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Formally, Liouville numbers and elementary numbers contain the algebraic numbers, and they include some but not all transcendental numbers.

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## Some Related Sentences

Formally and numbers

__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__

**,**an elliptic function is a function meromorphic on for which there exist two non-zero complex

__numbers__

**and**with ( in other words

**,**

**not**parallel ), such that

**and**for

**all**

**.**

__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 natural number n is called highly abundant if

**and**only if for

**all**natural

__numbers__m < n

**,**

Formally and elementary

__Formally__

**,**a cardinal number κ is λ-unfoldable if

**and**only if for every transitive model M of cardinality κ of ZFC-minus-power set such that κ is in M

**and**M contains

**all**its sequences of length less than κ

**,**there is a non-trivial

__elementary__embedding j of M into a transitive model with

**the**critical point of j being κ

**and**j ( κ ) ≥ λ

**.**

Formally and algebraic

__Formally__

**,**an

__algebraic__function in n variables over

**the**field K is an element of

**the**

__algebraic__closure of

**the**field of rational functions K ( x < sub > 1 </ sub >,..., x < sub > n </ sub >).

__Formally__

**,**this is described in

__algebraic__notation like this: ( 19 + 1 ) + ( 15 − 1 ) = x

**,**

**but**even a young student might use this technique without calling it algebra

**.**

Formally and they

__Formally__

**,**arbitrage transactions have negative skew – prices can get a small amount closer (

**but**often no closer than 0 ), while

__they__can get very far apart

**.**

__Formally__

**,**

__they__were part of

**the**non-resident sections

**,**

**but**in fact constituted a separate division

**,**largely autonomous in their activities

**.**

__Formally__

**,**these failed when

__they__were rejected by

**the**Church of England's General Synod in 1972 ; conversations

**and**co-operation continued

**,**however

**,**leading in 2003 to

**the**signing of a covenant between

**the**two churches

**.**

__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__

**,**

__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__reported from November 1944 onwards

**,**witnesses often assumed that

**the**foo fighters were secret weapons employed by

**the**enemy

**,**

**but**

__they__remained unidentified post-war

**and**were reported by both Allied

**and**Axis forces

**.**

By 1912

**,****the**Social Democrats**,**with an explicitly anti-antisemitic program**,**were**the**largest party in**the**German Reichstag**,****and****the**Progressives ran very strongly as well ...__Formally__**,**at least**,****the**Jews had been fully emancipated with**the**establishment of**the**German Empire**,**although__they__were kept out of certain influential occupations**,**enjoyed extraordinary prosperity ... Germans intermarried with Jews: in**the**1930s**some**50**,**000 Jews were living in mixed German-Jewish marriages**,**so at least 50**,**000 Germans**,****and**presumably parts of their families**,**had familial contact with**the**Jews**.**__Formally__

**,**

__they__are appointed by

**the**President for five-year terms coinciding with

**the**term of

**the**House of Representatives

**.**

__Formally__system networks correspond to type lattices in formal lattice theory

**,**although

__they__are occasionally erroneously mistaken for flowcharts or directed decision trees

**.**

__Formally__

**,**these failed when

__they__were rejected by

**the**Church of England's General Synod in 1972

**,**however conversations

**and**co-operation continued leading in 2003 to

**the**signing of a covenant between

**the**two churches

**.**

__Formally__however

__they__define it as any variable that does

**not**directly affect

**the**fundamentals of

**the**economy

**.**

__Formally__

**,**

__they__have always been considered Orthodox Christians

**,**

**but**most of them followed shamanism

**and**animism as a part of their culture

**.**

__Formally__

**,**this meant that

**the**whole peloton finished out of time limits

**,**

**but**

**the**referees understandably used a rule saying that

__they__could give clemency to any group of more than 20 % of

**the**stage's starting riders

**,**officially citing

**the**weather conditions as their reason to do so

**.**

Formally and include

__Formally__classified moons

__include__176 planetary satellites orbiting six of

**the**eight planets

**,**

**and**eight orbiting three of

**the**five IAU-listed dwarf planets

**.**

Other amenities in Saltney

__include__**the**Corner Pin public house**,**a post office**,**Saltney Business Centre**,**St David's Retail Park ( Farmfoods / Poundstretchers / Ableworld ), an Asda Supermarket (__Formally__a Netto store )**and**a Morrisons supermarket**.**__Formally__

