the quantum heisenberg group h 1 q the structure of the quantum heisenberg group is studied in the two different frameworks of the lie algebra deformations and of the quantum matrix pseudogroups the r matrix connecting the two approaches together with its classical limit r are explicitly calculated by using the contraction technique and the problems connected with the limiting procedure discussed some unusual properties of the quantum enveloping heisenberg algebra are shown

exponential convergence and tractability of multivariate integration for korobov spaces in this paper we study multivariate integration for a weighted korobov space for which the fourier coefficients of the functions decay exponentially fast this implies that the functions of this space are infinitely times differentiable weights of the korobov space monitor the influence of each variable and each group of variables we show that there are numerical integration rules which achieve an exponential convergence of the worst case integration error we also investigate the dependence of the worst case error on the number of variables s and show various tractability results under certain conditions on weights of the korobov space tractability means that the dependence on s is never exponential and sometimes the dependence on s is polynomial or there is no dependence on s at all

quantization of poisson lie groups and applications letg be a connected poisson lie group we discuss aspects of the question of drinfel d can g be quantized and give some answers wheng is semisimple a case where the answer isyes we introduce quantizable poisson subalgebras ofc g related to harmonic analysis ong they are a generalization of f r t models of quantum groups and provide new examples of quantized poisson algebras

groupoid equivariant prequantization in their 2005 paper c laurent gengoux and p xu define prequantization for pre hamiltonian actions of quasi presymplectic lie groupoids in terms of s1 central extensions of lie groupoids the definition requires that the quasi presymplectic structure be exact i e the closed 3 form on the unit space of the lie groupoid must be exact in the present paper we define prequantization for pre hamiltonian actions of not necessarily exact quasi presymplectic lie groupoids in terms of dixmier douady bundles the definition is a natural adaptation of e meinrenken s treatment of prequantization for quasi hamiltonian lie group actions with group valued moment map the definition given in this paper is shown to be compatible with the definition of laurent gengoux and xu when the underlying quasi preysmplectic structure is exact properties related to morita invariance and symplectic reduction are established

classical solutions of the quantum yang baxter equation the classical analogue is developed here for part of the construction in which knot and link invariants are produced from representations of quantum groups whereas previous work begins with a quantum group obtained by deforming the multiplication of functions on a poisson lie group we work directly with a poisson lie groupg and its associated symplectic groupoid the classical analog of the quantumr matrix is a lagrangian submanifold open image in new window in the cartesian square of the symplectic groupoid for any symplectic leafs ing open image in new window induces a symplectic automorphism ofs s which satisfies the set theoretic yang baxter equation when combined with the flip map exchanging components and suitably implanted in each cartesian powers n generates a symplectic action of the braid groupb n ons n application of a symplectic trace formula to the fixed point set of the action of braids should lead to link invariants but work on this last step is still in progress

an r matrix approach to the quantization of the euclidean group e 2 the r matrices for two different deformations of the euclidean group e 2 calculated in a two dimensional representation are used to determine the deformed hopf algebra of the representative functions the duality of the latter with the initial quantum algebras is explicitly proved and the relationship between the two quantum groups is discussed and clarified

unified products for leibniz algebras applications abstract let g be a leibniz algebra and e a vector space containing g as a subspace all leibniz algebra structures on e containing g as a subalgebra are explicitly described and classified by two non abelian cohomological type objects hl g 2 v g provides the classification up to an isomorphism that stabilizes g and hl 2 v g will classify all such structures from the view point of the extension problem here v is a complement of g in e a general product called the unified product is introduced as a tool for our approach the crossed resp bicrossed products between two leibniz algebras are introduced as special cases of the unified product the first one is responsible for the extension problem while the bicrossed product is responsible for the factorization problem the description and the classification of all complements of a given extension g e of leibniz algebras are given as a converse of the factorization problem they are classified by another cohomological object denoted by ha 2 h g where is the canonical matched pair associated to a given complement h several examples are worked out in details

poisson lie group of pseudodifferential symbols we introduce a lie bialgebra structure on the central extension of the lie algebra of differential operators with scalar or matrix coefficients on the line and on the circle this defines a poisson lie structure on the dual group of pseudodifferential symbols of an arbitrary real or complex order we show that the usual second benney gl n kdv or gl n adler gelfand dickey and kp poisson structures are naturally realized as restrictions of this poisson structure to submanifolds of this universal poisson lie group moreover the reduced sl n versions of these manifolds orw n algebras in physical terminology can be viewed as certain subspaces of the quotient of this poisson lie group by the dressing action of the group of functions on the circle or as a result of a poisson reduction finally we define an infinite set of commuting functions on the poisson lie group that give the standard families of hamiltonians when restricted to the submanifolds mentioned above the poisson structure and hamiltonians on the whole group interpolate between the poisson structures and hamiltonians of benney kp and kdv flows we also discuss the geometrical meaning ofw as a limit of poisson algebraswe as e 0

quantization of poisson manifolds from the integrability of the modular function we discuss a framework for quantizing a poisson manifold via the quantization of its symplectic groupoid combining the tools of geometric quantization with the results of renault s theory of groupoid c algebras this setting allows very singular polarizations in particular we consider the case when the modular function is multiplicatively integrable i e when the space of leaves of the polarization inherits a groupoid structure if suitable regularity conditions are satisfied then one can define the quantum algebra as the convolution algebra of the subgroupoid of leaves satisfying the bohr sommerfeld conditions

