conformal covariantization of moyal lax operators a covariant approach to the conformal property associated with moyal lax operators is given by identifying the conformal covariance with the second gelfand dickey flow we covariantize moyal lax operators to construct the primary fields of one parameter deformation of classical w algebras

generalized drinfel d sokolov hierarchies a general approach is adopted to the construction of integrable hierarchies of partial differential equations a series of hierarchies associated to untwisted kac moody algebras and conjugacy classes of the weyl group of the underlying finite lie algebra is obtained the generalized kdv hierarchies of v g drinfel d and v v sokolov are obtained as the special case for the coxeter element various examples of the general formalism are treated in some detail including the fractional kdv hierarchies

constrained kp hierarchy and bi hamiltonian structures the kakomtsev petviashvili kp hierarchy is considered together with the evolutions of eigenfunctions and adjoint eigenfunctions constraining the kp flows in terms of squared eigenfunctions one obtains 1 1 dimensional integrable equations with scattering problems given by pseudo differential lax operators the bi hamiltonian nature of these systems is shown by a systematic construction of two general poisson brackets on the algebra of associated lax operators gauge transformations provide miura links to modified equations these systems are constrained flows of the modified kp hierarchy for which again a general description of their bi hamiltonian nature is given the gauge transformations are shown to be poisson maps relating the bi hamiltonian structures of the constrained kp hierarchy and the modified kp hierarchy the simplest realization of this scheme yields the akns hierarchy and its miura link to the kaup broer hierarchy

divergence runge phenomenon for least squares polynomial approximation on an equispaced grid and mock chebyshev subset interpolation runge showed more than a century ago that polynomial interpolation of a function f x using points evenly spaced on email protected 1 1 could diverge on parts of this interval even if f x was analytic everywhere on the interval least squares fitting of a polynomial of degree n to an evenly spaced grid with p points should improve accuracy if email protected n we show through numerical experiments that such an overdetermined fit reduces but does not eliminate the runge phenomenon more precisely define b n 1 p the least squares fit will fail to converge everywhere on 1 1 as n for fixed b if f x has a singularity inside a curve d b in the complex plane the width of the region enclosed by the convergence boundary d shrinks as b diminishes i e more points for a fixed polynomial degree but shrinks to the interval 1 1 only when b 0 we also show that the runge phenomenon can be completely defeated by interpolation on a mock chebyshev grid a subset of n 1 points from an equispaced grid with o n 2 points chosen to mimic the non uniform n 1 point chebyshev lobatto grid

a comparison of numerical algorithms for fourier extension of the first second and third kinds the range of fourier methods can be significantly increased by extending a nonperiodic function f x to a periodic function f on a larger interval when f x is analytically known on the extended interval the extension is straightforward when f x is unknown outside the physical interval there is no standard recipe worse still like a radarless aircraft groping through fog the algorithm may wreck on the mountain in fog problem a function f x which is perfectly well behaved on the physical interval may very well have singularities in the extended domain in this article we compare several algorithms for successfully extending a function f x into the fog even when the analytic extension is singular the best third kind extension requires singular value decomposition with iterative refinement but achieves accuracy close to machine precision

improved multiquadric method for elliptic partial differential equations via pde collocation on the boundary the multiquadric radial basis function mq method is a recent meshless collocation method with global basis functions it was introduced for discretizing partial differential equations pdes by kansa in the early 1990s the mq method was originally used for interpolation of scattered data and it was shown to have exponential convergence for interpolation problems in 1 we have extended the kansa mq method to numerical solution and detection of bifurcations in 1d and 2d parameterized nonlinear elliptic pdes we have found there that the modest size nonlinear systems resulting from the mq discretization can be efficiently continued by a standard continuation software such as auto we have observed high accuracy with a small number of unknowns as compared with most known results from the literature in this paper we formulate an improved kansa mq method with pde collocation on the boundary mq pdecb we add an additional set of nodes which can lie inside or outside of the domain adjacent to the boundary and correspondingly add an additional set of collocation equations obtained via collocation of the pde on the boundary numerical results are given that show a considerable improvement in accuracy of the mq pdecb method over the kansa mq method with both methods having exponential convergence with essentially the same rates

relationship between the moyal kp and the sato kp hierarchies the formula describing the isomorphism between the moyal kp and the sato kp hierarchies is given we obtain the conserved densities of the dispersionless kp hierarchy by taking the limit k to 0 of the conserved quantities derived in the sato approach

approximation on non tensor domains including squircles part iii single domain spectral methods have been largely restricted to tensor product bases on a tensor product grid to break the tensor barrier we study approximation in a domain bounded by a squircle the zero isoline of b x y x 2 y 2 1 the boundary varies smoothly from a circle 1 to the square polynomial least squares hyperinterpolation converges geometrically as long as the number of points p is at least double the number of basis functions n the polynomial grid was made denser near the boundaries chebyshev like by depositing grid points along wisely chosen contours of b gaussian radial basis functions rbfs were more robust in the sense that they too converged geometrically but hyperinterpolation p n and a chebyshevized grid were unnecessary a uniform grid truncated to include only those points within the squircle was satisfactory even without interpolation points on the boundary although boundary points are a cost effective improvement for a given number of points p however rbf interpolation was only slightly more accurate than polynomial hyperinterpolation and needed twice as many basis functions interpolation costs can be greatly reduced by exploiting the invariance of the squircle bounded domain to the eight elements d 4 dihedral group

