Homomorphisms between diffeomorphism groups

For r at least 3, p at least 2, we classify all actions of the groups Diff^r_c(R) and Diff^r_+(S1) by C^p -diffeomorphisms on the line and on the circle. This is the same as describing all nontrivial group homomorphisms between groups of compactly supported diffeomorphisms on 1- manifolds. We show that all such actions have an elementary form, which we call topologically diagonal. As an application, we answer a question of Ghys in the 1-manifold case: if M is any closed manifold, and Diff(M)_0 injects into the diffeomorphism group of a 1-manifold, must M be 1 dimensional? We show that the answer is yes, even under more general conditions. Several lemmas on subgroups of diffeomorphism groups are of independent interest, including results on commuting subgroups and flows.Comment: Contains corrections and additional references. A revised version will appear in Ergodic Theory and Dynamical System

Die Entfaltung der Wirbeltiere

Während die klassische Darstellungsweise der Paläontologie stets vom Niederen zum Höheren und vom Alteren zum Jüngeren führt, entspricht die Fragestellung der paläontologischen Forschung einem vergleichenden Hin und Her zwischen den Lebensformen der Gegenwart und der Vergangenheit. Die folgende Darstellung versucht dem dadurch Ausdruck zu verleihen, daß die Gegenwart in ihr nicht nur als Ziel des Ausblicks, sondern auch als Ausgangspunkt der Betrachtung fungiert

Review of Graßmann, Robert, Theory of Number or Arithmetic in Strict Scientific Presentation by Strict Use of Formulas (1891)

The author of this book pursues the objective of treating the whole of pure mathematics [die ganze reine Mathematik] in four sections [Abtheilungen]. One half of the first of these sections is dedicated to arithmetic and is already available. The other half of the first section “A heuristic treatise on number [Zahlenlehre in freier Gedankenentwicklung]” which treats the same discipline is supposed to follow. The author may have opted for such an unusual separation [of the treatment of arithme..

Notwendige Änderungen der Altersrentenversicherung als soziale Infrastruktur der Marktwirtschaft Rentenbemessung nach Kinderzahl

In seinen Ausführungen zu einer Rentenreform, die die Rentenbemessung in Zusammenhang mit der Kinderzahl definiert, unterstützt Egon Hölder, Präsident des Statistischen Bundesamtes a.D., einen von Prof. Hans-Werner Sinn entwickelten Vorschlag zum Umbau der Altersrentenversicherung.Gesetzliche Rentenversicherung, Alterssicherung, Rentenreform, Kinderbetreuung, Deutschland

Intuition and Reasoning in Geometry

The way in which geometrical knowledge has been obtained has always attracted the attention of philosophers. The fact that there is a science that concerns things outside our thinking and that proceeds inferentially appeared striking, and gave rise to specific theories of experience and space. Nonetheless, the geometrical method has not yet been sufficiently investigated. Philosophers who investigate the theory of knowledge discuss the question of whether geometry is an empirical science, but..

Adolph Mayer : Nekrolog / gesprochen in der öffentlichen Gesamtsitzung beider Klassen am 14. November 1908 von O. Hölder

Adolf Mayer wurde am 15. Februar 1839 zu Leipzig geboren. Er studierte zuerst in Heidelberg anfänglich Chemie, dann auch Mathematik und Mineralogie, danach in Göttingen, Leipzig, wieder in Heidelberg und in Königsberg. Mit einer Habilitationsschrift über Variationsrechnung erhielt er 1866 in Leipzig die Venia legendi. 1871 wurde er Extraordinarius, 1890 Ordinarius daselbst. Im Jahre 1900 setzte er wegen Krankheit vorübergehend aus, anfangs 1908 mußte er die ihm liebgewordene Tätigkeit ganz einstellen. Er suchte Heilung im Süden, starb aber schon am 11. April 1908 in Gries bei Bozen. Seine Arbeiten gehören den Gebieten der Differentialgleichungen, der Variationsrechnung und der Mechanik an. (Rezension von Felix Müller (1843-1928) im Jahrbuch über die Fortschritte der Mathematik, Band 39. 1908, S. 40

An explicit height bound for the classical modular polynomial

For a prime m, let Phi_m be the classical modular polynomial, and let h(Phi_m) denote its logarithmic height. By specializing a theorem of Cohen, we prove that h(Phi_m) <= 6 m log m + 16 m + 14 sqrt m log m. As a corollary, we find that h(Phi_m) <= 6 m log m + 18 m also holds. A table of h(Phi_m) values is provided for m <= 3607.Comment: Minor correction to the constants in Theorem 1 and Corollary 9. To appear in the Ramanujan Journal. 17 pages

Extending structures I: the level of groups

Let $H$ be a group and $E$ a set such that $H \subseteq E$. We shall describe and classify up to an isomorphism of groups that stabilizes $H$ the set of all group structures that can be defined on $E$ such that $H$ is a subgroup of $E$. A general product, which we call the unified product, is constructed such that both the crossed product and the bicrossed product of two groups are special cases of it. It is associated to $H$ and to a system $\bigl((S, 1_S,\ast), \triangleleft, \, \triangleright, \, f \bigl)$ called a group extending structure and we denote it by $H \ltimes S$. There exists a group structure on $E$ containing $H$ as a subgroup if and only if there exists an isomorphism of groups $(E, \cdot) \cong H \ltimes S$, for some group extending structure $\bigl((S, 1_S,\ast), \triangleleft, \, \triangleright, \, f \bigl)$. All such group structures on $E$ are classified up to an isomorphism of groups that stabilizes $H$ by a cohomological type set ${\mathcal K}^{2}_{\ltimes} (H, (S, 1_S))$. A Schreier type theorem is proved and an explicit example is given: it classifies up to an isomorphism that stabilizes $H$ all groups that contain $H$ as a subgroup of index 2.Comment: 17 pages; to appear in Algebras and Representation Theor

Nature of complex singularities for the 2D Euler equation

A detailed study of complex-space singularities of the two-dimensional incompressible Euler equation is performed in the short-time asymptotic r\'egime when such singularities are very far from the real domain; this allows an exact recursive determination of arbitrarily many spatial Fourier coefficients. Using high-precision arithmetic we find that the Fourier coefficients of the stream function are given over more than two decades of wavenumbers by \hat F(\k) = C(\theta) k^{-\alpha} \ue ^ {-k \delta(\theta)}, where \k = k(\cos \theta, \sin \theta). The prefactor exponent $\alpha$, typically between 5/2 and 8/3, is determined with an accuracy better than 0.01. It depends on the initial condition but not on $\theta$. The vorticity diverges as $s^{-\beta}$, where $\alpha+\beta= 7/2$ and $s$ is the distance to the (complex) singular manifold. This new type of non-universal singularity is permitted by the strong reduction of nonlinearity (depletion) which is associated to incompressibility. Spectral calculations show that the scaling reported above persists well beyond the time of validity of the short-time asymptotics. A simple model in which the vorticity is treated as a passive scalar is shown analytically to have universal singularities with exponent $\alpha =5/2$.Comment: 22 pages, 24 figures, published version; a version of the paper with higher-quality figures is available at http://www.obs-nice.fr/etc7/euler.pd