from kontsevich witten to linear hodge integrals via virasoro operators we give a proof of alexandrov s conjecture on a formula connecting the kontsevich witten and hodge tau functions using only the virasoro operators this formula has been confirmed up to an unknown constant factor in this paper we show that this factor is indeed equal to one by investigating series expansions for the lambert w function on different points we give a proof of alexandrov s conjecture on a formula connecting the kontsevich witten and hodge tau functions using only the virasoro operators this formula has been confirmed up to an unknown constant factor in this paper we show that this factor is indeed equal to one by investigating series expansions for the lambert w function on different points

hodge and prym tau functions jenkins strebel differentials and combinatorial model of mathcal m_ g n the goal of the paper is to apply the theory of integrable systems to construct explicit sections of line bundles over the combinatorial model of the moduli space of pointed riemann surfaces based on strebel differentials these line bundles are tensor products of the determinants of the hodge or prym vector bundles with the standard tautological line bundles mathcal l j and the sections are constructed in terms of tau functions the combinatorial model is interpreted as the real slice of a complex analytic moduli space of quadratic differentials where the phase of each tau function provides a section of a circle bundle the phase of the ratio of the prym and hodge tau functions gives a section of the kappa 1 circle bundle by evaluating the increment of the phase around co dimension 2 sub complexes we identify the poincare dual cycles to the chern classes of the corresponding line bundles they are expressed explicitly as combination of witten s cycle w 5 and kontsevich s boundary this provides combinatorial analogues of mumford s relations on mathcal m g n and penner s relations in the hyperbolic combinatorial model the free homotopy classes of loops around w 5 are interpreted as pentagon moves while those of loops around kontsevich s boundary as combinatorial dehn twists throughout the paper we exploit the classical description of the combinatorial model in terms of strebel differentials parametrized in terms of period or homological coordinates we show that they provide darboux coordinates for the symplectic structure introduced by kontsevich we also express the latter as the intersection pairing in the odd homology of the canonical double cover

associahedra cellular w construction and products of a_ infty algebras the aim of this paper is to construct a functorial tensor product of a algebras or equivalently an explicit diagonal for the operad of cellular chains over the integers of the stasheff associahedron these constructions in fact already appeared saneblidze and umble 2000 and 2002 we will try to give a more conceptual presentation we also prove that there does not exist a coassociative diagonal

givental formula in terms of virasoro operators we present a conjecture that the universal enveloping algebra of differential operators tk over c coincides in the origin with the universal enveloping algebra of the borel subalgebra of virasoro generators from the kontsevich model thus we can decompose any pseudo differential operator to a combination of the virasoro operators using this decomposition we present the right hand side of the givental formula math ag 0008067 as a constant part of the differential operator we introduce in the case of cp1 studied in the paper by song and song hep th 0103254 the left hand side of the givental formula is a unit which imposes certain constraints on this differential operator we explicitly check that these constraints are correct up to o q4 we also propose a conjecture of factorization modulo hirota equation of the differential operator introduced and check this conjecture with the same accuracy

a note on a result of sun ji guang sensitivity of the cs and gsv decompositions in siam j numer anal 20 1983 pp 611 625 sun ji guang gives a perturbation analysis of a generalized singular value decomposition this note rephrases some of his ideas and derives an improved version of one of his results using straightforward matrix ideas some elaboration is given on the meaning of the result and the analysis is extended to cover the case of most usual interest to the numerical analyst

towards a generalized singular value decomposition we suggest a form for and give a constructive derivation of the generalized singular value decomposition of any two matrices having the same number of columns we outline its desirable characteristics and compare it to an earlier suggestion by van loan siam j numer anal 13 1976 pp 76 83 the present form largely follows from the work of van loan but is slightly more general and computationally more amenable than that in the paper cited we also prove a useful extension of a theorem of stewart siam rev 19 1977 pp 634 662 on unitary decompositions of submatrices of a unitary matrix

compactified picard stacks over the moduli stack of stable curves with marked points abstract in this paper we give a construction of algebraic artin stacks endowed with a modular map onto the moduli stack of stable curves of genus g with n marked points the stacks we construct are smooth irreducible and have dimension 4 g 3 n yielding a geometrically meaningful compactification of the universal picard stack parametrizing n pointed smooth curves together with a line bundle

masur veech volumes frequencies of simple closed geodesics and intersection numbers of moduli spaces of curves we express the masur veech volume and the area siegel veech constant of the moduli space of meromorphic quadratic differential with simple poles as polynomials in the intersection numbers of psi classes supported on the boundary cycles of the deligne mumford compactification of the moduli space of curves our formulae are derived from lattice point count involving the kontsevich volume polynomials that also appear in mirzakhani s recursion for the weil petersson volumes of the moduli space of bordered hyperbolic riemann surfaces a similar formula for the masur veech volume though without explicit evaluation was obtained earlier by mirzakhani through completely different approach we prove further result up to an explicit normalization factor depending only on the genus and on the number of cusps the density of the orbit of any simple closed multicurve computed by mirzakhani coincides with the density of square tiled surfaces having horizontal cylinder decomposition associated to the simple closed multicurve we study the resulting densities in more detail in the special case when there are no cusps in particular we compute explicitly the asymptotic frequencies of separating and non separating simple closed geodesics on a closed hyperbolic surface of genus g for all small genera g and we show that in large genera the separating closed geodesics are exponentially less frequent we conclude with detailed conjectural description of combinatorial geometry of a random simple closed multicurve on a surface of large genus and of a random square tiled surface of large genus this description is conditional to the conjectural asymptotic formula for the masur veech volume in large genera and to the conjectural uniform asymptotic formula for certain sums of intersection numbers of psi classes in large genera

