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Formally, the definition only requires some invertibility, so we can substitute for Q any matrix M whose eigenvalues do not include − 1.

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## Some Related Sentences

Formally and definition

__Formally__

**,**a system is said to be observable if

**,**

**for**

**any**possible sequence of state and control vectors

**,**

**the**current state

**can**be determined in finite time using

**only**

**the**outputs ( this

__definition__is slanted towards

**the**state space representation ).

__Formally__

**,**if f is a harmonic function

**,**then f cannot exhibit a true local maximum within

**the**domain of

__definition__of f

**.**In other words

**,**either f is a constant function

**,**or

**,**

**for**

**any**point inside

**the**domain of f

**,**there exist other points arbitrarily close to at which f takes larger values

**.**

__Formally__

**,**let X be

**any**scheme and S be a sheaf of graded-algebras (

**the**

__definition__of which is similar to

**the**

__definition__of-modules on a locally ringed space ): that is

**,**a sheaf with a direct sum decomposition

__Formally__

**,**a biased graph Ω is a pair ( G

**,**B ) where B is a linear class of circles ; this by

__definition__is a class of circles that satisfies

**the**theta-graph property mentioned above

**.**

Formally and only

__Formally__

**the**Prince-Electors elected a King of

**the**Romans

**,**who was elected in Germany but became Holy Roman Emperor

__only__when crowned by

**the**Pope

**.**

Informally

**,**G has**the**above presentation if it is**the**" freest group " generated by S subject__only__to**the**relations R**.**__Formally__**,****the**group G is said to have**the**above presentation if it is isomorphic to**the**quotient of a free group on S by**the**normal subgroup generated by**the**relations R**.**__Formally__

**,**

**the**case where

__only__a subset of parameters is defined is still a composite hypothesis ; nonetheless

**,**

**the**term point hypothesis is often applied in such cases

**,**particularly where

**the**hypothesis test

**can**be structured in such a way that

**the**distribution of

**the**test statistic (

**the**distribution under

**the**null hypothesis ) does

**not**depend on

**the**parameters

**whose**values have

**not**been specified under

**the**point null hypothesis

**.**

__Formally__

**the**euro is

**the**

__only__legal tender in Büsingen ; however

**,**

**the**Swiss franc is far more popular in everyday use as shopping is generally done in nearby Schaffhausen instead of across

**the**border ( s ) in Germany

**.**

__Formally__

**,**Mbenga – Mbuti Pygmy music consists of at most

__only__four parts

**,**and

**can**be described as an

**,**" ostinato with variations ," or similar to a passacaglia

**,**in that it is cyclical

**.**

__Formally__

**,**

**the**complexity class MA is

**the**set of decision problems that

**can**be decided in polynomial time by an Arthur – Merlin protocol where Merlin's

__only__move precedes

**any**computation by Arthur

**.**

__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__Roman Kalyn is now

**the**

__only__member of

**the**group

**,**but de-facto GreenJolly exists in name

__only__

**.**

__Formally__

**,**a number n is colossally abundant if and

__only__if there is an ε > 0 such that

**for**all k >

**1**

**,**

__Formally__

**,**a natural number n is called highly abundant if and

__only__if

**for**all natural numbers m < n

**,**

__Formally__this mission was declared to be

__only__in pursuit of

**the**Russian vessels and ammunition taken to Anzali by

**the**White Russian counter-revolutionary General Denikin

**,**who had been given asylum by British forces in Anzali

**.**

__Formally__known as Crown Point Airport

**,**

**the**airport was renamed

**the**Arthur Napoleon Raymond Robinson International Airport on May 19

**,**2011 in honour of

**the**

__only__Prime Minister and President from

**the**island of Tobago

**.**

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

**.**

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__

**,**

**we**begin by considering

__some__family of distributions

**for**a random variable X

**,**that is indexed by

__some__θ

**.**

__Formally__

**,**Liouville numbers and elementary numbers contain

**the**algebraic numbers

**,**and they

**include**

__some__but

**not**all transcendental numbers

**.**

__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 so

__Formally__

**,**let G be a Coxeter group with reduced root system R and k < sub > v </ sub > a multiplicity function on R (

__so__k < sub > u </ sub > = k < sub > v </ sub > whenever

**the**reflections σ < sub > u </ sub > and σ < sub > v </ sub > corresponding to

**the**roots u and v are conjugate in G ).

__Formally__

**,**

**the**Ministry of Aeronautics was founded on January 20

**,**1941 and

__so__its military branch called " National Air Forces ", changed to " Brazilian Air Force " ( Força Aérea Brasileira-FAB ) on May

**,**22

**.**

__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

**.**

__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 we

__Formally__

**,**

__we__start with a category C with finite products ( i

**.**e

**.**C has a terminal object

**1**and

**any**two objects of C have a product ).

__Formally__it is precisely in allowing quantification over class variables α

**,**β

**,**etc., that

__we__assume a range of values

**for**these variables to refer to

**.**

__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__

**,**

__we__have

**for**

**the**approximation to

**the**full solution A

**,**a series in

**the**small parameter ( here called ), like

**the**following:

__Formally__

**,**if

__we__write F < sub > Δ </ sub >( x ) to mean

**the**f-polynomial of Δ

**,**then

**the**h-polynomial of Δ is

__Formally__

**,**

**for**a countable set of events A < sub >

**1**</ sub >, A < sub > 2 </ sub >, A < sub > 3 </ sub >, ...,

__we__have

__Formally__

**,**let A be a real

**matrix**of which

__we__want to compute

**the**

**eigenvalues**

**,**and let A < sub > 0 </ sub >:= A

**.**

__Formally__

**,**an antihomomorphism between X and Y is a homomorphism

**,**where equals Y as a set

**,**but has multiplication reversed: denoting

**the**multiplication on Y as and

**the**multiplication on as

**,**

__we__have

**.**

__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__

**,**

__we__define a bad field as a structure of

**the**form ( K

**,**T ), where K is an algebraically closed field and T is an infinite

**,**proper

**,**distinguished subgroup of K

**,**such that ( K

**,**T ) is of finite Morley rank in its full language

**.**

__Formally__

**,**if

__we__denote

**the**set of stable functions by S ( D ) and

**the**stability radius by r ( f

**,**D ), then:

0.218 seconds.