Notion Of Height Research Articles

Fibres non r\xe9duites d\u2019un sch\xe9ma arithm\xe9tique

(Non-reduced fibers of an arithmetic scheme) For a reduced projective scheme over the ring of integers of a number field, the set of places over which the fibres of the scheme are not reduced is a finite set. We give an explicit upper bound for the product of the norms of places in this set. For this purpose, we introduce a generalization of the notion of height over the adelic ring. We reduce the general case of a scheme of pure dimension to the case of a hypersurface by using the theory of Chow varieties. The case of a hypersurface is then treated with the help of the resultant of the equation of the hypersurface with some partial derivatives of the equation.

Mit dem Schreibblock in die Berge. Zur Höhen- und Bergmetaphorik in der Prosa Thomas Bernhards

This article analyses the metaphors of height and mountains in the prose work of Thomas Bernhard, interlinking them with his concept of writing. It argues that the vertical axis is marked by a historical index, which succinctly articulates the dilemma of the post-war writer, who is entangled in a genealogical and dynastic relationship with a generation of perpetrators. In Bernhard’s prose, the notion of height as material and cultural heritage is occupied by a “deadly parental inheritance” (›Beton‹) that renders the rhetoric of overcoming and sublimation futile. This essay thus examines the sociohistorical and poetological motivation of Bernhard’s critique of height, resulting in a deconstructivist form of writing.

Estimates for maximal functions associated with hypersurfaces in ℝ3 and related problems of harmonic analysis

We study the boundedness problem for maximal operators $ \mathcal{M} $ associated with averages along smooth hypersurfaces S of finite type in 3-dimensional Euclidean space. For p > 2, we prove that if no affine tangent plane to S passes through the origin and S is analytic, then the associated maximal operator is bounded on $ {L^p}\left( {{\mathbb{R}^3}} \right) $ if and only if p > h(S), where h(S) denotes the so-called height of the surface S (defined in terms of certain Newton diagrams). For non-analytic S we obtain the same statement with the exception of the exponent p = h(S). Our notion of height h(S) is closely related to A. N. Varchenko’s notion of height h(ϕ) for functions ϕ such that S can be locally represented as the graph of ϕ after a rotation of coordinates. Several consequences of this result are discussed. In particular we verify a conjecture by E. M. Stein and its generalization by A. Iosevich and E. Sawyer on the connection between the decay rate of the Fourier transform of the surface measure on S and the Lp-boundedness of the associated maximal operator $ \mathcal{M} $, and a conjecture by Iosevich and Sawyer which relates the Lp-boundedness of $ \mathcal{M} $ to an integrability condition on S for the distance to tangential hyperplanes, in dimension 3. In particular, we also give essentially sharp uniform estimates for the Fourier transform of the surface measure on S, thus extending a result by V. N. Karpushkin from the analytic to the smooth setting and implicitly verifying a conjecture by V. I. Arnold in our context. As an immediate application of this, we obtain an $ {L^p}\left( {{\mathbb{R}^3}} \right) - {L^2}(S) $ Fourier restriction theorem for S.

Prebasic submodules over valuation rings

Basic concepts in the theory of modules over valuation rings are introduced. The notion of height is used to define indicators of elements, whose irregularities are investigated. The indicator leads to the new notion of smoothness, a property which does not originate from abelian groups. Invariants generalizing the finite Ulm-Kaplansky invariants of abelian p-groups, as well as the Baer invariants for completely decomposable torsion-free abelian groups, are defined, and several results relating these invariants of a module to those of submodules are proved. All these concepts lead to the notion of prebasic submodules, which seems to be the right analogue of the basic subgroups in abelian groups.