on the neumann problem for pde s with a small parameter and the corresponding diffusion processes the diffusion process in a region g subset mathbb r 2 governed by the operator tilde l varepsilon frac 1 2 u xx frac1 2 varepsilon u zz inside the region and undergoing instantaneous co normal reflection at the boundary is considered we show that the slow component of this process converges to a diffusion process on a certain graph corresponding to the problem this allows to find the main term of the asymptotics for the solution of the corresponding neumann problem in g the operator tilde l varepsilon is up to the factor e 1 the result of small perturbation of the operator frac 1 2 u zz our approach works for other operators diffusion processes in any dimension if the process corresponding to the non perturbed operator has a first integral and the e process is non degenerate on non singular level sets of this first integral

divergent square averages in this paper we answer a question of j bourgain which was motivated by questions a bellow and h furstenberg we show that the sequencefn 2 g 1d1 is l 1 universally bad this implies that it is not true that given a dynamical system x u t andf 2l 1 the ergodic means

spectra of graph neighborhoods and scattering let g e e 0 be a family of e thin riemannian manifolds modeled on a finite metric graph g for example the e neighborhood of an embedding of g in some euclidean space with straight edges we study the asymptotic behavior of the spectrum of the laplace beltrami operator on g e as e 0 for various boundary conditions we obtain complete asymptotic expansions for the kth eigenvalue and the eigenfunctions uniformly for k ce 1 in terms of scattering data on a non compact limit space we then use this to determine the quantum graph which is to be regarded as the limit object in a spectral sense of the family g e our method is a direct construction of approximate eigenfunctions from the scattering and graph data and the use of a priori estimates to show that all eigenfunctions are obtained in this way

on ergodic diffusions on continuous graphs whose centered resolvent admits a trace abstract we consider ergodic and reversible diffusions on continuous and connected graphs g with a finite number of bifurcation vertices and some rays going to infinity a necessary and sufficient condition is presented for the spectrum of the associated generator l to be without continuous part and for the sum of the inverses of its eigenvalues except 0 to be finite this criterion is easily computable in terms of the coefficients of l and does not depend on the transition kernels at the vertices its motivation is that it is conjectured to be also a necessary and sufficient condition for the diffusion to admit strong stationary times whatever its initial distribution this is known to be true if g is the real line the above criterion for the centered resolvent to be of trace class is next extended to markov processes on denumerable connected graphs with only a finite number of vertices of degree larger than or equal to 3

averaging principle for second order approximation of heterogeneous models with homogeneous models typically models with a heterogeneous property are considerably harder to analyze than the corresponding homogeneous models in which the heterogeneous property is replaced by its average value in this study we show that any outcome of a heterogeneous model that satisfies the two properties of differentiability and symmetry is o 2 equivalent to the outcome of the corresponding homogeneous model where is the level of heterogeneity we then use this averaging principle to obtain new results in queuing theory game theory auctions and social networks marketing

snakes and spiders brownian motion on ℝ trees we consider diffusion processes on a class of trees the processes are defined in a manner similar to that of le gall s brownian snake each point in the tree has a real valued height or generation and the height of the diffusion process evolves as a brownian motion when the height process decreases the diffusion retreats back along a lineage whereas when the height process increases the diffusion chooses among branching lineages according to relative weights given by a possibly infinite measure on the family of lineages the class of trees we consider can have branch points with countably infinite branching and lineages along which the branch points have points of accumulation

convergence of spectra of uniformly fattened open book structures abstract we consider a stratified compact surface m in r 3 which we call an open book structure whose singular points form simple closed smooth curves and its neighborhood m which we call a fattened open book structure assuming a transversality condition at the singularities and absence of zero dimensional strata i e corners we show that the discrete spectrum of the neumann laplacian in m converges when 0 to the spectrum of a differential operator on m similar results have been obtained before for the case of fattened graphs i e m being one dimensional in the case of a smooth surfaces m the problem has been studied well however having singularities along strata of lower dimensions significantly complicates considerations as in the quantum graph case such considerations are triggered by various applications

distributed self localization for relative position sensing networks in 2d space this paper studies the 2d localization problem of a sensor network given anchor node positions in a common global coordinate frame and relative position measurements in local coordinate frames between node pairs it is assumed that the local coordinate frames of different sensors have different orientations and the orientation difference with respect to the global coordinate frame are not known in terms of graph connectivity a necessary and sufficient condition is obtained for self localizability that leads to a fully distributed localization algorithm moreover a distributed verification algorithm is developed to check the graph connectivity condition which can terminate successfully when the sensor network is self localizable finally a fully distributed linear and iterative algorithm based on the complex valued laplacian associated with the sensor network is proposed which converges globally and gives the correct localization result

