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Jacques Philippe Marie Binet
Jacques Philippe Marie Binet (; 2 February 1786 – 12 May 1856) was a French mathematician, physicist and astronomer born in Rennes; he died in Paris, France, in 1856. He made significant contributions to number theory, and the mathematical foundations of matrix algebra which would later lead to important contributions by Cayley and others. In his memoir on the theory of the conjugate axis and of the moment of inertia of bodies he enumerated the principle now known as ''Binet's theorem''. He is also recognized as the first to describe the rule for multiplying matrices in 1812, and ''Binet's formula'' expressing Fibonacci numbers in closed form is named in his honour, although the same result was known to Abraham de Moivre a century earlier. Career Binet graduated from l'École Polytechnique in 1806, and returned as a teacher in 1807. He advanced in position until 1816 when he became an inspector of studies at l'École. He held this post until 13 November 1830, when he was dismi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
King Louis-Philippe
Louis Philippe (6 October 1773 – 26 August 1850) was King of the French from 1830 to 1848, and the penultimate monarch of France. As Louis Philippe, Duke of Chartres, he distinguished himself commanding troops during the Revolutionary Wars and was promoted to lieutenant general by the age of nineteen, but he broke with the Republic over its decision to execute King Louis XVI. He fled to Switzerland in 1793 after being connected with a plot to restore France's monarchy. His father Louis Philippe II, Duke of Orléans (Philippe Égalité) fell under suspicion and was executed during the Reign of Terror. Louis Philippe remained in exile for 21 years until the Bourbon Restoration. He was proclaimed king in 1830 after his cousin Charles X was forced to abdicate by the July Revolution (and because of the Spanish renounciation). The reign of Louis Philippe is known as the July Monarchy and was dominated by wealthy industrialists and bankers. He followed conservative policies, e ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
19th-century French Mathematicians
The 19th (nineteenth) century began on 1 January 1801 ( MDCCCI), and ended on 31 December 1900 ( MCM). The 19th century was the ninth century of the 2nd millennium. The 19th century was characterized by vast social upheaval. Slavery was abolished in much of Europe and the Americas. The First Industrial Revolution, though it began in the late 18th century, expanding beyond its British homeland for the first time during this century, particularly remaking the economies and societies of the Low Countries, the Rhineland, Northern Italy, and the Northeastern United States. A few decades later, the Second Industrial Revolution led to ever more massive urbanization and much higher levels of productivity, profit, and prosperity, a pattern that continued into the 20th century. The Islamic gunpowder empires fell into decline and European imperialism brought much of South Asia, Southeast Asia, and almost all of Africa under colonial rule. It was also marked by the collapse of the large ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Matrix Multiplication
In mathematics, particularly in linear algebra, matrix multiplication is a binary operation that produces a matrix from two matrices. For matrix multiplication, the number of columns in the first matrix must be equal to the number of rows in the second matrix. The resulting matrix, known as the matrix product, has the number of rows of the first and the number of columns of the second matrix. The product of matrices and is denoted as . Matrix multiplication was first described by the French mathematician Jacques Philippe Marie Binet in 1812, to represent the composition of linear maps that are represented by matrices. Matrix multiplication is thus a basic tool of linear algebra, and as such has numerous applications in many areas of mathematics, as well as in applied mathematics, statistics, physics, economics, and engineering. Computing matrix products is a central operation in all computational applications of linear algebra. Notation This article will use the following notati ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cauchy–Binet Formula
In mathematics, specifically linear algebra, the Cauchy–Binet formula, named after Augustin-Louis Cauchy and Jacques Philippe Marie Binet, is an identity for the determinant of the product of two rectangular matrices of transpose shapes (so that the product is well-defined and square). It generalizes the statement that the determinant of a product of square matrices is equal to the product of their determinants. The formula is valid for matrices with the entries from any commutative ring. Statement Let ''A'' be an ''m''×''n'' matrix and ''B'' an ''n''×''m'' matrix. Write 'n''for the set , and \tbinomm for the set of ''m''-combinations of 'n''(i.e., subsets of 'n''of size ''m''; there are \tbinom nm of them). For S\in\tbinomm, write ''A'' 'm''''S'' for the ''m''×''m'' matrix whose columns are the columns of ''A'' at indices from ''S'', and ''B''''S'', 'm''/sub> for the ''m''×''m'' matrix whose rows are the rows of ''B'' at indices from ''S''. The ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Binet Equation
The Binet equation, derived by Jacques Philippe Marie Binet, provides the form of a central force given the shape of the orbital motion in plane polar coordinates. The equation can also be used to derive the shape of the orbit for a given force law, but this usually involves the solution to a second order nonlinear ordinary differential equation. A unique solution is impossible in the case of circular motion about the center of force. Equation The shape of an orbit is often conveniently described in terms of relative distance r as a function of angle \theta . For the Binet equation, the orbital shape is instead more concisely described by the reciprocal u=1/r as a function of \theta. Define the specific angular momentum as h=L/m where L is the angular momentum and m is the mass. The Binet equation, derived in the next section, gives the force in terms of the function u(\theta) : :F(^)=-mh^u^\left(\frac+u\right). Derivation Newton's Second Law for a purely central force is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Binet–Cauchy Identity
