a new phenomenon in the critical exponent for structurally damped semi linear evolution equations abstract in this paper we find the critical exponent for global small data solutions to the cauchy problem in r n for dissipative evolution equations with power nonlinearities u p or u t p u t t u t u u p u t p here n 0 with 2 we show that the critical exponent for each of the two nonlinearities is related to each of the two possible asymptotic profiles of the linear part of the equation which are described by the diffusion equations v t v 0 w t w 0 the nonexistence of global solutions in the critical and subcritical cases is proved by using the test function method under suitable sign assumptions on the initial data and lifespan estimates are obtained by assuming small initial data in sobolev spaces we prove the existence of global solutions in the supercritical case up to some maximum space dimension n and we derive l q estimates for the solution for q 1 for 2 the result holds in any space dimension n 1 the existence result also remains valid if and or are fractional

the critical exponent for the dissipative plate equation with power nonlinearity abstract in this paper we find the critical exponent of global small data solutions for a damped plate equation with power nonlinearity u t t u t t 2 u u t u p t 0 x r 2 and for a system of two weakly coupled damped plate equations we show how assuming small data in the energy space h 2 h 1 and in l 1 is sufficient to compensate the regularity loss type decay effect created by the rotational inertia term u t t

time independent invariants of motion for the quadratic system a hamiltonian method is developed to obtain first integrals of the form p qmu and p qmu rnu for a general system of two dimensional autonomous ordinary differential equations with quadratic terms where p q and r are linear or quadratic polynomials and mu nu are two real numbers it is found that there is an intimate relationship between the polynomials and the equilibrium points of the system which is useful for determining the existence of periodic orbits and asymptotic behaviour

nonlinear scattering theory at low energy abstract we study the scattering theory of a conservative nonlinear one parameter group of operators on a hilbert space x relative to a group of linear unitary operators under certain hypotheses the scattering operator carries a neighborhood of 0 in x into x the theory is designed to apply to the semilinear schrodinger and klein gordon equations

quasipolynomial generalization of lotka volterra mappings in recent years it has been shown that lotka volterra mappings constitute a valuable tool from both the theoretical and the applied points of view with developments in very diverse fields such as physics population dynamics chemistry and economy the purpose of this work is to demonstrate that many of the most important ideas and algebraic methods that constitute the basis of the quasipolynomial formalism originally conceived for the analysis of ordinary differential equations can be extended into the mapping domain the extension of the formalism into the discrete time context is remarkable as far as the quasipolynomial methodology had never been shown to be applicable beyond the differential case it will be demonstrated that lotka volterra mappings play a central role in the quasipolynomial formalism for the discrete time case moreover the extension of the formalism into the discrete time domain allows a significant generalization of lotka volterra mappings as well as a whole transfer of algebraic methods into the discrete time context the result is a novel and more general conceptual framework for the understanding of lotka volterra mappings as well as a new range of possibilities that become open not only for the theoretical analysis of lotka volterra mappings and their generalizations but also for the development of new applications

convergence to nonlinear diffusion waves for solutions of a system of hyperbolic conservation laws with damping we consider a model of hyperbolic conservation laws with damping and show that the solutions tend to those of a nonlinear parabolic equation time asymptotically the hyperbolic model may be viewed as isentropic euler equations with friction term added to the momentum equation to model gas flow through a porous media in this case our result justifies darcy s law time asymptotically our model may also be viewed as an elastic model with damping

global existence of small data solutions for wave models with super exponential propagation speed abstract we consider semi linear classical damped wave models with time dependent speed of propagation and time dependent dissipation the right hand side is a power nonlinearity which can be interpreted only as a source term we are interested in the interplay of time dependent coefficients on the global existence of small data solutions the considerations are divided into two cases depending on the behavior of the propagation speed the super exponential and the sub exponential case in this paper we study the super exponential case one main tool of our approach is the use of matsumura type estimates for a family of cauchy problems depending on a parameter

weighted strichartz estimates and global existence for semilinear wave equations in this paper we prove a sharp global existence theorem in all dimensions for nonlinear wave equations with power type nonlinearities the proof is based on a weighted strichartz estimate involving powers of the lorentz distance

long time asymptotic expansion for the damped semilinear wave equation the first initial boundary value problem is considered for the damped semilinear wave equation with the quadratic nonlinearity for small initial data and homogeneous boundary conditions its solution is constructed in the form of a series in the eigenfunctions of the laplace operator the long time asymptotic expansion is obtained which shows the nonlinear effects of amplitude and frequency multiplication the same results hold for the admissible initial data that are not small

