spread of information and infection on finite random networks the modeling of epidemic like processes on random networks has received considerable attention in recent years while these processes are inherently stochastic most previous work has been focused on deterministic models that ignore important fluctuations that may persist even in the infinite network size limit in a previous paper for a class of epidemic and rumor processes we derived approximate models for the full probability distribution of the final size of the epidemic as opposed to only mean values in this paper we examine via direct simulations the adequacy of the approximate model to describe stochastic epidemics and rumors on several random network topologies homogeneous networks erd os r enyi er random graphs barabasi albert scale free networks and random geometric graphs we find that the approximate model is reasonably accurate in predicting the probability of spread however the position of the threshold and the conditional mean of the final size for processes near the threshold are not well described by the approximate model even in the case of homogeneous networks we attribute this failure to the presence of other structural properties beyond degree degree correlations and in particular clustering which are present in any finite network but are not incorporated in the approximate model in order to test this hypothesis we perform additional simulations on a set of er random graphs where degree degree correlations and clustering are separately and independently introduced using recently proposed algorithms from the literature our results show that even strong degree degree correlations have only weak effects on the position of the threshold and the conditional mean of the final size on the other hand the introduction of clustering greatly affects both the position of the threshold and the conditional mean similar analysis for the barabasi albert scale free network confirms the significance of clustering on the dynamics of rumor spread for this network though with its highly skewed degree distribution the addition of positive correlation had a much stronger effect on the final size distribution than was found for the simple random graph

consensus dynamics on random rectangular graphs abstract a random rectangular graph rrg is a generalization of the random geometric graph rgg in which the nodes are embedded into a rectangle with side lengths a and b 1 a instead of on a unit square 0 1 2 two nodes are then connected if and only if they are separated at a euclidean distance smaller than or equal to a certain threshold radius r when a 1 the rrg is identical to the rgg here we apply the consensus dynamics model to the rrg our main result is a lower bound for the time of consensus i e the time at which the network reaches a global consensus state to prove this result we need first to find an upper bound for the algebraic connectivity of the rrg i e the second smallest eigenvalue of the combinatorial laplacian of the graph this bound is based on a tight lower bound found for the graph diameter our results prove that as the rectangle in which the nodes are embedded becomes more elongated the rrg becomes a large world i e the diameter grows to infinity and a poorly connected graph i e the algebraic connectivity decays to zero the main consequence of these findings is the proof that the time of consensus in rrgs grows to infinity as the rectangle becomes more elongated in closing consensus dynamics in rrgs strongly depend on the geometric characteristics of the embedding space and reaching the consensus state becomes more difficult as the rectangle is more elongated

remarks on the semilinear schrodinger equation abstract we investigate the initial value problem for some semi linear schrodinger equations in two space dimensions with exponential growth

global well posedness and scattering for the defocusing l 2 critical nonlinear schrodinger equation when d 2 in this article we prove that the defocusing cubic nonlinear schrodinger initial value problem is globally well posed and scattering for u0 l2 r2 the proof uses the bilinear estimates of planchon and vega and a frequency localized interaction morawetz estimate similar to the high frequency estimate of colliander keel staffilani takaoka and tao and especially the low frequency estimate of dodson

the cauchy problem of a focusing energy critical nonlinear schrodinger equation abstract in this paper we combine variational methods and harmonic analysis to discuss the cauchy problem of a focusing nonlinear schrodinger equation we study the global well posedness finite time blowup and asymptotic behavior of this problem by hamiltonian property we establish two types of invariant evolution flows then from one flow and the stability of classical energy critical nonlinear schrodinger equation we find that the solution exists globally and scattering occurs finally we get a precise blowup criterion of this problem for positive energy initial data via the other flow

multilinear estimates for periodic kdv equations and applications we prove an endpoint multilinear estimate for the xs b spaces associated to the periodic airy equation as a consequence we obtain sharp local well posedness results for periodic generalized kdv equations as well as some global well posedness results below the energy norm

