a note on the invariants for the time dependent oscillator the author gives a procedure for the construction of equivalent families of invariants for ermakov systems starting directly from the equations of motion

order equivalent triangular norms in this paper an equivalence relation on the class of t norms induced by a t partial order is provided and discussed the equivalence classes linked to some special t norms are characterized as well as some properties preserved by the introduced equivalence defining the set of incomparable elements with respect to the t partial order this set is deeply investigated finally we give an answer to a recently posed open problem

wigner functions for time dependent coupled linear oscillators via linear and quadratic invariant processes at exact resonance we derive the linear and quadratic invariants for time dependent coupled linear oscillators in the presence of second harmonic generation employing these invariants we introduce an accurate definition of the dirac operators from which the wavefunctions in both coherent and fock number states representations are calculated we also derive the w wigner function of an arbitrary state at time t evolving in the system under consideration moreover we calculate the correlation coefficient between position and momentum and we find that it is identical in both fields whatever the state used

a theory of exact and approximate configurational invariants abstract a theory of integrable hamiltonian systems in two dimensions is formulated and applied the four dimensional phase space problem that is the vanishing of the poisson bracket between another invariant and the hamiltonian is here reduced to the solution of a series of two dimensional configuration space equations the theory is also applicable to more than two dimensions the constraints are found which admit to integrability of the orbits for magnetic or coriolis forces as well as for forces derivable from a potential when a system admits a given invariant the invariant is found for example by quadrature a number of examples including known and apparently previously unknown invariants are given the theory of exact integrals of the motion also can be extended to the derivation of approximate invariants the orbital structure of integrable or approximately integrable systems correlates with the degree maximum power of the velocity of a standardized invariant the theory admits a variational principle among other approximation techniques for the computation of a best approximate invariant the problem of the general cubic potential with one symmetric coordinate v 1 2 ax 2 1 2 by 2 cx 2 y 1 3 dy 3 of which the well studied henon heiles potential is the special case for a b and c d is examined in detail

a direct approach to finding exact invariants for one dimensional time dependent classical hamiltonians for a classical hamiltonian h 1 2 p2 v q t with an arbitrary time dependent potential v q t exact invariants that can be expressed as series in positive powers of p i q p t n 0pnfn q t are examined the method is based on direct use of the equation di dt i t i h 0 a recursion relation for the coefficients fn q t is obtained all potentials that admit an invariant quadratic in p are found and for those potentials all invariants quadratic in p are determined the feasibility of extending the analysis to find invariants that are polynomials in p of higher degree than quadratic is discussed the systems for which invariants quadratic in p have been found are transformed to autonomous systems by a canonical transformation

linear and quadratic invariants for the transformed tavis cummings model in the present communication we introduce the transformed tavis cummings problem as a physical model to discuss constants of the motion invariants for such a system the hamiltonian we have used can be regarded as a most general time dependent frequency converter model the advantage of the present work is to handle a real physical problem which represents the interaction between two coupled oscillators in this context we obtained real and complex classes of linear and quadratic invariants we have employed the real quadratic invariants to define a new dirac operator from which the wavefunction in the coherent states is obtained

classification scheme for two dimensional ermakov type systems and generalizations we describe a classification scheme for certain pairs of coupled nonlinear second order ordinary differential equations having a first integral which can be obtained by an ermakov type elimination procedure the classification produces a hierarchy of systems having different levels of generality different cases within the same level are distinguished by the integrability of a certain one form which is related to a nonlinear superposition law between the two equations the study of these one forms and the classification are made systematic by using the general reduction theorem of pfaffian forms the paper contains illustrations of the way previous examples fit into the classification scheme and a discussion of its generalization to higher dimensions

homogeneous quadratic invariants for one dimensional classical homogeneous quadratic hamiltonians some properties of the auxiliary equation using a previously derived characterization of the homogeneous quadratic invariants for an arbitrary one dimensional classical homogeneous quadratic hamiltonian we obtain various nonlinear superposition laws we establish a close connection between one of these the quadratic superposition law and the original characterization

