tyz expansion for the kepler manifold the main goal of the paper is to address the issue of the existence of kempf s distortion function and the tian yau zelditch tyz asymptotic expansion for the kepler manifold an important example of non compact manifold motivated by the recent results for compact manifolds we construct kempf s distortion function and derive a precise tyz asymptotic expansion for the kepler manifold we get an exact formula finite asymptotic expansion of n 1 terms and exponentially small error terms uniformly with respect to the discrete quantization parameter m rightarrow infty hbar m 1 rightarrow 0 standing for planck s constant and x rightarrow infty x in mathbb c n moreover the coefficients are calculated explicitly and they turned out to be homogeneous functions with respect to the polar radius in the kepler manifold we show that our estimates are sharp by analyzing the nonharmonic behaviour of tm for m rightarrow infty the arguments of the proofs combine geometrical methods quantization tools and functional analytic techniques for investigating asymptotic expansions in the framework of analytic gevrey spaces

discontinuous spectral element methods for time and space fractional advection equations we develop spectral element methods for a time and space fractional advection equation of the form prescript 0 mathcal d tau t u x t theta prescript 0 mathcal d nu x u x t f x t of order tau in 0 1 nu in 0 1 subject to dirichlet initial boundary conditions we present two spectrally accurate and efficient methods for global discretization of both temporal and spatial terms instead of employing traditional low order time integration methods to this end we first develop a petrov galerkin in time and discontinuous galerkin in space pg dg method where we carry out the time integration using a single time domain spectral method sm and we perform the spatial discretization using the discontinuous spectral hp element method dsem this scheme also leads to a more efficient time integration when the time derivative is integer order i e tau 1 we develop the sm dsem scheme based on a new spectral theory for fractional sturm liouville problems

moment maps scalar curvature and quantization of kahler manifolds building on donaldson s work on constant scalar curvature metrics we study the space of regular kahler metrics e i e those for which deformation quantization has been defined by cahen gutt and rawnsley after giving in sects 2 and 3 a review of donaldson s moment map approach we study the essential uniqueness of balanced basis i e of coherent states in a more general setting theorem 2 5 we then study the space e in sect 4 and we show in sect 5 how all the tools needed can be defined also in the case of non compact manifolds

scalar curvature and projective embeddings i we prove that a metric of constant scalar curvature on a polarised kahler manifold is the limit of metrics induced from a specific sequence of projective embeddings satisfying a condition introduced by h luo this gives as a corollary the uniqueness of constant scalar curvature kahler metrics in a given rational cohomology class the proof uses results in the literature on the asymptotics of the bergman kernel the arguments are presented in a general framework involving moment maps for two different group actions

convergence analysis of the chebyshev legendre spectral method for a class of fredholm fractional integro differential equations abstract in this paper we propose and analyze a spectral chebyshev legendre approximation for fractional order integro differential equations of fredholm type the fractional derivative is described in the caputo sense our proposed method is illustrated by considering some examples whose exact solutions are available we prove that the error of the approximate solution decays exponentially in l 2 norm

the space of kahler metrics this paper the second of a series deals with the function space mathcal h of all smooth kahler metrics in any given n dimensional closed complex manifold v these metrics being restricted to a given fixed real cohomology class called a polarization of v this function space is equipped with a pre hilbert metric structure introduced by t mabuchi 10 who also showed that formally this metric has nonpositive curvature in the first paper of this series 4 the second author showed that the same space is a path length space he also proved that mathcal h is geodesically convex in the sense that for any two points of mathcal h there is a unique geodesic path joining them which is always length minimizing and of class c1 1 this partially verifies two conjectures of donaldson 8 on the subject in the present paper we show first of all that the space is as expected a path length space of nonpositive curvature in the sense of a d aleksandrov a second result is related to the theory of extremal kahler metrics namely that the gradient flow in mathcal h of the k energy of v has the property that it strictly decreases the length of all paths in mathcal h except those induced by one parameter families of holomorphic automorphisms of m

