maximal symmetry group of the hamilton jacobi equation relativistic particle in flat spacetime lie s extended group method has been used to obtain the maximal symmetry group of the hamilton jacobi equation for a relativistic particle moving in flat spacetime for a massive particle it is the 21 parameter inhomogeneous pseudo orthogonal group io 4 1 of signature 4 1 for a zero mass particle like a photon or a neutrino this group is an infinite parameter lie group with an infinite parameter invariant subgroup such that the factor group is isomorphic to the inhomogeneous pseudo orthogonal group io 3 1 of signature 3 1

information processing in complex networks graph entropy and information functionals this paper introduces a general framework for defining the entropy of a graph our definition is based on a local information graph and on information functionals derived from the topological structure of a given graph more precisely an information functional quantifies structural information of a graph based on a derived probability distribution such a probability distribution leads directly to an entropy of a graph then the structural information content of a graph will be is interpreted and defined as the derived graph entropy another major contribution of this paper is the investigation of relationships between graph entropies in addition to this we provide numerical results demonstrating not only the feasibility of our method which has polynomial time complexity but also its usefulness with regard to practical applications aiming to an understanding of information processing in complex networks

studies in numerical nonlinear instability ii a new look at u t uu x 0 abstract it is shown that in general leap frog schemes any particular unstable solution behaves as an attractor of other solutions for a leap frog discretization of u t uu x 0 a particular kind of unstable solution is constructed which generically attracts any other solution estimates of the overflow time are presented and related to the notions of stability threshold and restricted stability

localization and coherence in nonintegrable systems we study the irreversible dynamics of nonlinear nonintegrable hamiltonian oscillator chains approaching their statistical asymptotic states in systems constrained by more than one conserved quantity the partitioning of the conserved quantities leads naturally to localized and coherent structures if the phase space is compact the final equilibrium state is governed by entropy maximization and the coherent structures are stable lumps in systems where the phase space is not compact the coherent structures can be collapsed represented in phase space by a heteroclinic connection of some unstable saddle to infinity

the algebraic structure of generalized ermakov systems in three dimensions the characteristic algebra of generalized ermakov systems is sl 2 r the structure of these systems in three dimensions is obtained a subset in the form of an equation of motion with the additional requirement of so 3 symmetry is studied it includes the classical equation of the magnetic monopole the existence of three vectors of poincare type is established consideration is given to weak generalized ermakov systems in which the symmetry breaking occurs in the radial equation

maximal symmetry groups of quantum relativistic equations the author has used lie s extended group method to obtain the maximal symmetry groups of the dirac equation for finite mass spin 1 2 particles and the weyl equation for zero mass spin 1 2 particles in both cases the maximal symmetry group is an infinite parameter lie group having an invariant subgroup also of an infinite number of generators the corresponding factor group for the dirac equation is an eleven parameter lie group isomorphic to the weyl group in the case of the weyl equation the corresponding factor group is a sixteen parameter lie group containing a proper subgroup isomorphic to the conformal group

n dimensional equations with the maximum number of symmetry generators the authors obtain the general class of ordinary differential equations that can be reduced by a point transformation to the free particle equation and given the general form of the lie symmetry generators for these equations they apply these results to obtain the symmetry generators for the n dimensional harmonic oscillator and for a particle in a constant magnetic field

a class of integrable potentials a class of time independent two dimensional integrable potentials all possessing an invariant of the same general form is constructed one of these potentials is superintegrable its invariants realize the symmetry algebra so 3 for negative energies e 2 for zero energy and so 2 1 for positive energies a transformation of coupling constants reveals that in parabolic coordinates this potential is the harmonic oscillator acted on by constant forces this and another potential in the class may be considered as successive extensions of the kepler potential the analytic properties of these integrable systems in the complex time plane are also discussed