an explicit two step quantization of poisson structures and lie bialgebras we develop a new approach to deformation quantizations of lie bialgebras and poisson structures which goes in two steps in the first step one associates to any poisson resp lie bialgebra structure a so called quantizable poisson resp lie bialgebra structure we give explicit transcendental formulae for this correspondence in the second step one deformation quantizes a quantizable poisson resp lie bialgebra structure we show again explicit transcendental formulae for this second step correspondence as a byproduct we obtain configuration space models for biassociahedron and bipermutohedron in the poisson case the first step is the most non trivial one and requires a choice of an associator while the second step quantization is essentially unique it is independent of a choice of an associator and can be done by a trivial induction we conjecture that similar statements hold true in the case of lie bialgebras the main new result is a surprisingly simple explicit universal formula which uses only smooth differential forms for universal quantizations of finite dimensional lie bialgebras

tau functions as widom constants we define a tau function for a generic riemann hilbert problem posed on a union of non intersecting smooth closed curves with jump matrices analytic in their neighborhood the tau function depends on parameters of the jumps and is expressed as the fredholm determinant of an integral operator with block integrable kernel constructed in terms of elementary parametrices its logarithmic derivatives with respect to parameters are given by contour integrals involving these parametrices and the solution of the riemann hilbert problem in the case of one circle the tau function coincides with widom s determinant arising in the asymptotics of block toeplitz matrices our construction gives the jimbo miwa ueno tau function for riemann hilbert problems of isomonodromic origin painleve vi v iii garnier system etc and the sato segal wilson tau function for integrable hierarchies such as gelfand dickey and drinfeld sokolov

principal angles between subspaces in an a based scalar product algorithms and perturbation estimates computation of principal angles between subspaces is important in many applications e g in statistics and information retrieval in statistics the angles are closely related to measures of dependency and covariance of random variables when applied to column spaces of matrices the principal angles describe canonical correlations of a matrix pair we highlight that all popular software codes for canonical correlations compute only cosine of principal angles thus making impossible because of round off errors finding small angles accurately we review a combination of sine and cosine based algorithms that provide accurate results for all angles we generalize the method to the computation of principal angles in an a based scalar product for a symmetric and positive definite matrix a we provide a comprehensive overview of interesting properties of principal angles we prove basic perturbation theorems for absolute errors for sine and cosine of principal angles with improved constants numerical examples and a detailed description of our code are given

history and generality of the cs decomposition abstract it is almost a quarter of a century since chandler davis and william kahan brought together the key ideas of what stewart later completed and defined to be the cs decomposition csd of a partitioned unitary matrix this paper outlines some germane points in the history of the csd pointing out the contributions of jordan of davis and kahan and of stewart and the relationship of the csd to the direct rotation of davis and kato the paper provides an easy to memorize constructive proof of the csd reviews one of its important uses and suggests a motivation for the csd which emphasizes how generally useful it is it shows the relation between the csd and generalized singular value decompositions and points out some useful nullity properties one form of the csd trivially reveals finally it shows how via the qr factorization the csd can be used to obtain interesting results for partitioned nonsingular matrices we suggest the csd be taught in its most general form with no restrictions on the two by two partition and initially with no mention of angles between subspaces

lambert w solves the noncommutative varphi 4 φ 4 model the closed dyson schwinger equation for the 2 point function of the noncommutative lambda phi 4 2 model is rearranged into the boundary value problem for a sectionally holomorphic function in two variables we prove an exact formula for a solution in terms of lambert s w function this solution is holomorphic in lambda inside a domain which contains 1 log 4 infty our methods include the hilbert transform perturbation series and lagrange burmann resummation

the decorated teichmuller space of punctured surfaces a principal 5 bundle over the usual teichmuller space of ans times punctured surface is introduced the bundle is mapping class group equivariant and admits an invariant foliation several coordinatizations of the total space of the bundle are developed there is furthermore a natural cell decomposition of the bundle finally we compute the coordinate action of the mapping class group on the total space the total space is found to have a rich equivariant geometric structure we sketch some connections with arithmetic groups diophantine approximations and certain problems in plane euclidean geometry furthermore these investigations lead to an explicit scheme of integration over the moduli spaces and to the construction of a universal teichmuller space which we hope will provide a formalism for understanding some connections between the teichmuller theory the kp hierarchy and the virasoro algebra these latter applications are pursued elsewhere

singular value and generalized singular value decompositions and the solution of linear matrix equations abstract the solution of the linear matrix equations i axb cyd e and ii axb fxg e h are considered new necessary and sufficient conditions for the consistency of the equations are derived some using the generalized singular value decomposition special cases iii ax yd e and iv axb e are treated using the singular value decomposition numerical algorithms for the solutions are also suggested

computing thecs decomposition of a partitioned orthonormal matrix this paper describes an algorithm for simultaneously diagonalizing by orthogonal transformations the blocks of a partitioned matrix having orthonormal columns

