comment on on the general solution for the modified emden type equation ddot x alpha x pdot x beta x 3 0 it is shown that the modified emden type equation in the title belongs to a class of lienard s equations that may be mapped into separable abel s equations and are therefore integrable first integrals for equations of this class are easily derived there are three of them corresponding to three distinct dynamical regimes and integrated to yield parametric representations of the corresponding solutions

nonstandard conserved hamiltonian structures in dissipative damped systems nonlinear generalizations of damped harmonic oscillator in this paper we point out the existence of a remarkable nonlocal transformation between the damped harmonic oscillator and a modified emden type nonlinear oscillator equation with linear forcing x xx x3 x 0 which preserves the form of the time independent integral conservative hamiltonian and the equation of motion generalizing this transformation we prove the existence of nonstandard conservative hamiltonian structure for a general class of damped nonlinear oscillators including lienard type systems further using the above hamiltonian structure for a specific example namely the generalized modified emden equation x xqx x2q 1 0 where and q are arbitrary parameters the general solution is obtained through appropriate canonical transformations we also present the conservative hamiltonian structure of the damped mathews lakshmanan oscillator equation the associated lagrangian description for all the above systems is also briefly discussed

mappings approximately preserving orthogonality in normed spaces abstract we answer many open questions regarding approximately orthogonality preserving mappings in birkhoff james sense in normed spaces in particular we show that every approximately orthogonality preserving linear mapping in chmielinski sense is necessarily a scalar multiple of an e isometry thus whenever e isometries are close to isometries we obtain stability an example is given showing that approximately orthogonality preserving mappings are in general far from scalar multiples of isometries that is stability does not hold

analysis and solution of a nonlinear second order differential equation through rescaling and through a dynamical point of view the solutions of the equation y yy y3 0 where is a free parameter are investigated for 1 9 the equation is linearizable through an eight parameter symmetry group and is completely integrable for 1 9 only two symmetries subsist but through a dynamical description the analytical asymptotic solutions and their behavior are given according to the value of and according to the initial conditions

symmetry and singularity properties of second order ordinary differential equations of lie s type iii abstract we extend the work of abraham shrauner b abraham shrauner hidden symmetries and linearization of the modified painleve ince equation j math phys 34 1993 4809 4816 on the linearization of the modified painleve ince equation to a wider class of nonlinear second order ordinary differential equations invariant under the symmetries of time translation and self similarity in the process we demonstrate a remarkable connection with the parameters obtained in the singularity analysis of this class of equations

a nonlocal connection between certain linear and nonlinear ordinary differential equations oscillators we explore a nonlocal connection between certain linear and nonlinear ordinary differential equations odes representing physically important oscillator systems and identify a class of integrable nonlinear odes of any order we also devise a method to derive explicit general solutions of the nonlinear odes interestingly many well known integrable models can be accommodated into our scheme and our procedure thereby provides further understanding of these models

hidden and contact symmetries of ordinary differential equations the lie algebra for the maximal contact symmetries of third order ordinary differential equations odes is examined for type i and ii hidden symmetries where the analysis of hidden symmetries for point symmetries is extended to contact symmetries ones invariant under the group associated with the ten dimensional maximal lie algebra may produce type i hidden symmetries for two parameter subgroups and type ii hidden symmetries for certain solvable non abelian three parameter subgroups in the third order odes when they are reduced in order a new class of type ii hidden symmetries is recognized in which contact symmetries transform to point symmetries in some reduction paths two examples of odes invariant under subgroups of the ten parameter group under which y 0 is invariant demonstrate the new class of type ii hidden symmetries

orthogonality to matrix subspaces and a distance formula abstract we obtain a necessary and sufficient condition for a matrix a to be birkhoff james orthogonal to any subspace w of m n c using this we obtain an expression for the distance of a from any unital c subalgebra of m n c

equipartition and mean square responses in large undamped structures it is shown that the smoothed spectral energy density at one point in a large complex structure may be approximated in a simple manner as proportional to the frequency smoothed admittance at the receiver and at the source and inversely proportional to the global modal density thus the mean square response may be estimated with little computational burden knowing only the local properties at source and receiver and the size of the system the approximate predictions are compared with the results of direct numerical simulations and found to be accurate except in the presence of anderson localization work supported by nsf