global existence and scattering for a class of nonlinear fourth order schrodinger equation below the energy space in this paper we consider a class of nonlinear fourth order schr odinger equation namely left begin array rcl i partial t u delta 2 u u nu 1 u quad 1 frac 8 d nu 1 frac 8 d 4 u 0 u 0 in h gamma mathbb r d quad 5 leq d leq 11 end array right using the i method combined with the interaction morawetz inequality we establish the global well posedness and scattering in h gamma mathbb r d with gamma d nu 0

tensor products and correlation estimates with applications to nonlinear schrodinger equations we prove new interaction morawetz type correlation estimates in one and two dimensions in dimension two the estimate corresponds to the nonlinear diagonal analogue of bourgain s bilinear refinement of strichartz for the 2d case we provide a proof in two different ways first we follow the original approach of lin and strauss but applied to tensor products of solutions we then demonstrate the proof using commutator vector operators acting on the conservation laws of the equation this method can be generalized to obtain correlation estimates in all dimensions in one dimension we use the gauss weierstrass summability method acting on the conservation laws we then apply the 2d estimate to nonlinear schrodinger equations and derive a direct proof of nakanishi s h 1 scattering result for every l 2 supercritical nonlinearity we also prove scattering below the energy space for a certain class of l 2 supercritical equations in this paper we obtain new 1 a priori estimates for solutions of the nonlinear schrodinger equation in one and two dimension we also provide a systematic way to obtain the known interaction a priori estimates for dimensions higher than three these estimates are monotonicity formulae that take advantage of the conservation of the momentum of the equation due to the pioneering work 19 estimates of this type are referred to as morawetz estimates in the literature we then apply these estimates to study the global behavior of solutions to the nonlinear schrodinger equation to be more precise we want to study the global in time behavior of solutions to the following initial value problem

remarks on the repulsive wigner poisson system we present an estimate for the space time integral of classical solutions to the repulsive wigner poisson system we use markowich s formalism between wigner possion and schrodinger poisson systems through this formalism and morawetz interaction potentials we derive the same a priori estimate given by chae and ha for the repulsive vlasov poisson system

h 1 scattering for systems of n defocusing weakly coupled nls equations in low space dimensions abstract we prove that the scattering operators and wave operators are well defined in the energy space for the system of defocusing schrodinger equations i t u u 1 n u p 1 u p 1 u 0 1 n u 0 1 n u 0 1 n h 1 r d n with n 2 0 0 for p 2 if d 1 p 1 if d 2 and 1 p 2 if d 3

random geometric graphs we analyze graphs in which each vertex is assigned random coordinates in a geometric space of arbitrary dimensionality and only edges between adjacent points are present the critical connectivity is found numerically by examining the size of the largest cluster we derive an analytical expression for the cluster coefficient which shows that the graphs are distinctly different from standard random graphs even for infinite dimensionality insights relevant for graph bipartitioning are included

on scattering for the defocusing high dimensional inter critical nls abstract in this paper we study the critical norm conjecture for the inter critical nonlinear schrodinger equation with critical index s c satisfying 1 2 s c 1 when d 5 under the assumption of uniform boundedness of the critical norm we prove the global well posedness and scattering for the cauchy problem we follow the standard concentration compactness rigidity method established in 15 16 and treat three scenarios for the critical element respectively moreover double duhamel method and interaction morawetz estimate are applied to exclude the critical element

a new proof of scattering theory for the 3d radial focusing energy critical nls with combined terms abstract in this paper we revisit the nonlinear schrodinger equation i t u u u 2 u u 4 u in dimension three and give a simple proof of the scattering results for radial solutions under the energy threshold that avoids the concentrate compactness method the main new ingredient is a scattering criterion evacuation of the energy critical potential energy implies scattering which extends the scattering criterion of tao 18 to the energy critical case