the evolution operator technique in solving the schrodinger equation and its application to disentangling exponential operators and solving the problem of a mass varying harmonic oscillator the authors present a method for finding the evolution operator for the schrodinger equation for the hamiltonian expressible as h t a1 t j4 a2 t j0 a3 t j where j j0 and j are the su 2 group generators such a method is applied to the disentangling technique for exponential operators which are not necessarily unitary as a demonstration of our general approach they solved the problem of a harmonic oscillator with a varying mass

hamiltonian structure and integrability of the stationary kuramoto sivashinsky equation in this article the stationary kuramoto sivashinsky ks equation is studied it is shown first that this equation can be transformed into a one dimensional hamiltonian system describing the motion of a particle in a time dependent potential second working with this formalism and coming back then to the initial ks equation a subset of initial conditions is derived for which the corresponding solutions u x will behave like the dominant pole 120 x x0 3 deduced from the painleve analysis and will explode in a finite time finally special attention is given to the well known kuramoto tsuzuki solution and it is shown that provided the independent variable x is initialized in a specified domain this solution explodes also like 120 x x0 3

representations of one dimensional hamiltonians in terms of their invariants a general formalism for representing the hamiltonian of a system with one degree of freedom in terms of its invariants is developed those hamiltonians h q p t are derived for which any particular function i q p t is an invariant for each of those hamiltonians a function canonically conjugate to i q p t is derived which is also an invariant of h q p t the formalism is also presented for the case in which i q p t is expressed as a function of two canonically conjugate functions the formalism is illustrated by applying it to the case of a particle moving in a time dependent potential some earlier results are recovered and an invariant is found for a new potential lines for further study are outlined that may be fruitful for finding more examples of integrable systems

uninorms on bounded lattices we study uninorms on bounded lattices we show the existence of uninorms with the neutral element which is different from the top element and the bottom element using the fact that the t norms and t conorms on arbitrary lattice l always exist as a by product we give a method of constructing uninorms using this method we obtain also the smallest uninorm and the greatest uninorm with a given neutral element e

one dimensional nonautonomous dynamical systems with exact transcendental invariants new classes of dynamical systems for nonautonomous particle motion in one dimension are exactly solved in terms of arbitrary functions of both space and time besides extending considerably the class of mechanical systems with known exact solutions the explicit dependence on space represents an improvement on previous results such as those obtained by lewis and leach this might be helpful for extending the lewis and symon technique of constructing the bgky type of solutions for the vlasov poisson equations

propagation of singularities and maximal functions in the plane in this work we mainly generalize bourgain s circular maximal function to include variable coefficient averages our techniques involve a combination of bourgain s basic ideas plus microlocal analysis in particular to see the role of curvature we rely heavily on methods used in studying propogation of singularities for hyperbolic differential equations we also show that forp 2 there is local smoothing inl p for solutions to the wave equation

a first integral for a class of time dependent anharmonic oscillators with multiple anharmonicities the integrability of the equation q f t q2 g t q3 h t q4 j t q5 is considered particular cases of this equation arise in the study of charged plasma in an axially symmetric magnetic field and in shear free spherically symmetric gravitational fields in general relativity the above equation with only a quadratic term arises in the study of shear free fluids in general a krasinski j math phys 30 433 1989 the equation with a cubic term is applicable when there is an electromagnetic field in special cases we reduce the solution to a quadrature that has solutions in terms of elliptic integrals a lie point symmetry analysis is performed and the different cases that arise are considered for the existence of a symmetry

unitary relations in time dependent harmonic oscillators for a harmonic oscillator with time dependent positive mass and frequency a unitary operator is shown to transform the quantum states of the system to those of a harmonic oscillator system of unit mass and time dependent frequency as well as operators for a driven harmonic oscillator a unitary transformation which relates the driven system and a system of the same mass and frequency without a driving force is given as a generalization of previous results in terms of the solution of classical equation of motion of the driven system thus transformations give a simple way of finding exact wavefunctions of a driven harmonic oscillator system provided the quantum states of the corresponding system of unit mass are given

