note on basic features of large time behaviour of heat kernels large time behaviour of heat semigroups and more gener ally of positive selfadjoint semigroups is studied convergence of the semigroup to the ground state and of averaged logarithms of kernels to the ground state energy is shown in the general framework of positivity improving selfadjoint semigroups this framework includes laplacians on manifolds metric graphs and discrete graphs

dynamics of latent voters we study the effect of latency on binary choice opinion formation models latency is introduced into the models as an additional dynamic rule after a voter changes its opinion it enters a waiting period of stochastic length where no further changes take place we first focus on the voter model and show that as a result of introducing latency the average magnetization is not conserved and the system is driven toward zero magnetization independently of initial conditions the model is studied analytically in the mean field case and by simulations in one dimension we also address the behavior of the majority rule model with added latency and show that the competition between imitation and latency leads to a rich phenomenology

variational properties and orbital stability of standing waves for nls equation on a star graph we study standing waves for a model of nonlinear schr odinger equation on a graph the graph is obtained joining n halflines at a vertex i e it is a star graph at the vertex an interaction occurs due to a boundary condition of delta type with strength alpha leqslant 0 which includes the free or kirchhoff case alpha 0 there imposed the nonlinearity is of focusing power type the dynamics is given by an equation of the form i frac d dt psi t h psi t psi t 2 mu psi t where h is the s a operator which generates the linear schr odinger dynamics on the graph in the hilbert space l 2 gg we show the existence of several families of standing waves solutions of the form phi t e i omega t psi omega where psi omega is a vector amplitude parametrized by a frequency omega they exist for every sign of the coupling at the vertex attractive alpha 0 and for every omega frac alpha 2 n 2 the number of the families depends on the number n of the edges of the graph we moreover determine the ground state which is the standing wave the amplitude of which minimizes the action on the natural nehari constraint and order the various families of standing waves according to their increasing action thus determining a nonlinear spectrum of the problem finally we show that the ground state is orbitally stable for every allowed omega if the nonlinearity is subcritical or critical for supercritical nonlinearities there exists an omega frac alpha 2 n 2 such that the ground state is orbitally stable for frac alpha 2 n 2 omega omega action minimization requires technical subtleties in particular an adaptation of decreasing rearrangements on a star graph is developed with a generalization of the p olya szeg h o inequality

perturbation of doubly periodic solution branches with applications to the cahn hilliard equation abstract in this paper we prove the existence of doubly periodic solutions of certain nonlinear elliptic problems on r 2 and study the geometry of their nodal domains in particular we will show that if we perturb a nonlinear elliptic equation exhibiting a small amplitude doubly periodic solution whose nodal domains form a checkerboard pattern then the perturbed equation will have a unique nearby solution which is still doubly periodic but for which the nodal line structure breaks up moreover we indicate what can happen if we start with a large amplitude doubly periodic solution whose nodal domains form a checkerboard pattern and we relate these solutions to the cahn hilliard equation and spinodal decomposition

essential self adjointness of the graph laplacian we study the operator theory associated with such infinite graphs g as occur in electrical networks in fractals in statistical mechanics and even in internet search engines our emphasis is on the determination of spectral data for a natural laplace operator associated with the graph in question this operator will depend not only on g but also on a prescribed positive real valued function c defined on the edges in g in electrical network models this function c will determine a conductance number for each edge we show that the corresponding laplace operator is automatically essential self adjoint by this we mean that is defined on the dense subspace d of all the real valued functions on the set of vertices g0 with finite support in the hilbert space l2 g0 the conclusion is that the closure of the operator is self adjoint in l2 g0 and so in particular that it has a unique spectral resolution determined by a projection valued measure on the borel subsets of the infinite half line we

a feynman kac ito formula for magnetic schrodinger operators on graphs in this paper we prove a feynman kac ito formula for magnetic schrodinger operators on arbitrary weighted graphs to do so we have to provide a natural and general framework both on the operator theoretic and the probabilistic side of the equation on the operator side we identify a very general class of potentials that allows the definition of magnetic schrodinger operators on the probabilistic side we introduce an appropriate notion of stochastic line integrals with respect to magnetic potentials apart from linking the world of discrete magnetic operators with the probabilistic world through the feynman kac ito formula the insights from this paper gained on both sides should be of an independent interest as applications of the feynman kac ito formula we prove a kato inequality a golden thompson inequality and an explicit representation of the quadratic form domains corresponding to a large class of potentials

existence of a tree of stieltjes strings corresponding to two given spectra transversal vibrations of a plane rooted tree of stieltjes strings are considered with dirichlet boundary conditions at each pendant vertex continuity and kirchhoff conditions are imposed at each interior vertex except the root the dirichlet problem is that with the dirichlet condition at the root and the neumann problem is that with continuity and kirchhoff conditions at the root it is shown that strict interlacing is a sufficient condition for two sequences of real numbers to be the spectra of neumann and dirichlet problems generated by stieltjes string recurrent relations on any prescribed tree

the equivalence of certain heat kernel and green function bounds abstract we show that certain upper and lower bounds on the green function and heat kernel of a second order elliptic operator in a bounded region are equivalent and imply a number of apparently stronger bounds we also show that these bounds are equivalent to the harnack inequality except for peculiar regions and are therefore almost always valid

