test configurations large deviations and geodesic rays on toric varieties this article contains a detailed study in the case of a toric variety of the geodesic rays t defined by phong and sturm corresponding to test configurations t in the sense of donaldson we show that the bergman approximations k t z of phong and sturm converge in c1 to the geodesic ray t and that the geodesic ray itself is c1 1 and no better in particular the kahler metrics t 0 i t associated to the geodesic ray of potentials are discontinuous across certain hypersurfaces and are degenerate on certain open sets a novelty in the analysis is the connection between bergman metrics bergman kernels and the theory of large deviations we construct a sequence of measures kz on the polytope of the toric variety show that they satisfy a large deviations principle and relate the rate function to the geodesic ray

exact solutions to nonlinear nonautonomous space fractional diffusion equations with absorption we analyze a nonlinear fractional diffusion equation with absorption by employing fractional spatial derivatives and obtain some more exact classes of solutions in particular the diffusion equation employed here extends some known diffusion equations such as the porous medium equation and the thin film equation we also discuss some implications by considering a diffusion coefficient d x t d t x theta theta in r and a drift force f k 1 t x k alpha x x alpha 1 in both situations we relate our solutions to those obtained within the maximum entropy principle by using the tsallis entropy

a criterion for the properness of the k energy in a general kahler class in this paper we give a criterion for the properness of the k energy in a general kahler class of a compact kahler manifold by using song weinkove s result in commun pure appl math 61 2 210 229 2008 as applications we give some kahler classes on mathbb c mathbb p 2 3 overline mathbb c mathbb p 2 and mathbb c mathbb p 2 8 overline mathbb c mathbb p 2 in which the k energy is proper finally we prove song weinkove s result on the existence of critical points of hat j functional by the continuity method

the moser trudinger inequality on kahler einstein manifolds we prove the conjecture of tian on the strong form of the moser trudinger inequality for k ahler einstein manifolds with positive first chern class when there are no holomorphic vector fields and more generally when the setting is invariant under a maximal compact subgroup of the automorphism group

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

a direct method for sparse least squares problems with lower and upper bounds a direct method is developed for solving linear least squares problems mathop min left ax b right 2 limits x wherea is large and sparse and the solution is subject to lower and upper boundsl x u the problem is initially transformed to upper triangular form by a sparseqr factorization an active set algorithm is then used the key step is the stable updating of ther factor associated with the columns ofa corresponding to the free variables when theq factor is not available for this a new method is developed which uses the semi normal equations and iterative refinement

uniform k stability for extremal metrics on toric varieties abstract in this paper we prove that for toric varieties the uniform k stability is a necessary condition for the existence of extremal metrics

numerical approach of fokker planck equation with caputo fabrizio fractional derivative using ritz approximation abstract in this manuscript a type of fokker planck equation fpe with caputo fabrizio fractional derivative is considered we present a numerical approach which is based on the ritz method with known basis functions to transform this equation into an optimization problem it leads to a nonlinear algebraic system then we obtain the coefficients of basis functions by solving the algebraic system the convergence of this technique is discussed extensively three examples are included to show the applicability and validity of this method

scalar curvature and stability of toric varieties we define a stability condition for a polarised algebraic variety and state a conjecture relating this to the existence of a kahler metric of constant scalar curvature the main result of the paper goes some way towards verifying this conjecture in the case of toric surfaces we prove that under the stability hypothesis the mabuchi functional is bounded below on invariant metrics and that minimising sequences have a certain convergence property in the reverse direction we give new examples of polarised surfaces which do not admit metrics of constant scalar curvature the proofs use a general framework developed by guillemin and abreu in which invariant kahler metrics correspond to convex functions on the moment polytope of a toric variety this paper is a step towards the solution of the general problem of finding conditions under which a complex projective variety admits a kahler metric of constant scalar curvature the pattern of the answer one expects is that this differential geometric condition should be equivalent to some notion of stability in the sense of geometric invariant theory this expectation is probably now an item of folklore going back to suggestions put forward by yau in the case of kahlereinstein metrics and the many results of tian and others in this case reinforced by a detailed formal picture which makes clear the analogy with the well established relation between the stability of vector bundles and yang mills connections 5 here we begin the investigation of

modified futaki invariant and equivariant riemann roch formula abstract in this paper we give a new version of the modified futaki invariant for a test configuration associated to the soliton action on a fano manifold our version will naturally come from toric test configurations defined by donaldson for toric manifolds as an application we show that the modified k energy is proper for toric invariant kahler potentials on a toric fano manifold

k energy on polarized compactifications of lie groups abstract in this paper we study mabuchi s k energy on a compactification m of a reductive lie group g which is a complexification of its maximal compact subgroup k we give a criterion for the properness of k energy on the space of k k invariant kahler potentials in particular it turns to give an alternative proof of delcroix s theorem for the existence of kahler einstein metrics in case of fano manifolds m we also study the existence of minimizers of k energy for general kahler classes of m

