quantitative diophantine approximation on affine subspaces recently adiceam et al adv math 302 231 279 2016 proved a quantitative version of the convergence case of the khintchine groshev theorem for nondegenerate manifolds motivated by applications to interference alignment in the present paper we obtain analogues of their results for affine subspaces

kazhdan projections random walks and ergodic theorems in this paper we investigate generalizations of kazhdan s property t to the setting of uniformly convex banach spaces we explain the interplay between the existence of spectral gaps and that of kazhdan projections our methods employ markov operators associated to a random walk on the group for which we provide new norm estimates and convergence results this construction exhibits useful properties and flexibility and allows to reclassify the existence of kazhdan projections from a c algebraic phenomenon to a natural object associated to random walks on groups we give a number of applications of these results in particular we address several open questions we give a direct comparison of properties te and fe with lafforgue s reinforced banach property t we obtain shrinking target theorems for orbits of kazhdan groups and apply them to a question of kleinbock and margulis finally we construct non compact ghost projections for warped cones answering a question of willett and yu in this last case we conjecture that such warped cones provide new counterexamples to the coarse baum connes conjecture

a dynamically consistent method to solve nonlinear multidimensional advection reaction equations with fractional diffusion abstract this work is motivated by an extension of both the burgers fisher and the burgers huxley equations in multiple dimensions considering riesz fractional diffusion initial boundary conditions which are positive and bounded are imposed on a closed and bounded set and a finite difference method is proposed to approximate the solutions of the fractional model the methodology is a linear and implicit technique which is based on fractional centered differences we show in this manuscript that the method can be expressed in vector form using a minkowski matrix under suitable conditions the main properties of minkowski matrices are used then to establish the existence and the uniqueness of the solutions of the finite difference method as well as the capability of the technique to preserve the positivity and the boundedness additionally we show that the method is a consistent technique which is stable and convergent with first order of convergence in time and second order in space some illustrative simulations show that the scheme is capable of preserving the positivity and the boundedness of the numerical approximations

testing homogeneity of proportion ratios for stratified correlated bilateral data in two arm randomized clinical trials stratified data analysis is an important research topic in many biomedical studies and clinical trials in this article we develop five test statistics for testing the homogeneity of proportion ratios for stratified correlated bilateral binary data based on an equal correlation model assumption bootstrap procedures based on these test statistics are also considered to evaluate the performance of these statistics and procedures we conduct monte carlo simulations to study their empirical sizes and powers under various scenarios our results suggest that the procedure based on score statistic performs well generally and is highly recommended when the sample size is large procedures based on the commonly used weighted least square estimate and logarithmic transformation with mantel haenszel estimate are recommended as they do not involve any computation of maximum likelihood estimates requiring iterative algorithms we also derive approximate sample size formulas based on the recommended test procedures finally we apply the proposed methods to analyze a multi center randomized clinical trial for scleroderma patients

symplectic methods for the nonlinear schrodinger equation abstract in this paper we show that the spatial discretization of the nonlinear schrodinger equation leads to a hamiltonian system which can be simulated with symplectic numerical schemes in particular we apply two symplectic integrators to the nonlinear schrodinger equation and we demonstrate that they are able to produce accurate results and to preserve very well the invariants of the original system such as the energy and charge

two energy conserving numerical schemes for the sine gordon equation two explicit conservative numerical schemes for the sine gordon equation are proposed their stability and convergence are proved numerical simulation of a sine gordon soliton shows that the new schemes are very accurate and fast

design and numerical analysis of a logarithmic scheme for nonlinear fractional diffusion reaction equations abstract in this work we consider a parabolic partial differential equation with fractional diffusion that generalizes the well known fisher s and hodgkin huxley equations the spatial fractional derivatives are understood in the sense of riesz and initial boundary conditions on a closed and bounded interval are considered here the mathematical model is presented in an equivalent form and a finite difference discretization based on fractional order centered differences is proposed the scheme is the first explicit logarithmic model proposed in the literature to solve fractional diffusion reaction equations we establish rigorously the capability of the technique to preserve the positivity and the boundedness of the methodology moreover we propose conditions under which the monotonicity of the numerical model is also preserved the consistency the stability and the convergence of the scheme are also proved mathematically and some a priori bounds for the numerical solutions are proposed we provide some numerical simulations in order to confirm that the method is capable of preserving the positivity and the boundedness of the approximations and a numerical study of the convergence of the technique is carried out confirming thus the analytical results

a deterministic model for the distribution of the stopping time in a stochastic equation and its numerical solution in this work we consider a stochastic differential equation that generalizes the well known paris equation from fracture of materials the model describes the propagation of cracks on solids and it includes a deterministic summand and a stochastic component in terms of a brownian motion the use of ito s stochastic integral gives an equivalent stochastic integral equation that is further generalized here we note that the probability distribution of the stopping time of the general model satisfies a deterministic diffusion advection partial differential equation for which the solution is known only in a reduced number of particular cases motivated by these analytical results we develop a fast finite difference method to approximate the distribution of the stopping time the method is an explicit exponential like technique that preserves the main features of a probability distribution namely the non negativity the boundedness from above by 1 as well as the spatial monotonicity moreover the method is a monotone technique that is also capable of preserving the temporal monotonicity of the approximations these properties of the proposed methodology are thoroughly established in the present manuscript a continuity condition of the numerical solutions in terms of the initial conditions and the temporal computational parameter is established also together with a limiting property of the methodology when the free parameter tends to infinity for comparison purposes we are providing an implicit and stable discretization of the mathematical model which has a second order of convergence but for which conditions that guarantee the positivity the boundedness and the monotonicity of approximations are not available the numerical simulations obtained with implementations of our techniques show that the explicit method is an efficient scheme that preserves the characteristics of interest non negativity boundedness from above by 1 and monotonicity and that the numerical approximations are in good agreement with the known exact solutions

