the inverse gaussian distribution in wireless channels second order statistics and channel capacity in this paper we introduce the inverse gaussian ig fading distribution to model the envelope and the power of the received signal in radio propagation fading channels and free space optical fso channels the joint distribution of the ig process and its derivative and the marginal distribution of the derivative are developed the obtained formulas are then used to derive the second order statistics of the received signal envelope and the channel capacity under ig and nakagami ig nig fading distributions numerical results sustaining our analysis are provided and the impacts of various parameters on the system performance are investigated

symmetry preserving difference schemes for some heat transfer equations lie group analysis of differential equations is a generally recognized method which provides invariant solutions integrability conservation laws etc in this paper we present three characteristic examples of the construction of invariant difference equations and meshes where the original continuous symmetries are preserved in discrete models conservation of symmetries in difference modelling helps to retain qualitative properties of the differential equations in their difference counterparts

fade statistics in nakagami lognormal channels new expressions for the average level crossing rate lcr and average time fade duration afd of the signal envelope in microcellular mobile radio channels characterized by nakagami 1960 fading and lognormal shadowing so called nakagami lognormal nln channels are derived new lcr and afd are computed and the effects of the nakagami m factor and shadowing spread spl sigma sub s on these fade statistics are discussed the generality of the nln model is demonstrated by matching some lcr curves with previously reported measurement for the land mobile satellite channel commonly described as the rice lognormal channels

lie point symmetries of the discrete liouville equation the liouville equation is well known to be linearizable by a point transformation it has an infinite dimensional lie point symmetry algebra isomorphic to a direct sum of two virasoro algebras we show that it is not possible to discretize the equation keeping the entire symmetry algebra as point symmetries we do however construct a difference system approximating the liouville equation that is invariant under the maximal finite subgroup the invariant scheme is an explicit one and provides a much better approximation of exact solutions than a comparable standard noninvariant scheme and also than a scheme invariant under an infinite dimensional group of generalized symmetries

periodic solutions of a second order nonlinear system abstract we discuss the existence of periodic solutions of the system x t d dt grad f x t g t x t e t under various asymptotic behaviours of g

first integrals of difference hamiltonian equations in the present paper the well known noether s identity which represents the connection between symmetries and first integrals of euler lagrange equations is rewritten in terms of the hamiltonian function this approach based on the hamiltonian identity provides a simple and clear way to find first integrals of canonical hamiltonian equations without integration a discrete analog of the hamiltonian identity is developed it leads to a connection between symmetries and first integrals of difference hamiltonian equations that can be used to conserve the structural properties of hamiltonian equations under discretizaton the results are illustrated by a number of examples for both continuous and difference hamiltonian equations

invariant conservation law preserving discretizations of linear and nonlinear wave equations symmetry and conservation law preserving finite difference discretizations are obtained for linear and nonlinear one dimensional wave equations on five and nine point stencils using the theory of lie point symmetries of difference equations and the discrete direct multiplier method of conservation law construction in particular for the linear wave equation an explicit five point scheme is presented that preserves the discrete analogs of its basic geometric point symmetries and six of the corresponding conservation laws for a class of nonlinear wave equations arising in hyperelasticity a nine point implicit scheme is constructed preserving four point symmetries and three local conservation laws other discretizations of the nonlinear wave equations preserving different subsets of conservation laws are discussed

lie symmetries for integrable evolution equations on the lattice in this work we show how to construct symmetries for the differential difference equations associated with the discrete schrodinger spectral problem we find the whole set of symmetries which in the continuous limit go into the lie point symmetries of the corresponding partial differential equation i e the korteweg de vries equation among these of particular relevance is the non autonomous symmetry which in the continuous limit goes into the dilation symmetry for the corresponding equation unlike the continuous case this symmetry turns out to be a master symmetry thus belonging to the infinite dimensional group of generalized symmetries

