1.
Stephen Hawking
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Hawking was the first to set forth a theory of cosmology explained by a union of the general theory of relativity and quantum mechanics. He is a supporter of the many-worlds interpretation of quantum mechanics. In 2002, Hawking was ranked number 25 in the BBCs poll of the 100 Greatest Britons, Hawking has a rare early-onset, slow-progressing form of amyotrophic lateral sclerosis that has gradually paralysed him over the decades. He now communicates using a single cheek muscle attached to a speech-generating device, Hawking was born on 8 January 1942 in Oxford, England to Frank and Isobel Hawking. Despite their families financial constraints, both attended the University of Oxford, where Frank read medicine and Isobel read Philosophy. The two met shortly after the beginning of the Second World War at a research institute where Isobel was working as a secretary. They lived in Highgate, but, as London was being bombed in those years, Hawking has two younger sisters, Philippa and Mary, and an adopted brother, Edward. In 1950, when Hawkings father became head of the division of parasitology at the National Institute for Medical Research, Hawking and his moved to St Albans. In St Albans, the family were considered intelligent and somewhat eccentric. They lived an existence in a large, cluttered, and poorly maintained house. During one of Hawkings fathers frequent absences working in Africa, the rest of the family spent four months in Majorca visiting his mothers friend Beryl and her husband, Hawking began his schooling at the Byron House School in Highgate, London. He later blamed its progressive methods for his failure to learn to read while at the school, in St Albans, the eight-year-old Hawking attended St Albans High School for Girls for a few months. At that time, younger boys could attend one of the houses, the family placed a high value on education. Hawkings father wanted his son to attend the well-regarded Westminster School and his family could not afford the school fees without the financial aid of a scholarship, so Hawking remained at St Albans. From 1958 on, with the help of the mathematics teacher Dikran Tahta, they built a computer from clock parts, although at school Hawking was known as Einstein, Hawking was not initially successful academically. With time, he began to show aptitude for scientific subjects and, inspired by Tahta. Hawkings father advised him to medicine, concerned that there were few jobs for mathematics graduates. He wanted Hawking to attend University College, Oxford, his own alma mater, as it was not possible to read mathematics there at the time, Hawking decided to study physics and chemistry

2.
International Standard Book Number
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The International Standard Book Number is a unique numeric commercial book identifier. An ISBN is assigned to each edition and variation of a book, for example, an e-book, a paperback and a hardcover edition of the same book would each have a different ISBN. The ISBN is 13 digits long if assigned on or after 1 January 2007, the method of assigning an ISBN is nation-based and varies from country to country, often depending on how large the publishing industry is within a country. The initial ISBN configuration of recognition was generated in 1967 based upon the 9-digit Standard Book Numbering created in 1966, the 10-digit ISBN format was developed by the International Organization for Standardization and was published in 1970 as international standard ISO2108. Occasionally, a book may appear without a printed ISBN if it is printed privately or the author does not follow the usual ISBN procedure, however, this can be rectified later. Another identifier, the International Standard Serial Number, identifies periodical publications such as magazines, the ISBN configuration of recognition was generated in 1967 in the United Kingdom by David Whitaker and in 1968 in the US by Emery Koltay. The 10-digit ISBN format was developed by the International Organization for Standardization and was published in 1970 as international standard ISO2108, the United Kingdom continued to use the 9-digit SBN code until 1974. The ISO on-line facility only refers back to 1978, an SBN may be converted to an ISBN by prefixing the digit 0. For example, the edition of Mr. J. G. Reeder Returns, published by Hodder in 1965, has SBN340013818 -340 indicating the publisher,01381 their serial number. This can be converted to ISBN 0-340-01381-8, the check digit does not need to be re-calculated, since 1 January 2007, ISBNs have contained 13 digits, a format that is compatible with Bookland European Article Number EAN-13s. An ISBN is assigned to each edition and variation of a book, for example, an ebook, a paperback, and a hardcover edition of the same book would each have a different ISBN. The ISBN is 13 digits long if assigned on or after 1 January 2007, a 13-digit ISBN can be separated into its parts, and when this is done it is customary to separate the parts with hyphens or spaces. Separating the parts of a 10-digit ISBN is also done with either hyphens or spaces, figuring out how to correctly separate a given ISBN number is complicated, because most of the parts do not use a fixed number of digits. ISBN issuance is country-specific, in that ISBNs are issued by the ISBN registration agency that is responsible for country or territory regardless of the publication language. Some ISBN registration agencies are based in national libraries or within ministries of culture, in other cases, the ISBN registration service is provided by organisations such as bibliographic data providers that are not government funded. In Canada, ISBNs are issued at no cost with the purpose of encouraging Canadian culture. In the United Kingdom, United States, and some countries, where the service is provided by non-government-funded organisations. Australia, ISBNs are issued by the library services agency Thorpe-Bowker

3.
The Grand Design (book)
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The Grand Design is a popular-science book written by physicists Stephen Hawking and Leonard Mlodinow and published by Bantam Books in 2010. The book examines the history of knowledge about the universe. The authors of the point out that a Unified Field Theory may not exist. It argues that invoking God is not necessary to explain the origins of the universe, in response to criticism, Hawking has said, One cant prove that God doesnt exist, but science makes God unnecessary. When pressed on his own religious views by the Channel 4 documentary Genius of Britain, published in the United States on September 7,2010, the book became the number one bestseller on Amazon. com just a few days after publication. It was published in the United Kingdom on September 9,2010 and it topped the list of adult non-fiction books of The New York Times Non-fiction Best Seller list in Sept-Oct 2010. The book examines the history of knowledge about the universe. It starts with the Ionian Greeks, who claimed that nature works by laws and it later presents the work of Nicolaus Copernicus, who advocated the concept that the Earth is not located in the center of the universe. The authors then describe the theory of quantum mechanics using, as an example, the presentation has been described as easy to understand by some reviewers, but also as sometimes impenetrable, by others. The central claim of the book is that the theory of quantum mechanics, the book concludes with the statement that only some universes of the multiple universes support life forms. We, of course, are located in one of those universes, the laws of nature that are required for life forms to exist appear in some universes by pure chance, Hawking and Mlodinow explain. Evolutionary biologist and advocate for atheism Richard Dawkins welcomed Hawkings position and said that Darwinism kicked God out of biology, Hawking is now administering the coup de grace. Theoretical physicist Sean M. questions that are part of human curiosity. If our universe arose spontaneously from nothing at all, one might predict that its total energy should be zero, and when we measure the total energy of the universe, which could have been anything, the answer turns out to be the only one consistent with this possibility. But data like this coming in from our revolutionary new tools promise to turn much of what is now metaphysics into physics, whether God survives is anyones guess. James Trefil, a professor of physics at George Mason University, said in his Washington Post review and it gets into the deepest questions of modern cosmology without a single equation. The reader will be able to get through it without bogging down in a lot of detail and will. Maybe in the end the whole multiverse idea will actually turn out to be right, canada Press journalist Carl Hartman said, Cosmologists, the people who study the entire cosmos, will want to read British physicist and mathematician Stephen Hawkings new book

