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, 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 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, the magnetic field of an object is often described in terms of a multipole expansion.

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, spacetime symmetries are usually described by smooth vector fields on a smooth manifold.

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 formula above entails that all incomes are at least the lower bound x < sub > m </ sub >, which is positive.

Mathematically it is a manifold consisting of " events " which are described by some type of coordinate system.

Mathematically, the uncertainty relation between position and momentum arises because the expressions of the wavefunction in the two corresponding bases are Fourier transforms of one another ( i. e., position and momentum are conjugate variables ).

David Sklansky, in his book The Theory of Poker, states " Mathematically, the optimal bluffing strategy is to bluff in such a way that the chances against your bluffing are identical to the pot odds your opponent is getting.

Mathematically they are understood through parallel transport.

Mathematically, classical physics equations are ones in which Planck's constant does not appear.

Mathematically, corrosion rates of glasses are characterized by normalized corrosion rates of elements NR < sub > i </ sub > ( g / cm < sup > 2 </ sup >· d ) which are determined as the ratio of total amount of released species into the water M < sub > i </ sub > ( g ) to the water-contacting surface area S ( cm < sup > 2 </ sup >), time of contact t ( days ) and weight fraction content of the element in the glass f < sub > i </ sub >:

Mathematically, removing the rotational symmetry by placing the " 20 " at the top, there are 19 !, or 121, 645, 100, 408, 832, 000 possible dartboards.

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, majorans may be modeled by allowing them to propagate through a material while all other Standard Model forces are fixed to an orbifold point.

Mathematically, dislocations are a type of topological defect, sometimes called a soliton.

Mathematically, spin networks are related to group representation theory and can be used to construct knot invariants such as the Jones polynomial.

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 ideal drums with membranes of these two different shapes ( but otherwise identical ) would sound the same, because the eigenfrequency | eigenfrequencies are all equal, so the Timbre # Spectra | timbral spectra would contain the same overtones.

Mathematically, nominal numbering is a one-to-one function from a set of objects being named to the set of numerals: it is a function because each object is assigned a numeral, and it is one-to-one because different objects are assigned different numerals.

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, barrel and pincushion distortion are quadratic, meaning they increase as the square of distance from the center.

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 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 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 a polygonal mesh may be considered an unstructured grid, or undirected graph, with additional properties of geometry, shape and topology.