dirac nijenhuis structures in this paper we study the concept of dirac nijenhuis structures we consider them to be a pair where d is a dirac structure defined with respect to a lie bialgebroid a a and is a nijenhuis operator which defines a deformation of the lie algebroid structure of d in a compatible way the transformation can be considered to deform also the double of the lie bialgebroid which leads to the concept of dirac nijenhuis structures of type i or to not affect it leading to dirac nijenhuis structures of type ii we prove that the concept contains poisson nijenhuis structures as a particular case and provides another example for the case of kahler manifolds

lie bialgebroids and poisson groupoids lie bialgebras arise as infinitesimal invariants of poisson lie groups a lie bialgebra is a lie algebra g with a lie algebra structure on the dual g which is compatible with the lie algebra g in a certain sense for a poisson group g the multiplicative poisson structure induces a lie algebra structure on the lie algebra dual g which makes g g into a lie bialgebra in fact there is a one one correspondence between poisson lie groups and lie bialgebras if the lie groups are assumed to be simply connected 7 16 19 the importance of poisson lie groups themselves arises in part from their role as classical limits of quantum groups 8 and in part because they provide a class of poisson structures for which the realization problem is tractable 15 poisson groupoids were introduced by weinstein 24 as a generalization of both poisson lie groups and the symplectic groupoids which arise in the integration of arbitrary poisson manifolds 4 11 he noted that the lie algebroid dual a g of a poisson groupoid g itself has a lie algebroid structure but did not develop the infinitesimal structure further in this paper we introduce and study a natural infinitesimal invariant for poisson groupoids the lie bialgebroids of the title our ultimate purpose is to develop a lie theory for poisson groupoids parallel to that for poisson groups in this paper we are primarily concerned with the first half of this process that is with the construction of the lie bialgebroid of a poisson groupoid after the preliminary 2 in which we describe the generalization to arbitrary lie algebroids of the exterior calculus and schouten calculus in 3 we define a lie bialgebroid to be a lie algebroid a whose dual a is also equipped with a lie algebroid structure such that the coboundary operator d a a associated to a satisfies a cocycle equation with respect to a the lie algebra of sections of a this is clearly a straightforward extension of the concept of a lie bialgebra 16 but cannot be formalized in lie algebroid cohomological terms since there is no satisfactory adjoint representation for a general lie algebroid most of 3 is devoted to proving that this definition is self dual if a a is a lie bialgebroid then a a is also in 4 we briefly consider the special case of lie bialgebroids satisfying a triangularity condition which include some important examples such as the usual triangular lie bialgebras and lie bialgebroids associated to poisson manifolds the techniques used in 2 4 are similar to those known for lie bialgebras it would be possible by suitably generalizing the proof for poisson groups to prove

a differential complex for poisson manifolds construction du complexe canonique d une variete de poisson homologie canonique des varietes symplectiques application a l homologie de hochschild des algebres non commutatives

univalent harmonic mappings and linearly connected domains abstract we investigate the relationship between the univalence of f and of h in the decomposition f h g of a sense preserving harmonic mapping defined in the unit disk d c among other results we determine the holomorphic univalent maps h for which there exists c 0 such that every harmonic mapping of the form f h g with g c h is univalent the notion of a linearly connected domain appears in our study in a relevant way

on lengths areas and lipschitz continuity of polyharmonic mappings abstract in this paper we continue our investigation of polyharmonic mappings in the complex plane first we establish two landau type theorems we also show a three circles type theorem and an area version of the schwarz lemma finally we study lipschitz continuity of polyharmonic mappings with respect to the distance ratio metric

a review of the decomposition method in applied mathematics abstract the decomposition method can be an effective procedure for analytical solution of a wide class of dynamical systems without linearization or weak nonlinearity assumptions closure approximations perturbation theory or restrictive assumptions on stochasticitiy

on the quasi isometries of harmonic quasiconformal mappings abstract we prove versions of the ahlfors schwarz lemma for quasiconformal euclidean harmonic functions and harmonic mappings with respect to the poincare metric