a geometrical method towards first integrals for dynamical systems we develop a method based on darboux s and liouville s works to find first integrals and or invariant manifolds for a physically relevant class of dynamical systems without making any assumption on these elements forms we apply it to three dynamical systems lotka volterra lorenz and rikitake

wave equations with time dependent dissipation ii effective dissipation abstract this article is intended to present a construction of structural representations of solutions to the cauchy problem for wave equations with time dependent dissipation above scaling these representations are used to give estimates of the solution and its derivatives based on l q r n q 2 the article represents the second part within a series in jens wirth wave equations with time dependent dissipation i non effective dissipation j differential equations 222 2 2006 487 514 weak dissipations below scaling were discussed

on the existence and nonexistence of global solutions for certain semilinear exterior problems with nontrivial robin boundary conditions abstract we consider three types of semilinear equations elliptic parabolic and hyperbolic posed in the n dimensional exterior domain r n d where n 2 and d is the closed unit ball in r n a nontrivial robin boundary condition is imposed on the boundary of d using a test function approach with judicious choices of the test functions we show that the considered problems share a common critical behavior we discuss separately the cases n 2 and n 3 moreover in the case n 3 the dependence of the critical exponent on initial data is discussed to the best our knowledge the study of the critical behavior in an exterior domain with a nontrivial robin boundary condition has never been studied in the literature

families of invariants of the motion for the lotka volterra equations the linear polynomials family the modified carleman embedding method already introduced by the authors to find first integrals invariants of the motion of polynomial form to the lotka volterra system is described in detail and its efficiency to treat the n dimensional system proved using this method an extensive investigation is performed for polynomials of the first degree which allow a classification of the integrals in three families for some systems possessing one invariant it is possible to find a second invariant using rescaling methods they represent very restrictive solutions implying that there exists a great number of conditions among the equation s coefficients to satisfy a proof is given that the volterra invariants can be deduced as a limit finally the interesting properties of the solutions of these systems are studied in detail

critical exponent for the cauchy problem to the weakly coupled damped wave system abstract in this paper we consider a system of weakly coupled semilinear damped wave equations we determine the critical exponent for any space dimensions our proof of the global existence of solutions for supercritical nonlinearities is based on a weighted energy method whose multiplier is appropriately modified in the case where one of the exponent of the nonlinear term is less than the so called fujita s critical exponent we also give estimates of the lifespan of solutions from above for subcritical nonlinearities

compact differences of composition operators abstract a characterization of compact difference is given for composition operators acting on the standard weighted bergman spaces and necessary conditions are given on a larger scale of weighted dirichlet spaces conditions are given under which a composition operator can be written as a finite sum of composition operators modulo the compacts the additive structure of the space of composition operators modulo the compact operators is investigated further and a sufficient condition is given to insure that two composition operators lie in the same component

an application of l1 estimates for oscillating integrals to parabolic like semi linear structurally damped σ evolution models abstract we study the following cauchy problems for semi linear structurally damped evolution models u t t u u t f u u t u 0 x u 0 x u t 0 x u 1 x with 1 0 and 0 2 here the function f u u t stands for the power nonlinearities u p and u t p with a given number p 1 we are interested in investigating l 1 estimates for oscillating integrals in the presentation of the solutions to the corresponding linear models with vanishing right hand sides by applying the theory of modified bessel functions and faa di bruno s formula by assuming additional l m regularity on the initial data we use l m l q l q and l q l q estimates with q 1 and m 1 q to prove the global in time existence of small data sobolev solutions to the above semi linear models from suitable function spaces basing on l q spaces

decay of solutions for a semilinear system of elastic waves in an exterior domain with damping near infinity abstract we show that the solutions of a semilinear system of elastic waves in an exterior domain with a localized damping near infinity decay in an algebraic rate to zero we impose an additional condition on the lame coefficients it seems that this restriction cannot be overcome by using the two finite speed propagation of the elastic model since we do not assume compact support on the initial data and because the dissipation does not have compact support the decay rates obtained for the total energy of the linear problem and the l 2 norm of the solution improve previous results for the semilinear problem the decay rates in this paper seem to be the first contribution mainly in the context of initial data without compact support and localized dissipation

