a two step matrix splitting iteration for computing pagerank the pagerank algorithm plays an important role in determining the importance of web pages the core of this algorithm involves using the classical power method to compute the pagerank vector which is the principal eigenvector of the matrix representing the web link graph nevertheless it is well known that the power method may perform poorly when the second largest eigenvalue is close to the dominant one in this article we present a new approach that is based on the two step splitting iteration framework the description and convergence of the new algorithm are discussed in detail numerical examples are given to illustrate the performance of this algorithm

on the convergence of additive and multiplicative splitting iterations for systems of linear equations we study convergence conditions for the additive and the multiplicative splitting iteration methods i e two generalizations of the additive and the multiplicative schwarz iterations for hermitian and non hermitian systems of linear equations under an algebraic setting theoretical analyses show that when the coefficient and the splitting matrices are hermitian or non hermitian but diagonalizable satisfying mild conditions both additive and multiplicative splitting iteration methods are convergent even if the coefficient matrix is indefinite

refined iterative algorithms based on arnoldi s process for large unsymmetric eigenproblems abstract arnoldi s method has been popular for computing the small number of selected eigenvalues and the associated eigenvectors of large unsymmetric matrices however the approximate eigenvectors or ritz vectors obtained by arnoldi s method cannot be guaranteed to converge in theory even if the approximate eigenvalues or ritz values do in order to circumvent this potential danger a new strategy is proposed that computes refined approximate eigenvectors by small sized singular value decompositions it is shown that refined approximate eigenvectors converge to eigenvectors if ritz values do moreover the resulting refined algorithms converge more rapidly we report some numerical experiments and compare the refined algorithms with their counterparts the iterative arnoldi and arnoldi chebyshev algorithms the results show that the refined algorithms are considerably more efficient than their counterparts

duality under dependency inversion and noether theory for second order lagrangians abstract we consider the duality which is obtained by interchanging dependent and independent variables in a lagrangian this duality is studied here for a second order lagrangian which could describe e g a euler bernoulli beam more generally a one dimensional elastic body of higher grade however the essential theory follows purely formally for a general lagrangian next the corresponding noether theory of transformation groups and invariants is developed making use of duality as far as possible and showing the limits of duality theory particular emphasis is laid on the difference between relations equally valid on any curve and those valid only on the solution curve stressing the difference between true and apparent invariants

a reordering for the pagerank problem we describe a reordering particularly suited to the pagerank problem which reduces the computation of the pagerank vector to that of solving a much smaller system and then using forward substitution to get the full solution vector we compare the theoretical rates of convergence of the original pagerank algorithm to that of the new reordered pagerank algorithm showing that the new algorithm can do no worse than the original algorithm we present results of an experimental comparison on five datasets which demonstrate that the reordered pagerank algorithm can provide a speedup of as much as a factor of 6 we also note potential additional benefits that result from the proposed reordering

a note on the two step matrix splitting iteration for computing pagerank computing pagerank plays an important part in determining the importance of web pages based on the classical power method and the inner outer iteration proposed by gleich et al 2010 gu et al 2015 presented a two step splitting iteration i e the power inner outer pio iteration for the computation of pagerank in this paper we develop a variant of the pio iteration by applying multi step power method to combine with the inner outer iteration the new method is denoted as the mpio iteration its convergence is analyzed in detail numerical experiments on several pagerank problems are used to illustrate the effectiveness of the mpio iteration

parallel multisplitting iteration methods based on m splitting for the pagerank problem a class of iterative schemes based on the regular splitting is proposed for pagerank computation by some specific splitting these methods will reduce to some well known iterative schemes like jacobi method and successive overrelaxation method to suit computational requirements of the modern high speed multiprocessor environments we further study synchronous parallel counterparts for the new splitting iteration methods by using the m splitting of the coefficient matrix we establish the convergence theory of these synchronous multisplitting iteration methods finally some numerical experiments are presented to illustrate the feasibility stability and effectiveness of the proposed algorithms

