about the fastest growth of the order parameter in models of percolation abstract the growth of the average size s max of the largest component at the percolation threshold p c n on a graph of size n has been defined as s max p c n n n here we argue that the precise value of the growth exponent indicates the nature of percolation transition 1 or 1 determines if the transition is continuous or discontinuous we show that a related exponent 1 which describes how the average maximal jump sizes in the order parameter decays on increasing the system size is the single exponent that describes the finite size scaling of a number of distributions related to the fastest growth of the order parameter in these problems excellent quality scaling analysis are presented for the two single peak distributions corresponding to the order parameters at the two ends of the maximal jump the bimodal distribution constructed by the weighted average of these distributions and for the distribution of the maximal jump in the order parameter

a criterion for non existence of invariant tori for hamiltonian systems abstract a particularly interesting class of invariant tori for a hamiltonian system satisfying the legendre condition is the class of invariant lagrangian graphs a classical result of the calculus of variations is that for such systems every orbit on an invariant lagrangian graph has minimal action this provides a simple criterion for non existence of invariant lagrangian graphs namely if an orbit has a variation which decreases the action or it possesses conjugate points then there is no invariant lagrangian graph containing it as an example the criterion is applied to the motion of a particle in the field of two waves to prove that there are no invariant tori with initial velocity between those of the two waves if the amplitudes of the waves exceed certain values

analysis on the evolution process of bfw like model with discontinuous percolation of multiple giant components abstract recently the modified bfw model on random graphs w chen r m d souza phys rev lett 106 2011 115701 which shows a discontinuous percolation transition with multiple giant components has attracted much attention from physicists and statisticians in this paper by establishing the evolution equations on the modified bfw model the evolution process and steady states on both random graphs and finite dimensional lattices are analyzed on a random graph by varying the edge accepted rate the system stabilizes in a steady state with different numbers of giant components moreover a close correspondence is built between the values of and the number of giant components in steady states the efficiency of which is verified by the numerical simulations then the sizes of giant components for different evolution strategies can be obtained by solving some constraints derived from the evolution equations meanwhile a similar analysis is expanded to finite dimensional lattices and we find the bfw model on a finite dimensional lattice has different steady states from those on a random graph but they have the same evolution mechanism the analysis of the evolution process and steady state is of great help to explain the properties of discontinuous percolation and the role of nonlocality

degree dependent network growth from preferential attachment to explosive percolation we present a simple model of network growth and solve it by writing the dynamic equations for its macroscopic characteristics such as the degree distribution and degree correlations this allows us to study carefully the percolation transition using a generating functions theory the model considers a network with a fixed number of nodes wherein links are introduced using degree dependent linking probabilities pk to illustrate the techniques and support our findings using monte carlo simulations we introduce the exemplary linking rule pk k with between 1 and this parameter may be used to interpolate between different regimes for negative links are most likely attached to high degree nodes on the other hand in case 0 nodes with low degrees are connected and the model asymptotically approaches a process undergoing explosive percolation

transmission of packets on a hierarchical network statistics and explosive percolation we analyze an idealized model for the transmission or flow of particles or discrete packets of information in a weight bearing branching hierarchical two dimensional network and its variants the capacities add hierarchically down the clusters each node can accommodate a limited number of packets depending on its capacity and the packets hop from node to node following the links between the nodes the statistical properties of this system are given by the maxwell boltzmann distribution we obtain analytical expressions for the mean occupation numbers as functions of capacity for different network topologies the analytical results are shown to be in agreement with the numerical simulations the traffic flow in these models can be represented by the site percolation problem it is seen that the percolation transitions in the 2d model and in its variant lattices are continuous transitions whereas the transition is found to be explosive discontinuous for the v lattice the critical case of the 2d lattice the scaling behavior of the second order percolation case is studied in detail we discuss the implications of our analysis

