The insight of Langlands was to find the proper generalization of Dirichlet L-functions, which would allow the formulation of Artin's statement in this more general setting.

His generalization of the classical Riemann-Roch theorem launched the study of algebraic and topological K-theory.

The culmination of their investigations, the Arzelà – Ascoli theorem, was a generalization of the Bolzano – Weierstrass theorem to families of continuous functions, the precise conclusion of which was that it was possible to extract a uniformly convergent sequence of functions from a suitable family of functions.

This is a generalization of the Heine – Borel theorem, which states that any closed and bounded subspace S of R < sup > n </ sup > is compact and therefore complete.

If the conjecture were true, it would be a generalization of Fermat's last theorem, which could be seen as the special case n = 2: if, then.

The first generalization of the theorem is found in Gauss's second monograph ( 1832 ) on biquadratic reciprocity.

the Heawood conjecture, a generalization of the four color theorem, which would require seven.

Further, there is a generalization of the Stone – Weierstrass theorem to noncompact Tychonoff spaces, namely, any continuous function on a Tychonoff space is approximated uniformly on compact sets by algebras of the type appearing in the Stone – Weierstrass theorem and described below.

A different generalization of Weierstrass ' original theorem is Mergelyan's theorem, which generalizes it to functions defined on certain subsets of the complex plane.

This modern form of Stokes ' theorem is a vast generalization of a classical result first discovered by Lord Kelvin, who communicated it to George Stokes in July 1850.

Stokes ' theorem is a vast generalization of this theorem in the following sense.

This is a generalization of the inverse function theorem to maps between manifolds.

This can be used to prove Fermat's little theorem and its generalization, Euler's theorem.

Karl Jacobi's generalization of the identity to n bodies and to the present form of Laplace's identity closely resembles the classical virial theorem.

A generalization of classical Gödel completeness theorem is provable in EVŁ.

The proof of the global embedding theorem relies on Nash's far-reaching generalization of the implicit function theorem, the Nash – Moser theorem and Newton's method with postconditioning.

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.

Taylor's theorem also generalizes to multivariate and vector valued functions on any dimensions n and m. This generalization of Taylor's theorem is the basis for the definition of so-called jets which appear in differential geometry and partial differential equations.

This generalization suffered from many exceptions, and was subsequently replaced by the suggestion that valences were fixed at certain oxidation states.

Another generalization states that a faithfully flat morphism locally of finite type with X quasi-compact has a quasi-section, i. e. there exists affine and faithfully flat and quasi-finite over X together with an X-morphism.

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

The Artin reciprocity law, which is a high level generalization of the Gauss quadratic reciprocity law, states that the product vanishes on the multiplicative group of the number field.

* In quantum field theory and string theory, a generalization of coherent states to the case of infinitely many degrees of freedom is used to define a vacuum state with a different vacuum expectation value from the original vacuum.

The generalization of the time-independent perturbation theory to the multi-parameter case can be formulated more systematically using the language of differential geometry, which basically defines the derivatives of the quantum states and calculate the perturbative corrections by taking derivatives iteratively at the unperturbed point.

A generalization of this theorem is the Helmholtz decomposition which states that any vector field can be decomposed as a sum of a solenoidal vector field and an irrotational vector field.

In the mid 1990s, in the study of non-equilibrium dynamics of spin glass models, a generalization of the fluctuation-dissipation theorem was discovered that holds for asymptotic non-stationary states, where the temperature appearing in the equilibrium relation is substituted by an effective temperature with a non-trivial dependence on the time scales.

The generalization rule states that can be derived if y is not mentioned in Γ and x does not occur in φ.

A topological generalization of Radon's theorem states that, if ƒ is any continuous function from a ( d + 1 )- dimensional simplex to d-dimensional space, then the simplex has two disjoint faces whose images under ƒ are not disjoint.

The principle of corresponding states expresses the generalization that the properties of a gas which are dependent on intermolecular forces are related to the critical properties of the gas in a universal way.

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

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.

The generalization of this equation to three arbitrary regular singular points is given by Riemann's differential equation.

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.