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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.
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Mathematically and speaking
Mathematically speaking, magnetic fields and electric fields are convertible with relative motion as a 2nd-order tensor or bivector.
Mathematically speaking, however, there seems to be no reason for us to reject the negative-energy solutions.
Mathematically speaking, the color charge of a particle is the value of a certain quadratic Casimir operator in the representation of the particle.
Mathematically speaking, there are seventeen basic patterns, described as wallpaper groups, that can be used to tile an infinite plane.
Mathematically speaking, a sphere and a plane are not isometric, even locally.
Mathematically speaking it is an Iteration | Iterative process.
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 speaking, both in a standard and a TD approach, we would try to optimize some cost function, related to the error in our predictions of the expectation of some random variable, E. However, while in the standard approach we in some sense assume E = z ( the actual observed value ), in the TD approach we use a model.
: Mathematically speaking, charge transfer in semiconductor devices is due both to conduction driven by the electric field ( drift ) and by diffusion.
Mathematically speaking, the most apparent distinguishing feature of the energy conditions is that they are essentially restrictions on the eigenvalues and eigenvectors of the matter tensor.
Mathematically and state
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, the coherent state is defined to be the right eigenstate of the annihilation operator.
Mathematically, the state of each two-electron composite system can be described by the state vector
Mathematically, each cell can be modeled as a dissipative, nonlinear dynamical system where information is encoded via its initial state, inputs, and variables used to define its behavior.
Mathematically, quantum amplification can be represented with an unitary operator, which entangles the state of the optical field with internal degrees of freedom of the amplifier.
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, 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 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.
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 system
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 it is a manifold consisting of " events " which are described by some type of coordinate system.
Mathematically, if the coordinate system undergoes a transformation described by an invertible matrix M, so that a coordinate vector x is transformed to x ′
Mathematically, if the coordinate system undergoes a transformation described by an invertible matrix M, so that a coordinate vector x is transformed to x ′
Mathematically, the absolute entropy of any system at zero temperature is the natural log of the number of ground states times Boltzmann's constant k < sub > B </ sub >.
Mathematically, linear least squares is the problem of approximately solving an overdetermined system of linear equations, where the best approximation is defined as that which minimizes the sum of squared differences between the data values and their corresponding modeled values.
Mathematically, how the impulse is described depends on whether the system is modeled in discrete or continuous time.
Mathematically, there is an exact one-to-one mapping of impenetrable bosons ( in a one-dimensional system ) onto a system of fermions that do not interact at all.
It was here that he began his work with intellectually gifted youth, creating in 1971 the Study of Mathematically Precocious Youth ( SMPY ), which would eventually lead to a revolution in how gifted children were to be identified and treated within the education system.
Mathematically and number
Mathematically this means that the Fresnel reflection coefficient becomes a complex rather than a real number.
Mathematically, the commutator of the baryon number quantum operator with the ( perturbative ) Standard Model hamiltonian is zero:
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, in order to have a reasonable chance to obtain one molecule of the original extract, the patient would have to consume an amount of the remedy roughly 10 < sup > 321 </ sup > times the number of atoms in the observable universe.
Mathematically, degrees of freedom is the dimension of the domain of a random vector, or essentially the number of ' free ' components: how many components need to be known before the vector is fully determined.
Mathematically, the number of points in the inner circle is equal to the number of points in the outer circle, i. e. there is a bijection between them.
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