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Formally, Gaussian integers are the set

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## Some Related Sentences

Formally and Gaussian

Formally and integers

__Formally__

**,**

**the**theorem is stated as follows: There exist unique

__integers__q and r such that a = qd + r and 0 ≤ r < | d |, where | d | denotes

**the**absolute value of d.

Formally and are

However

**,**shortly after this positive result**,**Kurt Gödel published On__Formally__Undecidable Propositions of Principia Mathematica and Related Systems ( 1931 ), showing that in any sufficiently strong axiomatic system there__are__true statements which cannot be proved in**the**system.__Formally__

**,**as per

**the**2002 Memorandum of Understanding between

**the**BSI and

**the**United Kingdom Government

**,**British Standards

__are__defined as:

__Formally__speaking

**,**a collation method typically defines a total order on a

**set**of possible identifiers

**,**called sort keys

**,**which consequently produces a total preorder on

**the**

**set**of items of information ( items with

**the**same identifier

__are__not placed in any defined order ).

__Formally__

**,**this sharing of dynamics is referred to as universality

**,**and systems with precisely

**the**same critical exponents

__are__said to belong to

**the**same universality class.

Limits and colimits in a category C

__are__defined by means of diagrams in C.__Formally__**,**a diagram of type J in C is a functor from J to C:__Formally__

**,**collective noun forms such as “ a group of people ”

__are__represented by second-order variables

**,**or by first-order variables standing for sets ( which

__are__well-defined objects in mathematics and logic ).

__Formally__

**,**we

__are__given a

**set**of hypotheses and a

**set**of manifestations ; they

__are__related by

**the**domain knowledge

**,**represented by a function that takes as an argument a

**set**of hypotheses and gives as a result

**the**corresponding

**set**of manifestations.

__Formally__

**,**two variables

__are__inversely proportional ( or varying inversely

**,**or in inverse variation

**,**or in inverse proportion or in reciprocal proportion ) if one of

**the**variables is directly proportional with

**the**multiplicative inverse ( reciprocal ) of

**the**other

**,**or equivalently if their product is a constant.

__Formally__

**,**they

__are__partial derivatives of

**the**option price with respect to

**the**independent variables ( technically

**,**one Greek

**,**gamma

**,**is a partial derivative of another Greek

**,**called delta ).

__Formally__

**,**

**the**word is applied to persons who

__are__publicly accepted in a recognised capacity

**,**such as professional employment

**,**graduation from a course of study

**,**etc., to give critical commentaries in one or any of a number of specific fields of public or private achievement or endeavour.

(

__Formally__speaking**,**this then satisfies**the**premises of an axiom of well-founded induction**,**which asserts that these two conditions__are__sufficient for**the**proposition to hold for all x.__Formally__

**,**

**the**outcomes Y < sub > i </ sub >

__are__described as being Bernoulli-distributed data

**,**where each outcome is determined by an unobserved probability p < sub > i </ sub > that is specific to

**the**outcome at hand

**,**but related to

**the**explanatory variables.

__Formally__this approximation is founded on

**the**variational principle

**,**valid for Hamiltonians that

__are__bounded from below.

__Formally__

**,**an analytic function ƒ ( z ) of

**the**real or complex variables z < sub > 1 </ sub >,…, z < sub > n </ sub > is transcendental if z < sub > 1 </ sub >, …, z < sub > n </ sub >, ƒ ( z )

__are__algebraically independent

**,**i. e., if ƒ is transcendental over

**the**field C ( z < sub > 1 </ sub >, …, z < sub > n </ sub >).

__Formally__most of these approaches

__are__similar to an artificial neural network

**,**as inputs to a node

__are__summed up and

**the**result serves as input to a sigmoid function

**,**e. g., but proteins do often control gene expression in a synergistic

**,**i. e. non-linear

**,**way.

__Formally__

**,**

**the**sets of free and bound names of a process in π – calculus

__are__defined inductively as follows.

Formally and set

__Formally__

**,**

**the**

__set__of all context-free languages is identical to

**the**

__set__of languages accepted by pushdown automata ( PDA ).

__Formally__

**,**if M is a

__set__

**,**

**the**identity function f on M is defined to be that function with domain and codomain M which satisfies

__Formally__

**,**according to

**the**Constitution

**,**citizens of Turkmenistan have

**the**right to

__set__up political parties and other public associations

**,**acting within

**the**framework of

**the**Constitution and laws

**,**and public associations and groups of citizens have

**the**right to nominate their candidates in accordance with

**the**election law.

__Formally__

**,**

**the**convex hull may be defined as

**the**intersection of all convex sets containing X or as

**the**

__set__of all convex combinations of points in X.

__Formally__

**,**

**the**discrete cosine transform is a linear

**,**invertible function ( where denotes

**the**

__set__of real numbers ), or equivalently an invertible N × N square matrix.

__Formally__

**,**a function ƒ is real analytic on an open

__set__D in

**the**real line if for any x < sub > 0 </ sub > in D one can write

__Formally__

**,**

**the**movement is a rondo that acts as

**the**theme for a

__set__of eight variations

**,**capped off by a dramatic coda.

__Formally__

**,**

**the**discrete sine transform is a linear

**,**invertible function F: R < sup > N </ sup > < tt >-></ tt > R < sup > N </ sup > ( where R denotes

**the**

__set__of real numbers ), or equivalently an N × N square matrix.

__Formally__

**,**

**the**discrete Hartley transform is a linear

**,**invertible function H: R < sup > n </ sup > < tt >-></ tt > R < sup > n </ sup > ( where R denotes

**the**

__set__of real numbers ).

__Formally__

**,**a constraint satisfaction problem is defined as a triple

**,**where is a

__set__of variables

**,**is a domain of values

**,**and is a

__set__of constraints.

Gaussian and integers

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.
This paper introduced what is now called

**the**ring of__Gaussian____integers__**,****the****set**of all complex numbers a + bi where a and b**are**__integers__.
If R is a Euclidean domain in which euclidean division is given algorithmically ( as is

**the**case for instance when R = F where F is a field**,**or when R is**the**ring of__Gaussian____integers__), then greatest common divisors can be computed using a form of**the**Euclidean algorithm based on**the**division procedure.
An example of such a domain is

**the**__Gaussian____integers__Z**,**that is**,****the****set**of complex numbers of**the**form a + bi where i denotes**the**imaginary unit and a and b**are**arbitrary__integers__.
In number theory

**,**a__Gaussian__integer is a complex number whose real and imaginary part**are**both__integers__.
The

__Gaussian____integers__**,**with ordinary addition and multiplication of complex numbers**,**form an integral domain**,**usually written as Z.
Note that when they

**are**considered within**the**complex plane**the**__Gaussian____integers__may be seen to constitute**the**2-dimensional integer lattice.
The

__Gaussian____integers__form a principal ideal domain with units 1**,**− 1**,**i**,**and − i. If x is a__Gaussian__integer**,****the**four numbers x**,**ix**,**− x**,**and − ix**are**called**the**associates of x.
The ring of

__Gaussian____integers__is**the**integral closure of Z in**the**field of__Gaussian__rationals Q ( i ) consisting of**the**complex numbers whose real and imaginary part**are**both rational.
The ring of

__Gaussian____integers__was introduced by Carl Friedrich Gauss in his second monograph on quartic reciprocity ( 1832 ) ( see ).
Gauss discovered that

**the**law of biquadratic reciprocity and its supplements were more easily stated and proved as statements about " whole complex numbers " ( i. e.**the**__Gaussian____integers__) than they**are**as statements about ordinary whole numbers ( i. e.**the**__integers__).0.159 seconds.