the focusing energy critical nonlinear schrodinger equation in dimensions five and higher we consider the focusing energy critical nonlinear schr odinger equation iu t delta u u 4 over d 2 u in dimensions d geq 5 we prove that if a maximal lifespan solution u colon i times bbb r d to bbb c obeys sup t in i nabla u t 2 nabla w 2 then it is global and scatters both forward and backward in time here w denotes the ground state which is a stationary solution of the equation in particular if a solution has both energy and kinetic energy less than those of the ground state w at some point in time then the solution is global and scatters we also show that any solution that blows up with bounded kinetic energy must concentrate at least the kinetic energy of the ground state similar results were obtained by kenig and merle for spherically symmetric initial data and dimensions d 3 4 5

global well posedness and scattering for the defocusing l 2 critical nonlinear schrodinger equation when d 1 in this paper we prove global well posedness and scattering for the defocusing one dimensional mass critical nonlinear schr o dinger equation we make use of a long time strichartz estimate and a frequency localized morawetz estimate this continues work begun in earlier papers for dimensions d geq 3 and d 2

quadratic forms for the 1 d semilinear schrodinger equation this paper is concerned with 1 d quadratic semilinear schr6dinger equations we study local well posedness in classical sobolev space hs of the associated initial value problem and periodic boundary value problem our main interest is to obtain the lowest value of s which guarantees the desired local well posedness result we prove that at least for the quadratic cases these values are negative and depend on the structure of the nonlinearity considered

supercritical geometric optics for nonlinear schrodinger equations we consider the small time semi classical limit for nonlinear schrodinger equations with defocusing smooth nonlinearity for a super cubic nonlinearity the limiting system is not directly hyperbolic due to the presence of vacuum to overcome this issue we introduce new unknown functions which are defined nonlinearly in terms of the wave function itself this approach provides a local version of the modulated energy functional introduced by y brenier the system we obtain is hyperbolic symmetric and the justification of wkb analysis follows

global wellposedness of defocusing critical nonlinear schrodinger equation in the radial case is globally wellposed in time more precisely we obtain a unique solution u u ch1 0 such that for all time u t depends continuously on the data in fact the dependence is even real analytic here moreover there is scattering for t the same statement holds for radial data h s 1 and proves in particular global existence of classical solutions in the radially symmetric case also this issue was open thus this is the analogue for nls of the result for the wave equation with quintic nonlinearity obtained by m struwe str in the radial case and by m grillakis gr s s in general in the case of the wave equation the proof is based on the following two different facts

global existence for 2d nonlinear schrodinger equations via high low frequency decomposition method abstract we study global existence of solutions for the cauchy problem of the nonlinear schrodinger equation i u t u u 2 m u in the 2 dimension case where m is a positive integer m 2 using the high low frequency decomposition method we prove that if 10 m 6 10 m 5 s 1 then for any initial value h s r 2 the cauchy problem has a global solution in c r h s r 2 and it can be split into u t e i t y t with y c r h 1 r 2 satisfying y t h 1 1 t 2 1 s 10 m 5 s 10 m 6 where is an arbitrary sufficiently small positive number

global well posedness and scattering for the energy critical nonlinear schrodinger equation in r 3 we obtain global well posedness scattering and global l 10 spacetime bounds for energy class solutions to the quintic defocusing schrodinger equa tion in r 1 3 which is energy critical in particular this establishes global existence of classical solutions our work extends the results of bourgain 4 and grillakis 20 which handled the radial case the method is similar in spirit to the induction on energy strategy of bourgain 4 but we perform the induction analysis in both frequency space and physical space simultaneously and replace the morawetz inequality by an interaction variant first used in 12 13 the principal advantage of the interaction morawetz estimate is that it is not localized to the spatial origin and so is better able to handle nonradial solutions in particular this interaction estimate together with an almost conservation argument controlling the movement of l 2 mass in fre quency space rules out the possibility of energy concentration

