1.
Turin
–
Turin is a city and an important business and cultural centre in northern Italy, capital of the Piedmont region and was the first capital city of Italy. The city is located mainly on the bank of the Po River, in front of Susa Valley and surrounded by the western Alpine arch. The population of the city proper is 892,649 while the population of the area is estimated by Eurostat to be 1.7 million inhabitants. The Turin metropolitan area is estimated by the OECD to have a population of 2.2 million, in 1997 a part of the historical center of Torino was inscribed in the World Heritage List under the name Residences of the Royal House of Savoy. Turin is well known for its Renaissance, Baroque, Rococo, Neo-classical, many of Turins public squares, castles, gardens and elegant palazzi such as Palazzo Madama, were built between the 16th and 18th centuries. This was after the capital of the Duchy of Savoy was moved to Turin from Chambery as part of the urban expansion, the city used to be a major European political center. Turin was Italys first capital city in 1861 and home to the House of Savoy, from 1563, it was the capital of the Duchy of Savoy, then of the Kingdom of Sardinia ruled by the Royal House of Savoy and finally the first capital of the unified Italy. Turin is sometimes called the cradle of Italian liberty for having been the birthplace and home of notable politicians and people who contributed to the Risorgimento, such as Cavour. The city currently hosts some of Italys best universities, colleges, academies, lycea and gymnasia, such as the University of Turin, founded in the 15th century, in addition, the city is home to museums such as the Museo Egizio and the Mole Antonelliana. Turins attractions make it one of the worlds top 250 tourist destinations, Turin is ranked third in Italy, after Milan and Rome, for economic strength. With a GDP of $58 billion, Turin is the worlds 78th richest city by purchasing power, as of 2010, the city has been ranked by GaWC as a Gamma World city. Turin is also home to much of the Italian automotive industry, the Taurini were an ancient Celto-Ligurian Alpine people, who occupied the upper valley of the Po River, in the center of modern Piedmont. In 218 BC, they were attacked by Hannibal as he was allied with their long-standing enemies, the Taurini chief town was captured by Hannibals forces after a three-day siege. As a people they are mentioned in history. It is believed that a Roman colony was established in 27 BC under the name of Castra Taurinorum, both Livy and Strabo mention the Taurinis country as including one of the passes of the Alps, which points to a wider use of the name in earlier times. In the 1st century BC, the Romans created a military camp, the typical Roman street grid can still be seen in the modern city, especially in the neighborhood known as the Quadrilatero Romano. Via Garibaldi traces the path of the Roman citys decumanus which began at the Porta Decumani. The Porta Palatina, on the side of the current city centre, is still preserved in a park near the Cathedral
Turin
–
From top to bottom, left to right: panorama of the
Mole Antonelliana,
Valentino Park with the medieval village, Piazza Castello with Palazzo Reale and Palazzo Madama, San Carlo Plaza with the Caval ëd Bronz, the Arco Olimpico and the Lingotto, the sarcophagus of Oki at the
Egyptian Museum, a view of the hills, the
Po, the Gran Madre, the Monte of Cappuccini and
Palatine Towers.
Turin
–
The Roman
Palatine Towers.
Turin
–
Siege of Turin
Turin
–
Turin in the 17th century.
2.
Kingdom of Sardinia
–
The Kingdom of Sardinia was a state in Southern Europe which existed from the early 14th until the mid-19th century. It was the state of todays Italy. When it was acquired by the Duke of Savoy in 1720, however, the Savoyards united it with their possessions on the Italian mainland and, by the time of the Crimean War in 1853, had built the resulting kingdom into a strong power. The formal name of the entire Savoyard state was the States of His Majesty the King of Sardinia and its final capital was Turin, the capital of Savoy since the Middle Ages. Beginning in 1324, James and his successors conquered the island of Sardinia, in 1420 the last competing claim to the island was bought out. After the union of the crowns of Aragon and Castile, Sardinia became a part of the burgeoning Spanish Empire, in 1720 it was ceded by the Habsburg and Bourbon claimants to the Spanish throne to Duke Victor Amadeus II of Savoy. While in theory the traditional capital of the island of Sardinia and seat of its viceroys was Cagliari, the Congress of Vienna, which restructured Europe after Napoleons defeat, returned to Savoy its mainland possessions and augmented them with Liguria, taken from the Republic of Genoa. In 1847–48, in a fusion, the various Savoyard states were unified under one legal system, with the capital in Turin, and granted a constitution. There followed the annexation of Lombardy, the central Italian states and the Two Sicilies, Venetia, in 238 BC Sardinia became, along with Corsica, a province of the Roman Empire. The Romans ruled the island until the middle of the 5th century, when it was occupied by the Vandals, in 534 AD it was reconquered by the Romans, but now from the Eastern Roman Empire, Byzantium. It remained a Byzantine province until the Arab conquest of Sicily in the 9th century, after that, communications with Constantinople became very difficult, and powerful families of the island assumed control of the land. Starting from 705–706, Saracens from north Africa harassed the population of the coastal cities, information about the Sardinian political situation in the following centuries is scarce. There is a record of another massive Saracen sea attack in 1015–16 from the Balearics, the Saracen attempt to invade the island was stopped by the Judicatus with the support of the fleets of the maritime republics of Pisa and Genoa, free cities of the Holy Roman Empire. Pope Benedict VIII also requested aid from the republics of Pisa. Even the title of Judices was a Byzantine reminder of the Greek church and state, of these sovereigns only two names are known, Turcoturiu and Salusiu, who probably ruled in the 10th century. The Archons still wrote in Greek or Latin, but one of the first documents of the Judex of Cagliari, their successor, was written in romance Sardinian language. The realm was divided into four kingdoms, the Judicati, perfectly organized as was the previous realm, but was now under the influence of the Pope. That was the cause of leading to a long war between the Judices, who regarded themselves as kings fighting against rebellious nobles
Kingdom of Sardinia
–
A map of the Kingdom of Sardinia in 1856, after the fusion of all its provinces into a single jurisdiction
Kingdom of Sardinia
–
The final flag used by the kingdom under the "
Perfect Fusion " (1848–1861)
Kingdom of Sardinia
–
Flag of the Kingdom of Sardinia in the middle of the 16th century
Kingdom of Sardinia
–
Kingdom of Sardinia 16th-century map
3.
Greater French Empire
–
The First French Empire, Note 1 was the empire of Napoleon Bonaparte of France and the dominant power in much of continental Europe at the beginning of the 19th century. Its name was a misnomer, as France already had colonies overseas and was short lived compared to the Colonial Empire, a series of wars, known collectively as the Napoleonic Wars, extended French influence over much of Western Europe and into Poland. The plot included Bonapartes brother Lucien, then serving as speaker of the Council of Five Hundred, Roger Ducos, another Director, on 9 November 1799 and the following day, troops led by Bonaparte seized control. They dispersed the legislative councils, leaving a rump legislature to name Bonaparte, Sieyès, although Sieyès expected to dominate the new regime, the Consulate, he was outmaneuvered by Bonaparte, who drafted the Constitution of the Year VIII and secured his own election as First Consul. He thus became the most powerful person in France, a power that was increased by the Constitution of the Year X, the Battle of Marengo inaugurated the political idea that was to continue its development until Napoleons Moscow campaign. Napoleon planned only to keep the Duchy of Milan for France, setting aside Austria, the Peace of Amiens, which cost him control of Egypt, was a temporary truce. He gradually extended his authority in Italy by annexing the Piedmont and by acquiring Genoa, Parma, Tuscany and Naples, then he laid siege to the Roman state and initiated the Concordat of 1801 to control the material claims of the pope. Napoleon would have ruling elites from a fusion of the new bourgeoisie, on 12 May 1802, the French Tribunat voted unanimously, with exception of Carnot, in favour of the Life Consulship for the leader of France. This action was confirmed by the Corps Législatif, a general plebiscite followed thereafter resulting in 3,653,600 votes aye and 8,272 votes nay. On 2 August 1802, Napoleon Bonaparte was proclaimed Consul for life, pro-revolutionary sentiment swept through Germany aided by the Recess of 1803, which brought Bavaria, Württemberg and Baden to Frances side. The memories of imperial Rome were for a time, after Julius Caesar and Charlemagne. The Treaty of Pressburg, signed on 26 December 1805, did little other than create a more unified Germany to threaten France. On the other hand, Napoleons creation of the Kingdom of Italy, the occupation of Ancona, to create satellite states, Napoleon installed his relatives as rulers of many European states. The Bonapartes began to marry into old European monarchies, gaining sovereignty over many nations, in addition to the vassal titles, Napoleons closest relatives were also granted the title of French Prince and formed the Imperial House of France. Met with opposition, Napoleon would not tolerate any neutral power, Prussia had been offered the territory of Hanover to stay out of the Third Coalition. With the diplomatic situation changing, Napoleon offered Great Britain the province as part of a peace proposal and this, combined with growing tensions in Germany over French hegemony, Prussia responded by forming an alliance with Russia and sending troops into Bavaria on 1 October 1806. In this War of the Fourth Coalition, Napoleon destroyed the armies of Frederick William at Jena-Auerstedt, the Eylau and the Friedland against the Russians finally ruined Frederick the Greats formerly mighty kingdom, obliging Russia and Prussia to make peace with France at Tilsit. The Treaties of Tilsit ended the war between Russia and the French Empire and began an alliance between the two empires that held power of much of the rest of Europe, the two empires secretly agreed to aid each other in disputes
Greater French Empire
–
The Battle of Austerlitz
Greater French Empire
–
Flag
Greater French Empire
–
The
Arc de Triomphe, ordered by
Napoleon in honour of his
Grande Armée, is one of the several landmarks whose construction was started in Paris during the First French Empire.
Greater French Empire
–
Napoleon reviews the
Imperial Guard before the
Battle of Jena, 1806
4.
Piedmont
–
Piedmont is one of the 20 regions of Italy. It has an area of 25,402 square kilometres and a population of about 4.6 million, the capital of Piedmont is Turin. The name Piedmont comes from medieval Latin Pedemontium or Pedemontis, i. e. ad pedem montium, meaning “at the foot of the mountains”. Other towns of Piedmont with more than 20,000 inhabitants sorted by population and it borders with France, Switzerland and the Italian regions of Lombardy, Liguria, Aosta Valley and for a very small fragment with Emilia Romagna. The geography of Piedmont is 43. 3% mountainous, along with areas of hills. Piedmont is the second largest of Italys 20 regions, after Sicily and it is broadly coincident with the upper part of the drainage basin of the river Po, which rises from the slopes of Monviso in the west of the region and is Italy’s largest river. The Po collects all the waters provided within the semicircle of mountains which surround the region on three sides, from the highest peaks the land slopes down to hilly areas, and then to the upper, and then to the lower great Padan Plain. 7. 6% of the territory is considered protected area. There are 56 different national or regional parks, one of the most famous is the Gran Paradiso National Park located between Piedmont and the Aosta Valley, Piedmont was inhabited in early historic times by Celtic-Ligurian tribes such as the Taurini and the Salassi. They were later subdued by the Romans, who founded several colonies there including Augusta Taurinorum, after the fall of the Western Roman Empire, the region was repeatedly invaded by the Burgundians, the Goths, Byzantines, Lombards, Franks. In the 9th–10th centuries there were incursions by the Magyars. At the time Piedmont, as part of the Kingdom of Italy within the Holy Roman Empire, was subdivided into several marks, in 1046, Oddo of Savoy added Piedmont to their main territory of Savoy, with a capital at Chambéry. Other areas remained independent, such as the powerful comuni of Asti and Alessandria, the County of Savoy was elevated to a duchy in 1416, and Duke Emanuele Filiberto moved the seat to Turin in 1563. In 1720, the Duke of Savoy became King of Sardinia, founding what evolved into the Kingdom of Sardinia, the Republic of Alba was created in 1796 as a French client republic in Piedmont. A new client republic, the Piedmontese Republic, existed between 1798 and 1799 before it was reoccupied by Austrian and Russian troops, in June 1800 a third client republic, the Subalpine Republic, was established in Piedmont. It fell under full French control in 1801 and it was annexed by France in September 1802, in the congress of Vienna, the Kingdom of Sardinia was restored, and furthermore received the Republic of Genoa to strengthen it as a barrier against France. Piedmont was a springboard for Italys unification in 1859–1861, following earlier unsuccessful wars against the Austrian Empire in 1820–1821 and this process is sometimes referred to as Piedmontisation. However, the efforts were countered by the efforts of rural farmers
Piedmont
–
A
Montferrat landscape, with the distant
Alps in the background.
Piedmont
–
Piedmont Piemonte
Piedmont
–
The
Palazzina di caccia of Stupinigi, in
Nichelino, is a
UNESCO World Heritage Site.
Piedmont
–
The
Kingdom of Sardinia in 1856.
5.
Prussia
–
Prussia was a historic state originating out of the Duchy of Prussia and the Margraviate of Brandenburg, and centred on the region of Prussia. For centuries, the House of Hohenzollern ruled Prussia, successfully expanding its size by way of an unusually well-organised, Prussia, with its capital in Königsberg and from 1701 in Berlin, shaped the history of Germany. In 1871, German states united to create the German Empire under Prussian leadership, in November 1918, the monarchies were abolished and the nobility lost its political power during the German Revolution of 1918–19. The Kingdom of Prussia was thus abolished in favour of a republic—the Free State of Prussia, from 1933, Prussia lost its independence as a result of the Prussian coup, when the Nazi regime was successfully establishing its Gleichschaltung laws in pursuit of a unitary state. Prussia existed de jure until its liquidation by the Allied Control Council Enactment No.46 of 25 February 1947. The name Prussia derives from the Old Prussians, in the 13th century, the Teutonic Knights—an organized Catholic medieval military order of German crusaders—conquered the lands inhabited by them. In 1308, the Teutonic Knights conquered the region of Pomerelia with Gdańsk and their monastic state was mostly Germanised through immigration from central and western Germany and in the south, it was Polonised by settlers from Masovia. The Second Peace of Thorn split Prussia into the western Royal Prussia, a province of Poland, and the part, from 1525 called the Duchy of Prussia. The union of Brandenburg and the Duchy of Prussia in 1618 led to the proclamation of the Kingdom of Prussia in 1701, Prussia entered the ranks of the great powers shortly after becoming a kingdom, and exercised most influence in the 18th and 19th centuries. During the 18th century it had a say in many international affairs under the reign of Frederick the Great. During the 19th century, Chancellor Otto von Bismarck united the German principalities into a Lesser Germany which excluded the Austrian Empire. At the Congress of Vienna, which redrew the map of Europe following Napoleons defeat, Prussia acquired a section of north western Germany. The country then grew rapidly in influence economically and politically, and became the core of the North German Confederation in 1867, and then of the German Empire in 1871. The Kingdom of Prussia was now so large and so dominant in the new Germany that Junkers and other Prussian élites identified more and more as Germans and less as Prussians. In the Weimar Republic, the state of Prussia lost nearly all of its legal and political importance following the 1932 coup led by Franz von Papen. East Prussia lost all of its German population after 1945, as Poland, the main coat of arms of Prussia, as well as the flag of Prussia, depicted a black eagle on a white background. The black and white colours were already used by the Teutonic Knights. The Teutonic Order wore a white coat embroidered with a cross with gold insert
Prussia
–
... during the
Renaissance period
Prussia
–
Flag (1892–1918)
Prussia
–
... according to the design of 1702
Prussia
–
Prussian King's Crown (
Hohenzollern Castle Collection)
6.
