singular kahler einstein metrics we study degenerate complex monge ampere equations of the form omega dd c f n e t f mu where omega is a big semi kahler form on a compact kahler manifold x of dimension n t in r and mu f omega n is a positive measure with density f in l p x omega n p 1 we prove the existence and unicity of continuous o plurisubharmonic solutions in case x is projective and omega psi omega where psi x to v is a proper birational morphism to a normal projective variety omega in ns r v is an ample class and mu has only algebraic singularities we prove that the solution is smooth in the regular locus of the equation we use these results to construct singular kahler einstein metrics of non positive curvature on projective klt pairs in particular on canonical models of algebraic varieties of general type

spatiotemporal chaos in mixed linear nonlinear coupled logistic map lattice abstract we investigate the spatiotemporal dynamics with mixing degrees of nonlinear chaotic maps in coupling connections here the coupling methods between lattices are both linear neighborhood coupling and the nonlinear chaotic map coupling of lattices while strictly nearest neighborhood coupling is only a special case in the proposed system we employed the criteria such as kolmogorov sinai entropy density and universality bifurcation diagrams space amplitude and space time diagrams to investigate the chaotic behaviors of the proposed system in this paper in fact the proposed system contains new features for applications of cryptography such as the larger range of parameters for chaotic behaviors the higher percentage of lattices in chaotic behaviors for most of parameters and less periodic windows in bifurcation diagrams furthermore we also show the parameter ranges of the proposed system which hold those features in cryptography compared with those of the cml system finally we design the encryption scheme based on the proposed system for an explicit illustration

robust emergent activity in dynamical networks we study the evolution of a random weighted network with complex nonlinear dynamics at each node whose activity may cease as a result of interactions with other nodes starting from a knowledge of the microlevel behavior at each node we develop a macroscopic description of the system in terms of the statistical features of the subnetwork of active nodes we find that very different networks evolve to active subnetworks with similar asymptotic characteristics the size of the active set is independent of the total number of nodes in the network and the average degree of the active nodes is independent of both the network size and its connectivity this robustness has strong implications for dynamical networks observed in the natural world notably the existence of a characteristic range of links per species across ecological systems

twisted fourier mukai functors due to a theorem by orlov every exact fully faithful functor between the bounded derived categories of coherent sheaves on smooth projective varieties is of fourier mukai type we extend this result to the case of bounded derived categories of twisted coherent sheaves and at the same time we weaken the hypotheses on the functor as an application we get a complete description of the exact functors between the abelian categories of twisted coherent sheaves on smooth projective varieties

solvability of the quaternionic monge ampere equation on compact manifolds with a flat hyperkahler metric abstract a quaternionic version of the calabi problem was formulated by alesker and verbitsky 2010 6 it conjectures a solvability of a quaternionic monge ampere equation on a compact hkt manifold hkt stays for hyperkahler with torsion in this paper this problem is solved under the extra assumption that the manifold admits a flat hyperkahler metric compatible with the underlying hypercomplex structure the proof uses the continuity method and a priori estimates

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 priori estimates and a liouville theorem for complex monge ampere equations on considere des solutions plurisousharmoniques u 2 des equations complexes de monge ampere definies sur un sous ensemble ouvert cc u u 2

spatiotemporal chaos in mixed linear nonlinear two dimensional coupled logistic map lattice abstract we investigate a new spatiotemporal dynamics with mixing degrees of nonlinear chaotic maps for spatial coupling connections based on 2dcml here the coupling methods are including with linear neighborhood coupling and the nonlinear chaotic map coupling of lattices and the former 2dcml system is only a special case in the proposed system in this paper the criteria such kolmogorov sinai entropy density and universality bifurcation diagrams space amplitude and snapshot pattern diagrams are provided in order to investigate the chaotic behaviors of the proposed system furthermore we also investigate the parameter ranges of the proposed system which holds those features in comparisons with those of the 2dcml system and the mlncml system theoretical analysis and computer simulation indicate that the proposed system contains features such as the higher percentage of lattices in chaotic behaviors for most of parameters less periodic windows in bifurcation diagrams and the larger range of parameters for chaotic behaviors which is more suitable for cryptography