invariants for the time dependent harmonic oscillator ii cubic and quartic invariants for pt i see ibid vol 16 no 16 p 3805 1983 the existence is established of homogeneous time dependent invariants of arbitrary degree in the coordinate and momentum for the harmonic oscillator with variable mass and frequency in the constant mass constant frequency case the cubic and quartic invariants are calculated and some comments are made on the connection with the linear and quadratic invariants

quantum mechanics as a multidimensional ermakov theory i time independent systems presents a multidimensional ermakov theory applicable to both separable and nonseparable time independent quantum mechanical systems the separability of the system being determined by the separability of the wavefunction in configuration space a consequence of the theory is the existence of a new exact invariant for multidimensional quantum systems in one dimension the author shows that this new invariant reduces to the well known ermakov lewis invariant two applications of the theory are given in the first case a new exact invariant is obtained for planar optical and particle channelling systems in the second case he obtains the ermakov invariants and electron ray path equations for the hydrogen atom

feynman propagator for time dependent lagrangians possessing an invariant quadratic in momentum a class of time dependent classical lagrangians possessing an invariant quadratic in momentum is considered from a quantal point of view quantum mechanics is introduced through the feynman propagator defined as a path integral involving the classical action it is shown without carrying out an explicit path integration that the propagator for such a time dependent system is related to the propagator of an associated time independent problem the expansion of the propagator in terms of the eigenfunctions of the invariant operator is derived and the equivalence of the present theory to that of lewis and reisenfeld 1969 is discussed explicit analytic forms of propagators are obtained for some cases to illustrate the application of the present approach

a further note on the henon heiles problem the method of the lie theory of extended groups is applied to the henon heiles problem only one generator for a one parameter group is found the corresponding first integral is the energy it is inferred that no other exact integral exists

local smoothing for operators failing the cinematic curvature condition in this paper we examine a class of averaging operators which exhibit local smoothing that is viewed as a function of space and time variables the operators yield more smoothing than the fixed time estimates sogge showed in a more general setting that if these operators satisfy a cinematic curvature condition they will exhibit some local smoothing c d sogge propagation of singularities and maximal functions in the plane invent math 104 1991 231 251 here we translate this condition into the setting of averaging operators in the plane we prove that cinematic curvature is not necessary for local smoothing to occur exhibiting a class of operators which fail the cinematic curvature condition but still satisfy a local smoothing estimate furthermore the amount of local smoothing exhibited by these operators is strictly less than that conjectured for operators satisfying the cinematic curvature condition

berry s phase as the asymptotic limit of an exact evolution an example the authors exhibit for arbitrary time variations of the parameters of the generalised harmonic oscillator hamiltonian a canonical transformation which facilitates an exact analysis of quantal phase and classical angle formulae for berry s phase and hannay s angle are readily obtained by taking the adiabatic limit of the exact theory

berry s phase and wavefunctions for time dependent hamilton systems an alternative derivation of berry s phase 1985 to that given by morales 1988 is given for non autonomous hamiltonian systems which admit an energy like first integral the generality of the problems treated is greater

invariants of particle motion in one dimensional time dependent potentials the author considers one dimensional time dependent hamiltonians of the form h p2 2 v q t it is shown that the potentials determined in an implicit way by the relation phi v a1v2 a2v a3 t q where phi is an arbitrary function and a1 a2 and a3 are arbitrary parameters admit a single valued constant of motion this constant of motion is a higher transcendental function in the momentum

on the relationships between the orders induced by uninorms and nullnorms abstract in this paper some relationships between the orders induced by uninorms and nullnorms are discussed some relations between the sets admitting some incomparability w r t those orders are investigated also some relationships between the algebraic structures obtained from those orders are determined

lie symmetries nonlinear equations of motion and new ermakov systems it is shown that a lagrangian of the form l 1 1 rho 2 omega 2 t rho 2 g t f k t rho said to be in factored form yields an equation of motion that is equivalent to the most general equation derivable via noether s theorem from the unfactored lagrangian l 1 2 rho 2 p rho t in view of this equivalence the theory of extended lie groups is applied to the factored nonlinear equation of motion p omega 2 t rho g t f k t rho to obtain its lie symmetries the latter are obtained when g t and k t initially arbitrary are determined in terms of a function x t which satisfies the auxiliary equation x omega 2 t x k x3 it is then possible with the auxiliary equation and the equation of motion to form a coupled pair of nonlinear equations an ermakov system whose first integral is not invariant under the action of the symmetry group in contrast to previous ermakov systems

