an algorithmic method for showing existence of nontrivial non classical symmetries of partial differential equations without solving determining equations abstract in this paper based on differential characteristic set theory and the associated algorithm also called wu s method an algorithmic method is presented to decide on the existence of a nontrivial non classical symmetry of a given partial differential equation without solving the corresponding nonlinear determining system the theory and algorithm give a partial answer for the open problem posed by p a clarkson and e l mansfield in 21 on non classical symmetries of partial differential equations as applications of our algorithm non classical symmetries and corresponding invariant solutions are found for several evolution equations

first order compatibility for a 2 1 dimensional diffusion equation a class of first order partial differential equations is obtained compatible with a 2 1 dimensional linear diffusion equation with a nonlinear source term we will show that if the source term is quasilinear then compatible equations are quasilinear furthermore we obtain diffusion equations with source terms that admit non quasilinear compatible equations

generalized conditional symmetries of evolution equations we analyze the relationship of generalized conditional symmetries of evolution equations to the formal compatibility and passivity of systems of differential equations as well as to systems of vector fields in involution earlier results on the connection between generalized conditional invariance and generalized reduction of evolution equations are revisited this leads to a no go theorem on determining equations for operators of generalized conditional symmetry it is also shown that up to certain equivalences there exists a one to one correspondence between generalized conditional symmetries of an evolution equation and parametric families of its solutions

symmetry properties of a system of euler type equations for magnetized plasmas we provide all lie point symmetries of a system of partial differential equations of great interest to plasma physics we deduce the corresponding invariant solutions and discuss their physical interpretation we also consider some of its conditional and partial symmetries and obtain other interesting solutions finally we point out that although the system can be cast in divergence form and admits conserved currents it does not admit potential symmetries

conditional lie backlund symmetries and solutions to n 1 dimensional nonlinear diffusion equations this paper discusses a class of n 1 dimensional nonlinear diffusion equations with source term which arises in nonlinear shear flows of non newtonian fluids it is shown that some radially symmetric equations admit certain types of conditional lie backlund symmetries as a result exact solutions to the resulting equations are obtained those solutions extend the known ones such as instantaneous source solutions of the porous medium equation with absorption term the phenomena of extinction and blow up and behavior to many of the solutions are described

potential symmetries and direct reduction methods of order two for partial differential equations written in conservative form a remarkable link between potential symmetries and direct reduction methods of order two is enlightened

conditional lie backlund symmetries and differential constraints for inhomogeneous nonlinear diffusion equations due to linear determining equations abstract the method of linear determining equations is developed to study conditional lie backlund symmetries for evolution equations which is more general than the classical determining equations for lie s generators as an application of this approach the complete classification is presented for the inhomogeneous nonlinear diffusion equations which admit the second order and third order conditional lie backlund symmetries several examples are given to illustrate the corresponding symmetry reductions due to the compatibility of the invariant conditions and the governing equations

asymptotic symmetries and asymptotically symmetric solutions of partial differential equations symmetry methods for differential equations are a powerful tool to attack nonlinear problems in particular for determining solutions with given symmetries to nonlinear pdes since in real applications one is often interested in solutions which are asymptotically symmetric we propose here an approach to asymptotic symmetry based on the methods of lie theory we adopt translate in geometric language and develop the renormalization group approach recently proposed by bricmont and kupiainen 1992 for the ginzburg landau equation

reduction operators of burgers equation abstract the solution of the problem on reduction operators and nonclassical reductions of the burgers equation is systematically treated and completed a new proof of the theorem on the special no go case of regular reduction operators is presented and the representation of the coefficients of operators in terms of solutions of the initial equation is constructed for this case all possible nonclassical reductions of the burgers equation to single ordinary differential equations are exhaustively described any lie reduction of the burgers equation proves to be equivalent via the hopf cole transformation to a parameterized family of lie reductions of the linear heat equation

lattice scale free networks with weighted linking recently models of the scale free sf networks on lattices were investigated which consider the influence of the embedded space on the networks since a lot of real networks exist on the 2d global surface it is helpful to discuss these models in this paper based on the lattice sf networks model a linking weight is added thus an additional parameter which can control the clustering coefficient is introduced depending on the linking weight the properties of the model change smoothly from the lattice sf model to the sf random graphs

symmetry reductions and exact solutions of a class of nonlinear heat equations abstract classical and nonclassical symmetries of the nonlinear heat equation ut uxx f u are considered the method of differential grobner bases is used both to find the conditions on f u under which symmetries other than the trivial spatial and temporal translational symmetries exist and to solve the determining equations for the infinitesimals a catalogue of symmetry reductions is given including some new reductions for the linear heat equation and a catalogue of exact solutions of the nonlinear heat equation for cubic f u in terms of the roots of f u 0

