maass operators and eisenstein series let g k be a hermitian symmetric space of the non compact type and let f c g be an arithmetic subgroup it is obvious that the enveloping algebra u g 0 of the complexified lie algebra of g gives rise to operators which transform automorphic forms on n with respect to f to f automorphic forms of a different type however it is in general not easy to write these differential operators explicitly in the coordinates of in the case g sp n ir when n is the siegel upper half plane of genus n maass constructed certain explicit operators of this type as a consequence of his construction he was able to show that his operators transform certain simple eisenstein series for f to other specific simple eisenstein series for f for a summary of his results and for references see 6 in this paper we generalize these results of maass to arbitrary hermitian symmetric spaces of the non compact type namely we show that the enveloping algebra u g 0 gives rise to differential operators a such that for certain pairs of canonical automorphy factors j j on g we have aj j cf theorem 2 4 and corollary 3 4 we do this abstractly using the transformation properties of the automorphy factors in particular we do not give explicit formulas for our differential operators which we call maass operators it would be interesting to do so in general we have also restricted our attention to cases in which one of the automorphy factors j j is scalar valued the same methods probably work more generally our interest in these operators arose from their use by shimura in his study of rationality properties for special values of zeta functions see 8 9 and subsequent papers katz 4 has also introduced these operators in another form the author has extended shimura s method to the case of siegel modular forms in 3 and intends to use the results of the present paper in future work of the same kind for this reason a brief and somewhat obtuse discussion of the rationality of maass operators has been included in sect 4 in any specific i e coordinatized example the results of sect 4 can be improved upon

multiplicity one theorems the archimedean case let g be one of the classical lie groups gln 1 r gln 1 c u p q 1 o p q 1 on 1 c so p q 1 son 1 c and let g 0 be respectively the subgroup gln r gln c u p q o p q on c so p q son c embedded in g in the standard way we show that every irreducible casselman wallach representation of g 0 occurs with multiplicity at most one in every irreducible casselman wallach representation of g similar results are proved for the jacobi groups gln r nh2n 1 r gln c n h2n 1 c u p q nh2p 2q 1 r sp2n r nh2n 1 r sp2n c nh2n 1 c with their respective subgroups gln r gln c u p q sp2n r and sp2n c

convex solutions of polynomial like iterative equations in this paper convex solutions and concave solutions of polynomial like iterative equations are investigated a result for non monotonic solutions is given first and applied then to prove the existence of convex continuous solutions and concave ones furthermore another condition for convex solutions which is weaker in some aspects is also given the uniqueness and stability of those solutions are also discussed

principal curve algorithms for partitioning high dimensional data spaces most partitioning algorithms iteratively partition a space into cells that contain underlying linear or nonlinear structures using linear partitioning strategies the compactness of each cell depends on how well the locally linear partitioning strategy approximates the intrinsic structure to partition a compact structure for complex data in a nonlinear context this paper proposes a nonlinear partition strategy this is a principal curve tree pc tree which is implemented iteratively given that a pc passes through the middle of the data distribution it allows for partitioning based on the arc length of the pc to enhance the partitioning of a given space a residual version of the pc tree algorithm is developed denoted here as the principal component analysis tree pcr tree algorithm because of its residual property the pcr tree can yield the intrinsic dimension of high dimensional data comparisons presented in this paper confirm that the proposed pc tree and pcr tree approaches show a better performance than several other competing partitioning algorithms in terms of vector quantization error and nearest neighbor search the comparison also shows that the proposed algorithms outperform competing linear methods in total average coverage which measures the nonlinear compactness of partitioning algorithms

critical points on the boundaries of siegel disks let be a polynomial map of the riemann sphere of degree at least two we prove that if has a siegel disk g on which the rotation number satisfies a diophantine condition then the boundary of g contains a critical point