on compactness of the difference of composition operators abstract let and be analytic self maps of the unit disc and denote by c and c the induced composition operators the compactness and weak compactness of the difference t c c are studied on h p spaces of the unit disc and l p spaces of the unit circle it is shown that the compactness of t on h p is independent of p 1 the compactness of t on l 1 and m the space of complex measures is characterized and examples of and are constructed such that t is compact on h 1 but non compact on l 1 other given results deal with l weakly compact counterparts of the previous results and a conjecture of j e shapiro

an application of atomic decomposition in bergman spaces to the study of differences of composition operators abstract in this paper we characterize bounded differences of composition operators from standard weighted bergman spaces a p into lebesgue spaces l q when 1 and p q in the process we also find an interesting connection between the difference operator and certain weighted composition operators answering a question raised in saukko 2011 18 by combining the results of this paper with those of saukko 2011 18 we obtain a complete characterization of bounded and compact differences between standard weighted bergman spaces

nonexistence of global solutions to semilinear wave equations in high dimensions on considere le probleme de cauchy pour l equation d onde semi lineaire 2 t 2 u x t u x t p p 1 u x o f x u t x o g x avec f g c o r n

compact linear combination of composition operators on bergman spaces abstract motivated by the question of shapiro and sundberg raised in 1990 study on linear combinations of composition operators has been a topic of growing interest in this paper we completely characterize the compactness of any finite linear combination of composition operators with general symbols on the weighted bergman spaces in two classical terms one is a function theoretic characterization of julia caratheodory type and the other is a measure theoretic characterization of carleson type our approach is completely different from what has been known so far

weakly coupled systems of semilinear effectively damped waves with time dependent coefficient different power nonlinearities and different regularity of the data abstract we study the global existence of small data solutions to the cauchy problem for semilinear damped wave equations with an effective dissipation term where the data are supposed to belong to different classes of regularity we apply these results to the cauchy problem for weakly coupled systems of semilinear effectively damped waves with respect to the defined classes of regularity for different power nonlinearities

global existence for semi linear structurally damped σ evolution models abstract the main purpose of this paper is to study the global existence of small data solutions for semi linear structurally damped evolution models of the form u t t u u t f d a u u t u 0 x u 0 x u t 0 x u 1 x with 1 0 and 2 this is a family of structurally damped evolution models interpolating between models with exterior damping 0 and those with visco elastic type damping the function f d a u u t represents power non linearities d a u p for a 0 or u t p our goal is to propose a fujita type exponent diving the admissible range of powers p into those allowing global existence of small data solutions stability of zero solution and those producing a blow up behavior even for small data on the one hand we use new results from harmonic analysis for fractional gagliardo nirenberg inequality or for superposition operators see appendix a on the other hand our approach bases on l p l q estimates not necessarily on the conjugate line for solutions to the corresponding linear models assuming additional l 1 regularity for the data the linear models we have here in mind are v t t v v t 0 v 0 x v 0 x v t 0 x v 1 x with 1 0 and 0

compact linear combinations of composition operators induced by linear fractional maps it has been known that the difference of two composition operators induced by linear fractional self maps of a ball cannot be nontrivially compact on either the hardy space or any standard weighted bergman space in this paper we extend this result in two significant directions the difference is extended to general linear combinations and inducing maps are extended to linear fractional maps taking a ball into another possibly of different dimension

a maximum principle applied to quasi geostrophic equations we study the initial value problem for dissipative 2d quasi geostrophic equations proving local existence global results for small initial data in the super critical case decay of lp norms and asymptotic behavior of viscosity solution in the critical case our proofs are based on a maximum principle valid for more general flows

linear relations in the calkin algebra for composition operators we consider this and related questions when is a finite linear combination of composition operators acting on the hardy space or the standard weighted bergman spaces on the unit disk a compact operator

dispersion of small amplitude solutions of the generalized korteweg de vries equation abstract we study the long time behavior of small solutions of the initial value problem for the generalized korteweg de vries equation t u x 3 u x f u 0 gkdv u x 0 g x for the case where f w w s with s 1 4 23 57 3 8625 our results imply that if g l11 g l22 is sufficiently small then sup r 1 t 1 3 u t l in particular the solution tends to zero in the supremum norm the proofs make use of duhamel s formula and dispersion estimates for the linear propagator as well as chain and leibniz rules for fractional derivatives of compositions d f u lp and products d fg lp 0

