blowing up kahler manifolds with constant scalar curvature ii in this paper we prove the existence of kahler metrics of constant scalar curvature on the blow up at finitely many points of a compact manifold that already carries a constant scalar curvature kahler metric in the case where the manifold has nontrivial holomorphic vector fields with zeros we give necessary conditions on the number and locations of the blow up points for the blow up to carry constant scalar curvature kahler metrics

the bernstein problem for affine maximal hypersurfaces in this paper we prove the validity of the chern conjecture in affine geometry 18 namely that an affine maximal graph of a smooth locally uniformly convex function on two dimensional euclidean space r 2 must be a paraboloid more generally we shall consider the n dimensional case r n showing that the corresponding result holds in higher dimensions provided that a uniform strict convexity condition holds we also extend the notion of affine maximal to non smooth convex graphs and produce a counterexample showing that the bernstein result does not hold in this generality for dimension n 10

structural properties of large scale systems the role of the structural properties of large scale systems for the synthesis of feasible control schemes is investigated the feasibility of considering a state variable as an output control objective of a system is expressed via the concepts of multi input connectability and structural functional controllability it is shown that these concepts can be formulated as mixed integer linear programming milp problems and efficient algorithms are presented for the determination of the above notions as well as for accessibility structural controllability and observability several examples illustrate the concepts and the proposed approach

toric surfaces k stability and calabi flow let x be a toric surface and u be a normalized symplectic potential on the corresponding polygon p suppose that the riemannian curvature is bounded by a constant c 1 and int partial p u d sigma 0 depending only on c 1 c 2 and p such that donaldson s m condition holds for u as an application we show that if x p is analytic relative k stable then the modified calabi flow converges to an extremal metric exponentially fast by assuming that the calabi flow exists for all time and the riemannian curvature is uniformly bounded along the calabi flow

geometric properties of boundary sections of solutions to the monge ampere equation and applications abstract in this paper we establish several geometric properties of boundary sections of convex solutions to the monge ampere equation the engulfing and separating properties and volume estimates as applications we prove a covering lemma of besicovitch type a covering theorem and a strong type p p estimate for the maximal function corresponding to boundary sections moreover we show that the monge ampere setting forms a space of homogeneous type

optimal control of harvesting in a nonlinear elliptic system arising from population dynamics the aim of this article is to study an optimal control problem for a system of nonlinear elliptic equations with dirichlet boundary conditions u x a x x c x u x eu x u x x x 1 1 x b x u x d x x f x u x x x u x x 0 on 1 2 the authors were supported in part by cicyt ministry of education and culture spain under grant pb98 1343 by junta de andalucia fqm116 and by eec contract human capital and mobility program erbchrxct 940494 also they thank o arino for suggesting the study of this control problem

robust impulse control of uncertain singular systems by decentralized output feedback this correspondence considers a problem of robust impulse control of uncertain singular systems by decentralized output feedback it derives a perturbation upper bound that ensures the existence of decentralized output feedback control law such that the closed loop system remains both regularity and impulse free for any allowable perturbations algebraic sufficient conditions are also presented that guarantee the existence of so called decentralized robust impulse controllers

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

on the generic controllability of continuous generalized state space systems abstract in this paper using a duality property between regular and generalized state space systems the necessary and sufficient conditions for the generic controllability of continuous generalized state space systems are established the necessary and sufficient conditions for the generic observability of such systems are also derived

moduli spaces of critical riemannian metrics in dimension four abstract we obtain a compactness result for various classes of riemannian metrics in dimension four in particular our method applies to anti self dual metrics kahler metrics with constant scalar curvature and metrics with harmonic curvature with certain geometric non collapsing assumptions the moduli space can be compactified by adding metrics with orbifold like singularities similar results were obtained for einstein metrics in j amer math soc 2 3 1989 455 invent math 97 2 1989 313 invent math 101 1 1990 101 but our analysis differs substantially from the einstein case in that we do not assume any pointwise ricci curvature bound

