Hermite Polynomials and the Zero-Distribution of Riemann's Zeta Function

AMS Subject Classification 2010: 11M26, 33C45, 42A38.

Some Properties of Mittag-Leffler Functions and Matrix-Variate Analogues: A Statistical Perspective

Mathematical Subject Classification 2010:26A33, 33E99, 15A52, 62E15.

Special Functions and Pathways for Problems in Astrophysics: An Essay in Honor of A.M. Mathai

The paper provides a review of A.M. Mathai's applications of the theory of special functions, particularly generalized hypergeometric functions, to problems in stellar physics and formation of structure in the Universe and to questions related to reaction, diffusion, and reaction-diffusion models. The essay also highlights Mathai's recent work on entropic, distributional, and differential pathways to basic concepts in statistical mechanics, making use of his earlier research results in information and statistical distribution theory. The results presented in the essay cover a period of time in Mathai's research from 1982 to 2008 and are all related to the thematic area of the gravitationally stabilized solar fusion reactor and fractional reaction-diffusion, taking into account concepts of non-extensive statistical mechanics. The time period referred to above coincides also with Mathai's exceptional contributions to the establishment and operation of the Centre for Mathematical Sciences, India, as well as the holding of the United Nations (UN)/European Space Agency (ESA)/National Aeronautics and Space Administration (NASA) of the United States/ Japanese Aerospace Exploration Agency (JAXA) Workshops on basic space science and the International Heliophysical Year 2007, around the world. Professor Mathai's contributions to the latter, since 1991, are a testimony for his social con-science applied to international scientific activity.

Dependence Structure of some Bivariate Distributions

Dependence in the world of uncertainty is a complex concept. However, it exists, is asymmetric, has magnitude and direction, and can be measured. We use some measures of dependence between random events to illustrate how to apply it in the study of dependence between non-numeric bivariate variables and numeric random variables. Graphics show what is the inner dependence structure in the Clayton Archimedean copula and the Bivariate Poisson distribution. We know this approach is valid for studying the local dependence structure for any pair of random variables determined by its empirical or theoretical distribution. And it can be used also to simulate dependent events and dependent r/v/’s, but some restrictions apply. ACM Computing Classification System (1998): G.3, J.2.

Bayesian Prediction of Weibull Distribution Based on Fixed and Random Sample Size

2000 Mathematics Subject Classification: 62E16, 65C05, 65C20.

Indication strength of coenological similarity patterns based on genus-level taxon lists and prevalence distribution

Several methods and indicators can be used to evaluate the coenological state of a given habitat, the ones which can be created simply, quickly, standardizably and reliably and which can be used to exactly quantify the state of a given habitat in point of numbers can be of outstanding practical importance in ecology. One possible method is the examination of the genera which can be found in a given habitat in great abundance and have little number of species and various ecological characteristics. For this purpose one of the most appropriate groups is that of ground-dwelling oribatid mites (Acari: Oribatida). In our research, joining the bioindication methodological project of the “Adaptation to Climate Change” Research Group of the Hungarian Academy of Sciences, the indication strength of genus-level taxon lists and the effects of the main pattern-generating factors creating similarity patterns were analysed with the help of data series on oribatid mites collected by us and originating from literature. Our aim was to develop a method with the help of which the difference expressed with distance functions between two oribatid mite genus lists originating from any sources can correspond to spatial and temporal scales. Our results prove that these genus lists are able to express the spatial distance of the habitats. With the help of this base of comparison changes in disturbed or transformed habitats can be expressed by means of oribatid mite communities, with spatial and temporal distances.

Phylogenetic and ontogenetic influences on the distribution of anthocyanins and betacyanins in leaves of tropical plants

We examined the anatomy of expanding, mature, and senescing leaves of tropical plants for the presence of red pigments: anthocyanins and betacyanins. We studied 463 species in total, 370 genera, belonging to 94 families. This included 21 species from five families in the Caryophyllales, where betacyanins are the basis for red color. We also included 14 species of ferns and gymnosperms in seven families and 29 species with undersurface coloration at maturity. We analyzed 399 angiosperm species (74 families) for factors (especially developmental and evolutionary) influencing anthocyanin production during expansion and senescence. During expansion, 44.9% produced anthocyanins and only 13.5% during senescence. At both stages, relatively few patterns of tissue distributions developed, primarily in the mesophyll, and very few taxa produced anthocyanins in dermal and ground tissue simultaneously. Of the 35 species producing anthocyanins both in development and senescence, most had similar cellular distributions. Anthocyanin distributions were identical in different developing leaves of three heteroblastic taxa. Phylogeny has influenced the distribution of anthocyanins in the epidermis and mesophyll of expanding leaves and the palisade parenchyma during senescence, although these influences are not strong. Betacyanins appear to have similar distributions in leaves of taxa within the Caryophyllales and, perhaps, similar functions. The presence of anthocyanins in the mesophyll of so many species is inconsistent with the hypothesis of protection against UV damage or fungal pathogens, and the differing tissue distributions indicate that the pigments may function in different ways, as in photoprotection and freeradical scavenging.

