the spectrum of a magnetic schrodinger operator with randomly located delta impurities we consider a single band approximation to the random schrodinger operator in an external magnetic field the spectrum of such an operator has been characterized in the case where delta impurities are located on the sites of a lattice in this paper we generalize these results by letting the delta impurities have random positions as well as strengths they are located in squares of a lattice with a general bounded distribution we characterize the entire spectrum of this operator when the magnetic field is sufficiently high we show that the whole spectrum is pure point the energy coinciding with the first landau level is infinitely degenerate and that the eigenfunctions corresponding to other landau band energies are exponentially localized

decay rate of the green function in a random potential on the bethe lattice and a criterion for localization it is shown that in the tight binding anderson model on the bethe lattice the exponential decay rate of the green function can be obtained for arbitrary energies and arbitrary disorder analytical results in the case of the lorentzian distribution of site energies are presented as an application of these results a criterion for the localized region on the corresponding regular lattice on the level of the bethe peierls approximation is proposed our criterion yields exact results in the one dimensional limit and yields correct band edges for the hypercubic lattice in the vanishing limit of disorder the mobility edge trajectory obtained by our criterion is given by an elliptic curve in the case of the lorentzian distribution and its shape is found to be in qualitatively good agreement with that obtained by the finite size scaling method in the three dimensional system

moment analysis for localization in random schrodinger operators we study localization effects of disorder on the spectral and dynamical properties of schrodinger operators with random potentials the new results include exponentially decaying bounds on the transition amplitude and related projection kernels including in the mean these are derived through the analysis of fractional moments of the resolvent which are finite due to the resonance diffusing effects of the disorder the main difficulty which has up to now prevented an extension of this method to the continuum can be traced to the lack of a uniform bound on the lifshitz krein spectral shift associated with the local potential terms the difficulty is avoided here through the use of a weak l 1 estimate concerning the boundary value distribution of resolvents of maximally dissipative operators combined with standard tools of relative compactness theory

periodic solutions of some infinite dimensional hamiltonian systems associated with non linear partial difference equations ii this is our second paper devoted to the study of some non linear schrodinger equations with random potential we study the non linear eigenvalue problems corresponding to these equations we exhibit a countable family of eigenfunctions corresponding to simple eigenvalues densely embedded in the band tails contrary to our results in the first paper the results established in the present paper hold for an arbitrary strength of the non linear cubic term in the non linear schrodinger equation

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

real space renormalization group and totalitarian paradox of majority rule voting abstract the effect of majority rule voting in hierarchical structures is studied using the basic concepts from real space renormalization group it shows in particular that a huge majority can be self eliminated while climbing up the hierarchy levels this majority democratic self elimination articulates around the existence of fixed points in the voting flow an unstable fixed point determines the critical threshold to full and total power it can be varied from 50 up to 77 of initial support our model could shed new light on the last century eastern european communist collapse

impact of critical mass on the evolution of cooperation in spatial public goods games we study the evolution of cooperation under the assumption that the collective benefits of group membership can only be harvested if the fraction of cooperators within the group i e their critical mass exceeds a threshold value considering structured populations we show that a moderate fraction of cooperators can prevail even at very low multiplication factors if the critical mass is minimal for larger multiplication factors however the level of cooperation is highest at an intermediate value of the critical mass the latter is robust to variations of the group size and the interaction network topology applying the optimal critical mass threshold we show that the fraction of cooperators in public goods games is significantly larger than in the traditional linear model where the produced public good is proportional to the fraction of cooperators within the group

localization length of stationary states in the nonlinear schrodinger equation for the nonlinear schrodinger equation nlse in the presence of disorder exponentially localized stationary states are found we demonstrate analytically that the localization length is typically independent of the strength of the nonlinearity and is identical to the one found for the corresponding linear equation the analysis makes use of the correspondence between the stationary nlse and the langevin equation as well as of the resulting fokker planck equation the calculations are performed for the white noise random potential and an exact expression for the exponential growth of the eigenstates is obtained analytically it is argued that the main conclusions are robust

anderson model with decaying randomness mobility edge in this paper we consider the anderson model with decaying randomness a nq omega n a n 0 n in mathbb z nu and q omega n i i d random variables with an absolutely continuous distribution mu for a class of mu we show the following results on a set omega of full measure

