Relaxation of alternating iterative algorithms for the Cauchy problem associated with the modified Helmholtz equation

We propose two algorithms involving the relaxation of either the given Dirichlet data or the prescribed Neumann data on the over-specified boundary, in the case of the alternating iterative algorithm of ` 12 ` 12 `$12 `&12 `#12 `^12 `_12 `%12 `~12 *Kozlov91 applied to Cauchy problems for the modified Helmholtz equation. A convergence proof of these relaxation methods is given, along with a stopping criterion. The numerical results obtained using these procedures, in conjunction with the boundary element method (BEM), show the numerical stability, convergence, consistency and computational efficiency of the proposed methods.

A relaxation method of an alternating iterative algorithm for the Cauchy problem in linear isotropic elasticity

We propose two algorithms involving the relaxation of either the given Dirichlet data (boundary displacements) or the prescribed Neumann data (boundary tractions) on the over-specified boundary in the case of the alternating iterative algorithm of Kozlov et al. [16] applied to Cauchy problems in linear elasticity. A convergence proof of these relaxation methods is given, along with a stopping criterion. The numerical results obtained using these procedures, in conjunction with the boundary element method (BEM), show the numerical stability, convergence, consistency and computational efficiency of the proposed method.

An alternating method for Cauchy problems for Helmholtz-type operators in non-homogeneous medium

Kozlov & Maz'ya (1989, Algebra Anal., 1, 144–170) proposed an alternating iterative method for solving Cauchy problems for general strongly elliptic and formally self-adjoint systems. However, in many applied problems, operators appear that do not satisfy these requirements, e.g. Helmholtz-type operators. Therefore, in this study, an alternating procedure for solving Cauchy problems for self-adjoint non-coercive elliptic operators of second order is presented. A convergence proof of this procedure is given.

An alternating boundary integral based method for a Cauchy problem for the Laplace equation in semi-infinite regions

We consider a Cauchy problem for the Laplace equation in a two-dimensional semi-infinite region with a bounded inclusion, i.e. the region is the intersection between a half-plane and the exterior of a bounded closed curve contained in the half-plane. The Cauchy data are given on the unbounded part of the boundary of the region and the aim is to construct the solution on the boundary of the inclusion. In 1989, Kozlov and Maz'ya [10] proposed an alternating iterative method for solving Cauchy problems for general strongly elliptic and formally self-adjoint systems in bounded domains. We extend their approach to our setting and in each iteration step mixed boundary value problems for the Laplace equation in the semi-infinite region are solved. Well-posedness of these mixed problems are investigated and convergence of the alternating procedure is examined. For the numerical implementation an efficient boundary integral equation method is proposed, based on the indirect variant of the boundary integral equation approach. The mixed problems are reduced to integral equations over the (bounded) boundary of the inclusion. Numerical examples are included showing the feasibility of the proposed method.

An alternating potential based approach for a Cauchy problem for the Laplace equation in a planar domain with a cut

We consider a Cauchy problem for the Laplace equation in a bounded region containing a cut, where the region is formed by removing a sufficiently smooth arc (the cut) from a bounded simply connected domain D. The aim is to reconstruct the solution on the cut from the values of the solution and its normal derivative on the boundary of the domain D. We propose an alternating iterative method which involves solving direct mixed problems for the Laplace operator in the same region. These mixed problems have either a Dirichlet or a Neumann boundary condition imposed on the cut and are solved by a potential approach. Each of these mixed problems is reduced to a system of integral equations of the first kind with logarithmic and hypersingular kernels and at most a square root singularity in the densities at the endpoints of the cut. The full discretization of the direct problems is realized by a trigonometric quadrature method which has super-algebraic convergence. The numerical examples presented illustrate the feasibility of the proposed method.

Long period grating directional bend sensor based on asymmetric index modification of cladding

A long period grating (LPG) written in a standard optical fibre was modified by using a femtosecond laser to induce an asymmetric change in the cladding's refractive index. This device produced blue and red wavelength shifts depending on the orientation of applied curvature, with maximum sensitivities of -1.6 nm m and +3.8 nm m, suggesting that this type of LPG may be useful as a shape sensor.

Characterization of asymmetric filtered 40 Gb/s RZ-DPSK system-strong filtering considerations

We present the impact of frequency offsetting of strong (e.g. 35 GHz) optical filters on the performance of 42.7 Gb/s 50% RZ-DPSK systems. The performance is evaluated when offsetting the filter by substantial amounts and it is found that with an offset of almost half the bit rate there is a significant improvement in the calculated 'Q' (> 1 dB). We deployed balanced, constructive single ended and destructive single ended detection, so that we could investigate the physical origins of the penalty reduction of asymmetric filtering of 42.7 Gb/s 50% RZ-DPSK system.

