“Are the genre and the Geschlecht one and the same number?” An inquiry into Alfred Clebsch's Geschlecht

This article is aimed at throwing new light on the history of the notion of genus, whose paternity is usually attributed to Bernhard Riemann while its original name Geschlecht is often credited to Alfred Clebsch. By comparing the approaches of the two mathematicians, we show that Clebsch's act of naming was rooted in a projective geometric reinterpretation of Riemann's research, and that his Geschlecht was actually a different notion than that of Riemann. We also prove that until the beginning of the 1880s, mathematicians clearly distinguished between the notions of Clebsch and Riemann, the former being mainly associated with algebraic curves, and the latter with surfaces and Riemann surfaces. In the concluding remarks, we discuss the historiographic issues raised by the use of phrases like “the genus of a Riemann surface”—which began to appear in some works of Felix Klein at the very end of the 1870s—to describe Riemann's original research.     

Zermelo in the mirror of the Baer correspondence, 1930–1931

Around 1931 Zermelo had an extended correspondence with the young Reinhold Baer concerning the edition of Cantor's collected works. Some of the letters also deal with Skolem's paradox and Gödel's first incompleteness theorem. Whereas Zermelo's letters are lost, most of Baer's letters are contained in the Zermelo Nachlass. Besides giving insight into Zermelo's reaction to Skolem's and Gödel's results, the letters also demonstrate Baer's clear understanding of the behavior of models of set theory and of the relevance of Gödel's first incompleteness theorem.     

Constructing an international library: The collections of journals in Turin's Special Mathematics Library

The Special Mathematics Library of Turin University, founded in 1883, was fundamental in the development of two research schools under the leadership of C. Segre and G. Peano. First founded to house a growing collection of international journals acquired through both purchase and exchange from publishing centres worldwide, it later evolved into a ‘presence library’ modelled on the legendary Lesezimmer in Göttingen. A systematic study of the library's history and its directors' policies provides interesting insights into the various aspects of the international circulation of journals and their use at different times and in various contexts in Turin (Turin Academy of Sciences, Società di cultura, national university library, etc.).     

Two letters from Otto Neugebauer to Thomas Eric Peet on ancient Egyptian mathematics

We describe two letters of 1926 from the historian of mathematics Otto Neugebauer to the Egyptologist Thomas Eric Peet (Griffith Institute Archive, Oxford: Peet MSS 4.9). The letters concern Neugebauer's study of certain parts of the Rhind Mathematical Papyrus, as found in the doctoral dissertation that he had completed in Göttingen that year, for which Peet's 1923 edition of the papyrus was a major source. In the letters, we see the young Neugebauer establishing himself within the wider Egyptological community. The letters open up a discussion of Peet's own work on ancient Egyptian mathematics.     

General Relativity in the Making, 1916–1918: Rudolf J. Humm in Göttingen and Berlin

Mathematische Semesterberichte - In 1916, the Swiss student Rudolf J. Humm arrived in Göttingen to study relativity theory under Hilbert. He enrolled in his courses, attended Klein’s...     

Helmut Hasse in 1934

In 1934 Helmut Hasse became Professor of Mathematics at Göttingen. Hasse's attitudes and behavior during the Nazi period were representative of the ambiguous position of much of the mathematical community at that time. Various aspects of Hasse's situation make him appear to be almost the ideal example of the apolitical conservative and ideologically naive German academic in extremis. The purpose of this article is to present him as such an example.     

The astronomical work of Carl Friedrich Gauss (1777–1855)