a group theoretical identification of integrable cases of the lienard type equation x f x x g x 0 i equations having nonmaximal number of lie point symmetries we carry out a detailed lie point symmetry group classification of the lienard type equation x f x x g x 0 where f x and g x are arbitrary smooth functions of x we divide our analysis into two parts in the present first part we isolate equations that admit lesser parameter lie point symmetries namely one two and three parameter symmetries and in the second part we identify equations that admit maximal eight parameter lie point symmetries in the former case the invariant equations form a family of integrable equations and in the latter case they form a class of linearizable equations under point transformations further we prove the integrability of all the equations obtained in the present paper through equivalence transformations either by providing the general solution or by constructing time independent hamiltonians several of these equations are being identified for the first time from the group theoretical analysis

on the complete integrability and linearization of nonlinear ordinary differential equations ii third order equations we introduce a method for finding general solutions of third order nonlinear differential equations by extending the modified prelle singer method we describe a procedure to deduce all the integrals of motion associated with the given equation so that the general solution follows straightforwardly from these integrals the method is illustrated with several examples further we propose a powerful method of identifying linearizing transformations the proposed method not only unifies all the known linearizing transformations systematically but also introduces a new and generalized linearizing transformation in addition to the above we provide an algorithm to invert the non local linearizing transformation through this procedure the general solution for the original nonlinear equation can be obtained from the solution of the linear ordinary differential equation

ensemble energy average and energy flow relationships for nonstationary vibrating systems abstract this paper attempts to introduce a new point of view on energy analysis in structural dynamics with particular emphasis to its link with uncertainty and complexity a linear elastic system undergoing free vibrations is considered the system is subdivided into two subsystems and their respective energies together with the shared energy flow are analysed first the ensemble energy average of the two subsystems assuming uncertain the natural frequencies is investigated it is shown how the energy averages follow a simple law when observing the long term response of the system obtained by a suitable asymptotic expansion the second part of the analysis shows how the ensemble energy average of a set of random samples is representative even of the single case if the system is complex enough the two previous points combined produce a result that applies to the energy sharing between two subsystems even independently of uncertainty for complex systems a simple energy sharing law is indeed stated moreover in the case of absence of damping a nonlinear relation between the energy flow and the energy weighted difference between the two subsystems is derived on the other hand when damping is present this relationship becomes linear including two terms one is proportional to the energy weighted difference between the two subsystems the other being proportional to its time derivative therefore the approach suggests a way for deriving a general approach to energy sharing in vibration with results that in some cases are reminiscent of those met in statistical energy analysis finally computational experiments performed on systems of increasing complexity validate the theoretical results

approximate birkhoff james orthogonality in the space of bounded linear operators there are two notions of approximate birkhoff james orthogonality in a normed space we characterize both the notions of approximate birkhoff james orthogonality in the space of bounded linear operators defined on a normed space a complete characterization of approximate birkhoff james orthogonality in the space of bounded linear operators defined on hilbert space of any dimension is obtained which improves on the recent result by chmielinski et al 2017 4 in which they characterized approximate birkhoff james orthogonality of linear operators on finite dimensional hilbert space and also of compact operators on any hilbert space c 2017 elsevier inc all rights reserved

on the general solution for the modified emden type equation in this paper we demonstrate that the modified equation of emden type mee is integrable either explicitly or by quadrature for any value of and we also prove that the mee possesses appropriate time independent hamiltonian functions for the full range of parameters and in addition we show that the mee is intimately connected with two well known nonlinear models namely the equation of force free duffing type oscillator and the two dimensional lotka volterra equation and thus the complete integrability of the latter two models can also be understood in terms of the mee

on lie systems and kummer schwarz equations a lie system is a system of first order differential equations admitting a superposition rule i e a map that expresses its general solution in terms of a generic family of particular solutions and certain constants in this work we use the geometric theory of lie systems to prove that the explicit integration of second and third order kummer schwarz equations is equivalent to obtaining a particular solution of a lie system on sl 2 r this same result can be extended to riccati milne pinney and to the here defined generalised kummer schwarz equations which include several types of kummer schwarz equations as particular cases we demonstrate that all the above mentioned equations related to the same lie system on sl 2 r can be integrated simultaneously which retrieves and generalizes in a unified and simpler manner previous results appearing in the literature as a byproduct we recover various properties of the schwarzian derivative

