Bieberbach's Conjecture, the de Branges and Weinstein Functions and the Askey-Gasper Inequality

The Bieberbach conjecture about the coefficients of univalent functions of the unit disk was formulated by Ludwig Bieberbach in 1916 [Bieberbach1916]. The conjecture states that the coefficients of univalent functions are majorized by those of the Koebe function which maps the unit disk onto a radially slit plane. The Bieberbach conjecture was quite a difficult problem, and it was surprisingly proved by Louis de Branges in 1984 [deBranges1985] when some experts were rather trying to disprove it. It turned out that an inequality of Askey and Gasper [AskeyGasper1976] about certain hypergeometric functions played a crucial role in de Branges' proof. In this article I describe the historical development of the conjecture and the main ideas that led to the proof. The proof of Lenard Weinstein (1991) [Weinstein1991] follows, and it is shown how the two proofs are interrelated. Both proofs depend on polynomial systems that are directly related with the Koebe function. At this point algorithms of computer algebra come into the play, and computer demonstrations are given that show how important parts of the proofs can be automated.

A generalization of Student’s t-distribution from the viewpoint of special functions

Student’s t-distribution has found various applications in mathematical statistics. One of the main properties of the t-distribution is to converge to the normal distribution as the number of samples tends to infinity. In this paper, by using a Cauchy integral we introduce a generalization of the t-distribution function with four free parameters and show that it converges to the normal distribution again. We provide a comprehensive treatment of mathematical properties of this new distribution. Moreover, since the Fisher F-distribution has a close relationship with the t-distribution, we also introduce a generalization of the F-distribution and prove that it converges to the chi-square distribution as the number of samples tends to infinity. Finally some particular sub-cases of these distributions are considered.

Some New Classes of Orthogonal Polynomials and Special Functions

In dieser Dissertation präsentieren wir zunächst eine Verallgemeinerung der üblichen Sturm-Liouville-Probleme mit symmetrischen Lösungen und erklären eine umfassendere Klasse. Dann führen wir einige neue Klassen orthogonaler Polynome und spezieller Funktionen ein, welche sich aus dieser symmetrischen Verallgemeinerung ableiten lassen. Als eine spezielle Konsequenz dieser Verallgemeinerung führen wir ein Polynomsystem mit vier freien Parametern ein und zeigen, dass in diesem System fast alle klassischen symmetrischen orthogonalen Polynome wie die Legendrepolynome, die Chebyshevpolynome erster und zweiter Art, die Gegenbauerpolynome, die verallgemeinerten Gegenbauerpolynome, die Hermitepolynome, die verallgemeinerten Hermitepolynome und zwei weitere neue endliche Systeme orthogonaler Polynome enthalten sind. All diese Polynome können direkt durch das neu eingeführte System ausgedrückt werden. Ferner bestimmen wir alle Standardeigenschaften des neuen Systems, insbesondere eine explizite Darstellung, eine Differentialgleichung zweiter Ordnung, eine generische Orthogonalitätsbeziehung sowie eine generische Dreitermrekursion. Außerdem benutzen wir diese Erweiterung, um die assoziierten Legendrefunktionen, welche viele Anwendungen in Physik und Ingenieurwissenschaften haben, zu verallgemeinern, und wir zeigen, dass diese Verallgemeinerung Orthogonalitätseigenschaft und -intervall erhält. In einem weiteren Kapitel der Dissertation studieren wir detailliert die Standardeigenschaften endlicher orthogonaler Polynomsysteme, welche sich aus der üblichen Sturm-Liouville-Theorie ergeben und wir zeigen, dass sie orthogonal bezüglich der Fisherschen F-Verteilung, der inversen Gammaverteilung und der verallgemeinerten t-Verteilung sind. Im nächsten Abschnitt der Dissertation betrachten wir eine vierparametrige Verallgemeinerung der Studentschen t-Verteilung. Wir zeigen, dass diese Verteilung gegen die Normalverteilung konvergiert, wenn die Anzahl der Stichprobe gegen Unendlich strebt. Eine ähnliche Verallgemeinerung der Fisherschen F-Verteilung konvergiert gegen die chi-Quadrat-Verteilung. Ferner führen wir im letzten Abschnitt der Dissertation einige neue Folgen spezieller Funktionen ein, welche Anwendungen bei der Lösung in Kugelkoordinaten der klassischen Potentialgleichung, der Wärmeleitungsgleichung und der Wellengleichung haben. Schließlich erklären wir zwei neue Klassen rationaler orthogonaler hypergeometrischer Funktionen, und wir zeigen unter Benutzung der Fouriertransformation und der Parsevalschen Gleichung, dass es sich um endliche Orthogonalsysteme mit Gewichtsfunktionen vom Gammatyp handelt.

Duplication coefficients via generating functions

In this paper, we solve the duplication problem P_n(ax) = sum_{m=0}^{n}C_m(n,a)P_m(x) where {P_n}_{n>=0} belongs to a wide class of polynomials, including the classical orthogonal polynomials (Hermite, Laguerre, Jacobi) as well as the classical discrete orthogonal polynomials (Charlier, Meixner, Krawtchouk) for the specific case a = −1. We give closed-form expressions as well as recurrence relations satisfied by the duplication coefficients.