Directional bend sensor based on an asymmetric modification of the fiber cladding by femtosecond laser

A femtosecond laser was used to modify a part of the cladding of a standard LPG bend sensor. The device produced wavelength shifts depending upon the direction of bend, thus making a shape sensor. © 2005 Optical Society of America.

Duty cycle optimization in a wavelength allocated WDM transmission with narrow asymmetric VSB filtering

We investigate the impact of a duty cycle on a wavelength allocated transmission at 40 Gbit/s with narrow, off-centre, optical filtering. We also study how the shape of the off-centred VSB filter affects the performance of the optical system. © 2004 Elsevier Inc. All rights reserved.

Asymmetric reactions of organosilicon compounds and properties of novel optically active organosilanes

Partial reduction of racemic methoxysilanes by 1:1 complexes of lithium aluminium hydride with optically active cinchona and ephedra alkaloids give optically active silanes and methoxysilanes. Optical yields depend on the groups attached to silicon and the alkaloid used but in some cases approach 50%, The method has been used to prepare novel optically active organosilanes, possessing an asymmetric silicon centre, which are either inaccessible by any of the other available routes or would require a time consuming preparation. Such compounds are of use in the study of the mechanism of substitutions at silicon. Attempts have been made to rationalize the results of the asymmetric reductions in terms of differences in sterio and electronic interactions in diastereoisomeric transition states. Circular dichroism and optical rotatory dispersion spectra have been obtained for the optically active products in an attempt to elucidate the absolute configurations of the novel asymmetric organosilanes. The results from these studies provide a useful addition to the data so far accumulated for asymmetrically perturbed aromatic chromophores. Nuclear magnetic resonanoe studies of diastereoisomaric (-)-menthoxysilanes show that these compounds possess resonances extremely useful in the determination of optical purities for asymmetric organosilanes which possess an aromatic group. The effect of variable temperature on the spectra has revealed evidence for the conformational preferences in these compounds. Other diastereoisomeric alkoxysilanes have been prepared and their n.m.r.spectra studied in the hope of establishing trends. Exploratory studies for other asymmetric reactions proceeding at silicon have proved unfruitful.

An alternating method for the stationary Stokes system

An alternating procedure for solving a Cauchy problem for the stationary Stokes system is presented. A convergence proof of this procedure and numerical results are included.

Design of FBG-based ultra-narrow asymmetric filter for transmission with 0.8 bit/s/Hz spectral efficiency without polarisation multiplexing

We propose a novel approach to ultra-narrow optical filtering based on a specially designed slightly asymmetric filter, which can be fabricated using fibre Bragg gratings. A feasibility of 8×40 Gbit/s DWDM RZ transmission with 0.8 bit/s/Hz spectral efficiency (without polarisation multiplexing) over 1280 km of SMF/DCF link without FEC has been confirmed by numerical modelling. © 2004 Elsevier Inc. All rights reserved.

Build-up of auditory stream segregation induced by tone sequences of constant or alternating frequency and the resetting effects of single deviants

A sequence of constant-frequency tones can promote streaming in a subsequent sequence of alternating-frequency tones, but why this effect occurs is not fully understood and its time course has not been investigated. Experiment 1 used a 2.0-s-long constant-frequency inducer (10 repetitions of a low-frequency pure tone) to promote segregation in a subsequent, 1.2-s test sequence of alternating low- and high-frequency tones. Replacing the final inducer tone with silence substantially reduced reported test-sequence segregation. This reduction did not occur when either the 4th or 7th inducer was replaced with silence. This suggests that a change at the induction/test-sequence boundary actively resets build-up, rather than less segregation occurring simply because fewer inducer tones were presented. Furthermore, Experiment 2 found that a constant-frequency inducer produced its maximum segregation-promoting effect after only three tones—this contrasts with the more gradual build-up typically observed for alternating-frequency sequences. Experiment 3 required listeners to judge continuously the grouping of 20-s test sequences. Constant-frequency inducers were considerably more effective at promoting segregation than alternating ones; this difference persisted for ~10 s. In addition, resetting arising from a single deviant (longer tone) was associated only with constant-frequency inducers. Overall, the results suggest that constant-frequency inducers promote segregation by capturing one subset of test-sequence tones into an ongoing, preestablished stream, and that a deviant tone may reduce segregation by disrupting this capture. These findings offer new insight into the dynamics of stream segregation, and have implications for the neural basis of streaming and the role of attention in stream formation. (PsycINFO Database Record (c) 2013 APA, all rights reserved)