a sparsity exploiting variant of the bartels golub decomposition for linear programming bases we describe a sparsity exploiting variant of the bartels golub decomposition for linear programming bases it includes interchanges that whenever this is possible avoid the use of any eliminations with consequent fill ins when revising the factorization at an iteration test results on some medium scale problems are presented and comparisons made with the algorithm of forrest and tomlin

kahler ricci solitons on toric manifolds with positive first chern class abstract in this paper we prove there exists a kahler ricci soliton unique up to holomorphic automorphisms on any toric kahler manifold with positive first chern class and the kahler ricci soliton is a kahler einstein metric if and only if the futaki invariant vanishes

extremal kahler metrics on projective bundles over a curve let m p e be the complex manifold underlying the total space of the projectivization of a holomorphic vector bundle e over a compact complex curve of genus 2 building on ideas of fujiki 1992 27 we prove that m admits a kahler metric of constant scalar curvature if and only if e is polystable we also address the more general existence problem of extremal kahler metrics on such bundles and prove that the splitting of e as a direct sum of stable subbundles is necessary and sufficient condition for the existence of extremal kahler metrics in kahler classes sufficiently far from the boundary of the kahler cone the methods used to prove the above results apply to a wider class of manifolds called rigid toric bundles over a semisimple base which are fibrations associated to a principal torus bundle over a product of constant scalar curvature kahler manifolds with fibres isomorphic to a given toric kahler variety we discuss various ramifications of our approach to this class of manifolds

the kahler geometry of bott manifolds abstract we study the kahler geometry of stage n bott manifolds which can be viewed as n dimensional generalizations of hirzebruch surfaces we show using a simple induction argument and the generalized calabi construction from 3 6 that any stage n bott manifold m n admits an extremal kahler metric we also give necessary conditions for m n to admit a constant scalar curvature kahler metric we obtain more precise results for stage 3 bott manifolds including in particular some interesting relations with c projective geometry and some explicit examples of almost kahler structures to place these results in context we review and develop the topology complex geometry and symplectic geometry of bott manifolds in particular we study the kahler cone the automorphism group and the fano condition we also relate the number of conjugacy classes of maximal tori in the symplectomorphism group to the number of biholomorphism classes compatible with the symplectic structure

a sparse counterpart of reichel and gragg s package qrup we describe how to maintain the triangular factor of a sparse qr factorization when columns are added and deleted and q cannot be stored for sparsity reasons the updating procedures could be thought of as a sparse counterpart of reichel and gragg s package qrup they allow us to solve a sequence of sparse linear least squares subproblems in which each matrix b k is an independent subset of the columns of a fixed matrix a and b k 1 is obtained by adding or deleting one column like coleman and hulbert t coleman l hulbert a direct active set algorithm for large sparse quadratic programs with simple bounds math program 45 1989 373 406 we adapt the sparse direct methodology of bjorck and oreborn of the late 1980s but without forming a ta which may be only positive semidefinite our matlab 5 implementation works with a suitable row and column numbering within a static triangular sparsity pattern that is computed in advance by symbolic factorization of a ta and preserved with placeholders

extremal metrics on toric surfaces a continuity method the paper develops an existence theory for solutions of the abreu equation which include extremal metrics on toric surfaces the technique employed is a continuity method combined with blow up arguments general existence results are obtained assuming a hypothesis the m condition on the solutions which is shown to be related to the injectivity radius

newton s method for large bound constrained optimization problems we analyze a trust region version of newton s method for bound constrained problems our approach relies on the geometry of the feasible set not on the particular representation in terms of constraints the convergence theory holds for linearly constrained problems and yields global and superlinear convergence without assuming either strict complementarity or linear independence of the active constraints we also show that the convergence theory leads to an efficient implementation for large bound constrained problems

a numerical technique for solving a class of fractional variational problems this paper presents a numerical method for solving a class of fractional variational problems fvps with multiple dependent variables multi order fractional derivatives and a group of boundary conditions the fractional derivative in the problem is in the caputo sense in the presented method the given optimization problem reduces to a system of algebraic equations using polynomial basis functions an approximate solution for the fvp is achieved by solving the system the choice of polynomial basis functions provides the method with such a flexibility that initial and boundary conditions can be easily imposed we extensively discuss the convergence of the method and finally present illustrative examples to demonstrate validity and applicability of the new technique

extremal metrics on toric surfaces abstract in this paper we study the abreu equation on toric surfaces with prescribed scalar curvatures on delzant polytope in particular we prove the existence of the extremal metric when relative k stability is assumed

affine techniques on extremal metrics on toric surfaces abstract this paper consists of a package of real and complex affine techniques for the abreu equation on toric surfaces manifolds in particular as an application of this package we give an interior estimate for the ricci tensor of toric surfaces