reduction of poisson manifolds and jacobi manifolds via dirac theory we give two kinds of reduction relating poisson manifolds and jacobi manifolds by classifying reducible dirac and jacobi dirac structures into two classes respectively the proof needs only some direct calculation without using the existence of momentum mappings or the introducing of admissible functions etc meanwhile we present some examples and applications

harmonic univalent mappings into a half plane with nonreal vertical slits a class of harmonic univalent mappings is constructed by applying the method of clunie and sheil small these mappings assuming their values in a half plane with a vertical boundary omit two vertical half lines symmetric w r t the real axis several basic properties are proved

a schwarz pick inequality for harmonic quasiconformal mappings and its applications abstract the main result of this paper is the sharp generalized schwarz pick inequality for euclidean harmonic quasiconformal mappings with convex ranges which generalizes a result given by mateljevic as its applications we obtain the property of quasi isometry with respect to the poincare distance and an analogue of the koebe theorem for this class of mappings

applying the double side method to solution nonlinear pendulum problem the double side approach method is using mathematical programming and method of weighted residual to the solution nonlinear differential equation the method is a semi exact solution it is based on collocation method and mathematical programming to create the bilateral inequality for this process the differential equation problem will be changed to mathematical programming problem by using the optimization method of genetic algorithms gas we can get minimum and maximum solution to satisfy inequality the advantage of the method is that the computer storage cell requirements are less than with the finite element method a nonlinear differential equation problem is studied as an example the efficiency accuracy and simplicity of this approach are illustrated indicating that the proposed method can be easily extended to solve a wide range of physical engineering problems

use residual correction method to calculate the upper and lower solutions of initial value problem abstract this article attempts to establish solutions monotonic relation with residuals of initial value problem through the concept of maximum principle for differential equations then discretizes the equation by the simple spline approximation for initial value problem converts the once complex inequation constraint mathematical programming problem into a simple problem of equation iteration by applying the new residual correction method for initial value problem put forth by this article and obtains the upper and lower solutions of initial value problem results from numerical validation indicate this method is accurate simple and fast the approximate solutions obtained can correctly analyze error range and the new method put forth hereby helps increase the accuracy of mean approximate solutions

estimates on bloch constants for planar harmonic mappings the schwarz lemma and bloch constants for planar bounded harmonic mappings are considered sharper form and better estimates are obtained our results improve the one made by dorff and nowak as well as by chen gauthier and hengartner

supersymmetry and morse theory it is shown that the morse inequalities can be obtained by consideration of a certain supersymmetric quantum mechanics hamiltonian some of the implications of modern ideas in mathematics for supersymmetric theories are discussed

quasiconformal and harmonic mappings between jordan domains let and 1 be jordan domains let 0 1 and let f omega mapsto omega 1 be a harmonic homeomorphism the object of the paper is to prove the following results a if f is q c and 1 c 1 then f is lipschitz b if f is q c 1 c 1 and 1 is convex then f is bi lipschitz and c if is the unit disk 1 is convex and 1 c 1 then f is quasiconformal if and only if its boundary function is bi lipschitz and the hilbert transform of its derivative is in l these extend the results of pavlovic ann acad sci fenn 27 365 372 2002

on certain geometric properties of polyharmonic mappings abstract the main purpose of this paper is twofold first the three quarter theorem is established for a large class of polyharmonic mappings second a class of polyharmonic mappings with positive jacobian and being univalent in the unit disk d are constructed

on the univalence of the log biharmonic mappings in this paper we establish the univalence and starlikeness connection between log biharmonic mappings and logharmonic mappings

on some properties of solutions of the biharmonic equation in the present paper the properties of the linear complex operator l f zfz z fz which is defined on the class of complex valued c1 functions in the plane are investigated it is shown that harmonicity and biharmonicity are invariant under the linear operatorl results concerning starlikeness and convexity of biharmonic functions versus the corresponding harmonic functions are considered the operator l can be manipulated to express the conditions in the definitions of starlikeness and convexity in a convenient way

decompositions oflpand hardy spaces of polyharmonic functions abstract let h p q k 0 p q f of order k on the unit n ball for which the function r 1 r 1 q m p f r belongs to l q 0 1 our main result is that when k 2 and 1 the operator f pf f where pf is the poisson integral of f acts as an isomorphism of h p q k onto the direct sum of h p q 1 and h p q 2 k 1 another decomposition theorem closely related to the almansi representation theorem is also given