In algebra, the Binet–Cauchy identity, named after Jacques Philippe Marie Binet and Augustin-Louis Cauchy, states that \left(\sum_^n a_i c_i\right) \left(\sum_^n b_j d_j\right) = \left(\sum_^n a_i d_i\right) \left(\sum_^n b_j c_j\right) + \sum_ (a_i b_j - a_j b_i ) (c_i d_j - c_j d_i ) for every choice of real or complex numbers (or more generally, elements of a commutative ring). Setting and , it gives Lagrange's identity, which is a stronger version of the Cauchy–Schwarz inequality for the Euclidean space \R^n. The Binet-Cauchy identity is a special case of the Cauchy–Binet formula for matrix determinants. The Binet–Cauchy identity and exterior algebra When , the first and second terms on the right hand side become the squared magnitudes of dot and cross products respectively; in dimensions these become the magnitudes of the dot and wedge products. We may write it (a \cdot c)(b \cdot d) = (a \cdot d)(b \cdot c) + (a \wedge b) \cdot (c \wedge d) where , , , and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Closed-form Expression
In mathematics, a closed-form expression is a mathematical expression that uses a finite number of standard operations. It may contain constants, variables, certain well-known operations (e.g., + − × ÷), and functions (e.g., ''n''th root, exponent, logarithm, trigonometric functions, and inverse hyperbolic functions), but usually no limit, differentiation, or integration. The set of operations and functions may vary with author and context. Example: roots of polynomials The solutions of any quadratic equation with complex coefficients can be expressed in closed form in terms of addition, subtraction, multiplication, division, and square root extraction, each of which is an elementary function. For example, the quadratic equation :ax^2+bx+c=0, is tractable since its solutions can be expressed as a closed-form expression, i.e. in terms of elementary functions: :x=\frac. Similarly, solutions of cubic and quartic (third and fourth degree) equations can be expresse ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fibonacci Sequence
In mathematics, the Fibonacci numbers, commonly denoted , form a integer sequence, sequence, the Fibonacci sequence, in which each number is the sum of the two preceding ones. The sequence commonly starts from 0 and 1, although some authors start the sequence from 1 and 1 or sometimes (as did Fibonacci) from 1 and 2. Starting from 0 and 1, the first few values in the sequence are: :0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144. The Fibonacci numbers were first described in Indian mathematics, as early as 200 BC in work by Pingala on enumerating possible patterns of Sanskrit poetry formed from syllables of two lengths. They are named after the Italian mathematician Leonardo of Pisa, later known as Fibonacci, who introduced the sequence to Western European mathematics in his 1202 book ''Liber Abaci''. Fibonacci numbers appear unexpectedly often in mathematics, so much so that there is an entire journal dedicated to their study, the ''Fibonacci Quarterly''. Applications of Fibonacci ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Académie Des Sciences
The French Academy of Sciences (French: ''Académie des sciences'') is a learned society, founded in 1666 by Louis XIV at the suggestion of Jean-Baptiste Colbert, to encourage and protect the spirit of French scientific research. It was at the forefront of scientific developments in Europe in the 17th and 18th centuries, and is one of the earliest Academies of Sciences. Currently headed by Patrick Flandrin (President of the Academy), it is one of the five Academies of the Institut de France. History The Academy of Sciences traces its origin to Colbert's plan to create a general academy. He chose a small group of scholars who met on 22 December 1666 in the King's library, near the present-day Bibliothèque Nationals, and thereafter held twice-weekly working meetings there in the two rooms assigned to the group. The first 30 years of the Academy's existence were relatively informal, since no statutes had as yet been laid down for the institution. In contrast to its British ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Légion D'Honneur
The National Order of the Legion of Honour (french: Ordre national de la Légion d'honneur), formerly the Royal Order of the Legion of Honour ('), is the highest French order of merit, both military and civil. Established in 1802 by Napoleon Bonaparte, it has been retained (with occasional slight alterations) by all later French governments and regimes. The order's motto is ' ("Honour and Fatherland"); its seat is the Palais de la Légion d'Honneur next to the Musée d'Orsay, on the left bank of the Seine in Paris. The order is divided into five degrees of increasing distinction: ' (Knight), ' (Officer), ' (Commander), ' (Grand Officer) and ' (Grand Cross). History Consulate During the French Revolution, all of the French orders of chivalry were abolished and replaced with Weapons of Honour. It was the wish of Napoleon Bonaparte, the First Consul, to create a reward to commend civilians and soldiers. From this wish was instituted a , a body of men that was not an order of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Knight
A knight is a person granted an honorary title of knighthood by a head of state (including the Pope) or representative for service to the monarch, the church or the country, especially in a military capacity. Knighthood finds origins in the Greek ''hippeis'' and '' hoplite'' (ἱππεῖς) and Roman '' eques'' and ''centurion'' of classical antiquity. In the Early Middle Ages in Europe, knighthood was conferred upon mounted warriors. During the High Middle Ages, knighthood was considered a class of lower nobility. By the Late Middle Ages, the rank had become associated with the ideals of chivalry, a code of conduct for the perfect courtly Christian warrior. Often, a knight was a vassal who served as an elite fighter or a bodyguard for a lord, with payment in the form of land holdings. The lords trusted the knights, who were skilled in battle on horseback. Knighthood in the Middle Ages was closely linked with horsemanship (and especially the joust) from its origins in th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
