Problems and Exercises of Geodesy
It is a collection of problems and exercises of geodesy and the theory of errors.
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24 paper(s) to start with
Leibniz vs Ishiguro: Closing a quarter-century of syncategoremania
Did Leibniz exploit infinitesimals and infinities `a la rigueur, or only as shorthand for quantified propositions that refer to ordinary Archimedean magnitudes? Chapter 5 in (Ishiguro 1990) is a defense of the latter position, which she reformulates in terms of Russellian logical fictions. Ishiguro does not explain how to reconcile this interpretation with Leibniz's repeated assertions that infinitesimals violate the Archimedean property, viz., Euclid's Elements, V.4. We present textual evidence from Leibniz, as well as historical evidence from the early decades of the calculus, to undermine Ishiguro's interpretation. Leibniz frequently writes that his infinitesimals are useful fictions, and we agree; but we shall show that it is best not to understand them as logical fictions; instead, they are better understood as pure fictions. Keywords: Archimedean property; infinitesimal; logical fiction; pure fiction; quantified paraphrase; law of homogeneity
Conformally correct tilings
We discuss the art and science of producing conformally correct euclidean and hyperbolic tilings of compact surfaces. As an example, we present a tiling of the Chmutov surface by hyperbolic (2, 4, 6) triangles.
Infinite-Dimensional Generalizations of Orthogonal Groups over Hilbert Spaces : Constructions and Properties
In real Hilbert spaces, this paper generalizes the orthogonal groups $\mathrm{O}(n)$ in two ways. One way is by finite multiplications of a family of operators from reflections which results in a group denoted as $Θ(κ)$, the other is by considering the automorphism group of the Hilbert space denoted as $O(κ)$. We also try to research the algebraic relationship between the two generalizations and their relationship to the stable~orthogonal~group~$\mathrm{O}=\varinjlim\mathrm{O}(n)$ in terms of topology. In this paper we mainly show that : (a) $Θ(κ)$ is a topological and normal subgroup of $O(κ)$; (b) $O^{(n)}(κ) \to O^{(n+1)}(κ) \stackrelπ{\to} S^κ$ is a fibre bundle where $O^{(n)}(κ)$ is a subgroup of $O(κ)$ and $S^κ$ is a generalized sphere.
Revisiting Eisenstein-type criterion over integers
The following result, a consequence of Dumas criterion for irreducibility of polynomials over integers, is generally proved using the notion of Newton diagram: Let $f(x)$ be a polynomial with integer coefficients and $k$ be a positive integer relatively prime to the degree of $f(x)$. Suppose that there exists a prime number $p$ such that the leading coefficient of $f(x)$ is not divisible by $p$, all the remaining coefficients are divisible by $p^k$, and the constant term of $f(x)$ is not divisible by $p^{k+1}$. Then $f(x)$ is irreducible over $\mathbb{Z}$. For $k=1$, this is precisely the Eisenstein criterion. The aim of this article is to give an alternate proof, accessible to the undergraduate students, of this result for $k\in \{2,3,4\}$ using basic divisibility properties of integers.
Reflections in a cup of coffee
Allegedly, Brouwer discovered his famous fixed point theorem while stirring a cup of coffee and noticing that there is always at least one point in the liquid that does not move. In this paper, based on a talk in honour of Brouwer at the University of Amsterdam, we will explore how Brouwer's ideas about this phenomenon spilt over in a lot of different areas of mathematics and how this eventually led to an intriguing geometrical theory we now know as mirror symmetry.
Faà di Bruno's note on eponymous formula, trilingual version
About 160 years ago, the Italian mathematician Faà di Bruno published two notes dealing about the now eponymous formula giving the derivative of any order of a composition of two functions. We reproduce here the two original notes, Faà di Bruno (1855, 1857), written respectively in Italian and in French, and propose a translation in English.
The Prehistory of the Subsystems of Second-Order Arithmetic
This paper presents a systematic study of the prehistory of the traditional subsystems of second-order arithmetic that feature prominently in the reverse mathematics program of Friedman and Simpson. We look in particular at: (i) the long arc from Poincaré to Feferman as concerns arithmetic definability and provability, (ii) the interplay between finitism and the formalization of analysis in the lecture notes and publications of Hilbert and Bernays, (iii) the uncertainty as to the constructive status of principles equivalent to Weak König's Lemma, and (iv) the large-scale intellectual backdrop to arithmetical transfinite recursion in descriptive set theory and its effectivization by Borel, Lusin, Addison, and others.
