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Mathematically, the theory is modelled after Bernhard Riemann's metric geometry, but the Lorentz group of spacetime symmetries ( an essential ingredient of Einstein's own theory of special relativity ) replaces the group of rotational symmetries of space.
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Mathematically and theory
In 1827, the German scientist Georg Ohm expressed his law in this famous book " Die galvanische Kette, mathematisch bearbeitet " ( The Galvanic Circuit Investigated Mathematically ) in which he gave his complete theory of electricity.
Mathematically, QED is an abelian gauge theory with the symmetry group U ( 1 ).
Mathematically, QCD is a non-Abelian gauge theory based on a local ( gauge ) symmetry group called SU ( 3 ).
Mathematically, this condition is also approached exponentially ; in theory it takes infinite time, but in practice it is over, for all intents and purposes, in a much shorter period.
Mathematically, a Yang – Mills instanton is a self-dual or anti-self-dual connection in a principal bundle over a four-dimensional Riemannian manifold that plays the role of physical space-time in non-abelian gauge theory.
Mathematically, spin networks are related to group representation theory and can be used to construct knot invariants such as the Jones polynomial.
Mathematically and is
Mathematically it is defined as follows:
Mathematically, the motion of a continuum using the Eulerian description is expressed by the mapping function
Mathematically, the catenary curve is the graph of the hyperbolic cosine function.
Mathematically, the conjecture states that the maximal Cauchy development of generic compact or asymptotically flat initial data is locally inextendible as a regular Lorentzian manifold.
Mathematically, density is defined as mass divided by volume:
Mathematically this is described as the convolution of a time-varying input signal x ( t ) with the filter's impulse response h, defined as:
Mathematically, the unification is accomplished under an SU ( 2 ) × U ( 1 ) gauge group.
Mathematically, incompressibility is expressed by saying that the density ρ of a fluid parcel does not change as it moves in the flow field, i. e.,
Mathematically, this is because in reversible processes, the change in entropy of the cold reservoir is the negative of that of the hot reservoir ( i. e., ), keeping the overall change of entropy zero.
Mathematically, this says that the state vector is confined to one of the two eigenspaces of P, and is not allowed to range over the entire Hilbert space.
Mathematically, a sequence of coin flips ( fair or not ) is an example of a Bernoulli process, and its entropy is given by the binary entropy function.
Mathematically, this is expressed by saying that the curl of the flow field is everywhere equal to zero.
Mathematically, the logistic map is written
Mathematically, a neuron's network function is defined as a composition of other functions, which can further be defined as a composition of other functions.
Mathematically, the total electromagnetic energy density ( radiation energy density ) in thermodynamic equilibrium from Planck's law is
Mathematically, the light received from stars as a function of star distance in a hypothetical fractal cosmos is:
Mathematically, the formula above entails that all incomes are at least the lower bound x < sub > m </ sub >, which is positive.
Mathematically, this correspondendence is supported by the second term, on the r. h. s.
Mathematically, this tunneling current is given by
Mathematically and after
Mathematically, to ensure that the spacing after evaluating is proportional to the spacing prior to that evaluation, if is and our new triplet of points is,, and then we want:
Mathematically and metric
Mathematically, the Chebyshev distance is a metric induced by the supremum norm or uniform norm.
Mathematically, spacetime is represented by a 4-dimensional differentiable manifold M and the metric is given as a covariant, second-rank, symmetric tensor on M, conventionally denoted by g. Moreover the metric is required to be nondegenerate with signature (< tt >-+++</ tt >).
Mathematically and geometry
Mathematically a polygonal mesh may be considered an unstructured grid, or undirected graph, with additional properties of geometry, shape and topology.
Mathematically and Lorentz
Mathematically, under the treatment of the Stokes parameters as a Minkowski 4-vector, the transformation is a scaled Lorentz boost ( due to the isomorphism of SL ( 2, C ) and the restricted Lorentz group, SO ( 3, 1 )).
Mathematically, the Lorentz group may be described as the generalized orthogonal group O ( 1, 3 ), the matrix Lie group which preserves the quadratic form
Mathematically it is a four-vector, and is Lorentz covariant.
Mathematically it may look as though all of the fields are vector-valued ( in addition to being operator-valued ), since they all have several components, can be multiplied by matrices, etc., but physicists assign a more specific meaning to the word: a vector is something which transforms like a four-vector under Lorentz transformations, and a scalar is something which does not transform under Lorentz transformations.
Mathematically and group
Mathematically, this replacement may be described by elements of the SU ( 3 ) group.
Mathematically, the Ricci decomposition is the decomposition of the space of all tensors having the symmetries of the Riemann tensor into its irreducible representations for the action of the orthogonal group.
Mathematically and spacetime
Mathematically, each physical law can be expressed with respect to the coordinates given by an inertial frame of reference by a mathematical equation ( for instance, a differential equation ) which relates the various coordinates of the various objects in the spacetime.
Mathematically, spacetime symmetries are usually described by smooth vector fields on a smooth manifold.
Mathematically and symmetries
Mathematically, continuous symmetries are described by continuous or smooth functions.
Mathematically and special
Mathematically, this means that they are asymptotically locally Euclidean ( or perhaps asymptotically locally flat ) hyperkähler 4-manifolds, and in this sense, they are special examples of Einstein manifolds.
Mathematically and rotational
Mathematically, removing the rotational symmetry by placing the " 20 " at the top, there are 19 !, or 121, 645, 100, 408, 832, 000 possible dartboards.
Mathematically and space
Mathematically this invariance can be ensured in one of two ways: by treating the four-vectors as Euclidean vectors and multiplying time by the square root of ; or by keeping time a real quantity and embedding the vectors in a Minkowski space.
Mathematically, the E field can be thought of as a function that associates a vector with every point in space.
Mathematically, the factor of two relating physical angles to their counterparts in Stokes space derives from the use of second-order moments and correlations, and incorporates the loss of information due to absolute phase invariance.
Mathematically, a scalar field on a region U is a real or complex-valued function or distribution on U. The region U may be a set in some Euclidean space, Minkowski space, or more generally a subset of a manifold, and it is typical in mathematics to impose further conditions on the field, such that it be continuous or often continuously differentiable to some order.
Mathematically speaking, the simplification reduces the state space of the system to a finite number, and the partial differential equations ( PDEs ) of the continuous ( infinite-dimensional ) time and space model of the physical system into ordinary differential equations ( ODEs ) with a finite number of parameters.
Mathematically, the ( real valued ) amplitude of one wave component is represented by a scalar wave function u that depends on both space and time:
Mathematically, a pseudoscalar is an element of the top exterior power of a vector space, or the top power of a Clifford algebra ; see pseudoscalar ( Clifford algebra ).
Mathematically, a vector x in an n-dimensional Euclidean space can be defined as an ordered list of n real numbers ( the Cartesian coordinates of P ): x = x < sub > 2 </ sub >, ..., x < sub > n </ sub >.
Mathematically, a multiple criteria design problem can be represented in the decision space as follows:
Mathematically four-dimensional space is simply a space with four spatial dimensions, that is a space that needs four parameters to specify a point in it.
Mathematically, the Willmore energy of a smooth closed surface embedded in three-dimensional Euclidean space is defined to be the integral of the square of the mean curvature minus the Gaussian curvature.
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