representations for the first associated q classical orthogonal polynomials let pk x q be any system of the q classical orthogonal polynomials and let be the corresponding weight function satisfying the q difference equation dq where and are polynomials of degree at most 2 and exactly 1 respectively further let p 1 k x q be associated polynomials of the polynomials pk x q explicit forms of the coefficients bn k and cn k in the expansions p 1 n 1 x q k 0n 1bn k k x p 1 n 1 x q k 0n 1cn kpk x q are given in terms of basic hypergeometric functions here k x equals xk if 0 0 or x q k if 1 0 where x x q 1 x x the most important representatives of those two classes are the families of litte q jacobi and big q jacobi polynomials respectively writing the second order nonhomogeneous q difference equation satisfied by p 1 n 1 x q in a special form recurrence relations in k for bn k and cn k are obtained in terms of and

analytic solutions of iterative functional equations abstract let r be a given positive number denote by d dr the closed disc in the complex plane c whose center is the origin and radius is r for any subset k of c and any integer m 1 write a d m k f f d m k is a continuous map and f dm is analytic suppose g a d n 1 c and h k a d k c k 2 n in this paper we study the iterative functional equation g z f z f2 h2 z f z fn hn z f z fn 1 z 0 and give some conditions for the equation to have a solution and a unique solution in a d d

constraint local principal curve existing principal curve algorithms have some drawbacks such as time consuming and narrow application scope in practice since these algorithms are mainly based on global optimization in this paper we present the concept of constraint local principal curve clpc which uses local optimization methods and restricts the principal curve with two fixed endpoints to reduce the computational complexity in addition we propose three clpc algorithms by local optimization and adaptive radius to expand the range of applications and increase the solution quality the first algorithm i e clpcg is based on greedy thinking the second algorithm i e clpcs uses one dimensional search and the last algorithm clpcc combines the greedy thinking and one dimensional search then we define six performance indices to evaluate the performance of the clpc algorithms finally we present some numerical experiments with three simulation data sets and two gps measured data sets in both highway and railway the results indicate that all of the three clpc algorithms can obtain high accuracy data from multiple low accuracy data efficiently the clpc algorithms can improve the accuracy and computational speed compared with the existing k segment principal curve kpc algorithm in addition clpcc outperforms clpcg and clpcs according to the comprehensive experiments while clpcg runs much faster than other ones we present the concept of constraint local principal curve clpc to reduce the computational complexity we propose three clpc algorithms that combine local optimization and adaptive radius to expand the range of applications and increase the solution quality we present some numerical experiments with three simulation data sets and two measured gps data sets in highway and railway the clpc algorithms can improve the accuracy and computational speed compared with the existing kpc algorithms the features of the each clpc algorithm are analyzed according to the comprehensive experiments

bessel models for gsp 4 methods of theta correspondence are used to analyze local and global bessel models for gsp 4 proving a conjecture of gross and prasad which describes these models in terms of local epsilon factors in the local case and the non vanishing of central critical l value in the global case

relating invariant linear form and local epsilon factors via global methods we use the recent proof of jacquet s conjecture due to harris and kudla hk and the burger sarnak principle see bs to give a proof of the relationship between the existence of trilinear forms on representations of gl 2 k u for a non archimedean local field k u and local epsilon factors which was earlier proved only in the odd residue characteristic by this author in p1 theorem 1 4 the method used is very flexible and gives a global proof of a theorem of saito and tunnell about characters of gl 2 using a theorem of waldspurger w theorem 2 about period integrals for gl 2 and also an extension of the theorem of saito and tunnell by this author in p3 theorem 1 2 which was earlier proved only in odd residue characteristic in the appendix to this article h saito gives a local proof of lemma 4 which plays an important role in the article

kneser s theorem in q calculus while difference equations deal with discrete calculus and differential equations with continuous calculus so called q difference equations are considered when studying q calculus in this paper we obtain certain oscillation criteria for second order q difference equations among them a q calculus version of the famous kneser theorem

the saito kurokawa lifting and functoriality certain nontempered liftings from pgl 2 pgl 2 to pgsp 4 are constructed using the theory of local and global theta lifts the resulting representations on pgsp 4 are the saito kurokawa representations the lifting is shown to be functorial under certain reasonable assumptions on the local langlands correspondence for pgsp 4