lie and noether symmetry groups of nonlinear equations it is found that the nonlinear ordinary differential equation rho omega 2 t rho 1 rho 3 has a three parameter lie group of symmetries which are also noether symmetries the invariants associated with the group are calculated the authors discuss a new way of generating n parameter lie symmetry groups which are associated with n parameter nonlinear differential equations

solutions in closed form and as power series to the real lorenz equations using the method of the lie theory of extended groups and for the parameter values b 1 and r 0 we construct explicitly the general exact solution to the real lorenz equations in terms of jacobian elliptic functions in the context of our approach further possible completely integrable cases of the lorenz system are discussed by considering the result of the painleve analysis for 1 b 2 and r 1 9 and negative values of r the latter case r 0 and 0 not following from the painleve test for other positive parameter values and in the form of appropriate power series we find some particular exact solutions which do not possess the painleve property

on exact solutions for damped anharmonic oscillators for arbitrary functions f1 f2 and f3 the anharmonic oscillator x f1 t x f2 t x f3 t x3 0 cannot be solved in closed form i e the general solution cannot be expressed as elliptic functions the authors apply the painleve test to obtain the constraint on the functions f1 f2 and f3 for which the equation passes the test the constraint on f1 f2 and f3 i e the differential equation which f1 f2 and f3 obey is discussed and solutions are given

invariants for the time dependent harmonic oscillator i two equivalent families of linear invariants and two equivalent families of quadratic invariants are obtained for a harmonic oscillator with variable mass or with variable frequency explicit formulae are obtained for the linear and quadratic invariants in the case of a damped oscillator and the hamiltonian is seen as a special quadratic invariant

the modularity condition for uninorms and t operators in this work we study the functional equation given by the modularity condition f x g y z g f x y z for all z x where the unknown functions f g are uninorms and or t operators we characterize all the solutions in the four possible cases i when f g are t operators ii when f is a t operator and g a uninorm iii when f is a uninorm and g a t operator and finally iv when f g are uninorms

rational functions of momentum as invariants for one dimensional time dependent potentials basic theory a framework for the momentum resonance formulation of lewis and leach ann phys ny 164 47 1985 that casts new light into the nature of exact explicitly time dependent invariants for one dimensional time dependent potentials and produces additional examples of such invariants is presented the momentum resonance formulation postulates that the invariant be a rational function of momentum with simple poles which are called momentum resonances it is shown that an invariant of resonance type can be written as a functional of the potential in terms of the solution of a system of linear algebraic equations and a single necessary and sufficient condition for a potential to admit an invariant of resonance type is obtained these results are obtained by reformulating the problem in terms of a set of discrete moments that satisfy two separate recursion formulas invariants for new time dependent potentials can be obtained and previously known invariants are recovered

first integrals and symmetries of time dependent hamiltonian systems in this paper first integrals of one dimensional time dependent hamiltonians h q p t are studied first of all it is shown that the second first integral j q h t t dq hp valid for autonomous hamiltonians h q p can be extended to the case of nonautonomous hamiltonians h q p t provided hp resp t is replaced with ip where i q p t is a first integral of h q p t resp a new time t j q i t t dq ip in a second part we derive a technique leading to a first integral i q p t from a lie symmetry as an example this technique together with the expression of the second first integral j q i t is used to completely integrate a special case of the linear harmonic oscillator with a time dependent frequency as a result lewis first integral is recovered from a lie symmetry in a way that generalizes the recent work of shivamoggi and muilenberg phys lett a 154 1 1991

equivalence relations induced by the u partial order abstract in this paper an equivalence relation on the class of uninorms with the same neutral element induced by a u partial order is provided and discussed the equivalence classes linked to some special uninorms are characterized furthermore the set of incomparable elements with respect to the u partial order is deeply investigated finally another equivalence relation on the class of uninorms with the same neutral element is introduced and studied

