Mathematics includes the study of such topics as quantity, structure and change. Mathematicians use patterns to formulate new conjectures; when mathematical structures are good models of real phenomena mathematical reasoning can provide insight or predictions about nature. Through the use of abstraction and logic, mathematics developed from counting, calculation and the systematic study of the shapes and motions of physical objects. Practical mathematics has been a human activity from as far back; the research required to solve mathematical problems can take years or centuries of sustained inquiry. Rigorous arguments first appeared in Greek mathematics, most notably in Euclid's Elements. Since the pioneering work of Giuseppe Peano, David Hilbert, others on axiomatic systems in the late 19th century, it has become customary to view mathematical research as establishing truth by rigorous deduction from appropriately chosen axioms and definitions. Mathematics developed at a slow pace until the Renaissance, when mathematical innovations interacting with new scientific discoveries led to a rapid increase in the rate of mathematical discovery that has continued to the present day.
Mathematics is essential in many fields, including natural science, medicine and the social sciences. Applied mathematics has led to new mathematical disciplines, such as statistics and game theory. Mathematicians engage in pure mathematics without having any application in mind, but practical applications for what began as pure mathematics are discovered later; the history of mathematics can be seen as an ever-increasing series of abstractions. The first abstraction, shared by many animals, was that of numbers: the realization that a collection of two apples and a collection of two oranges have something in common, namely quantity of their members; as evidenced by tallies found on bone, in addition to recognizing how to count physical objects, prehistoric peoples may have recognized how to count abstract quantities, like time – days, years. Evidence for more complex mathematics does not appear until around 3000 BC, when the Babylonians and Egyptians began using arithmetic and geometry for taxation and other financial calculations, for building and construction, for astronomy.
The most ancient mathematical texts from Mesopotamia and Egypt are from 2000–1800 BC. Many early texts mention Pythagorean triples and so, by inference, the Pythagorean theorem seems to be the most ancient and widespread mathematical development after basic arithmetic and geometry, it is in Babylonian mathematics that elementary arithmetic first appear in the archaeological record. The Babylonians possessed a place-value system, used a sexagesimal numeral system, still in use today for measuring angles and time. Beginning in the 6th century BC with the Pythagoreans, the Ancient Greeks began a systematic study of mathematics as a subject in its own right with Greek mathematics. Around 300 BC, Euclid introduced the axiomatic method still used in mathematics today, consisting of definition, axiom and proof, his textbook Elements is considered the most successful and influential textbook of all time. The greatest mathematician of antiquity is held to be Archimedes of Syracuse, he developed formulas for calculating the surface area and volume of solids of revolution and used the method of exhaustion to calculate the area under the arc of a parabola with the summation of an infinite series, in a manner not too dissimilar from modern calculus.
Other notable achievements of Greek mathematics are conic sections, trigonometry (Hipparchus of Nicaea, the beginnings of algebra. The Hindu–Arabic numeral system and the rules for the use of its operations, in use throughout the world today, evolved over the course of the first millennium AD in India and were transmitted to the Western world via Islamic mathematics. Other notable developments of Indian mathematics include the modern definition of sine and cosine, an early form of infinite series. During the Golden Age of Islam during the 9th and 10th centuries, mathematics saw many important innovations building on Greek mathematics; the most notable achievement of Islamic mathematics was the development of algebra. Other notable achievements of the Islamic period are advances in spherical trigonometry and the addition of the decimal point to the Arabic numeral system. Many notable mathematicians from this period were Persian, such as Al-Khwarismi, Omar Khayyam and Sharaf al-Dīn al-Ṭūsī. During the early modern period, mathematics began to develop at an accelerating pace in Western Europe.
