The result can be considered as a type of generalized geometry, projective geometry, but it can also be used to produce proofs in ordinary Euclidean geometry in which the number of special cases is reduced.

This generalized Euclidean algorithm can be put to many of the same uses as Euclid's original algorithm in the ring of integers: in any Euclidean domain, one can apply the Euclidean algorithm to compute the greatest common divisor of any two elements.

Some authors also require the domain of the Euclidean function be the entire ring R ; this can always be accommodated by adding 1 to the values at all nonzero elements, and defining the function to be 0 at the zero element of R, but the result is somewhat awkward in the case of K. The definition is sometimes generalized by allowing the Euclidean function to take its values in any well-ordered set ; this weakening does not affect the most important implications of the Euclidean property.

The Euclidean algorithm has been generalized further to other mathematical structures, such as knots and multivariate polynomials.

This generalized definition implies that the above mentioned geometric entities are a special kind of vectors, as they are elements of a special kind of vector space called Euclidean space.

With the latter definition, convex hulls may be extended from Euclidean spaces to arbitrary real vector spaces ; they may also be generalized further, to oriented matroids.

It was also during this period that the differentiation was generalized to Euclidean space and the complex plane.

This concept can be generalized to n-dimensional Euclidean space, with the singular values of any n × n square matrix being viewed as the semiaxes of an n-dimensional ellipsoid.

Though this page chiefly treats del in three dimensions, this definition can be generalized to the n-dimensional Euclidean space R < sup > n </ sup >.

The concept has been generalized to differential manifolds of arbitrary dimension embedded in a Euclidean space.

This can be generalized to model an n + 1 dimensional hyperbolic space by replacing the real number x by a vector in an n dimensional Euclidean vector space.

This idea is easily generalized to three dimensional Euclidean space, where a point is represented by an ordered triplet (,, ) with the additional third number representing depth and often denoted by.

However, the Garden of Eden theorem can be generalized beyond Euclidean spaces, to cellular automata defined on the elements of an amenable group or a sofic group ; the proof of this generalization uses the Ax – Grothendieck theorem, an analogous relation between injectivity and bijectivity in algebraic geometry.

The Potts model consists of spins that are placed on a lattice ; the lattice is usually taken to be a two-dimensional rectangular Euclidean lattice, but is often generalized to other dimensions or other lattices.

Frequently this is used in 2d or 3d, but can be easily generalized to Euclidean spaces in any number of dimensions.

* w ( x ) is the number of variations of the generalized Sturm chain obtained from and ( by successive Euclidean divisions ) where for a real y.

In differential geometry, the Laplace operator, named after Pierre-Simon Laplace, can be generalized to operate on functions defined on surfaces in Euclidean space and, more generally, on Riemannian and pseudo-Riemannian manifolds.

* w ( x ) is the number of variations of the generalized Sturm chain obtained from and by applying the Euclidean algorithm ;

Church subsequently modified his methods to include use of Herbrand – Gödel recursion and then proved ( 1936 ) that the Entscheidungsproblem is unsolvable: There is no generalized " effective calculation " ( method, algorithm ) that can determine whether or not a formula in either the recursive-or λ-calculus is " valid " ( more precisely: no method to show that a well formed formula has a " normal form ").

The original algorithm was described only for natural numbers and geometric lengths ( real numbers ), but the algorithm was generalized in the 19th century to other types of numbers, such as Gaussian integers and polynomials in one variable.

generalized the algorithm to digital waveguide synthesis, which could also be used to model acoustic waves in tubes and on drum membranes.

This generalized version of the game is NP-complete ; it is unlikely that any algorithm more efficient than a brute-force search exists that can find solutions for arbitrary generalized FreeCell configurations.

replaced this with a more efficient idea that later generalized as the " subgroup algorithm " ( in which form it works just as well for permutations and rotations ).

The Rete algorithm provides a generalized logical description of an implementation of functionality responsible for matching data tuples (" facts ") against productions (" rules ") in a pattern-matching production system ( a category of rule engine ).

Its original version, due to Gary L. Miller, is deterministic, but the determinism relies on the unproven generalized Riemann hypothesis ; Michael O. Rabin modified it to obtain an unconditional probabilistic algorithm.

One drawback is that, like Freenet, DHTs only directly support exact-match search, rather than keyword search, although Freenet's routing algorithm can be generalized to any key type where a closeness operation can be defined.

A generalized algorithm to correct for the pressure level difference between the finger and brachial sites within an individual patient was developed and this correction worked under all circumstances that it was tested, even when it was not designed for it, since it applied general physiological principles.

Bruun's algorithm is a fast Fourier transform ( FFT ) algorithm based on an unusual recursive polynomial-factorization approach, proposed for powers of two by G. Bruun in 1978 and generalized to arbitrary even composite sizes by H. Murakami in 1996.

The Bruun factorization, and thus the Bruun FFT algorithm, was generalized to handle arbitrary even composite lengths, i. e. dividing the polynomial degree by an arbitrary radix ( factor ), as follows.

Lloyd's Method I algorithm, originally described in 1957, can be generalized in a straighforward way for application to vector data.

