# Volume 4 (2012), Number 1

## DUAL and BIPOLAR Mathematical Models Derived from the Solutions of Two Very Simple Mathematical Equations and Their Application in Physics

### Author(s): MALVINA BAICA AND MIRCEA CARDU

**Abstract:** In this paper we develop a “Dual” model for the use in for solving wave and corpuscle
problems in Quantum Mechanics relative to behavior of light in some diverse processes in
Physics. For example, in the course of emissions and absorption in the Photo Electric
Effect and in the Compton Light Effect there are observations of a corpuscular nature.
Conversely, while observing the phenomena of interference, diffraction, interference and
polarization, light exhibits an oscillatory character. The dual model will, applied to
these phenomena, be able to distinguish among these separate effects. We further perform
a “Bipolar” model for problems common to electric fields and the magnetic fields (these
fields must occur simultaneously). Each propagation direction of the electromagnetic wave
can be represented by two sinusoid curves situated in two perpendicular planes. One of
them represents the oscillations of the electrical intensity vector and the other represents
the oscillations of magnetic intensity vector. These can be accorded a unified treatment
by the proposed new bipolar model. There are many such phenomena in Physics which exhibit
either a dual behavior or a bipolar behavior. The work is to be based upon algebra,
mathematical logic, quantum theory and electromagnetic theory. Both models will emerge
from the solutions of similar but not identical equations.

## An Application of Univalent Solutions to Fractional Volterra Equation in Complex Plane

### Author(s): AISHA AHMED AMER AND MASLINA DARUS

**Abstract:** In this article, we discuss the existence and uniqueness of solution to
fractional Volterra equation in complex plane. We apply our results on the single species
model of Volterra type. Fixed point theorems are the main tool used here to establish the
existence and uniqueness results. First we use Banach contraction principle and then
Krasnoselskii‘s fixed point theorem under certain conditions. Moreover, we prove that
the solution can be extended to maximal interval of existence.

## Convergence of a Generalized Iterations for a Countable Family of Nonexpansive Mappings

### Author(s): MOHSEN ALIMOHAMMADY AND VAHID DADASHI

**Abstract:** In this paper we propose a new modified Mann iterations with certain control
conditions for a countable family of nonexpansive mappings $\left\{{T}_{n}\right\}$ in Banach spaces. The sequence $\left\{{x}_{n}\right\}$ generated by the iteration converges strongly to a common fixed point
in $\bigcap _{n=1}^{\infty}F\left({T}_{n}\right)$ which is a solution of certain variational inequality. Moreover,
we get a specific conclusion from main results.

## Polynomial Coefficients and Approximation Errors of Entire Series

### Author(s): HUZOOR H. KHAN AND RIFAQAT ALI

**Abstract:** In this paper we consider the maximum of entire function $f\left(z\right)$ over a
certain lemniscate instead of considering the maximum of $f\left(z\right)$ on
$\left|f\right(z\left)\right|=r$ and obtained analogous results for entire functions of the
form $f\left(z\right)=\sum _{k=1}^{\infty}{q}_{k}\left(z\right){\left[\gamma \right(z\left)\right]}^{k-1}$, where
$\gamma \left(z\right)$ is a polynomial of degree $m$ and ${q}_{k}\left(z\right)$ is of degree $(m-1)$. The
$(p,q)$-order and generalized
$(p,q)$-type have been characterized
in terms of Polynomial Coefficients and ${L}^{s}-$approximation errors,
$1\le s\le \infty $. Finally, a saturation theorem for
$f\left(z\right)$ which can be extended to a entire function of
$(p,q)$-order 0 or 1 and for entire
functions of minimal generalized $(p,q)$-type have been obtained.

## On Polynomial Instability of Variational Difference Equations in Banach Spaces

### Author(s): MIHAELA AURELIA TOMESCU

**Abstract:** The object of this paper is to study two concepts of polynomial instability for
variational nonautonomous difference equations in Banach spaces. Obtained characterizations for
these concepts are generalizations of the classical results due to E.Barbashin ([1]), R.Datko ([4])
and A. Lyapunov ([6]) for variational difference equations.

