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, it refers to a property of solutions to the Schrödinger equation ; since the Schrödinger equation is linear, any linear combination of solutions to a particular equation will also be a solution of it.

Mathematically, the extremely wideband excitation provided by the spark gap was bandpass filtered by the self-oscillating antenna side circuit, which because of its simple construction, also had a relatively broad and badly controlled filter characteristic.

Mathematically, applying a Gaussian blur to an image is the same as convolving the image with a Gaussian function ; this is also known as a two-dimensional Weierstrass transform.

Mathematically, terminal velocity — without considering buoyancy effects — is given by

§ — Mathematically eliminated from the chase.

Mathematically speaking, magnetic fields and electric fields are convertible with relative motion as a 2nd-order tensor or bivector.

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, a baseband modulated signal may be approximated by a sinusoidal continuous wave signal with a frequency f < sub > m </ sub >.

Mathematically, QED is an abelian gauge theory with the symmetry group U ( 1 ).

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

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, 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.

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.

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.

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, this can be represented for n species as:

Mathematically, for rotation about a fixed axis through the center of mass,

Mathematically, the equation may be rearranged to compute torque for a given power output.

Mathematically speaking, however, there seems to be no reason for us to reject the negative-energy solutions.

Mathematically, the D4σ beam width in the x-dimension for the beam profile is expressed as

Mathematically, the criterion for whether spontaneous fission can occur in a time short enough to be observed by present methods, is approximately:

The fundamental requirement for incompressible flow is that the density,, is constant within an infinitesimal volume, dV, which moves at the velocity of the fluid, v. Mathematically, this constraint implies that the material derivative ( discussed below ) of the density must vanish to ensure incompressible flow.

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, 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, apart from discretization ( whole euros both for income and for tax ), the tax is a continuous, convex, piecewise linear function of income.

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, the iterated logarithm is well-defined not only for base 2 and base e, but for any base greater than.

Mathematically, it would be said that the forward / backward algorithm computes for < math > 0 < k < t </ math >.