trigonometry of complex hermitian type homogeneous symmetric spaces this paper contains a thorough study of the trigonometry of the homogeneous symmetric spaces in the cayley klein dickson family of spaces of complex hermitian type and rank 1 the complex hermitian elliptic pn su n 1 u 1 su n and hyperbolic hn su n 1 u 1 su n spaces their analogues with indefinite hermitian metric su p 1 q u 1 su p q and the non compact symmetric space sl n 1 so 1 1 sl n are the generic members in this family the remaining spaces are some contractions of the former the method encapsulates trigonometry for this whole family of spaces into a single basic trigonometric group equation and has universality and self duality as its distinctive traits all previously known results on the trigonometry of pn and hn follow as particular cases of our general equations the following topics are covered rather explicitly 0 description of the complete cayley klein dickson family of rank 1 spaces of complex type 1 derivation of the single basic group trigonometric equation 2 translation to the basic complex hermitian cosine sine and dual cosine laws 3 comprehensive exploration of the bestiarium of complex hermitian trigonometric equations 4 uncovering of a cartan sector of hermitian trigonometry related to triangle symplectic area and co area 5 existence conditions for a triangle in these spaces as inequalities and 6 restriction to the two special cases of complex collinear and purely real triangles the physical quantum space of states of any quantum system belongs as the complex hermitian space member to this parametrized family hence its trigonometry appears as a rather particular case of the equations we obtain

superintegrable systems on the two dimensional sphere s2 and the hyperbolic plane h2 the existence of superintegrable systems with n 2 degrees of freedom possessing three independent globally defined constants of motion which are quadratic in the velocities is studied on the two dimensional sphere s2 and on the hyperbolic plane h2 the approach used is based on enforcing the conditions for the existence of two independent integrals further than the energy this is done in a way which allows us to discuss at once the cases of the sphere s2 and the hyperbolical plane h2 by considering the curvature as a parameter different superintegrable potentials are obtained as the solutions of certain systems of two dependent second order partial differential equations the euclidean results are directly recovered for 0 and the superintegrable potentials on either the standard unit sphere radius r 1 or the unit lobachewski plane radius r 1 appear as the particular values of the dependent superintegrable potentials for the values 1 and 1 some new superintegrable potentials are f

derivation of lagrangian and hermitian shape functions for quadrilateral elements this paper introduces a general theory for the derivation of the shape functions for the quadrilateral family of finite elements the first section deals with the lagrangian shape functions for the cases of uniform and boundary described elements two basic procedures are introduced the first by linear combinations of side interpolations and the second by superposition the remainder of the paper introduces a theory for the general uniform hermitian element of any order details for quadrilateral elements with first order derivatives are explained all of the shape functions presented here were derived in the interval 0 1 the shape functions developed by such an engineering approach have been used successfully in the absea finite element system of cranfield institute of technology

rotopulsators of the curved n body problem we consider the n body problem in spaces of constant curvature and study its rotopulsators i e solutions for which the configuration of the bodies rotates and changes size during the motion rotopulsators fall naturally into five groups positive elliptic positive elliptic elliptic negative elliptic negative hyperbolic and negative elliptic hyperbolic depending on the nature and number of their rotations and on whether they occur in spaces of positive or negative curvature after obtaining existence criteria for each type of rotopulsator we derive their conservation laws we further deal with the existence and uniqueness of some classes of rotopulsators in the 2 and 3 body case and prove two general results about the qualitative behaviour of rotopulsators more precisely for positive curvature we show that there is no foliation of the 3 sphere with clifford tori such that the motion of each body is confined to some clifford torus for negative curvature a similar result is proved relative to foliations of the hyperbolic 3 sphere with hyperbolic cylinders

block regularization of the kepler problem on surfaces of revolution with positive constant curvature abstract we consider the kepler problem on surfaces of revolution that are homeomorphic to s 2 and have constant gaussian curvature we show that the system is maximally superintegrable finding constants of motion that generalize the runge lenz vector then using such first integrals we determine the class of surfaces that lead to block regularizable collision singularities in particular we show that the singularities are always regularizable if the surfaces are spherical orbifolds of revolution with constant curvature

