poisson involutions spin calogero moser systems associated with symmetric lie subalgebras and the symmetric space spin ruijsenaars schneider models we develop a general scheme to construct integrable systems starting from realizations in symmetric coboundary dynamical lie algebroids and symmetric coboundary dynamical poisson groupoids the method is based on the successive use of dirac reduction and poisson reduction then we show that certain spin calogero moser systems associated with symmetric lie subalgebras can be studied in this fashion we also consider some spin generalized ruisjenaars schneider equations which correspond to the n soliton solutions of a 1 n affine toda field theory in this case we show how the equations are obtained from the dirac reduction of some hamiltonian system on a symmetric coboundary dynamical poisson groupoid

poisson groups and differential galois theory of schroedinger equation on the circle we combine the projective geometry approach to schroedinger equations on the circle and differential galois theory with the theory of poisson lie groups to construct a natural poisson structure on the space of wave functions at the zero energy level applications to kdv like nonlinear equations are discussed the same approach is applied to 2nd order difference operators on a one dimensional lattice yielding an extension of the lattice poisson virasoro algebra

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

auditory perceptual interpretation of the vowel an account of the simple vowels of american english will be given in terms of the author s auditory perceptual theory data from many sources will be used to illustrate and support this view in addition to theoretical suggestions concerning an auditory perceptual transformation and a segmentation rule the perceptualized spectral patterns associated with the nonretroflex vowels will be shown to fall into regions within a slab in a three dimensional auditory perceptual space these regions form a vowel map that by simple rotations can be related to various vowel charts such as those of jones pike or fant while the presentheory is an elaboration of traditional formant ratio theory it will be shown that a concept of a perceptual reference is needed not only for talker normalization but also to disambiguate vowels not distinguished by simple formant ratio theory additionally loci for the retroflex and nasalized vowels illustrate the utility of describing vowel spectra by five numbers combined into three dimensions simple f1 by f2 descriptions are not adequate except for enigmatophiles work supported by nincds and afosr

material point methods applied to one dimensional shock waves and dual domain material point method with sub points using a simple one dimensional shock problem as an example the present paper investigates numerical properties of the original material point method mpm the generalized interpolation material point gimp method the convected particle domain interpolation cpdi method and the dual domain material point ddmp method for a weak isothermal shock of ideal gas the mpm cannot be used with accuracy with a small number of particles per cell gimp and cpdi produce reasonable results however as the number of particles increases the methods fail to converge and produce pressure spikes the ddmp method behaves in an opposite way with a small number of particles per cell ddmp results are unsatisfactory as the number of particles increases the ddmp results converge to correct solutions but the large number of particles needed for convergence makes the method very expensive to use in these types of shock wave problems in two or three dimensional cases the cause for producing the unsatisfactory ddmp results is identified a simple improvement to the method is introduced by using sub points with this improvement the ddmp method produces high quality numerical solutions with a very small number of particles although in the present paper the numerical examples are one dimensional all derivations are for multidimensional problems with the technique of approximately tracking particle domains of cpdi the extension of this sub point method to multidimensional problems is straightforward this new method preserves the conservation properties of the ddmp method which conserves mass and momentum exactly and conserves energy to the second order in both spatial and temporal discretizations

an implicit particle in cell method for granular materials an implicit in time method for granular materials is described the method combines the material point method a first order contact algorithm and a newton krylov equation solver to give improved energy conservation stabilization of the finite grid instability and the correct description of collisions between grains

a quantum duality principle for coisotropic subgroups and poisson quotients we develop a quantum duality principle for coisotropic subgroups of a formal poisson group and its dual namely starting from a quantum coisotropic subgroup for a quantization of a given poisson group we provide functorial recipes to produce quantizations of the dual coisotropic subgroup in the dual formal poisson group by the natural link between subgroups and homogeneous spaces we argue a quantum duality principle for poisson homogeneous spaces which are poisson quotients i e have at least one zero dimensional symplectic leaf as an application we provide an explicit quantization of the homogeneous sln space of stokes matrices with the poisson structure given by dubrovin and ugaglia

