hamiltonian stationary cones and self similar solutions in higher dimension in lw we construct examples of two dimensional hamiltonian stationary self shrinkers and self expanders for lagrangian mean curvature flows which are asymptotic to the union of two schoen wolfson cones these self shrinkers and self expanders can be glued together to yield solutions of the brakke flow a weak formulation of the mean curvature flow moreover there is no mass loss along the brakke flow in this paper we generalize these results to higher dimension we construct new higher dimensional hamiltonian stationary cones of different topology as generalizations of the schoen wolfson cones hamiltonian stationary self shrinkers and self expanders that are asymptotic to these hamiltonian stationary cones are also constructed they can also be glued together to produce eternal solutions of the brakke flow without mass loss finally we show the same conclusion holds for those lagrangian self similar examples recently found by joyce tsui and the first author in jlt

second variation and stabilities of minimal lagrangian submanifolds in kahler manifolds first we derive a new second variation formula which holds for any minimal lagrangian submanifold in k ihler manifolds using this we prove that any minimal lagrangian submanifold in kfihler manifolds with nonpositive chern class is stable we also prove that in kfihler manifolds with positive chern class any stable minimal lagrangian submanifold l satisfies that h l 0 next we introduce the notion of hamiltonian stability either local or global of minimal lagrangian submanifolds in einstein kfihler manifolds using our second variation formula we find a general criterion for the local hamiltonian stability for minimal lagrangian submanifolds in einstein k ihler manifolds this criterion is satisfied for all real forms which are einstein of irreducible hermitian symmetric spaces e g the cannonical rp c cp it is also satisfied by the clifford torus t c cp

heat equations on minimal submanifolds and their applications soit m n une sous variete a n dimensions immergee de facon minimale de m n l l 1 soit d m un domaine compact on demontre des theoremes de comparaisons pour les noyaux de la chaleur h x y t et k x y t pour les conditions aux limites de dirichet et de neumann

immersions minimales premiere valeur propre du laplacien et volume conforme generalisation de quelques resultats de li yau en toute dimension caracterisation de la metrique admettant des immersions minimales par ses premieres fonctions propres

competitive lotka volterra population dynamics with jumps abstract this paper considers competitive lotka volterra population dynamics with jumps the contributions of this paper are as follows a we show that a stochastic differential equation sde with jumps associated with the model has a unique global positive solution b we discuss the uniform boundedness of the p th moment with p 0 and reveal the sample lyapunov exponents c using a variation of constants formula for a class of sdes with jumps we provide an explicit solution for one dimensional competitive lotka volterra population dynamics with jumps and investigate the sample lyapunov exponent for each component and the extinction of our n dimensional model

critical behavior of propagation on small world networks we report numerical evidence that an epidemiclike model which can be interpreted as the propagation of a rumor exhibits critical behavior at a finite randomness of the underlying small world network the transition occurs between a regime where the rumor dies in a small neighborhood of its origin and a regime where it spreads over a finite fraction of the whole population critical exponents are evaluated through finite size scaling analysis and the dependence of the critical randomness with the network connectivity is studied the behavior of this system as a function of the small network randomness bears noticeable similarities with an epidemiological model reported recently m kuperman and g abramson phys rev lett 86 2909 2001 in spite of substantial differences in the respective dynamical rules

number of guards needed by a museum a phase transition in vertex covering of random graphs in this letter we study the np complete vertex cover problem on finite connectivity random graphs when the allowed size of the cover set is decreased a discontinuous transition in solvability and typical case complexity occurs this transition is characterized by means of exact numerical simulations as well as by analytical replica calculations the replica symmetric phase diagram is in excellent agreement with numerical findings up to average connectivity e where replica symmetry becomes locally unstable

stationary distribution and extinction of a stochastic seir epidemic model with standard incidence abstract in this paper we consider a stochastic seir epidemic model with standard incidence by constructing suitable stochastic lyapunov functions we establish sufficient conditions for the existence of ergodic stationary distribution to the model our result does not depend on the positive endemic equilibrium e of the deterministic system which improves the previous result to a greater extent moreover we also establish sufficient conditions for extinction of the disease