a short proof of a kupershmidt wilson theorem this is a short elementary proof of a statement originally observed by adler then pursued by the author by kupershmidt and wilson and in a more general setting by drinfeld and sokolov

unsymmetric and symmetric meshless schemes for the unsteady convection diffusion equation in this paper we investigate the application of unsymmetric and symmetric meshless collocation techniques with radial basis functions for solving the unsteady convection diffusion equation we employ the method of lines approach to discretize the governing operator equation the stability of both explicit and implicit time stepping schemes are analyzed numerical results are presented for 1d and 2d problems to compare the performance of the unsymmetric and symmetric collocation techniques we compare the performance of various globally supported radial basis functions such as multiquadric inverse multiquadric gaussian thin plate splines and quintics numerical studies suggest that symmetric collocation is only marginally better than the unsymmetric approach further it appears that both collocation techniques require a very dense set of collocation points in order to achieve accurate results for high pe clet numbers

generalized drinfeld sokolov reductions and kdv type hierarchies generalized drinfeld sokolov ds hierarchies are constructed through local reductions of hamiltonian flows generated by monodromy invariants on the dual of a loop algebra following earlier work of de groot et al reductions based upon graded regular elements of arbitrary heisenberg subalgebras are considered we show that in the case of the nontwisted loop algebra l gl n graded regular elements exist only in those heisenberg subalgebras which correspond either to the partitions ofn into the sum of equal numbersn pr or to equal numbers plus onen pr 1 we prove that the reduction belonging to the grade 1 regular elements in the casen pr yields thep p matrix version of the gelfand dickeyr kdv hierarchy generalizing the scalar casep 1 considered by ds the methods of ds are utilized throughout the analysis but formulating the reduction entirely within the hamiltonian framework provided by the classical r matrix approach leads to some simplifications even forp 1

an analytic approximation to the cardinal functions of gaussian radial basis functions on an infinite lattice gaussian radial basis functions rbfs have been very useful in computer graphics and for numerical solutions of partial differential equations where these rbfs are defined on a grid with uniform spacing h as translates of the master function f x a h exp a 2 h 2 x 2 where a is a user choosable constant unfortunately computing the coefficients of f x jh a h requires solving a linear system with a dense matrix it would be much more efficient to rearrange the basis functions into the equivalent lagrangian or cardinal basis because the interpolation matrix in the new basis is the identity matrix the cardinal basis c j x a h is defined by the set of linear combinations of the gaussians such that c j kh 1 when k j and c j kh 0 for all integers k j we show that the cardinal functions for the uniform grid are c j x h a c x h j a where c x a a 2 p sin px sinh a 2x the relative error is only about 4exp email protected 2 a 2 as demonstrated by the explicit second order approximation it has long been known that the error in a series of gaussian rbfs does not converge to zero for fixed a as h 0 but only to an error saturation proportional to exp email protected 2 a 2 because the error in our approximation to the master cardinal function c x a is the square of the error saturation there is no penalty for using our new approximations to obtain matrix free interpolating rbf approximations to an arbitrary function f x the master cardinal function on a uniform grid in d dimensions is just the direct product of the one dimensional cardinal functions thus in two dimensions c x y a a 4 p 2 sin px sin py sinh a 2x sinh a 2y we show that the matrix free interpolation can be extended to non uniform grids by a smooth change of coordinates

braid matrices and structure constants for minimal conformal models using the feigin fuchs representation of minimal conformal models in a form introduced recently by one of us the braid group representation matrices describing the analytic continuation properties of conformal blocks are computed in a suitable normalization their matrix elements are shown to essentially factorize into pairs of boltzmann weights of critical rsos models in a certain limit of the spectral parameter these boltzmann weights are related to quantum groupr matrices by the vertex sos transformation we show that the crossing symmetry of the four point function in left right symmetric models follows from a quantum group relation also called crossing symmetry this observation gives a simple way to evaluate the structure constants

the runge phenomenon and spatially variable shape parameters in rbf interpolation many studies mostly empirical have been devoted to finding an optimal shape parameter for radial basis functions rbf when exploring the underlying factors that determine what is a good such choice we are led to consider the runge phenomenon rp best known in cases of high order polynomial interpolation as a key error mechanism this observation suggests that it can be advantageous to let the shape parameter vary spatially rather than assigning a single value to it benefits typically include improvements in both accuracy and numerical conditioning still another benefit arises if one wishes to improve local accuracy by clustering nodes in selected areas this idea is routinely used when working with splines or finite element methods however local refinement with rbfs may cause rp type errors unless we use a spatially variable shape paremeter with this enhancement rbf approximations combine freedom from meshes with spectral accuracy on irregular domains and furthermore permit local node clustering to improve the resolution wherever this might be needed

quantum line operators from lax pairs motivated by the realization of the yang baxter equation of 2d integrable models in the 4d gauge theory of costello witten yamazaki cwy we study the embedding of integrable 2d toda field models inside this construction this is done by using the lax formulation of 2d integrable systems and by thinking of the standard lax pair l in terms of components of cwy gauge connection propagating along particular directions in the gauge bundle we also use the results of the cwy theory to build quantum line operators for 2d toda models and compute the one loop contribution of two intersecting lines exchanging one gluon other features such as local symmetries and comments on extension of our method to other 2d integrable models are also discussed we also comment on some basic points that still need a refinement before discussing about a fully consistent embedding of lax pairs into cwy theory