symmetry adapted moving mesh schemes for the nonlinear schrodinger equation in this paper we consider symmetry preserving difference schemes for the nonlinear schrodinger equation where n is the number of space dimensions this equation describes one dimensional waves in n space dimensions in many physical situations including phenomena in plasma physics and nonlinear optics we will consider the nonintegrable case n 2 for which the equation admits solutions that blow up in a finite time and construct discretizations based upon moving mesh schemes that have the same lie group properties and lagrangian structures as the continuous counterpart

lie symmetries of multidimensional difference equations a method is presented for calculating the lie point symmetries of a scalar difference equation on a two dimensional lattice the symmetry transformations act on the equations and on the lattice they take solutions into solutions and can be used to perform symmetry reduction the method generalizes the one presented in a recent publication for the case of ordinary difference equations in turn it can easily be generalized to difference systems involving an arbitrary number of dependent and independent variables

on symmetries conservation laws and variational problems for partial differential equations the problem of the correspondence between symmetries and conservation laws for partial differential equations is considered for lagrangian systems the set of noether variational symmetries can be shown to lead to the set of all local conservation laws for partial differential equations without well defined lagrangian functions there is no universal correspondence between symmetries and conservation laws in this article it is shown that for a large class of differential equations there is a natural way to associate conservation laws with symmetries the class consists of many interesting equations e g korteweg de vries equation kadomtsev petviashvili equation boussinesq equation nonlinear diffusion equation monge ampere equation regularized long wave equation and navier stokes equations characteristics of the corresponding conservation laws are calculated and examples are given for lagrangian systems the consistency of the approach with the standard noether results is discussed

are there contact transformations for discrete equations we define infinitesimal contact transformations for ordinary difference schemes as transformations that depend on k 1 lattice points k 1 and can be integrated to form a local or global lie group we then prove that such contact transformations do not exist

the adjoint equation method for constructing first integrals of difference equations a new method for finding first integrals of discrete equations is presented it can be used for discrete equations which do not possess a variational lagrangian or hamiltonian formulation the method is based on a newly established identity which links symmetries of the underlying discrete equations solutions of the discrete adjoint equations and first integrals the method is applied to an invariant mapping and to discretizations of second order and third order ordinary differential equations in examples the set of independent first integrals makes it possible to find the general solution of the discrete equations

difference schemes with point symmetries and their numerical tests symmetry preserving difference schemes approximating second and third order ordinary differential equations are presented they have the same three or four dimensional symmetry groups as the original differential equations the new difference schemes are tested as numerical methods the obtained numerical solutions are shown to be much more accurate than those obtained by standard methods without an increase in cost for an example involving a solution with a singularity in the integration region the symmetry preserving scheme contrary to standard ones provides solutions valid beyond the singular point

periodic solutions of linear and quasilinear neutral functional differential equations abstract in this paper we will use the fourier expansions to generalize hale s alternative results for the existence of periodic solutions of linear retarded functional differential equations to neutral equations using some a priori estimation and the leray schauder degree theory we will obtain some existence theorems of periodic solutions for a class of quasilinear neutral equations the results also generalize the works of hatvani and krisztin on retarded equations

lie algebra contractions and symmetries of the toda hierarchy the lie algebra l of generalized and point symmetries of the equations in the toda hierarchy is shown to be a semidirect sum of two infinite dimensional lie algebras one perfect the other abelian in the continuous limit the structure of the lie algebra changes a contraction occurs with the lattice spacing as the contraction parameter in particular for the toda equation itself a set of five elements involving both point symmetries and generalized ones contracts to the point symmetry algebra of the potential kdv equation

contact transformations for difference schemes we define a class of transformations of the dependent and independent variables in an ordinary difference scheme the transformations leave the solution set of the system invariant and reduces to a group of contact transformations in the continuous limit we use a simple example to show that the class is not empty and that such contact transformations for discrete systems genuinely exist

wavelets and quantum algebras wavelets known to be useful in nonlinear multiscale processes and in multiresolution analysis are shown to have a q deformed algebraic structure the translation and dilation operators associate to any scaling equation a nonlinear two parameter algebra this structure can be mapped onto the quantum group slq 2 in one limit and approaches a fourier series generating algebra in another limit a duality between any scaling function and its corresponding nonlinear algebra is obtained examples for the haar and b wavelets are worked out in detail

