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Pierre Frédéric Sarrus
Pierre Frédéric Sarrus (; 10 March 1798, Saint-Affrique – 20 November 1861) was a French mathematician. Sarrus was a professor at the University of Strasbourg, France (1826–1856) and a member of the French Academy of Sciences in Paris (1842). He is the author of several treatises, including one on the solution of numeric equations with multiple unknowns (1842); one on multiple integrals and their integrability conditions; and one on the determination of the orbits of the comets. He also discovered a mnemonic rule for solving the determinant of a 3-by-3 matrix, named Sarrus' scheme. Sarrus also demonstrated the fundamental lemma of the calculus of variations. Sarrus numbers are pseudoprime A pseudoprime is a probable prime (an integer that shares a property common to all prime numbers) that is not actually prime. Pseudoprimes are classified according to which property of primes they satisfy. Some sources use the term pseudoprime to ...s to base 2. Sarrus also develo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Saint-Affrique
Saint-Affrique (; Languedocien: ''Sant Africa'') is a commune in the Aveyron department in Southern France. History Saint-Affrique grew in the 6th century around the tomb of St. Africain, bishop of Comminges. In the 12th century a fortress was built on the neighboring rock of Caylus. The possession of Saint-Affrique was vigorously contested during the French Wars of Religion. It was eventually occupied by the Huguenots until 1629, when it was seized and dismantled by a royal army. Geography The Sorgues, a tributary of the Dourdou de Camarès, flows through the commune and crosses the town. The Dourdou de Camarès flows northwestward through the western part of the commune and forms part of its northwestern border. Population Sights An old bridge over the Sorgue and some megaliths in the neighborhood, especially, the dolmen of Tiergues, are of antiquarian interest. Personalities Saint-Affrique was the birthplace of: * Pierre Frédéric Sarrus (1798–1861), mathematici ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rule Of Sarrus
In linear algebra, the Rule of Sarrus is a mnemonic device for computing the determinant of a 3 \times 3 matrix named after the French mathematician Pierre Frédéric Sarrus. Consider a 3 \times 3 matrix :M=\begin a_ & a_ & a_ \\ a_ & a_ & a_ \\ a_ & a_ & a_ \end, then its determinant can be computed by the following scheme. Write out the first two columns of the matrix to the right of the third column, giving five columns in a row. Then add the products of the diagonals going from top to bottom (solid) and subtract the products of the diagonals going from bottom to top (dashed). This yieldsPaul Cohn: ''Elements of Linear Algebra''. CRC Press, 1994, p. 69/ref> : \begin \det(M) &= \det \begin a_ & a_ & a_ \\ a_ & a_ & a_ \\ a_ & a_ & a_ \end\\ pt &= a_a_a_+a_a_a_+a_a_a_-a_a_a_- a_a_a_-a_a_a_. \end A similar scheme based on diagonals works for 2 \times 2 matrices: :\det(M)= \det \begin a_ & a_ \\ a_ & a_ \end = a_a_ - a_a_{12}. Bot ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Linear Algebraists
Linearity is the property of a mathematical relationship (''function'') that can be graphically represented as a straight line. Linearity is closely related to '' proportionality''. Examples in physics include rectilinear motion, the linear relationship of voltage and current in an electrical conductor (Ohm's law), and the relationship of mass and weight. By contrast, more complicated relationships are ''nonlinear''. Generalized for functions in more than one dimension, linearity means the property of a function of being compatible with addition and scaling, also known as the superposition principle. The word linear comes from Latin ''linearis'', "pertaining to or resembling a line". In mathematics In mathematics, a linear map or linear function ''f''(''x'') is a function that satisfies the two properties: * Additivity: . * Homogeneity of degree 1: for all α. These properties are known as the superposition principle. In this definition, ''x'' is not necessarily a real nu ... [...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] |
Sarrus Linkage
The Sarrus linkage, invented in 1853 by Pierre Frédéric Sarrus, is a mechanical linkage to convert a limited circular motion to a linear motion or vice versa without reference guideways. It is a spatial six-bar linkage (6R) with two groups of three parallel adjacent joint-axes. Although Charles-Nicolas Peaucellier was widely recognized for being the first to invent such a straight-line mechanism, the Sarrus linkage had been invented earlier; however, it was largely unnoticed for a time. – Sarrus' mechanism Description The Sarrus linkage consists of four links in two identical groups that are perpendicular to each other, with all links having equal lengths. In the examples shown, the linkage uses two horizontal plates (cyan) ...[...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pseudoprime
A pseudoprime is a probable prime (an integer that shares a property common to all prime numbers) that is not actually prime. Pseudoprimes are classified according to which property of primes they satisfy. Some sources use the term pseudoprime to describe all probable primes, both composite numbers and actual primes. Pseudoprimes are of primary importance in public-key cryptography, which makes use of the difficulty of factoring large numbers into their prime factors. Carl Pomerance estimated in 1988 that it would cost $10 million to factor a number with 144 digits, and $100 billion to factor a 200-digit number (the cost today is dramatically lower but still prohibitively high). But finding two large prime numbers as needed for this use is also expensive, so various probabilistic primality tests are used, some of which in rare cases inappropriately deliver composite numbers instead of primes. On the other hand, deterministic primality tests, such as the AKS primality test, do not g ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fermat Pseudoprime