4.
Leopold Kronecker
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Leopold Kronecker was a German mathematician who worked on number theory, algebra and logic. He criticized Cantors work on set theory, and was quoted by Weber as having said, Die ganzen Zahlen hat der liebe Gott gemacht, Kronecker was a student and lifelong friend of Ernst Kummer. Leopold Kronecker was born on 7 December 1823 in Liegnitz, Prussia in a wealthy Jewish family, Kronecker then went to the Liegnitz Gymnasium where he was interested in a wide range of topics including science, history and philosophy, while also practicing gymnastics and swimming. At the gymnasium he was taught by Ernst Kummer, who noticed and encouraged the boys interest in mathematics, in 1841 Kronecker became a student at the University of Berlin where his interest did not immediately focus on mathematics, but rather spread over several subjects including astronomy and philosophy. He spent the summer of 1843 at the University of Bonn studying astronomy, back in Berlin, Kronecker studied mathematics with Peter Gustav Lejeune Dirichlet and in 1845 defended his dissertation in algebraic number theory written under Dirichlets supervision. After obtaining his degree, Kronecker did not follow his interest in research on a career path. He went back to his hometown to manage a large farming estate built up by his mothers uncle, in 1848 he married his cousin Fanny Prausnitzer, and the couple had six children. For several years Kronecker focused on business, and although he continued to study mathematics as a hobby and corresponded with Kummer, in 1853 he wrote a memoir on the algebraic solvability of equations extending the work of Évariste Galois on the theory of equations. Due to his business activity, Kronecker was financially comfortable, Dirichlet, whose wife Rebecka came from the wealthy Mendelssohn family, had introduced Kronecker to the Berlin elite. He became a friend of Karl Weierstrass, who had recently joined the university. Over the following years Kronecker published numerous papers resulting from his previous years independent research, as a result of this published research, he was elected a member of the Berlin Academy in 1861. Although he held no official university position, Kronecker had the right as a member of the Academy to hold classes at the University of Berlin and he decided to do so, starting in 1862. In 1866, when Riemann died, Kronecker was offered the chair at the University of Göttingen. Only in 1883, when Kummer retired from the University, was Kronecker invited to succeed him, Kronecker was the supervisor of Kurt Hensel, Adolf Kneser, Mathias Lerch, and Franz Mertens, amongst others. Kronecker died on 29 December 1891 in Berlin, several months after the death of his wife, in the last year of his life, he converted to Christianity. He is buried in the Alter St Matthäus Kirchhof cemetery in Berlin-Schöneberg, an important part of Kroneckers research focused on number theory and algebra. In an 1853 paper on the theory of equations and Galois theory he formulated the Kronecker–Weber theorem and he also introduced the structure theorem for finitely-generated abelian groups. Kronecker studied elliptic functions and conjectured his liebster Jugendtraum, a generalization that was put forward by Hilbert in a modified form as his twelfth problem

5.
Integer
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An integer is a number that can be written without a fractional component. For example,21,4,0, and −2048 are integers, while 9.75, 5 1⁄2, the set of integers consists of zero, the positive natural numbers, also called whole numbers or counting numbers, and their additive inverses. This is often denoted by a boldface Z or blackboard bold Z standing for the German word Zahlen, ℤ is a subset of the sets of rational and real numbers and, like the natural numbers, is countably infinite. The integers form the smallest group and the smallest ring containing the natural numbers, in algebraic number theory, the integers are sometimes called rational integers to distinguish them from the more general algebraic integers. In fact, the integers are the integers that are also rational numbers. Like the natural numbers, Z is closed under the operations of addition and multiplication, that is, however, with the inclusion of the negative natural numbers, and, importantly,0, Z is also closed under subtraction. The integers form a ring which is the most basic one, in the following sense, for any unital ring. This universal property, namely to be an object in the category of rings. Z is not closed under division, since the quotient of two integers, need not be an integer, although the natural numbers are closed under exponentiation, the integers are not. The following lists some of the properties of addition and multiplication for any integers a, b and c. In the language of algebra, the first five properties listed above for addition say that Z under addition is an abelian group. As a group under addition, Z is a cyclic group, in fact, Z under addition is the only infinite cyclic group, in the sense that any infinite cyclic group is isomorphic to Z. The first four properties listed above for multiplication say that Z under multiplication is a commutative monoid. However, not every integer has an inverse, e. g. there is no integer x such that 2x =1, because the left hand side is even. This means that Z under multiplication is not a group, all the rules from the above property table, except for the last, taken together say that Z together with addition and multiplication is a commutative ring with unity. It is the prototype of all objects of algebraic structure. Only those equalities of expressions are true in Z for all values of variables, note that certain non-zero integers map to zero in certain rings. The lack of zero-divisors in the means that the commutative ring Z is an integral domain

6.
Euclid
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Euclid, sometimes called Euclid of Alexandria to distinguish him from Euclides of Megara, was a Greek mathematician, often referred to as the father of geometry. He was active in Alexandria during the reign of Ptolemy I, in the Elements, Euclid deduced the principles of what is now called Euclidean geometry from a small set of axioms. Euclid also wrote works on perspective, conic sections, spherical geometry, number theory, Euclid is the anglicized version of the Greek name Εὐκλείδης, which means renowned, glorious. Very few original references to Euclid survive, so little is known about his life, the date, place and circumstances of both his birth and death are unknown and may only be estimated roughly relative to other people mentioned with him. He is rarely mentioned by name by other Greek mathematicians from Archimedes onward, the few historical references to Euclid were written centuries after he lived by Proclus c.450 AD and Pappus of Alexandria c.320 AD. Proclus introduces Euclid only briefly in his Commentary on the Elements, Proclus later retells a story that, when Ptolemy I asked if there was a shorter path to learning geometry than Euclids Elements, Euclid replied there is no royal road to geometry. This anecdote is questionable since it is similar to a story told about Menaechmus, a detailed biography of Euclid is given by Arabian authors, mentioning, for example, a birth town of Tyre. This biography is generally believed to be completely fictitious, however, this hypothesis is not well accepted by scholars and there is little evidence in its favor. The only reference that historians rely on of Euclid having written the Elements was from Proclus, although best known for its geometric results, the Elements also includes number theory. The geometrical system described in the Elements was long known simply as geometry, today, however, that system is often referred to as Euclidean geometry to distinguish it from other so-called non-Euclidean geometries that mathematicians discovered in the 19th century. In addition to the Elements, at least five works of Euclid have survived to the present day and they follow the same logical structure as Elements, with definitions and proved propositions. Data deals with the nature and implications of information in geometrical problems. On Divisions of Figures, which only partially in Arabic translation. It is similar to a first-century AD work by Heron of Alexandria, catoptrics, which concerns the mathematical theory of mirrors, particularly the images formed in plane and spherical concave mirrors. The attribution is held to be anachronistic however by J J OConnor, phaenomena, a treatise on spherical astronomy, survives in Greek, it is quite similar to On the Moving Sphere by Autolycus of Pitane, who flourished around 310 BC. Optics is the earliest surviving Greek treatise on perspective, in its definitions Euclid follows the Platonic tradition that vision is caused by discrete rays which emanate from the eye. One important definition is the fourth, Things seen under a greater angle appear greater, proposition 45 is interesting, proving that for any two unequal magnitudes, there is a point from which the two appear equal. Other works are attributed to Euclid, but have been lost

