The approach based on the Einstein's mobility and Teorell formula gives the following generalization of Fick's equation for the multicomponent diffusion of the perfect components:

This is a generalization of Einstein's equation above, since the momentum of a photon is given by p

Because relativity treats space and time as a whole, a relativistic generalization of this equation requires that space and time derivatives must enter symmetrically, as they do in the Maxwell equations that govern the behavior of light — the equations must be differentially of the same order in space and time.

Thus we cannot get a simple generalization of the Schrödinger equation under the naive assumption that the wave function is a relativistic scalar, and the equation it satisfies, second order in time.

Although it is not a successful relativistic generalization of the Schrödinger equation, this equation is resurrected in the context of quantum field theory, where it is known as the Klein – Gordon equation, and describes a spinless particle field ( e. g. pi meson ).

Although physical modelling was not a new concept in acoustics and synthesis, having been implemented using finite difference approximations of the wave equation by Hiller and Ruiz in 1971, it was not until the development of the Karplus-Strong algorithm, the subsequent refinement and generalization of the algorithm into the extremely efficient digital waveguide synthesis by Julius O. Smith III and others, and the increase in DSP power in the late 1980s that commercial implementations became feasible.

A historically useful generalization supported by the Arrhenius equation is that, for many common chemical reactions at room temperature, the reaction rate doubles for every 10 degree Celsius increase in temperature.

is appropriate for generalization to quantum field theory and the Dirac equation.

In 1927, soon after the Schrödinger equation was introduced, Vladimir Fock wrote an article about its generalization for the case of magnetic fields, where forces were dependent on velocity, and independently derived this equation.

These states are unified by the fact that their behavior is described by a wave equation or some generalization thereof.

This establishes that the above equation, clearly a generalization of the previous one, defines the Zariski topology on any affine variety.

This is a generalization of the Euler – Lagrange equation: indeed, the functional derivative was introduced in physics within the derivation of the Lagrange equation of the second kind from the principle of least action in Lagrangian mechanics ( 18th century ).

The following functional equations are as a generalization of the b-parts functional equation for semigroups and groups, even in a binary system ( magma ), that are introduced by him:

A Cartan geometry is a generalization of a smooth Klein geometry, in which the structure equation is not assumed, but is instead used to define a notion of curvature.

The abelian hidden subgroup problem is a generalization of many problems that can be solved by a quantum computer, such as Simon's problem, solving Pell's equation, testing the principal ideal of a ring R and factoring.

If drift must be taken into account, the Smoluchowski equation provides an appropriate generalization.

This is called the Gauss equation, as it may be viewed as a generalization of Gauss's Theorema Egregium.

A generalization of the replicator equation which incorporates mutation is given by the replicator-mutator equation, which takes the following form in the continuous version:

This equation is a simultaneous generalization of the replicator equation and the quasispecies equation, and is used in the mathematical analysis of language.

No generalization of these results to spaces of more than three dimensions has so far been found possible.

A generalization of Artin's theorem states that whenever three elements in an alternative algebra associate ( i. e. ) the subalgebra generated by those elements is associative.

Euler discussed a generalization of Euclidean geometry called affine geometry, which retains the fifth postulate unmodified while weakening postulates three and four in a way that eliminates the notions of angle ( whence right triangles become meaningless ) and of equality of length of line segments in general ( whence circles become meaningless ) while retaining the notions of parallelism as an equivalence relation between lines, and equality of length of parallel line segments ( so line segments continue to have a midpoint ).

The generalization of these three properties to more abstract vector spaces leads to the notion of norm.

The torsion and curvature are related by the Frenet – Serret formulas ( in three dimensions ) and their generalization ( in higher dimensions ).

Peirce's appreciation of these three dimensions serves to flesh out a physiognomy of inquiry far more solid than the flatter image of inductive generalization < span lang = la > simpliciter </ span >, which is merely the relabeling of phenomenological patterns.

The peculiarity of the tetration among these operations is that the first three ( addition, multiplication and exponentiation ) are generalized for complex values of n, while for tetration, no such regular generalization is yet established ; and tetration is not considered an elementary function.

At the beginning is the well-known generalization of Euclid I. 47, then follow various theorems on the circle, leading up to the problem of the construction of a circle which shall circumscribe three given circles, touching each other two and two.

