conditional lie backlund symmetry second order differential constraint and direct reduction of diffusion systems abstract the equivalence relation between conditional lie backlund symmetry higher order differential constraint and direct reduction is presented in the case of evolution systems the second order nonlinear conditional lie backlund symmetries and corresponding induced differential constraints admitted by nonlinear diffusion systems are identified as an application of the conditional lie backlund symmetry of evolution systems consequently the corresponding symmetry reductions are obtained due to the compatibility of the resulting differential constraints and the governing systems

differential stackel matrices we show that additive separation of variables for linear homogeneous equations of all orders is characterized by differential stackel matrices generalizations of the classical stackel matrices used for multiplicative separation of second order schrodinger equations and additive separation of hamilton jacobi equations we work out the principal properties of these matrices and demonstrate that even for second order laplace equations additive separation may occur when multiplicative separation does not

on the possible multiplicities of the eigenvalues of a hermitian matrix whose graph is a tree we consider the general problem of determining which lists of multiplicities for the eigenvalues occur among hermitian matrices the graph of whose off diagonal entries is a given tree several restrictions are cited and a construction strategy is given together these are sufficient to characterize all lists for each tree in two infinite classes the double paths and generalized stars and to tabulate all lists for trees on fewer than nine vertices such tables should be useful for formulating and dispelling general conjectures

on some differential invariants for a family of diffusion equations the equivalence transformation algebra le and some of its differential invariants for the class of equations ut h u ux x f x u ux h 0 are obtained using these invariants we characterize subclasses which can be mapped by means of an equivalence transformation into the well studied family of equations vt vkvx x

multiple emitter location and signal parameter estimation processing the signals received on an array of sensors for the location of the emitter is of great enough interest to have been treated under many special case assumptions the general problem considers sensors with arbitrary locations and arbitrary directional characteristics gain phase polarization in a noise interference environment of arbitrary covariance matrix this report is concerned first with the multiple emitter aspect of this problem and second with the generality of solution a description is given of the multiple signal classification music algorithm which provides asymptotically unbiased estimates of 1 number of incident wavefronts present 2 directions of arrival doa or emitter locations 3 strengths and cross correlations among the incident waveforms 4 noise interference strength examples and comparisons with methods based on maximum likelihood ml and maximum entropy me as well as conventional beamforming are included an example of its use as a multiple frequency estimator operating on time series is included

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

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

functional separation of variables for laplace equations in two dimensions the authors say that a solution psi of a partial differential equation in two real variables x1 x2 is functionally separable in these variables if psi x1 x2 phi a x1 b x2 for single variable functions phi a b such that phi a b not 0 they classify all possibilities for regular functional separation in local coordinates for equations of the form delta 2 psi f psi x1 x2 where delta 2 is the laplace beltrami operator on a two dimensional riemannian or pseudo riemannian space if the dependence of f on x1 x2 is non trivial then separation can occur for conformal cartesian coordinates on any space if f g psi then for orthogonal coordinates they find that true functional separation i e separation other than additive or multiplicative occurs precisely for cartesian coordinates in the euclidean and pseudo euclidean planes the sine gordon equation provides an example of this separation for many of these cases the separated solutions a b can be expressed in terms of elliptic functions for non orthogonal coordinates and f g psi true functional separation occurs precisely for cartesian coordinates in the pseudo euclidean plane and for a coordinate system on the hyperboloid of one sheet a pseudo riemannian space of constant curvature

exact similarity solutions to some nonlinear diffusion equations the author gives new closed form similarity solutions to n dimensional radially symmetric nonlinear diffusion equations with power law and exponential diffusivities these solutions are generalisations of known similarity solutions

new variable separation approach application to nonlinear diffusion equations the concept of the derivative dependent functional separable solution ddfss as a generalization to the functional separable solution is proposed as an application it is used to discuss the generalized nonlinear diffusion equations based on the generalized conditional symmetry approach as a consequence a complete list of canonical forms for such equations which admit the ddfss is obtained and some exact solutions to the resulting equations are described

exact polynomial solutions to some nonlinear diffusion equations abstract a class of non group invariant exact solutions to a number of nonlinear diffusion equations is constructed expressed in the appropriate variables these solutions are polynomials in the spatial variable the method is mainly applicable to equations which may be written in an appropriate quadratically nonlinear form a number of equivalence transformations between various nonlinear diffusion equations are also derived

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

some non self similar solutions to a nonlinear diffusion equation the author generates exact solutions to the nonlinear diffusion equation ut del u 1 2 del u which are not similarity solutions some applications and generalizations are indicated

nonclassical symmetry reductions of a porous medium equation with convection in this paper new symmetry reductions and exact solutions are presented for the porous medium equation with convection those spatial forms for which the equation can be reduced to an ordinary differential equation are studied the symmetry reductions and exact solutions presented are derived by using the nonclassical method developed by bluman and cole and are unobtainable by lie classical method the asymptotic behaviour of some of the new solutions is analysed

classification of coupled systems with two component nonlinear diffusion equations by the invariant subspace method the invariant subspace method is developed to perform classification of systems with two component nonlinear diffusion equations which was carried out with respect to the invariant subspaces defined by linear ordinary differential equations as a result the corresponding exact solutions generated by invariant subspaces to the resulting systems are obtained in most cases two components of these exact solutions belong to different scalar subspaces behaviour to several exact solutions of the systems is described

classical point symmetries of a porous medium equation the lie group formalism is applied to deduce symmetries of the porous medium equation we study those spatial forms that admit the classical symmetry group the reduction obtained from the optimal system of subalgebras are derived some new exact solutions can be obtained

direct reduction and differential constraints direct reductions of partial differential equations to systems of ordinary differential equations are in one to one correspondence with compatible differential constraints the differential constraint method is applied to prove that a parabolic evolution equation admits infinitely many characteristic second order reductions but admits a non characteristic second order reduction if and only if it is linearizable

potential symmetries of a porous medium equation potential symmetries which are not local symmetries are carried out for the porous medium equation where when it can be written in a conserved form these symmetries are realized as local symmetries of a related auxiliary system and lead to the construction of corresponding invariant solutions as well as to the linearization of the equation by non invertible mappings

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

functional separable solutions of nonlinear reaction diffusion equations with variable coefficients abstract the paper presents a number of new functional separable solutions to nonlinear reaction diffusion equations of the form c x u t a x u x x b x u x p x f u where f u is an arbitrary function it is shown that any three of the four variable coefficients a x b x c x p x of such equations can be chosen arbitrarily and the remaining coefficient can be expressed through the others examples of specific equations and their exact solutions are given the results obtained are generalized to more complex multidimensional nonlinear reaction diffusion equations with variable coefficients also some functional separable solutions to nonlinear reaction diffusion equations with delay u t u x x a x f u w w u x t where 0 is the delay time and f u w is an arbitrary function of two arguments are obtained it is important to note that the exact solutions of nonlinear pdes and delay pdes that contain arbitrary functions and therefore have sufficient generality are of the greatest practical interest for testing and evaluating the accuracy of various numerical and approximate analytical methods for solving corresponding initial boundary value problems

new non lie ansatze and exact solutions of nonlinear reaction diffusion convection equations a constructive method for obtaining new exact solutions of nonlinear evolution equations is further developed the method is based on the consideration of a fixed nonlinear partial differential equation together with an additional generating condition in the form of a linear high order ordinary differential equation using this method new non lie ansatze and exact solutions are obtained for two classes of diffusion equations with power and exponential nonlinearities which describe real processes in physics chemistry and biology the analysis of the found solutions and the relation of the proposed method to some approaches which have been suggested in several recently published papers are presented

nonclassical symmetry solutions and the methods of bluman cole and clarkson kruskal nonclassical symmetry solutions of physically relevant partial differential equations are considered via the reduction methods of bluman and cole and clarkson and kruskal consistency conditions will be provided to show that if satisfied these two methods are equivalent in the sense that they lead to the same symmetry solutions the boussinesq equation and burgers equation are used as illustrative examples exact solutions one of which is new will be presented for burgers equation obtained from the bluman and cole method yet not obtainable by clarkson and kruskal s method

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

linearly independent vertices and minimum semidefinite rank we study the minimum semidefinite rank of a graph using vector representations of the graph and of certain subgraphs we present a sufficient condition for when the vectors corresponding to a set of vertices of a graph must be linearly independent in any vector representation of that graph and conjecture that the resulting graph invariant is equal to minimum semidefinite rank rotation of vector representations by a unitary matrix allows us to find the minimum semidefinite rank of the join of two graphs we also improve upon previous results concerning the effect on minimum semidefinite rank of the removal of a vertex

