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Method Of Least Squares
The method of least squares is a mathematical optimization technique that aims to determine the best fit function by minimizing the sum of the squares of the differences between the observed values and the predicted values of the model. The method is widely used in areas such as regression analysis, curve fitting and data modeling. The least squares method can be categorized into linear and nonlinear forms, depending on the relationship between the model parameters and the observed data. The method was first proposed by Adrien-Marie Legendre in 1805 and further developed by Carl Friedrich Gauss. History Founding The method of least squares grew out of the fields of astronomy and geodesy, as scientists and mathematicians sought to provide solutions to the challenges of navigating the Earth's oceans during the Age of Discovery. The accurate description of the behavior of celestial bodies was the key to enabling ships to sail in open seas, where sailors could no longer rely on la ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Linear Least Squares2
In mathematics, the term ''linear'' is used in two distinct senses for two different properties: * linearity of a ''function (mathematics), function'' (or ''mapping (mathematics), mapping''); * linearity of a ''polynomial''. An example of a linear function is the function defined by f(x)=(ax,bx) that maps the real line to a line in the Euclidean plane R2 that passes through the origin. An example of a linear polynomial in the variables X, Y and Z is aX+bY+cZ+d. Linearity of a mapping is closely related to ''Proportionality (mathematics), proportionality''. Examples in physics include the linear relationship of voltage and Electric current, current in an electrical conductor (Ohm's law), and the relationship of mass and weight. By contrast, more complicated relationships, such as between velocity and kinetic energy, are ''Nonlinear system, nonlinear''. Generalized for functions in more than one dimension (mathematics), dimension, linearity means the property of a function of b ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Bendixen - Carl Friedrich Gauß, 1828
Bendixen is a surname. Notable people with the surname include: * Aage Bendixen (1887–1973), Danish actor * Alfred Bendixen, American literatary scholar * Fanny Bendixen ( – 1899), Canadian gold-rush hotelier * Finn Bendixen (born 1949), Norwegian long-jumper * Friedrich Bendixen (1864–1920), German banker * Harry Bendixen (1901–1954), Danish footballer and journalist * Niclas Bendixen (born 1972), Danish theatre director * Ole Bendixen (1869–1958), Danish explorer * Ole Christian Bendixen (born 1947), Norwegian sailor * Ulla Bendixen, Danish folktronic musician {{surname Danish-language surnames ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Franz Xaver Von Zach
Baron Franz Xaver von Zach (''Franz Xaver Freiherr von Zach''; 4 June 1754 – 2 September 1832) was an Austrian astronomer born at Pest, Hungary (now Budapest in Hungary). Biography Zach studied physics at the Royal University of Pest, and served for some time in the Austrian army. He taught at the University of Lemberg (now Lviv, Ukraine) and worked in its observatory. He lived in Paris in 1780–83, and in London from 1783 to 1786 as tutor in the house of the Saxony, Saxon ambassador, Hans Moritz von Brühl. In Paris and London he entered the circles of astronomers like Jérôme Lalande, Pierre-Simon Laplace and William Herschel. In 1786 he was appointed by Ernest II, Duke of Saxe-Gotha-Altenburg director of the new Gotha Observatory, observatory on Seeberg hill at Gotha (town), Gotha, which was finished in 1791. At the close of the 18th century, he organised the first European congress of astronomers in 1798 and established the "celestial police", a group of twenty-four ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Kepler's Laws Of Planetary Motion
In astronomy, Kepler's laws of planetary motion, published by Johannes Kepler in 1609 (except the third law, which was fully published in 1619), describe the orbits of planets around the Sun. These laws replaced circular orbits and epicycles in the heliocentric theory of Nicolaus Copernicus with elliptical orbits and explained how planetary velocities vary. The three laws state that: # The orbit of a planet is an ellipse with the Sun at one of the two foci. # A line segment joining a planet and the Sun sweeps out equal areas during equal intervals of time. # The square of a planet's orbital period is proportional to the cube of the length of the semi-major axis of its orbit. The elliptical orbits of planets were indicated by calculations of the orbit of Mars. From this, Kepler inferred that other bodies in the Solar System, including those farther away from the Sun, also have elliptical orbits. The second law establishes that when a planet is closer to the Sun, it travels fa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Giuseppe Piazzi
Giuseppe Piazzi ( , ; 16 July 1746 – 22 July 1826) was an Italian Catholic Church, Catholic priest of the Theatines, Theatine order, mathematician, and astronomer. He established an observatory at Palermo, now the ''Palermo Astronomical Observatory, Osservatorio Astronomico di Palermo – Giuseppe S. Vaiana''. He is perhaps most famous for his discovery of the first dwarf planet, Ceres (dwarf planet), Ceres. Early life No documented account of Piazzi's scientific education exists in any of the astronomer’s biographies, even the earliest ones. However, it is certain that Piazzi pursued studies in Turin, likely attending lessons by Giovan Battista Beccaria. Between 1768 and 1770, he resided at the Theatines' Home in Sant'Andrea della Valle, Rome, where he studied mathematics under François Jacquier. In July 1770, Piazzi was appointed to the chair of Mathematics at the University of Malta. In December 1773, he moved to Ravenna, where he served as "prefetto degli student ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Ceres (dwarf Planet)
Ceres (minor-planet designation: 1 Ceres) is a dwarf planet in the middle main asteroid belt between the orbits of Mars and Jupiter. It was the first known asteroid, discovered on 1 January 1801 by Giuseppe Piazzi at Palermo Astronomical Observatory in Sicily, and announced as a new planet. Ceres was later classified as an asteroid and then a dwarf planet, the only one not beyond Neptune's orbit. Ceres's diameter is about a quarter that of the Moon. Its small size means that even at its brightest it is too dim to be seen by the naked eye, except under extremely dark skies. Its apparent magnitude ranges from 6.7 to 9.3, peaking at opposition (planets), opposition (when it is closest to Earth) once every 15- to 16-month synodic period. As a result, its surface features are barely visible even with the most powerful telescopes, and little was known about it until the robotic NASA spacecraft Dawn (spacecraft), ''Dawn'' approached Ceres for its orbital mission in 2015. ''Dawn'' fo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Gauss's Method