a unified petrov galerkin spectral method for fractional pdes existing numerical methods for fractional pdes suffer from low accuracy and inefficiency in dealing with three dimensional problems or with long time integrations we develop a unified and spectrally accurate petrov galerkin pg spectral method for a weak formulation of the general linear fractional partial differential equations fpdes of the form 0dt2 u j 1dcj ajdxj2 ju u f where 2 2 j 0 1 j 0 1 in a 1 d1 d dimensional space time domain subject to dirichlet initial and boundary conditions we perform the stability analysis in 1 d and the corresponding convergence study of the scheme in multi d the unified pg spectral method applies to the entire family of linear hyperbolichyperbolic parabolicparabolic and ellipticelliptic like equations we develop the pg method based on a new spectral theory for fractional sturm liouville problems fslps recently introduced in zayernouri and karniadakis 2013 specifically we employ the eigenfunctions of the fslp of first kind fslp i called jacobi poly fractonomials as temporal spatial bases next we construct a different space for test functions from poly fractonomial eigenfunctions of the fslp of second kind fslp ii besides the high order spatial accuracy of the pg method we demonstrate its efficiency and spectral accuracy in time integration schemes for solving time dependent fpdes as well rather than employing algebraically accurate traditional methods especially when 2 12 1 finally we formulate a general fast linear solver based on the eigenpairs of the corresponding temporal and spatial mass matrices with respect to the stiffness matrices which reduces the computational cost drastically we demonstrate that this framework can reduce to hyperbolic fpdes such as time and space fractional advection tsfa parabolic fpdes such as time and space fractional diffusion tsfd model and elliptic fpdes such as fractional helmholtz poisson equations with the same ease and cost several numerical tests confirm the efficiency and spectral convergence of the unified pg spectral method for the aforementioned families of fpdes moreover we demonstrate the computational efficiency of the new approach in higher dimensions e g 1 3 1 5 and 1 9 dimensional problems

symmetries quotients and kahler einstein metrics we consider fano manifolds m that admit a collection of finite automorphism groups g1 gk such that the quotients m gi are smooth

on the convergence of spline collocation methods for solving fractional differential equations in the first part of this paper we study the regularity properties of solutions of initial value problems of linear multi term fractional differential equations we then use these results in the convergence analysis of a polynomial spline collocation method for solving such problems numerically using an integral equation reformulation and special non uniform grids global convergence estimates are derived from these estimates it follows that the method has a rapid convergence if we use suitable nonuniform grids and the nodes of the composite gaussian quadrature formulas as collocation points theoretical results are verified by some numerical examples

on the lower order terms of the asymptotic expansion of tian yau zelditch we compute the first three coefficients of the asymptotic expansion of tian yau zelditch we also prove that in general the fcth coefficient is a polynomial of the curvature and its derivative of weight k

continuous quantum jumps and infinite dimensional stochastic equations from a mathematical point of view a class of infinite dimensional stochastic differential equations describing continuous spontaneous localization in quantum dynamics will be studied existence and uniqueness of weak and strong solutions of respective equations are proven via cameron martin girsanov transformation the case of gaussian initial states is explicitly solved

a new operational matrix for solving fractional order differential equations abstract fractional calculus has been used to model physical and engineering processes that are found to be best described by fractional differential equations for that reason we need a reliable and efficient technique for the solution of fractional differential equations this paper deals with the numerical solution of a class of fractional differential equations the fractional derivatives are described in the caputo sense our main aim is to generalize the legendre operational matrix to the fractional calculus in this approach a truncated legendre series together with the legendre operational matrix of fractional derivatives are used for numerical integration of fractional differential equations the main characteristic behind the approach using this technique is that it reduces such problems to those of solving a system of algebraic equations thus greatly simplifying the problem the method is applied to solve two types of fractional differential equations linear and nonlinear illustrative examples are included to demonstrate the validity and applicability of the presented technique

lower and upper bounds on csl parameters from latent image formation and igm heating we study lower and upper bounds on the parameters for stochastic state vector reduction focusing on the mass proportional continuous spontaneous localization csl model we show that the assumption that the state vector is reduced when a latent image is formed in photography or etched track detection requires a csl reduction rate parameter that is larger than conventionally assumed by a factor of roughly 2 109 2 for a correlation length rc of 10 5cm we reanalyse existing upper bounds on the reduction rate and conclude that all are compatible with such an increase in the best bounds that we have obtained come from a consideration of heating of the intergalactic medium igm which shows that can be at most 108 1 times as large as the standard csl value again for rc 10 5cm for both the lower and upper bounds quoted errors are not purely statistical errors but rather are estimates reflecting modelling uncertainties we discuss modifications in our analysis corresponding to a larger value of rc with a substantially enlarged rate parameter csl effects may be within range of experimental detection or refutation with current technologies