an energy conservative difference scheme for the nonlinear fractional schrodinger equations abstract in this paper an energy conservative crank nicolson difference scheme for nonlinear riesz space fractional schrodinger equations is studied we give a rigorous analysis of the conservation properties including mass conservation and energy conservation in the discrete sense based on brouwer fixed point theorem the existence of the difference solution is proved by virtue of the energy method the difference solution is shown to be unique and convergent at the order of o 2 h 2 in the l 2 norm with time step and mesh size h finally a linearized iterative algorithm is presented and numerical experiments are given to confirm the theoretical results

continuous limit of discrete systems with long range interaction discrete systems with long range interactions are considered continuous medium models as continuous limit of discrete chain system are defined long range interactions of chain elements that give the fractional equations for the medium model are discussed the chain equations of motion with long range interaction are mapped into the continuum equation with the riesz fractional derivative we formulate the consistent definition of continuous limit for the systems with long range interactions in this paper we consider a wide class of long range interactions that give fractional medium equations in the continuous limit the power law interaction is a special case of this class

estimation methods for the join distribution of repeated binary observations the joint distribution of repeated binary observations is multinomial and can be specified using a representation first suggested by bahadur 1961 in studies in item analysis and prediction 158 168 stanford california stanford university press and later by cox 1972 applied statistics 21 13 120 using the bahadur representation the marginal probabilities of success can be related to a set of covariates using the logistic link function or any other suitable link function besides the parameters of the marginal regression model we may also have interest in the probability of success on any of the repeated measures for example in the six cities study a longitudinal study of the health effects of air pollution we have interest in both the marginal probability of a child wheezing at age t t 10 11 12 and the union probability of wheezing at any of the three ages this union probability can be specified in terms of the joint probabilities and the second higher order correlations we discuss several methods of estimating the parameters of the bahadur model

space time spectral method for the cattaneo equation with time fractional derivative abstract this paper introduces a high order accurate numerical method for solving the cattaneo equation with time fractional derivative it is based on the galerkin legendre spectral method in space and the chebyshev collocation method in time arbitrarily high order accurate can be made in both space and time optimal priori error bound of the semi discrete method and the stability and convergence of the full discrete method are strictly given extensive experimental results confirm the theoretical claims of this method in both space and time

a caveat concerning independence estimating equations with multivariate binary data clustered binary data occur commonly in both the biomedical and health sciences in this paper we consider logistic regression models for multivariate binary responses where the association between the responses is largely regarded as a nuisance characteristic of the data in particular we consider the estimator based on independence estimating equations iee which assumes that the responses are independent this estimator has been shown to be nearly efficient when compared with maximum likelihood ml and generalized estimating equations gee in a variety of settings the purpose of this paper is to highlight a circumstance where assuming independence can lead to quite substantial losses of efficiency in particular when the covariate design includes within cluster covariates assuming independence can lead to a considerable loss of efficiency in estimating the regression parameters associated with those covariates

random effects models with non parametric priors we discuss the performance of non parametric maximum likelihood npml estimators for the distribution of a univariate random effect in the analysis of longitudinal data for continuous data we analyse generated and real data sets and compare the npml method to those that assume a gaussian random effects distribution and to ordinary least squares for binary outcomes we use generated data to study the moderate and large sample performance of the npml compared with a method based on a gaussian random effect distribution in logistic regression we find that estimated fixed effects are compatible for all approaches but that appropriate standard errors for the npml require adjusting the likelihood based standard errors we conclude that the non parametric approach provides an attractive alternative to gaussian based methods though additional evaluations are necessary before it can be recommended for general use

stable multi domain spectral penalty methods for fractional partial differential equations we propose stable multi domain spectral penalty methods suitable for solving fractional partial differential equations with fractional derivatives of any order first a high order discretization is proposed to approximate fractional derivatives of any order on any given grids based on orthogonal polynomials the approximation order is analyzed and verified through numerical examples based on the discrete fractional derivative we introduce stable multi domain spectral penalty methods for solving fractional advection and diffusion equations the equations are discretized in each sub domain separately and the global schemes are obtained by weakly imposed boundary and interface conditions through a penalty term stability of the schemes are analyzed and numerical examples based on both uniform and nonuniform grids are considered to highlight the flexibility and high accuracy of the proposed schemes

a structure preserving method for a class of nonlinear dissipative wave equations with riesz space fractional derivatives abstract in this manuscript we consider an initial boundary value problem governed by a 1 1 dimensional hyperbolic partial differential equation with constant damping that generalizes many nonlinear wave equations from mathematical physics the model considers the presence of a spatial laplacian of fractional order which is defined in terms of riesz fractional derivatives as well as the inclusion of a generic continuously differentiable potential it is known that the undamped regime has an associated positive energy functional and we show here that it is preserved throughout time under suitable boundary conditions to approximate the solutions of this model we propose a finite difference discretization based on fractional centered differences some discrete quantities are proposed in this work to estimate the energy functional and we show that the numerical method is capable of conserving the discrete energy under the same boundary conditions for which the continuous model is conservative moreover we establish suitable computational constraints under which the discrete energy of the system is positive the method is consistent of second order and is both stable and convergent the numerical simulations shown here illustrate the most important features of our numerical methodology