scattering theory for nonlinear schrodinger equations with inverse square potential abstract we study the long time behavior of solutions to nonlinear schrodinger equations with some critical rough potential of a x 2 type the new ingredients are the interaction morawetz type inequalities and sobolev norm property associated with p a a x 2 we use such properties to obtain the scattering theory for the defocusing energy subcritical nonlinear schrodinger equation with inverse square potential in energy space h 1 r n

on the cauchy problem of 3 d energy critical schrodinger equations with subcritical perturbations abstract we investigate the global well posedness scattering and blow up phenomena when the 3 d quintic nonlinear schrodinger equation which is energy critical is perturbed by a subcritical nonlinearity 1 u p u we find when the quintic term is defocussing then the solution is always global no matter what the sign of 1 is scattering will occur either when the perturbation is defocussing and 4 3 p 4 or when the mass of the solution is small enough and 4 3 p 4 when the quintic term is focusing we show the blow up for certain solutions

low regularity global solutions for nonlinear evolution equations of kirchhoff type we study the global solvability of the cauchy dirichlet problem for two second order in time nonlinear integro differential equations 1 the extensible beam plate equation utt 2u m u 2dx u 0 x t r 2 a special case of the kirchhoff equation utt a b u 2dx 2 u 0 x t r by exploiting the i method of j colliander m keel g staffilani h takaoka t tao we prove that both equations admit global in time infinite energy solutions in case 1 the energy is the mechanical energy in case 2 it is a second order invariant introduced by s i pokhozhaev for the extensible beam equation 1 we also consider the effect of linear dissipation on such low regularity solutions and we prove their exponential decay as t

random rectangular graphs a generalization of the random geometric graph rgg model is proposed by considering a set of points uniformly and independently distributed on a rectangle of unit area instead of on a unit square 0 1 2 the topological properties of the random rectangular graphs rrgs generated by this model are then studied as a function of the rectangle sides lengths a and b 1 a and the radius r used to connect the nodes when a 1 we recover the rgg and when a the very elongated rectangle generated resembles a one dimensional rgg we obtain here analytical expressions for the average degree degree distribution connectivity average path length and clustering coefficient for rrg these results provide evidence that show that most of these properties depend on the connection radius and the side length of the rectangle usually in a monotonic way the clustering coefficient however increases when the square is transformed into a slightly elongated rectangle and after this maximum it decays with the increase of the elongation of the rectangle we support all our findings by computational simulations that show the goodness of the theoretical models proposed for rrgs

endpoint strichartz estimates we prove an abstract strichartz estimate which implies previously unknown endpoint strichartz estimates for the wave equation in dimension n 4 and the schrodinger equation in dimension n 3 three other applications are discussed local existence for a nonlinear wave equation and strichartz type estimates for more general dispersive equations and for the kinetic transport equation

semilinear schrodinger flows on hyperbolic spaces scattering in h 1 we prove global well posedness and scattering in h1 for the defocusing nonlinear schrodinger equations left begin array ll i partial t delta g u u u 2 sigma u 0 phi end array right on the hyperbolic spaces mathbb h d d 2 for exponents sigma in 0 2 d 2 the main unexpected conclusion is scattering to linear solutions in the case of small exponents for comparison on euclidean spaces scattering in h1 is not known for any exponent sigma in 1 d 2 d and is known to fail for sigma in 0 1 d our main ingredients are certain noneuclidean global in time strichartz estimates and noneuclidean morawetz inequalities

random spherical graphs this work addresses a modification of the random geometric graph rgg model by considering a set of points uniformly and independently distributed on the surface of a d 1 sphere with radius r in a d dimensional euclidean space instead of on an unit hypercube 0 1 d then two vertices are connected by a link if their great circle distance is at most s in the case of d 3 the topological properties of the random spherical graphs rsgs generated by this model are studied as a function of s we obtain analytical expressions for the average degree degree distribution connectivity average path length diameter and clustering coefficient for rsgs by setting r sqrt pi 2 pi we also show the differences between the topological properties of rsgs and those of two dimensional rggs and random rectangular graphs rrgs surprisingly in terms of the average clustering coefficient rsgs look more similar to the analytical estimation for rggs than rggs themselves when their boundary effects are considered we support all our findings by computer simulations that corroborate the accuracy of the theoretical models proposed for rsgs