invariant difference schemes and their application to sl 2 hbox mathbb r invariant ordinary differential equations we present an exposition of a method of discretizing ordinary differential equations while preserving their lie point symmetries this method is very general and can be applied to any ordinary differential equations ode with a nontrivial symmetry group the method is applied to obtain numerical solutions of second and third order odes invariant under two different realizations of the symmetry preserving method is shown to provide a better qualitative description of solutions than standard methods in particular it provides solutions that are valid close to singularities and beyond them

symmetries of discrete dynamical systems involving two species the lie point symmetries of a coupled system of two nonlinear differential difference equations are investigated it is shown that in special cases the symmetry group can be infinite dimensional in other cases up to ten dimensional the equations can describe the interaction of two long molecular chains each involving one type of atoms

symmetry group of partial differential equations and of differential difference equations the toda lattice versus the korteweg de vries equation the authors correlate the symmetry group of the continuous transformations of the toda lattice to that of the korteweg de vries equation they show how by taking into account the continuous limit of the toda lattice the four parameter symmetry group of the toda lattice is contained in that of the kdv equation by an inverse process discretization of the symmetry group of the kdv equation they find a discrete element of the symmetry group of the toda lattice which gives by symmetry reduction its soliton solution

conservation laws of partial difference equations with two independent variables this paper introduces a technique for obtaining the conservation laws of a given scalar partial difference equation with two independent variables unlike methods that are based on nother s theorem this new technique does not use symmetries neither does it require the difference equation to have any special structure such as a lagrangian hamiltonian or multisymplectic formulation instead it uses a discrete analogue of the variational complex

on the existence of periodic solutions to rayleigh differential equation of neutral type in the critical case abstract in this paper the authors study the existence of periodic solutions for a second order neutral functional differential equation x t c x t f x t g t x t t e t in the critical case c 1 by analyzing some properties of the linear difference operator a a x t x t c x t and using mawhin s continuation theorem some new results are obtained

level crossing rate and average outage duration of free space optical links the level crossing rate lcr and the average outage duration aod are two important second order statistics that allow a deeper understanding of the behavior of the channel in this paper we study these metrics in order to assess the performance of free space optical fso communication links in the presence of weak atmospheric turbulence and rice induced pointing errors more specifically we derive an integral and a gauss laguerre quadrature representation for both the lcr and the aod in the single hop case and for their respective bounds in the multihop case selected numerical simulations are presented to show the accuracy of the derived results and to study the effect of certain system parameters on these two performance metrics

the korteweg de vries equation and its symmetry preserving discretization the korteweg de vries equation is one of the most important nonlinear evolution equations in the mathematical sciences in this article invariant discretization schemes are constructed for this equation both in the lagrangian and in the eulerian form we also propose invariant schemes that preserve the momentum numerical tests are carried out for all invariant discretization schemes and related to standard numerical schemes we find that the invariant discretization schemes give generally the same level of accuracy as the standard schemes with the added benefit of preserving galilean transformations which is demonstrated numerically as well

symmetries and first integrals of ordinary difference equations this paper describes a new symmetry based approach to solving a given ordinary difference equation by studying the local structure of the set of solutions we derive a systematic method for determining one parameter lie groups of symmetries in closed form such groups can be used to achieve successive reductions of order if there are enough symmetries the difference equation can be completely solved several examples are used to illustrate the technique for transitive and intransitive symmetry groups it is also shown that every linear second order ordinary difference equation has a lie algebra of symmetry generators that is isomorphic to sl 3 the paper concludes with a systematic method for constructing first integrals directly which can be used even if no symmetries are known

lie group classification of second order ordinary difference equations a group classification of invariant difference models i e difference equations and meshes is presented in the continuous limit the results go over into lie s classification of second order ordinary differential equations the discrete model is a three point one and we show that it can be invariant under lie groups of dimension 0 n 6

