In mathematics, a Diophantine equation is an equation of the form P ( x < sub > 1 </ sub >, ..., x < sub > j </ sub >, y < sub > 1 </ sub >, ..., y < sub > k </ sub >)= 0 ( usually abbreviated P (,)= 0 ) where P (,) is a polynomial with integer coefficients.

For historical reasons and in order to have application to the solution of Diophantine equations, results in number theory have been scrutinised more than in other branches of mathematics to see if their content is effectively computable.

In mathematics, diophantine geometry is one approach to the theory of Diophantine equations, formulating questions about such equations in terms of algebraic geometry over a ground field K that is not algebraically closed, such as the field of rational numbers or a finite field, or more general commutative ring such as the integers.

In mathematics, the Littlewood conjecture is an open problem () in Diophantine approximation, proposed by John Edensor Littlewood around 1930.

In mathematics, a Thue equation is a Diophantine equation of the form

In mathematics, in the theory of Diophantine approximation, Weyl's criterion states that a sequence of real numbers is equidistributed mod 1 if and only if for any non-zero integer

In mathematics, in the field of number theory, the Ramanujan – Nagell equation is a particular exponential Diophantine equation.

Initially a study of systems of polynomial equations in several variables, the subject of algebraic geometry starts where equation solving leaves off, and it becomes even more important to understand the intrinsic properties of the totality of solutions of a system of equations, than to find a specific solution ; this leads into some of the deepest areas in all of mathematics, both conceptually and in terms of technique.

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:

In mathematics, it is closely related to the Poisson kernel, which is the fundamental solution for the Laplace equation in the upper half-plane.

The arbitrary assumptions made by Fresnel to arrive at the Huygens – Fresnel equation emerge automatically from the mathematics in this derivation.

which looks the same as in an inertial frame, but now the force F ′ is the resultant of not only F, but also additional terms ( the paragraph following this equation presents the main points without detailed mathematics ):

In mathematics, statistics, and the mathematical sciences, a parameter is a quantity that serves to relate functions and variables using a common variable when such a relationship would be difficult to explicate with an equation.

In mathematics, a quadratic equation is a univariate polynomial equation of the second degree.

In mathematics, a transcendental number is a ( possibly complex ) number that is not algebraic — that is, it is not a root of a non-constant polynomial equation with rational coefficients.

In mathematics, Laplace's equation is a second-order partial differential equation named after Pierre-Simon Laplace who first studied its properties.

His dissertation, entitled, " Solutions of the Mathieu equation of period 4 pi and certain related functions ", was beyond the comprehension of the chemistry and physics faculty, and only when some members of the mathematics department, including the chairman, insisted that the work was good enough that they would grant the doctorate if the chemistry department would not, was he granted a Ph. D. in chemistry in 1935.

Although Dirac did not at first fully appreciate what his own equation was telling him, his resolute faith in the logic of mathematics as a means to physical reasoning, his explanation of spin as a consequence of the union of quantum mechanics and relativity, and the eventual discovery of the positron, represents one of the great triumphs of theoretical physics, fully on a par with the work of Newton, Maxwell, and Einstein before him.

In mathematics, a partial differential equation ( PDE ) is a differential equation that contains unknown multivariable functions and their partial derivatives.

In mathematics, mathematical physics and the theory of stochastic processes, a harmonic function is a twice continuously differentiable function f: U → R ( where U is an open subset of R < sup > n </ sup >) which satisfies Laplace's equation, i. e.

More mundanely, an identity in mathematics may be an equation that holds true for all values of a variable.

In mathematics, Legendre functions are solutions to Legendre's differential equation:

In mathematics, a hyperboloid is a quadric – a type of surface in three dimensions – described by the equation

In mathematics, a recurrence relation is an equation that recursively defines a sequence, once one or more initial terms are given: each further term of the sequence is defined as a function of the preceding terms.

In mathematics, LHS is informal shorthand for the left-hand side of an equation.

This is an unsolved problem which probably has never been seriously investigated, although one frequently hears the comment that we have insufficient specialists of the kind who can compete with the Germans or Swiss, for example, in precision machinery and mathematics, or the Finns in geochemistry.

Next September, after receiving a degree from Yale's Master of Arts in Teaching Program, I will be teaching somewhere -- that much is guaranteed by the present shortage of mathematics teachers.

But because science is based on mathematics doesn't mean that a hot rodder must necessarily be a mathematician.

Like primitive numbers in mathematics, the entire axiological framework is taken to rest upon its operational worth.

In the new situation, philosophy is able to provide the social sciences with the same guidance that mathematics offers the physical sciences, a reservoir of logical relations that can be used in framing hypotheses having explanatory and predictive value.

So, too, is the mathematical competence of a college graduate who has majored in mathematics.

The principal of the school announced that -- despite the help of private tutors in Hollywood and Philadelphia -- Fabian is a 10-o'clock scholar in English and mathematics.

In mathematics and statistics, the arithmetic mean, or simply the mean or average when the context is clear, is the central tendency of a collection of numbers taken as the sum of the numbers divided by the size of the collection.

The term " arithmetic mean " is preferred in mathematics and statistics because it helps distinguish it from other means such as the geometric and harmonic mean.

In addition to mathematics and statistics, the arithmetic mean is used frequently in fields such as economics, sociology, and history, though it is used in almost every academic field to some extent.

The use of the soroban is still taught in Japanese primary schools as part of mathematics, primarily as an aid to faster mental calculation.

In mathematics and computer science, an algorithm ( originating from al-Khwārizmī, the famous Persian mathematician Muḥammad ibn Mūsā al-Khwārizmī ) is a step-by-step procedure for calculations.

In mathematics, the axiom of choice, or AC, is an axiom of set theory equivalent to the statement that " the product of a collection of non-empty sets is non-empty ".

The axiom of choice is avoided in some varieties of constructive mathematics, although there are varieties of constructive mathematics in which the axiom of choice is embraced.

The axiom of choice has also been thoroughly studied in the context of constructive mathematics, where non-classical logic is employed.

:" A choice function exists in constructive mathematics, because a choice is implied by the very meaning of existence.

Although the axiom of countable choice in particular is commonly used in constructive mathematics, its use has also been questioned.

One of the most interesting aspects of the axiom of choice is the large number of places in mathematics that it shows up.

There is no prize awarded for mathematics, but see Abel Prize.