the hamiltonian structures of the super kp hierarchy associated with an even parity super lax operator the even parity super lax operator is considered for the supersymmetric kp hierarchy of the form l d2 i 0 ui 2d i 1 and two hamiltonian structures are obtained following the standard method of gel fand and dikii it is observed that the first hamiltonian structure is local and linear whereas the second hamiltonian structure is nonlocal and nonlinear among the superfields appearing in the lax operator their connections with the super w algebra are discussed briefly

a supersymmetric extension of the kadomtsev petviashvili hierarchy an extension of the kadomtsev petviashvili hierarchy by odd variables is given conservation laws and formal integrability are proved

the moyal bracket and the dispersionless limit of the kp hierarchy a new lax equation is introduced for the kp hierarchy which avoids the use of pseudo differential operators as used in the sate approach this lax equation is closer to that used in the study of the dispersionless kp hierarchy and is obtained by replacing the poisson bracket with the moyal bracket the dispersionless limit under which the moyal bracket collapses to the poisson bracket is particularly simple

the hamiltonian structure of the nonabelian toda hierarchy we show that a subset of the whole class of nonlinear differential difference equations associated with the discrete analog of the matrix schrodinger operator is endowed with a hamiltonian structure and exhibits an infinite number of integrals of motion in involution we also establish the relation between these integrals of motion and the transmission coefficient of the underlying linear problem and show that such a relation implies that for the whole class previously introduced there exists an infinite number of conserved quantities

the hamiltonian structures associated with a generalized lax operator it is shown that with every lax operator which is a pseudodifferential operator of nonzero leading order is associated a kp hierarchy for each such operator we construct the second gelfand dikii bracket associated with the lax equation and show that it defines a hamiltonian structure when the leading order is positive the corresponding compatible first hamiltonian structure which turns out in general to be different from the naive first gelfand dikii bracket is derived the corresponding hamiltonian structures for the constrained lax operator where the next to leading order term vanishes or has a constant coefficient is discussed

refining the shifted topological vertex we study aspects of the refining and shifting properties of the 3d macmahon function c3 q used in topological string theory and bkp hierarchy we derive the explicit expressions of the shifted topological vertex s q and its refined version t q t these vertices complete results in literature

hamiltonian structures of generalized manin radul super kdv and constrained super kp hierarchies a study of hamiltonian structures associated with supersymmetric lax operators is presented following a constructive approach the hamiltonian structures of inami kanno super kdv hierarchy and constrained modified super kp hierarchy are investigated from the reduced supersymmetric gelfand dickey brackets by applying a gauge transformation on the hamiltonian structures associated with these two nonstandard super lax hierarchies we obtain the hamiltonian structures of generalized manin radul super kdv and constrained super kp hierarchies we also work out a few examples and compare them with the known results

on the constrained mock chebyshev least squares the algebraic polynomial interpolation on n 1 uniformly distributed nodes can be affected by the runge phenomenon also when the function f to be interpolated is analytic among all techniques that have been proposed to defeat this phenomenon there is the mock chebyshev interpolation which produces a polynomial p that interpolates f on a subset of m 1 of the given nodes whose elements mimic as well as possible the chebyshev lobatto points of order m in this work we use the simultaneous approximation theory to produce a polynomial p of degree r greater than m which still interpolates f on the m 1 mock chebyshev nodes minimizing at the same time the approximation error in a least squares sense on the other points of the sampling grid we give indications on how to select the degree r in order to obtain polynomial approximant good in the uniform norm furthermore we provide a sufficient condition under which the accuracy of the mock chebyshev interpolation in the uniform norm is improved numerical results are provided

the integrable hierarchy constructed from a pair of kdv type hierarchies and its associated w algebra for any two arbitrary positive integers n and m using them th kdv hierarchy and the n m th kdv hierarchy as building blocks we are able to construct another integrable hierarchy referred to as the n m th kdv hierarchy thew algebra associated to the second hamiltonian structure of the n m th kdv hierarchy calledw n m algebra is isomorphic via a miura map to the direct sum of aw m algebra aw n m algebra and an additionalu 1 current algebra in turn from the latter we can always construct a representation of aw algebra

on diff s1 covariantization of pseudodifferential operator a study of diff s1 covariant properties of a pseudodifferential operator of integer degree is presented first it is shown that the action of diff s1 defines a hamiltonian flow defined by the second gelfand dickey bracket if and only if the pseudodifferential operator transforms covariantly second the covariant form of a pseudodifferential operator of degree n 0 1 is constructed by exploiting the inverse of the covariant derivative this in particular implies the existence of a primary basis for wkp n n 0 1

classical n 1 w superalgebras from hamiltonian reduction a combinatorial proof is presented of the fact that the space of supersymmetric lax operators admits a poisson structure analogous to the second gel fand dickey bracket of the generalized kdv hierarchies this allows us to prove that the space of lax operators of odd order has a symplectic submanifold defined by anty symmetric operators which inherits a poisson structure defining classicalw superalgebras extending then 1 supervirasoro algebra this construction thus yields an infinite series of extended superconformal algebras

