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

Mathematically, the potential of the ion cores can be treated by various models, the simplest being the nearly free electron model.

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 restoring force F is given by

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, the most basic wave is the ( spatially ) one-dimensional sine wave ( or harmonic wave or sinusoid ) with an amplitude described by the equation:

Mathematically, division by zero within reals is explicitly undefined if the problem is zero divided by zero, and is defined as either positive or negative infinity depending on the signs of the numbers involved, or is undefined.

Mathematically, convection can be described by the convection – diffusion equation or the generic scalar transport equation.

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, we write: " 40: 60 " = " 2: 3 " ( dividing both quantities by 20 ).

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, 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, this can be expressed by means of the signal to noise ratio ( SNR or S / N ).

Mathematically, the climatological annual cycle is commonly estimated from observational data or model output by taking the average of all Januaries, all Februaries, and so forth.

Mathematically, the cyclomatic complexity of a structured program is defined with reference to the control flow graph of the program, a directed graph containing the basic blocks of the program, with an edge between two basic blocks if control may pass from the first to the second.

Mathematically the switching mixer is not much different from a multiplying mixer, just because instead of the LO sine wave term we would use the signum function.

Mathematically a bijective function is used on the characters ' positions to encrypt and an inverse function to decrypt.

Mathematically such a symmetry is represented as a flow, φ, which acts on the variables as follows

However, in what newspapers such as the Wall Street Journal later call the " math wars ", organizations such as Mathematically Correct complain that some elementary texts based on the standards, including Mathland, have almost entirely abandoned any instruction of traditional arithmetic in favor of cutting, coloring, pasting, and writing.

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 Marshallian PED was based on a point-price definition, using differential calculus to calculate elasticities.

Mathematically, defining down to be positive, the net force acting on an object falling near the surface of Earth is ( according to the drag equation ):

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, this corresponds to the fact that spherical coordinates do not define a coordinate chart on the sphere at zenith and nadir.

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 parallel projection of the point on the-plane gives.

Mathematically oriented quantitative analysts tend to have more of a reliance on numerical analysis, and less of a reliance on statistics and econometrics.

Mathematically, how the impulse is described depends on whether the system is modeled in discrete or continuous time.

Mathematically, a quantum operation is a linear map Φ between spaces of trace class operators on Hilbert spaces H and G such that

Mathematically, texture filtering is a type of anti-aliasing, but it filters out high frequencies from the texture fill whereas other AA techniques generally focus on visual edges.

Mathematically it can be explained by considering the normal modes of electromagnetic field which explicitly depend on the boundary ( or matching ) conditions on the surfaces of the interacting bodies.

Mathematically, he is most well known for his treatise Logarithmo-technica on logarithms, published in 1668.

Mathematically, spacetime is represented by a 4-dimensional differentiable manifold M and the metric is given as a covariant, second-rank, symmetric tensor on M, conventionally denoted by g. Moreover the metric is required to be nondegenerate with signature (< tt >-+++</ tt >).