First French Empire
–
The First French Empire, Note 1 was the empire of Napoleon Bonaparte of France and the dominant power in much of continental Europe at the beginning of the 19th century. Its name was a misnomer, as France already had colonies overseas and was short lived compared to the Colonial Empire, a series of wars, known collectively as the Napoleonic Wars, extended French influence over much of Western Europe and into Poland. The plot included Bonapartes brother Lucien, then serving as speaker of the Council of Five Hundred, Roger Ducos, another Director, on 9 November 1799 and the following day, troops led by Bonaparte seized control. They dispersed the legislative councils, leaving a rump legislature to name Bonaparte, Sieyès, although Sieyès expected to dominate the new regime, the Consulate, he was outmaneuvered by Bonaparte, who drafted the Constitution of the Year VIII and secured his own election as First Consul. He thus became the most powerful person in France, a power that was increased by the Constitution of the Year X, the Battle of Marengo inaugurated the political idea that was to continue its development until Napoleons Moscow campaign. Napoleon planned only to keep the Duchy of Milan for France, setting aside Austria, the Peace of Amiens, which cost him control of Egypt, was a temporary truce. He gradually extended his authority in Italy by annexing the Piedmont and by acquiring Genoa, Parma, Tuscany and Naples, then he laid siege to the Roman state and initiated the Concordat of 1801 to control the material claims of the pope. Napoleon would have ruling elites from a fusion of the new bourgeoisie, on 12 May 1802, the French Tribunat voted unanimously, with exception of Carnot, in favour of the Life Consulship for the leader of France. This action was confirmed by the Corps Législatif, a general plebiscite followed thereafter resulting in 3,653,600 votes aye and 8,272 votes nay. On 2 August 1802, Napoleon Bonaparte was proclaimed Consul for life, pro-revolutionary sentiment swept through Germany aided by the Recess of 1803, which brought Bavaria, Württemberg and Baden to Frances side. The memories of imperial Rome were for a time, after Julius Caesar and Charlemagne. The Treaty of Pressburg, signed on 26 December 1805, did little other than create a more unified Germany to threaten France. On the other hand, Napoleons creation of the Kingdom of Italy, the occupation of Ancona, to create satellite states, Napoleon installed his relatives as rulers of many European states. The Bonapartes began to marry into old European monarchies, gaining sovereignty over many nations, in addition to the vassal titles, Napoleons closest relatives were also granted the title of French Prince and formed the Imperial House of France. Met with opposition, Napoleon would not tolerate any neutral power, Prussia had been offered the territory of Hanover to stay out of the Third Coalition. With the diplomatic situation changing, Napoleon offered Great Britain the province as part of a peace proposal and this, combined with growing tensions in Germany over French hegemony, Prussia responded by forming an alliance with Russia and sending troops into Bavaria on 1 October 1806. In this War of the Fourth Coalition, Napoleon destroyed the armies of Frederick William at Jena-Auerstedt, the Eylau and the Friedland against the Russians finally ruined Frederick the Greats formerly mighty kingdom, obliging Russia and Prussia to make peace with France at Tilsit. The Treaties of Tilsit ended the war between Russia and the French Empire and began an alliance between the two empires that held power of much of the rest of Europe, the two empires secretly agreed to aid each other in disputes
First French Empire
–
The Battle of Austerlitz
First French Empire
–
Flag
First French Empire
–
The
Arc de Triomphe, ordered by
Napoleon in honour of his
Grande Armée, is one of the several landmarks whose construction was started in Paris during the First French Empire.
First French Empire
–
Napoleon reviews the
Imperial Guard before the
Battle of Jena, 1806
7.
University of Turin
–
The University of Turin is a university in the city of Turin in the Piedmont region of north-western Italy. It is one of the oldest universities in Europe, and continues to play an important role in research, the University of Turin was founded as a studium in 1404, under the initiative of Prince Ludovico di Savoia. From 1427 to 1436 the seat of the university was transferred to Chieri and it was closed in 1536, and reestablished by Duke Emmanuel Philibert thirty years later. With the reforms carried out by Victor Amadeus II, the University of Turin became a new model for many other universities. During the 18th century, the University faced an enormous growth in faculty and endowment size, notable scholars of this period include Cesare Lombroso, Carlo Forlanini and Arturo Graf. In the 20th century, the University of Turin was one of the centers of the Italian anti-fascism, the new impulse was performed in collaboration with other national and international research centers, as well as with local organizations and the Italian Minister of Public Instruction. By the end of the 1990s, the campuses of Alessandria, Novara. The new institution, which only held courses in civil. The Bishop, as Rector of Studies, proclaimed and conferred the title on the new doctors, after a series of interruptions in its activities, the university was moved to Chieri and later, in 1434, to Savigliano. In 1436, when the returned to Turin, Ludovico di Savoia. The ducal licenses of 6 October 1436 set up the three faculties of Theology, Arts and Medicine, and Law, and twenty-five lectureships or chairs. The growth and development of the role of Turin as the capital led to the consolidation of the University. The Study, closed at the beginning of 1536 with the French occupation, reopened in 1558 with lecturers at Mondovì, it was re-established in Turin in 1566. With Emmanuel Philibert and Charles Emmanuel I, the University enjoyed a season of great prosperity due to the presence of illustrious teachers, the opening of the new premises marked a major turning point in the history of the greatest Piedmontese educational institution. This had an effect on the cultural linguistic models of the Duchy. At the time, the Piedmontese Studium became a point of reference for university reforms at Parma and Modena, Charles Emmanuel III continued the policy of innovation and consolidation commenced by Victor Amadeus II and created a University Museum in 1739. However, in the last decades of the 18th century, the course of events at the University, closely connected to international developments, led to urban unrest. The revolt of university students in 1791 joined by artisans who stormed the Collegio delle Province in 1792 causing numerous victims, was an instance of this conflict
University of Turin
–
Hall of the Rectorate Palace of the University of Turin
University of Turin
–
Seal of the University of Turin
University of Turin
–
The
Minerva Statue in front of the Rectorate Palace at the University of Turin.
University of Turin
–
The revolt of the students of Turin University, 1821
8.
Analytical mechanics
–
In theoretical physics and mathematical physics, analytical mechanics, or theoretical mechanics is a collection of closely related alternative formulations of classical mechanics. It was developed by scientists and mathematicians during the 18th century and onward. A scalar is a quantity, whereas a vector is represented by quantity, the equations of motion are derived from the scalar quantity by some underlying principle about the scalars variation. Analytical mechanics takes advantage of a systems constraints to solve problems, the constraints limit the degrees of freedom the system can have, and can be used to reduce the number of coordinates needed to solve for the motion. The formalism is well suited to arbitrary choices of coordinates, known in the context as generalized coordinates and it does not always work for non-conservative forces or dissipative forces like friction, in which case one may revert to Newtonian mechanics or use the Udwadia–Kalaba equation. Two dominant branches of mechanics are Lagrangian mechanics and Hamiltonian mechanics. There are other such as Hamilton–Jacobi theory, Routhian mechanics. All equations of motion for particles and fields, in any formalism, one result is Noethers theorem, a statement which connects conservation laws to their associated symmetries. Analytical mechanics does not introduce new physics and is not more general than Newtonian mechanics, rather it is a collection of equivalent formalisms which have broad application. In fact the principles and formalisms can be used in relativistic mechanics and general relativity. Analytical mechanics is used widely, from physics to applied mathematics. The methods of analytical mechanics apply to particles, each with a finite number of degrees of freedom. They can be modified to describe continuous fields or fluids, which have infinite degrees of freedom, the definitions and equations have a close analogy with those of mechanics. Generalized coordinates and constraints In Newtonian mechanics, one customarily uses all three Cartesian coordinates, or other 3D coordinate system, to refer to a position during its motion. In physical systems, however, some structure or other system usually constrains the motion from taking certain directions. In the Lagrangian and Hamiltonian formalisms, the constraints are incorporated into the motions geometry and these are known as generalized coordinates, denoted qi. Difference between curvillinear and generalized coordinates Generalized coordinates incorporate constraints on the system, there is one generalized coordinate qi for each degree of freedom, i. e. each way the system can change its configuration, as curvilinear lengths or angles of rotation. Generalized coordinates are not the same as curvilinear coordinates, DAlemberts principle The foundation which the subject is built on is DAlemberts principle
Analytical mechanics
–
As the system evolves, q traces a path through
configuration space (only some are shown). The path taken by the system (red) has a stationary action (δ S = 0) under small changes in the configuration of the system (δ q).
9.
Mathematical analysis
–
Mathematical analysis is the branch of mathematics dealing with limits and related theories, such as differentiation, integration, measure, infinite series, and analytic functions. These theories are studied in the context of real and complex numbers. Analysis evolved from calculus, which involves the elementary concepts and techniques of analysis, analysis may be distinguished from geometry, however, it can be applied to any space of mathematical objects that has a definition of nearness or specific distances between objects. Mathematical analysis formally developed in the 17th century during the Scientific Revolution, early results in analysis were implicitly present in the early days of ancient Greek mathematics. For instance, a geometric sum is implicit in Zenos paradox of the dichotomy. The explicit use of infinitesimals appears in Archimedes The Method of Mechanical Theorems, in Asia, the Chinese mathematician Liu Hui used the method of exhaustion in the 3rd century AD to find the area of a circle. Zu Chongzhi established a method that would later be called Cavalieris principle to find the volume of a sphere in the 5th century, the Indian mathematician Bhāskara II gave examples of the derivative and used what is now known as Rolles theorem in the 12th century. In the 14th century, Madhava of Sangamagrama developed infinite series expansions, like the power series and his followers at the Kerala school of astronomy and mathematics further expanded his works, up to the 16th century. The modern foundations of analysis were established in 17th century Europe. During this period, calculus techniques were applied to approximate discrete problems by continuous ones, in the 18th century, Euler introduced the notion of mathematical function. Real analysis began to emerge as an independent subject when Bernard Bolzano introduced the definition of continuity in 1816. In 1821, Cauchy began to put calculus on a firm logical foundation by rejecting the principle of the generality of algebra widely used in earlier work, instead, Cauchy formulated calculus in terms of geometric ideas and infinitesimals. Thus, his definition of continuity required a change in x to correspond to an infinitesimal change in y. He also introduced the concept of the Cauchy sequence, and started the theory of complex analysis. Poisson, Liouville, Fourier and others studied partial differential equations, the contributions of these mathematicians and others, such as Weierstrass, developed the -definition of limit approach, thus founding the modern field of mathematical analysis. In the middle of the 19th century Riemann introduced his theory of integration, the last third of the century saw the arithmetization of analysis by Weierstrass, who thought that geometric reasoning was inherently misleading, and introduced the epsilon-delta definition of limit. Then, mathematicians started worrying that they were assuming the existence of a continuum of numbers without proof. Around that time, the attempts to refine the theorems of Riemann integration led to the study of the size of the set of discontinuities of real functions, also, monsters began to be investigated
Mathematical analysis
–
A
strange attractor arising from a
differential equation. Differential equations are an important area of mathematical analysis with many applications to
science and
engineering.
10.
Number theory
–
Number theory or, in older usage, arithmetic is a branch of pure mathematics devoted primarily to the study of the integers. It is sometimes called The Queen of Mathematics because of its place in the discipline. Number theorists study prime numbers as well as the properties of objects out of integers or defined as generalizations of the integers. Integers can be considered either in themselves or as solutions to equations, questions in number theory are often best understood through the study of analytical objects that encode properties of the integers, primes or other number-theoretic objects in some fashion. One may also study real numbers in relation to rational numbers, the older term for number theory is arithmetic. By the early century, it had been superseded by number theory. The use of the arithmetic for number theory regained some ground in the second half of the 20th century. In particular, arithmetical is preferred as an adjective to number-theoretic. The first historical find of a nature is a fragment of a table. The triples are too many and too large to have been obtained by brute force, the heading over the first column reads, The takiltum of the diagonal which has been subtracted such that the width. The tables layout suggests that it was constructed by means of what amounts, in language, to the identity 2 +1 =2. If some other method was used, the triples were first constructed and then reordered by c / a, presumably for use as a table. It is not known what these applications may have been, or whether there could have any, Babylonian astronomy, for example. It has been suggested instead that the table was a source of examples for school problems. While Babylonian number theory—or what survives of Babylonian mathematics that can be called thus—consists of this single, striking fragment, late Neoplatonic sources state that Pythagoras learned mathematics from the Babylonians. Much earlier sources state that Thales and Pythagoras traveled and studied in Egypt, Euclid IX 21—34 is very probably Pythagorean, it is very simple material, but it is all that is needed to prove that 2 is irrational. Pythagorean mystics gave great importance to the odd and the even, the discovery that 2 is irrational is credited to the early Pythagoreans. This forced a distinction between numbers, on the one hand, and lengths and proportions, on the other hand, the Pythagorean tradition spoke also of so-called polygonal or figurate numbers
Number theory
–
A
Lehmer sieve, which is a primitive
digital computer once used for finding
primes and solving simple
Diophantine equations.
Number theory
–
The Plimpton 322 tablet
Number theory
–
Title page of the 1621 edition of Diophantus' Arithmetica, translated into
Latin by
Claude Gaspard Bachet de Méziriac.
Number theory
–
Leonhard Euler
11.
Mathematics
–
Mathematics is the study of topics such as quantity, structure, space, and change. There is a range of views among mathematicians and philosophers as to the exact scope, Mathematicians seek out patterns and use them to formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proof, when mathematical structures are good models of real phenomena, then mathematical reasoning can provide insight or predictions about nature. Through the use of abstraction and logic, mathematics developed from counting, calculation, measurement, practical mathematics has been a human activity from as far back as written records exist. The research required to solve mathematical problems can take years or even centuries of sustained inquiry, rigorous arguments first appeared in Greek mathematics, most notably in Euclids Elements. Galileo Galilei said, The universe cannot be read until we have learned the language and it is written in mathematical language, and the letters are triangles, circles and other geometrical figures, without which means it is humanly impossible to comprehend a single word. Without these, one is wandering about in a dark labyrinth, carl Friedrich Gauss referred to mathematics as the Queen of the Sciences. Benjamin Peirce called mathematics the science that draws necessary conclusions, David Hilbert said of mathematics, We are not speaking here of arbitrariness in any sense. Mathematics is not like a game whose tasks are determined by arbitrarily stipulated rules, rather, it is a conceptual system possessing internal necessity that can only be so and by no means otherwise. Albert Einstein stated that as far as the laws of mathematics refer to reality, they are not certain, Mathematics is essential in many fields, including natural science, engineering, medicine, finance and the social sciences. Applied mathematics has led to entirely new mathematical disciplines, such as statistics, Mathematicians also engage in pure mathematics, or mathematics for its own sake, without having any application in mind. There is no clear line separating pure and applied mathematics, the history of mathematics can be seen as an ever-increasing series of abstractions. The earliest uses of mathematics were in trading, land measurement, painting and weaving patterns, in Babylonian mathematics elementary arithmetic first appears in the archaeological record. Numeracy pre-dated writing and numeral systems have many and diverse. Between 600 and 300 BC the Ancient Greeks began a study of mathematics in its own right with Greek mathematics. Mathematics has since been extended, and there has been a fruitful interaction between mathematics and science, to the benefit of both. Mathematical discoveries continue to be made today, the overwhelming majority of works in this ocean contain new mathematical theorems and their proofs. The word máthēma is derived from μανθάνω, while the modern Greek equivalent is μαθαίνω, in Greece, the word for mathematics came to have the narrower and more technical meaning mathematical study even in Classical times
Mathematics
–
Euclid (holding
calipers), Greek mathematician, 3rd century BC, as imagined by
Raphael in this detail from
The School of Athens.
Mathematics
–
Greek mathematician
Pythagoras (c. 570 – c. 495 BC), commonly credited with discovering the
Pythagorean theorem
Mathematics
–
Leonardo Fibonacci, the
Italian mathematician who established the Hindu–Arabic numeral system to the Western World
Mathematics
–
Carl Friedrich Gauss, known as the prince of mathematicians
12.