superintegrability in a two dimensional space of nonconstant curvature a hamiltonian with two degrees of freedom is said to be superintegrable if it admits three functionally independent integrals of the motion this property has been extensively studied in the case of two dimensional spaces of constant possibly zero curvature when all the independent integrals are either quadratic or linear in the canonical momenta in this article the first steps are taken to solve the problem of superintegrability of this type on an arbitrary curved manifold in two dimensions this is done by examining in detail one of the spaces of revolution found by g koenigs we determine that there are essentially three distinct potentials which when added to the free hamiltonian of this space have this type of superintegrability separation of variables for the associated hamilton jacobi and schrodinger equations is discussed the classical and quantum quadratic algebras associated with each of these potentials are determined

the master integrable two dimensional system with a quartic second integral in this paper we construct a new integrable natural conservative mechanical system admitting a second invariant of the fourth degree in velocities this system is quite general and involves 21 parameters we also show that all systems with a quartic integral known up to date can be obtained from it as special cases by a relevant choice of the values of parameters the results are applied to problems of particle and rigid body dynamics new integrable cases are obtained as special versions of the new system these cases include motions in a plane lobachevsky plane sphere and surfaces of variable curvature they also include generalizations of the classical cases of kovalevskaya chaplygin and goriatchev with the addition of certain singular terms to the potential

trigonometry of spacetimes a new self dual approach to a curvature signature in dependent trigonometry a new method to obtain trigonometry for the real spaces of constant curvature and metric of any even degenerate signature is presented the method could be described as curvature signature in dependent trigonometry and encapsulates trigonometry for all these spaces into a single basic trigonometric group equation this brings to its logical end the idea of an absolute trigonometry and provides equations which hold true for the nine two dimensional spaces of constant curvature and any signature this family of spaces includes both relativistic and non relativistic spacetimes therefore a complete discussion of trigonometry in the six de sitter minkowskian newton hooke and galilean spacetimes follow as particular instances of the general approach distinctive traits of the method are universality and self duality every equation is meaningful for the nine spaces at once and displays invariance explicitly under a duality transformation relating the nine spaces amongst themselves these basic structural properties allow a complete study of trigonometry and in fact any equation previously known for the three classical riemannian spaces also has a version for the remaining six spacetimes in most cases these equations are new

the super separability of the three body inverse square calogero system the geometrical theory of the variable separation for the hamilton jacobi equation is applied to the classical three body inverse square calogero system it is proved that this system is separable in infinitely many inequivalent ways related to five different kinds of separable webs in the euclidean three space and the corresponding systems of independent first integrals in involution are computed

numerical linear algebra for reconstruction inverse problems our goal in this paper is to discuss various issues we have encountered in trying to find and implement efficient solvers for a boundary integral equation bie formulation of an iterative method for solving a reconstruction problem we survey some methods from numerical linear algebra which are relevant for the solution of this class of inverse problems we motivate the use of our constructing algorithm discuss its implementation and mention the use of preconditioned krylov methods

quadratic poisson algebras of two dimensional classical superintegrable systems and quadratic associative algebras of quantum superintegrable systems the integrals of motion of the classical two dimensional superintegrable systems with quadratic integrals of motion close in a restrained quadratic poisson algebra whose the general form is investigated each classical superintegrable problem has a quantum counterpart a quantum superintegrable system the quadratic poisson algebra is deformed into a quantum associative algebra the finite dimensional representations of this algebra are calculated by using a deformed parafermion oscillator technique it is shown that the finite dimensional representations of the quadratic algebra are determined by the energy eigenvalues of the superintegrable system the calculation of energy eigenvalues is reduced to the solution of algebraic equations which are universal that is for all two dimensional superintegrable systems with quadratic integrals of motion

completeness of superintegrability in two dimensional constant curvature spaces we classify the hamiltonians h px2 py2 v x y of all classical superintegrable systems in two dimensional complex euclidean space with two additional second order constants of the motion we similarly classify the superintegrable hamiltonians h j12 j22 j32 v x y z on the complex two sphere where x2 y2 z2 1 this is achieved in all generality using properties of the complex euclidean group and the complex orthogonal group

a local boundary integral equation method for potential problems boundary integral equation boundary element methods have the advantage over other commonly used numerical methods that they do not require values of the unknowns at points within the solution domain to be computed further benefits would be obtained if attention could be confined to information at one small part of the boundary the particular region of interest in a given problem a local boundary integral equation method based on a taylor series expansion of the unknown function is developed to do this for two dimensional potential problems governed by laplace s equation very accurate local values of the function and its derivatives can be obtained the method should find particular application in the efficient refinement of approximate solutions obtained by other numerical techniques