spinodal decomposition for the cahn hilliard equation in higher dimensions nonlinear dynamics this paper addresses the phenomenon of spinodal decomposition for the cahn hilliard equation where n n 1 2 3 is a bounded domain with sufficiently smooth boundary and f is cubic like for example f u u u 3 based on the results of 26 the nonlinear cahn hilliard equation will be discussed this equation generates a nonlinear semiflow in certain affine subspaces of h 2 in a neighborhood u e with size proportional to e n around the constant solution where lies in the spinodal region we observe the following behavior within a local inertial manifold containing there exists a finite dimensional invariant manifold which dominates the behavior of all solutions starting with initial conditions from a small ball around with probability almost 1 the dimension of is proportional to e n and the elements of exhibit a common geometric quantity which is strongly related to a characteristic wavelength proportional to e

large time behavior of the heat equation on complete manifolds with non negative ricci curvature sur une variete de riemann complete m on considere le noyau de la chaleur h x y t qui resout l equation de la chaleur t f x t 0 sur mx 0 avec la donnee initiale f x 0 f x en posant f x t m h x y t f y dy

distribution theory for discontinuous test functions and differential operators with generalized coefficients investigation of the differential operators with the generalized coefficients having singular support on a disjoint set of points requires the consideration of the distribution theory with the set of discontinuous test functions such a distribution theory for test functions having discontinuity at one point is developed a four parameter family of schrodinger operators formed by the operators with singular potential singular metrics and singular gauge field is considered it is proved that this family of singular interactions describes all possible selfadjoint extensions of the second derivative operator defined on the functions vanishing in a neighbourhood of the point approximation by operators with smooth coefficients is discussed c 1996 academic press inc

application of statistical physics to politics abstract the concept and technics of real space renormalization group are applied to study majority rule voting in hierarchical structures it is found that democratic voting can lead to totalitarianism by keeping in power a small minority conditions of this paradox are analyzed and singled out indeed majority rule produces critical thresholds to absolute power values of these thresholds can vary from 50 up to at least 77 the associated underlying mechanism could provide an explanation for both former apparent eternity of communist leaderships and their sudden collapse

majority versus minority dynamics phase transition in an interacting two state spin system we introduce a simple model of opinion dynamics in which binary state agents evolve due to the influence of agents in a local neighborhood in a single update step a fixed size group is defined and all agents in the group adopt the state of the local majority with probability p or that of the local minority with probability 1 p for group size g 3 there is a phase transition at p c 2 3 in all spatial dimensions for p p c the global majority quickly predominates while for p p c the system is driven to a mixed state in which the densities of agents in each state are equal for p p c the average magnetization the difference in the density of agents in the two states is conserved and the system obeys classical voter model dynamics in one dimension and within a kirkwood decoupling scheme the final magnetization in a finite length system has a nontrivial dependence on the initial magnetization for all p not equal p c in agreement with numerical results at p c the exact two spin correlation functions decay algebraically toward the value 1 and the system coarsens as in the classical voter model

coalescing random walks and voter model consensus times on the torus in mathbb z d voter models on z d and on the torus of site n in z d are considered the second ones get trapped with probability 1 as t tends to infinity at all 0 s or all 1 s the asymptotic growth of these trapping times are examined as n tends to infinity this is done with limit thewems for coalescing random walk systems on the torus of side n in z d

pattern formation in a nonlinear model for animal coats abstract several models have been proposed for describing the formation of animal coat patterns we consider reaction diffusion models due to murray which rely on a turing instability for the pattern selection in this paper we describe the early stages of the pattern formation process for large domain sizes this includes the selection mechanism and the geometry of the patterns generated by the nonlinear system on one two and three dimensional base domains these results are obtained by an adaptation of results explaining the occurrence of spinodal decomposition in materials science as modeled by the cahn hilliard equation we use techniques of dynamical systems viewing solutions of the reaction diffusion model in terms of nonlinear semiflows our results are applicable to any parabolic system exhibiting a turing instability

quantum graphs ii some spectral properties of quantum and combinatorial graphs the paper deals with some spectral properties of mostly infinite quantum and combinatorial graphs quantum graphs have been intensively studied lately due to their numerous applications to mesoscopic physics nanotechnology optics and other areas a schnol type theorem is proven that allows one to detect that a point belongs to the spectrum when a generalized eigenfunction with an sub exponential growth integral estimate is available a theorem on spectral gap opening for decorated quantum graphs is established its analogue is known for the combinatorial case it is also shown that if a periodic combinatorial or quantum graph has a point spectrum it is generated by compactly supported eigenfunctions scars

voting and catalytic processes with inhomogeneities we consider the dynamics of the voter model and of the monomer monomer catalytic process in the presence of many competing inhomogeneities and show through exact calculations and numerical simulations that their presence results in a nontrivial fluctuating steady state whose properties are studied and turn out to specifically depend on the dimensionality of the system the strength of the inhomogeneities and their separating distances in fact in arbitrary dimensions we obtain an exact yet formal expression of the order parameters magnetization and concentration of adsorbed particles in the presence of an arbitrary number n of inhomogeneities zealots in the voter language and formal similarities with suitable electrostatic systems are pointed out in the nontrivial cases n 1 2 we explicitly compute the static and long time properties of the order parameters and therefore capture the generic features of the systems when ng2 the problems are studied through numerical simulations in one spatial dimension we also compute the expressions of the stationary order parameters in the completely disordered case where n is arbitrary large particular attention is paid to the spatial dependence of the stationary order parameters and formal connections with electrostatics

integrable models from twisted half loop algebras this paper is devoted to the construction of new integrable quantum mechanical models based on certain subalgebras of the half loop algebra of various results about these subalgebras are proven by presenting them in the notation of the st petersburg school these results are then used to demonstrate the integrability and find the symmetries of two types of physical system twisted gaudin magnets and calogero type models of particles on several half lines meeting at a point