algebraic structure of multiparameter quantum groups let g be a connected semi simple complex lie group we define and study the multi parameter quantum group cq p g in the case where q is a complex parameter that is not a root of unity using a method of twisting bigraded hopf algebras by a cocycle 2 we develop a unified approach to the construction of cq p g and of the multi parameter drinfeld double dq p using a general method of deforming bigraded pairs of hopf algebras we construct a hopf pairing between these algebras from which we deduce a peterweyl type theorem for cq p g we then describe the prime and primitive spectra of cq p g generalizing a result of joseph in the one parameter case this description was conjectured and established in the sl n case by the first and second authors in 15 16 it was proved in the general case by joseph in 18 19 in particular the orbits in prim cq p g under the natural action of the maximal torus h are indexed as in the one parameter case by the elements of the double weyl group w w unlike the one parameter case there is not in general a bijection between symp g and prim cq p g however in the case when

double products and hypersymplectic structures on ℝ4n in this paper we give a procedure to construct hypersymplectic structures on 4n beginning with affine symplectic data on 2n these structures are shown to be invariant by a 3 step nilpotent double lie group and the resulting metrics are complete and not necessarily flat explicit examples of this construction are exhibited

symplectic structure of the moduli space of flat connection on a riemann surface we consider the canonical symplectic structure on the moduli space of flat g connections on a riemann surface of genus g with a marked points for a being a semisimple lie algebra we obtain an explicit efficient formula for this symplectic form and prove

classification of three dimensional lie bialgebras the paper contains a classification of the three dimensional real and complex lie bialgebras with a description of the corresponding drinfel d doubles

leaf preserving quantizations of poisson su 2 are not coalgebra homomorphisms although it has been found that some deformation quantizations of the poissonsu 2 preserve symplectic leaves and some preserve the group i e coalgebra operation this paper shows that a quantization ofsu 2 cannot be both leaf preserving and group preserving

prime spectra of quantum semisimple groups we study the prime ideal spaces of the quantized function algebras rq g for g a semisimple lie group and q an indeterminate our method is to examine the structure of algebras satisfying a set of seven hypotheses and then to demonstrate using work of joseph hodges and levasseur that the algebras rq g satisfy this list of assumptions rings satisfying the assumptions are shown to satisfy normal separation and therefore jategaonkar s strong second layer condition for such rings much representation theoretic information is carried by the graph of links of the prime spectrum and so we proceed to a detailed study of the prime links of algebras satisfying the list of assumptions homogeneity is a key feature it is proved that the clique of any prime ideal coincides with its orbit under a finite rank free abelian group of automorphisms bounds on the ranks of these groups are obtained in the case of rq g in the final section the results are specialized to the case g sln c where detailed calculations can be used to illustrate the general results as a preliminary set of examples we show also that the multiparameter quantum coordinate rings of affine n space satisfy our axiom scheme when the group generated by the parameters is torsionfree 0 introduction and background 0 1 let p and q be prime ideals of a noetherian ring r we say that p is linked to q and write p q if there is an ideal a of r with pq c a c p n q such that p n q a is a nonzero torsionfree left r p module and a torsionfree right r q module this condition can be simplified in case r satisfies the second layer condition see below in that case p q if and only if p n q pq is faithful as a left r p module and as a right r q module 39 10 1 4 the graph of links of r is the directed graph whose vertices are the points of spec r the set of prime ideals of r with a directed edge from p to q if and only if p q the primes in the connected component of the graph of links containing p constitute the clique of p denoted clique p there is a close connection between the graph of links of r and the representation theory of r cf 20 4 this connection is particularly strong when r satisfies the strong second layer condition which requires that whenever u is a cyclic uniform r module with the annihilator received by the editors november 4 1994 and in revised form september 5 1995 1991 mathematics subject classification primary 16d30 16d60 16p40 17b37 the research of the second author was partially supported by a grant from the national science foundation usa part of the work was carried out while he visited the university of glasgow mathematics department during october 1993 supported by the edinburgh and london mathematical societies work on a revised version of the paper was completed in summer 1995 during a visit by both authors to the department of mathematics of the university of washington whom both thank for its hospitality the travel costs of the first author were in part covered by a grant from the carnegie trust for the universities of scotland 1996 american mathematical society

a class of quadratic matrix algebras arising from the quantized enveloping algebra uq a2n 1 a natural family of quantized matrix algebras is introduced it includes the two best studied such located inside uq a2n 1 it consists of quadratic algebras with the same hilbert series as polynomials in n2 variables we discuss their general properties and investigate some members of the family in great detail with respect to associated varieties degrees centers and symplectic leaves finally the space of rank r matrices becomes a poisson submanifold and there is an associated tensor category of rank r matrices

on poisson actions of compact lie groups on symplectic manifolds let g p be a compact simple poisson lie group equipped with a poisson structure p and m omega be a symplectic manifold assume that m carries a poisson action of g p and there is an equivariant moment map in the sense of lu and weinstein which maps

quantum grassmann manifolds orbits of the quantum dressing transformation forsu q n acting on its solvable dual are introduced the case is considered when the corresponding classical orbits coincide with grassmann manifolds quantization of the poisson bracket on a zariski open subset of the grassmann manifold yields a algebra generated by the quantum coordinate functions the commutation relations are written in a compact form with the help of ther matrix finite dimensional irreducible representations ofu h mathfrak s mathfrak l n mathbb c are derived from the algebra structure

quantum lobachevsky planes we classify all sl 2 r covariant poisson structures on the lobachevsky plane with respect to all multiplicative poisson structures on sl 2 r and describe quantizations for all these poisson structures