a bounded and efficient scheme for multidimensional problems with anomalous convection and diffusion abstract this work is motivated by a generalization of the well known burgers fisher and burgers huxley equations in multiple dimensions considering riesz fractional diffusion and convection initial boundary conditions which are positive and bounded are imposed on a closed and bounded rectangular domain in this manuscript we propose a finite difference method to approximate the positive and bounded solutions of the fractional model the methodology is a linear three steps crank nicolson technique which is based on the use of fractional centered differences the properties of fractional centered differences are employed to establish the existence and the uniqueness of solutions of the finite difference method as well as the capability of the technique to preserve the positivity and the boundedness of the approximations we show in this work that the method is capable of preserving some of the constant solutions of the continuous model additionally we prove that our technique is a second order consistent stable and quadratically convergent scheme suitable bounds for the numerical solutions are also derived in this work finally some illustrative simulations show that the method is able to preserve the positivity and the boundedness of the numerical approximations in agreement with the analytic results proved in this work numerical comparisons provided in this work confirm the rate of convergence of the numerical technique

two boundedness and monotonicity preserving methods for a generalized fisher kpp equation two finite difference methods are proposed for a generalized fisher kpp equation these methods are capable of preserving the skew symmetry of the equation these methods preserve the positivity boundedness and monotonicity of solutions these methods are solvable these methods are convergent a semi explicit finite difference method and an implicit finite difference method are proposed for a generalized fisher kpp equation with two space variables it is proved that these two methods could preserve the skew symmetry of this equation moreover under a condition on the step sizes the semi explicit method is capable of preserving the positivity the boundedness and the spatial and temporal monotonicity of initial approximations and the implicit method is able to preserve these properties with no restriction on the step sizes the stability and convergence of these two methods are also analyzed respectively finally some numerical simulations are provided to verify the validity of our analytical results

multivariate logistic models for incomplete binary responses abstract in this article we describe a likelihood based regression model appropriate for analyzing incomplete multivariate binary responses we focus on marginal models that is models where the marginal mean or expectation of the binary response is related to a set of covariates the association between the binary responses is modeled in terms of conditional log odds ratios when the nonresponse mechanism is ignorable it is not necessary to specify a nonresponse model and valid inferences can be obtained provided that the likelihood for the responses has been correctly specified but when the nonresponse mechanism is nonignorable valid inferences can only be obtained by incorporating a model for nonresponse an unresolved issue with nonignorable models concerns the identifiability of the parameters so far no general and practically useful necessary and sufficient conditions for identifiability are available here we suggest some simple procedures for examining the identifiability status of nonign

the analysis of binary longitudinal data with time dependent covariates we consider longitudinal studies with binary outcomes that are measured repeatedly on subjects over time the goal of our analysis was to fit a logistic model that relates the expected value of the outcomes with explanatory variables that are measured on each subject however additional care must be taken to adjust for the association between the repeated measurements on each subject we propose a new maximum likelihood method for covariates that may be fixed or time varying we also implement and make comparisons with two other approaches generalized estimating equations which may be more robust to misspecification of the true correlation structure and alternating logistic regression which models association via odds ratios that are subject to less restrictive constraints than are correlations the proposed estimation procedure will yield consistent and asymptotically normal estimates of the regression and correlation parameters if the correlation on consecutive measurements on a subject is correctly specified simulations demonstrate that our approach can yield improved efficiency in estimation of the regression parameter for equally spaced and complete data the gains in efficiency were greatest for the parameter associated with a time by group interaction term and for stronger values of the correlation for unequally spaced data and with dropout according to a missing at random mechanism mark1ml with correctly specified consecutive correlations yielded substantial improvements in terms of both bias and efficiency we present an analysis to demonstrate application of the methods we consider we also offer an r function for easy implementation of our approach copyright 2012 john wiley sons ltd

a numerically efficient hamiltonian method for fractional wave equations abstract in this work we consider a partial differential equation that extends the well known wave equation the model under consideration is a multidimensional equation which includes the presence of both a damping term and a fractional laplacian of the riesz type homogeneous dirichlet boundary conditions on a closed and bounded spatial interval are considered in this work the mathematical model has a fractional hamiltonian which is conserved when the damping coefficient is equal to zero and dissipated otherwise motivated by these facts we propose a finite difference method to approximate the solutions of the continuous model the method is an implicit scheme which is based on the use of fractional centered differences to approximate the spatial fractional derivatives of the model a discretized form of the hamiltoninan is also proposed in this work and we prove analytically that the method is capable of preserving dissipating the discrete energy when the continuous model preserves dissipates the energy we establish rigorously the properties of consistency stability and convergence of the method and provide some a priori bounds for the numerical solutions moreover we prove the existence and the uniqueness of the numerical solutions as well as the unconditional stability of the method in the linear regime some computer simulations that assess the capability of the method to preserve dissipate the energy are carried out for illustration purposes

a structure preserving scheme for the kolmogorov fokker planck equation in this paper we introduce a numerical scheme which preserves the behavior of solutions to the kolmogorov equation as time tends to infinity the method presented is based on a self similar change of variables technique to transform the kolmogorov equation into a new form such that the problem of designing structure preserving schemes for the original equation amounts to building a standard scheme for the transformed equation this transformation also has the added benefit of allowing for an exact operator splitting scheme whereas in the original form a standard operator splitting was only second order finally we verify the preservation of long time behavior through numerical simulations

diophantine approximation for products of linear maps logarithmic improvements this paper is devoted to the study of a problem of cassels in multiplicative diophantine approximation which involves minimising values of a product of affine linear forms computed at integral points it was previously known that values of this product become arbitrary close to zero and we establish that in fact they approximate zero with an explicit rate our approach is based on investigating quantitative density of orbits of higher rank abelian groups