the maslov cocycle smooth structures and real analytic complete integrability this paper proves two main results first it is shown that if is a smooth manifold homeomorphic to the standard n torus t n r n z n and h is a real analytically completely integrable convex hamiltonian on t then is diffeomorphic to tn second it is proven that for some topological 7 manifolds the cotangent bundle of each smooth structure admits a real analytically completely integrable riemannian metric hamiltonian

optical hamiltonians and symplectic twist maps abstract this paper concentrates on optical hamiltonian systems of t t n i e those for which hpp is a positive definite matrix and their relationship with symplectic twist maps we present theorems of decomposition by symplectic twist maps and existence of periodic orbits for these systems the novelty of these results resides in the fact that no explicit asymptotic condition is imposed on the system

a new route to explosive percolation abstract the biased link occupation rule in the achlioptas process ap discourages the large clusters from growing much ahead of others and encourages faster growth of clusters which lag behind in this paper we propose a model where this tendency is sharply reflected in the gamma distribution of the cluster sizes unlike the power law distribution in ap in this model single edges between pairs of clusters of sizes s i and s j are occupied with a probability s i s j the parameter is continuously tunable over the entire real axis numerical studies indicate that for c the transition is first order c 0 for a square lattice and c 1 2 for random graphs in the limits of this model coincides with models well established in the literature

a new obstruction to embedding lagrangian tori n let 2n be endowed with the canonical symplectic form a i 1 dxi dy i and consider a lagrangian embedding of a compact manifold j l r 2 a a well known result due to g r o m o v g states that the cohomology class of a in h z e 2 l is non zero we identify l with its image in r2 an equivalent assertion is that the liouville form 2 y 7 1 xldy i pulls back to a closed form j 2 on l whose cohomology class does not vanish g r o m o v s proof uses subtle properties of holomorphic maps into c he shows that there is always a holomorphic disk a in c with boundary in l on such a disk a is positive hence sa a soa 2 0 in this paper we shall only consider the case l t and give an elementary proof of gromov s result elementary should be understood in the sense of number theory that is without using holomorphic functions in fact our result is somewhat more precise and can be stated as follows

lagrangian singularities of invariant tori of hamiltonian systems with two degrees of freedom on considere des projections lagrangiennes du 2 tore invariant sous un flot hamiltonien g t

cluster aggregation model for discontinuous percolation transitions the evolution of the erd ifmmode mbox h o else h o fi s r enyi er network by adding edges is a basis model for irreversible kinetic aggregation phenomena such er processes can be described by a rate equation for the evolution of the cluster size distribution with the connection kernel k ij ensuremath sim ij where ij is the product of the sizes of two merging clusters here we study that when the giant cluster is discouraged to develop by a sublinear kernel k ij ensuremath sim ij ensuremath omega with 0 ensuremath le ensuremath omega l1 2 the percolation transition pt is discontinuous such discontinuous pt can occur even when the er dynamics evolves from proper initial conditions the obtained evolutionary properties of the simple model sheds light on the origin of the discontinuous pt in other nonequilibrium kinetic systems

aubry mather sets and birkhoff s theorem for geodesic flows on the two dimensional torus in this paper we state the graph property for incompressible continuouse tori invariant under goedesic flows of riemannian metrics on the two dimensional torus also our method gives a new proof of birkhoff s theorem for twist maps of the cylinder we prove that if there exist an invariant incompressible torus of geodesic flow with irrational rotation number then it necessarily contains the aubry mather set with this rotation number

unstable supercritical discontinuous percolation transitions percolation is a pervasive concept 1 which has applications in a wide variety of natural technological and social systems 2 3 4 5 ranging from conductivity of composite materials 6 7 and polymerization 8 to epidemic spreading 9 10 11 and information diffusion 12 13

classical diffusion on eden trees we study an aggregation process which gives rise to compact clusters with no loops a generalization of the node counting theorem applicable to such branched graphs trees is proved this is used to determine the spectral dimension of these clusters on square and cubic embedding lattices by monte carlo simulations the results are explained in terms of the geometric structure of these trees

