The main result in this thesis is the generalisation of Bergman’s diamond lemma for ring theory to power series rings. This generalisation makes it possible to treat problems in which there arise infinite descending chains. Several results in the literature are shown to be special cases of this diamond lemma and examples are given of interesting problems which could not previously be treated. O...

Various concepts of orthogonality on the real line are reviewed that arise in connection with quadrature rules. Orthogonality relative to a positive measure and Gauss-type quadrature rules are classical. More recent types of orthogonality include orthogonality relative to a sign-variable measure, which arises in connection with Gauss-Kronrod quadrature, and power (or implicit) orthogonality enc...

' J. von Neumann, these PROCEEDINGS, 18, 70-82 (1932). 4E. Hopf, these PROCEEDINGS, 18, 93-100 (1932). 0G. D. Birkhoff, these PROCEEDINGS, 17, 656-660 (1931). 6 G. D. Birkhoff and P. Smith, Journal de Math., 7 [9],(1928), p. 365. 7 H. Poincar6, Calcul des Probabilitils, 2nd Edition, Paris, 1912, 320-333. 8 This remark is due to B. 0. Koopman who privately communicated it to the author two month...

We show that for almost every map in a transversal one-parameter family of piecewise expanding unimodal maps the Birkhoff sum of suitable observables along the forward orbit of the turning point satisfies the law of iterated logarithm. This result will follow from an almost sure invariance principle for the Birkhoff sum, as a function on the parameter space. Furthermore, we obtain a similar res...

We will prove that the Pierce-Birkhoff Conjecture holds for non-singular two-dimensional affine real algebraic varieties over real closed fields, i.e., if W is such a variety, then every piecewise polynomial function on W can be written as suprema of infima of polynomial functions on W . More precisely, we will give a proof of the so-called Connectedness Conjecture for the coordinate rings of s...

An algebra with two binary operations · and + that are commutative, associative, and idempotent is called a bisemilattice. A bisemilattice satisfying Birkhoff’s equation x · (x + y) = x + (x · y) is a Birkhoff system. Each bisemilattice determines, and is determined by, two semilattices, one for the operation + and one for the operation ·. A bisemilattice for which each of these semilattices is...

In this work we provide a framework for dynamic secret sharing and present the first dynamic and verifiable hierarchical secret sharing scheme based on Birkhoff interpolation. Since the scheme is dynamic it allows, without reconstructing the message distributed, to add and remove shareholders, to renew shares, and to modify the conditions for accessing the message. Furthermore, each shareholder...

It follows from Birkhoff’s Ergodic Theorem that the irregular set of points for which the Birkhoff averages of a given continuous function diverge has zero measure with respect to any finite invariant measure. In strong contrast, for systems with the weak specification property, we show here that if the irregular set is nonempty, then it is residual. This includes topologically transitive topol...