longitudinal data analysis using generalized linear models summary this paper proposes an extension of generalized linear models to the analysis of longitudinal data we introduce a class of estimating equations that give consistent estimates of the regression parameters and of their variance under mild assumptions about the time dependence the estimating equations are derived without specifying the joint distribution of a subject s observations yet they reduce to the score equations for multivariate gaussian outcomes asymptotic theory is presented for the general class of estimators specific cases in which we assume independence m dependence and exchangeable correlation structures from each subject are discussed efficiency of the proposed estimators in two simple situations is considered the approach is closely related to quasi likelih ood some key ironh estimating equation generalized linear model longitudinal data quasi likelihood repeated measures

comparison of alternative regression models for paired binary data we used simulated data derived from real ophthalmologic examples to evaluate the performance of alternative logistic regression approaches for paired binary data approaches considered were standard logistic regression ignoring the correlation between fellow eyes treating individuals classified on the basis of their more impaired eye as the unit of analysis or considering only right eyes marginal logistic regression models fitted by the maximum likelihood approach of lipsitz laird and harrington or the estimating equation approach of liang and zeger and conditional logistic regression models fitted by the maximum likelihood approach of rosner or the estimating equation approach of connolly and liang taylor series approximations were used to compare conditional and marginal parameter estimates consideration of type i and ii error rates found application of standard logistic regression to be inferior to methods that treated the eye as the unit of analysis and accounted for the correlation between fellow eyes among these latter approaches none was uniformly superior to the others across the range of condition considered

estimating logistic regression parameters for bivariate binary data summary consider bivariate binary data with possibly different covariates for each marginal binary observation suppose that the correlation between paired observations is a nuisance and the interest lies in estimating logistic regression parameters associated with the covariates we consider the estimator obtained by using the independence estimating equation with corrected variance this estimator can be very efficient and robust compared with maximum likelihood or the estimator of liang and zeger some comments are made concerning higher dimensional multivariate data

diophantine approximation on rational quadrics we compute the hausdorff dimension of sets of very well approximable vectors on rational quadrics we use ubiquitous systems and the geometry of locally symmetric spaces as a byproduct we obtain the hausdorff dimension of the set of rays with a fixed maximal singular direction which move away into one end of a locally symmetric space at linear depth infinitely many times

finite difference schemes for nonlinear wave equation that inherit energy conservation property abstract we propose two general finite difference schemes that inherit energy conservation property from nonlinear wave equations such as the nonlinear klein gordon equation nlkge one of proposed schemes is implicit and another is explicit many studies exist on fdss that inherit energy conservation property from nlkge and we can derive all of their schemes from the proposed general schemes in this paper the most important feature of our procedure is a rigorous discretization of variational derivatives using summation by parts which implies that the inherited properties are satisfied exactly because of this the derived schemes are expected to be numerically stable and yield solutions converging to pde solutions we make new fdss for fermi pasta ulam equation string vibration equation shimoji kawai equation ske and ebihara equation and verify numerically the inheritance of the energy conservation property for nlkge and ske

efficient energy preserving integrators for oscillatory hamiltonian systems in this paper we focus our attention on deriving and analyzing an efficient energy preserving formula for the system of nonlinear oscillatory or highly oscillatory second order differential equations q t mq t f q t where m is a symmetric positive semi definite matrix with m 1 and f q q u q is the negative gradient of a real valued function u q this system is a hamiltonian system with the hamiltonian h p q 1 2 p t p 1 2 q t mq u q the energy preserving formula exactly preserves the hamiltonian we analyze in detail the properties of the energy preserving formula and propose new efficient energy preserving integrators in the sense of numerical implementation the convergence analysis of the fixed point iteration is presented for the implicit integrators proposed in this paper it is shown that the convergence of implicit average vector field methods is dependent on m whereas the convergence of the new energy preserving integrators is independent of m the fermi pasta ulam problem and the sine gordon equation are carried out numerically to show the competence and efficiency of the novel integrators in comparison with the well known average vector field methods in the scientific literature

crank nicolson galerkin spectral method for solving two dimensional time space distributed order weakly singular integro partial differential equation abstract the fractional pdes based upon the distributed order fractional derivative have several applications in physics the two dimensional time space distributed order weakly singular integro partial differential model is investigated by a combination of finite difference and galerkin spectral methods a second order finite difference formula is employed to approximate the temporal variable in this stage the stability and convergence of the semi discrete scheme are proved then the galerkin spectral method based on the modified jacobi polynomials is applied to discrete the space variable also in this step the error estimate of the full discrete scheme is studied finally two test problems have been presented to confirm the theoretical results

a linearly implicit conservative difference scheme for the space fractional coupled nonlinear schrodinger equations abstract in this paper a linearly implicit conservative difference scheme for the coupled nonlinear schrodinger equations with space fractional derivative is proposed this scheme conserves the mass and energy in the discrete level and only needs to solve a linear system at each step the existence and uniqueness of the difference solution are proved the stability and convergence of the scheme are discussed and it is shown to be convergent of order o 2 h 2 in the discrete l 2 norm with the time step and mesh size h when the fractional order is two all those results are in accord with the difference scheme proposed for the classical non fractional coupled nonlinear schrodinger equations some numerical examples are also reported

error estimate of finite element finite difference technique for solution of two dimensional weakly singular integro partial differential equation with space and time fractional derivatives abstract in the current investigation an error estimate has been proposed to solve the two dimensional weakly singular integro partial differential equation with space and time fractional derivatives based on the finite element finite difference scheme the time and space derivatives are based on the riemann liouville and riesz fractional derivatives respectively at first the temporal variable has been discretized by a second order difference scheme and then the space variable has been approximated by the finite element method fem the analytical study shows that the presented scheme is unconditionally stable and convergent finally some examples have been introduced to confirm the theoretical results and efficiency of the proposed technique

