negative results for universal forecasting scheme of ergodic processes

affine scaling interior levenberg marquardt method for bound constrained semismooth equations under local error bound conditions we develop and analyze a new affine scaling levenberg marquardt method with nonmonotonic interior backtracking line search technique for solving bound constrained semismooth equations under local error bound conditions the affine scaling levenberg marquardt equation is based on a minimization of the squared euclidean norm of linear model adding a quadratic affine scaling matrix to find a solution that belongs to the bounded constraints on variable the global convergence results are developed in a very general setting of computing trial directions by a semismooth levenberg marquardt method where a backtracking line search technique projects trial steps onto the feasible interior set we establish that close to the solution set the affine scaling interior levenberg marquardt algorithm is shown to converge locally q superlinearly depending on the quality of the semismooth and levenberg marquardt parameter under an error bound assumption that is much weaker than the standard nonsingularity condition that is bd regular condition under nonsmooth case a nonmonotonic criterion should bring about speed up the convergence progress in the contours of objective function with large curvature

erratum to levenberg marquardt methods with strong local convergence properties for solving nonlinear equations with convex constraints we consider the problem of finding a solution of a constrained and not necessarily square system of equations i e we consider systems of nonlinear equations and want to find a solution that belongs to a certain feasible set to this end we present two levenberg marquardt type algorithms that differ in the way they compute their search directions the first method solves a strictly convex minimization problem at each iteration whereas the second one solves only one system of linear equations in each step both methods are shown to converge locally quadratically under an error bound assumption that is much weaker than the standard nonsingularity condition both methods can be globalized in an easy way some numerical results for the second method indicate that the algorithm works quite well in practice

universal lax pairs for spin calogero moser models and spin exchange models for any root system and a set of vectors which form a single orbit of the reflection weyl group g generated by a spin calogero moser model can be defined for each of the potentials rational hyperbolic trigonometric and elliptic for each member of to be called a site we associate a vector space v whose element is called a spin its dynamical variables are the canonical coordinates of a particle in rr r rank of and spin exchange operators which exchange the spins at the sites and s here s is the reflection generated by for each and a spin exchange model can be defined the hamiltonian of a spin exchange model is a linear combination of the spin exchange operators only it is obtained by freezing the canonical variables at the equilibrium point of the corresponding classical calogero moser model for ar and set of vector weights it reduces to the well known haldane shastry model universal lax pair operators for both spin calogero moser models and spin exchange models are presented which enable us to construct as many conserved quantities as the number of sites for degenerate potentials

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 convergence of modified newton methods for solving equations containing a non differentiable term we provide a semilocal convergence analysis for certain modified newton methods for solving equations containing a non differentiable term the sufficient convergence conditions of the corresponding newton methods are often taken as the sufficient conditions for the modified newton methods that is why the latter methods are not usually treated separately from the former however here we show that weaker conditions as well as a finer error analysis than before can be obtained for the convergence of modified newton methods numerical examples are also provided

karush kuhn tucker systems regularity conditions error bounds and a class of newton type methods we consider optimality systems of karush kuhn tucker kkt type which arise for example as primal dual conditions characterizing solutions of optimization problems or variational inequalities in particular we discuss error bounds and newton type methods for such systems an exhaustive comparison of various regularity conditions which arise in this context is given we obtain a new error bound under an assumption which we show to be strictly weaker than assumptions previously used for kkt systems such as quasi regularity or semistability equivalently the r0 property error bounds are useful among oth er things for identifying active constraints and developing efficient local algorithms we propose a family of local newton type algorithms this family contains some known active set newton methods as well as some new methods regularity conditions required for local superlinear convergence compare favorably with convergence conditions of nonsmooth newton methods and sequential quadratic programming methods

quantum versus classical integrability in ruijsenaars schneider systems the relationship resemblance and or contrast between quantum and classical integrability in ruijsenaars schneider systems which are one parameter deformations of calogero moser systems is addressed many remarkable properties of classical calogero and sutherland systems based on any root system at equilibrium are reported in a previous paper corrigan sasaki for example the minimum energies frequencies of small oscillations and the eigenvalues of lax pair matrices at equilibrium are all integer valued in this paper we report that similar features and results hold for the ruijsenaars schneider type of integrable systems based on the classical root systems

