Generalization of a Conjecture in the Geometry of Polynomials

In this paper we survey work on and around the following conjecture, which was first stated about 45 years ago: If all the zeros of an algebraic polynomial p (of degree n ≥ 2) lie in a disk with radius r, then, for each zero z1 of p, the disk with center z1 and radius r contains at least one zero of the derivative p′ . Until now, this conjecture has been proved for n ≤ 8 only. We also put the conjecture in a more general framework involving higher order derivatives and sets defined by the zeros of the polynomials.

Optical quality and visual performance with customised soft contact lenses for keratoconus

Purpose: This study investigated how aberration-controlling, customised soft contact lenses corrected higher-order ocular aberrations and visual performance in keratoconic patients compared to other forms of refractive correction (spectacles and rigid gas-permeable lenses). Methods: Twenty-two patients (16 rigid gas-permeable contact lens wearers and six spectacle wearers) were fitted with standard toric soft lenses and customised lenses (designed to correct 3rd-order coma aberrations). In the rigid gas-permeable lens-wearing patients, ocular aberrations were measured without lenses, with the patient's habitual lenses and with the study lenses (Hartmann-Shack aberrometry). In the spectacle-wearing patients, ocular aberrations were measured both with and without the study lenses. LogMAR visual acuity (high-contrast and low-contrast) was evaluated with the patient wearing their habitual correction (of either spectacles or rigid gas-permeable contact lenses) and with the study lenses. Results: In the contact lens wearers, the habitual rigid gas-permeable lenses and customised lenses provided significant reductions in 3rd-order coma root-mean-square (RMS) error, 3rd-order RMS and higher-order RMS error (p ≤ 0.004). In the spectacle wearers, the standard toric lenses and customised lenses significantly reduced 3rd-order RMS and higher-order RMS errors (p ≤ 0.005). The spectacle wearers showed no significant differences in visual performance measured between their habitual spectacles and the study lenses. However, in the contact lens wearers, the habitual rigid gas-permeable lenses and standard toric lenses provided significantly better high-contrast acuities compared to the customised lenses (p ≤ 0.006). Conclusions: The customised lenses provided substantial reductions in ocular aberrations in these keratoconic patients; however, the poor visual performances achieved with these lenses are most likely to be due to small, on-eye lens decentrations. © 2014 The College of Optometrists.

Optical quality for keratoconic eyes with conventional RGP lens and simulated, customised contact lens corrections:a comparison

Purpose: To compare monochromatic aberrations of keratoconic eyes when uncorrected, corrected with spherically-powered RGP (rigid gas-permeable) contact lenses and corrected using simulations of customised soft contact lenses for different magnitudes of rotation (up to 15°) and translation (up to 1mm) from their ideal position. Methods: The ocular aberrations of examples of mild, moderate and severe keratoconic eyes were measured when uncorrected and when wearing their habitual RGP lenses. Residual aberrations and point-spread functions of each eye were simulated using an ideal, customised soft contact lens (designed to neutralise higher-order aberrations, HOA) were calculated as a function of the angle of rotation of the lens from its ideal orientation, and its horizontal and vertical translation. Results: In agreement with the results of other authors, the RGP lenses markedly reduced both lower-order aberrations and HOA for all three patients. When compared with the RGP lens corrections, the customised lens simulations only provided optical improvements if their movements were constrained within limits which appear to be difficult to achieve with current technologies. Conclusions: At the present time, customised contact lens corrections appear likely to offer, at best, only minor optical improvements over RGP lenses for patients with keratoconus. If made in soft materials, however, these lenses may be preferred by patients in term of comfort. © 2012 The College of Optometrists.

