Numerical Integration of Differential Equations. Part I.: Katherine Johnson and the Euler’s Method.

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This article was inspired by the beautiful 2016 movie Hidden Figures (based on the book of the same name by M. L. Shetterley) which tell the dramatic story of three talented black women scientist that worked as “human computers” for NASA in 1961 for the Mercury project.

Figure 1: Official theatrical poster of the movie and the phFoto of the real protagonist. From left to right. Mary Jackson, Katherine Goble and Dorothy Vaughan. (source: wikipedia)

In the movie, the mathematician Creola Katherine Johnson (or Globe) (interpreted by Taraji P. Henson), had a brilliant intuition on how to numerically solve the complex problem to find the transfer trajectory for the reentry into the Earth atmosphere of the Friendship 7 capsule with the astronaut John Glenn on board. In the particular scene, she was standing together with other engineers and the director of the Langley Research Center (a fictional character interpreted by Kevin Coster) in front of the vast blackboard looking to graph and equations when she says that the solution might be in the “old math” and she runs to take an old book from a bookshelf with the description of the Euler method [UPDATE, May 2022: I just come across another excellent youtube video by the Tibees about the mathematical work of K. Johnson at NASA, and at the end of the video she reveal that the book shown in the movie is the classic H Margenau & G. M Murphy, The Mathematics of Physics and Chemistry. Published by Van Nostrand, 1956]. The scene is also nicely described in the youtube video lesson by Prof. Alan Garfinkel of the UCLA. A detailed description of the numerical solution based on the original derivation of K. Johnson is in the Wolfram blog website.

Katherine Globe was using for these complex calculation her brilliant brain with the support of a mechanical calculator (the Friden STW-10, in the movie, this machine is visible in different scenes). In a scene of the film, she revealed that her typical computing performance was of 10000 calculations per day and probably for calculations, she was not referring to single arithmetic operations! These exceptional mathematical skills have given a significative contribution at the beginning of the American space program, but it became insufficient to handle the more complex mathematics necessary to land the man on the Moon, and the other fantastic NASA achievements.

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The Calculation of the Lattice Energy: The Born-Haber Cycle

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My blog in italian on this topics is very popular and for this reason I decided to add an English translation (when I have some free time, I will also translate the text in the Figure and Table). So be tune and more will come!

The stability of a crystal lattice at constant T and P conditions is linked to the Gibbs free energy of lattice formation by the relations

M^+ (g) + X^- (g) \rightarrow MX (s) \hfill (1)

\displaystyle \Delta G^{\circ} = \Delta H^{\circ} - T \Delta S^{\circ} \hfill (2)

If {\Delta G^{\circ}} is more negative for the formation of the {I} structure than for the {II} structure, the {II \rightarrow I} transition will be spontaneous and the solid will have that structure.

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Easter 2019: Modelling Patterns and Forms in Nature II. The Egg

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When you start with a portrait and search for a pure form, 
a clear volume, through successive eliminations, you arrive 
inevitably at the egg. Likewise, starting with the egg and 
following the same process in reverse, one finishes with 
the portrait.

PABLO PICASSO

Easter is coming and what better time to talk about eggs!

During my recent mathematical explorations of natural shapes and forms, my attention has been caught by the shape of birds’ eggs. In the exciting book by J. Adams, A Mathematical Walk in Nature [1], you can find a short review of the different mathematical modeling approaches to describe the shape of an egg. Among them, the geometrical one by Baker [2] is revealed as one of the most versatile as it can accurately reproduce the shapes of a large variety of bird eggs [2]. More recently, the model was used to perform a systematic and comparative study of the shape of bird eggs. This study, published in Science magazine [3], a two-dimensional morphological space defined by the parameters of Baker’s equation, has been used to show the diversity of the shape of 1400 species of birds. Combining this information with a mechanical model and phylogenetics information, the authors have shown that egg shape correlates with flight ability on broad taxonomic scales. They concluded that adaptations for flight may have been critical drivers of egg-shape variation in birds [3].

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I Primi 150 Anni della Tavola Periodica degli Elementi

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Che la nobiltà dell’Uomo, acquisita in cento secoli di prove e di errori, era consistita nel farsi signore della materia, e che io mi ero iscritto a Chimica perché a questa nobiltà mi volevo mantenere fedele. Che vincere la materia è comprenderla, e comprendere la materia è necessario per comprendere l’universo e noi stessi: e che quindi il Sistema Periodico di Mendeleev, che proprio in quelle settimane imparavamo laboriosamente a dipanare, era una poesia, più alta e più solenne di tutte le poesie digerite in liceo: a pensarci bene, aveva perfino le rime! 

Primo Levi, Il sistema periodico

Il 6 Marzo del 1869 il chimico russo Dmitri Ivanovich Mendeleyev presento’ alla Societa’ di Chimica Russa, una comunicazione dal titolo La dipendenza delle proprieta’ degli elementi chimica dal peso atomico. In questa storica comunicazione, Mendeleev pesento’ una tabella in cui organizzava gli elementi chimici allora noti. Questa tabella segno’ anche la fama del suo autore poiche’ fu la prima versione della moderna tavola periodica degli elementi chimici.