7.
Archimedes
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Archimedes of Syracuse was a Greek mathematician, physicist, engineer, inventor, and astronomer. Although few details of his life are known, he is regarded as one of the scientists in classical antiquity. He was also one of the first to apply mathematics to physical phenomena, founding hydrostatics and statics and he is credited with designing innovative machines, such as his screw pump, compound pulleys, and defensive war machines to protect his native Syracuse from invasion. Archimedes died during the Siege of Syracuse when he was killed by a Roman soldier despite orders that he should not be harmed. Cicero describes visiting the tomb of Archimedes, which was surmounted by a sphere and a cylinder, unlike his inventions, the mathematical writings of Archimedes were little known in antiquity. Archimedes was born c.287 BC in the city of Syracuse, Sicily, at that time a self-governing colony in Magna Graecia. The date of birth is based on a statement by the Byzantine Greek historian John Tzetzes that Archimedes lived for 75 years, in The Sand Reckoner, Archimedes gives his fathers name as Phidias, an astronomer about whom nothing is known. Plutarch wrote in his Parallel Lives that Archimedes was related to King Hiero II, a biography of Archimedes was written by his friend Heracleides but this work has been lost, leaving the details of his life obscure. It is unknown, for instance, whether he married or had children. During his youth, Archimedes may have studied in Alexandria, Egypt and he referred to Conon of Samos as his friend, while two of his works have introductions addressed to Eratosthenes. Archimedes died c.212 BC during the Second Punic War, according to the popular account given by Plutarch, Archimedes was contemplating a mathematical diagram when the city was captured. A Roman soldier commanded him to come and meet General Marcellus but he declined, the soldier was enraged by this, and killed Archimedes with his sword. Plutarch also gives an account of the death of Archimedes which suggests that he may have been killed while attempting to surrender to a Roman soldier. According to this story, Archimedes was carrying mathematical instruments, and was killed because the thought that they were valuable items. General Marcellus was reportedly angered by the death of Archimedes, as he considered him a valuable asset and had ordered that he not be harmed. Marcellus called Archimedes a geometrical Briareus, the last words attributed to Archimedes are Do not disturb my circles, a reference to the circles in the mathematical drawing that he was supposedly studying when disturbed by the Roman soldier. This quote is given in Latin as Noli turbare circulos meos. The phrase is given in Katharevousa Greek as μὴ μου τοὺς κύκλους τάραττε

8.
Diophantus
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Diophantus of Alexandria, sometimes called the father of algebra, was an Alexandrian Greek mathematician and the author of a series of books called Arithmetica, many of which are now lost. These texts deal with solving algebraic equations and this led to tremendous advances in number theory, and the study of Diophantine equations and of Diophantine approximations remain important areas of mathematical research. Diophantus coined the term παρισότης to refer to an approximate equality and this term was rendered as adaequalitas in Latin, and became the technique of adequality developed by Pierre de Fermat to find maxima for functions and tangent lines to curves. Diophantus was the first Greek mathematician who recognized fractions as numbers, thus he allowed positive rational numbers for the coefficients, in modern use, Diophantine equations are usually algebraic equations with integer coefficients, for which integer solutions are sought. Diophantus also made advances in mathematical notation, little is known about the life of Diophantus. He lived in Alexandria, Egypt, probably from between AD200 and 214 to 284 or 298, much of our knowledge of the life of Diophantus is derived from a 5th-century Greek anthology of number games and puzzles created by Metrodorus. One of the states, Here lies Diophantus, the wonder behold. Alas, the child of master and sage After attaining half the measure of his fathers life chill fate took him. After consoling his fate by the science of numbers for four years and this puzzle implies that Diophantus age x can be expressed as x = x/6 + x/12 + x/7 +5 + x/2 +4 which gives x a value of 84 years. However, the accuracy of the information cannot be independently confirmed, the Arithmetica is the major work of Diophantus and the most prominent work on algebra in Greek mathematics. It is a collection of problems giving numerical solutions of both determinate and indeterminate equations, of the original thirteen books of which Arithmetica consisted only six have survived, though there are some who believe that four Arab books discovered in 1968 are also by Diophantus. Some Diophantine problems from Arithmetica have been found in Arabic sources and it should be mentioned here that Diophantus never used general methods in his solutions. Hermann Hankel, renowned German mathematician made the following remark regarding Diophantus, “Our author not the slightest trace of a general, comprehensive method is discernible, each problem calls for some special method which refuses to work even for the most closely related problems. The portion of the Greek Arithmetica that survived, however, was, like all ancient Greek texts transmitted to the modern world, copied by. In addition, some portion of the Arithmetica probably survived in the Arab tradition. ”Arithmetica was first translated from Greek into Latin by Bombelli in 1570, however, Bombelli borrowed many of the problems for his own book Algebra. The editio princeps of Arithmetica was published in 1575 by Xylander, the best known Latin translation of Arithmetica was made by Bachet in 1621 and became the first Latin edition that was widely available. Pierre de Fermat owned a copy, studied it, and made notes in the margins. I have a marvelous proof of this proposition which this margin is too narrow to contain. ”Fermats proof was never found

9.
Isaac Newton
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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

10.
Leonhard Euler
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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