energy conserving galerkin approximations for quasigeostrophic dynamics abstract a method is presented for constructing energy conserving galerkin approximations in the vertical coordinate of the full quasigeostrophic model with active surface buoyancy the derivation generalizes the approach of rocha et al 2016 1 to allow for general bases details are then presented for a specific set of bases legendre polynomials for potential vorticity and a recombined legendre basis from shen 1994 2 for the streamfunction the method is tested in the context of linear baroclinic instability calculations where it is compared to the standard second order finite difference method and to a chebyshev collocation method the galerkin scheme is quite accurate even for a small number of degrees of freedom n and growth rates converge much more quickly with increasing n for the galerkin scheme than for the finite difference scheme the galerkin scheme is at least as accurate as finite differences and can in some cases achieve the same accuracy as the finite difference scheme with ten times fewer degrees of freedom the energy conserving galerkin scheme is of comparable accuracy to the chebyshev collocation scheme in most linear stability calculations but not in the eady problem where the chebyshev scheme is significantly more accurate finally the three methods are compared in the context of a simplified version of the nonlinear equations the two surface model with zero potential vorticity the chebyshev scheme is the most accurate followed by the galerkin scheme and then the finite difference scheme all three methods conserve energy with similar accuracy despite not having any a priori guarantee of energy conservation for the chebyshev scheme further nonlinear tests with non zero potential vorticity to assess the merits of the methods will be performed in a future work

total destruction of lagrangian tori for an integrable tonelli hamiltonian with d d 2 degrees of freedom we show that all of the lagrangian tori can be destroyed by analytic perturbations which are arbitrarily small in the cd topology

correlated percolation and tricriticality the recent proliferation of correlated percolation models models where the addition of edges and or vertices is no longer independent of other edges and or vertices has been motivated by the quest to find discontinuous percolation transitions the leader in this proliferation is what is known as explosive percolation a recent proof demonstrates that a large class of explosive percolation type models does not in fact exhibit a discontinuous transition riordan and warnke science 333 322 2011 here we discuss two lesser known correlated percolation models the k 3 core model on random graphs and the counter balance model in two dimensions both exhibiting discontinuous transitions to search for tricriticality we construct mixtures of these models with other percolation models exhibiting the more typical continuous transition using a powerful rate equation approach we demonstrate that a mixture of k 2 core and k 3 core vertices on the random graph exhibits a tricritical point however for a mixture of k core and counter balance vertices in two dimensions as the fraction of counter balance vertices is increased numerics and heuristic arguments suggest that there is a line of continuous transitions with the line ending at a discontinuous transition i e when all vertices are counter balanced interestingly these heuristic arguments may help identify the ingredients needed for a discontinuous transition in low dimensions in addition our results may have potential implications for glassy and jamming systems

the maslov class of the lagrange surfaces and gromov s pseudo holomorphic curves for an immersed lagrange submanifold w c t x one can define a nonnegative integer topologic invariant m w such that the image of h1 w z under the maslov class is equal to m w z in this paper the value of m w is calculated for the case of a two dimensional oriented manifold x with the universal cover homeomorphic to r2 and an embedded lagrange torus w it is proved that if x t2 and w is homologic to the zero section then m w 0 in all the other cases m w 2 the last result is true also for a wide class of oriented properly embedded lagrange surfaces in t r the proof is based on the gromov s theory of pseudo holomorphic curves some applications to the hamiltonian mechanics are mentioned