Visual performance and optical quality with soft lenses in keratoconus patients

Purpose: To assess visual performance and ocular aberrations in keratoconic patients using toric soft contact lenses (SCL), rigid-gas-permeable (RGP) contact lenses and spectacle lens correction. Methods: Twenty-two keratoconus patients (16 RGP lens wearers and six spectacle wearers) were fitted with toric SCL. Ocular aberrations were measured with and without the patient's habitual RGP lenses and with the SCL in place. In the spectacle wearers, aberrations were measured with and without the SCL. Visual performance (high- and low-contrast visual acuity) was evaluated with the patient's habitual correction and with the SCL. Results: In the RGP lens wearers both the habitual lenses and the toric SCL significantly reduced coma, trefoil, 3rd-order, 4th-order cylinder and higher-order root-mean-square (RMS) aberrations (p≤0.015). In the spectacle wearers the toric SCL significantly reduced coma, 3rd-order and higher-order RMS aberrations (p≤0.01). The patients' habitual RGP lenses gave better low-contrast acuity (p≤0.006) compared to the toric SCL; however, no significant difference was found between lens types for high-contrast acuity (p=0.10). In the spectacle wearers no significant differences in visual performance measurements were found between the patients' spectacles and the toric SCL (p≥0.06). Conclusion: The results show that RGP lenses provided superior visual performances and greater reduction of 3rd-order aberrations compared to toric SCL in this group of keratoconic patients. In the spectacle-wearing group, visual performance with the toric SCL was found to be comparable to that measured with spectacles. Nevertheless, with the exception of spherical aberration, the toric SCL were successful in significantly reducing uncorrected higher-order aberrations. Ophthalmic & Physiological Optics © 2012 The College of Optometrists.

Repeatability of ocular aberration measurements in patients with keratoconus

Purpose: To explore the repeatability of lower-order and higher-order ocular aberrations measured in patients with keratoconus. Methods: The IRX-3 (Imagine Eyes, Paris, France) aberrometer was used to record lower-order and higher-order aberrations in 31 eyes of 31 patients with keratoconus. Four monocular measurements were taken consecutively for each patient. The aberrometry data were analysed up to the 5th Zernike order for a 4-mm pupil diameter. The data were evaluated using repeated-measures anova and Friedman analyses. Repeatability was analysed using within-subject standard deviation (SW) and the repeatability limit (r) calculated as 1.96 ×√2×Sw. Results: Of the 11 aberration terms evaluated, the repeatability of Z (2,0) (mean= 1.36μm; SW=0.09μm; r=0.26μm); Z (2,±2) RMS (mean=1.05μm; SW= 0.09μm; r=0.24μm) and Z (4,0) aberrations (mean=0.34μm; SW=0.09 μm; r=0.24μm) showed the highest variability. In contrast, Z (3,±1) RMS aberrations (mean=0.85μm; SW=0.06μm; r=0.16μm) and Z (4,±2) RMS aberrations (mean=0.40μm; SW=0.07μm; r=0.18μm) showed comparatively better repeatability. Conclusions: The lower-order and higher-order aberrations measured in this group of keratoconic patients showed higher levels of variability compared to previous investigations of visually-normal subjects. These results may be of interest to eyecare practitioners involved in the design and fitting of aberration-controlling contact lenses for patients with keratoconus. © 2011 The College of Optometrists.

What's in a broken promissory obligation? Developing and testing a multiple component form measure of psychological contract breach

While the literature has suggested the possibility of breach being composed of multiple facets, no previous study has investigated this possibility empirically. This study examined the factor structure of typical component forms in order to develop a multiple component form measure of breach. Two studies were conducted. In study 1 (N = 420) multi-item measures based on causal indicators representing promissory obligations were developed for the five potential component forms (delay, magnitude, type/form, inequity and reciprocal imbalance). Exploratory factor analysis showed that the five components loaded onto one higher order factor, namely psychological contract breach suggesting that breach is composed of different aspects rather than types of breach. Confirmatory factor analysis provided further evidence for the proposed model. In addition, the model achieved high construct reliability and showed good construct, convergent, discriminant and predictive validity. Study 2 data (N = 189), used to validate study 1 results, compared the multiple-component measure with an established multiple item measure of breach (rather than a single item as in study 1) and also tested for discriminant validity with an established multiple item measure of violation. Findings replicated those in study 1. The findings have important implications for considering alternative, more comprehensive and elaborate ways of assessing breach.