holomorphic motions and teichmuller spaces we prove an equivariant form of slodkowski s theorem that every holomorphic motion of a subset of the extended complex plane c extends to a holomorphic motion of c as a consequence we prove that every holomorphic map of the unit disc into teichmuller space lifts to a holomorphic map into the space of beltrami forms we use this lifting theorem to study the teichmuller metric

adaptive methods for the computation of pagerank we observe that the convergence patterns of pages in the pagerank algorithm have a nonuniform distribution specifically many pages converge to their true pagerank quickly while relatively few pages take a much longer time to converge furthermore we observe that these slow converging pages are generally those pages with high pagerank we use this observation to devise a simple algorithm to speed up the computation of pagerank in which the pagerank of pages that have converged are not recomputed at each iteration after convergence this algorithm which we call adaptive pagerank speeds up the computation of pagerank by nearly 30

integrability of the 2d lotka volterra system via polynomial first integrals and polynomial inverse integrating factors we present new first integrals of the two dimensional lotka volterra systems which have a polynomial inverse integrating factor moreover we characterize all the polynomial first integrals of the two dimensional lotka volterra systems

stable quasiconformal mapping class groups and asymptotic teichmuller spaces the stable quasiconformal mapping class group is a group of quasiconformal mapping classes of a riemann surface that are homotopic to the identity outside some topologically finite subsurface its analytic counterpart is a group of mapping classes that act on the asymptotic teich m ul ler space trivially we prove that the stable quasiconformal mapping class group is coincident with the asymptotically trivial mapping class group for every riemann surface satisfying a certain geometric condition consequently the intermediate teich m ul ler space which is the quotient space of the teich m ul ler space by the asymptotically trivial mapping class group has a complex manifold structure and its automorphism group is geometrically isomorphic to the asymptotic teich m ul ler modular group the proof utilizes a condition for an asymptotic teich m ul ler modular transformation to be of finite order and this is given by the consideration of hyperbolic geometry of topologically infinite surfaces and its deformation under quasiconformal homeomorphisms also these arguments enable us to show that every asymptotic teich m ul ler modular transformation of finite order has a fixed point on the asymptotic teich m ul ler space which can be regarded as an asymptotic version of the nielsen theorem

rational extrapolation for the pagerank vector an important problem in web search is to determine the importance of each page from the mathematical point of view this problem consists in finding the nonnegative left eigenvector of a matrix corresponding to its dominant eigenvalue 1 since this matrix is neither stochastic nor irreducible the power method has convergence problems so the matrix is replaced by a convex combination depending on a parameter c with a rank one matrix its left principal eigenvector now depends on c and it is the pagerank vector we are looking for however when c is close to 1 the problem is ill conditioned and the power method converges slowly so the idea developed in this paper consists in computing the pagerank vector for several values of c and then to extrapolate them by a conveniently chosen rational function at a point near 1 the choice of this extrapolating function is based on the mathematical expression of the pagerank vector as a function of c numerical experiments end the paper

continuous nonpoint symmetries of ordinary difference equations a method to derive the continuous nonpoint symmetries of ordinary difference equations o e of order two and higher is presented a partial classification of second and fourth order difference equations that admit nonpoint symmetries both rational and polynomial forms which are quadratic in each variable is reported also exploiting the obtained symmetries it is shown how to construct integrals of motion or invariant for each of the considered equations the question of integrability of the fourth order difference equations possessing the above type of nonpoint symmetries has also been briefly discussed

extended tangent bundle formalism for time dependent lagrangian systems the lagrangian formalism for time dependent systems is developed using vector fields defined in the extended tangent bundle t q r the definition of the time independent extended lagrangian function l associated to a time dependent lagrangian l is given and then the techniques of symplectic mechanics are used to prove some properties of the equations determining the euler lagrange vector field the relations between the time independent and the time dependent helmholtz conditions are analyzed from a geometric perspective and finally the noether theorem for the extended lagrangian l is considered

