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Bernoulli Stochastics
Bernoulli can refer to: People *Bernoulli family of 17th and 18th century Swiss mathematicians: ** Daniel Bernoulli (1700–1782), developer of Bernoulli's principle **Jacob Bernoulli (1654–1705), also known as Jacques, after whom Bernoulli numbers are named ** Jacob II Bernoulli (1759–1789) **Johann Bernoulli (1667–1748) **Johann II Bernoulli (1710–1790) **Johann III Bernoulli (1744–1807), also known as Jean, astronomer **Nicolaus I Bernoulli (1687–1759) **Nicolaus II Bernoulli (1695–1726) * Elisabeth Bernoulli (1873–1935), Swiss temperance campaigner *Hans Benno Bernoulli (1876–1959), Swiss architect * Ludwig Bernoully (1873–1928), German architect Mathematics * Bernoulli differential equation * Bernoulli distribution and Bernoulli random variable * Bernoulli's inequality * Bernoulli's triangle * Bernoulli number * Bernoulli polynomials * Bernoulli process * Bernoulli trial * Lemniscate of Bernoulli Science * 2034 Bernoulli, minor planet * Bernoulli's princi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bernoulli Family
The Bernoulli family () of Basel was a 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 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 ''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 Spanish persecution of the Protestants. Jacob's grandson, a spice trader, also named Jacob, moved to Basel, Switzerland in 1620, and was granted citizenship in 1622. His son, (Nicolaus, 1623–1708), Leon's great-great-grandson, ma ... [...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: * (1 + x)^r \geq 1 + rx for every integer ''r'' ≥ 0 and real number ''x'' > −1. The inequality is strict if ''x'' ≠ 0 and ''r'' ≥ 2. * (1 + x)^r \geq 1 + rx for every even integer ''r'' ≥ 0 and every real number ''x''. * (1 + x)^r \geq 1 + rx for every integer ''r'' ≥ 0 and every real number ''x'' ≥ −2.Excluding the case and , or assuming that . * (1 + x)^r \geq 1 + rx for every real number ''r'' ≥ 1 and ''x'' ≥ −1. The inequalities are strict if ''x'' ≠ 0 and ''r'' ≠ 0, 1. * (1 + x)^r \leq 1 + rx for every real number 0 ≤ ''r'' ≤ 1 and ''x'' ≥ −1. History Jacob Bernoulli first ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Maniac Mansion
''Maniac Mansion'' is a 1987 graphic adventure video game developed and published by Lucasfilm Games. It follows teenage protagonist Dave Miller as he attempts to rescue his girlfriend Sandy Pantz from a mad scientist, whose mind has been enslaved by a sentient meteor. The player uses a point-and-click interface to guide Dave and two of his six playable friends through the scientist's mansion while solving puzzles and avoiding dangers. Gameplay is non-linear, and the game must be completed in different ways based on the player's choice of characters. Initially released for the Commodore 64 and Apple II, ''Maniac Mansion'' was Lucasfilm Games' first self-published product. The game was conceived in 1985 by Ron Gilbert and Gary Winnick, who sought to tell a comedic story based on horror film and B-movie clichés. They mapped out the project as a paper-and-pencil game before coding commenced. While earlier adventure titles had relied on command lines, Gilbert disliked such syst ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bernoulli Box
The Bernoulli Box (or simply Bernoulli, named after Daniel Bernoulli) is a high-capacity (for the time) removable floppy disk storage system that is Iomega's first widely known product. It was released in 1982. Overview The drive spins a PET film floppy disk at about 3000 rpm, 1 μm over a read-write head, using Bernoulli's principle to pull the flexible disk towards the head as long as the disk is spinning. In theory this makes the Bernoulli drive more reliable than a contemporary hard disk drive, since a head crash is impossible. The original Bernoulli disks came in capacities of 5, 10, and 20 MB. They are roughly 21 cm by 27.5 cm, similar to the size of a sheet of A4 paper. The most popular system was the Bernoulli Box II, whose disk cases are 13.6 cm wide, 14 cm long and 0.9 cm thick, somewhat resembling a 3 -inch standard floppy disk but in 5 -inch form factor. Bernoulli Box II disks came in the following capacities: 20 MB, 35&nbs ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Euler–Bernoulli Beam Theory
Euler–Bernoulli beam theory (also known as engineer's beam theory or classical beam theory) is a simplification of the linear theory of elasticity which provides a means of calculating the load-carrying and deflection characteristics of beams. It covers the case corresponding to small deflections of a beam that is subjected to lateral loads only. By ignoring the effects of shear deformation and rotatory inertia, it is thus a special case of Timoshenko–Ehrenfest beam theory. It was first enunciated circa 1750, but was not applied on a large scale until the development of the Eiffel Tower and the Ferris wheel in the late 19th century. Following these successful demonstrations, it quickly became a cornerstone of engineering and an enabler of the Second Industrial Revolution. Additional mathematical models have been developed, such as plate theory, but the simplicity of beam theory makes it an important tool in the sciences, especially structural and mechanical engineering. Hi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bernoulli (crater)