necessary conditions for the existence of quasi polynomial invariants the quasi polynomial and lotka volterra systems abstract we show that any quasi polynomial invariant of a quasi polynomial dynamical system can be transformed into a quasi polynomial invariant of a homogeneous quadratic lotka volterra dynamical system we show how this quasi polynomial invariant can be decomposed in a simple manner this decomposition permits to conclude that the existence of polynomial semi invariants in lotka volterra systems is a necessary condition for the existence of quasi polynomial invariants we derive a method which allows to construct the necessary conditions for existence of semi invariants on lotka volterra dynamical systems applications are given

joint carleson measure and the difference of composition operators on aαp bn we introduce a concept of joint carleson measure and characterize when the difference of two composition operators on a p bn the weighted bergman space over the unit ball bn in cn is bounded or compact we apply this joint carleson measure characterization to composition operators with smooth symbols and construct an interesting example which shows that the boundedness or the compactness depends on p when n 2 this is in sharp contrast with the single composition operator case where the boundedness or the compactness is independent of p 0 moreover the compact difference on the weighted bergman spaces over the unit disc is known to be independent of p 0 and the compact difference on a p bn is known to be independent of p 0 if each composition operator is bounded on a p bn for some 1 2

symmetries of differential equations from sophus lie to computer algebra the topic of this article is the symmetry analysis of differential equations and the applications of computer algebra to the extensive analytical calculations which are usually involved in it the whole area naturally decomposes into two parts depending on whether ordinary or partial differential equations are considered we show how a symmetry may be applied to lower the order of an ordinary differential equation or to obtain similarity solutions of partial differential equations the computer algebra packages sods and spde respectively which have been developed to perform almost all algebraic manipulations necessary to determine the symmetry group of a given differential equation are presented furthermore it is argued that the application of computer algebra systems has qualitatively changed this area of applied mathematics

stability properties of a general class of nonlinear dynamical systems we establish sufficient conditions for the boundedness of the trajectories and the stability of the fixed points in a class of general nonlinear systems the so called quasi polynomial vector fields with the help of a natural embedding of such systems in a family of generalized lotka volterra lv equations a purely algebraic procedure is developed to determine such conditions we apply our method to obtain new results for lv systems by a reparametrization in time variable and to study general nonlinear vector fields originally far from the lv format

difference of composition operators between standard weighted bergman spaces we obtain estimates for the norm and essential norm of the difference of two composition operators between certain bergman spaces in particular a necessary and sufficient condition for boundedness and compactness of the operator is established finally we give a sufficient condition for boundedness and compactness of the difference operator between hardy spaces

new blow up results for nonlinear boundary value problems in exterior domains abstract we consider exterior boundary value problems with nontrivial boundary conditions including three types of semilinear equations heat equation wave equation and damped wave equation using a test function approach with a choice of a suitable family of test functions we obtain the critical exponents in the sense of fujita for the considered problems our obtained results close some open questions from the literature

strichartz estimates in the hyperbolic space and global existence for the semilinear wave equation the aim of this article is twofold first we consider the wave equation in the hyperbolic space hi and obtain the counterparts of the strichartz type estimates in this context next we examine the relationship between semilinear hyperbolic equations in the minkowski space and in the hyperbolic space this leads to a simple proof of the recent result of georgiev lindblad and sogge on global existence for solutions to semilinear hyperbolic problems with small data shifting the space time strichartz estimates from the hyperbolic space to the minkowski space yields weighted strichartz estimates in rn x 1r which extend the ones of georgiev lindblad and sogge

an application of lp lq decay estimates to the semi linear wave equation with parabolic like structural damping abstract in this paper we study the global existence of small data solutions to u t t u 2 a u t u p u 0 x u 0 x u t 0 x u 1 x where a 0 0 1 2 and p 1 assuming small data in some sobolev spaces we obtain the global existence for p 1 2 n 2 in space dimension n n where n n as 1 2 in particular our result holds in any space dimension n 2 if 1 2

l1 l1 estimates for the strongly damped plate equation abstract in this paper we derive l 1 l 1 long time estimates for the strongly damped plate equation u t t 2 u 2 u t 0 x r n t r u 0 x u 0 x u t 0 x u 1 x in particular we prove that u t l 1 c 1 t n 4 u 0 l 1 1 t 1 2 u 1 l 1 for any t 0 in space dimension n 5