material point method applied to multiphase flows the particle in cell method pic especially the latest version of it the material point method mpm has shown significant advantage over the pure lagrangian method or the pure eulerian method in numerical simulations of problems involving large deformations it avoids the mesh distortion and tangling issues associated with lagrangian methods and the advection errors associated with eulerian methods its application to multiphase flows or multi material deformations however encounters a numerical difficulty of satisfying continuity requirement due to the inconsistence of the interpolation schemes used for different phases it is shown in section 3 that current methods of enforcing this requirement either leads to erroneous results or can cause significant accumulation of errors in the present paper a different numerical method is introduced to ensure that the continuity requirement is satisfied with an error consistent with the discretization error and will not grow beyond that during the time advancement in the calculation this method is independent of physical models its numerical implementation is quite similar to the common method used in eulerian calculations of multiphase flows examples calculated using this method are presented

material point method with enriched shape function for crack problems abstract a material point method mpm generalized interpolation material point method gimp with enriched shape function empm egimp for short is proposed for modelling crack problems in the mpm gimp framework the empm egimp enriches the nodal degrees of freedom based on the idea of the extended finite element method xfem this improvement allows the crack problem whose displacement and velocity are discontinuous to be simulated by only one set of background grid meshes so multigrid and multiple velocity fields are unnecessary in addition the technique we developed to lump the mass matrix makes the empm egimp can be implemented easily in a conventional mpm gimp code if there is no crack the empm egimp degenerates to the conventional mpm gimp the level set method lsm is employed in the empm egimp for easily tracking crack surfaces the crack moves with the material points physically so that the lsm function is carried by particles in each time step the lsm function value is mapped to the grid nodes from the particles and the nodes to be enriched can be identified conveniently from the grid nodal lsm function values numerical experiments for stress fields distribution fracture parameters calculation and crack propagation are provided to validate the proposed empm egimp

an adaptive finite element material point method and its application in extreme deformation problems abstract taking advantages of both lagrangian and eulerian methods material point method mpm is suitable for modeling problems with extreme deformation however mpm is less accurate and less efficient than finite element method fem for small deformation problems due to particle quadrature and mappings between particles and background grid applied in mpm to take advantages of both fem and mpm an adaptive finite element material point method is developed for modeling the dynamic behavior of material under extreme loading bodies are initially discretized by finite elements and then the elements with large strain are adaptively converted into mpm particles based on their degree of distortion or plastic strain during the solution process the interaction between the remaining finite elements and mpm particles is implemented based on the background grid in mpm framework several numerical examples are presented to validate the efficiency and accuracy of the proposed method and the numerical results are in good agreement with experiments while the efficiency of the method is higher than that of both mpm and fem

an assessment of acoustic contrast between long and short vowels using convex hulls an alternative to the spectral overlap assessment metric soam first introduced by wassink 2006 j acoust soc am 119 4 2334 2350 is introduced the soam quantifies the intra and inter language differences between long short vowel pairs through a comparison of spectral f1 f2 and temporal properties modeled with best fit ellipses f1 f2 space and ellipsoids f1 f2 duration however the soam ellipses and ellipsoids rely on a gaussian distribution of vowel data and a dense dataset neither of which can be assumed in endangered languages or languages with limited available data the method presented in this paper called the vowel overlap assessment with convex hulls voach method improves upon the earlier metric through the use of best fit convex shapes the voach method reduces the incorporation of empty data into calculations of vowel space both methods are applied to numu oregon northern paiute an endangered language of the western united states calculations from the voac

