A Note on the Legendre Series Solution of the Biharmonic Equation for Cylindrical Problems

The solution of cylindrical problems is addressed. A series solution is considered of the biharmonic equation, in which the series terms of the stress function Φ are expressions based upon Legendre polynomials and logarithmically singular functions. An explicit form of a polynomial supplementing each logarithmically singular part of the series solution is obtained.     

An Incompressible Fluid in a Weakly Deformable Shell. Part I: Analysis of the Model

We consider the flow of a viscous incompressible fluid in a pipe when the boundary is a deformable shell of Naghdi type. We prove that the corresponding system of partial differential equations has a solution when the deformation of the shell is smooth and small enough. We propose an algorithm that uncouples the unknowns and prove its convergence.     

Nonlinear analysis of laser droplet generation by means of 0–1 test for chaos

We apply the recently improved version of the 0–1 test for chaos to real experimental time series of laser droplet generation process. In particular two marginal regimes of dripping are considered: spontaneous and forced dripping. The outcomes of the test reveal that both spontaneous and forced dripping time series can be characterized as chaotic, which coincides with the previous analysis based on nonlinear time series analysis.     

Electroweak tests with τ leptons at LEP

We report on two sensitive tests of lepton universality carried out by the 4 LEP experiments at the Z⁰ pole. From measurements of the τ polarization in e⁺ e^?→τ ⁺ τ ^?, the ratios of the vector and axial vector coupling constants of the electron and the tau lepton to the weak neutral current are obtained to beg _{v e} /g _{a e} =0.066±0.015 andg _{V τ} /g _{A τ} =0.070±0.009 respectively. From measurement of the τ lifetime and the τ leptonic branching ratios, the ratio of the coupling constants describing weak leptonic decays of the τ and the μ is measured to beG _τ /G _μ =0.996±0.008.     

Standard<Emphasis Type="Italic">SU</Emphasis>(5) supergravity grand unification model and predictions

A review is given of the StandardSU(5) supergravity model. This model has passed an important check regarding unification of the electro-weak and the strong couplings using high precision LEP data. It is shown that for a significant domain of the parameter space the model also satisfies constraints on the SUSY spectrum from CDF and LEP, as well as proton stability and cosmological relic density constraints.     

Physics with large extra dimensions

The recent understanding of string theory opens the possibility that the string scale can be as low as a few TeV. The apparent weakness of gravitational interactions can then be accounted by the existence of large internal dimensions, in the sub-millimeter region. Furthermore, our world must be confined to live on a brane transverse to these large dimensions, with which it interacts only gravitationally. In my lecture, I describe briefly this scenario which gives a new theoretical framework for solving the gauge hierarchy problem and the unification of all interactions. I also discuss a minimal embedding of the standard model, gauge coupling unification and proton stability. On leave from: Centre de Physique Théorique, Ecole Polytechnique, 91128 Palaiseau, Cedex, France.     

Simulation and design of a bandpass filter on metasubstrates

This work presents an analysis of the microstrip bandpass filter on metamaterial substrates. The filter is composed of two ring resonators with quarter-wavelength side-coupled sections. The filter input is provided, as well as the output port, using a quarter-wavelength side-coupled microstrip line section. Simulations by finite element method have been carried out to verify the effect of the metamaterial substrate properties on the filter performance, and to compare these results to those obtained considering isotropic substrate.     

An ultra-small heterostructure wavelength division multiplexer (WDM) with the ability to select two wavelengths from the s-band

In this article, a photonic crystal channel drop filter (CDF) based on \(2\times 3\) ring resonators is presented. At first, the effects of changing the radius of lattice rods and the lattice dielectric constant on the dropping efficiency of a 3-port CDF with one resonator are investigated. Then by developing this base structure, a new 4-port heterostructure CDF composed of two regions with rods made of silicone and germanium is presented, which can operate in the ‘S’ band of the communication window. The photonic crystal heterostructure CDF consists of a horizontal waveguide and two ring resonators that have been installed, in symmetry to the horizontal axis, in two regions with different refractive indices. These ring resonators act as energy couplers and capture at their resonant frequencies the electromagnetic energy which is propagated in the bus waveguide. For the analysis of transmission characteristics and the band structure of the filter, two methods have been employed: the two-dimensional Finite–difference time domain method and the Plane Wave Expansion method. In the final structure, dropping efficiencies of 97 and 89 % can be achieved at ports D and B, respectively, and also an acceptable quality factor can be obtained in the communication window. The overall size of this device is 174.14  \(\upmu \hbox {m}^{2}\) . Due to its small size, this structure can be used in Wavelength Division Multiplexer applications in the Optical Integrated Circuits.     

Mathematical modelling and numerical simulation of the morphological development of neurons

Background

The morphological development of neurons is a very complex process involving both genetic and environmental components. Mathematical modelling and numerical simulation are valuable tools in helping us unravel particular aspects of how individual neurons grow their characteristic morphologies and eventually form appropriate networks with each other.

Methods

A variety of mathematical models that consider (1) neurite initiation (2) neurite elongation (3) axon pathfinding, and (4) neurite branching and dendritic shape formation are reviewed. The different mathematical techniques employed are also described.

Results

Some comparison of modelling results with experimental data is made. A critique of different modelling techniques is given, leading to a proposal for a unified modelling environment for models of neuronal development.

Conclusion

A unified mathematical and numerical simulation framework should lead to an expansion of work on models of neuronal development, as has occurred with compartmental models of neuronal electrical activity.

     

Energy spectrum of multi-radiation of X-rays in a low energy Mather-type plasma focus device

The multi-radiation of X-rays was investigated with special attention to their energy spectrum in a Mather-type plasma focus device （operated with argon gas）. The analysis is based on the effect of anomalous resistances. To study the energy spectrum, a four-channel diode X-ray spectrometer was used along with a special set of filters. The filters were suitable for detection of medium range X-rays as well as hard X-rays with energy exceeding 30 keV. The results indicate that the anomalous resistivity effect during the post pinch phase may cause multi-radiation of X-rays with a total duration of 300 ± 50 ns. The significant contribution of Cu-Kα was due to the medium range X-rays, nonetheless, hard X-rays with energies greater than 15 keV also participate in the process. The total emitted X-ray energy in the forms of Cu-K and Cu-K/3 was around 0.14 ± 0.02 （J/Sr） and 0.04 ±0.01 （J/Sr）, respectively. The total energy of the emitted hard X-ray （〉 15 keV） was around 0.12± 0.02 （J/Sr）.