autonomous self similar ordinary differential equations and the painleve connection we demonstrate an intimate connection between nonlinear higher order ordinary differential equations possessing the two symmetries of autonomy and self similarity and the leading order behaviour and resonances determined in the application of the painleve test similar behaviour is seen for systems of first order differential equations several examples illustrate the theory in an integrable case of the abc system the singularity analysis reveals a positive and a negative resonance and the method of leading order behaviour leads naturally to a laurent expansion containing both

equivalence classes of second order ordinary differential equations with only a three dimensional lie algebra of point symmetries and linearisation there are seven equivalence classes of second order ordinary differential equations possessing only three lie point symmetries and hence not linearisible by means of a point transformation we examine the representatives of these classes for linearisibility by means of other types of transformation in particular we compare the potential for linearisibility and the possession of the painleve property the complete symmetry group is realised in the standard algebra for each of the equivalence classes

on the equivalence of linear equations under nonlocal transformation abstract linear n th order n 3 ordinary differential equations have been shown to possess n 1 n 2 or n 4 lie point symmetries each class contains equations which are equivalent under point transformation by taking the example of third order equations we show that all linear equations are equivalent if the class of transformation is broadened to include nonlocal transformations and hence the representative of this class of equations is y n 0

complete symmetry groups of ordinary differential equations and their integrals some basic considerations abstract the concept of the complete symmetry group of a differential equation introduced by j krause 1994 j math phys 35 5734 5748 is extended to integrals of such equations this paper is devoted to some aspects characterising complete symmetry groups the algebras of the symmetries of both differential equations and integrals are studied in the context of equations for which the elements are represented by point or contact symmetries so that there is no ambiguity about the group both algebras and groups are found to be nonunique

the painleve test hidden symmetries and the equation y yy ky3 0 for general values of the parameter k the equation y yy ky3 0 can be reduced to quadrature via a lie algebraic approach either direct or through hidden symmetries for specific values of k mostly in 1 9 1 8 the solution can be expressed in parametric form for these values of k the equation passes the weak painleve test for some other values of k the equation passes the painleve test but the solution cannot in general be expressed parametrically

induced damping by a nearly continuous distribution of nearly undamped oscillators linear analysis abstract the induced damping in a master oscillator contributed by a set of satellite oscillators is obtained in terms of a summation over a discrete distribution of the set the distribution is with respect to the resonance frequencies of the satellite oscillators in the set the distribution is cast in an ascending order and is assumed to be centered about the resonance frequency of the master oscillator in isolation if the modal overlap parameters are less than unity significant undulations are present in the induced damping the less the modal overlap parameters are compared with unity the more prominent are the undulations the undulations are largely suppressed when the local modal overlap parameters exceed unity moreover appropriately averaging the undulations yields values for the induced damping that coincide with those obtained when the modal overlap parameters exceed unity further it transpires that these common values are independent of the individual modal overlap parameters when the summation is replaced by an integration the first order results are undulations free and the values so obtained again coincide with those pertaining to modal overlap parameters that exceed unity without defining the excursions in the undulations the transition from a discrete to a continuous distribution that is implied by a summation to an integration must assume that the modal overlap parameters exceed unity in particular without careful and meaningful qualifications it may be misleading to assume a priori that the modal overlap parameters are equal to zero

the complete symmetry group of the generalised hyperladder problem abstract we further consider the n dimensional ladder system that is the homogeneous quadratic system of first order differential equations of the form x i x i j 1 n a ij x j i 1 n where aij i 1 j i j 1 n introduced by imai and hirata nlin si 0212007 we establish the most general system of first order ordinary differential equations invariant under the algebra which characterises the ladder system of imai and hirata and the algebra of minimal dimension required to specify completely this most general system we provide the complete symmetry group of the generalised hyperladder system and discuss its integrability

on the complete symmetry group of the classical kepler system a rather strong concept of symmetry is introduced in classical mechanics in the sense that some mechanical systems can be completely characterized by the symmetry laws they obey accordingly a complete symmetry group realization in mechanics must be endowed with the following two features 1 the group acts freely and transitively on the manifold of all allowed motions of the system 2 the given equations of motion are the only ordinary differential equations that remain invariant under the specified action of the group this program is applied successfully to the classical kepler problem since the complete symmetry group for this particular system is here obtained the importance of this result for the quantum kinematic theory of the kepler system is emphasized