a geometric representation of spectral and temporal vowel features quantification of vowel overlap in three linguistic varieties a geometrical method for computing overlap between vowel distributions the spectral overlap assessment metric soam is applied to an investigation of spectral f1 f2 and temporal duration relations in three different types of systems one claimed to exhibit primary quality american english one primary quantity jamaican creole and one about which no claims have been made jamaican english shapes orientations and proximities of pairs of vowel distributions involved in phonological oppositions are modeled using best fit ellipses in f1 x f2 space and ellipsoids f1 x f2 x duration overlap fractions computed for each pair suggest that spectral and temporal features interact differently in the three varieties and oppositions under a two dimensional analysis two of three american english oppositions show no overlap the third shows partial overlap all jamaican creole oppositions exhibit complete overlap when f1 and f2 alone are modeled but no or partial overlap with incorporation of a factor for duration jamaican english three dimensional overlap fractions resemble two dimensional results for american english a multidimensional analysis tool such as soam appears to provide a more objective basis for simultaneously investigating spectral and temporal relations within vowel systems normalization methods and the soam method are described in an extended appendix

material point method enhanced by modified gradient of shape function a numerical scheme of computing quantities involving gradients of shape functions is introduced for the material point method mpm so that the quantities are continuous as material points move across cell boundaries the noise and instability caused by cell crossing of the material points are then eliminated in this scheme the formulas used to compute these quantities can be expressed in the same forms as in the original material point method but with the gradient of the shape function modified for one dimensional cases the gradient of the shape function used in the generalized interpolation material point gimp method is a special case of the modified gradient if the characteristic function of a material point is introduced the characteristic function of a material point is not otherwise needed in this scheme therefore difficulties in tracking its evolution are avoided although the support of the modified gradient of a shape function is enlarged from the cell containing the material point to also include the immediate neighbor cells all the non local effects of a material point can be accounted for by two consecutive local operations therefore this scheme can be used in calculations with unstructured grids this scheme is proved to satisfy mass and momentum conservations exactly the error in energy conservation is shown to be second order on both spatial and temporal discretizations although the error in energy conservation is the same order as that in the original material point method numerical examples show that this scheme has significantly better energy conservation properties than those of the original material point method

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

from klein to painleve via fourier laplace and jimbo we describe a method for constructing explicit algebraic solutions to the sixth painleve equation generalising that of dubrovin and mazzocco there are basically two steps first we explain how to construct finite braid group orbits of triples of elements of rm sl 2 rm c out of triples of generators of three dimensional complex reflection groups this involves the fourier laplace transform for certain irregular connections then we adapt a result of jimbo to produce the painleve vi solutions in particular this solves a riemann hilbert problem explicitly each step is illustrated using the complex reflection group associated to klein s simple group of order 168 this leads to a new algebraic solution with seven branches we also prove that unlike the algebraic solutions of dubrovin and mazzocco and hitchin this solution is not equivalent to any solution coming from a finite subgroup of rm sl 2 rm c the results of this paper also yield a simple proof of a recent theorem of inaba iwasaki and saito on the action of okamoto s affine rm d 4 symmetry group as well as the correct connection formulae for generic painleve vi equations

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

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

a rigid irregular connection on the projective line in this paper we construct a connection on the trivial g bundle on p 1 for any simple complex algebraic group g which is regular outside of the points 0 and has a regular singularity at the point 0 with principal unipotent monodromy and has an irregular singularity at the point with slope 1 h the reciprocal of the coxeter number of g the connection which admits the structure of an oper in the sense of beilinson and drinfeld appears to be the characteristic 0 counterpart of a hypothetical family of l adic representations which should parametrize a specific automorphic representation under the global langlands correspondence these l adic representations and their characteristic 0 counterparts have been constructed in some cases by deligne and katz our connection is constructed uniformly for any simple algebraic group and characterized using the formalism of opers it provides an example of the geometric langlands correspondence with wild ramification we compute the de rham cohomology of our connection with values in a representation v of g and describe the differential galois group of as a subgroup of g

distribution coefficient algorithm for small mass nodes in material point method when using the time explicit material point method to simulate interaction of materials accompanied by large deformations and fragmentation one often encounters a numerical instability caused by small node mass because acceleration on a mesh node is obtained by dividing the total force on the node by the mass of the node when the material points are in the far sides of the cells containing the node typically happening near material interfaces the node mass can be very small leading to artificially large acceleration and then numerical instability for the case of small material deformations this instability is typically avoided by placing the material points away from cell boundaries for cases with large deformations with the exception of initial conditions there is no control on locations of the material points the instability caused by small mass nodes is often encountered to avoid this instability tiny time steps are usually required in a numerical calculation in this work we present a numerical algorithm to treat this instability we show that this algorithm satisfies mass and momentum conservation laws the error in energy conservation is proportional to the second order of the time step consistent with the explicit material point method numerical implementation of the algorithm is described numerical examples show effectiveness of the algorithm