global bifurcation and nodal solutions for a sturm liouville problem with a nonsmooth nonlinearity abstract in this paper we shall establish the unilateral global bifurcation which bifurcates from the trivial solutions axis or from infinity of a class of nonlinear sturm liouville problems with nondifferentiable nonlinearity respectively as applications of the above results we shall determine the interval of r in which there exist nodal solutions for the following problem u t r a t f u 0 t 0 1 u 0 u 1 0 where a t is a positive continuous function on 0 1 f c r r but is not necessarily differentiable at the origin and infinity moreover as a special case of the above problem we shall prove more details about the existence of nodal solutions for a class of half linear eigenvalue problems

solvability controllability and observability of continuous descriptor systems in this paper we investigate the properties of the continuous descriptor system e dot x t ax t bu t 0 leqt leqb where e a and b are complex and possibly singular matrices and u t is a complex function differentiable sufficiently many times the traditional approach to such systems is to separate the state equations from the algebraic equations however such algorithms usually destroy the natural physically based sparsity and structure of the original system therefore we consider descriptor systems in their original form such systems possess numerous properties not shared by the well known state variable systems first we relate classical theories of matrix pencils to the solvability of descriptor systems then we extend the concepts of reachability controllability and observability of state variable systems to descriptor systems and describe the set of reachable states for descriptor systems

on regularizing singular systems by decentralized output feedback this paper considers linear time invariant decentralized singular systems which are either nonregular or if they are regular they have impulsive modes it derives algebraic necessary and sufficient conditions for making a singular system both regular and impulse free by decentralized output feedback control laws and decentralized proportional plus derivative output feedback control laws

some recent results in combinatorial approaches to dynamical systems abstract this paper is a unified presentation of recent results obtained by the author on the structural analysis of dynamical systems by matroid theoretic combinatorial methods the first part consists of the results on combinatorial and algorithmic characterizations of control theoretic generic properties the analysis is based on a physically reasonable mathematical model which assumes that the coefficients in the equations are classified into independent physical parameters and dimensionless fixed constants the concept of mixed matrix and its canonical form turn out to be convenient mathematical tools the second part describes a combinatorial analogue of the dynamical system theory developed in the matroid theoretic framework the matrices a and b in the state space equations ax bu are replaced by bimatroids or linking systems

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

robust stabilizability verification of polynomially uncertain lti systems this paper investigates the stabilizability of uncertain linear time invariant lti systems via structurally constrained controllers first an lti uncertain system is considered whose state space matrices depend polynomially on the uncertainty vector defined over some region it is shown that if the system is stabilizable by a structurally constrained controller in one point belonging to the region then it is stabilizable by a controller with the same structure in all points belonging to the region except for those located on an algebraic variety thus if a system is stabilizable via a constrained controller at the nominal point then it is almost always stabilizable at any operating point around the nominal model it is also shown how this algebraic variety or the dominant subvariety of it can be computed efficiently the results obtained in this paper encompass a broad range of the existing results in the literature on robust stability of the lti systems in addition to new ones

generic controllability theorems for descriptor systems this paper presents an extension of the structural controllability theorem to the case of structured descriptor systems sds a major difficulty in this problem is the loss of independence among the entries of the system matrices this difficulty is resolved by introducing the idea of column structured matrices csm first the paper defines and characterizes c and r controllability for descriptor systems then the generic controllability theorem is proved extending the structural controllability theorem to column structured systems css finally based on this result necessary conditions and sufficient conditions for generic r and c controllability are obtained the result gives a complete characterization of generic controllability for sds with nonsingular e and a partial solution for the case of singular e

w4 p solution to the second boundary value problem of the prescribed affine mean curvature and abreu s equations abstract the second boundary value problem of the prescribed affine mean curvature equation is a nonlinear fourth order geometric partial differential equation it was introduced by trudinger and wang in 2005 in their investigation of the affine plateau problem in affine geometry the previous works of trudinger wang chau weinkove and the author solved this global problem in w 4 p under some restrictions on the sign or integrability of the affine mean curvature we remove these restrictions in this paper and obtain w 4 p solution to the second boundary value problem when the affine mean curvature belongs to l p with p greater than the dimension our self contained analysis also covers the case of abreu s equation

interior estimates for the n dimensional abreuʼs equation abstract we study the abreu s equation in n dimensional polytopes and derive interior estimates of solutions under the assumption of the uniform k stability