extended conformal symmetries and u 1 currents the authors investigate the structure of extended conformal algebras generated by local chiral fields of spin 1 2 n using the gelfand dickey algebra of formal pseudodifferential operators for the gl n algebra the commutation relations of zamolodchikov s spin 3 operator algebra 1985 are derived in the presence of an additional u 1 conserved current

the n component kp hierarchy and representation theory it is the aim of the present article to give all formulations of the n component kp hierarchy and clarify connections between them the generalization to the n component kp hierarchy is important because it contains many of the most popular systems of soliton equations like the davey stewartson system for n 2 the two dimensional toda lattice for n 2 the n wave system for n 3 and the darboux egoroff system it also allows us to construct natural generalizations to the davey stewartson and toda lattice systems of course the inclusion of all these systems in the n component kp hierarchy allows us to construct their solutions by making use of vertex operators

adaptive residual subsampling methods for radial basis function interpolation and collocation problems we construct a new adaptive algorithm for radial basis functions rbfs method applied to interpolation boundary value and initial boundary value problems with localized features nodes can be added and removed based on residuals evaluated at a finer point set we also adapt the shape parameters of rbfs based on the node spacings to prevent the growth of the conditioning of the interpolation matrix the performance of the method is shown in numerical examples in one and two space dimensions with nontrivial domains

existence and uniqueness of the universal w algebra a precise definition of the universal classical w algebra for the wn series is given and its existence and uniqueness are proved the main observation is that there is a natural reduction from wn to wn 1 that allows us to define the universal w algebra as an inverse limit this universal w algebra is in a sense the smallest w algebra from which all wn can be obtained by reduction these results extend to other w algebras obtained by reducing the gel fand dickey brackets as well as to w superalgebras obtained from the supersymmetric gel fand dickey brackets

poisson and hamiltonian superpairs over polarized associative algebras a poisson superpair is a pair of poisson superalgebra structures on a supercommutative associative algebra whose any linear combination is also a poisson superalgebra structure in this paper we first construct certain linear and quadratic poisson superpairs over a semi finitely filtered polarized 2 graded associative algebra then we give a construction of certain hamiltonian superpairs in the formal variational calculus over any finite dimensional 2 graded associative algebra with a supersymmetric nondegenerate associative bilinear form our constructions are based on the adler mapping in a general sense our results in this paper can be viewed as noncommutative generalizations of the adler gel fand dikii hamiltonian pair

quartic gaussian and inverse quartic gaussian radial basis functions the importance of a nonnegative fourier transform we catalogue the numerical properties of approximations using two novel types of radial basis functions f r the qg species is a basis of exponentials of quartic argument f x f r b f x a h j 1 na jexp a h 4 x x j 4 where the x j are the rbf centers and also the interpolation points we show that quartic gaussian rbfs fail at many discrete values of the shape parameter a we show through a detailed analysis that these singularities are directly related to zeros of q k the fourier transform of exp x 4 if we reverse the roles and take q x as the rbf all difficulties disappear because these iqg rbfs have a fourier transform which is nonnegative for all real k we explain that although the quartic gaussian exp x 4 is positive definite in the physics dynamical systems sense of being zero free and nonnegative it lacks the crucial property of being positive definition in the rbf analysis sense

the dispersionless lax equations and topological minimal models it is shown that perturbed rings of the primary chiral fields of the topological minimal models coincide with some particular solutions of the dispersionless lax equations the exact formulae for the tree level partition functions of an topological minimal models are found the virasoro constraints for the analogue of the function of the dispersionless lax equation correspond ing to these models are proved

spectral methods in non tensor geometry part ii single domain spectral methods have been largely restricted to tensor product bases on a tensor product grid to break the tensor barrier we studied approximation in two idealized families of domains one family is bounded by a squircle the zero isoline of b x y x 2 y 2 1 the boundary varies smoothly from a circle 1 to the square the other family is bounded by a perturbed quadrifolium the plane algebraic curve x 2 y 2 x 2 y 2 3 x 2 y 2 2 this varies smoothly from the singular self intersecting curve known as the quadrifolium to a circle as varies from zero to infinity we compared two different bivariate polynomial bases truncating by total degree zernike polynomials are natural for the disk tensor products of chebyshev polynomials are equally sensible for the square both yield an exponential rate of convergence for our non tensor neither disk nor square domains indeed the chebyshev basis worked well for the disk and the zernike polynomials were good for the square the expected differences due to numerical ill conditioning did not emerge much to our surprise the price for the nontensor domain was that hyperinterpolation was necessary that is least squares fitting with more interpolation constraints than unknowns denoting the number of interpolation points by p and the basis size by n a ratio of p n around two to three was optimum while p near one was very inacccurate a uniform grid truncated to include only those points within the squircle or other boundary curve was satisfactory even without interpolation points on the boundary although boundary points are a cost effective improvement interpolation costs were greatly reduced by exploiting the invariance of the squircle bounded and perturbed quadrifolium domains to the eight element d4 dihedral group