The development of calculus by Newton and Leibniz in the 17th century revolutionized mathematics. Leonhard Euler was the most notable mathematician of the 18th century, contributing numerous theorems and discoveries; the foremost mathematician of the 19th century was the German mathematician Carl Friedrich Gauss, who made numerous contributions to fields such as algebra, differential geometry, matrix theory, number theory, statistics. In the early 20th century, Kurt Gödel transformed mathematics by publishing his incompleteness theorems, which show that any axiomatic system, consistent will contain unprovable propositions. Mathematics has since been extended, there has been a fruitful interaction between mathematics and science, to
École Centrale Paris
École Centrale Paris was a French postgraduate-level institute of research and higher education in engineering and science. It was known by its official name École Centrale des Arts et Manufactures. Founded in 1829, it was among the most selective grandes écoles. Rooted in rich entrepreneurial tradition since the industrial revolution era, it served as the cradle for top-level engineers and executives who continue to constitute a major part of the industry leadership in France. Since the 19th century, its model of education for training generalist engineers inspired the establishment of several engineering institutes around the world, such as the École Polytechnique Fédérale de Lausanne in Switzerland, Faculté polytechnique de Mons in Belgium, as well as other member schools of the Ecole Centrales Group alliance in France, Morocco and India. In 2015, École Centrale Paris merged with Supélec to form CentraleSupélec, a constituent institute of the University of Paris-Saclay. "Between 1832 and 1870, the Central School of Arts and Manufactures produced 3,000 engineers, served as a model for most of the industrialized countries."
École Centrale des Arts et Manufactures was founded in 1829 as a private institute by Alphonse Lavallée, a lawyer and a prominent businessman from Nantes, who put forward most of his personal capital into founding the school, together with three top scientists who became its founding associates: Eugène Peclet, Jean-Baptiste Dumas, Théodore Olivier. Notably, Lavallée was a shareholder of Le Globe, which became in 1831 the official organ of the Saint-Simonian movement; the founding vision of École Centrale was to train multidisciplinary engineers who will become the first "doctors of factories and mills" of the then-emerging industrial sector in France, at a time when most of the other engineering schools trained students for public service. As the scientific discoveries in this era were beginning to have a major impact on industrial development in Europe, a new breed of engineers with a broad and rigorous knowledge of sciences and mathematics were needed in order for France to develop its industry and compete amongst the world's superpowers.
The school was located in various premises in Paris, including Hotel Salé and buildings which now belong to Conservatoire National des Arts et Métiers. Lavallée served as the first president of École Centrale. In 1857, Lavallée transferred the ownership of the school to the French state in order to ensure its sustainability. Under Napoleon's initiative for an imperial university, the school was temporarily renamed as École Impériale des Arts et Manufactures. In 1862, graduates of the school were awarded accredited graduate diplomas in engineering, with the official academic title of'ingénieur des arts et manufactures', the first of its kind in France; the school was transferred in 1969 to a new campus located in Châtenay-Malabry. The Châtenay-Malabry campus was designed by architect Jean Fayeton, was inaugurated by President Georges Pompidou, accompanied on this occasion by Robert Galley; the school was renamed as École Centrale des Arts et Manufactures. In 2015, the school formed a strategic alliance with Supélec to create CentraleSupélec, part of the University of Paris-Saclay.
The new campus is located in Gif-sur-Yvette 20 km from the center of Paris. École Centrale Paris was one of the Centrale Graduate Schools associated as the Groupe Centrale network with its sister institutes. Since 1837, the school had established several international partnerships with the world's leading universities, such as California Institute of Technology, University of Cambridge, ETH Zurich, Georgia Institute of Technology, Harvard University, Indian Institutes of Technology, KAIST, Princeton University, Universidad Politécnica de Madrid, Massachusetts Institute of Technology, Politecnico di Milano, National University of Singapore, Stanford University, University of Toronto, Tsinghua University, TU Delft and Technische Universität München, it was a founding member of the TIME network among top engineering schools in Europe, a member of the UniverSud Paris and the CESAER association of European engineering schools. Located in the Hôtel de Juigné, the main campus of the school was transferred to rue Montgolfier in 1884, where it stayed until 1969.