bloch constant and landau s theorem for planar p harmonic mappings abstract in this paper our main aim is to introduce the concept of planar p harmonic mappings and investigate the properties of these mappings first we discuss the p harmonic bloch mappings two estimates on the bloch constant are obtained which are generalizations of the main results in colonna 1989 9 as a consequence of these investigations we establish a bloch and landau s theorem for p harmonic mappings

a biharmonic maximum principle for hyperbolic surfaces let w be a simply connected two dimensional riemannian manifold with a cysmooth metric ds we can always introduce local isothermal coordinates near any point of w that is such coordinates x y that the metric is represented in the form ds x y 1 4 o x y dx dy for some positive weight function o and since w is orientable as a two dimensional simply connected manifold these local isothermal coordinates can serve as conformal charts for a complex structure on w see 1 pp 124 126 the k be uniformization theorem then says that w is conformally equivalent to one of the three sets the riemann sphere s 1 4 cw fyg the complex plane c or the open unit disk d this equivalence together with the choice of isothermal coordinates allows us to identify w with one of the above three sets w supplied with the isothermal riemannian metric

twisted jacobi manifolds twisted dirac jacobi structures and quasi jacobi bialgebroids we study twisted jacobi manifolds a concept that we had introduced in a previous note twisted jacobi manifolds can be characterized using twisted dirac jacobi which are sub bundles of courant jacobi algebroids we show that each twisted jacobi manifold has an associated lie algebroid with a 1 cocycle we introduce the notion of quasi jacobi bialgebroid and prove that each twisted jacobi manifold has a quasi jacobi bialgebroid canonically associated moreover the double of a quasi jacobi bialgebroid is a courant jacobi algebroid several examples of twisted jacobi manifolds and twisted dirac jacobi structures are presented

jacobi structures revisited jacobi algebroids i e graded lie brackets on the grassmann algebra associated with a vector bundle which satisfy a property similar to that of the jacobi brackets are introduced they turn out to be equivalent to generalized lie algebroids in the sense of iglesias and marrero jacobi bialgebroids are defined in the same manner a lifting procedure of elements of this grassmann algebra to multivector fields on the total space of the vector bundle which preserves the corresponding lie brackets is developed this gives the possibility of associating canonically a lie algebroid with any local lie algebra in the sense of kirillov

landau s theorem for log p harmonic mappings abstract the main aim of this paper is to prove the existence of landau bloch constant for log p harmonic mappings

norm of the pre schwarzian derivative bloch s constant and coefficient bounds in some classes of harmonic mappings abstract the aim of the paper is a study of a family of sense preserving complex valued harmonic functions f that are normalized in the open unit disk and such that its analytic part is convex in one direction we prove the univalence and establish estimates on pre schwarzian derivative and the bloch constant for co analytic part of harmonic mapping the bounds of the coefficients of co analytic part are also given

lengths areas and lipschitz type spaces of planar harmonic mappings abstract in this paper we give bounds for length and area distortion for harmonic k quasiconformal mappings and investigate certain lipschitz type spaces on harmonic mappings

representation formula and bi lipschitz continuity of solutions to inhomogeneous biharmonic dirichlet problems in the unit disk abstract the aim of this paper is twofold first we establish the representation formula and the uniqueness of the solutions to a class of inhomogeneous biharmonic dirichlet problems and then prove the bi lipschitz continuity of the solutions

a note on harmonic quasiconformal mappings abstract in this note we show that a harmonic quasiconformal mapping f u i v with respect to the poincare metric of the upper half plane r 2 onto itself such that v x y v y or u x y u x is a conformal mapping

jacobi nijenhuis algebroids and their modular classes jacobi nijenhuis algebroids are defined as a natural generalization of poisson nijenhuis algebroids in the case where there exists a nijenhuis operator on a jacobi algebroid which is compatible with it we study modular classes of jacobi and jacobi nijenhuis algebroids