connectivity distribution of spatial networks we study spatial networks constructed by randomly placing nodes on a manifold and joining two nodes with an edge whenever their distance is less than a certain cutoff we derive the general expression for the connectivity distribution of such networks as a functional of the distribution of the nodes we show that for regular spatial densities the corresponding spatial network has a connectivity distribution decreasing faster than an exponential in contrast we also show that scale free networks with a power law decreasing connectivity distribution are obtained when a certain information measure of the node distribution integral of higher powers of the distribution diverges we illustrate our results on a simple example for which we present simulation results finally we speculate on the role played by the limiting case p k proportional variant k 1 which appears empirically to be relevant to spatial networks of biological origin such as the ones constructed from gene expression data

bilinear estimates and applications to 2d nls the three bilinearities uv uv uv for functions u v r2 0 t 7 c are sharply estimated in function spaces xs b associated to the schrodinger operator i t these bilinear estimates imply local wellposedness results for schrodinger equations with quadratic nonlinearity improved bounds on the growth of spatial sobolev norms of finite energy global in time and blow up solutions of the cubic nonlinear schrodinger equation and certain generalizations are also obtained

quintic nls in the exterior of a strictly convex obstacle we consider the defocusing energy critical nonlinear schr odinger equation in the exterior of a smooth compact strictly convex obstacle in three dimensions for the initial value problem with dirichlet boundary condition we prove global well posedness and scattering for all initial data in the energy space

the dynamics of the 3d radial nls with the combined terms in this paper we show the scattering and blow up result of the radial solution with the energy below the threshold for the nonlinear schrodinger equation nls with the combined terms iu t delta u u 4 u u 2 u qquad qquad qquad qquad rm cnls in the energy space h 1 mathbb r 3 the threshold is given by the ground state w for the energy critical nls iu t u u 4 u this problem was proposed by tao visan and zhang in comm pdes 32 1281 1343 2007 the main difficulty is lack of the scaling invariance illuminated by ibrahim et al in analysis pde 4 3 405 460 2011 we need to give the new radial profile decomposition with the scaling parameter then apply it to the scattering theory our result shows that the defocusing dot h 1 subcritical perturbation u 2 u does not affect the determination of the threshold of the scattering solution of cnls in the energy space

a sharp condition for scattering of the radial 3d cubic nonlinear schrodinger equation we consider the problem of identifying sharp criteria under which radial h 1 finite energy solutions to the focusing 3d cubic nonlinear schrodinger equation nls i t u u u 2 u 0 scatter i e approach the solution to a linear schrodinger equation as t the criteria is expressed in terms of the scale invariant quantities u 0 l 2 nabla u 0 l 2 and m u e u where u 0 denotes the initial data and m u and e u denote the conserved in time mass and energy of the corresponding solution u t the focusing nls possesses a soliton solution e it q x where q is the ground state solution to a nonlinear elliptic equation and we prove that if m u e u q l 2 nabla q l 2 then the solution blows up in finite time the technique employed is parallel to that employed by kenig merle 17 in their study of the energy critical nls

random geometric graphs with general connection functions in the original 1961 gilbert model of random geometric graphs nodes are placed according to a poisson point process and links formed between those within a fixed range motivated by wireless ad hoc networks soft or probabilistic connection models have recently been introduced involving a connection function h r that gives the probability that two nodes at distance r are linked directly connect in many applications not only wireless networks it is desirable that the graph is connected that is every node is linked to every other node in a multihop fashion here the connection probability of a dense network in a convex domain in two or three dimensions is expressed in terms of contributions from boundary components for a very general class of connection functions it turns out that only a few quantities such as moments of the connection function appear good agreement is found with special cases from previous studies and with numerical simulations