stokes matrices poisson lie groups and frobenius manifolds the purpose of this paper is to point out and then draw some consequences of the fact that the poisson lie group g dual to g gln c may be identified with a certain moduli space of meromorphic connections over the unit disc having an irregular singularity at the origin g will be fully described in sect 2 the key feature of this point of view is that there is a holomorphic map g g from the dual of the lie algebra to the group g for each choice of diagonal matrix a0 with distinct eigenvalues the irregular type this map is essentially the riemann hilbert map or de rham morphism for such connections we will call it the monodromy map it is generically a local analytic isomorphism the main result is

computational geometry an introduction from the reviews this book offers a coherent treatment at the graduate textbook level of the field that has come to be known in the last decade or so as computational geometry the book is well organized and lucidly written a timely contribution by two founders of the field it clearly demonstrates that computational geometry in the plane is now a fairly well understood branch of computer science and mathematics it also points the way to the solution of the more challenging problems in dimensions higher than two

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

stability conditions and stokes factors let mathcal a be the category of modules over a complex finite dimensional algebra we show that the space of stability conditions on mathcal a parametrises an isomonodromic family of irregular connections on 1 with values in the hall algebra of mathcal a the residues of these connections are given by the holomorphic generating function for counting invariants in mathcal a constructed by d joyce in geom topol 11 667 725 2007

comment on a geometric representation of spectral and temporal vowel features quantification of vowel overlap in three linguistic varieties j acoust soc am 119 2334 2350 2006 in a recent paper by wassink j acoust soc am 119 2334 2350 2006 the spectral overlap assessment metric soam was proposed for quantifying the degree of acoustic overlap between vowels the soam does not fully take account of probability densities an alternative metric is proposed which is based on quadratic discriminant analysis and takes account of probability densities in the form of a posteriori probabilities unlike the soam the a posteriori probability based metric allows for a direct comparison of vowel overlaps calculated using different numbers of dimensions e g three dimensions fl f2 and duration versus two dimensions fl and f2

convexity properties of the moment mapping ii be its associated momen t mapping see w for definitions the set x is the union of co adjoint orbits the main result of this paper is a description of the orbit structure of this set to describe this result we first note that the coadjoint orbits in q can be parametr ized as follows let t be a caf tan subalgebra of g and let b be a positive definite g invariant bilinear form on by means of b we get a g equivariant identification q let t be the subspace of q corresponding to t then every co adjoint orbit intersects i in a weyl group orbit so there is a one one correspondence between co adjoint orbits in q and weyl group orbits in t to get an even sharper parametrization let t be a weyl chamber in then every weyl group orbit intersects t in a single point so t is a modul i space for the co adjoint orbits two orbits in x are said to be of the same type if they are isomorphic as homogeneous g spaces since x is compact there are at most a finite number of different types of orbits see 1113 let h be a closed subgroup of t of codimension r and let cit be the set of all orbits in x of type g h

quasi hamiltonian geometry of meromorphic connections for each connected complex reductive group g we find a family of new examples of complex quasi hamiltonian g spaces with g valued moment maps these spaces arise naturally as moduli spaces of suitably framed meromorphic connections on principal g bundles over a disk and they generalise the conjugacy class example of alekseev malkin and meinrenken 3 which appears in the simple pole case using the fusion product in the theory this gives a finite dimensional construction of the natural symplectic structures on the spaces of monodromy stokes data of meromorphic connections over arbitrary genus riemann surfaces together with a new proof of the symplectic nature of isomonodromic deformations of such connections

geometry and braiding of stokes data fission and wild character varieties a family of new algebraic poisson varieties will be constructed generalising the complex character varieties of riemann surfaces then the well known poisson mapping class group actions on the character varieties will be generalised