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

a bernstein theorem for affine maximal type hypersurfaces under decaying convexity abstract in this paper we will study a fourth order fully nonlinear pde u i j d i j w 0 w det d 2 u 0 of affine maximal type by a formula of quasi norm of third derivatives derived from pure pde tricks we will prove that any entire convex solution must be a quadratic function provided 0 1 2 n 1 2 n 8 1 2 n 1 2 n 8 and the minimal convex constant r decays slower than r n for some positive constant n given explicitly below

a tutorial on the geometric analysis of linear time invariant implicit systems abstract a background of recent work in nonregular and regular implicit systems of the form e x ax bu is provided then the output nulling unknown input and composite subspaces are defined for singular systems with recursions given for their computation inner subspaces that is those in the domain of e and a are distinguished from outer subspaces those in the codomain of e and a the notions of composite preimage and composite image are introduced the result is a framework that unifies some of the recent work in the geometric theory of implicit systems system properties are discussed for both regular and nonregular systems and duality is investigated

technical communique a note on the action of constant pseudostate feedback on the internal properness of an arma model in this note we study the effect of constant pseudostate feedback on the internal properness of a linear multivariable system described by an arma model it is shown that the existence of a constant peudostate feedback control law which makes the closed loop system internally proper is equivalent to the absence of decoupling zeros at infinity of the open loop system a well known result from the theory of descriptor systems

bernstein problem of affine maximal type hypersurfaces on dimension n 3 abstract bernstein problem for affine maximal type equation 0 1 u i j d i j w 0 w det d 2 u x r n has been a core problem in affine geometry a conjecture proposed firstly by chern 1977 6 for entire graph and then extended by trudinger wang 2000 14 to its fully generality asserts that any euclidean complete affine maximal type locally uniformly convex c 4 hypersurface in r n 1 must be an elliptic paraboloid at the same time this conjecture was solved completely by trudinger wang for dimension n 2 and 3 4 and later extended by jia li 2009 12 to n 2 3 4 1 see also zhou 2012 16 for a different proof on the past twenty years many efforts were done toward higher dimensional issues but not really successful yet even for the case of dimension n 3 in this paper we will construct non quadratic affine maximal type hypersurfaces which are euclidean compete for n 3 1 2 n 1 n

unilateral global bifurcation phenomena and nodal solutions for p laplacian in this paper we establish a dancer type unilateral global bifurcation result for one dimensional p laplacian problem p u m t p u g t u t 0 1 u 0 u 1 0 under some natural hypotheses on the perturbation function g 0 1 r r r we show that k p is a bifurcation point of the above problem and there are two distinct unbounded continua ck and ck consisting of the bifurcation branch ck from k p 0 where k p is the k th eigenvalue of the linear problem corresponding to the above problem as the applications of the above result we study the existence of nodal solutions for the following problem p u f t u 0 t 0 1 u 0 u 1 0 moreover based on the bifurcation result of girg and takac 2008 13 we prove that there exist at least a positive solution and a negative one for the following problem div p u f x u in u 0 on

algorithms to verify generic causality and controllability of descriptor systems graph theoretic algorithms are developed for determining generic causality and controllability of structured descriptor systems sds s for the case of nonsingular e a completely graph theoretic characterization of generic controllability of sds s is developed the case of singular e is partially resolved with graph theoretic characterization of necessary conditions and sufficient conditions for generic causality and controllability developed separately although not stated explicitly these results can be easily translated into a graph theoretic algorithm to test for generic observability of sds s