hamiltonian stationary shrinkers and expanders for lagrangian mean curvature flows we construct examples of shrinkers and expanders for lagrangian mean curvature flows these examples are hamiltonian stationary and asymptotic to the union of two hamiltonian stationary cones found by schoen and wolfson in sw the schoen wolfson cones cp q are obstructions to the existence problems of special lagrangians or lagrangian minimal surfaces in the variational approach it is known that these cone singularities cannot be resolved by any smooth oriented lagrangian submanifolds the shrinkers and expanders that we found can be glued together to yield solutions of the brakke motiona weak formulation of the mean curvature flow for any coprime pair p q other than 2 1 we construct such a solution that resolves any single schoen wolfson cone cp q this thus provides an evidence to schoen wolfson s conjecture that the 2 1 cone is the only area minimizing cone higher dimensional generalizations are also obtained

deformations of special legendrian submanifolds in sasaki einstein manifolds in this paper we study the deformation theory of submanifolds characterized by a system of differential forms and provide a criterion for deformations of such submanifolds to be unobstructed we apply this deformation theory to special legendrian submanifolds in sasaki einstein manifolds in general special legendrian deformations have the obstruction however we show that the deformation space of special legendrian submanifolds is the intersection of two larger smooth deformation spaces of different types we also prove that any special legendrian submanifold admits smooth deformations which are not special legendrian deformations given by harmonic 1 forms

nodal sets for solutions of elliptic equations on etudie sur un domaine connexe r n l ensemble zero u 1 0 d une solution u d une equation elliptique a ij d i d j u b j d j u cu 0 ou a ij b j c sont bornes et a ij est continu on demontre que la mesure de hausdorff a n 1 dimensions de u 1 0 est finie dans un voisinage d un point x 0 ou u a un ordre fini d annulation

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

asymptotic behavior of stochastic multi group epidemic models with distributed delays abstract in this paper we introduce stochasticity into multi group epidemic models with distributed delays and general kernel functions the stochasticity in the model is a standard technique in stochastic population modeling when the perturbations are small by using the method of stochastic lyapunov functions we carry out a detailed analysis on the asymptotic behavior of the stochastic model regarding of the basic reproduction number r 0 if r 0 1 the solution of the stochastic system oscillates around the disease free equilibrium e 0 while if r 0 1 the solution of the stochastic model fluctuates around the endemic equilibrium e moreover we also establish sufficient conditions of these results

special lagrangian cones we study special lagrangian cones in inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 01i with isolated singularities especially the case n 3 our main result constructs an infinite family of special lagrangian cones in inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 02i each of which has a toroidal link we obtain a detailed geometric description of these tori we prove a regularity result for special lagrangian cones in inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 03i with a spherical link any such cone must be a plane we also construct a one parameter family of asymptotically conical special lagrangian submanifolds from any special lagrangian cone

the threshold of a stochastic sis epidemic model with vaccination this paper is concerned with the long time behavior of a stochastic sis epidemic model with vaccination sufficient conditions for extinction and persistence in mean are obtained we find a threshold of the stochastic model which determines the outcome of the disease in case the white noises are small at last some numerical simulations are carried out to support our results

a stage structured predator prey model of beddington deangelis type we formulate and study a robust stage structured predator prey model of beddington deangelis type functional response the time delay is the time taken from birth to maturity the beddington deangelis functional response is similar to the holling type 2 functional response but contains an extra term describing mutual interference by predators first we show that the predator coexists with prey permanently if and only if the predator s recruitment rate at the peak of prey abundance is larger than its death rate second we show that if the system is permanent then a sufficiently large degree of the predator interference can not only stabilize the system but also guarantee the stability of the system against the increase of the carrying capacity of prey and the increase of birth rate of the adult predator third we show both analytically and numerically that stability switches of interior equilibrium may occur as maturation time delay increases stability may change from stable to unstable to finally stable implying that a large delay can be stabilizing

duality of positive currents and plurisubharmonic functions in calibrated geometry recently the authors showed that there is a robust potential theory attached to any calibrated manifold x phi in particular on x there exist phi plurisubharmonic functions phi convex domains phi convex boundaries etc all inter related and having a number of good properties in this paper we show that in a strong sense the plurisubharmonic functions are the polar duals of the phi submanifolds or more generally the phi currents studied in the original paper on calibrations in particular we establish an analogue of duval sibony duality which characterizes points in the phi convex hull of a compact set k subset x in terms of phi positive green s currents on x and jensen measures on k we also characterize boundaries of phi currents entirely in terms of phi plurisubharmonic functions specific calibrations are used as examples throughout analogues of the hodge conjecture in calibrated geometry are considered

