A missing link: a reappraisal of the date, architectural context and significance of the great tower of Dudley Castle

Abstract The great tower of Dudley Castle, in the West Midlands, is re-examined in order to situate it within the evolutionary sequence of great tower designs. In so doing, it is argued that the origins of its plan are to be found in the works of the early to mid-thirteenth century, and that the tower itself was probably begun during the 1260s. Furthermore, it is asserted that the tower represents a milestone in the thinking that underpinned the redevelopment of castle mottes, and that it is to be seen as the prime connection between the circle-based plans that dominated motte redevelopments in the twelfth and thirteenth centuries, and later developments that led, ultimately, to the radically different, but architecturally successful scheme adopted by the builder of the donjon of Warkworth Castle in Northumberland. Résumé La grande tour du château de Dudley, dans la région de West Midlands, est réexaminée dans le but de la situer dans la séquence évolutionnaire des grands modèles de tours. Ce faisant, on soutient que les origines de son plan pourront se trouver dans les travaux exécutés entre le début et le milieu du treizième siècle, et que la tour elle-même avait probablement été commencée au cours des années 1260. En outre, on affirme que la tour représente un point marquant des idées à la base de la restructuration des mottes de châteaux, et qu’on peut la voir comme lien primordial entre les plans basés sur le cercle qui dominaient les restructurations des mottes au douzième et au treizième siècles, et les développements ultérieurs dont l’aboutissement final est le plan totalement différent, mais néanmoins une réussite au niveau de l’architecture, adopté par le bâtisseur du donjon du château de Warkworth au Northumberland. Zusammenfassung Der Großturm der Burg von Dudley in der Grafschaft West Midlands wird erneut untersucht um ihn in eine Entwicklungsabfolge von Großturmdesigns einzuordnen. Demzufolge sind die Ursprünge seines Grundrisses in den Arbeiten des frühen bis mittleren dreizehnten Jahrhunderts zu finden und der Turmbau wurde wahrscheinlich um 1260 begonnen. Darüber hinaus wird behauptet, daß der Turm ein Meilenstein in der Denkweise war, die zur Wiederentwicklung von Turmhügelburgen führte, und daß es als ein wichtigers Verbindungsglied zu den kreisförmigen Plänen fungiert, die für die Wiederentwicklung von Turmhügelburgen im zwölften und dreizehnten Jahrhunderten typisch war und späteren Entwicklungen, die zu radikal verschiedenen Entwürfen führten, und von den Erbauern des Burgfrieds der Warkworth Burg in der Grafschaft Northumberland übernommen wurde

The spectral shift function for compactly supported perturbations of Schr\"odinger operators on large bounded domains

We study the asymptotic behavior as L \to \infty of the finite-volume spectral shift function for a positive, compactly-supported perturbation of a Schr\"odinger operator in d-dimensional Euclidean space, restricted to a cube of side length L with Dirichlet boundary conditions. The size of the support of the perturbation is fixed and independent of L. We prove that the Ces\`aro mean of finite-volume spectral shift functions remains pointwise bounded along certain sequences L_n \to \infty for Lebesgue-almost every energy. In deriving this result, we give a short proof of the vague convergence of the finite-volume spectral shift functions to the infinite-volume spectral shift function as L \to\infty . Our findings complement earlier results of W. Kirsch [Proc. Amer. Math. Soc. 101, 509 - 512 (1987), Int. Eqns. Op. Th. 12, 383 - 391 (1989)] who gave examples of positive, compactly-supported perturbations of finite-volume Dirichlet Laplacians for which the pointwise limit of the spectral shift function does not exist for any given positive energy. Our methods also provide a new proof of the Birman--Solomyak formula for the spectral shift function that may be used to express the measure given by the infinite-volume spectral shift function directly in terms of the potential.Comment: Minor changes and some rearrangements; version as publishe

Edge Currents for Quantum Hall Systems, II. Two-Edge, Bounded and Unbounded Geometries

Devices exhibiting the integer quantum Hall effect can be modeled by one-electron Schroedinger operators describing the planar motion of an electron in a perpendicular, constant magnetic field, and under the influence of an electrostatic potential. The electron motion is confined to bounded or unbounded subsets of the plane by confining potential barriers. The edges of the confining potential barriers create edge currents. This is the second of two papers in which we review recent progress and prove explicit lower bounds on the edge currents associated with one- and two-edge geometries. In this paper, we study various unbounded and bounded, two-edge geometries with soft and hard confining potentials. These two-edge geometries describe the electron confined to unbounded regions in the plane, such as a strip, or to bounded regions, such as a finite length cylinder. We prove that the edge currents are stable under various perturbations, provided they are suitably small relative to the magnetic field strength, including perturbations by random potentials. The existence of, and the estimates on, the edge currents are independent of the spectral type of the operator.Comment: 57 page