compact representations of kohn sham invariant subspaces we present a method to compute a hierarchical approximate representation of the solutions of the kohn sham equations the method is based on a recursive bisection algorithm and yields one particle wave functions localized on subdomains of varying sizes the accuracy of the representation is set a priori by specifying the highest acceptable error in 2 norm for any solution applications to large systems are used to illustrate the achievable reduction in data size implications for the development of linear scaling methods and for the acceleration of conventional iterative methods are discussed

the kontsevich matrix integral convergence to the painleve hierarchy and stokes phenomenon we show that the kontsevich integral on n times n matrices n infty is the isomonodromic tau function associated to a 2 times 2 riemann hilbert problem the approach allows us to gain control of the analysis of the convergence as n to infty by an appropriate choice of the external source matrix in kontsevich s integral we show that the limit produces the isomonodromic tau function of a special tronquee solution of the first painleve hierarchy and we identify the solution in terms of the stokes data of the associated linear problem we also show that there are several tau functions that are analytic in appropriate sectors of the space of parameters and that the formal witten kontsevich tau function is the asymptotic expansion of each of them in their respective sectors thus providing an analytic tool to analyze its nonlinear stokes phenomenon

isomonodromic deformations of a rational differential system and reconstruction with the topological recursion the sl2 case in this paper we show that it is always possible to deform a differential equation x x l x x with l x sl2 c x by introducing a small formal parameter in such a way that it satisfies the topological type properties of bergere borot and eynard annales henri poincare 16 12 2713 2782 2015 this is obtained by including the former differential equation in an isomonodromic system and using some homogeneity conditions to introduce the topological recursion is then proved to provide a formal series expansion of the corresponding tau function whose coefficients can thus be expressed in terms of intersections of tautological classes in the deligne mumford compactification of the moduli space of surfaces we present a few examples including any fuchsian system of sl2 c x as well as some elements of painleve hierarchies

perturbation analysis of generalized singular subspaces this paper as a continuation of the paper 20 in numerische mathematik studies the subspaces associated with the generalized singular value decomposition second order perturbation expansions frechet derivatives and condition numbers and perturbation bounds for the subspaces are derived

geometry of batalin vilkovisky quantization the geometry ofp manifolds odd symplectic manifolds andsp manifolds p manifolds provided with a volume element is studied a complete classification of these manifolds is given this classification is used to prove some results about batalin vilkovisky procedure of quantization in particular to obtain a very general result about gauge independence of this procedure

all loop topological string amplitudes from chern simons theory we demonstrate the equivalence of all loop closed topological string amplitudes on toric local calabi yau threefolds with computations of certain knot invariants for chern simons theory we use this equivalence to compute the topological string amplitudes in certain cases to very high degree and to all genera in particular we explicitly compute the topological string amplitudes for open image in new window 2 up to degree 12 and open image in new window 1 open image in new window 1 up to total degree 10 to all genera this also leads to certain novel large n dualities in the context of ordinary superstrings involving duals of type ii superstrings on local calabi yau three folds without any fluxes

comments on risk sensitive adaptive trackers for strict feedback systems with output measurements there were some omissions in the cited paper when establishing the filter and getting the control input using backstepping methodology the recursive procedure was revised and the possibility of using such method in risk sensitive output adaptive trackers was considered here

topological recursion in enumerative geometry and random matrices we review the method of symplectic invariants recently introduced to solve matrix models loop equations in the so called topological expansion and further extended beyond the context of matrix models for any given spectral curve one defines a sequence of differential forms and a sequence of complex numbers fg called symplectic invariants we recall the definition of fg s and we explain their main properties in particular symplectic invariance integrability modularity as well as their limits and their deformations then we give several examples of applications in particular matrix models enumeration of discrete surfaces maps algebraic geometry and topological strings and non intersecting brownian motions

graph homology koszul and verdier duality we show that verdier duality for certain sheaves on the moduli spaces of graphs associated to differential graded operads corresponds to the cobar duality of operads which specializes to koszul duality for koszul operads this in particular gives a conceptual explanation of the appearance of graph cohomology of both the commutative and lie types in computations of the cohomology of the outer automorphism group of a free group another consequence is an explicit computation of dualizing sheaves on spaces of metric graphs thus characterizing to which extent these spaces are different from oriented orbifolds we also provide a relation between the cohomology of the space of metric ribbon graphs known to be homotopy equivalent to the moduli space of riemann surfaces and the cohomology of a certain sheaf on the space of usual metric graphs

synthesis of multi qudit hybrid and d valued quantum logic circuits by decomposition recent research in generalizing quantum computation from 2 valued qudits to d valued qudits has shown practical advantages for scaling up a quantum computer a further generalization leads to quantum computing with hybrid qudits where two or more qudits have different finite dimensions advantages of hybrid and d valued gates circuits and their physical realizations have been studied in detail by muthukrishnan and stroud multi valued logic gates for quantum computation phys rev a 62 2000 052309 10 daboul et al quantum gates on hybrid qudits j phys a math gen 36 2003 2525 2536 5 and bartlett et al quantum encodings in spin systems and harmonic oscillators phys rev a 65 2002 052316 17 in both cases a quantum computation is performed when a unitary evolution operator acting as a quantum logic gate transforms the state of qudits in a quantum system unitary operators can be represented by square unitary matrices if the system consists of a single qudit then tilma et al generalized euler angle parameterization for su n j phys a math gen 35 2002 10467 10501 15 have shown that the unitary evolution matrix gate can be synthesized in terms of its euler angle parametrization however if the quantum system consists of multiple qudits then a gate may be synthesized by matrix decomposition techniques such as qr factorization and the cosine sine decomposition csd in this article we present a csd based synthesis method for n qudit hybrid quantum gates and as a consequence derive a csd based synthesis method for n qudit gates where all the qudits have the same dimension

