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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.
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generalized and Euclidean
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.
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.
The generalized Euclidean algorithm identifies the greatest common right or left divisor of two Hurwitz quaternions, where the " size " of the remainder is measured by the norm.
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 ;
generalized and 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.
In 1992, Deutsch and Jozsa produced a deterministic algorithm which was generalized to a function which takes bits for its input.
generalized and can
Miriam had not yet goaded him into mentioning her directly, but one can feel the generalized anger in Wright's remarks to reporters when he was asked, one morning on arrival in Chicago, what he thought of the city as a whole.
Although the Af calculation is obvious by analogy with that for gravitational field and osmotic pressure, it is interesting to confirm it by a method which can be generalized to include related effects.
Note that the acid can be the charged species and the conjugate base can be neutral in which case the generalized reaction scheme could be written as HA < sup >+</ sup > H < sup >+</ sup > + A.
The definition works without any changes if instead of vector spaces over a field F, we use modules over a commutative ring R. It also can be easily generalized to n-ary functions, where the proper term is multilinear.
The effects of this study can be observed in humans with amnesia, indicating the role of the hippocampus in developing episodic memories that can be generalized to similar situations.
It can also be generalized to explain variables across the economy, for example, total output ( estimated as real GDP ) and the general price level, as studied in macroeconomics.
η is the symbol for the linear predictor of a generalized linear model, and can also be used to denote the median of a population.
In the presence of non-spherical errors, the generalized least squares estimator can be shown to be BLUE.
What can be even more confusing is that certain syndromes, such as West syndrome, featuring seizures such as infantile spasms, can be classified as idiopathic, syndromic, or cryptogenic depending on cause and can arise from both focal or generalized epileptic lesions.