random time dependent quantum walks we consider the discrete time unitary dynamics given by a quantum walk on the lattice mathbb z d performed by a quantum particle with internal degree of freedom called coin state according to the following iterated rule a unitary update of the coin state takes place followed by a shift on the lattice conditioned on the coin state of the particle we study the large time behavior of the quantum mechanical probability distribution of the position observable in mathbb z d when the sequence of unitary updates is given by an i i d sequence of random matrices when averaged over the randomness this distribution is shown to display a drift proportional to the time and its centered counterpart is shown to display a diffusive behavior with a diffusion matrix we compute a moderate deviation principle is also proven to hold for the averaged distribution and the limit of the suitably rescaled corresponding characteristic function is shown to satisfy a diffusion equation a generalization to unitary updates distributed according to a markov process is also provided

nonlocal symmetries and formulae for generation of solutions for a class of diffusion convection equations abstract new formulae of nonlocal nonlinear superposition and generation of solutions are proposed for nonlinear diffusion convection equations which are linearizable or are invariant with respect to a generalized hodograph transformation or connected by this transformation we study in what particular ways additional lie symmetries of diffusion convection equations induce nonlocal symmetries of equations obtained from the initial ones by nonlocal transformations the formulae derived are used for the construction of exact solutions

scope for further analytical solutions for constant flux infiltration into a semi infinite soil profile or redistribution in a finite soil profile we attempt to generate new solutions for the moisture content form of the one dimensional richards 1931 equation using the lisle 1992 equivalence mapping this mapping is used as no more general set of transformations exists for mapping the one dimensional richards equation into itself starting from a given solution the mapping has the potential to generate an infinite number of new solutions for a series of nonlinear diffusivity and hydraulic conductivity functions we first seek new analytical solutions satisfying richards equation subject to a constant flux surface boundary condition for a semi infinite dry soil starting with the burgers model the first iteration produces an existing solution while subsequent iterations are shown to endlessly reproduce this same solution next we briefly consider the problem of redistribution in a finite length soil in this case lisle s equivalence mapping is generalized to account for arbitrary initial conditions as was the case for infiltration however it is found that new analytical solutions are not generated using the equivalence mapping although existing solutions are recovered

exact transient solutions to nonlinear diffusion convection equations in higher dimensions the complete lie algebra of classical infinitesimal symmetries of the nonlinear diffusion convection equation in two and three dimensions is presented except for some cases involving constant diffusivity a complete reduction to an ordinary differential equation is not possible however closed form solutions are obtained for special forms of both the 2d and 3d nonlinear diffusion convection equations using a symmetry reduction and an additional physical constraint this extends the small list of closed form transient solutions already known

quantum graphs a simple model for chaotic scattering we connect quantum graphs with infinite leads and turn them into scattering systems we show that they display all the features which characterize quantum scattering systems with an underlying classical chaotic dynamics typical poles delay time and conductance distributions ericson fluctuations and when considered statistically the ensemble of scattering matrices reproduces quite well the predictions of the appropriately defined random matrix ensembles the underlying classical dynamics can be defined and it provides important parameters which are needed for the quantum theory in particular we derive exact expressions for the scattering matrix and an exact trace formula for the density of resonances in terms of classical orbits analogous to the semiclassical theory of chaotic scattering we use this in order to investigate the origin of the connection between random matrix theory and the underlying classical chaotic dynamics being an exact theory and due to its relative simplicity it offers new insights into this problem which is at the forefront of the research in chaotic scattering and related fields

linearisation and potential symmetries of certain systems of diffusion equations abstract we consider systems of two pure one dimensional diffusion equations that have considerable interest in soil science and mathematical biology we construct non local symmetries for these systems these are determined by expressing the equations in a partially and wholly conserved form and then by performing a potential symmetry analysis on those systems that can be linearised we give several examples of such systems and in a specific case we show how linearising and hodograph type mappings can lead to new solutions of the diffusion system

from quantum cellular automata to quantum lattice gases a natural architecture for nanoscale quantum computation is that of a quantum cellular automaton motivated by this observation we begin an investigation of exactly unitary cellular automata after proving that there can be no nontrivial homogeneous local unitary scalar cellular automaton in one dimension we weaken the homogeneity condition and show that there are nontrivial exactly unitary partitioning cellular automata we find a one parameter family of evolution rules which are best interpreted as those for a one particle quantum automaton this model is naturally reformulated as a two component cellular automaton which we demonstrate to limit to the dirac equation we describe two generalizations of this automaton the second of which to multiple interacting particles is the correct definition of a quantum lattice gas