asymptotic behavior of jost functions near resonance points for wigner von neumann type potentials in this work are considered radial schrodinger operators v r e where v r a sin br r w r with w r bounded w r o r 2 at infinity a b real the asymptotic behavior of the jost function and the scattering matrix near the resonance point e0 b2 4 are studied if a b then this point may be an eigenvalue embedded in the continuous spectrum the leading behavior of the jost function for all values of a and b were determined somewhat surprisingly situations were found where the jost function becomes singular as e e0 even if e0 is an embedded eigenvalue moreover it was found that the scattering matrix is always discontinuous at e0 except in a few special cases it is also shown that the asymptotics for the jost function and the scattering matrix hold under weaker assumptions on w r in particular an angular momentum term l l 1 r 2 may be incorporated into w r the results were also applied to a whole line problem with a potential v x such that v x 0 for x 0 and v x of wigner von neumann

unexpectedly linear behavior for the cahn hilliard equation this paper gives theoretical results on spinodal decomposition for the cahn hillard equation we prove a mechanism which explains why most solutions for the cahn hilliard equation starting near a homogeneous equilibrium within the spinodal interval exhibit phase separation with a characteristic wavelength when exiting a ball of radius r namely most solutions are driven into a region of phase space in which linear behavior dominates for much longer than expected the cahn hilliard equation depends on a small parameter epsilon modeling the atomic scale interaction length we quantify the behavior of solutions as epsilon rightarrow 0 specifically we show that most solutions starting close to the homogeneous equilibrium remain close to the corresponding solution of the linearized equation with relative distance o epsilon 2 n 2 up to a ball of radius r in the h 2 omega norm where r is proportional to epsilon 1 rho n 4 as epsilon to 0 here n in 1 2 3 denotes the

opinion dynamics on an adaptive random network we revisit the classical model for voter dynamics in a two party system with two basic modifications in contrast to the original voter model studied in regular lattices we implement the opinion formation process in a random network of agents in which interactions are no longer restricted by geographical distance in addition we incorporate the rapidly changing nature of the interpersonal relations in the model at each time step agents can update their relationships this update is determined by their own opinion and by their preference to make connections with individuals sharing the same opinion or rather with opponents in this way the network is built in an adaptive manner in the sense that its structure is correlated and evolves with the dynamics of the agents the simplicity of the model allows us to examine several issues analytically we establish criteria to determine whether consensus or polarization will be the outcome of the dynamics and on what time scales these states will be reached in finite systems consensus is typical while in infinite systems a disordered metastable state can emerge and persist for infinitely long time before consensus is reached

slow dynamics for the cahn hilliard equation in higher space dimensions the motion of bubbles it is known that the van der waals cahn hilliard w c h dynamics can be approximated by a quasi static stefan problem with surface tension it turns out that the stefan problem has a manifold of equilibria equal in dimension to that of the domain any sphere of fixed radius with interface contained in the domain is an equilibrium indistinguishable from the point of view of the perimeter functional we resolve this degeneracy by showing that at the w c h level this manifold is replaced by a quasi invariant stable manifold on which the typical solution preserves its bubble like shape until it reaches the boundary moreover we show that the bubble moves superslowly we also obtain an equation that determines those special spheres that correspond to equilibria at the w c h level our work establishes the phenomenon of superslow motion in higher space dimensions in the class of single interface solutions

absorption semigroups their generators and schrodinger semigroups on presente des resultats de perturbations pour des c 0 semi groupes on considere le probleme aux valeurs initiales pour l equation de la chaleur avec absorption avec une solution dans l p r v avec une mauvaise fonction v r r on considere le meme probleme dans l 2 r pour l equation de schrodinger

high energy analysis and levinson s theorem for the selfadjoint matrix schrodinger operator on the half line the matrix schrodinger equation with a selfadjoint matrix potential is considered on the half line with the general selfadjoint boundary condition at the origin when the matrix potential is integrable the high energy asymptotics are established for the related jost matrix the inverse of the jost matrix and the scattering matrix under the additional assumption that the matrix potential has a first moment levinson s theorem is derived relating the number of bound states to the change in the argument of the determinant of the scattering matrix

the free dirac operator on compact and noncompact graphs the free dirac operator defined on composite one dimensional structures consisting of finitely many half lines and intervals is investigated the influence of the connection points between the constituents is modeled by transition conditions for the wave functions or equivalently by different self adjoint extensions of the dirac operator general relations between the parameters of the extensions and the eigenvalues resp the scattering coefficients are derived and then applied to the cases of a bundle of half lines a point defect a branching line and an eye shaped structure

four parameter point interaction in 1d quantum systems the author constructs a four parameter point interaction for a nonrelativistic particle moving on a line as the limit of a short range interaction with a range tending toward zero for particular choices of the parameters one can obtain a delta interaction or the so called delta interaction the hamiltonian corresponding to the four parameter point interaction is shown to correspond to the four parameter self adjoint hamiltonian of the free particle moving on the line with the origin excluded

semilinear evolution equations in banach spaces with application to parabolic partial differential equations a theory for a class of semilinear evolution equations in banach spaces is developed which when applied to certain parabolic partial differential equations with nonlinear terms in divergence form gives strong solutions even for nondifferentiable data

