Some Analytic Aspects Concerning the Collatz Problem

A series of relatively simple equivalences to the Collatz conjecture, concerning the Collatz mapping [...] are presented. [...] The main topic of this paper consists in investigating a certain linear equation in the space of special Dirichlet series. The conjecture that this equation possesses a null space of dimension 1, generated by the Riemann zeta function, is equivalent to the Collatz conjecture. A number of analytic properties of the operator, which defines the linear equation, is given, some of them concern problems of analytic continuation in the complex domain. A few remarks with respect to generalizations of those problems conclude the paper

Ägypten vor den Wahlen : Historische Entscheidung am Nil

In Ägypten, dem bevölkerungsreichsten arabischen Land, sind für den 28. November Parlamentswahlen angesetzt. Erstmals in ihrer langen Geschichte werden die Menschen in dem Land am Nil die Gelegenheit haben, in freien und pluralistischen Wahlen ihre Vertreter zu bestimmen. Das Wahlverfahren ist mehrstufig und höchstkompliziert. Neben den Abgeordneten des Unterhauses können die Ägypter in getrennten Wahlen auch die Mitglieder des Oberhauses bestimmen. Wenn alles nach Plan verläuft, soll im März des neuen Jahres das endgültige Ergebnis des historischen Urnengangs vorliegen. Die historische Öffnung des politischen Systems ist das wichtigste Ergebnis der „Revolution von 25. Januar“, die das Ende der langjährigen Herrschaft des Präsidenten Hosni Mubarak besiegelte. Gleichwohl mehren sich die Stimmen, die von der unvollendeten, ja verratenen Revolution sprechen. Hierzu passt das Bild von Ägyptens Liberalen, die in einem Klima fortschreitender gesellschaftspolitischer Islamisierung um das politische Überleben kämpfen

More Results on B-Splines via Recurrence Relations

The present paper is to be understood as a revision and continuation of a recent series of publications on recursively defined B-splines, due to C. deBoor and K. Höllig [4] and G. Ciascola [6, 7]. We define B-splines as solutions of a certain difference equation of evolution type, which is equivalent to the well-known B-spline recurrence relation. It turns out that this approach leads to very elementary and direct proofs, in particular without using the Marsden identity, of many important properties of the B-splines. Special emphasis is also laid on the probability theoretic aspects of these functions. Among other results, we prove explicit formulas for the kth moments, the variance and the distribution function of a B-spline

A typology of church attitudes towards the state with special reference to the German church controversy, 1933-1945.

Thesis (Ph.D.)--Boston UniversityThe purpose of this investigation is to trace the development of the various attitudes of the church towards the state and to establish some definite attitudinal types with special reference to the church controversy in Germany from 1933 - 1945. On account of the complexity of the subject, the study makes use of three disciplines: theology, history of religion and sociology of religion. The church controversy in Germany will enter the history of the universal church as a twentieth century revelation of the original problem of the attitude of the Christian Church towards the state. The attitudes of opposition, accommodation, co-ordination and identification express each in its own way the desire to find a solution to the perplexing problem of the relationship of the two institutions. [TRUNCATED

On the Chebyshev Norm of Polynomial B-Splines

Polynomial B-splines of given order m and with knots of arbitrary multiplicity are investigated with respect to their Chebyshev norm. We present a complete characterization of those B-splines with maximal and with minimal norm, compute these norms explicitly and study their behavior as m tends to infinity. Furthermore, the norm of the B-spline corresponding to the equidistant distribution of knots is studied. Finally, we analyse those types of knot distributions, for which the norms of the corresponding B-splines converge to zero as m ∞