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, 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 ).

Mathematically, torque is defined as the cross product of force and the lever-arm distance, which tends to produce rotation.

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, 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 pressure of a mixture of gases can be defined as the summation

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

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, 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 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 Mandelbrot set consists of the set of points c in the complex plane for which the iteratively defined sequence

Mathematically, dot gain is defined as:

Mathematically, the mass deficit is defined as

Mathematically, it is defined as.

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 relationship between a cell and its neighbors, located within an area of influence, can be defined by a coupling law, and this is what primarily determines the behavior of the processor.

Mathematically ( using the Laplace transform ) the impedance of this network can be written as:

Mathematically this is expressed using the virial theorem, which states that, to maintain equilibrium, the gravitational potential energy must equal twice the internal thermal energy.

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

r is the coefficient to represent x using D. Mathematically, sparse dictionary learning means the following

Mathematically, tagged unions correspond to disjoint or discriminated unions, usually written using +.

Mathematically, two Luenberger observers can be used, if is properly selected, using, for example, positive systems properties: one for the upper bound ( that ensures that converges to zero from above when, in the absence of noise and uncertainty ), and a lower bound ( that ensures that converges to zero from below ).