numerical solution of semidifferential equations by collocation method we study the numerical solution of semidifferential equation of order 4 by collocation spline method we derive a system of equations that characterizing the numerical solution some numerical examples are also presented to illustrate our results

balanced metrics on cartan and cartan hartogs domains this paper consists of two results dealing with balanced metrics in donaldson terminology on noncompact complex manifolds in the first one we describe all balanced metrics on cartan domains in the second one we show that the only cartan hartogs domain which admits a balanced metric is the complex hyperbolic space by combining these results with those obtained in loi and zedda mathematische annalen 2011 to appear we also provide the first example of complete kahler einstein and projectively induced metric g such that g is not balanced for all 0

comparative study of three numerical schemes for fractional integro differential equations this paper presents a comparative study of three numerical schemes such as linear quadratic and quadratic linear scheme for the fractional integro differential equations defined in terms of the caputo fractional derivatives the error estimates of the respective approximations are also established numerical tests of the discussed schemes show that all schemes work well and when the number of terms approximating the solution is increased the desired solution is achieved the accuracy of the numerical schemes with respect to the step size h is analysed and illustrated through various tables finally comparative performances of the schemes are discussed comparative study of three schemes such as linear quadratic and quadratic linear schemes is presented for the fractional integro differential equations the convergences of the schemes are also established the performances of the schemes are compared through the test examples

collocation method with convergence for generalized fractional integro differential equations abstract in this paper we study a numerical approach for some class of generalized fractional integro differential equations gfides defined in terms of the b operators presented recently we develop collocation method for linear and nonlinear forms of gfides the numerical approach uses the idea of collocation methods for solving integral equations legendre polynomials are used to approximate the solution in finite dimensional space with convergence analysis the obtained approximate solution recovers the solution of the fractional integro differential equation fide defined using caputo derivatives in a special case fides containing convolution type kernels appear in diverse area of science and engineering applications therefore some test examples varying the kernel in the b operator are considered to perform the numerical investigations the numerical results validate the presented scheme and provide good accuracy using few legendre basis functions

numerical algorithm to solve system of nonlinear fractional differential equations based on wavelets method and the error analysis in this paper a system of nonlinear fractional differential equations fdes are considered they have been solved by legendre wavelets method combining with its operational matrix however there are no articles about solving this system using wavelets method the main purpose of this technique is to transform the initial equations into a nonlinear system of algebraic equations which can be solved easily the convergence and error analysis are presented to show the correctness and feasibility of method proposed for solving the above mentioned problem finally the applicability and efficiency of the mentioned approach are demonstrated by three numerical examples

application of the collocation method for solving nonlinear fractional integro differential equations in this paper using the collocation method we solve the nonlinear fractional integro differential equations nfide of the form f t y t a cd t a 0y t a cd t a ry t email protected t y t a tk t s f s y s ds y k a d k k 0 1 m 0 1 we study the convergence and the stability analysis of this method for f t y t a cd t a 0y t a cd t a ry t y t email protected j 0 rb j a cd t a jy t g t some numerical examples are given to show the efficiency of the presented method

finite tycz expansions and csck metrics abstract let m g be a kahler manifold whose associated kahler form is integral and let l h m be a quantization hermitian line bundle in this paper we study those kahler manifolds m g admitting a finite tycz expansion namely those for which the associated kempf distortion function t m g is of the form t m g p f s p m s f s 1 p m s 1 f r p m r f j c m s r z we show that if the tycz expansion is finite then t m g is indeed a polynomial in m of degree n n dim c m and the log term of the szego kernel of the disc bundle d l vanishes where l is the dual bundle of l moreover we provide a complete classification of the kahler manifolds admitting finite tycz expansion either when m is a complex curve or when m is a complex surface with a csck metric which admits a radial kahler potential