nonlocal symmetries of a class of scalar and coupled nonlinear ordinary differential equations of any order in this paper we devise a systematic procedure to obtain nonlocal symmetries of a class of scalar nonlinear ordinary differential equations odes of arbitrary order related to linear odes through nonlocal relations the procedure makes use of the lie point symmetries of the linear odes and the nonlocal connection to deduce the nonlocal symmetries of the corresponding nonlinear odes using these nonlocal symmetries we obtain reduction transformations and reduced equations to specific examples we find that the reduced equations can be explicitly integrated to deduce the general solutions for these cases we also extend this procedure to coupled higher order nonlinear odes with specific reference to second order nonlinear odes

shaping of a system s frequency response using an array of subordinate oscillators the frequency response of an oscillating structure can be tailored by attaching one or more subordinate oscillators this paper shows how the magnitude and phase of the frequency response can be deliberately shaped by prescribing the distributions of the dynamic properties in an array of such subordinate oscillators exact analytic governing equations of motion are derived for the coupled system composed of the primary system and the subordinate array for a relatively small number 100 of attached oscillators whose total mass is small 1 relative to the primary structure it is possible to engineer frequency response functions of the primary oscillator to have for example nearly linear phase or constant amplitude over a frequency band of interest the frequency range over which response shaping is achieved is determined by the band of the attached oscillators it is shown that the common analytic methodology for designing a dynamic vibration absorber represents the limiting case of a single oscill

the ermanno bernoulli constants and representations of the complete symmetry group of the kepler problem the expression of the components of the equation of motion of the classical kepler problem in terms of the natural variables associated with the ermanno bernoulli constants leads naturally to the same equations as are obtained by the technique of reduction of order developed by nucci j math phys 37 1772 1996 reported by nucci and leach j math phys 42 746 2001 three representations of the complete symmetry group of the kepler problem are obtained from the three standard representations of the complete symmetry group of the simple harmonic oscillator the algebra of the complete symmetry group of the two dimensional kepler problem is identified to be a1 a3 3 the applicability of the results to other classes of problem such as the kepler problem with drag which possess a conserved vector of laplace runge lenz type is indicated the three dimensional kepler problem is shown to be completely specified by six symmetries rather than the eight previously reported by krause j math phys

a nonlocal connection between certain linear and nonlinear ordinary differential equations part ii complex nonlinear oscillators in this paper we present a method to identify integrable complex nonlinear oscillator systems and construct their solutions for this purpose we introduce two types of nonlocal transformations which relate specific classes of nonlinear complex ordinary differential equations odes with complex linear odes thereby proving the integrability of the former we also show how to construct the solutions using the two types of nonlocal transformations with several physically interesting cases as examples

ordinary differential equations invariant under translation in the independent variable and rescaling the lagrangian formulation abstract the lagrangian formulation of the class of general second order ordinary differential equations invariant under translation in the independent variable and rescaling is presented the differential equations arising from this analysis are analysed using the painleve test the well known differential equation y y y k y 3 0 is a unique member of this class when k 3 since it is linearisable by a point transformation a wider subset is shown to be linearisable by means of a nonlocal transformation

orthogonality of matrices abstract let a and b be rectangular matrices then a is orthogonal to b if a b a for every scalar some approximation theory and convexity results on matrices are used to study orthogonality of matrices and answer an open problem of bhatia and semrl

operator norm attainment and birkhoff james orthogonality abstract recently in sain d paul k operator norm attainment and inner product spaces linear algebra appl 439 2013 2448 2452 it was proved that if t is a linear operator on a finite dimensional real normed linear space x such that t attains norm only on d where d is a connected closed subset of s x then t satisfies the bhatia semrl bs property bhatia r semrl p orthogonality of matrices and distance problem linear algebra appl 287 1999 77 85 i e for a l x t b a implies that there exists x d such that t x b a x here we explore the converse of the above result we prove that in a real normed linear space x of dimension 2 a linear operator t satisfies the bs property if and only if the set of unit vectors on which t attains norm is connected in the corresponding projective space r p 1 s x x x motivated by the result in 2 dimensions we conjecture that this characterization of bs property is true in general n dimensional real normed linear spaces we further prove that if the space x is strictly convex then the set of operators in l x which satisfy the bs property is dense in l x

exponential nonlocal symmetries and nonnormal reduction of order the conventional approach to double reduction of the order of an ordinary differential equation using lie symmetries is via the normal subgroups of point symmetries we show that provided that one is prepared to use nonlocal symmetries initial reduction by the nonnormal subgroup does not prevent the double reduction we further illustrate our results with the general third order equations invariant under the nonsolvable algebras sl 2 r of which the chazy equation is a noted example and so 3