on the backlund transformation for the gel fand dickey equations we study the backlund transformation of the gel fand dickey equations and in particular how the factorization ofnth order differential operators leads to lax type equations for first order operators generalizing work of adler and moser 1 in a similar fashion we study the toda equations

solving the kp hierarchy by gauge transformations we show that it is convenient to use gauge transformations sometimes called explicit backlund transformations to generate new solutions for the kp hierarchy two particular kinds of gauge transformation operators constructed out of the initial wave functions are of fundamental importance in this approach through such gauge transformations a very simple formula for the tau function is obtained encompassing and unifying all kinds of existing solutions the corresponding free fermion representation and baker functions for the new function can also be constructed

integrability of the frobenius algebra valued kadomtsev petviashvili hierarchy we introduce a frobenius algebra valued kadomtsev petviashvili kp hierarchy and show the existence of frobenius algebra valued function for this hierarchy in addition we construct its hamiltonian structures by using the adler dickey gelfand method as a byproduct of these constructions we show that the coupled kp hierarchy defined by casati and ortenzi j geom phys 56 418 449 2006 has at least n basic different local bi hamiltonian structures finally via the construction of the second hamiltonian structures we obtain some local matrix or frobenius algebra valued generalizations of classical w algebras

fourier embedded domain methods extending a function defined on an irregular region to a rectangle so that the extension is spatially periodic and c a simple way to solve a partial differential equation in a non rectangular domain w is to embed the domain in a rectangle b and solve the problem more easily in the rectangle to apply a fourier spectral method on the rectangle it is necessary to extend the inhomogeneous term in the pde f x y to b in such a way that the extended function g x y is periodic and infinitely differentiable and yet is equal to f x y everywhere in the irregular domain w if the boundary of w email protected is defined as the zero isoline of a function f x y then a suitable extension is g x y f x y h email protected x y constant where h is a smoothed approximation to the step function which is defined in the main text

a stable algorithm for flat radial basis functions on a sphere when radial basis functions rbfs are made increasingly flat the interpolation error typically decreases steadily until some point when runge type oscillations either halt or reverse this trend because the most obvious method to calculate an rbf interpolant becomes a numerically unstable algorithm for a stable problem in the case of near flat basis functions there will typically also be a separate point at which disastrous ill conditioning enters we introduce here a new method rbf qr which entirely eliminates such ill conditioning and we apply it in the special case when the data points are distributed over the surface of a sphere this algorithm works even for thousands of node points and it allows the rbf shape parameter to be optimized without the limitations imposed by stability concerns since interpolation in the flat rbf limit on a sphere is found to coincide with spherical harmonics interpolation new insights are gained as to why the rbf approach with nonflat basis functions often is the more accurate of the two methods

group systems groupoids and moduli spaces of parabolic bundles moduli spaces of homomorphisms or more generally twisted homomorphisms from fundamental groups of surfaces to compact connected lie groups were connected with geometry through their identification with moduli spaces of holomorphic vector bundles see 29 atiyah and bott 2 initiated a new approach to the study of these moduli spaces by identifying them with moduli spaces of projectively fiat constant central curvature connections on principal bundles over riemann surfaces which they analyzed by methods of gauge theory in particular they showed that an invariant inner product on the lie algebra of the lie group in question induces a natural symplectic structure on a certain smooth open stratum although this moduli space is a finite dimensional object generally a stratified space which is locally semialgebraic 19 but sometimes a manifold its symplectic structure on the stratum just mentioned was obtained by applying the method of symplectic reduction to the action of an infinite dimensional group the group of gauge transformations on an infinite dimensional symplectic manifold the space of all connections on a principal bundle

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

newton iteration with multiquadrics for the solution of nonlinear pdes abstract newton iteration is a standard tool for the numerical solution of nonlinear partial differential equations we show how globally supported multiquadric radial basis functions can be used for this task one of the insights gained is that the use of coarse meshes during the initial iterations along with a multiquadric parameter which is adjusted with the meshsize increases the efficiency and stability of the resulting algorithm some experiments with nash iteration are also included

polynomials and potential theory for gaussian radial basis function interpolation we explore a connection between gaussian radial basis functions and polynomials using standard tools of potential theory we find that these radial functions are susceptible to the runge phenomenon not only in the limit of increasingly flat functions but also in the finite shape parameter case we show that there exist interpolation node distributions that prevent such phenomena and allow stable approximations using polynomials also provides an explicit interpolation formula that avoids the difficulties of inverting interpolation matrices while not imposing restrictions on the shape parameter or number of points

the n m th korteweg de vries hierarchy and the associated w algebra a differential integrable hierarchy which is called the n m th korteweg de vries kdv hierarchy whose lax operator is obtained by properly adding m pseudo differential terms to the lax operator of the nth kdv hierarchy is discussed herein this new hierarchy contains both the higher kdv hierarchy and multifield representation of the kadomtsev petviashvili kp hierarchy as subsystems and naturally appears in multimatrix models the n 2m 1 coordinates or fields of this hierarchy satisfy two algebras of compatible poisson brackets which are local and polynomial each poisson structure generates an extended w1 and w algebras respectively w n m is called the generating algebra of the extended w algebra this algebra which corresponds with the second poisson structure shares many features of the usual wn algebra it is shown that there exist m distinct reductions of the n m th kdv hierarchy which are obtained by imposing suitable second class constraints the most drastic reduction corresponds