The coefficients can be defined with a generalization of Pascal's triangle to three dimensions, called Pascal's pyramid or Pascal's tetrahedron.

In geometry, polytope means the generalization to any dimension of the sequence: polygon in two dimensions, polyhedron in three dimensions, and polychoron in four dimensions.

There is no natural generalization to more than three players which divides the cake without extra cuts.

This conjecture is part of a generalization of Fermat's polygonal number theorem to three dimensional figurate numbers, also called polyhedral numbers.

The axiom of choice can be seen as asserting the generalization of this property, already evident for finite collections, to arbitrary collections.

So in a sense, functors between arbitrary categories are a kind of generalization of monoid homomorphisms to categories with more than one object.

The study and the generalization of this problem by Tait, Heawood, Ramsey and Hadwiger led to the study of the colorings of the graphs embedded on surfaces with arbitrary genus.

In geometry, a simplex ( plural simplexes or simplices ) is a generalization of the notion of a triangle or tetrahedron to arbitrary dimension.

In mathematics, an n-sphere is a generalization of the surface of an ordinary sphere to arbitrary dimension.

A natural generalization of the inverse semigroup is to define an ( arbitrary ) unary operation ° such that ( a °)°= a for all a in S ; this endows S with a type < 2, 1 > algebra.

Formal generalization of the pi – theorem for the case of arbitrary number of quantities was for the first time given by A. Vaschy in 1892, and later and, apparently, independently, by A. Federman, D. Riabouchinsky in 1911 and by Buckingham in 1914.

In mathematics, formal power series are a generalization of polynomials as formal objects, where the number of terms is allowed to be infinite ; this implies giving up the possibility to substitute arbitrary values for indeterminates.

Let us examine some other cases ; we shall find that Peacock's principle is not a solution of the difficulty ; the great logical process of generalization cannot be reduced to any such easy and arbitrary procedure.

It is a natural generalization of the topology of the real numbers to arbitrary totally ordered sets.

It is a generalization of the Cauchy formula for repeated integration to arbitrary order.

In abstract algebra, the concept of a module over a ring is a generalization of the notion of vector space, wherein the corresponding scalars are allowed to lie in an arbitrary ring.

The uniformization theorem is a generalization of the Riemann mapping theorem from proper simply connected open subsets of the plane to arbitrary simply connected Riemann surfaces.

Leibniz picks up on the generalization used by Locke and adopts a less rigid approach: clearly there is no perfect correspondence between words and things, but neither is the relationship completely arbitrary, although he seems vague about what that relationship might be.

In category theory, a coequalizer ( or coequaliser ) is a generalization of a quotient by an equivalence relation to objects in an arbitrary category.

This equivalence can be generalized to pass between groups of multiplicative type ( a distinguished class of formal groups ) and arbitrary abelian groups, and such a generalization can be convenient if one wants to work in a well-behaved category, since the category of tori doesn't have kernels or filtered colimits.

A generalization of this method and extension to arbitrary precision is provided by Fog ( 2008 ).

This method appeals to the definition, and allows generalization to arbitrary dimensions.

The BV formalism, based on an action that contains both fields and " antifields ", can be thought of as a vast generalization of the original BRST formalism for pure Yang – Mills theory to an arbitrary Lagrangian gauge theory.

Tutte also conjectured a generalization of the snark theorem to arbitrary graphs: every bridgeless graph with no Petersen minor has a nowhere zero 4-flow.

In complex analysis, a branch of mathematics, a generalized continued fraction is a generalization of regular continued fractions in canonical form, in which the partial numerators and partial denominators can assume arbitrary real or complex values.

Jacobi fields give a natural generalization of this phenomenon to arbitrary Riemannian manifolds.

More generally the Cartan-Helgason theorem gives the decomposition when G / H is a compact symmetric space, in which case all multiplicities are one ; a generalization to arbitrary σ has since been obtained by.

One generalization of this result to arbitrary dimension, < big > ℝ </ big >< sup > d </ sup >, was found by Agarwal and Aronov.

This was generalized by Hermann Grassmann to arbitrary r and n using a generalization of Plücker's coordinates, sometimes called Grassmann coordinates.