La tavola periodic degli elementi di Mendeleev. I trattini rappresentano elelmenti sconosciuti nel 1871. (fonte della figura: wikipedia)

Mendeleyev, preparando una seconda edizione del suo libro di chimica, stava cercando un modo per classificare gli elementi chimici allora conosciuti (53 ovvero meno della meta’ di quelli che conosciamo oggi) per fare chiarezza sulle loro proprieta’. In una nota, Mendeleyev racconta che l’ispirazione gli sia venuta in sogno (non e’ la prima volta che Orfeo suggerisce a chimici le loro grandi scoperte scientifici!) [2]:

I saw in a dream a table where all the elements fell into place as required. Awakening, I immediately wrote it down on a piece of paper.

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Calculus in a Nutshell: the Definite Integral of Monovariate Functions

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The definite integral is the key tool in calculus for defining and calculating quantities important to mathematics and science, such as areas, volumes, lengths of curved paths, probabilities, and the weights of various objects, just to mention a few.

The idea behind the integral is that we can effectively compute such quantities by breaking them into small pieces and then summing the contributions from each piece.

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The Lissajous-Bowditch Curves

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Try to glue a small mirror to the end of a bent piece of wire fixed to a stable platform and let the laser beam of a laser pointer reflect on it. Entangled spires of an ephemeral red dragon will perform a hypnotic dance on the wall of your room. This voluptuous dance results from two mutually perpendicular harmonic oscillations produced by the oscillations of the elastic wire. 

The curved patterns are called Lissajous-Bowditch figures and named after the French physicist Jules Antoine Lissajous who did a detailed study of them (published in his Mémoire sur l’étude optique des mouvements vibratoires, 1857). The American mathematician Nathaniel Bowditch (1773 – 1838) conducted earlier and independent studies on the same curves and for this reason, the figures are also called  Lissajous-Bowditch curves. Lissajous invented different mechanical devices consisting of two mirrors attached to two oriented diapasons (or other oscillators) by double reflecting a collimated ray of light on a screen, produce these figures upon oscillations of the diapasons.  The diapason can be substituted with elastic wires, speakers, pendulum or electronic circuits. In the last case, the light is the electron beam of a cathodic tube (or its digital equivalent)  of an oscilloscope. This article is about the mathematical theory behind these curves introduced with a demonstration program and an example of education application proposed in 1827 by C. Wheatstone . Finally, we will give a look to the equivalent of the L-B curves in 3D by exploring the spherical L-B curves.

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The Magic Imaginary Numbers

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Complex numbers may appear a difficult subject given the name. However, there is nothing of really complicated about complex numbers. However, they definitively add a pinch of \em magic \em in the mathematics manipulations that you can do with them!

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Berechnung der Konstante von Madelung

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Die gesamte Coulomb-Potentialenergie eines Kristalls ist die Summe der einzelnen Terme der elektrostatischen Potentialenergie

\displaystyle V_{AB} = \frac{e^2}{4\pi\epsilon_0} \frac{Z_AZ_B}{r_{AB}} \hfill (1)

zum Laden von Ionen  {q_A} e {q_B} und  getrennt nach Entfernung {r_{AB}}.

Die Summe erstreckt sich auf alle im Festkörper vorhandenen Ionenpaare für alle kristallinen Strukturen.

Die Summe konvergiert sehr langsam, weil die ersten Nachbarn des Zentralatoms einen substanziellen Beitrag zur Summe mit einem negativen Term liefern, während die benachbarten Sekunden nur mit einem etwas weicheren positiven Term beitragen, und so weiter. Auf diese Weise wird der Gesamteffekt sicherstellen, dass eine totale Initation der Anziehung zwischen Kationen und Anionen vorherrscht mit einem (negativen) Beitrag, der für die Gesamtenergie günstig ist.

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Calculus in a Nutshell: Functions and their Derivatives

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When I was about thirteen, the library was going to get ‘Calculus for the Practical Man.’ By this time I knew, from reading the encyclopedia, that calculus was an important and interesting subject, and I ought to learn it.  

Richard P. Feynman, from What Do You Care What Other People Think?

Introduction

Calculus is an important branch of mathematics that deals with the methods for calculating derivatives and integrals of functions and using this information to study the properties of functions. It was independently invented by I. Newton and W. Leibniz in the 18{^{th}} century and it was further developed by other great mathematicians in the centuries that follows (see Figure below).

Figure 1: Some of the great mathematician that invented the Calculus.

It comprises two areas:

  • Differential calculus {\rightarrow} It concerns the study of the rate of variation of functions.
  • Integral calculus {\rightarrow} It concern the study of the area under functions.

Depending on the nature of the functions involved in the calculations, we can further distinguish between the single- and multi-variable calculus. In this chapter, the main concepts and methods of the single-variable calculus are summarised.

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