quantum graphs with singular two particle interactions we construct quantum models of two particles on a compact metric graph with singular two particle interactions the hamiltonians are self adjoint realizations of laplacians acting on functions defined on pairs of edges in such a way that the interaction is provided by boundary conditions in order to find such hamiltonians closed and semi bounded quadratic forms are constructed from which the associated self adjoint operators are extracted we provide a general characterization of such operators and furthermore produce certain classes of examples we then consider identical particles and project to the bosonic and fermionic subspaces finally we show that the operators possess purely discrete spectra and that the eigenvalues are distributed following an appropriate weyl asymptotic law

dispersion for the schrodinger equation on networks in this paper we consider the schrodinger equation on a network formed by a tree with the last generation of edges formed by infinite strips we give an explicit description of the solution of the linear schrodinger equation with constant coefficients this allows us to prove dispersive estimates which in turn are useful for solving the nonlinear schrodinger equation the proof extends also to the laminar case of positive step function coefficients having a finite number of discontinuities

index theorems for quantum graphs in geometric analysis an index theorem relates the difference of the numbers of solutions of two differential equations to the topological structure of the manifold or bundle concerned sometimes using the heat kernels of two higher order differential operators as an intermediary in this paper the case of quantum graphs is addressed a quantum graph is a graph considered as a singular one dimensional variety and equipped with a second order differential hamiltonian h a laplacian with suitable conditions at vertices for the case of scale invariant vertex conditions i e conditions that do not mix the values of functions and of their derivatives the constant term of the heat kernel expansion is shown to be proportional to the trace of the internal scattering matrix of the graph this observation is placed into the index theory context by factoring the laplacian into two first order operators h a a and relating the constant term to the index of a an independent consideration provides an index formula for any differential operator on a finite quantum graph in terms of the vertex conditions it is also found that the algebraic multiplicity of 0 as a root of the secular determinant of h is the sum of the nullities of a and a

on δ like potential scattering on star graphs we discuss the potential scattering on the noncompact star graph the schrodinger operator with the short range potential localized in a neighborhood of the graph vertex is considered we study the asymptotic behavior of the corresponding scattering matrix in the zero range limit it has been known for a long time that in dimension 1 there is no non trivial hamiltonian with the distributional potential i e the potential acts as a totally reflecting wall several authors have in recent years studied the scattering properties of the regularizing potentials e 2q x e approximating the first derivative of the dirac delta function a non zero transmission through the regularized potential has been shown to exist as e 0 we extend these results to star graphs with the point interaction which is an analog of the potential on the line we prove that generically such a potential on the graph is opaque we also show that there exists a countable set of resonant intensities for which a partial transmission through the potential occurs this set of resonances is referred to as the resonant set and is determined as the spectrum of an auxiliary sturm liouville problem associated with q on the graph

how to introduce temperature to the 1d sznajd model abstract we investigate the possibility of introducing temperature to the one dimensional sznajd model and propose a natural extension of the original model by including other types of interactions we characterise different kinds of equilibria into which the extended system can evolve we determine the consequences of fulfilling the detailed balance condition and we prove that in some cases it is equivalent to microscopic reversibility we found the equivalence of the model to the standard inflow model with interactions up to next nearest neighbors it is shown that under some constraints there exists a hamiltonian compatible with the dynamics and its form resembles that of the 1d annni model it appears however that the standard approach of constructing temperature from the hamiltonian fails in this situation we propose a simple definition of the temperature like quantity that measures the size of fluctuations in the system at equilibrium the complete list of zero temperature degenerated cases as well as the list of ground states of the derived hamiltonian are provided

a factorization of the scattering matrix for the schrodinger equation and for the wave equation in one dimension for the one dimensional schrodinger equation with a potential decaying at both ends of the real axis the scattering matrix of the potential is given explicitly in terms of the scattering matrices corresponding to the fragments of this potential a similar result also holds for the wave equation in a nonhomogeneous nondispersive medium where the wave speed has the same asymptotics at both ends of the real line

maximum norms of random sums and transient pattern formation many interesting and complicated patterns in the applied sciences are formed through transient pattern formation processes in this paper we concentrate on the phenomenon of spinodal decomposition in metal alloys as described by the cahn hilliard equation this model depends on a small parameter and one is generally interested in establishing sharp lower bounds on the amplitudes of the patterns as the parameter approaches zero recent results on spinodal decomposition have produced such lower bounds unfortunately for higher dimensional base domains these bounds are orders of magnitude smaller than what one would expect from simulations and experiments the bounds exhibit a dependence on the dimension of the domain which from a theoretical point of view seemed unavoidable but which could not be observed in practice in this paper we resolve this apparent paradox by employing probabilistic methods we can improve the lower bounds for certain domains and remove the dimension dependence we thereby obtain optimal results which close the gap between analytical methods and numerical observations and provide more insight into the nature of the decomposition process we also indicate how our results can be adapted to other situations

comment on factorization of scattering matrices due to partitioning of potentials in one dimensional schrodinger type equations j math phys 37 5897 1996

appendix resonances on a simple graph we study the scattering problem on a graph consisting of a line with a finite length appendix the two parts are coupled through boundary conditions depending on three parameters the motion on the line is free while the appendix supports a potential the appendix bound states give rise to a ladder of resonances we construct the resolvent and solve the corresponding pole condition for a weak coupling in general the condition only admits an analytic solution in particular cases we find the pole positions numerically for a linear potential and show that the poles eventually return to the real axis when the coupling strength increases

