Hyperbolic Fourth-R Quadratic Equation and Holomorphic Fourth-R Polynomials

MSC 2010: 30C10, 32A30, 30G35

Class Number Two for Real Quadratic Fields of Richaud-Degert Type

2000 Mathematics Subject Classification: Primary: 11D09, 11A55, 11C08, 11R11, 11R29; Secondary: 11R65, 11S40; 11R09.

Equiconvergence and equisummability of Jacobi series

2010 Mathematics Subject Classification: 33C45, 40G05.

Development of safety performance functions for empirical Bayes estimation of crash reduction factors

Crash reduction factors (CRFs) are used to estimate the potential number of traffic crashes expected to be prevented from investment in safety improvement projects. The method used to develop CRFs in Florida has been based on the commonly used before-and-after approach. This approach suffers from a widely recognized problem known as regression-to-the-mean (RTM). The Empirical Bayes (EB) method has been introduced as a means to addressing the RTM problem. This method requires the information from both the treatment and reference sites in order to predict the expected number of crashes had the safety improvement projects at the treatment sites not been implemented. The information from the reference sites is estimated from a safety performance function (SPF), which is a mathematical relationship that links crashes to traffic exposure. The objective of this dissertation was to develop the SPFs for different functional classes of the Florida State Highway System. Crash data from years 2001 through 2003 along with traffic and geometric data were used in the SPF model development. SPFs for both rural and urban roadway categories were developed. The modeling data used were based on one-mile segments that contain homogeneous traffic and geometric conditions within each segment. Segments involving intersections were excluded. The scatter plots of data show that the relationships between crashes and traffic exposure are nonlinear, that crashes increase with traffic exposure in an increasing rate. Four regression models, namely, Poisson (PRM), Negative Binomial (NBRM), zero-inflated Poisson (ZIP), and zero-inflated Negative Binomial (ZINB), were fitted to the one-mile segment records for individual roadway categories. The best model was selected for each category based on a combination of the Likelihood Ratio test, the Vuong statistical test, and the Akaike's Information Criterion (AIC). The NBRM model was found to be appropriate for only one category and the ZINB model was found to be more appropriate for six other categories. The overall results show that the Negative Binomial distribution model generally provides a better fit for the data than the Poisson distribution model. In addition, the ZINB model was found to give the best fit when the count data exhibit excess zeros and over-dispersion for most of the roadway categories. While model validation shows that most data points fall within the 95% prediction intervals of the models developed, the Pearson goodness-of-fit measure does not show statistical significance. This is expected as traffic volume is only one of the many factors contributing to the overall crash experience, and that the SPFs are to be applied in conjunction with Accident Modification Factors (AMFs) to further account for the safety impacts of major geometric features before arriving at the final crash prediction. However, with improved traffic and crash data quality, the crash prediction power of SPF models may be further improved.

Switching patterns and steady-state analysis of grid-connected and stand-alone single-stage boost-inverters for PV applications

Renewable or sustainable energy (SE) sources have attracted the attention of many countries because the power generated is environmentally friendly, and the sources are not subject to the instability of price and availability. This dissertation presents new trends in the DC-AC converters (inverters) used in renewable energy sources, particularly for photovoltaic (PV) energy systems. A review of the existing technologies is performed for both single-phase and three-phase systems, and the pros and cons of the best candidates are investigated. In many modern energy conversion systems, a DC voltage, which is provided from a SE source or energy storage device, must be boosted and converted to an AC voltage with a fixed amplitude and frequency. A novel switching pattern based on the concept of the conventional space-vector pulse-width-modulated (SVPWM) technique is developed for single-stage, boost-inverters using the topology of current source inverters (CSI). The six main switching states, and two zeros, with three switches conducting at any given instant in conventional SVPWM techniques are modified herein into three charging states and six discharging states with only two switches conducting at any given instant. The charging states are necessary in order to boost the DC input voltage. It is demonstrated that the CSI topology in conjunction with the developed switching pattern is capable of providing the required residential AC voltage from a low DC voltage of one PV panel at its rated power for both linear and nonlinear loads. In a micro-grid, the active and reactive power control and consequently voltage regulation is one of the main requirements. Therefore, the capability of the single-stage boost-inverter in controlling the active power and providing the reactive power is investigated. It is demonstrated that the injected active and reactive power can be independently controlled through two modulation indices introduced in the proposed switching algorithm. The system is capable of injecting a desirable level of reactive power, while the maximum power point tracking (MPPT) dictates the desirable active power. The developed switching pattern is experimentally verified through a laboratory scaled three-phase 200W boost-inverter for both grid-connected and stand-alone cases and the results are presented.