two conjectures on ricci flat kahler metrics we propose two conjectures about ricci flat kahler metrics conjecture 1 a ricci flat projectively induced metric is flat conjecture 2 a ricci flat metric on an n dimensional complex manifold such that the a n 1 coefficient of the tyz expansion vanishes is flat we verify conjecture 1 see theorem 1 1 under the assumptions that the metric is radial and stable projectively induced and conjecture 2 see theorem 1 2 for complex surfaces whose metric is either radial or complete and ale we end the paper by showing by means of the simanca metric that the assumption of ricci flatness in conjecture 1 and in theorem 1 2 cannot be weakened to scalar flatness see theorem 1 3

szego kernel regular quantizations and spherical cr structures we compute the szego kernel of the unit circle bundle of a negative line bundle dual to a regular quantum line bundle over a compact kahler manifold as a corollary we provide an infinite family of smoothly bounded strictly pseudoconvex domains on complex manifolds disk bundles over homogeneous hodge manifolds for which the log terms in the fefferman expansion of the szego kernel vanish and which are not locally cr equivalent to the sphere we also give a proof of the fact that for homogeneous hodge manifolds the existence of a locally spherical cr structure on the unit circle bundle alone implies that the manifold is biholomorphic to a projective space our results generalize those obtained by englis math z 264 4 901 912 2010 for hermitian symmetric spaces of compact type

quantization of kahler manifolds ii we use berezin s dequantization procedure to define a formal product on a dense subalgebra of the algebra ofsmooth functions on a compact homogeneous kahler manifold m we prove that this formal product isconvergent when m isa hermitian symmetric space 1993 american mathematical society

reconstruction of exponentially rate of convergence to legendre collocation solution of a class of fractional integro differential equations in this paper legendre collocation method an easy to use variant of the spectral methods for the numerical solution of a class of fractional integro differential equations fide s is researched in order to obtain high order accuracy for the approximation the integral term in the resulting equation is approximated by using legendre gauss quadrature formula an efficient convergence analysis of the proposed method is given and rate of convergence is established in the l 2 norm due to the fact that the solutions of fide s usually have a weak singularity at origin we use a variable transformation to change the original equation into a new equation with a smooth solution we prove that after this regularization technique numerical solution of the new equation by adopting the legendre collocation method has exponentially rate of convergence numerical results are presented which clarify the high accuracy of the proposed method

collocation method for linear and nonlinear fredholm and volterra integral equations a collocation procedure is developed for the linear and nonlinear fredholm and volterra integral equations using the globally defined b spline and auxiliary basis functions the solution is collocated by cubic b spline and then the integral equation is approximated by the 5 points gauss turan quadrature formula with respect to the legendre weight function combination of these two approaches is the main idea of this paper to reduce the cost of computation and complexity the error analysis of proposed numerical method is studied theoretically numerical results are given to illustrate the efficiency of the proposed method the results are compared with the results obtained by other methods to verify that this method is accurate and efficient

bergman kernels tyz expansions and hankel operators on the kepler manifold for a class of o n 1 r invariant measures on the kepler manifold possessing finite moments of all orders we describe the reproducing kernels of the associated bergman spaces discuss the corresponding asymptotic expansions of tian yau zelditch and study the relevant hankel operators with conjugate holomorphic symbols related reproducing kernels on the minimal ball are also discussed finally we observe that the kepler manifold either does not admit balanced metrics or such metrics are not unique

application of the fractional differential transform method to fractional order integro differential equations with nonlocal boundary conditions in this paper the fractional differential transform method is developed to solve fractional integro differential equations with nonlocal boundary conditions the method is described and illustrated with numerical examples the results reveal that the method is accurate and easy to apply