dispersion of small amplitude solutions of the generalized korteweg de vries equation abstract we study the long time behavior of small solutions of the initial value problem for the generalized korteweg de vries equation t u x 3 u x f u 0 gkdv u x 0 g x for the case where f w w s with s 1 4 23 57 3 8625 our results imply that if g l11 g l22 is sufficiently small then sup r 1 t 1 3 u t l in particular the solution tends to zero in the supremum norm the proofs make use of duhamel s formula and dispersion estimates for the linear propagator as well as chain and leibniz rules for fractional derivatives of compositions d f u lp and products d fg lp 0

on the interpolation with the potential bound for global solutions of the defocusing cubic wave equation on t2 abstract consider the solutions of the defocusing cubic wave equation t t u u u 3 with real data in h s t 2 h s 1 t 2 s 2 5 we prove that the solutions exist globally in time by contradiction assuming that one of the maximal times of existence is finite we prove that the sobolev norm of each of these solutions is bounded in an open neighborhood of it by using the i method see e g 4 it consists of introducing a mollified energy and estimating its growth in this neighborhood the growth is first estimated on small subintervals by means of dispersive bounds then it is estimated on the whole interval by an iteration process in order to enlarge the size of subintervals on which the dispersive bounds hold we interpolate with the potential bound for the low frequency part our results complement those obtained in 1

multilinear weighted convolution of l 2 functions and applications to nonlinear dispersive equations the x s b spaces as used by beals bourgain kenig ponce vega klainerman machedon and others are fundamental tools to study the low regularity behavior of nonlinear dispersive equations it is of particular interest to obtain bilinear or multilinear estimates involving these spaces by plancherel s theorem and duality these estimates reduce to estimating a weighted convolution integral in terms of the l 2 norms of the component functions in this paper we systematically study weighted convolution estimates on l 2 as a consequence we obtain sharp bilinear estimates for the kdv wave and schrodinger x s b spaces

smoothing properties and retarded estimates for some dispersive evolution equations smoothing properties in the form of space time integrability properties play an important role in the study of dispersive evolution equations a number of them follow from a combination of general arguments and specific estimates we present a general formulation which makes the separation between the two types of ingredients as clear as possible and we illustrate it with the examples of the schrodinger equation of the wave equation and of a class of 1 1 dimensional equations related to the benjamin ono equation of special interest for the cauchy problem are retarded estimates expressed in terms of those properties we derive a number of such estimates associated with the last example and we mention briefly an application of those estimates to the cauchy problem for the generalized benjamin ono equation

global existence and scattering for rough solutions of a nonlinear schr odinger equation on r 3 we prove global existence and scattering for the defocusing cubic nonlinear schrodinger equation in h s r 3 for s 4 the main new estimate in the argument is a morawetz type inequality for the solution this estimate bounds k x t k l 4 r3 r whereas the well known morawetz type estimate of lin strauss controls r 1 0 r r3 x t 4 x dxdt

degree correlations in random geometric graphs spatially embedded networks are important in several disciplines the prototypical spatial network we assume is the random geometric graph of which many properties are known here we present new results for the two point degree correlation function in terms of the clustering coefficient of the graphs for two dimensional space in particular with extensions to arbitrary finite dimensions

global well posedness and scattering for the defocusing hartree equation in r3 abstract we consider the defocusing h 1 2 subcritical hartree equation i t u u x u 2 u 2 3 in r 3 we prove its global well posedness and scattering property in h s r 3 for s 7 2 2 when 5 2 s 3 2 2 1 when 5 2 this improves the previous work of miao xu and zhao 2009 18 a new ingredient used here is that we apply some bilinear strichartz estimates