a harmonic restarted arnoldi algorithm for calculating eigenvalues and determining multiplicity abstract a restarted arnoldi algorithm is given that computes eigenvalues and eigenvectors it is related to implicitly restarted arnoldi but has a simpler restarting approach harmonic and regular rayleigh ritz versions are possible for multiple eigenvalues an approach is proposed that first computes eigenvalues with the new harmonic restarted arnoldi algorithm then uses random restarts to determine multiplicity this avoids the need for a block method or for relying on roundoff error to produce the multiple copies

off diagonal low rank preconditioner for difficult pagerank problems abstract pagerank problem is the cornerstone of google search engine and is usually stated as solving a huge linear system moreover when the damping factor approaches 1 the spectrum properties of this system deteriorate rapidly and this system becomes difficult to solve in this paper we demonstrate that the coefficient matrix of this system can be transferred into a block form by partitioning its rows into special sets in particular the off diagonal part of the block coefficient matrix can be compressed by a simple low rank factorization which can be beneficial for solving the pagerank problem hence a matrix partition method is proposed to discover the special sets of rows for supporting the low rank factorization then a preconditioner based on the low rank factorization is proposed for solving difficult pagerank problems numerical experiments are presented to support the discussions and to illustrate the effectiveness of the proposed methods

block triangular and skew hermitian splitting methods for positive definite linear systems by further generalizing the concept of hermitian or normal and skew hermitian splitting for a non hermitian and positive definite matrix we introduce a new splitting called positive definite and skew hermitian splitting pss and then establish a class of pss methods similar to the hermitian or normal and skew hermitian splitting hss or nss method for iteratively solving the positive definite systems of linear equations theoretical analysis shows that the pss method converges unconditionally to the exact solution of the linear system with the upper bound of its convergence factor dependent only on the spectrum of the positive definite splitting matrix and independent of the spectrum of the skew hermitian splitting matrix as well as the eigenvectors of all matrices involved when we specialize the pss to block triangular or triangular and skew hermitian splitting btss or tss the pss method naturally leads to a btss or tss iteration method which may be more practical and efficient than the hss and nss iteration methods applications of the btss method to the linear systems of block two by two structures are discussed in detail numerical experiments further show the effectiveness of our new methods

a gmres power algorithm for computing pagerank problems abstract the pagerank algorithm for determining the importance of web pages has become a central technique in web search we propose a new method to speed up the convergence performance for computing pagerank when the damping factor is close to one called as gmres power which is based on a periodic combination of the power method with the gmres algorithm the description and convergence analysis of the new algorithm are discussed in detail numerical results are reported to confirm the efficiency of the new algorithm

integrable three particle systems hidden symmetries and deformations of the calogero moser system the properties of some three particle systems are studied using as an approach the theory of generalized or hidden symmetries it is proven that the existence of a family that possesses in addition to the energy function two nonlinear constants of motion in involution this family of integrable systems is obtained by considering deformations of the lagrangian of the calogero moser system finally the generalization of the results to the general case of n particles is discussed

symmetries separability and volume forms in order to clarify certain misconceptions in the literature we discuss the details of the way determining equations for general symmetries of a lagrangian system differ from determining equations for noether symmetries and establish the minimal set of extra conditions which have to be imposed in practical situations for the former to reduce to the latter we further derive properties by which in situations where the system is integrable through quadratic integrals in involution the components of the corresponding noether symmetries themselves can be used to compute the separation variables if they exist for the hamilton jacobi equation

the general inner outer iteration method based on regular splittings for the pagerank problem abstract in this paper combined the regular splittings of the coefficient matrix i p with the inner outer iteration framework 9 a general inner outer gio iteration method is presented for solving the pagerank problem firstly the aor and modified aor maor methods for solving the pagerank problem are constructed and several comparison results are also given next the gio iteration scheme is developed and its overall convergence is analyzed in detail furthermore the preconditioner derived from the gio iteration can be used to accelerate the krylov subspace methods such as gmres method finally some numerical experiments on several pagerank problems are provided to illustrate the efficiency of the proposed algorithm

