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Formally, given complex-valued functions f and g of a natural number variable n, one writes

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## Some Related Sentences

Formally and given

__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__

**,**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__

**,**the statement that " value decreases over time " is

__given__by defining the linear differential operator as:

__Formally__

**,**it is

**a**norm defined on the space

**of**bounded linear operators between two

__given__normed vector spaces.

__Formally__

**,**the problem can be stated as follows:

__given__

**a**desired property

**,**expressed as

**a**temporal logic formula p

**,**

**and**

**a**structure M with initial state s

**,**decide if.

__Formally__

**,**

__given__two categories C

**and**D

**,**an equivalence

**of**categories consists

**of**

**a**functor F: C → D

**,**

**a**functor G: D → C

**,**

**and**two

**natural**isomorphisms ε: FG → I < sub > D </ sub >

**and**η: I < sub > C </ sub >→ GF.

__Formally__

**,**an absolute coequalizer

**of**

**a**pair in

**a**category C is

**a**coequalizer as defined above but with the added property that

__given__any functor F ( Q ) together with F ( q ) is the coequalizer

**of**F (

**f**)

**and**F (

**g**) in the category D. Split coequalizers are examples

**of**absolute coequalizers.

__Formally__

**,**

__given__

**a**partially ordered set ( P

**,**≤), then an element

**g**

**of**

**a**subset S

**of**P is the greatest element

**of**S if

__Formally__

**,**

**a**deterministic algorithm computes

**a**mathematical function ;

**a**function has

**a**unique value for any

__given__input

**,**

**and**the algorithm is

**a**process that produces this particular value as output.

__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__

**,**

__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__

**,**

__given__

**a**graph G

**,**

**a**vertex labeling is

**a**function mapping vertices

**of**G to

**a**set

**of**labels.

__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__

**,**

**a**married or widowed woman can be called by the

__given__name

**of**her husband ( Madame (

__given__name

**of**husband ) family name or Madame veuve (

__given__name

**of**husband ) family name ); this is now slightly out

**of**fashion.

Formally and complex-valued

__Formally__

**,**the bounded Borel functional calculus

**of**

**a**self adjoint operator T on Hilbert space H is

**a**mapping defined on the space

**of**bounded

__complex-valued__Borel

**functions**

**f**on the real line

**,**

Formally and functions

__Formally__

**,**the Congress serves two

__functions__: to approve changes to the Party constitution regarding policy

**and**to elect

**a**Central Committee

**,**about 300 strong.

__Formally__

**,**this means that the probability density

__functions__or probability mass

__functions__in this class have the form

__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__

**,**if we denote the set

**of**stable

__functions__by S ( D )

**and**the stability radius by r (

**f**

**,**D ), then:

Formally and f

__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__

**,**

**a**cardinal κ is defined to be weakly compact if it is uncountable

**and**for every function

__f__: < sup > 2 </ sup > →

__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__

**,**

__f__< sub > X

**,**Y </ sub >( x

**,**y ) is the probability density function

**of**( X

**,**Y ) with respect to the product measure on the respective supports

**of**X

**and**Y.

__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 and g

__Formally__

**,**if T is

**a**complete sufficient statistic for θ

**and**E (

__g__( T )) = τ ( θ ) then

__g__( T ) is the minimum-variance unbiased estimator ( MVUE )

**of**τ ( θ ).

__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 and natural

__Formally__

**,**

**a**rigged Hilbert space consists

**of**

**a**Hilbert space H

**,**together with

**a**subspace Φ which carries

**a**finer topology

**,**that is

**one**for which the

__natural__inclusion

__Formally__

**,**the use

**of**

**a**reduction is the function that sends each

__natural__

**number**

**n**to the largest

__natural__

**number**m whose membership in the set B was queried by the reduction while determining the membership

**of**

**n**in A.

__Formally__

**,**

**a**

__natural__

**number**

**n**is called highly abundant if

**and**only if for all

__natural__numbers m <

**n**

**,**

__Formally__

**,**citizen science has been defined as " the systematic collection

**and**analysis

**of**data ; development

**of**technology ; testing

**of**

__natural__phenomena ;

**and**the dissemination

**of**these activities by researchers on

**a**primarily avocational basis ".

__Formally__

**,**the right Kan extension

**of**along consists

**of**

**a**functor

**and**

**a**

__natural__transformation which is couniversal with respect to the specification

**,**in the sense that for any functor

**and**

__natural__transformation

**,**

**a**unique

__natural__transformation is defined

**and**fits into

**a**commutative diagram

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