deformation of lie derivative and μ symmetries we introduce in the spirit of witten s gauging of exterior differential a deformed lie derivative that allows a geometrical interpretation of and symmetries in complete analogy with standard symmetries the case of variational symmetries both for odes and for pdes is also considered in this approach leading to and conservation laws

lie algebroid foliations and script e1 m dirac structures we prove some general results about the relation between the 1 cocycles of an arbitrary lie algebroid a over m and the leaves of the lie algebroid foliation on m associated with a using these results we show that a 1 m dirac structure l induces on every leaf f of its characteristic foliation a 1 f dirac structure lf which comes from a precontact structure or from a locally conformal presymplectic structure on f in addition we prove that a dirac structure on m can be obtained from l and we discuss the relation between the leaves of the characteristic foliations of l and

the schrodinger operator as a generalized laplacian the schrodinger operators on the newtonian spacetime are defined in a way which make them independent of the class of inertial observers in this picture the schrodinger operators act not on functions on the spacetime but on sections of a certain one dimensional complex vector bundle the schrodinger line bundle this line bundle has trivializations indexed by inertial observers and is associated with an u 1 principal bundle with an analogous list of trivializations the schrodinger principal bundle if an inertial frame is fixed the schrodinger bundle can be identified with the trivial bundle over spacetime but as there is no canonical trivialization inertial frame these sections interpreted as wavefunctions cannot be viewed as actual functions on the spacetime in this approach the change of an observer results not only in the change of actual coordinates in the spacetime but also in a change of the phase of wavefunctions for the schrodinger principal bundle a natural differential calculus for wave forms is developed that leads to a natural generalization of the concept of the laplace beltrami operator associated with a pseudo riemannian metric the free schrodinger operator turns out to be the laplace beltrami operator associated with a naturally distinguished invariant pseudo riemannian metric on the schrodinger principal bundle the presented framework does not involve any ad hoc or axiomatically introduced geometrical structures it is based on the traditional understanding of the schrodinger operator in a given reference frame which is supported by producing right physics predictions and it is proven to be strictly related to the frame independent formulation of analytical newtonian mechanics and hamilton jacobi equations that makes a bridge between the classical and quantum theory

manin triples for lie bialgebroids in his study of dirac structures a notion which includes both poisson structures and closed 2 forms t courant introduced a bracket on the direct sum of vector fields and 1 forms this bracket does not satisfy the jacobi identity except on certain subspaces in this paper we systematize the properties of this bracket in the definition of a courant algebroid this structure on a vector bundle e rightarrow m consists of an antisymmetric bracket on the sections of e whose jacobi anomaly has an explicit expression in terms of a bundle map e rightarrow tm and a field of symmetric bilinear forms on e when m is a point the definition reduces to that of a lie algebra carrying an invariant nondegenerate symmetric bilinear form for any lie bialgebroid a a over m a notion defined by mackenzie and xu there is a natural courant algebroid structure on a oplus a which is the drinfel d double of a lie bialgebra when m is a point conversely if a and a are complementary isotropic subbundles of a courant algebroid e closed under the bracket such a bundle with dimension half that of e is called a dirac structure there is a natural lie bialgebroid structure on a a whose double is isomorphic to e the theory of manin triples is thereby extended from lie algebras to lie algebroids our work gives a new approach to bihamiltonian structures and a new way of combining two poisson structures to obtain a third one we also take some tentative steps toward generalizing drinfel d s theory of poisson homogeneous spaces from groups to groupoids

landau s theorem for planar harmonic mappings chen gauthier and hengartner obtained two versions of landau s theorem for bounded planar harmonic mappings later dorff and nowak improved their estimates furthermore grigoryan and huang independently improved their results in this note we improve these last results by obtaining better coefficient estimates for bounded and normalized planar harmonic mappings in particular our theorems are sharp when m 1 which are consistent with landau s theorem when m 1

landau s theorem for certain biharmonic mappings abstract in this paper we show the existence of landau and bloch constants for biharmonic mappings of the form l f here l represents the linear complex operator l z z z z defined on the class of complex valued c 1 functions in the plane and f belongs to the class of biharmonic mappings of the form f z z 2 g z k z z 1 where g and k are harmonic

