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Twist (mathematics)
In differential geometry Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra and multili ..., the twist of a ''Ribbon (mathematics), ribbon'' is its rate of change (mathematics), rate of axial rotation. Let a ribbon (X,U) be composted of space curve X=X(s), where s is the arc length of X, and U=U(s) the a unit normal vector, perpendicular at each point to X. Since the ribbon (X,U) has edges X and X'=X+\varepsilon U, the twist (or ''total twist number'') Tw measures the average winding number, winding of the edge curve X' around and along the axial curve X. According to Love (1944) twist is defined by : Tw = \dfrac \int \left( U \times \dfrac \right) \cdot \dfrac ds \; , where dX/ds is the unit tangent vector to X. The total twist number Tw can be decomposed (Moffatt & Ricca 1992) into ''normalized ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Differential Geometry
Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra and multilinear algebra. The field has its origins in the study of spherical geometry as far back as antiquity. It also relates to astronomy, the geodesy of the Earth, and later the study of hyperbolic geometry by Lobachevsky. The simplest examples of smooth spaces are the plane and space curves and surfaces in the three-dimensional Euclidean space, and the study of these shapes formed the basis for development of modern differential geometry during the 18th and 19th centuries. Since the late 19th century, differential geometry has grown into a field concerned more generally with geometric structures on differentiable manifolds. A geometric structure is one which defines some notion of size, distance, shape, volume, or other rigidifying structu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hydrodynamical Helicity
:''This page is about helicity in fluid dynamics. For helicity of magnetic fields, see magnetic helicity. For helicity in particle physics, see helicity (particle physics).'' In fluid dynamics, helicity is, under appropriate conditions, an invariant of the Euler equations of fluid flow, having a topological interpretation as a measure of linkage and/or knottedness of vortex lines in the flow. This was first proved by Jean-Jacques Moreau in 1961 and Moffatt derived it in 1969 without the knowledge of Moreau's paper. This helicity invariant is an extension of Woltjer's theorem for magnetic helicity. Let \mathbf(x,t) be the velocity field and \nabla\times\mathbf the corresponding vorticity field. Under the following three conditions, the vortex lines are transported with (or 'frozen in') the flow: (i) the fluid is inviscid; (ii) either the flow is incompressible (\nabla\cdot\mathbf = 0), or it is compressible with a barotropic relation p = p(\rho) between pressure p and density \rh ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Renzo L
Renzo, the diminutive of Lorenzo, is an Italian masculine given name and a surname. Given name Notable people named Renzo include the following: *Renzo Alverà (1933–2005), Italian bobsledder *Renzo Arbore (born 1937), Italian TV host, showman, singer, musician, film actor, and film director *Renzo Barbieri (1940–2007), Italian author and editor of Italian comics *Renzo Caldara (born 1943), Italian bobsledder *Renzo Cesana (1907–1970), Italian-American actor, writer, composer, and songwriter *Renzo Cramerotti (born 1947), Italian male javelin thrower *Renzo Dalmazzo (1886–?), Italian lieutenant general *Renzo De Felice (1929–1996), Italian historian *Renzo De Vecchi (1894–1967), Italian football player and coach * Renzo Fenci (1914–1999), Italian-American sculptor based in Southern California. *Renzo Furlan (born 1970), Italian tennis player * Renzo Fujiwara (born 1973), A minor character in the movie The End of Cygnus *Renzo Gobbo (born 1961), Italian associat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Keith Moffatt
Henry Keith Moffatt, FRS FRSE (born 12 April 1935) is a Scottish mathematician with research interests in the field of fluid dynamics, particularly magnetohydrodynamics and the theory of turbulence. He was Professor of Mathematical Physics at the University of Cambridge from 1980 to 2002. Early life and education Moffatt was born on 12 April 1935 to Emmeline Marchant and Frederick Henry Moffatt''.'' He was schooled at George Watson's College, Edinburgh, going on to study Mathematical Sciences at the University of Edinburgh, graduating in 1957. He then went to Trinity College, Cambridge, where he studied mathematics and, 1959, he was a Wrangler. In 1960, he was awarded a Smith's Prize while preparing his PhD. He received his PhD in 1962, the title of his dissertation was ''Magnetohydrodynamic Turbulence.'' Career After completing his PhD, Moffatt joined the staff of the Mathematics Faculty in Cambridge as an Assistant Lecturer and became a Fellow of Trinity College. He was ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Augustus Edward Hough Love
Augustus Edward Hough Love FRS (17 April 1863, Weston-super-Mare – 5 June 1940, Oxford), often known as A. E. H. Love, was a mathematician famous for his work on the mathematical theory of elasticity. He also worked on wave propagation and his work on the structure of the Earth in ''Some Problems of Geodynamics'' won for him the Adams prize in 1911 when he developed a mathematical model of surface waves known as Love waves. Love also contributed to the theory of tidal locking and introduced the parameters known as Love numbers, used in problems related to Earth tides, the tidal deformation of the solid Earth due to the gravitational attraction of the Moon and Sun. He was educated at Wolverhampton Grammar School and in 1881 won a scholarship to St John's College, Cambridge, where he was at first undecided whether to study classics or mathematics. His successful progress (he was placed Second Wrangler) vindicated his choice of mathematics, and in 1886 he was elected Fellow of t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Twisted Sheaf
In mathematics, a twisted sheaf is a variant of a coherent sheaf. Precisely, it is specified by: an open covering in the étale topology ''U''''i'', coherent sheaves ''F''''i'' over ''U''''i'', a Čech 2-cocycle ''θ'' on the covering ''U''''i'' as well as the isomorphisms :g_: F_j, _ \overset\to F_i, _ satisfying *g_ = \operatorname_, *g_ = g_^, *g_ \circ g_ \circ g_ = \theta_ \operatorname_. The notion of twisted sheaves was introduced by Jean Giraud. The above definition due to Căldăraru is down-to-earth but is equivalent to a more sophisticated definition in terms of gerbe; see § 2.1.3 of . See also * Reflexive sheaf In algebraic geometry, a reflexive sheaf is a coherent sheaf that is isomorphic to its second dual (as a sheaf of modules) via the canonical map. The second dual of a coherent sheaf is called the reflexive hull of the sheaf. A basic example of a re ... * Torsion sheaf References * * Geometry {{algebraic-geometry-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Twist (rational Trigonometry)
