Système universitaire de documentation
The système universitaire de documentation or SUDOC is a system used by the libraries of French universities and higher education establishments to identify and manage the documents in their possession. The catalog, which contains more than 10 million references, allows students and researcher to search for bibliographical and location information in over 3,400 documentation centers, it is maintained by the Bibliographic Agency for Higher Education. Official website
The Euler–Mascheroni constant is a mathematical constant recurring in analysis and number theory denoted by the lowercase Greek letter gamma. It is defined as the limiting difference between the harmonic series and the natural logarithm: γ = lim n → ∞ = ∫ 1 ∞ d x. Here, ⌊ x ⌋ represents the floor function; the numerical value of the Euler–Mascheroni constant, to 50 decimal places, is: 0.57721566490153286060651209008240243104215933593992... The constant first appeared in a 1734 paper by the Swiss mathematician Leonhard Euler, titled De Progressionibus harmonicis observationes. Euler used the notations O for the constant. In 1790, Italian mathematician Lorenzo Mascheroni used the notations a for the constant; the notation γ appears nowhere in the writings of either Euler or Mascheroni, was chosen at a time because of the constant's connection to the gamma function. For example, the German mathematician Carl Anton Bretschneider used the notation γ in 1835 and Augustus De Morgan used it in a textbook published in parts from 1836 to 1842.
The Euler–Mascheroni constant appears, among other places, in the following: Expressions involving the exponential integral* The Laplace transform* of the natural logarithm The first term of the Laurent series expansion for the Riemann zeta function*, where it is the first of the Stieltjes constants* Calculations of the digamma function A product formula for the gamma function An inequality for Euler's totient function The growth rate of the divisor function In Dimensional regularization of Feynman diagrams in Quantum Field Theory The calculation of the Meissel–Mertens constant The third of Mertens' theorems* Solution of the second kind to Bessel's equation In the regularization/renormalization of the Harmonic series as a finite value The mean of the Gumbel distribution The information entropy of the Weibull and Lévy distributions, implicitly, of the chi-squared distribution for one or two degrees of freedom. The answer to the coupon collector's problem* In some formulations of Zipf's law A definition of the cosine integral* Lower bounds to a prime gap An upper bound on Shannon entropy in quantum information theory The number γ has not been proved algebraic or transcendental.
In fact, it is not known whether γ is irrational. Continued fraction analysis reveals that if γ is rational, its denominator must be greater than 10242080; the ubiquity of γ revealed by the large number of equations below makes the irrationality of γ a major open question in mathematics. See Sondow. Γ is related to the digamma function Ψ, hence the derivative of the gamma function Γ, when both functions are evaluated at 1. Thus: − γ = Γ ′ = Ψ; this is equal to the limits: − γ = lim z → 0 = lim z → 0. Further limit results are: lim z → 0 1 z = 2 γ lim z → 0 1 z ( 1 Ψ ( 1
Straightedge and compass construction
Straightedge and compass construction known as ruler-and-compass construction or classical construction, is the construction of lengths and other geometric figures using only an idealized ruler and compass. The idealized ruler, known as a straightedge, is assumed to be infinite in length, have only one edge, no markings on it; the compass is assumed to "collapse" when lifted from the page, so may not be directly used to transfer distances. More formally, the only permissible constructions are those granted by Euclid's first three postulates, it turns out to be the case that every point constructible using straightedge and compass may be constructed using compass alone. The ancient Greek mathematicians first conceived straightedge and compass constructions, a number of ancient problems in plane geometry impose this restriction; the ancient Greeks developed many constructions. Gauss showed that most are not; some of the most famous straightedge and compass problems were proven impossible by Pierre Wantzel in 1837, using the mathematical theory of fields.
