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Jacob Bernoulli
Jacob Bernoulli (also known as James in English or Jacques in French; – 16 August 1705) was a Swiss mathematician. He sided with Gottfried Wilhelm Leibniz during the Leibniz–Newton calculus controversy and was an early proponent of Leibnizian calculus, to which he made numerous contributions. A member of the Bernoulli family, he, along with his brother Johann, was one of the founders of the calculus of variations. He also discovered the fundamental mathematical constant . However, his most important contribution was in the field of probability, where he derived the first version of the law of large numbers in his work '' Ars Conjectandi''.Jacob (Jacques) Bernoulli [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Basel
Basel ( ; ), also known as Basle ( ), ; ; ; . is a city in northwestern Switzerland on the river Rhine (at the transition from the High Rhine, High to the Upper Rhine). Basel is Switzerland's List of cities in Switzerland, third-most-populous city (after Zurich and Geneva), with 177,595 inhabitants within the city municipality limits. The official language of Basel is Swiss Standard German and the main spoken language is the local Basel German dialect. Basel is commonly considered to be the cultural capital of Switzerland and the city is famous for its many Museums in Basel, museums, including the Kunstmuseum Basel, Kunstmuseum, which is the first collection of art accessible to the public in the world (1661) and the largest museum of Swiss art, art in Switzerland, the Fondation Beyeler (located in Riehen), the Museum Tinguely and the Museum of Contemporary Art (Basel), Museum of Contemporary Art, which is the first public museum of contemporary art in Europe. Forty museums ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bernoulli Scheme
In mathematics, the Bernoulli scheme or Bernoulli shift is a generalization of the Bernoulli process to more than two possible outcomes. Bernoulli schemes appear naturally in symbolic dynamics, and are thus important in the study of dynamical systems. Many important dynamical systems (such as Axiom A systems) exhibit a repellor that is the product of the Cantor set and a smooth manifold, and the dynamics on the Cantor set are isomorphic to that of the Bernoulli shift. This is essentially the Markov partition. The term ''shift'' is in reference to the shift operator, which may be used to study Bernoulli schemes. The Ornstein isomorphism theorem shows that Bernoulli shifts are isomorphic when their entropy is equal. Definition A Bernoulli scheme is a discrete-time stochastic process where each independent random variable may take on one of ''N'' distinct possible values, with the outcome ''i'' occurring with probability p_i, with ''i'' = 1, ..., ''N'', and : ... [...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 Function (mathematics), functions and functional (mathematics), functionals, to find maxima and minima of functionals: Map (mathematics), mappings from a set of Function (mathematics), 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: li ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bernoulli Family
The Bernoulli family ( ; ; ) of Basel was a Patrician (post-Roman Europe), patrician family, notable for having produced eight mathematically gifted academics who, among them, contributed substantially to the development of mathematics and physics during the Early Modern Switzerland, early modern period. History Originally from Antwerp, a branch of the family relocated to Basel in 1620. While their origin in Antwerp is certain, proposed earlier connections with the Dutch family of Italian ancestry called ''Bornouilla'' (''Bernoullie''), or with the Castilian family ''de Bernuy'' (''Bernoille'', ''Bernouille''), are uncertain. The first known member of the family was Leon Bernoulli (d. 1561), a doctor in Antwerp, at that time part of the Spanish Netherlands. His son, Jacob, emigrated to Frankfurt am Main in 1570 to escape from the Inquisition in the Netherlands, Spanish persecution of the Protestants. Jacob's grandson, a spice trader, also named Jacob, moved to Basel, Switzerlan ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Calculus
Calculus is the mathematics, mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizations of arithmetic operations. Originally called infinitesimal calculus or "the calculus of infinitesimals", it has two major branches, differential calculus and integral calculus. The former concerns instantaneous Rate of change (mathematics), rates of change, and the slopes of curves, while the latter concerns accumulation of quantities, and areas under or between curves. These two branches are related to each other by the fundamental theorem of calculus. They make use of the fundamental notions of convergence (mathematics), convergence of infinite sequences and Series (mathematics), infinite series to a well-defined limit (mathematics), limit. It is the "mathematical backbone" for dealing with problems where variables change with time or another reference variable. Infinitesimal calculus was formulated separately ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Leibniz–Newton Calculus Controversy
