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Sine-Gordon Equation
The sine-Gordon equation is a second-order nonlinear partial differential equation for a function \varphi dependent on two variables typically denoted x and t, involving the wave operator and the sine of \varphi. It was originally introduced by in the course of study of surfaces of constant negative curvature as the Gauss–Codazzi equation for surfaces of constant Gaussian curvature −1 in 3-dimensional space. The equation was rediscovered by in their study of crystal dislocations known as the Frenkel–Kontorova model. This equation attracted a lot of attention in the 1970s due to the presence of soliton solutions, and is an example of an integrable PDE. Among well-known integrable PDEs, the sine-Gordon equation is the only ''relativistic'' system due to its Lorentz invariance. Realizations of the sine-Gordon equation Differential geometry This is the first derivation of the equation, by Bour (1862). There are two equivalent forms of the sine-Gordon equation. I ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nonlinear Partial Differential Equation
In mathematics and physics, a nonlinear partial differential equation is a partial differential equation with nonlinear system, nonlinear terms. They describe many different physical systems, ranging from gravitation to fluid dynamics, and have been used in mathematics to solve problems such as the Poincaré conjecture and the Calabi conjecture. They are difficult to study: almost no general techniques exist that work for all such equations, and usually each individual equation has to be studied as a separate problem. The distinction between a linear and a nonlinear partial differential equation is usually made in terms of the properties of the Operator (mathematics), operator that defines the PDE itself. Methods for studying nonlinear partial differential equations Existence and uniqueness of solutions A fundamental question for any PDE is the existence and uniqueness of a solution for given boundary conditions. For nonlinear equations these questions are in general very hard: ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Arc Length
Arc length is the distance between two points along a section of a curve. Development of a formulation of arc length suitable for applications to mathematics and the sciences is a problem in vector calculus and in differential geometry. In the most basic formulation of arc length for a vector valued curve (thought of as the trajectory of a particle), the arc length is obtained by integrating speed, the magnitude of the velocity vector over the curve with respect to time. Thus the length of a continuously differentiable curve (x(t),y(t)), for a\le t\le b, in the Euclidean plane is given as the integral L = \int_a^b \sqrt\,dt, (because \sqrt is the magnitude of the velocity vector (x'(t),y'(t)), i.e., the particle's speed). The defining integral of arc length does not always have a closed-form expression, and numerical integration may be used instead to obtain numerical values of arc length. Determining the length of an irregular arc segment by approximating the arc segment as ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sine Gordon 5
In mathematics, sine and cosine are trigonometric functions of an angle. The sine and cosine of an acute angle are defined in the context of a right triangle: for the specified angle, its sine is the ratio of the length of the side opposite that angle to the length of the longest side of the triangle (the hypotenuse), and the cosine is the ratio of the length of the adjacent leg to that of the hypotenuse. For an angle \theta, the sine and cosine functions are denoted as \sin(\theta) and \cos(\theta). The definitions of sine and cosine have been extended to any real value in terms of the lengths of certain line segments in a unit circle. More modern definitions express the sine and cosine as infinite series, or as the solutions of certain differential equations, allowing their extension to arbitrary positive and negative values and even to complex numbers. The sine and cosine functions are commonly used to model periodic phenomena such as sound and light waves, the position and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lorentz Boost
In physics, the Lorentz transformations are a six-parameter family of linear transformations from a coordinate frame in spacetime to another frame that moves at a constant velocity relative to the former. The respective inverse transformation is then parameterized by the negative of this velocity. The transformations are named after the Dutch physicist Hendrik Lorentz. The most common form of the transformation, parametrized by the real constant v, representing a velocity confined to the -direction, is expressed as \begin t' &= \gamma \left( t - \frac \right) \\ x' &= \gamma \left( x - v t \right)\\ y' &= y \\ z' &= z \end where and are the coordinates of an event in two frames with the spatial origins coinciding at , where the primed frame is seen from the unprimed frame as moving with speed along the -axis, where is the speed of light, and \gamma = \frac is the Lorentz factor. When speed is much smaller than , the Lorentz factor is negligibly different from 1, but a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sophus Lie
Marius Sophus Lie ( ; ; 17 December 1842 – 18 February 1899) was a Norwegian mathematician. He largely created the theory of continuous symmetry and applied it to the study of geometry and differential equations. He also made substantial contributions to the development of algebra. Life and career Marius Sophus Lie was born on 17 December 1842 in the small town of Nordfjordeid. He was the youngest of six children born to Lutheran pastor Johann Herman Lie and his wife, who came from a well-known Trondheim family. He had his primary education in the south-eastern coast of Moss, before attending high school in Oslo (known then as Christiania). After graduating from high school, his ambition towards a military career was dashed when the army rejected him due to poor eyesight. He then enrolled at the University of Christiania. Sophus Lie's first mathematical work, ''Repräsentation der Imaginären der Plangeometrie'', was published in 1869 by the Academy of Sciences in Chris ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Squeeze Mapping
