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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.
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Mathematically and –
Mathematically, convection can be described by the convection – diffusion equation or the generic scalar transport equation.
Mathematically, the wavelet will resonate if the unknown signal contains information of similar frequency – just as a tuning fork physically resonates with sound waves of its specific tuning frequency.
* Mathematically derived scale – Researchers infer respondents ’ evaluations mathematically.
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, QED is an abelian gauge theory with the symmetry group U ( 1 ).
Mathematically, this tunneling current is given by
Mathematically, QCD is a non-Abelian gauge theory based on a local ( gauge ) symmetry group called SU ( 3 ).
Mathematically and over
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 stock can be seen as an accumulation or integration of flows over time-with outflows subtracting from the stock.
Mathematically, Florida's 25 electoral votes became the key to an election win, despite the fact that both New Mexico and Oregon were declared in favor of Gore over the next few days.
Mathematically and four-dimensional
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 and manifold
Mathematically it is a manifold consisting of " events " which are described by some type of coordinate system.
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, 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, spacetime symmetries are usually described by smooth vector fields on a smooth manifold.
Mathematically and physical
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 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 speaking, two quantum states and are separated by a selection rule if for any given Hamiltonian, while they are separated by a superselection rule if for all physical observables.
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, this is represented as a ' hopping integral ' or ' transfer integral ' between neighboring atoms, which can be viewed as the physical principle that creates electron bands in crystalline materials, due to overlapping between atomic orbitals.
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, 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.
Mathematically, spin networks are related to group representation theory and can be used to construct knot invariants such as the Jones polynomial.
Mathematically and .
Mathematically, we shall not distinguish the experiment of three dice tossed once from that of one die tossed three times.
Mathematically, this means the coefficient for the projection of onto.
Mathematically, this means that for a causal linear system to be stable all of the poles of its transfer function must have negative-real values, i. e. the real part of all the poles are less than zero.
Mathematically, the conjecture states that, for generic initial data, the maximal Cauchy development possesses a complete future null infinity.
Mathematically speaking, magnetic fields and electric fields are convertible with relative motion as a 2nd-order tensor or bivector.
Mathematically, the potential of the ion cores can be treated by various models, the simplest being the nearly free electron model.
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, quasicrystals have been shown to be derivable from a general method, which treats them as projections of a higher-dimensional lattice.
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