uniqueness of invariant means for measure preserving transformations for some compact abelian groups x e g t n 2 and hii 1 z2 the group g of topological automorphisms of x has the haar integral as the unique g invariant mean on l x ax this gives a new characterization of lebesgue measure on the bounded lebesgue measurable subsets 8 of rn n 3 it is the unique normalized positive finitely additive measure on 8 which is invariant under isometries and the transformation of rn xi xn xi x2 x2 i x other examples of as well as necessary and sufficient conditions for the uniqueness of a mean on l q x 3 p which is invariant by some group of measure preserving transformations of the probability space x 3 p are described 0 introduction let 8 be the ring of bounded lebesgue measurable sets in rn or in sn the n dimensional unit sphere in rn and let xn be the lebesgue measure on 8 normalized by xn j 1 where j 0 1 or by xn sn 1 respectively the classical characterization by lebesgue of is that it is the unique positive real valued functionf on 8 which satisfies these three conditions a f jn 1 orf sn 1 respectively b f is invariant under isometries c f is countably additive in 1923 banach 1 studied the question of ruziewicz as to whetherf is still unique when c is replaced by c0 f is finitely additive banach gave a negative answer to this question for r 1 r2 and s1 but for r n 3 or sn n 2 this question is still unanswered in this paper we will prove a theorem which comes close to a solution of this problem for rn n 3 theorem 3 10 if f satisfies a b and c0 and if f is invariant under the transformation of r n n 3 given by xi x2 x f xi x2 x2 9 xn then f xn our method of proof will give several other theorems similar to this one in theorem 3 10 we do not need the full strength of b we need only know that f is absolutely continuous with respect to x i e for n e 3 f n 0 whenever n 0 the fact that for rn n 3 or sn n 2 a b and c0 imply this absolute continuity of f is an observation of tarski which is proved like this received by the editors march 3 1980 presented to the society january 10 1981 ams mos subject classifications 1970 primary 28a70 43a07 43a40 1981 american mathematical society 0002 9947 81 0000 0268 04 50

crank nicolson method for the fractional diffusion equation with the riesz fractional derivative we examine a numerical method to approximate to a fractional diffusion equation with the riesz fractional derivative in a finite domain which has second order accuracy in time and space level in order to approximate the riesz fractional derivative we use the fractional centered derivative approach we determine the error of the riesz fractional derivative to the fractional centered difference we apply the crank nicolson method to a fractional diffusion equation which has the riesz fractional derivative and obtain that the method is unconditionally stable and convergent numerical results are given to demonstrate the accuracy of the crank nicolson method for the fractional diffusion equation with using fractional centered difference approach

some new structure preserving algorithms for general multi symplectic formulations of hamiltonian pdes many partial differential equations pdes can be written as a multi symplectic hamiltonian system which has three local conservation laws namely multi symplectic conservation law local energy conservation law and local momentum conservation law in this paper we give several systematic methods for discretizing general multi symplectic formulations of hamiltonian pdes including a local energy preserving algorithm a class of global energy preserving methods and a local momentum preserving algorithm the methods are illustrated by the nonlinear schrodinger equation and the korteweg de vries equation numerical experiments are presented to demonstrate the conservative properties of the proposed numerical methods

construction of quasi periodic solutions of state dependent delay differential equations by the parameterization method ii analytic case abstract we construct analytic quasi periodic solutions of a state dependent delay differential equation with quasi periodically forcing we show that if we consider a family of problems that depends on one dimensional parameters with some non degeneracy conditions there is a positive measure set of parameters for which the system admits analytic quasi periodic solutions the main difficulty to be overcome is the appearance of small divisors and this is the reason why we need to exclude parameters our main result is proved by a nash moser fast convergent method and is formulated in the a posteriori format of numerical analysis that is given an approximate solution of a functional equation which satisfies some non degeneracy conditions we can find a true solution close to it this is in sharp contrast with the finite regularity theory developed in 18 we conjecture that the exclusion of parameters is a real phenomenon and not a technical difficulty more precisely for generic families of perturbations the quasi periodic solutions are only finitely differentiable in open sets in the complement of parameters set

fractional generalization of gradient and hamiltonian systems we consider a fractional generalization of hamiltonian and gradient systems we use differential forms and exterior derivatives of fractional orders we derive fractional generalization of helmholtz conditions for phase space examples of fractional gradient and hamiltonian systems are considered the stationary states for these systems are derived

algorithm for some anomalously diffusive hyperbolic systems in molecular dynamics theoretical analysis and pattern formation abstract departing from a two dimensional hyperbolic system that describes the interaction between some activator and inhibitor substances in chemical reactions we investigate a general form of that model using a finite difference approach the model under investigation is a nonlinear system consisting of two coupled partial differential equations with generalized reaction terms the presence of two dimensional diffusive terms consisting of fractional operators of the riesz type is considered here using spatial differentiation orders in the set 0 1 1 2 we impose initial conditions on a closed and bounded rectangle and a finite difference methodology based on the use of fractional centered differences is proposed among the most important results of this work we prove the existence and the uniqueness of the solutions of the numerical method and establish analytically the second order consistency of our scheme moreover the discrete energy method is employed to prove the stability and the quadratic convergence of the technique some numerical simulations obtained through our method show the presence of turing patterns and wave instabilities in agreement with some reports found in the literature on superdiffusive hyperbolic activator inhibitor systems we show numerically that the presence of turing patterns is independent of the size of the spatial domain as a new application we show that turing patterns are also present in subdiffusive scenarios