lie symmetry classification analysis for nonlinear coupled diffusion the group properties of the 1 1 dimensional matrix diffusion equation of the form with respect to point symmetries are given it is shown that in particular cases the group properties of this equation are similar to those of the corresponding scalar equation namely the lie algebra is extended for power and exponential functions with additional extensions when powers of and 2 exist however a specific form of is shown to exist for cases admitting both infinite and finite groups the result obtained is used to construct new group invariant solutions for impulsive boundary inputs using two similarity variables a reduction is applied to the coupled diffusion of temperature and volumetric moisture content in porous media under periodic boundary conditions a perturbation method is employed to obtain an explicit solution for the case when can be expressed exponentially in terms of the similarity variables

potential symmetries of nonlinear diffusion convection equations in this paper potential symmetries are sought for the nonlinear diffusion convection equations the functional forms of f u and k u that admit such symmetries are completely classified

inversion of hierarchical bayesian models using gaussian processes over the past decade computational approaches to neuroimaging have increasingly made use of hierarchical bayesian models hbms either for inferring on physiological mechanisms underlying fmri data e g dynamic causal modelling dcm or for deriving computational trajectories from behavioural data which serve as regressors in general linear models however an unresolved problem is that standard methods for inverting the hierarchical bayesian model are either very slow e g markov chain monte carlo methods mcmc or are vulnerable to local minima in non convex optimisation problems such as variational bayes vb this article considers gaussian process optimisation gpo as an alternative approach for global optimisation of sufficiently smooth and efficiently evaluable objective functions gpo avoids being trapped in local extrema and can be computationally much more efficient than mcmc here we examine the benefits of gpo for inverting hbms commonly used in neuroimaging including dcm for fmri and the hierarchical gaussian filter hgf importantly to achieve computational efficiency despite high dimensional optimisation problems we introduce a novel combination of gpo and local gradient based search methods the utility of this gpo implementation for dcm and hgf is evaluated against mcmc and vb using both synthetic data from simulations and empirical data our results demonstrate that gpo provides parameter estimates with equivalent or better accuracy than the other techniques but at a fraction of the computational cost required for mcmc we anticipate that gpo will prove useful for robust and efficient inversion of high dimensional and nonlinear models of neuroimaging data

spectral properties of quantum walks on rooted binary trees we define coined quantum walks on the infinite rooted binary tree given by unitary operators u c on an associated infinite dimensional hilbert space depending on a unitary coin matrix c in u 3 and study their spectral properties for circulant unitary coin matrices c we derive an equation for the caratheodory function associated to the spectral measure of a cyclic vector for u c this allows us to show that for all circulant unitary coin matrices the spectrum of the quantum walk has no singular continuous component furthermore for coin matrices c which are orthogonal circulant matrices we show that the spectrum of the quantum walk is absolutely continuous except for four coin matrices for which the spectrum of u c is pure point

population dynamics under the laplace assumption in this paper we describe a generic approach to modelling dynamics in neuronal populations this approach models a full density on the states of neuronal populations but finesses this high dimensional problem by re formulating density dynamics in terms of ordinary differential equations on the sufficient statistics of the densities considered c f the method of moments the particular form for the population density we adopt is a gaussian density c f the laplace assumption this means population dynamics are described by equations governing the evolution of the population s mean and covariance we derive these equations from the fokker planck formalism and illustrate their application to a conductance based model of neuronal exchanges one interesting aspect of this formulation is that we can uncouple the mean and covariance to furnish a neural mass model which rests only on the populations mean this enables us to compare equivalent mean field and neural mass models of the same populations and evaluate quantitatively the contribution of population variance to the expected dynamics the mean field model presented here will form the basis of a dynamic causal model of observed electromagnetic signals in future work

quantum chaos on discrete graphs adapting a method developed for the study of quantum chaos on quantum metric graphs kottos and smilansky 1997 phys rev lett 79 4794 kottos and smilansky 1999 ann phys ny 274 76 spectral functions and trace formulae for discrete laplacians on graphs are derived this is achieved by expressing the spectral secular equation in terms of the periodic orbits of the graph and obtaining functions which belong to the class of functions proposed originally by ihara 1966 j mat soc japan 18 219 and expanded by subsequent authors stark and terras 1996 adv math 121 124 kotani and sunada 2000 j math sci univ tokyo 7 7 finally a model of classical dynamics on the discrete graph is proposed it is analogous to the corresponding classical dynamics derived for quantum graphs kottos and smilansky 1997 phys rev lett 79 4794 kottos and smilansky 1999 ann phys ny 274 76

spectral transition for random quantum walks on trees we define and analyze random quantum walks on homogeneous trees of degree q geq 3 such walks describe the discrete time evolution of a quantum particle with internal degree of freedom in mathbb c q hopping on the neighboring sites of the tree in the presence of static disorder the one time step random unitary evolution operator of the particle depends on a unitary matrix c in u q which monitors the strength of the disorder we prove for any q that there exist open sets of matrices in u q for which the random evolution has either pure point spectrum almost surely or purely absolutely continuous spectrum thereby showing the existence of a spectral transition driven by c in u q for q in 3 4 we establish properties of the spectral diagram which provide a description of the spectral transition

a scattering approach to the quantization of hamiltonians in two dimensions application to the wedge billiard we extend the scattering approach to quantize general hamiltonian systems in two dimensions and demonstrate it for the wedge billiard the resulting energy levels are given by the zeros of a secular equation the corresponding wavefunctions may also be constructed we verify analytically that this quantization condition is exact in the case of the integrable wedges this method has been applied successfully to computing numerically a few thousand levels for various values of the angle using these data we can check the gutzwiller trace formula and other semiclassical relations involving the scattering matrix and periodic orbits for which we find excellent agreement

quantum markov chains a new approach to quantum markov chains is presented we first define a transition operation matrix tom as a matrix whose entries are completely positive maps whose column sums form a quantum operation a quantum markov chain is defined to be a pair g e where g is a directed graph and e eij is a tom whose entry eij labels the edge from vertex j to vertex i we think of the vertices of g as sites that a quantum system can occupy and eij is the transition operation from site j to site i in one time step the discrete dynamics of the system is obtained by iterating the tom e we next consider a special type of tom called a transition effect matrix in this case there are two types of dynamics a state dynamics and an operator dynamics although these two types are not identical they are statistically equivalent we next give examples that illustrate various properties of quantum markov chains we conclude by showing that our formalism generalizes the usual framework for quantum random walks

search via quantum walk we propose a new method for designing quantum search algorithms for finding a marked element in the state space of a classical markov chain the algorithm is based on a quantum walk a la szegedy quantum speed up of markov chain based algorithms in proceedings of the 45th ieee symposium on foundations of computer science ieee computer society press 2004 pp 32 41 that is defined in terms of the markov chain the main new idea is to apply quantum phase estimation to the quantum walk in order to implement an approximate reflection operator this operator is then used in an amplitude amplification scheme as a result we considerably expand the scope of the previous approaches of ambainis quantum walk algorithm for element distinctness in proceedings of the 45th ieee symposium on foundations of computer science ieee computer society press 2004 pp 22 31 and szegedy 2004 our algorithm combines the benefits of these approaches in terms of being able to find marked elements incurring the smaller cost of the two and being applicable to a larger class of markov chains in addition it is conceptually simple and avoids some technical difficulties in the previous analyses of several algorithms based on quantum walk

asymptotic evolution of quantum walks with random coin we study the asymptotic position distribution of general quantum walks on a lattice including walks with a random coin which is chosen from step to step by a general markov chain in the unitary i e nonrandom case we allow any unitary operator which commutes with translations and couples only sites at a finite distance from each other for example a single step of the walk could be composed of any finite succession of different shift and coin operations in the usual sense with any lattice dimension and coin dimension we find ballistic scaling and establish a direct method for computing the asymptotic distribution of position divided by time namely as the distribution of the discrete time analog of the group velocity in the random case we let a markov chain control process pick in each step one of finitely many unitary walks in the sense described above in ballistic order we find a nonrandom drift which depends only on the mean of the control process and not on the initial state in diffusive scaling the limiting distribution is asymptotically gaussian with a covariance matrix diffusion matrix depending on momentum the diffusion matrix depends not only on the mean but also on the transition rates of the control process in the nonrandom limit i e when the coins chosen are all very close or the transition rates of the control process are small leading to long intervals of ballistic evolution the diffusion matrix diverges our method is based on spatial fourier transforms and the first and second order perturbation theory of the eigenvalue 1 of the transition operator for each value of the momentum