vector field approximation using radial basis functions in this article we investigate the performance of rbf pde methods for approximating solenoidal fields it is well known that global rbf collocation methods present a trade off principle which means that smoothness implies high convergence order plus ill conditioning on the other hand local methods for solving this problem have recently appeared in the literature in this paper we perform a numerical investigation of the differences between rbf global and local methods in order to investigate the possible advantage of using local methods for the approximation of vector fields more precisely we compare the local hermite interpolation technique using inverse multiquadrics against the non symmetric collocation method of kansa

lie group valued moment maps we develop a theory of quasi hamiltonian g spaces for which the moment map takes values in the group g itself rather than in the dual of the lie algebra the theory includes counterparts of hamiltonian reductions the guillemin sternberg symplectic cro

non local matrix generalizations of w algebras there is a standard way to define two symplectic hamiltonian structures the first and second gelfand dikii brackets on the space of ordinarymth order linear differential opeatorsl d m u1dm 1 u2dm 2 u m in this paper i consider in detail the case where theu k aren n matrix valued functions with particular emphasis on the more interesting second gelfand dikii bracket of particular interest is the reduction to the symplectic submanifoldu1 0 this reduction gives rise to matrix generalizations of the classical version of thenon linear w m algebras calledv n m algebras the non commutativity of the matrices leads tonon local terms in thesev n m algebra s i show that these algebras contain a conformal virasoro subalgebra and that combinationsw k of theu k can be formed that aren n matrices of conformally primary fields of spink in analogy with the scalar casen 1 in general however thev m n algebras have a much richer structure than thew m algebras as can be seen on the examples of thenon linear andnon local poisson brackets u2 ab u2 cd u2 ab w3 cd and w3 ab w3 cd which i work out explicitly for allm andn a matrix miura transformations is derived mapping these complicated second gelfand dikii brackets of theu k to a set of much simpler poisson brackets providing the analogoue of the free field representation of thew m algebras

the hidden algebraic structure of the kaup broer hierarchy in this paper we investigate the conformal algebra associated with the coupled kaup broer hierarchy which is obtained from the coupled akns hierarchy by gauge transformation we find that the algebraic structure hidden behind the coupled kaup broer hierarchy can be revealed after mapping the lax operator of the coupled kaup broer hierarchy to a yajima oikawa type lax operator furthermore we show that the second hamiltonian structure of the coupled kaup broer hierarchy can be simplified by factorizing the lax operator in multiplicative form and thus obtain the free field realization of the associated algebras

error saturation in gaussian radial basis functions on a finite interval radial basis function rbf interpolation is a meshless strategy with great promise for adaptive approximation one restriction is error saturation which occurs for many types of rbfs including gaussian rbfs of the form f x a h exp email protected 2 x h 2 in the limit h 0 for fixed a the error does not converge to zero but rather to e s a previous studies have theoretically determined the saturation error for gaussian rbf on an infinite uniform interval and for the same with a single point omitted the gap enormously increases e s a we show experimentally that the saturation error on the unit interval email protected 1 1 is about 0 06exp 0 47 a 2 email protected huge compared to the o email protected a 2 exp email protected 2 email protected 2 saturation error for a grid with one point omitted we show that the reason the saturation is so large on a finite interval is that it is equivalent to an infinite grid which is uniform except for a gap of many points the saturation error can be avoided by choosing email protected 1 the flat limit but the condition number of the interpolation matrix explodes as o exp p 2 email protected 2 the best strategy is to choose the largest a which yields an acceptably small saturation error if the user chooses an error tolerance d then a o p t i m u m d 1 2log d 0 06

on the huge lie superalgebra of pseudo superdifferential operators and super kp hierarchies lie superalgebraic methods are used to establish a connection between the huge lie superalgebra of super pseudo differential operators and various super kp hierarchies we show in particular that splits into 5 2 2 1 graded algebras expected to correspond to five classes of super kp hierarchies generalizing the well known manin radul and figueroa mas ramos supersymmetric kp hierarchies

six strategies for defeating the runge phenomenon in gaussian radial basis functions on a finite interval radial basis function rbf interpolation is a meshless strategy with great promise for adaptive approximation because it is meshless there is no canonical grid to act as a starting point for function adaptive grid modification uniform grids are therefore common with rbfs like polynomial interpolation which is the limit of rbf interpolation when the rbf functions are very wide the flat limit rbf interpolants are vulnerable to the runge phenomenon whether the grid is uniform or irregular the n point interpolant may diverge as n on the interval spanned by the interpolation points even for a function f x that is analytic everywhere on and near this interval in this work we discuss six strategies for defeating the runge phenomenon specializing to a uniform grid on a finite interval and one particular species of rbfs in one space dimension f x exp a 2 h 2 x 2 where h is the grid spacing three strategies fail but three are successful one good strategy is to vary a with the number of interpolation points n as n 1 4 unfortunately this gives both a subgeometric rate of convergence for the approximation error falling as exp qn for some constant q and a matrix condition number that grows as exp pn for some p 0 in order to explain why fixed a independent of n is not a convergent strategy we digress to discuss error saturation and show experimentally that the saturation error on the finite interval is about 0 6exp 0 47 a 2 f if a user chosen error tolerance d is acceptable then the optimum choice is a o p t i m u m d 1 2log d 0 06 the second good strategy is rbf extension which uses rbf functions with centers on a interval slightly larger than the target interval to approximate f x a third strategy is to split the interval into two boundary layers and a large middle interval and apply separate rbf approximations on each the slow decrease of a strategy is much cheaper but rbf extension is much more resistant to ill conditioning and therefore can achieve much lower errors the three interval method is the least accurate but is robust and inexpensive