global well posedness for the nonlinear wave equation with a cubic nonlinearity in two space dimensions abstract in this paper we prove the global well posedness for the nonlinear wave equation with a cubic nonlinearity u t t u u 3 in two space dimensions with initial data u 0 u 1 h x s h x s 1 1 s 5 11 0 4545

global well posedness and scattering for the mass critical nonlinear schrodinger equation with mass below the mass of the ground state abstract in this paper we prove that the focusing d dimensional mass critical nonlinear schrodinger initial value problem is globally well posed and scattering for u 0 l 2 r d u 0 l 2 r d q l 2 r d where q is the ground state and d 1 we first establish an interaction morawetz estimate that is positive definite when u 0 l 2 r d q l 2 r d and has the appropriate scaling next we will prove an interaction morawetz estimate localized to low frequencies similar to the estimates made in 13 14 15 see also 12 for localization to high frequencies in the energy critical case

the defocusing energy supercritical nls in four space dimensions abstract we consider a class of defocusing energy supercritical nonlinear schrodinger equations in four space dimensions following a concentration compactness approach we show that for 1 s c 3 2 any solution that remains bounded in the critical sobolev space h x s c r 4 must be global and scatter key ingredients in the proof include a long time strichartz estimate and a frequency localized interaction morawetz inequality

global existence for semilinear schrodinger equations in 2 1 dimensions abstract this paper is concerned with global well posedness of the 2 dimensional defocusing semilinear schrodinger equation i u t u u 2 m u in the sobolev space h s r 2 in a previous work of guo and cui c guo s cui global existence for 2d nonlinear schrodinger equations via high low frequency decomposition method j math anal appl 324 2006 882 907 it was proved that global well posedness holds in h s r 2 for s 10 m 6 10 m 5 that result is obtained by using the high low frequency decomposition method in this paper we apply the i method to improve that result and prove that global well posedness holds in h s r 2 for s 1 5 17 4 m

a new approach to the discretization of continuous time controllers a new method for discretizing continuous time controllers is derived using principles of controller approximation it focuses on the closed loop use of the discrete time controller the resulting approximation criterion is a measure for stability of the control system provides an upper bound on the sampling time for which stability can be guaranteed and because it is based on continuous time controller approximation it indicates the cost of discretization in terms of performance degradation the discrete time controller is obtained through minimization of this criterion which can be performed with standard software used in h sub infinity controller design

global well posedness and scattering for the defocusing energy critical nonlinear schrodinger equation in r1 4 we obtain global well posedness scattering uniform regularity and global l6t x spacetime bounds for energy space solutions to the defocusing energy critical nonlinear schrodinger equation in a 4 our arguments closely follow those of colliander keel et al though our derivation of the frequency localized interaction morawetz estimate is somewhat simpler as a consequence our method yields a better bound on the l6t x norm

random geometric graphs in high dimension many machine learning algorithms used for dimensional reduction and manifold learning leverage on the computation of the nearest neighbors to each point of a data set to perform their tasks these proximity relations define a so called geometric graph where two nodes are linked if they are sufficiently close to each other random geometric graphs where the positions of nodes are randomly generated in a subset of r d offer a null model to study typical properties of data sets and of machine learning algorithms up to now most of the literature focused on the characterization of low dimensional random geometric graphs whereas typical data sets of interest in machine learning live in high dimensional spaces d 10 2 in this work we consider the infinite dimensions limit of hard and soft random geometric graphs and we show how to compute the average number of subgraphs of given finite size k e g the average number of k cliques this analysis highlights that local observables display different behaviors depending on the chosen ensemble soft random geometric graphs with continuous activation functions converge to the naive infinite dimensional limit provided by erdos renyi graphs whereas hard random geometric graphs can show systematic deviations from it we present numerical evidence that our analytical results exact in infinite dimensions provide a good approximation also for dimension d 10