degree based entropies of networks revisited studies on the information content of graphs and networks have been initiated in the late fifties based on the seminal work due to shannon and rashevsky various graph parameters have been used for the construction of entropy based measures to characterize the structure of complex networks based on shannon s entropy in cao et al extremality of degree based graph entropies inform sci 278 2014 22 33 we studied graph entropies which are based on vertex degrees by using so called information functionals as a matter of fact there has been very little work to find their extremal values when considered shannon entropy based graph measures we pursue with this line of research by proving further extremal properties of the degree based graph entropies

on graph entropy measures based on the number of independent sets and matchings abstract in this paper we consider the graph entropy measures based on the number of independent sets and matchings the reason to study these measures relates to the fact that the independent set and matching problem is computationally demanding however these features can be calculated for smaller networks in case one can establish efficient estimations those measures may be also used for larger graphs so we establish some upper and lower bounds as well as some information inequalities for these information theoretic quantities in order to give further evidence we also generate numerical results to study these measures such as list the extremal graphs for these entropies those results reveal the two entropies possess some new features

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

symmetries of linear systems of second order ordinary differential equations in this paper several problems concerning the lie algebra structure of symmetries and variational symmetries of a general linear system of second order ordinary differential equations are studied in particular a necessary and sufficient condition is obtained in terms of the coefficients of the system for the system s symmetry algebra to be of maximal dimension i e n2 4n 3 and isomorphic to sl n 2 r the well known symmetry algebra of the free particle equation x 0 when this condition is satisfied it is proved that the system is lagrangian and that its variational symmetry algebra is isomorphic to a fixed n2 3n 6 2 dimensional lie algebra whose structure levi mal cev decomposition and realization by means of a matrix algebra is determined for the particular case of isotropic systems which includes as far as is known all the examples treated in the literature explicit formulas for the generators of both the symmetry algebra and the variational symmetry algebra are obtained

a numerical energy conserving method for the dnls equation an implicit numerical energy conserving method is developed for the derivative nonlinear schrodinger dnls equation for periodic boundary conditions we find no numerical high frequency modulational instabilities in addition to the modulational instability from a linear analysis around a nonlinear state for the dnls equation if the modulation is small and k0 a22 2 t k0 is the wavenumber and a the amplitude the numerical scheme is used to follow the nonlinear behavior of the dnls modulational instability the numerical code is also tested by the evolution for one soliton initial data these tests show that if the modulation is not small compared to the background wave amplitude new nonlinear numerical instabilities are introduced

extremality of degree based graph entropies abstract many graph invariants have been used for the construction of entropy based measures to characterize the structure of complex networks based on shannon s entropy we study graph entropies which are based on vertex degrees by using so called information functionals when considering shannon entropy based graph measures there has been very little work to find their extremal values the main contribution of this paper is to prove some extremal values for the underlying graph entropy of certain families of graphs and to find the connection between the graph entropy and the sum of degree powers further conjectures to determine extremal values of graph entropies are given

on modulational instabilities in discretisations of the korteweg devries equation abstract w l briggs a c newell and t sarie j comput phys 51 83 1983 have studied nonlinear instabilities in discretisations of the equation u 1 u u u x 0 where u is a constant they demonstrated that solutions of leap frog discretisations of this equation could be destabilised by a dynamical process which can localise noise on the spatial grid d m sloan and a r mitchell j comput phys 67 1986 examined these instabilities by considering perturbations of the basic solution which take the form of side band fourier modes here this latter work is extended to cover analogous discretisations of the korteweg de vries equation

nordhaus gaddum type results for graph irregularities abstract a graph whose vertices have the same degree is called regular otherwise the graph is irregular in fact various measures of irregularity have been proposed and examined for a given graph g v e with v v 1 v 2 v n and edge set e g di is the vertex degree where 1 i n the irregularity of g is defined by i r r g v i v j e g d i d j a similar measure can be defined by i r r 2 g v i v j e g d i d j 2 the total irregularity of g is defined by i r r t g 1 2 v i v j v g d i d j the variance of the vertex degrees is defined v a r g 1 n i 1 n d i 2 2 m n 2 in this paper we present some nordhaus gaddum type results for these measures and draw conclusions

