The Drake equation is closely related to the Fermi paradox in that Drake suggested that a large number of extraterrestrial civilizations would form, but that the lack of evidence of such civilizations ( the Fermi paradox ) suggests that technological civilizations tend to disappear rather quickly.

As the Universe expands, the density of radiation and ordinary and dark matter declines more quickly than the density of dark energy ( see equation of state ) and, eventually, dark energy dominates.

The equation can be tested rather quickly, which can be used for probabilistic primality testing.

Because tree farms are managed to enhance rapid growth, a tree farm tends to sequester carbon more quickly than an unmanaged forest, considering only the sequestration side of the equation and not the carbon release due to rot, fire, or harvest.

This equivalence can be used to quickly solve for the recurrence relationship for the coefficients in the power series solution of a linear differential equation.

The equation can be tested rather quickly, which can be used for probabilistic primality testing.

Results from a nomogram are obtained very quickly and reliably by simply drawing one or more lines, and the user does not even need to know the actual equation being calculated.

On the other hand, solutions to this equation for large initial values cause the rhs to quickly approach zero.

The equation of hydrostatic equilibrium may need to be modified by adding a radial acceleration term if the radius of the star is changing very quickly, for example if the star is radially pulsating.

Further simplifications of this equation enables one to derive all the properties listed above thus far.

Developed by Ettore Majorana, this Clifford module enables the construction of a Dirac-like equation without complex numbers, and its elements are called Majorana spinors.

The Frobenius method enables us to create a power series solution to such a differential equation, provided that p ( z ) and q ( z ) are themselves analytic at 0 or, being analytic elsewhere, both their limits at 0 exist ( and are finite ).

This equation is incorrect for the calculation of drag in most cases.

For example, most ab initio calculations make the Born – Oppenheimer approximation, which greatly simplifies the underlying Schrödinger equation by assuming that the nuclei remain in place during the calculation.

The Redlich – Kwong equation is adequate for calculation of gas phase properties when the ratio of the pressure to the critical pressure ( reduced pressure ) is less than about one-half of the ratio of the temperature to the critical temperature ( reduced temperature ):

The answer to this initial calculation is then substituted back into the equation, resulting in a new answer which is again substituted.

The equation is monochromatic, so color radiosity rendering requires calculation for each of the required colors.

Other standard iterative methods for matrix equation solutions can also be used, for example the Gauss – Seidel method, where updated values for each patch are used in the calculation as soon as they are computed, rather than all being updated synchronously at the end of each sweep.

A similar calculation shows that v also satisfies the Laplace equation.

Once a new RTT is calculated, it is entered into the equation above to obtain an average RTT for that connection, and the procedure continues for every new calculation.

While precise calculation of positronium energy levels uses the Bethe-Salpeter equation, the similarity between positronium and hydrogen allows for a rough estimate.

The Starling equation defines the forces across a semipermeable membrane and allows calculation of the net flux:

Similarly, uncertainty is propagated through calculations so that the calculated value has some degree of uncertainty depending upon the uncertainties of the measured values and the equation used in the calculation .</ li >

Use this equation for quick calculation, at room temperature:

Two long papers published in 1840 — one on attractions and Clairaut's theorem, and the other on the variation of gravity at the surface of the earth — also demand notice, as do his mathematical memoirs on the critical values of sums of periodic series ( 1847 ) and on the numerical calculation of a class of definite integrals and infinite series ( 1850 ) and his discussion of a differential equation relating to the breaking of railway bridges ( 1849 ), research related to his evidence given to the Royal Commission on the Use of Iron in Railway structures after the Dee bridge disaster of 1847.

* Quadratic formula, calculation to solve a quadratric equation for the independent variable ( x )

As it is only simplistic and unclassified scaling laws that are commonly encountered, that do not take important things like varying land topography into account to ease calculation time and equation length.

( The Bohr equation is used to justify the inclusion of half the second phase in this calculation.

By the time of Isaac Newton's research, paper or vellum was an important computing resource, and even in our present time, researchers like Enrico Fermi would cover random scraps of paper with calculation, to satisfy their curiosity about an equation.

* True example of using Henderson – Hasselbalch equation for calculation net charge of proteins

Direct calculation shows that this is a solution of the differential equation at every point, including and.

Although it is still used for representation of kinetic data, non-linear regression or alternative linear forms of the Michaelis – Menten equation such as the Hanes-Woolf plot or Eadie – Hofstee plot are generally used for the calculation of parameters.

For a more accurate calculation at a higher concentration, Ge and Wang ( 2010 ) proposed a new equation:

He also built on the work of Wright with the Fokker-Planck equation by introducing the Kolmogorov backward equation to population genetics, allowing the calculation of the probability of a gene to become fixed in a population.

The velocities are not explicitly given in the basic Störmer equation, but often they are necessary for the calculation of certain physical quantities like the kinetic energy.

Gibbs showed that if we use the equation Z = ξ < sup > N </ sup >, the entropy of a classical ideal gas is

In the same way he also obtained what is now known as the " Gibbs – Duhem equation.

Instead of also using E < sub > a </ sub >, however, the Eyring equation uses the concept of Gibbs free energy and the symbol * to denote the energy of the transition state.

It can be shown using the Gibbs – Duhem equation that if Raoult's law holds over the entire concentration range x = 0 – 1 in a binary solution then, for the second component, the same must also hold.

In 1873, Willard Gibbs published A Method of Geometrical Representation of the Thermodynamic Properties of Substances by Means of Surfaces, in which he introduced the preliminary outline of the principles of his new equation able to predict or estimate the tendencies of various natural processes to ensue when bodies or systems are brought into contact.

Applying Gibbs free energy equation for reactions:

The Gibbs – Helmholtz equation is a thermodynamic equation useful for calculating changes in the Gibbs energy of a system as a function of temperature.

where H is the enthalpy, T the absolute temperature and G the Gibbs free energy of the system, all at constant pressure p. The equation states that the change in the G / T ratio at constant pressure as a result of an infinitesimally small change in temperature is a factor H / T < sup > 2 </ sup >.

relating the Gibbs energy to a chemical equilibrium constant, the van't Hoff equation can be derived.

The Gibbs – Helmholtz equation can be derived by this second master equation, and the chain rule for partial derivatives.

The fundamental equation of chemical thermodynamics for a system containing n constituent species, with the i-th species having N < sub > i </ sub > particles is, in terms of Gibbs energy

In his 1873 paper A Method of Geometrical Representation of the Thermodynamic Properties of Substances by Means of Surfaces, Gibbs introduced the preliminary outline of the principles of his new equation able to predict or estimate the tendencies of various natural processes to ensue when bodies or systems are brought into contact.

In 1873, Willard Gibbs published A Method of Geometrical Representation of the Thermodynamic Properties of Substances by Means of Surfaces, in which he introduced the preliminary outline of the principles of his new equation able to predict or estimate the tendencies of various natural processes to ensue when bodies or systems are brought into contact.

This is one form of Gibbs fundamental equation.

The temperature dependence of the Gibbs energy for an ideal gas is given by the Gibbs-Helmholtz equation and its pressure dependence is given by:

The entropy of the gas in the two-container system could be immediately calculated, but if the equation is not extensive, the entropy would not be 2 * S. In fact, Gibbs ' non-extensive entropy equation would predict additional entropy.

As understood by Gibbs, and reemphasized more recently, this is a misuse of the entropy equation.

In particular, Gibbs ' non-extensive entropy equation of an ideal gas was not intended for varying numbers of particles.