The Fourier Series

Pure mathematics is much more than an armory of tools and techniques for the applied mathematician. On the other hand, the pure mathematician has ever been grateful to applied mathematics for stimulus and inspiration. From the vibrations of the violin string they have drawn enchanting harmonies of Fourier Series, and to study the triode valve they have invented a whole theory of non-linear oscillations.

George Frederick James Temple In 100 Years of Mathematics: a Personal Viewpoint (1981).


Figure 1: Jean-Baptiste Joseph Fourier(source wikipedia)

The Fourier Series is a very important mathematics tool discovered by Jean-Baptiste Joseph Fourier in the 18th century. The Fourier series is used in many important areas of science and engineering. They are used to give an analytical approximate description of complex periodic function or series of data.  In this blog, I am going to give a short introduction to it.

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La Serie​ di Taylor

La serie di Taylor è un utilissimo strumento matematico. In questo blog, ne darò una breve descrizione dando qualche esempio di applicazione.

Chi è il signor Taylor?

Brook Taylor (1685 – 1731) era un matematico britannico del XVII secolo che ha dimostrato la formula che porta il suo nome, e l’argomento di questo blog, nel volume Methodus Incrementorum Directa et Inversa (1715). Maggiori informazioni si possono trovare nella corrispondente pagina della wikipedia.

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The Taylor Series

The Taylor series is a mathematical tool that, sometimes, it is not easy to immediately grasp by freshman students. In this blog, I will give a short review of it giving some examples of applications.

Who is Mr. Taylor?

Brook Taylor (1685 – 1731) was a 17th-century British mathematician. He demonstrated the celebrated Taylor formula, the topics of this blog, in his masterwork Methodus Incrementorum Directa et Inversa (1715). For more information, just give a read to the following wiki page.

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Retro Programming II: the Amiga and the Computational Beauty of the Leaf

In my archaeological exploration of old computer files, I came across another simple but exciting Amiga Basic program I programmed in 1989. It is named “Foglie”, the Italian name for leaves. It was an attempt to explore some ideas of functional plant morphology modelling. The stimulus comes after the reading of the paper by Karl J. Niklas on issue 213 of Le Science (the Italian edition of the Scientific American magazine [1]). The article titled “Computer-simulated plant evolution” described the modelling of plants to study their interaction with the environment. It was a fascinating paper; still, simple and primitive graphics caught my imagination. Nowadays, the field of digital morphology has come to an age (just to mention one, Avatar), and we can have an idea of this progress in the level of realism in movies, video games, and TV programs. However, the organism’s form and shape have always caught my curiosity and interest. The structure of leaf nervation was an intriguing pattern related to my acquaintance with the fascinating fractals objects, another recurrent topic in the pages of scientific magazines of the period.

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La programmazione in Awk II: Life in a Shell

Il gioco Life fu inventato negli anni ’70 dal prolifico matematico John H. Conway (vedi [5] per la sua biografia) ed è diventato famoso dopo la pubblicazione di Martin Gardner nella sua rubrica di matematica amatoriale sulla rivista Scientific American [1,2]. Il gioco è basato sugli automi cellulari concepiti da Konrad Zuse e Stanislaw M. Ulam all’inizio degli anni ’50, e poi adottati da John von Neumann per il suo studio sugli automi auto-replicanti [2,3]. Un automa cellulare è composto da unità (celle) interagenti disposte in una griglia quadrataIl sistema si evolve in cicli di vita in cui ogni cella cambia stato e nuove celle possono nascere e altre possono sopravvivere o, eventualmente, morire. Lo stato di ogni cella nel ciclo successivo è definito dall’interazione con le celle adiacenti in base a delle regole. L’interazione avviene con i primi vicini di ciascuna cella. Come mostrato nella Figura 1, è possibile utilizzare due tipi di intorni (cerchi) della cella centrale. Il gioco Life usa il tipo di proposto da Moore. 

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Awk Programming II: Life in a Shell

The game of Life was invented in the ’70 by the prolific mathematician John H. Conway (on the 11/4/2020 sadly J.H. Conway passed away at the age of 82 after having contracted the COVID-19, see [5] for his biography). The game becomes popular after Martin Gardner described it in his famous column in the Scientific American magazine [1,2].  The game is based on cellular automata conceived by Konrad Zuse and Stanislaw M. Ulam at beginning of the ’50 and then adopted by John von Neumann for his study on self-replicating automata [2,3]. A cellular automaton is composed of interacting units (cells) arranged in a square grid. The system evolves in life cycles where each cell change status and new cells can be born, and others can survive or eventually die. The status of each cell in the next cycle is defined by the interaction with their neighbor cells according to a given set of rules. The interaction occurs with the first neighbors of each cell. As shown in Figure 1, two type of neighbor’s cells (circles) can be used, the game of Life uses the Moore type neighborhood.

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