the gromov witten potential of a point hurwitz numbers and hodge integrals hurwitz numbers which count certain covers of the projective line or equivalently factorizations of permutations into transpositions have been extensively studied for over a century the gromov witten potential f of a point the generating series for descendent integrals on the moduli space of curves is a central object of study in gromov witten theory we define a slightly enriched gromov witten potential g including integrals involving one lambda class and show that after a non trivial change of variables g h in positive genus where h is a generating series for hurwitz numbers we prove a conjecture of goulden and jackson on higher genus hurwitz numbers which turns out to be an analogue of a genus expansion ansatz of itzykson and zuber as consequences we have new combinatorial constraints on f and a much more direct proof of the ansatz of itzykson and zuber we can produce recursions and explicit formulas for hurwitz numbers the algorithm presented proves all such recursions as examples we present surprisingly simple new recursions in genus 0 to 3 similar recursions should exist for all genera as we expect this paper also to be of interest to combinatorialists we have tried to make it as self contained as possible including reviewing some results and definitions well known in algebraic and symplectic geometry and mathematical physics 2000 mathematical subject classification primary 14h10 81t40 secondary 05c30 58d29

on the symmetric solutions of linear matrix equations abstract the symmetric solutions of linear matrix equations ax c and axb c are considered the necessary and sufficient conditions for the consistency of the equations with a symmetric condition on solutions are derived using the singular value decomposition and the generalized singular value decomposition numerical algorithms for solving the equations are suggested

least squares solutions to the matrix equations ax b and xc d abstract the least squares solution and the least squares symmetric solution with the minimum norm of the matrix equations ax b and xc d are considered in this paper by the matrix differentiation and the spectral decomposition of matrices an explicit representation of such solution is given

sato s backlund transformations additional symmetries and asvm formula for the discrete kp hierarchy two kinds of symmetries sato s backlund transformations and additional symmetries for the discrete kp dkp hierarchy are introduced and the asvm formula which demonstrates the equivalence of these two kinds of symmetries is obtained in this process the fay identity and the difference fay identity of the dkp hierarchy are introduced and the asvm formula in the form of tau function is calculated

the spaces of laurent polynomials gromov witten theory of ℙ1 orbifolds and integrable hierarchies let mk m be the space of laurent polynomials in one variable x t1x tk mx where k m 1 are fixed integers and tk m 6 0 according to b dubrovin 11 mk m can be equipped with a semi simple frobenius structure in this paper we prove that the corresponding descendent and ancestor potentials of mk m defined as in 16 satisfy hirota quadratic equations hqe for short let ck m be the orbifold obtained from p by cutting small discs d1 z and d2 z 1 around z 0 and z and gluing back the orbifolds d1 zk and d2 zm in the obvious way we show that the orbifold quantum cohomology of ck m coincides with mk m as frobenius manifolds modulo some yet to be clarified details this implies that the descendent respectively the ancestor potential of mk m is a generating function for the descendent respectively ancestor orbifold gromov witten invariants of ck m there is a certain similarity between our hqe and the lax operators of the extended bi graded toda hierarchy introduced by g carlet in 7 therefore it is plausible that our hqe characterize the tau functions of this hierarchy and we expect that the extended bi graded toda hierarchy governs the gromov witten theory of ck m

connecting the kontsevich witten and hodge tau functions by the widehat gl infty operators in this paper we present an explicit formula that connects the kontsevich witten tau function and the hodge tau function by differential operators belonging to the widehat gl infty group indeed we show that the two tau functions can be connected using virasoro operators this proves a conjecture posted by alexandrov in from hurwitz numbers to kontsevich witten tau function a connection by virasoro operators letters in mathematical physics doi 10 1007 s11005 013 0655 0 2014

triangulated riemann surfaces with boundary and the weil petersson poisson structure given a hyperbolic surface with geodesic boundary s the lengths of a maximal system of disjoint simple geodesic arcs on s that start and end at s perpendicularly are coordinates on the teichmuller space t s we express the weil petersson poisson structure of t s in this system of coordinates and we prove that it limits pointwise to the piecewise linear poisson structure defined by kontsevich on the arc complex of s at the same time we obtain a formula for the first order variation of the distance between two closed geodesics under fenchel nielsen deformation

the symmetric solution of the matrix equations ax ya c axat bybt c and atxa btxb c d abstract the necessary and sufficient conditions for the existence of and the expressions for the symmetric solutions of matrix equations i ax ya c ii axa t byb t c and iii a t xa b t xb c d are derived in addition the minimum 2 norm least squares symmetric solution of equation i the minimum 2 norm symmetric solution of equation ii and the least squares solution of equation iii are obtained