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

symmetry and singularity properties of the generalised kummer schwarz and related equations abstract we examine the generalised kummer schwarz equation and some of its generalisations from the viewpoints of symmetry and singularity analyses we determine the complete symmetry group of the general equation and show that different forms of the fourth order representative illustrate the three possible classes of laurent series to be expected in the course of the singularity analysis

a unification in the theory of linearization of second order nonlinear ordinary differential equations in this letter we introduce a new generalized linearizing transformation glt for second order nonlinear ordinary differential equations snodes the well known invertible point ipt and non point transformations npt can be derived as sub cases of the glt a wider class of nonlinear odes that cannot be linearized through npt and ipt can be linearized by this glt we also illustrate how to construct glts and to identify the form of the linearizable equations and propose a procedure to derive the general solution from this glt for the snodes we demonstrate the theory with two examples which are of contemporary interest

macroscopic description of microscopically strongly inhomogenous systems a mathematical basis for the synthesis of higher gradients metamaterials we consider the time evolution of a one dimensional n gradient continuum our aim is to construct and analyze discrete approximations in terms of physically realizable mechanical systems referred to as microscopic because they are living on a smaller space scale we validate our construction by proving a convergence theorem of the microscopic system to the given continuum as the scale parameter goes to zero

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

on the properties of a variant of the riccati system of equations a variant of the generalized riccati system of equations is considered it is shown that for 2n 3 the system admits a bilagrangian description and the dynamics has a node at the origin whereas for much smaller than a critical value the dynamics is periodic the origin being a centre it is found that the solution changes from being periodic to aperiodic at a critical point which is independent of the initial conditions this behaviour is explained by finding a scaling argument via which the phase trajectories corresponding to different initial conditions collapse onto a single universal orbit numerical evidence for the transition is shown further using a perturbative renormalization group argument it is conjectured that the oscillator exhibits isochronous oscillations the correctness of the conjecture is established numerically

nonlocal symmetries and integrable ordinary differential equations x 3xx x3 0 and its generalizations the equation x 3xx x3 0 is well known in many areas of mathematics and physics it possesses the algebra sl 3 r of lie point symmetries hence is equivalent to the equation for a free particle and both left and right painleve series we investigate two higher dimensional analogs in terms of their symmetry and singularity structures we find a drastic reduction in symmetry and a loss of some of the singularity properties from the nonlocal symmetries we are able to determine the complete symmetry group as being represented by a five dimensional abelian algebra

connection between the existence of first integrals and the painleve property in two dimensional lotka volterra and quadratic systems taking advantage of the considerable amount of work done in the search for first integrals invariants for the two dimensional lotka volterra system and the quadratic system lvs and qs we compare the relations needed to exhibit invariants one for the lvs at least three for the qs to the two conditions of the painleve test index and compatibility we find that eventually restricting the invariants to those which are analytic all exponents integers and thereby adding new constraints these constraints always coalesce with the two painleve conditions we conclude that straightforward application of the painleve test picks up only these simple analytic invariants and that possession of the painleve property is too strong a condition for the existence of the invariants

energy equipartition and frequency distribution in complex attachments as reported in several recent publications an undamped simple oscillator with a complex attachment that consists of a set of undamped parallel resonators can exhibit unusual energy sharing properties the conservative set of oscillators of the attachment can absorb nearly all the impulsive energy applied to the primary oscillator to which it is connected the key factor in the ability of the attachment to absorb energy with near irreversibility correlates with the natural frequency distribution of the resonators within it the reported results also show that a family of optimal frequency distributions can be determined on the basis of a variational approach minimizing a certain functional related to the system response the present paper establishes a link between these optimal frequency distributions and the energy equipartition principle optimal frequency distributions are those that spread the injected energy as uniformly as possible over the degrees of freedom or over the modes of the system theoretical as well as numerical results presented support this point of view

orthogonality of matrices and some distance problems abstract if a and b are matrices such that a zb a for all complex numbers z then a is said to be orthogonal to b we find necessary and sufficient conditions for this to be the case some applications and generalisations are also discussed