local interaction scale controls the existence of a nontrivial optimal critical mass in opinion spreading we study a model of opinion formation where the collective decision of a group is said to happen if the fraction of agents having the most common opinion exceeds a threshold value a critical mass we find that there exists a unique nontrivial critical mass giving the most efficient convergence to consensus in addition we observe that for small critical masses the characteristic time scale for the relaxation to consensus splits into two the shorter time scale corresponds to a direct relaxation and the longer one can be explained by the existence of intermediate metastable states similar to those found in p chen and s redner phys rev e 71 036101 2005 this longer time scale is dependent on the precise condition for consensus with a modification of the condition it can go away

improved impedance matrix localization method em problems moment method calculations have the well known limitations of requiring excessive storage and execution times for even modestly large electromagnetics problems the impedance matrix localization iml method was introduced as a modification to standard moment method calculations to ease these limitations it utilizes a matrix transformation which effectively changes the basis testing functions into ones resembling traveling waves an improved method that uses an orthogonal transformation to generate standing wave like basis functions is presented here remarkable improvements are achieved in the numerical stability of the method and in its compatibility with iterative solvers furthermore the correspondence of the large elements in this matrix to geometrical theory of diffraction gtd terms is strengthened as is the possibility of further increasing the speed of iterative solutions by constructing preconditioners based on the pattern of nonzero matrix elements

nonlinear q voter model we introduce a nonlinear variant of the voter model the q voter model in which q neighbors with possible repetition are consulted for a voter to change opinion if the q neighbors agree the voter takes their opinion if they do not have a unanimous opinion still a voter can flip its state with probability epsilon we solve the model on a fully connected network i e in mean field and compute the exit probability as well as the average time to reach consensus by employing the backward fokker planck formalism and scaling arguments we analyze the results in the perspective of a recently proposed langevin equation aimed at describing generic phase transitions in systems with two z2 symmetric absorbing states in particular by deriving explicitly the coefficients of such a langevin equation as a function of the microscopic flipping probabilities we find that in mean field the q voter model exhibits a disordered phase for high epsilon and an ordered one for low epsilon with three possible ways to go from one to the other i a unique generalized voter like transition ii a series of two consecutive transitions one ising like in which the z2 symmetry is broken and a separate one in the directed percolation class in which the system falls into an absorbing state and iii a series of two transitions including an intermediate regime in which the final state depends on initial conditions this third so far unexplored scenario in which a type of ordering dynamics emerges is rationalized and found to be specific of mean field i e fluctuations are explicitly shown to wash it out in spatially extended systems

singular continuous spectrum is generic in a variety of contexts we prove that singular continuous spectrum is generic in the sense that for certain natural complete metric spaces of operators those with singular spectrum are a dense g delta

extended states in the anderson model on the bethe lattice abstract we prove that the anderson hamiltonianh von the bethe lattice has extended states for small disorder more precisely given any closed intervalicontained in the interior of the spectrum of the laplacian on the bethe lattice we prove that for small disorderh has purely absolutely continuous spectrum iniwith probability one i e ac h i iand pp h i sc h i with probability one and its integrated density of states is continuously differentiable on the intervali

local fluctuation of the spectrum of a multidimensional anderson tight binding model we consider the anderson tight binding modelh v acting inl 2 z d and its restrictionh to finite hypercubes z d herev v x x z d is a random potential consisting of independent identically distributed random variables let e j j be the eigenvalues ofh and let j e e j e j 1 be its rescaled eigenvalues then assuming that the exponential decay of the fractional moment of the green function holds for complex energies neare and that the density of statesn e exists ate we shall prove that the random sequence j e j considered as a point process onr 1 converges weakly to the stationary poisson point process with intensity measuren e dx as gets large thus extending the result of molchanov proved for a one dimensional continuum random schrodinger operator on the other hand the exponential decay of the fractional moment of the green function was established recently by aizenman molchanov and graf as a technical lemma for proving anderson localization at large disorder or at extreme energy thus our result in this paper can be summarized as follows near the energye where anderson localization is expected there is no correlation between eigenvalues ofh if is large

majority model on a network with communities we focus on the majority model in a topology consisting of two coupled fully connected networks thereby mimicking the existence of communities in social networks we show that a transition takes place at a value of the interconnectivity parameter above this value only symmetric solutions prevail where both communities agree with each other and reach consensus below this value in contrast the communities can reach opposite opinions and an asymmetric state is attained the importance of the interface between the subnetworks is shown