miura map between lattice kadomtsev petviashvili and its modification is canonical we consider the miura map between the lattice kadomtsev petviashvili hierarchy and the lattice modified kp hierarchy and prove that the map is canonical not only between the first hamiltonian structures but also between the second hamiltonian structures

supersymmetric extension of the korteweg de vries equation it is shown that among a one parameter family of supersymmetric extensions of the korteweg de vries equation there is a special system that has an infinite number of conservation laws which can be formulated in the second hamiltonian structure and which has a nontrivial lax representation its modified version is also discussed

classical w algebras we reconsider the relation between classicalw algebras and deformations of differential operators emphasizing the consistency with diffeomorphisms generators of thew algebra that arek differentials are constructed by a systematic procedure the method extends following drinfeld and sokolov tow algebras based on arbitrary simple lie algebras

the constrained modified kp hierarchy and the generalized miura transformations in this letter we consider the second hamiltonian structure of the constrained modified kp hierarchy after mapping the lax operator to a pure differential operator the second structure becomes the sum of the second and the third gelfand dickey brackets defined by this differential operator we simplify this hamiltonian structure by factorizing the lax operator into linear terms

higher spin fields and the gelfand dickey algebra we show that in 2 dimensional field theory higher spin algebras are contained in the algebra of formal pseudodifferential operators introduced by gelfand and dickey to describe integrable nonlinear differential equations in lax form the spin 2 and 3 algebras are discussed in detail and the generalization to all higher spins is outlined this provides a conformal field theory approach to the representation theory of gelfand dickey algebras

on the gelfand dickey algebra gd sln and the wn symmetry i the bosonic case the algebra of nonlinear local and nonlocal differential operators acting on the ring of analytic functions is studied it is shown in particular that this space splits into 3 2 special subalgebras jr j 0 1 r 1 each subalgebra is completely specified by quantum numbers s and p q describing the conformal spin and the lowest and the highest degrees respectively the algebra and its dual of local pure nonlocal differential operators is used to calculate the general expression of the gelfand dickey bracket and the wn symmetry poisson one in terms of a set of spin j canonical fields uj 2 j n and a nonlinear u cubic dependent differential operator d n i j z u the explicit form of this operator is worked out other remarkable features are also discussed

on the backlund transformation for the moyal korteweg de vries hierarchy we study the b cklund symmetry for the moyal korteweg de vries kdv hierarchy based on the kupershmidt wilson theorem associated with second gelfand dickey structure with respect to the moyal bracket which generalizes the result of adler for the ordinary kdv

the zero curvature formulation of the skdv equations the fermionic extensions of the kdv equation are derived from the zero curvature condition associated with the superalgebra osp 2 1 this derivation clarifies why there are only two such extensions possible and why only one of them is supersymmetric a lenard type of derivation of these equations is also presented

fully meshless solution of the one dimensional multigroup neutron transport equation with the radial basis function collocation method abstract in this paper a fully meshless method is proposed for the numerical solution of the one dimensional multigroup neutron transport equation with anisotropic scattering both first order and even parity forms of the transport equation are studied the radial basis function collocation method is chosen for the spatial treatment and legendre polynomials are used to approximate the angular variable the selection of the legendre polynomials instead of discrete ordinates approach resulted with a fully meshless algorithm in both independent variables multiquadric is utilized as the radial function seven problems are considered to evaluate the performance of the method the results show that the method converges exponentially and it is possible to obtain high levels of accuracies for the multiplication factor and neutron flux with a good stability in both spatial and angular domains for the one group isotropic benchmark problem discrete ordinates solutions employing discontinuous linear finite elements for the spatial variable are also provided and a comparison of the methods revealed that the fully meshless method produced more accurate results than the discrete ordinates finite element scheme when the shape parameter is properly chosen

limit problems for interpolation by analytic radial basis functions interpolation problems for analytic radial basis functions like the gaussian and inverse multiquadrics can degenerate in two ways the radial basis functions can be scaled to become increasingly flat or the data points coalesce in the limit while the radial basis functions stay fixed both cases call for a careful regularization which if carried out explicitly yields a preconditioning technique for the degenerating linear systems behind these interpolation problems this paper deals with both cases for the increasingly flat limit we recover results by larsson and fornberg together with lee yoon and yoon concerning convergence of interpolants towards polynomials with slight modifications the same technique can also handle scenarios with coalescing data points for fixed radial basis functions the results show that the degenerating local lagrange interpolation problems converge towards certain hermite birkhoff problems this is an important prerequisite for dealing with approximation by radial basis functions adaptively using freely varying data sites