exponential extinction of a stochastic predator prey model with allee effect abstract in this paper we study a stochastic predator prey model with beddington deangelis functional response and allee effect whose coefficients are dependent on expectations we prove that there is a unique global positive solution to the system with the positive initial value moreover sufficient conditions for exponential extinction are established

random graph coloring statistical physics approach the problem of vertex coloring in random graphs is studied using methods of statistical physics and probability our analytical results are compared to those obtained by exact enumeration and monte carlo simulations we critically discuss the merits and shortcomings of the various methods and interpret the results obtained we present an exact analytical expression for the two coloring problem as well as general replica symmetric approximated solutions for the thermodynamics of the graph coloring problem with p colors and k body edges

stochastic delay lotka volterra model we reveal in this paper that the environmental noise will not only suppress a potential population explosion in the stochastic delay lotka volterra model but will also make the solutions to be stochastically ultimately bounded to reveal these interesting facts we stochastically perturb the delay lotka volterra model x t diag x1 t xn t b ax t into the ito form dx t diag x1 t xn t b ax t dt x t dw t and show that although the solution to the original delay equation may explode to infinity in a finite time with probability one that of the associated stochastic delay equation does not we also show that the solution of the stochastic equation will be stochastically ultimately bounded without any additional condition on the matrix a

three limit cycles in a leslie gower predator prey model with additive allee effect in this work a bidimensional continuous time differential equations system is analyzed which is derived of leslie type predator prey schemes by considering a nonmonotonic functional response and allee effect on population prey for the system obtained we describe the bifurcation diagram of limit cycles that appears in the first quadrant the only quadrant of interest for the sake of realism we show that under certain conditions over the parameters the system allows the existence of three limit cycles the first two cycles are infinitesimal ones generated by hopf bifurcation the third one arises from a homoclinic bifurcation furthermore we give conditions over the parameters such that the model allows long term extinction or survival of both populations in particular the presence of a weak allee effect does not imply extinction of populations necessarily for our model

a note on a stage specific predator prey stochastic model abstract in this work we analyze a modified version of a stage specific structure in a predator prey model recently introduced in the literature we assume that the predator death rate is subject to randomness and we discuss through both a theoretical and a computational approach about the influence of this assumption in the behavior of the process

asymptotic behavior of a stochastic delayed seir epidemic model with nonlinear incidence abstract this paper is concerned with a stochastic delayed seir epidemic model with nonlinear incidence firstly we verify that there exists a unique global positive solution of the system then by constructing some suitable lyapunov functions we study the asymptotic behaviors of the disease free equilibrium and the endemic equilibrium respectively

dynamics of a stochastic tuberculosis model with constant recruitment and varying total population size abstract in this paper we develop a mathematical model for a tuberculosis model with constant recruitment and varying total population size by incorporating stochastic perturbations by constructing suitable stochastic lyapunov functions we establish sufficient conditions for the existence of an ergodic stationary distribution as well as extinction of the disease to the stochastic system

the geometric complexity of special lagrangian t2 cones we prove a number of results on the geometric complexity of special lagrangian slg t2 cones in 3 every slg t2 cone has a fundamental integer invariant its spectral curve genus we prove that the spectral curve genus of an slg t2 cone gives a lower bound for its geometric complexity i e the area the stability index and the legendrian index of any slg t2 cone are all bounded below by explicit linearly growing functions of the spectral curve genus we prove that the cone on the clifford torus which has spectral curve genus zero in s5 is the unique slg t2 cone with the smallest possible legendrian index and hence that it is the unique stable slg t2 cone this leads to a classification of all rigid index 1 slg cone types in dimension three for cones with spectral curve genus two we give refined lower bounds for the area the legendrian index and the stability index one consequence of these bounds is that there exist s1 invariant slg torus cones of arbitrarily large area legendrian and stability indices we explain some consequences of our results for the programme due to joyce to understand the most common three dimensional isolated singularities of generic families of slg submanifolds in almost calabi yau manifolds