In spite of existing proofs of impossibility, some persist in trying to solve these problems. Many of these problems are solvable provided that other geometric transformations are allowed: for example, doubling the cube is possible using geometric constructions, but not possible using straightedge and compass alone. In terms of algebra, a length is constructible if and only if it represents a constructible number, an angle is constructible if and only if its cosine is a constructible number. A number is constructible if and only if it can be written using the four basic arithmetic operations and the extraction of square roots but of no higher-order roots; the "straightedge" and "compass" of straightedge and compass constructions are idealizations of rulers and compasses in the real world: The straightedge is infinitely long, but it has no markings on it and has only one straight edge, unlike ordinary rulers. It can only be used to extend an existing segment; the compass can be opened arbitrarily wide.
Circles can only be drawn starting from two given points: a point on the circle. The compass may not collapse when it is not drawing a circle. Actual compasses do not collapse and modern geometric constructions use this feature. A'collapsing compass' would appear to be a less powerful instrument. However, by the compass equivalence theorem in Proposition 2 of Book 1 of Euclid's Elements, no power is lost by using a collapsing compass. Although the proposition is correct, its proofs have a checkered history; each construction must be exact. "Eyeballing" it and getting close does not count as a solution. Each construction must terminate; that is, it must have a finite number of steps, not be the limit of closer approximations. Stated this way and compass constructions appear to be a parlour game, rather than a serious practical problem; the ancient Greek mathematicians first attempted straightedge and compass constructions, they discovered how to construct sums, products and square roots of given lengths.
They could construct half of a given angle, a square whose area is twice that of another square, a square having the same area as a given polygon, a regular polygon with 3, 4, or 5 sides. But they could not construct one third of a given angle except in particular cases, or a square with the same area as a given circle, or a regular polygon with other numbers of sides. Nor could they construct the side of a cube whose volume would be twice the volume of a cube with a given side. Hippocrates and Menaechmus showed that the volume of the cube could be doubled by finding the intersections of hyperbolas and parabolas, but these cannot be constructed by straightedge and compass. In the fifth century BCE, Hippias used a curve that he called a quadratrix to both trisect the general angle and square the circle, Nicomedes in the second century BCE showed how to use a conchoid to trisect an arbitrary angle. No progress on the unsolved problems was made for two millennia, until in 1796 Gauss showed that a regular polygon with 17 sides could be constructed.
In 1837 Pierre Wantzel published a proof of the impossibility of trisecting an arbitrary angle or of doubling the volume of a cube, based on the impossibility of constructing cube roots of lengths. He showed that Gauss's sufficient constructibility condition for regular polygons is necessary. In 1882 Lindemann showed that π is a transcendental number, thus that it is impossible by straightedge and compass to construct a square with the same area as a given circle. All straightedge and compass constructions consist of repeated application of five basic constructions using the points and circles that have been constructed; these are: Creating the line through two existing points Creating the circle through one point with centre another point Creating the point, th
Pavia is a town and comune of south-western Lombardy, northern Italy, 35 kilometres south of Milan on the lower Ticino river near its confluence with the Po. It has a population of c. 73,000. The city was the capital of the Kingdom of the Lombards from 572 to 774. Pavia is the capital of the fertile province of Pavia, known for agricultural products including wine, rice and dairy products. Although there are a number of industries located in the suburbs, these tend not to disturb the peaceful atmosphere of the town, it is home to the ancient University of Pavia, which together with the IUSS, Ghislieri College, Borromeo College, Nuovo College, Santa Caterina College and the EDiSU, belongs to the Pavia Study System. Pavia is the episcopal seat of the Roman Catholic Bishop of Pavia; the city possesses many artistic and cultural treasures, including several important churches and museums, such as the well-known Certosa di Pavia. The Central Hospital of Pavia is one of the most important hospitals in Italy.
Dating back to pre-Roman times, the town of Pavia known as Ticinum, was a municipality and an important military site under the Roman Empire. It was said by Pliny the Elder to have been founded by the Laevi and Marici, two Ligurian tribes, while Ptolemy attributes it to the Insubres; the Roman city most began as a small military camp, built by the consul Publius Cornelius Scipio in 218 BC to guard a wooden bridge he had built over the river Ticinum, on his way to search for Hannibal, rumoured to have managed to lead an army over the Alps and into Italy. The forces of Rome and Carthage ran into each other soon thereafter, the Romans suffered the first of many crushing defeats at the hands of Hannibal, with the consul himself losing his life; the bridge was destroyed, but the fortified camp, which at the time was the most forward Roman military outpost in the Po Valley, somehow survived the long Second Punic War, evolved into a garrison town. Its importance grew with the extension of the Via Aemilia from Ariminum to the Po River, which it crossed at Placentia and there forked, one branch going to Mediolanum and the other to Ticinum, thence to Laumellum where it divided once more, one branch going to Vercellae - and thence to Eporedia and Augusta Praetoria - and the other to Valentia - and thence to Augusta Taurinorum.
