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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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## Some Related Sentences

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**.**0.195 seconds.