multi subject analyses with dynamic causal modeling currently most studies that employ dynamic causal modeling dcm use random effects rfx analysis to make group inferences applying a second level frequentist test to subjects parameter estimates in some instances however fixed effects ffx analysis can be more appropriate such analyses can be implemented by combining the subjects posterior densities according to bayes theorem either on a multivariate bayesian parameter averaging or bpa or univariate basis posterior variance weighted averaging or pvwa or by applying dcm to time series averaged across subjects beforehand temporal averaging or ta while all these ffx approaches have the advantage of allowing for bayesian inferences on parameters a systematic comparison of their statistical properties has been lacking so far based on simulated data generated from a two region network we examined the effects of signal to noise ratio snr and population heterogeneity on group level parameter estimates data sets were simulated assuming either a homogeneous large population n 60 with constant connectivities across subjects or a heterogeneous population with varying parameters ta showed advantages at lower snr but is limited in its applicability because bpa and pvwa take into account posterior co variance structure they can yield non intuitive results when only considering posterior means this problem is relevant for high snr data pronounced parameter interdependencies and when ffx assumptions are violated i e inhomogeneous groups it diminishes with decreasing snr and is absent for models with independent parameters or when ffx assumptions are appropriate group results obtained with these ffx approaches should therefore be interpreted carefully by considering estimates of dependencies among model parameters

a remark on discrete wave scattering we consider the spectral problem for the adjacency matrix of a discrete star graph composed of a compact cluster and a few semi infinite periodic leads attached based on the spectral properties of the adjacency matrix we develop the lax phillips scattering theory for the corresponding discrete wave equation

comparing dynamic causal models this article describes the use of bayes factors for comparing dynamic causal models dcms dcms are used to make inferences about effective connectivity from functional magnetic resonance imaging fmri data these inferences however are contingent upon assumptions about model structure that is the connectivity pattern between the regions included in the model given the current lack of detailed knowledge on anatomical connectivity in the human brain there are often considerable degrees of freedom when defining the connectional structure of dcms in addition many plausible scientific hypotheses may exist about which connections are changed by experimental manipulation and a formal procedure for directly comparing these competing hypotheses is highly desirable in this article we show how bayes factors can be used to guide choices about model structure both concerning the intrinsic connectivity pattern and the contextual modulation of individual connections the combined use of bayes factors and dcm thus allows one to evaluate competing scientific theories about the architecture of large scale neural networks and the neuronal interactions that mediate perception and cognition

exterior interior duality for discrete graphs the exterior interior duality expresses a deep connection between the laplace spectrum in bounded and connected domains in and the scattering matrices in the exterior of the domains here this link is extended to the study of the spectrum of the discrete laplacian on finite graphs for this purpose two methods are devised for associating scattering matrices to the graphs the exterior interior duality is derived for both methods

bayesian estimation of dynamical systems an application to fmri this paper presents a method for estimating the conditional or posterior distribution of the parameters of deterministic dynamical systems the procedure conforms to an em implementation of a gauss newton search for the maximum of the conditional or posterior density the inclusion of priors in the estimation procedure ensures robust and rapid convergence and the resulting conditional densities enable bayesian inference about the model parameters the method is demonstrated using an input state output model of the hemodynamic coupling between experimentally designed causes or factors in fmri studies and the ensuing bold response this example represents a generalization of current fmri analysis models that accommodates nonlinearities and in which the parameters have an explicit physical interpretation second the approach extends classical inference based on the likelihood of the data given a null hypothesis about the parameters to more plausible inferences about the parameters of the model given the data this inference provides for confidence intervals based on the conditional density 2002 elsevier science usa

potential symmetries and new solutions of a simplified model for reacting mixtures in this paper we investigate potential symmetries of a simplified model for reacting mixtures we find new similarity reductions and wider class of solutions through this approach further we explore an invertible mapping which linearizes the reacting mixture model

variational free energy and the laplace approximation this note derives the variational free energy under the laplace approximation with a focus on accounting for additional model complexity induced by increasing the number of model parameters this is relevant when using the free energy as an approximation to the log evidence in bayesian model averaging and selection by setting restricted maximum likelihood reml in the larger context of variational learning and expectation maximisation em we show how the reml objective function can be adjusted to provide an approximation to the log evidence for a particular model this means reml can be used for model selection specifically to select or compare models with different covariance components this is useful in the context of hierarchical models because it enables a principled selection of priors that under simple hyperpriors can be used for automatic model selection and relevance determination ard deriving the reml objective function from basic variational principles discloses the simple relationships among variational bayes em and reml furthermore we show that em is formally identical to a full variational treatment when the precisions are linear in the hyperparameters finally we also consider briefly dynamic models and how these inform the regularisation of free energy ascent schemes like em and reml

