Solutions of the Dirac equation contained negative energy quantum states.

where is the Boltzmann constant, T is temperature ( assumed to be a well-defined quantity ), is the degeneracy ( meaning, the number of levels having energy ; sometimes, the more general ' states ' are used instead of levels, to avoid using degeneracy in the equation ), N is the total number of particles and Z ( T ) is the partition function.

In words, this equation states that for faster speeds ( bigger | v |) the road must be banked more steeply ( a larger value for θ ), and for sharper turns ( smaller R ) the road also must be banked more steeply, which accords with intuition.

The Drake equation states that:

An equation states that two expressions are equal using the symbol for equality, ( the equals sign ).

The solution of the Schrödinger equation goes much further than the Bohr model, because it also yields the shape of the electron's wave function (" orbital ") for the various possible quantum-mechanical states, thus explaining the anisotropic character of atomic bonds.

In terms of field lines, this equation states that magnetic field lines neither begin nor end but make loops or extend to infinity and back.

However, Bernoulli's equation states:

In algebra, the rational root theorem ( or rational root test ) states a constraint on rational solutions ( or roots ) of the polynomial equation

One of the most important equations in Statistical mechanics ( analogous to in mechanics, or the Schroedinger equation in quantum mechanics ) is the definition of the partition function, which is essentially a weighted sum of all possible states available to a system.

The equation states that two nucleophiles react with the same relative reactivity regardless of the nature of the electrophile, which is in violation of the Reactivity – selectivity principle.

The theorem loosely states that if we have a physically meaningful equation involving a certain number, n, of physical variables, and these variables are expressible in terms of k independent fundamental physical quantities, then the original expression is equivalent to an equation involving a set of p = n − k dimensionless parameters constructed from the original variables: it is a scheme for nondimensionalization.

The Cauchy – Kowalevski theorem states that the Cauchy problem for any partial differential equation whose coefficients are analytic in the unknown function and its derivatives, has a locally unique analytic solution.

The general equation for such a system of rules is k < sup > k < sup > s </ sup ></ sup >, where k is the number of possible states for a cell, and s is the number of neighboring cells ( including the cell to be calculated itself ) used to determine the cell's next state.

* 1926 Erwin Schrödinger states his nonrelativistic quantum wave equation and formulates quantum wave mechanics

* 1928 Paul Dirac states his relativistic electron quantum wave equation

* 1872 – Ludwig Boltzmann states the Boltzmann equation for the temporal development of distribution functions in phase space, and publishes his H-theorem

* 1920 – Megh Nad Saha states his ionization equation

For example, the following equation, from a hypothetical cipher, states the XOR sum of the first and third plaintext bits ( as in a block cipher's block ) and the first ciphertext bit is equal to the second bit of the key:

The time-dependent Schrödinger equation predicts that wavefunctions can form standing waves, called stationary states ( also called " orbitals ", as in atomic orbitals or molecular orbitals ).

These states are important in their own right, and moreover if the stationary states are classified and understood, then it becomes easier to solve the time-dependent Schrödinger equation for any state.

The time-independent Schrödinger equation is the equation describing stationary states.

The following equation expresses an affine transformation in GF ( 2 < sup > 8 </ sup >) ( with "+" representing XOR ):

While the complex form has an imaginary component, after the necessary calculations are performed in the complex plane, its real value can be extracted giving a real valued equation representing an actual plane wave.

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.

Their accuracy does not change when new theories are worked out, but rather the scope of application, since the equation ( if any ) representing the law does not change.

This is done by defining a sequence of value functions V < sub > 1 </ sub >, V < sub > 2 </ sub >, ..., V < sub > n </ sub >, with an argument y representing the state of the system at times i from 1 to n. The definition of V < sub > n </ sub >( y ) is the value obtained in state y at the last time n. The values V < sub > i </ sub > at earlier times i = n − 1, n − 2, ..., 2, 1 can be found by working backwards, using a recursive relationship called the Bellman equation.

For instance, this occurs in linear systems if one equation is a simple multiple of the other ( representing the same line, e. g. 2x + y

# representing the general structure to an equation,

For instance, in the equation below, the growth of population is a function of the minimum of three Michaelis-Menten terms representing limitation by factors, and.