a note on generalization of the lewis invariant and the ermakov systems in this paper we examine several possible generalizations of the well known lewis invariant and the ermakov systems the analysis relies on the application of the theorem of emmy noether the problem of constructing lewis type invariants is reduced to a problem of solving a single partial differential equation the so called potential equation within this approach the role of the auxiliary equation is clarified by an analysis of linear systems generalization of the ermakov type procedure for derivation of the lewis invariant is also discussed moreover different forms of general solution of the problem will be provided which extend the range of possible applications of the analysis

generalised symmetries of some nonlinear finite dimensional systems an approach based on the theorem that every variational symmetry of a variational problem is also a symmetry of the euler lagrange equation has been applied to find the generalised symmetries for the generalised henon heiles system a system possessing non polynomial type potential and a super integrable system the constants of motion of these systems have also been found

the determination of nonlocal symmetries by the technique of reduction of order autonomous systems of ordinary differential equations can be rewritten as systems of first order ordinary differential equations and one of the dependent variables chosen as a new independent variable some of the variables are eliminated to give a mixed system of first and second order equations for which the determination of point symmetries can be automated without having to make an ansatz on the detailed structure of the symmetry because the coefficient function for the original independent variable appears only as its derivative in the reduced system symmetries which are nonlocal in this variable become local symmetries of the reduced system and can be computed algorithmically

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

first integrals for some nonlinear time dependent hamiltonian systems an observation of a simple property of canonical transformations leads to a procedure for determining first integrals for classes of hamiltonians in illustrative examples the most general result presently known from other methods is recovered a new result presented and a generalization to more than one degree of freedom discussed

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

nonlinear canonical transformations i the linear canonical transformation q aq bp p cq dp ad bc 1 q p constant is extended to include terms of the second third and fourth degree products of second degree transformations are used to construct third and fourth degree transformations the question of factorisability of fourth degree transformations is discussed

ermakov and non ermakov systems in quantum dissipative models via the hydrodynamical formulation of quantum mechanics a unified protocol to treat the quantum time dependent harmonic oscillator with friction is presented described by two different models an explicitly time dependent linear schrodinger equation caldirola kanai model and a logarithmic nonlinear schrodinger equation kostin model for the former model an ermakov system that makes it possible to obtain an invariant of ermakov lewis type is derived for the latter model a non ermakov system is derived instead and it is shown that neither an exact nor an approximate invariant of ermakov lewis type exists

darboux s problem of quadratic integrals the problem of determining all standard classical hamiltonians in two dimensions with euclidean metric which admit constants of motion quadratic in the momenta is resolved several general results are given which make it obvious that the systems found do possess such integrals

lie transformation group solutions of the nonlinear one dimensional vlasov equation the solution of the exact nonlinear one dimensional vlasov equation with a space and time dependent electric field is reduced to the solution of a nonlinear first order ordinary differential equation with two subsidiary equations the reduction holds for any electric field nonlinear in the spatial coordinate or for a subclass of electric fields linear in the spatial coordinate and is equivalent to the solution of a generalized bernstein greene kruskal bgk vlasov equation with a velocity dependent lorentz force the lie method for the solution of differential equations invariant under a transformation group has been used to calculate the group generator the canonical variables and the generalized bgk vlasov equation analytical forms for the functional dependence of the vlasov equation one particle distribution function are given

lie backlund groups and the linearisation of differential equations it is shown that groups of lie backlund lb transformations which depend on non local variables are related by a change of variables to the lb tangent transformations of ibragimov and anderson 1979 involving no more than arbitrary order derivatives the transformation of any lb symmetry operator by an invertible change of variables is discussed it is pointed out that once a differential equation admits an lb operator then a large number of secondary equations will admit the same operator the lb theory involving non local variables and the notion of secondary equations are used to characterise group theoretically the linearisation of the burgers equation ut uux uxx 0 and of the ode uxx omega 2 x u ku 3 0

