Holonomic Theory
Holonomic (introduced by Heinrich Hertz in 1894 from the Greek ὅλος meaning whole, entire and νόμ-ος meaning law) may refer to: Mathematics * Holonomic basis, a set of basis vector fields such that some coordinate system exists for which e_k = * Holonomic constraints, which are expressible as a function of the coordinates x_j\,\! and time t\,\! * Holonomic module in the theory of D-modules * Holonomic function, a smooth function that is a solution of a linear homogeneous differential equation with polynomial coefficients Other uses * Holonomic brain theory, model of cognitive function as being guided by a matrix of neurological wave interference patterns See also * Holonomy in differential geometry * Holon (other) Holon is a city on the central coastal strip south of Tel Aviv, Israel. Holon may also refer to: * ''Holon'' (Nik Bärtsch album), 2008 * ''Holon'' (Equinox album), 1998 * Holon (philosophy), something that is simultaneously a whole and a ...
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Heinrich Hertz
Heinrich Rudolf Hertz ( ; ; 22 February 1857 – 1 January 1894) was a German physicist who first conclusively proved the existence of the electromagnetic waves predicted by James Clerk Maxwell's Maxwell's equations, equations of electromagnetism. The unit of frequency, cycle per second, was named the "hertz" in his honor.IEC History
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Biography

Heinrich Rudolf Hertz was born in 1857 in Hamburg, then a sovereign state of the German Confederation, into a prosperous and cultured Hanseatic (class), Hanseatic family. His father was Gustav Ferdinand Hertz. His mother was Anna Elisabeth Pfefferkorn. While studying at the Gelehrtenschule des Johanneums in Hamburg, Hertz showed an aptitude for sciences as well as languages, learning Arabic. He studied sciences and engineering in th ...
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Holonomic Basis
In mathematics and mathematical physics, a coordinate basis or holonomic basis for a differentiable manifold is a set of basis vector fields defined at every point of a region of the manifold as :\mathbf_ = \lim_ \frac , where is the displacement vector between the point and a nearby point whose coordinate separation from is along the coordinate curve (i.e. the curve on the manifold through for which the local coordinate varies and all other coordinates are constant). It is possible to make an association between such a basis and directional derivative operators. Given a parameterized curve on the manifold defined by with the tangent vector , where , and a function defined in a neighbourhood of , the variation of along can be written as :\frac = \frac\frac = u^ \frac f . Since we have that , the identification is often made between a coordinate basis vector and the partial derivative operator , under the interpretation of vectors as operators acting on functions ...
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Holonomic Constraints
In classical mechanics, holonomic constraints are relations between the position variables (and possibly time) that can be expressed in the following form: :f(u_1, u_2, u_3,\ldots, u_n, t) = 0 where \ are the ''n'' generalized coordinates that describe the system. For example, the motion of a particle constrained to lie on the surface of a sphere is subject to a holonomic constraint, but if the particle is able to fall off the sphere under the influence of gravity, the constraint becomes non-holonomic. For the first case, the holonomic constraint may be given by the equation :r^2-a^2=0 where r is the distance from the centre of a sphere of radius a, whereas the second non-holonomic case may be given by :r^2 - a^2 \geq 0 Velocity-dependent constraints (also called semi-holonomic constraints) such as :f(u_1,u_2,\ldots,u_n,\dot_1,\dot_2,\ldots,\dot_n,t)=0 are not usually holonomic. Holonomic system In classical mechanics a system may be defined as holonomic if all constraints ...
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Holonomic Module
In mathematics, a ''D''-module is a module over a ring ''D'' of differential operators. The major interest of such ''D''-modules is as an approach to the theory of linear partial differential equations. Since around 1970, ''D''-module theory has been built up, mainly as a response to the ideas of Mikio Sato on algebraic analysis, and expanding on the work of Sato and Joseph Bernstein on the Bernstein–Sato polynomial. Early major results were the Kashiwara constructibility theorem and Kashiwara index theorem of Masaki Kashiwara. The methods of ''D''-module theory have always been drawn from sheaf theory and other techniques with inspiration from the work of Alexander Grothendieck in algebraic geometry. The approach is global in character, and differs from the functional analysis techniques traditionally used to study differential operators. The strongest results are obtained for over-determined systems (holonomic systems), and on the characteristic variety cut out by the symbols, ...
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Holonomic Function
In mathematics, and more specifically in analysis, a holonomic function is a smooth function of several variables that is a solution of a system of linear homogeneous differential equations with polynomial coefficients and satisfies a suitable dimension condition in terms of D-modules theory. More precisely, a holonomic function is an element of a holonomic module of smooth functions. Holonomic functions can also be described as differentiably finite functions, also known as D-finite functions. When a power series in the variables is the Taylor expansion of a holonomic function, the sequence of its coefficients, in one or several indices, is also called ''holonomic''. Holonomic sequences are also called P-recursive sequences: they are defined recursively by multivariate recurrences satisfied by the whole sequence and by suitable specializations of it. The situation simplifies in the univariate case: any univariate sequence that satisfies a linear homogeneous recurrence relat ...
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Holonomic Brain Theory
Holonomic brain theory, also known as The Holographic Brain, is a branch of neuroscience investigating the idea that human consciousness is formed by quantum effects in or between brain cells. Holonomic refers to representations in a Hilbert phase space defined by both spectral and space-time coordinates. The Holonomic Brain Theory is opposed by traditional neuroscience, which investigates the brain's behavior by looking at patterns of neurons and the surrounding chemistry. The entire field of quantum consciousness is often criticized as pseudoscience. This specific theory of quantum consciousness was developed by neuroscientist Karl Pribram initially in collaboration with physicist David Bohm building on the initial theories of holograms originally formulated by Dennis Gabor. It describes human cognition by modeling the brain as a holographic storage network. Pribram suggests these processes involve electric oscillations in the brain's fine-fibered dendritic webs, which are dif ...
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Holonomy
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 holonomy of a connection (mathematics), connection on a smooth manifold is a general geometrical consequence of the curvature of the connection measuring the extent to which parallel transport around closed loops fails to preserve the geometrical data being transported. For flat connections, the associated holonomy is a type of monodromy and is an inherently global notion. For curved connections, holonomy has nontrivial local and global features. Any kind of connection on a manifold gives rise, through its parallel transport maps, to some notion of holonomy. The most common forms of holonomy are for connections possessing some kind of symmetry. Important examples include: holonomy of the Levi-Civit ...
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Holon (other)
Holon is a city on the central coastal strip south of Tel Aviv, Israel. Holon may also refer to: * ''Holon'' (Nik Bärtsch album), 2008 * ''Holon'' (Equinox album), 1998 * Holon (philosophy), something that is simultaneously a whole and a part * Holon (physics), a quasiparticle that electrons can split into during the process of spin–charge separation * ''Holon'' (sculpture), a sculpture by Donald Wilson in Portland, Oregon See also * Holo (other) * Holonomic (other) * Holonomy, a concept in differential geometry * Holonymy In linguistics, meronymy () is a semantic relation between a meronym denoting a part and a holonym denoting a whole. In simpler terms, a meronym is in a ''part-of'' relationship with its holonym. For example, ''finger'' is a meronym of ''hand' ...