first integrals for a general linear system of two second order odes via a partial lagrangian the partial noether operators and first integrals of a general system of two linear second order ordinary differential equations odes with variable coefficients are studied by means of a partial lagrangian the canonical form for the general system of two second order ordinary differential equations is invoked and all cases of this system are discussed with respect to partial noether operators we also tabulate the results for the special case b x c x of the system which was considered elsewhere using a lagrangian and a partial lagrangian the first integrals are obtained explicitly by exploiting a noether like theorem with the help of partial noether operators this study gives a new way to construct first integrals for systems without a variational principle as not all linear equations have a lagrangian physical applications to conservative and oscillator mechanical systems are given

time dependent lagrangian systems a geometric approach using semibasic forms a new presentation of the geometric formalism in r tq of the time dependent lagrangian systems is given the formalism is used to prove some properties of the equations determining the euler lagrange vector field xl for relating the theory with the time independent case and for studying the time dependent lagrangian inverse problem

biholomorphic maps between teichmuller spaces in this paper we study biholomorphic maps between teichmuller spaces and the induced linear isometries between the corresponding tangent spaces the first main result in this paper is the following classification theorem if m and n are two riemann surfaces that are not of exceptional type and if there exists a biholomorphic map between the corresponding teichmuller spaces teich m and teich n then m and n are quasiconformally related also every such biholomorphic map is geometric in particular we have that every automorphism of the teichmuller space teich m must be geometric this result generalizes the previously known results see 2 5 7 and enables us to prove the well known conjecture that states that the group of automorphisms of teich m is isomorphic to the mapping class group of m whenever the surface m is not of exceptional type in order to prove the above results we develop a method for studying linear isometries between l 1 type spaces our focus is on studying linear isometries between banach spaces of integrable holomorphic quadratic differentials which are supported on riemann surfaces our main result in this direction theorem 1 1 states that if m and n are riemann surfaces of nonexceptional type then every linear isometry between a 1 m and a 1 n is geometric that is every such isometry is induced by a conformal map between m and n

on computing pagerank via lumping the google matrix computing google s pagerank via lumping the google matrix was recently analyzed in i c f ipsen t m selee pagerank computation with special attention to dangling nodes siam j matrix anal appl 29 2007 1281 1296 it was shown that all of the dangling nodes can be lumped into a single node and the pagerank could be obtained by applying the power method to the reduced matrix furthermore the stochastic reduced matrix had the same nonzero eigenvalues as the full google matrix and the power method applied to the reduced matrix had the same convergence rate as that of the power method applied to the full matrix therefore a large amount of operations could be saved for computing the full pagerank vector in this note we show that the reduced matrix obtained by lumping the dangling nodes can be further reduced by lumping a class of nondangling nodes called weakly nondangling nodes to another single node and the further reduced matrix is also stochastic with the same nonzero eigenvalues as the google matrix

on the determination of non local symmetries the importance of non local symmetries of differential equations lies in their manifestation as lie point symmetries of the equations resulting from reduction of order the reason for the determination of these symmetries in second order equations with only one lie point symmetry is self evident however the disadvantage of non local symmetries is that no systematic approach to their determination exists we present such an approach applicable to differential equations of any order and apply it to some second order ordinary differential equations and show that they have a rich occurrence we also look at possible generalizations of the concept of non local symmetries

lie symmetries and invariants of the lotka volterra system in this paper we use the lie symmetry method for finding rational and transcendental symmetry transformations and invariants for the 3d lotka volterra system

an arnoldi inout algorithm for computing pagerank problems the pagerank algorithm plays an important role in the web search engines to speed up the convergence behavior for computing pagerank we propose a new method called as arnoldi inout which is the inner outer iteration method modified with the thick restarted arnoldi method the description and convergence of the new algorithm are discussed in detail numerical results are given to illustrate the efficiency of the new algorithm

