aggregation and mixed integer rounding to solve mips in this paper we discuss the use of mixed integer rounding mir inequalities to solve mixed integer programs mir inequalities are essentially gomory mixed integer cuts however as we wish to use problem structure we insist that mir inequalities be generated from constraints or simple aggregations of constraints of the original problem this idea is motivated by the observation that several strong valid inequalities based on specific problem structure can be derived as mir inequalities here we present and test a separation routine for such mir inequalities that includes a heuristic row aggregation procedure to generate a single knapsack plus continuous variables constraint complementation of variables and finally the generation of an mir inequality inserted in a branch and cut system the results suggest that such a routine is a useful additional tool for tackling a variety of mixed integer programming problems

safe and tight linear estimators for global optimization global optimization problems are often approached by branch and bound algorithms which use linear relaxations of the nonlinear constraints computed from the current variable bounds this paper studies how to derive safe linear relaxations to account for numerical errors arising when computing the linear coefficients it first proposes two classes of safe linear estimators for univariate functions class 1 estimators generalize previously suggested estimators from quadratic to arbitrary functions while class 2 estimators are novel when they apply class 2 estimators are shown to be tighter theoretically in a certain sense and almost always tighter numerically the paper then generalizes these results to multivariate functions it shows how to derive estimators for multivariate functions by combining univariate estimators derived for each variable independently moreover the combination of tight class 1 safe univariate estimators is shown to be a tight class 1 safe multivariate estimator finally multivariate class 2 estimators are shown to be theoretically tighter in a certain sense than multivariate class 1 estimators

stable bundles and integrable systems on considere la geometrie symplectique des fibres cotangents aux espaces de modules de fibres vectoriels stables sur une surface de riemann on montre que ce sont des systemes dynamiques hamiltoniens algebriquement completement integrables

spectra of wess zumino witten models with arbitrary simple groups we consider the wess zumino witten two dimensional sigma models with fields taking values in an arbitrary connected but not necessarily simply connected simple lie groupg the quantum states of the theory are realized geometrically as sections of a line bundle over the loop grouplg the action of the current algebra of the theory is decomposed into highest weight representations by explicit construction of the highest weight states this solves for the spectrum of the models as a by product we obtain modular invariant partition functions of the theory on tori the present paper extends the results of 7 where the casesg su 2 andso 3 were treated

maximal inner boxes in parametric ae solution sets with linear shape we consider linear systems of equations a p x b p where the parameters p are linearly dependent and come from prescribed boxes and the sets of solutions defined in various ways which have linear boundary one fundamental problem is to compute a box being inside a parametric solution set we first consider parametric tolerable solution sets being convex polyhedrons for such solution sets we prove that finding a maximal inner box is an np hard problem this justifies our exponential linear programming methods for computing maximal inner boxes we also propose a polynomial heuristic that yields a large but not necessarily the maximal inner box next we discuss how to apply the presented linear programming methods for finding large inner estimations of general parametric ae solution sets with linear shape numerical examples illustrate the properties of the methods and their application

noncommutative extensions of elliptic integrable euler arnold tops and painleve vi equation in this paper we suggest generalizations of elliptic integrable tops to matrix valued variables our consideration is based on the r matrix description which provides lax pairs in terms of quantum and classical r matrices first we prove that for relativistic and non relativistic tops such lax pairs with spectral parameters follow from the associative yang baxter equation and its degenerations then we proceed to matrix extensions of the models and find out that some additional constraints are required for their construction we describe a matrix version of the reduced elliptic top and verify that the latter constraints are fulfilled in this case the construction of matrix extensions is naturally generalized to the monodromy preserving equation in this way we get matrix extensions of the painleve vi equation and its multidimensional analogues written in the form of non autonomous elliptic tops finally it is mentioned that the matrix valued variables can be replaced by elements of noncommutative associative algebra at the end of the paper we also describe special elliptic gaudin models which can be considered as matrix extensions of the reduced elliptic top