Mathematical physics
–
Mathematical physics refers to development of mathematical methods for application to problems in physics. It is a branch of applied mathematics, but deals with physical problems, there are several distinct branches of mathematical physics, and these roughly correspond to particular historical periods. The rigorous, abstract and advanced re-formulation of Newtonian mechanics adopting the Lagrangian mechanics, both formulations are embodied in analytical mechanics. These approaches and ideas can be and, in fact, have extended to other areas of physics as statistical mechanics, continuum mechanics, classical field theory. Moreover, they have provided several examples and basic ideas in differential geometry, the theory of partial differential equations are perhaps most closely associated with mathematical physics. These were developed intensively from the half of the eighteenth century until the 1930s. Physical applications of these developments include hydrodynamics, celestial mechanics, continuum mechanics, elasticity theory, acoustics, thermodynamics, electricity, magnetism, and aerodynamics. The theory of atomic spectra developed almost concurrently with the fields of linear algebra. Nonrelativistic quantum mechanics includes Schrödinger operators, and it has connections to atomic, Quantum information theory is another subspecialty. The special and general theories of relativity require a different type of mathematics. This was group theory, which played an important role in quantum field theory and differential geometry. This was, however, gradually supplemented by topology and functional analysis in the description of cosmological as well as quantum field theory phenomena. In this area both homological algebra and category theory are important nowadays, statistical mechanics forms a separate field, which includes the theory of phase transitions. It relies upon the Hamiltonian mechanics and it is related with the more mathematical ergodic theory. There are increasing interactions between combinatorics and physics, in statistical physics. The usage of the mathematical physics is sometimes idiosyncratic. Certain parts of mathematics that arose from the development of physics are not, in fact, considered parts of mathematical physics. The term mathematical physics is sometimes used to research aimed at studying and solving problems inspired by physics or thought experiments within a mathematically rigorous framework
Mathematical physics
–
An example of mathematical physics: solutions of
Schrödinger's equation for
quantum harmonic oscillators (left) with their
amplitudes (right).
13.
Leonhard Euler
–
He also introduced much of the modern mathematical terminology and notation, particularly for mathematical analysis, such as the notion of a mathematical function. He is also known for his work in mechanics, fluid dynamics, optics, astronomy, Euler was one of the most eminent mathematicians of the 18th century, and is held to be one of the greatest in history. He is also considered to be the most prolific mathematician of all time. His collected works fill 60 to 80 quarto volumes, more than anybody in the field and he spent most of his adult life in Saint Petersburg, Russia, and in Berlin, then the capital of Prussia. A statement attributed to Pierre-Simon Laplace expresses Eulers influence on mathematics, Read Euler, read Euler, Leonhard Euler was born on 15 April 1707, in Basel, Switzerland to Paul III Euler, a pastor of the Reformed Church, and Marguerite née Brucker, a pastors daughter. He had two sisters, Anna Maria and Maria Magdalena, and a younger brother Johann Heinrich. Soon after the birth of Leonhard, the Eulers moved from Basel to the town of Riehen, Paul Euler was a friend of the Bernoulli family, Johann Bernoulli was then regarded as Europes foremost mathematician, and would eventually be the most important influence on young Leonhard. Eulers formal education started in Basel, where he was sent to live with his maternal grandmother. In 1720, aged thirteen, he enrolled at the University of Basel, during that time, he was receiving Saturday afternoon lessons from Johann Bernoulli, who quickly discovered his new pupils incredible talent for mathematics. In 1726, Euler completed a dissertation on the propagation of sound with the title De Sono, at that time, he was unsuccessfully attempting to obtain a position at the University of Basel. In 1727, he first entered the Paris Academy Prize Problem competition, Pierre Bouguer, who became known as the father of naval architecture, won and Euler took second place. Euler later won this annual prize twelve times, around this time Johann Bernoullis two sons, Daniel and Nicolaus, were working at the Imperial Russian Academy of Sciences in Saint Petersburg. In November 1726 Euler eagerly accepted the offer, but delayed making the trip to Saint Petersburg while he applied for a physics professorship at the University of Basel. Euler arrived in Saint Petersburg on 17 May 1727 and he was promoted from his junior post in the medical department of the academy to a position in the mathematics department. He lodged with Daniel Bernoulli with whom he worked in close collaboration. Euler mastered Russian and settled life in Saint Petersburg. He also took on a job as a medic in the Russian Navy. The Academy at Saint Petersburg, established by Peter the Great, was intended to improve education in Russia, as a result, it was made especially attractive to foreign scholars like Euler
Leonhard Euler
–
Portrait by
Jakob Emanuel Handmann (1756)
Leonhard Euler
–
1957
Soviet Union stamp commemorating the 250th birthday of Euler. The text says: 250 years from the birth of the great mathematician, academician Leonhard Euler.
Leonhard Euler
–
Stamp of the former
German Democratic Republic honoring Euler on the 200th anniversary of his death. Across the centre it shows his
polyhedral formula, nowadays written as " v − e + f = 2".
Leonhard Euler
–
Euler's grave at the
Alexander Nevsky Monastery
14.
Joseph Fourier
–
The Fourier transform and Fouriers law are also named in his honour. Fourier is also credited with the discovery of the greenhouse effect. Fourier was born at Auxerre, the son of a tailor and he was orphaned at age nine. Fourier was recommended to the Bishop of Auxerre, and through this introduction, the commissions in the scientific corps of the army were reserved for those of good birth, and being thus ineligible, he accepted a military lectureship on mathematics. He took a prominent part in his own district in promoting the French Revolution and he was imprisoned briefly during the Terror but in 1795 was appointed to the École Normale, and subsequently succeeded Joseph-Louis Lagrange at the École Polytechnique. Fourier accompanied Napoleon Bonaparte on his Egyptian expedition in 1798, as scientific adviser, cut off from France by the English fleet, he organized the workshops on which the French army had to rely for their munitions of war. He also contributed several papers to the Egyptian Institute which Napoleon founded at Cairo. After the British victories and the capitulation of the French under General Menou in 1801, in 1801, Napoleon appointed Fourier Prefect of the Department of Isère in Grenoble, where he oversaw road construction and other projects. However, Fourier had previously returned home from the Napoleon expedition to Egypt to resume his academic post as professor at École Polytechnique when Napoleon decided otherwise in his remark. The Prefect of the Department of Isère having recently died, I would like to express my confidence in citizen Fourier by appointing him to this place, hence being faithful to Napoleon, he took the office of Prefect. It was while at Grenoble that he began to experiment on the propagation of heat and he presented his paper On the Propagation of Heat in Solid Bodies to the Paris Institute on December 21,1807. He also contributed to the monumental Description de lÉgypte, Fourier moved to England in 1816. Later, he returned to France, and in 1822 succeeded Jean Baptiste Joseph Delambre as Permanent Secretary of the French Academy of Sciences, in 1830, he was elected a foreign member of the Royal Swedish Academy of Sciences. In 1830, his health began to take its toll, Fourier had already experienced, in Egypt and Grenoble. At Paris, it was impossible to be mistaken with respect to the cause of the frequent suffocations which he experienced. A fall, however, which he sustained on the 4th of May 1830, while descending a flight of stairs, shortly after this event, he died in his bed on 16 May 1830. His name is one of the 72 names inscribed on the Eiffel Tower, a bronze statue was erected in Auxerre in 1849, but it was melted down for armaments during World War II. Joseph Fourier University in Grenoble is named after him and this book was translated, with editorial corrections, into English 56 years later by Freeman
Joseph Fourier
–
Jean-Baptiste Joseph Fourier
Joseph Fourier
–
1820 watercolor
caricatures of French mathematicians
Adrien-Marie Legendre (left) and Joseph Fourier (right) by French artist
Julien-Leopold Boilly, watercolor portrait numbers 29 and 30 of Album de 73 Portraits-Charge Aquarelle’s des Membres de I’Institute.
Joseph Fourier
–
Sketch of Fourier, circa 1820.
Joseph Fourier
–
Bust of Fourier in Grenoble
15.
Giovanni Antonio Amedeo Plana
–
Giovanni Antonio Amedeo Plana was an Italian astronomer and mathematician. Plana was born in Voghera, Italy to Antonio Maria Plana and Giovanna Giacoboni, at the age of 15 he was sent to live with his uncles in Grenoble to complete his education. In 1800 he entered the École Polytechnique, and was one of the students of Joseph-Louis Lagrange, in 1811 he was appointed to the chair of astronomy at the University of Turin thanks to the influence of Lagrange. He spent the remainder of his teaching at that institution. Planas contributions included work on the motions of the Moon, as well as integrals, elliptic functions, heat, electrostatics, and geodesy. In 1820 he was one of the winners of an awarded by the Académie des Sciences in Paris based on the construction of lunar tables using the law of gravity. In 1832 he published the Théorie du mouvement de la lune, in 1834 he was awarded with the Copley Medal by the Royal Society for his studies on lunar motion. He became astronomer royal, and then in 1844 a Baron, at the age of 80 he was granted membership in the prestigious Académie des Sciences. He is considered one of the premiere Italian scientists of his age, the crater Plana on the Moon is named in his honor. Biography and a source for this page, oConnor, John J. Robertson, Edmund F. Giovanni Antonio Amedeo Plana, MacTutor History of Mathematics archive, University of St Andrews
Giovanni Antonio Amedeo Plana
–
Giovanni Antonio Amedeo Plana.
16.
Enlightenment Era
–
The Enlightenment was an intellectual movement which dominated the world of ideas in Europe during the 18th century, The Century of Philosophy. In France, the doctrines of les Lumières were individual liberty and religious tolerance in opposition to an absolute monarchy. French historians traditionally place the Enlightenment between 1715, the year that Louis XIV died, and 1789, the beginning of the French Revolution, some recent historians begin the period in the 1620s, with the start of the scientific revolution. Les philosophes of the widely circulated their ideas through meetings at scientific academies, Masonic lodges, literary salons, coffee houses. The ideas of the Enlightenment undermined the authority of the monarchy and the Church, a variety of 19th-century movements, including liberalism and neo-classicism, trace their intellectual heritage back to the Enlightenment. The Age of Enlightenment was preceded by and closely associated with the scientific revolution, earlier philosophers whose work influenced the Enlightenment included Francis Bacon, René Descartes, John Locke, and Baruch Spinoza. The major figures of the Enlightenment included Cesare Beccaria, Voltaire, Denis Diderot, Jean-Jacques Rousseau, David Hume, Adam Smith, Benjamin Franklin visited Europe repeatedly and contributed actively to the scientific and political debates there and brought the newest ideas back to Philadelphia. Thomas Jefferson closely followed European ideas and later incorporated some of the ideals of the Enlightenment into the Declaration of Independence, others like James Madison incorporated them into the Constitution in 1787. The most influential publication of the Enlightenment was the Encyclopédie, the ideas of the Enlightenment played a major role in inspiring the French Revolution, which began in 1789. After the Revolution, the Enlightenment was followed by an intellectual movement known as Romanticism. René Descartes rationalist philosophy laid the foundation for enlightenment thinking and his attempt to construct the sciences on a secure metaphysical foundation was not as successful as his method of doubt applied in philosophic areas leading to a dualistic doctrine of mind and matter. His skepticism was refined by John Lockes 1690 Essay Concerning Human Understanding and his dualism was challenged by Spinozas uncompromising assertion of the unity of matter in his Tractatus and Ethics. Both lines of thought were opposed by a conservative Counter-Enlightenment. In the mid-18th century, Paris became the center of an explosion of philosophic and scientific activity challenging traditional doctrines, the political philosopher Montesquieu introduced the idea of a separation of powers in a government, a concept which was enthusiastically adopted by the authors of the United States Constitution. Francis Hutcheson, a philosopher, described the utilitarian and consequentialist principle that virtue is that which provides, in his words. Much of what is incorporated in the method and some modern attitudes towards the relationship between science and religion were developed by his protégés David Hume and Adam Smith. Hume became a figure in the skeptical philosophical and empiricist traditions of philosophy. Immanuel Kant tried to reconcile rationalism and religious belief, individual freedom and political authority, as well as map out a view of the sphere through private
Enlightenment Era
–
German philosopher
Immanuel Kant
Enlightenment Era
–
History of
Western philosophy
Enlightenment Era
–
Cesare Beccaria, father of classical criminal theory (1738–1794)
Enlightenment Era
–
Like other Enlightenment philosophers,
Rousseau was critical of the
Atlantic slave trade.
17.
Astronomer
–
An astronomer is a scientist in the field of astronomy who concentrates their studies on a specific question or field outside of the scope of Earth. They look at stars, planets, moons, comets and galaxies, as well as other celestial objects — either in observational astronomy. Examples of topics or fields astronomers work on include, planetary science, solar astronomy, there are also related but distinct subjects like physical cosmology which studies the Universe as a whole. Astronomers usually fit into two types, Observational astronomers make direct observations of planets, stars and galaxies, and analyze the data, theoretical astronomers create and investigate models of things that cannot be observed. They use this data to create models or simulations to theorize how different celestial bodies work, there are further subcategories inside these two main branches of astronomy such as planetary astronomy, galactic astronomy or physical cosmology. Today, that distinction has disappeared and the terms astronomer. Professional astronomers are highly educated individuals who typically have a Ph. D. in physics or astronomy and are employed by research institutions or universities. They spend the majority of their time working on research, although quite often have other duties such as teaching, building instruments. The number of astronomers in the United States is actually quite small. The American Astronomical Society, which is the organization of professional astronomers in North America, has approximately 7,000 members. This number includes scientists from other such as physics, geology. The International Astronomical Union comprises almost 10,145 members from 70 different countries who are involved in research at the Ph. D. level. Before CCDs, photographic plates were a method of observation. Modern astronomers spend relatively little time at telescopes usually just a few weeks per year, analysis of observed phenomena, along with making predictions as to the causes of what they observe, takes the majority of observational astronomers time. Astronomers who serve as faculty spend much of their time teaching undergraduate and graduate classes, most universities also have outreach programs including public telescope time and sometimes planetariums as a public service to encourage interest in the field. Those who become astronomers usually have a background in maths, sciences. Taking courses that teach how to research, write and present papers are also invaluable, in college/university most astronomers get a Ph. D. in astronomy or physics. Keeping in mind how few there are it is understood that graduate schools in this field are very competitive
Astronomer
–
The Astronomer by
Johannes Vermeer
Astronomer
–
Galileo is often referred to as the Father of
modern astronomy
Astronomer
–
Guy Consolmagno (Vatikan observatory), analyzing a meteorite, 2014
Astronomer
–
Emily Lakdawalla at the Planetary Conference 2013
18.
Classical mechanics
–
In physics, classical mechanics is one of the two major sub-fields of mechanics, along with quantum mechanics. Classical mechanics is concerned with the set of physical laws describing the motion of bodies under the influence of a system of forces. The study of the motion of bodies is an ancient one, making classical mechanics one of the oldest and largest subjects in science, engineering and technology. Classical mechanics describes the motion of objects, from projectiles to parts of machinery, as well as astronomical objects, such as spacecraft, planets, stars. Within classical mechanics are fields of study that describe the behavior of solids, liquids and gases, Classical mechanics also provides extremely accurate results as long as the domain of study is restricted to large objects and the speeds involved do not approach the speed of light. When both quantum and classical mechanics cannot apply, such as at the level with high speeds. Since these aspects of physics were developed long before the emergence of quantum physics and relativity, however, a number of modern sources do include relativistic mechanics, which in their view represents classical mechanics in its most developed and accurate form. Later, more abstract and general methods were developed, leading to reformulations of classical mechanics known as Lagrangian mechanics and these advances were largely made in the 18th and 19th centuries, and they extend substantially beyond Newtons work, particularly through their use of analytical mechanics. The following introduces the concepts of classical mechanics. For simplicity, it often models real-world objects as point particles, the motion of a point particle is characterized by a small number of parameters, its position, mass, and the forces applied to it. Each of these parameters is discussed in turn, in reality, the kind of objects that classical mechanics can describe always have a non-zero size. Objects with non-zero size have more complicated behavior than hypothetical point particles, because of the degrees of freedom. However, the results for point particles can be used to such objects by treating them as composite objects. The center of mass of a composite object behaves like a point particle, Classical mechanics uses common-sense notions of how matter and forces exist and interact. It assumes that matter and energy have definite, knowable attributes such as where an object is in space, non-relativistic mechanics also assumes that forces act instantaneously. The position of a point particle is defined with respect to a fixed reference point in space called the origin O, in space. A simple coordinate system might describe the position of a point P by means of a designated as r. In general, the point particle need not be stationary relative to O, such that r is a function of t, the time
Classical mechanics
–
Sir
Isaac Newton (1643–1727), an influential figure in the history of physics and whose
three laws of motion form the basis of classical mechanics
Classical mechanics
–
Diagram of orbital motion of a satellite around the earth, showing perpendicular velocity and acceleration (force) vectors.