In orbital mechanics (a subfield of celestial mechanics), Gauss's method is used for preliminary orbit determination from at least three observations (more observations increases the accuracy of the determined orbit) of the orbiting body of interest at three different times. The required information are the times of observations, the position vectors of the observation points (in Equatorial Coordinate System), the direction cosine vector of the orbiting body from the observation points (from Topocentric Equatorial Coordinate System) and general physical data. Working in 1801, Carl Friedrich Gauss developed important mathematical techniques (summed up in Gauss's methods) which were specifically used to determine the orbit of Ceres. The method shown following is the orbit determination of an orbiting body about the focal body where the observations were taken from, whereas the method for determining Ceres' orbit requires a bit more effort because the observations were taken from Eart ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Probability Density
In probability theory, a probability density function (PDF), density function, or density of an absolutely continuous random variable, is a function whose value at any given sample (or point) in the sample space (the set of possible values taken by the random variable) can be interpreted as providing a '' relative likelihood'' that the value of the random variable would be equal to that sample. Probability density is the probability per unit length, in other words, while the ''absolute likelihood'' for a continuous random variable to take on any particular value is 0 (since there is an infinite set of possible values to begin with), the value of the PDF at two different samples can be used to infer, in any particular draw of the random variable, how much more likely it is that the random variable would be close to one sample compared to the other sample. More precisely, the PDF is used to specify the probability of the random variable falling ''within a particular range o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Arithmetic Mean
In mathematics and statistics, the arithmetic mean ( ), arithmetic average, or just the ''mean'' or ''average'' is the sum of a collection of numbers divided by the count of numbers in the collection. The collection is often a set of results from an experiment, an observational study, or a Survey (statistics), survey. The term "arithmetic mean" is preferred in some contexts in mathematics and statistics because it helps to distinguish it from other types of means, such as geometric mean, geometric and harmonic mean, harmonic. Arithmetic means are also frequently used in economics, anthropology, history, and almost every other academic field to some extent. For example, per capita income is the arithmetic average of the income of a nation's Human population, population. While the arithmetic mean is often used to report central tendency, central tendencies, it is not a robust statistic: it is greatly influenced by outliers (Value (mathematics), values much larger or smaller than ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Normal Distribution
In probability theory and statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real-valued random variable. The general form of its probability density function is f(x) = \frac e^\,. The parameter is the mean or expectation of the distribution (and also its median and mode), while the parameter \sigma^2 is the variance. The standard deviation of the distribution is (sigma). A random variable with a Gaussian distribution is said to be normally distributed, and is called a normal deviate. Normal distributions are important in statistics and are often used in the natural and social sciences to represent real-valued random variables whose distributions are not known. Their importance is partly due to the central limit theorem. It states that, under some conditions, the average of many samples (observations) of a random variable with finite mean and variance is itself a random variable—whose distribution c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Carl Friedrich Gauss
Johann Carl Friedrich Gauss (; ; ; 30 April 177723 February 1855) was a German mathematician, astronomer, geodesist, and physicist, who contributed to many fields in mathematics and science. He was director of the Göttingen Observatory and professor of astronomy from 1807 until his death in 1855. While studying at the University of Göttingen, he propounded several mathematical theorems. As an independent scholar, he wrote the masterpieces '' Disquisitiones Arithmeticae'' and ''Theoria motus corporum coelestium''. Gauss produced the second and third complete proofs of the fundamental theorem of algebra. In number theory, he made numerous contributions, such as the composition law, the law of quadratic reciprocity and the Fermat polygonal number theorem. He also contributed to the theory of binary and ternary quadratic forms, the construction of the heptadecagon, and the theory of hypergeometric series. Due to Gauss' extensive and fundamental contributions to science ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Prussia
Prussia (; ; Old Prussian: ''Prūsija'') was a Germans, German state centred on the North European Plain that originated from the 1525 secularization of the Prussia (region), Prussian part of the State of the Teutonic Order. For centuries, the House of Hohenzollern ruled Prussia, expanding its size with the Prussian Army. Prussia, with its capital at Königsberg and then, when it became the Kingdom of Prussia in 1701, History of Berlin, Berlin, decisively shaped the history of Germany. Prussia formed the German Empire when it united the German states in 1871. It was ''de facto'' dissolved by 1932 Prussian coup d'état, an emergency decree transferring powers of the Prussian government to German Chancellor Franz von Papen in 1932 and ''de jure'' by Abolition of Prussia, an Allied decree in 1947. The name ''Prussia'' derives from the Old Prussians who were conquered by the Teutonic Knightsan organized Catholic medieval Military order (religious society), military order of Pru ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |