k polystability of q fano varieties admitting kahler einstein metrics it is shown that any possibly singular fano variety x admitting a kahler einstein metric is k polystable thus confirming one direction of the yau tian donaldson conjecture in the setting of mathbb q fano varieties equipped with their anti canonical polarization the proof is based on a new formula expressing the donaldson futaki invariants in terms of the slope of the ding functional along a geodesic ray in the space of all bounded positively curved metrics on the anti canonical line bundle of x one consequence is that a toric fano variety x is k polystable iff it is k polystable along toric degenerations iff 0 is the barycenter of the canonical weight polytope p associated to x the results also extend to the logarithmic setting and in particular to the setting of kahler einstein metrics with edge cone singularities applications to geodesic stability bounds on the ricci potential and perelman s lambda entropy functional on k unstable fano manifolds are also given

unstable products of smooth curves we give examples of smooth manifolds with negative first chern class which are slope unstable with respect to certain polarisations and so have kahler classes that do not admit any constant scalar curvature kahler metrics these manifolds also have unstable hilbert and chow points we compare this to the work of song weinkove on the j flow

minimal rotationally invariant bases for hyperelasticity rotationally invariant polynomial bases of the hyperelastic strain energy function are rederived using methods of group theory invariant theory and computational algebra a set of minimal basis functions is given for each of the 11 laue groups with a complete set of rewriting syzygies the ideal generated from this minimal basis agrees with the classic work of smith and rivlin trans amer math soc 88 1958 pp 175 193 however the structure of the invariant algebra described here calls for fewer terms beginning with the fourth degree in strain for most groups

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

convexity of the k energy on the space of kahler metrics and uniqueness of extremal metrics we establish the convexity of mabuchi s k energy functional along weak geodesics in the space of kahler potentials on a compact kahler manifold thus confirming a conjecture of chen and give some applications in kahler geometry including a proof of the uniqueness of constant scalar curvature metrics or more generally extremal metrics modulo automorphisms the key ingredient is a new local positivity property of weak solutions to the homogeneous monge ampere equation on a product domain whose proof uses plurisubharmonic variation of bergman kernels

residuation of linear series and the effective cone of cd we obtain new information about divisors on the dth symmetric power cd of a general curve c of genus g 4 this includes a complete description of the effective cone of cg 1 and a partial computation of the volume function on one of its non nef subcones as well as new bounds for the effective and movable cones of cd in the range g 1 d g 2 we also obtain for each g 5 a divisor on cg 1 with non equidimensional stable base locus for a general hyperelliptic curve c of genus g we obtain a complete description of the effective cone of cd for 2 d g and an integral divisor on cg 1 which has non integral volume whenever g is not a power of 2 1 introduction let c be a smooth complex projective algebraic curve and let d 1 be an integer the dth symmetric power cd c d sd is a smooth d dimensional complex projective variety which is a fine moduli space parametriz ing effective divisors of degree d on c in this paper we study the cone of effective divisors on cd and its refinements there are two natural divisor classes on cd for each p c the image of the embedding

a conservative three dimensional eulerian method for coupled solid fluid shock capturing a new method is presented for the explicit eulerian finite difference computation of shock capturing problems involving multiple resolved material phases in three dimensions we solve separately for each phase the equations of fluid dynamics or solid mechanics using as interface boundary conditions artificially extended representations of the individual phases for fluids we use a new 3d spatially unsplit implementation of the piecewise parabolic ppm method of colella and woodward for solids we use the 3d spatially unsplit eulerian solid mechanics method of miller and colella vacuum and perfectly incompressible obstacles may also be employed as phases a separate problem is the time evolution of material interfaces which are represented by planar segments constructed with a volume of fluid method the volume fractions are advanced in time using a second order 3d spatially unsplit advection routine with a velocity field determined by solution of interface normal two phase riemann problems from the riemann problem solutions we also determine cross interface momentum and energy fluxes the volume fractions in mixed cells may be arbitrarily small which would ordinarily make the courant friedrichs lewy time step stability limit arbitrarily small as well we overcome this limitation using the mass redistribution formalism to conservatively redistribute generalized mass in the neighborhood of the split cells we present an application of this method to an explosion contained in a metal can a reactive fluid approximating pbx 9404 is encased within an elastic plastic solid approximating copper surrounded by vacuum our implementation is in parallel and with adaptive mesh refinement

relative k stability for kahler manifolds we study the existence of extremal kahler metrics on kahler manifolds after introducing a notion of relative k stability for kahler manifolds we prove that kahler manifolds admitting extremal kahler metrics are relatively k stable along the way we prove a general l p lower bound on the calabi functional involving test configurations and their associated numerical invariants answering a question of donaldson when the kahler manifold is projective our definition of relative k stability is stronger than the usual definition given by szekelyhidi in particular our result strengthens the known results in the projective case even for constant scalar curvature kahler metrics and rules out a well known counterexample to the naive version of the yau tian donaldson conjecture in this setting