approximate orthogonality in normed spaces and its applications abstract in a normed space we consider an approximate orthogonality relation related to the birkhoff orthogonality we give some properties of this relation as well as applications in particular we characterize the approximate orthogonality in the class of linear bounded operators on a hilbert space

on the generalizations of the kummer schwarz equation abstract we carry out complete group classifications of two non homogeneous generalizations of the kummer schwarz equation in particular we prove that the symmetry algebra for one of them is s l 2 r s l 2 r and that it can be mapped by an equivalence transformation to the homogeneous kummer schwarz equation for this purpose we calculate the continuous equivalence group for that equation we also study the linearization of the considered equations

operator norm attainment and inner product spaces abstract in this paper we prove that a finite dimensional real normed linear space x is an inner product space iff for any linear operator t on x t attains its norm at e 1 e 2 s x implies t attains its norm at span e 1 e 2 s x

orthogonality of matrices abstract let x be a real finite dimensional normed space with unit sphere sx and let l x be the space of linear operators from x into itself it is proved that x is an inner product space if and only if for a c l x a c u s x a au au cu where denotes birkhoff orthogonality

transient energy exchange between a primary structure and a set of oscillators return time and apparent damping in this paper we examine the conditions that influence the return time the time it takes before energy returns from a set of satellite oscillators attached to a primary structure two methods are presented to estimate the return time one estimate is based on an analysis of the reaction force on a rigid base by a finite number of oscillators as compared with an infinite number of continuously distributed oscillators the result gives a lower bound estimate for the return time a more accurate estimation results from considering the dynamic behavior of a set of oscillators as waves in a waveguide such an analogy explains energy flow between a primary structure and the oscillators in terms of pseudowaves and shows that a nonlinear frequency distribution of the oscillators leads to pseudodispersive waves the resulting approximate expressions show the influence of the natural frequency distribution within the set of oscillators and of their number on the return time as compared with the asymptotic case of a continuous set with infinite oscillators in the paper we also introduce a new method based on a hilbert envelope to estimate the apparent damping loss factor of the primary structure during the return time considering transient energy flow from the primary structure before any energy reflects back from the attached oscillators the expressions developed for return time and damping factor show close agreement with direct numerical simulations the paper concludes with a discussion of the return time and its relation to apparent damping and optimum frequency distribution within a set of oscillators that maximize these quantities

on the uniqueness of the schwarzian and linearisation by nonlocal contact transformation abstract we investigate the uniqueness of the schwarzian from a group theoretic point of view the kummer schwarz equation obtained from the schwarzian is also analysed for its group properties related equations arising in the study of geodesic curves on a surface of constant curvature are constructed these equations are generalised by reducing the symmetry imposed the unique generalisation is linearised under a nonlocal contact transformation

symmetry lie algebras of nth order ordinary differential equations abstract we show that an nth n 3 order linear ordinary differential equation has exactly one of n 1 n 2 or n 4 the maximum point symmetries the lie algebras corresponding to the respective numbers of point symmetries are obtained then it is shown that a necessary and sufficient conditon for an nth n 3 order equation to be linearizable via a point transformation is that it must admit the n dimensional abelian algebra na1 a1 a1 a1 we discuss in detail the symmetry realizations of n 1 a1 s a1 finally we prove that an nth n 3 order equation q n h t q qn 1 cannot admit exactly an n 3 dimensional algebra of point symmetries which is a subalgebra of na1 gl 2 r

symmetry singularities and integrability in complex dynamics ii rescaling and time translation in two dimensional systems the explicit integrability of second order ordinary differential equations invariant under time translation and rescaling is investigated quadratic systems generated from the linearisable version of this class of equations are analysed to determine the relationship between the painleve and singularity properties of the different systems the transformation contains a parameter and for critical values intimately related to the possession of the painleve property in the parent second order equation one finds a difference from the generic behaviour this study is a prelude to a full discussion of the class of transformations which preserve the painleve property in the construction of quadratic systems from scalar nth order ordinary differential equations invariant under time invariant under time translation and rescaling

