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

distance dissimilarity index and network community structure we address the question of finding the community structure of a complex network in an earlier effort h zhou phys rev e 67 041908 2003 the concept of network random walking is introduced and a distance measure defined here we calculate based on this distance measure the dissimilarity index between nearest neighboring vertices of a network and design an algorithm to partition these vertices into communities that are hierarchically organized each community is characterized by an upper and a lower dissimilarity threshold the algorithm is applied to several artificial and real world networks and excellent results are obtained in the case of artificially generated random modular networks this method outperforms the algorithm based on the concept of edge betweenness centrality for yeast s protein protein interaction network we are able to identify many clusters that have well defined biological functions

lie symmetry analysis and exact solutions of fractional ordinary differential equations with neutral delay abstract in this paper lie symmetry method is employed to solve the fractional neutral ordinary differential equations the symmetry analysis was carried out and infinite dimension symmetry algebras are obtained an application of lie symmetry techniques is used to obtain the exact solutions

moving mesh methods for problems with blow up in this paper we consider the numerical solution of pdes with blow up for which scaling invariance plays a natural role in describing the underlying solution structures it is a challenging numerical problem to capture the qualitative behaviour in the blow up region and the use of nonuniform meshes is essential we consider moving mesh methods for which the mesh is determined using so called moving mesh partial differential equations mmpdes specifically the underlying pde and the mmpde are solved for the blow up solution and the computational mesh simultaneously motivated by the desire for the mmpde to preserve the scaling invariance of the underlying problem we study the effect of different choices of mmpdes and monitor functions it is shown that for suitable ones the mmpde solution evolves towards a moving mesh which close to the blow up point automatically places the mesh points in such a manner that the ignition kernel which is well known to be a natural coordinate in describing the behaviour of blow up approaches a constant as t to t the blow up time several numerical examples are given to verify the theory for these mmpde methods and to illustrate their efficacy

solvable lie algebras of dimension six six dimensional solvable lie algebras over the field of real numbers that possess nilradicals of dimension four are classified into equivalence classes this completes mubarakzyanov s classification of the real six dimensional solvable lie algebras

detecting community structure using label propagation with weighted coherent neighborhood propinquity abstract community detection has become an important methodology to understand the organization and function of various real world networks the label propagation algorithm lpa is an almost linear time algorithm proved to be effective in finding a good community structure however lpa has a limitation caused by its one hop horizon specifically each node in lpa adopts the label shared by most of its one hop neighbors much network topology information is lost in this process which we believe is one of the main reasons for its instability and poor performance therefore in this paper we introduce a measure named weighted coherent neighborhood propinquity weighted cnp to represent the probability that a pair of vertices are involved in the same community in label update a node adopts the label that has the maximum weighted cnp instead of the one that is shared by most of its neighbors we propose a dynamic and adaptive weighted cnp called entropic cnp by using the principal of entropy to modulate the weights furthermore we propose a framework to integrate the weighted cnp in other algorithms in detecting community structure we test our algorithm on both computer generated networks and real world networks the experimental results show that our algorithm is more robust and effective than lpa in large scale networks

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

advanced modularity specialized label propagation algorithm for detecting communities in networks abstract a modularity specialized label propagation algorithm lpam for detecting network communities was recently proposed this promising algorithm offers some desirable qualities however lpam favors community divisions where all communities are similar in total degree and thus it is prone to get stuck in poor local maxima in the modularity space to escape local maxima we employ a multistep greedy agglomerative algorithm msg that can merge multiple pairs of communities at a time combining lpam and msg we propose an advanced modularity specialized label propagation algorithm lpam experiments show that lpam successfully detects communities with higher modularity values than ever reported in two commonly used real world networks moreover lpam offers a fair compromise between accuracy and speed

conservation laws of semidiscrete hamiltonian equations many evolution partial differential equations pdes can be cast into hamiltonian form conservation laws of these equations are related to one parameter hamiltonian symmetries admitted by the pdes p j olver applications of lie groups to differential equations springer new york 1986 in this paper we consider symmetries and noether s theorem for semidiscrete hamiltonian equations which are obtained by space discretization of hamiltonian pdes using symmetries one can find conservation laws of these equations several applications including a transfer equation and the korteweg de vries equation are presented