an iteration method for solving the linear system ax b abstract in this paper based on a convergence splitting of the matrix a we present an inner outer iteration method for solving the linear system a x b we analyze the overall convergence of this method without any other restriction on its parameters moreover we give the accelerated inner outer iteration method and discuss how to apply the inner outer iterations as a preconditioner for the krylov subspace methods the inner outer iteration method can also be used for the solution of a x b c finally several numerical examples are given to validate the performance of our proposed algorithms

an exact invariant for a class of time dependent anharmonic oscillators with cubic anharmonicity an exact invariant is constructed for a class of time dependent anharmonic oscillators using the method of the lie theory of extended groups the presence of the anharmonic term imposes a constraint on the nature of the time dependence for a subclass it is possible to obtain an energy like integral and a condition under which the motion is bounded

extension of giacomini s results concerning invariants for one dimensional time dependent potentials one dimensional time dependent potentials are considered for which an invariant can be expressed in terms of the potential and the momentum according to the formulation of giacomini new solutions of giacomini s equations are derived in addition possibilities are discussed for extending giacomini s approach to more general systems

on equations with universal invariance a general discussion of equations with universal invariance for a scalar field is provided in the framework of lagrangian theory of first order systems the condition of universal invariance is that the lagrange souriau form of the system is invariant up to a function factor with respect to these transformations we consider in detail the case when this function is homogeneous in the first order derivatives when this function is non trivial we have an example of non noetherian symmetries and obtain some equations intensively studied in the literature

differential geometric aspects of the cartan form symmetry theory this paper demonstrates that the classical cartan form 1l is not adequate for the determination of all the natural symmetries and conservation laws for a lagrangian l it is shown that the various extensions 2l rl of the classical cartan form introduced in recent papers give larger symmetry groups g1 g2 gr this paper also introduces the notion of contact equivalent lagrangians which serves to clarify the idea that different lagrangians can give rise to the same variational and symmetry theories

on the lie symmetries of the classical kepler problem the standard version of noether s theorem when applied to the classical kepler problem leads to the constants of energy and angular momentum but does not give the hidden symmetry known as the runge lenz vector lie s theory of differential equations is used to obtain all three constants of motion the transformations of solutions under the point transformations to which these constants correspond are studied the results are generalised to n dimensions

dynamics on teichmuller spaces and self covering of riemann surfaces a non injective holomorphic self cover of a riemann surface induces a non surjective holomorphic self embedding of its teichmuller space we investigate the dynamics of such self embeddings by applying our structure theorem of self covering of riemann surfaces and examine the distribution of its isometric vectors on the tangent bundle over the teichmuller space we also extend our observation to quasiregular self covers of riemann surfaces and give an answer to a certain problem on quasiconformal equivalence to a holomorphic self cover

lie and noether symmetries and a result of logan the notions of lie and conformal invariance symmetry for a regular second order equation field are shown to be essentially identical it is pointed out that the way in which conformal invariance symmetries are traditionally defined is not invariant and that this leads to difficulties in interpreting a recent result of logan 1985 however the introduction of coordinate free geometrical apparatus leads to a simple proof of logan s result and interpretation of the examples considered by him

an arnoldi extrapolation algorithm for computing pagerank the arnoldi type algorithm proposed by golub and greif g golub c greif an arnoldi type algorithm for computing pagerank bit 46 2006 759 771 is a restarted krylov subspace method for computing pagerank however this algorithm may not be efficient when the damping factor is high and the dimension of the search subspace is small in this paper we first develop an extrapolation method based on ritz values we then consider how to periodically knit this extrapolation method together with the arnoldi type algorithm the resulting algorithm is the arnoldi extrapolation algorithm the convergence of the new algorithm is analyzed numerical experiments demonstrate the numerical behavior of this algorithm