diffusion phenomena for the wave equation with space dependent damping in an exterior domain abstract in this paper we consider the asymptotic behavior of solutions to the wave equation with space dependent damping in an exterior domain we prove that when the damping is effective the solution is approximated by that of the corresponding heat equation as time tends to infinity our proof is based on semigroup estimates for the corresponding heat equation and weighted energy estimates for the damped wave equation the optimality of the decay late for solutions is also established

integrals of quadratic ordinary differential equations in r3 the lotka volterra system abstract a method already introduced by the last two authors for finding the integrable cases of three dimensional autonomous ordinary differential equations based on the frobenius integrability theorem is described in detail using this method and computer algebra the so called three dimensional lotka volterra system is studied many cases of integrability are thus found the study of this system is completed by the application of painleve analysis and the jacobi last multiplier method the methods used are of general interest and can be applied to many other systems

on the connected component of compact composition operators on the hardy space we show that there exist non compact composition operators in the connected component of the compact ones on the classical hardy space h2 this answers a question posed by shapiro and sundberg in 1990 we also establish an improved version of a theorem of maccluer giving a lower bound for the essential norm of a difference of composition operators in terms of the angular derivatives of their symbols as a main tool we use aleksandrov clark measures

compact double differences of composition operators on the bergman spaces abstract as is well known on the weighted bergman spaces over the unit disk compactness of differences of two composition operators is characterized by certain cancellation property of the inducing maps at every bad boundary point which makes each composition operator in the difference fail to be compact recently the second and third authors obtained a result implying that double difference cancellation is not possible for linear combinations of three composition operators in this paper we obtain a complete characterization for compact double differences formed by four composition operators applying our characterization we easily recover known results on linear combinations of two or three composition operators as another application we also show that double difference cancellation is possible for linear combinations of four composition operators by constructing an explicit example of a compact double difference formed by two noncompact differences in spite of such an example our characterization also shows that double difference cancellation may occur in the global sense only and that genuine double difference cancellation is not possible in a certain local sense

on the complete and partial integrability of non hamiltonian systems abstract the methods of singularity analysis are applied to several third order non hamiltonian systems of physical significance including the lotka volterra equations the three wave interaction and the rikitake dynamo model complete integrability is defined and new completely integrable systems are discovered by means of the painleve property in all these cases we obtain integrals which reduce the equations either to a final quadrature or to an irreducible second order ordinary differential equation ode solved by painleve transcendents relaxing the painleve property we find many partially integrable cases whose movable singularities are poles at leading order with in t t 0 terms entering at higher orders in an n th order generalized rossler model a precise relation is established between the partial fulfillment of the painleve conditions and the existence of n 2 integrals of the motion

critical exponents for nonlinear diffusion equations with nonlinear boundary sources abstract this paper is concerned with the large time behavior of solutions to two types of nonlinear diffusion equations with nonlinear boundary sources on the exterior domain of the unit ball we are interested in the critical global exponent q 0 and the critical fujita exponent q c for the problems considered and show that q 0 q c for the multi dimensional porous medium equation and non newtonian filtration equation with nonlinear boundary sources this is quite different from the known results that q 0 q c for the one dimensional case

lie symmetries of the lorenz model abstract we study the generalized symmetries of the lorenz model to find the parameter values at which one or more time dependent integrals of motion exist in these cases the integrals are found trivially from the symmetries themselves a complete study of the one completely algebraically integrable case shows a the dynamics can be integrated exactly by reducing it first to a lower dimensional system b the symmetry vector field is hamiltonian these facts hold for other dissipative completely integrable dynamical systems as well the analytic study of a natural two form reveals that it is an entire function of time the foliation of phase space induced by the two form for the partially integrable cases has a simple description in terms of the coefficients occurring in the laurent series expansions of the dependent variables

large time behavior and lp lq estimate of solutions of 2 dimensional nonlinear damped wave equations abstract we show the asymptotic behavior of the solution to the cauchy problem of the two dimensional damped wave equation it is shown that the solution of the linear damped wave equation asymptotically decompose into a solution of the heat and wave equations and the difference of those solutions satisfies the lp lq type estimate this is a two dimensional generalization of the three dimensional result due to nishihara math z 244 2003 631 to show this we use the fourier transform and observe that the evolution operators of the damped wave equation can be approximated by the solutions of the heat and wave equations by using the lp lq estimate we also discuss the asymptotic behavior of the semilinear problem of the damped wave equation with the power nonlinearity u u our result covers the whole super critical case 1 where the 1 is well known as the fujita exponent when n 2