jet fields connections and second order differential equations jet fields are defined on fibred manifolds and represented as type 1 1 tensor fields corresponding to cartan ehresmann connections when defined on the first jet bundle they may be used to generalise the second order differential equation fields used in theoretical mechanics and to extend several existing results to field theories

bounds for degree based network entropies in this paper we continue studying degree based entropies for networks the quantities represent information theoretic network measures which are based on using information functionals involving vertex degrees we prove bounds for entropies which are based on information functionals using degree powers and come up with interrelations between different measures such interrelations are important to study connections between the measures required to understand the measures in depth

integrals and symmetries the bernoulli laplace lenz vector the notions of independent constants of motion are clarified emphasising the role of precise definition of the background space using a well defined algorithm that relates symmetries of equations of motion of a classical system newtonian lagrangian or hamiltonian to a conserved object in one to one fashion the symmetries of the coulomb and linear force problems are determined

symmetries first integrals and the inverse problem of lagrangian mechanics deals with the following question given a symmetry vector field y of a system of second order ordinary differential equations and an associated constant of the motion f is it possible to find a lagrangian l for the system such that y becomes a noether symmetry with respect to l and f its implied noether constant for one degree of freedom systems the answer to this question is affirmative in addition attention is paid to the construction of a suitable constant of the motion f for given symmetry y and vice versa several examples are discussed

complete solution to a problem on the maximal energy of unicyclic bipartite graphs abstract the energy of a simple graph g denoted by e g is defined as the sum of the absolute values of all eigenvalues of its adjacency matrix denote by c n the cycle and p n 6 the unicyclic graph obtained by connecting a vertex of c 6 with a leaf of p n 6 caporossi et al conjectured that the unicyclic graph with maximal energy is p n 6 for n 8 12 14 and n 16 in hou et al 2002 y hou i gutman c woo unicyclic graphs with maximal energy linear algebra appl 356 2002 27 36 the authors proved that e p n 6 is maximal within the class of the unicyclic bipartite n vertex graphs differing from c n and they also claimed that the energies of c n and p n 6 is quasi order incomparable and left this as an open problem in this paper by utilizing the coulson integral formula and some knowledge of real analysis especially by employing certain combinatorial techniques we show that the energy of p n 6 is greater than that of c n for n 8 12 14 and n 16 which completely solves this open problem and partially solves the above conjecture

focusing a mechanism for instability of nonlinear finite difference equations abstract a mechanism for the destabilization of numerical algorithms for partial differential equations is suggested the novelty of the work is that it attempts to explain the dynamical process by which noise can localize on a spatial grid and cause finite amplitude instability thresholds to be exceeded at distinct locations

the influence of differencing and cfl number on implicit time dependent non linear calculations abstract this paper is concerned with the stability and precision of transient implicit finite difference algorithms the test problem is the burgers equation the influence of space discretisation and initial values is studied linear analysis provides incomplete stability conditions for both continuous and weak solutions and we derive a simple criterion of augmented stability taking into account the non linearity of the problem the real stability limits appear to be a combination of the linear stability analysis criterion and of this augmented stability criterion

parametric excitation of computational modes inherent to leap frog scheme applied to the korteweg de vries equation abstract the numerical instability in the usage of the leap frog scheme applied to the korteweg de vries equation is studied the leap frog scheme generates computational modes which are not included in the original partial differential equation it is shown that the numerical instability comes from the parametric excitation of the computational modes by physical waves a method to suppress the computational modes is given the method enables us to make long time integration of the korteweg de vries equation even if the amplitude of wave is large

lie symmetries and invariants for the time dependent generalizations of the equation r c1rnl c2rmr 0 in this comment the authors use an invertible point transformation for the study of time dependent equations derived from the equation r c1rn l c2rmr 0 this procedure generalizes some results obtained recently by leach and gorringe 1990