iterative equations in banach spaces abstract let x be a banach space let k x f x x lipschitz f id sup and let p k x k x f k x by applying the banach contraction principle we prove that if p is sufficiently close in a certain sense to the identity then the equation p f f has a unique solution f as a corollary we obtain results on iterative equations of the types i a i f i x f x or i a i f i x f x with operator coefficients in banach spaces

learning and design of principal curves principal curves have been defined as self consistent smooth curves which pass through the middle of a d dimensional probability distribution or data cloud they give a summary of the data and also serve as an efficient feature extraction tool we take a new approach by defining principal curves as continuous curves of a given length which minimize the expected squared distance between the curve and points of the space randomly chosen according to a given distribution the new definition makes it possible to theoretically analyze principal curve learning from training data and it also leads to a new practical construction our theoretical learning scheme chooses a curve from a class of polygonal lines with k segments and with a given total length to minimize the average squared distance over n training points drawn independently convergence properties of this learning scheme are analyzed and a practical version of this theoretical algorithm is implemented in each iteration of the algorithm a new vertex is added to the polygonal line and the positions of the vertices are updated so that they minimize a penalized squared distance criterion simulation results demonstrate that the new algorithm compares favorably with previous methods both in terms of performance and computational complexity and is more robust to varying data models

local e factors and characters of gl 2 introduction the representation theory of gl 2 k for k a nonarchimedean local field is somewhat more complicated than the corresponding theory for gl 2 r in particular the formulas for the characters of discrete series representation of gl 2 r have a concise closed form while the published character formulas for supercuspidal representations of gl 2 k occupy several pages 10 14 the purpose of this paper is to give a concise formula for the character of certain infinite dimensional representations of gl 2 k the representations considered are admissible in the sense that the stabilizer of a vector in the representation space is open in gl 2 k and the subspace stabilized by an open compact subgroup of gl 2 k is finite dimensional irreducible admissible representations ir of gl 2 k satisfy schur s lemma so that the center k of gl 2 k acts by multiplication by a quasicharacter there exists a locally constant character function ch defined on a dense open subgroup of gl 2 k which determines the isomorphism class of ir 7 section 7 jacquet and langlands 7 section 2 introduced factors e 10 x attached to the twist of ir by quasicharacters x of k which describe the action of r in a specific model for each separable quadratic extension l of k there exists an irreducible admissible representation bcl k 1r of gl 2 l the base change of ir to l 9 section 2 this representation of gl 2 l enters into the expression for the character the main result of this paper is the following formula recall that conjugacy classes of nonsplit cartan subgroups are parameterized by quadratic separable field extensions l of k via embeddings of l in gl 2 k

quasiconformal homeomorphisms and dynamics i solution of the fatou julia problem on wandering domains on demontre un resultat dynamique sur l iteration d une application analytique complexe sur la sphere de riemann

multiplicity one theorems for fourier jacobi models for every genuine irreducible admissible smooth representation pi of the metaplectic group widetilde rm sp 2n over a p adic field and every smooth oscillator representation omega psi of widetilde rm sp 2n we prove that the tensor product pi otimes omega psi is multiplicity free as a smooth representation of the symplectic group rm sp 2n similar results are proved for general linear groups and unitary groups

leading coefficient problem for polynomial like iterative equations abstract because of difficulties in applying fixed point theorems most of known results on the polynomial like iterative equation j 1 n j f j x f x were given by assuming 1 0 and the existence of solutions under the most natural assumption n 0 is an interesting problem called leading coefficient problem for this problem locally expansive invertible c 1 solutions are obtained in the expansive case and the non hyperbolic case in w zhang on existence for polynomial like iterative equations results math 45 2004 185 194 and c 0 increasing solutions are constructed in b xu w zhang construction of continuous solutions and stability for the polynomial like iterative equation j math anal appl 325 2007 1160 1170 in this paper we discuss c 1 solutions for more combinations between expansive mappings and contractive ones and combinations between increasing mappings and decreasing ones