geometric isomorphisms between infinite dimensional teichmuller spaces let x and y be the interiors of bordered riemann surfaces with finitely generated fundamental groups and nonempty borders we prove that every holomorphic isomorphism of the teichmiiller space of x onto the teichmiiller space of y is induced by a quasiconformal homeomorphism of x onto y these teichmiiller spaces are not finite dimensional and their groups of holomorphic automorphisms do not act properly discontinuously so the proof presents difficulties not present in the classical case to overcome them we study weak continuity properties of isometries of the tangent spaces to teichmiiller space and special properties of teichmiiller disks

fiber spaces over teichmuller spaces for fuchsian groups with elliptic elements we determine the biholomorphic fiber preserving isomorphisms of fiber spaces over teichmuller spaces for fuchsian groups with elliptic elements we show that except in some special cases a biholomorphic fiber preserving isomorphism between two bers fiber spaces is always an allowable mapping we find that the situation is different for teichmuller curves showing that in general there are some other biholomorphic fiber preserving isomorphisms between teichmuller curves besides the allowable mappings

infinite dimensional second order ordinary differential equations via t2m abstract the vector bundle structure obtained on the second order acceleration tangent bundle t 2 m of a smooth manifold m by means of a linear connection on the base provides an alternative way for the study of second order ordinary differential equations on manifolds of finite and infinite dimension second order vector fields and their integral curves could provide a new way of representing and solving a wide class of evolutionary equations for states on frechet manifolds of sections that arise naturally as inequivalent configurations of a physical field the technique is illustrated by examples in the framework of banach and frechet spaces and on lie groups in particular discussing the case of autoparallel curves which include geodesics if the connection is induced by a riemannian structure

a unified framework for the construction of various matrix multisplitting iterative methods for large sparse system of linear equations abstract a unified framework for the construction of various synchronous and asynchronous parallel matrix multisplitting iterative methods suitable to the simd and mimd multiprocessor systems respectively is presented and its convergence theory is established under rather weak conditions these afford general method models and systematical convergence criterions for studying the parallel iterations in the sense of matrix multisplitting in addition how the known parallel matrix multisplitting iterative methods can be classified into this new framework and what novel ones can be generated by it are shown in detail

dynamics of the polynomial differential systems with homogeneous nonlinearities and a star node abstract we consider the class of polynomial differential equations x x p n x y y y q n x y in r 2 where p n x y and q n x y are homogeneous polynomials of degree n 1 and 0 i e the class of polynomial differential systems with homogeneous nonlinearities with a star node at the origin we prove that these systems are darboux integrable moreover for these systems we study the existence and non existence of limit cycles surrounding the equilibrium point located at the origin

generalizations of noether s theorem in classical mechanics in this paper a review is presented of various approaches to the generalization of the version of noether s theorem which is presented in most textbooks on classical mechanics its motivation is the controversy still persisting around the possible scope of a noether type theorem allowing for velocity dependent transformations our analysis is centered around the one factor common to all known treatments namely the structure of the related first integral we first discuss the most general framework in which a function of the above mentioned structure constitutes a first integral of a given lagrangian system and show that one cannot really talk about an interrelationship between symmetries and first integrals there we then compare different proposed generalizations of noether s theorem by describing the nature of the restrictions which characterize them when they are situated within the broadest framework we prove a seemingly new equivalence result between the two main approaches that of invaria

frobenius method and invariants for one dimensional time dependent hamiltonian systems we apply the frobenius integrability theorem in the search for invariants for one dimensional hamiltonian systems with a time dependent potential we obtain several classes of potential functions for which the frobenius method ensures the existence of a two dimensional foliation to which the motion is constrained in particular we derive a new infinite class of potentials for which the motion is assuredly restricted to a two dimensional foliation in some cases the frobenius method allows the explicit construction of an associated invariant the inverse result is proven that if an invariant is known then it can always be furnished by the frobenius method