11.
Pierre-Simon Laplace
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Pierre-Simon, marquis de Laplace was an influential French scholar whose work was important to the development of mathematics, statistics, physics and astronomy. He summarized and extended the work of his predecessors in his five-volume Mécanique Céleste and this work translated the geometric study of classical mechanics to one based on calculus, opening up a broader range of problems. In statistics, the Bayesian interpretation of probability was developed mainly by Laplace, Laplace formulated Laplaces equation, and pioneered the Laplace transform which appears in many branches of mathematical physics, a field that he took a leading role in forming. The Laplacian differential operator, widely used in mathematics, is named after him. Laplace is remembered as one of the greatest scientists of all time, sometimes referred to as the French Newton or Newton of France, he has been described as possessing a phenomenal natural mathematical faculty superior to that of any of his contemporaries. Laplace became a count of the Empire in 1806 and was named a marquis in 1817, Laplace was born in Beaumont-en-Auge, Normandy on 23 March 1749, a village four miles west of Pont lEveque in Normandy. According to W. W. Rouse Ball, His father, Pierre de Laplace and his great-uncle, Maitre Oliver de Laplace, had held the title of Chirurgien Royal. It would seem that from a pupil he became an usher in the school at Beaumont, however, Karl Pearson is scathing about the inaccuracies in Rouse Balls account and states, Indeed Caen was probably in Laplaces day the most intellectually active of all the towns of Normandy. It was here that Laplace was educated and was provisionally a professor and it was here he wrote his first paper published in the Mélanges of the Royal Society of Turin, Tome iv. 1766–1769, at least two years before he went at 22 or 23 to Paris in 1771, thus before he was 20 he was in touch with Lagrange in Turin. He did not go to Paris a raw self-taught country lad with only a peasant background, the École Militaire of Beaumont did not replace the old school until 1776. His parents were from comfortable families and his father was Pierre Laplace, and his mother was Marie-Anne Sochon. The Laplace family was involved in agriculture until at least 1750, Pierre Simon Laplace attended a school in the village run at a Benedictine priory, his father intending that he be ordained in the Roman Catholic Church. At sixteen, to further his fathers intention, he was sent to the University of Caen to read theology, at the university, he was mentored by two enthusiastic teachers of mathematics, Christophe Gadbled and Pierre Le Canu, who awoke his zeal for the subject. Here Laplaces brilliance as a mathematician was recognised and while still at Caen he wrote a memoir Sur le Calcul integral aux differences infiniment petites et aux differences finies. About this time, recognizing that he had no vocation for the priesthood, in this connection reference may perhaps be made to the statement, which has appeared in some notices of him, that he broke altogether with the church and became an atheist. Laplace did not graduate in theology but left for Paris with a letter of introduction from Le Canu to Jean le Rond dAlembert who at time was supreme in scientific circles. According to his great-great-grandson, dAlembert received him rather poorly, and to get rid of him gave him a mathematics book

12.
Joseph Fourier
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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

13.
Carl Friedrich Gauss
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Johann Carl Friedrich Gauss was born on 30 April 1777 in Brunswick, in the Duchy of Brunswick-Wolfenbüttel, as the son of poor working-class parents. Gauss later solved this puzzle about his birthdate in the context of finding the date of Easter and he was christened and confirmed in a church near the school he attended as a child. A contested story relates that, when he was eight, he figured out how to add up all the numbers from 1 to 100, there are many other anecdotes about his precocity while a toddler, and he made his first ground-breaking mathematical discoveries while still a teenager. He completed Disquisitiones Arithmeticae, his opus, in 1798 at the age of 21. This work was fundamental in consolidating number theory as a discipline and has shaped the field to the present day, while at university, Gauss independently rediscovered several important theorems. Gauss was so pleased by this result that he requested that a regular heptadecagon be inscribed on his tombstone, the stonemason declined, stating that the difficult construction would essentially look like a circle. The year 1796 was most productive for both Gauss and number theory and he discovered a construction of the heptadecagon on 30 March. He further advanced modular arithmetic, greatly simplifying manipulations in number theory, on 8 April he became the first to prove the quadratic reciprocity law. This remarkably general law allows mathematicians to determine the solvability of any quadratic equation in modular arithmetic, the prime number theorem, conjectured on 31 May, gives a good understanding of how the prime numbers are distributed among the integers. Gauss also discovered that every integer is representable as a sum of at most three triangular numbers on 10 July and then jotted down in his diary the note, ΕΥΡΗΚΑ. On October 1 he published a result on the number of solutions of polynomials with coefficients in finite fields, in 1831 Gauss developed a fruitful collaboration with the physics professor Wilhelm Weber, leading to new knowledge in magnetism and the discovery of Kirchhoffs circuit laws in electricity. It was during this time that he formulated his namesake law and they constructed the first electromechanical telegraph in 1833, which connected the observatory with the institute for physics in Göttingen. In 1840, Gauss published his influential Dioptrische Untersuchungen, in which he gave the first systematic analysis on the formation of images under a paraxial approximation. Among his results, Gauss showed that under a paraxial approximation an optical system can be characterized by its cardinal points and he derived the Gaussian lens formula. In 1845, he became associated member of the Royal Institute of the Netherlands, in 1854, Gauss selected the topic for Bernhard Riemanns Habilitationvortrag, Über die Hypothesen, welche der Geometrie zu Grunde liegen. On the way home from Riemanns lecture, Weber reported that Gauss was full of praise, Gauss died in Göttingen, on 23 February 1855 and is interred in the Albani Cemetery there. Two individuals gave eulogies at his funeral, Gausss son-in-law Heinrich Ewald and Wolfgang Sartorius von Waltershausen and his brain was preserved and was studied by Rudolf Wagner who found its mass to be 1,492 grams and the cerebral area equal to 219,588 square millimeters. Highly developed convolutions were also found, which in the early 20th century were suggested as the explanation of his genius, Gauss was a Lutheran Protestant, a member of the St. Albans Evangelical Lutheran church in Göttingen

14.
Augustin-Louis Cauchy
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Baron Augustin-Louis Cauchy FRS FRSE was a French mathematician who made pioneering contributions to analysis. He was one of the first to state and prove theorems of calculus rigorously and he almost singlehandedly founded complex analysis and the study of permutation groups in abstract algebra. A profound mathematician, Cauchy had an influence over his contemporaries. His writings range widely in mathematics and mathematical physics, more concepts and theorems have been named for Cauchy than for any other mathematician. Cauchy was a writer, he wrote approximately eight hundred research articles. Cauchy was the son of Louis François Cauchy and Marie-Madeleine Desestre, Cauchy married Aloise de Bure in 1818. She was a relative of the publisher who published most of Cauchys works. By her he had two daughters, Marie Françoise Alicia and Marie Mathilde, Cauchys father was a high official in the Parisian Police of the New Régime. He lost his position because of the French Revolution that broke out one month before Augustin-Louis was born, the Cauchy family survived the revolution and the following Reign of Terror by escaping to Arcueil, where Cauchy received his first education, from his father. After the execution of Robespierre, it was safe for the family to return to Paris, there Louis-François Cauchy found himself a new bureaucratic job, and quickly moved up the ranks. When Napoleon Bonaparte came to power, Louis-François Cauchy was further promoted, the famous mathematician Lagrange was also a friend of the Cauchy family. On Lagranges advice, Augustin-Louis was enrolled in the École Centrale du Panthéon, most of the curriculum consisted of classical languages, the young and ambitious Cauchy, being a brilliant student, won many prizes in Latin and Humanities. In spite of successes, Augustin-Louis chose an engineering career. In 1805 he placed second out of 293 applicants on this exam, one of the main purposes of this school was to give future civil and military engineers a high-level scientific and mathematical education. The school functioned under military discipline, which caused the young, nevertheless, he finished the Polytechnique in 1807, at the age of 18, and went on to the École des Ponts et Chaussées. He graduated in engineering, with the highest honors. After finishing school in 1810, Cauchy accepted a job as an engineer in Cherbourg. Cauchys first two manuscripts were accepted, the one was rejected