umbral calculus difference equations and the discrete schrodinger equation in this paper we discuss umbral calculus as a method of systematically discretizing linear differential equations while preserving their point symmetries as well as generalized symmetries the method is then applied to the schrodinger equation in order to obtain a realization of nonrelativistic quantum mechanics in discrete space time in this approach a quantum system on a lattice has a symmetry algebra isomorphic to that of the continuous case moreover systems that are integrable superintegrable or exactly solvable preserve these properties in the discrete case

symmetries of the discrete burgers equation a discrete cole hopf transformation is used to derive a discrete burgers equation that is linearizable to a discrete heat equation a five dimensional symmetry algebra is obtained that reduces to the lie point symmetry algebra of the usual burgers equation in the continuous limit this lie algebra is used to obtain explicit invariant solutions

relation between backlund transformations and higher continuous symmetries of the toda equation in this paper we study one aspect of the continuous symmetries of the toda equation namely we establish a correspondence between backlund transformations and continuous symmetries of the toda equation a symmetry transformation acting on a solution of the toda equation can be seen as a superposition of backlund transformations conversely a backlund transformation can be written at least formally as a composition of infinitely many higher symmetry transformations this result reinforces the opinion that the presence of sufficiently many continuous symmetries for discrete equations is an indication of their integrability

lie algebras of vector fields in the real plane finite dimensional real analytic lie algebras of vector fields on r 2 are completely classified up to changes of local coordinates

lie symmetries of finite difference equations discretizations of the helmholtz heat and wave equations on uniform lattices are considered in various space time dimensions the symmetry properties of these finite difference equations are determined and it is found that they retain the same lie symmetry algebras as their continuum limits solutions with definite transformation properties are obtained identities and formulas for these functions are then derived using the symmetry algebra

continuous symmetries of difference equations lie group theory was originally created more than 100 years ago as a tool for solving ordinary and partial differential equations in this article we review the results of a much more recent program the use of lie groups to study difference equations we show that the mismatch between continuous symmetries and discrete equations can be resolved in at least two manners one is to use generalized symmetries acting on solutions of difference equations but leaving the lattice invariant the other is to restrict them to point symmetries but to allow them to also transform the lattice

discretization of nonlinear evolution equations over associative function algebras abstract a general approach is proposed for discretizing nonlinear dynamical systems and field theories on suitable functional spaces defined over a regular lattice of points in such a way that both their symmetry and integrability properties are preserved a class of discrete kdv equations is introduced also new hierarchies of discrete evolution equations of gelfand dickey type are defined

lie symmetry preservation and shock capturing methods it is known that some traditional shock capturing schemes including godunov or roe type schemes generate downstream oscillatory wave patterns that cannot be effectively dampened by the dissipation methods they provide the theme of this paper is to understand the formation and behavior at these upstream downstream patterns from the point view of the symmetry inherent in the pde this paper investigates symmetry breaking and spatial oscillations in finite difference solutions for the inviscid burgers and one dimensional navier stokes shock equations using the lax wendroff tvd and nonoscillatory nonfree parameter dissipative schemes symmetry breaking is considered as a candidate for the inherent cause of the nonphysical oscillations in the vicinity of the shock

a new conservation theorem a general theorem on conservation laws for arbitrary differential equations is proved the theorem is valid also for any system of differential equations where the number of equations is equal to the number of dependent variables the new theorem does not require existence of a lagrangian and is based on a concept of an adjoint equation for non linear equations suggested recently by the author it is proved that the adjoint equation inherits all symmetries of the original equation accordingly one can associate a conservation law with any group of lie lie backlund or non local symmetries and find conservation laws for differential equations without classical lagrangians

point symmetries of generalized toda field theories ii symmetry reduction the lie symmetries of a large class of generalized toda field theories are studied and used to perform symmetry reduction reductions lead to generalized toda lattices on one hand and to periodic systems on the other boundary conditions are introduced to reduce theories on an infinite lattice to those on semi infinite or finite ones

periodic solutions to a second order nonlinear neutral functional differential equation in the critical case abstract in this paper the authors study the existence of periodic solutions for a second order neutral functional differential equation x t cx t f x t x t g t x t t e t in the critical case c 1 by employing mawhin s continuation theorem and some analysis techniques sufficient conditions are given for the existence of periodic solutions