diophantine conditions imply critical points on the boundaries of siegel disks of polynomials let f be a polynomial map of the riemann sphere of degree at least two we prove that if f has a siegel disk g on which the rotation number satisfies a diophantine condition then either the boundary b of g contains a critical point or b is a lakes of wada indecomposable continuum with one of the lakes containing a critical point consequently if the boundary b of g has only 2 complementary domains then b contains a critical point we also show without any assumption on the rotation number that each proper nondegenerate subcontinuum of the boundary b of g is tree like and any other bounded complementary domain of b is a preperiodic component of the grand orbit of g finally we establish some conditions under which b contains no periodic point

discrete q derivatives and symmetries of q difference equations in this paper we extend the umbral calculus developed to deal with difference equations on uniform lattices to q difference equations we show that many properties considered for shift invariant difference operators satisfying the umbral calculus can be implemented to the case of the q difference operators this q umbral calculus can be used to provide solutions to linear q difference equations and q differential delay equations to illustrate the method we will apply the obtained results to the construction of symmetry solutions for the q heat equation

the geometry of julia sets the long term analysis of dynamical systems inspired the study of the dynamics of families of mappings many of these investigations led to the study of the dynamics of mappings on cantor sets and on intervals julia sets play a critical role in the understanding of the dynamics of families of mappings in this paper we introduce another class of objects called hairy objects which share many properties with the cantor set and the interval they are topologically unique and admit only one embedding in the plane these uniqueness properties explain the regular occurrence of hairy objects in pictures of julia sets hairy objects are ubiquitous hairy arcs will be used to give a complete topological description of the julia sets of many members of the exponential family

discrete derivatives and symmetries of difference equations we show with an example of the discrete heat equation that for any given discrete derivative we can construct a nontrivial leibniz rule suitable for finding the symmetries of discrete equations in this way we obtain a symmetry lie algebra defined in terms of shift operators isomorphic to that of the continuous heat equation

fourth order q difference equation for the first associated of the q classical orthogonal polynomials abstract we derive the fourth order q difference equation satisfied by the first associated of the q classical orthogonal polynomials the coefficients of this equation are given in terms of the polynomials and which appear in the q pearson difference equation dq defining the weight of the q classical orthogonal polynomials inside the q hahn tableau

solutions of equivariance for a polynomial like iterative equation using fixed point theorems we discuss continuous solutions of equivariance for a polynomial like iterative equation on the real line where is a closed subgroup of the general linear group gl r our main results guarantee the existence of solutions with certain kinds of symmetry we show that under restrictive hypotheses similar results can be proved in a higher dimensional case where the symmetry group is a topologically finitely generated subgroup of the group generated by rotations and dilations in n dimensional euclidean space

hypergeometric type q difference equations rodrigues type representation for the second kind solution a rodrigues type representation for the second kind solution of a second order q difference equation of hypergeometric type is given this representation contains some q extensions of integrals related with relevant special functions for these integrals a general recurrence relation which only involves the coefficients of the q difference equation is also presented

small divisor problem for an analytic q difference equation abstract similar to the problem of linearization the small divisor problem also arises in the discussion of invertible analytic solutions of a class of q difference equations in this paper we give the existence of such solutions under the brjuno condition and prove that the equation may not have a nontrivial analytic solution when the brjuno condition is violated these results are applied to discussing a nonlinear iterative equation

analytic solutions of a nonlinear iterative equation near neutral fixed points and poles in this paper existence of analytic solutions of a nonlinear iterative equations is studied when given functions are all analytic and when given functions have poles as well as in many previous works we reduce this problem to finding analytic solutions of a functional equation without iteration of the unknown function f for technical reasons in previous works an indeterminate constant related to the eigenvalue of the linearized f at its fixed point o is required to fulfill the diophantine condition that o is an irrationally neutral fixed point of f in this paper the case of rationally neutral fixed points is also discussed where the diophantine condition is not required