15.
Nikolai Lobachevsky
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Nikolai Ivanovich Lobachevsky was a Russian mathematician and geometer, known primarily for his work on hyperbolic geometry, otherwise known as Lobachevskian geometry. William Kingdon Clifford called Lobachevsky the Copernicus of Geometry due to the character of his work. He was one of three children and his father, a clerk in a land surveying office, died when he was seven, and his mother moved to Kazan. Lobachevsky attended Kazan Gymnasium from 1802, graduating in 1807 and then received a scholarship to Kazan University, at Kazan University, Lobachevsky was influenced by professor Johann Christian Martin Bartels, a former teacher and friend of German mathematician Carl Friedrich Gauss. Lobachevsky received a degree in physics and mathematics in 1811. He served in administrative positions and became the rector of Kazan University in 1827. In 1832, he married Varvara Alexeyevna Moiseyeva and they had a large number of children. He was dismissed from the university in 1846, ostensibly due to his health, by the early 1850s, he was nearly blind. He died in poverty in 1856, Lobachevskys main achievement is the development of a non-Euclidean geometry, also referred to as Lobachevskian geometry. Before him, mathematicians were trying to deduce Euclids fifth postulate from other axioms, Euclids fifth is a rule in Euclidean geometry which states that for any given line and point not on the line, there is one parallel line through the point not intersecting the line. Lobachevsky would instead develop a geometry in which the fifth postulate was not true and this idea was first reported on February 23,1826 to the session of the department of physics and mathematics, and this research was printed in the UMA in 1829–1830. The non-Euclidean geometry that Lobachevsky developed is referred to as hyperbolic geometry and he developed the angle of parallelism which depends on the distance the point is off the given line. In hyperbolic geometry the sum of angles in a triangle must be less than 180 degrees. Non-Euclidean geometry stimulated the development of geometry which has many applications. Hyperbolic geometry is referred to as Lobachevskian geometry or Bolyai–Lobachevskian geometry. Some mathematicians and historians have claimed that Lobachevsky in his studies in non-Euclidean geometry was influenced by Gauss. Gauss himself appreciated Lobachevskys published works very highly, but they never had personal correspondence between them prior to the publication, Lobachevskys magnum opus Geometriya was completed in 1823, but was not published in its exact original form until 1909, long after he had died. Lobachevsky was also the author of New Foundations of Geometry and he also wrote Geometrical Investigations on the Theory of Parallels and Pangeometry

16.
George Boole
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George Boole was an English mathematician, educator, philosopher and logician. He worked in the fields of differential equations and algebraic logic, Boolean logic is credited with laying the foundations for the information age. Boole was born in Lincoln, Lincolnshire, England, the son of John Boole Sr and he had a primary school education, and received lessons from his father, but had little further formal and academic teaching. William Brooke, a bookseller in Lincoln, may have helped him with Latin and he was self-taught in modern languages. At age 16 Boole became the breadwinner for his parents and three siblings, taking up a junior teaching position in Doncaster at Heighams School. Boole participated in the Mechanics Institute, in the Greyfriars, Lincoln, without a teacher, it took him many years to master calculus. At age 19, Boole successfully established his own school in Lincoln, four years later he took over Halls Academy in Waddington, outside Lincoln, following the death of Robert Hall. In 1840 he moved back to Lincoln, where he ran a boarding school, Boole became a prominent local figure, an admirer of John Kaye, the bishop. He took part in the campaign for early closing. With E. R. Larken and others he set up a society in 1847. He associated also with the Chartist Thomas Cooper, whose wife was a relation, from 1838 onwards Boole was making contacts with sympathetic British academic mathematicians and reading more widely. He studied algebra in the form of symbolic methods, as far as these were understood at the time, Booles status as mathematician was recognised by his appointment in 1849 as the first professor of mathematics at Queens College, Cork in Ireland. He met his wife, Mary Everest, there in 1850 while she was visiting her uncle John Ryall who was Professor of Greek. They married some years later in 1855 and he maintained his ties with Lincoln, working there with E. R. Larken in a campaign to reduce prostitution. Boole was awarded the Keith Medal by the Royal Society of Edinburgh in 1855 and was elected a Fellow of the Royal Society in 1857 and he received honorary degrees of LL. D. from the University of Dublin and the University of Oxford. In late November 1864, Boole walked, in rain, from his home at Lichfield Cottage in Ballintemple to the university. He soon became ill, developing a cold and high fever. As his wife believed that remedies should resemble their cause, she put her husband to bed and poured buckets of water over him – the wet having brought on his illness, Booles condition worsened and on 8 December 1864, he died of fever-induced pleural effusion

17.
Bernhard Riemann
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Georg Friedrich Bernhard Riemann was a German mathematician who made contributions to analysis, number theory, and differential geometry. In the field of analysis, he is mostly known for the first rigorous formulation of the integral, the Riemann integral. His contributions to complex analysis include most notably the introduction of Riemann surfaces, breaking new ground in a natural, through his pioneering contributions to differential geometry, Bernhard Riemann laid the foundations of the mathematics of general relativity. Riemann was born on September 17,1826 in Breselenz, a village near Dannenberg in the Kingdom of Hanover in what is the Federal Republic of Germany today and his father, Friedrich Bernhard Riemann, was a poor Lutheran pastor in Breselenz who fought in the Napoleonic Wars. His mother, Charlotte Ebell, died before her children had reached adulthood, Riemann was the second of six children, shy and suffering from numerous nervous breakdowns. Riemann exhibited exceptional skills, such as calculation abilities, from an early age but suffered from timidity. During 1840, Riemann went to Hanover to live with his grandmother, after the death of his grandmother in 1842, he attended high school at the Johanneum Lüneburg. In high school, Riemann studied the Bible intensively, but he was distracted by mathematics. His teachers were amazed by his ability to perform complicated mathematical operations. In 1846, at the age of 19, he started studying philology and Christian theology in order to become a pastor and help with his familys finances. During the spring of 1846, his father, after gathering enough money, sent Riemann to the University of Göttingen, however, once there, he began studying mathematics under Carl Friedrich Gauss. Gauss recommended that Riemann give up his work and enter the mathematical field, after getting his fathers approval. During his time of study, Jacobi, Lejeune Dirichlet, Steiner and he stayed in Berlin for two years and returned to Göttingen in 1849. Riemann held his first lectures in 1854, which founded the field of Riemannian geometry, in 1857, there was an attempt to promote Riemann to extraordinary professor status at the University of Göttingen. Although this attempt failed, it did result in Riemann finally being granted a regular salary, in 1859, following Lejeune Dirichlets death, he was promoted to head the mathematics department at Göttingen. He was also the first to suggest using dimensions higher than three or four in order to describe physical reality. In 1862 he married Elise Koch and had a daughter, Riemann fled Göttingen when the armies of Hanover and Prussia clashed there in 1866. He died of tuberculosis during his journey to Italy in Selasca where he was buried in the cemetery in Biganzolo