unification of extremal length geometry on teichmuller space via intersection number in this paper we give a framework for the study of the extremal length geometry of teichmuller space after s kerckhoff f gardiner and h masur there is a natural compactification using extremal length geometry introduced by gardiner and masur the compactification is realized in a certain projective space we develop the extremal length geometry in the cone which is defined as the inverse image of the compactification via the quotient mapping the compactification is identified with a subset of the cone by taking an appropriate lift the cone contains canonically the space of measured foliations in the boundary we first extend the geometric intersection number on the space of measured foliations to the cone and observe that the restriction of the intersection number to teichmuller space is represented by an explicit formula in terms of the gromov product with respect to the teichmuller distance from this observation we deduce that the gromov product extends continuously to the compactification as an application we obtain an alternative approach to a characterization of the isometry group of teichmuller space we also obtain a new realization of teichmuller space a hyperboloid model of teichmuller space with respect to the teichmuller distance

an inner outer iteration for computing pagerank we present a new iterative scheme for pagerank computation the algorithm is applied to the linear system formulation of the problem using inner outer stationary iterations it is simple can be easily implemented and parallelized and requires minimal storage overhead our convergence analysis shows that the algorithm is effective for a crude inner tolerance and is not sensitive to the choice of the parameters involved the same idea can be used as a preconditioning technique for nonstationary schemes numerical examples featuring matrices of dimensions exceeding 100 000 000 in sequential and parallel environments demonstrate the merits of our technique our code is available online for viewing and testing along with several large scale examples

two stage iterations based on composite splittings for rectangular linear systems abstract in this paper we introduce a two stage method to solve rectangular linear systems that exhibits faster convergence than typical stationary iterative methods under suitable conditions we prove convergence of the new method the number of outer iterations can be reduced by using a few significant number of inner iterations for efficient computations further we perform a comparison analysis and establish that a higher number of inner iterations ensures a smaller spectral radius of the global iteration matrix we also discuss the uniqueness of a proper splitting and illustrate different comparison theorems for different subclasses of proper splittings

higher order noether symmetries and constants of the motion the authors identify a class of symmetries for lagrangian systems called higher order noether symmetries which yield a corresponding first integral without further integrations first a version of noether s theorem is recalled in which noether symmetries are considered to be symmetries of the two form d theta theta being the cartan form nth order noether symmetries are then defined by the requirement lynd theta 0 the picture is further generalised by exploring various ways in which the computation of repeated lie derivatives of d theta can lead to the identification of first integrals a number of illustrative examples are discussed

generalized lie symmetries and complete integrability of certain nonlinear hamiltonian systems with three degrees of freedom a systematic method for investigating the existence of nontrivial generalized lie symmetries is presented and the associated integrals of motion for nonlinear oscillator systems with three degrees of freedom defined in terms of the lagrangian by l 1 2 x2 y2 z2 v x y z are constructed then the method is applied to study the integrability properties of quartically and cubically coupled nonlinear oscillators with three degrees of freedom compatibility with the painleve property is also investigated

first integrals for the modified emden equation q α t q qn 0 it is shown that the modified emden equation q t q qn 0 possesses first integrals for functions t other than kt 1 the function t is obtained explicitly in the case n 3 and parametrically for other n 2 the case n 2 is seen to be particularly difficult to solve

variational principles for locally variational forms we present the theory of higher order local variational principles in fibered manifolds in which the fundamental global concept is a locally variational dynamical form any two lepage forms defining a local variational principle for this form differ on intersection of their domains by a variationally trivial form in this sense but in a different geometric setting the local variational principles satisfy analogous properties as the variational functionals of the chern simons type the resulting theory of extremals and symmetries extends the first order theories of the lagrange souriau form presented by grigore and popp and closed equivalents of the first order euler lagrange forms of hakova and krupkova conceptually our approach differs from prieto who uses the poincare cartan forms which do not have higher order global analogues