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

finding community structure in networks using the eigenvectors of matrices we consider the problem of detecting communities or modules in networks groups of vertices with a higher than average density of edges connecting them previous work indicates that a robust approach to this problem is the maximization of the benefit function known as modularity over possible divisions of a network here we show that this maximization process can be written in terms of the eigenspectrum of a matrix we call the modularity matrix which plays a role in community detection similar to that played by the graph laplacian in graph partitioning calculations this result leads us to a number of possible algorithms for detecting community structure as well as several other results including a spectral measure of bipartite structure in networks and a centrality measure that identifies vertices that occupy central positions within the communities to which they belong the algorithms and measures proposed are illustrated with applications to a variety of real world complex networks

stepping community detection algorithm based on label propagation and similarity abstract community or module structure is one of the most common features in complex networks the label propagation algorithm lpa is a near linear time algorithm that is able to detect community structure effectively nevertheless when labeling a node the lpa adopts the label belonging to the majority of its neighbors which means that it treats all neighbors equally in spite of their different effects on the node another disadvantage of lpa is that the results it generates are not unique in this paper we propose a modified lpa called stepping lpa s in which labels are propagated by similarity furthermore our algorithm divides networks using a stepping framework and uses an evaluation function proposed in this paper to select the final unique partition we tested this algorithm on several artificial and real world networks the results show that stepping lpa s can obtain accurate and meaningful community structure without priori information

symmetries of nonlinear differential equations and linearisation a non linear ordinary differential equation is linearisable if it possesses sl 3 r symmetry the conditions under which the abelian two dimensional subalgebras of sl 3 r are sufficient for linearisation are established

symmetry and classes of transport equations abstract a method for classification and determination of local transport equations is presented based on the use of lie symmetries the starting point is the choice of a specific symmetry group and the definition of a class of partial differential equations results associated with the fokker planck equation in 1 1 and 2 1 dimensions are given

nonlinear differential difference equations associated backlund transformations and lax technique the lax technique can be successfully employed to derive the class of nonlinear differential difference equations associated with the discrete analogue of the matrix schrodinger spectral problem and solvable by the spectral transform

non classical symmetry reduction example of the boussinesq equation a symmetry of an equation will leave the set of all solutions invariant a conditional symmetry will leave only a subset of solutions defined by some differential condition invariant the authors show how a specific class of conditional symmetries can be used to reduce a partial differential equation to an ordinary one in particular for the boussinesq equation these conditional symmetries together with the ordinary ones provide all possible reductions to ordinary differential equations a group theoretical explanation of the recently obtained new reductions is provided

normalized cuts and image segmentation we propose a novel approach for solving the perceptual grouping problem in vision rather than focusing on local features and their consistencies in the image data our approach aims at extracting the global impression of an image we treat image segmentation as a graph partitioning problem and propose a novel global criterion the normalized cut for segmenting the graph the normalized cut criterion measures both the total dissimilarity between the different groups as well as the total similarity within the groups we show that an efficient computational technique based on a generalized eigenvalue problem can be used to optimize this criterion we applied this approach to segmenting static images as well as motion sequences and found the results to be very encouraging

invariantization of the crank nicolson method for burgers equation abstract in this article a geometric technique to construct numerical schemes for partial differential equations pdes that inherit lie symmetries is proposed the moving frame method enables one to adjust the numerical schemes in a geometric manner and systematically construct proper invariant versions of them to illustrate the method we study invariantization of the crank nicolson scheme for burgers equation with careful choice of normalization equations the invariantized schemes are shown to surpass the standard scheme successfully removing numerical oscillation around sharp transition layers