18.
Karl Weierstrass
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Karl Theodor Wilhelm Weierstrass was a German mathematician often cited as the father of modern analysis. Despite leaving university without a degree, he studied mathematics and trained as a teacher, eventually teaching mathematics, physics, botany, Weierstrass was born in Ostenfelde, part of Ennigerloh, Province of Westphalia. Weierstrass was the son of Wilhelm Weierstrass, a government official and his interest in mathematics began while he was a gymnasium student at the Theodorianum in Paderborn. He was sent to the University of Bonn upon graduation to prepare for a government position, because his studies were to be in the fields of law, economics, and finance, he was immediately in conflict with his hopes to study mathematics. He resolved the conflict by paying little heed to his course of study. The outcome was to leave the university without a degree, after that he studied mathematics at the Münster Academy and his father was able to obtain a place for him in a teacher training school in Münster. Later he was certified as a teacher in that city, during this period of study, Weierstrass attended the lectures of Christoph Gudermann and became interested in elliptic functions. In 1843 he taught in Deutsch Krone in West Prussia and since 1848 he taught at the Lyceum Hosianum in Braunsberg, besides mathematics he also taught physics, botanics and gymnastics. Weierstrass may have had a child named Franz with the widow of his friend Carl Wilhelm Borchardt. After 1850 Weierstrass suffered from a period of illness, but was able to publish papers that brought him fame. The University of Königsberg conferred an honorary degree on him on 31 March 1854. In 1856 he took a chair at the Gewerbeinstitut, which became the Technical University of Berlin. In 1864 he became professor at the Friedrich-Wilhelms-Universität Berlin, which became the Humboldt Universität zu Berlin. He was immobile for the last three years of his life, and died in Berlin from pneumonia, delta-epsilon proofs are first found in the works of Cauchy in the 1820s. Cauchy did not clearly distinguish between continuity and uniform continuity on an interval, notably, in his 1821 Cours danalyse, Cauchy argued that the limit of continuous functions was itself continuous, a statement interpreted as being incorrect by many scholars. The correct statement is rather that the limit of continuous functions is continuous. Weierstrass saw the importance of the concept, and both formalized it and applied it widely throughout the foundations of calculus, using this definition, he proved the Intermediate Value Theorem. He also proved the Bolzano–Weierstrass theorem and used it to study the properties of functions on closed and bounded intervals

19.
Richard Dedekind
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Julius Wilhelm Richard Dedekind was a German mathematician who made important contributions to abstract algebra, algebraic number theory and the definition of the real numbers. Dedekinds father was Julius Levin Ulrich Dedekind, an administrator of Collegium Carolinum in Braunschweig, as an adult, he never used the names Julius Wilhelm. He was born, lived most of his life, and died in Braunschweig and he first attended the Collegium Carolinum in 1848 before transferring to the University of Göttingen in 1850. There, Dedekind was taught number theory by professor Moritz Stern, Gauss was still teaching, although mostly at an elementary level, and Dedekind became his last student. Dedekind received his doctorate in 1852, for a thesis titled Über die Theorie der Eulerschen Integrale and this thesis did not display the talent evident by Dedekinds subsequent publications. At that time, the University of Berlin, not Göttingen, was the facility for mathematical research in Germany. Thus Dedekind went to Berlin for two years of study, where he and Bernhard Riemann were contemporaries, they were awarded the habilitation in 1854. Dedekind returned to Göttingen to teach as a Privatdozent, giving courses on probability and he studied for a while with Peter Gustav Lejeune Dirichlet, and they became good friends. Because of lingering weaknesses in his knowledge, he studied elliptic. Yet he was also the first at Göttingen to lecture concerning Galois theory, about this time, he became one of the first people to understand the importance of the notion of groups for algebra and arithmetic. In 1858, he began teaching at the Polytechnic school in Zürich, when the Collegium Carolinum was upgraded to a Technische Hochschule in 1862, Dedekind returned to his native Braunschweig, where he spent the rest of his life, teaching at the Institute. He retired in 1894, but did occasional teaching and continued to publish and he never married, instead living with his sister Julia. Dedekind was elected to the Academies of Berlin and Rome, and he received honorary doctorates from the universities of Oslo, Zurich, and Braunschweig. While teaching calculus for the first time at the Polytechnic school, Dedekind developed the now known as a Dedekind cut. The idea of a cut is that an irrational number divides the rational numbers into two classes, with all the numbers of one class being strictly greater than all the numbers of the other class. Every location on the number line continuum contains either a rational or an irrational number, thus there are no empty locations, gaps, or discontinuities. Dedekind published his thoughts on irrational numbers and Dedekind cuts in his pamphlet Stetigkeit und irrationale Zahlen, in modern terminology, Vollständigkeit, Dedekinds theorem states that if there existed a one-to-one correspondence between two sets, then Dedekind said that the two sets were similar. Thus the set N of natural numbers can be shown to be similar to the subset of N whose members are the squares of every member of N, N12345678910

20.
Georg Cantor
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Georg Ferdinand Ludwig Philipp Cantor was a German mathematician. He invented set theory, which has become a theory in mathematics. In fact, Cantors method of proof of this theorem implies the existence of an infinity of infinities and he defined the cardinal and ordinal numbers and their arithmetic. Cantors work is of great philosophical interest, a fact of which he was well aware, E. J. Brouwer, while Ludwig Wittgenstein raised philosophical objections. Cantor, a devout Lutheran, believed the theory had been communicated to him by God, Kronecker objected to Cantors proofs that the algebraic numbers are countable, and that the transcendental numbers are uncountable, results now included in a standard mathematics curriculum. The harsh criticism has been matched by later accolades, in 1904, the Royal Society awarded Cantor its Sylvester Medal, the highest honor it can confer for work in mathematics. David Hilbert defended it from its critics by declaring, From his paradise that Cantor with us unfolded, we hold our breath in awe, knowing, we shall not be expelled. Georg Cantor was born in the merchant colony in Saint Petersburg, Russia. Georg, the oldest of six children, was regarded as an outstanding violinist and his grandfather Franz Böhm was a well-known musician and soloist in a Russian imperial orchestra. In 1860, Cantor graduated with distinction from the Realschule in Darmstadt, his skills in mathematics. In 1862, Cantor entered the Swiss Federal Polytechnic and he spent the summer of 1866 at the University of Göttingen, then and later a center for mathematical research. Cantor submitted his dissertation on number theory at the University of Berlin in 1867, after teaching briefly in a Berlin girls school, Cantor took up a position at the University of Halle, where he spent his entire career. He was awarded the habilitation for his thesis, also on number theory. In 1874, Cantor married Vally Guttmann and they had six children, the last born in 1886. Cantor was able to support a family despite modest academic pay, during his honeymoon in the Harz mountains, Cantor spent much time in mathematical discussions with Richard Dedekind, whom he had met two years earlier while on Swiss holiday. Cantor was promoted to Extraordinary Professor in 1872 and made full Professor in 1879, however, his work encountered too much opposition for that to be possible. Worse yet, Kronecker, a figure within the mathematical community and Cantors former professor. Cantor came to believe that Kroneckers stance would make it impossible for him ever to leave Halle, in 1881, Cantors Halle colleague Eduard Heine died, creating a vacant chair