Bernoulli is a lunar impact crater that is located in the northeast part of the Moon. It lies to the south of the crater Messala, and east of Geminus. This formation is nearly circular with several slight outward bulges around the perimeter. There is a sunken depression along part of the southern wall, forming an outward triangular bulge in the rim. The rim is highest along the eastern side, climbing to 4 km. At the midpoint of the crater floor is a central peak formation. Satellite craters By convention these features are identified on lunar maps by placing the letter on the side of the crater midpoint that is closest to Bernoulli. See also * 2034 Bernoulli, minor planet * Bernoulli family The Bernoulli family () of Basel was a 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 period. ... References * * * * ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
2034 Bernoulli
2034 Bernoulli (), provisional designation , is a stony asteroid from the inner regions of the asteroid belt, approximately 9 kilometers in diameter. The asteroid was discovered on 5 March 1973, by Swiss astronomer Paul Wild at Zimmerwald Observatory near Bern, Switzerland, and named for the members of the Bernoulli family. Orbit and classification ''Bernoulli'' orbits the Sun in the inner main-belt at a distance of 1.8–2.7 AU once every 3 years and 4 months (1,230 days). Its orbit has an eccentricity of 0.18 and an inclination of 9 ° with respect to the ecliptic. The first used precovery was taken at Palomar Observatory in 1951, extending the asteroid's observation arc by 22 years prior to its official discovery, while the first unused observation was made ten years earlier at Uccle Observatory in 1941. Physical characteristics ''Bernoulli'' is an assumed, common, stony S-type asteroid. Rotation period A rotational lightcurve of ''Bernoulli'' was obtained fro ... [...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 (bisector o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bernoulli Trial
In the theory of probability and statistics, a Bernoulli trial (or binomial trial) is a random experiment with exactly two possible outcomes, "success" and "failure", in which the probability of success is the same every time the experiment is conducted. It is named after Jacob Bernoulli, a 17th-century Swiss mathematician, who analyzed them in his ''Ars Conjectandi'' (1713). The mathematical formalisation of the Bernoulli trial is known as the Bernoulli process. This article offers an elementary introduction to the concept, whereas the article on the Bernoulli process offers a more advanced treatment. Since a Bernoulli trial has only two possible outcomes, it can be framed as some "yes or no" question. For example: *Is the top card of a shuffled deck an ace? *Was the newborn child a girl? (See human sex ratio.) Therefore, success and failure are merely labels for the two outcomes, and should not be construed literally. The term "success" in this sense consists in the result ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bernoulli Process
In probability and statistics, a Bernoulli process (named after Jacob Bernoulli) is a finite or infinite sequence of binary random variables, so it is a discrete-time stochastic process that takes only two values, canonically 0 and 1. The component Bernoulli variables ''X''''i'' are identically distributed and independent. Prosaically, a Bernoulli process is a repeated coin flipping, possibly with an unfair coin (but with consistent unfairness). Every variable ''X''''i'' in the sequence is associated with a Bernoulli trial or experiment. They all have the same Bernoulli distribution. Much of what can be said about the Bernoulli process can also be generalized to more than two outcomes (such as the process for a six-sided die); this generalization is known as the Bernoulli scheme. The problem of determining the process, given only a limited sample of Bernoulli trials, may be called the problem of checking whether a coin is fair. Definition A Bernoulli process is a fini ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bernoulli Polynomials
In mathematics, the Bernoulli polynomials, named after Jacob Bernoulli, combine the Bernoulli numbers and binomial coefficients. They are used for series expansion of functions, and with the Euler–MacLaurin formula. These polynomials occur in the study of many special functions and, in particular, the Riemann zeta function and the Hurwitz zeta function. They are an Appell sequence (i.e. a Sheffer sequence for the ordinary derivative operator). For the Bernoulli polynomials, the number of crossings of the ''x''-axis in the unit interval does not go up with the degree. In the limit of large degree, they approach, when appropriately scaled, the sine and cosine functions. A similar set of polynomials, based on a generating function, is the family of Euler polynomials. Representations The Bernoulli polynomials ''B''''n'' can be defined by a generating function. They also admit a variety of derived representations. Generating functions The generating function for the Bernoulli ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bernoulli Number
In mathematics, the Bernoulli numbers are a sequence of rational numbers which occur frequently in analysis. The Bernoulli numbers appear in (and can be defined by) the Taylor series expansions of the tangent and hyperbolic tangent functions, in Faulhaber's formula for the sum of ''m''-th powers of the first ''n'' positive integers, in the Euler–Maclaurin formula, and in expressions for certain values of the Riemann zeta function. The values of the first 20 Bernoulli numbers are given in the adjacent table. Two conventions are used in the literature, denoted here by B^_n and B^_n; they differ only for , where B^_1=-1/2 and B^_1=+1/2. For every odd , . For every even , is negative if is divisible by 4 and positive otherwise. The Bernoulli numbers are special values of the Bernoulli polynomials B_n(x), with B^_n=B_n(0) and B^+_n=B_n(1). The Bernoulli numbers were discovered around the same time by the Swiss mathematician Jacob Bernoulli, after whom they are named, and indepe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