the general linear model and the generalized singular value decomposition abstract a new derivation is given for the generalized singular value decomposition of two matrices x and f having the same number of rows it is shown how this decomposition reveals the structure of the general gauss markov linear model y x 2 ff and exhibits the structure and solution of the generalized linear least squares problem used to provide the best linear unbiased estimator for the model the decomposition is used to prove optimality of the estimator and to reveal the structure of the covariance matrix of the error of the estimator

intersection theory on the moduli space of curves and the matrix airy function we show that two natural approaches to quantum gravity coincide this identity is nontrivial and relies on the equivalence of each approach to kdv equations we also investigate related mathematical problems

risk sensitive adaptive trackers for strict feedback systems with output measurements this note investigates the control of stochastic nonlinear systems with parametric uncertainty the class of systems considered are single input single output and in strict feedback form with the performance measured with respect to a risk sensitive cost criterion the uncertainty in the system description is assumed to be linearly parameterized where the unmeasured parameters are generated by stochastic differential equations by employing the backstepping design technique on the estimates of the unmeasured states provided by a simple state estimator an output feedback adaptive controller is constructed which maintains an arbitrarily small average value for the risk sensitive cost the controller designed achieves boundedness in probability for all closed loop signals and under certain conditions the tracking error converges to zero almost surely

on quantum cohomology rings for hypersurfaces in cp n 1 using the torus action method we construct a one variable polynomial representation of quantum cohomology ring for degree k hypersurface in cpn 1 the results interpolate the well known result of cpn 2 model and the one of calabi yau hypersuface in cpn 1 we find in the k n 2 case the principal relation of this ring has a very simple form compatible with toric compactification of moduli space of holomorphic maps from cp1 to cpn 1

towards the geometry of double hurwitz numbers abstract double hurwitz numbers count branched covers of cp 1 with fixed branch points with simple branching required over all but two points 0 and and the branching over 0 and specified by partitions of the degree with m and n parts respectively single hurwitz numbers or more usually hurwitz numbers have a rich structure explored by many authors in fields as diverse as algebraic geometry symplectic geometry combinatorics representation theory and mathematical physics the remarkable elsv formula relates single hurwitz numbers to intersection theory on the moduli space of curves this connection has led to many consequences including okounkov and pandharipande s proof of witten s conjecture in this paper we determine the structure of double hurwitz numbers using techniques from geometry algebra and representation theory our motivation is geometric we give evidence that double hurwitz numbers are top intersections on a moduli space of curves with a line bundle a universal picard variety in particular we prove a piecewise polynomiality result analogous to that implied by the elsv formula in the case m 1 complete branching over one point and n is arbitrary we conjecture an elsv type formula and show it to be true in genus 0 and 1 the corresponding witten type correlation function has a richer structure than that for single hurwitz numbers and we show that it satisfies many geometric properties such as the string and dilaton equations and an itzykson zuber style genus expansion ansatz we give a symmetric function description of the double hurwitz generating series which leads to explicit formulae for double hurwitz numbers with given m and n as a function of genus in the case where m is fixed but not necessarily 1 we prove a topological recursion on the corresponding generating series which leads to closed form expressions for double hurwitz numbers and an analogue of the goulden jackson polynomiality conjecture an early conjectural variant of the elsv formula in a later paper faber s intersection number conjecture and genus 0 double hurwitz numbers 2005 in preparation the formulae in genus 0 will be shown to be equivalent to the formulae for top intersections on the moduli space of smooth curves m g for example three formulae we give there will imply faber s intersection number conjecture in moduli of curves and abelian varieties aspects of mathematics vol e33 vieweg braunschweig 1999 pp 109 129 in arbitrary genus with up to three points

how covariant closed string theory solves a minimal area problem for a given genusg riemann surface withn 0 punctures n 3 forg 0 we consider the problem of finding the metric of minimal area under the condition that the length of any nontrivial closed curve be greater or equal to 2 the minimal area metrics are found for the case of all punctured genus zero surfaces and for many of the higher genus surfaces both with and without punctures these metrics are induced by jenkins strebel quadratic differentials they arise from the string diagrams corresponding to restricted feynman graphs of a closed string field theory action containing classical and quantum restricted polyhedra

the matrix nearness problem for symmetric matrices associated with the matrix equation atxa btxb c d abstract a direct method based on the projection theorem in inner products spaces the generalized singular value decomposition and the canonical correlation decomposition is presented for finding the optimal approximate solution x in the set s e to a given matrix x where s e denotes the least squares symmetric solution set of the matrix equation atxa btxb c d the analytical expression of the optimal approximate solution x is obtained and an algorithm for finding this solution is also suggested

weil petersson volumes and intersection theory on the moduli space of curves in this paper we establish a relationship between the weil petersson volume vgin b of the moduli space mg n b of hyperbolic riemann surfaces with geodesic boundary components of lengths b bn and the intersection numbers of tauto logical classes on the moduli space mg n of stable curves as a result by using the recursive formula for vg n b obtained in 22 we derive a new proof of the virasoro constraints for a point this result is equivalent to the witten kontsevich formula 14

on a variational formulation of the qsvd and the rsvd abstract recently m t chu r f funderlic and g h golub siam j matrix anal appl 18 1997 1082 1092 presented a variational formulation for the quotient singular value decomposition qsvd of two matrices a r n m c r p m which is a generalization to two matrices of the ordinary singular value decomposition svd and characterizes the role of two orthogonal matrices in qsvd in this paper we give an alternative derivation of this variational formulation and extend it to establish an analogous variational formulation for the restricted singular value decomposition rsvd of a matrix triplet a r n m b r n l c r p m which provides a new understanding of the orthogonal matrices appearing in this decomposition