on the existence of periodic solutions for neutral functional differential equation abstract by means of mawhin s continuation theorem existence criteria are established for the periodic solutions of neutral functional differential equation

periodic solutions to a kind of neutral functional differential equation in the critical case in this paper we consider a kind of neutral functional differential equation as follows x t cx t gt xt t e t in the critical case c 1 by employing the topological degree theory and some analysis techniques we obtain some new results on the existence of periodic solutions

discretization of partial differential equations preserving their physical symmetries a procedure for obtaining a minimal discretization of a partial differential equation preserving all of its lie point symmetries is presented minimal in this case means that the differential equation is replaced by a partial difference scheme involving n difference equations where n is the number of independent and dependent variables we restrict ourselves to one scalar function of two independent variables as examples invariant discretizations of the heat burgers and korteweg de vries equations are presented some exact solutions of the discrete schemes are obtained

continuous symmetries of lagrangians and exact solutions of discrete equations one of the difficulties encountered when studying physical theories in discrete space time is that of describing the underlying continuous symmetries like lorentz or galilei invariance one of the ways of addressing this difficulty is to consider point transformations acting simultaneously on difference equations and lattices in a previous article we have classified ordinary difference schemes invariant under lie groups of point transformations the present article is devoted to an invariant lagrangian formalism for scalar single variable difference schemes the formalism is used to obtain first integrals and explicit exact solutions of the schemes equations invariant under two and three dimensional groups of lagrangian symmetries are considered

conservation laws of semidiscrete canonical hamiltonian equations there are many evolution partial differential equations which can be cast into hamiltonian form conservation laws of these equations are related to one parameter hamiltonian symmetries admitted by the pdes the same result holds for semidiscrete hamiltonian equations in this paper we consider semidiscrete canonical hamiltonian equations using symmetries we find conservation laws for the semidiscretized nonlinear wave equation and schrodinger equation

the rician inverse gaussian distribution a new model for non rayleigh signal amplitude statistics in this paper we introduce a new statistical distribution for modeling non rayleigh amplitude statistics which we have called the rician inverse gaussian riig distribution it is a mixture of the rice distribution and the inverse gaussian distribution the probability density function pdf is given in closed form as a function of three parameters this makes the pdf very flexible in the sense that it may be fitted to a variety of shapes ranging from the rayleigh shaped pdf to a noncentral spl chi sup 2 shaped pdf the theoretical basis of the new model is quite thoroughly discussed and we also give two iterative algorithms for estimating its parameters from data finally we include some modeling examples where we have tested the ability of the distribution to represent locale amplitude histograms of linear medical ultrasound data and single look synthetic aperture radar data we compare the goodness of fit of the riig model with that of the k model and in most cases the new model turns out as a better statistical model for the data we also include a series of log likelihood tests to evaluate the predictive performance of the proposed model

lie point symmetries of differential difference equations we present an algorithm for determining the lie point symmetries of differential equations on fixed non transforming lattices i e equations involving both continuous and discrete independent variables the symmetries of a specific integrable discretization of the krichever novikov equation the toda lattice and toda field theory are presented as examples of the general method

symmetries of discrete dynamical systems differential difference equations of the form un fn t un 1 un un 1 are classified according to their continuous lie point symmetry groups it is shown that for nonlinear equations the symmetry group can be at most seven dimensional the integrable toda lattice is a member of this class and has a four dimensional symmetry group

symmetry classification of diatomic molecular chains a symmetry classification of possible interactions in a diatomic molecular chain is provided for nonlinear interactions the group of lie point transformations leaving the lattice invariant and taking solutions into solutions is at most five dimensional an example is considered in which subgroups of the symmetry group are used to reduce the dynamical differential difference equations to purely difference ones

point symmetries of generalized toda field theories a class of two dimensional field theories with exponential interactions is introduced the interaction depends on two coupling matrices and is sufficiently general to include all toda field theories existing in the literature lie point symmetries of these theories are found for an infinite semi infinite and finite number of fields special attention is accorded to conformal invariance and its breaking