## On Optimizing Linear Multiple Regression Models Using Stepwise Regression

### Author(s): CONSTANTIN ZĂVOIANU AND FELICIA ZĂVOIANU

**Abstract:** This article is focused on the problem of optimizing linear multiple regression
models using the stepwise regression method. Thus, the technique for constructing the computation
matrix is initially presented, then, the optimization algorithm is described, and finally, the method
for determining the elements that are not contained in the optimal regression model is presented. The
last part of this paper contains a case study where the optimization algorithm is applied on a dataset
containing authentic territorial statistical data.

## Green’s Tensor for an Elastic Circle and Its Application in Micromechanics of Defects in Solids

### Author(s): SEREMET VICTOR

**Abstract:** The Green’s tensor for the first (displacement given) boundary value
problem of elasticity for a circular domain is computed under a closed form expression. The method
of solution uses the “incompressible influence element” for which the Green’s tensor is
given by representation using Green’s function for Poisson’s equation. Using such a
representation, is shown that the main problem is to find the dilatation along the boundary induced
by the displacements Green’s function. The volume dilatation is than obtained by solving an
integral equation along the circular boundary. Explicit expressions are obtained for the Green’s
displacements tensor and for the traction along the circular boundary, allowing expressing the
solution for any kind of “displacement” boundary condition and body forces. On the basis of the
constructed Green’s tensor is given the integral formula which presents a generalization
of the well known Poisson’s integral formula from the theory of harmonic potentials onto
the theory of elasticity. An example of application of Green’s tensor in micromechanics
of defects in solids as radial Volterra’s slip dislocation in an elastic circle is
presented. These results were obtained in explicit form and for the first time. Applied here
the “incompressible influence element method” (IIEM) can be used to derive the Green’s
tensor for a wide classis of different boundary value problems for canonical domains of many
systems of coordinates. So, IIEM will increase considerable the possibilities to solve new
complicate boundary value problems in bounded and “unbounded” solids, acted by different inner
actions: body forces, temperature dislocations, Volterra’s dislocations, eigenstrains,
inclusions etc and any boundary displacements.

## Thermoelastostatic Equilibrium of Some Spherical Semi-wedges: Green’s Functions and Integral Formulas

### Author(s): SEREMET VICTOR

**Abstract:** In this study new exact Green’s functions and new exact Green-type
integral formula for a boundary value problem (BVP) in thermoelasticity for some spherical
semi-wedges with mixed homogeneous mechanical boundary conditions (zero normal stresses and
tangential displacements on quarter-plane ${\Gamma}_{\phi 0}$
and on the marginal circular infinite sector ${\Gamma}_{\beta \left(\pi \u22152\right)}$;
zero normal displacements and tangential stresses on quarter-plane ${\Gamma}_{\phi \alpha}$)
are derived. The thermoelastic displacements are subjected to a heat source applied in the inner
points of the spherical semi-wedges and to a mixed non-homogeneous boundary heat conditions
(temperatures on quarter-plane ${\Gamma}_{\phi 0}$ and on the marginal
circular infinite sector ${\Gamma}_{\beta \left(\pi \u22152\right)}$;
heat flux on quarter-plane ${\Gamma}_{\phi \alpha}$). When thermoelastic
Green’s functions are derived, the thermoelastic displacements are created by an inner
unit point heat source, described by $\delta $-Dirac’s function. All results are obtained in
terms of elementary functions that are formulated in a special theorem.

## Regarding the Stability of Rotors Running in Variable Stiffness Bearings

### Author(s): CRISTIAN PAVEL, AMELITTA LEGENDI AND RADU PANAITESCU-LIESS

**Abstract:** Low bearings stiffness gives rise to the rotor steady state response,
this state being superior to that obtained with moderate and high bearing stiffness. The paper
describes a combined theoretical- experimental investigation into the stability of a flexible
rotor running in flexible bearings, with particular references to low stiffness bearings.
Results show that the bearings type proposed can be used effectively without resulting in
system instability.