The Ramanujan master theorem and its implications for special functions

We study a number of possible extensions of the Ramanujan master theorem, which is formulated here by using methods of Umbral nature. We discuss the implications of the procedure for the theory of special functions, like the derivation of formulae concerning the integrals of products of families of Bessel functions and the successive derivatives of Bessel type functions. We stress also that the procedure we propose allows a unified treatment of many problems appearing in applications, which can formally be reduced to the evaluation of exponential- or Gaussian-like integrals.Comment: 12 page

A Generalization of Chetaev's Principle for a Class of Higher Order Non-holonomic Constraints

The constraint distribution in non-holonomic mechanics has a double role. On one hand, it is a kinematic constraint, that is, it is a restriction on the motion itself. On the other hand, it is also a restriction on the allowed variations when using D'Alembert's Principle to derive the equations of motion. We will show that many systems of physical interest where D'Alembert's Principle does not apply can be conveniently modeled within the general idea of the Principle of Virtual Work by the introduction of both kinematic constraints and variational constraints as being independent entities. This includes, for example, elastic rolling bodies and pneumatic tires. Also, D'Alembert's Principle and Chetaev's Principle fall into this scheme. We emphasize the geometric point of view, avoiding the use of local coordinates, which is the appropriate setting for dealing with questions of global nature, like reduction.Comment: 27 pages. Journal of Mathematical Physics (to zappear

The helium atom in a strong magnetic field

We investigate the electronic structure of the helium atom in a magnetic field b etween B=0 and B=100a.u. The atom is treated as a nonrelativistic system with two interactin g electrons and a fixed nucleus. Scaling laws are provided connecting the fixed-nucleus Hamiltonia n to the one for the case of finite nuclear mass. Respecting the symmetries of the electronic Ham iltonian in the presence of a magnetic field, we represent this Hamiltonian as a matrix with res pect to a two-particle basis composed of one-particle states of a Gaussian basis set. The corresponding generalized eigenvalue problem is solved numerically, providing in the present paper results for vanish ing magnetic quantum number M=0 and even or odd z-parity, each for both singlet and triplet spin symmetry. Total electronic energies of the ground state and the first few excitations in each su bspace as well as their one-electron ionization energies are presented as a function of the magnetic fie ld, and their behaviour is discussed. Energy values for electromagnetic transitions within the M=0 sub space are shown, and a complete table of wavelengths at all the detected stationary points with respect to their field dependence is given, thereby providing a basis for a comparison with observed ab sorption spectra of magnetic white dwarfs.Comment: 21 pages, 4 Figures, acc.f.publ.in J.Phys.

Lattice Green functions in all dimensions

We give a systematic treatment of lattice Green functions (LGF) on the $d$-dimensional diamond, simple cubic, body-centred cubic and face-centred cubic lattices for arbitrary dimensionality $d \ge 2$ for the first three lattices, and for $2 \le d \le 5$ for the hyper-fcc lattice. We show that there is a close connection between the LGF of the $d$-dimensional hypercubic lattice and that of the $(d-1)$-dimensional diamond lattice. We give constant-term formulations of LGFs for all lattices and dimensions. Through a still under-developed connection with Mahler measures, we point out an unexpected connection between the coefficients of the s.c., b.c.c. and diamond LGFs and some Ramanujan-type formulae for $1/\pi.$Comment: 30 page

New solutions of Heun general equation

We show that in four particular cases the derivative of the solution of Heun general equation can be expressed in terms of a solution to another Heun equation. Starting from this property, we use the Gauss hypergeometric functions to construct series solutions to Heun equation for the mentioned cases. Each of the hypergeometric functions involved has correct singular behavior at only one of the singular points of the equation; the sum, however, has correct behavior

Nuclear Effects on Heavy Boson Production at RHIC and LHC

We predict W and Z transverse momentum distributions from proton-proton and nuclear collisions at RHIC and LHC. A resummation formalism with power corrections to the renormalization group equations is used. The dependence of the resummed QCD results on the non-perturbative input is very weak for the systems considered. Shadowing effects are discussed and found to be unimportant at RHIC, but important for LHC. We study the enhancement of power corrections due to multiple scattering in nuclear collisions and numerically illustrate the weak effects of the dependence on the nuclear mass.Comment: 21 pages, 11 figure