exponential decay of green s functions for a class of long range hamiltonians we consider a class of long range hamiltonians with diagonal disorder onl2 z for anyergodic potentialv with non empty essential range we prove the exponential decay of the green s functions for energies in the essential range ifv is independent identically distributed we obtain the exponential decay of the green s functions for all coupling constant 0 moreover the hamiltonian has only pure point spectrum

localization for random and quasiperiodic potentials a survey is made of some recent mathematical results and techniques for schrodinger operators with random and quasiperiodic potentials a new proof of localization for random potentials established in collaboration with h von dreifus is sketched

localization in general one dimensional random systems i jacobi matrices we consider random discrete schrodinger operators in a strip with a potentialv n n a label in and a finite label across the strip andv an ergodic process we prove thath0 v has only point spectrum with probability one under two assumptions 1 theconditional distribution of v n n 0 1 all conditioned on v n 0 1 all has an absolutely continuous component with positive probability 2 for a e e no lyaponov exponent is zero

dynamical localization for discrete and continuous random schrodinger operators we show for a large class of random schrodinger operators h on and on that dynamical localization holds i e that with probability one for a suitable energy interval i and for q a positive real here is a function of sufficiently rapid decrease and p i h is the spectral projector of h corresponding to the interval i the result is obtained through the control of the decay of the eigenfunctions of h and covers in the discrete case the anderson tight binding model with bernoulli potential dimension 1 or singular potential 1 and in the continuous case anderson as well as random landau hamiltonians

absence of diffusion in the anderson tight binding model for large disorder or low energy we prove that the green s function of the anderson tight binding hamiltonian decays exponentially fast at long distances on v with probability 1 we must assume that either the disorder is large or the energy is sufficiently low our proof is based on perturbation theory about an infinite sequence of block hamiltonians and is related to kam methods

franco american meeting on the mathematics of random and almost periodic potentials we present the abstracts of papers given at a conference held at the ecole polytechnique in june 1983 on the mathematics of random and almost periodic potentials

anderson localization for bernoulli and other singular potentials we prove exponential localization in the anderson model under very weak assumptions on the potential distribution in one dimension we allow any measure which is not concentrated on a single point and possesses some finite moment in particular this solves the longstanding problem of localization for bernoulli potentials i e potentials that take only two values in dimensions greater than one we prove localization at high disorder for potentials with holder continuous distributions and for bounded potentials whose distribution is a convex combination of a holder continuous distribution with high disorder and an arbitrary distribution these include potentials with singular distributions we also show that for certain bernoulli potentials in one dimension the integrated density of states has a nontrivial singular component

rapid solution of integral equations of classical potential theory an algorithm is described for rapid solution of classical boundary value problems dirichlet an neumann for the laplace equation based on iteratively solving integral equations of potential theory cpu time requirements for previously published algorithms of this type are proportional to n2 where n is the number of nodes in the discretization of the boundary of the region the cpu time requirements for the algorithm of the present paper are proportional to n making it considerably more practical for large scale problems

anderson localization for multi dimensional systems at large disorder or large energy we prove that discrete schrodinger operators on d with a random potential have almost surely only pure point spectrum and exponentially decaying eigenfunctions for large disorder or large energy this is the first proof of localization for multi dimensional anderson models

localization for off diagonal disorder and for continuous schrodinger operators we extend the proof of localization by delyon levy and souillard to accommodate the anderson model with off diagonal disorder and the continuous schrodinger equation with a random potential

the quantum hall effect for electrons in a random potential we assume the existence of sufficiently localised states near the edges of each landau band we then prove that the hall conductivity is quantised and the parallel conductivity vanishes when the filling factor stays close to an integer the hall integer is a topological invariant given by the landau band index