It was at Pavia in 476 AD that the reign of Romulus Augustulus, the last emperor of the Western Roman Empire ended and Roman rule ceased in Italy. Romulus Augustulus, while considered the last emperor of the Western Roman Empire, was a usurper of the imperial throne. Though being the emperor, Romulus Augustulus was the mouthpiece for his father Orestes, the person who exercised power and governed Italy during Romulus Augustulus's short reign. Ten months after Romulus Augustulus's reign began, Orestes's soldiers under the command of one of his officers named Odoacer and killed Orestes in the city of Pavia in 476; the rioting that took place as part of Odoacer's uprising against Orestes sparked fires that burnt much of Pavia to the point that Odoacer, as the new king of Italy, had to suspend the taxes for the city for five years so that it could finance its recovery. Without his father, Romulus Augustulus was powerless. Instead of killing Romulus Augustulus, Odoacer pensioned him off at 6,000 solidi a year before declaring the end of the Western Roman Empire and himself king of the new Kingdom of Italy.
Odoacer's reign as king of Italy did not last long, because in 488 the Ostrogothic peoples led by their king Theoderic invaded Italy and waged war against Odoacer. After fighting for 5 years, Theoderic defeated Odoacer and on March 15, 493, assassinated Odoacer at a banquet meant to negotiate a peace between the two rulers. With the establishment of the Ostrogoth kingdom based in northern Italy, Theoderic began his vast program of public building. Pavia was among several cities that Theodoric chose to expand, he began the construction of the vast palace complex that would become the residence of Lombard monarchs several decades later. Theoderic commissioned the building of the Roman-styled amphitheatre and bath complex in Pavia. Near the end of Theoderic's reign the Christian philosopher Boethius was imprisoned in one of Pavia's churches from 522 to 525 before his execution for treason, it was during Boethius's captivity in Pavia that he wrote his seminal work the Consolation of Philosophy. Pavia played an important role in the war between the Eastern Roman Empire and the Ostrogoths that began in 535.
After the Eastern Roman general Belisarius's victory over the Ostrogothic leader Wittigis in 540 and the loss of most of the Ostrogoth lands in Italy, Pavia was among the last centres of Ostrogothic resistance that continued the war and opposed Eastern Roman rule. After the capitulation of the Ostrogothic leadership in 540 more than a thousand men remained garrisoned in Pavia and Verona dedicated to opposing Eastern Roman rule; the resilience of Ostrogoth strongholds like Pavia against invading forces allowed pockets of Ostrogothic rule to limp along until being defeated in 561. Pavia and the peninsula of Italy didn't remain long under the rule of the Eastern Roman Empire, for in 568, a new people invaded Italy; this new invading people in 568
Bergamo is a city in the alpine Lombardy region of northern Italy 40 km northeast of Milan, about 30 km from Switzerland, the alpine lakes Como and Iseo and 70 km from Garda and Maggiore. The Bergamo Alps begin north of the city. With a population of around 120,000, Bergamo is the fourth-largest city in Lombardy. Bergamo is the seat of the Province of Bergamo; the metropolitan area of Bergamo extends beyond the administrative city limits, spanning over a densely urbanized area with less than 500,000 inhabitants. The Bergamo metropolitan area is itself part of the broader Milan metropolitan area, home to over 8 million people; the city of Bergamo is composed of an old walled core, known as Città Alta, nestled within a system of hills, the modern expansion in the plains below. The upper town is encircled by massive Venetian defensive systems that are a UNESCO World Heritage Site since 9 July 2017. Bergamo is well connected to several cities in Italy, thanks to the motorway A4 stretching on the axis between Turin, Verona and Trieste.