On Virtual Displacement and Virtual Work in Lagrangian Dynamics

The confusion and ambiguity encountered by students, in understanding virtual displacement and virtual work, is discussed in this article. A definition of virtual displacement is presented that allows one to express them explicitly for holonomic (velocity independent), non-holonomic (velocity dependent), scleronomous (time independent) and rheonomous (time dependent) constraints. It is observed that for holonomic, scleronomous constraints, the virtual displacements are the displacements allowed by the constraints. However, this is not so for a general class of constraints. For simple physical systems, it is shown that, the work done by the constraint forces on virtual displacements is zero. This motivates Lagrange's extension of d'Alembert's principle to system of particles in constrained motion. However a similar zero work principle does not hold for the allowed displacements. It is also demonstrated that d'Alembert's principle of zero virtual work is necessary for the solvability of a constrained mechanical problem. We identify this special class of constraints, physically realized and solvable, as {\it the ideal constraints}. The concept of virtual displacement and the principle of zero virtual work by constraint forces are central to both Lagrange's method of undetermined multipliers, and Lagrange's equations in generalized coordinates.Comment: 12 pages, 10 figures. This article is based on an earlier article physics/0410123. It includes new figures, equations and logical conten

A transient network of telechelic polymers and microspheres : structure and rheology

We study the structure and dynamics of a transient network composed of droplets of microemulsion connected by telechelic polymers. The polymer induces a bridging attraction between droplets without changing their shape. A viscoelastic behaviour is induced in the initially liquid solution, characterised in the linear regime by a stretched exponential stress relaxation. We analyse this relaxation in the light of classical theories of transient networks. The role of the elastic reorganisations in the deformed network is emphasized. In the non linear regime, a fast relaxation dynamics is followed by a second one having the same rate as in the linear regime. This behaviour, under step strain experiments, should induce a non monotonic behaviour in the elastic component of the stress under constant shear rate. However, we obtain in this case a singularity in the flow curve very different from the one observed in other systems, that we interpret in terms of fracture behaviour.Comment: 9 pages, 4 figure

Nonparametric Information Geometry

The differential-geometric structure of the set of positive densities on a given measure space has raised the interest of many mathematicians after the discovery by C.R. Rao of the geometric meaning of the Fisher information. Most of the research is focused on parametric statistical models. In series of papers by author and coworkers a particular version of the nonparametric case has been discussed. It consists of a minimalistic structure modeled according the theory of exponential families: given a reference density other densities are represented by the centered log likelihood which is an element of an Orlicz space. This mappings give a system of charts of a Banach manifold. It has been observed that, while the construction is natural, the practical applicability is limited by the technical difficulty to deal with such a class of Banach spaces. It has been suggested recently to replace the exponential function with other functions with similar behavior but polynomial growth at infinity in order to obtain more tractable Banach spaces, e.g. Hilbert spaces. We give first a review of our theory with special emphasis on the specific issues of the infinite dimensional setting. In a second part we discuss two specific topics, differential equations and the metric connection. The position of this line of research with respect to other approaches is briefly discussed.Comment: Submitted for publication in the Proceedings od GSI2013 Aug 28-30 2013 Pari

A matrix recurrence for systems of Clifford algebra-valued orthogonal polynomials

Recently, the authors developed a matrix approach to multivariate polynomial sequences by using methods of Hypercomplex Function Theory ("Matrix representations of a basic polynomial sequence in arbitrary dimension". Comput. Methods Funct. Theory, 12 (2012), no. 2, 371-391). This paper deals with an extension of that approach to a recurrence relation for the construction of a complete system of orthogonal Clifford-algebra valued polynomials of arbitrary degree. At the same time the matrix approach sheds new light on results about systems of Clifford algebra-valued orthogonal polynomials obtained by Guerlebeck, Bock, Lavicka, Delanghe et al. during the last five years. In fact, it allows to prove directly some intrinsic properties of the building blocks essential in the construction process, but not studied so far.Fundação para a Ciência e a Tecnologia (FCT