Classical mechanics
–
Hamilton 's greatest contribution is perhaps the reformulation of
Newtonian mechanics, now called
Hamiltonian mechanics.
19.
Prussian Academy of Sciences
–
The Royal Prussian Academy of Sciences was an academic academy established in Berlin on 11 July 1700, four years after the Akademie der Künste or Arts Academy, to which Berlin Academy may also refer. In the 18th century it was a French-language institution, and its most active members were Huguenots who had fled persecution in France. Unlike other academies, the Prussian Academy was not directly funded out of the state treasury, Frederick granted it the monopoly on producing and selling calendars in Brandenburg, a suggestion by Leibniz. As Frederick was crowned King in Prussia in 1701, creating the Kingdom of Prussia, while other academies focused on a few topics, the Prussian Academy was the first to teach both sciences and humanities. In 1710, the statute was set, dividing the academy in two sciences and two humanities classes. This was not changed until 1830, when the physics-mathematics and the philosophy-history classes replaced the four old classes, the reign of King Frederick II saw major changes to the academy. In 1744, the Nouvelle Société Littéraire and the Society of Sciences were merged into the Königliche Akademie der Wissenschaften, an obligation from the new statute were public calls for ideas on unsolved scientific questions with a monetary reward for solutions. However, those were taken over by the University of Berlin Aarsleff notes that before Frederick came to the throne in 1740, Frederick made French the official language and speculative philosophy the most important topic of study. The membership was strong in mathematics and philosophy and included Immanuel Kant, Jean DAlembert, Pierre-Louis de Maupertuis, by contrast d Alembert took a republican rather than monarchical approach and emphasized the international Republic of Letters as the vehicle for scientific advance. By 1789, however, the academy had gained an international repute while making contributions to German culture. Frederick invited Joseph-Louis Lagrange to succeed Leonhard Euler as director, both were world-class mathematicians, other intellectuals attracted to the philosophers kingdom were Francesco Algarotti, dArgens, and Julien Offray de La Mettrie. Immanuel Kant published religious writings in Berlin which would have been censored elsewhere in Europe, beginning in 1815, research businesses led by academy committees were founded at the academy. They employed mostly scientists to work alongside the corresponding committees members, University departments emanated from some of these businesses after 1945. Under Nazi rule, the academy was subject to the Gleichschaltung, the new academy statute went in effect on 8 June 1939, reorganizing the academy according to the Nazi leader principle. Following World War II, the Soviet Military Administration in Germany or SMAD reorganized the academy under the name of Deutsche Akademie der Wissenschaften zu Berlin on 1 July 1946, on 25 November 1915 Albert Einstein presented his field equations of general relativity to the Academy. In 1972, it was renamed Akademie der Wissenschaften der DDR or AdW, at its height, the AdW had 400 researchers and 24,000 employees in locations across East Germany. 60 of the AdW members broke off and created the private Leibniz Society in 1993,1763 Immanuel Kant, foreign member 1786 Voltaire, c. Berlin-Brandenburg Academy of Sciences and Humanities, formerly the Prussian Academy of Sciences
Prussian Academy of Sciences
–
Entrance to the former Prussian Academy of Sciences on
Unter Den Linden 8. Today it houses the
Berlin State Library.
20.
French Academy of Sciences
–
The French Academy of Sciences is a learned society, founded in 1666 by Louis XIV at the suggestion of Jean-Baptiste Colbert, to encourage and protect the spirit of French scientific research. It was at the forefront of developments in Europe in the 17th and 18th centuries. Currently headed by Sébastien Candel, it is one of the five Academies of the Institut de France, the Academy of Sciences makes its origin to Colberts plan to create a general academy. He chose a group of scholars who met on 22 December 1666 in the Kings library. The first 30 years of the Academys existence were relatively informal, in contrast to its British counterpart, the Academy was founded as an organ of government. The Academy was expected to remain apolitical, and to avoid discussion of religious, on 20 January 1699, Louis XIV gave the Company its first rules. The Academy received the name of Royal Academy of Sciences and was installed in the Louvre in Paris, following this reform, the Academy began publishing a volume each year with information on all the work done by its members and obituaries for members who had died. This reform also codified the method by which members of the Academy could receive pensions for their work, on 8 August 1793, the National Convention abolished all the academies. Almost all the old members of the previously abolished Académie were formally re-elected, among the exceptions was Dominique, comte de Cassini, who refused to take his seat. In 1816, the again renamed Royal Academy of Sciences became autonomous, while forming part of the Institute of France, in the Second Republic, the name returned to Académie des sciences. During this period, the Academy was funded by and accountable to the Ministry of Public Instruction, the Academy came to control French patent laws in the course of the eighteenth century, acting as the liaison of artisans knowledge to the public domain. As a result, academicians dominated technological activities in France, the Academy proceedings were published under the name Comptes rendus de lAcadémie des sciences. The Comptes rendus is now a series with seven titles. The publications can be found on site of the French National Library, in 1818 the French Academy of Sciences launched a competition to explain the properties of light. The civil engineer Augustin-Jean Fresnel entered this competition by submitting a new theory of light. Siméon Denis Poisson, one of the members of the judging committee, being a supporter of the particle-theory of light, he looked for a way to disprove it. The Poisson spot is not easily observed in every-day situations, so it was natural for Poisson to interpret it as an absurd result. However, the head of the committee, Dominique-François-Jean Arago, and he molded a 2-mm metallic disk to a glass plate with wax
French Academy of Sciences
–
A heroic depiction of the activities of the Academy from 1698
French Academy of Sciences
–
Colbert Presenting the Members of the Royal Academy of Sciences to Louis XIV in 1667
French Academy of Sciences
–
The
Institut de France in Paris where the Academy is housed
21.
Isaac Newton
–
His book Philosophiæ Naturalis Principia Mathematica, first published in 1687, laid the foundations of classical mechanics. Newton also made contributions to optics, and he shares credit with Gottfried Wilhelm Leibniz for developing the infinitesimal calculus. Newtons Principia formulated the laws of motion and universal gravitation that dominated scientists view of the universe for the next three centuries. Newtons work on light was collected in his influential book Opticks. He also formulated a law of cooling, made the first theoretical calculation of the speed of sound. Newton was a fellow of Trinity College and the second Lucasian Professor of Mathematics at the University of Cambridge, politically and personally tied to the Whig party, Newton served two brief terms as Member of Parliament for the University of Cambridge, in 1689–90 and 1701–02. He was knighted by Queen Anne in 1705 and he spent the last three decades of his life in London, serving as Warden and Master of the Royal Mint and his father, also named Isaac Newton, had died three months before. Born prematurely, he was a child, his mother Hannah Ayscough reportedly said that he could have fit inside a quart mug. When Newton was three, his mother remarried and went to live with her new husband, the Reverend Barnabas Smith, leaving her son in the care of his maternal grandmother, Newtons mother had three children from her second marriage. From the age of twelve until he was seventeen, Newton was educated at The Kings School, Grantham which taught Latin and Greek. He was removed from school, and by October 1659, he was to be found at Woolsthorpe-by-Colsterworth, Henry Stokes, master at the Kings School, persuaded his mother to send him back to school so that he might complete his education. Motivated partly by a desire for revenge against a bully, he became the top-ranked student. In June 1661, he was admitted to Trinity College, Cambridge and he started as a subsizar—paying his way by performing valets duties—until he was awarded a scholarship in 1664, which guaranteed him four more years until he would get his M. A. He set down in his notebook a series of Quaestiones about mechanical philosophy as he found it, in 1665, he discovered the generalised binomial theorem and began to develop a mathematical theory that later became calculus. Soon after Newton had obtained his B. A. degree in August 1665, in April 1667, he returned to Cambridge and in October was elected as a fellow of Trinity. Fellows were required to become ordained priests, although this was not enforced in the restoration years, however, by 1675 the issue could not be avoided and by then his unconventional views stood in the way. Nevertheless, Newton managed to avoid it by means of a special permission from Charles II. A and he was elected a Fellow of the Royal Society in 1672. Newtons work has been said to distinctly advance every branch of mathematics then studied and his work on the subject usually referred to as fluxions or calculus, seen in a manuscript of October 1666, is now published among Newtons mathematical papers
Isaac Newton
–
Portrait of Isaac Newton in 1689 (age 46) by
Godfrey Kneller
Isaac Newton
–
Newton in a 1702 portrait by
Godfrey Kneller
Isaac Newton
–
Isaac Newton (Bolton, Sarah K. Famous Men of Science. NY: Thomas Y. Crowell & Co., 1889)
Isaac Newton
–
Replica of Newton's second
Reflecting telescope that he presented to the
Royal Society in 1672
22.
History of the metric system
–
Concepts similar to those behind the metric system had been discussed in the 16th and 17th centuries. Simon Stevin had published his ideas for a decimal notation and John Wilkins had published a proposal for a system of measurement based on natural units. The work of reforming the old system of weights and measures was sponsored by the revolutionary government, the metric system was to be, in the words of philosopher and mathematician Condorcet, for all people for all time. Reference copies for both units were manufactured and placed in the custody of the French Academy of Sciences, by 1812, due to the unpopularity of the new metric system, France had reverted to units similar to those of their old system. In 1837 the metric system was re-adopted by France, and also during the first half of the 19th century was adopted by the scientific community, maxwell proposed three base units, length, mass and time. This concept worked well with mechanics, but attempts to describe electromagnetic forces in terms of these units encountered difficulties. This impasse was resolved by Giovanni Giorgi, who in 1901 proved that a coherent system that incorporated electromagnetic units had to have a unit as a fourth base unit. The mole was added as a base unit in 1971. Since the end of the 20th century, an effort has been undertaken to redefine the ampere, kilogram, mole, the first practical implementation of the metric system was the system implemented by French Revolutionaries towards the end of the 18th century. Its key features were that, It was decimal in nature and it derived its unit sizes from nature. Units that have different dimensions are related to other in a rational manner. Prefixes are used to denote multiples and sub-multiples of its units and these features had already been explored and expounded by various scholars and academics in the two centuries prior to the French metric system being implemented. Simon Stevin is credited with introducing the system into general use in Europe. Twentieth-century writers such Bigourdan and McGreevy credit the French cleric Gabriel Mouton as the originator of the metric system, in 2007 a proposal for a coherent decimal system of measurement by the English cleric John Wilkins received publicity. During the early era, Roman numerals were used in Europe to represent numbers, but the Arabs represented numbers using the Hindu numeral system. In about 1202, Fibonacci published his book Liber Abaci which introduced the concept of positional notation into Europe and these symbols evolved into the numerals 0,1,2 etc. At that time there was dispute regarding the difference between numbers and irrational numbers and there was no consistency in the way in which decimal fractions were represented. In 1586, Simon Stevin published a pamphlet called De Thiende which historians credit as being the basis of modern notation for decimal fractions
History of the metric system
–
Frontispiece of the publication where John Wilkins proposed a metric system of units in which
length,
mass,
volume and
area would be related to each other
History of the metric system
–
James Watt, British inventor and advocate of an international decimalized system of measure
History of the metric system
–
A clock of the republican era showing both
decimal and
standard time
History of the metric system
–
Repeating circle – the instrument used for triangulation when measuring the meridian
23.
French First Republic
–
In the history of France, the First Republic, officially the French Republic, was founded on 21 September 1792 during the French Revolution. The First Republic lasted until the declaration of the First Empire in 1804 under Napoleon, under the Legislative Assembly, which was in power before the proclamation of the First Republic, France was engaged in war with Prussia and Austria. The foreign threat exacerbated Frances political turmoil amid the French Revolution and deepened the passion, in the violence of 10 August 1792, citizens stormed the Tuileries Palace, killing six hundred of the Kings Swiss guards and insisting on the removal of the king. A renewed fear of action prompted further violence, and in the first week of September 1792, mobs of Parisians broke into the citys prisons. This included nobles, clergymen, and political prisoners, but also numerous common criminals, such as prostitutes and petty thieves, many murdered in their cells—raped, stabbed and this became known as the September Massacres. The resulting Convention was founded with the purpose of abolishing the monarchy. The Conventions first act, on 10 August 1792, was to establish the French First Republic, the King, by then a private citizen bearing his family name of Capet, was subsequently put on trial for crimes of high treason starting in December 1792. On 16 January 1793 he was convicted, and on 21 January, throughout the winter of 1792 and spring of 1793, Paris was plagued by food riots and mass hunger. The new Convention did little to remedy the problem until late spring of 1793, despite growing discontent with the National Convention as a ruling body, in June the Convention drafted the Constitution of 1793, which was ratified by popular vote in early August. The Committees laws and policies took the revolution to unprecedented heights, after the arrest and execution of Robespierre in July 1794, the Jacobin club was closed, and the surviving Girondins were reinstated. A year later, the National Convention adopted the Constitution of the Year III and they reestablished freedom of worship, began releasing large numbers of prisoners, and most importantly, initiated elections for a new legislative body. On 3 November 1795, the Directory was established, the period known as the French Consulate began with the coup of 18 Brumaire in 1799. Members of the Directory itself planned the coup, indicating clearly the failing power of the Directory, Napoleon Bonaparte was a co-conspirator in the coup, and became head of the government as the First Consul. He would later proclaim himself Emperor of the French, ending the First French Republic and ushering in the French First Empire
French First Republic
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Napoleon Bonaparte seizes power during the Coup of 18 Brumaire
French First Republic
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Flag
24.
Bureau des Longitudes
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During the 19th century, it was responsible for synchronizing clocks across the world. It was headed during this time by François Arago and Henri Poincaré, the Bureau now functions as an academy and still meets monthly to discuss topics related to astronomy. The Bureau was founded by the National Convention after it heard a report drawn up jointly by the Committee of Navy, the Committee of Finances, as a result, the Bureau was established with authority over the Paris Observatory and all other astronomical establishments throughout France. The Bureau was charged with taking control of the seas away from the English and improving accuracy when tracking the longitudes of ships through astronomical observations, by a decree of 30 January 1854, the Bureaus mission was extended to embrace geodesy, time standardisation and astronomical measurements. This decree granted independence to the Paris Observatory, separating it from the Bureau, the Bureau was successful at setting a universal time in Paris via air pulses sent through pneumatic tubes. It later worked to synchronize time across the French colonial empire by determining the length of time for a signal to make a trip to. The French Bureau of Longitude established a commission in the year 1897 to extend the system to the measurement of time. They planned to abolish the antiquated division of the day hours, minutes, and seconds, and replace it by a division into tenths, thousandths. This was a revival of a dream that was in the minds of the creators of the system at the time of the French Revolution a hundred years earlier. Some members of the Bureau of Longitude commission introduced a proposal, retaining the old-fashioned hour as the basic unit of time. Poincaré served as secretary of the commission and took its work very seriously and he was a fervent believer in a universal metric system. The rest of the world outside France gave no support to the proposals. After three years of work, the commission was dissolved in 1900. Since 1970, the board has been constituted with 13 members,3 nominated by the Académie des Sciences, since 1998, practical work has been carried out by the Institut de mécanique céleste et de calcul des éphémérides. Institut de mécanique céleste et de calcul des éphémérides Bureau Des Longitudes Galison, einsteins Clocks, Poincarés Maps, Empires of Time
Bureau des Longitudes
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ABBE GREGOIRE (1750-1831).