exact solution of a linear molecular motor model driven by two step fluctuations and subject to protein friction we investigate by analytical means the stochastic equations of motion of a linear molecular motor model based on the concept of protein friction solving the coupled langevin equations originally proposed by mogilner et al phys lett a 237 297 1998 and averaging over both the two step internal conformational fluctuations and the thermal noise we present explicit analytical expressions for the average motion and the velocity force relationship our results allow for a direct interpretation of details of this motor model which are not readily accessible from numerical solutions in particular we find that the model is able to predict physiologically reasonable values for the load free motor velocity and the motor mobility

stochastic averaging of energy envelope of preisach hysteretic systems the method of stochastic averaging of energy envelope is applied to predict the response of bilinear hysteretic systems to white noise excitation the statistics of stationary response for displacement velocity and energy envelope as functions of excitation level secondary slope and viscous damping are analyzed in detail comparisons with digital simulation show that the method is superior to other approximate methods have been applied to random response of hysteretic systems

random perturbations of dynamical systems and diffusion processes with conservation laws in this paper we consider random perturbations of dynamical systems and diffusion processes with a first integral we calculate under some assumptions the limiting behavior of the slow component of the perturbed system in an appropriate time scale for a general class of perturbations the phase space of the slow motion is a graph defined by the first integral this is a natural generalization of the results concerning random perturbations of hamiltonian systems considering diffusion processes as the unperturbed system allows to study the multidimensional case and leads to a new effect the limiting slow motion can spend non zero time at some points of the graph in particular such delay at the vertices leads to more general gluing conditions our approach allows one to obtain new results on singular perturbations of pde s

skew cir process conditional characteristic function moments and bond pricing abstract this paper is concerned with one general feller s branching diffusion called skew cir process we derive the conditional characteristic function and moment of this general diffusion process first then with the same computing idea we handle with its application in bond pricing all the results we adopt are closed forms

quantum graphs as holonomic constraints we consider the dynamics on a quantum graph as the limit of the dynamics generated by a one particle hamiltonian in r2 with a potential having a deep strict minimum on the graph when the width of the well shrinks to zero for a generic graph we prove convergence outside the vertices to the free dynamics on the edges for a simple model of a graph with two edges and one vertex we prove convergence of the dynamics to the one generated by the laplacian with dirichlet boundary conditions in the vertex

stationary mean field games systems defined on networks we consider a stationary mean field games system constrained on a network according to the optimal control interpretation of the problem some transition conditions on the vertices are imposed we prove separately the well posedness for each of the two equations composing the system finally we prove existence and uniqueness of the solution of the mean field games system

spectral density analysis of nonlinear hysteretic systems a method for the analysis of spectral densities of hysteretic nonlinearities driven by diffusion processes is presented this method is based on the preisach formalism for the description of hysteresis and the mathematical machinery of diffusion processes on graphs the calculations are appreciably simplified by the introduction of the effective distribution function the implementation of the method for the case of the ornstein uhlenbeck input process is presented in detail and analytical expressions for spectral noise densities for various hysteretic systems are obtained the general qualitative features of these spectral densities are examined and their dependence on various parameters is discussed because of the universality of the preisach model this approach can be used to compute spectra in hysteresis nonlinearities of various physical origins

distributed localization with mixed measurements under switching topologies abstract this paper investigates the distributed localization problem of sensor networks with mixed measurements each node holds a local coordinate system without a common orientation and is capable of measuring only one type of information either distance bearing or relative position to near by nodes thus three types of measurements are mixed in the sensor networks moreover the communication topologies in the sensor networks may be time varying due to unreliable communications this paper develops a fully distributed algorithm called bcdl barycentric coordinate based distributed localization where each node starts from a random initial guess about its true coordinate and converges to the true coordinate via only local node interactions the key idea in bcdl is to establish a unified linear equation constraints for the sensor coordinates by using the barycentric coordinates of each node with respect to its neighbors though the sensor nodes may have different types of measurements then a distributed iterative algorithm is proposed to solve the linear equations under time varying communication networks a necessary and sufficient graphical condition is obtained to ensure global convergence of the distributed algorithm

random response of preisach hysteretic systems an approximate method for analyzing the response of preisach hysteretic systems with non local memory under stationary gaussian excitation is proposed the covariance matrix equation of system response is derived the cross correlation function of preisach hysteretic force and response in the covariance equation is evaluated based on the switching probability analysis and the gaussian approximation of response process and an explicit expression for the cross correlation function is given for the case of symmetric preisach weighting function it is shown that the numerical result obtained by using the proposed method is in good agreement with that from digital simulation