Functions satisfying holonomic q-differential equations

In a similar manner as in some previous papers, where explicit algorithms for finding the differential equations satisfied by holonomic functions were given, in this paper we deal with the space of the q-holonomic functions which are the solutions of linear q-differential equations with polynomial coefficients. The sum, product and the composition with power functions of q-holonomic functions are also q-holonomic and the resulting q-differential equations can be computed algorithmically.

Thermodynamic functions of element 105 in neutral and ionized states

The basic thermodynamic functions, the entropy, free energy, and enthalpy, for element 105 (hahnium) in electronic configurations d^3 s^2, d^3 sp, and d^4s^1 and for its +5 ionized state (5f^14) have been calculated as a function of temperature. The data are based on the results of the calculations of the corresponding electronic states of element 105 using the multiconfiguration Dirac-Fock method.

Priors Stabilizers and Basis Functions: From Regularization to Radial, Tensor and Additive Splines

We had previously shown that regularization principles lead to approximation schemes, as Radial Basis Functions, which are equivalent to networks with one layer of hidden units, called Regularization Networks. In this paper we show that regularization networks encompass a much broader range of approximation schemes, including many of the popular general additive models, Breiman's hinge functions and some forms of Projection Pursuit Regression. In the probabilistic interpretation of regularization, the different classes of basis functions correspond to different classes of prior probabilities on the approximating function spaces, and therefore to different types of smoothness assumptions. In the final part of the paper, we also show a relation between activation functions of the Gaussian and sigmoidal type.

Hilbert space on probability density functions with Aitchison geometry

Compositional data analysis motivated the introduction of a complete Euclidean structure in the simplex of D parts. This was based on the early work of J. Aitchison (1986) and completed recently when Aitchinson distance in the simplex was associated with an inner product and orthonormal bases were identified (Aitchison and others, 2002; Egozcue and others, 2003). A partition of the support of a random variable generates a composition by assigning the probability of each interval to a part of the composition. One can imagine that the partition can be refined and the probability density would represent a kind of continuous composition of probabilities in a simplex of infinitely many parts. This intuitive idea would lead to a Hilbert-space of probability densities by generalizing the Aitchison geometry for compositions in the simplex into the set probability densities

Comparing methods for dimensionality reduction when data are density functions

Functional Data Analysis (FDA) deals with samples where a whole function is observed for each individual. A particular case of FDA is when the observed functions are density functions, that are also an example of infinite dimensional compositional data. In this work we compare several methods for dimensionality reduction for this particular type of data: functional principal components analysis (PCA) with or without a previous data transformation and multidimensional scaling (MDS) for diferent inter-densities distances, one of them taking into account the compositional nature of density functions. The difeerent methods are applied to both artificial and real data (households income distributions)

Special Functions

Exam questions and solutions in LaTex

Special Functions pdf

Exam questions and solutions in PDF

Vector functions and curves

Exercises and solutions about vector functions and curves.

Mnesic performance and executive functions in patients with multiple sclerosis (MS)

The Episodic Memory (EM) and the Executive Functions (EF) are cognitive areas that are affected in patients with diagnosis of Multiple Sclerosis (MS). Nowadays there exists scarce works destined to explore the infl uence of the EF on measures of mnesic performance in MS. For this reason, we analyze the effect of the EF on the performance in a set of memory measures. We worked with a clinical group (n=36) and with a control group (n=36) compared by age and educational level. The results show that the clinical group obtained significantly low average values in all the mnesic indexes (with exception of recognition) and in all the executive measures. All the executive indexes showed significant associations with some of the indexes of mnesic performance. These findings suggest that the problems in the episodic memory in EM patients could be analyzed as the manifestation of a global disorder that could be similar to the one that involves the EF. 

Rehabilitation of executive functions: Implications and strategies

Executive Functions (EF) concern a range of abilitiesincluding problem-solving, planning, initiation, selfmonitoring,conscious attention, cope with new situationsand the ability to modify plans if necessary. It’s ahigh cognitive function that is crucial for a person to getengaged and maintain daily activities whilst keeping agood quality of life. Problems in the EF were formerlyknown as Dysexecutive Syndrome (DS). There are manymodels concerning DS, although the literature on thesubject still remains unclear. Several works appoint theeffects brought by elderly life, as well as abuse of drugsand some psychopathologies. These factors are knownto increase the distress of the frontal circuits and thatcould be associated to executive deficits. The effects ofDS would compromise individuals in day-to-day routine,academic, social and labor fields. There is a growingbody of studies trying to determine the causes, implications,associations and the best way to take care of theseeffects. This work intends to review DS, focusing on themost important fields related to this area, such as psychopathologyassociations, cognitive reserve, assessmentand cognitive rehabilitation programs.

Executive functions in school-aged children: age and socioeconomic status effects

Lately, the study of prefrontal executive functions in grade scholars has noticeably increased. The aim of this study is to investigate the influence of age and socioeconomic status (sEs) on executive tasks performance and to analyze those socioeconomic variables that predict a better execution. A sample of 254 children aged between 7 and 12 years from the city of santa Fe, Argentina and belonging to different socioeconomic status were tested. A bat- tery of executive functions sensitive to prefrontal function was used to obtain the results. These in- dicate a significant influence of age and SES on executive functions. The cognitive patterns follow a different path according to the development and sEs effect. Besides, it is revealed a pattern of low cognitive functioning in low-sEs children in all executive functions. Finally, from the variables included in this study, it was found that only the educational level of the mother and the housing conditions are associated to the children’s executive function. The results are discussed in terms of the influence of the cerebral maturation and the envi- ronmental variables in the executive functioning.