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# IMA Journal of Applied Mathematics Current Issue

Published: Tue, 12 Dec 2017 00:00:00 GMT

Last Build Date: Wed, 13 Dec 2017 01:48:14 GMT

On control strategies for avoiding loss of information in non-destructive testing

Tue, 12 Dec 2017 00:00:00 GMT

Abstract
This study takes place in the framework of non-destructive testing. Based on energy invariants, one can derive boundary quantities which could help to detect damages in a structure. Since only a small part of a structure is equipped with sensors, a lot of information may be lost through the boundary where there is no sensor. In this article, we study several control strategies in order to focus most of the information contained in the energy invariant used, to the sensors.

Riemann’s zeta function and the broadband structure of pure harmonics

Mon, 20 Nov 2017 00:00:00 GMT

Abstract
Let $a\in (0,1)$ and let $F_s(a)$ be the periodized zeta function that is defined as $F_s(a) = \sum n^{-s} \exp (2\pi \,{\rm i}na)$ for $\Re s >1$, and extended to the complex plane via analytic continuation. Let $s_n = \sigma_n + {\rm i}t_n, \, t_n >0$, denote the sequence of non-trivial zeros of the Riemann zeta function in the upper halfplane ordered according to non-decreasing ordinates. We demonstrate that, assuming the Riemann Hypothesis, the Cesàro means of the sequence $F_{s_n} (a)$ converge to the first harmonic $\exp (2\pi \,{\rm i} a)$ in the sense of periodic distributions. This reveals a natural broadband structure of the pure tone. The proof involves Fujii’s refinement of the classical Landau theorem related to the uniform distribution modulo one of the non-trivial zeros of $\zeta(s)$. We also discuss the applicable aspects of this phenomenon, focusing on broadband encodings of signals.

Investigation of an explosive food chain model with interference and inhibitory effects

Sat, 18 Nov 2017 00:00:00 GMT

Abstract
In the current manuscript, we have investigated the temporal as well as spatio-temporal dynamics of a three species modified Leslie–Gower food chain model with Holling type IV and Crowley–Martin function responses. We have shown that explosion in the top predator population can be prevented if group defence is sufficiently strong at the lowest trophic levels. This demonstrates that group defence can act as a damping mechanism, and prevent population explosion of apex predators. We also show that the spatially explicit model can exhibit diffusion-driven instability, that depends strongly on the intensity of the group defence, in the prey population. Standard bifurcation analysis and the period doubling route to chaos are also investigated.

Mathematical modelling of a magnetic immunoassay

Fri, 17 Nov 2017 00:00:00 GMT

Abstract
A mathematical model is developed to describe the action of a novel form of fluidic biosensor that uses paramagnetic particles (PMPs) that have been pre-coated with target-specific antibodies. In an initial phase the particles are introduced to a sample solution containing the target which then binds to the particles via antigen–antibody reactions. During the test phase a magnet is used to draw the PMPs to the sensor surface which is similarly coated with specific antibodies. During this process, cross-links are formed by the antigens thereby binding the PMPs to the sensor surface. After the magnetic field is removed, a voltage change across an inductor below the sensor surface is recorded, which is deemed to depend on the number of magnetic particles that have been bound to the sensor surface. The fundamental question addressed is to explain the range of experimentally observed dose–response curves, and how this depends on the various parameters of the problem. In particular, observations have shown both rising and falling dose–response curves, as well as ‘hooked’ dose–response curves possessing local maxima. Initially a particle-dynamics computational model is produced to determine the time scales of the key processes involved, but is shown to be unable to produce differently shaped dose–response curves. The computational model suggests spatio-temporal effects are unimportant, therefore a homogenized rate-equation model is developed for each of the key phases of the immunoassay process. Binding rates are shown to depend on various geometric factors related to the diameter of the PMPs and the size of the sensor surface. The dose–response is shown to depend crucially on various saturation effects during each phase, and conditions can be derived, in some cases analytically, for each of the three qualitatively different curve types. Furthermore, non-dimensionalization reveals 5 key dimensionless parameters and the dependence of these curve shapes on each is revealed. The results point to future quantitative approaches to sensor design and calibration.

A filtering based multi-innovation gradient estimation algorithm and performance analysis for nonlinear dynamical systems

Mon, 06 Nov 2017 00:00:00 GMT

Abstract
This article studies the problem for parameter identification of nonlinear dynamical systems (i.e., the Hammerstein–Wiener systems) with additive coloured noises. Based on the gradient search and the key term separation, a generalized extended stochastic gradient (GESG) algorithm is given for estimating the system parameters. To improve the computational efficiency, a data filtering based GESG algorithm and a data filtering based multi-innovation GESG algorithm are derived by applying the data filtering technique and the multi-innovation identification theory. Moreover, the proposed algorithms are proved to be convergent under proper conditions. Finally, the simulation results verify the theoretical analysis.

On the reduction of coupled NLS equations to non-linear phase equations via modulation of a two-phase wavetrain

Tue, 26 Sep 2017 00:00:00 GMT

Abstract
The phase dynamics of two phase wavetrains in the coupled non-linear Schrödinger (NLS) equations are investigated as an example of the dispersion arising from singular wave action. It is shown that when the wavetrain becomes singular, there is a reduction from coupled NLS to a scalar Korteweg–de Vries (KdV) equation, and if there is a further degeneracy the scalar two-way Boussinesq emerges. This is the first such derivation of the two-way Boussinesq reduction in this setting. A novelty in the theory is that the coefficients in the resulting equations are determined from properties of the wavetrain and underlying conservation laws. This theory generalizes the reduction from a single defocussing NLS equation to the KdV equation, and introduces Boussinesq dynamics to finite amplitude states in this family. A discussion of the effect of the phase dynamics on the wavetrain solution shows that the reductions provide an insight into a mechanism for the bifurcation of periodic wavetrains to dark and bright solitary waves.

On the numerical solution of a T-Sylvester type matrix equation arising in the control of stochastic partial differential equations

Mon, 25 Sep 2017 00:00:00 GMT

Abstract
We outline a derivation of a nonlinear system of equations, which finds the entries of an $m \times N$ matrix $K$, given the eigenvalues of a matrix $D$, a diagonal $N \times N$ matrix $A$ and an $N \times m$ matrix $B$. These matrices are related through the matrix equation $D = 2A+BK+K^tB^t$, which is sometimes called a t-Sylvester equation. The need to prescribe the eigenvalues of the matrix $D$ is motivated by the control of the surface roughness of certain nonlinear SPDEs (e.g., the stochastic Kuramoto–Sivashinsky equation) using non-trivial controls. We implement the methodology to solve numerically the nonlinear system for various test cases, including matrices related to the control of the stochastic Kuramoto–Sivashinsky equation and for randomly generated matrices. We study the effect of increasing the dimensions of the system and changing the size of the matrices $B$ and $K$ (which correspond to using more or less controls) and find good convergence of the solutions.

A pipe organ-inspired ultrasonic transducer

Tue, 05 Sep 2017 00:00:00 GMT

Abstract
This article considers a number of backplate designs for the bandwidth improvement of electrostatic ultrasonic transducers in both transmission and reception modes. Motivated by the design of pipe organs, transducers with backplates which incorporate a number of acoustically resonating conduits are modelled using a transmission line mathematical model which describes the displacement of the electrostatic membrane. The model illustrates that by increasing the number and varying the length of these conduits, the transmission voltage response and the reception force response can be improved over the traditional design by around 50 and 35%, respectively.