a metropolis hastings algorithm for dynamic causal models abstract dynamic causal modelling dcm is a modelling framework used to describe causal interactions in dynamical systems it was developed to infer the causal architecture of networks of neuronal populations in the brain friston k j harrison l penny w 2003 dynamic causal modelling neuroimage aug 19 4 1273 302 in current formulations of dcm the mean structure of the likelihood is a nonlinear and numerical function of the parameters which precludes exact or analytic bayesian inversion to date approximations to the posterior depend on the assumption of normality i e the laplace assumption in particular two arguments have been used to motivate normality of the prior and posterior distributions first gaussian priors on the parameters are specified carefully to ensure that activity in the dynamic system of neuronal populations converges to a steady state i e the dynamic system is dissipative secondly normality of the posterior is an approximation based on general asymptotic results regarding the form of the posterior under infinite data friston k j harrison l penny w 2003 dynamic causal modelling neuroimage aug 19 4 1273 302 here we provide a critique of these assumptions and evaluate them numerically we use a bayesian inversion scheme the metropolis hastings algorithm that eschews both assumptions this affords an independent route to the posterior and an external means to assess the performance of conventional schemes for dcm it also allows us to assess the sensitivity of the posterior to different priors first we retain the conventional priors and compare the ensuing approximate posterior laplace to the exact posterior mcmc our analyses show that the laplace approximation is appropriate for practical purposes in a second independent set of analyses we compare the exact posterior under conventional priors with an exact posterior under newly defined uninformative priors reassuringly we observe that the posterior is for all practical purposes insensitive of the choice of prior

open quantum random walks a new model of quantum random walks is introduced on lattices as well as on finite graphs these quantum random walks take into account the behavior of open quantum systems they are the exact quantum analogues of classical markov chains we explore the quantum trajectory point of view on these quantum random walks that is we show that measuring the position of the particle after each time step gives rise to a classical markov chain on the lattice times the state space of the particle this quantum trajectory is a simulation of the master equation of the quantum random walk the physical pertinence of such quantum random walks and the way they can be concretely realized is discussed differences and connections with the already well known quantum random walks such as the hadamard random walk are established

hierarchy of conservation laws of diffusion convection equations we introduce notions of equivalence of conservation laws with respect to lie symmetry groups for fixed systems of differential equations and with respect to equivalence groups or sets of admissible transformations for classes of such systems we also revise the notion of linear dependence of conservation laws and define the notion of local dependence of potentials to construct conservation laws we develop and apply the most direct method which is effective to use in the case of two independent variables admitting possibility of dependence of conserved vectors on a number of potentials we generalize the iteration procedure proposed by bluman and doran wu for finding nonlocal potential conservation laws as an example we completely classify potential conservation laws including arbitrary order local ones of diffusion convection equations with respect to the equivalence group and construct an exhaustive list of locally inequivalent potential systems corresponding to these equations

dynamical localization of quantum walks in random environments the dynamics of a one dimensional quantum walker on the lattice with two internal degrees of freedom the coin states is considered the discrete time unitary dynamics is determined by the repeated action of a coin operator in u 2 on the internal degrees of freedom followed by a one step shift to the right or left conditioned on the state of the coin for a fixed coin operator the dynamics is known to be ballistic

scattering on compact manifolds with infinitely thin horns the quantum mechanical scattering on a compact manifold with semi axes attached to the manifold hedgehog shaped manifold is considered the complete description of the spectral structure of schrodinger operators on such a manifold is done the proof of existence and uniqueness of scattering states is presented an explicit form for the scattering matrix is obtained and unitarity of this matrix is proven it is shown that the positive part of the spectrum of the schrodinger operator on the initial compact manifold as well as the spectrum of a point perturbation of such an operator may be recovered from the scattering amplitude for one attached half line moreover the positive part of the spectrum of the initial schrodinger operator is fully determined by the conductance properties of an electronic device consisting of the initial manifold and two wires attached to it