25.
Functional (mathematics)
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In mathematics, and particularly in functional analysis and the calculus of variations, a functional is a function from a vector space into its underlying field of scalars. Commonly the vector space is a space of functions, thus the functional takes a function for its input argument and its use originates in the calculus of variations, where one searches for a function that minimizes a given functional. A particularly important application in physics is searching for a state of a system that minimizes the energy functional, the mapping x 0 ↦ f is a function, where x0 is an argument of a function f. At the same time, the mapping of a function to the value of the function at a point f ↦ f is a functional, here, integrals such as f ↦ I = ∫ Ω H μ form a special class of functionals. They map a function f into a number, provided that H is real-valued. The set of vectors x → such that x → ⋅ y → is zero is a subspace of X. If a functionals value can be computed for small segments of the curve and then summed to find the total value. For example, F = ∫ x 0 x 1 y d x is local while F = ∫ x 0 x 1 y d x ∫ x 0 x 1 d x is non-local. This occurs commonly when integrals occur separately in the numerator and denominator of a such as in calculations of center of mass. In such equations there may be several sets of variable unknowns, Functional derivatives are used in Lagrangian mechanics. They are derivatives of functionals, i. e. they carry information on how a functional changes when the function changes by a small amount. Richard Feynman used functional integrals as the idea in his sum over the histories formulation of quantum mechanics. This usage implies an integral taken over some function space, linear functional Optimization Tensor Hazewinkel, Michiel, ed. Functional, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4 Rowland, Todd. The dual space and dual module, Algebra, Graduate Texts in Mathematics,211, New York, Springer-Verlag, pp. 142–146, ISBN 978-0-387-95385-4, Zbl 0984.00001, MR1878556
Functional (mathematics)
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The
Riemann integral is a
linear functional on the vector space of Riemann-integrable functions from to.
26.
Lagrange multipliers
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In mathematical optimization, the method of Lagrange multipliers is a strategy for finding the local maxima and minima of a function subject to equality constraints. For instance, consider the optimization problem maximize f subject to g = c and we need both f and g to have continuous first partial derivatives. We introduce a new variable called a Lagrange multiplier and study the Lagrange function defined by L = f − λ ⋅, if f is a maximum of f for the original constrained problem, then there exists λ0 such that is a stationary point for the Lagrange function. However, not all stationary points yield a solution of the original problem, thus, the method of Lagrange multipliers yields a necessary condition for optimality in constrained problems. Sufficient conditions for a minimum or maximum also exist, one of the most common problems in calculus is that of finding maxima or minima of a function, but it is often difficult to find a closed form for the function being extremized. Such difficulties often arise when one wishes to maximize or minimize a function subject to fixed outside equality constraints, the method of Lagrange multipliers is a powerful tool for solving this class of problems without the need to explicitly solve the conditions and use them to eliminate extra variables. Consider the two-dimensional problem introduced above maximize f subject to g =0, the method of Lagrange multipliers relies on the intuition that at a maximum, f cannot be increasing in the direction of any neighboring point where g =0. If it were, we could walk along g =0 to get higher and we can visualize contours of f given by f = d for various values of d, and the contour of g given by g =0. Suppose we walk along the line with g =0. We are interested in finding points where f does not change as we walk, there are two ways this could happen, First, we could be following a contour line of f, since by definition f does not change as we walk along its contour lines. This would mean that the lines of f and g are parallel here. The second possibility is that we have reached a part of f. Thus we want points where g =0 and ∇ x, y f = λ ∇ x, y g, the constant λ is required because although the two gradient vectors are parallel, the magnitudes of the gradient vectors are generally not equal. This constant is called the Lagrange multiplier, notice that this method also solves the second possibility, if f is level, then its gradient is zero, and setting λ =0 is a solution regardless of g. To incorporate these conditions into one equation, we introduce an auxiliary function L = f − λ ⋅ g, note that this amounts to solving three equations in three unknowns. This is the method of Lagrange multipliers, note that ∇ λ L =0 implies g =0. The constrained extrema of f are points of the Lagrangian L. One may reformulate the Lagrangian as a Hamiltonian, in case the solutions are local minima for the Hamiltonian
Lagrange multipliers
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Figure 1: Find x and y to maximize f (x, y) subject to a constraint (shown in red) g (x, y) = c.
27.
Differential equation
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A differential equation is a mathematical equation that relates some function with its derivatives. In applications, the functions usually represent physical quantities, the derivatives represent their rates of change, because such relations are extremely common, differential equations play a prominent role in many disciplines including engineering, physics, economics, and biology. In pure mathematics, differential equations are studied from different perspectives. Only the simplest differential equations are solvable by explicit formulas, however, if a self-contained formula for the solution is not available, the solution may be numerically approximated using computers. Differential equations first came into existence with the invention of calculus by Newton, jacob Bernoulli proposed the Bernoulli differential equation in 1695. This is a differential equation of the form y ′ + P y = Q y n for which the following year Leibniz obtained solutions by simplifying it. Historically, the problem of a string such as that of a musical instrument was studied by Jean le Rond dAlembert, Leonhard Euler, Daniel Bernoulli. In 1746, d’Alembert discovered the wave equation, and within ten years Euler discovered the three-dimensional wave equation. The Euler–Lagrange equation was developed in the 1750s by Euler and Lagrange in connection with their studies of the tautochrone problem. This is the problem of determining a curve on which a particle will fall to a fixed point in a fixed amount of time. Lagrange solved this problem in 1755 and sent the solution to Euler, both further developed Lagranges method and applied it to mechanics, which led to the formulation of Lagrangian mechanics. Contained in this book was Fouriers proposal of his heat equation for conductive diffusion of heat and this partial differential equation is now taught to every student of mathematical physics. For example, in mechanics, the motion of a body is described by its position. Newtons laws allow one to express these variables dynamically as an equation for the unknown position of the body as a function of time. In some cases, this equation may be solved explicitly. An example of modelling a real world problem using differential equations is the determination of the velocity of a ball falling through the air, considering only gravity, the balls acceleration towards the ground is the acceleration due to gravity minus the acceleration due to air resistance. Gravity is considered constant, and air resistance may be modeled as proportional to the balls velocity and this means that the balls acceleration, which is a derivative of its velocity, depends on the velocity. Finding the velocity as a function of time involves solving a differential equation, Differential equations can be divided into several types
Differential equation
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Navier–Stokes differential equations used to simulate airflow around an obstruction.
28.
Differential calculus
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In mathematics, differential calculus is a subfield of calculus concerned with the study of the rates at which quantities change. It is one of the two divisions of calculus, the other being integral calculus. The primary objects of study in differential calculus are the derivative of a function, related notions such as the differential, the derivative of a function at a chosen input value describes the rate of change of the function near that input value. The process of finding a derivative is called differentiation, geometrically, the derivative at a point is the slope of the tangent line to the graph of the function at that point, provided that the derivative exists and is defined at that point. For a real-valued function of a real variable, the derivative of a function at a point generally determines the best linear approximation to the function at that point. Differential calculus and integral calculus are connected by the theorem of calculus. Differentiation has applications to nearly all quantitative disciplines, for example, in physics, the derivative of the displacement of a moving body with respect to time is the velocity of the body, and the derivative of velocity with respect to time is acceleration. The derivative of the momentum of a body equals the applied to the body. The reaction rate of a reaction is a derivative. In operations research, derivatives determine the most efficient ways to transport materials, derivatives are frequently used to find the maxima and minima of a function. Equations involving derivatives are called differential equations and are fundamental in describing natural phenomena, derivatives and their generalizations appear in many fields of mathematics, such as complex analysis, functional analysis, differential geometry, measure theory, and abstract algebra. Suppose that x and y are real numbers and that y is a function of x and this relationship can be written as y = f. If f is the equation for a line, then there are two real numbers m and b such that y = mx + b. In this slope-intercept form, the m is called the slope and can be determined from the formula, m = change in y change in x = Δ y Δ x. It follows that Δy = m Δx, a general function is not a line, so it does not have a slope. Geometrically, the derivative of f at the point x = a is the slope of the tangent line to the function f at the point a and this is often denoted f ′ in Lagranges notation or dy/dx|x = a in Leibnizs notation. Since the derivative is the slope of the approximation to f at the point a. If every point a in the domain of f has a derivative, for example, if f = x2, then the derivative function f ′ = dy/dx = 2x
Differential calculus
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The graph of a function, drawn in black, and a tangent line to that function, drawn in red. The slope of the tangent line equals the derivative of the function at the marked point.
29.
Probability theory
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Probability theory is the branch of mathematics concerned with probability, the analysis of random phenomena. It is not possible to predict precisely results of random events, two representative mathematical results describing such patterns are the law of large numbers and the central limit theorem. As a mathematical foundation for statistics, probability theory is essential to human activities that involve quantitative analysis of large sets of data. Methods of probability theory also apply to descriptions of complex systems given only partial knowledge of their state, a great discovery of twentieth century physics was the probabilistic nature of physical phenomena at atomic scales, described in quantum mechanics. Christiaan Huygens published a book on the subject in 1657 and in the 19th century, initially, probability theory mainly considered discrete events, and its methods were mainly combinatorial. Eventually, analytical considerations compelled the incorporation of continuous variables into the theory and this culminated in modern probability theory, on foundations laid by Andrey Nikolaevich Kolmogorov. Kolmogorov combined the notion of space, introduced by Richard von Mises. This became the mostly undisputed axiomatic basis for modern probability theory, most introductions to probability theory treat discrete probability distributions and continuous probability distributions separately. The more mathematically advanced measure theory-based treatment of probability covers the discrete, continuous, consider an experiment that can produce a number of outcomes. The set of all outcomes is called the space of the experiment. The power set of the space is formed by considering all different collections of possible results. For example, rolling an honest die produces one of six possible results, one collection of possible results corresponds to getting an odd number. Thus, the subset is an element of the set of the sample space of die rolls. In this case, is the event that the die falls on some odd number, If the results that actually occur fall in a given event, that event is said to have occurred. Probability is a way of assigning every event a value between zero and one, with the requirement that the event made up of all possible results be assigned a value of one, the probability that any one of the events, or will occur is 5/6. This is the same as saying that the probability of event is 5/6 and this event encompasses the possibility of any number except five being rolled. The mutually exclusive event has a probability of 1/6, and the event has a probability of 1, discrete probability theory deals with events that occur in countable sample spaces. Modern definition, The modern definition starts with a finite or countable set called the sample space, which relates to the set of all possible outcomes in classical sense, denoted by Ω
Probability theory
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The
normal distribution, a continuous probability distribution.
Probability theory
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The
Poisson distribution, a discrete probability distribution.
30.
Group theory
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In mathematics and abstract algebra, group theory studies the algebraic structures known as groups. Groups recur throughout mathematics, and the methods of group theory have influenced many parts of algebra, linear algebraic groups and Lie groups are two branches of group theory that have experienced advances and have become subject areas in their own right. Various physical systems, such as crystals and the hydrogen atom, thus group theory and the closely related representation theory have many important applications in physics, chemistry, and materials science. Group theory is central to public key cryptography. The first class of groups to undergo a systematic study was permutation groups, given any set X and a collection G of bijections of X into itself that is closed under compositions and inverses, G is a group acting on X. If X consists of n elements and G consists of all permutations, G is the symmetric group Sn, in general, an early construction due to Cayley exhibited any group as a permutation group, acting on itself by means of the left regular representation. In many cases, the structure of a group can be studied using the properties of its action on the corresponding set. For example, in this way one proves that for n ≥5 and this fact plays a key role in the impossibility of solving a general algebraic equation of degree n ≥5 in radicals. The next important class of groups is given by matrix groups, here G is a set consisting of invertible matrices of given order n over a field K that is closed under the products and inverses. Such a group acts on the vector space Kn by linear transformations. In the case of groups, X is a set, for matrix groups. The concept of a group is closely related with the concept of a symmetry group. The theory of groups forms a bridge connecting group theory with differential geometry. A long line of research, originating with Lie and Klein, the groups themselves may be discrete or continuous. Most groups considered in the first stage of the development of group theory were concrete, having been realized through numbers, permutations, or matrices. It was not until the nineteenth century that the idea of an abstract group as a set with operations satisfying a certain system of axioms began to take hold. A typical way of specifying an abstract group is through a presentation by generators and relations, a significant source of abstract groups is given by the construction of a factor group, or quotient group, G/H, of a group G by a normal subgroup H. Class groups of algebraic number fields were among the earliest examples of factor groups, of much interest in number theory
Group theory
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Water molecule with symmetry axis
Group theory
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The popular puzzle
Rubik's cube invented in 1974 by
Ernő Rubik has been used as an illustration of
permutation groups.
31.
Calculus
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Calculus is the mathematical study of continuous change, in the same way that geometry is the study of shape and algebra is the study of generalizations of arithmetic operations. It has two branches, differential calculus, and integral calculus, these two branches are related to each other by the fundamental theorem of calculus. Both branches make use of the notions of convergence of infinite sequences. Generally, modern calculus is considered to have developed in the 17th century by Isaac Newton. Today, calculus has widespread uses in science, engineering and economics, Calculus is a part of modern mathematics education. A course in calculus is a gateway to other, more advanced courses in mathematics devoted to the study of functions and limits, Calculus has historically been called the calculus of infinitesimals, or infinitesimal calculus. Calculus is also used for naming some methods of calculation or theories of computation, such as calculus, calculus of variations, lambda calculus. The ancient period introduced some of the ideas that led to integral calculus, the method of exhaustion was later discovered independently in China by Liu Hui in the 3rd century AD in order to find the area of a circle. In the 5th century AD, Zu Gengzhi, son of Zu Chongzhi, indian mathematicians gave a non-rigorous method of a sort of differentiation of some trigonometric functions. In the Middle East, Alhazen derived a formula for the sum of fourth powers. He used the results to carry out what would now be called an integration, Cavalieris work was not well respected since his methods could lead to erroneous results, and the infinitesimal quantities he introduced were disreputable at first. The formal study of calculus brought together Cavalieris infinitesimals with the calculus of finite differences developed in Europe at around the same time, pierre de Fermat, claiming that he borrowed from Diophantus, introduced the concept of adequality, which represented equality up to an infinitesimal error term. The combination was achieved by John Wallis, Isaac Barrow, and James Gregory, in other work, he developed series expansions for functions, including fractional and irrational powers, and it was clear that he understood the principles of the Taylor series. He did not publish all these discoveries, and at this time infinitesimal methods were considered disreputable. These ideas were arranged into a calculus of infinitesimals by Gottfried Wilhelm Leibniz. He is now regarded as an independent inventor of and contributor to calculus, unlike Newton, Leibniz paid a lot of attention to the formalism, often spending days determining appropriate symbols for concepts. Leibniz and Newton are usually credited with the invention of calculus. Newton was the first to apply calculus to general physics and Leibniz developed much of the used in calculus today
Calculus
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Isaac Newton developed the use of calculus in his
laws of motion and
gravitation.
Calculus
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Gottfried Wilhelm Leibniz was the first to publish his results on the development of calculus.
Calculus
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Maria Gaetana Agnesi
Calculus
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The
logarithmic spiral of the
Nautilus shell is a classical image used to depict the growth and change related to calculus
32.