global w 2 p estimates for solutions to the linearized monge ampere equations in this paper we establish global w 2 p estimates for solutions to the linearized monge ampere equations under natural assumptions on the domain monge ampere measures and boundary data our estimates are affine invariant analogues of the global w 2 p estimates of winter for fully nonlinear uniformly elliptic equations and also linearized counterparts of savin s global w 2 p estimates for the monge ampere equations

on a system of reaction diffusion equations describing a population with two age groups abstract a system of reaction diffusion equations modelling a population divided into juvenile and adult age groups is studied the system is not co operative but its linear part is and this makes it possible to establish existence non existence and stability results for non negative solutions of the system in terms of the principal eigenvalue of the corresponding linearized system

the global existence and convergence of the calabi flow on cn zn izn in this note we study the long time existence of the calabi flow on x cn zn izn assuming the uniform bound of the total energy we establish the non collapsing property of the calabi flow by using donaldson s estimates and streets regularity theorem next we show that the curvature is uniformly bounded along the calabi flow on x when the dimension is 2 partially confirming chen s conjecture moreover we show that the calabi flow exponentially converges to the flat kahler metric for arbitrary dimension if the curvature is uniformly bounded partially confirming donaldson s conjecture

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

deadbeat control of discrete descriptor systems with inaccessible descriptor vector the problem of deadbeat compensator design is considered for a discrete descriptor system with inaccessible descriptor variables the full compensator design is reduced to separate designs of two regular deadbeat controllers of a lower order driven by this compensator the descriptor variables of the system starting from given initial conditions can reach the origin in a minimum of c 0 time steps where c and c are the largest right kronecker indices of the pencils ze a b and ze a c respectively

global bifurcation of positive radial solutions for an elliptic system in reactor dynamics in this paper we are concerned with the existence of positive radial solutions of the elliptic system email protected email protected f x u r 1 0 is a constant m 0 is a parameter and 0 0 is continuous and f t s 0 for all t s r 1 r 2 x 0 under some appropriate conditions on the nonlinearity f we show that the above system possesses at least one positive radial solution for any email protected 0 the proof of our main results is based upon bifurcation techniques

remarks on controllability of implicit linear discrete time systems abstract some results concerning controllability of implicit linear discrete time system of the form ex k 1 fx k gu k are given it is not assumed that the corresponding pencil se f is regular the concept of x i x f controllability is defined and it is shown that it generalizes various definitions of controllability used in the existing literature of the subject computable geometric conditions for an implicit system to be x 1 x f controllable are given some particular cases are discussed the existence of a universal time of controllability is also proven

the calabi flow on kahler surfaces with bounded sobolev constant i we consider the formation of singularities along the calabi flow by assuming the uniformly bounded sobolev constants on kahler surfaces we prove that if curvature tensor is not uniformly bounded then one can form a singular model called deepest bubble such deepest bubble has to be a scalar flat ale kahler metric in certain kahler classes on toric fano surfaces the sobolev constants are a priori bounded along the calabi flow with small calabi energy we can also show in certain cases no deepest bubble can form along the flow it follows that the curvature tensor is uniformly bounded and the flow exists for all time and converges to an extremal metric subsequently to illustrate our results more clearly we focus on an example on mathbb cp 2 blown up three points at generic position our result also implies existence of constant scalar curvature metrics on mathbb cp 2 blown up three points at generic position in the kahler classes where the exceptional divisors have the same area

positive solutions of an elliptic system modeling a population with two age groups we study the existence of positive solutions for an elliptic system modeling two subpopulations of the same species competing for resources under some suitable assumptions we find the necessary and sufficient conditions for the existence of positive solutions of the system by using bifurcation techniques moreover an open problem proposed by bouguima et al 2008 is answered

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 affine plateau problem in this paper we study a second order variational problem for locally convex hypersurfaces which is the affine invariant analogue of the classical plateau problem for minimal surfaces we prove existence regularity and uniqueness results for hypersurfaces maximizing affine area under appropriate boundary conditions