localization at large disorder and at extreme energies an elementary derivation the work presents a short proof of localization under the conditions of either strong disorder 0 or extreme energies for a wide class of self adjoint operators with random matrix elements acting inl 2 spaces a prototypical example is the discrete schrodinger operatorh u 0 x v x onz d d 1 withu 0 x a specified background potential and v x generated as random variables the general results apply to operators with replaced by a non local self adjoint operatort whose matrix elements satisfy y t x y s const uniformly inx for somes 1 localization means here that within a specified energy range the spectrum ofh is of the pure point type or equivalently the wave functions do not spread indefinitely under the unitary time evolution generated byh the effect is produced by strong disorder in either the potential or in the off diagonal matrix elementst x y under rapid decay oft x y the corresponding eigenfunctions are also proven to decay exponentially the method is based on resolvent techniques the central technical ideas include the use of low moments of the resolvent kernel i e g e x y s withs small enough 1 to avoid the divergence caused by the distribution s cauchy tails and an effective use of the simple form of the dependence ofg e x y on the individual matrix elements ofh in elucidating the implications of the fundamental equation h e g e x x 0 x x0 this approach simplifies previous derivations of localization results avoiding the small denominator difficulties which have been hitherto encountered in the subject it also yields some new results which include localization under the following sets of conditions i potentials with an inhomogeneous non random partu 0 x ii the bethe lattice iii operators with very slow decay in the off diagonal terms t x y 1 x y d e and iv localization produced by disordered boundary conditions

the schrodinger equation with a quasi periodic potential we consider the schrodinger equation dx2 v e cosx cos cix 0 v eqi where e is small and a satisfies the diophantine inequality lp qcl c q2 forp qez q 0 we look for solutions of the form ikx ikx in x im ax 0 m x e q x e 2m ne inxne e if we try to solve for v v mn we are led to the schrodinger equation on the lattice z2 h k q ba v k q en where a is the discrete laplacian without diagonal terms and v k is some 2 potential on z we have two main results 4 for e sufficiently small h k has pure point spectrum for almost every k 2 for e sufficiently small the operator d dx 2 e cosx cos ax has no point spectrum to prove our results we must get decay estimates on the green s function e h 1 the decay of the eigenfunction follows from this in general we must keep track of small divisors which can make the green s function large this is accomplished by a kam kolmogorov arnold moser type of multiscale perturbation analysis

on the use of wavelet expansions in the method of moments em scattering an approach which incorporates the theory of wavelet transforms in method of moments solutions for electromagnetic wave interaction problems is presented the unknown field or response is expressed as a twofold summation of shifted and dilated forms of a properly chosen basis function which is often referred to as the mother wavelet the wavelet expansion can adaptively fit itself to the various length scales associated with the scatterer by distributing the localized functions near the discontinuities and the more spatially diffused ones over the smooth expanses of the scatterer the approach is thus best suited for the analysis of scatterers which contain a broad spectrum of length scales ranging from a subwavelength to several wavelengths using a galerkin method and subsequently applying a threshold procedure the moment method matrix is rendered sparsely populated the structure of the matrix reveals the localized scale fitting distribution long before the matrix equation is solved the performance of the proposed discretization scheme is illustrated by a numerical study of electromagnetic coupling through a double slot aperture

perturbation theory for periodic orbits in a class of infinite dimensional hamiltonian systems we consider a class of hamiltonian systems describing an infinite array of coupled anharmonic oscillators and we study the bifurcation of periodic orbits off the equilibrium point the family of orbits we construct can be parametrized by their periods which belong to cantor sets of large measure containing certain periods of the linearized problem as accumulation points the infinitely many holes forming a dense set on which the existence of a periodic orbit cannot be proven originate from a dense set of resonances that are present in the system we also have a result concerning the existence of solutions of arbitrarily large amplitude

localization for random schrodinger operators with correlated potentials we prove localization at high disorder or low energy for lattice schrodinger operators with random potentials whose values at different lattice sites are correlated over large distances the class of admissible random potentials for our multiscale analysis includes potentials with a stationary gaussian distribution whose covariance functionc x y decays as x y where 0 can be arbitrarily small and potentials whose probability distribution is a completely analytical gibbs measure the result for gaussian potentials depends on a multivariable form of nelson s best possible hypercontractive estimate

fast wavelet transforms and numerical algorithms i a class of algorithms is introduced for the rapid numerical application of a class of linear operators to arbitrary vectors previously published schemes of this type utilize detailed analytical information about the operators being applied and are specific to extremely narrow classes of matrices in contrast the methods presented here are based on the recently developed theory of wavelets and are applicable to all calderon zygmund and pseudo differential operators the algorithms of this paper require order o n or o n log n operations to apply an n n matrix to a vector depending on the particular operator and the version of the algorithm being used and our numerical experiments indicate that many previously intractable problems become manageable with the techniques presented here