quantum symmetries of q difference equations a general method is presented to determine the symmetry operators of linear q difference equations it is applied to q difference analogs of the helmoltz heat and wave equations in diverse dimensions the symmetries are found to generate q deformations of classical lie algebras the method of separation of variables is used to obtain solutions which are seen to involve many basic special functions this allows the derivation of various identities and formulas for these q functions

decreasing solutions and convex solutions of the polynomial like iterative equation abstract in this paper we prove the existence and uniqueness of decreasing solutions for the polynomial like iterative equation so as to answer problem 2 in j zhang l yang w zhang some advances on functional equations adv math china 24 1995 385 405 or problem 3 in w zhang j a baker continuous solutions of a polynomial like iterative equation with variable coefficients ann polon math 73 2000 29 36 furthermore we completely investigate increasing convex or concave solutions and decreasing convex or concave solutions of this equation so that the results obtained in w zhang k nikodem b xu convex solutions of polynomial like iterative equations j math anal appl 315 2006 29 40 are improved

regularity of solutions for the navier stokes system of incompressible flows on a polygon abstract we are concerned with singularities and regularities of solutions for the navier stokes system of incompressible flows on a polygonal domain with a concave vertex we subtract the corner singularities by the stokes operator from the solution velocity and pressure functions of the system it is shown that the stress intensity factors are functions of time variable belong to a fractional sobolev space on the time interval and can be expressed in terms of given data an increased regularity for the remainder is obtained

uniformization of attracting basins for exponential maps soit la famille exponentielle complexe e z exp z il y a un ensemble ouvert de valeurs de dans le plan complexe pour lesquelles e a un point fixe attractif unique est le bassin d attraction pour ces valeurs on considere l uniformisation de et on etudie la dynamique induite par sur le disque unite

construction of continuous solutions and stability for the polynomial like iterative equation abstract most of known results such as existence uniqueness and stability for polynomial like iterative equations were given under the assumption that the coefficient of the first order iteration term does not vanish the existence with a non zero leading coefficient was therefore raised as an open problem it was positively answered for local c 1 solutions later in this paper this problem is answered further by constructing c 0 solutions moreover we discuss the stability of those c 0 solutions which consequently implies a result of the stability for iterative roots

extremal properties of principal curves in the plane principal curves were introduced to formalize the notion of a curve passing through the middle of a dataset vaguely speaking a curve is said to pass through the middle of a dataset if every point on the curve is the average of the observations projecting onto it this idea can be made precise by defining principal curves for probability densities in this paper we study principal curves in the plane like linear principal components principal curves are critical points of the expected squared distance from the data however the largest and smallest principal components are extrema of the distance whereas all principal curves are saddle points this explains why cross validation does not appear to be a viable method for choosing the complexity of principal curve estimates

umbral calculus difference equations and the discrete schrodinger equation in this paper we discuss umbral calculus as a method of systematically discretizing linear differential equations while preserving their point symmetries as well as generalized symmetries the method is then applied to the schrodinger equation in order to obtain a realization of nonrelativistic quantum mechanics in discrete space time in this approach a quantum system on a lattice has a symmetry algebra isomorphic to that of the continuous case moreover systems that are integrable superintegrable or exactly solvable preserve these properties in the discrete case

non monotonic solutions and continuously differentiable solutions of conjugacy equations in this paper solutions of conjugacy equation f f x g f x for a strictly decreasing continuous given function f and a continuous given function g maybe non monotonic are constructed by piecewise defining we determine the conditions for piecewise continuously differentiable solutions of conjugacy equations with a strictly decreasing continuously differentiable given function f and a continuously differentiable given function g finally the recursive algorithm is implemented in matlab software and two examples are respectively presented for a non monotonic solution and a continuously differentiable one

analytic solutions of a general nonlinear functional equations near resonance existence of analytic solutions of a general class of nonlinear functional equations is discussed this general class includes some specific functional equations studied recently moreover we can generalize this problem to finding analytic solutions of a general class of iterative equations

