The best vertex degree characterization of Hamiltonian graphs was provided in 1972 by the Bondy – Chvátal theorem, which generalizes earlier results by G. A. Dirac ( 1952 ) and Øystein Ore.

The Deutsch – Jozsa Algorithm generalizes earlier ( 1985 ) work by David Deutsch, which provided a solution for the simple case.

The theorem is named after Witold Hurewicz, and generalizes earlier results of Henri Poincaré.

This generalizes an analogous fact for normal operators.

the distribution of Frobenius conjugacy classes in Galois groups over Q ( or, more generally, Galois groups over any number field ) generalizes Dirichlet's classical result about primes in arithmetic progressions.

In Fredholm theory, this result generalizes to integral operators on multi-dimensional spaces, including, for example, Riemannian manifolds.

This 2007 result by Katz, Schaps, and Vishne generalizes the results of Peter Sarnak and Peter Buser in the case of arithmetic groups defined over Q, from their seminal 1994 paper ( see below ).

Model theory generalizes the notion of algebraic extension to arbitrary theories: an embedding of M into N is called an algebraic extension if for every x in N there is a formula p with parameters in M, such that p ( x ) is true and the set

Compactness generalizes many important properties of closed and bounded intervals in the real line ; that is, intervals of the form for real numbers and.

Faà di Bruno's formula for higher-order derivatives of single-variable functions generalizes to the multivariable case.

In this, it generalizes maximization approaches developed to analyze market actors such as in the supply and demand model and allows for incomplete information of actors.

The above development generalizes in a more-or-less straightforward fashion to general principal G-bundles for some arbitrary Lie group G taking the place of U ( 1 ).

For instance, Fermat's little theorem for the nonzero integers modulo a prime generalizes to Euler's theorem for the invertible numbers modulo any nonzero integer, which generalizes to Lagrange's theorem for finite groups.

Since there is a one-to-one correspondence between Borel regular measures in the interval and functions of bounded variation ( that assigns to each function of bounded variation the corresponding Lebesgue-Stieltjes measure, and the integral with respect to the Lebesgue-Stieltjes measure agrees with the Riemann-Stieltjes integral for continuous functions ), the above stated theorem generalizes the original statement of F. Riesz.

Thinking of matrices as tensors, the tensor rank generalizes to arbitrary tensors ; note that for tensors of order greater than 2 ( matrices are order 2 tensors ), rank is very hard to compute, unlike for matrices.

This criticism of the application of the model of supply and demand generalizes, particularly to all markets for factors of production.

The notion of paracompactness generalizes ordinary compactness ; a key motivation for the notion of paracompactness is that it is a sufficient condition for the existence of partitions of unity.

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 generalizes the notion of geodesic for Riemannian manifolds.

Thinking of curvature as a measure, rather than as a function, Descartes ' theorem is Gauss – Bonnet where the curvature is a discrete measure, and Gauss – Bonnet for measures generalizes both Gauss – Bonnet for smooth manifolds and Descartes ' theorem.

Then the existence of an eigenvalue is equivalent to the ideal generated by ( the relations satisfied by ) being non-empty, which exactly generalizes the usual proof of existence of an eigenvalue existing for a single matrix over an algebraically closed field by showing that the characteristic polynomial has a zero.

The coefficient of determination generalizes the correlation coefficient for relationships beyond simple linear regression.

In its simplest form, ANOVA provides a statistical test of whether or not the means of several groups are all equal, and therefore generalizes t-test to more than two groups.

* Formally, when working over the reals, as here, this is accomplished by considering the limit as ε → 0 ; but the " infinitesimal " language generalizes directly to Lie groups over general rings.

The Poincaré series generalizes the theta series to automorphic forms with respect to arbitrary Fuchsian groups.

The theorem generalizes the Jordan – Hölder decomposition for finite groups ( in which the primes are the finite simple groups ), to all finite transformation semigroups ( for which the primes are again the finite simple groups plus all subsemigroups of the " flip-flop " ( see above ).

The solution for the hidden subgroup problem in the abelian case generalizes to finding the normal core in case of subgroups of arbitrary groups.

This theorem generalizes to projective groups of higher dimensions and gives an important infinite family PSL ( n, q ) of finite simple groups.

But unlike the constraint-based gauge fixing procedures above, the R < sub > ξ </ sub > gauge generalizes well to non-abelian gauge groups such as the SU ( 3 ) of QCD.

This definition generalizes to direct sums of finitely many abelian groups.

In algebra, association schemes generalize groups, and the theory of association schemes generalizes the character theory of linear representations of groups.

ANOVA generalizes to the study of the effects of multiple factors.

Thus, what is known as the extreme value theorem in calculus generalizes to compact spaces.

Faà di Bruno's formula generalizes the chain rule to higher derivatives.

This proof has the advantage that it generalizes to several variables.

Two ideals A and B in the commutative ring R are called coprime ( or comaximal ) if A + B = R. This generalizes Bézout's identity: with this definition, two principal ideals ( a ) and ( b ) in the ring of integers Z are coprime if and only if a and b are coprime.

The construction also generalizes to locally ringed spaces.

Riemannian geometry generalizes Euclidean geometry to spaces that are not necessarily flat, although they still resemble the Euclidean space at each point " infinitesimally ", i. e. in the first order of approximation.

Pseudo-Riemannian geometry generalizes Riemannian geometry to the case in which the metric tensor need not be positive-definite.

Industrial organization generalizes from that special case to study the strategic behavior of firms that do have significant control of price.

This approach brings the tools of algebra and calculus to bear on questions of geometry, and has the advantage that it generalizes easily to Euclidean spaces of more than three dimensions.

The Hausdorff dimension generalizes the notion of the dimension of a real vector space.

The notion of irreducible fraction generalizes to the field of fractions of any unique factorization domain: any element of such a field can be written as a fraction in which denominator and numerator are coprime, by dividing both by their greatest common divisor.

However, the principle of special relativity generalizes the notion of inertial frame to include all physical laws, not simply Newton's first law.

This principle generalizes the notion of an inertial frame.

Such a definition generalizes to the concept of an identity morphism in category theory, where the endomorphisms of M need not be functions.

The above discussion generalizes readily to the case of N particles.

FIPS PUB 198 generalizes and standardizes the use of HMACs.