a tangent algorithm for computing the generalized singular value decomposition we present two new algorithms for floating point computation of the generalized singular values of a real pair a b of full column rank matrices and for floating point solution of the generalized eigenvalue problem hx lambda mx with symmetric positive definite matrices h and m the pair a b is replaced with an equivalent pair a b and the generalized singular values are computed as the singular values of the explicitly computed matrix f a b 1 the singular values of f are computed using the jacobi method the relative accuracy of the computed singular value approximations does not depend on column scalings of a and b that is the accuracy is nearly the same for all pairs ad 1 bd 2 with d 1 d 2 arbitrary diagonal nonsingular matrices similarly the pencil h lambda m is replaced with an equivalent pencil h lambda m and the eigenvalues of h lambda m are computed as the squares of the singular values of g l h l m 1 where l h l m are the cholesky factors of h m respectively and the matrix g is explicitly computed as the solution of a linear system of equations for the computed approximation lambda delta lambda of any exact eigenvalue lambda the relative error delta lambda lambda is of order p n mbox boldmath varepsilon max min delta in cal d kappa 2 delta h delta min delta in cal d kappa 2 delta m delta where p n is a modestly growing polynomial of the dimension of the problem mbox boldmath varepsilon is the round off unit of floating point arithmetic cal d denotes the set of diagonal nonsingular matrices and kappa 2 cdot is the spectral condition number furthermore floating point computation corresponds to an exact computation with h delta h m delta m where for all i j delta h ij sqrt h ii h jj and delta m ij sqrt m ii m jj are of order of mbox boldmath varepsilon times a modest function of n

classes on the moduli space of riemann surfaces through a noncommutative batalin vilkovisky formalism abstract using the machinery of the batalin vilkovisky formalism we construct cohomology classes on compactifications of the moduli space of riemann surfaces from the data of a contractible differential graded frobenius algebra we describe how evaluating these cohomology classes upon a well known construction producing homology classes in the moduli space can be expressed in terms of the feynman diagram expansion of some functional integral by computing these integrals for specific examples we are able to demonstrate that this construction produces families of nontrivial classes

perturbative series and the moduli space of riemann surfaces on utilise des techniques de la theorie quantique des champs pour calculer des quantites reliees aux groupes de symetrie de paires f g ou f est une surface et g une epine de f

correlation functions of the kdv hierarchy and applications to intersection numbers over m g n we derive an explicit generating function of correlations functions of an arbitrary tau function of the kdv hierarchy in particular applications our formulation gives closed formulae of a new type for the generating series of intersection numbers of classes as well as of mixed and classes in full genera

symmetric solutions of linear matrix equations by matrix decompositions abstract the symmetric solutions of linear matrix equations like ax b have been considered using the kronecker product formulation 17 and the commutation matrix k 5 11 in this paper a novel approach is followed using the singular value generalized singular value real schur and real generalized schur decompositions solvability conditions for the symmetric and nonsymmetric solutions are obtained and the numerical properties of the related solution procedures are discussed briefly

volume optimization normal surfaces and thurston s equation on triangulated 3 manifolds we propose a finite dimensional variational principle on triangulated 3 manifolds so that its critical points are related to solutions to thurston s gluing equation and haken s normal surface equation the action functional is the volume

on simplified order one factorizations of paraunitary filterbanks this paper revisits the order one factorization of causal finite impulse response fir paraunitary filterbanks pu fbs the basic form of the factorization was proposed by vaidyanathan et al in 1987 which is a cascade of general unitary matrices separated by diagonal delay matrices with arbitrary number of delay elements recently gao et al have proved the completeness of this factorization and developed a more efficient structure that only uses approximately half number of free parameters in this paper by briefly analyzing gao et al s derivation we first point out that gao et al s factorization contains redundant free parameters two simplified structures of vaidyanathan s factorization are then developed i e a post filtering based structure and a prefiltering based structure our simplification relies on consecutive removal of extra degrees of freedom in adjacent stages which is accomplished through the c s decomposition of a general unitary matrix since the conventional c s decomposition leads to a redundant representation a new c s decomposition is developed to minimize the number of free parameters by further incorporating the givens rotation factorization the proposed structures can maintain the completeness and the minimality of the original lattice compared with gao et al s factorization our derivations are much simpler while the resulting structures contain fewer free parameters and less implementation cost besides these new factorizations indicate that for a pu fb with a given filter length the symmetric delay factorization offers the largest degrees of design freedom several design examples are presented to confirm the validity of the theory

equations of the moduli of pointed curves in the infinite grassmannian the main result of this paper is the explicit computa tion of the equations defining the moduli space of triples c p where c is an integral and complete algebraic curve p a smooth rational point anda certain isomorphism this is achieved by introducing algebraically infinite grassmannians tau and baker ahkiezer functions and by proving an addition formula for tau functions