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.

Winter sea surface temperature reconstruction from planktic foraminiferal transfer functions in the northeastern Arabian Sea over the last two millennia

The Indian monsoon system is an important climate feature of the northern Indian Ocean. Small variations of the wind and precipitation patterns have fundamental influence on the societal, agricultural, and economic development of India and its neighboring countries. To understand current trends, sensitivity to forcing, or natural variation, records beyond the instrumental period are needed. However, high-resolution archives of past winter monsoon variability are scarce. One potential archive of such records are marine sediments deposited on the continental slope in the NE Arabian Sea, an area where present-day conditions are dominated by the winter monsoon. In this region, winter monsoon conditions lead to distinctive changes in surface water properties, affecting marine plankton communities that are deposited in the sediment. Using planktic foraminifera as a sensitive and well-preserved plankton group, we first characterize the response of their species distribution on environmental gradients from a dataset of surface sediment samples in the tropical and sub-tropical Indian Ocean. Transfer functions for quantitative paleoenvironmental reconstructions were applied to a decadal-scale record of assemblage counts from the Pakistan Margin spanning the last 2000?years. The reconstructed temperature record reveals an intensification of winter monsoon intensity near the year 100 CE. Prior to this transition, winter temperatures were >1.5°C warmer than today. Conditions similar to the present seem to have established after 450 CE, interrupted by a singular event near 950 CE with warmer temperatures and accordingly weak winter monsoon. Frequency analysis revealed significant 75-, 40-, and 37-year cycles, which are known from decadal- to centennial-scale resolution records of Indian summer monsoon variability and interpreted as solar irradiance forcing. Our first independent record of Indian winter monsoon activity confirms that winter and summer monsoons were modulated on the same frequency bands and thus indicates that both monsoon systems are likely controlled by the same driving force.

Dynamic fator Models for bivariate Count Data: an application to fire activity

The study of forest re activity, in its several aspects, is essencial to understand the phenomenon and to prevent environmental public catastrophes. In this context the analysis of monthly number of res along several years is one aspect to have into account in order to better comprehend this tematic. The goal of this work is to analyze the monthly number of forest res in the neighboring districts of Aveiro and Coimbra, Portugal, through dynamic factor models for bivariate count series. We use a bayesian approach, through MCMC methods, to estimate the model parameters as well as to estimate the common latent factor to both series.

STATISTICAL METHODS FOR ADVANCED ECONOMETRIC MODELS WITH APPLICATIONS TO VEHICLE HOLDING AND SPEED QUANTILES DISTRIBUTION

This dissertation proposes statistical methods to formulate, estimate and apply complex transportation models. Two main problems are part of the analyses conducted and presented in this dissertation. The first method solves an econometric problem and is concerned with the joint estimation of models that contain both discrete and continuous decision variables. The use of ordered models along with a regression is proposed and their effectiveness is evaluated with respect to unordered models. Procedure to calculate and optimize the log-likelihood functions of both discrete-continuous approaches are derived, and difficulties associated with the estimation of unordered models explained. Numerical approximation methods based on the Genz algortithm are implemented in order to solve the multidimensional integral associated with the unordered modeling structure. The problems deriving from the lack of smoothness of the probit model around the maximum of the log-likelihood function, which makes the optimization and the calculation of standard deviations very difficult, are carefully analyzed. A methodology to perform out-of-sample validation in the context of a joint model is proposed. Comprehensive numerical experiments have been conducted on both simulated and real data. In particular, the discrete-continuous models are estimated and applied to vehicle ownership and use models on data extracted from the 2009 National Household Travel Survey. The second part of this work offers a comprehensive statistical analysis of free-flow speed distribution; the method is applied to data collected on a sample of roads in Italy. A linear mixed model that includes speed quantiles in its predictors is estimated. Results show that there is no road effect in the analysis of free-flow speeds, which is particularly important for model transferability. A very general framework to predict random effects with few observations and incomplete access to model covariates is formulated and applied to predict the distribution of free-flow speed quantiles. The speed distribution of most road sections is successfully predicted; jack-knife estimates are calculated and used to explain why some sections are poorly predicted. Eventually, this work contributes to the literature in transportation modeling by proposing econometric model formulations for discrete-continuous variables, more efficient methods for the calculation of multivariate normal probabilities, and random effects models for free-flow speed estimation that takes into account the survey design. All methods are rigorously validated on both real and simulated data.