exponentially accurate spectral and spectral element methods for fractional odes current discretizations of fractional differential equations lead to numerical solutions of low order of accuracy here we present different methods for fractional odes that lead to exponentially fast decay of the error first we develop a petrov galerkin pg spectral method for fractional initial value problems fivps of the form d t 0 u t f t and fractional final value problems ffvps d t t u t g t where 0 1 subject to dirichlet initial final conditions these schemes are developed based on a new spectral theory for fractional sturm liouville problems fslps which has been recently developed in 1 specifically we obtain solutions to fivps and ffvps in terms of the new fractional non polynomial basis functions called jacobi polyfractonomials which are the eigenfunctions of the fslp of first kind fslp i correspondingly we employ another space of test functions as the span of polyfractonomial eigenfunctions of the fslp of second kind fslp ii subsequently we develop a discontinuous spectral method dsm of petrov galerkin sense for the aforementioned fivps and ffvps where the basis functions do not satisfy the initial final conditions finally we extend the dsm scheme to a discontinuous spectral element method dsem for efficient longer time integration and adaptive refinement in these discontinuous schemes we employ the asymptotic eigensolutions to fslp i ii which are of jacobi polynomial forms as basis and test functions our numerical tests confirm the exponential algebraic convergence respectively in p and h refinements for various test cases with integer and fractional order solutions

cubic b spline approximation for linear stochastic integro differential equation of fractional order abstract in this paper the cubic b spline collocation method is used for solving the stochastic integro differential equation of fractional order we show that stochastic integro differential equation of fractional order is equivalent to a modified stochastic integral equation then we apply the proposed method to obtain a numerical scheme of the modified stochastic integral equation using this method the problem solving turns into a linear system solution of equations also the convergence analysis of this numerical approach has been discussed in the end examples are given to test the accuracy and the implementation of the method the results are compared with the results obtained by other methods to verify that this method is accurate and efficient

spline collocation methods for linear multi term fractional differential equations abstract some regularity properties of the solution of linear multi term fractional differential equations are derived based on these properties the numerical solution of such equations by piecewise polynomial collocation methods is discussed the results obtained in this paper extend the results of pedas and tamme 2011 15 where we have assumed that in the fractional differential equation the order of the highest derivative of the unknown function is an integer in the present paper we study the attainable order of convergence of spline collocation methods for solving general linear fractional differential equations using caputo form of the fractional derivatives and show how the convergence rate depends on the choice of the grid and collocation points theoretical results are verified by some numerical examples

numerical approach for solving stochastic volterra fredholm integral equations by stochastic operational matrix in this paper we obtain stochastic operational matrix of block pulse functions on interval 0 1 to solve stochastic volterra fredholm integral equations by using block pulse functions and their stochastic operational matrix of integration the stochastic volterra fredholm integral equation can be reduced to a linear lower triangular system which can be directly solved by forward substitution we prove that the rate of convergence is o h furthermore the results show that the approximate solutions have a good degree of accuracy

numerical solution of fractional integro differential equations by collocation method abstract this paper deals with the numerical solution of fractional integro differential equations of the type d q y t p t y t f t 0 t k t s y s d s t i 0 1 by polynomial spline functions we derive a system of equations that characterizing the numerical solution some numerical examples are also provided to illustrated our results

kahler einstein metrics with positive scalar curvature in this paper we prove that the existence of kahler einstein metrics implies the stability of the underlying kahler manifold in a suitable sense in particular this disproves a long standing conjecture that a compact kahler manifold admits kahler einstein metrics if it has positive first chern class and no nontrivial holomorphic vector fields we will also establish an analytic criterion for the existence of kahler einstein metrics our arguments also yield that the analytic criterion is satisfied on stable kahler manifolds provided that the partial c 0 estimate posed in t6 is true

hyperkahler metrics and supersymmetry we describe two constructions of hyperkahler manifolds one based on a legendre transform and one on a sympletic quotient these constructions arose in the context of supersymmetric nonlinear models but can be described entirely geometrically in this general setting we attempt to clarify the relation between supersymmetry and aspects of modern differential geometry along the way reviewing many basic and well known ideas in the hope of making them accessible to a new audience

euler polynomial solutions of nonlinear stochastic ito volterra integral equations abstract in this study a practical matrix method based on operational matrices of integration and collocation points is presented to find the approximate solution of nonlinear stochastic ito volterra integral equations for this aim first we compute the stochastic operational matrix of euler polynomials then by using this operational matrix nonlinear stochastic ito volterra integral equation is reduced to a system of nonlinear algebraic equations with unknown euler coefficients which can be solved by using a convenient numerical method such as newton s method also convergence analysis and error estimation of the approach is discussed and an upper error bound is provided under mild conditions finally some numerical examples are provided to confirm the accuracy and efficiency of the proposed method all of the numerical computations have been performed on a pc by running some programs written in matlab software