on the geometric quantization of jacobi manifolds the geometric quantization of jacobi manifolds is discussed a natural cohomology termed lichnerowicz jacobi on a jacobi manifold is introduced and using it the existence of prequantization bundles is characterized to do this a notion of contravariant derivatives is used in such a way that the procedure developed by vaisman for poisson manifolds is naturally extended a notion of polarization is discussed and the quantization problem is studied the existence of prequantization representations is also considered

bohr s inequalities for the analytic functions with lacunary series and harmonic functions abstract we determine the bohr radius for the class of all functions f of the form f z z m k 0 a k p z k p analytic in the unit disk z 1 and satisfy the condition f z 1 for all z 1 in particular our result also contains a solution to a recent conjecture of ali et al 9 for the bohr radius for odd analytic functions solved by the authors in 17 we consider a more flexible approach by introducing the p bohr radius for harmonic functions which in turn contains the classical bohr radius as special case also we prove several other new results and discuss p bohr radius for the class of odd harmonic bounded functions

dirac structures for generalized lie bialgebroids we study dirac structures for generalized courant algebroids which are doubles of generalized lie bialgebroids the cases investigated include graphs of bivector fields and characteristic pairs of some sub bundles

on the univalence of polyharmonic mappings abstract a 2p times continuously differentiable complex valued function f u i v in a simply connected domain is polyharmonic or p harmonic if it satisfies the polyharmonic equation p f 0 every polyharmonic mapping f can be written as f z k 1 p z 2 p 1 g p k 1 z where each g p k 1 is harmonic in this paper we investigate the univalence of polyharmonic mappings on linearly connected domains and the relation between univalence of f z and that of g p z the notion of stable univalence and logpolyharminc mappings are also considered

poisson jacobi reduction of homogeneous tensors the notion of homogeneous tensors is discussed we show that there is a one to one correspondence between multivector fields on a manifold m homogeneous with respect to a vector field on m and first order polydifferential operators on a closed submanifold n of codimension 1 such that is transversal to n this correspondence relates the schouten nijenhuis bracket of multivector fields on m to the schouten jacobi bracket of first order polydifferential operators on n and generalizes the poissonization of jacobi manifolds actually it can be viewed as a super poissonization this procedure of passing from a homogeneous multivector field to a first order polydifferential operator can also be understood as a sort of reduction in the standard case a half of a poisson reduction a dual version of the above correspondence yields in particular the correspondence between homogeneous symplectic structures on m and contact structures on n

landau s theorem for biharmonic mappings abstract in this paper we show the existence of landau constant for biharmonic mappings of the form f z z 2 g z k z z 1 where g and k are harmonic

landau s theorem for polyharmonic mappings abstract in this paper we first investigate coefficient estimates for bounded polyharmonic mappings in the unit disk d then we obtain two versions of landau s theorem for polyharmonic mappings f and for the mappings of the type l f where l is the differential operator of abdulhadi abu muhanna and khuri examples and numerical estimates are given

some properties of a class of elliptic partial differential operators abstract we prove schwarz pick type estimates and coefficient estimates for a class of functions induced by the elliptic partial differential operators then we apply these results to obtain a landau type theorem

on quasiconformal self mappings of the unit disk satisfying poisson s equation let qc k g be a family of k quasiconformal mappings of the open unit disk onto itself satisfying the pde w g g c u w 0 0 it is proved that qc k g is a uniformly lipschitz family moreover if g is small enough then the family is uniformly bi lipschitz the estimations are asymptotically sharp as k 1 and g 0 so w qc k g behaves almost like a rotation for sufficiently small k and g