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

density shrinking algorithm for community detection with path based similarity abstract community structure is ubiquitous in real world complex networks finding the communities is the key to understand the functions of those networks a lot of works have been done in designing algorithms for community detection but it remains a challenge in the field traditional modularity optimization suffers from the resolution limit problem recent researches show that combining the density based technique with the modularity optimization can overcome the resolution limit and an efficient algorithm named denshrink was provided the main procedure of denshrink is repeatedly finding and merging micro communities broad sense into super nodes until they cannot merge analyses in this paper show that if the procedure is replaced by finding and merging only dense pairs both of the detection accuracy and runtime can be obviously improved thus an improved density based algorithm imds is provided since the time complexity path based similarity indexes are difficult to be applied in community detection for high performance in this paper the path based katz index is simplified and used in the imds algorithm

community detection in complex networks using extremal optimization we propose a method to find the community structure in complex networks based on an extremal optimization of the value of modularity the method outperforms the optimal modularity found by the existing algorithms in the literature giving a better understanding of the community structure we present the results of the algorithm for computer simulated and real networks and compare them with other approaches the efficiency and accuracy of the method make it feasible to be used for the accurate identification of community structure in large complex networks

lie algebras associated with scalar second order ordinary differential equations second order ordinary differential equations are classified according to their lie algebra of point symmetries the existence of these symmetries provides a way to solve the equations or to transform them to simpler forms canonical forms of generators for equations with three point symmetries are established it is further shown that an equation cannot have exactly r 4 5 6 7 point symmetries representative s of equivalence class es of equations possessing s 1 2 3 8 point symmetry generator s are then obtained

benchmark graphs for testing community detection algorithms community structure is one of the most important features of real networks and reveals the internal organization of the nodes many algorithms have been proposed but the crucial issue of testing i e the question of how good an algorithm is with respect to others is still open standard tests include the analysis of simple artificial graphs with a built in community structure that the algorithm has to recover however the special graphs adopted in actual tests have a structure that does not reflect the real properties of nodes and communities found in real networks here we introduce a class of benchmark graphs that account for the heterogeneity in the distributions of node degrees and of community sizes we use this benchmark to test two popular methods of community detection modularity optimization and potts model clustering the results show that the benchmark poses a much more severe test to algorithms than standard benchmarks revealing limits that may not be apparent at a first analysis

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

sensitivity of spin glass order to temperature changes by a loop expansion around parisi s mean field theory for an ising spin glass it is shown that the overlap of the magnetization patterns belonging to two different temperatures t and t vanishes to any order si t si t 0 while the correlation overlap sisj t sisj t calculated to first loop order and for technical reasons for dimensions d 8 only is found to decay exponentially with a characteristic length approximately t t 1

new similarity reductions of the boussinesq equation some new similarity reductions of the boussinesq equation which arises in several physical applications including shallow water waves and also is of considerable mathematical interest because it is a soliton equation solvable by inverse scattering are presented these new similarity reductions including some new reductions to the first second and fourth painleve equations cannot be obtained using the standard lie group method for finding group invariant solutions of partial differential equations they are determined using a new and direct method that involves no group theoretical techniques

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

angular momentum theory umbral calculus and combinatorics abstract the theory of the unitary irreducible representations of the unitary group su 2 is reviewed with the objective of demonstrating the fundamental role that the umbral calculus and combinatorics play for the physicist the study of the group su 2 is synonymous with the theory of angular momentum in quantum theory and kronecker products of irreducible unitary representations comprise the mathematical apparatus for building composite systems from simpler constituents the clebsch gordon coefficients which are essential to the binary theory of composite systems are shown to have an umbral calculus origin macmahon s master theorem is shown to be the basic result for generating the mathematical quantities needed for bringing comprehension to the properties of composite systems