some rigorous results for random and quasi periodic potentials abstract we review some mathematical results and conjectures for random and quasi periodic schrodinger operators the role of localization in certain nonlinear systems is also discussed in addition we present some general arguments which show that the correlation length exponent v 2 d for a wide class of d dimensional random systems including the dilute ising ferro magnet

regularity of the density of states of random schrodinger operators in this paper we solve a long standing open problem for random schrodinger operators on l 2 mathbb r d with i i d single site random potentials we allow a large class of free operators including magnetic potential however our method of proof works only for the case when the random potentials satisfy a complete covering condition we require that the supports of the random potentials cover mathbb r d and the bump functions that appear in the random potentials form a partition of unity for such models we show that the density of states dos is m times differentiable in the part of the spectrum where exponential localization is valid if the single site distribution has compact support and has holder continuous m 1 st derivative the required holder continuity depends on the fractional moment bounds satisfied by appropriate operator kernels our proof of the random schrodinger operator case is an extensions of our proof for anderson type models on ell 2 mathbb g mathbb g a countable set with the property that the cardinality of the set of points at distance n from any fixed point grows at some rate in n alpha alpha 0 this condition rules out the bethe lattice where our method of proof works but the degree of smoothness also depends on the localization length a result we do not present here even for these models the random potentials need to satisfy a complete covering condition the anderson model on the lattice for which regularity results were known earlier also satisfies the complete covering condition

anomalous thermostat and intraband discrete breathers abstract we investigate the dynamics of a macroscopic system which consists of an anharmonic subsystem embedded in an arbitrary harmonic lattice including quenched disorder the coupling between both parts is bilinear elimination of the harmonic degrees of freedom leads to a nonlinear langevin equation with memory kernels t and noise term t for the anharmonic coordinates q t q t for zero temperature i e for t 0 we prove that the support of the fourier transform of t and of the time averaged velocity velocity correlation functions k t of the anharmonic system cannot overlap as a consequence the asymptotic solutions can be constant periodic quasiperiodic or almost periodic and possibly weakly chaotic for a sinusoidal trajectory q t with frequency we find that the energy e t transferred to the harmonic system up to time t is proportional to t if equals one of the phonon frequencies it is 2 we prove that there is a zero measure set l such that for in its full measure complement r l it is 0 i e there is no energy dissipation under certain conditions l contains a subset l such that for l the dissipation rate is nonzero and may be subdissipative 0 1 or superdissipative 1 2 compared to ordinary dissipation 1 consequently the harmonic bath does act as an anomalous thermostat in variance with the common belief that elimination of a macroscopically large number of degrees of freedom always generates dissipation forcing convergence to equilibrium intraband discrete breathers are such solutions which do not relax we prove for arbitrary anharmonicity and small but finite coupling that intraband discrete breathers with frequency exist for all in a cantor set c k of finite lebesgue measure this is achieved by estimating the contribution of small denominators appearing for g t related to t for c k the small denominators do not lead to divergencies such that g t is a smooth and bounded function in t

spectral properties of random schrodinger operators with unbounded potentials we investigate spectral properties of random schrodinger operators h n 1 n acting onl 2 z d where n are independent random variables uniformly distributed on 0 1

impedance matrix compression using adaptively constructed basis functions wavelet expansions have been employed recently in numerical solutions of commonly used frequency domain integral equations in this paper we propose a novel method for integrating wavelet based transforms into existing numerical solvers the newly proposed method differs from the presently used ones in two ways first the transformation is affected by means of a digital filtering approach this approach renders the transform algorithm adaptive and facilitates the derivation of a basis which best suits the problem at hand second the conventional thresholding procedure applied to the impedance matrix is substituted for by a compression process in which only the significant terms in the expansion of the yet unknown current are retained and subsequently derived numerical results for a few tm scattering problems are included to demonstrate the advantages of the proposed method over the presently used ones

majority rule dynamics in finite dimensions we investigate the long time behavior of a majority rule opinion dynamics model in finite spatial dimensions each site of the system is endowed with a two state spin variable that evolves by majority rule in a single update event a group of spins with a fixed odd size is specified and all members of the group adopt the local majority state repeated application of this update step leads to a coarsening mosaic of spin domains and ultimate consensus in a finite system the approach to consensus is governed by two disparate time scales with the longer time scale arising from realizations in which spins organize into coherent single opinion bands the consequences of this geometrical organization on the long time kinetics are explored