Lagrange interpolation
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In numerical analysis, Lagrange polynomials are used for polynomial interpolation. For a given set of points x j and numbers y j. Although named after Joseph Louis Lagrange, who published it in 1795 and it is also an easy consequence of a formula published in 1783 by Leonhard Euler. Uses of Lagrange polynomials include the Newton–Cotes method of numerical integration, Lagrange interpolation is susceptible to Runges phenomenon of large oscillation. And changing the points x j requires recalculating the entire interpolant, note how, given the initial assumption that no two x i are the same, x j − x m ≠0, so this expression is always well-defined. On the other hand, if also y i = y j, then those two points would actually be one single point. It follows that y i ℓ i = y i, so at each point x i, L = y i +0 +0 + ⋯ +0 = y i, showing that L interpolates the function exactly. ℓ j = ∏ m =0, m ≠ j k x i − x m x j − x m We consider what happens when this product is expanded. Because the product skips m = j, if i = j then all terms are x j − x m x j − x m =1. So ℓ j = δ j i = {1, if j = i 0, so, L = ∑ j =0 k y j ℓ j = ∑ j =0 k y j δ j i = y i. Thus the function L is a polynomial with degree at most k, additionally, the interpolating polynomial is unique, as shown by the unisolvence theorem at the polynomial interpolation article. Solving an interpolation problem leads to a problem in linear algebra amounting to inversion of a matrix. Using a standard basis for our interpolation polynomial L = ∑ j =0 k x j m j. This construction is analogous to the Chinese Remainder Theorem, instead of checking for remainders of integers modulo prime numbers, we are checking for remainders of polynomials when divided by linears. We wish to interpolate ƒ = x2 over the range 1 ≤ x ≤3, the Lagrange form of the interpolation polynomial shows the linear character of polynomial interpolation and the uniqueness of the interpolation polynomial. Therefore, it is preferred in proofs and theoretical arguments, uniqueness can also be seen from the invertibility of the Vandermonde matrix, due to the non-vanishing of the Vandermonde determinant. But, as can be seen from the construction, each time a node xk changes, a better form of the interpolation polynomial for practical purposes is the barycentric form of the Lagrange interpolation or Newton polynomials. Lagrange and other interpolation at equally spaced points, as in the example above, yield a polynomial oscillating above and below the true function
Lagrange interpolation
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Example of Lagrange polynomial interpolation divergence.
Lagrange interpolation
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This image shows, for four points ((−9, 5), (−4, 2), (−1, −2), (7, 9)), the (cubic) interpolation polynomial L (x) (in black), which is the sum of the scaled basis polynomials y 0 ℓ 0 (x), y 1 ℓ 1 (x), y 2 ℓ 2 (x) and y 3 ℓ 3 (x). The interpolation polynomial passes through all four control points, and each scaled basis polynomial passes through its respective control point and is 0 where x corresponds to the other three control points.
33.
Taylor series
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In mathematics, a Taylor series is a representation of a function as an infinite sum of terms that are calculated from the values of the functions derivatives at a single point. The concept of a Taylor series was formulated by the Scottish mathematician James Gregory, a function can be approximated by using a finite number of terms of its Taylor series. Taylors theorem gives quantitative estimates on the error introduced by the use of such an approximation, the polynomial formed by taking some initial terms of the Taylor series is called a Taylor polynomial. The Taylor series of a function is the limit of that functions Taylor polynomials as the degree increases, a function may not be equal to its Taylor series, even if its Taylor series converges at every point. A function that is equal to its Taylor series in an interval is known as an analytic function in that interval. The Taylor series of a real or complex-valued function f that is differentiable at a real or complex number a is the power series f + f ′1. Which can be written in the more compact sigma notation as ∑ n =0 ∞ f n, N where n. denotes the factorial of n and f denotes the nth derivative of f evaluated at the point a. The derivative of order zero of f is defined to be f itself and 0 and 0. are both defined to be 1, when a =0, the series is also called a Maclaurin series. The Maclaurin series for any polynomial is the polynomial itself. The Maclaurin series for 1/1 − x is the geometric series 1 + x + x 2 + x 3 + ⋯ so the Taylor series for 1/x at a =1 is 1 − +2 −3 + ⋯. The Taylor series for the exponential function ex at a =0 is x 00, + ⋯ =1 + x + x 22 + x 36 + x 424 + x 5120 + ⋯ = ∑ n =0 ∞ x n n. The above expansion holds because the derivative of ex with respect to x is also ex and this leaves the terms n in the numerator and n. in the denominator for each term in the infinite sum. The Greek philosopher Zeno considered the problem of summing an infinite series to achieve a result, but rejected it as an impossibility. It was through Archimedess method of exhaustion that a number of progressive subdivisions could be performed to achieve a finite result. Liu Hui independently employed a similar method a few centuries later, in the 14th century, the earliest examples of the use of Taylor series and closely related methods were given by Madhava of Sangamagrama. The Kerala school of astronomy and mathematics further expanded his works with various series expansions, in the 17th century, James Gregory also worked in this area and published several Maclaurin series. It was not until 1715 however that a method for constructing these series for all functions for which they exist was finally provided by Brook Taylor. The Maclaurin series was named after Colin Maclaurin, a professor in Edinburgh, if f is given by a convergent power series in an open disc centered at b in the complex plane, it is said to be analytic in this disc
Taylor series
–
As the degree of the Taylor polynomial rises, it approaches the correct function. This image shows sin(x) and its Taylor approximations, polynomials of degree 1, 3, 5, 7, 9, 11 and 13.
34.
Three-body problem
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The three-body problem is a special case of the n-body problem. Historically, the first specific three-body problem to receive extended study was the one involving the Moon, the Earth, in an extended modern sense, a three-body problem is a class of problems in classical or quantum mechanics that model the motion of three particles. The gravitational problem of three bodies in its traditional sense dates in substance from 1687, when Isaac Newton published his Principia. The physical problem was addressed by Amerigo Vespucci and subsequently by Galileo Galilei, however the accuracy of the lunar theory was low, due to the perturbing effect of the Sun and planets on the motion of the Moon around the Earth. They submitted their competing first analyses to the Académie Royale des Sciences in 1747 and it was in connection with these researches, in Paris, in the 1740s, that the name three-body problem began to be commonly used. An account published in 1761 by Jean le Rond dAlembert indicates that the name was first used in 1747, in 1887, mathematicians Heinrich Bruns and Henri Poincaré showed that there is no general analytical solution for the three-body problem given by algebraic expressions and integrals. The motion of three bodies is generally non-repeating, except in special cases, a prominent example of the classical three-body problem is the movement of a planet with a satellite around a star. In this case, the problem is simplified to two instances of the two-body problem, however, the effect of the star on the movement of the satellite around the planet can be considered as a perturbation. While a spaceflight involving a gravity assist tends to be at least a problem, once far away from the Earth when Earths gravity becomes negligible. The general statement for the three body problem is as follows, in the circular restricted three-body problem, two massive bodies move in circular orbits around their common center of mass, and the third mass is negligible with respect to the other two. It can be useful to consider the effective potential, in 1767 Leonhard Euler found three families of periodic solutions in which the three masses are collinear at each instant. In 1772 Lagrange found a family of solutions in which the three form an equilateral triangle at each instant. Together, these form the central configurations for the three-body problem. These solutions are valid for any mass ratios, and the move on Keplerian ellipses. These five families are the only solutions for which there are explicit analytic formulae. In 1893 Meissel stated what is called the Pythagorean three-body problem. Burrau further investigated this problem in 1913, in 1967 Victor Szebehely and coworkers established eventual escape for this problem using numerical integration, while at the same time finding a nearby periodic solution. In 1911, United States scientist William Duncan MacMillan found one special solution, in 1961, Russian mathematician Sitnikov improved this solution
Three-body problem
–
Figure 1: Configuration of the Sitnikov Problem
Three-body problem
35.
Lagrangian point
–
The Lagrange points mark positions where the combined gravitational pull of the two large masses provides precisely the centrifugal force required to orbit with them. There are five points, labeled L1 to L5, all in the orbital plane of the two large bodies. The first three are on the line connecting the two bodies, the last two, L4 and L5, each form an equilateral triangle with the two large bodies. The two latter points are stable, which implies that objects can orbit around them in a coordinate system tied to the two large bodies. Several planets have satellites near their L4 and L5 points with respect to the Sun, the three collinear Lagrange points were discovered by Leonhard Euler a few years before Lagrange discovered the remaining two. In 1772, Joseph-Louis Lagrange published an Essay on the three-body problem, in the first chapter he considered the general three-body problem. From that, in the chapter, he demonstrated two special constant-pattern solutions, the collinear and the equilateral, for any three masses, with circular orbits. The five Lagrangian points are labeled and defined as follows, The L1 point lies on the line defined by the two large masses M1 and M2, and between them. It is the most intuitively understood of the Lagrangian points, the one where the attraction of M2 partially cancels M1s gravitational attraction. Explanation An object that orbits the Sun more closely than Earth would normally have an orbital period than Earth. If the object is directly between Earth and the Sun, then Earths gravity counteracts some of the Suns pull on the object, the closer to Earth the object is, the greater this effect is. At the L1 point, the period of the object becomes exactly equal to Earths orbital period. L1 is about 1.5 million kilometers from Earth, the L2 point lies on the line through the two large masses, beyond the smaller of the two. Here, the forces of the two large masses balance the centrifugal effect on a body at L2. Explanation On the opposite side of Earth from the Sun, the period of an object would normally be greater than that of Earth. The extra pull of Earths gravity decreases the orbital period of the object, like L1, L2 is about 1.5 million kilometers from Earth. The L3 point lies on the line defined by the two masses, beyond the larger of the two. Explanation L3 in the Sun–Earth system exists on the side of the Sun
Lagrangian point
–
Visualisation of the relationship between the Lagrangian points (red) of a planet (blue) orbiting a star (yellow) anticlockwise, and the
effective potential in the plane containing the orbit (grey rubber-sheet model with purple contours of equal potential).
Click for animation.
Lagrangian point
–
Lagrange points in the Sun–Earth system (not to scale)
36.
Newtonian mechanics
–
In physics, classical mechanics is one of the two major sub-fields of mechanics, along with quantum mechanics. Classical mechanics is concerned with the set of physical laws describing the motion of bodies under the influence of a system of forces. The study of the motion of bodies is an ancient one, making classical mechanics one of the oldest and largest subjects in science, engineering and technology. Classical mechanics describes the motion of objects, from projectiles to parts of machinery, as well as astronomical objects, such as spacecraft, planets, stars. Within classical mechanics are fields of study that describe the behavior of solids, liquids and gases, Classical mechanics also provides extremely accurate results as long as the domain of study is restricted to large objects and the speeds involved do not approach the speed of light. When both quantum and classical mechanics cannot apply, such as at the level with high speeds. Since these aspects of physics were developed long before the emergence of quantum physics and relativity, however, a number of modern sources do include relativistic mechanics, which in their view represents classical mechanics in its most developed and accurate form. Later, more abstract and general methods were developed, leading to reformulations of classical mechanics known as Lagrangian mechanics and these advances were largely made in the 18th and 19th centuries, and they extend substantially beyond Newtons work, particularly through their use of analytical mechanics. The following introduces the concepts of classical mechanics. For simplicity, it often models real-world objects as point particles, the motion of a point particle is characterized by a small number of parameters, its position, mass, and the forces applied to it. Each of these parameters is discussed in turn, in reality, the kind of objects that classical mechanics can describe always have a non-zero size. Objects with non-zero size have more complicated behavior than hypothetical point particles, because of the degrees of freedom. However, the results for point particles can be used to such objects by treating them as composite objects. The center of mass of a composite object behaves like a point particle, Classical mechanics uses common-sense notions of how matter and forces exist and interact. It assumes that matter and energy have definite, knowable attributes such as where an object is in space, non-relativistic mechanics also assumes that forces act instantaneously. The position of a point particle is defined with respect to a fixed reference point in space called the origin O, in space. A simple coordinate system might describe the position of a point P by means of a designated as r. In general, the point particle need not be stationary relative to O, such that r is a function of t, the time
Newtonian mechanics
–
Sir
Isaac Newton (1643–1727), an influential figure in the history of physics and whose
three laws of motion form the basis of classical mechanics
Newtonian mechanics
–
Diagram of orbital motion of a satellite around the earth, showing perpendicular velocity and acceleration (force) vectors.
Newtonian mechanics
–
Hamilton 's greatest contribution is perhaps the reformulation of
Newtonian mechanics, now called
Hamiltonian mechanics.
37.
Lagrangian mechanics
–
Lagrangian mechanics is a reformulation of classical mechanics, introduced by the Italian-French mathematician and astronomer Joseph-Louis Lagrange in 1788. No new physics is introduced in Lagrangian mechanics compared to Newtonian mechanics, Newtons laws can include non-conservative forces like friction, however, they must include constraint forces explicitly and are best suited to Cartesian coordinates. Lagrangian mechanics is ideal for systems with conservative forces and for bypassing constraint forces in any coordinate system, dissipative and driven forces can be accounted for by splitting the external forces into a sum of potential and non-potential forces, leading to a set of modified Euler-Lagrange equations. Generalized coordinates can be chosen by convenience, to exploit symmetries in the system or the geometry of the constraints, Lagrangian mechanics also reveals conserved quantities and their symmetries in a direct way, as a special case of Noethers theorem. Lagrangian mechanics is important not just for its applications. It can also be applied to systems by analogy, for instance to coupled electric circuits with inductances and capacitances. Lagrangian mechanics is used to solve mechanical problems in physics. Lagrangian mechanics applies to the dynamics of particles, fields are described using a Lagrangian density, Lagranges equations are also used in optimisation problems of dynamic systems. In mechanics, Lagranges equations of the second kind are used more than those of the first kind. Suppose we have a bead sliding around on a wire, or a simple pendulum. This choice eliminates the need for the constraint force to enter into the resultant system of equations, there are fewer equations since one is not directly calculating the influence of the constraint on the particle at a given moment. For a wide variety of systems, if the size and shape of a massive object are negligible. For a system of N point particles with masses m1, m2, MN, each particle has a position vector, denoted r1, r2. Cartesian coordinates are often sufficient, so r1 =, r2 =, in three dimensional space, each position vector requires three coordinates to uniquely define the location of a point, so there are 3N coordinates to uniquely define the configuration of the system. These are all points in space to locate the particles. The velocity of particle is how fast the particle moves along its path of motion. In Newtonian mechanics, the equations of motion are given by Newtons laws, the second law net force equals mass times acceleration, Σ F = m d2r/dt2, applies to each particle. For an N particle system in 3d, there are 3N second order differential equations in the positions of the particles to solve for
Lagrangian mechanics
–
Joseph-Louis Lagrange (1736—1813)
Lagrangian mechanics
–
Isaac Newton (1642—1726)
Lagrangian mechanics
–
Jean d'Alembert (1717—1783)
38.
W.W. Rouse Ball
–
Walter William Rouse Ball, known as W. W. Rouse Ball, was a British mathematician, lawyer, and fellow at Trinity College, Cambridge from 1878 to 1905. He was also an amateur magician, and the founding president of the Cambridge Pentacle Club in 1919. Ball was the son and heir of Walter Frederick Ball, of 3, St Johns Park Villas, South Hampstead, London. Educated at University College School, he entered Trinity College, Cambridge in 1870, became a scholar and first Smiths Prizeman and he became a Fellow of Trinity in 1875, and remained one for the rest of his life. He is buried at the Parish of the Ascension Burial Ground in Cambridge and he is commemorated in the naming of the small pavilion, now used as changing rooms and toilets, on Jesus Green in Cambridge. A History of the Study of Mathematics at Cambridge, Cambridge University Press,1889 A Short Account of the History of Mathematics at Project Gutenberg, dover 1960 republication of fourth edition. Mathematical Recreations and Essays at Project Gutenberg A History of the First Trinity Boat Club Cambridge Papers at Project Gutenberg, string Figures, Cambridge, W. Heffer & Sons Rouse Ball Professor of Mathematics Rouse Ball Professor of English Law Martin Gardner, another author of recreational mathematics. Singmaster, David,1892 Walter William Rouse Ball, Mathematical recreations and problems of past and present times, in Grattan-Guinness, W. W. Rouse Ball at the Mathematics Genealogy Project W. W. Rouse Ball at Find a Grave
W.W. Rouse Ball
–
W.W. Rouse Ball
39.