fredholm hodge and liouville theorems on noncompact manifolds fredholm liouville hodge and l2 cohomology theorems are proved for laplacians associated with a class of metrics defined on manifolds that have finitely many ends the metrics are conformal to ones that are asymptotically translation invariant they are not necessarily complete the fredholm results are of necessity with respect to weighted sobolev spaces embedding and compact embedding theorems are also proved for these spaces two of the most useful facts in analysis on a compact riemannian manifold are that the laplacian is fredholm and its kernel consists of closed and coclosed forms that provide unique representatives for all the de rham cohomology classes naturally one would like to extend these results to noncompact manifolds the first such result in this direction is due to kodaira 9 12 p 165 it is that l2 a m g is the orthogonal direct sum of dco aq 1m dg co aq lm and k2 awm g ae l2 am g ida dga 0 unfortunately if no restriction is made on the manifold or the metric one cannot improve on this the laplacian need not be fredholrn l2 harmonic forms need not be closed or coclosed even if they are closed and coclosed the space of harmonic l2 forms need not be finite dimensional see 7 and even if the laplacian is fredholm and l2 harmonic forms are closed and coclosed those forms need not provide unique or total representation of de rham cohomology thus one of the main questions in analysis on noncompact manifolds is what conditions on m and g allow one to carry over the fredholm and hodge theorems for compact manifolds and if they cannot be carried over completely to what extent can they be in the case of hodge s theorem i e properties of the space of l2 harmonic forms this question has been quite actively investigated recently see 2 8 11 and 13 for instance one of the results of atiyah patodi and singer in 2 is that if a manifold has cylindrical ends then k 2 a m g is naturally isomorphic to the image of hcomp m in hfdr m see 2 proposition 4 9 in 11 muller investigates the spectrum of the laplacian on manifolds that outside a compact set are q x r with received by the editors august 30 1985 and in revised form april 10 1986 1980 mathematics subject classification 1985 revision primary 58gi0 35j05 1987 american mathematical society 0002 9947 87 1 00 25 per page this content downloaded from 157 55 39 45 on tue 19 jul 2016 04 20 24 utc all use subject to http about jstor org terms

special lagrangian cones with higher genus links for every odd natural number g 2d 1 we prove the existence of a countably infinite family of special lagrangian cones in mathbb c 3 over a closed riemann surface of genus g using a geometric pde gluing method

special lagrangian submanifolds with isolated conical singularities v survey and applications this is the last in a series of five papers studying compact special lagrangian submanifolds sl m folds x in almost calabi yau m folds m with singularities x1 x n locally modelled on special lagrangian cones c1 c n in c m with isolated singularities at 0 readers are advised to begin with this paper we survey the major results of the previous four papers giving brief explanations of the proofs we apply the results to describe the boundary of a moduli space of compact nonsingular sl m folds n in m we prove the existence of special lagrangian connected sums n1 nk of sl mfolds n1 n k in m we also study sl 3 folds with t 2 cone singularities proving results related to ideas of the author on invariants of calabi yau 3 folds and the syz conjecture let x be a compact sl m fold with isolated conical singularities xi and cones ci for i 1 n the first paper studied the regularity of x near its singular points and the the second the moduli space of deformations of x the third and fourth papers construct desingularizations of x realizing x as a limit of a family of compact nonsingular sl m folds n t in m for small t 0 let li be an asymptotically conical sl m fold in cm asymptotic to ci at infinity we make n t by gluing tli into x at xi for i 1 n

minimal immersions of surfaces by the first eigenfunctions and conformal area let 0 m s be a minimal immersion of a compact surface into a unit sphere then the linear functions of 0 are eigenfunctions for the laplacian of m corresponding to the eigenvalue 2 2 the main purpose of this paper is to study those minimal surfaces for which 2 is exactly the first non zero eigenvalue of its laplacian this kind of immersions have a peculiar behaviour among all compact minimal surfaces of the sphere and they appear naturally when one considers different geometric problems as li and yau have shown in 6 the methods that we use in this paper are based for the most part on 6 it is known that the only metric on a 2 dimensional sphere admitting a minimal immersion into s by the first eigenfunctions is the standard one this follows for example from the fact that the multiplicity of the first eigenvalue for such a metric is at most three see the cheng work 3 our first result shows that it is possible to extend this property for an arbitrary compact surface in the following way