an efficient and safe framework for solving optimization problems interval methods have shown their ability to locate and prove the existence of a global optima in a safe and rigorous way unfortunately these methods are rather slow efficient solvers for optimization problems are based on linear relaxations however the latter are unsafe and thus may overestimate or worst underestimate the very global minima this paper introduces quadopt an efficient and safe framework to rigorously bound the global optima as well as its location quadopt uses consistency techniques to speed up the initial convergence of the interval narrowing algorithms a lower bound is computed on a linear relaxation of the constraint system and the objective function all these computations are based on a safe and rigorous implementation of linear programming techniques first experimental results are very promising

holomorphic bundles and many body systems we show that spin generalization of elliptic calogero moser system elliptic extension of gaudin model and their cousins are the degenerations of hitchin systems applications to the constructions of integrals of motion angle action variables and quantum systems are discussed the constructions of classical systems are motivated by conformal field theory and their quantum counterparts can be thought of as being the degenerations of the critical level knizhnik zamolodchikov bernard equations

infinite dimensional algebras and a trigonometric basis for the classical lie algebras this paper explores features of the infinite dimensional algebras that have been previously introduced in particular it is shown that the classical simple lie algebras an bn cn dn may be expressed in an egalitarian basis with trigonometric structure constants the transformation to the standard cartan weyl basis and the particularly transparent n limit that this formulation allows is provided

bihamiltonian structures and quadratic algebras in hydrodynamics and on non commutative torus we demonstrate the common bihamiltonian nature of several integrable systems the first one is an elliptic rotator that is an integrable euler arnold top on the complex group gl n open image in new window for any n whose inertia ellipsiod is related to a choice of an elliptic curve its bihamiltonian structure is provided by the compatible linear and quadratic poisson brackets both of which are governed by the belavin drinfeld classical elliptic r matrix we also generalize this bihamiltonian construction of integrable euler arnold tops to several infinite dimensional groups appearing as certain large n limits of gl n open image in new window these are the group of a non commutative torus nct and the group of symplectomorphisms sdiff t2 of the two dimensional torus the elliptic rotator on symplectomorphisms gives an elliptic version of an ideal 2d hydrodynamics which turns out to be an integrable system in particular we define the quadratic poisson algebra on the space of hamiltonians on t2 depending on two irrational numbers in conclusion we quantize the infinite dimensional quadratic poisson algebra in a fashion similar to the corresponding finite dimensional case

safe bounds in linear and mixed integer linear programming current mixed integer linear programming solvers are based on linear programming routines that use floating point arithmetic occasionally this leads to wrong solutions even for problems where all coefficients and all solution components are small integers an example is given where many state of the art milp solvers fail it is then shown how using directed rounding and interval arithmetic cheap pre and postprocessing of the linear programs arising in a branch and cut framework can guarantee that no solution is lost at least for mixed integer programs in which all variables can be bounded rigorously by bounds of reasonable size

classical r matrices and the feigin odesskii algebra via hamiltonian and poisson reductions we present a formula for a classical r matrix of an integrable system obtained by hamiltonian reduction of some free field theories using pure gauge symmetries the framework of the reduction is restricted only by the assumption that the respective gauge transformations are lie group ones our formula is in terms of dirac brackets and some new observations on these brackets are made we apply our method to derive a classical r matrix for the elliptic calogero moser system with spin starting from the higgs bundle over an elliptic curve with marked points in the paper we also derive a classical feigin odesskii algebra by a poisson reduction of some modification of the higgs bundle over an elliptic curve this allows us to include integrable lattice models in a hitchin type construction

rigorous error bounds for the optimal value in semidefinite programming a wide variety of problems in global optimization combinatorial optimization as well as systems and control theory can be solved by using linear and semidefinite programming sometimes due to the use of floating point arithmetic in combination with ill conditioning and degeneracy erroneous results may be produced the purpose of this article is to show how rigorous error bounds for the optimal value can be computed by carefully postprocessing the output of a linear or semidefinite programming solver it turns out that in many cases the computational costs for postprocessing are small compared to the effort required by the solver numerical results are presented including problems from the sdplib and the netlib lp library these libraries contain many ill conditioned and real life problems

a class of integrable spin calogero moser systems we introduce a class of spin calogero moser systems associated with root systems of simple lie algebras and give the associated lax representations with spectral parameter and fundamental poisson bracket relations the associated integrable models called integrable spin calogero moser systems in the paper and their lax pairs are then obtained via poisson reduction and gauge transformations for lie algebras of a n type this new class of integrable systems includes the usual calogero moser systems as subsystems our method is guided by a general framework which we develop here using dynamical lie algebroids