Benthic foraminiferal distribution on the southeastern Brazilian shelf and upper slope

Foraminiferal data were obtained from 66 samples of box cores on the southeastern Brazilian upper margin (between 23.8A degrees-25.9A degrees S and 42.8A degrees-46.13A degrees W) to evaluate the benthic foraminiferal fauna distribution and its relation to some selected abiotic parameters. We focused on areas with different primary production regimes on the southern Brazilian margin, which is generally considered as an oligotrophic region. The total density (D), richness (R), mean diversity (H) over bar', average living depth (ALD(X) ) and percentages of specimens of different microhabitats (epifauna, shallow infauna, intermediate infauna and deep infauna) were analyzed. The dominant species identified were Uvigerina spp., Globocassidulina subglobosa, Bulimina marginata, Adercotryma wrighti, Islandiella norcrossi, Rhizammina spp. and Brizalina sp.. We also established a set of mathematical functions for analyzing the vertical foraminiferal distribution patterns, providing a quantitative tool that allows correlating the microfaunal density distributions with abiotic factors. In general, the cores that fit with pure exponential decaying functions were related to the oligotrophic conditions prevalent on the Brazilian margin and to the flow of the Brazilian Current (BC). Different foraminiferal responses were identified in cores located in higher productivity zones, such as the northern and the southern region of the study area, where high percentages of infauna were encountered in these cores, and the functions used to fit these profiles differ appreciably from a pure exponential function, as a response of the significant living fauna in deeper layers of the sediment. One of the main factors supporting the different foraminiferal assemblage responses may be related to the differences in primary productivity of the water column and, consequently, in the estimated carbon flux to the sea floor. Nevertheless, also bottom water velocities, substrate type and water depth need to be considered.

Modeling Flood Stage-Duration-Frequency: A Risk Assessment of Critical Infrastructure in the Tidal Potomac

The service of a critical infrastructure, such as a municipal wastewater treatment plant (MWWTP), is taken for granted until a flood or another low frequency, high consequence crisis brings its fragility to attention. The unique aspects of the MWWTP call for a method to quantify the flood stage-duration-frequency relationship. By developing a bivariate joint distribution model of flood stage and duration, this study adds a second dimension, time, into flood risk studies. A new parameter, inter-event time, is developed to further illustrate the effect of event separation on the frequency assessment. The method is tested on riverine, estuary and tidal sites in the Mid-Atlantic region. Equipment damage functions are characterized by linear and step damage models. The Expected Annual Damage (EAD) of the underground equipment is further estimated by the parametric joint distribution model, which is a function of both flood stage and duration, demonstrating the application of the bivariate model in risk assessment. Flood likelihood may alter due to climate change. A sensitivity analysis method is developed to assess future flood risk by estimating flood frequency under conditions of higher sea level and stream flow response to increased precipitation intensity. Scenarios based on steady and unsteady flow analysis are generated for current climate, future climate within this century, and future climate beyond this century, consistent with the WWTP planning horizons. The spatial extent of flood risk is visualized by inundation mapping and GIS-Assisted Risk Register (GARR). This research will help the stakeholders of the critical infrastructure be aware of the flood risk, vulnerability, and the inherent uncertainty.

La distribution des zéros des fonctions L

Selon la philosophie de Katz et Sarnak, la distribution des zéros des fonctions $L$ est prédite par le comportement des valeurs propres de matrices aléatoires. En particulier, le comportement des zéros près du point central révèle le type de symétrie de la famille de fonctions $L$. Une fois la symétrie identifiée, la philosophie de Katz et Sarnak conjecture que plusieurs statistiques associées aux zéros seront modélisées par les valeurs propres de matrices aléatoires du groupe correspondant. Ce mémoire étudiera la distribution des zéros près du point central de la famille des courbes elliptiques sur $\mathbb{Q}[i]$. Brumer a effectué ces calculs en 1992 sur la famille de courbes elliptiques sur $\mathbb{Q}$. Les nouvelles problématiques reliées à la généralisation de ses travaux vers un corps de nombres seront mises en évidence

A Study on the Correlation of Bivariate And Trivariate Normal Models

Suppose two or more variables are jointly normally distributed. If there is a common relationship between these variables it would be very important to quantify this relationship by a parameter called the correlation coefficient which measures its strength, and the use of it can develop an equation for predicting, and ultimately draw testable conclusion about the parent population. This research focused on the correlation coefficient ρ for the bivariate and trivariate normal distribution when equal variances and equal covariances are considered. Particularly, we derived the maximum Likelihood Estimators (MLE) of the distribution parameters assuming all of them are unknown, and we studied the properties and asymptotic distribution of . Showing this asymptotic normality, we were able to construct confidence intervals of the correlation coefficient ρ and test hypothesis about ρ. With a series of simulations, the performance of our new estimators were studied and were compared with those estimators that already exist in the literature. The results indicated that the MLE has a better or similar performance than the others.