sl 3 r as the group of symmetry transformations for all one dimensional linear systems the converse problem of similarity analysis is solved in general for the finite symmetry transformations of any inhomogeneous ordinary linear differential equation of the second order x f2 t x f1 t x f0 t the eight parameter realizations of the symmetry group are obtained in the form f 1p2 f where f stands for transformations of t x that depend exclusively on the fundamental solutions of the equation and where p2 is an arbitrary projective transformation in the plane thus it is shown that the full point symmetry group corresponds to sl 3 r indeed without recourse to the lie algebra also a technique is obtained for calculating the finite point symmetry realization of sl 3 r for any given one dimensional linear system some miscellaneous examples are given

grid resonances focusing and benjamin feir instabilities in leapfrog time discretizations abstract w l briggs a c newell and t sarie j comput phys 50 83 1983 studied the solutions of a class of discretizations of u t au x uu x 0 where a is a constant they discovered that the numerical solution of the discretization becomes unbounded despite the fact that theoretically the solution should remain bounded they attributed this anomalous behavior to a focusing mechanism in this paper we make use of a multiple scales analysis and show that the instabilities are caused by a resonance effect introduced by the discretizations the contributions from the space and time discretizations are analysed separately and in detail thus the structure of the focusing mechanism becomes transparent

some remarks on the generalized noether theory of point symmetry transformations of the lagrangian the noether theory of infinitesimal point symmetry transformations is revisited and generalized under the broad scope of the general theory of transformations of lagrangian mechanics the basic generalized point symmetries of a lagrangian function are thus obtained and briefly discussed it is proved that under very general and physically reasonable provisions the point symmetry group of the lagrangian is necessarily a finite lie group and a rather simple technique is then introduced for the explicit calculation of the associated lie algebra next the algebra obeyed by the set of basic noether quantities which also includes the traditional noether constants of motion is examined in this way it is shown that not all the generalized point symmetries of a lagrangian yield an associated conservation law the author presents three simple examples of this not so well known aspect of the noether theory

new classes of conversed quantities associated with non noether symmetries a result of hojman and harleston 1981 has been used to extend the known relationship between conserved quantities and non noether symmetries it is shown that n constants of the motion may be derived from each non noether symmetry of an n dimensional lagrangian system this applies whether the symmetry is continuous or discrete in addition the properties of the cartan form are utilised to to give a new proof of the hojman harleston theorem

on nonlinear instabilities in leap frog finite difference schemes abstract briggs newell and sarie j comput phys 51 83 1983 have discussed a mechanism for the destabilisation of finite difference approximations to nonlinear partial differential equations their ideas were developed using the leap frog approximation to the advection equation here the same situation is examined in a manner which compares the basic solution to a periodic wavetrain an investigation is made into the stability of the basic solution to small disturbances which take the form of side band fourier modes the relation between side band growth and envelope modulation is discussed

a complete basis set of functionally independent invariants of continuous lie groups using the methods of differential equations the author has obtained the complete set of functionally independent bases of invariants for some continuous lie groups of mathematical physics i the conformal group ii the semi direct product of the weyl group and the one parameter group of psi translation iii the direct product of the poincare group and the one parameter group of psi scaling iv the weyl group v the 14 parameter maximal symmetry group of the schrodinger equation for the coulombic system of atoms and molecules and vi su 3 of elementary particle physics the first four groups are the only groups which together with the poincare group are the maximal symmetry groups of all forms of the non linear klein gordon equation

some new aspects of first integrals and symmetries for central force dynamics for the general central force equations of motion in n 1 dimensions a complete set of 2n first integrals is derived in an explicit algorithmic way without the use of dynamical symmetries or noether s theorem the derivation uses the polar formulation of the equations of motion and yields energy angular momentum a generalized laplace runge lenz vector and a temporal quantity involving the time variable explicitly a variant of the general laplace runge lenz vector which generalizes hamilton s eccentricity vector is also obtained the physical meaning of the general laplace runge lenz vector its variant and the temporal quantity is discussed for general central forces their properties are compared for precessing bounded trajectories versus non precessing bounded trajectories as well as unbounded trajectories by considering an inverse square force kepler problem and a cubically perturbed inverse square force newtonian revolving orbit problem

