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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.
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Mathematically and four-dimensional
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 and space
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 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, 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, a multiple criteria design problem can be represented in the decision space as follows:
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.
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 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, 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 simply
Mathematically, configuration simply describes the linear combination of Slater determinants used for the wave function.
Mathematically and with
Mathematically speaking, magnetic fields and electric fields are convertible with relative motion as a 2nd-order tensor or bivector.
Mathematically, a baseband modulated signal may be approximated by a sinusoidal continuous wave signal with a frequency f < sub > m </ sub >.
Mathematically, the most basic wave is the ( spatially ) one-dimensional sine wave ( or harmonic wave or sinusoid ) with an amplitude described by the 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, the LM curve is defined by the equation, where the supply of money is represented as the real amount M / P ( as opposed to the nominal amount M ), with P representing the price level, and L being the real demand for money, which is some function of the interest rate i and the level Y of real income.
Mathematically, the rate of change of X ( its derivative with respect to time ) vanishes,
Mathematically, assuming the terminator lies on a great circle, such a crescent will actually be the figure bounded by a half-ellipse and a half-circle, with the major axis of the ellipse coinciding with a diameter of the semicircle.
Mathematically talented, Lamarr also co-invented — with composer George Antheil — an early technique for spread spectrum communications and frequency hopping, necessary for wireless communication from the pre-computer age to the present day.
Mathematically, the modulated Dirac comb is equivalent to the product of the comb function with s ( t ).
Mathematically, a braid with that property is called a Brunnian braid.
Mathematically, the process is the same as that of quantum tunneling, except with electromagnetic waves instead of quantum-mechanical wavefunctions.
McDuffie argued that the 40 % tariff on cotton finished goods meant that “ the manufacturer actually invades your barns, and plunders you of 40 out of every 100 bales that you produce .” Mathematically incorrect, this argument still struck a nerve with his constituency.
Mathematically, the commutator of the baryon number quantum operator with the ( perturbative ) Standard Model hamiltonian is zero:
Mathematically, the gradient of a two-variable function ( here the image intensity function ) is at each image point, a 2D vector with the components given by the derivatives in the horizontal and vertical directions.
Mathematically, the operator uses two 3 × 3 kernels which are convolved with the original image to calculate approximations of the derivatives-one for horizontal changes, and one for vertical.
Mathematically, the function is expressed as the derivative of the consumption function with respect to disposable income.
Mathematically, the inverse relation may be stated with causal relation as:
Mathematically the RDW is calculated with the following formula:
Mathematically, the number of strands is the greatest common divisor of the number of leads and the number of bights ; the knot may be tied with a single strand if and only if the two numbers are coprime.
Mathematically a polygonal mesh may be considered an unstructured grid, or undirected graph, with additional properties of geometry, shape and topology.
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