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, 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 ).

In 1827, the German scientist Georg Ohm expressed his law in this famous book " Die galvanische Kette, mathematisch bearbeitet " ( The Galvanic Circuit Investigated Mathematically ) in which he gave his complete theory of electricity.

Mathematically, this is expressed as 1 minus unavailability.

Mathematically, the D4σ beam width in the x-dimension for the beam profile is expressed 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, this can be expressed by means of the signal to noise ratio ( SNR or S / N ).

Mathematically this is expressed as:

Mathematically, the FT is expressed as:

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, the function is expressed as the derivative of the consumption function with respect to disposable income.

Mathematically, the modulus of resilience can be expressed by the product of the square of the yield stress divided by two times the Young's modulus.

Mathematically, the image is expressed as:

Mathematically the von Mises yield criterion is expressed as:

Mathematically, it can be expressed as:

Mathematically, each physical law can be expressed with respect to the coordinates given by an inertial frame of reference by a mathematical equation ( for instance, a differential equation ) which relates the various coordinates of the various objects in the spacetime.

Mathematically, this may be expressed as,

Mathematically, this is expressed by the following condition:

Mathematically, the marginal propensity to import ( MPM ) function is expressed as the derivative of the import ( I ) function with respect to disposable income ( Y ).

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, 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.