elliptic schlesinger system and painleve vi we consider an elliptic generalization of the schlesinger system ess with positions of marked points on an elliptic curve and its modular parameter as independent variables the parameters in the moduli space of the complex structure this system was originally discovered by takasaki hep th 9711095 in the quasi classical limit of the sl n vertex model our derivation is purely classical ess is defined as a symplectic quotient of the space of connections of bundles of degree 1 over the elliptic curves with marked points the ess is a non autonomous hamiltonian system with pairwise commuting hamiltonians the system is bi hamiltonian with respect to the linear and introduced here quadratic poisson brackets the latter are the multi colour form of the sklyanin feigin odesski classical algebras the ess is the monodromy independence condition on the complex structure for the linear systems related to the flat bundle the case of four points for a special initial data is reduced to the painleve vi equation in the form of the zhukovsky volterra gyrostat proposed in our previous paper

a comparison of complete global optimization solvers results are reported of testing a number of existing state of the art solvers for global constrained optimization and constraint satisfaction on a set of over 1000 test problems in up to 1000 variables collected from the literature the test problems are available online in ampl and were translated into the input formats of the various solvers using routines from the coconut environment these translators are available online too

characteristic classes and hitchin systems general construction we consider topologically non trivial higgs g bundles over riemann surfaces g with marked points and the corresponding hitchin systems we show that if g is not simply connected then there exists a finite number of different sectors of the higgs bundles endowed with the hitchin hamiltonians they correspond to different characteristic classes of the underlying bundles defined as elements of h 2 sigma g mathcal z g mathcal z g is a center of g we define the conformal version cg of g an analog of gl n for sl n and relate the characteristic classes with degrees of cg bundles we describe explicitly bundles in the genus one g 1 case if 1 has one marked point and the bundles are trivial then the hitchin systems coincide with calogero moser cm systems for the nontrivial bundles we call the corresponding systems the modified calogero moser mcm systems their phase space has the same dimension as the phase space of the cm systems with spin variables but less number of particles and greater number of spin variables starting with these bundles we construct lax operators quadratic hamiltonians and define the phase spaces and the poisson structure using dynamical r matrices the latter are completion of the classification list of etingof varchenko corresponding to the trivial bundles to describe the systems we use a special basis in the lie algebras that generalizes the basis of t hooft matrices for sl n we find that the mcm systems contain the standard cm subsystems related to some unbroken subalgebras the configuration space of the cm particles is the moduli space of the stable holomorphic bundles with non trivial characteristic classes

calogero moser systems and hitchin systems we exhibit the elliptic calogero moser system as a hitchin system of g principal higgs pairs the group g though naturally associated to any root system is not semi simple we then interpret the lax pairs with spectral parameter of d hoker and phong dp1 and bordner corrigan and sasaki bcs1 in terms of equivariant embeddings of the hitchin system of g into that of gl n

painleve vi rigid tops and reflection equation we show that the painleve vi equation has an equivalent form of the non autonomous zhukovsky volterra gyrostat this system is a generalization of the euler top in mathbb c 3 and includes the additional constant gyrostat momentum the quantization of its autonomous version is achieved by the reflection equation the corresponding quadratic algebra generalizes the sklyanin algebra as by product we define integrable xyz spin chain on a finite lattice with new boundary conditions

hitchin systems symplectic hecke correspondence and two dimensional version the aim of this paper is two fold first we define symplectic maps between hitchin systems related to holomorphic bundles of different degrees we call these maps the symplectic hecke correspondence shc of the corresponding higgs bundles they are constructed by means of the modification of the underlying holomorphic bundles shc allows to construct backlund transformations in the hitchin systems defined over riemann curves with marked points we apply the general scheme to the elliptic calogero moser cm system and construct shc to an integrable sl n euler arnold top the elliptic sl n rotator next we propose a generalization of the hitchin approach to 2d integrable theories related to the higgs bundles of infinite rank the main example is an integrable two dimensional version of the two body elliptic cm system the previous construction allows us to define shc between the two dimensional elliptic cm system and the landau lifshitz equation