optimal lower bounds for donaldson s j functional abstract in this paper we provide an explicit formula for the optimal lower bound of donaldson s j functional in the sense of finding explicitly the optimal constant in the definition of coercivity which always exists and takes negative values in general this constant is positive precisely if the j equation admits a solution and the explicit formula has a number of applications first this leads to new existence criteria for constant scalar curvature kahler csck metrics in terms of tian s alpha invariant moreover we use the above formula to discuss calabi dream manifolds and an analogous notion for the j equation and show that for surfaces the optimal bound is an explicitly computable rational function which typically tends to minus infinity as the underlying class approaches the boundary of the kahler cone even when the underlying kahler classes admit csck metrics as a final application we show that if the lejmi szekelyhidi conjecture holds then the optimal bound coincides with its algebraic counterpart the set of j semistable classes equals the closure of the set of uniformly j stable classes in the kahler cone and there exists an optimal degeneration for uniform j stability

divisors of secant planes to curves abstract inside the symmetric product of a very general curve we consider the codimension one subvarieties of symmetric tuples of points imposing exceptional secant conditions on linear series on the curve of fixed degree and dimension we compute the classes of such divisors and thus obtain improved bounds for the slope of the cone of effective divisor classes on symmetric products of a very general curve by letting the moduli of the curve vary we study more generally the classes of the related divisors inside the moduli space of stable pointed curves

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

volumes of complex analytic subvarieties of hermitian symmetric spaces we give lower bounds of volumes of k dimensional complex analytic subvarieties of certain naturally defined domains in n dimensional complex space forms of constant positive zero or negative holomorphic sectional curvature for each 1 k n the lower bounds are sharp in the sense that these bounds are attained by k dimensional complete totally geodesic complex submanifolds such lower bounds are obtained by constructing singular potential functions corresponding to blow ups of the kahler metrics involved similar lower bounds are also obtained in the case of hermitian symmetric spaces of noncompact type in this case the lower bounds are sharp for those values of k at which the hermitian symmetric space contains k dimensional complete totally geodesic complex submanifolds which are complex hyperbolic spaces of minimum holomorphic sectional curvature

on symmetric products of curves let c be a smooth complex projective curve of genus g and let c be its second symmetric product this paper concerns the study of some attempts at extending to c the notion of gonality in particular we prove that the degree of irrationality of c is at least g 1 when c is generic and that the minimum gonality of curves through the generic point of c equals the gonality of c in order to produce the main results we deal with correspondences on the k fold symmetric product of c with some interesting linear subspaces of p enjoying a condition of cayley bacharach type and with monodromy of rational maps as an application we also give new bounds on the ample cone of c when c is a generic curve of genus 6 g 8

on the convergence and singularities of the j flow with applications to the mabuchi energy the j flow of s k donaldson and x x chen is a parabolic flow on kahler manifolds with two kahler metrics it is the gradient flow of the j functional that appears in chen s formula for the mabuchi energy we find a positivity condition in terms of the two metrics that is both necessary and sufficient for the convergence of the j flow to a critical metric we use this result to show that on manifolds with ample canonical bundle the mabuchi energy is proper on all kahler classes in an open neighborhood of the canonical class defined by a positivity condition this improves previous results of chen and of the second author we discuss the implications of this for the problem of the existence of constant scalar curvature kahler metrics we also study the singularities of the j flow and under certain conditions which always hold for dimension 2 derive some estimates away from a subvariety we discuss the conjectural remark of donaldson that if the j flow does not converge on a kahler surface then it should blow up over some curves of negative self intersection 2007 wiley periodicals inc

on the period matrix of a riemann surface of large genus with an appendix by j h conway and n j a sloane riemann showed that a period matrix of a compact riemann surface of genusg 1 satisfies certain relations we give a further simple combinatorial property related to the length of the shortest non zero lattice vector satisfied by such a period matrix see 1 13 in particular it is shown that for large genus the entire locus of jacobians lies in a very small neighborhood of the boundary of the space of principally polarized abelian varieties

assortativity and act degree distribution of some collaboration networks empirical investigation results on weighted and un weighted assortativity act degree distribution degree distribution and node strength distribution of nine real world collaboration networks have been presented the investigations propose that act degree distribution degree distribution and node strength distribution usually show so called shifted power law spl function forms which can continuously vary from an ideal power law to an ideal exponential decay two parameters a and z can be used for description of the distribution functions another conclusion is that assortativity coefficient and the parameter a or z is monotonously dependent on each other by the collaboration network evolution model introduced in a reference p zhang et al physica a 360 2006 599 we analytically derived the spl distributions which typically appeared in general situations where nodes are selected partially randomly with a probability p and partially by linear preferential principle with the probability 1 p the analytic discussion gives an explicit expression on the relationship between the random selection proportion p and the parameters a and z the numerical simulation results by the model show a monotonic dependence of assortativity on the random selection proportion p the empirically obtained assortativity versus a or z curve for the four collaboration networks with small maximal act size t max shows a good agreement with the model prediction according to the curves one can qualitatively judge the random selection proportion of the real world network in its evolution process r 2007 elsevier b v all rights reserved

