On the ternary complex analysis and its applications

Previouly a possible extension of the complex number, together with its connected trigonometry was introduced. In this paper we focuss on the simplest case of ternary complex numbers. Then, some types of holomorphicity adapted to the ternary complex numbers and the corresponding results upon integration of differential forms are given. Several physical applications are given, and in particuler one type of holomorphic function gives rise to a new form of stationary magnetic field. The movement of a monopole type object in this field is then studied and shown to be integrable. The monopole scattering in the ternary field is finally studied.Comment: LaTeX 28 page

On hypergeometric series reductions from integral representations, the Kampe de Feriet function, and elsewhere

Single variable hypergeometric functions pFq arise in connection with the power series solution of the Schrodinger equation or in the summation of perturbation expansions in quantum mechanics. For these applications, it is of interest to obtain analytic expressions, and we present the reduction of a number of cases of pFp and p+1F_p, mainly for p=2 and p=3. These and related series have additional applications in quantum and statistical physics and chemistry.Comment: 17 pages, no figure

Multi-Dimensional Hermite Polynomials in Quantum Optics

We study a class of optical circuits with vacuum input states consisting of Gaussian sources without coherent displacements such as down-converters and squeezers, together with detectors and passive interferometry (beam-splitters, polarisation rotations, phase-shifters etc.). We show that the outgoing state leaving the optical circuit can be expressed in terms of so-called multi-dimensional Hermite polynomials and give their recursion and orthogonality relations. We show how quantum teleportation of photon polarisation can be modelled using this description.Comment: 10 pages, submitted to J. Phys. A, removed spurious fil

A massive Feynman integral and some reduction relations for Appell functions

New explicit expressions are derived for the one-loop two-point Feynman integral with arbitrary external momentum and masses $m_1^2$ and $m_2^2$ in D dimensions. The results are given in terms of Appell functions, manifestly symmetric with respect to the masses $m_i^2$. Equating our expressions with previously known results in terms of Gauss hypergeometric functions yields reduction relations for the involved Appell functions that are apparently new mathematical results.Comment: 19 pages. To appear in Journal of Mathematical Physic

Controlling Effect of Geometrically Defined Local Structural Changes on Chaotic Hamiltonian Systems

An effective characterization of chaotic conservative Hamiltonian systems in terms of the curvature associated with a Riemannian metric tensor derived from the structure of the Hamiltonian has been extended to a wide class of potential models of standard form through definition of a conformal metric. The geodesic equations reproduce the Hamilton equations of the original potential model through an inverse map in the tangent space. The second covariant derivative of the geodesic deviation in this space generates a dynamical curvature, resulting in (energy dependent) criteria for unstable behavior different from the usual Lyapunov criteria. We show here that this criterion can be constructively used to modify locally the potential of a chaotic Hamiltonian model in such a way that stable motion is achieved. Since our criterion for instability is local in coordinate space, these results provide a new and minimal method for achieving control of a chaotic system

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

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

Geodesics around Weyl-Bach's Ring Solution

We explore some of the gravitational features of a uniform ring both in the Newtonian potential theory and in General Relativity. We use a spacetime associated to a Weyl static solution of the vacuum Einstein's equations with ring like singularity. The Newtonian motion for a test particle in the gravitational field of the ring is studied and compared with the corresponding geodesic motion in the given spacetime. We have found a relativistic peculiar attraction: free falling particle geodesics are lead to the inner rim but never hit the ring.Comment: 8 figures, 14 pages. LaTeX w/ subfigure, graphic

Projective dynamics and first integrals

We present the theory of tensors with Young tableau symmetry as an efficient computational tool in dealing with the polynomial first integrals of a natural system in classical mechanics. We relate a special kind of such first integrals, already studied by Lundmark, to Beltrami's theorem about projectively flat Riemannian manifolds. We set the ground for a new and simple theory of the integrable systems having only quadratic first integrals. This theory begins with two centered quadrics related by central projection, each quadric being a model of a space of constant curvature. Finally, we present an extension of these models to the case of degenerate quadratic forms.Comment: 39 pages, 2 figure

Morning versus Afternoon Body Mass in Free-Living or Controlled Euhydration

The standard protocol to assess hydration status is by measuring body mass in the early morning without controlling fluid intake. However, obtaining first-morning body mass is not necessarily feasible for many situations, for example, most physical activities take place in the afternoon. Thus, first-morning body mass might not be practical to assess hydration status. PURPOSE: To investigate first-morning body mass versus afternoon body mass in free- living and controlled euhydration. METHODS: 9 males (age: 21 ± 2; mass: 79.7 ± 17.8 kg) and 5 females (age: 22 ± 2; mass: 60.5 ± 13.6 kg) visited the laboratory in the morning (7:00-9:00am) and afternoon (2:00-4:00pm) for six days to measure their nude body mass and urine specific gravity (USG). Participants were in the free-living (FL) condition for the first three consecutive days, and then in a euhydrated (EUH) state (USGRESULTS: There were no interactions between FL and EUH with morning and afternoon in USG (Morning-FL, 1.017±0.005; Afternoon-FL, 1.012±0.006; Morning-EUH, 1.011±0.004; Afternoon-EUH, 1.007±0.004; p=0.390). No statistically significant differences were found between morning and afternoon in both FL and EUH controlled (Morning-FL, 72.7±18.3 kg; Afternoon-FL, 72.0±18.1 kg; Morning-EUH, 72.9±18.1 kg; Afternoon-EUH, 73.1±18.1 kg, p=0.661). CONCLUSION: There is no difference between morning and afternoon body mass, regardless of the hydration status. This means that first morning body mass is no more, or less, accurate than afternoon