correlated binary regression with covariates specific to each binary observation regression methods are considered for the analysis of correlated binary data when each binary observation may have its own covariates it is argued that binary response models that condition on some or all binary responses in a given block are useful for studying certain types of dependencies but not for the estimation of marginal response probabilities or pairwise correlations fully parametric approaches to these latter problems appear to be unduly complicated except in such special cases as the analysis of paired binary data hence a generalized estimating equation approach is advocated for inference on response probabilities and correlations illustrations involving both small and large block sizes are provided

on the solution of a generalized higgs boson equation in the de sitter space time through an efficient and hamiltonian scheme abstract the present work is the first paper in the literature to report on a hamiltonian discretization of the fractional higgs boson equation in the de sitter space time and its theoretical analysis more precisely we design herein a numerically efficient finite difference hamiltonian technique for the solution of a fractional extension of the higgs boson equation in the de sitter space time the model under investigation is a multidimensional equation with generalized potential and riesz space fractional derivatives of orders in 1 2 an energy integral for the model is readily available and we propose a nonlinear implicit and consistent numerical technique based on fractional order centered differences with similar hamiltonian properties in the discrete scenario a fractional energy approach is used then to prove the properties of stability and convergence of the technique for simulation purposes we consider both the classical and the fractional higgs real valued scalar fields in the de sitter space time and find results qualitatively similar to those available in the literature for the sake of convenience we provide the matlab code of an alternative linear discretization of the method presented in this work this linear implicit approach is thoroughly analyzed also

fractional schrodinger equation some properties of the fractional schrodinger equation are studied we prove the hermiticity of the fractional hamilton operator and establish the parity conservation law for fractional quantum mechanics as physical applications of the fractional schrodinger equation we find the energy spectra of a hydrogenlike atom fractional bohr atom and of a fractional oscillator in the semiclassical approximation an equation for the fractional probability current density is developed and discussed we also discuss the relationships between the fractional and standard schrodinger equations

the ergodic theory of shrinking targets to any dynamical system equipped with a metric we associate a class of well approximable sets in the case of an expanding rational map of the riemann sphere acting on its julia set we estimate and in some cases compute the hausdorff dimension of the associated well approximable sets the methods used show a clear link between distortion properties and the type of results obtained in this paper via ergodic theory and ubiquity

an analysis of implicit conservative difference solver for fractional klein gordon zakharov system abstract in this paper we propose an efficient linearly implicit conservative difference solver for the fractional klein gordon zakharov system first of all we present a detailed derivation of the energy conservation property of the system in the discrete setting then by using the mathematical induction it is proved that the proposed scheme is uniquely solvable subsequently by virtue of the discrete energy method and a cut off function technique it is shown that the proposed solver possesses the convergence rates of o t 2 h 2 in the sense of l and l2 norms respectively and is unconditionally stable finally numerical results testify the effectiveness of the proposed scheme and exhibit the correctness of theoretical results

limit theorems for partially hyperbolic systems we consider a large class of partially hyperbolic systems containing among others affine maps frame flows on negatively curved manifolds and mostly contracting diffeomorphisms if the rate of mixing is sufficiently high the system satisfies many classical limit theorems of probability theory

on the solution of hyperbolic two dimensional fractional systems via discrete variational schemes of high order of accuracy abstract in this work we consider a general class of damped wave equations in two spatial dimensions the model considers the presence of weyl space fractional derivatives as well as a generic nonlinear potential the system has an associated positive energy functional when damping is not present in which case the model is capable of preserving the energy throughout time meanwhile the energy of the system is dissipated in the damped scenario in this work the weyl space fractional derivatives are approximated through second order accurate fractional centered differences a high order compact difference scheme with fourth order accuracy in space and second order in time is proposed some associated discrete quantities are introduced to estimate the energy functional we prove that the numerical method is capable of conserving the discrete variational structure under the same conditions for which the continuous model is conservative the positivity of the discrete energy of the system is also discussed the properties of consistency solvability stability and convergence of the proposed method are rigorously proved we provide some numerical simulations that illustrate the agreement between the physical properties of the continuous and the discrete models

simultaneously modeling joint and marginal distributions of multivariate categorical responses abstract we discuss model fitting methods for analyzing simultaneously the joint and marginal distributions of multivariate categorical responses the models are members of a broad class of generalized logit and loglinear models we fit them by improving a maximum likelihood algorithm that uses lagrange s method of undetermined multipliers and a newton raphson iterative scheme we also discuss goodness of fit tests and adjusted residuals and give asymptotic distributions of model parameter estimators for this class of models inferences are equivalent for poisson and multinomial sampling assumptions simultaneous models for joint and marginal distributions may be useful in a variety of applications including studies dealing with longitudinal data multiple indicators in opinion research cross over designs social mobility and inter rater agreement the models are illustrated for one such application using data from a recent general social survey regarding opinions about various types of government s

kam theory for quasi periodic equilibria in 1d quasi periodic media ii long range interactions we consider frenkel kontorova models corresponding to one dimensional quasi crystal with non nearest neighbor and many body interactions we formulate and prove a kam type theorem which establishes the existence of quasi periodic solutions the interactions we consider do not need to be of finite range or involve finitely many particles but have to decay sufficiently fast with respect to the distance of the position of the atoms the kam theorem we present has an a posteriori format we do not need to assume that the system is close to integrable we just assume that there is an approximate solution for the functional equation which satisfies some non degeneracy conditions