random coupling of chaotic maps leads to spatiotemporal synchronization we investigate the spatiotemporal dynamics of a network of coupled chaotic maps with varying degrees of randomness in coupling connections while strictly nearest neighbor coupling never allows spatiotemporal synchronization in our system randomly rewiring some of those connections stabilizes entire networks at x where x is the strongly unstable fixed point solution of the local chaotic map in fact the smallest degree of randomness in spatial connections opens up a window of stability for the synchronized fixed point in coupling parameter space further the coupling b i f r at which the onset of spatiotemporal synchronization occurs scales with the fraction of rewired sites p as a power law for 0 1 p 1 is we also show that the regularizing effect of random connections can be understood from stability analysis of the probabilistic evolution equation for the system and approximate analytical expressions for the range and b i f r are obtained

externally controlled attractor selection in a high dimensional system a high dimensional dynamical system with global couplings that can serve as a prototype for systems with very large numbers of attractors like memory is investigated and shown to be controllable by external inputs by changing the duration that noise is added final attractors are selected as the number of degrees of freedom of the nonsynchronized elements decreases one by one over time furthermore it is found that this selection of attractors is also possible by controlling the sweeping speed of a parameter the mechanism for this controlled selection is explained and shown to be rather general applications to attractor switching are given

d equivalence and k equivalence let x and y be smooth projective varieties over they are called d equivalent if their derived categories of bounded complexes of coherent sheaves are equivalent as triangulated categories and k equivalent if they are birationally equivalent and the pull backs of their canonical divisors to a common resolution coincide we expect that the two equivalences coincide for birationally equivalent varieties we shall provide a partial answer to the above problem in this paper

geodesics in the space of kahler cone metrics i in this paper we study the dirichlet problem of the geodesic equation in the space of k ahler cone metrics cal h cal b that is equivalent to a homogeneous complex monge amp ere equation whose boundary values consist of k ahler metrics with cone singularities our approach concerns the generalization of the function space defined by donaldson to the case of k ahler manifolds with boundary moreover we introduce a subspace cal h c of cal h cal b which we define by prescribing appropriate geometric conditions our main result is the existence uniqueness and regularity of c 1 1 cal b geodesics whose boundary values lie in cal h c moreover we prove that such geodesic is the limit of a sequence of c 2 alpha cal b approximate geodesics under the c 1 1 cal b norm as a geometric application we prove the metric space structure of cal h c

interior w 2 p estimates for solutions of the monge ampere equation in this work we adapt the techniques developed in c1 to prove interior estimates for solutions of perturbations of the monge ampere equation

propagation of desynchronous disturbances in synchronized chaotic one way coupled map lattices propagations of desynchronous perturbations in synchronization processes of spatiotemporal chaos are investigated by considering chaotic one way coupled map lattices under large coupling approximation the desynchronous area in time space is analytically calculated based on the concepts of comoving lyapunov exponent and absolute largest lyapunov exponent the ideas used in this paper are expected to be applicable to synchronizations of spatiotemporally chaotic systems of coupled maps and coupled oscillators with convective instability

regularity estimates of solutions to complex monge ampere equations on hermitian manifolds in this paper we obtain the bedford taylor interior c2 estimate and local calabi c3 estimate for the solutions to complex monge ampere equations on hermitian manifolds

uniqueness in e x ω abstract we prove uniqueness for the dirichlet problem for the complex monge ampere equation on compact kahler manifolds in the case of probability measures vanishing on pluripolar sets the proof uses the mass concentration technique due to kolodziej coupled with inequalities for mixed monge ampere measures and the comparison principle

a local regularity of the complex monge ampere equation we prove a local regularity and a corresponding a priori estimate for plurisubharmonic solutions of the nondegenerate complex monge ampere equation assumingthattheir w 2 p normisundercontrolforsome p n n 1 thiscondition isoptimal weuseinparticularsomemethodsdevelopedbytrudingerandanestimate for the complex monge ampere equation due to kooodziej

a c2 α estimate of the complex monge ampere equation abstract in this paper we prove a c 2 estimate for the solution to the complex monge ampere equation det u i j f with 0 f c under the assumption that u c 1 for some 1 which depends on n and

reconstruction and finiteness results for fourier mukai partners abstract we develop some methods for studying the fourier mukai partners of an algebraic variety as applications we prove that abelian varieties have finitely many fourier mukai partners and that they are uniquely determined by their derived category of coherent d modules we also generalize a famous theorem due to bondal and orlov