random sequential adsorption line segments on the square lattice we study kinetics of the rmgle layer random sequenrisl adsorption of lint segments of ked kn5h on the square iatijcc by a monte carlo srmulatton the area covered b the line segments grows with time and finally reacher a jamming limit when no more adsorption is possible the jamming coverage depends on the segment length and 11s vanation is studted at the late stage approach of the coverage to the jamming limit is arymptotieally exponential with a rate found to be independent of the segment length based on our monte carlo data an act expression for the late stage deposition kinettcs 15 conjectured random sequential adsorption rsa has been the subject of much recent experimental and theoretical investigation objects of figte size are randomly deposited adsorbed on an initially empty substrate so that no two objects overlap the quantity of interest is the fraction of the total area b f in time i covered by the depositing particles due to the blocking of the area by the already randomly adsorbed particles the limiting 3amming coverage co is less than the close packing the emergence of this jammed state is iduenced by the infiuite memory effects consequently its formation cannot he described by mean field theory except for very early times when 6 t af experimental studies 11 41 include e g adhesion of colloidal particles 131 and proteins 2 on homogeneous substrates with relaxation times much longer than the formation time of deposits theoretical studies s 171 of rsa include series expansions is l j numerical monte carlo mc simulations 9 14 16 and some analytical results 1171 mostly for one dimensional systems in such studies it was shown that the precise form of the long time behaviour of b r depends on the shape and orientational freedom of the adsorbing particles earlier studies have focused on continuum deposition models for which a power law behaviour of the late stage deposition 1171

explosive percolation a numerical analysis percolation is one of the most studied processes in statistical physics a recent paper by achlioptas et al science 323 1453 2009 showed that the percolation transition which is usually continuous becomes discontinuous explosive if links are added to the system according to special cooperative rules achlioptas processes in this paper we present a detailed numerical analysis of achlioptas processes with product rule on various systems including lattices random networks a la erdos renyi and scale free networks in all cases we recover the explosive transition by achlioptas et al however the explosive percolation transition is kind of hybrid as despite the discontinuity of the order parameter at the threshold one observes traces of analytical behavior such as power law distributions of cluster sizes in particular for scale free networks with degree exponent 3 all relevant percolation variables display power law scaling just as in continuous second order phase transitions

continuous percolation phase transitions of random networks under a generalized achlioptas process using finite size scaling we have investigated the percolation phase transitions of evolving random networks under a generalized achlioptas process gap during this gap the edge with a minimum product of two connecting cluster sizes is taken with a probability p from two randomly chosen edges this model becomes the erdos renyi network at p 0 5 and the random network under the achlioptas process at p 1 using both the fixed point of the size ratio s 2 s 1 and the straight line of ln s 1 where s 1 and s 2 are the reduced sizes of the largest and the second largest cluster we demonstrate that the phase transitions of this model are continuous for 0 5 0 9 beta upsilon and s 2 s 1 at critical point vary with p and the universality class of phase transitions depends on p

percolation transitions with nonlocal constraint we investigate percolation transitions in a nonlocal network model numerically in this model each node has an exclusive partner and a link is forbidden between two nodes whose r neighbors share any exclusive pair the r neighbor of a node x is defined as a set of at most n r neighbors of x where n is the total number of nodes the parameter r controls the strength of a nonlocal effect the system is found to undergo a percolation transition belonging to the mean field universality class for r 1 2 the system undergoes a peculiar phase transition from a nonpercolating phase to a quasicritical phase where the largest cluster size g scales as g n with 0 74 1 in the marginal case with r 1 2 the model displays a percolation transition that does not belong to the mean field universality class

hamiltonian approach for explosive percolation we present a cluster growth process that provides a clear connection between equilibrium statistical mechanics and an explosive percolation model similar to the one recently proposed by d achlioptas et al science 323 1453 2009 we show that the following two ingredients are sufficient for obtaining an abrupt first order transition in the fraction of the system occupied by the largest cluster i the size of all growing clusters should be kept approximately the same and ii the inclusion of merging bonds i e bonds connecting vertices in different clusters should dominate with respect to the redundant bonds i e bonds connecting vertices in the same cluster moreover in the extreme limit where only merging bonds are present a complete enumeration scheme based on treelike graphs can be used to obtain an exact solution of our model that displays a first order transition finally the presented mechanism can be viewed as a generalization of standard percolation that discloses a family of models with potential application in growth and fragmentation processes of real network systems