best constants in rosenthal type inequalities and the kruglov operator let x be a symmetric banach function space on 0 1 with the kruglov property and let f f k n k 1 n 1 be an arbitrary sequence of independent random variables in x this paper presents sharp estimates in the deterministic characterization of the quantities

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

schrodinger equation with time dependent boundary conditions a schrodinger equation for a well potential with varying width is studied generalized canonical transformations are shown to transform the problem into a time dependent harmonic oscillator problem submitted to fixed boundary conditions this transformed problem is solved by a perturbation technique and gives the evolution of the average energy of the system according to the motion of the well motions corresponding to a renormalization or compaction group are shown to be solvable by separation of variables

inequalities for noncommutative differentially subordinate martingales abstract the classical differential subordination of martingales introduced by burkholder in the eighties is generalized to the noncommutative setting working under this domination we establish the strong type inequalities with the constants of optimal order as p 1 and p and the corresponding endpoint weak type 1 1 estimate in contrast to the classical case we need to introduce two different versions of noncommutative differential subordination depending on the range of the exponents for the l p estimate 2 p a certain weaker version is sufficient on the other hand the strong type p p inequality 1 p 2 and the weak type 1 1 estimate require a stronger version as an application we present a new proof of noncommutative burkholder gundy inequalities the main technical advance is a noncommutative version of the good inequality and a certain summation argument we expect that these techniques will be useful in other situation as well

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

a class of finite element methods for singularly perturbed second order differential equations abstract standard galerkin finite element methods give poor accuracy when applied to second order elliptic problems with a significant convective term an upwind finite element was introduced to overcome this difficulty for constant coefficient problems with zero source term this paper extends the use of this type of element to variable coefficient problems with nonzero source term by introducing a class of generalised upwind elements called comparison upwind finite elements two elements from this class are presented in detail in this paper we obtain nodal error estimates and global l1 and l2 error estimates for both methods finally some numerical results are presented which demonstrate the methods accuracy

φ moment inequalities for independent and freely independent random variables abstract this paper is devoted to the study of moments of sums of independent freely independent random variables more precisely let f k k 1 n be a sequence of positive symmetrically distributed independent random variables and let be an orlicz function with 2 condition we provide an equivalent expression for the quantity e k 1 n f k in term of the sum of disjoint copies of the sequence f k k 1 n we also prove an analogous result in the setting of free probability furthermore we provide an equivalent characterization of sup 1 k n x k for positive freely independent random variables and also present some new results on free johnson schechtman inequalities in the quasi banach symmetric operator space

a method of approximating expectations of functions of sums of independent random variables let x 1 x 2 cdots be a sequence of independent random variables with s n sum n i 1 x i fix alpha 0 let phi cdot be a continuous strictly increasing function on lbrack 0 infty such that phi 0 0 and phi cx leq c alpha phi x for all x 0 and all c geq 2 suppose a is a real number and j is a finite nonempty subset of the positive integers in this paper we are interested in approximating e max j in j phi a s j we construct a number b j a from the one dimensional distributions of the x s such that the ratio e max j in j phi a s j phi b j a is bounded above and below by positive constants which depend only on alpha bounds for these constants are given

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

rosenthal type inequalities for free chaos let a denote the reduced amalgamated free product of a family a 1 a 2 an of von neumann algebras over a von neumann subalgebra b with respect to normal faithful conditional expectations e k a k b we investigate the norm in lp a of homogeneous polynomials of a given degree d we first generalize voiculescu s inequality to arbitrary degree d 1 and indices 1 p this can be regarded as a free analogue of the classical rosenthal inequality our second result is a length reduction formula from which we generalize recent results of pisier ricard and the authors all constants in our estimates are independent of n so that we may consider infinitely many free factors as applications we study square functions of free martingales more precisely we show that in contrast with the khintchine and rosenthal inequalities the free analogue of the burkholder gundy inequalities does not hold in l a at the end of the paper we also consider khintchine type inequalities for shlyakhtenko s generalized circular systems

johnson schechtman inequalities for noncommutative martingales abstract in this paper we study johnson schechtman inequalities for noncommutative martingales more precisely disjointification inequalities of noncommutative martingale difference sequences are proved in an arbitrary symmetric operator space e m of a finite von neumann algebra m without making any assumption on the boyd indices of e we show that we can obtain johnson schechtman inequalities for arbitrary martingale difference sequences and that in contrast with the classical case of independent random variables or the noncommutative case of freely independent random variables the inequalities are one sided except when e l 2 0 1 as an application we partly resolve a problem stated by randrianantoanina and wu in 46 we also show that we can obtain sharp moment analogues for orlicz functions satisfying p convexity and q concavity for 1 p 2 q 2 and p 2 2 q this is new even for the classical case we also extend and strengthen the noncommutative burkholder gundy inequalities in symmetric spaces and in the moment case

johnson schechtman inequalities in the free probability theory abstract we introduce and study a free kruglov operator as an application of this study we prove an analogue of the johnson schechtman inequalities in the setting of free probability theory and an arbitrary symmetric operator space this extends and complements earlier l p results of junge et al 2007 12