a numerical criterion for very ample line bundles let x be a projective algebraic manifold of dimension n and let l be an ample line bundle over x we give a numerical criterion ensuring that the adjoint bundle kx l is very ample the sufficient conditions are expressed in terms of lower bounds for the intersection numbers l y over subvarieties y of x in the case of surfaces our criterion gives universal bounds and is only slightly weaker than i reider s criterion when dimx 3 and codimy 2 the lower bounds for l y involve a numerical constant which depends on the geometry of x by means of an iteration process it is finally shown that 2kx ml is very ample form 12n our approach is mostly analytic and based on a combination of hormander s l estimates for the operator lelong number theory and the aubin calabi yau theorem

blowing up and desingularizing constant scalar curvature kahler manifolds this paper is concerned with the existence of constant scalar curvature kahler metrics on blow ups at finitely many points of compact manifolds which already carry constant scalar curvature kahler metrics we also consider the desingularization of isolated quotient singularities of compact orbifolds which carry constant scalar curvature kahler metrics

a spatial model for social networks abstract we study spatial embeddings of random graphs in which nodes are randomly distributed in geographical space we let the edge probability between any two nodes to be dependent on the spatial distance between them and demonstrate that this model captures many generic properties of social networks including the small world properties skewed degree distribution and most distinctively the existence of community structures

divisors on symmetric products of curves for a curve with general moduli the neron severi group of its symmetric products is generated by the classes of two divisors x and in this paper we give bounds for the cones of effective and ample divisors in the x plane

on the nef cone of symmetric products of a generic curve the nef cone of the symmetric product c of a curve c with general moduli is easily determined when k is at least equal to the gonality gon c of the curve in this paper we describe the nef cone for k gon c 1 when the genus of c is even in this case the boundary of the nef cone has rational slope and is determined using curves on c associated to the g s of c 1 introduction let c be a smooth irreducible curve of genus g 1 for any k 2 denote by c its mi symmetric product that parametrizes effective degree k divisors z x x2 on c let ni c k resp nl c k be the set of r linear combinations of 1 cycles resp divisors on k modulo numerical equivalence nx k is called the n ron severi group of c and denoted ns c k n k and nl k are finite dimensional vector spaces and the bilinear form given by the intersection pairing is nondegenerate if c has general moduli n c and nl c have rank two an interesting problem is to try to determine inside n c the convex cone of classes of effective 1 cycles denoted by ne c and called the cone of curves or mori cone dually one can study inside nl c the nef cone nef k i e the closure of the convex cone of classes of ample divisors by kleiman s criterion of ampleness

convergence of the j flow on toric manifolds we show that on a kahler manifold whether the j flow converges or not is independent of the chosen background metric in its kahler class on toric manifolds we give a numerical characterization of when the j flow converges verifying a conjecture in 19 m lejmi and g szekelyhidi the j flow and stability in this case we also strengthen existing results on more general inverse sigma k equations on kahler manifolds

a new conformal invariant and its applications to the willmore conjecture and the first eigenvalue of compact surfaces let m be a compact riemannian manifold with a fixed conformal structure then we introduce the concept of conformal volume of m in the following manner for each branched conformal immersion q9 of m into the unit sphere s n we consider the set of all branched conformal immersions obtained by composi t ion of qo with the conformal automorphisms of s we let vc n qg be the max imum volume of these branched immersions the conformal volume of m is defined to be the infimum of v n q0 where qo ranges over all branched conformal immersions of m into the unit sphere s in this paper we study the case when m is a compact surface and we call the conformal volume of m to be the conformal area of m we demonst ra te that this conformal invariant is non trivial in fact we prove that if there exists a minimal immersion of m into s where coordinate functions are first eigenfunctions then the conformal area of m is given by the area of m with respect to the induced metric this enables us to compute the conformal area for several surfaces for example the conformal area of r p z is 6n and the conformal area of the square torus is 27c 2 we believe that the computa t ion of the conformal area for general surfaces will be very impor tant in studying the geometry of compact surfaces we demonstrate this claim by applying the concept of conformal area to two different branches of surface theory the first application is to study the total curvature of a compact surface in r this problem has a long history fenchel and fary 9 proved that for a closed curve o in r slk 27c where k is its curvature then milnor 12

