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Formally, this means that we want a function to be monotonic.

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## Some Related Sentences

Formally and means

__Formally__

**,**

**this**

__means__

**that**the probability density functions or probability mass functions in

**this**class have the form

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__

**,**

**this**

__means__symmetry under

**a**sub-group of the Euclidean group of isometries in two or three dimensional Euclidean space

**.**

__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__

**,**

**this**

__means__classifying finitely generated groups with their word metric up

**to**quasi-isometry

**.**

__Formally__

**,**however

**,**the role also carries the title of " Klingon supreme commander " ( TNG's " Reunion "), which presumably

__means__commander-in-chief of the military

**.**

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__

**,**

__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 function

__Formally__

**,**there is

**a**clear distinction: " DFT " refers

**to**

**a**mathematical transformation or

__function__

**,**regardless of how it is computed

**,**whereas " FFT " refers

**to**

**a**specific family of algorithms for computing DFTs

**.**

__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__

**,**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__

**,**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__

**,**if is an open subset of the complex plane

**,**

**a**point of

**,**and is

**a**holomorphic

__function__

**,**then is called

**a**removable singularity for if there exists

**a**holomorphic

__function__which coincides with on

**.**

__Formally__

**,**the problem of supervised pattern recognition can

**be**stated as follows: Given an unknown

__function__( the ground truth )

**that**maps input instances

**to**output labels

**,**along with training data assumed

**to**represent accurate examples of the mapping

**,**produce

**a**

__function__

**that**approximates as closely as possible the correct mapping

**.**

__Formally__

**,**

**a**statistic s is

**a**measurable

__function__of X ; thus

**,**

**a**statistic s is evaluated on

**a**random variable X

**,**taking the value s ( X ), which is itself

**a**random variable

**.**

__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**cardinal κ is defined

**to**

**be**weakly compact if it is uncountable and for every

__function__f: < sup > 2 </ sup > →

__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__

**,**let

**be**

**a**stochastic process and let represent the cumulative distribution

__function__of the joint distribution of at times

**.**

__Formally__

**,**an ultrametric space is

**a**set of points with an associated distance

__function__( also called

**a**metric )

0.185 seconds.