on the nonlinear convexity theorem of kostant a classical result of schur and horn sc ho states that the set of diagonal elements of all n x n hermitian matrices with fixed eigenvalues is a convex set in irn kostant kt has generalized this result to the case of any semisimple lie group this is often referred to as the linear convexity theorem of kostant picking up the diagonal of a hermitian matrix is a linear operation this result was later put into the framework of symplectic geometry by atiyah ay guillemin sternberg gu st and duistermaat du1 the key argument being that the map that picks up the diagonal of a hermitian matrix is a moment map for a hamiltonian torus action and that the image of such a moment map is always a convex polytope there is also a nonlinear version of the convexity theorem of kostant it has not been connected so far with symplectic geometry the purpose of this paper is to make this connection we first describe the nonlinear convexity theorem for the case of sl n q see 2 for more details the gram schmidt orthonormalization process in linear algebra asserts that every nonsingular n x n complex matrix g can be uniquely written as the product g kan for some unitary matrix k positive diagonal matrix a and strictly upper triangular matrix n we use a to denote the group of all positive diagonal matrices and we call the matrix a in the above decomposition the a component of g the nonlinear convexity theorem of kostant now asserts that the set of a components of g as g runs through all positive definite hermitian matrices with fixed eigenvalues a aj an is the convex hull of the set of all points obtained by permuting the coordinates of a here we identify a with its lie algebra a via the exponential map thus convexity in a makes sense for a general semisimple lie group g the gram schmidt decomposition is replaced by an iwasawa decomposition g kan of g and the permutation group that appeared above is replaced by the weyl group w of k a if an element g egis written as the product g kan for k e k a e a

symplectic manifolds and isomonodromic deformations abstract we study moduli spaces of meromorphic connections with arbitrary order poles over riemann surfaces together with the corresponding spaces of monodromy data involving stokes matrices natural symplectic structures are found and described both explicitly and from an infinite dimensional viewpoint generalising the atiyah bott approach this enables us to give an intrinsic symplectic description of the isomonodromic deformation equations of jimbo miwa and ueno thereby putting the existing results for the six painleve equations and schlesinger s equations into a uniform framework

structures in bcn ruijsenaars schneider models we construct the classical r matrix structure for the lax formulation of bcn ruijsenaars schneider systems proposed in commun math phys 115 127 1988 the r matrix structure takes a quadratic form similar to the an ruijsenaars schneider poisson bracket behavior although the dynamical dependence is more complicated commuting hamiltonians stemming from the bcn ruijsenaars schneider lax matrix are shown to be linear combinations of particular koornwinder van diejen external fields ruijsenaars schneider models for specific values of the exponential one body couplings uniqueness of such commuting hamiltonians is established once the first of them and the general analytic structure are given

deformed integrable σ models classical r matrices and classical exchange algebra on drinfel d doubles we describe a unifying framework for the systematic construction of integrable deformations of integrable models within the hamiltonian formalism it applies equally to both the yang baxter type as well as gauged wzw type deformations which were considered recently in the literature as a byproduct these two families of integrable deformations are shown to be poisson lie t dual of one another

isomonodromy aspects of the tt equations of cecotti and vafa ii riemann hilbert problem in guest et al arxiv 1209 2045 part i we computed the stokes data for the smooth solutions of the tt toda equations whose existence we had previously established by p d e methods here we formulate the existence problem as a riemann hilbert problem based on this stokes data and solve it under certain conditions theorem 5 4 in the process we compute the connection matrix for all smooth solutions thus completing the computation of the monodromy data theorem 5 5 we also give connection formulae relating the asymptotics at zero and infinity of all smooth solutions theorem 4 1 clarifying the region of validity of the formulae established earlier by tracy and widom finally we resolve some conjectures of cecotti and vafa concerning the positivity of s st where s is the stokes matrix and the unimodularity of the eigenvalues of the monodromy matrix theorem 5 6 in particular we show that unitarity implies regularity for the tt toda equations

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