clustering properties of a generalized critical euclidean network many real world networks exhibit a scale free feature have a small diameter and a high clustering tendency we study the properties of a growing network which has all these features in which an incoming node is connected to its ith predecessor of degree k i with a link of length l using a probability proportional to k beta i l alpha for alpha 0 5 the network is scale free at beta 1 with the degree distribution p k proportional to k gamma and gamma 3 0 as in the barabasi albert model alpha 0 beta 1 we find a phase boundary in the alpha beta plane along which the network is scale free interestingly we find a scale free behavior even for beta 1 for alpha 0 5 where the existence of a different universality class is indicated from the behavior of the degree distribution and the clustering coefficients the network has a small diameter in the entire scale free region the clustering coefficients emulate the behavior of most real networks for increasing negative values of alpha on the phase boundary

evolving networks with disadvantaged long range connections we consider a growing network whose growth algorithm is based on the preferential attachment typical for scale free constructions but where the long range bonds are disadvantaged thus the probability of getting connected to a site at distance d is proportional to d alpha where alpha is a tunable parameter of the model we show that the properties of the networks grown with alpha 1 the structure of the network is quite different thus in this regime the node degree distribution is no longer a power law and it is well represented by a stretched exponential on the other hand the small world property of the growing networks is preserved at all values of alpha

structure properties of evolutionary spatially embedded networks abstract this work is a modeling of evolutionary networks embedded in one or two dimensional configuration space the evolution is based on two attachments depending on degree and spatial distance the probability for a new node n to connect with a previous node i at distance r n i follows a k i j k j 1 a r n i j r n j where k i is the degree of node i and a are tunable parameters in spatial driven model a 0 the spatial distance distribution follows the power law feature the mean topological distance l and the clustering coefficient c exhibit phase transitions at same critical values of which change with the dimensionality d of the embedding space when a 0 the degree distribution follows the shifted power law spl which interpolates between exponential and scale free distributions depending on the value of a

the mabuchi geometry of finite energy classes abstract we introduce different finsler metrics on the space of smooth kahler potentials that will induce a natural geometry on various finite energy classes e x motivated by questions raised by r berman v guedj and y rubinstein we characterize the underlying topology of these spaces in terms of convergence in energy and give applications of our results to existence of kahler einstein metrics on fano manifolds

from regular to growing small world networks abstract we propose a growing model which interpolates between one dimensional regular lattice and small world networks the model undergoes an interesting phase transition from large to small worlds we investigate the structural properties by both theoretical predictions and numerical simulations our growing model is a complementarity for the important static watts strogatz network model

an iterative riemann solver for systems of hyperbolic conservation laws with application to hyperelastic solid mechanics in this paper we present a general iterative method for the solution of the riemann problem for hyperbolic systems of pdes the method is based on the multiple shooting method for free boundary value problems we demonstrate the method by solving one dimensional riemann problems for hyperelastic solid mechanics even for conditions representative of routine laboratory conditions and military ballistics dramatic differences are seen between the exact and approximate riemann solution the greatest discrepancy arises from misallocation of energy between compressional and thermal modes by the approximate solver resulting in nonphysical entropy and temperature estimates several pathological conditions arise in common practice and modifications to the method to handle these are discussed these include points where genuine nonlinearity is lost degeneracies and eigenvector deficiencies that occur upon melting

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

k stability and kahler einstein metrics we prove that if a fano manifold m is k stable then it admits a kahler einstein metric it affirms a longstanding conjecture for fano manifolds 2015 wiley periodicals inc

injectivity radius and gonality of a compact riemann surface we obtain a sharp lower bound for the volumes of purely 1 dimensional complex analytic subvarieties in a geodesic tubular neighborhood of the diagonal of the cartesian product of a compact riemann surface with itself this leads to a lower bound of the seshadri number of the canonical line bundle of the cartesian product with respect to the diagonal as a consequence we obtain an upper bound for the hyperbolic injectivity radii of compact riemann surfaces of a fixed gonality in particular we obtain the limiting behavior of the gonalities of a tower of compact riemann surfaces we also give an application of our results to an invariant related to the ample cone of the symmetric product of a riemann surface

modulated scale free network in euclidean space a random network is grown by introducing at unit rate randomly selected nodes on the euclidean space a node is randomly connected to its ith predecessor of degree k i with a directed link of length l using a probability proportional to k i l alpha our numerical study indicates that the network is scale free for all values of alpha alpha c and the degree distribution decays stretched exponentially for the other values of alpha the link length distribution follows a power law d l approximately l delta where delta is calculated exactly for the whole range of values of alpha

the j flow and stability we study the j flow from the point of view of an algebro geometric stability condition in terms of this we give a lower bound for the natural associated energy functional and we show that the blowup behavior found by fang and lai 11 is reflected by the optimal destabilizer finally we prove a general existence result on complex tori