local structure preserving algorithms for the good boussinesq equation in this paper we derive a series of local structure preserving algorithms for the good boussinesq equation including multisymplectic geometric structure preserving algorithms local energy preserving algorithms and local momentum preserving algorithms the outstanding advantage of the proposed algorithms is that they conserve these local structures in any time space region exactly for example the proposed local energy preserving algorithms preserve the local energy conservation law in any local domain therefore the local structure preserving algorithms overcome the shortage of global structure preserving algorithms on the boundary conditions especially with suitable boundary conditions such as periodic or homogeneous boundary conditions the local structure preserving algorithms will be global structure preserving algorithms numerical results verify the theoretical analysis

dissipative or conservative finite difference schemes for complex valued nonlinear partial differential equations abstract we propose a new procedure for designing finite difference schemes that inherit energy conservation or dissipation property from complex valued nonlinear partial differential equations pdes such as the nonlinear schrodinger equation the ginzburg landau equation and the newell whitehead equation the procedure is a complex version of the procedure that furihata has recently presented for real valued nonlinear pdes furthermore we show that the proposed procedure can be modified for designing linearly implicit finite difference schemes that inherit energy conservation or dissipation property

maximum likelihood regression methods for paired binary data we discuss maximum likelihood methods for analysing binary responses measured at two times such as in a cross over design we construct a 2 2 table for each individual with cell probabilities corresponding to the cross classification of the responses at the two times the underlying likelihood for each individual is multinomial with four cells the three dimensional parameter space of the multinomial distribution is completely specified by the two marginal probabilities of success of the 2 2 table and an association parameter between the binary responses at the two times we examine a logistic model for the marginal probabilities of the 2 2 table for individual i the association parameters we consider are either the correlation coefficient the odds ratio or the relative risk simulations show that the parameter estimates for the logistic regression model for the marginal probabilities are not very sensitive to the parameters used to describe the association between the binary responses at the two times thus we suggest choosing the measure of association for ease of interpretation

central limit theorems for the shrinking target problem suppose b i b p r i are nested balls of radius r i about a point p in a dynamical system t x the question of whether t i x b i infinitely often i o for a e x is often called the shrinking target problem in many dynamical settings it has been shown that if e n sum i 1 n mu b i diverges then there is a quantitative rate of entry and lim n to infty frac 1 e n sum j 1 n 1 b i t i x to1 for a e x x this is a self norming type of strong law of large numbers we establish self norming central limit theorems clt of the form lim n to infty frac 1 a n sum i 1 n 1 b i t i x mu b i to n 0 1 in distribution for a variety of hyperbolic and non uniformly hyperbolic dynamical systems the normalization constants are a 2 n sim e sum i 1 n 1 b i t i x mu b i 2 dynamical systems to which our results apply include smooth expanding maps of the interval rychlik type maps gibbs markov maps rational maps and in higher dimensions piecewise expanding maps for such central limit theorems the main difficulty is to prove that the non stationary variance has a limit in probability

positivity preserving dg and central dg methods for ideal mhd equations ideal mhd equations arise in many applications such as astrophysical plasmas and space physics and they consist of a system of nonlinear hyperbolic conservation laws the exact density r and pressure p should be non negative numerically such positivity property is not always satisfied by approximated solutions one can encounter this when simulating problems with low density high mach number or much large magnetic energy compared with internal energy when this occurs numerical instability may develop and the simulation can break down in this paper we propose positivity preserving discontinuous galerkin and central discontinuous galerkin methods for solving ideal mhd equations by following x zhang c w shu journal of computational physics 229 2010 8918 8934 in one dimension the positivity preserving property is established for both methods under a reasonable assumption the performance of the proposed methods in terms of accuracy stability and positivity preserving property is demonstrated through a set of one and two dimensional numerical experiments the proposed methods formally can be of any order of accuracy

inhomogeneous theory of dual diophantine approximation on manifolds the theory of inhomogeneous diophantine approximation on manifolds is developed in particular the notion of nice manifolds is introduced and the divergence part of the groshev type theory is established for all such manifolds our results naturally incorporate and generalize the homogeneous measure and dimension theorems for non degenerate manifolds established to date the results have natural applications beyond the standard inhomogeneous theory such as diophantine approximation by algebraic integers

an extension of quantitative nondivergence and applications to diophantine exponents we present a sharpening of nondivergence estimates for unipotent or more gener ally polynomial like flows on homogeneous spaces applied to metric diophantine approxima tion it yields precise formulas for diophantine exponents of affine subspaces ofr n and their nondegenerate submanifolds

properties of ideal point classification models for bivariate binary data the ideal point classification ipc model was originally proposed for analysing multinomial data in the presence of predictors in this paper we studied properties of the ipc model for analysing bivariate binary data with a specific focus on three quantities 1 the marginal probabilities 2 the association structure between the two binary responses and 3 the joint probabilities we found that the ipc model with a specific class point configuration represents either the marginal probabilities or the association structure however the ipc model is not able to represent both quantities at the same time we then derived a new parametrization of the model the bivariate ipc bipc model which is able to represent both the marginal probabilities and the association structure like the standard ipc model the results of the bipc model can be displayed in a biplot from which the effects of predictors on the binary responses and on their association can be read we will illustrate our findings with a psychological example relating personality traits to depression and anxiety disorders

a bivariate logistic regression model based on latent variables bivariate observations of binary and ordinal data arise frequently and require a bivariate modeling approach in cases where one is interested in aspects of the marginal distributions as separate outcomes along with the association between the two we consider methods for constructing such bivariate models based on latent variables with logistic marginals and propose a model based on the ali mikhail haq bivariate logistic distribution we motivate the model as an extension of that based on the gumbel type 2 distribution as considered by other authors and as a bivariate extension of the logistic distribution which preserves certain natural characteristics basic properties of the obtained model are studied and the proposed methods are illustrated through analysis of two data sets a basic science cognitive experiment of visual recognition and awareness and a clinical data set describing assessments of walking disability among multiple sclerosis patients