Self-affinity in financial asset returns

We test for departures from normal and independent and identically distributed (NIID) log returns, for log returns under the alternative hypothesis that are self-affine and either long-range dependent, or drawn randomly from an L-stable distribution with infinite higher-order moments. The finite sample performance of estimators of the two forms of self-affinity is explored in a simulation study. In contrast to rescaled range analysis and other conventional estimation methods, the variant of fluctuation analysis that considers finite sample moments only is able to identify both forms of self-affinity. When log returns are self-affine and long-range dependent under the alternative hypothesis, however, rescaled range analysis has higher power than fluctuation analysis. The techniques are illustrated by means of an analysis of the daily log returns for the indices of 11 stock markets of developed countries. Several of the smaller stock markets by capitalization exhibit evidence of long-range dependence in log returns. © 2012 Elsevier Inc. All rights reserved.

Nonlinear soliton propagation in a few mode optical fibre

We experimentally demonstrate adiabatic soliton propagation in the fundamental mode of a few mode optical fibre and more complex behaviour in a higher order mode, indicating that the impact of nonlinearities differs for each mode.

Phase noise tolerance study in coherent optical circular QAM transmissions with Viterbi- Viterbi carrier phase estimation

We present a performance evaluation of a non-conventional approach to implement phase noise tolerant optical systems with multilevel modulation formats. The performance of normalized Viterbi-Viterbi carrier phase estimation (V-V CPE) is investigated in detail for circular m-level quadrature amplitude modulation (C-mQAM) signals. The intrinsic property of C-mQAM constellation points with a uniform phase separation allows a straightforward employment of V-V CPE without the need to adapt constellation. Compared with conventional feed-forward CPE for square QAM signals, the simulated results show an enhanced tolerance of linewidth symbol duration product (ΔvTs) at a low sensitivity penalty by using feed-forward CPE structure with C-mQAM. This scheme can be easily upgraded to higher order modulations without inducing considerable complexity.

Time-Dependent Differential Inclusions and Viability

AMS subject classification: Primary 34A60, Secondary 49J52.

Direction Finding Estimators of Cyclostationary Signals in Array Processing for Microwave Power Transmission

A solar power satellite is paid attention to as a clean, inexhaustible large- scale base-load power supply. The following technology related to beam control is used: A pilot signal is sent from the power receiving site and after direction of arrival estimation the beam is directed back to the earth by same direction. A novel direction-finding algorithm based on linear prediction technique for exploiting cyclostationary statistical information (spatial and temporal) is explored. Many modulated communication signals exhibit a cyclostationarity (or periodic correlation) property, corresponding to the underlying periodicity arising from carrier frequencies or baud rates. The problem was solved by using both cyclic second-order statistics and cyclic higher-order statistics. By evaluating the corresponding cyclic statistics of the received data at certain cycle frequencies, we can extract the cyclic correlations of only signals with the same cycle frequency and null out the cyclic correlations of stationary additive noise and all other co-channel interferences with different cycle frequencies. Thus, the signal detection capability can be significantly improved. The proposed algorithms employ cyclic higher-order statistics of the array output and suppress additive Gaussian noise of unknown spectral content, even when the noise shares common cycle frequencies with the non-Gaussian signals of interest. The proposed method completely exploits temporal information (multiple lag ), and also can correctly estimate direction of arrival of desired signals by suppressing undesired signals. Our approach was generalized over direction of arrival estimation of cyclostationary coherent signals. In this paper, we propose a new approach for exploiting cyclostationarity that seems to be more advanced in comparison with the other existing direction finding algorithms.