Dynamic image precompensation for improving visual performance of computer users with ocular aberrations

With the progress of computer technology, computers are expected to be more intelligent in the interaction with humans, presenting information according to the user's psychological and physiological characteristics. However, computer users with visual problems may encounter difficulties on the perception of icons, menus, and other graphical information displayed on the screen, limiting the efficiency of their interaction with computers. In this dissertation, a personalized and dynamic image precompensation method was developed to improve the visual performance of the computer users with ocular aberrations. The precompensation was applied on the graphical targets before presenting them on the screen, aiming to counteract the visual blurring caused by the ocular aberration of the user's eye. A complete and systematic modeling approach to describe the retinal image formation of the computer user was presented, taking advantage of modeling tools, such as Zernike polynomials, wavefront aberration, Point Spread Function and Modulation Transfer Function. The ocular aberration of the computer user was originally measured by a wavefront aberrometer, as a reference for the precompensation model. The dynamic precompensation was generated based on the resized aberration, with the real-time pupil diameter monitored. The potential visual benefit of the dynamic precompensation method was explored through software simulation, with the aberration data from a real human subject. An "artificial eye'' experiment was conducted by simulating the human eye with a high-definition camera, providing objective evaluation to the image quality after precompensation. In addition, an empirical evaluation with 20 human participants was also designed and implemented, involving image recognition tests performed under a more realistic viewing environment of computer use. The statistical analysis results of the empirical experiment confirmed the effectiveness of the dynamic precompensation method, by showing significant improvement on the recognition accuracy. The merit and necessity of the dynamic precompensation were also substantiated by comparing it with the static precompensation. The visual benefit of the dynamic precompensation was further confirmed by the subjective assessments collected from the evaluation participants.

On point estimation of the abnormality of a Mahalanobis index

Open Access funded by Medical Research Council Acknowledgment The work reported here was funded by a grant from the Medical Research Council, UK, grant number: MR/J013838/1.

Irreducibility of Random Hilbert Schemes

We prove that a random Hilbert scheme that parametrizes the closed subschemes with a fixed Hilbert polynomial in some projective space is irreducible and nonsingular with probability greater than $0.5$. To consider the set of nonempty Hilbert schemes as a probability space, we transform this set into a disjoint union of infinite binary trees, reinterpreting Macaulay's classification of admissible Hilbert polynomials. Choosing discrete probability distributions with infinite support on the trees establishes our notion of random Hilbert schemes. To bound the probability that random Hilbert schemes are irreducible and nonsingular, we show that at least half of the vertices in the binary trees correspond to Hilbert schemes with unique Borel-fixed points.

Geometric Hamiltonian formulation of a variational problem depending on the covariant acceleration

We consider a second-order variational problem depending on the covariant acceleration, which is related to the notion of Riemannian cubic polynomials. This problem and the corresponding optimal control problem are described in the context of higher order tangent bundles using geometric tools. The main tool, a presymplectic variant of Pontryagin’s maximum principle, allows us to study the dynamics of the control problem.

Series representations of the remainders in the expansions for certain functions with applications

We present a summary of the series representations of the remainders in the expansions in ascending powers of t of 2/(et+1)2/(et+1) , sech t and coth t and establish simple bounds for these remainders when t>0t>0 . Several applications of these expansions are given which enable us to deduce some inequalities and completely monotonic functions associated with the ratio of two gamma functions. In addition, we derive a (presumably new) quadratic recurrence relation for the Bernoulli numbers Bn.

Switching Patterns and Steady-State Analysis of Grid-Connected and Stand-Alone Single-Stage Boost-Inverters for PV Applications

Renewable or sustainable energy (SE) sources have attracted the attention of many countries because the power generated is environmentally friendly, and the sources are not subject to the instability of price and availability. This dissertation presents new trends in the DC-AC converters (inverters) used in renewable energy sources, particularly for photovoltaic (PV) energy systems. A review of the existing technologies is performed for both single-phase and three-phase systems, and the pros and cons of the best candidates are investigated. In many modern energy conversion systems, a DC voltage, which is provided from a SE source or energy storage device, must be boosted and converted to an AC voltage with a fixed amplitude and frequency. A novel switching pattern based on the concept of the conventional space-vector pulse-width-modulated (SVPWM) technique is developed for single-stage, boost-inverters using the topology of current source inverters (CSI). The six main switching states, and two zeros, with three switches conducting at any given instant in conventional SVPWM techniques are modified herein into three charging states and six discharging states with only two switches conducting at any given instant. The charging states are necessary in order to boost the DC input voltage. It is demonstrated that the CSI topology in conjunction with the developed switching pattern is capable of providing the required residential AC voltage from a low DC voltage of one PV panel at its rated power for both linear and nonlinear loads. In a micro-grid, the active and reactive power control and consequently voltage regulation is one of the main requirements. Therefore, the capability of the single-stage boost-inverter in controlling the active power and providing the reactive power is investigated. It is demonstrated that the injected active and reactive power can be independently controlled through two modulation indices introduced in the proposed switching algorithm. The system is capable of injecting a desirable level of reactive power, while the maximum power point tracking (MPPT) dictates the desirable active power. The developed switching pattern is experimentally verified through a laboratory scaled three-phase 200W boost-inverter for both grid-connected and stand-alone cases and the results are presented.