21.
Henri Lebesgue
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His theory was published originally in his dissertation Intégrale, longueur, aire at the University of Nancy during 1902. Henri Lebesgue was born on 28 June 1875 in Beauvais, Oise, Lebesgues father was a typesetter and his mother was a school teacher. His parents assembled at home a library that the young Henri was able to use and his father died of tuberculosis when Lebesgue was still very young and his mother had to support him by herself. In 1894 Lebesgue was accepted at the École Normale Supérieure, where he continued to focus his energy on the study of mathematics, graduating in 1897. At the same time he started his studies at the Sorbonne. In 1899 he moved to a position at the Lycée Central in Nancy. In 1902 he earned his Ph. D. from the Sorbonne with the thesis on Integral, Length, Area, submitted with Borel, four years older. Lebesgue married the sister of one of his students, and he. After publishing his thesis, Lebesgue was offered in 1902 a position at the University of Rennes, lecturing there until 1906, in 1910 Lebesgue moved to the Sorbonne as a maître de conférences, being promoted to professor starting with 1919. In 1921 he left the Sorbonne to become professor of mathematics at the Collège de France, in 1922 he was elected a member of the Académie des Sciences. Henri Lebesgue died on 26 July 1941 in Paris, Lebesgues first paper was published in 1898 and was titled Sur lapproximation des fonctions. It dealt with Weierstrass theorem on approximation to continuous functions by polynomials, between March 1899 and April 1901 Lebesgue published six notes in Comptes Rendus. The first of these, unrelated to his development of Lebesgue integration, Lebesgues great thesis, Intégrale, longueur, aire, with the full account of this work, appeared in the Annali di Matematica in 1902. The first chapter develops the theory of measure, in the second chapter he defines the integral both geometrically and analytically. The next chapters expand the Comptes Rendus notes dealing with length, area, the final chapter deals mainly with Plateaus problem. This dissertation is considered to be one of the finest ever written by a mathematician and his lectures from 1902 to 1903 were collected into a Borel tract Leçons sur lintégration et la recherche des fonctions primitives. The problem of integration regarded as the search for a function is the keynote of the book. Lebesgue presents the problem of integration in its context, addressing Augustin-Louis Cauchy, Peter Gustav Lejeune Dirichlet

22.
Alan Turing
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Alan Mathison Turing OBE FRS was an English computer scientist, mathematician, logician, cryptanalyst and theoretical biologist. Turing is widely considered to be the father of computer science. During the Second World War, Turing worked for the Government Code and Cypher School at Bletchley Park, for a time he led Hut 8, the section responsible for German naval cryptanalysis. After the war, he worked at the National Physical Laboratory and he wrote a paper on the chemical basis of morphogenesis, and predicted oscillating chemical reactions such as the Belousov–Zhabotinsky reaction, first observed in the 1960s. Turing was prosecuted in 1952 for homosexual acts, when by the Labouchere Amendment and he accepted chemical castration treatment, with DES, as an alternative to prison. Turing died in 1954,16 days before his 42nd birthday, an inquest determined his death as suicide, but it has been noted that the known evidence is also consistent with accidental poisoning. In 2009, following an Internet campaign, British Prime Minister Gordon Brown made a public apology on behalf of the British government for the appalling way he was treated. Queen Elizabeth II granted him a pardon in 2013. The Alan Turing law is now a term for a 2017 law in the United Kingdom that retroactively pardons men cautioned or convicted under historical legislation that outlawed homosexual acts. Turings father was the son of a clergyman, the Rev. John Robert Turing, from a Scottish family of merchants that had based in the Netherlands. Turings mother, Julius wife, was Ethel Sara, daughter of Edward Waller Stoney, the Stoneys were a Protestant Anglo-Irish gentry family from both County Tipperary and County Longford, while Ethel herself had spent much of her childhood in County Clare. Julius work with the ICS brought the family to British India and he had an elder brother, John. At Hastings, Turing stayed at Baston Lodge, Upper Maze Hill, St Leonards-on-Sea, very early in life, Turing showed signs of the genius that he was later to display prominently. His parents purchased a house in Guildford in 1927, and Turing lived there during school holidays, the location is also marked with a blue plaque. Turings parents enrolled him at St Michaels, a day school at 20 Charles Road, St Leonards-on-Sea, the headmistress recognised his talent early on, as did many of his subsequent educators. From January 1922 to 1926, Turing was educated at Hazelhurst Preparatory School, in 1926, at the age of 13, he went on to Sherborne School, an independent school in the market town of Sherborne in Dorset. Turings natural inclination towards mathematics and science did not earn him respect from some of the teachers at Sherborne and his headmaster wrote to his parents, I hope he will not fall between two stools. If he is to stay at school, he must aim at becoming educated

23.
Hawking radiation
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Hawking radiation is blackbody radiation that is predicted to be released by black holes, due to quantum effects near the event horizon. Hawking radiation reduces the mass and energy of black holes and is also known as black hole evaporation. Because of this, black holes that do not gain mass through other means are expected to shrink, micro black holes are predicted to be larger net emitters of radiation than larger black holes and should shrink and dissipate faster. In June 2008, NASA launched the Fermi space telescope, which is searching for the terminal gamma-ray flashes expected from evaporating primordial black holes. In the event that speculative large extra dimension theories are correct, CERNs Large Hadron Collider may be able to create black holes. In September 2010, a signal that is related to black hole Hawking radiation was claimed to have been observed in a laboratory experiment involving optical light pulses. However, the results remain unverified and debatable, other projects have been launched to look for this radiation within the framework of analog gravity. Black holes are sites of immense gravitational attraction, classically, the gravitation is so powerful that nothing, not even electromagnetic radiation, can escape from the black hole. It is yet unknown how gravity can be incorporated into quantum mechanics, nevertheless, far from the black hole the gravitational effects can be weak enough for calculations to be reliably performed in the framework of quantum field theory in curved spacetime. Hawking showed that quantum effects allow black holes to emit exact black body radiation, the electromagnetic radiation is produced as if emitted by a black body with a temperature inversely proportional to the mass of the black hole. Physical insight into the process may be gained by imagining that particle–antiparticle radiation is emitted from just beyond the event horizon. This radiation does not come directly from the hole itself. As the particle–antiparticle pair was produced by the holes gravitational energy. An alternative view of the process is that vacuum fluctuations cause a particle–antiparticle pair to appear close to the event horizon of a black hole, one of the pair falls into the black hole while the other escapes. In order to preserve total energy, the particle that fell into the hole must have had a negative energy. This causes the black hole to lose mass, and, to an outside observer, in another model, the process is a quantum tunnelling effect, whereby particle–antiparticle pairs will form from the vacuum, and one will tunnel outside the event horizon. This leads to the black hole information paradox, however, according to the conjectured gauge-gravity duality, black holes in certain cases are equivalent to solutions of quantum field theory at a non-zero temperature. This means that no information loss is expected in black holes, if this is correct, then Hawkings original calculation should be corrected, though it is not known how