lie symmetries for the charge monopole problem the authors obtain the lie symmetries for the equation of motion of an electric charge interacting with a magnetic monopole fixed at the origin they also discuss the construction of first integrals for this problem

dynamical noether symmetries the authors use noether s theorem within a restricted class of dynamical transformations involving velocity dependence to obtain first integral constants of the motion not available as noether point transformation constants

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

the relationship between the symmetries of and the existence of conserved vectors for the equation r f r l g r r 0 the existence of explicit expressions for conserved vectors for the charge monopole problem and the kepler problem is well known the lie algebras of the point transformations under which the equations of motion are invariant have been found more recently the recent discovery of an explicit expression for a laplace runge lenz like vector for the equation of motion r h l r hh kr 2 r o h h r from which one of the equations for the orbit is easily obtained has prompted the question of what is the lie algebra of the point symmetries of the equation r f r l g r r bold o of which each of the above problems is a member in the case that f r identical to o it is well known that laplace runge lenz like vectors exist the existence of such conserved vectors does not imply a particular algebraic structure of the lie point symmetries of the equation of motion however the existence of such symmetries provides a systematic method for constructing the vectors

the weak nonlinear instability of euler explicit scheme for the convective equation abstract the weak nonlinear instability of the euler explicit scheme for the convective equation u uux vxx is studied analytically using perturbation expansion and discrete fourier transform techniques short time weak nonlinear growth of the fourier modes for the noise is found provided that the linear stability conditions are satisfied as long as the perturbation expansion is valid the maximum growth is approximately proportional to the inverse of the grid spacing

exact analysis of nonlinear instability in a discrete burgers equation abstract a family of explicit nonlinear numerical schemes for burgers equation is derived by means of a discrete version of the hopf cole transformation exact nonlinear stability conditions for these schemes are then found and for one particular scheme the exact stability criteria are compared to the conventional linearized stability condition

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

digraphs with large maximum wiener index recently the concept of wiener index was extended to digraphs which are not necessarily strongly connected and it was shown that some fundamental results extend naturally within this concept this extension could be applicable in the topics of directed large networks particularly because with this measure one assigns finite values to the average distance and betweenness centrality of the nodes in a directed network it is not hard to show that among digraphs on n vertices the directed cycle c n achieves the maximum wiener index next we investigate digraphs with the second maximum wiener index one can consider this problem in the realm of all digraphs or restricted to those obtained by directing undirected graphs so called antisymmetric digraphs in both situations we obtain that such digraphs are constructed from c n n 6 by adding a single arc we conclude the paper with consideration for possible further works

a canonical reduction of order for the kepler problem using a new redudion of order technique we obtain four conserved quantities of the darsical planar kepler problem via the symmetries of the equations of motion and an extra vector field by passing to a suitable quotient manifold of the evolution space we deduce that negative energy orbits are closed and periodic without having to solve the differential equation in this paper we look at the classical kepler problem through the application of a new reduction of order technique l we believe that this technique provides a systematic means of producing first integrals from group actions more general than symmetries without relying on the existence of a variational principle moreover it is a powerful tool for investigating global properties such as periodicity the kepler problem is a good vehicle for a demonstration of these features the planar equations of motion are defined by a second order differential equation field on five dimensional evolution space there are three point symmetries which take solutions of the equation of motion into other solutions time translation which leaves the solution curves invariant but alters their starting times rotation about the origin which again leaves the curves invariant but alters the initial angular position and the transformation associated with the runge leuz vector which allows the orientation and eccentricity of the orbits to remain the same but changes the semi latus rectum we wish to show that conserved quantities of the system i e the objects which remain invariant along solution curves can be obtained using alternative techniques to noether s theorem 2 or solving the differential equation 3 41 lie s method 2 provides us with the three point symmetries mentioned above to effect our technique a fourth vector field is required which is a symmetry of the integrable distribution made up of the second order differential equation field and its three symmetries to find this field we modify the usual reduction of order via symmetry method by taking successive quotients of the extended phase space by the three actions for example we project down to a four dimensional manifold by setting the time coordinate to a constant thus identifying all the orbits which only differ by a time translation and we reduce our differential equation field to first order on a two dimensional quotient