computing the cs and the generalized singular value decompositions if the columns of a matrix are orthonormal and it is partitioned into a 2 by 1 block matrix then the singular value decompositions of the blocks are related this is the essence of the cs decomposition the computation of these related svd s requires some care stewart has given an algorithm that uses the linpack svd algorithm together with a jacobitype clean up operation on a cross product matrix our technique is equally stable and fast but avoids the cross product matrix the simplicity of our technique makes it more amenable to parallel computation on systolic type computer architectures these developments are of interest because a good way to compute the generalized singular value decomposition of a matrix pair a b is to compute the cs decomposition of a certain orthogonal column matrix related toa andb

analyticity of the total ancestor potential in singularity theory abstract k saito s theory of primitive forms gives a natural semi simple frobenius manifold structure on the space of miniversal deformations of an isolated singularity on the other hand givental introduced the notion of a total ancestor potential for every semi simple point of a frobenius manifold and conjectured that in the settings of singularity theory his definition extends analytically to non semisimple points as well in this paper we prove givental s conjecture by using the eynard orantin recursion

a qr type reduction for computing the svd of a general matrix product quotient in this paper a qr type reduction technique is developed for the computation of the svd of a general matrix product quotient a a 1 s 1 a 2 s 2 cdots a m s m with a i in bf r n times n hbox and s i 1 or s i 1 first the matrix a is reduced by at most m qr factorizations to the form q 1 11 q 1 21 1 where q 1 11 q 1 21 in bf r n times n and q 1 11 tq 1 11 q 1 21 tq 1 21 i then the svd of a is obtained by computing the csd cosine sine decomposition of q 1 11 and q 1 21 using the matlab command gsvd the performance of the proposed method is verified by some numerical examples

authors reply to comments on risk sensitive adaptive trackers for strict feedback systems with output measurements the authors acknowledge the comments made by dai and shi see ibid p 717 19 2006 and for pointing out two omissions in their paper arslan and basar see ibid vol 47 no 10 p 1754 8 2002 they outline the corrections along the lines suggested 1 the unknown parameter spl theta in the system dynamics must be constant 2 in one of the equations the terms inside the boxes that were inadvertently omitted in the backstepping design presented by arslan and basar 2002 must be included

matrix integration and combinatorics of modular groups integration over gaussian matrix ensembles was used to obtain the virtual euler characteristics of mapping class groups we present some simplifications in the combinatorial part of the calculation

gromov witten theory of orbicurves the space of tri polynomials and symplectic field theory of seifert fibrations we compute with symplectic field theory sft techniques the gromov witten theory of mathbb p 1 alpha 1 ldots alpha a i e the complex projective line with a orbifold points a natural subclass of these orbifolds the ones with polynomial quantum cohomology gives rise to a family of polynomial frobenius manifolds and integrable systems of hamiltonian pdes which extend the dispersionless bigraded toda hierarchy carlet the extended bigraded toda hierarchy arxiv preprint arxiv math ph 0604024 we then define a frobenius structure on the spaces of polynomials in three complex variables of the form f x y z xyz p1 x p2 y p3 z which contains as special cases the ones constructed on the space of laurent polynomials dubrovin geometry of 2d topologica field theories integrable systems and quantum groups springer lecture notes in mathematics 1620 120 348 1996 milanov and tseng the space of laurent polynomials mathbb p 1 orbifolds and integrable hierarchies preprint arxiv math 0607012v3 math ag we prove a mirror theorem stating that these frobenius structures are isomorphic to the ones found before for polynomial mathbb p 1 orbifolds finally we link rational sft of seifert fibrations over mathbb p 1 a b c with orbifold gromov witten invariants of the base extending a known result bourgeois a morse bott approach to contact homology ph d dissertation stanford university 2002 valid in the smooth case

the partition function of the two matrix model as an isomonodromic τ function we consider the itzykson zuber eynard mehta two matrix model and prove that the partition function is an isomonodromic function in a sense that generalizes that of jimbo et al monodromy preserving deformation of linear ordinary differential equations with rational coefficients physica d 2 306 1981 in order to achieve the generalization we need to define a notion of function for isomonodromic systems where the adregularity of the leading coefficient is not a necessary requirement

two dimensional quantum gravity and quasiclassical integrable hierarchies the main results for the two dimensional quantum gravity conjectured from the matrix model or integrable approach are presented in the form to be compared with the worldsheet or liouville approach in a spherical limit the integrable side for minimal string theories is completely formulated using simple manipulations with two polynomials based on residue formulae from quasiclassical hierarchies explicit computations for particular models are performed and certain delicate issues of nontrivial relations among them are discussed they concern the connections between different theories obtained as expansions of basically the same stringy solution to dispersionless kadomtsev petviashvili hierarchy in different backgrounds characterized by nonvanishing background values of different times being the simplest known example of the change of the quantum numbers of physical observables when moving to a different point in the moduli space of the theory

estimates of weil petersson volumes via effective divisors we study the asymptotics of the weil petersson volumes of the moduli spaces of compact riemann surfaces of genus g with n punctures for fixed n as g

the unstable integral homology of the mapping class groups of a surface with boundary we construct a graph complex calculating the integral homology of the bordered mapping class groups we compute the homology of the bordered mapping class groups of the surfaces s 1 1 s 1 2 and s 2 1 using the circle action on this graph complex we build a double complex and a spectral sequence converging to the homology of the unbordered mapping class groups we compute the homology of the punctured mapping class groups associated to the surfaces s 1 1 and s 2 1 finally we use miller s operad to get the first kudo araki and browder operations on our graph complex we also consider an unstable version of the higher kudo araki dyer lashoff operations