spectral duality for planar billiards for a bounded open domain with connected complement inr2 and piecewise smooth boundary we consider the dirichlet laplacian on and the s matrix on the complement c we show that the on shell s matricessk have eigenvalues converging to 1 ask k0 exactly when has an eigenvalue at energyk02 this includes multiplicities and proves a weak form of transparency atk k0 we also show that stronger forms of transparency such assk0 having an eigenvalue 1 are not expected to hold in general

gradient based mcmc samplers for dynamic causal modelling in this technical note we derive two mcmc markov chain monte carlo samplers for dynamic causal models dcms specifically we use a hamiltonian mcmc hmc e where sampling is simulated using hamilton s equation of motion and b langevin monte carlo algorithm lmc r and lmc e that simulates the langevin diffusion of samples using gradients either on a euclidean e or on a riemannian r manifold while lmc r requires minimal tuning the implementation of hmc e is heavily dependent on its tuning parameters these parameters are therefore optimised by learning a gaussian process model of the time normalised sample correlation matrix this allows one to formulate an objective function that balances tuning parameter exploration and exploitation furnishing an intervention free inference scheme using neural mass models nmms a class of biophysically motivated dcms we find that hmc e is statistically more efficient than lmc r with a riemannian metric yet both gradient based samplers are far superior to the random walk metropolis algorithm which proves inadequate to steer away from dynamical instability

group theoretical framework for potential symmetries of evolution equations we develop algebraic approach to the problem of classification of potential symmetries of nonlinear evolution equations it is essentially based on the recently discovered fact r zhdanov j math phys 50 053522 2009 that any such symmetry is mapped into a contact symmetry the approach enables using the classical results on classification of contact symmetries of nonlinear evolution equations by sokolov and magadeev to classify evolution equations admitting potential symmetries we construct several examples of new nonlinear fourth order evolution equations admitting potential symmetries since the symmetries obtained depend on nonlocal variables they cannot be derived by the infinitesimal lie approach

systematic construction of hidden nonlocal symmetries for the inhomogeneous nonlinear diffusion equation we consider a class of inhomogeneous nonlinear diffusion equations inde that arise in solute transport theory hidden nonlocal symmetries that seem not to be recorded in the literature are systematically determined by considering an integrated equation obtained using the general integral variable rather than a system of first order partial differential equations pdes associated with the concentration and flux of a conservation law reductions for the inde to ordinary differential equations odes are performed and some invariant solutions are constructed

new classes of symmetries for partial differential equations new classes of symmetries for partial differential equations are introduced by writing a given partial differential equation s in a conserved form a related system t with potentials as additional dependent variables is obtained the lie group of point transformations admitted by t induces a symmetry group of s new symmetries may be obtained for s that are neither point nor lie backlund symmetries they are determined by a completely algorithmic procedure significant new symmetries are found for the wave equation with a variable wave speed and the nonlinear diffusion equation

the characterization of reaction convection diffusion processes by travelling waves it has long been known that the heat equation displays infinite speed of propagation this is to say that if the initial data are nonnegative and have nonempty compact support the solution of an initial value problem is positive everywhere after any infinitesimal time however since the nineteen fifties it has also been known that certain nonlinear diffusion equations of degenerate parabolic type do not display this phenomenon for these equations the generalized solution of an initial value problem with compactly supported initial data will have bounded support with respect to the spatial variable at all times in this paper the necessary and sufficient criterion for finite speed of propagation for the general nonlinear reaction convection diffusion equation formula is determined the assumptions on the coefficientsa bandcare such that the classification unifies and generalizes previously known results the technique employed is comparision of an arbitrary solution of the equation with suitably constructed travelling wave solutions and subsolutions basically the central conclusion is that the equation exhibits finite speed of propagation if and only if it admits a travelling wave solution with bounded support concurrently the search for a travelling wave solution with bounded support can be reduced to the study of a singular nonlinear integral equation whose solution must satisfy a certain constraint

the dirichlet to neumann map for a resistor network relations that characterize the dirichlet to neumann map for a square network of resistors are given using these relations a parametrization of the set of dirichlet to neumann maps is given

disordered quantum walks in one lattice dimension we study a spin 12 particle moving on a one dimensional lattice subject to disorder induced by a random space dependent quantum coin the discrete time evolution is given by a family of random unitary quantum walk operators where the shift operation is assumed to be deterministic each coin is an independent identically distributed random variable with values in the group of two dimensional unitary matrices we derive sufficient conditions on the probability distribution of the coins such that the system exhibits dynamical localization put differently the tunneling probability between two lattice sites decays rapidly for almost all choices of random coins and after arbitrary many time steps with increasing distance our findings imply that this effect takes place if the coin is chosen at random from the haar measure or some measure continuous with respect to it but also for a class of discrete probability measures which support consists of two coins one of them being the hadamard coin