existence of an upper critical dimension in the majority voter model we study the critical properties of the majority voter model on d dimensional hypercubic lattices in two dimensions the majority voter model belongs to the same universality class as that of the ising model however the critical behaviors of the majority voter model on four dimensions do not exhibit mean field behavior using the monte carlo simulation on d dimensional hypercubic lattices we obtain the critical exponents up to d 7 and find that the upper critical dimension is 6 for the majority voter model we also confirm our results using mean field calculation

large time behavior of the heat kernel and the behavior of the green function near criticality for nonsymmetric elliptic operators abstract let p be a second order elliptic operator which is defined on a domain r n and let w x be a nonzero nonnegative function assume that p p w x admits a positive minimal green function g p x y for all lim 0 g p x y f x y where f 0 for x y unless p and p admit ground states x and x respectively with eigenvalue zero such that z w z z dz in this case f x y x y z w z z dx moreover if z z dz then the heat kernel k x y t of the parabolic equation lu u t pu 0 satisfies lim t k x y t x y z z dz

on diagonal lower estimate of heat kernels on graphs abstract we establish a new on diagonal lower estimate of continuous time heat kernels for large time on graphs to achieve the goal we first introduce an upper estimate of heat kernels in natural graph metric then use the upper estimate and the volume growth condition to show the validity of the on diagonal lower estimate

cheeger constants growth and spectrum of locally tessellating planar graphs in this article we study relations between the local geometry of planar graphs combinatorial curvature and global geometric invariants namely the cheeger constants and the exponential growth we also discuss spectral applications

diffusion on delone sets we consider graphs associated to delone sets in euclidean space such graphs arise in various ways from tilings here we provide a unified framework in this context we study the associated laplace operators and show gaussian heat kernel bounds for their semigroups these results apply to both metric and discrete graphs

the number of eigenvalues of the matrix schrodinger operator on the half line with general boundary conditions we prove a bound of the bargmann birman schwinger type on the number of eigenvalues of the matrix schrodinger operator on the half line with the most general self adjoint boundary condition at the origin and with selfadjoint matrix potentials that are integrable and have a finite first moment

monte carlo simulations for spinodal decomposition this paper addresses the phenomenon of spinodal decomposition for the cahn hilliard equation namely we are interested in why most solutions to the cahn hilliard equation which start near a homogeneous equilibrium u 0 in the spinodal interval exhibit phase separation with a characteristic wavelength when exiting a ball of radius r in a hilbert space centered at u 0 there are two mathematical explanations for spinodal decomposition due to grant and to maier paape and wanner in this paper we numerically compare these two mathematical approaches in fact we are able to synthesize the understanding we gain from our numerics with the approach of maier paape and wanner leading to a better understanding of the underlying mechanism for this behavior with this new approach we can explain spinodal decomposition for a longer time and larger radius than either of the previous two approaches a rigorous mathematical explanation is contained in a separate paper our approach is to use monte carlo simulations to examine the dependence of r the radius to which spinodal decomposition occurs as a function of the parameter e of the governing equation we give a description of the dominating regions on the surface of the ball by estimating certain densities of the distributions of the exit points we observe and can show rigorously that the behavior of most solutions originating near the equilibrium is determined completely by the linearization for an unexpectedly long time we explain the mechanism for this unexpectedly linear behavior and show that for some exceptional solutions this cannot be observed we also describe the dynamics of these exceptional solutions

the essential spectrum of the laplacian on rapidly branching tessellations in this paper we characterize absence of the essential spectrum of the laplacian under a hyperbolicity assumption for general graphs moreover we present a characterization for absence of the essential spectrum for planar tessellations in terms of curvature

on the number of states bound by one dimensional finite periodic potentials bound states and zero energy resonances of one dimensional finite periodic potentials are investigated by means of levinson s theorem for finite range potentials supporting no bound states a lower bound for the reduced time delay at threshold is derived

uniform existence of the integrated density of states for random schrodinger operators on metric graphs over zd abstract we consider ergodic random schrodinger operators on the metric graph z d with random potentials and random boundary conditions taking values in a finite set we show that normalized finite volume eigenvalue counting functions converge to a limit uniformly in the energy variable this limit the integrated density of states can be expressed by a closed shubin pastur type trace formula it supports the spectrum and its points of discontinuity are characterized by existence of compactly supported eigenfunctions among other examples we discuss random magnetic fields and percolation models

spinodal decomposition for the cahn hilliard equation in higher dimensions part i probability and wavelength estimate this paper is the first in a series of two papers addressing the phenomenon of spinodal decomposition for the cahn hilliard equation where is a bounded domain with sufficiently smooth boundary and f is cubic like for example f u u u3 we will present the main ideas of our approach and explain in what way our method differs from known results in one space dimension due to grant 26 furthermore we derive certain probability and wavelength estimates the probability estimate is needed to understand why in a neighborhood of a homogeneous equilibrium u0 of the cahn hilliard equation with mass in the spinodal region a strongly unstable manifold has dominating effects this is demonstrated for the linearized equation but will be essential for the nonlinear setting in the second paper 37 as well moreover we introduce the notion of a characteristic wavelength for the strongly unstable directions

factorization of scattering matrices due to partitioning of potentials in one dimensional schrodinger type equations the one dimensional schrodinger equation and two of its generalizations are considered as they arise in quantum mechanics wave propagation in a nonhomogeneous medium and wave propagation in a nonconservative medium where energy may be absorbed or generated generically the zero energy transmission coefficient vanishes when the potential is nontrivial but in the exceptional case this coefficient is nonzero resulting in tunneling through the potential it is shown that any nontrivial exceptional potential can always be fragmented into two generic pieces furthermore any nontrivial potential generic or exceptional can be fragmented into generic pieces in infinitely many ways the results remain valid when dirac delta functions are included in the potential and other coefficients are added to the schrodinger equation for such schrodinger equations factorization formulas are obtained that relate the scattering matrices of the fragments to the scattering matrix of the full problem