the m n symmetric procrustes problem abstract an p q matrix a is said to be m n symmetric if man man t for given m r n p n r q n in this paper the following m n symmetric procrustes problem is studied find the m n symmetric matrix a which minimizes the frobenius norm of ax b where x and b are given rectangular matrices we use project theorem the singular value decomposition and the generalized singular value decomposition of matrices to analysis the problem and to derive a stable method for its solution the related optimal approximation problem to a given matrix on the solution set is solved furthermore the algorithm to compute the optimal approximate solution and the numerical experiment are given

the euler characteristic of the moduli space of curves let fg 1 g 1 be the mapping class group consisting of all isotopy classes of base point and orientation preserving homeomorphisms of a closed oriented surface f of genus g let 1 be its euler characteristic in the sense of wall that is z f i fgl f l z e f where f is any torsion free subgroup of finite index in f 1 and e is a contractible space on which f acts freely and properly discontinuously an example of such a space is the teichmiiller space 1 and g f can be interpreted as the orbifold euler characteristic of f i join the moduli space of curves of genus g with base point the purpose of this paper is to prove the following formula for f i

a formula of two partition hodge integrals jmg n where tpi ci l is the first chern class of l and xj cj e is the j th chern class of the hodge bundle the study of hodge integrals is an important part of intersection theory on mg n hodge integrals also naturally arise when one computes gromov witten invariants by localization techniques for example the following generating series of hodge integrals arises when one computes local invariants of a toric fano surface in a calabi yau 3 fold by virtual localization 33

the witten equation mirror symmetry and quantum singularity theory for any nondegenerate quasi homogeneous hypersurface singularity we describe a family of moduli spaces a virtual cycle and a corresponding cohomological eld theory associated to the singularity this theory is analogous to gromov witten theory and generalizes the theory of r spin curves which corresponds to the simple singularity ar 1 we also resolve two outstanding conjectures of witten the rst conjecture is that ade singularities are self dual and the second conjecture is that the total potential functions of ade singularities satisfy corresponding ade integrable hierarchies other cases of integrable hierarchies are also discussed

generalized reflexive solutions of the matrix equation axb d and an associated optimal approximation problem let r c m x m and s c n x n be nontrivial unitary involutions i e r h r r 1 i m and s h s s 1 i n we say that g c m x n is a generalized reflexive matrix if rgs g the set of all mxn generalized reflexive matrices is denoted by grc m x n in this paper a sufficient and necessary condition for the matrix equation axb d where a c p x m b c n x q and d c p x q to have a solution x grc m x n is established and if it exists a representation of the solution set s x is given an optimal approximation between a given matrix x c m x n and the affine subspace s x is discussed an explicit formula for the unique optimal approximation solution is presented and a numerical example is provided

tau functions and virasoro symmetries for drinfeld sokolov hierarchies abstract for each drinfeld sokolov integrable hierarchy associated to affine kac moody algebra we obtain a uniform construction of tau function by using tau symmetric hamiltonian densities moreover we represent its virasoro symmetries as linear nonlinear actions on the tau function the relations between the tau function constructed in this paper and those defined for particular cases of drinfeld sokolov hierarchies in the literature are clarified we also show that whenever the affine kac moody algebra is simply laced or twisted the tau function of the drinfeld sokolov hierarchy coincides with the solution of the corresponding kac wakimoto hierarchy constructed from the principal vertex operator realization of the affine algebra

advances in random matrix theory zeta functions and sphere packing over four hundred years ago sir walter raleigh asked his mathematical assistant to find formulas for the number of cannonballs in regularly stacked piles these investigations aroused the curiosity of the astronomer johannes kepler and led to a problem that has gone centuries without a solution why is the familiar cannonball stack the most efficient arrangement possible here we discuss the solution that hales found in 1998 almost every part of the 282 page proof relies on long computer verifications random matrix theory was developed by physicists to describe the spectra of complex nuclei in particular the statistical fluctuations of the eigenvalues the energy levels follow certain universal laws based on symmetry types we describe these and then discuss the remarkable appearance of these laws for zeros of the riemann zeta function which is the generating function for prime numbers and is the last special function from the last century that is not understood today explaining this phenomenon is a central problem these topics are distinct so we present them separately with their own introductory remarks

least squares solution with the minimum norm for the matrix equation axb gxh c d based on the projection theorem in hilbert space by making use of the generalized singular value decomposition and the canonical correlation decomposition an analytical expression of the least squares solution for the matrix equation axb gxh c d with the minimum norm is derived an algorithm for finding the minimum norm solution is described and some numerical results have been given

on certain infinite dimensional aspects of arakelov intersection theory letk y x be an embedding of compact complex manifolds bismut and lebeau have calculated the quillen norm of the canonical isomorphism identifying the determinant of the cohomology of a holomorphic vector bundle overy and the determinant of the cohomology of a resolution by a complex of holomorhic vector bundles overx the purpose of this paper is to show that the formula of bismut lebeau can be viewed as an equivariant intersection formula over the loop space of the considered manifolds in the presence of an infinite dimensional excess normal bundle this excess normal bundle is responsible for the appearance of the additive genusr of gillet and soule in the formula of bismut and lebeau