"… I seem […] only like a boy playing on the sea-shore, and diverting myself in now and then finding a smoother pebble or a prettier shell than ordinary, whilst the great ocean of truth lay all undiscovered before me". – Isaac Newton.

Il 6 di 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.

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.

Try to glue a small mirror to an 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 dragon of light will perform a hypnotic dance on the wall of your room. This voluptuous dance is the results of 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. I the last case, the light is the electron beam of a cathodic tube (or its digital equivalent) of an oscilloscope. This blog is about these curves and shows demonstrations and applications.

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!

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

zum Laden von Ionen e und getrennt nach Entfernung .

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.

I am European of Italian origin. I very proud of my origin but, unfortunately, my carrier put me in a orbit that does intersect my country only during the vacation time. In these close encounters, my landing site is Frosinone. When not Italian acquaintance want to know about my Heimat, most of them are puzzled about the name of my hometown Frosinone and its location. Usually, I help them to overcome the comprehensible impasse by giving as reference Rome and telling them that my birthplace is somewhere 80 km in the South of the Caput Mundi. Last year (2018) my hometown soccer team (Frosinone Calcio, nickenamed Canarini, The Canaries, for their home colours)) moved in the first league (A) of the national soccer championship. So, let see if this success will help to raise its notoriety! In this blog, I won’t write about the success of the Canaries Canarini (although I am proud of that even if I am not such a soccer game fan) but about my hometown and the bucolic place in the central Italy where it is located.

Frosinone is situated in the south of the Lazio region and it is also one of its 5 provinces (together with Roma, Viterbo, Rieti, and Latina). It is in the heart of the Ciociaria a beautyful area of central Italy named after an ancient type of footwear called Ciocia. The old town is situated on the top of a hill 291 meter above sea level. From its panoramic position, it overlooks the valley of the Sacco river. The following picture was taken from the Belvedere on of the most spectacular panoramic spot in the town.

The photo shows the new part of the town with his 60-meter tall skyscraper L’Edera in De Mattheis’ Square (the tall black and white building on the right side) that just this year turned 50 years old. In the central part of the figure, there is the commercial area of the town with its “high street” (via Aldo Moro) fitted with shops that can be reached from the Belvedere using a cable lift that arrive just where the photo was taken. On the right side background, it is visible the pre-Apennines mountain chains that in December are covered by snow. The mountains on the left side are the called Lepini frames the boundary of the Sacco valley. The Sacco river flows just under the base of Frosinone hill and continues its course in the valley bearing its name between the Lepini and Ernici mountains.

Frosinone was an ancient center of the Volsci Frusino population. They were proud opponents of Rome but the town was then conquered, and it became a Roman municipium in 386 BC. In medieval times it was in the hands of the Byzantines and Lombards and in the 817, the Franks gave it to the Church.

The Coat of Arm of the city of Frosinone (Figure below on the left) was created in the 1928. It show a lion with a the motto Bellator (Frusino (Frosinone the warrior) while in the banner of the coat of arm of the provence of Frosinone (Figure below on the right), the motto is ferocior ad bellandum (the most fierce in fighting). The motto is in honor of the eroic attitude shown by th e inhabitants to contrast the invasion by the Hannibal’s army.

The most important monuments,located in the old part of the city on the top hill (Frosinone alto) are the Church of S. Benedetto (19th century Baroque), the Cathedral (originally Romanesque but it was damage during the war II and rebuilt after the War), and finally the little Church of S. Lucia of Neo-classical style.

The ciociaria is a green region and origianally the traditional agriculture was mainly based on forests and pastures. Over the last fifty years, the population has considerably increased and industry has developed, with aid from the Mezzogiorno Fund and with the construction of the Rome-Naples stretch of motorway. Today there are many factories, mainly sited along the principal roads and railway lines, operating in the engineering, textile, plastics and foodstuff sectors.

In the provincia of Frosinone, there are many historical important town such as Anagni (Museum of Southern Latium), Alatri (Acropolis, folklore festival in August), Ferentino (Acropolis, Romanesque Duomo), Veroli (Abbey Museum).

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 century and it was further developed by other great mathematicians in the centuries that follows (see Figure below).

It comprises two areas:

Differential calculus It concerns the study of the rate of variation of functions.

Integral calculus 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.

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).

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.