''Divine Proportions: Rational Trigonometry to Universal Geometry'' is a 2005 book by the mathematician Norman J. Wildberger on a proposed alternative approach to Euclidean geometry and trigonometry, called rational trigonometry. The book advocates replacing the usual basic quantities of trigonometry, Euclidean distance and angle measure, by squared distance and the square of the sine of the angle, respectively. This is logically equivalent to the standard development (as the replacement quantities can be expressed in terms of the standard ones and vice versa). The author claims his approach holds some advantages, such as avoiding the need for irrational numbers. The book was "essentially self-published" by Wildberger through his publishing company Wild Egg. The formulas and theorems in the book are regarded as correct mathematics but the claims about practical or pedagogical superiority are primarily promoted by Wildberger himself and have received mixed reviews. Overview Th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Twist (screw Theory)
Screw theory is the algebraic calculation of pairs of vectors, such as forces and moments or angular and linear velocity, that arise in the kinematics and dynamics of rigid bodies. The mathematical framework was developed by Sir Robert Stawell Ball in 1876 for application in kinematics and statics of mechanisms (rigid body mechanics). Screw theory provides a mathematical formulation for the geometry of lines which is central to rigid body dynamics, where lines form the screw axes of spatial movement and the lines of action of forces. The pair of vectors that form the Plücker coordinates of a line define a unit screw, and general screws are obtained by multiplication by a pair of real numbers and addition of vectors. An important result of screw theory is that geometric calculations for points using vectors have parallel geometric calculations for lines obtained by replacing vectors with screws. This is termed the ''transfer principle.'' Screw theory has become an im ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Structural Complexity (applied Mathematics)
Structural complexity is a science of applied mathematics, that aims at relating fundamental physical or biological aspects of a complex system with the mathematical description of the morphological complexity that the system exhibits, by establishing rigorous relations between mathematical and physical properties of such system. Structural complexity emerges from all systems that display morphological organization. Filamentary structures, for instance, are an example of coherent structures that emerge, interact and evolve in many physical and biological systems, such as mass distribution in the Universe, vortex filaments in turbulent flows, neural networks in our brain and genetic material (such as DNA) in a cell. In general information on the degree of morphological disorder present in the system tells us something important about fundamental physical or biological processes. Structural complexity methods are based on applications of differential geometry and topology (and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Physical Knot Theory
Physical may refer to: *Physical examination In a physical examination, medical examination, or clinical examination, a medical practitioner examines a patient for any possible medical signs or symptoms of a medical condition. It generally consists of a series of questions about the patien ..., a regular overall check-up with a doctor * ''Physical'' (Olivia Newton-John album), 1981 ** "Physical" (Olivia Newton-John song) * ''Physical'' (Gabe Gurnsey album) * "Physical" (Alcazar song) (2004) * "Physical" (Enrique Iglesias song) (2014) * "Physical" (Dua Lipa song) (2020) *"Physical (You're So)", a 1980 song by Adam & the Ants, the B side to " Dog Eat Dog" * ''Physical'' (TV series), an American television series See also {{disambiguation ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Magnetic Helicity
In plasma physics, magnetic helicity is a measure of the linkage, twist, and writhe of a magnetic field. In ideal magnetohydrodynamics, magnetic helicity is conserved. When a magnetic field contains magnetic helicity, it tends to form large-scale structures from small-scale ones. This process can be referred as an inverse transfer in Fourier space. This second property makes magnetic helicity special: three-dimensional turbulent flows tend to "destroy" structure, in the sense that large-scale vortices break-up in smaller and smaller ones (a process called "direct energy cascade", described by Lewis Fry Richardson and Andrey Nikolaevich Kolmogorov). At the smallest scales, the vortices are dissipated in heat through viscous effects. Through a sort of "inverse cascade of magnetic helicity", the opposite happens: small helical structures (with a non-zero magnetic helicity) lead to the formation of large-scale magnetic fields. This is for example visible in the heliospheric curren ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Topological Fluid Dynamics
Topological ideas are relevant to fluid dynamics (including magnetohydrodynamics) at the kinematic level, since any fluid flow involves continuous deformation of any transported scalar or vector field. Problems of stirring and mixing are particularly susceptible to topological techniques. Thus, for example, the Thurston–Nielsen classification has been fruitfully applied to the problem of stirring in two-dimensions by any number of stirrers following a time-periodic 'stirring protocol' (Boyland, Aref & Stremler 2000). Other studies are concerned with flows having chaotic particle paths, and associated exponential rates of mixing (Ottino 1989). At the dynamic level, the fact that vortex lines are transported by any flow governed by the classical Euler equations implies conservation of any vortical structure within the flow. Such structures are characterised at least in part by the helicity of certain sub-regions of the flow field, a topological invariant of the equations. Helicit ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