replica bounds for optimization problems and diluted spin systems in this paper we generalize to the case of diluted spin models and random combinatorial optimization problems a technique recently introduced by guerra cond mat 0205123 to prove that the replica method generates variational bounds for disordered systems we analyze a family of models that includes the viana bray model the diluted p spin model or random xor sat problem and the random k sat problem showing that the replica cavity method at the various levels of approximation provides systematic schemes to obtain lower bounds of the free energy at all temperatures and of the ground state energy in the case of k sat and xor sat it thus gives upper bounds of the satisfiability threshold our analysis underlines deep connections with the cavity method which are not evident in the long range case

non standard quantum 1 1 poincare group a t matrix approach the hopf algebra dual form for the non standard uniparametric deformation of the 1 1 poincare algebra iso 1 1 is deduced in this framework the quantum coordinates that generate funw iso 1 1 define an infinite dimensional lie algebra the t matrix formalism is used to derive a universal r matrix for both uwiso 1 1 and funw iso 1 1 it is also shown how these results can be generalized for the triangular deformations of 1 1 poincare and galilei algebras that include a spacetime dilation generator

free motion on the poisson su n group is the phase space of the poisson su n we explicitly calculate the symplectic structure of define an analogue of the hamiltonian of the free motion on su n and solve the corresponding equations of motion velocity is related to the momentum by a nonlinear legendre transformation

hochschild and cyclic homology of quantum groups for an arbitrary complex linear semisimple lie groupg we consider hopf algebras of the deformations of the formal and algebraic functions ong the hochschild and cyclic homology of these hopf algebras are computed when the value of the deformation parameter is generic

quantization of the poisson su 2 and its poisson homogeneous space the 2 sphere we show that deformation quantizations of the poisson structures on the poisson lie groupsu 2 and its homogeneous space the 2 sphere are compatible with woronowicz s deformation quantization ofsu 2 s group structure and podles deformation quantization of 2 sphere s homogeneous structure respectively so in a certain sense the multiplicativity of the lie poisson structure onsu 2 at the classical level is preserved under quantization

quadratic poisson algebras of two dimensional classical superintegrable systems and quadratic associative algebras of quantum superintegrable systems the integrals of motion of the classical two dimensional superintegrable systems with quadratic integrals of motion close in a restrained quadratic poisson algebra whose the general form is investigated each classical superintegrable problem has a quantum counterpart a quantum superintegrable system the quadratic poisson algebra is deformed into a quantum associative algebra the finite dimensional representations of this algebra are calculated by using a deformed parafermion oscillator technique it is shown that the finite dimensional representations of the quadratic algebra are determined by the energy eigenvalues of the superintegrable system the calculation of energy eigenvalues is reduced to the solution of algebraic equations which are universal that is for all two dimensional superintegrable systems with quadratic integrals of motion

generalized quantization scheme for central extensions of lie algebras we present the method for finding nonlinear poisson lie group structures on the vector spaces and for their quantization explicit quantization formulas are proposed for arbitrary central extension of a lie algebra

construction of interlaced polynomial lattice rules for infinitely differentiable functions we study multivariate integration over the s dimensional unit cube in a weighted space of infinitely differentiable functions it is known from a recent result by suzuki that there exists a good quasi monte carlo qmc rule which achieves a super polynomial convergence of the worst case error in this function space and moreover that this convergence behavior is independent of the dimension under a certain condition on the weights in this paper we provide a constructive approach to finding a good qmc rule achieving such a dimension independent super polynomial convergence of the worst case error specifically we prove that interlaced polynomial lattice rules with an interlacing factor chosen properly depending on the number of points n and the weights can be constructed using a fast component by component algorithm in at most o sn log n 2 arithmetic operations to achieve a dimension independent super polynomial convergence the key idea for the proof of the worst case error bound is to use a variant of jensen s inequality with a purposely designed concave function

a systematic construction of completely integrable hamiltonians from coalgebras a universal algorithm to construct n particle classical and quantum completely integrable hamiltonian systems from representations of coalgebras with casimir elements is presented in particular this construction shows that quantum deformations can be interpreted as generating structures for integrable deformations of hamiltonian systems with coalgebra symmetry in order to illustrate this general method the algebra and the oscillator algebra are used to derive new classical integrable systems including a generalization of gaudin calogero systems and oscillator chains quantum deformations are then used to obtain some explicit integrable deformations of the previous long range interacting systems and a non coboundary deformation of the 1 1 poincare algebra is shown to provide a new ruijsenaars schneider like hamiltonian

hopf von neumann algebra bicrossproducts kac algebra bicrossproducts and the classical yang baxter equations two groups g1 g2 are said to be a matched pair if each acts on the space of the other and these actions say obey a certain compatibility condition for every matched pair of locally compact groups g1 g2 we construct an associated coinvolutive hopf von neumann algebra m g1 l g2 by simultaneous cross product and cross coproduct for non trivial these bicrossproduct hopf von neumann algebras are non commutative and non cocommutative if the modules for the actions are also matched then these bicrossproducts are kac algebras in this case we show that the dual kac algebra is of the same form with the roles of g1 g2 and of interchanged examples exist with g1 a simply connected lie group and choices of g2 determined by suitable solutions of the classical yang baxter equations on the complexification of the lie algebra of g1

broken replica symmetry bounds in the mean field spin glass model by using a simple interpolation argument in previous work we have proven the existence of the thermodynamic limit for mean field disordered models including the sherrington kirkpatrick model and the derrida p spin model here we extend this argument in order to compare the limiting free energy with the expression given by the parisi ansatz and including full spontaneous replica symmetry breaking our main result is that the quenched average of the free energy is bounded from below by the value given in the parisi ansatz uniformly in the size of the system moreover the difference between the two expressions is given in the form of a sum rule extending our previous work on the comparison between the true free energy and its replica symmetric sherrington kirkpatrick approximation we give also a variational bound for the infinite volume limit of the ground state energy per site