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

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

an 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

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

invariants for dissipative nonlinear systems by using rescaling a rescaling transformation of space and time is introduced in the study of nonlinear dissipative systems that are described by a second order differential equation with a friction term proportional to the velocity t v the transformation is of the form x t where x c t t d dt a2 t this rescaling is used to find each potential for which there exists an exact invariant quadratic in the velocity and to find the invariant the invariants are found explicitly for a power law potential t xm 1 m 1 and an arbitrary coefficient of friction t we show in an example how the rescaling transformation can be chosen to give an asymptotic solution of the equation in cases where the exact invariant does not exist for certain parameters the asymptotic solution is a self similar solution that is an attractor for all initial conditions the technique of applying a rescaling transformation has been useful in other problems and may have additional practical applications

a brief survey on numerical methods for solving singularly perturbed problems in the present paper a brief survey on computational techniques for the different classes of singularly perturbed problems is given this survey is a continuation of work performed earlier by the first author and contains the literature of the work done by the researchers during the years 2000 2009 however some older important relevant papers are also included in this survey we also mentioned those papers which are not surveyed in the previous survey papers by the first author of this paper see appl math comput 30 1989 223 259 130 2002 457 510 134 2003 371 429 for details thus this survey paper contains a surprisingly large amount of literature on singularly perturbed problems and indeed can serve as an introduction to some of the ideas and methods for the singular perturbation problems

equidistributed random variables in lp q abstract we consider linear combinations of independent identically distributed random variables in l p q in fact we provide several norm inequalities for sums from a larger class of equidistributed random variables

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

interpolation and φ moment inequalities of noncommutative martingales this paper is devoted to the study of moment inequalities for noncommutative martingales in particular we prove the noncommutative moment analogues of martingale transformations stein s inequalities khintchine s inequalities for rademacher s random variables and burkholder gundy s inequalities the key ingredient is a noncommutative version of marcinkiewicz type interpolation theorem for orlicz spaces which we establish in this paper

petrov galerkin finite element methods with a hinged test space for singularly perturbed problems on introduit des methodes d elements finis du type de petrov galerkin pour la solution approchee des problemes a deux points limites relatifs a des equations differentielles lineaires du second ordre a perturbations singulieres

noncommutative kothe duality using techniques drawn from the classical theory of rearrangement invariant banach function spaces we develop a duality theory in the sense of kothe for symmetric banach spaces of measurable operators affiliated with a semifinite von neumann algebra equipped with a distinguished trace a principal result of the paper is the identification of the kothe dual of a given banach space of measurable operators in terms of normality

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

scattering s matrix derived from invariants of the ermakov lewis type an amplitude phase formula for the s matrix due to a central potential is derived the derivation makes use of invariants of the ermakov lewis type

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

on the multiplication of free n tuples of noncommutative random variables let a1 a n b 1 b n be random variables in a noncommutative probability space such that a1 a n is free from b1 b n we show how the joint distribution of the n tuple a1b1 a n b n can be described in terms of the joint distributions of a1 a n a nd b 1 b n by using the combinatorics of the n dimensional r transform we point out a few applications that can be easily derived from our result concerning the left and right translation with a semicircular element see sections 1 6 1 10 and the compression with a projection see sections 1 11 1 14 of an n tuple of noncommutative random variables a different approach to two of these applications is presented by dan voiculescu in an appendix to the paper introduction the theory of free random variables was developed in a se quence of papers of d voiculescu see 17 or the recent survey in 16 as an instrument for approaching free products of operator algebras its particular aspect addressed in the present paper is the one concerning the addition and multiplica tion of free random variables as shown by voiculescu in 13 14 a powerful method in the study of these operations is the use of transforms that convert them respectively into addition and multiplication of complex analytic func tions or in an algebraic framework of formal power series in an indeterminate z the precise definitions of these transforms called r transform for the addition problem and s transform for the multiplication problem will be reviewed in the sections 1 2 1 3 below in the present paper we are pursuing a combination of two ideas that have appeared recently in the study of the r and s transforms the first idea is that the connection between the r and the s transform is closer than one might suspect at first glance a way of making this precise was pointed out in our paper 6 in the form of the equation s r for a distribution with nonvan ishing mean and where is a combinatorial object with a precise significance the fourier transform for multiplicative functions on noncrossing partitions a byproduct of our result in 6 is the remark that the multiplication of free