zimmer s conjecture for actions of mathrm sl m mathbb z we prove zimmer s conjecture for c 2 actions by finite index subgroups of mathrm sl m mathbb z provided m 3 the method utilizes many ingredients from our earlier proof of the conjecture for actions by cocompact lattices in mathrm sl m mathbb r brown et al in zimmer s conjecture subexponential growth measure rigidity and strong property t 2016 arxiv 1608 04995 but new ideas are needed to overcome the lack of compactness of the space g times m gamma admitting the induced g action non compactness allows both measures and lyapunov exponents to escape to infinity under averaging and a number of algebraic geometric and dynamical tools are used control this escape new ideas are provided by the work of lubotzky mozes and raghunathan on the structure of nonuniform lattices and in particular of mathrm sl m mathbb z providing a geometric decomposition of the cusp into rank one directions whose geometry is more easily controlled the proof also makes use of a precise quantitative form of non divergence of unipotent orbits by kleinbock and margulis and an extension by de la salle of strong property t to representations of nonuniform lattices

on a positivity preserving numerical model for a linearized hyperbolic fisher kolmogorov petrovski piscounov equation abstract in this work we introduce a nonstandard finite difference scheme to approximate positive solutions of a modified hyperbolic fisher kolmogorov petrovski piscounov equation appearing in the investigation of the dynamics of some populations the technique has a consistency of second order and it provides nonnegative approximations for nonnegative initial profiles the stability analysis of the method is carried out in detail and the scheme is validated against known analytical solutions for some initial boundary value problems

quantitative recurrence results letx be a probability measure spacex x endowed with a compatible metricd so that x d has a countable base it is well known that ift x x is measure preserving then almost all pointsx x are recurrent i e lim begin array 20 c inf n geqq 1 end array d x t n x 0 we show that under the additional assumption that the hausdorff measureh x ofx is finite for some 0 this result can be strengthened lim begin array 20 c inf n geqq 1 end array left n 1 alpha d x t n x right infty for almost all pointsx x a number of applications are considered

flows on homogeneous spaces and diophantine approximation on manifolds we present a new approach to metric diophantine approximation on manifolds based on the correspondence between approximation properties of numbers and orbit properties of certain flows on homogeneous spaces this approach yields a new proof of a conjecture of mahler originally settled by v g sprindzuk in 1964 we also prove several related hypotheses of baker and sprindzuk formulated in 1970s the core of the proof is a theorem which generalizes and sharpens earlier results on nondivergence of unipotent flows on the space of lattices

a fourth order compact adi scheme for two dimensional nonlinear space fractional schrodinger equation in this paper a novel compact operator is derived for the approximation of the riesz derivative with order alpha in 1 2 the compact operator is proved with fourth order accuracy combining the compact operator in space discretization a linearized difference scheme is proposed for a two dimensional nonlinear space fractional schrodinger equation it is proved that the difference scheme is uniquely solvable stable and convergent with order o tau 2 h 4 where tau is the time step size h max h 1 h 2 and h 1 h 2 are space grid sizes in the x direction and the y direction respectively based on the linearized difference scheme a compact alternating direction implicit scheme is presented and analyzed numerical results demonstrate that the compact operator does not bring in extra computational cost but improves the accuracy of the scheme greatly

positivity and boundedness preserving schemes for space time fractional predator prey reaction diffusion model the semi implicit schemes for the nonlinear predator prey reaction diffusion model with the space time fractional derivatives are discussed where the space fractional derivative is discretized by the fractional centered difference and wsgd scheme the stability and convergence of the semi implicit schemes are analyzed in the l norm we theoretically prove that the numerical schemes are stable and convergent without the restriction on the ratio of space and time stepsizes and numerically further confirm that the schemes have first order convergence in time and second order convergence in space then we discuss the positivity and boundedness properties of the analytical solutions of the discussed model and show that the numerical solutions preserve the positivity and boundedness the presented numerical examples confirm the theoretical results and convergence orders

numerical solution of the sine gordon equation two new difference schemes are proposed for the sine gordon equation they have the advantage that there is a discrete energy which is conserved their stability and convergence is proved also we show some numerical computations using one of these schemes

logarithm laws for flows on homogeneous spaces in this paper we generalize and sharpen d sullivan s logarithm law for geodesics by specifying conditions on a sequence of subsets at t n of a homogeneous space g g a semisimple lie group an irreducible lattice and a sequence of elements ft of g under which t n ftx at is infinite for a e x g the main tool is exponential decay of correlation coefficients of smooth functions ong besides the general higher rank version of sullivan s result as a consequence we obtain a new proof of the classical khinchin groshev theorem on simultaneous diophantine approximation and settle a conjecture recently made by m skriganov

a compact fourth order in space energy preserving method for riesz space fractional nonlinear wave equations in this work we investigate numerically a nonlinear hyperbolic partial differential equation with space fractional derivatives of the riesz type the model under consideration generalizes various nonlinear wave equations including the sine gordon and the nonlinear klein gordon models the system considered in this work is conservative when homogeneous dirichlet boundary conditions are imposed motivated by this fact we propose a finite difference method based on fractional centered differences that is capable of preserving the discrete energy of the system the method under consideration is a nonlinear implicit scheme which has various numerical properties among the most interesting numerical features we show that the methodology is consistent of second order in time and fourth order in space moreover we show that the technique is stable and convergent some numerical simulations show that the method is capable of preserving the energy of the discrete system this characteristic of the technique is in obvious agreement with the properties of its continuous counterpart