classical extended conformal algebras associated with constrained kp hierarchy the conformal property of the second hamiltonian structure of constrained kp hierarchy derived by oevel and strampp is examined it is found that it naturally gives a family of nonlocal extended conformal algebras two examples of such algebras are given and it is found that they are similar to bilal s v algebra by taking a gauge transformation one can map the constrained kp hierarchy to kupershmidt s nonstandard lax hierarchy the second hamiltonian structure in this representation is considered it is shown that after mapping the lax operator to a pure differential operator the second structure becomes the sum of the second and the third gelfand dickey brackets defined by this differential operator the corresponding conformally covariant form of this differential operator is also given

modifying the kp the n th constrained kp hierarchies and their hamiltonian structures the kadomtsev petviashvili kp hierarchy has infinitely many hamiltonian pairs then th pair of them is associated withl n wherel is the pseudodifferential operator pdo 3 4 in this paper by the factorizationl n l n l 1 withl j j 1 n being the independent pdos we construct the miura transformation for the kp which leads to a decomposition of the second hamiltonian structure in then th pair to a direct sum each term in the sum is the second structure in the initial pair associated withl j when we impose a constraint 1 9 i e a new type of reduction to the kp hierarchy we obtain the similar results for the constrained kp hierarchy in particular the second hamiltonian structure for this hierarchy is transformed to a vastly simpler one

stratified symplectic spaces and reduction let m w be a hamiltonian g space with proper momentum map j m g it is well known that if zero is a regular value of j and g acts freely on the level set j 0 then the reduced space mo j 0 g is a symplectic manifold we show that if the regularity assumptions are dropped the space mo is a union of symplectic manifolds i e it is a stratified symplectic space arms et al 2 proved that mo possesses a natural poisson bracket using their result we study hamiltonian dynamics on the reduced space in particular we show that hamiltonian flows are strata preserving and give a recipe for lifting a reduced hamiltonian flow to the level set j 0 finally we give a detailed description of the stratification of mo and prove the existence of a connected

on the bi hamiltonian structures of the skdv the first and the second hamiltonian structures of the supersymmetric kdv equation are derived following the conventional approach of gel fand and dickey in particular it is shown that the nonlocal nature of the first bracket is a direct consequence of the constrained nature of the lax operator

a one parameter family of hamiltonian structures for the kp hierarchy and a continuous deformation of the nonlinear wkp algebra the kp hierarchy is hamiltonian relative to a one parameter family of poisson structures obtained from a generalized adler map in the space of formal pseudodifferential symbols with noninteger powers the resulting w algebra is a one parameter deformation of wkp admitting a central extension for generic values of the parameter reducing naturally to w n for special values of the parameter and contracting to the centrally extended w1 w and further truncations in the classical limit all algebras in the one parameter family are equivalent and isomorphic towkp the reduction induced by setting the spin one field to zero yields a one parameter deformation of hat w infty which contracts to a new nonlinear algebra of the w type

classical n 2 w superalgebras from superpseudodifferential operators we study the supersymmetric gelfand dickey algebras associated with the superpseudodifferential operators of positive as well as negative leading order we show that upon the usual constraint these algebras contain the n 2 super virasoro algebra as a subalgebra when the leading order is odd the decompositions of the coefficient functions into n 1 primary fields are then obtained by covariantizing the superpseudodifferential operators we discuss the problem of identifying n 2 supermultiplets and work out a couple of supermultiplets by explicit computations

on the variation of the poisson structures of certain moduli spaces let pi be the fundamental group of a compact connected orientable topological surface with finitely many boundary circles our aim is to construct poisson algebras of continuous functions on certain open subspaces cal r pi g vec w of the space of representations of pi in a lie group g whose lie algebra is supposed to be endowed with a nondegenerate invariant symmetric bilinear form when g is compact and connected the spaces cal r pi g vec w may be taken to be dense in the space of all representations the spaces cal r pi g vec w contain spaces of representations rm rep pi g vec c where the values of those generators of the fundamental group which correspond to the boundary circles are constrained to lie in fixed conjugacy classes and the representation spaces rm rep pi g vec c being endowed with universal poisson algebras constructed elsewhere earlier the restrictions from the spaces cal r pi g vec w to any of the spaces rm rep pi g vec c are poisson maps thus the poisson algebra on each cal r pi g vec w gives a description of the variation of the universal poisson structures on the spaces rm rep pi g vec c as the chosen conjugacy classes move across g we also give twisted versions of these results

superconformal covariantization of superdifferential operator on 1 1 superspace and classical n 2 w superalgebras a study of the superconformal covariantization of superdifferential operators defined on 1 1 superspace is presented it is shown that a superdifferential operator with a particular type of constraint can be covariantized only when it is of odd order in such a case the action of superconformal transformation on the superdifferential operator is nothing but a hamiltonian flow defined by the corresponding supersymmetric second gelfand dickey bracket the covariant form of a superdifferential operator of odd order is given

miura and auto backlund transformations for the ckp and cmkp hierarchies we construct the miura and auto backlund transformations for the ckp and cmkp hierarchies both the eigenfunctions and the adjoint eigenfunctions of the hierarchies are used to trigger the transformations the canonical properties of the constructed miura and auto backlund transformations are also investigated

multi component bi super hamiltonian kdv systems it is shown that a new class of classical multi component super kdv equations is bi super hamiltonian by extending the method for the verification of graded jacobi identity the multi component extension of super mkdv equations is obtained by using the super miura transformation