Kingdom of France
–
The Kingdom of France was a medieval and early modern monarchy in Western Europe. It was one of the most powerful states in Europe and a great power since the Late Middle Ages and it was also an early colonial power, with possessions around the world. France originated as West Francia, the half of the Carolingian Empire. A branch of the Carolingian dynasty continued to rule until 987, the territory remained known as Francia and its ruler as rex Francorum well into the High Middle Ages. The first king calling himself Roi de France was Philip II, France continued to be ruled by the Capetians and their cadet lines—the Valois and Bourbon—until the monarchy was overthrown in 1792 during the French Revolution. France in the Middle Ages was a de-centralised, feudal monarchy, in Brittany and Catalonia the authority of the French king was barely felt. Lorraine and Provence were states of the Holy Roman Empire and not yet a part of France, during the Late Middle Ages, the Kings of England laid claim to the French throne, resulting in a series of conflicts known as the Hundred Years War. Subsequently, France sought to extend its influence into Italy, but was defeated by Spain in the ensuing Italian Wars, religiously France became divided between the Catholic majority and a Protestant minority, the Huguenots, which led to a series of civil wars, the Wars of Religion. France laid claim to large stretches of North America, known collectively as New France, Wars with Great Britain led to the loss of much of this territory by 1763. French intervention in the American Revolutionary War helped secure the independence of the new United States of America, the Kingdom of France adopted a written constitution in 1791, but the Kingdom was abolished a year later and replaced with the First French Republic. The monarchy was restored by the great powers in 1814. During the later years of the elderly Charlemagnes rule, the Vikings made advances along the northern and western perimeters of the Kingdom of the Franks, after Charlemagnes death in 814 his heirs were incapable of maintaining political unity and the empire began to crumble. The Treaty of Verdun of 843 divided the Carolingian Empire into three parts, with Charles the Bald ruling over West Francia, the nucleus of what would develop into the kingdom of France. Viking advances were allowed to increase, and their dreaded longboats were sailing up the Loire and Seine rivers and other waterways, wreaking havoc. During the reign of Charles the Simple, Normans under Rollo from Norway, were settled in an area on either side of the River Seine, downstream from Paris, that was to become Normandy. With its offshoots, the houses of Valois and Bourbon, it was to rule France for more than 800 years. Henry II inherited the Duchy of Normandy and the County of Anjou, and married Frances newly divorced ex-queen, Eleanor of Aquitaine, after the French victory at the Battle of Bouvines in 1214, the English monarchs maintained power only in southwestern Duchy of Guyenne. The death of Charles IV of France in 1328 without male heirs ended the main Capetian line, under Salic law the crown could not pass through a woman, so the throne passed to Philip VI, son of Charles of Valois
Kingdom of France
–
The Kingdom of France in 1789.
Ancien Régime provinces in
1789.
Kingdom of France
–
Royal Standard^{a}
Kingdom of France
–
Henry IV, by
Frans Pourbus the younger, 1610.
Kingdom of France
–
Louis XIII, by
Philippe de Champaigne, 1647.
40.
Roman Catholic
–
The Catholic Church, also known as the Roman Catholic Church or the Universal Church, is the largest Christian church, with more than 1.28 billion members worldwide. As one of the oldest religious institutions in the world, it has played a prominent role in the history, headed by the Bishop of Rome, known as the Pope, the churchs doctrines are summarised in the Nicene Creed and the Apostles Creed. Its central administration is located in Vatican City, enclaved within Rome, the Catholic Church is notable within Western Christianity for its sacred tradition and seven sacraments. It teaches that it is the one church founded by Jesus Christ, that its bishops are the successors of Christs apostles. The Catholic Church maintains that the doctrine on faith and morals that it declares as definitive is infallible. The Latin Church, the Eastern Catholic Churches, as well as such as mendicant orders and enclosed monastic orders. Among the sacraments, the one is the Eucharist, celebrated liturgically in the Mass. The church teaches that through consecration by a priest the sacrificial bread and wine become the body, the Catholic Church practises closed communion, with only baptised members in a state of grace ordinarily permitted to receive the Eucharist. The Virgin Mary is venerated in the Catholic Church as Queen of Heaven and is honoured in numerous Marian devotions. The Catholic Church has influenced Western philosophy, science, art and culture, Catholic spiritual teaching includes spreading the Gospel while Catholic social teaching emphasises support for the sick, the poor and the afflicted through the corporal and spiritual works of mercy. The Catholic Church is the largest non-government provider of education and medical services in the world, from the late 20th century, the Catholic Church has been criticised for its doctrines on sexuality, its refusal to ordain women and its handling of sexual abuse cases. Catholic was first used to describe the church in the early 2nd century, the first known use of the phrase the catholic church occurred in the letter from Saint Ignatius of Antioch to the Smyrnaeans, written about 110 AD. In the Catechetical Discourses of Saint Cyril of Jerusalem, the name Catholic Church was used to distinguish it from other groups that call themselves the church. The use of the adjective Roman to describe the Church as governed especially by the Bishop of Rome became more widespread after the Fall of the Western Roman Empire and into the Early Middle Ages. Catholic Church is the name used in the Catechism of the Catholic Church. The Catholic Church follows an episcopal polity, led by bishops who have received the sacrament of Holy Orders who are given formal jurisdictions of governance within the church. Ultimately leading the entire Catholic Church is the Bishop of Rome, commonly called the pope, in parallel to the diocesan structure are a variety of religious institutes that function autonomously, often subject only to the authority of the pope, though sometimes subject to the local bishop. Most religious institutes only have male or female members but some have both, additionally, lay members aid many liturgical functions during worship services
Roman Catholic
–
Saint Peter's Basilica,
Vatican City
Roman Catholic
–
St. Peter's Basilica,
Vatican City
Roman Catholic
–
Pope Francis, elected in the
papal conclave, 2013
Roman Catholic
–
Traditional graphic representation of the Trinity: The earliest attested version of the diagram, from a manuscript of
Peter of Poitiers ' writings, c. 1210
41.
Charles Emmanuel III of Sardinia
–
Charles Emmanuel III was the Duke of Savoy and King of Sardinia from 1730 until his death. He was born a Prince of Savoy in Turin to Victor Amadeus II of Savoy and his maternal grandparents were Prince Philippe of France and his first wife Princess Henrietta Anne, the youngest daughter of King Charles I of England and Henrietta Maria of France. From his birth he was styled as the Duke of Aosta, at the time of his birth, Charles Emmanuel was not the heir to the Duchy of Savoy, his older brother Prince Victor Amadeus John Philip, Prince of Piedmont, was the heir apparent. Charles Emmanuel was the second of three males that would be born to his parents and his older brother died in 1715 and Charles Emmanuel then became heir apparent. As a result of his aid in the War of the Spanish Succession, Victor Amadeus was forced to exchange Sicily for the less important kingdom of Sardinia in 1720 after objections from an alliance of four nations, including several of his former allies. Yet he retained his new title of King, however, Victor Amadeus in his late years was dominated by shyness and sadness, probably under the effect of some mental illness. In the end, on 3 September 1730, he abdicated and he was not loved by Victor Amadeus, and consequently received an incomplete education. He however acquired noteworthy knowledge in the field along his father. In summer,1731, after having recovered from a fatal illness. The old king was confined to the Castle of Rivoli, where he died without any further harm to Charles. In the War of the Polish Succession Charles Emmanuel sided with the French- backed king Stanislaw I, after the treaty of alliance signed in Turin, on 28 October 1733 he marched on Milan and occupied Lombardy without significant losses. However, when France tried to convince Philip V of Spain to join the coalition, he asked to receive Milan and this was not acceptable for Charles Emmanuel, as it would recreate a Spanish domination in Italy as it had been in the previous centuries. While negotiations continued about the matter, the Savoy-French-Spanish troops attacked Mantua under the command of Charles Emmanuel himself. Sure that in the end Mantua would be assigned to Spain, the Franco-Piedmontese army was victorious in two battles at Crocetta and Guastalla. In the end, when Austria and France signed a peace, in exchange, he was given some territories, including Langhe, Tortona and Novara. Charles Emmanuel was involved in the War of the Austrian Succession, in which he sided with Maria Theresa of Austria, with financial and naval support from England. After noteworthy but inconclusive initial successes, he had to face the French-Spanish invasion of Savoy and, after a failed allied attempt to conquer the Kingdom of Naples, when the enemy army invaded Piedmont, in 1744 he personally defended Cuneo against the Spanish-French besiegers. The following year, with some 20,000 men, he was facing an invasion of two armies with a total of some 60,000 troops, the important strongholds of Alessandria, Asti and Casale fell
Charles Emmanuel III of Sardinia
–
Charles Emmanuel III
Charles Emmanuel III of Sardinia
–
The young Charles Emmanuel as duke of Aosta, by an unknown artist.
Charles Emmanuel III of Sardinia
–
A portrait of a young Charles Emmanuel
Charles Emmanuel III of Sardinia
–
The children of Charles and his second wife; (L-R)
Eleonora;
Victor Amadeus;
Maria Felicita and
Maria Luisa Gabriella.
42.
Edmond Halley
–
Edmond Halley, FRS was an English astronomer, geophysicist, mathematician, meteorologist, and physicist who is best known for computing the orbit of Halleys Comet. He was the second Astronomer Royal in Britain, succeeding John Flamsteed, Halley was born in Haggerston, in east London. His father, Edmond Halley Sr. came from a Derbyshire family and was a wealthy soap-maker in London, as a child, Halley was very interested in mathematics. He studied at St Pauls School, and from 1673 at The Queens College, while still an undergraduate, Halley published papers on the Solar System and sunspots. While at Oxford University, Halley was introduced to John Flamsteed, influenced by Flamsteeds project to compile a catalog of northern stars, Halley proposed to do the same for the Southern Hemisphere. In 1676, Halley visited the south Atlantic island of Saint Helena, while there he observed a transit of Mercury, and realised that a similar transit of Venus could be used to determine the absolute size of the Solar System. He returned to England in May 1678, in the following year he went to Danzig on behalf of the Royal Society to help resolve a dispute. Because astronomer Johannes Hevelius did not use a telescope, his observations had been questioned by Robert Hooke, Halley stayed with Hevelius and he observed and verified the quality of Hevelius observations. In 1679, Halley published the results from his observations on St. Helena as Catalogus Stellarum Australium which included details of 341 southern stars and these additions to contemporary star maps earned him comparison with Tycho Brahe, e. g. the southern Tycho as described by Flamsteed. Halley was awarded his M. A. degree at Oxford, in 1686, Halley published the second part of the results from his Helenian expedition, being a paper and chart on trade winds and monsoons. The symbols he used to represent trailing winds still exist in most modern day weather chart representations, in this article he identified solar heating as the cause of atmospheric motions. He also established the relationship between pressure and height above sea level. His charts were an important contribution to the field of information visualisation. Halley spent most of his time on lunar observations, but was interested in the problems of gravity. One problem that attracted his attention was the proof of Keplers laws of planetary motion, Halleys first calculations with comets were thereby for the orbit of comet Kirch, based on Flamsteeds observations in 1680-1. Although he was to calculate the orbit of the comet of 1682. They indicated a periodicity of 575 years, thus appearing in the years 531 and 1106 and it is now known to have an orbital period of circa 10,000 years. In 1691, Halley built a bell, a device in which the atmosphere was replenished by way of weighted barrels of air sent down from the surface
Edmond Halley
–
Bust of Halley (
Royal Observatory, Greenwich)
Edmond Halley
–
Portrait by
Thomas Murray, c. 1687
Edmond Halley
–
Halley's grave
Edmond Halley
–
Plaque in South Cloister of
Westminster Abbey
43.
Charles Emmanuel III
–
Charles Emmanuel III was the Duke of Savoy and King of Sardinia from 1730 until his death. He was born a Prince of Savoy in Turin to Victor Amadeus II of Savoy and his maternal grandparents were Prince Philippe of France and his first wife Princess Henrietta Anne, the youngest daughter of King Charles I of England and Henrietta Maria of France. From his birth he was styled as the Duke of Aosta, at the time of his birth, Charles Emmanuel was not the heir to the Duchy of Savoy, his older brother Prince Victor Amadeus John Philip, Prince of Piedmont, was the heir apparent. Charles Emmanuel was the second of three males that would be born to his parents and his older brother died in 1715 and Charles Emmanuel then became heir apparent. As a result of his aid in the War of the Spanish Succession, Victor Amadeus was forced to exchange Sicily for the less important kingdom of Sardinia in 1720 after objections from an alliance of four nations, including several of his former allies. Yet he retained his new title of King, however, Victor Amadeus in his late years was dominated by shyness and sadness, probably under the effect of some mental illness. In the end, on 3 September 1730, he abdicated and he was not loved by Victor Amadeus, and consequently received an incomplete education. He however acquired noteworthy knowledge in the field along his father. In summer,1731, after having recovered from a fatal illness. The old king was confined to the Castle of Rivoli, where he died without any further harm to Charles. In the War of the Polish Succession Charles Emmanuel sided with the French- backed king Stanislaw I, after the treaty of alliance signed in Turin, on 28 October 1733 he marched on Milan and occupied Lombardy without significant losses. However, when France tried to convince Philip V of Spain to join the coalition, he asked to receive Milan and this was not acceptable for Charles Emmanuel, as it would recreate a Spanish domination in Italy as it had been in the previous centuries. While negotiations continued about the matter, the Savoy-French-Spanish troops attacked Mantua under the command of Charles Emmanuel himself. Sure that in the end Mantua would be assigned to Spain, the Franco-Piedmontese army was victorious in two battles at Crocetta and Guastalla. In the end, when Austria and France signed a peace, in exchange, he was given some territories, including Langhe, Tortona and Novara. Charles Emmanuel was involved in the War of the Austrian Succession, in which he sided with Maria Theresa of Austria, with financial and naval support from England. After noteworthy but inconclusive initial successes, he had to face the French-Spanish invasion of Savoy and, after a failed allied attempt to conquer the Kingdom of Naples, when the enemy army invaded Piedmont, in 1744 he personally defended Cuneo against the Spanish-French besiegers. The following year, with some 20,000 men, he was facing an invasion of two armies with a total of some 60,000 troops, the important strongholds of Alessandria, Asti and Casale fell
Charles Emmanuel III
–
Charles Emmanuel III
Charles Emmanuel III
–
The young Charles Emmanuel as duke of Aosta, by an unknown artist.
Charles Emmanuel III
–
A portrait of a young Charles Emmanuel
Charles Emmanuel III
–
The children of Charles and his second wife; (L-R)
Eleonora;
Victor Amadeus;
Maria Felicita and
Maria Luisa Gabriella.
44.
