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.

A Study of a Comprehensive Career Services System in Chinese Universities

Since 1999, with the adoption of expansion policy in higher education by the Chinese government, enrollment and graduate numbers have been increasing at an unprecedented speed. Accustomed to a system in which university graduates were placed, many students are not trained in “selling themselves”, which exacerbates the situation leading to a skyrocketing unemployment rate among new graduates. The idea of emphasizing career services comes with increasing employment pressure among university graduates in recent years. The 1998 “Higher Education Act” made it a legislative requirement. Thereafter, the Ministry of Education issued a series of documents in order to promote the development of career services. All higher education institutions are required to set up special career service centers and to set a ratio of 1:500 between career staff and the total number of students. Related career management courses, especially career planning classes, are required to be clearly included as specific modules into the teaching plan with a requirement of no less than 38 sessions in one semester at all universities. Developing career services in higher education has thus become a hot issue. One of the more notable trends in higher education in recent years has been the transformation of university career service centers from merely being the coordinators of on-campus placement into full service centers for international career development. The traditional core of career services in higher education had been built around guidance, information and placements (Watts, 1997). This core was still in place, but the role of higher education career services has changed considerably in recent years and the nature of each part is being transformed (Watts, 1997). Most services are undertaking a range of additional activities, and the career guidance issue is emphasized much more than before. Career management courses, especially career planning classes, are given special focus in developing career services in the Chinese case. This links career services clearly and directly with the course provision function. In China, most career service centers are engaging in the transformation period from a “management-oriented” organization to a “service-oriented” organization. Besides guidance services, information services and placement activities, there is a need to blend them together with the new additional teaching function, which follows the general trend as regulated by the government. The role of career services has been expanding and this has brought more challenges to its development in Chinese higher education. Chinese universities still remain in the period of exploration and establishment in developing their own career services. In the face of the new situation, it is very important and meaningful to explore and establish a comprehensive career services system to address student needs in the universities. A key part in developing this system is the introduction of career courses and delivering related career management skills to the students. So there is the need to restructure the career service sectors within the Chinese universities in general. The career service centers will operate as a hub and function as a spoke in the wheel of this model system, providing support and information to staff located in individual teaching departments who are responsible for the delivery of career education, information, advice and guidance. The career service centers will also provide training and career planning classes. The purpose of establishing a comprehensive career services system is to provide a strong base for student career development. The students can prepare themselves well in psychology, ideology and ability before employment with the assistance of effective career services. To conclude, according to the different characteristics and needs of students, there will be appropriate services and guidance in different stages and different ways. In other words, related career services and career guidance activities would be started for newly enrolled freshmen and continue throughout their whole university process. For the operation of a comprehensive services system, there is a need for strong support by the government in the form of macro-control and policy guarantee, but support by the government in the form of macro-control and policy guarantee, but also a need for close cooperation with the academic administration and faculties to be actively involved in career planning and employment programs. As an integral function within the universities, career services must develop and maintain productive relationships with relevant campus offices and key stakeholders both within the universities and externally.