dirac method and symplectic submanifolds in the cotangent bundle of a factorizable lie group we study some symplectic submanifolds in the cotangent bundle of a factorizable lie group defined by second class constraints by applying the dirac method we study many issues of these spaces as fundamental dirac brackets symmetries and collective dynamics as the main application we study integrable systems on these submanifolds as inherited from a system on the whole cotangent bundle meeting in a natural way with the adler kostant symes theory of integrability

the poisson geometry of su 1 1 we study the natural poisson structure on the group su 1 1 and related questions in particular we give an explicit description of the ginzburg weinstein isomorphism for the sets of admissible elements we also establish an analog of thompson s conjecture for this group

three dimensional quantum algebras a cartan like point of view a perturbative quantization procedure for lie bialgebras is introduced the relevance of the choice of a completely symmetrized basis of the quantum universal enveloping algebra is stressed sets of elements of the quantum algebra that play a role similar to generators in the case of lie algebras are considered and a cartan like procedure applied to find a representative for each class of quantum algebras the method is used to construct and classify all three dimensional complex quantum algebras that are compatible with a given type of coproduct the quantization of all lie algebras that in the classical limit belong to the most relevant sector in the classification for three dimensional lie bialgebras given in 1 is thus performed new quantizations of solvable algebras whose simplicity makes them suitable for possible physical applications are obtained and already known related quantum algebras recovered

the local structure of poisson manifolds varietes de poisson et applications decomposition structures de poisson lineaires approximation lineaire systemes hamiltoniens le probleme de linearisation groupes de fonction realisations et applications impulsion paires duales et groupes de jauge existence des realisations unicite des realisations les problemes des 3 corps restreints et autres exemples

quantum line bundles on s2 and method of orbits for suq 2 the quantum sphere is covered by two quantum complex planes quantum line bundle on s2 is introduced by specifying a transition function the structure of the quantum complex planes enables one to construct representations of suq 2 acting in spaces of holomorphic sections in the quantum case dressing transformation replaces coadjoint action

results related to threshold phenomena research in satisfiability lower bounds we present a history of results related to the threhold phenomena which arise in the study of random conjunctive normal form cnf formulas in a companion paper d achlioptas theoret comput sci this volume the major ideas used to achieve many of the lower bounds results on the location of the threshold are described in an informal intuitive manner

classification of real three dimensional poisson lie groups all real three dimensional poisson lie pl groups are explicitly constructed and fully classified under group automorphisms by making use of their one to one correspondence with the complete classification of real three dimensional lie bialgebras given in gomez 2000 j math phys 41 4939 many of these 3d pl groups are non coboundary structures whose poisson brackets are given here for the first time casimir functions for all three dimensional pl groups are given and some features of several pl structures are commented

classical deformations poisson lie contractions and quantization of dual lie bialgebras a poisson hopf algebra of smooth functions is simultaneously constructed on the two dimensional euclidean poincare and heisenberg groups by using a classical r matrix which is invariant under contraction the quantization for this algebra of functions is developed and its dual hopf algebra is also computed contractions on these quantum groups are studied it is shown that within this setting classical deformations are transformed into quantum ones by hopf algebra duality and the quantum heisenberg algebra is derived by means of a dual poisson lie quantization that deforms the standard moyal weyl product

the hyperbolic moduli space of flat connections and the isomorphism of symplectic multiplicity spaces the hyperbolic moduli space of flat connections and the isomorphism of symplectic multiplicity spaces

fast algorithms for component by component construction of rank 1 lattice rules in shift invariant reproducing kernel hilbert spaces we reformulate the original component by component algorithm for rank 1 lattices in a matrix vector notation so as to highlight its structural properties for function spaces similar to a weighted korobov space we derive a technique which has construction cost o snlog n in contrast with the original algorithm which has construction cost o sn 2 herein s is the number of dimensions and n the number of points taken prime in contrast to other approaches to speed up construction our fast algorithm computes exactly the same quantity as the original algorithm the presented algorithm can also be used to construct randomly shifted lattice rules in weighted sobolev spaces

decomposable representations and lagrangian submanifolds of moduli spaces associated to surface groups the importance of explicit examples of lagrangian submanifolds of moduli spaces is revealed by papers such as dostoglou and salamon ann of math 2 139 3 581 640 1994 and salamon proceedings of the international congress of mathematicians vol 1 2 zurich 1994 pp 526 536 birkhauser basel 1995 given a 3 manifold m with boundary m dostoglou and salamon use such examples to obtain a proof of the atiyah floer conjecture relating the symplectic floer homology of the representation space hom 1 m u u associated to an explicit pair of lagrangian submanifolds of this representation space and the instanton homology of the 3 manifold m in the present paper we construct a lagrangian submanifold of the space of representations mathcal m g l hom mathcal c pi g l u u of the fundamental group g lof a punctured riemann surface g l into an arbitrary compact connected lie group u this lagrangian submanifold is obtained as the fixed point set of an anti symplectic involution hat beta defined on mathcal m g l we show that the involution hat beta is induced by a form reversing involution defined on the quasi hamiltonian space u times u g times mathcal c 1 times cdots times mathcal c l the fact that hat beta has a non empty fixed point set is a consequence of the real convexity theorem for group valued momentum maps proved in schaffhauser a real convexity theorem for quasi hamiltonian actions submitted 25 p 2007 http arxiv org abs math 0705 0858 the notion of decomposable representation provides a geometric interpretation of the lagrangian submanifold thus obtained