The city is served by Il Caravaggio International Airport, the third-busiest airport in Italy with 12.3 million passengers in 2017. Bergamo is the second most visited city in Lombardy after Milan. Bergamo occupies the site of the ancient town of Bergomum, founded as a settlement of the Celtic tribe of Cenomani. In 49 BC it became a Roman municipality. An important hub on the military road between Friuli and Raetia, it was destroyed by Attila in the 5th century. From the 6th century Bergamo was the seat of one of the most important Lombard duchies of northern Italy, together with Brescia and Cividale del Friuli: its first Lombard duke was Wallaris. After the conquest of the Lombard Kingdom by Charlemagne, it became the seat of a county under one Auteramus. An important Lombardic hoard dating from the 6th to 7th centuries was found in the vicinity of the city in the 19th century and is now in the British Museum. From the 11th century onwards, Bergamo was an independent commune, taking part in the Lombard League which defeated Frederick I Barbarossa in 1165.
The local Guelph and Ghibelline factions were the Suardi, respectively. Feuding between the two caused the family of Omodeo Tasso to flee north c. 1250, but he returned to Bergamo in the 13th century to organize the city's couriers: this would lead to the Imperial Thurn und Taxis dynasty credited with organizing the first modern postal service. After a short period under the House of Malatesta starting from 1407, Bergamo was ceded in 1428 by the Duchy of Milan to the Republic of Venice in the context of the Wars in Lombardy and the aftermath of the 1427 Battle of Maclodio. Despite the brief interlude granted by the Treaty of Lodi in 1454, the uneasy balance of power among the Northern Italian states precipitated the Italian Wars, a series of conflicts from 1494 to 1559 that involved, at various times the Papal States and the Holy Roman Empire; the wars, which were both a result and cause of Venetian involvement in the power politics of mainland Italy, prompted Venice to assert its direct rule over its mainland domains.
As much of the fighting during the Italian Wars took place during sieges, increasing levels of fortification were adopted, using such new developments as detached bastions that could withstand sustained artillery fire. The Treaty of Campo Formio formally recognized the inclusion of Bergamo and other parts of Northern Italy into the Cisalpine Republic, a "sister republic" of the French First Republic, superseded in 1802 by the short-lived Napoleonic Italian Republic and in 1805 by the Napoleonic Kingdom of Italy. At the 1815 Congress of Vienna, Bergamo was assigned to the Kingdom of Lombardy–Venetia, a crown land of the Austrian Empire; the visit of Ferdinand I in 1838 coincided with the opening of the new boulevard stretching into the plains, leading to the railway station, inaugurated in 1857. The Austrian rule was at first welcomed, but challenged by Italian independentist insurrections in 1848. Giuseppe Garibaldi conquered Bergamo in 1859, during the Second Italian War of Independence; as a result, the city was incorporated into the newly founded Kingdom of Italy.
For its contribution to the Italian unification movement, Bergamo is known as Città dei Mille, because a significant part of the rank-and-file supporting Giuseppe Garibaldi in his expedition against the Kingdom of the Two Sicilies came from Bergamo and its environs. During the twentieth century, Bergamo became one of Italy's most industrialized areas. In 1907, Marcello Piacentini devised a new urban master plan, implemented between 1912 and 1927, in a style reminiscent of Novecento Italiano and Modernist Rationalism; the 2017 43rd G7 summit on agriculture was held in Bergamo, in the context of the broader international meeting organized in Taormina. The "Charter of Bergamo" is an international commitment, signed during the summit, to reduce hunger worldwide by 2030, strengthen cooperation for agricultural development in Africa, ensure price transparency; the town has two centres: Città alta, a hilltop medieval town, surrounded by 16th-century defensive walls, the Città bassa. The two parts of the town are connected by funicular and footpaths.
The upper city, surrounded by Venetian walls built in the 16th century, forms the historic centre of Bergamo. Walking along the narrow medieval streets, you can visi