Its current location neighbours the Parc de Sceaux. Former location of the École Centrale, rue Montgolfier in Paris: The school is now located at Châtenay-Malabry, Hauts-de-Seine, a southern suburb of Paris, next to the Parc de Sceaux and its Château de Sceaux. Within the main campus at Châtenay Malabry, ECP hosts eight laboratories: Molecular and Macroscopic Energy, Combustion System Analysis and Macroeconomics Modeling Industrial Engineering Chemical Engineering and Materials Processing Laboratory Applied Mathematics Soil and Structure Mechanics Technology and Strategy Solids Structure and PropertiesMost of the 2000 students at École Centrale Paris stay in dedicated on-campus student residences, located near the research labs and accessible via public transport. Following the merger of the school with Supelec, now forming CentraleSupelec, the progressive move of the campus has started from Chatenay-Malabry to Gif-sur-Yvette. Most French students who were admitted to École Centrale Paris had completed 2 to 3 years of post high school education in sciences through the classes préparatoires or
Pierre Louis Dulong
Pierre Louis Dulong FRS FRSE was a French physicist and chemist. He is remembered today for the law of Dulong and Petit, although he was much-lauded by his contemporaries for his studies into the elasticity of steam, conduction of heat, specific heats of gases, he worked most extensively on the specific heat capacity and the expansion and refractive indices of gases. He collaborated several times with fellow scientist Alexis Petit, the co-creator of the Dulong–Petit law. Dulong was born in France. An only child, he was orphaned at the age of 4, he was brought up by his aunt in Auxerre, he gained his secondary education in Auxerre and the Lycée Pierre Corneille in Rouen before entering the École Polytechnique, Paris in 1801, only for his studies to be impeded by poor health. He began studying medicine, but gave this up because of a lack of financial means, to concentrate on science, working under the direction of Thénard. In chemistry, he contributed to knowledge on: the double decomposition of salts nitrous acid the oxides of phosphorus the oxides of nitrogen catalysis by metals.
Dulong discovered the dangerously sensitive nitrogen trichloride in 1811, losing three fingers and an eye in the process. The fact that Dulong kept the accident a secret meant that Sir Humphrey Davy's investigation of the compound had the same unfortunate consequence, although Davy's injuries were less severe. In addition to his accomplishments in chemistry, Dulong has been hailed as an interdisciplinary expert, his contemporaries in the Royal Society of London acknowledged his "command of every department of physical science". In 1815, Dulong collaborated for the first time with Alexis Petit, in publishing a paper on heat expansion; the two would continue to collaborate. In 1819, Dulong and Petit showed that the mass heat capacity of metallic elements are inversely proportional to their atomic masses, this being now known as the Dulong–Petit law; this law, though discredited in modern times, helped develop the periodic table and, more broadly, the examination of atomic masses. In 1818, Dulong was honored by the French Academy for work that would contribute to his co-discovery of the Dulong–Petit law.
In 1820, Dulong succeeded Petit, who retired due to poor health, as professor of physics at École Polytechnique. Dulong studied the elasticity of steam, the measurement of temperatures, the behavior of elastic fluids, he studied. He made the first precise comparison of the mercury- and air-temperature scales. In 1830, he was elected a foreign member of the Royal Swedish Academy of Sciences, he died of stomach cancer in Paris. His is one of the names of 72 scientists inscribed on the Eiffel Tower. At the time of his death, he was working on the development of precise methods in calorimetry, his last paper, published the year of his death, examined the heat released from chemical reactions. Roberto Piazza’s 2016 paper on the Dulong–Petit law provides biographical and temperament details by contemporary and fellow physicist, Jules Jamin. “Petit had a lively intelligence, an elegant and easy speech, he seduced with an amiable look, got attached, surrendered himself to his tendencies rather than governing them.