In linear algebra, a squeeze mapping, also called a squeeze transformation, is a type of linear map that preserves Euclidean area of regions in the Cartesian plane, but is ''not'' a rotation (mathematics), rotation or shear mapping. For a fixed positive real number , the mapping :(x, y) \mapsto (ax, y/a) is the ''squeeze mapping'' with parameter . Since :\ is a hyperbola, if and , then and the points of the image of the squeeze mapping are on the same hyperbola as is. For this reason it is natural to think of the squeeze mapping as a hyperbolic rotation, as did Émile Borel in 1914, by analogy with ''circular rotations'', which preserve circles. Logarithm and hyperbolic angle The squeeze mapping sets the stage for development of the concept of logarithms. The problem of finding the area bounded by a hyperbola (such as is one of quadrature (mathematics), quadrature. The solution, found by Grégoire de Saint-Vincent and Alphonse Antonio de Sarasa in 1647, required the natura ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bäcklund Transformation
Backlund is a Swedish surname. Notable people with the surname include: * Albert Victor Bäcklund (1845-1922), mathematician * Bengt Backlund (1926–2006), Swedish flatwater canoer * Bob Backlund (born 1949), American professional wrestler * Filip Backlund (born 1990), Swedish motorcycle road racer * Göran Backlund (born 1957), Swedish sprint canoer * Gordon Backlund (born 1940), American politician and electrical engineer * Gösta Backlund (1893—1918), Swedish footballer * Gotthard Backlund, Swedish chess master * Ivar Backlund (1892—1969), Swedish officer * Johan Backlund (born 1981), Swedish ice hockey goaltender * Jukka Backlund (born 1982), Finnish music producer * Kaj Backlund (1945–2013), Finnish jazz trumpeter, composer, and bandleader * Mikael Backlund (born 1989), Swedish ice hockey player * Nils Backlund (1896–1964), Swedish water polo player * Oskar Backlund (1846–1916), Swedish-Russian astronomer * Sven Backlund (1917–1997), Swedish diplomat * Victor L. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Albert Victor Bäcklund
Albert Victor Bäcklund (11 January 1845 – 23 February 1922) was a Sweden, Swedish mathematician and physicist. He was a professor at Lund University and its Rector (academia), rector from 1907 to 1909. He was born in Malmöhus County, now Skåne County, in southern Sweden and became a student at the nearby University of Lund in 1861. In 1864 he started to work for the observatory, and received his Ph.D. in 1868 working on methods to obtain the latitude of a place by astronomical observations. He was awarded the title of associate professor in Mechanics and Mathematical Physics in 1878, and elected Fellow of the Swedish Academy of Science in 1888. In 1897 he became a full professor. In 1874 he was awarded a travel grant to study abroad for six months. He went to the universities at Leipzig and Erlangen to work with Felix Klein and Ferdinand von Lindemann. His most important work was in the field of transformations pioneered by Sophus Lie. He improved greatly the understanding o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Luigi Bianchi
Luigi Bianchi (18 January 1856 – 6 June 1928) was an Italian mathematician. He was born in Parma, Emilia-Romagna, and died in Pisa. He was a leading member of the vigorous geometric school which flourished in Italy during the later years of the 19th century and the early years of the twentieth century. Biography Like his friend and colleague Gregorio Ricci-Curbastro, Bianchi studied at the Scuola Normale Superiore in Pisa under Enrico Betti, a leading differential geometer who is today best remembered for his seminal contributions to topology, and Ulisse Dini, a leading expert on function theory. Bianchi was also greatly influenced by the geometrical ideas of Bernhard Riemann and by the work on transformation groups of Sophus Lie and Felix Klein. Bianchi became a professor at the Scuola Normale Superiore in Pisa in 1896, where he spent the remainder of his career. At Pisa, his colleagues included the talented Ricci. In 1890, Bianchi and Dini supervised the dissertation of t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Deforming A Pseudosphere To Dini's Surface
Deformation can refer to: * Deformation (engineering), changes in an object's shape or form due to the application of a force or forces. ** Deformation (physics), such changes considered and analyzed as displacements of continuum bodies. * Deformation (meteorology), a measure of the rate at which the shapes of clouds and other fluid bodies change. * Deformation (mathematics), the study of conditions leading to slightly different solutions of mathematical equations, models and problems. * Deformation (volcanology), a measure of the rate at which the shapes of volcanoes change. * Deformation (biology), a harmful mutation or other deformation in an organism. See also * Deformity (medicine), a major difference in the shape of a body part or organ compared to its common or average shape. * Plasticity (physics) In physics and materials science, plasticity (also known as plastic deformation) is the ability of a solid material to undergo permanent Deformation (engineering), defo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rigid Transformation
In mathematics, a rigid transformation (also called Euclidean transformation or Euclidean isometry) is a geometric transformation of a Euclidean space that preserves the Euclidean distance between every pair of points. The rigid transformations include rotations, translations, reflections, or any sequence of these. Reflections are sometimes excluded from the definition of a rigid transformation by requiring that the transformation also preserve the handedness of objects in the Euclidean space. (A reflection would not preserve handedness; for instance, it would transform a left hand into a right hand.) To avoid ambiguity, a transformation that preserves handedness is known as a rigid motion, a Euclidean motion, or a proper rigid transformation. In dimension two, a rigid motion is either a translation or a rotation. In dimension three, every rigid motion can be decomposed as the composition of a rotation and a translation, and is thus sometimes called a rototranslation. In ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hilbert's Theorem (differential Geometry)
In differential geometry, Hilbert's theorem (1901) states that there exists no complete Smooth surface, regular surface S of constant negative gaussian curvature K immersion (mathematics), immersed in \mathbb^. This theorem answers the question for the negative case of which surfaces in \mathbb^ can be obtained by isometrically immersing complete manifolds with constant curvature. History * Hilbert's theorem was first treated by David Hilbert in "Über Flächen von konstanter Krümmung" (Trans. Amer. Math. Soc. 2 (1901), 87–99). * A different proof was given shortly after by E. Holmgren in "Sur les surfaces à courbure constante négative" (1902). * A far-leading generalization was obtained by Nikolai Efimov in 1975.Ефимов, Н. В. Непогружаемость полуплоскости Лобачевского. Вестн. МГУ. Сер. мат., мех. — 1975. — No 2. — С. 83—86. Proof The proof (mathematics), proof of Hilbert's theorem is elaborate and re ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