hedging with a correlated asset solution of a nonlinear pricing pde hedging a contingent claim with an asset which is not perfectly correlated with the underlying asset results in unhedgeable residual risk even if the residual risk is considered diversifiable the option writer is faced with the problem of uncertainty in the estimation of the drift rates of the underlying and the hedging instrument if the residual risk is not considered diversifiable then this risk can be priced using an actuarial standard deviation principle in infinitesimal time in both cases these models result in the same nonlinear partial differential equation pde a fully implicit monotone discretization method is developed for solution of this pricing pde this method is shown to converge to the viscosity solution certain grid conditions are required to guarantee monotonicity an algorithm is derived which given an initial grid inserts a finite number of nodes in the grid to ensure that the monotonicity condition is satisfied at each timestep the nonlinear discretized algebraic equations are solved using an iterative algorithm which is shown to be globally convergent monte carlo hedging examples are given to illustrate the profit and loss distribution at the expiry of the option

a regularization semismooth newton method based on the generalized fischer burmeister function for p0 ncps we consider a regularization method for nonlinear complementarity problems with f being a p 0 function which replaces the original problem with a sequence of the regularized complementarity problems in this paper this sequence of regularized complementarity problems are solved approximately by applying the generalized newton method for an equivalent augmented system of equations constructed by the generalized fischer burmeister fb ncp functions f p with p 1 we test the performance of the regularization semismooth newton method based on the family of ncp functions through solving all test problems from mcplib numerical experiments indicate that the method associated with a smaller p for example email protected 1 1 2 usually has better numerical performance and the generalized fb functions f p with email protected 1 1 2 can be used as the substitutions for the fb function f 2

a nonsmooth version of newton s method newton s method for solving a nonlinear equation of several variables is extended to a nonsmooth case by using the generalized jacobian instead of the derivative this extension includes the b derivative version of newton s method as a special case convergence theorems are proved under the condition of semismoothness it is shown that the gradient function of the augmented lagrangian forc2 nonlinear programming is semismooth thus the extended newton s method can be used in the augmented lagrangian method for solving nonlinear programs

equilibria of discrete integrable systems and deformation of classical orthogonal polynomials the ruijsenaars schneider systems are discrete version of the calogero moser cm systems in the sense that the momentum operator p appears in the hamiltonians as a polynomial in e p is a deformation parameter instead of an ordinary polynomial in p in the hierarchies of c m systems we determine the polynomials describing the equilibrium positions of the rational and trigonometric ruijsenaars schneider systems based on classical root systems these are deformation of the classical orthogonal polynomials the hermite laguerre and jacobi polynomials which describe the equilibrium positions of the corresponding calogero and sutherland systems the orthogonality of the original polynomials is inherited by the deformed ones which satisfy three term recurrence and certain functional equations the latter reduce to the celebrated second order differential equations satisfied by the classical orthogonal polynomials

second order necessary conditions in semismooth optimization first and second order conditions are given which are necessary for a functionf to have a local minimal value atx inrn it is assumed thatf is locally lipschitzian nearx and semismooth atx the necessary conditions are expressed in terms of the generalized gradients of nonsmooth analysis and certain second order directional derivatives the method of proof bears no resemblance to standard methods three special cases are discussed here but applications to constrained problems are made elsewhere

the relaxed nonlinear phss like iteration method for absolute value equations finding the solution of the absolute value equation ave a x x b has attracted much attention in recent years in this paper we propose a relaxed nonlinear phss like iterative method which is more efficient than the picard hss iterative method for the ave and is a generalization of the nonlinear hss like iteration method for the ave by using the theory of nonsmooth analysis we prove the convergence of the relaxed nonlinear phss like iterative method for the ave numerical experiments are given to demonstrate the feasibility robustness and effectiveness of the relaxed nonlinear hss like method

a new non interior continuation method for solving the second order cone complementarity problem abstract in this paper based on a symmetrically perturbed smoothing fischer burmeister function a non interior continuation method is proposed for solving the second order cone complementarity problem soccp the proposed algorithm solves only one linear system of equations and performs only one line search at each iteration under monotonicity it is shown that our algorithm is globally and locally superlinearly convergent without requiring strict complementarity assumption at the soccp solution furthermore the proposed algorithm has local quadratic convergence under mild conditions some numerical results are reported which indicate the effectiveness of the proposed algorithm