He was credited with an instinctive scientific intuition, a power of premature invention, certain presages of an assured future that everyone foresaw and desired, so great was the benevolence which he inspired. Dulong was the opposite: His language was thoughtful, his attitude serious and his appearance cold He worked but with certainty, with a continuity and a power of will that nothing stopped, I should say with a courage that no danger could push back. In the absence of that vivacity of the mind which invents but likes to rest, he had the sense of scientific exactness, the gusto for precision experiments, the talent of combining them, the patience of completing them, the art, unknown before him, to carry them to the limits of accuracy Petit had more mathematical tendency, Dulong was more experimental, he was married to Emelie Augustine Riviere in 1803. Dulong was dismissed as a dry, standoffish individual, his few friends disagreed with this view. Dulong was noted both for his devotion to science and the stolid casual, bravery he displayed in prosecuting his experiments.
One such experiment involved the construction of a glass tubular apparatus atop the tower at the Abbey of St. Genevieve; the tower was unsteady enough that an explosion of the experimental materials likely considering their volatility, could have toppled the tower and killed the researching physicists, including Dulong. The experiment though "full of danger and difficulty", was completed under Dulong's leadership. Another example of Dulong's indifference to danger amid scientific pursuit came about in his studies into nitrogen trichloride. Despite losing two fingers and one eye in his initial experiments, Dulong continued to research the unknown substance, his inquiry led to more injuries, after which he turned over the results of his studies to Sir Humphrey Davy. In life, Dulong poured the bulk of his finances into his scientific experiments, he was destitute. As a result, he died without leaving his family any significant inheritance, he is buried in Père Lachaise Cemetery. His monument was paid for by his scientific peers.
Petit, Alexis-Thérèse. "Recherche
National Library of the Czech Republic
The National Library of the Czech Republic is the central library of the Czech Republic. It is directed by the Ministry of Culture; the library's main building is located in the historical Clementinum building in Prague, where half of its books are kept. The other half of the collection is stored in the district of Hostivař; the National Library is the biggest library in the Czech Republic, in its funds there are around 6 million documents. The library has around 60,000 registered readers; as well as Czech texts, the library stores older material from Turkey and India. The library houses books for Charles University in Prague; the library won international recognition in 2005 as it received the inaugural Jikji Prize from UNESCO via the Memory of the World Programme for its efforts in digitising old texts. The project, which commenced in 1992, involved the digitisation of 1,700 documents in its first 13 years; the most precious medieval manuscripts preserved in the National Library are the Codex Vyssegradensis and the Passional of Abbes Kunigunde.
In 2006 the Czech parliament approved funding for the construction of a new library building on Letna plain, between Hradčanská metro station and Sparta Prague's football ground, Letná stadium. In March 2007, following a request for tender, Czech architect Jan Kaplický was selected by a jury to undertake the project, with a projected completion date of 2011. In 2007 the project was delayed following objections regarding its proposed location from government officials including Prague Mayor Pavel Bém and President Václav Klaus. Plans for the building had still not been decided in February 2008, with the matter being referred to the Office for the Protection of Competition in order to determine if the tender had been won fairly. In 2008, Minister of Culture Václav Jehlička announced the end of the project, following a ruling from the European Commission that the tender process had not been carried out legally; the library was affected by the 2002 European floods, with some documents moved to upper levels to avoid the excess water.
Over 4,000 books were removed from the library in July 2011 following flooding in parts of the main building. There was a fire at the library in December 2012. List of national and state libraries Official website
In physics, the kinetic energy of an object is the energy that it possesses due to its motion. It is defined as the work needed to accelerate a body of a given mass from rest to its stated velocity. Having gained this energy during its acceleration, the body maintains this kinetic energy unless its speed changes; the same amount of work is done by the body when decelerating from its current speed to a state of rest. In classical mechanics, the kinetic energy of a non-rotating object of mass m traveling at a speed v is 1 2 m v 2. In relativistic mechanics, this is a good approximation only when v is much less than the speed of light; the standard unit of kinetic energy is the joule. The imperial unit of kinetic energy is the foot-pound; the adjective kinetic has its roots in the Greek word κίνησις kinesis, meaning "motion". The dichotomy between kinetic energy and potential energy can be traced back to Aristotle's concepts of actuality and potentiality; the principle in classical mechanics that E ∝ mv2 was first developed by Gottfried Leibniz and Johann Bernoulli, who described kinetic energy as the living force, vis viva.
