Quasi-separation of the biharmonic partial differential equation

In this paper, we consider analytical and numerical solutions to the Dirichlet boundary-value problem for the biharmonic partial differential equation on a disc of finite radius in the plane. The physical interpretation of these solutions is that of the harmonic oscillations of a thin, clamped plate. For the linear, fourth-order, biharmonic partial differential equation in the plane, it is well known that the solution method of separation in polar coordinates is not possible, in general. However, in this paper, for circular domains in the plane, it is shown that a method, here called quasi-separation of variables, does lead to solutions of the partial differential equation. These solutions are products of solutions of two ordinary linear differential equations: a fourth-order radial equation and a second-order angular differential equation. To be expected, without complete separation of the polar variables, there is some restriction on the range of these solutions in comparison with the corresponding separated solutions of the second-order harmonic differential equation in the plane. Notwithstanding these restrictions, the quasi-separation method leads to solutions of the Dirichlet boundary-value problem on a disc with centre at the origin, with boundary conditions determined by the solution and its inward drawn normal taking the value 0 on the edge of the disc. One significant feature for these biharmonic boundary-value problems, in general, follows from the form of the biharmonic differential expression when represented in polar coordinates. In this form, the differential expression has a singularity at the origin, in the radial variable. This singularity translates to a singularity at the origin of the fourth-order radial separated equation; this singularity necessitates the application of a third boundary condition in order to determine a self-adjoint solution to the Dirichlet boundary-value problem. The penultimate section of the paper reports on numerical solutions to the Dirichlet boundary-value problem; these results are also presented graphically. Two specific cases are studied in detail and numerical values of the eigenvalues are compared with the results obtained in earlier studies.

Blank beam fabrication of regenerated type IA gratings

We demonstrate an idealized method for the fabrication of regenerated type IA fibre Bragg gratings using commonly available apparatus. We use this technique to show that gratings written in the same fibre with the same period may have central wavelengths which are 14.4 nm apart and have an + 11.5% and - 1.2% difference in temperature and strain coefficients, respectively. We use these results to show that temperature compensated dual grating sensor heads, of an arbitrary length, may be quickly and consistently manufactured. © 2004 IOP Publishing Ltd.

Subject identification via ECG fiducial-based systems:influence of the type of QT interval correction

Electrocardiography (ECG) has been recently proposed as biometric trait for identification purposes. Intra-individual variations of ECG might affect identification performance. These variations are mainly due to Heart Rate Variability (HRV). In particular, HRV causes changes in the QT intervals along the ECG waveforms. This work is aimed at analysing the influence of seven QT interval correction methods (based on population models) on the performance of ECG-fiducial-based identification systems. In addition, we have also considered the influence of training set size, classifier, classifier ensemble as well as the number of consecutive heartbeats in a majority voting scheme. The ECG signals used in this study were collected from thirty-nine subjects within the Physionet open access database. Public domain software was used for fiducial points detection. Results suggested that QT correction is indeed required to improve the performance. However, there is no clear choice among the seven explored approaches for QT correction (identification rate between 0.97 and 0.99). MultiLayer Perceptron and Support Vector Machine seemed to have better generalization capabilities, in terms of classification performance, with respect to Decision Tree-based classifiers. No such strong influence of the training-set size and the number of consecutive heartbeats has been observed on the majority voting scheme.

On the use of the type I and II scheme for classifying ultrafast laser direct-write photonics

The use of the Type I and Type II scheme, first introduced and used by fiber Bragg grating researchers, has recently been adopted by the ultrafast laser direct-write photonics community to classify the physical geometry of waveguides written into glasses and crystals. This has created confusion between the fiber Bragg grating and direct-write photonics community. Here we propose a return to the original basis of the classification based on the characteristics of the material modification rather than the physical geometry of the waveguide.

Polynomials of Pellian Type and Continued Fractions

We investigate infinite families of integral quadratic polynomials {fk (X)} k∈N and show that, for a fixed k ∈ N and arbitrary X ∈ N, the period length of the simple continued fraction expansion of √fk (X) is constant. Furthermore, we show that the period lengths of √fk (X) go to infinity with k. For each member of the families involved, we show how to determine, in an easy fashion, the fundamental unit of the underlying quadratic field. We also demonstrate how the simple continued fraction ex- pansion of √fk (X) is related to that of √C, where √fk (X) = ak*X^2 +bk*X + C. This continues work in [1]–[4].

A Random Evolution Inclusion of Subdifferential Type in Hilbert Spaces

In this paper we study a nonlinear evolution inclusion of subdifferential type in Hilbert spaces. The perturbation term is Hausdorff continuous in the state variable and has closed but not necessarily convex values. Our result is a stochastic generalization of an existence theorem proved by Kravvaritis and Papageorgiou in [6].

Generalizing of Neural Nets: Functional Nets of Special Type

Special generalizing for the artificial neural nets: so called RFT – FN – is under discussion in the report. Such refinement touch upon the constituent elements for the conception of artificial neural network, namely, the choice of main primary functional elements in the net, the way to connect them(topology) and the structure of the net as a whole. As to the last, the structure of the functional net proposed is determined dynamically just in the constructing the net by itself by the special recurrent procedure. The number of newly joining primary functional elements, the topology of its connecting and tuning of the primary elements is the content of the each recurrent step. The procedure is terminated under fulfilling “natural” criteria relating residuals for example. The functional proposed can be used in solving the approximation problem for the functions, represented by its observations, for classifying and clustering, pattern recognition, etc. Recurrent procedure provide for the versatile optimizing possibilities: as on the each step of the procedure and wholly: by the choice of the newly joining elements, topology, by the affine transformations if input and intermediate coordinate as well as by its nonlinear coordinate wise transformations. All considerations are essentially based, constructively and evidently represented by the means of the Generalized Inverse.

Complexation in composite solutions of melanin with 2,4,7-trinitrofluorenone

A feasibility of formation of donor-acceptor charge-transfer (CT) complexes between melanin and 2,4,7-trinitrofluorenone (TNF) being good electron acceptor has been studied in solutions by means of the absorption and photoluminescence (PL) spectra. The model of electronic transitions in a melanin-TNF composite solution has been proposed. © 2014 Copyright Taylor & Francis Group, LLC.

Algorithm BIDIMS for Automated Systematization of Data Array. Case Study: Rediscovering Mendeleev’s Periodic Table of Chemical Elements

The method (algorithm BIDIMS) of multivariate objects display to bidimensional structure in which the sum of differences of objects properties and their nearest neighbors is minimal is being described. The basic regularities on the set of objects at this ordering become evident. Besides, such structures (tables) have high inductive opportunities: many latent properties of objects may be predicted on their coordinates in this table. Opportunities of a method are illustrated on an example of bidimentional ordering of chemical elements. The table received in result practically coincides with the periodic Mendeleev table.

Krätzel Function as a Function of Hypergeometric Type

2000 Mathematics Subject Classification: 33C60, 33C20, 44A15

Generalized Sobolev Spaces of Exponential Type Associated with the Dunkl Operators

2000 Mathematics Subject Classification: 35E45

Polynomial Expansions for Solutions of Higher-Order Bessel Heat Equation in Quantum Calculus

Mathematics Subject Class.: 33C10,33D60,26D15,33D05,33D15,33D90

Existence and Asymptotic Stability of Solutions of a Perturbed Quadratic Fractional Integral Equation

Mathematics Subject Classification: 45G10, 45M99, 47H09

Semi-Smooth Newton Methods for the Time Optimal Control of Nonautonomous Ordinary Differential Equations

AMS Subj. Classification: 49J15, 49M15