symplectic structures associated to lie poisson groups the lie poisson analogues of the cotangent bundle and coadjoint orbits of a lie group are considered for the natural poisson brackets the symplectic leaves in these manifolds are classified and the corresponding symplectic forms are described thus the construction of the kirillov symplectic form is generalized for lie poisson groups

stokes phenomenon gelfand zeitlin systems and relative ginzburg weinstein linearization abstract in 2007 alekseev meinrenken proved that there exists a ginzburg weinstein diffeomorphism from the dual lie algebra u n to the dual poisson lie group u n compatible with the gelfand zeitlin integrable systems in this paper we explicitly construct such diffeomorphisms via stokes phenomenon and boalch s dual exponential maps then we introduce a relative version of the ginzburg weinstein linearization motivated by irregular riemann hilbert correspondence and generalize the results of enriquez etingof marshall to this relative setting in particular we prove the connection matrix for a certain irregular riemann hilbert problem satisfies a relative gauge transformation equation of the alekseev meinrenken dynamical r matrices this gauge equation is then derived as the semiclassical limit of the relative drinfeld twist equation

multivariate integration of infinitely many times differentiable functions in weighted korobov spaces we study multivariate integration for a weighted korobov space of periodic infinitely many times differentiable functions for which the fourier coefficients decay exponentially fast the weights are defined in terms of two non decreasing sequences a ai and b bi of numbers no less than one and a parameter 0 1 let e n s be the minimal worst case error of all algorithms that use n function values in the s variate case we would like to check conditions on a b and such that e n s decays exponentially fast i e for some q 0 1 and p 0 we have e n s o q n p as n goes to infinity the factor in the o notation may depend on s in an arbitrary way we prove that exponential convergence holds iff b i 1 1 bi independently of a and furthermore the largest p of exponential convergence is 1 b we also study exponential convergence with weak polynomial and strong polynomial tractability this means that e n s c s q n p for all n and s and with log c s exp o s for weak tractability with a polynomial bound on log c s for polynomial tractability and with uniformly bounded c s for strong polynomial tractability we prove that the notions of weak polynomial and strong polynomial tractability are equivalent and hold iff b and ai are exponentially growing with i we also prove that the largest or the supremum of p for exponential convergence with strong polynomial tractability belongs to 1 2b 1 b

quantum structure of the motion groups of the two dimensional cayley klein geometries a simultaneous and global scheme of quantum deformation is defined for the set of algebras corresponding to the groups of motions of the two dimensional cayley klein geometries their central extensions are also considered under this unified pattern in both cases some fundamental properties characterizing the classical ck geometries as the existence of a set of commuting involutions contractions and dualities relationships remain in the quantum version

good interlaced polynomial lattice rules for numerical integration in weighted walsh spaces quadrature rules using higher order digital nets and sequences are known to exploit the smoothness of a function for numerical integration and to achieve an improved rate of convergence as compared to classical digital nets and sequences for smooth functions a construction principle of higher order digital nets and sequences based on a digit interlacing function was introduced in dick 2007 which interlaces classical digital nets or sequences whose number of components is a multiple of the dimension in this paper we study the use of polynomial lattice point sets for interlaced components we call quadrature rules using such point sets interlaced polynomial lattice rules we consider weighted walsh spaces containing smooth functions and derive two upper bounds on the worst case error for interlaced polynomial lattice rules both of which can be employed as a quality criterion for the construction of interlaced polynomial lattice rules we investigate the component by component construction and the korobov construction as a means of explicit constructions of good interlaced polynomial lattice rules that achieve the optimal rate of the worst case error through this approach we are able to obtain a good dependence of the worst case error bounds on the dimension under certain conditions on the weights while significantly reducing the construction cost as compared to higher order polynomial lattice rules

double bracket equations and geodesic flows on symmetric spaces in this paper we consider the geometry of hamiltonian flows on the cotangent bundle of coadjoint orbits of compact lie groups and on symmetric spaces a key idea here is the use of the normal metric to define the kinetic energy this leads to hamiltonian flows of the double bracket type we analyze the integrability of geodesic flows according to the method of thimm we obtain via the double bracket formalism a quite explicit form of the relevant commuting flows and a correspondingly transparent proof of involutivity we demonstrate for example integrability of the geodesic flow on the real and complex grassmannians we also consider right invariant systems and the generalized rigid body equations in this setting

minimal contagious sets in random regular graphs the bootstrap percolation or threshold model is a dynamic process modelling the propagation of an epidemic on a graph where inactive vertices become active if their number of active neighbours reach some threshold we study an optimization problem related to it namely the determination of the minimal number of active sites in an initial configuration that leads to the activation of the whole graph under this dynamics with and without a constraint on the time needed for the complete activation this problem encompasses in special cases many extremal characteristics of graphs like their independence decycling or domination number and can also be seen as a packing problem of repulsive particles we use the cavity method including the effects of replica symmetry breaking an heuristic technique of statistical mechanics many predictions of which have been confirmed rigorously in the recent years we have obtained in this way several quantitative conjectures on the size of minimal contagious sets in large random regular graphs the most striking being that 5 regular random graph with a threshold of activation of 3 resp 6 regular with threshold 4 have contagious sets containing a fraction 1 6 resp 1 4 of the total number of vertices equivalently these numbers are the minimal fraction of vertices that have to be removed from a 5 regular resp 6 regular random graph to destroy its 3 core we also investigated survey propagation like algorithmic procedures for solving this optimization problem on single instances of random regular graphs