constrained opinion dynamics freezing and slow evolution we study opinion formation in a population of leftists centrists and rightist in an interaction between neighbouring agents a centrist and a leftist can become both centrists or leftists and similarly for a centrist and a rightist while leftists and rightists do not affect each other the evolution is controlled by the initial density of centrists 0 for any spatial dimension the system reaches a centrist consensus with probability 0 while with probability 1 0 the final state is either an extremist consensus or a frozen population of leftists and rightists in one dimension we determine the opinion evolution by mapping the system onto a spin 1 ising model with zero temperature glauber kinetics the approach to the final state is governed by a t long time tail with 2 0 as 0 0 in the one dimensional frozen state the length distribution of single opinion domains has an algebraic small size tail x 2 1 and the average domain length is l2 where l is the length of the system

difference equations isoperimetric inequality and transience of certain random walks the difference laplacian on a square lattice in rn has been stud ied by many authors in this paper an analogous difference operator is studied for an arbitrary graph it is shown that many properties of the laplacian in the continuous setting e g the maximum principle the harnack inequality and cheeger s bound for the lowest eigenvalue hold for this difference oper ator the difference laplacian governs the random walk on a graph just as the laplace operator governs the brownian motion as an application of the theory of the difference laplacian it is shown that the random walk on a class of graphs is transient the random walks we consider are defined as follows let k be a connected graph i e a one dimensional simplicial complex for a vertex x e k let m x denote the number of edges emanating from x the probability that a particle moves from x to another vertex y e k is l m x if x and y are connected by an edge and it is zero otherwise as observed by courant friedrichs and lewy cfl for the case of a square lattice in the plane this random walk is intimately related to the difference analog of the laplacian

second phase spinodal decomposition for the cahn hilliard cook equation we consider the dynamics of a nonlinear partial differential equation perturbed by additive noise assuming that the underlying deterministic equation has an unstable equilibrium we show that the nonlinear stochastic partial differential equation exhibits essentially linear dynamics far from equilibrium more precisely we show that most trajectories starting at the unstable equilibrium are driven away in two stages after passing through a cylindrical region most trajectories diverge from the deterministic equilibrium through a cone shaped region which is centered around a finite dimensional subspace corresponding to strongly unstable eigenfunctions of the linearized equation and on which the influence of the nonlinearity is surprisingly small this abstract result is then applied to explain spinodal decomposition in the stochastic cahn hilliard cook equation on a domain g this equation depends on a small interaction parameter e 0 and one is generally interested in asymptotic results as e 0 specifically we show that linear behavior dominates the dynamics up to distances from the deterministic equilibrium which can reach e 2 dim g 2 with respect to the h 2 g norm

effect of initial concentration and spatial heterogeneity of active agent distribution on opinion dynamics abstract we analyze the effect of spatial heterogeneity in the initial spin distribution on spin dynamics in a three state square lattice divided into spatial cells districts in the spirit of the statistical mechanics of social impact we introduce a dominant influence rule dir according to which in a single update step a chosen node adopts the state determined by the influence of its discussion group formed by the node itself and its neighbors within one or more coordination spheres in contrast to models based on some form of majority rule mr a system governed by the dir is easily trapped in a stable non consensus state if all nodes of the discussion group have the same weight of influence to ensure that a consensus in the dir system is necessarily reached we need to put a stochastic process in the update rule further the stochastic dir model is used as a starting point for understanding the effect of spatial heterogeneity of active agent non zero spin distribution on the exit probabilities initially the positive and negative spins active agents are assigned to some nodes with non uniform spatial distributions while the rest of the nodes remain in the state with spin zero uncommitted voters by varying the relative means and skewness of the initial spin distributions we observe critical behaviors of exit probabilities in finite size systems the critical exponents are obtained by monte carlo simulations the results of numerical simulations are discussed in the context of social dynamics

large time behavior of the heat kernel in this paper we study the large time behavior of the minimal heat kernel kpm x y t of a general time independent parabolic operator lu ut p x x u which is defined on a noncompact manifold m more precisely we prove that limt e 0tkpm x y t always exists here 0 is the generalized principal eigenvalue of the operator p in m

exit probability in generalised kinetic ising model in this paper we study generalised ising glauber models with inflow of information in one dimension and derive expressions for the exit probability using well established analytical methods the analytical expressions agree very well with the results obtained from numerical simulation only when the interaction is restricted to the nearest neighbor but as the range of interaction is increased the analytical results deviate from simulation results systematically the reasons for the deviation as well as some related open questions are discussed

boundary conditions at the derivative of a delta function the author derives the joining conditions on psi at a point where the potential is the nth derivative of a dirac delta function

solutions of nonlinear planar elliptic problems with triangle symmetry abstract in this paper we continue the study of the nodal domain structure of doubly periodic solutions of certain nonlinear elliptic problems initiated in fifeet al physica d100 1997 257 278 more precisely we consider small amplitude solutions of u f u 0 in r 2whose nodal domains consist of equilateral triangles tiling the plane if this equation is suitably perturbed then for genericfwe prove the existence of unique nearby solutions with triangle symmetry and show how their nodal domain geometry breaks up furthermore we treat the non generic rectangular cases which had to be excluded in fifeet al physica d100 1997 257 278 as well as other nodal domain structures