asymptotic localization of stationary states in the nonlinear schrodinger equation the mapping of the nonlinear schrodinger equation with a random potential on the fokker planck equation is used to calculate the localization length of its stationary states the asymptotic growth rates of the moments of the wave function and its derivative for the linear schrodinger equation in a random potential are computed analytically and resummation is used to obtain the corresponding growth rate for the nonlinear schrodinger equation and the localization length of the stationary states

anderson localization for one and quasi one dimensional systems we prove almost sure exponential localization of all the eigenfunctions and nondegeneracy of the spectrum for random discrete schrodinger operators on one and quasi one dimensional lattices this paper provides a much simpler proof of these results than previous approaches and extends to a much wider class of systems we remark in particular that the singular continuous spectrum observed in some quasiperiodic systems disappears under arbitrarily small local perturbations of the potential our results allow us to prove that e g for strong disorder the smallest positive lyapunov exponent of some products of random matrices does not vanish as the size of the matrices increases to infinity

a central limit theorem for products of random matrices and goe statistics for the anderson model on long boxes we consider products of random matrices that are small independent identically distributed perturbations of a fixed matrix mathcal t 0 focusing on the eigenvalues of mathcal t 0 of a particular size we obtain a limit to a sde in a critical scaling previous results required mathcal t 0 to be a conjugated unitary matrix so it could not have eigenvalues of different modulus from the result we can also obtain a limit sde for the markov process given by the action of the random products on the flag manifold applying the result to random schrodinger operators we can improve some results by valko and virag showing goe statistics for the rescaled eigenvalue process of a sequence of anderson models on long boxes in particular we solve a problem posed in their work

localization in general one dimensional random systems we discuss two ways of extending the recent ideas of localization from discrete schrodinger operators jacobi matrices to the continuum case one case allows us to prove localization in the goldshade molchanov pastur model for a larger class of functions than previously the other method studies the model v wherev is a random constant in each hyper cube we extend wegner s result on the lipschitz nature of the ids to this model

adaptive multiscale moment method ammm for analysis of scattering from three dimensional perfectly conducting structures the adaptive multiscale moment method ammm is presented for the analysis of scattering from three dimensional 3d perfectly conducting bodies this algorithm employs the conventional moment method mm using the subsectional triangular patch basis functions and a special matrix transformation which is derived from solving the fredholm equation of the first kind by the multiscale technique this methodology is more suitable for problems where the matrix solution time is much greater than the matrix fill time the widely used triangular patch vector basis functions developed by rao et al 1982 are used for expansion and testing functions in the conventional mm the objective here is to compress the unknowns in existing mm codes which solves for the currents crossing the edges of the triangular patch basis functions by use of a matrix transformation the currents source terms and impedance matrix can be arranged in the form of different scales from one scale to another scale the initial guess for the solution can be predicted according to the properties of the multiscale technique ammm can reduce automatically the size of the linear equations so as to improve the efficiency of the conventional mm the basic difference between this methodology and the other wavelet based techniques that have been presented so far is that we apply the compression not to the impedance matrix but to the solution itself directly in an iterative fashion even though it is an unknown two numerical results are presented which demonstrate that the ammm is a useful method for analysis of electromagnetic scattering from arbitrary shaped 3d perfectly conducting bodies

on the spectrum of schrodinger operators with a random potential we investigate the spectrum of schrodinger operatorsh of the type h qi f x xi i qi and i independent identically distributed random variables i d we establish a strong connection between the spectrum ofh and the spectra of deterministic periodic schrodinger operators from this we derive a condition for the existence of forbidden zones in the spectrum ofh for random one and three dimensional kronig penney potentials the spectrum is given explicitly