existence and nonexistence of global solutions for a nonlinear hyperbolic system with damping abstract this paper concerns with the cauchy problem for the damped nonlinear hyperbolic system u t t u u t v p t x r 1 r n v t t v v t u q t x r 1 r n u 0 x u 0 x u t 0 x u 1 x x r n v 0 x v 0 x v t 0 x v 1 x x r n where p q 1 and satisfy p q 1 we shall prove that if max 1 p p q 1 1 q p q 1 n 2 for n 1 or 3 then the solution with small initial data exists globally in time and satisfies the decay estimates u t c 1 t and v t c 1 t for some positive constants and while if max 1 p p q 1 1 q p q 1 n 2 for n 1 then every solution with initial data having positive average value does not exist globally

algebraic structures and invariant manifolds of differential systems algebraic tools are applied to find integrability properties of odes bilinear nonassociative algebras are associated to a large class of polynomial and nonpolynomial systems of differential equations since all equations in this class are related to a canonical quadratic differential system the lotka volterra system these algebras are classified up to dimension 3 and examples for dimension 4 and 5 are given their subalgebras are associated to nonlinear invariant manifolds in the phase space these manifolds are calculated explicitly more general algebraic invariant surfaces are also obtained by combining a theorem of walcher and the lotka volterra canonical form applications are given for lorenz model lotka may leonard and rikitake systems

exact smoothing properties of schrodinger semigroups we study schrodinger semigroups in the scale of sobolev spaces and show that for kato class potentials the range of such semigroups in lp has exactly two more derivatives than the potential this proves a conjecture of b simon we show that eigenfunctions of schrodinger operators are generically smoother by exactly two derivatives in given sobolev spaces than their potentials we give applications to the relation between the potential s smoothness and particle kinetic energy in the context of quantum mechanics and characterize kinetic energies in coulomb systems the techniques of proof involve leibniz and chain rules for fractional derivatives which are of independent interest as well as a new characterization of the kato class 1 introduction in this paper we attempt a precise study of the action of schrodinger semigroups in the scale of sobolev spaces work in this area was begun by b simon si4 in 1985 he proved a number of positive and negative results on such smoothing and also presented an open question in the form of a conjecture regarding necessary and sufficient conditions on the potential v for smoothing of order s by the semigroup of functions in l2 rd study of schrodinger semigroups is of interest for a number of reasons see

global existence of solutions for semilinear damped wave equations in rn with noncompactly supported initial data abstract we consider the n dimensional cauchy problem in r n for a semilinear damped wave equation with a power type nonlinearity u p for a noncompactly supported initial data which has a small energy we shall derive a global in time existence result in the case when the power of the nonlinear term satisfies 1 2 n p n n 2 this generalizes a previous result due to todorova yordanov j differential equations 174 2001 464 489 which dealt with a solution in the framework of compactly supported initial data

cancellation properties of composition operators on bergman spaces abstract the compact difference of two composition operators on the bergman spaces over the unit disc is characterized in 11 in terms of certain cancellation property of the inducing maps at every bad boundary points which make each single composition operator not to be compact in this paper we completely characterize the compactness of a linear combination of three composition operators on the bergman space as one consequence of this characterization we show that there is no cancellation property for the compactness of double difference of composition operators more precisely we show that if i are distinct and none of c i is compact then c 1 c 2 c 3 c 1 is compact if and only if both c 1 c 2 and c 3 c 1 are compact

nonexistence and existence of various order integrals for two and three dimensional polynomial potentials the nonexistence of integrals of the motion which are sixth and fourth degree polynomials in the velocities was established for a range of polynomial potentials the first known systematic search for sixth order integrals was performed for cubic and quartic polynomial potentials it revealed that there exist no nondegenerate cases with such integrals for either potential similarly there exists no nondegenerate fifth or sixth degree polynomial potentials possessing quartic invariants a complete list of three dimensional cubic potentials with quadratic integrals is given all these integrable three dimensional potentials can be interpreted as orthogonal superpositions of known integrable two dimensional potentials possessing quadratic integrals they correspond to 3 of 11 possible coordinate systems in which three dimensional potentials separate

components of linear fractional composition operators abstract we determine when two linear fractional composition operators on the hardy space h 2 belong to the same component in the collection of all composition operators on h 2 we show that two such composition operators in the same component may fail to have compact difference which answers a question raised by joel shapiro and carl sundberg