24.
Black-hole thermodynamics
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In physics, black hole thermodynamics is the area of study that seeks to reconcile the laws of thermodynamics with the existence of black hole event horizons. The second law of thermodynamics requires that black holes have entropy, if black holes carried no entropy, it would be possible to violate the second law by throwing mass into the black hole. The increase of the entropy of the hole more than compensates for the decrease of the entropy carried by the object that was swallowed. Starting from theorems proved by Stephen Hawking, Jacob Bekenstein conjectured that the black hole entropy was proportional to the area of its event horizon divided by the Planck area. In 1973 Bekenstein suggested /4 π as the constant of proportionality, asserting that if the constant was not exactly this, the next year in 1974, Hawking showed that black holes emit thermal Hawking radiation corresponding to a certain temperature. This is often referred to as the Bekenstein–Hawking formula, the subscript BH either stands for black hole or Bekenstein-Hawking. The black hole entropy is proportional to the area of its event horizon A, the fact that the black hole entropy is also the maximal entropy that can be obtained by the Bekenstein bound was the main observation that led to the holographic principle. In fact, so called no hair theorems appeared to suggest that black holes could have only a single microstate, various studies are in progress, but this has not yet been elucidated. In Loop quantum gravity it is possible to associate a geometrical interpretation to the microstates, LQG offers a geometric explanation of the finiteness of the entropy and of the proportionality of the area of the horizon. It is possible to derive, from the covariant formulation of quantum theory the correct relation between energy and area, the Unruh temperature and the distribution that yields Hawking entropy. The calculation makes use of the notion of dynamical horizon and is done for non-extremal black holes, there seems to be also discussed the calculation of Bekenstein-Hawking entropy from the point of view of LQG. The four laws of black hole mechanics are physical properties that black holes are believed to satisfy, the laws, analogous to the laws of thermodynamics, were discovered by Brandon Carter, Stephen Hawking, and James Bardeen. The laws of black hole mechanics are expressed in geometrized units, the horizon has constant surface gravity for a stationary black hole. The horizon area is, assuming the weak condition, a non-decreasing function of time. This law was superseded by Hawkings discovery that black holes radiate and it is not possible to form a black hole with vanishing surface gravity. κ =0 is not possible to achieve, the zeroth law is analogous to the zeroth law of thermodynamics which states that the temperature is constant throughout a body in thermal equilibrium. It suggests that the gravity is analogous to temperature. T constant for thermal equilibrium for a system is analogous to κ constant over the horizon of a stationary black hole

25.
A Brief History of Time
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A Brief History of Time, From the Big Bang to Black Holes is a popular-science book on cosmology by British physicist Stephen Hawking. It was first published in 1988, Hawking wrote the book for nonspecialist readers with no prior knowledge of scientific theories. He talks about basic concepts like space and time, basic building blocks that make up the universe and he writes about cosmological phenomena such as the Big Bang and the black holes. He discusses two major theories, general relativity and quantum mechanics, that scientists use to describe the universe. Finally, he talks about the search for a theory that describes everything in the universe in a coherent manner. The book became a bestseller and sold more than 10 million copies in 20 years and it was also on the London Sunday Times bestseller list for more than four years and was translated into 35 languages by 2001. Early in 1983, Hawking first approached Simon Mitton, the editor in charge of books at Cambridge University Press. Mitton was doubtful about all the equations in the draft manuscript, with some difficulty, he persuaded Hawking to drop all but one equation. The author himself notes in the books acknowledgements that he was warned that for every equation in the book, the book does employ a number of complex models, diagrams, and other illustrations to detail some of the concepts it explores. In A Brief History of Time, Stephen Hawking attempts to explain a range of subjects in cosmology, including the Big Bang, black holes and light cones and his main goal is to give an overview of the subject, but he also attempts to explain some complex mathematics. In the first chapter, Hawking discusses the history of studies, including the ideas of Aristotle. Aristotle, unlike other people of his time, thought that the Earth was round. Aristotle also thought that the sun and stars went around the Earth in perfect circles, second-century Greek astronomer Ptolemy also pondered the positions of the sun and stars in the universe and made a planetary model that described Aristotles thinking in more detail. Today, it is known that the opposite is true, the earth goes around the sun, the Aristotelian and Ptolemaic ideas about the position of the stars and sun were disproved in 1609. The first person to present an argument that the earth revolves around the sun was the Polish priest Nicholas Copernicus. To fit the observations, Kepler proposed an elliptical orbit model instead of a circular one, in his 1687 book on gravity, Principia Mathematica, Isaac Newton used complex mathematics to further support Copernicuss idea. Newtons model also meant that stars, like the sun, were not fixed but, rather, nevertheless, Newton believed that the universe was made up of an infinite number of stars which were more or less static. Many of his contemporaries, including German philosopher Heinrich Olbers, disagreed, the origin of the universe represented another great topic of study and debate over the centuries

26.
The Universe in a Nutshell
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The Universe in a Nutshell is a 2001 book about theoretical physics by Stephen Hawking. In it, he explains to a general audience various matters relating to the Lucasian professors work, such as Gödels Incompleteness Theorem and he tells the history and principles of modern physics. He seeks to combine Einsteins General Theory of Relativity and Richard Feynmans idea of multiple histories into one complete unified theory that will describe everything that happens in the universe, the Universe in a Nutshell is winner of the Aventis Prizes for Science Books 2002. It is generally considered a sequel and was created to update the public concerning developments since the multi-million-copy bestseller A Brief History of Time published in 1988, roger Penrose Kip Thorne Physical cosmology Positivism

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George's Secret Key to the Universe
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Georges Secret Key to the Universe is a 2007 childrens book written by Lucy and Stephen Hawking with Christophe Galfard. The book was followed by four sequels, Georges Cosmic Treasure Hunt in 2009, George and the Big Bang in 2011, George and the Unbreakable Code in 2014 and George and it is intended for readers aged 9 and up. The main characters in the book are Susan, Daisy, Terrance, George, Eric, Annie, Dr. Reeper, and Cosmos, Cosmos can draw windows allowing people to look into outer space, as well as doors which act as portals allowing travel into outer space. It is written like a story and aims to describe aspects of the universe in a manner that is accessible to children. It starts by describing atoms, stars, planets and their moons and it then goes on to describe black holes, which remains the topic of focus in the last part of the book. At frequent intervals throughout the book, there are pictures and fact files of the different references to universal objects, the books title was announced in June 2007, and was released on October 23,2007. The Independent gave the book a review, calling it an excellent book that will do wonders to raise enthusiasm for physics among young readers. It did, however, add that the storytelling has some rough edges, about. com gave the book 3½ out of 5, stating Recommended for kids, but not for adults. The reviewer from Kirkus Reviews was more critical, accusing the authors of setting aside the laws of physics whenever convenient to the story, the reviewer concluded that they expected the book to sell well, but that it doesn’t show much respect for its target audience. Common Sense Media gave the book 2 stars out of 5, stating The nonfiction parts are fine, good information, clearly told, but surprisingly, much of the fictional story isnt scientifically accurate. This might be forgivable in straight sci-fi or fantasy, but in a book that purports to teach the basics of astronomy and physics, its just confusing -- how are young readers to know whats true, whats theoretical, and whats just plain nonsense