partial lie point symmetries of differential equations when we consider a differential equation 0 whose set of solutions is a lie point exact symmetry of this is a lie point invertible transformation t such that t i e such that any solution to 0 is transformed into a generally different solution to the same equation here we define partial symmetries of 0 as lie point invertible transformations t such that there is a non empty subset such that t i e such that there is a subset of solutions to 0 which are transformed into one another we discuss how to determine both partial symmetries and the invariant set and show that our procedure is effective by means of concrete examples we also discuss relations with conditional symmetries and how our discussion applies to the special case of dynamical systems our discussion will focus on continuous lie point partial symmetries but our approach would also be suitable for more general classes of transformations the discussion is indeed extended to partial generalized or lie backlund symmetries along the same lines and in the appendix we will discuss the case of discrete partial symmetries

on the weak symmetry groups of partial differential equations abstract we define an algorithm to characterize all weak symmetry groups of a partial differential equation new invariant solutions to the fokker planck equation are found

non classical symmetries and the singular manifold method revisited the connection between the singular manifold method painleve expansions truncated at the constant term and symmetry reductions of two members of a family of cahn hilliard equations is considered the conjecture that similarity information for a nonlinear partial differential equation may always be fully recovered from the singular manifold method is violated for these equations and is thus shown to be invalid in general given that several earlier examples demonstrate the connection between the two techniques in some cases it now becomes necessary to establish when such a relationship exists a question related to a deeper understanding of painleve analysis this issue is also briefly discussed

similarity reductions of partial differential equations the author determine the relation between two methods of reduction the similarity reduction method using nonclassical symmetry groups and the direct approach used by clarkson and kruskal 1989 they prove that the solutions which are obtained by similarity in correspondence to nonclassical groups constitute a larger family than the one obtained by the method of clarkson and kruskal the two procedures are equivalent only if the generators zeta x t u tau x t u eta x t u of the nonclassical groups are such that zeta tau is independent of u to explain these results he proves the existence of families of solutions of the burgers equation which are found by means of nonclassical symmetry reduction and which cannot be determined via the general reduction form of clarkson and kruskal

on the relation between standard and μ symmetries for pdes we give a geometrical interpretation of the notion of prolongations of vector fields and of the related concept of symmetry for partial differential equations extending to pdes the notion of symmetry for odes we give in particular a result concerning the relationship between symmetries and standard exact symmetries the notion is also extended to the case of conditional and partial symmetries and we analyse the relation between local symmetries and nonlocal standard symmetries

reduction operators of linear second order parabolic equations the reduction operators i e the operators of nonclassical conditional symmetry of 1 1 dimensional second order linear parabolic partial differential equations and all the possible reductions of these equations to ordinary differential ones are exhaustively described this problem proves to be equivalent in some sense to solving initial equations the no go result is extended to the investigation of point transformations admissible transformations equivalence transformations lie symmetries and lie reductions of the determining equations for the nonclassical symmetries transformations linearizing the determining equations are obtained in the general case and under different additional constraints a nontrivial example illustrating applications of reduction operators to finding exact solutions of equations from the class under consideration is presented an observed connection between reduction operators and darboux transformations is discussed

lie and conditional symmetries of the three component diffusive lotka volterra system lie and q conditional symmetries of the classical three component diffusive lotka volterra system in the case of one space variable are studied the group classification problems for finding lie symmetries and q conditional symmetries of the first type are completely solved notably non lie symmetries q conditional symmetry operators for a multi component nonlinear reaction diffusion system are constructed for the first time the results are compared with those derived for the two component diffusive lotka volterra system the conditional symmetry obtained for the non lie reduction of the three component system used for modeling competition between three species in population dynamics is applied and the relevant exact solutions are found particularly the exact solution describing different scenarios of competition between three species is constructed

group invariant solutions of differential equations we introduce the concept of a weak symmetry group of a system of partial differential equations that generalizes the nonclassical method introduced by bluman and cole for finding group invariant

simple waves and invariant solutions of quasilinear systems the solutions of a system of homogeneous first order quasilinear evolution equations for which the propagation velocity depends only on the field value are characterized by their invariance under a particular type of flow all simple wave solutions are shown to have this property the local images of simple wave solutions are the integral curves of a system of ordinary differential equations on the field space the model for barotropic compressible flow is an example of a system that has invariant solutions of higher rank a solution of this system is invariant under the flow of the convective derivative if and only if it is divergence free the evolution of initial data is shown to be divergence free if and only if its differential is nilpotent