The equation of time is also the east or west component of the analemma, a curve representing the angular offset of the Sun from its mean position on the celestial sphere as viewed from Earth.

In 1969, Velo and Zwanziger showed that the Rarita – Schwinger lagrangian coupled to electromagnetism leads to equation with solutions representing wavefronts, some of which propagate faster than light.

Because the last part of the equation is non-zero only if, the equation is usually solved by representing it as a matrix for, thus getting equation

Between the perception of target and cursor and the construction of the signal representing the distance between them there is a delay of Τ milliseconds, so that the working perceptual signal at time t represents the target-to-cursor distance at a prior time, t – Τ. Consequently, the equation used in the model is

Despite these large assumptions, the groundwater flow equation does a good job of representing the distribution of heads in aquifers due to a transient distribution of sources and sinks.

where is a constant, and the Einstein tensor on the left side of the equation is equated to the stress-energy tensor representing the energy and momentum present in the spacetime.

Pourbaix Diagrams are thermodynamic charts constructed using the Nernst equation and visualize the relationship between possible phases of a system, bounded by lines representing the reactions that transport between them.

Stone gives an example of this: when computing the roots of a quadratic equation the computor must know how to take a square root.

This equation shows that output is proportional to the square of the length of a side.

So for the gravitational force — or, more generally, for any inverse square force law — the right hand side of the equation becomes a constant and the equation is seen to be the harmonic equation ( up to a shift of origin of the dependent variable ).

Formulas for expressing the roots of polynomials of degree 2 in terms of square roots have been known since ancient times ( see quadratic equation ), and for polynomials of degree 3 or 4 similar formulas ( using cube roots in addition to square roots ) were found in the 16th century ( see cubic function and quartic function for the formulas and Niccolo Fontana Tartaglia, Lodovico Ferrari, Gerolamo Cardano, and Vieta for historical details ).

Lagrange proved that for any natural number n that is not a perfect square there are x and y > 0 that satisfy Pell's equation.

These solutions yield good rational approximations of the form x / y to the square root of n. In Cartesian coordinates, the equation has the form of a hyperbola ; it can be seen that solutions occur where the curve has integral ( x, y ) coordinates.

408 are two solutions to the Pell equation, and gave very close approximations to the square root of two.

Later, Archimedes used a similar equation to approximate the square root of 3, and found 1351 / 780.

The intensity ( in watts per square meter ) of the stimulated emission is governed by the following differential equation:

The converse is not true: not all irrational numbers are transcendental, e. g. the square root of 2 is irrational but not a transcendental number, since it is a solution of the polynomial equation x < sup > 2 </ sup > − 2

For example, the square root of 2 is irrational and not transcendental ( because it is a solution of the polynomial equation x < sup > 2 </ sup > − 2 = 0 ).

When v is larger than c, the denominator in the equation for the energy is " imaginary ", as the value inside the square root is negative.

replace p by its operator equivalent, expand the square root in an infinite series of derivative operators, set up an eigenvalue problem, then solve the equation formally by iterations.

: Why is there no formula for the roots of a fifth ( or higher ) degree polynomial equation in terms of the coefficients of the polynomial, using only the usual algebraic operations ( addition, subtraction, multiplication, division ) and application of radicals ( square roots, cube roots, etc )?

As has long been known, the equation has a solution in the unknowns precisely when the parameter is 0 or not a perfect square.

This observation means that if is a square matrix and has no vanishing singular value, the equation has no non-zero as a solution.

The solutions of any second-degree polynomial equation can be expressed in terms of addition, subtraction, multiplication, division, and square roots, using the familiar quadratic formula: The roots of the following equation are shown below:

In equation ( 1 ), and ( 2 ), the first terms fall off as the inverse square of the distance from the particle, and this first term is called the generalized Coulomb field or velocity field.

For sharp-cornered bluff bodies, like square cylinders and plates held transverse to the flow direction, this equation is applicable with the drag coefficient as a constant value when the Reynolds number is greater than 1000 .< ref >