mathematical models for hysteresis the various existing classical models for hysteresis preisach ishlinskii duhem madelung are surveyed as well a more modern treatments by contemporary workers the emphasis is on a clear mathematical description of the formulation and properties of each model in addition the authors try to make the reader aware of the many open questions in the study of hysteresis

distributed localization for 2 d sensor networks with bearing only measurements under switching topologies this paper investigates the problem of bearing measurement based distributed localization for sensor networks that are undirected and switching each node holds a local coordinate system with no knowledge about the global coordinate system and measures the bearing angle information about its neighbors in its local coordinate system a novel scheme for localization is developed using a complex laplacian to overcome the challenges due to the absence of a global coordinate system and the presence of topology switching in communication first by using bearing only measurements an algorithm is proposed to establish linear equation constraints for the coordinates of sensor nodes in the global coordinate frame the main idea is that each node uses its own bearing and its neighbors bearing information to construct a similar configuration though it is not able to recover the true configuration by using only bearing measurements second a distributed iterative algorithm is proposed such that all the sensor nodes can cooperatively find the true coordinates of themselves it is shown that the algorithm exponentially converges provided that the communication network jointly satisfies certain connectivity properties the simulation results validate our proposed algorithm

reaction diffusion on metric graphs from 3d to 1d abstract we provide stochastic foundations for the analysis of a class of reaction diffusion systems using as an example the known temporal analysis of products tap experiments showing how to effectively obtain explicit solutions to the associated equations by approximating the 3 dimensional domain of diffusion u the reactor by 1 dimensional network models in a typical tap experiment a pulse of reactant gas of species a is injected into u which is filled with chemically inert material permeable to gas diffusion particles of catalyst are placed amid this inert medium forming active sites where the reaction a b may occur on part of the boundary of u designated as the exit a mixture of a and b can escape we study the problem of determining the molar fraction of product gas in the mixture after u is fully evacuated this fraction is identified with the reaction probability that is the probability of a single diffusing molecule reacting before leaving u specifically we are interested in how this probability depends on the reaction rate constant k after giving a stochastic formulation of the problem and the boundary value problem whose solution is this probability we study a class of domains called fat graphs comprising a network of thin tubes with active sites at junctures the main result of the paper is that in the limit as the thin tubes approach curves in 3 dimensional space reaction probability converges to functions of the point of gas injection that can be computed explicitly in terms of a rescaled parameter k by this 3d to 1d reduction the simpler processes on metric graphs can be used as model systems for more realistic 3 dimensional configurations this is illustrated with analytic and numerical examples one example in particular illustrates an important application of our method finding the catalyst configuration that maximizes reaction probability at a given reaction rate constant

a distributed continuous time algorithm for network localization using angle of arrival information this paper studies the angle of arrival aoa localization problem namely localizing networks based on the angle of arrival measurements between certain neighboring network nodes together with the absolute locations of some anchor nodes we propose the concepts of stiffness matrix and fixability for the anchored formation graphs modeling the networks and show that they provide a complete characterization of the aoa localizability as well as an explicit formula for the localization result moreover a distributed continuous time algorithm is proposed that converges globally to the correct localization result on fixable formation graphs performances of the proposed algorithm e g convergence rate and robustness to communication delay are characterized and optimized sensitivities of the localization results with respect to errors in aoa measurements and anchor node positions are also analyzed

the vanishing viscosity limit for hamilton jacobi equations on networks abstract for a hamilton jacobi equation defined on a network we introduce its vanishing viscosity approximation the elliptic equation is given on the edges and coupled with kirchhoff type conditions at the transition vertices we prove that there exists exactly one solution of this elliptic approximation and mainly that as the viscosity vanishes it converges to the unique solution of the original problem

restricted weak type endpoint estimates for k spherical maximal functions in this paper we use the approximation formula for the fourier transform of the solution set of lattice points on k spheres and methods of bourgain and ionescu to refine the ell p mathbb z d boundedness results for discrete k spherical maximal functions to a restricted weak type result at the endpoint we introduce a density parameter which may be viewed as a discrete version of minkowski dimension used in related works on the continuous analgoue in order to exploit recent progress of wooley and bourgain demeter guth on the vinogradov mean value conjectures via a novel approximation formula for a single average and obtain improved bounds for lacunary discrete k spherical maximal functions when k ge 3

discrete analogues in harmonic analysis spherical averages in this paper we prove an analogue in the discrete setting of d o fthe spherical maximal theorem for d the methods used are two fold the application of certain sampling techniques and ideas arising in the study of the number of representations of an integer as a sum of d squares in particular the circle method the results we obtained are by necessity limited to d 5 and moreover the range of p for the l p estimates differs from its analogue in d