a history of graph entropy measures this survey seeks to describe methods for measuring the entropy of graphs and to demonstrate the wide applicability of entropy measures setting the scene with a review of classical measures for determining the structural information content of graphs we discuss graph entropy measures which play an important role in a variety of problem areas including biology chemistry and sociology in addition we examine relationships between selected entropy measures illustrating differences quantitatively with concrete examples

a finite element model of two layer ocean circulation a mixed finite element formulation is applied to solve the quasi geostrophic equations in a two layer ocean domain enclosed by solid boundaries a combination of leap frog and schemes is used to integrate the equations in time wood s formula of the wall vorticity is derived in our formulation for the case of square elements adjacent to the boundaries long term computations with and without lumping have been carried out

exact invariants for a class of time dependent nonlinear hamiltonian systems a method of generalizing a class of invariants for a time dependent linear oscillator is developed for the motion of a mass point in one dimension with a general time dependent nonlinear potential formulas are derived for the allowable time dependent potentials and for the corresponding invariants the method by which these conclusions are reached is interesting theoretically and is explained in detail

hosoya entropy of fullerene graphs abstract entropy based methods are useful tools for investigating various problems in mathematical chemistry computational physics and pattern recognition in this paper we introduce a general framework for applying shannon entropy to fullerene graphs and used it to investigate their properties we show that important physical properties of these molecules can be determined by applying hosoya entropy to their corresponding graphs

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

reflections on the symmetry conservation law duality and the runge lenz vector following recent interest in an earlier work on the kepler problem prince and eliezer 1981 a number of remarks are made concerning the many faceted symmetry conservation law duality in classical mechanics in particular the advantages of associating a number of symmetries with the runge lenz vector are discussed

network entropies based on independent sets and matchings entropy based measures have been used to characterize the structure of complex networks various graph entropies based on different graph parameters have been proposed and studied so it has been intricate to decide which measure is the right one for solving a particular problem in this paper we introduce graph entropy measures based on independent sets and matchings of graphs the values of entropies of some special graphs are calculated and we draw several conclusions regrading the usability of the measures based on our findings we conclude that it should be demanding studying extremal values of the novel entropies

runge kutta smoother for suppression of computational mode instability of leapfrog scheme abstract the runge kutta smoother is applied to suppress nonlinear numerical instabilities in the leapfrog scheme for time integration of the korteweg de vries equation the accuracy of integration is compared with that by the use of the second order smoother the runge kutta smoother enables us to make long time integration of the korteweg de vries equation for large amplitudes

a linear focusing mechanism for dispersive and non dispersive wave problems a linear focusing mechanism for a wave packet propagation in a non periodic domain is explained here using fourier laplace spectral theory the global analysis method is used here to obtain numerical properties at each node of the full domain in this work we show the spectacular growth of an error packet at a particular node and wavenumber a phenomenon termed as focusing which depends on a cfl number n c and b numerical method used for discretization in this study we have shown the focusing phenomenon for the numerical solution of the 1d convection equation and linearized rotating shallow water equation

sl 3 r as the group of symmetry transformations for all one dimensional linear systems iii equivalent lagrangian formalisms the sl 3 r theory of projective transformations of the plane is applied to the lagrangians of all one dimensional newtonian linear systems noether and non noether equivalent lagrangians as well as the associated noether and non noether constants of motion are thus obtained in a completely general and systematic way complete unification is achieved by this group theoretic approach to lagrangians of one dimensional linear systems

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