A Primer of Mathematical Writing, Second Edition
This is a tract on the art and practice of mathematical writing. Not only does the book cover basic principles of grammar, syntax, and usage, but it takes into account developments of the last twenty years that have been inspired by the Internet. There is considerable discussion of TeX and other modern writing environments. We also consider electronic journals, print-on-demand books, Open Access Journals, preprint servers, and many other aspects of modern publishing life.
Gregory's sixth operation
In relation to a thesis put forward by Marx Wartofsky, we seek to show that a historiography of mathematics requires an analysis of the ontology of the part of mathematics under scrutiny. Following Ian Hacking, we point out that in the history of mathematics the amount of contingency is larger than is usually thought. As a case study, we analyze the historians' approach to interpreting James Gregory's expression ultimate terms in his paper attempting to prove the irrationality of pi. Here Gregory referred to the last or ultimate terms of a series. More broadly, we analyze the following questions: which modern framework is more appropriate for interpreting the procedures at work in texts from the early history of infinitesimal analysis? as well as the related question: what is a logical theory that is close to something early modern mathematicians could have used when studying infinite series and quadrature problems? We argue that what has been routinely viewed from the viewpoint of classical analysis as an example of an "unrigorous" practice, in fact finds close procedural proxies in modern infinitesimal theories. We analyze a mix of social and religious reasons tha
On the history of the Ulam's Conjugacy
We show the results on the history of the invention of the conjugacy $h(x)=\frac{2}π\arcsin\sqrt{x}$ of one-dimensional $[0,\, 1]\rightarrow [0,\, 1]$ maps $f(x)=4x(1-x)$ and $g(x)=1-|1-2x|$.
Few more Comments on Benford's Law
It is pointed out that the language of quotient groups and wrapped distributions allows an elementary discussion of Benford's Law, and adds arguments supporting wide-spread observability of this statistics.
A shortcut for evaluating some log integrals from products and limits
In this short paper, I introduce an elementary method for exactly evaluating the definite integrals $\, \int_0^π{\ln{(\sinθ)}\,dθ}$, $\int_0^{π/2}{\ln{(\sinθ)}\,dθ}$, $\int_0^{π/2}{\ln{(\cosθ)}\,dθ}$, and $\int_0^{π/2}{\ln{(\tanθ)}\,dθ} \,$ in finite terms. The method consists in to manipulate the sums obtained from the logarithm of certain products of trigonometric functions at rational multiples of $π$, putting them in the form of Riemann sums. As this method does not involve any search for primitives, it represents a good alternative to more involved integration techniques. As a bonus, I show how to apply the method for easily evaluating $\,\int_0^1{\ln{Γ(x)} \, d x}$.
Yuri Safarov (1958-2015)
This is the editor's preface to the special issue of Journal of Spectral Theory, in memory of Yuri Safarov.
Nicolas-Auguste Tissot: A link between cartography and quasiconformal theory
Nicolas-Auguste Tissot (1824--1897) published a series of papers on cartography in which he introduced a tool which became known later on, among geographers, under the name of the "Tissot indicatrix." This tool was broadly used during the twentieth century in the theory and in the practical aspects of the drawing of geographical maps. The Tissot indicatrix is a graphical representation of a field of ellipses on a map that describes its distortion. Tissot studied extensively, from a mathematical viewpoint, the distortion of mappings from the sphere onto the Euclidean plane that are used in drawing geographical maps, and more generally he developed a theory for the distorsion of mappings between general surfaces. His ideas are at the heart of the work on quasiconformal mappings that was developed several decades after him by Gr{ö}tzsch, Lavrentieff, Ahlfors and Teichm{ü}ller. Gr{ö}tzsch mentions the work of Tissot and he uses the terminology related to his name (in particular, Gr{ö}tzsch uses the Tissot indicatrix). Teichm{ü}ller mentions the name of Tissot in a historical section in one of his fundamental papers where he claims that quasiconformal mappings were used by geogr
The Effect of Gender in the Publication Patterns in Mathematics
Despite the increasing number of women graduating in mathematics, a systemic gender imbalance persists and is signified by a pronounced gender gap in the distribution of active researchers and professors. Especially at the level of university faculty, women mathematicians continue being drastically underrepresented, decades after the first affirmative action measures have been put into place. A solid publication record is of paramount importance for securing permanent positions. Thus, the question arises whether the publication patterns of men and women mathematicians differ in a significant way. Making use of the zbMATH database, one of the most comprehensive metadata sources on mathematical publications, we analyze the scholarly output of ~150,000 mathematicians from the past four decades whose gender we algorithmically inferred. We focus on development over time, collaboration through coautorships, presumed journal quality and distribution of research topics -- factors known to have a strong impact on job perspectives. We report significant differences between genders which may put women at a disadvantage when pursuing an academic career in mathematics.