existence of analytic solutions of an iterative functional equation abstract this paper is concern analytic solutions of an iterative functional equation of the form f p z q f z h f z z c byconstructing a convergent power series solution of an auxiliary equation g 2 z p z h g z p z z c analytic solutions of the original equation are obtained we discuss not only these appeared in the auxiliary equation at the hyperbolic case 0

continuous symmetries of difference equations lie group theory was originally created more than 100 years ago as a tool for solving ordinary and partial differential equations in this article we review the results of a much more recent program the use of lie groups to study difference equations we show that the mismatch between continuous symmetries and discrete equations can be resolved in at least two manners one is to use generalized symmetries acting on solutions of difference equations but leaving the lattice invariant the other is to restrict them to point symmetries but to allow them to also transform the lattice

locally defined principal curves and surfaces principal curves are defined as self consistent smooth curves passing through the middle of the data and they have been used in many applications of machine learning as a generalization dimensionality reduction and a feature extraction tool we redefine principal curves and surfaces in terms of the gradient and the hessian of the probability density estimate this provides a geometric understanding of the principal curves and surfaces as well as a unifying view for clustering principal curve fitting and manifold learning by regarding those as principal manifolds of different intrinsic dimensionalities the theory does not impose any particular density estimation method can be used with any density estimator that gives continuous first and second derivatives therefore we first present our principal curve surface definition without assuming any particular density estimation method afterwards we develop practical algorithms for the commonly used kernel density estimation kde and gaussian mixture models gmm results of these algorithms are presented in notional data sets as well as real applications with comparisons to other approaches in the principal curve literature all in all we present a novel theoretical understanding of principal curves and surfaces practical algorithms as general purpose machine learning tools and applications of these algorithms to several practical problems

a unified model for probabilistic principal surfaces principal curves and surfaces are nonlinear generalizations of principal components and subspaces respectively they can provide insightful summary of high dimensional data not typically attainable by classical linear methods solutions to several problems such as proof of existence and convergence faced by the original principal curve formulation have been proposed in the past few years nevertheless these solutions are not generally extensible to principal surfaces the mere computation of which presents a formidable obstacle consequently relatively few studies of principal surfaces are available we previously 2000 proposed the probabilistic principal surface pps to address a number of issues associated with current principal surface algorithms pps uses a manifold oriented covariance noise model based on the generative topographical mapping gtm which can be viewed as a parametric formulation of kohonen s self organizing map building on the pps we introduce a unified covariance model that implements pps 0 1 by varying the clamping parameter spl alpha then we comprehensively evaluate the empirical performance of pps gtm and the manifold aligned gtm on three popular benchmark data sets it is shown in two different comparisons that the pps outperforms the gtm under identical parameter settings convergence of the pps is found to be identical to that of the gtm and the computational overhead incurred by the pps decreases to 40 percent or less for more complex manifolds these results show that the generalized pps provides a flexible and effective way of obtaining principal surfaces

twisted conformal algebra so 4 2 a new twisted deformation uz so 4 2 of the conformal algebra of the 3 1 dimensional minkowskian spacetime is presented this construction is provided by a classical r matrix spanned by ten weyl poincare generators which generalizes non standard quantum deformations previously obtained for so 2 2 and so 3 2 however by introducing a conformal null plane basis it is found that the twist can indeed be supported by an eight dimensional carrier subalgebra by construction the weyl poincare subalgebra remains as a hopf subalgebra after deformation non relativistic limits of uz so 4 2 are shown to be well defined and they give rise to new twisted conformal algebras of galilean and carroll spacetimes furthermore a difference differential massless klein gordon or wave equation with twisted conformal symmetry is constructed through deformed momenta and position operators the deformation parameter is interpreted as the lattice step on a uniform minkowskian spacetime lattice discretized along two basic null plane directions

q exponential and q gamma functions i q exponential functionsa some properties of the q exponential functions standard and symmetric are investigated for general complex nonzero q in particular for q 1 the asymptotic behavior of the q exponential functions of a complex argument z is studied as z approaches through an infinite geometric sequence zn qn n z of complex numbers with common ratio q a generalized q exponential function is defined and some of its properties are also discussed for q 0 including the singular case of q a root of unity or more generally q 1

