Metrical theory for α\alpha-Rosen fractions

The Rosen fractions form an infinite family which generalizes the nearest-integer continued fractions. In this paper we introduce a new class of continued fractions related to the Rosen fractions, the $\alpha$-Rosen fractions. The metrical properties of these $\alpha$-Rosen fractions are studied. We find planar natural extensions for the associated interval maps, and show that these regions are closely related to similar region for the 'classical' Rosen fraction. This allows us to unify and generalize results of diophantine approximation from the literature

Entropy quotients and correct digits in number-theoretic expansions

Expansions that furnish increasingly good approximations to real numbers are usually related to dynamical systems. Although comparing dynamical systems seems difficult in general, Lochs was able in 1964 to relate the relative speed of approximation of decimal and regular continued fraction expansions (almost everywhere) to the quotient of the entropies of their dynamical systems. He used detailed knowledge of the continued fraction operator. In 2001, a generalization of Lochs' result was given by Dajani and Fieldsteel in \citeDajF, describing the rate at which the digits of one number-theoretic expansion determine those of another. Their proofs are based on covering arguments and not on the dynamics of specific maps. In this paper we give a dynamical proof for certain classes of transformations, and we describe explicitly the distribution of the number of digits determined when comparing two expansions in integer bases. Finally, using this generalization of Lochs' result, we estimate the unknown entropy of certain number theoretic expansions by comparing the speed of convergence with that of an expansion with known entropy.Comment: Published at http://dx.doi.org/10.1214/074921706000000202 in the IMS Lecture Notes--Monograph Series (http://www.imstat.org/publications/lecnotes.htm) by the Institute of Mathematical Statistics (http://www.imstat.org

Tong's spectrum for Rosen continued fractions

The Rosen fractions are an infinite set of continued fraction algorithms, each giving expansions of real numbers in terms of certain algebraic integers. For each, we give a best possible upper bound for the minimum in appropriate consecutive blocks of approximation coefficients (in the sense of Diophantine approximation by continued fraction convergents). We also obtain metrical results for large blocks of ``bad'' approximations.Comment: 22 pages, 5 figure

Statistic and ergodic properties of Minkowski's diagonal continued fraction

AbstractRecently the author introduced a new class of continued fraction expansions, the S-expansions. Here it is shown that Minkowski's diagonal continued fraction (DCF) is an S-expansion. Due to this, statistic and ergodic properties of the DCF can be given

Town planning in the Netherlands since 1800

This article summarizes part of the gist of Town Planning in the Netherlands since 1800. Responses to Enlightenment Ideas and Geopolitical Realities, a book that discusses Dutch urbanism in its international setting, dividing its contents in a series of clusters that are presented as being determined by geopolitics, ideology, and planning. The timeframe of over 200 years (400 years if the prologue is included) highlights continuities and discontinuities that otherwise would have been lost – a strong motive in favor or writing books instead of articles. It defines urbanism as a combination of spatial planning (distributing human activities across space in cities, regions and on the global level) and design (one of its uses being that of a billboard for local identity, the community, the nation or political ideologies).In the two centuries of urban planning presented here, the Netherlands had to re-invent itself several times. Dutch urban history is marked by changes on the international scene, the prevailing political ideals, the development of modern planning as a distinct discipline and the continuous changes of the main countries that inspired Dutch planning – France until the 1830s, Germany from the 1830s to the 1930s, and since the 1930s increasingly the United States. The years between 1795 and 1815 marked the end of an era: though the nation still clung to the idea that it could play an important role in the world’s political affairs, its days as a ‘hyper power’ - to quote Amy Chua - were numbered. Nothing illustrated this more vividly than the run-down state of most of its cities. Especially in the province of Holland, many had become a faded imagine of their former self: comparing maps made during the so-called Golden Age with the brand new cadaster maps that had been ordered by the national government in the early nineteenth century, nobody could escape the impression that for almost two hundred years, nothing had changed. Some cities, for instance Enkhuizen, had even lost a large part of their inhabitants and demolished part of their buildings... In 1815 the Netherlands definitely abandoned the political structure that had characterized it in its heydays: the federal republic was replaced by a unitary state headed by a king, William I, a representative of the Orange family the fate of which was closely connected to the Netherlands since it gained independence in the late sixteenth century. William I, who had spent many years in England, embarked upon an ambitious campaign that should restore the country to its former glory, an ambition sparkled by the merger with the Southern Netherlands (now Belgium). Canals were dug and new roads, subdivided according to a French inspired classification system, were built that connected the two re-united states, accepting the distinct nature of both: the former Republic was expected to revitalize an economy based on trade, banking and agriculture, whereas the Southern half continued to promote industry. Combined with many other differences - the North was Calvinist and bourgeois, the South was Catholic and here the aristocracy never lost its privileged position - the merger proved untenable and after a short civil war, Belgium became independent (and subsequently developed into the continent’s first industrial state). William’s revanchist policies caused economic stagnation in the Northern half, the huge investments in its infrastructure proved futile

Modern Hospitals and Cultural Heritage

The decades between 1950 and 1980 mark the heydays of modern hospital architecture. It represents an ideal merger between Modernism and medicine and a highly specific approach to health and illness as medical qualities. Since the 1990s, public health experts have recognized that aspects that have been discarded both by medicine and by modern architecture should be re-integrated in all policies that target health: the modern hospital has become a relic of the past. This essay is a plea to incorporate the changing views on health and illness in the value assessment of the modern hospital