symmetry preserving discretization of ordinary differential equations large symmetry groups and higher order equations ordinary differential equations odes and ordinary difference systems o ss invariant under the actions of the lie groups and of projective transformations of the independent variables x and dependent variables y are constructed the odes are continuous limits of the o ss or conversely the o ss are invariant discretizations of the odes the invariant o ss are used to calculate numerical solutions of the invariant odes of order up to five the solutions of the invariant numerical schemes are compared to numerical solutions obtained by standard runge kutta methods and to exact solutions when available the invariant method performs at least as well as standard ones and much better in the vicinity of singularities of solutions

lie group formalism for difference equations the methods of lie group analysis of differential equations are generalized so as to provide an infinitesimal formalism for calculating symmetries of difference equations several examples are analysed one of them being a nonlinear difference equation for the linear equations the symmetry algebra of the discrete equation is found to be isomorphic to that of its continuous limit

geometric integrators for odes geometric integration is the numerical integration of a differential equation while preserving one or more of its geometric properties exactly i e to within round off error many of these geometric properties are of crucial importance in physical applications preservation of energy momentum angular momentum phase space volume symmetries time reversal symmetry symplectic structure and dissipation are examples in this paper we present a survey of geometric numerical integration methods for ordinary differential equations our aim has been to make the review of use for both the novice and the more experienced practitioner interested in the new developments and directions of the past decade to this end the reader who is interested in reading up on detailed technicalities will be provided with numerous signposts to the relevant literature

lie symmetries and exact solutions of first order difference schemes we show that any first order ordinary differential equation with a known lie point symmetry group can be discretized into a difference scheme with the same symmetry group in general the lattices are not regular ones but must be adapted to the symmetries considered the invariant difference schemes can be so chosen that their solutions coincide exactly with those of the original differential equation

quasifinite highest weight modules over the lie algebra of differential operators on the circle we classify positive energy representations with finite degeneracies of the lie algebraw 1 and construct them in terms of representation theory of the lie algebra hat gl infty r m of infinites matrices with finite number of non zero diagonals over the algebrar m t t m 1 the unitary ones are classified as well similar results are obtained for the sin algebras

construction of partial difference schemes i the clairaut schwarz young theorem on the lattice difference operators in several variables do not always commute their commutation depends on the lattice equations in this paper we study the conditions over a lattice to allow commutativity of these difference operators and their implications for the discretization of partial differential equations

dilation symmetries and equations on the lattice we discuss the role of dilation symmetries for differential difference equations depending on nearest neighbour interactions in particular we show that for a simple class of differential difference equations of this kind symmetries which depend linearly on time are only compatible with linearizable equations

the discretized boussinesq equation and its conditional symmetry reduction in this article we show that we can carry out the symmetry preserving discretization of the boussinesq equation with respect to three of its more significant conditional symmetries we perform the symmetry reduction of the obtained nonlinear discrete schemes with respect to the conditional symmetries and obtain the reduced discrete equations which unlike in the continuous case are not always reducible to second order difference equations a numerical comparison with the exact continuous solution given by weierstrass elliptic functions is carried out

nonlinear self adjointness and conservation laws the general concept of nonlinear self adjointness of differential equations is introduced it includes the linear self adjointness as a particular case moreover it embraces the strict self adjointness definition 1 and quasi self adjointness introduced earlier by the author it is shown that the equations possessing nonlinear self adjointness can be written equivalently in a strictly self adjoint form by using appropriate multipliers all linear equations possess the property of nonlinear self adjointness and hence can be rewritten in a nonlinear strictly self adjoint form for example the heat equation ut u 0 becomes strictly self adjoint after multiplying by u 1 conservation laws associated with symmetries are given in an explicit form for all nonlinearly self adjoint partial differential equations and systems

quantum symmetries of q difference equations a general method is presented to determine the symmetry operators of linear q difference equations it is applied to q difference analogs of the helmoltz heat and wave equations in diverse dimensions the symmetries are found to generate q deformations of classical lie algebras the method of separation of variables is used to obtain solutions which are seen to involve many basic special functions this allows the derivation of various identities and formulas for these q functions