The equation was simple: wealth brought them happiness, and their united front to the world was their warning that they meant to keep everything they had, let no one in on the secrets.

The radiation loss from the anode surface was computed according to Af where Af is the mean of the fourth powers of the temperatures Af and Af calculated analogously to equation ( 1 ).

Apart from the personal equation, another reason advanced in favor of the meeting was that too often in the past the U.S. appeared to have been dragged reluctantly to the summit.

Three of the four persons present, all foreign students in Tokyo, had been playing a game of judging popular Japanese foods by the In and Out system, an equation in which Zen philosophy was used as the modifier.

) TAI was henceforth a realisation of TT, with the equation TT ( TAI )

The study of altruism was the initial impetus behind George R. Price's development of the Price equation, which is a mathematical equation used to study genetic evolution.

With de Broglie's suggestion of the existence of electron matter waves in 1924, and for a short time before the full 1926 Schrödinger equation treatment of hydrogen like atom, a Bohr electron " wavelength " could be seen to be a function of its momentum, and thus a Bohr orbiting electron was seen to orbit in a circle at a multiple of its half-wavelength ( this historically incorrect Bohr model is still occasionally taught to students ).

One simple version of adaptive expectations is stated in the following equation, where is the next year's rate of inflation that is currently expected ; is this year's rate of inflation that was expected last year ; and is this year's actual rate of inflation:

In 1930, Stoner derived the internal energy-density equation of state for a Fermi gas, and was then able to treat the mass-radius relationship in a fully relativistic manner, giving a limiting mass of approximately ( for μ < sub > e </ sub >= 2. 5 ) 2. 19 · 10 < sup > 30 </ sup > kg.

The mathematical properties of the catenary curve were first studied by Robert Hooke in the 1670s, and its equation was derived by Leibniz, Huygens and Johann Bernoulli in 1691.

His work was a key aspect of Hermann Weyl and John von Neumann's work on the mathematical equivalence of Werner Heisenberg's matrix mechanics and Erwin Schrödinger's wave equation and his namesake Hilbert space plays an important part in quantum theory.

Hilbert's tenth problem was to determine whether a given polynomial Diophantine equation with integer coefficients has an integer solution.

The equation was eventually solved by Euler in the early 18th century, who also solved a number of other Diophantine equations.

One solution was all he looked for in a quadratic equation.

Using these modelling techniques, Malthus ' population principle of growth was later transformed into a model known as the logistic equation:

At 15, he was reading the original papers of Joseph Louis Lagrange, such as the landmark Réflexions sur la résolution algébrique des équations which likely motivated his later work on equation theory, and Leçons sur le calcul des fonctions, work intended for professional mathematicians, yet his classwork remained uninspired, and his teachers accused him of affecting ambition and originality in a negative way.

He submitted his memoir on equation theory several times, but it was never published in his lifetime due to various events.

Boyle's Law was perhaps the first expression of an equation of state.

His new formula revolutionized the study of equations of state, and was most famously continued via the Redlich – Kwong equation of state and the Soave modification of Redlich – Kwong.

Proposed in 1873, the van der Waals equation of state was one of the first to perform markedly better than the ideal gas law.

Introduced in 1949 the Redlich – Kwong equation of state was a considerable improvement over other equations of the time.

In 1923, while writing the appendix for the Italian edition of the book The Mathematical Theory of Relativity by A. Kopff, Enrico Fermi pointed out, for the first time, that hidden inside the famous Einstein equation (), there was an enormous amount of nuclear potential energy to be exploited.

“ The symbolism that Diophantus introduced for the first time, and undoubtedly devised himself, provided a short and readily comprehensible means of expressing an equation ...

This “ field gradient pulse ” method was initially devised for NMR by Stejskal and Tanner who derived the reduction in signal due to the application of the pulse gradient related to the amount of diffusion that is occurring through the following equation:

The equation was formulated by Dr. Frank Drake in 1961, a decade after the objections raised by Enrico Fermi, in an attempt to find a systematic means to evaluate the numerous probabilities involved in alien life.

Bitter over Soviet anti-semitism, of the five pioneer investigators of the field, Cocconi, Morrison, Cornell University's Frank Drake ( of the 1961 Project Ozma and the Drake equation ), Sagan, and Shklovsky, Shklovsky was quite aware of sharing his Jewish identity with both Morrison and Sagan.

Multiplication, subtraction, and addition can then be accomplished as they appear in the equation by starting at the left end of the equation and working toward the right.

The propagation of sound waves in a fluid ( such as water ) can be modeled by an equation of motion ( conservation of momentum ) and an equation of continuity ( conservation of mass ).

The equation for the conservation of mass in a fluid volume ( without any mass sources or sinks ) is given by

Intermediate results were binary, written onto paper sheets by electrostatically modifying the resistance at 1500 locations to represent 30 of the 50 bit numbers ( one equation ).

If it is assumed that all color values are premultiplied by their alpha values (), we can rewrite the equation for output color as:

The oblique asymptote, for the function f ( x ), will be given by the equation y = mx + n.

Subsequently, the accelerative force on any given ion is controlled by the electrostatic equation, where n is the ionisation state of the ion, and e is the fundamental electric charge.

If the heat of vaporization and the vapor pressure of a liquid at a certain temperature is known, the normal boiling point can be calculated by using the Clausius-Clapeyron equation thus:

In mathematics, Bessel functions, first defined by the mathematician Daniel Bernoulli and generalized by Friedrich Bessel, are canonical solutions y ( x ) of Bessel's differential equation:

The Bessel functions of the second kind, denoted by Y < sub > α </ sub >( x ), occasionally denoted instead by N < sub > α </ sub >( x ), are solutions of the Bessel differential equation.

Another important formulation of the two linearly independent solutions to Bessel's equation are the Hankel functions H < sub > α </ sub >< sup >( 1 )</ sup >( x ) and H < sub > α </ sub >< sup >( 2 )</ sup >( x ), defined by:

Because Bessel's equation becomes Hermitian ( self-adjoint ) if it is divided by x, the solutions must satisfy an orthogonality relationship for appropriate boundary conditions.

The number of atoms on the left and the right in the equation for a chemical transformation is equal ( when unequal, the transformation by definition is not chemical, but rather a nuclear reaction or radioactive decay ).

Despite the failure of this derivation, the equilibrium constant for a reaction is indeed a constant, independent of the activities of the various species involved, though it does depend on temperature as observed by the van't Hoff equation.

The effect on cosmology of the dark energy that these models describe is given by the dark energy's equation of state, which varies depending upon the theory.

These two specifications are related through the material density by the equation.

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 total energy is determined by approximate solutions of the time-dependent Schrödinger equation, usually with no relativistic terms included, and by making use of the Born – Oppenheimer approximation, which allows for the separation of electronic and nuclear motions, thereby simplifying the Schrödinger equation.