marginal values and second order necessary conditions for optimality second order necessary conditions in nonlinear programming are derived by a new method that does not require the usual sort of constraint qualification in terms of the multiplier vectors appearing in such second order conditions an estimate is obtained for the generalized subgradients of the optimal value function associated with a parameterized nonlinear programming problem this yields estimates for marginal values with respect to the parameters the main theoretical tools are the augmented lagrangian and despite the assumption of second order smoothness of objective constraints the subdifferential calculus that has recently been developed for nonsmooth nonconvex functions

enhancing robustness and immunization in geographical networks we find that different geographical structures of networks lead to varied percolation thresholds although these networks may have similar abstract topological structures thus strategies for enhancing robustness and immunization of a geographical network are proposed using the generating function formalism we obtain an explicit form of the percolation threshold q c for networks containing arbitrary order cycles for three cycles the dependence of q c on the clustering coefficients is ascertained the analysis substantiates the validity of the strategies with analytical evidence

stability in stochastic programming with recourse estimated parameters in this paper stability of the optimal solution of stochastic programs with recourse with respect to parameters of the given distribution of random coefficients is studied provided that the set of admissible solutions is defined by equality constraints only asymptotical normality of the optimal solution follows by standard methods if nonnegativity constraints are taken into account the problem is solved under assumption of strict complementarity known from the theory of nonlinear programming theorem 1 the general results are applied to the simple recourse problem with random right hand sides under various assumptions on the underlying distribution theorems 2 4

a remarkable class of non classical symmetries of the steady two dimensional boundary layer equations we study the steady two dimensional boundary layer equations in the flat and axisymmetric case to show that some similarity reductions already found in the literature with ad hoc new methods of reduction are indeed invariant solutions under the action of non classical symmetries as introduced by bluman and cole moreover we show that the celebrated von mises transformation that reduces the two dimensional flat boundary layer equations to a second order evolution equation is indeed a backlund transformation related to a non classical symmetry

evolution equations invariant surface conditions and functional separation of variables abstract this paper is devoted to a discussion of the reduction methods for evolution equations based on invariant surface conditions related to functional separation of variables the relationship of these methods with nonclassical and weak point symmetries is stressed applications to diffusion equations with an inhomogeneous reaction term or with saturating dissipation are provided

second order sensitivity analysis and asymptotic theory of parametrized nonlinear programs in this paper we study second order differential properties of an optimal value function x it is shown that under certain conditions x possesses second order directional derivatives which can be calculated by solving corresponding quadratic programs also upper and lower bounds on these derivatives are introduced under weaker assumptions in particular we show that the second order directional derivative is infinite if the corresponding quadratic program is unbounded finally sensitivity results are applied to investigate asymptotics of estimators in parametrized nonlinear programs

growing networks under geographical constraints inspired by the structure of technological weblike systems we discuss network evolution mechanisms which give rise to topological properties found in real spatial networks thus we suggest that the peculiar structure of transport and distribution networks is fundamentally determined by two factors these are the dependence of the spatial interaction range of vertices on the vertex attractiveness or importance within the network and on the inhomogeneous distribution of vertices in space we propose and analyze numerically a simple model based on these generating mechanisms which seems for instance to be able to reproduce known structural features of the internet

the network of commuters in london abstract we have studied the directed and weighted network in which the wards of london are vertices and two vertices are connected whenever there is at least one person commuting to work from one ward to another remarkably the in strength and in degree distribution tail is a power law with exponent around 2 while the out strength and out degree distribution tail is exponential we propose a simple square lattice model to explain the observed empirical behaviour

new conditional symmetries and exact solutions of reaction diffusion convection equations with exponential nonlinearities abstract a complete description of q conditional symmetries for a class of reaction diffusion convection equations with exponential diffusivities is derived it is shown that all the known results for reaction diffusion equations with exponential diffusivities follow as particular cases from those obtained here but not vice versa the symmetries obtained for constructing exact solutions of the relevant equations are successfully applied the exact solutions are compared with those found earlier by means of different approaches an application of the solutions for solving a boundary value problem arising in population dynamics is presented

symmetry reductions of partial differential equations related to singular manifold expansions it is shown that all the information yielded by the reduction methods of bluman and cole 1974 and clarkson and kruskal 1989 can be obtained using the singular manifold expansion the burgers equation modified korteweg de vries equation caudrey dodd gibbon equation and the fitzhugh nagumo equation are used as illustrative examples several new exact solutions are presented

all solutions of standard symmetric linear partial differential equations have classical lie symmetry abstract it is proven that every solution of any linear partial differential equation with an independent variable deforming classical lie point symmetry is invariant under some classical lie point symmetry this is true for any number of independent variables and for equations of any order higher than one although this result makes use of the infinite dimensional component of the lie symmetry algebra due to linear superposition it is shown that new similarity solutions previously thought not to be classical can be recovered prospectively by allowing symmetries to include superposition of similarity solutions already known from the finite part of the symmetry algebra this result applies to all constant coefficient equations and to many variable coefficient equations such as fokker planck equations