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

detection of community structure in networks based on community coefficients abstract determining community structure in networks is fundamental to the analysis of the structural and functional properties of those networks including social networks computer networks and biological networks modularity function q which was proposed by newman and girvan was once the most widely used criterion for evaluating the partition of a network into communities however modularity q is subject to a serious resolution limit in this paper we propose a new function for evaluating the partition of a network into communities this is called community coefficient c using community coefficient c we can automatically identify the ideal number of communities in the network without any prior knowledge we demonstrate that community coefficient c is superior to the modularity q and does not have a resolution limit we also compared the two widely used community structure partitioning methods the hierarchical partitioning algorithm and the normalized cuts ncut spectral partitioning algorithm we tested these methods on computer generated networks and real world networks whose community structures were already known the ncut algorithm and community coefficient c were found to produce better results than hierarchical algorithms unlike several other community detection methods the proposed method effectively partitioned the networks into different community structures and indicated the correct number of communities

affine linear and d4 symmetric lattice equations symmetry analysis and reductions we consider lattice equations on which are autonomous affine linear and possess the symmetries of the square some basic properties of equations of this type are derived as well as a sufficient linearization condition and a conservation law a systematic analysis of the lie point and the generalized three and five point symmetries is presented it leads to the generic form of the symmetry generators of all the equations in this class which satisfy a certain non degeneracy condition finally symmetry reductions of certain lattice equations to discrete analogs of the painleve equations are considered

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 reduction for the kadomtsev petviashvili equation using a loop algebra the kadomtsev petviashvili kp equation ut 3uux 2 1 4 uxxx x 3 uyy 4 0 allows an infinite dimensional lie group of symmetries i e a group transforming solutions amongst each other the lie algebra of this symmetry group depends on three arbitrary functions of time t and is shown to be related to a subalgebra of the loop algebra a 1 4 low dimensional subalgebras of the symmetry algebra are identified specifically all those of dimension n 3 and also a physically important six dimensional lie algebra containing translations dilations galilei transformations and quasirotations new solutions of the kp equation are obtained by symmetry reduction using the one dimensional subalgebras of the symmetry algebra these solutions contain up to three arbitrary functions of t

on the identification of a lie algebra given by its structure constants i direct decompositions levi decompositions and nilradicals abstract methods are given for identifying a lie algebra l given by its structure constants the identification involves a transformation to a canonical basis in which the structure becomes obvious thus decomposable lie algebras are already decomposed into direct sums of indecomposable ones an indecomposable lie algebra that is not simple or solvable has its radical exhibited a solvable algebra has its nilradical displayed the methods all lead to simple algorithms that have been implemented as computer programs involving some symbolic manipulations

simplifying the form of lie groups admitted by a given differential equation rather general results are obtained for determining the nature of infinitesimal generators which can be admitted by a given differential equation simple criteria are given to determine whether or not 1 the infinitesimals of independent variables can depend only on independent variables and 2 the infinitesimal of the dependent variable can depend at most linearly on the dependent variable many examples are given a trivial consequence of this paper is that an admitted lie group for any linear partial differential equation of at least second order must have infinitesimals of independent variables depending only on independent variables and the infinitesimal of the dependent variable depends at most linearly on the dependent variable this latter result previously has only been proved for a second order linear pde in the last century sophus lie developed the theory of continuous groups lie groups of transformations he showed that such transformations can be characterized in terms of infinitesimal generators moreover lie gave an algorithm to find the infinitesimal generators admitted by a given differential equation de for any de the admittance of a lie group leads to the construction of a family of new solutions from any known solution the admittance of a lie group for any ordinary differential equation ode leads constructively to a reduction of its order plus quadratures if a partial differential equation pde is invariant under a lie group one can construct special families of solutions called invariant solutions or similarity solutions if a de can be derived from a variational principle then admittance of a lie group is a necessary condition in order to find conservation laws by means of noether s theorem for recent references on lie s algorithm to find infinitesimal generators admitted by a given de and various uses of lie groups for des see l 4 71 52

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 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

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

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

equivalence classes and linearization of the riccati and abel chain abstract the problem of linearization by point transformations is solved for equations in the generalized riccati and abel chain of order not exceeding the fourth it is shown in particular that nonlinear third order and fourth order equations from the chain are not linearizable by any point transformations the lie pseudo group of equivalence transformations for equations of arbitrary orders from the chain are then found together with expressions for the transformed parameter functions an important subgroup of the group of equivalence transformations found is considered and some associated equivalence classes are exhibited