In number theory, the Fermat pseudoprimes make up the most important class of pseudoprimes that come from Fermat's little theorem. Definition Fermat's little theorem states that if ''p'' is prime and ''a'' is coprime to ''p'', then ''a''''p''−1 − 1 is divisible by ''p''. For an integer ''a'' > 1, if a composite integer ''x'' divides ''a''''x''−1 − 1, then ''x'' is called a Fermat pseudoprime to base ''a''. In other words, a composite integer is a Fermat pseudoprime to base ''a'' if it successfully passes the Fermat primality test for the base ''a''. The false statement that all numbers that pass the Fermat primality test for base 2, are prime, is called the Chinese hypothesis. The smallest base-2 Fermat pseudoprime is 341. It is not a prime, since it equals 11·31, but it satisfies Fermat's little theorem: 2340 ≡ 1 (mod 341) and thus passes the Fermat primality test for the base 2. Pseudoprimes to base 2 are sometimes called Sarrus numbers, afte ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Calculus Of Variations
The calculus of variations (or Variational Calculus) is a field of mathematical analysis that uses variations, which are small changes in functions and functionals, to find maxima and minima of functionals: mappings from a set of functions to the real numbers. Functionals are often expressed as definite integrals involving functions and their derivatives. Functions that maximize or minimize functionals may be found using the Euler–Lagrange equation of the calculus of variations. A simple example of such a problem is to find the curve of shortest length connecting two points. If there are no constraints, the solution is a straight line between the points. However, if the curve is constrained to lie on a surface in space, then the solution is less obvious, and possibly many solutions may exist. Such solutions are known as ''geodesics''. A related problem is posed by Fermat's principle: light follows the path of shortest optical length connecting two points, which depends up ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lemma (mathematics)
In mathematics, informal logic and argument mapping, a lemma (plural lemmas or lemmata) is a generally minor, proven proposition which is used as a stepping stone to a larger result. For that reason, it is also known as a "helping theorem" or an "auxiliary theorem". In many cases, a lemma derives its importance from the theorem it aims to prove; however, a lemma can also turn out to be more important than originally thought. The word "lemma" derives from the Ancient Greek ("anything which is received", such as a gift, profit, or a bribe). Comparison with theorem There is no formal distinction between a lemma and a theorem, only one of intention (see Theorem terminology). However, a lemma can be considered a minor result whose sole purpose is to help prove a more substantial theorem – a step in the direction of proof. Well-known lemmas A good stepping stone can lead to many others. Some powerful results in mathematics are known as lemmas, first named for their originally min ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Matrix (mathematics)
In mathematics, a matrix (plural matrices) is a rectangular array or table of numbers, symbols, or expressions, arranged in rows and columns, which is used to represent a mathematical object or a property of such an object. For example, \begin1 & 9 & -13 \\20 & 5 & -6 \end is a matrix with two rows and three columns. This is often referred to as a "two by three matrix", a "-matrix", or a matrix of dimension . Without further specifications, matrices represent linear maps, and allow explicit computations in linear algebra. Therefore, the study of matrices is a large part of linear algebra, and most properties and operations of abstract linear algebra can be expressed in terms of matrices. For example, matrix multiplication represents composition of linear maps. Not all matrices are related to linear algebra. This is, in particular, the case in graph theory, of incidence matrices, and adjacency matrices. ''This article focuses on matrices related to linear algebra, and, unle ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
French First Republic
In the history of France, the First Republic (french: Première République), sometimes referred to in historiography as Revolutionary France, and officially the French Republic (french: République française), was founded on 21 September 1792 during the French Revolution. The First Republic lasted until the declaration of the First Empire on 18 May 1804 under Napoléon Bonaparte, although the form of the government changed several times. This period was characterized by the fall of the monarchy, the establishment of the National Convention and the Reign of Terror, the Thermidorian Reaction and the founding of the Directory, and, finally, the creation Creation may refer to: Religion *''Creatio ex nihilo'', the concept that matter was created by God out of nothing * Creation myth, a religious story of the origin of the world and how people first came to inhabit it * Creationism, the belief tha ... of the French Consulate, Consulate and Napoleon's rise to power. End of the m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Determinant
In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if and only if the matrix is invertible and the linear map represented by the matrix is an isomorphism. The determinant of a product of matrices is the product of their determinants (the preceding property is a corollary of this one). The determinant of a matrix is denoted , , or . The determinant of a matrix is :\begin a & b\\c & d \end=ad-bc, and the determinant of a matrix is : \begin a & b & c \\ d & e & f \\ g & h & i \end= aei + bfg + cdh - ceg - bdi - afh. The determinant of a matrix can be defined in several equivalent ways. Leibniz formula expresses the determinant as a sum of signed products of matrix entries such that each summand is the product of different entries, and the number of these summands is n!, the factorial of (t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