holder continuity of solutions to the complex monge ampere equation with the right hand side in l p the case of compact kahler manifolds we prove that on compact kahler manifolds solutions to the complex monge ampere equation with the right hand side in lp p 1 are holder continuous

synchronized clusters in coupled map networks i numerical studies we study the synchronization of coupled maps on a variety of networks including regular one and two dimensional networks scale free networks small world networks tree networks and random networks for small coupling strengths nodes show turbulent behavior but form phase synchronized clusters as coupling increases when nodes show synchronized behavior we observe two interesting phenomena first there are some nodes of the floating type that show intermittent behavior between getting attached to some clusters and evolving independently second we identify two different ways of cluster formation namely self organized clusters which have mostly intracluster couplings and driven clusters which have mostly intercluster couplings the synchronized clusters may be of dominant self organized type dominant driven type or mixed type depending on the type of network and the parameters of the dynamics we define different states of the coupled dynamics by considering the number and type of synchronized clusters for the local dynamics governed by the logistic map we study the phase diagram in the plane of the coupling constant and the logistic map parameter ensuremath mu for large coupling strengths and nonlinear coupling we find that the scale free networks and the caley tree networks lead to better cluster formation than the other types of networks with the same average connectivity for most of our study we use the number of connections of the order of the number of nodes as the number of connections increases the number of nodes forming clusters and the size of the clusters in general increase

gromov hausdorff limits of kahler manifolds and algebraic geometry we prove a general result about the geometry of holomorphic line bundles over kahler manifolds

braid group actions on derived categories of coherent sheaves this paper gives a construction of braid group actions on the derived category of coherent sheaves on a variety x the motivation for this is m kontsevich s homological mirror conjecture together with the occurrence of certain braid group actions in symplectic geometry one of the main results is that when dim x geq 2 our braid group actions are always faithful we describe conjectural mirror symmetries between smoothings and resolutions of singularities which lead us to find examples of braid group actions arising from crepant resolutions of various singularities relations with the mckay correspondence and with exceptional sheaves on fano manifolds are given moreover the case of an elliptic curve is worked out in some detail

interior a priori estimates for solutions of fully non linear equations on etend une theorie de perturbations aux solutions d equations uniformement elliptiques d ordre 2 totalement non lineaires

plurisubharmonic approximation and boundary values of plurisubharmonic functions we study the problem of approximating plurisubharmonic functions on a bounded domain omega by continuous plurisubharmonic functions defined on neighborhoods of omega over bar it turns out that t

fourier mukai transformations on k3 surfaces with ρ 1 and atkin lehner involutions abstract we show that there is a surjection from the fourier mukai transformations on projective k3 surfaces with the picard number x 1 to so called the group of atkin lehner involutions this was expected in hosono lian oguiso and yau s paper

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

boundary regularity for the monge ampere and affine maximal surface equations in this paper we prove global second derivative estimates for solutions of the dirichlet problem for the monge ampere equation when the inhomoge neous term is only assumed to be holder continuous as a consequence of our approach we also establish the existence and uniqueness of globally smooth solutions to the second boundary value problem for the affine maximal surface equation and affine mean curvature equation

the complex monge ampere equation with infinite boundary value abstract in this article we consider the complex monge ampere equation with infinite boundary value in bounded pseudoconvex domains we prove the existence of strictly plurisubharmonic solution to the problem in convex domains under suitable growth conditions we also obtain for general pseudoconvex domains some nonexistence results which show that these growth conditions are nearly optimal

the complex monge ampere equation on compact kahler manifolds we consider the complex monge ampere equation on a compact kahler manifold m g when the right hand side f has rather weak regularity in particular we prove that estimate of triangle phi and the gradient estimate hold when f is in w 1 p 0 for any p 0 2n as an application we show that there exists a classical solution in w 3 p 0 for the complex monge ampere equation when f is in w 1 p 0

the dirichlet problem for nonlinear second order elliptic equations ii complex monge ampere and uniformaly elliptic equations on considere le probleme de dirichlet pour des equations elliptiques non lineaires d ordre 2 pour une fonction reelle dans un domaine borne de r n a frontiere lisse

the extremal function associated to intrinsic norms through the study of the degenerate complex monge ampere equation we establish the optimal regularity of the extremal function associated to intrinsic norms of chern levine nirenberg and bedford taylor we prove a conjecture of chern levine nirenberg on the extended intrinsic norms on complex manifolds and verify bedford taylor s representation formula for these norms in general