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

detecting network communities by propagating labels under constraints we investigate the recently proposed label propagation algorithm lpa for identifying network communities we reformulate the lpa as an equivalent optimization problem giving an objective function whose maxima correspond to community solutions by considering properties of the objective function we identify conceptual and practical drawbacks of the label propagation approach most importantly the disparity between increasing the value of the objective function and improving the quality of communities found to address the drawbacks we modify the objective function in the optimization problem producing a variety of algorithms that propagate labels subject to constraints of particular interest is a variant that maximizes the modularity measure of community quality performance properties and implementation details of the proposed algorithms are discussed bipartite as well as unipartite networks are considered

community mining with new node similarity by incorporating both global and local topological knowledge in a constrained random walk abstract detection of community is a crucial step to understand the structure and dynamics of complex networks most of conventional community detection methods focus on optimizing a certain objective function or on clustering nodes based on their similarities which leads to a phenomenon that they have preference for specific types of networks but are not general using constrained random walk we exploit global and local topology structures of network to propose a modified transition matrix and further to define a new similarity metric named isim between two nodes in contrast to the existing similarities isim does not work directly on the observed data but in a convergent stable space this feature makes isim robust to the observed noisy data in real world networks isim not only measures node s distance but also captures node s topology structure in network experiments on synthetic and real world networks demonstrate that isim can be successfully applied to community detection in broader types of networks and outperforms other community detection methods

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

modularity from fluctuations in random graphs and complex networks the mechanisms by which modularity emerges in complex networks are not well understood but recent reports have suggested that modularity may arise from evolutionary selection we show that finding the modularity of a network is analogous to finding the ground state energy of a spin system moreover we demonstrate that due to fluctuations stochastic network models give rise to modular networks specifically we show both numerically and analytically that random graphs and scale free networks have modularity we argue that this fact must be taken into consideration to define statistically significant modularity in complex networks

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

new self similar solutions of the nonlinear schrodinger equation with moving mesh computations we study the blow up self similar solutions of the radially symmetric nonlinear schrodinger equation nls given by iut urr d 1rur u u 2 with dimension d 2 these solutions become infinite in a finite time t by a series of careful numerical computations partly supported by analytic results we demonstrate that there is a countably infinite set of blow up self similar solutions which satisfy a second order complex ordinary differential equation with an integral constraint these solutions are characterised by the number of oscillations in their amplitude when d is close to 2 the solutions are computed as functions of d and their behaviour in the critical limit as d 2 is investigated the stability of these solutions is then studied by solving the nls by using an adaptive numerical method this method uses moving mesh partial differential equations and exploits the scaling invariance properties of the underlying equation we demonstrate that the single humped self similar solution is globally stable whereas the multi humped solutions all appear to be unstable

nonlinear differential difference equations as backlund transformations shows that the best known nonlinear differential difference equations associated with the discrete schrodinger spectral problem and also with the discrete zakharov shabat spectral problem can be interpreted as backlund transformations for some continuous nonlinear evolution equations

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

singularity confinement and chaos in discrete systems we present a number of second order maps which pass the singularity confinement test commonly used to identify integrable discrete systems but which nevertheless are non integrable as a more sensitive integrability test we propose the analysis of the complexity algebraic entropy of the map using the growth of the degree of its iterates integrability is associated with polynomial growth while the generic growth is exponential for chaotic systems

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

the multiscale expansions of difference equations in the small lattice spacing regime and a vicinity and integrability test i we propose an algorithmic procedure i to study the distance between an integrable pde and any discretization of it in the small lattice spacing regime and at the same time ii to test the asymptotic integrability properties of such discretization this method should provide in particular useful and concrete information on how good is any numerical scheme used to integrate a given integrable pde the procedure illustrated on a fairly general ten parameter family of discretizations of the nonlinear schr dinger equation consists of the following three steps i the construction of the continuous multiscale expansion of a generic solution of the discrete system at all orders in following degasperis et al 1997 physica d 100 187 211 ii the application to such an expansion of the degasperis procesi dp integrability test degasperis a and procesi m 1999 asymptotic integrability symmetry and perturbation theory spt98 ed a degasperis and g gaeta singapore world scientific pp 23 37 degasperis a 2001 multiscale expansion and integrability of dispersive wave equations lectures given at the euro summer school what is integrability isaac newton institute cambridge uk 13 24 august integrability lecture notes in physics vol 767 ed a mikhailov berlin springer to test the asymptotic integrability properties of the discrete system and its distance from its continuous limit iii the use of the main output of the dp test to construct infinitely many approximate symmetries and constants of motion of the discrete system through novel and simple formulas