potential equivalence transformations for nonlinear diffusion convection equations potential equivalence transformations pets are effectively applied to a class of nonlinear diffusion convection equations for this class all possible potential symmetries are classified and a theorem on their connection with point symmetries via pets is also proved it is shown that the known nonlocal transformations between equations under consideration are nothing but pets the action of pets on sets of exact solutions of a fast diffusion equation is investigated

approximate lie group analysis and solutions of 2d nonlinear diffusion convection equations approximate lie symmetries of the 2 1 dimensional nonlinear diffusion equation with a small convection are completely classified it is known that the invariance principle furnishes a systematic method of solving initial value problems the solutions of instantaneous source type of the 2d diffusion convection equation are obtained for the case of power law diffusivity using a symmetry reduction

immiscible two phase flow in a porous medium utilization of a laplace transform boost a boost operator is defined for the laplace transform it is used in conjunction with the laplace transform to solve the diffusion equation on a domain with multiple moving boundaries this problem arises in the study of immiscible multiphase flow in a finite medium including end effects extensions of this method to the solution of other linear differential equations are conceivable

nodal statistics on quantum graphs it has been suggested that the distribution of the suitably normalized number of zeros of laplacian eigenfunctions contains information about the geometry of the underlying domain we study this distribution more precisely the distribution of the nodal surplus for laplacian eigenfunctions of a metric graph the existence of the distribution is established along with its symmetry one consequence of the symmetry is that the graph s first betti number can be recovered as twice the average nodal surplus of its eigenfunctions furthermore for graphs with disjoint cycles it is proven that the distribution has a universal form it is binomial over the allowed range of values of the surplus to prove the latter result we introduce the notion of a local nodal surplus and study its symmetry and dependence properties establishing that the local nodal surpluses of disjoint cycles behave like independent bernoulli variables

new results on group classification of nonlinear diffusion convection equations using a new method and additional conditional and partial equivalence transformations we performed group classification in a class of variable coefficient 1 1 dimensional nonlinear diffusion convection equations of the general form f x ut d u ux x k u ux we obtain new interesting cases of such equations with the density f localized in space which have non trivial invariance algebra exact solutions of these equations are constructed we also consider the problem of investigation of the possible local transformations for an arbitrary pair of equations from the class under consideration i e of describing all the possible partial equivalence transformations in this class

on the exactly solvable equation s_t beta s gamma 2 s_x _x alpha beta s gamma 2 s_x occurring in two phase flow in porous media the symmetry approach of fokas j math phys 21 1980 pp 1318 1325 is used to show that the most general equation of the form s t g s s xx f s s x dg ds ne 0 which is exactly solvable is equivalent to the equation s t beta s gamma 2 s x x alpha beta s gamma 2 s x this equation is then mapped to the linear heat equation through a series of backlund transformations the above result is used to solve an initial boundary value problem modeling the process of waterflooding i e the displacement of oil initially in a porous reservoir by continuously injected water

scattering theory for graphs isomorphic to a regular tree at infinity we describe the spectral theory of the adjacency operator of a graph which is isomorphic to a regular tree at infinity using some combinatorics we reduce the problem to a scattering problem for a finite rank perturbation of the adjacency operator on a regular tree we develop this scattering theory using the classical recipes for schrodinger operators in euclidian spaces

non classical and potential symmetry analysis of richard s equation for moisture flow in soil this paper focuses upon the derivation of the non classical symmetries of bluman and cole as they apply to richard s equation for water flow in an unsaturated uniform soil it is shown that the determining equations for the non classical case lead to four highly non linear equations which have been solved in five particular cases in each case the corresponding similarity ansatz has been derived and richard s equation is reduced to an ordinary differential equation explicit solutions are produced when possible richard s equation is also expressed as a potential system and in reviewing the classical lie solutions a new symmetry is derived together with its similarity ansatz determining equations are then produced for the potential system using the non classical algorithm this results in an under determined set of equations and an example symmetry that reveals a missing classical case is presented an example of a classical and a non classical symmetry reduction applied to the infiltration of moisture in soil is presented the condition for surface invariance is used to demonstrate the equivalence of a classical lie and a potential symmetry