In the history of calculus, the calculus controversy () was an argument between mathematicians Isaac Newton and Gottfried Wilhelm Leibniz over who had first discovered calculus. The question was a major intellectual controversy, beginning in 1699 and reaching its peak in 1712. Leibniz had published his work on calculus first, but Newton's supporters accused Leibniz of plagiarizing Newton's unpublished ideas. The modern consensus is that the two men independently developed their ideas. Their creation of calculus has been called "the greatest advance in mathematics that had taken place since the time of Archimedes." Newton stated he had begun working on a form of calculus (which he called " The Method of Fluxions and Infinite Series") in 1666, at the age of 23, but did not publish it until 1737 as a minor annotation in the back of one of his works decades later (a relevant Newton manuscript of October 1666 is now published among his mathematical papers). Gottfried Leibniz began wo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lemniscate Of Bernoulli
In geometry, the lemniscate of Bernoulli is a plane curve defined from two given points and , known as foci, at distance from each other as the locus of points so that . The curve has a shape similar to the numeral 8 and to the ∞ symbol. Its name is from , which is Latin for "decorated with hanging ribbons". It is a special case of the Cassini oval and is a rational algebraic curve of degree 4. This lemniscate was first described in 1694 by Jakob Bernoulli as a modification of an ellipse, which is the locus of points for which the sum of the distances to each of two fixed ''focal points'' is a constant. A Cassini oval, by contrast, is the locus of points for which the ''product'' of these distances is constant. In the case where the curve passes through the point midway between the foci, the oval is a lemniscate of Bernoulli. This curve can be obtained as the inverse transform of a hyperbola, with the inversion circle centered at the center of the hyperbola (bisect ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bernoulli's Inequality
In mathematics, Bernoulli's inequality (named after Jacob Bernoulli) is an inequality that approximates exponentiations of 1+x. It is often employed in real analysis. It has several useful variants: Integer exponent * Case 1: (1 + x)^r \geq 1 + rx for every integer r\geq 1 and real number x\geq-1. The inequality is strict if x\neq 0 and r\geq 2. * Case 2: (1 + x)^r \geq 1 + rx for every integer r\geq 0 and every real number x\geq -2. * Case 3: (1 + x)^r \geq 1 + rx for every even integer r\geq 0 and every real number x. Real exponent * (1 + x)^r \geq 1 + rx for every real number r\geq 1 and x\geq -1. The inequality is strict if x\neq 0 and r\neq 1. * (1 + x)^r \leq 1 + rx for every real number 0\leq r\leq 1 and x\geq -1. History Jacob Bernoulli first published the inequality in his treatise "Positiones Arithmeticae de Seriebus Infinitis" (Basel, 1689), where he used the inequality often. According to Joseph E. Hofmann, Über die Exercitatio Geometrica des M. A. Ricci (1963 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bernoulli's Golden Theorem
In probability theory, the law of large numbers is a Law (mathematics), mathematical law that states that the average of the results obtained from a large number of independent random samples converges to the true value, if it exists. More formally, the law of large numbers states that given a sample of independent and identically distributed values, the Sample mean and covariance, sample mean converges to the true mean. The law of large numbers is important because it guarantees stable long-term results for the averages of some Randomness, random Event (probability theory), events. For example, while a casino may lose money in a single spin of the roulette wheel, its earnings will tend towards a predictable percentage over a large number of spins. Any winning streak by a player will eventually be overcome by the parameters of the game. Importantly, the law applies (as the name indicates) only when a ''large number'' of observations are considered. There is no principle that ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bernoulli Random Variable
In probability theory and statistics, the Bernoulli distribution, named after Swiss mathematician Jacob Bernoulli, is the discrete probability distribution of a random variable which takes the value 1 with probability p and the value 0 with probability q = 1-p. Less formally, it can be thought of as a model for the set of possible outcomes of any single experiment that asks a yes–no question. Such questions lead to outcomes that are Boolean-valued: a single bit whose value is success/yes/true/one with probability ''p'' and failure/no/ false/zero with probability ''q''. It can be used to represent a (possibly biased) coin toss where 1 and 0 would represent "heads" and "tails", respectively, and ''p'' would be the probability of the coin landing on heads (or vice versa where 1 would represent tails and ''p'' would be the probability of tails). In particular, unfair coins would have p \neq 1/2. The Bernoulli distribution is a special case of the binomial distribution where a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