smoothness of inertial manifolds the theory of invariant manifolds plays an important role in the study of dynamics of nonlinear systems in finite dimensional or infinite dimensional spaces such invariant manifolds include stable manifolds unstable manifolds and center manifolds see pliss 36 for example recently the theory of inertial manifolds has been developed for some dissipative evolution equations the inertial manifold introduced by foias sell and ttmam 141 is a finite dimensional lipschitz invariant manifold attracting solutions exponentially one of the important properties of inertial manifolds is that they contain the global attractors thus the study of dynamics of infinite dimensional nonlinear systems can be reduced to the study of dynamics of flows on the inertial manifold which in turn is described by the dynamics of an ordinary differential equation there are extensive works on this subject see for example constantin 3 constantin foias nicolaenko

commitment versus persuasion in the three party constrained voter model in the framework of the three party constrained voter model where voters of two radical parties a and b interact with centrists c and c we study the competition between a persuasive majority and a committed minority in this model a s and b s are incompatible voters that can convince centrists or be swayed by them here radical voters are more persuasive than centrists whose sub population comprises susceptible agents c and a fraction of centrist zealots c whereas c s may adopt the opinions a and b with respective rates 1 a and 1 b with a b 0 c s are committed individuals that always remain centrists furthermore a and b voters can become susceptible centrists c with a rate 1 the resulting competition between commitment and persuasion is studied in the mean field limit and for a finite population on a complete graph at mean field level there is a continuous transition from a coexistence phase when b whereas all species coexist when a b when c and the initial density of centrists is low one finds lnn when n 1 our analytical findings are corroborated by stochastic simulations

nonequilibrium transitions in complex networks a model of social interaction we analyze the nonequilibrium order disorder transition of axelrod s model of social interaction in several complex networks in a small world network we find a transition between an ordered homogeneous state and a disordered state the transition point is shifted by the degree of spatial disorder of the underlying network the network disorder favoring ordered configurations in random scale free networks the transition is only observed for finite size systems showing system size scaling while in the thermodynamic limit only ordered configurations are always obtained thus in the thermodynamic limit the transition disappears however in structured scale free networks the phase transition between an ordered and a disordered phase is restored

kirchhoff s rule for quantum wires we formulate and discuss one particle quantum scattering theory on an arbitrary finite graph with n open ends and where we define the hamiltonian to be minus the laplace operator with general boundary conditions at the vertices this results in a scattering theory with n channels the corresponding on shell s matrix formed by the reflection and transmission amplitudes for incoming plane waves of energy e 0 is given explicitly in terms of the boundary conditions and the lengths of the internal lines it is shown to be unitary which may be viewed as the quantum version of kirchhoff s law we exhibit covariance and symmetry properties it is symmetric if the boundary conditions are real also there is a duality transformation on the set of boundary conditions and the lengths of the internal lines such that the low energy behaviour of one theory gives the high energy behaviour of the transformed theory finally we provide a composition rule by which the on shell s matrix of a graph is factorizable in terms of the s matrices of its subgraphs all proofs use only known facts from the theory of self adjoint extensions standard linear algebra complex function theory and elementary arguments from the theory of hermitian symplectic forms

fixation in a cyclic lotka volterra model we study a cyclic lotka volterra model of n interacting species populating a d dimensional lattice in the realm of a kirkwood approximation a critical number of species nc d above which the system fixates is determined analytically we find nc d 5 14 23 in dimensionsdd 1 2 3 in remarkably good agreement with simulation results in two dimensions a cyclic variant of the lotka volterra model of interacting populations originally introduced by vito volterra for the description of the struggle for existence among species 1 2 has then appeared in a number of apparently unrelated fields ranging from plasma physics 3 to integrable systems 4 5 recently the cyclic lotka volterra model also known as the cyclic voter model has attracted considerable interest as it was realized that introduction of the spatial structure drastically enriches the dynamics 6 11 namely if species live on a one dimensional 1d lattice a homogeneous initial state evolves into a coarsening mosaic of interacting species this heterogeneous spatial structure spontaneously develops when the number of species is sufficiently small n nc fixation occurs i e the system approaches a frozen state little is known in higher dimensions not even the existence of nc has yet been established theoretically or numerically in simulations on two dimensional 2d lattices with n 6 10 species no sign of fixation has been found and instead a reactive steady state has been observed 6 11 in this letter we investigate the cyclic lotka volterra model in the framework of a kirkwood like approximation this approach predicts a finite nc in all spatial dimensions in the following we shall use the language of the voter model 12 consider the cyclic voter model with n possible opinions each site of a d dimensional cubic lattice is occupied by a voter which has an opinion labelled by with d 1 n voters can change their opinions in a cyclic manner 1 modulo n according to the opinions of their neighbourhood specifically the following sequential dynamics is implemented i we randomly choose a site of opinion say and one of its 2d nearest neighbours of opinion ii if d 1 then the chosen site changes its opinion from to d 1 and iii otherwise opinion does not change we set the timescale so that in unit time each site of the lattice is chosen once on average when nd 2 the cyclic voter model is

instability of two phase flows a lower bound on the dimension of the global attractor of the cahn hilliard navier stokes system abstract we consider a model for the flow of a mixture of two viscous and incompressible fluids in a two or three dimensional channel like domain the model consists of the navier stokes equations governing the fluid velocity coupled with a convective cahn hilliard equation for the relative density of atoms of one of the fluids we prove the instability of certain stationary solutions for such a system endowed with periodic boundary conditions on elongated domains 0 2 0 0 2 or 0 2 0 0 2 0 2 0 for a special class of periodic body forces provided that 0 and 0 are small enough as a consequence we deduce a lower bound for the hausdorff dimension of the global attractor