an exact invariant for the time dependent double well anharmonic oscillators lie theory and quasi invariance groups the author derives an exact energy type invariant for the double well anharmonic oscillators this invariant holds for certain time dependent potential parameters a nonlinear superposition principle is used to solve the nonlinear constraint equations which appear in the resolution

time dependent schrodinger equations having isomorphic symmetry algebras i classes of interrelated equations in this paper we focus on a general class of schrodinger equations that are time dependent and quadratic in x and p we transform schrodinger equations in this class via a class of time dependent mass equations to a class of solvable time dependent oscillator equations this transformation consists of a unitary transformation and a change in the time variable we derive mathematical constraints for the transformation and introduce two examples

adiabatic invariants of some time dependent oscillators two types of time dependent 1d oscillators are considered parametrically excited described by the mathieu equation and parametrically pumped with a periodic driving force the adopted field method approach is applied and the complete sets of their linear invariants as well as the corresponding quadratic invariants are derived

on the t partial order and properties for a t norm t on 0 1 a partial order t was recently defined and studied in this study for any fixed email protected 0 1 we define the set of incomparable elements according to t and this set is deeply investigated by means of the equivalence relation defined in 14 it is shown the set 0 1 denoting the set of all equivalence classes on t norms on 0 1 is uncountably infinite finally with the help of any t norm t on 0 1 it is obtained that the family t l l 0 1 of t norms on 0 1 if t is a divisible t norm then it is obtained that 0 1 t l is a lattice thus we give an answer to an open problem in 10

on the distributivity property for uninorms abstract distributivity between two operations is a property that was already posed many years ago and that is especially interesting in the framework of logical connectives for this reason the distributivity property has been extensively studied for several families of operations like triangular norms and conorms some kinds of uninorms and nullnorms also called t operators and even for some generalizations of them in this paper we investigate the distributivity equation involving two uninorms lying in any one of the most studied classes of uninorms leading to many new solutions

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

on time dependent quadratic quantum hamiltonians we apply the techniques of canonical transforms to equations of the type left a t mathbb p 2 b t mathbb p mathbb q mathbb q mathbb p c t mathbb q 2 d t mathbb q e t mathbb p f t mathbb 1 right psi q t i partial t psi q t where mathbb q and mathbb p are the quantum position and momentum operators the time dependent parameters of the text w wedge text sl text 2 r evolution operator are found through linear differential equations in terms of these we give explicitly the green s function all separating coordinates and similarity solutions of the equation we analyze the behavior of gaussian and coherent state initial conditions in closed form and present a new interpretation of all the lewis riesenfeld constants of motion

constants of the motion and diagonalization of the hamiltonian for some scattering processes complex invariants for sho modes or pairs of real invariants for rho modes are obtained for the frequency converter the parametric amplifier and for a three mode system that describes raman or brillouin scattering all possible constants of the motion and the diagonalization of the hamiltonian follow as a consequence under the assumption made in this paper the hamiltonian has also been considered by moshinsky and winternitz who have established the existence of its reduction to diagonal form however it is not easy to select the correct transformation from the totality comprising the symplectic group to reduce a given h this has been done explicitly showing among other results that the constants of the motion are similar often identical in the stable frequency converter domain and in the unstable amplifier domain

on the symmetries and invariants of the harmonic oscillator a general formulation of noether s theorem is applied to the equation of a harmonic oscillator the definition of symmetry includes the usual lie invariance as a special case and unlike standard formulations generates the full set of invariants i e gives closure under functional composition the analysis for a time dependent oscillator casts doubt on the importance of a known class of invariants the existence of a lagrangian function is shown to be inessential to the analysis

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

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

construction of new integrable hamiltonians in two degrees of freedom a new procedure for deriving integrable hamiltonians and their constants of the motion is introduced we term this procedure the truncation program integrable hamiltonians occurring in the truncation program possess constants of the motion which are polynomials in a perturbation parameter e the relationship between this program and the whittaker program in two degrees of freedom is discussed integrable hamiltonians occurring in the whittaker program a generalization of whittaker s work possess constants of the motion which are polynomials in the momentum coordinates many previously known integrable hamiltonians are derived a new family of integrable double resonance hamiltonians and a new family of integrable hamiltonians of the form p21 p22 2 v q1 q2 are derived