Frederick William Gehring, Life and Mathematics
Frederick William Gehring was a hugely influential mathematician who spent most of his career at the University of Michigan. Gehring's major research contributions were to Geometric Function Theory, particularly in higher dimensions $\IR^n$, $n\geq 3$. This field he developed in close coordination with colleagues, primarily in Finland, over three decades 1960 -- 1990. Gehring's seminal work drove this field forward initiating important connections with geometry and nonlinear partial differential equations, while addressing and solving major problems. During his career Gehring received many honours from the international mathematical community. He was invited three times to address the International Congress of Mathematicians, at Moscow in 1966, at Vancouver in 1974, and at Berkeley (a plenary lecture) in 1986. He was awarded honorary degrees from the University of Helsinki (1979), the University of Jyväskylä (1990), and the Norwegian University of Science and Technology (1997). In 1989, he was elected to the American Academy of Arts and Sciences and the National Academy of Sciences. This is an extended obituary of his life and mathematical contributions.
Unified Music Theories for General Equal-Temperament Systems
Why are white and black piano keys in an octave arranged as they are today? This article examines the relations between abstract algebra and key signature, scales, degrees, and keyboard configurations in general equal-temperament systems. Without confining the study to the twelve-tone equal-temperament (12-TET) system, we propose a set of basic axioms based on musical observations. The axioms may lead to scales that are reasonable both mathematically and musically in any equal-temperament system. We reexamine the mathematical understandings and interpretations of ideas in classical music theory, such as the circle of fifths, enharmonic equivalent, degrees such as the dominant and the subdominant, and the leading tone, and endow them with meaning outside of the 12-TET system. In the process of deriving scales, we create various kinds of sequences to describe facts in music theory, and we name these sequences systematically and unambiguously with the aim to facilitate future research.
Richard Dedekind: Brief an Keferstein / Letter to Keferstein
This is an edition of the famous letter by Richard Dedekind to H. Keferstein dated 27 February 1890
Did Lobachevsky Have A Model Of His "imaginary Geometry"?
The invention of non-Euclidean geometries is often seen through the optics of Hilbertian formal axiomatic method developed later in the 19th century. However such an anachronistic approach fails to provide a sound reading of Lobachevsky's geometrical works. Although the modern notion of model of a given theory has a counterpart in Lobachevsky's writings its role in Lobachevsky's geometrical theory turns to be very unusual. Lobachevsky doesn't consider various models of Hyperbolic geometry, as the modern reader would expect, but uses a non-standard model of Euclidean plane (as a particular surface in the Hyperbolic 3-space). In this paper I consider this Lobachevsky's construction, and show how it can be better analyzed within an alternative non-Hilbertian foundational framework, which relates the history of geometry of the 19th century to some recent developments in the field.
Le lettere di Enrico Betti (dal 1860 al 1886) conservate presso l'Istituto Mazziniano di Genova
The content of this paper is the noted transcription of the 83 letters written by Enrico Betti between 1860 and 1886 stored at the Istituto Mazziniano, Museo del Risorgimento in Genova and now available at www.luigi-cremona.it The letters are addressed to: Luigi Cremona (79), Eugenio Beltrami (2), Pietro Blaserna (1) and one son of Gaetano Giorgini (1). ----- La corrispondenza qui riprodotta, composta dalle 83 lettere scritte da Enrico Betti tra il 1860 e il 1886, si trova a Genova (Istituto Mazziniano, Museo del Risorgimento) e ora anche sul sito www.luigi-cremona.it Destinatari sono: Luigi Cremona (79), Eugenio Beltrami (2), Pietro Blaserna (1) e uno dei figli di Gaetano Giorgini (1).
Fiboquadratic Sequences and Extensions of the Cassini Identity Raised From the Study of Rithmomachia
In this paper, we introduce fiboquadratic sequences as an extension to infinity of the board of Rithmomachia and we prove that this extension gives raise to fiboquadratic sequences which we define here. Also, fiboquadratic sequences provide extensions of Cassini's Identity.
Fibonacci numbers and the golden ratio
In this expository paper written to commemorate Fibonacci Day 2016, we discuss famous relations involving the Fibonacci sequence, the golden ratio, continued fractions and nested radicals, and show how these fit into a more general framework stemming from the quadratic formula.
Revisiting the quadrisection problem of Jacob Bernoulli
Two perpendicular segments which divide a given triangle into 4 regions of equal area is called a quadrisection of the triangle. Leonhard Euler proved in 1779 that every scalene triangle has a quadrisection with its triangular part on the middle leg. We provide a complete description of the quadrisections of a triangle. For example, there is only one isosceles triangle which has exactly two quadrisections.
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Yong Zhang
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