dynamics of majority rule in two state interacting spin systems we introduce a two state opinion dynamics model where agents evolve by majority rule in each update a group of agents is specified whose members then all adopt the local majority state in the mean field limit where a group consists of randomly selected agents consensus is reached in a time that scales ln n where n is the number of agents on finite dimensional lattices where a group is a contiguous cluster the consensus time fluctuates strongly between realizations and grows as a dimension dependent power of n the upper critical dimension appears to be larger than 4 the final opinion always equals that of the initial majority except in one dimension

two body scattering on a graph and application to simple nanoelectronic devices the two body system on a graph with one junction is considered the effective three body scattering problem turns out to be exactly solvable for pointwise interactions additional degrees of freedom corresponding to a dynamics of some structure e g an atomic cluster located in the junction point of common contact of three thin electrodes are considered these degrees of freedom bring effective energy dependent interaction into the effective schrodinger equation in the scattering channel the wave function of the system is constructed in the explicit form using the extension theory methods the obtained results are applied to the qualitative description of a simple three electrode nanoelectronic device the perturbation theory approach based on the analysis of the liouville equation is suggested for calculation of the conductivity for such a device in terms of the obtained wave function

an evolutionary strategy based on partial imitation for solving optimization problems abstract in this work we introduce an evolutionary strategy to solve combinatorial optimization tasks i e problems characterized by a discrete search space in particular we focus on the traveling salesman problem tsp i e a famous problem whose search space grows exponentially increasing the number of cities up to becoming np hard the solutions of the tsp can be codified by arrays of cities and can be evaluated by fitness computed according to a cost function e g the length of a path our method is based on the evolution of an agent population by means of an imitative mechanism we define partial imitation in particular agents receive a random solution and then interacting among themselves may imitate the solutions of agents with a higher fitness since the imitation mechanism is only partial agents copy only one entry randomly chosen of another array i e solution in doing so the population converges towards a shared solution behaving like a spin system undergoing a cooling process i e driven towards an ordered phase we highlight that the adopted partial imitation mechanism allows the population to generate solutions over time before reaching the final equilibrium results of numerical simulations show that our method is able to find in a finite time both optimal and suboptimal solutions depending on the size of the considered search space

exact results for kinetics of catalytic reactions the kinetics of an irreversible catalytic reaction on substrate of arbitrary dimension is examined in the limit of infinitesimal reaction rate reaction controlled limit we solve the dimer dimer surface reaction model or voter model exactly in arbitrary dimension d the density of reactive interfaces is found to exhibit a power law decay for d 2 and a slow logarithmic decay in two dimensions we discuss the relevance of these results for the monomer monomer surface reaction model

classifying the general family of 1d point interactions a scattering approach we propose a classification scheme for the complete family of 1d point interactions to do so we first review the solutions of the wavefunctions and green functions of the problem second we derive the exact time dependent propagators in such a way that we can write the expressions for the k s in a very compact form as they should such expressions do reduce to the known results in the literature according to the potential parameter values then we analyse in general terms how the different point interactions scatter off arbitrary initially localized wave packets finally we show that the physical features associated with the scattering process can be used to establish a classification procedure moreover these physical characteristics are directly related to the potential parameters leading to the many formulae for the k s as an application we present numerical calculations for gaussian wave packets

existence and uniqueness of solutions to singular cahn hilliard equations with nonlinear viscosity terms and dynamic boundary conditions abstract we prove global existence and uniqueness of solutions to a cahn hilliard system with nonlinear viscosity terms and nonlinear dynamic boundary conditions the problem is highly nonlinear characterized by four nonlinearities and two separate diffusive terms all acting in the interior of the domain or on its boundary through a suitable approximation of the problem based on abstract theory of doubly nonlinear evolution equations existence and uniqueness of solutions are proved using compactness and monotonicity arguments the asymptotic behaviour of the solutions as the diffusion operator on the boundary vanishes is also shown

analysis of the physical laplacian and the heat flow on a locally finite graph we study the physical laplacian and the corresponding heat flow on an infinite locally finite graph with possibly unbounded valence

expansion in generalized eigenfunctions for laplacians on graphs and metric measure spaces we consider an arbitrary selfadjoint operator on a separable hilbert space to this operator we construct an expansion in generalized eigenfunctions in which the original hilbert space is decomposed as a direct integral of hilbert spaces consisting of general eigenfunctions this automatically gives a plancherel type formula for suitable operators on metric measure spaces we discuss some growth restrictions on the generalized eigenfunctions for laplacians on locally finite graphs the generalized eigenfunctions are exactly the solutions of the cor responding difference equation

higher order cahn hilliard equations with dynamic boundary conditions abstract our aim in this paper is to study the well posedness and the dissipativity of higher order cahn hilliard equations with dynamic boundary conditions more precisely we prove the existence and uniqueness of solutions and the existence of the global attractor

invariant manifolds for flows in banach spaces on presente une theorie des varietes invariantes lisses basee sur la methode classique de lyapunov penon pour des semi flots continus dans des espaces de banach

spinodal decomposition for the cahn hilliard cook equation this paper gives theoretical results on spinodal decomposition for the stochastic cahn hilliard cook equation which is a cahn hilliard equation perturbed by additive stochastic noise we prove that most realizations of the solution which start at a homogeneous state in the spinodal interval exhibit phase separation leading to the formation of complex patterns of a characteristic size