rigorous lower and upper bounds in linear programming we consider the computation of rigorous lower and upper error bounds for the optimal value of linear programming problems the input data of the lp problem may be exactly given or may vary between given lower and upper bounds the results are then verified for the family of lp problems with input data inside these bounds in many cases only a small computational effort is required for problems with finite simple bounds the rigorous lower bound of the optimal value can be computed with o n2 operations the error bounds can be used as well to perform a sensitivity analysis provided the width of the uncertainties is not too large some numerical examples are presented

automorphism groups of finite dimensional simple algebras we show that if a field k contains sufficiently many elements for instance if k is infinite and k is an algebraically closed field containing k then every linear algebraic k group over k is k isomorphic to aut a k k where a is a finite dimensional simple algebra over k in this paper algebra over a field means nonassociative algebra i e av ector space a over this field with multiplication given by a linear map a a a a1 a2 a1a2 subject to no a priori conditions cf sc fix a field k and an algebraically closed field extension k k our point of view of algebraic groups is that of bor h sp the underlying varieties of linear algebraic groups will be the affine algebraic varieties over k a susual algebraic group resp subgroup homomorphism defined over k is abbreviated to k group resp k subgroup k homomorphism if e f is a field extension and v is a vector space over f w edenote by ve the vector space v f e over e let a be a finite dimensional algebra over k and let v be its underlying vector space the k structure v on vk defines the k structure on the linear algebraic group gl vk as aut ak the full automorphism group of ak is a closed subgroup of gl vk it has the structure of a linear algebraic group if aut ak i sdefined over k that is always the case if k is perfect cf sp 12 1 2 then for each field extension f k the full automorphism group aut af o ff algebra af is the group aut ak f o ff rational points of the algebraic group aut ak

rational top and its classical r matrix we construct a rational integrable system the rational top on a co adjoint orbit of sl n lie group it is described by the lax operator with spectral parameter and classical non dynamical skew symmetric r matrix in the case of the orbit of minimal dimension the model is gauge equivalent to the rational calogero moser cm system to obtain the results we represent the lax operator of the cm model in two different factorized forms without spectral parameter related to the spinless case and another one with the spectral parameter the latter gives rise to the rational top while the first one is related to generalized cremmer gervais r matrices the gauge transformation relating the rational top and cm model provides the classical rational version of the irf vertex correspondence from the geometrical point of view it describes the modification of bundles over degenerated elliptic curve in view of the symplectic hecke correspondence the rational top is related to the rational spin cm model possible applications and generalizations of the suggested construction are discussed in particular the obtained r matrix defines a class of kzb equations

efficient and safe global constraints for handling numerical constraint systems numerical constraint systems are often handled by branch and prune algorithms that combine splitting techniques local consistencies and interval methods this paper first recalls the principles of tt quad a global constraint that works on a tight and safe linear relaxation of quadratic subsystems of constraints then it introduces a generalization of tt quad to polynomial constraint systems it also introduces a method to get safe linear relaxations and shows how to compute safe bounds of the variables of the linear constraint system different linearization techniques are investigated to limit the number of generated constraints tt quadsolver a new branch and prune algorithm that combines tt quad local consistencies and interval methods is introduced tt quadsolver has been evaluated on a variety of benchmarks from kinematics mechanics and robotics on these benchmarks it outperforms classical interval methods as well as constraint satisfaction problem solvers and it compares well with state of the art optimization solvers

geometry and classificatin of solutions of the classical dynamical yang baxter equation the classical yang baxter equation cybe is an algebraic equation central in the theory of integrable systems its nondegenerate solutions were classified by belavin and drinfeld quantization of cybe led to the theory of quantum groups a geometric interpretation of cybe was given by drinfeld and gave rise to the theory of poisson lie groups

a generalisation of the calogero moser system abstract a generalised calogero moser many body system in which the particles possess extra internal degrees of freedom is introduced and solved by the method of olshanetsky and perelomov further the hierarchy of flows commuting with this system is obtained from a hierarchy of linear systems possessing sl n symmetry by the usual method of reduction thus they are shown to be integrable