While Peterson's original formulation worked with only two processes, the algorithm can be generalized for more than two.

A generalized Feistel algorithm can be used to create strong permutations on small domains of size not a power of two ( see Format-Preserving Encryption ).

This is further generalized by DNA sequence alignment algorithms such as the Smith – Waterman algorithm, which make an operation's cost depend on where it is applied.

* 1996-Bruun's algorithm generalized to arbitrary even composite sizes by H. Murakami

In 1992, Deutsch and Jozsa produced a deterministic algorithm which was generalized to a function which takes bits for its input.

It is distinguishable from the formal written contract of employment which, for the most part, only identifies mutual duties and responsibilities in a generalized form.

One source identifies five very generalized steps, increasing from least use of force to greatest.

Luxemburg's idea of democracy, which Stanley Aronowitz calls " generalized democracy in an unarticulated form ", represents Luxemburgism's greatest break with " mainstream communism ", since it effectively diminishes the role of the Communist Party, but is in fact very similar to the views of Karl Marx (" The emancipation of the working classes must be conquered by the working classes themselves ").

The psychologist David H. Barlow of Boston University conducted a study that showed three common characteristics of people suffering from chronic anxiety, which he characterized as " a generalized biological vulnerability ," " a generalized psychological vulnerability ," and " a specific psychological vulnerability.

Although many board games have a jargon all their own, there is a generalized terminology to describe concepts applicable to basic game mechanics and attributes common to nearly all board games.

The most common seizures are myoclonic jerks, although generalized tonic-clonic seizures and absence seizures may occur as well.

Surgery ( thyroidectomy to remove the whole thyroid or a part of it ) is not extensively used because most common forms of hyperthyroidism are quite effectively treated by the radioactive iodine method, and because there is a risk of also removing the parathyroid glands, and of cutting the recurrent laryngeal nerve, making swallowing difficult, and even simply generalized staphylococcal infection as with any major surgery.

The most common such generalized counting function is the Chebyshev function, defined by

To that, Durkheim answered that any apparent cultural diversity is overridden by a larger, common, and more generalized cultural system, and the law.

The following is a generalized description of how a lawsuit may proceed in a common law jurisdiction:

Sustained generalized seizure activity and angioedema are other common causes of life-threatening airway obstruction which may require tracheal intubation to secure the airway.

Clinical depression, obsessive-compulsive disorder ( OCD ), phobias, and somatization disorder are the most common accompanying conditions in people with hypochondriasis, as well as a generalized anxiety disorder diagnosis at some point in their life.

Among relatives of people suffering from hypochondriasis only somatization disorder and generalized anxiety disorder were more common than in average families.

Lower mould and upper mould are more generalized descriptors than more common and specific terms such as male side, female side, a-side, b-side, tool side, bowl, hat, mandrel, etc.

The fact is that even in arithmetic the two processes of multiplication and division are generalized into a common multiplication ; and the difficulty consists in passing from the original idea of multiplication to the generalized idea of a tensor, which idea includes compressing the magnitude as well as stretching it.

In his view, Sartre gives such a generalized account of revolutionary organization that he ignores the real difficulties of groups engaging in common action without infringing the freedom of their individual members.

A bundle map from the base space itself ( with the identity mapping as projection ) to E is called a section of E. Fiber bundles can be generalized in a number of ways, the most common of which is requiring that the transition between the local trivial patches should lie in a certain topological group, known as the structure group, acting on the fiber F.

Given a set S of matrices, each of which is diagonalizable, and any two of which commute, it is always possible to simultaneously diagonalize all of the elements of S. Equivalently, for any set S of mutually commuting semisimple linear transformations of a finite-dimensional vector space V there exists a basis of V consisting of simultaneous eigenvectors of all elements of S. Each of these common eigenvectors v ∈ V, defines a linear functional on the subalgebra U of End ( V ) generated by the set of endomorphisms S ; this functional is defined as the map which associates to each element of U its eigenvalue on the eigenvector v. This " generalized eigenvalue " is a prototype for the notion of a weight.

For instance, they often make use of " generalized cultural comparisons ," taking various artefacts and monuments from one society, and highlighting similarities with those of another to support a conclusion that both had a common source — typically an ancient lost civilisation like Atlantis, Mu, or an extraterrestrial influence.

Gaston Deferre ’ s loi-cadre of the 23 June 1956 generalized universal suffrage throughout the territories d ’ outre-mer and based their assemblies on a common voting-roll.

* Contrary to common opinion Edison made use of the scientific method but in a way that was limited to the task at hand and did not seek to develop generalized theories.

The method of usable feasible directions, Rosen ’ s gradient projection ( generalized reduce gradient ) method, sequential unconstrained minimization techniques, sequential linear programming and eventually sequential quadratic programming methods were common choices.

The realization that soft matter contains innumerable examples of symmetry breaking, generalized elasticity, and many fluctuating degrees of freedom has re-invigorated classical fields of physics such as fluids ( now generalized to non-Newtonian and structured media ) and elasticity ( membranes, filaments, and anisotropic networks are all important and have common aspects ).