kahler geometry of horosymmetric varieties and application to mabuchi s k energy functional we introduce a class of almost homogeneous varieties contained in the class of spherical varieties and containing horospherical varieties as well as complete symmetric varieties we develop kahler geometry on these varieties with applications to canonical metrics in mind as a generalization of the guillemin abreu donaldson geometry of toric varieties namely we associate convex functions with hermitian metrics on line bundles and express the curvature form in terms of this function as well as the corresponding monge ampere volume form and scalar curvature we then provide an expression for the mabuchi functional and derive as an application a combinatorial sufficient condition of properness similar obtained by li zhou and zhu on group compactifications

active set algorithms for isotonic regression a unifying framework in this and subsequent papers we will show that several algorithms for the isotonic regression problem may be viewed as active set methods the active set approach provides a unifying framework for studying algorithms for isotonic regression simplifies the exposition of existing algorithms and leads to several new efficient algorithms we also investigate the computational complexity of several algorithms

relative k stability and modified k energy on toric manifolds in this paper we give a sufficient condition for both the relative k stability and the properness of modified k energy associated to calabi s extremal metrics on toric manifolds in addition several examples of toric manifolds which satisfy the sufficient condition are presented

from continuous time random walks to the fractional fokker planck equation we generalize the continuous time random walk ctrw to include the effect of space dependent jump probabilities when the mean waiting time diverges we derive a fractional fokker planck equation ffpe this equation describes anomalous diffusion in an external force field and close to thermal equilibrium we discuss the domain of validity of the fractional kinetic equation for the force free case we compare between the ctrw solution and that of the ffpe

the existence of kahler einstein metrics on manifolds with positive anticanonical line bundle and a suitable finite symmetry group on utilise la methode de continuite pour demontrer l existence de metriques de kahler einstein pour des varietes de kahler de fibre linge positif anticanonique sous l hypothese additionnelle de l existence d un bon groupe de symetries fini ou compact

application of a fractional advection dispersion equation abstract a transport equation that uses fractional order dispersion derivatives hasfundamental solutions that are le vy s a stable densities these densities represent plumesthat spread proportional to time 1 a have heavy tails and incorporate any degree ofskewness the equation is parsimonious since the dispersion parameter is not a functionof time or distance the scaling behavior of plumes that undergo le vy motion isaccounted for by the fractional derivative a laboratory tracer test is described by adispersion term of order 1 55 while the cape cod bromide plume is modeled by anequation of order 1 65 to 1 8 1 introduction anomalous or non fickian dispersion has been an activearea of research in the physics community since the introduc tion of continuous time random walks ctrw by montrolland weiss 1965 these random walks extended the predictivecapability of models built on the stochastic process of brown ian motion which is the basis for the classical advection dispersion equation ade the ctrw assign a joint space time distribution called the transition density to individualparticle motions when the tails are heavy enough i e powerlaw non fickian dispersion results for all time scales andspace scales

kahler einstein metrics on complex surfaces with c 1 0 various estimates of the lower bound of the holomorphic invariant m defined in t are given here by using branched coverings potential estimates and lelong numbers of positive d closed 1 1 currents of certain type etc these estimates are then applied to produce kahler einstein metrics on complex surfaces withc1 0 in particular we prove that there are kahler einstein structures withc1 0 on any manifold of differential type cp 2 overline ncp 2 3 leqq n leqq 8

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

on the existence and nonexistence of extremal metrics on toric kahler surfaces abstract in this paper we study the existence of extremal metrics on toric kahler surfaces we show that on every toric kahler surface there exists a kahler class in which the surface admits an extremal metric of calabi we found a toric kahler surface of 9 t c 2 fixed points which admits an unstable kahler class and there is no extremal metric of calabi in it moreover we prove a characterization of the k stability of toric surfaces by simple piecewise linear functions as an application we show that among all toric kahler surfaces with 5 or 6 t c 2 fixed points cp 2 3 cp 2 is the only one which allows vanishing futaki invariant and admits extremal metrics of constant scalar curvature

a direct active set algorithm for large sparse quadratic programs with simple bounds we show how a direct active set method for solving definite and indefinite quadratic programs with simple bounds can be efficiently implemented for large sparse problems all of the necessary factorizations can be carried out in a static data structure that is set up before the numeric computation begins the space required for these factorizations is no larger than that required for a single sparse cholesky factorization of a matrix with the same sparsity structure as the hessian of the quadratic we propose several improvements to this basic algorithm a new way to find a search direction in the indefinite case that allows us to free more than one variable at a time and a new heuristic method for finding a starting point these ideas are motivated by the two norm trust region problem additionally we also show how projection techniques can be used to add several constraints to the active set at each iteration our experimental results show that an algorithm with these improvements runs much faster than the basic algorithm for positive definite problems and finds local minima with lower function values for indefinite problems

variational solutions to extremal metrics on toric surfaces in this paper we prove the interior regularity of minimizers of donaldson s version of k energy associated to calabi s extremal metrics on toric surfaces a new idea is to use solutions to the first boundary value problem of abreu s equation to approximate a minimizer of k energy by the regularity we also prove the uniqueness of minimizers in dimension 2