Tautochrone
–
A tautochrone or isochrone curve is the curve for which the time taken by an object sliding without friction in uniform gravity to its lowest point is independent of its starting point. The curve is a cycloid, and the time is equal to π times the root of the radius over the acceleration of gravity. The tautochrone curve is the same as the curve for any given starting point. The tautochrone problem, the attempt to identify this curve, was solved by Christiaan Huygens in 1659 and he proved geometrically in his Horologium Oscillatorium, originally published in 1673, that the curve was a cycloid. Huygens also proved that the time of descent is equal to the time a body takes to fall vertically the same distance as the diameter of the circle that generates the cycloid, multiplied by π/2. In modern terms, this means that the time of descent is π r / g, where r is the radius of the circle which generates the cycloid and this solution was later used to attack the problem of the brachistochrone curve. Jakob Bernoulli solved the problem using calculus in a paper that saw the first published use of the term integral and these attempts proved to not be useful for a number of reasons. First, the bending of the string causes friction, changing the timing, second, there were much more significant sources of timing errors that overwhelmed any theoretical improvements that traveling on the tautochrone curve helps. Finally, the error of a pendulum decreases as length of the swing decreases. Later, the mathematicians Joseph Louis Lagrange and Leonhard Euler provided a solution to the problem. If the particles position is parametrized by the s from the lowest point. The potential energy is proportional to the height y, one way the curve can be an isochrone is if the Lagrangian is that of a simple harmonic oscillator, the height of the curve must be proportional to the arclength squared. Y = s 2, where the constant of proportionality has been set to 1 by changing units of length. The differential form of relation is d y =2 s d s, d y 2 =4 s 2 d s 2 =4 y, which eliminates s. To find the solution, integrate for x in terms of y, d x d y =1 −4 y 2 y, x = ∫1 −4 u 2 d u, where u = y. This integral is the area under a circle, which can be cut into a triangle. To see that this is a strangely parametrized cycloid, change variables to disentangle the transcendental, the simplest solution to the tautochrone problem is to note a direct relation between the angle of an incline and the gravity felt by a particle on the incline. A particle on a 90° vertical incline feels the effect of gravity
Tautochrone
–
Schematic of a cycloidal pendulum.
Tautochrone
–
Four balls slide down a cycloid curve from different positions, but they arrive at the bottom at the same time. The blue arrows show the points' acceleration along the curve. On the top is the time-position diagram.
45.
Pierre Louis Maupertuis
–
Pierre Louis Moreau de Maupertuis was a French mathematician, philosopher and man of letters. He became the Director of the Académie des Sciences, and the first President of the Prussian Academy of Science, Maupertuis made an expedition to Lapland to determine the shape of the Earth. He is often credited with having invented the principle of least action and his work in natural history is interesting in relation to modern science, since he touched on aspects of heredity and the struggle for life. Maupertuis was born at Saint-Malo, France, to a wealthy family of merchant-corsairs. His father, Renė, had involved in a number of enterprises that were central to the monarchy so that he thrived socially and politically. The son was educated in mathematics by a tutor, Nicolas Guisnée. In 1723 he was admitted to the Académie des Sciences and his early mathematical work revolved around the vis viva controversy, for which Maupertuis developed and extended the work of Isaac Newton and argued against the waning Cartesian mechanics. In the 1730s, the shape of the Earth became a flashpoint in the battle among rival systems of mechanics, Maupertuis, based on his exposition of Newton predicted that the Earth should be oblate, while his rival Jacques Cassini measured it astronomically to be prolate. In 1736 Maupertuis acted as chief of the French Geodesic Mission sent by King Louis XV to Lapland to measure the length of a degree of arc of the meridian and his results, which he published in a book detailing his procedures, essentially settled the controversy in his favor. The book included a narrative of the expedition, and an account of the Käymäjärvi Inscriptions. On his return home he became a member of almost all the societies of Europe. He also expanded into the realm, anonymously publishing a book that was part popular science, part philosophy. In 1740 Maupertuis went to Berlin at the invitation of Frederick II of Prussia, and took part in the Battle of Mollwitz, where he was taken prisoner by the Austrians. On his release he returned to Berlin, and thence to Paris, where he was elected director of the Academy of Sciences in 1742, and in the following year was admitted into the Académie française. His position became extremely awkward with the outbreak of the Seven Years War between his country and his patrons, and his reputation suffered in both Paris and Berlin. Finding his health declining, he retired in 1757 to the south of France, but went in 1758 to Basel, Maupertuis difficult disposition involved him in constant quarrels, of which his controversies with Samuel König and Voltaire during the latter part of his life are examples. The brilliance of much of what he did was undermined by his tendency to leave work unfinished and it was the insight of genius that led him to least-action principle, but a lack of intellectual energy or rigour that prevented his giving it the mathematical foundation that Lagrange would provide. He reveals remarkable powers of perception in heredity, in understanding the mechanism by which developed, even in immunology
Pierre Louis Maupertuis
–
Maupertuis, wearing " lapmudes " from his Lapland expedition.
Pierre Louis Maupertuis
–
Lettres
46.
Vibrating string
–
A vibration in a string is a wave. Resonance causes a string to produce a sound with constant frequency. If the length or tension of the string is correctly adjusted, vibrating strings are the basis of string instruments such as guitars, cellos, and pianos. Let Δ x be the length of a piece of string, m its mass, and μ its linear density. If the horizontal component of tension in the string is a constant, T, if both angles are small, then the tensions on either side are equal and the net horizontal force is zero. This is the equation for y, and the coefficient of the second time derivative term is equal to v −2, thus v = T μ. Once the speed of propagation is known, the frequency of the produced by the string can be calculated. The speed of propagation of a wave is equal to the wavelength λ divided by the period τ, or multiplied by the frequency f, v = λ τ = λ f. If the length of the string is L, the harmonic is the one produced by the vibration whose nodes are the two ends of the string, so L is half of the wavelength of the fundamental harmonic. Hence one obtains Mersennes laws, f = v 2 L =12 L T μ where T is the tension, μ is the linear density, and L is the length of the vibrating part of the string. This effect is called the effect, and the rate at which the string seems to vibrate is the difference between the frequency of the string and the refresh rate of the screen. The same can happen with a fluorescent lamp, at a rate that is the difference between the frequency of the string and the frequency of the alternating current. In daylight and other non-oscillating light sources, this effect does not occur and the string appears still but thicker, a similar but more controllable effect can be obtained using a stroboscope. This device allows matching the frequency of the flash lamp to the frequency of vibration of the string. In a dark room, this shows the waveform. Otherwise, one can use bending or, perhaps more easily, by adjusting the machine heads, to obtain the same, or a multiple, of the AC frequency to achieve the same effect. For example, in the case of a guitar, the 6th string pressed to the third gives a G at 97.999 Hz. A slight adjustment can alter it to 100 Hz, exactly one octave above the current frequency in Europe and most countries in Africa
Vibrating string
–
This article needs additional citations for
verification. Please help improve this article by
adding citations to reliable sources. Unsourced material may be challenged and removed. (October 2010)
Vibrating string
–
Vibration,
standing waves in a string. The
fundamental and the first 5
overtones in the
harmonic series.
47.
Brook Taylor
–
Brook Taylor FRS was an English mathematician who is best known for Taylors theorem and the Taylor series. Brook Taylor was born in Edmonton to John Taylor of Bifrons House in Patrixbourne, Kent and he entered St Johns College, Cambridge, as a fellow-commoner in 1701, and took degrees of LL. B. and LL. D. in 1709 and 1714, respectively. Taylors Methodus Incrementorum Directa et Inversa added a new branch to higher mathematics, among other ingenious applications, he used it to determine the form of movement of a vibrating string, by him first successfully reduced to mechanical principles. From 1715 his studies took a philosophical and religious bent and he corresponded in that year with the Comte de Montmort on the subject of Nicolas Malebranches tenets. Unfinished treatises, On the Jewish Sacrifices and On the Lawfulness of Eating Blood, written on his return from Aix-la-Chapelle in 1719, were afterwards found among his papers. His marriage in 1721 with Miss Brydges of Wallington, Surrey, led to an estrangement from his father, which ended in 1723 after her death in giving birth to a son, by the date of his fathers death in 1729 he had inherited the Bifrons estate. Taylors fragile health gave way, he fell into a decline and he was buried in London on 2 December 1731, near his first wife, in the churchyard of St Annes, Soho. A posthumous work entitled Contemplatio Philosophica was printed for private circulation in 1793 by Taylors grandson, Sir William Young, prefaced by a life of the author, and with an appendix containing letters addressed to him by Bolingbroke, Bossuet, and others. Several short papers by Taylor were published in Phil, vols. xxvii to xxxii, including accounts of some interesting experiments in magnetism and capillary attraction. A French translation was published in 1757, in Methodus Incrementorum, Taylor gave the first satisfactory investigation of astronomical refraction. Taylor, Brook, Methodus Incrementorum Directa et Inversa, London, Taylor is an impact crater located on the Moon, named in honour of Brook Taylor. Brook Taylor’s Work on Linear Perspective, anderson, Marlow, Katz, Victor, Wilson, Robin. Sherlock Holmes in Babylon, And Other Tales of Mathematical History, Brook Taylor and the Method of Increments. Archive for History of Exact Sciences, oConnor, John J. Robertson, Edmund F. Brook Taylor, MacTutor History of Mathematics archive, University of St Andrews, beningbrough Hall has a painting by John Closterman of Taylor aged about 12 with his brothers and sisters. See also NPG5320, The Children of John Taylor of Bifrons Park Brook Taylors pedigree Taylor, a crater on the Moon named after Brook Taylor
Brook Taylor
–
Brook Taylor (1685-1731)
Brook Taylor
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Methodus incrementorum directa et inversa, 1715
Brook Taylor
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Brook Taylor
48.
Echo (phenomenon)
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In audio signal processing and acoustics, echo is a reflection of sound that arrives at the listener with a delay after the direct sound. The delay is proportional to the distance of the surface from the source. Typical examples are the echo produced by the bottom of a well, by a building, or by the walls of an enclosed room, a true echo is a single reflection of the sound source. The word echo derives from the Greek ἠχώ, itself from ἦχος, echo in the folk story of Greek is a mountain nymph whose ability to speak was cursed, only able to repeat the last words anyone spoke to her. Some animals use echo for location sensing and navigation, such as cetaceans, acoustic waves are reflected by walls or other hard surfaces, such as mountains and privacy fences. The reason of reflection may be explained as a discontinuity in the propagation medium and this can be heard when the reflection returns with sufficient magnitude and delay to be perceived distinctly. When sound, or the echo itself, is reflected multiple times from multiple surfaces, the human ear cannot distinguish echo from the original direct sound if the delay is less than 1/15 of a second. The velocity of sound in dry air is approximately 343 m/s at a temperature of 25 °C, therefore, the reflecting object must be more than 17. 2m from the sound source for echo to be perceived by a person located at the source. When a sound produces an echo in two seconds, the object is 343m away. In nature, canyon walls or rock cliffs facing water are the most common settings for hearing echoes. The strength of echo is frequently measured in dB sound pressure level relative to the transmitted wave. Echoes may be desirable or undesirable, in music performance and recording, electric echo effects have been used since the 1950s. The Echoplex is a delay effect, first made in 1959 that recreates the sound of an acoustic echo. Designed by Mike Battle, the Echoplex set a standard for the effect in the 1960s and was used by most of the guitar players of the era. While Echoplexes were used heavily by guitar players, many recording studios used the Echoplex. Beginning in the 1970s, Market built the solid-state Echoplex for Maestro, in the 2000s, most echo effects units use electronic or digital circuitry to recreate the echo effect. Hamilton Mausoleum, Hamilton, South Lanarkshire, Scotland, Its high stone holds the record for the longest echo in the world, gol Gumbaz of Bijapur, India, Any whisper, clap or sound gets echoed repeatedly. The gazebo of Napier Museum in Trivandrum, Kerala, India, listen to Duck echoes and an animated demonstration of how an echo is formed
Echo (phenomenon)
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This illustration depicts the principle of sediment echo sounding, which uses a narrow beam of high energy and low frequency
49.
Beat (acoustics)
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In acoustics, a beat is an interference pattern between two sounds of slightly different frequencies, perceived as a periodic variation in volume whose rate is the difference of the two frequencies. When tuning instruments that can produce sustained tones, beats can readily be recognized, tuning two tones to a unison will present a peculiar effect, when the two tones are close in pitch but not identical, the difference in frequency generates the beating. The volume varies like in a tremolo as the sounds alternately interfere constructively and destructively, as the two tones gradually approach unison, the beating slows down and may become so slow as to be imperceptible. It can be proven that the envelope of the maxima and minima form a wave frequency is half the difference between the frequencies of the two original waves. Instead, it is perceived as a variation in the amplitude of the first term in the expression above. It can be said that the lower frequency cosine term is an envelope for the higher frequency one, the frequency of the modulation is f1 + f2/2, that is, the average of the two frequencies. It can be noted that every second burst in the pattern is inverted. Each peak is replaced by a trough and vice versa, however, because the human ear is not sensitive to the phase of a sound, only its amplitude or intensity, only the magnitude of the envelope is heard. A physical interpretation is that when cos =1 the two waves are in phase and they interfere constructively, when it is zero, they are out of phase and interfere destructively. Beats occur also in more complex sounds, or in sounds of different volumes, beating can also be heard between notes that are near to, but not exactly, a harmonic interval, due to some harmonic of the first note beating with a harmonic of the second note. For example, in the case of perfect fifth, the harmonic of the bass note beats with the second harmonic of the other note. Musicians commonly use interference beats to objectively check tuning at the unison, perfect fifth, piano and organ tuners even use a method involving counting beats, aiming at a particular number for a specific interval. The composer Alvin Lucier has written many pieces that feature interference beats as their main focus, composer Phill Niblocks music is entirely based on beating caused by microtonal differences. Binaural beats are heard when the right ear listens to a different tone than the left ear. Here, the tones do not interfere physically, but are summed by the brain in the olivary nucleus and this effect is related to the brains ability to locate sounds in three dimensions. Combination tone Gamelan tuning Heterodyne Consonance and dissonance Moiré pattern, a form of interference that generates new frequencies
Beat (acoustics)
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Diagram of beat frequency
50.
Series (mathematics)
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In mathematics, a series is, informally speaking, the sum of the terms of an infinite sequence. The sum of a sequence has defined first and last terms. To emphasize that there are a number of terms, a series is often called an infinite series. In order to make the notion of an infinite sum mathematically rigorous, given an infinite sequence, the associated series is the expression obtained by adding all those terms together, a 1 + a 2 + a 3 + ⋯. These can be written compactly as ∑ i =1 ∞ a i, by using the summation symbol ∑. The sequence can be composed of any kind of object for which addition is defined. A series is evaluated by examining the finite sums of the first n terms of a sequence, called the nth partial sum of the sequence, and taking the limit as n approaches infinity. If this limit does not exist, the infinite sum cannot be assigned a value, and, in this case, the series is said to be divergent. On the other hand, if the partial sums tend to a limit when the number of terms increases indefinitely, then the series is said to be convergent, and the limit is called the sum of the series. An example is the series from Zenos dichotomy and its mathematical representation, ∑ n =1 ∞12 n =12 +14 +18 + ⋯. The study of series is a part of mathematical analysis. Series are used in most areas of mathematics, even for studying finite structures, in addition to their ubiquity in mathematics, infinite series are also widely used in other quantitative disciplines such as physics, computer science, statistics and finance. For any sequence of numbers, real numbers, complex numbers, functions thereof. By definition the series ∑ n =0 ∞ a n converges to a limit L if and this definition is usually written as L = ∑ n =0 ∞ a n ⇔ L = lim k → ∞ s k. When the index set is the natural numbers I = N, a series indexed on the natural numbers is an ordered formal sum and so we rewrite ∑ n ∈ N as ∑ n =0 ∞ in order to emphasize the ordering induced by the natural numbers. Thus, we obtain the common notation for a series indexed by the natural numbers ∑ n =0 ∞ a n = a 0 + a 1 + a 2 + ⋯. When the semigroup G is also a space, then the series ∑ n =0 ∞ a n converges to an element L ∈ G if. This definition is usually written as L = ∑ n =0 ∞ a n ⇔ L = lim k → ∞ s k, a series ∑an is said to converge or to be convergent when the sequence SN of partial sums has a finite limit
Series (mathematics)
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Illustration of 3
geometric series with partial sums from 1 to 6 terms. The dashed line represents the limit.