Acceleration feedback is used as the main tool to symmetrize the nonsymmetric part of the system inertia matrix.
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In theoretical physics, the nonsymmetric gravitational theory (NGT) of John Moffat is a classical theory of gravitation that tries to explain the observation of the flat rotation curves of galaxies. In general relativity, the gravitational field is characterized by a symmetric rank-2 tensor, the metric tensor. The possibility of generalizing the metric tensor has been considered by many, including Albert Einstein and others. A general (nonsymmetric) tensor can always be decomposed into a symmetric and an antisymmetric part.
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A newer version of MSTG, in which the skew symmetric tensor field was replaced by a vector field, is scalar–tensor–vector gravity (STVG). STVG, like Milgrom's Modified Newtonian Dynamics (MOND), can provide an explanation for flat rotation curves of galaxies. Recently, Hammond showed the nonsymmetric part of the metric tensor was shown to be equal the torsion potential, a result following the metricity condition, that the length of a vector is invariant under parallel transport. In addition, the energy momentum tensor is not symmetric, and both the symmetric and nonsymmetric parts are those of a string.
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Since 2017 she has been a Visiting Miller Professor at the University of California, Berkeley. Along with colleagues O. Mandelshtam and L. Williams, in 2018 Corteel developed a new characterization of both symmetric and nonsymmetric Macdonald polynomials using the combinatorial exclusion process.
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The price one pays for avoiding inner products is that the method requires enough knowledge about spectrum of the coefficient matrix A, that is an upper estimate for the upper eigenvalue and lower estimate for the lower eigenvalue. There are modifications of the method for nonsymmetric matrices A.
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In numerical linear algebra, the biconjugate gradient stabilized method, often abbreviated as BiCGSTAB, is an iterative method developed by H. A. van der Vorst for the numerical solution of nonsymmetric linear systems. It is a variant of the biconjugate gradient method (BiCG) and has faster and smoother convergence than the original BiCG as well as other variants such as the conjugate gradient squared method (CGS). It is a Krylov subspace method.
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Moffat's work culminates in his nonsymmetric gravitational theory and scalar–tensor–vector gravity (now called MOG). His theory explains galactic rotation curves without invoking dark matter. He proposes a variable speed of light approach to cosmological problems, which posits that G/c is constant through time, but G and c separately have not been. Moreover, the speed of light c may have been much higher (at least trillion trillion times faster than the normal speed of light) during early moments of the Big Bang.
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John W. Moffat (born 24 May 1932) is a Danish-born British-Canadian physicist. He is currently Professor Emeritus in physics at the University of Toronto and is also an adjunct Professor in physics at the University of Waterloo and a resident affiliate member of the Perimeter Institute for Theoretical Physics. Moffat is best known for his work on gravity and cosmology, culminating in his nonsymmetric gravitational theory and scalar–tensor–vector gravity (now called MOG), and summarized in his 2008 book for general readers, Reinventing Gravity. His theory explains galactic rotation curves without invoking dark matter.
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Discretization in the space produces a system of ordinary differential equations for unsteady problems and algebraic equations for steady problems. Implicit or semi-implicit methods are generally used to integrate the ordinary differential equations, producing a system of (usually) nonlinear algebraic equations. Applying a Newton or Picard iteration produces a system of linear equations which is nonsymmetric in the presence of advection and indefinite in the presence of incompressibility. Such systems, particularly in 3D, are frequently too large for direct solvers, so iterative methods are used, either stationary methods such as successive overrelaxation or Krylov subspace methods.
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The Jacobi Method has been generalized to complex Hermitian matrices, general nonsymmetric real and complex matrices as well as block matrices. Since singular values of a real matrix are the square roots of the eigenvalues of the symmetric matrix S = A^T A it can also be used for the calculation of these values. For this case, the method is modified in such a way that S must not be explicitly calculated which reduces the danger of round-off errors. Note that J S J^T = J A^T A J^T = J A^T J^T J A J^T = B^T B with B \, := J A J^T .
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Continuing Einstein's search for a unified field theory, Moffat proposed a nonsymmetric gravitational theory that, like Einstein's unified field, incorporated a symmetric field (gravity) and an antisymmetric field. Unlike Einstein, however, Moffat made no attempt to identify the latter with electromagnetism, instead proposing that the antisymmetric component is another manifestation of gravity. As investigation progressed, the theory evolved in a variety of ways; most notably, Moffat postulated that the antisymmetric field may be massive. The current version of his modified gravity (MOG) theory, which grew out of this investigation, modifies Einstein's gravity with the addition of a vector field, while also promoting the constants of the theory to scalar fields.
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In the (usual) setting of operads with an action of the symmetric group on topological spaces, an operad A is said to be an A∞-operad if all of its spaces A(n) are Σn-equivariantly homotopy equivalent to the discrete spaces Σn (the symmetric group) with its multiplication action (where n ∈ N). In the setting of non-Σ operads (also termed nonsymmetric operads, operads without permutation), an operad A is A∞if all of its spaces A(n) are contractible. In other categories than topological spaces, the notions of homotopy and contractibility have to be replaced by suitable analogs, such as homology equivalences in the category of chain complexes.
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As the electromagnetic field is characterized by an antisymmetric rank-2 tensor, there is an obvious possibility for a unified theory: a nonsymmetric tensor composed of a symmetric part representing gravity, and an antisymmetric part that represents electromagnetism. Research in this direction ultimately proved fruitless; the desired classical unified field theory was not found. In 1979, Moffat made the observation that the antisymmetric part of the generalized metric tensor need not necessarily represent electromagnetism; it may represent a new, hypothetical force. Later, in 1995, Moffat noted that the field corresponding with the antisymmetric part need not be massless, like the electromagnetic (or gravitational) fields.
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Inspired by Einstein's approach to a unified field theory and Eddington's idea of the affine connection as the sole basis for differential geometric structure for space-time, Erwin Schrödinger from 1940 to 1951 thoroughly investigated pure-affine formulations of generalized gravitational theory. Although he initially assumed a symmetric affine connection, like Einstein he later considered the nonsymmetric field. Schrödinger's most striking discovery during this work was that the metric tensor was induced upon the manifold via a simple construction from the Riemann curvature tensor, which was in turn formed entirely from the affine connection. Further, taking this approach with the simplest feasible basis for the variational principle resulted in a field equation having the form of Einstein's general- relativistic field equation with a cosmological term arising automatically.
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