quantum dressing orbits on compact groups the quantum double is shown to imply the dressing transformation on quantum compact groups and the quantum iwasawa decompositon in the general case quantum dressing orbits are described explicitly as algebras the dual coalgebras consisting of differential operators are related to the quantum weyl elements besides the differential geometry on a quantum leaf allows a remarkably simple construction of irreducible representations of the algebras of quantum functions representation spaces then consist of analytic functions on classical phase spaces these representations are also interpreted in the framework of quantization in the spirit of berezin applied to symplectic leaves on classical compact groups convenient coherent states are introduced and a correspondence between classical and quantum observables is given

lie poisson structure on some poisson lie groups poisson lie groups appeared in the work of drinfel d see e g drl dr2 as classical objects corresponding to quantum groups going in the other direction we may say that a poisson lie group is a group of symmetries of a phase space that are allowed to twist in a certain sense the symplectic or poisson structure the poisson structure on the group controls this twisting in a precise way quantizing both the phase space and the symmetries one may obtain a quantum group acting on a quantum phase space in recent work lu and ratiu lr used so called standard poisson structures on a compact semisimple lie group k and on its poisson dual k in order to give a new proof of the nonlinear convexity theorem of kostant ko their method is analogous to the famous symplectic proof of the linear convexity theorem given by atiyah a and guillemin and stemnberg gs the nonlinear convexity theorem like the linear one follows from a very general result on convexity of the image of the momentum map a gs however in lr it is applied not to a coadjoint orbit in t but to a symplectic leaf in k the main result of this paper is that the standard poisson structure on the poisson dual k to a compact semisimple poisson lie group k is actually isomorphic to the linear one on t this theorem seems to be related to some facts in the theory of quantum groups namely for generic q the universal enveloping algebra u t and its quantum deformation uq t are isomorphic as algebras though not as coalgebras of course since the quantum version is not cocommutative in particular there is a bijective correspondence between their representations a direct connection between our work and its quantum analogues though is still to be found the present work supplies a positive answer to question 5 1 in lr and strongly depends on that paper to simplify reading we keep the notations of lr wherever possible on one hand but give all necessary definitions on the other our work is also related to du in which the nonlinear convexity theorem is reduced to the linear one by a deformation argument not unlike the one that we use in 5 the paper is organized as follows in 2 we define poisson lie groups discuss

higher order qmc petrov galerkin discretization for affine parametric operator equations with random field inputs we construct quasi monte carlo methods to approximate the expected values of linear functionals of petrov galerkin discretizations of parametric operator equations which depend on a possibly infinite sequence of parameters such problems arise in the numerical solution of differential and integral equations with random field inputs we analyze the regularity of the solutions with respect to the parameters in terms of the rate of decay of the fluctuations of the input field if p in 0 1 denotes the summability exponent corresponding to the fluctuations in affine parametric families of operators then we prove that deterministic interlaced polynomial lattice rules of order alpha lfloor 1 p rfloor 1 in s dimensions with n points can be constructed using a fast component by component algorithm in mathcal o alpha s n log n alpha 2 s 2 n operations to achieve a convergence rate of mathcal o n 1 p with the implied constant independent of s this dimension independen

quaternionic and poisson lie structures in three dimensional gravity the cosmological constant as deformation parameter each of the local isometry groups arising in three dimensional 3d gravity can be viewed as a group of unit split quaternions over a ring which depends on the cosmological constant in this paper we explain and prove this statement and use it as a unifying framework for studying poisson structures associated with the local isometry groups we show that in all cases except for the case of euclidean signature with positive cosmological constant the local isometry groups are equipped with the poisson lie structure of a classical double we calculate the dressing action of the factor groups on each other and find among others a simple and unified description of the symplectic leaves of su 2 and sl 2 r we also compute the poisson structure on the dual poisson lie groups of the local isometry groups and on their heisenberg doubles together they determine the poisson structure of the phase space of 3d gravity in the so called combinatorial description

random weights robust lattice rules and the geometry of the cbcrc algorithm in this paper we study lattice rules which are cubature formulae to approximate integrands over the unit cube 0 1 s from a weighted reproducing kernel hilbert space we assume that the weights are independent random variables with a given mean and variance for two reasons stemming from practical applications i it is usually not known in practice how to choose the weights thus by assuming that the weights are random variables we obtain robust constructions with respect to the weights of lattice rules this to some extend removes the necessity to carefully choose the weights ii in practice it is convenient to use the same lattice rule for many different integrands the best choice of weights for each integrand may vary to some degree hence considering the weights random variables does justice to how lattice rules are used in applications in this paper the worst case error is therefore a random variable depending on random weights we show how one can construct lattice rules which perform well for weights taken from a set with large measure such lattice rules are therefore robust with respect to certain changes in the weights the construction algorithm uses the component by component cbc idea based on two criteria one using the mean of the worst case error and the second criterion using a bound on the variance of the worst case error we call the new algorithm the cbc2c component by component with 2 constraints algorithm we also study a generalized version which uses r constraints which we call the cbcrc component by component with r constraints algorithm we show that lattice rules generated by the cbcrc algorithm simultaneously work well for all weights in a subspace spanned by the chosen weights 1 r thus in applications instead of finding one set of weights it is enough to find a convex polytope in which the optimal weights lie the price for this method is a factor r in the upper bound on the error and in the construction cost of the lattice rule thus the burden of determining one set of weights very precisely can be shifted to the construction of good lattice rules numerical results indicate the benefit of using the cbc2c algorithm for certain choices of weights

