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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.

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## Some Related Sentences

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__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__

**,**

__we__begin

**by**considering some family

**of**distributions for

**a**random variable X

**,**

**that**is indexed

**by**some θ

**.**

__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__

**,**

**the**definition only requires some invertibility

**,**so

__we__can substitute for Q any matrix M whose eigenvalues do not include − 1

**.**

__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:

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__

**,**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

**.**

0.096 seconds.