weighted integrability of polyharmonic functions abstract to address the uniqueness issues associated with the dirichlet problem for the n harmonic equation on the unit disk d in the plane we investigate the l p integrability of n harmonic functions with respect to the standard weights 1 z 2 the question at hand is the following if u solves n u 0 in d where stands for the laplacian and d u z p 1 z 2 d a z must then u z 0 here n is a positive integer is real and 0 p d a is the usual area element the answer will generally speaking depend on the triple n p the most interesting case is 0 p 1 for a given n we find an explicit critical curve p n p a piecewise affine function such that for n p there exist nontrivial functions u with n u 0 of the given integrability while for n p only u z 0 is possible we also investigate the obstruction to uniqueness for the dirichlet problem that is we study the structure of the functions in ph n p d when this space is nontrivial we find a new structural decomposition of the polyharmonic functions the cellular decomposition which decomposes the polyharmonic weighted l p space in a canonical fashion corresponding to the cellular expansion is a tiling of part of the p plane into cells the above uniqueness for the dirichlet problem may be considered for any elliptic operator of order 2n however the above mentioned critical integrability curve will depend rather strongly on the given elliptic operator even in the constant coefficient case for n 1

on a new relation between jacobi and homogeneous poisson manifolds we establish a new very close relationship which links jacobi structures and homogeneous poisson structures defined on the same manifold and study the characteristic foliations of the related structures several examples of this construction are also given

on the geometry of lambda symmetries and pde reduction we give a geometrical characterization of prolongations of vector fields and hence of symmetries of odes this allows an extension to the case of pdes and systems of pdes in this context the central object is a horizontal 1 form and we speak of prolongations of vector fields and symmetries of pdes we show that these are as good as standard symmetries in providing symmetry reduction of pdes and systems and explicit invariant solutions

distortion theorems for bloch functions we establish various distortion theorems for both normalized locally schlicht bloch functions and normalized bloch function with branch points these distortion theorems give lower bounds on either f z or re f z most of our distortion theorems are sharp and all extremal functions identified the main tools used in establishing these distortion theorems are the classical form of julia s lemma and a new version of julia s lemma that applies to certain multiple valued analytic functions

on the linear combinations of harmonic univalent mappings abstract in this paper we derive several sufficient conditions of the linear combinations of harmonic univalent mappings to be univalent and convex in the direction of the real axis furthermore some illustrative examples and imagine domains of the linear combinations satisfying the desired conditions are enumerated

study on monotonicity of boundary value problems of differential equations this article presents two new test methods to decide monotonic increasing or decreasing properties of differential equations for simple problems the eigenfunction test method can be applied to obtain their exact monotonicity or their loosest conditional expression and can contribute to probes into accumulated effect of various terms in equations on their monotonicity secondly the trial function test method sets relatively strict conditions for discrimination of monotonicity but it can verify complex non linear problems in a simple and rapid way therefore these two similar test methods will help to further understand theories on monotonicity thus making it possible to apply monotonic equations more widely to other problems

inequalities for zero balanced hypergeometric functions the authors study certain monotoneity and convexity properties of the gaussian hypergeometric function and those of the euler gamma function

the graded jacobi algebras and co homology jacobi algebroids i e jacobi versions of lie algebroids are studied in the context of graded jacobi brackets on graded commutative algebras this unifies various concepts of graded lie structures in geometry and physics a method of describing such structures by classical lie algebroids via certain gauging in the spirit of e witten s gauging of the exterior derivative is developed one constructs a corresponding cartan differential calculus graded commutative one in a natural manner this in turn gives canonical generating operators for triangular jacobi algebroids one gets in particular the lichnerowicz jacobi homology operators associated with classical jacobi structures courant jacobi brackets are obtained in a similar way and used to define an abstract notion of a courant jacobi algebroid and dirac jacobi structure

biorthogonal series solution of stokes flow problems in sectorial regions in this paper we develop an eigenfunction expansion method for solving stokes flow problems in sectorial cavities that arise in fluid dynamics the method leads to the development of a set of eigenfunctions adjoint eigenfunctions biorthogonality conditions and an algorithm for the computation of the coefficients of the eigenfunction expansion the resulting infinite system of linear equations is then solved by truncation these biorthogonality conditions are properties satisfied by the eigenfunctions and adjoint eigenfunctions which are used to compute the coefficients of the eigenfunction expansion solution the biorthogonality conditions are derived for a class of fourth order boundary value problems with variable coefficients that arise from separating variables of the governing stokes equation the method is applied to the slow steady flow in a two dimensional sectorial cavity the fluid is set into motion by the uniform translation of a covering plate or belt