homographic solutions of the curved 3 body problem abstract in the 2 dimensional curved 3 body problem we prove the existence of lagrangian and eulerian homographic orbits and provide their complete classification in the case of equal masses we also show that the only non homothetic hyperbolic eulerian solutions are the hyperbolic eulerian relative equilibria a result that proves their instability

superintegrability of the calogero moser system constants of motion master symmetries and time dependent symmetries the classical n dimensional calogero moser system is a maximally superintegrable system endowed with a rich variety of symmetries and constants of motion in the first part of the article some properties related with the existence of several families of constants of motion are analyzed in the second part the master symmetries and the time dependent symmetries of this system are studied

a new 2d integrable system with a quartic second invariant the construction of all 2d lagrangian systems which admit besides the energy another integral of motion that is quartic in velocities was reduced in our previous article yehia 2006 j phys a math gen 39 5807 24 to a single nonlinear pde in this paper we introduce a new solution of this equation leading to a new integrable system with a quartic integral which involves 16 free parameters a special case of the new system admits interpretation in a problem of rigid body dynamics it gives a new integrable variation of the cases due to kowalevski 1889 acta math 12 177 232 chaplygin 1903 tr otdel phys nauk obsh liub estest 11 7 10 goriatchev 1916 varshav univ izv 1 13 and yehia 2006 j phys a math gen 39 5807 24

a meshless method for solving the nonlinear inverse cauchy problem of elliptic type equation in a doubly connected domain abstract in the paper a nonlinear inverse cauchy problem of nonlinear elliptic type partial differential equation in an arbitrary doubly connected plane domain is solved using a novel meshless numerical method the unknown dirichlet data on an inner boundary are recovered by over specifying the cauchy data on an outer boundary a homogenization function is derived to annihilate the cauchy data on the outer boundary and then a homogenization technique generates a transformed equation in terms of a transformed variable whose outer cauchy boundary conditions are homogeneous when the numerical solution is expanded by a sequence of boundary functions which automatically satisfy the homogeneous cauchy boundary conditions on the outer boundary we can solve the transformed equation by the domain type meshless collocation method for the nonlinear inverse cauchy problems we require to iteratively solve the linear systems with the right hand sides varying per iteration step the accuracy and robustness of the homogenization boundary function method hbfm are examined through seven numerical examples where we compare the exact dirichlet data on the inner boundary to the ones recovered by the hbfm under a large noisy disturbance

response to comment on central potentials on spaces of constant curvature the kepler problem on the two dimensional sphere s2 and the hyperbolic plane h2 j math phys 46 052702 2005

quantum structure of the motion groups of the two dimensional cayley klein geometries a simultaneous and global scheme of quantum deformation is defined for the set of algebras corresponding to the groups of motions of the two dimensional cayley klein geometries their central extensions are also considered under this unified pattern in both cases some fundamental properties characterizing the classical ck geometries as the existence of a set of commuting involutions contractions and dualities relationships remain in the quantum version

on the superintegrability of calogero moser sutherland model the superintegrability of calogero moser sutherland hyperbolic model and nonsuperintegrability of trigonometric one are shown

integrable potentials on spaces with curvature from quantum groups a family of classical integrable systems defined on a deformation of the two dimensional sphere hyperbolic and anti de sitter spaces is constructed through hamiltonians defined on the non standard quantum deformation of a sl 2 poisson coalgebra all these spaces have a non constant curvature that depends on the deformation parameter z as particular cases the analogues of the harmonic oscillator and kepler coulomb potentials on such spaces are proposed another deformed hamiltonian is also shown to provide superintegrable systems on the usual sphere hyperbolic and anti de sitter spaces with a constant curvature that exactly coincides with z according to each specific space the resulting potential is interpreted as the superposition of a central harmonic oscillator with either two more oscillators or centrifugal barriers the non deformed limit z 0 of all these hamiltonians can then be regarded as the zero curvature limit contraction which leads to the corresponding super integrable systems on the flat euclidean and minkowskian spaces