exact results for the nonlinear diffusion equations delta u delta t delta delta x u 4 3 delta u delta x and delta u delta t delta delta x u 2 3 delta u delta x the author exploits local and non local symmetries of the two equations of the title and of some related equations some new exact solutions are derived and some specific boundary value problems of physical significance are considered

improved kinetic renormalisation group approach to diffusion limited aggregation an improved kinetic renormalisation group approach to diffusion limited aggregation is presented this approach is based on the growth process itself and accounts for the dispersity in the growth probabilities it yields the multifractal spectrum d q with better values at smaller q on the 2d square lattice the authors find df 1 694 for the fractal dimensions of the cluster and that of its interface and d 1 1 01 for the information dimension the former agrees with the simulation results df approximately 1 70 and the latter compares very well with the exact value d 1 1

the number of distinct eigenvalues for which an index decreases multiplicity abstract for an hermitian matrix a whose graph is a tree t we study the number of eigenvalues of a whose multiplicity decreases when a particular vertex is deleted from t explicit results are given when that number of eigenvalues is less than 4 and an inductive result thereafter the work is based in part on classical results about multiplicities but also on some new facts including a useful identity this allows us to give strong bounds based on simple facts about the location of the vertex in the tree some facts about matrices whose graphs are not trees are included and the classical diameter bound about the number of distinct eigenvalues for a tree follows

differential constraints and exact solutions of nonlinear diffusion equations the differential constraints are applied to obtain explicit solutions of nonlinear diffusion equations certain linear determining equations with parameters are used to find such differential constraints they generalize the determining equations used in the search for classical lie symmetries

multiplicity lists for symmetric matrices whose graphs have few missing edges abstract we characterize the possible lists of multiplicities occurring among the eigenvalues of real symmetric or hermitian matrices whose graph is one of k n k n less an edge or both possibilities for k n less two edges the lists are quite different from those for trees some construction techniques are developed here and additional results with more missing edges are given including the case of several independent edges

separation of variables of a generalized porous medium equation with nonlinear source this paper considers a general form of the porous medium equation with nonlinear source term ut d u u n x f u n 1 the functional separation of variables of this equation is studied by using the generalized conditional symmetry approach we obtain a complete list of canonical forms for such equations which admit the functional separable solutions as a consequence some exact solutions to the resulting equations are constructed and their behavior are also investigated 2002 elsevier science usa all rights reserved

a unified approach to painleve expansions abstract the painleve test for partial differential equations developed by weiss tabor and carnevale wtc is examined in detail and shown to provide a unified approach to the integrable properties of both ordinary and partial differential equations a simple modification of the wtc procedure used for partial differential equations enables us to determine the lax pairs hirota equations and auto backlund transformations for ordinary differential equations including a new lax pair for an integrable case of the henon heiles system a detailed study of the kdv hierarchy is made and a complete picture of the pattern of resonances for all solution branches is obtained the role of the singular branches is examined in detail and important new insights obtained in particular we find that each singular branch is simply a re expansion of the principal branch about a point on the pole manifold at which several isolated poles coalesce a parallel analysis is carried out for the akns hierarchy

algorithms for the nonclassical method of symmetry reductions in this article the authors first present an algorithm for calculating the determining equations associated with so called nonclassical method of symmetry reductions a la bluman and cole for systems of partial differential equations this algorithm requires significantly less computation time than that standardly used and avoids many of the difficulties commonly encountered the proof of correctness of the algorithm is a simple application of the theory of grobner bases in the second part they demonstrate some algorithms which may be used to analyse and often to solve the resulting systems of overdetermined nonlinear pdes the authors take as their principal example a generalised boussinesq equation which arises in shallow water theory although the equation appears to be nonintegrable the authors obtain an exact two soliton solution from a nonclassical reduction

symmetries periodic plane waves and blow up of l ω systems abstract nonclassical symmetries of one dimensional reaction diffusion equations of the type have been studied the functional forms of and for which the system admits nonclassical symmetries have been determined and the corresponding reduced systems have been obtained some of these reduced systems admit symmetries which lead to further reductions among the several classes of exact solutions that have been obtained asymptotically periodic plane waves appear as similarity solutions of systems we also have obtained a family of solutions that exhibit a blow up process

the heat equation shrinks embedded plane curves to round points soit c 0 s 1 r 2 une courbe lisse plongee dans le plan alors c s 1 0 t r 2 existe en satisfaisant c t k n ou k est la courbure de c et n est son vecteur unite normal entrant c t est lisse pour tout t il converge vers un point quand t t et sa forme limite quand t t est un cercle rond avec convergence dans norme c