Dynamic Image Precompensation for Improving Visual Performance of Computer Users with Ocular Aberrations

With the progress of computer technology, computers are expected to be more intelligent in the interaction with humans, presenting information according to the user's psychological and physiological characteristics. However, computer users with visual problems may encounter difficulties on the perception of icons, menus, and other graphical information displayed on the screen, limiting the efficiency of their interaction with computers. In this dissertation, a personalized and dynamic image precompensation method was developed to improve the visual performance of the computer users with ocular aberrations. The precompensation was applied on the graphical targets before presenting them on the screen, aiming to counteract the visual blurring caused by the ocular aberration of the user's eye. A complete and systematic modeling approach to describe the retinal image formation of the computer user was presented, taking advantage of modeling tools, such as Zernike polynomials, wavefront aberration, Point Spread Function and Modulation Transfer Function. The ocular aberration of the computer user was originally measured by a wavefront aberrometer, as a reference for the precompensation model. The dynamic precompensation was generated based on the resized aberration, with the real-time pupil diameter monitored. The potential visual benefit of the dynamic precompensation method was explored through software simulation, with the aberration data from a real human subject. An "artificial eye'' experiment was conducted by simulating the human eye with a high-definition camera, providing objective evaluation to the image quality after precompensation. In addition, an empirical evaluation with 20 human participants was also designed and implemented, involving image recognition tests performed under a more realistic viewing environment of computer use. The statistical analysis results of the empirical experiment confirmed the effectiveness of the dynamic precompensation method, by showing significant improvement on the recognition accuracy. The merit and necessity of the dynamic precompensation were also substantiated by comparing it with the static precompensation. The visual benefit of the dynamic precompensation was further confirmed by the subjective assessments collected from the evaluation participants.

E-polynomial of SL(2,C)-character varieties of complex curves of genus 3.

We compute the E-polynomials of the moduli spaces of representations of the fundamental group of a complex curve of genus g = 3 into SL(2, C), and also of the moduli space of twisted representations. The case of genus g = 1, 2 has already been done in [12]. We follow the geometric technique introduced in [12], based on stratifying the space of representations, and on the analysis of the behaviour of the E-polynomial under fibrations.

Multivariate orthogonal polynomials and integrable systems

Multivariate orthogonal polynomials in D real dimensions are considered from the perspective of the Cholesky factorization of a moment matrix. The approach allows for the construction of corresponding multivariate orthogonal polynomials, associated second kind functions, Jacobi type matrices and associated three term relations and also Christoffel-Darboux formulae. The multivariate orthogonal polynomials, their second kind functions and the corresponding Christoffel-Darboux kernels are shown to be quasi-determinants as well as Schur complements of bordered truncations of the moment matrix; quasi-tau functions are introduced. It is proven that the second kind functions are multivariate Cauchy transforms of the multivariate orthogonal polynomials. Discrete and continuous deformations of the measure lead to Toda type integrable hierarchy, being the corresponding flows described through Lax and Zakharov-Shabat equations; bilinear equations are found. Varying size matrix nonlinear partial difference and differential equations of the 2D Toda lattice type are shown to be solved by matrix coefficients of the multivariate orthogonal polynomials. The discrete flows, which are shown to be connected with a Gauss-Borel factorization of the Jacobi type matrices and its quasi-determinants, lead to expressions for the multivariate orthogonal polynomials and their second kind functions in terms of shifted quasi-tau matrices, which generalize to the multidimensional realm, those that relate the Baker and adjoint Baker functions to ratios of Miwa shifted tau-functions in the 1D scenario. In this context, the multivariate extension of the elementary Darboux transformation is given in terms of quasi-determinants of matrices built up by the evaluation, at a poised set of nodes lying in an appropriate hyperplane in R^D, of the multivariate orthogonal polynomials. The multivariate Christoffel formula for the iteration of m elementary Darboux transformations is given as a quasi-determinant. It is shown, using congruences in the space of semi-infinite matrices, that the discrete and continuous flows are intimately connected and determine nonlinear partial difference-differential equations that involve only one site in the integrable lattice behaving as a Kadomstev-Petviashvili type system. Finally, a brief discussion of measures with a particular linear isometry invariance and some of its consequences for the corresponding multivariate polynomials is given. In particular, it is shown that the Toda times that preserve the invariance condition lay in a secant variety of the Veronese variety of the fixed point set of the linear isometry.

On polynomials with given Hilbert function and applications

Using Macaulay's correspondence we study the family of Artinian Gorenstein local algebras with fixed symmetric Hilbert function decomposition. As an application we give a new lower bound for the dimension of cactus varieties of the third Veronese embedding. We discuss the case of cubic surfaces, where interesting phenomena occur.