scattering solutions in networks of thin fibers small diameter asymptotics small diameter asymptotics is obtained for scattering solutions in a network of thin fibers the asymptotics is expressed in terms of solutions of related problems on the limiting quantum graph we calculate the lagrangian gluing conditions at vertices v in gamma for the problems on the limiting graph if the frequency of the incident wave is above the bottom of the absolutely continuous spectrum the gluing conditions are formulated in terms of the scattering data for each individual junction of the network

robin type boundary conditions in transition from reaction diffusion equations in 3d domains to equations in 2d domains abstract we consider a singular limit of diffusion equations in 3d domains of thickness converging to zero in the 2d limit the resulting reaction diffusion equation has a source term resulting from the robin type boundary conditions imposed on boundaries of the original 3d domain the proposed approach can be applied to constructing approximate solutions of diffusion problems in thin planar cylindrical or spherical layers between two membranes as an example we refer to the problem of activation of b lymphocytes which typically have large nuclei and a thin cytoplasmic layer which can be considered as a spherical shell for this example assuming additionally axial symmetry we provide a rigorous convergence theorem in the language of semigroups of operators

convergence of spectra of mesoscopic systems collapsing onto a graph abstract let m be a finite graph in the plane and let me be a domain that looks like the e fattened graph m exact conditions on the domain are given it is shown that the spectrum of the neumann laplacian on me converges when e 0 to the spectrum of an ode problem on m the presence of an electromagnetic field is also allowed considerations of this kind arise naturally in mesoscopic physics and other areas of physics and chemistry the results of the paper extend the ones previously obtained by j rubinstein and m schatzman

diophantine equations and ergodic theorems let x be a probability measure space and t 1 t n be a family of commuting measure preserving invertible transformations on x let q m 1 m n be a homogeneous positive polynomial with integer coefficients and let s m e z n q m a denote the set of integer solutions m m 1 m n of the diophantine equation q m a we prove that under a certain nondegeneracy condition on the polynomial q m and an ergodic condition on the family of transformations t t 1 t n the images of the solution sets q x t m1 1 t m2 2 t m n n x m s become uniformly distributed on x w r t for a e x e x as that is the pointwise ergodic theorem holds when the standard averages are replaced by the ones where the exponents satisfy a diophantine equation the proof uses a variant of the hardy littlewood method of exponential sums developed by birch and davenport and techniques from harmonic analysis a key point is the corresponding maximal theorem which is a discrete analogue of a maximal theorem on r n corresponding to the level surfaces of the polynomial q x

reprint of a distributed algorithm for efficiently solving linear equations and its applications special issue jcw abstract a distributed algorithm is proposed for solving a linear algebraic equation a x b over a multi agent network where a r n n and the equation has a unique solution x r n each agent knows only a subset of the rows of a b controls a state vector x i t of size smaller than n and is able to receive information from its nearby neighbors neighbor relations are characterized by time dependent directed graphs it is shown that for a large class of time varying networks the proposed algorithm enables each agent to recursively update its own state by only using its neighbors states such that all x i t converge exponentially fast to a specific part of x of interest to agent i applications of the proposed algorithm include solving the least square solution problem and the network localization problem

singular perturbations involving fast diffusion abstract we study a couple of examples of pure and applied mathematics where interesting singular limits are obtained when diffusion s speed increases to infinity these include the recent models of kinase activity 17 29 and of early lung cancer 37 38 and the example of khasminskii 32

diffusion processes in thin tubes and their limits on graphs the present paper is concerned with diffusion processes running on tubular domains with conditions on nonreaching the boundary respectively reflecting at the boundary and corresponding processes in the limit where the thin tubular domains are shrinking to graphs the methods we use are probabilistic ones for shrinking we use big potentials respectively reflection on the boundary of tubes we show that there exists a unique limit process and we characterize the limit process by a second order differential generator acting on functions defined on the limit graph with kirchhoff boundary conditions at the vertices

on occurrence of spectral edges for periodic operators inside the brillouin zone the paper discusses the following frequently arising question on the spectral structure of periodic operators of mathematical physics e g schrodinger maxwell waveguide operators etc is it true that one can obtain the correct spectrum by using the values of the quasimomentum running over the boundary of the reduced brillouin zone only rather than the whole zone or do the edges of the spectrum occur necessarily at the set of corner high symmetry points this is known to be true in 1d while no apparent reasons exist for this to be happening in higher dimensions in many practical cases though this appears to be correct which sometimes leads to the claims that this is always true there seems to be no definite answer in the literature and one encounters different opinions about this problem in the community in this paper starting with simple discrete graph operators we construct a variety of convincing multiply periodic examples showing that the spectral edges might occur deeply inside the brillouin zone on the other hand it is also shown that in a generic case the situation of spectral edges appearing at high symmetry points is stable under small perturbations this explains to some degree why in many maybe even most practical cases the statement still holds