comments about frequency response plots of keller and anderson and rattan it is noticed that the method of comparison of frequency responses by using the nyquist plots of the closed loop system spectral transfer functions applied in keller anderson ibid vol 37 no 2 p 214 23 1992 and rattan ibid vol ac 29 p 282 5 1984 is not appropriate namely not only the shape of these plots but also the frequency distribution on them must be taken into account it is shown that significantly more legible is utilization of the bode plots of the control index presented by findeison 1978 which has a good physical insight

global regularity of wave maps ii small energy in two dimensions we show that wave maps from minkowski space 1 n to a sphere sm 1 are globally smooth if the initial data is smooth and has small norm in the critical sobolev space in all dimensions n 5 this generalizes the results in the prequel 40 of this paper which addressed the high dimensional case n 5 in particular in two dimensions we have global regularity whenever the energy is small and global regularity for large data is thus reduced to demonstrating non concentration of energy

scattering in the energy space for the nls with variable coefficients we consider the nls with variable coefficients in dimension n ge 3 begin aligned i partial t u lu f u 0 qquad lv nabla b cdot a x nabla b v c x v qquad nabla b nabla ib x end aligned on mathbb r n or more generally on an exterior domain with dirichlet boundary conditions for a gauge invariant defocusing nonlinearity of power type f u simeq u gamma 1 u we assume that l is a small long range perturbation of delta plus a potential with a large positive part the first main result of the paper is a bilinear smoothing interaction morawetz estimate for the solution as an application under the conditional assumption that strichartz estimates are valid for the linear flow e itl we prove global well posedness in the energy space for subcritical powers gamma 1 frac 4 n when the domain is mathbb r n by extending the strichartz estimates due to tataru am j math 130 3 571 634 2008 we prove that the conditional assumption is satisfied and deduce well posedness and scattering in the energy space

analyticity of the scattering operator for semilinear dispersive equations we present a general algorithm to show that a scattering operator associated to a semilinear dispersive equation is real analytic and to compute the coefficients of its taylor series at any point we illustrate this method in the case of the schrodinger equation with power like nonlinearity or with hartree type nonlinearity and in the case of the wave and klein gordon equations with power nonlinearity finally we discuss the link of this approach with inverse scattering and with complete integrability

existence of solutions for schrodinger evolution equations we study the existence uniqueness and regularity of the solution of the initial value problem for the time dependent schrodinger equationi u t 1 2 u v t x u u 0 u0 we provide sufficient conditions onv t x such that the equation generates a unique unitary propagatoru t s and such thatu t s u0ec1 l2 c0 h2 n foru0eh2 n the conditions are general enough to accommodate moving singularities of type x 2 n 4 or x n 2 n 3

impact of boundaries on fully connected random geometric networks many complex networks exhibit a percolation transition involving a macroscopic connected component with universal features largely independent of the microscopic model and the macroscopic domain geometry in contrast we show that the transition to full connectivity is strongly influenced by details of the boundary but observe an alternative form of universality our approach correctly distinguishes connectivity properties of networks in domains with equal bulk contributions it also facilitates system design to promote or avoid full connectivity for diverse geometries in arbitrary dimension

scattering for h 1 2 bounded solutions to the cubic defocusing nls in 3 dimensions we show that if a solution of the defocusing cubic nls in 3d remains bounded in the homogeneous sobolev norm of order 1 2 in its maximal interval of existence then the interval is infinite and the solution scatters no radial assumption is made

global existence for some radial low regularity nonlinear schrodinger equations abstract we prove the nonlinear schrodinger equation has a local solution for any energy subcritical nonlinearity when u 0 is the characteristic function of a ball in r n additionally we establish the existence of a global solution for n 3 when 2 n 1 4 n 2 and 2 finally we establish the existence of a global solution when the initial function is radial the nonlinear schrodinger equation has an energy subcritical nonlinearity and the initial function lies in h r n h 1 2 r n h 1 2 1 r n