on the energy increase in space collapse models a typical feature of spontaneous collapse models which aim at localizing wavefunctions in space is the violation of the principle of energy conservation in the models proposed in the literature the stochastic field which is responsible for the localization mechanism causes the momentum to behave like a brownian motion whose larger and larger fluctuations show up as a steady increase of the energy of the system in spite of the fact that in all situations such an increase is small and practically undetectable it is an undesirable feature that the energy of physical systems is not conserved but increases constantly in time diverging for t in this paper we show that this property of collapse models can be modified we propose a model of spontaneous wavefunction collapse sharing all most important features of usual models but such that the energy of isolated systems reaches an asymptotic finite value instead of increasing with a steady rate

an efficient computational method for solving nonlinear stochastic ito integral equations because of the nonlinearity closed form solutions of many important stochastic functional equations are virtually impossible to obtain thus numerical solutions are a viable alternative in this paper a new computational method based on the generalized hat basis functions together with their stochastic operational matrix of ito integration is proposed for solving nonlinear stochastic ito integral equations in large intervals in the proposed method a new technique for computing nonlinear terms in such problems is presented the main advantage of the proposed method is that it transforms problems under consideration into nonlinear systems of algebraic equations which can be simply solved error analysis of the proposed method is investigated and also the efficiency of this method is shown on some concrete examples the obtained results reveal that the proposed method is very accurate and efficient as two useful applications the proposed method is applied to obtain approximate solutions of the stochastic population growth models and stochastic pendulum problem

a novel method for solving second order fractional eigenvalue problems the paper presents a new numerical method for solving eigenvalue problems for fractional differential equations it combines two techniques the method of external excitation mee and the backward substitution method bsm the first one is a mathematical model of physical measurements when a mechanical electrical or acoustic system is excited by some source and resonant frequencies can be determined by using the growth of the amplitude of oscillations near these frequencies the bsm consists of replacing the original equation by an approximate equation which has an exact analytic solution with a set of free parameters these free parameters are determined by the use of the collocation procedure some examples are given to demonstrate the validity and applicability of the new method and a comparison is made with the existing results the numerical results show that the proposed method is of a high accuracy and is efficient for solving of a wide class of eigenvalue problems

towards quantum superpositions of a mirror an exact open systems analysis calculational details we give details of calculations analysing the proposed mirror superposition experiment of marshall simon penrose and bouwmeester within different stochastic models for state vector collapse we give two methods for exactly calculating the fringe visibility in these models one proceeding directly from the equation of motion for the expectation of the density matrix and the other proceeding from solving a linear stochastic unravelling of this equation we also give details of the calculation that identifies the stochasticity parameter implied by the small displacement taylor expansion of the csl model density matrix equation the implications of the two results are briefly discussed two pedagogical appendices review mathematical apparatus needed for the calculations

numerical solution of fractional integro differential equations by a hybrid collocation method fractional integro differential equations have been recently solved by many methods such as adomian decomposition method differential transform method collocation method and taylor expansion approach in this paper a hybrid collocation method is used which combines a non polynomial collocation used on the first subinterval and graded piecewise polynomial collocation used on the rest of the interval a theoretical analysis for the convergence order of the method is presented some numerical examples are given which confirm the theoretical results

radial balanced metrics on the unit ball of the kepler manifold abstract we show that there is no radial balanced metric on the unit ball of the kepler manifold with not too wild boundary behavior additionally we identify explicitly the weights corresponding to radial metrics with such boundary behavior which satisfy the balanced condition as far as germs at the boundary are concerned related results for poincare metrics are also established

approximate solution of fractional integro differential equations by taylor expansion method in this paper taylor expansion approach is presented for solving approximately a class of linear fractional integro differential equations including those of fredholm and of volterra types by means of the mth order taylor expansion of the unknown function at an arbitrary point the linear fractional integro differential equation can be converted approximately to a system of equations for the unknown function itself and its m derivatives under initial conditions this method gives a simple and closed form solution for a linear fractional integro differential equation in addition illustrative examples are presented to demonstrate the efficiency and accuracy of the proposed method