waiting time solutions of a nonlinear diffusion equation similarity solutions of the equation partial c partial t partial partial x c n partial c partial x n 0 are studied for both t 0 and t 0 and there is a stationary free boundary at x 0 and for t 0 have support in x at beta where beta 1 a 0 i e there is a moving free boundary x at beta by prescribing appropriate initial data c c 0 x x 0 at t t 0 these solutions demonstrate the existence of a waiting time t 0 before the free boundary moves a comparison theorem then enables a number of results to be obtained and conjectures to be made about the existence and length of waiting times for general initial data a three parameter family of solutions which exhibit blow up at t 0 is also constructed applications of these results to models for the spreading of a viscous blob on a hor

iterating the nonclassical symmetries method we show that iterations of the nonclassical symmetries method for any evolution equation give raise to new nonlinear equations which inherit the symmetry algebra of the given equation invariant solutions of these heir equations supply new solutions of the original equation the same method generates heir equations for hyperbolic and elliptic equations some examples are given

the heat equation shrinking convex plane curves soient m et m des varietes de riemann et f m m une application reguliere si m est une courbe convexe plongee dans le plan r 2 l equation de la chaleur contracte m a un point

music maximum likelihood and cramer rao bound the performance of the music and ml methods is studied and their statistical efficiency is analyzed the cramer rao bound crb for the estimation problems is derived and some useful properties of the crb covariance matrix are established the relationship between the music and ml estimators is investigated as well a numerical study is reported of the statistical efficiency of the music estimator for the problem of finding the directions of two plane waves using a uniform linear array an exact description of the results is included

positivity versus localization in degenerate diffusion equations on considere le probleme de cauchy u t u m f u 0 dans s r n o u x o u 0 x x r n ou m 1 n 1 et les fonctions f et u 0 satisfont les hypotheses 1 f c 1 o f o 0 2 u 0 est une fonction continue non negative definie sur r n a support borne

some non local transformations between nonlinear diffusion equations generalized backlund transformations are applied to derive links between a large number of different types of nonlinear diffusion equations including many which are of physical significance some new exactly linearizable forms are determined

nonclassical symmetries of a porous medium equation with absorption in this paper new symmetry reductions and exact solutions are presented for the porous medium equation with absorption those spatial forms for which the equation can be reduced to an ordinary differential equation are studied the symmetry reductions and exact solutions presented are derived by using the nonclassical method developed by bluman and cole and are unobtainable by lie classical method

undirected graphs of hermitian matrices that admit only two distinct eigenvalues abstract we consider the problem of determining those undirected n vertex graphs with a corresponding hermitian matrix that admits only two distinct eigenvalues with multiplicities k and n k after giving some general algebraic characterizations of these dual multiplicity graphs we then prove two major graph theoretic necessary conditions on such graphs construction techniques are then developed and these lead to a characterization of dual multiplicity graphs for which the lesser multiplicity is two

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

group invariant solutions and optimal systems for multidimensional hydrodynamics the group properties of the three dimensional 3 d one temperature hydrodynamic equations including nonlinear conduction and a thermal source are presented a subgroup corresponding to axisymmetric geometry is chosen and the details of the construction of the one and two dimensional optimal systems are shown the two dimensional optimal system is used to generate 23 intrinsically different reductions of the 2 d partial differential equations to ordinary differential equations these ordinary differential equations can be solved to provide analytic solutions to the original partial differential equations two example analytic solutions are presented a 2 d axisymmetric flow with a p2 asymmetry and a 3 d spiraling flow

a new eigenvector weighting method for stable high resolution array processing a stable high resolution estimator shire based on a new weighting function for eigenvectors of an array sensor covariance matrix is proposed for bearing estimation of closely spaced sources shire is data adaptive and requires no user defined parameter in its implementation numerical results are presented to illustrate the performance achievable

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

new explicit solutions of quasilinear heat equations with general first order sign invariants abstract for the one dimensional quasilinear heat equations ut u uxx f u with smooth functions u 0 and f u we derive conditions under which a general quadratic hamilton jacobi operator h u ut h2 u ux 2 h1 u ux h0 u with the functions h2 h1 h0 to be determined preserves both the signs 0 and 0 on the solutions u x t h u is then called a sign invariant of the given parabolic equation several types of different sign invariants are established which are governed by a linear second order ordinary differential equations new explicit solutions via finite dimensional dynamical systems corresponding to zero preserving of the first order operator are constructed