near irreversibility in a conservative linear structure with singularity points in its modal density through two complementary approaches using modal response and wave propagation the analyses presented here show the conditions under which a decaying impulse response or a nearly irreversible energy trapping takes place in a linear conservative continuous system the results show that the basic foundation of near irreversibility or apparent damping rests upon the presence of singularity points in the modal density of dynamic systems or analogously in the wave stopping properties associated with these singularities to illustrate the concept of apparent damping in detail a simple undamped beam is modified to introduce a singularity point in its modal density distribution simulations show that a suitable application of a compressive axial force to an undamped beam placed on an elastic foundation attenuates its impulse response with time and develops the characteristics of a nearly irreversible energy trap

the birkhoff james orthogonality in hilbert c modules abstract in this paper we characterize the birkhoff james orthogonality for elements of a hilbert c module in terms of states of the underlying c algebra we also show that the birkhoff james orthogonality in a hilbert c module over a c algebra a and orthogonality with respect to the a valued inner product coincide if and only if a is isomorphic to c in addition some new results concerning the case of equality in the triangle inequality for elements of a hilbert c module are obtained

properties of second order ordinary differential equations invariant under time translation and self similar transformation the general form of a second order ordinary differential equation invariant under time translation and self similarity is obtained together with its solution in parametric form next two representatives of two different families of such equations are studied in terms of rescaled phase space variables the first example has a simple physical interpretation and its limit cycle properties and period are easily derived similar results are found for the second example but the physical interpretation is less obvious the invariant equations are used as a basis to study the asymptotic behavior of related noninvariant equations thereby underlining the critical value nature of the parameters for which a self similar solution sss exists

approximate symmetry of birkhoff orthogonality abstract in a real normed space x we consider an approximate symmetry of the birkhoff orthogonality b and establish its connections with some properties of the space x moreover we introduce and study a new geometric constant for x connected with the considered property

a simple and unified approach to identify integrable nonlinear oscillators and systems in this paper we consider a generalized second order nonlinear ordinary differential equation ode of the form x k1xq k2 x k3x2q 1 k4xq 1 1x 0 where ki s i 1 2 3 4 1 and q are arbitrary parameters which includes several physically important nonlinear oscillators such as the simple harmonic oscillator anharmonic oscillator force free helmholtz oscillator force free duffing and duffing van der pol oscillators modified emden type equation and its hierarchy generalized duffing van der pol oscillator equation hierarchy and so on and investigate the integrability properties of this rather general equation we identify several new integrable cases for arbitrary value of the exponent q q r the q 1 and q 2 cases are analyzed in detail and the results are generalized to arbitrary q our results show that many classical integrable nonlinear oscillators can be derived as subcases of our results and significantly enlarge the list of integrable equations that exists in the contemporary literature t

theoretical foundations of apparent damping phenomena and nearly irreversible energy exchange in linear conservative systems this paper discusses a class of unexpected irreversible phenomena that can develop in linear conservative systems and provides a theoretical foundation that explains the underlying principles recent studies have shown that energy can be introduced to a linear system with near irreversibility or energy within a system can migrate to a subsystem nearly irreversibly even in the absence of dissipation provided that the system has a particular natural frequency distribution the present work introduces a general theory that provides a mathematical foundation and a physical explanation for the near irreversibility phenomena observed and reported in previous publications inspired by the properties of probability distribution functions the general formulation developed here is based on particular properties of harmonic series which form the common basis of linear dynamic system models the results demonstrate the existence of a special class of linear nondissipative dynamic systems that exhibit nearly irreversible energy exchange and possess a decaying impulse response in addition to uncovering a new class of dynamic system properties the results have far reaching implications in engineering applications where classical vibration damping or absorption techniques may not be effective furthermore the results also support the notion of nearly irreversible energy transfer in conservative linear systems which until now has been a concept associated exclusively with nonlinear systems

first integrals and reduction of a class of nonlinear higher order ordinary differential equations abstract we propose a method for constructing first integrals of higher order ordinary differential equations in particular third fourth and fifth order equations of the form x n h x x n 1 x are considered the relation of the proposed method to local and nonlocal symmetries are discussed

on the singularity analysis of ordinary differential equations invariant under time translation and rescaling the pt is applied to the general second and third order ordinary differential equations invariant under the two symmetries associated with time translation and rescaling in order to investigate their solvability and global integrability the effect of the two symmetries on the compatibility conditions is determined and we show that generally these conditions are automatically a consequence of the resonance condition use is made of truncated laurent series both in ascending and descending powers as an example the case of the generalized chazy equation is presented

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