Gauss's interest in astronomy dates from his student-days in Göttingen, and was stimulated by his reading of Franz Xavier von Zach's Monatliche Correspondenz… where he first read about Giuseppe Piazzi's discovery of the minor planet Ceres on 1 January 1801. He quickly produced a theory of orbital motion which enabled that faint star-like object to be rediscovered by von Zach and others after it emerged from the rays of the Sun. Von Zach continued to supply him with the observations of contemporary European astronomers from which he was able to improve his theory to such an extent that he could detect the effects of planetary perturbations in distorting the orbit from an elliptical form. To cope with the complexities which these introduced into the calculations of Ceres and more especially the other minor planet Pallas, discovered by Wilhelm Olbers in 1802, Gauss developed a new and more rigorous numerical approach by making use of his mathematical theory of interpolation and his method of least-squares analysis, which was embodied in his famous Theoria motus of 1809. His laborious researches on the theory of Pallas's motion, in which he enlisted the help of several former students, provided the framework of a new mathematical formulation of the problem whose solution can now be easily effected thanks to modern computational techniques.Up to the time of his appointment as Director of the Göttingen Observatory in 1807, Gauss had little opportunity for engaging himself in practical astronomical work. His first systematic observations were concerned with re-establishing the latitude of of that observatory, which had been well-determined by Tobias Mayer more than fifty years earlier. However, he found a small but not negligible discrepancy between results obtained independently from stellar and solar observations, as well as irregularities among later measurements of polar altitudes (made at the new observatory completed in 1816), which he was never able to explain, despite repeated attempts to do so using different instruments and observational techniques. Similar anomalies were also detected by a number of other astronomers at around this time. These may have been associated--at any rate, partially--with the phenomenon identified later in the century as a “variation of latitude” due to minor periodic fluctuations in the Earth's axis of rotation produced by meteorological and geological factors.     

Cultivating a research imperative: Mentoring mathematics at Stockholms Högskola, 1882–1887

Though the central role of Gösta Mittag-Leffler in the promotion of specialized, research-oriented mathematics at Stockholms Högskola is widely acknowledged, the specific social and technical means by which he sought to cultivate a fledgling research community there during the early- to mid-1880s have received little attention. In particular, a detailed study of the relationship of his own research activity to that of his first Swedish students is absent from the existing literature.Through the juxtaposition of their research activities and unpublished correspondence, this paper explores Mittag-Leffler's active and deliberate efforts to engage his students Ivar Bendixson and Edvard Phragmén in open problems within his own research agenda, support them through his institutional connections, and instill within them norms concerning research ideologies, practices of communication and criticism, and frameworks for shared knowledge. It also illuminates the extent to which his teachings took root in at least one student to emerge from his program, who would perpetuate the mathematical practices set in place by his teacher and set forth on the international stage to promote his newly-acquired system of values.     

A bound for diameter of arithmetic hyperbolic orbifolds

Weyl (Zur Infinitisimalgeometrie: Einordnung der projektiven und der konformen Auffasung, Nachrichten von der K. Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse, Göttinger Akademie der Wissenschaften, Göttingen, 1921) demonstrated that for a connected manifold of dimension greater than 1, if two Riemannian metrics are conformal and have the same geodesics up to a reparametrization, then one metric is a constant scaling of the other one. In the present paper we investigate the analogous property for sub-Riemannian metrics. In particular, we prove that the analogous statement, called the Weyl rigidity, holds either in real analytic category for all sub-Riemannian metrics on distributions with a specific property of their complex abnormal extremals, called minimal order, or in smooth category for all distributions such that all complex abnormal extremals of their nilpotent approximations are of minimal order. This also shows, in real analytic category, the genericity of distributions for which all sub-Riemannian metrics are Weyl rigid and genericity of Weyl rigid sub-Riemannian metrics on a given bracket generating distributions. Finally, this allows us to get analogous genericity results for projective rigidity of sub-Riemannian metrics, i.e., when the only sub-Riemannian metric having the same sub-Riemannian geodesics, up to a reparametrization, with a given one, is a constant scaling of this given one. This is the improvement of our results on the genericity of weaker rigidity properties proved in recent paper (Jean et al. in Geom Dedic 203(1):279–319, 2019).

     

A note on applicability of the incompleteness theorem to human mind

We shall present some relations between consistency and reflection principles which explain why is Gödel's incompleteness theorem wrongly used to argue that thinking machines are impossible.     

The Riemann solution in the one-dimensional inverse problem

It is shown that the introduction of the Riemann method in potential scattering is a natural device. Thereafter, the one-dimensional Schrödinger equation is considered using the Riemann method. An estimate is found for the associated translation kernel. This estimate reduces to Marchenko's when the domain of the variables is restricted to the positive quadrant of the (x, y) plane. The possibility of a Fourier transform of the Schrödinger equation solution is studied via Riemann's method. Extension of the investigations to the case where a dependence on threshold energies appears in the Schrödinger equation is discussed.     