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

on absence of diffusion near the bottom of the spectrum for a random schrodinger operator on l 2 bf r nu we consider a random schrodinger operator onl2 v of the form h omega delta v omega v omega x sigma chi c i x q i omega ci being a covering of v with unit cubes around the sites of v and qi i i d random variables with values in 0 1 we assume that theqi s are continuously distributed with bounded densityf q and that 0 p q0 1 2 1 then we show that an ergodic mean of the quantity dx x 2 exp ith x 2 t 1 vanishes provided ge h where is well localized around the origin andge is a positivec function with support in 0 e e e f our estimate ofe f is such that the set x v v x e f may contain with probability one an infinite cluster of cubes ci which are nearest neighbours the proof is based on the technique introduced by frohlich and spencer for the analysis of the anderson model

a localization principle for multiplicative perturbations abstract a certain resolvent estimate implies point spectrum for discrete linear wave equations with random local propagation speed the significance of point spectrum in this context is that scattering within a random medium produces only localized standing waves the implication is derived from a general result about multiplicative perturbations of a self adjoint operator the estimate implies point spectrum for almost every value of the perturbation parameter

effects of inhomogeneous influence of individuals on an order disorder transition in opinion dynamics we study the effects of inhomogeneous influence of individuals on collective phenomena we focus analytically on a typical model of the majority rule applied to the completely connected agents two types of individuals a and b with different influence activity are introduced the individuals a and b are distributed randomly with concentrations ensuremath nu and 1 ensuremath ensuremath nu at the beginning and fixed further on our main result is that the location of the order disorder transition is affected due to the introduction of the inhomogeneous influence this result highlights the importance of inhomogeneous influence between different types of individuals during the process of opinion updating

schrodinger operators in the twentieth century this paper reviews the past fifty years of work on spectral theory and related issues in nonrelativistic quantum mechanics

anderson localization and the space time characteristic of continuum states a proof of anderson localization is obtained by ruling out any continuous spectrum on the basis of the space time characteristic of its states

remark on the absence of absolutely continuous spectrum for d dimensional schrodinger operators with random potential for large disorder or low energy we show that there is no absolutely continuous part in the spectrum of the anderson tight binding model for large disorder or low energy the proof is based on the exponential decay of the green s function proved by frohlich and spencer the extension of this result to the continuous case is also discussed

one dimensional schrodinger operators with random or deterministic potentials new spectral types abstract we prove sufficient conditions involving only potential asymptotic near one of the infinities in order to have purely absolutely continuous components in the spectrum these deterministic results are then applied to random cases and exhibit classes of models for which with probability one one component of the spectrum is purely absolutely continuous and the rest is dense pure point with exponentially decaying eigenfunctions

dynamics of opinion formation in a small world network the dynamical process of opinion formation within a model using a local majority opinion updating rule is studied numerically in networks with the small world geometrical property the network is one in which shortcuts are added to randomly chosen pairs of nodes in an underlying regular lattice the presence of a small number of shortcuts is found to shorten the time to reach a consensus significantly the effects of having shortcuts in a lattice of fixed spatial dimension are shown to be analogous to that of increasing the spatial dimension in regular lattices the shortening of the consensus time is shown to be related to the shortening of the mean shortest path as shortcuts are added results can also be translated into that of the dynamics of a spin system in a small world network

generalized eigenvalue counting estimates for the anderson model we generalize minami s estimate for the anderson model and its extensions to n eigenvalues allowing for n arbitrary intervals and arbitrary single site probability measures with no atoms as an application we derive new results about the multiplicity of eigenvalues and mott s formula for the ac conductivity when the single site probability distribution is holder continuous

complex multipole beam approach to electromagnetic scattering problems a novel approach to reducing the matrix size associated with the method of moments mom solution of the problem of electromagnetic scattering from arbitrary shaped closed bodies is presented the key step in this approach is to represent the scattered field in terms of a series of beams produced by multipole sources located in a complex space on the scatterer boundary the fields generated by these multipole sources resemble the gabor basis functions by utilizing the properties of the gabor series guidelines for selecting the orders as well as locations of the multipole sources are developed it is shown that the present approach not only reduces the number of unknowns but also generates a generalized impedance matrix with a banded structure and a low condition number the accuracy of the proposed method is verified by comparing the numerical results with those derived by using the method of moments

rapid solution of integral equations of scattering theory in two dimensions abstract the present paper describes an algorithm for rapid solution of boundary value problems for the helmholtz equation in two dimensions based on iteratively solving integral equations of scattering theory cpu time requirements of previously published algorithms of this type are of the order n 2 where n is the number of nodes in the discretization of the boundary of the scatterer the cpu time requirements of the algorithm of the present paper are n 4 3 and can be further reduced making it considerably more practical for large scale problems