building blocks for quadratic julia sets we obtain results on the structure of the julia set of a quadratic polynomial p with an irrationally indifferent fixed point zo in the iterative dynamics of p in the cremer point case under the assumption that the julia set is a decomposable continuum we obtain a building block structure theorem for the corresponding julia set j j p there exists a nowhere dense subcontinuum b c j such that p b b b is the union of the impressions of a minimally invariant cantor set a of external rays b contains the critical point and b contains both the cremer point zo and its preimage in the siegel disk case under the assumption that no impression of an external ray contains the boundary of the siegel disk we obtain a similar result in this case b contains the boundary of the siegel disk properly if the critical point is not in the boundary and b contains no periodic points in both cases the julia set j is the closure of a skeleton s which is the increasing union of countably many copies of the building block b joined along preimages of copies of a critical continuum c containing the critical point in addition we prove that if p is any polynomial of degree d 2 with a siegel disk which contains no critical point on its boundary then the julia set j p is not locally connected we also observe that all quadratic polynomials which have an irrationally indifferent fixed point and a locally connected julia set have homeomorphic julia sets

stability of functional equations in single variable abstract this paper discusses hyers ulam stability for functional equations in single variable including the forms of linear functional equation nonlinear functional equation and iterative equation surveying many known and related results we clarify the relations between hyers ulam stability and other senses of stability such as iterative stability continuous dependence and robust stability which are used for functional equations applying results of nonlinear functional equations we give the hyers ulam stability of bottcher s equation we also prove a general result of hyers ulam stability for iterative equations

lie group formalism for difference equations the methods of lie group analysis of differential equations are generalized so as to provide an infinitesimal formalism for calculating symmetries of difference equations several examples are analysed one of them being a nonlinear difference equation for the linear equations the symmetry algebra of the discrete equation is found to be isomorphic to that of its continuous limit

hypergeometric type differential equations second kind solutions and related integrals a rodrigues type representation for the second kind solutions of a second order differential equation of hypergeometric type is given this representation contains some integrals related with relevant special functions for these integrals a general recurrence relation which only involves the coefficients of the differential equation is also presented finally an extension of the rodrigues type representation for the second solution of a second order difference equation of hypergeometric type is indicated

canonical commutation relation preserving maps we study maps preserving the heisenberg commutation relation ab ba 1 we find a one parameter deformation of the standard realization of the above algebra in terms of a coordinate and its dual derivative it involves a non local coordinate operator while the dual derivative is just the jackson finite difference operator substitution of this realization into any differential operator involving x and d dx results in an isospectral deformation of a continuous differential operator into a finite difference one we extend our results to the deformed heisenberg algebra ab qba 1 as an example of potential applications various deformations of the hahn polynomials are briefly discussed

analytic solutions of a functional equation for invariant curves abstract an iterative functional equation is deduced by c t ng and w zhang 1997 j differ equations appl 3 147 168 from the problem of invariant curves in this paper its analytic solutions are discussed by locally reducing the equation to another functional equation without iteration and by constructing solutions in uniformly convergent power series for the latter equation

existence and uniqueness of solutions in h1 δ of a general class of non linear functional equations abstract a functional analytic method is used to prove the existence and the uniqueness of a solution in the banach space h1 of a general class of non linear functional equations this general class includes some specific functional equations studied recently our results simplify and improve the existing results for these specific equations moreover for one of them we give an answer to an open problem

extensions of some results of p humbert on bezout s identity for classical orthogonal polynomials in this paper the bezout s identity is analyzed in the context of classical orthogonal polynomials solution of a second order differential equation of hypergeometric type differential equations relation with the starting family as well as recurrence relations and explicit representations are given for the bezout s pair extensions to classical orthogonal polynomials of a discrete variable and their q analogues are also presented applications of these results for the representation of the second kind functions are given