A New Variant of Ekeland's Variational Principle for Set-Valued Maps

In this article, following the idea used by Göpfert et al . [A. Göpfert, Chr. Tammer and C. Zalinescu (2000). On the vectorial Ekeland's variational principle and minimal points in product spaces. Nonlinear Analysis, Theory, Methods & Applications , 39 , 909-922] to derive an Ekeland's variational principle for vector-valued functions, we derive a new variant of Ekeland's variational principle for set-valued maps. Finally, we apply this variational principle to obtain an approximate necessary optimality condition for a class of set-valued optimization problems.     

Rings and Gödel algebras

In this paper, we study those rings whose semiring of ideals can be given the structure of a Gödel algebra. Such rings are called Gödel rings. We investigate such structures both from an algebraic and a topological point of view. Our main result states that every Gödel ring R is a subdirect product of prime Gödel rings R _i , and the Gödel algebra Id(R) associated to R is subdirectly embeddable as an algebraic lattice into ${{\prod_{i}}Id(R_{i})}$ , where each Id(R _i ) is the algebraic lattice of ideals of R _i that can be equipped with the structure of a Gödel algebra. We see that the mapping associating to each Gödel ring its Gödel algebra of ideals is functorial from the category of Gödel rings with epimorphisms into the full subcategory of frames whose objects are Gödel algebras and whose morphisms are complete epimorphisms.     

A note on the finitization of Abelian and Tauberian theorems

We present finitary formulations of two well known results concerning infinite series, namely Abel's theorem, which establishes that if a series converges to some limit then its Abel sum converges to the same limit, and Tauber's theorem, which presents a simple condition under which the converse holds. Our approach is inspired by proof theory, and in particular Gödel's functional interpretation, which we use to establish quantitative versions of both of these results.     

The roles of sets of nondensity points

A Riemann surfaceR and a nonnegative locally Hölder continuous 2-formP onR are constructed such that the nondensity points and the Green's energy nondensity points on the Royden harmonic boundary coincide yet the space of bounded energy finite solutions ofd*du=uP onR is properly contained in the space of bounded Dirichlet finite solutions.     

On the curvature of symmetric products of a compact Riemann surface

Let X be a compact connected Riemann surface of genus at least two. The main theorem of Bökstedt and Romão [3] says that for any positive integer n ≤ 2(genus(X) ? 1), the symmetric product S ⁿ (X) does not admit any Kähler metric satisfying the condition that all the holomorphic bisectional curvatures are nonnegative. Our aim here is to give a very simple and direct proof of this result of Bökstedt and Romão.     

Some consequences of the Riemann hypothesis for varieties over finite fields

We deduce from Deligne's form of the Riemann hypothesis and the hard Lefschetz theoremin l-adic cohomology the corresponding facts for any “reasonable” cohomology theory, in particular for crystalline cohomology, and give some applications to algebraic cycles.     

On the summation of Schlömilch's series

ABSTRACT

Schlömilch's series is named after the German mathematician Oscar Xavier Schlömilch, who derived it in 1857 as a Fourier series type expansion in terms of the Bessel function of the first kind. However, except for Bessel functions, here we consider an expansion in terms of Struve functions or Bessel and Struve integrals as well. The method for obtaining a sum of Schlömilch's series in terms of the Bessel or Struve functions is based on the summation of trigonometric series, which can be represented in terms of the Riemann zeta and related functions of reciprocal powers and in certain cases can be brought in the closed form, meaning that the infinite series are represented by finite sums. By using Krylov's method we obtain the convergence acceleration of the trigonometric series.     

The polytopologies of transfinite provability logic

Provability logics are modal or polymodal systems designed for modeling the behavior of Gödel’s provability predicate and its natural extensions. If Λ is any ordinal, the Gödel-Löb calculus GLP _Λ contains one modality [λ] for each λ < Λ, representing provability predicates of increasing strength. GLP _ω has no non-trivial Kripke frames, but it is sound and complete for its topological semantics, as was shown by Icard for the variable-free fragment and more recently by Beklemishev and Gabelaia for the full logic. In this paper we generalize Beklemishev and Gabelaia’s result to GLP _Λ for countable Λ. We also introduce provability ambiances, which are topological models where valuations of formulas are restricted. With this we show completeness of GLP _Λ for the class of provability ambiances based on Icard polytopologies.