hermitian cubic boundary elements for two dimensional potential problems a hermite interpolation based formulation is presented for the boundary element analysis of two dimensional potential problems two three noded hermitian cubic elements hce are introduced for the modelling of corners or points with non unique tangents on the boundary these elements along with the usual two noded hce are used in numerical examples the results obtained show that faster convergence can be achieved using hce compared with using lagrange interpolation type quadratic elements qe for about the same amount of computing resources

lagrangian formalism for nonlinear second order riccati systems one dimensional integrability and two dimensional superintegrability the existence of a lagrangian description for the second order riccati equation is analyzed and the results are applied to the study of two different nonlinear systems both related with the generalized riccati equation the lagrangians are non natural and the forces are not derivable from a potential the constant value e of a preserved energy function can be used as an appropriate parameter for characterizing the behavior of the solutions of these two systems in the second part the existence of two dimensional versions endowed with superintegrability is proved the explicit expressions of the additional integrals are obtained in both cases finally it is proved that the orbits of the second system that represents a nonlinear oscillator can be considered as nonlinear lissajous figures

on harmonic oscillators on the two dimensional sphere s2 and the hyperbolic plane h2 ii two harmonic oscillators isotropic and nonisotropic 2 1 are studied on the two dimensional sphere s2 and the hyperbolic plane h2 both systems are integrable and super integrable with constants of motion quadratic in the momenta these properties are shown to derive from a complex factorization for the constants of motion which holds for arbitrary values of the curvature and the dynamics of the euclidean harmonic 1 1 and 2 1 oscillators is directly recovered for 0 the harmonic oscillators on either the standard unit sphere radius r 1 or the unit lobachewski plane radius r 1 appear as the particular values of the dependent potentials for the values 1 and 1 finally a particular potential is proposed for representing the general spherical hyperbolic n 1 anisotropic harmonic oscillator on a two dimensional manifold of constant curvature

superintegrable n 2 systems quadratic constants of motion and potentials of drach the properties of superintegrable systems in two degrees of freedom possessing three independent globally defined constants of motion are studied using as an approach the existence of hidden symmetries and the generalized noether s theorem the potentials are obtained as solution of a system of two partial differential equations first the case of standard lagrangians is studied and then the method is applied to the case of lagrangians with a pseudo euclidean kinetic term finally the results are related with other approaches and with a family of potentials admitting a second integral of motion cubic in the velocities obtained by drach

quantum 2 1 kinematical algebras a global approach in this paper we give an approach to quantum deformations of the 2 1 kinematical lie algebras within a scheme that simultaneously describes all groups of motions of classical geometries in n 3 dimensions we cover at once all the kinematical geometries including the quantum versions of inonu wigner contractions which are defined in a natural way and relate q deformations as expected we thus obtain some q deformations previously known for the three dimensional euclidean and 2 1 poincare algebras and also some new q deformations for these and other kinematical algebras such as the 2 1 de sitter galilei and newton hooke algebras

the quantum harmonic oscillator on the sphere and the hyperbolic plane κ dependent formalism polar coordinates and hypergeometric functions a nonlinear model representing the quantum harmonic oscillator on the sphere and the hyperbolic plane is solved in polar coordinates r by making use of a curvature dependent formalism the curvature is considered as a parameter and then the radial schrodinger equation becomes a dependent gauss hypergeometric equation the energy spectrum and the wave functions are exactly obtained in both the sphere s2 0 and the hyperbolic plane h2 0 a comparative study between the spherical and the hyperbolic quantum results is presented

numerical method for solving a class of nonlinear elliptic inverse problems this paper discusses a method to solve a family of nonlinear inverse problems with cauchy conditions on a part of the boundary and no condition at all on another part an iterative boundary element procedure is proposed the scheme uses a dynamically estimated relaxation parameter on the under specified boundary various types of convergence boundary condition formulations and effects of added small perturbations into the input data are investigated the numerical results show that the method produces a stable reasonably approximate solution

exact solvability of superintegrable systems it is shown that all four superintegrable quantum systems on the euclidean plane possess the same underlying hidden algebra sl 3 the gauge rotated hamiltonians as well as their integrals of motion once rewritten in appropriate coordinates preserve a flag of polynomials this flag corresponds to highest weight finite dimensional representations of the sl 3 algebra realized by first order differential operators