sharp thresholds of graph properties and the sat problem consider g n p to be the probability space of random graphs on n vertices with edge probability p we will be considering subsets of this space defined by monotone graph properties a monotone graph property p is a property of graphs such that a p is invariant under graph automorphisims b if graph h has property p then so does any graph g having h as a subgraph a monotone symmetric family of graphs is a family defined by such a property one of the first observations made about random graphs by erdos and renyi in their seminal work on random graph theory 12 was the existence of threshold phenomena the fact that for many interesting properties p the probability of p appearing in g n p exhibits a sharp increase at a certain critical value of the parameter p bollobas and thomason proved the existence of threshold functions for all monotone set properties 6 and in 14 it is shown that this behavior is quite general and that all monotone graph properties exhibit threshold behavior i e the probability of their appearance increases from values very close to 0 to values close to 1 in a very small interval more precise analysis of the size of the threshold interval is done in 7 this threshold behavior which occurs in various settings which arise in combinatorics and computer science is an instance of the phenomenon of phase transitions which is the subject of much interest in statistical physics one of the main questions that arises in studying phase transitions is how sharp is the transition for example one of the motivations for this paper arose from the question of the sharpness of the phase transition for the property of satisfiability of a random kcnf boolean formula nati linial who introduced me to this problem suggested that although much concrete analysis was being performed on this problem the best approach would be to find general conditions for sharpness of the phase transition answering the question posed in 14 as to the relation between the length of the threshold interval and the value of the critical probability in this paper we indeed introduce a simple condition and prove it is sufficient stated roughly in the setting of random graphs the main theorem states that if a property has a coarse threshold then it can be approximated by the property of having certain given graphs as a subgraph this condition can be applied in a more

minimizing energy below the glass thresholds focusing on the optimization version of the random k satisfiability problem the max k sat problem we study the performance of the finite energy version of the survey propagation algorithm we show that a simple linear time backtrack decimation strategy is sufficient to reach configurations well below the lower bound for the dynamic threshold energy and very close to the analytic prediction for the optimal ground states a comparative numerical study on one of the most efficient local search procedures is also given

survey propagation at finite temperature application to a sourlas code as a toy model in this paper we investigate a finite temperature generalization of survey propagation by applying it to the problem of finite temperature decoding of a biased finite connectivity sourlas code for temperatures lower than the nishimori temperature we observe that the result is a shift of the location of the dynamical critical channel noise to larger values than the corresponding dynamical transition for belief propagation as suggested recently by migliorini and saad for ldpc codes we show how the finite temperature 1rsb sp gives accurate results in the regime where competing approaches fail to converge or fail to recover the retrieval state

phase transitions and the search problem we describe how techniques that were originally developed in statistical mechanics can be applied to search problems that arise commonly in artificial intelligence this approach is useful for understanding the typical behavior of classes of problems in particular these techniques predict that abrupt changes in computational cost analogous to physical phase transitions should occur universally as heuristic effectiveness or search space topology is varied we also present a number of open questions raised by these studies

quantized algebras of functions on homogeneous spaces with poisson stabilizers let g be a simply connected semisimple compact lie group with standard poisson structure k a closed poisson lie subgroup 0 q 1 we study a quantization c gq kq of the algebra of continuous functions on g k using results of soibelman and dijkhuizen stokman we classify the irreducible representations of c gq kq and obtain a composition series for c gq kq we describe closures of the symplectic leaves of g k refining the well known description in the case of flag manifolds in terms of the bruhat order we then show that the same rules describe the topology on the spectrum of c gq kq next we show that the family of c algebras c gq kq 0 q 1 has a canonical structure of a continuous field of c algebras and provides a strict deformation quantization of the poisson algebra mathbb c g k finally extending a result of nagy we show that c gq kq is canonically kk equivalent to c g k

quadratic poisson algebras and their infinite dimensional extensions a certain generalization of the classical modified yang baxter equation is presented also certain poisson algebras arising in the calculus of variations are investigated

explicit constructions of quasi monte carlo rules for the numerical integration of high dimensional periodic functions in this paper we give explicit constructions of point sets in the s dimensional unit cube yielding quasi monte carlo algorithms which achieve the optimal rate of convergence of the worst case error for numerically integrating high dimensional periodic functions in the classical measure p alpha of the worst case error introduced by korobov the convergence for every even integer alpha ge 1 is of cal o n min alpha d log n s alpha 2 where d is a parameter of the construction which can be chosen arbitrarily large and n is the number of quadrature points this convergence rate is known to be the best possible up to some log n factors we prove the result for the deterministic and also a randomized setting the construction is based on a suitable extension of digital t m s nets over the finite field mathbb z b

integrable potentials on spaces with curvature from quantum groups a family of classical integrable systems defined on a deformation of the two dimensional sphere hyperbolic and anti de sitter spaces is constructed through hamiltonians defined on the non standard quantum deformation of a sl 2 poisson coalgebra all these spaces have a non constant curvature that depends on the deformation parameter z as particular cases the analogues of the harmonic oscillator and kepler coulomb potentials on such spaces are proposed another deformed hamiltonian is also shown to provide superintegrable systems on the usual sphere hyperbolic and anti de sitter spaces with a constant curvature that exactly coincides with z according to each specific space the resulting potential is interpreted as the superposition of a central harmonic oscillator with either two more oscillators or centrifugal barriers the non deformed limit z 0 of all these hamiltonians can then be regarded as the zero curvature limit contraction which leads to the corresponding super integrable systems on the flat euclidean and minkowskian spaces

on the geometry of lie algebras and poisson tensors a geometric programme to analyse the structure of lie algebras is presented with special emphasis on the geometry of linear poisson tensors the notion of decomposable poisson tensors is introduced and an algorithm to construct all solvable lie algebras is presented poisson liouville structures are also introduced to discuss a new class of lie algebras which include as a subclass semi simple lie algebras a decomposition theorem for poisson tensors is proved for a class of poisson manifolds including linear ones simple lie algebras are also discussed from this viewpoint and lower dimensional real lie algebras are analysed