Statistical Analysis of Data on Linker Histones/DNA Interactions

2000 Mathematics Subject Classification: 62P10, 92C40

Joining the dots:using structured e-portfolio assignments to enhance reflection

Although there has been an increased interest in the use of electronic portfolios in higher education over the last five years, relatively little is known about the potential of such tools to support the development of higher order abilities for students, such as reflection, in a structured way that is suitable for assessment. This paper reports the findings from a small-scale research which sets out to compare the outcomes of reflective assignments in two cohorts of participants in a Postgraduate Certificate in Professional Practice in Higher Education in the UK. Participants in the programme were asked to submit reflective accounts using an e-portfolio system as part of their formal assessment. One cohort completed the assessment using some generic guidelines of how to reflect and construct an e-portfolio page without a given template or structure, whereas another cohort was given a specific template with clear assessment criteria to gauge the assembly of their reflections. The authors, who are also tutors in the programme, analysed the submitted reflections following open coding procedures. The analysis found a tendency for the reflection in the first cohort to be merely descriptive without progressing to speculating objectively about answers to relevant analytical questions about the process involved in the ability under scrutiny. In contrast the assignments of cohort two were found to be more insightful in terms of assimilating random bits of materials, thoughts and self-questions into complete reflective accounts. These findings bring some evidence to support and indeed promote a more structured approach to reflective practice, which can be further enhanced through a carefully created e-portfolio template and associated assessment criteria.

Brand personality complementatity:its effects on evaluations of extremely incongruent extensions

Prior research on brand extension has provided little evidence on enhancing the evaluation of extremely incongruent extension. Adopting the theoretical framework of schema congruity theory, the author posits that evaluations can be improved if brand personality impressions of both parent brand and extension are complementary. The author coins this as the brand personality complementarity (BPC) principle. Prior to examining BPC effect, cultural-specific brand personality scale was developed to identify universal and indigenous brand personality dimensions. The reason is BPC requires a reliable and valid brand personality scale in order to detect its effect. Following successful identification of the cultural-specific brand personality scale, a total of three experimental studies were done to investigate BPC effect. Specifically, one experimental study identified complementary levels amongst brand personality dimensions, whereas two experimental studies investigated the moderating effect of BPC. Findings from the scale development study reveal that Malaysian brand personality (MBP) scale is a second higher-order factor reflected by first higher-order factors of sophistication, youth, competence, and sincerity. Most importantly, findings from the experimental studies revealed; 1) different BPC levels amongst all possible pairs of MBP dimensions, 2) significant interaction effect of brand extension congruity x BPC, and 3) significant mediation effect of complementarity resolution. Specific findings indicated that when iv text-based stimuli were used to form brand personality impression, even low BPC level improves the evaluations of extremely incongruent extension. However, when visualbased stimuli were used, low BPC level worsen the extension evaluation compared those of the control condition (i.e. without brand personality impression). Implications for both academician and practitioner are discussed.

Stokes waves revisited:exact solutions in the asymptotic limit

The Stokes perturbative solution of the nonlinear (boundary value dependent) surface gravity wave problem is known to provide results of reasonable accuracy to engineers in estimating the phase speed and amplitudes of such nonlinear waves. The weakling in this structure though is the presence of aperiodic “secular variation” in the solution that does not agree with the known periodic propagation of surface waves. This has historically necessitated increasingly higher-ordered (perturbative) approximations in the representation of the velocity profile. The present article ameliorates this long-standing theoretical insufficiency by invoking a compact exact n-ordered solution in the asymptotic infinite depth limit, primarily based on a representation structured around the third-ordered perturbative solution, that leads to a seamless extension to higher-order (e.g., fifth-order) forms existing in the literature. The result from this study is expected to improve phenomenological engineering estimates, now that any desired higher-ordered expansion may be compacted within the same representation, but without any aperiodicity in the spectral pattern of the wave guides.