nonclassical symmetry reduction and riemann wave solutions abstract in this paper we employ the nonclassical symmetry method in order to obtain riemann multiple wave solutions of a system of first order quasilinear differential equations we show how to construct a lie module of vector fields which are symmetries of the system supplemented by certain first order differential constraints we demonstrate the usefulness of our approach on several examples

symmetry groups of differential difference equations and their compatibility abstract it is shown that the intrinsic determining equations of a given differential difference equation dde can be derived by the compatibility between the original equation and the intrinsic invariant surface condition the 2 1 dimensional toda lattice the special toda lattice and the dd kp equation serving as examples are used to illustrate this approach then backlund transformations of the 2 1 dimensional ddes including the special toda lattice the modified toda lattice and the dd kz equation are presented by using the non intrinsic direct method in addition the clarkson kruskal direct method is developed to find similarity reductions of the ddes

on the marginal function in nonlinear programming for several types of finite or infinite dimensional optimization problems the marginal function or optimal value function is characterized by different local approximations such as generalized gradients generalized directional derivatives directional hadamard or dini derivatives we give estimates for these terms which are determined by multipliers satisfying necessary optimality conditions when the functions which define the optimization problem are more than once continuously differentiable then higher order necessary conditions are employed to obtain refined estimates for the marginal function as a by product we give a new equivalent formulation of clarke s multiplier rule for nonsmooth optimization problems this shows that the set of all multipliers satisfying these necessary conditions is the union of a finite number of closed convex cones

on the existence of nonclassical symmetries of partial differential equations relations between determining equations for nonclassical symmetries of bluman and cole and integrability conditions of differential equations supplemented by invariance conditions are studied it is shown that every system of equations with p independent variables admits p dimensional groups of such symmetries nonclassical symmetries of the eikonal equation in a curved space are investigated they are related to shear free congruences of null geodesics

asymptotic distribution of test statistics in the analysis of moment structures under inequality constraints summary the analysis of moment structural models has become an important tool of investigation in behavioural educational and economic studies the chi squared largesample test is routinely employed to assess the goodness of fit of the model considered however in order to invoke the standard asymptotic distribution theory certain regularity conditions have to be met here we consider the case where the population value of the parameter vector is a boundary point of the feasible region we show that in this case the asymptotic distribution of test statistic is a mixture of chi squared distributions the problem of finding the corresponding weights is discussed

singular reduction modules of differential equations the notion of singular reduction modules i e of singular modules of nonclassical conditional symmetry of differential equations is introduced it is shown that the derivation of nonclassical symmetries for differential equations can be improved by an in depth prior study of the associated singular modules of vector fields the form of differential functions and differential equations possessing parameterized families of singular modules is described up to point transformations singular cases of finding reduction modules are related to lowering the order of the corresponding reduced equations as examples singular reduction modules of evolution equations and second order quasi linear equations are studied reductions of differential equations to algebraic equations and to first order ordinary differential equations are considered in detail within the framework proposed and are related to previous no go results on nonclassical symmetries

contact symmetries and solutions by reduction of partial differential equations the correspondence between a contact symmetry of a second order pde e and a point symmetry of the equivalent first order system s is used to determine a class of solutions for e which may not always be invariant under contact transformation we name these solutions pseudo invariant because they are determined from a point symmetry of s via a reduction method invariant solutions under the related contact transformation are included in the family of pseudo invariant solutions

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

potential symmetries and invariant solutions for the generalized one dimensional fokker planck equation abstract pucci and saccomandi j phys a 26 1993 681 have obtained invariant solutions for the fokker planck fp equation via potential symmetries herein we analyze the generalized one dimensional fp equation for various cases corresponding to the physically interesting situations through the application of the same technique some interesting outcomes of this study are the deductions of the new exact solutions that do not seem to have been reported in the literature

the nonclassical group analysis of the heat equation abstract the nonclassical method of reduction was devised originally by bluman and cole in 1969 to find new exact solutions of the heat equation much success has been had by many authors using the method to find new exact solutions of nonlinear equations of mathematical and physical significance however the defining equations for the nonclassical reductions of the heat equation itself have remained unsolved although particular solutions have been given recently arrigo goard and broadbridge showed that there are no nonclassical reduction solutions of constant coefficient linear equations that are not already classical lie symmetry reduction solutions their arguments leave open the problem of what is the general nonclassical group action and its effect on the relevant solution of the heat equation in this article both these problems are solved in the final section we use the methods developed to solve the remaining outstanding case of nonclassical reductions of burgers equation