walsh spaces containing smooth functions and quasi monte carlo rules of arbitrary high order we define a walsh space which contains all functions whose partial mixed derivatives up to order delta ge 1 exist and have finite variation in particular for a suitable choice of parameters this implies that certain sobolev spaces are contained in these walsh spaces for this walsh space we then show that quasi monte carlo rules based on digital t alpha s sequences achieve the optimal rate of convergence of the worst case error for numerical integration this rate of convergence is also optimal for the subspace of smooth functions explicit constructions of digital t alpha s sequences are given hence providing explicit quasi monte carlo rules which achieve the optimal rate of convergence of the integration error for arbitrarily smooth functions

random k satisfiability problem from an analytic solution to an efficient algorithm we study the problem of satisfiability of randomly chosen clauses each with k boolean variables using the cavity method at zero temperature we find the phase diagram for the k 3 case we show the existence of an intermediate phase in the satisfiable region where the proliferation of metastable states is at the origin of the slowdown of search algorithms the fundamental order parameter introduced in the cavity method which consists of surveys of local magnetic fields in the various possible states of the system can be computed for one given sample these surveys can be used to invent new types of algorithms for solving hard combinatorial optimizations problems one such algorithm is shown here for the k 3 satisfiability problem with very good performances

lattice rules for nonperiodic smooth integrands the aim of this paper is to show that one can achieve convergence rates of n alpha delta n for alpha 1 2 1 2 and for delta 0 0 arbitrarily small for nonperiodic alpha smooth cosine series using lattice rules without random shifting the smoothness of the functions can be measured by the decay rate of the cosine coefficients for a specific choice of the parameters the cosine series space coincides with the unanchored sobolev space of smoothness 1 we study the embeddings of various reproducing kernel hilbert spaces and numerical integration in the cosine series function space and show that by applying the so called tent transformation to a lattice rule one can achieve the almost optimal rate of convergence of the integration error the same holds true for symmetrized lattice rules for the tensor product of the direct sum of the korobov space and cosine series space but with a stronger dependence on the dimension in this case

irreducible decomposition for tensor product representations of jordanian quantum algebras tensor products of irreducible representations of the jordanian quantum algebras and are considered for both the highest weight finite dimensional representations of and the lowest weight infinite dimensional ones of it is shown that tensor product representations are reducible and that the decomposition rules to irreducible representations are exactly the same as those of corresponding lie algebras

deformation quantization of heisenberg manifolds form a smooth manifold equipped with a poisson bracket we formulate ac algebra framework for deformation quantization including the possibility of invariance under a lie group of diffeomorphisms preserving the poisson bracket we then show that the much studied non commutative tori give examples of such deformation quantizations invariant under the usual action of ordinary tori going beyond this the main results of the paper provide a construction of invariant deformation quantizations for those poisson brackets on heisenberg manifolds which are invariant under the action of the heisenberg lie group and for various generalizations suggested by this class of examples interesting examples are obtained of simplec algebras on which the heisenberg group acts ergodically

non coboundary poisson lie structures on the book group all possible poisson lie pl structures on the 3d real lie group generated by a dilation and two commuting translations are obtained their classification is fully performed by relating these pl groups to the corresponding lie bialgebra structures on the corresponding book lie algebra by construction all these poisson structures are quadratic poisson hopf algebras for which the group multiplication is a poisson map in contrast to the case of simple lie groups it turns out that most of the pl structures on the book group are non coboundary ones moreover from the viewpoint of poisson dynamics the most interesting pl book structures are just some of these non coboundaries which are explicitly analysed in particular we show that the two different q deformed poisson versions of the sl 2 r algebra appear as two distinguished cases in this classification as well as the quadratic poisson structure that underlies the integrability of a large class of 3d lotka volterra equations finally the quantization problem for these pl groups is sketched

ginzburg weinstein via gelfand zeitlin let u n be the unitary group and u n the dual of its lie al gebra equipped with the kirillov poisson structure in their 1983 paper guillemin sternberg introduced a densely defined hamil tonian action of a torus of dimension n 1 n 2 on u n with mo ment map given by the gelfand zeitlin coordinates a few years later flaschka ratiu described a similar multiplicative gelfand zeitlin system for the poisson lie group u n by the ginzburg weinstein theorem u n is isomorphic to u n as a poisson manifold flaschka ratiu conjectured that one can choose the ginzburg weinstein diffeomorphism in such a way that it intertwines the linear and nonlinear gelfand zeitlin systems our main result gives a proof of this conjecture and produces a canonical ginzburg weinstein diffeomorphism

instability of one step replica symmetry broken phase in satisfiability problems we reconsider the one step replica symmetry breaking 1rsb solutions of two random combinatorial problems k xorsat and k sat we present a general method for establishing the stability of these solutions with respect to further steps of replica symmetry breaking our approach extends the ideas of montanari and ricci tersenghi 2003 eur phys j b 33 339 to more general combinatorial problems it turns out that 1rsb is always unstable at sufficiently small clause density or high energy in particular the recent 1rsb solution to 3 sat is unstable at zero energy for m with m 4 153 on the other hand the sat unsat phase transition seems to be correctly described within 1rsb

a family of poisson structures on hermitian symmetric spaces we investigate the compatibility of symplectic kirillov kostant souriau structure and poisson lie structure on coadjoint orbits of semisimple lie group we prove that they are compatible for an orbit compact lie group iff the orbit is hermitian symmetric space we prove also the compatibility statement for non compact hermitian symmetric space as an example we describe a structure of symplectic leaves oncp n for this family these leaves may be considered as a perturbation of schubert cells possible applications to infinite dimensional examples are discussed