Willem's Gravesande of the Netherlands provided experimental evidence of this relationship. By dropping weights from different heights into a block of clay, Willem's Gravesande determined that their penetration depth was proportional to the square of their impact speed. Émilie du Châtelet published an explanation. The terms kinetic energy and work in their present scientific meanings date back to the mid-19th century. Early understandings of these ideas can be attributed to Gaspard-Gustave Coriolis, who in 1829 published the paper titled Du Calcul de l'Effet des Machines outlining the mathematics of kinetic energy. William Thomson Lord Kelvin, is given the credit for coining the term "kinetic energy" c. 1849–51. Energy occurs in many forms, including chemical energy, thermal energy, electromagnetic radiation, gravitational energy, electric energy, elastic energy, nuclear energy, rest energy; these can be categorized in two main classes: kinetic energy. Kinetic energy is the movement energy of an object.
Kinetic energy can be transformed into other kinds of energy. Kinetic energy may be best understood by examples that demonstrate how it is transformed to and from other forms of energy. For example, a cyclist uses chemical energy provided by food to accelerate a bicycle to a chosen speed. On a level surface, this speed can be maintained without further work, except to overcome air resistance and friction; the chemical energy has been converted into kinetic energy, the energy of motion, but the process is not efficient and produces heat within the cyclist. The kinetic energy in the moving cyclist and the bicycle can be converted to other forms. For example, the cyclist could encounter a hill just high enough to coast up, so that the bicycle comes to a complete halt at the top; the kinetic energy has now been converted to gravitational potential energy that can be released by freewheeling down the other side of the hill. Since the bicycle lost some of its energy to friction, it never regains all of its speed without additional pedaling.
The energy is not destroyed. Alternatively, the cyclist could connect a dynamo to one of the wheels and generate some electrical energy on the descent; the bicycle would be traveling slower at the bottom of the hill than without the generator because some of the energy has been diverted into electrical energy. Another possibility would be for the cyclist to apply the brakes, in which case the kinetic energy would be dissipated through friction as heat. Like any physical quantity, a function of velocity, the kinetic energy of an object depends on the relationship between the object and the observer's frame of reference. Thus, the kinetic energy of an object is not invariant. Spacecraft use chemical energy to launch and gain considerable kinetic energy to reach orbital velocity. In an circular orbit, this kinetic energy remains constant because there is no friction in near-earth space. However, it becomes apparent at re-entry. If the orbit is elliptical or hyperbolic throughout the orbit kinetic and potential energy are exchanged.
Without loss or gain, the sum of the kinetic and potential energy remains constant. Kinetic energy can be passed from one object to another. In the game of billiards, the player imposes kinetic energy on the cue ball by striking it with the cue stick. If the cue ball collides with another ball, it slows down and the ball it hit accelerates its speed as the kinetic energy is passed on to it. Collisions in billiards are elastic collisions, in which kinetic energy is preserved. In inelastic collisions, kinetic energy is dissipated in various forms of energy, such as heat, binding energy. Flywheels have been developed as a method of energy storage; this illustrates that kinetic energy is stored in rotational motion. Several mathematical descriptions of kinetic energy exist that describe it in the appropriate physical situation. For objects and processes in common human experience, the formula ½mv² given by Newtonian mechanics is suitable. However, if the speed of the object is comparabl
École polytechnique is a French public institution of higher education and research in Palaiseau, a suburb southwest of Paris. It is one of the most prestigious and selective French scientific and engineering schools, called grandes écoles in French, it is known for its ingénieur polytechnicien scientific degree program, equivalent to both a bachelor and master of science. Its entrance exam, the X-ENS exam, is renowned for its selectivity with a little over 500 admitted students out of the 53 848 students enrolled in the preparatory programs for the French scientific and engineering schools entrance exams; the school was established in 1794 by the mathematician Gaspard Monge during the French Revolution, was a military academy under Napoleon I in 1804. Although Polytechnique is no longer a military academy, the school is still supervised by the French ministry of defense, though only a small number of its students choose to pursue a military career. Located in the Latin Quarter of central Paris, the school's main buildings were moved in 1976 to Palaiseau on the Saclay Plateau.