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

vibration and stability of a cantilever column subject to a follower force passing through a fixed point abstract the paper describes the stability and natural vibration of column loaded by a force passing through a fixed point it is proved that the system is conservative and its loss of stability occurs via divergence on the other hand a change in modes occurring along the fundamental eigencurve which modifies its slope from positive to negative with the increasing load is found theoretically and confirmed by the modal analysis for that reason the system is proposed to be called the divergence pseudo flutter system experimental results concerning the natural vibration frequency agreed well with the numerical ones

free vibrations and the stability of a geometrically non linear column loaded by a follower force directed towards the positive pole abstract a slender system built as a geometrically non linear system and subjected to specific load by a follower force directed towards the positive pole is discussed the load is induced by heads composed of circular profile elements the problem of stability and free vibration was formulated on the basis of hamilton s principle and then owing to its non linearity solved using the straightforward expansion method this paper provides example results of numerical computations concerning the influence of selected parameters characterizing the considered column including pre stressing on the stability and free vibration the accuracy of the adopted mathematical model is also proved on the basis of experimental studies

the role of prestressing in establishing regions of instability for a compound column under conservative or non conservative load the influence of prestressing of a non linear two member column on its natural vibration and stability is studied the perturbation method is used for solving the problem regions of divergence and flutter instabilities for a column has been determined on the basis of courses of eigencurves in relation to prestress rate discontinuities of the critical force have been observed for the border value of the prestress leading to the instability of the column without the external load although each prestress lowers the critical force it can be used for passive vibration control

similarity reduction of a 2 1 volterra system the continuous point symmetries of a 2 1 discrete volterra system is derived and it forms a centreless infinite dimensional kac moody virasoro algebra using the symmetries a similarity reduction of the 2 1 volterra system to a partial differential difference equation with two independent variables is obtained also the lie point symmetries of the reduced partial differential difference equation is derived and its similarity reduction to an ordinary differential difference equation is obtained the ordinary differential difference equation reduces to the third painleve equation in the continuum limit further a lax pair for the reduced equation is derived indicating its complete integrability

symplectic structures their backlund transformations and hereditary symmetries abstract it is shown that compatible symplectic structures lead in a natural way to hereditary symmetries we recall that a hereditary symmetry is an operator valued function which immediately yields a hierarchy of evolution equations each having infinitely many commuting symmetries all generated by this hereditary symmetry furthermore this hereditary symmetry usually describes completely the soliton structure and the conservation laws of these equations this result then provide us with a method for constructing hereditary symmetries and hence exactly solvable evolution equations in addition we show how symplectic structures transform under backlund transformations this leads to a method for generating a whole class of symplectic structures from a given one several examples and applications are given illustrating the above results also the connection of our results with those of gelfand and dikii and of magri is briefly pointed out

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

evolution equations associated with the discrete analog of the matrix schrodinger spectral problem solvable by the inverse spectral transform through the generalized wronskian technique we derive the whole class of nonlinear differential difference equations associated with the discrete analog of the matrix schrodinger spectral problem for such equations we briefly discuss soliton solutions continuum limit and backlund transformations

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

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

symmetries and conditional symmetries of differential difference equations two different methods of finding lie point symmetries of differential difference equations are presented and applied to the two dimensional toda lattice continuous symmetries are combined with discrete ones to obtain various reductions to lower dimensional equations in particular to differential equations of the delay type the concept of conditional symmetries is extended from purely differential to differential difference equations and shown to incorporate backlund transformations

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 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