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

canonical transformations and exact invariants for dissipative systems a simple treatment to the problem of finding exact invariants and related auxiliary equations for time dependent oscillators with friction is presented the treatment is based on the use of a time dependent canonical transformation and an auxiliary transformation

invariants for time dependent potentials use of self similar techniques abstract we seek a generalization of the derivation by lewis and leach for invariants quadratic in p since the general equations for invariants cubic and quartic in p cannot be solved we specialize to self similar solutions of the equations consequently we can recover only a part of their solutions for quadratic invariants but new pairs of potentials and invariants are obtained where the leading term in the invariants is p3 or p4 more precisely all the potentials are of the form v q t at2 2 q t b q2 t2 where is prescribed and a is arbitrary different functional forms for b lead to different families of solutions where for each family the parameter is allowed to go from to finally a generalization to potentials with three terms is given

ordering based on uninorms in this paper an order induced by uninorms on bounded lattices is given and discussed by defining such an order the t partial order is extended to a more general form the relationship between the order induced by the underlying t norm and t conorm and the order induced by a uninorm is studied finally the order is extended to the order induced by n uninorms on chains

a single point characterization of representable uninorms we study uninorms of which both the underlying t norm and underlying t conorm are strict such uninorms are the only candidates for being representable by an additive generator we prove that the representability of such a uninorm depends solely on its value at a single arbitrary point in the remaining open part of the unit square more explicitly such a uninorm turns out to be representable if and only if this single value is located strictly between the minimum and the maximum of the corresponding arguments if this single value coincides with one of these bounds then the value of the uninorm at any point in the remaining open part is determined by the same bound

lie algebraic approach and quantum treatment of an anisotropic charged particle via the quadratic invariant we consider the problem of a charged harmonic oscillator under the influence of a constant magnetic field the system is assumed to be anisotropic and the magnetic field is applied along z axis a canonical transformation is invoked to remove the interaction term and the system is reduced to a model contains two uncoupled harmonic oscillators two classes of real and complex quadratic invariants constants of motion are obtained we employ the lie algebraic technique to find the most general solution for the wave function for both real and complex invariants the quadratic invariant is also used to derive two classes of creation and annihilation operators from which the wave functions in the coherent states and number states are obtained some discussion related to the advantage of using the quadratic invariants to solve the cauchy problem instead of the direct use of the hamiltonian itself is also given

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

nonlinear time dependent anharmonic oscillator asymptotic behavior and connected invariants the motion of a particle in a potential decreasing with time as x n is considered different time and space rescaling are considered in order to obtain the asymptotic solutions the validity of adiabatic invariants is discussed the classical critical case corresponds to the obtainment of self similar solutions for the quantum problem

maximal chains on the interval 0 1 with respect to t norm partial orders and uninorm partial orders abstract in this paper characterizations of maximal chains with respect to a t norm partial order a uninorm partial order or a partial order on the interval 0 1 are studied based on these properties some special t norms and uninorms can be constructed which have some given maximal chains moreover some partial orders on the interval 0 1 which can not be a t norm partial order or a uninorm partial order are investigated and characterized

uninorms in fuzzy systems modeling abstract fuzzy systems modeling provides a framework in which input output relationships can be modeled by partitioning the input into fuzzy regions for which we can describe the output the fuzzy modeling inference process the procedure used to obtain the output of a fuzzy systems model for a given input is shown to involve an aggregation step in which the contributions of the different components of the fuzzy systems model are combined it is shown that the uninorm operator provides a general class of operators to implement this step the uninorm it is recalled is an operator that provides a generalization of the t norm and t conorm by allowing the identity element to be any value in the unit interval another step in the inference process is the determination of the contribution of each component of the model based on its relevancy to the current input the performance of this step requires the use of an operator related to the implication operator which we call a relevancy transformation ret operator it is shown that the form of this ret operator is dependent upon the identity value of the uninorm used in the aggregation step we provide some general formulations for this class of operators we show how the well known forms of fuzzy inference are special cases of our uninorm based approach