Polytechnique has engaged in several partnerships to improve its international renown. It is a founding member of ParisTech, a grouping of leading engineering colleges in the Paris region established in 2007. In 2014 it became a founding member of the confederal University of Paris-Saclay. Among its alumni are three Nobel prize winners, three Presidents of France and many CEOs of French and international companies; as of 2018, it is associated with 4 Fields Medal winners and is currently ranked as the world's third-best small university by Times Higher Education's World University Rankings. Every year, many outstanding Polytechnique students earn admissions to the most prestigious academic institutions and graduate programs in the USA and in the UK demonstrating the recognition of the school and its best performing students internationally. During the 19th century, the specific model of École Polytechnique inspired the foundation of other well-known schools named "Polytechnic," such as Polytechnique Montréal, Athens Polytechnic, MIT, EPFL and Caltech.
The history of the École Polytechnique dates back over 200 years, to the time of the French Revolution. In 1794, the École centrale des travaux publics was founded by Lazare Carnot and Gaspard Monge at the time of the National Convention, it was renamed École polytechnique one year later. In 1805, Emperor Napoléon Bonaparte settled the École on Montagne Sainte-Geneviève, in the Quartier Latin, in central Paris, as a military academy and gave its motto Pour la Patrie, les Sciences et la Gloire. In 1814, students took part in the Battle of Paris against the Sixth Coalition. In 1830, fifty students participated in the July Revolution. In 1848, Polytechnique students were the leaders of the French Revolution of 1848, they were an important part of the post-revolutionary process, with one student becoming part of the post-revolution government. They were given the right to wear a sword as a recognition. During the First World War, students were mobilized and the school building was transformed into a hospital.
More than two hundred students were killed fighting for France during the war. During the Second World War, Polytechnique was moved away to Lyon in the free zone. More than four hundred polytechniciens were killed during the war, as part of Free French and French Resistance operations, or in Nazi camps. In 1970, École Polytechnique became a state-supported civilian institution, under the auspices of the Ministry of Defence. In 1972, women were admitted for the first time. In 1976, École Polytechnique moved from Paris to Palaiseau. In 1994, celebration of the bicentennial was chaired by President François Mitterrand. In 2000, a new cursus was set in place, passing to four years and reforming the polytechnicien curriculum. In 2005, École Polytechnique started awarding master's degrees. In 2007, it became a founding member of UniverSud ParisTech. In December 2014, it became a founding member of University of Paris-Saclay. In 1794, Polytechnique was hosted in the Palais Bourbon. One year it moved to Hôtel de Lassay, an hôtel particulier in the 7th arrondissement of Paris.
Napoleon moved Polytechnique to the Quartier Latin in 1805 when he set the school under a military administration. The Paris' campus is located near the Panthéon, in rue Descartes, 5, it is nicknamed "Carva" by the students. At 15 kilometres from Paris, the campus of the École Polytechnique is a privileged setting, it offers about 164 ha teaching facilities, student housing, food services and hospitality and an exceptional range of sports facilities to the 4,600 people who live on a daily basis campus. The nearest regional train station is Gare de Lozère. A number of buses connect the École Polytechnique with the larger RER and TGV station Massy-Palaiseau; the campus is close to other great scientific institutions in Saclay and Gif. The campus will be at the heart of the Engineering and Innovation sector of the confederal "University of Paris in Saclay". Major works are in progress to connect it to an automatic metro line direct to Paris. Polytechnique is a higher education establishment running under the supervision of the F