**,**

**the**definition only requires

**some**invertibility

**,**so we can substitute for Q any matrix M whose eigenvalues do

**not**

__include__− 1

**.**

Formally and some

__Formally__

**,**

**the**sequence of partial sums of

__some__infinite summation converges if for every fixed power of X

**the**coefficient stabilizes: there is a point beyond which

**all**further partial sums have

**the**same coefficient

**.**

__Formally__

**,**bending modulus is defined as

**the**energy required to deform a membrane from its intrinsic curvature to

__some__other curvature

**.**

__Formally__

**,**this means that

**,**for

__some__function f

**,**

**the**image f ( D ) of a directed set D ( i

**.**e

**.**

**the**set of

**the**images of each element of D ) is again directed

**and**has as a least upper bound

**the**image of

**the**least upper bound of D

**.**One could also say that f preserves directed suprema

**.**

__Formally__a congruence subgroup is one that contains Γ ( N ) for

__some__N

**,**

**and**

**the**least such N is

**the**level or Stufe of

**the**subgroup

**.**

__Formally__

**,**we begin by considering

__some__family of distributions for a random variable X

**,**that is indexed by

__some__θ

**.**

__Formally__

**,**rotational symmetry is symmetry with respect to

__some__or

**all**rotations in m-dimensional Euclidean space

**.**

__Formally__

**,**

**all**members of

**the**Ghelani family are Hindu Leuva Patel

**,**although

__some__have since chosen other religious beliefs

**.**

Bābak Khorram-Din (

__Formally__known as " Pāpak " meaning " Young Father ") (, alternative spelling: Bābak Xorramdin ; 795**,**according to__some__other sources 798 — January 838 ) was one of**the**main PersianArmenian historian Vardan Areweltsʻi**,**ca**.**__Formally__

**,**a strongly continuous semigroup is a representation of

**the**semigroup ( R < sub >+</ sub >,+) on

__some__Banach space X that is continuous in

**the**strong operator topology

**.**

Formally and all

__Formally__

**,**

**the**set of

__all__context-free languages is identical to

**the**set of languages accepted by pushdown automata ( PDA ).

__Formally__

**,**oxidation state is

**the**hypothetical charge that an atom would have if

__all__bonds to atoms of different elements were 100 % ionic

**.**

__Formally__given to bishop Philip of Senj

**,**

**the**permission to use

**the**Glagolitic liturgy (

**the**Roman Rite conducted in Slavic language instead of Latin

**,**

**not**

**the**Byzantine rite ), actually extended to

__all__Croatian lands

**,**mostly along

**the**Adriatic coast

**.**

__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__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**.**
; Electronics:

__Formally__a fixture in radio shacks**,**now used for assembly**and**repair of__all__sorts of electronic equipment including communication**,**computer**,****and**home entertainment items**.**__Formally__

**,**a ring is an Abelian group ( R

**,**+), together with a second binary operation * such that for

__all__a

**,**b

**and**c in R

**,**

__Formally__speaking

**,**

**the**opportunity was extended to

__all__classes to join China's project on its Leninist terms

**.**

__Formally__

**,**

__all__of them are entitled El Camarón de la Isla con la colaboración especial de Paco de Lucía with

**the**exception of Canastera

**.**

__Formally__titled ' An Act to protect

__all__Persons in

**the**United States in their Civil Rights

**,**

**and**furnish

**the**Means of their vindication

**,**

**the**Act declared that people born in

**the**United States

**and**

**not**subject to any foreign power are entitled to be citizens

**,**without regard to race

**,**color

**,**or previous condition of slavery or involuntary servitude

**.**

__Formally__

**,**

**the**game ( coalitional game ) consists of a finite set of players

**,**called

**the**grand coalition

**,**

**and**a characteristic function denotes

**the**power set of .</ ref > from

**the**set of

__all__possible coalitions of players to a set of payments that satisfies

**.**

__Formally__

**,**given two partially ordered sets ( S

**,**≤)

**and**( T

**,**≤), a function f: S → T is an order-embedding if f is both order-preserving

**and**order-reflecting

**,**i

**.**e

**.**for

__all__x

**and**y in S

**,**one has

__Formally__

**,**it is

**the**solid figure formed by

**the**locus of

__all__straight line segments that join

**the**apex to

**the**base

**.**

__Formally__

**,**given a finite set X

**,**a collection C of subsets of X

**,**

__all__of size n

**,**has Property B if we can partition X into two disjoint subsets Y

**and**Z such that every set in C meets both Y

**and**Z

**.**

__Formally__

**,**

**the**semantic of influence diagram is based on sequential construction of nodes

**and**arcs

**,**which implies a specification of

__all__conditional independencies in

**the**diagram

**.**

__Formally__

**,**a redistribution of

__all__agents ’ initial endowments of goods among

__all__agents in

**the**system is a core allocation if no group of agents can redistribute their own initial endowments among themselves so as to improve

**the**satisfaction of someone in

**the**group without impairing that of any other in

**the**group

**.**

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