Mathematically, the coherent state is defined to be the right eigenstate of the annihilation operator.

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 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, if represents the vector x then

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 E field can be thought of as a function that associates a vector with every point in space.

) Mathematically, if everything in the universe undergoes a rotation described by a rotation matrix R, so that a displacement 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, evanescent waves can be characterized by a wave vector where one or more of the vector's components has an imaginary value.

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, 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.

If a curve γ represents the path of a particle then the instantaneous velocity of the particle at a given point P is expressed by a vector, called the tangent vector to the curve at P. Mathematically, given a parametrized C < sup > 1 </ sup > curve γ = γ ( t ), for every value t

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, we can describe the reciprocal lattice as the set of all vectors K that satisfy the above identity for all lattice point position vectors R. This reciprocal lattice is itself a Bravais lattice, and the reciprocal of the reciprocal lattice is the original lattice, which reveals the Pontryagin duality of their respective vector spaces.

Mathematically, helicity is the sign of the projection of the spin vector onto the momentum vector: left is negative, right is positive.

Mathematically, there are exactly q < sup > m </ sup > possible messages of length m, and each such message can be regarded as a vector of length m. The encoding scheme converts an m-dimensional vector into an n-dimensional vector.

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 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, the condition for a vector field to be projective is equivalent to the existence of a one-form satisfying

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 first vector is the orthogonal, or least-squares, projection of the data vector onto the subspace spanned by the vector of 1's.

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 ).