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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 >).

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## Some Related Sentences

Formally and analytic

__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 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__

**,**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__

**,**

**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__

**,**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__

**,**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

**)**

Formally and ƒ

__Formally__

**,**in Euclidean space

**,**

**the**wave front set

**of**

__ƒ__

**is**defined as

**the**complement

**of**

**the**set

**of**all pairs

**(**x

**<**

**sub**

**>**0

**</**

**sub**

**>,**v

**)**such that there exists a test

**function**with φ

**(**x

**<**

**sub**

**>**0

**</**

**sub**>) ≠ 0 and

**an**open cone Γ containing v such that

**the**estimate

Formally and real

__Formally__

**,**

**the**singular value decomposition

**of**

**an**m ×

**n**

__real__

**or**

**complex**matrix M

**is**a factorization

**of**

**the**form

__Formally__

**,**Minkowski space

**is**a four-dimensional

__real__vector space equipped with a nondegenerate

**,**symmetric bilinear form with signature

**<**tt >(−,+,+,+)</ tt

**>**

**(**Some may also prefer

**the**alternative signature

**<**tt >(+,−,−,−)</ tt >; in general

**,**mathematicians and general relativists prefer

**the**former while particle physicists tend to use

**the**latter

**.**

__Formally__

**,**let A be a

__real__matrix

**of**which we want to compute

**the**eigenvalues

**,**and let A

**<**

**sub**

**>**0

**</**

**sub**>:= A

**.**

__Formally__

**,**complexification

**is**a functor Vect

**<**

**sub**

**>**R

**</**sup

**>**→ Vect

**<**

**sub**

**>**

**C**

**</**sup

**>,**from

**the**category

**of**

__real__vector spaces to

**the**category

**of**

**complex**vector spaces

**.**

__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 complex

__Formally__

**,**these reside in a

__complex__separable Hilbert space-variously called

**the**" state space "

**or**

**the**" associated Hilbert space "

**of**

**the**system-that

**is**well defined up to a

__complex__number

**of**norm

**1**

**(**

**the**phase factor ).

__Formally__

**,**a

__complex__projective space

**is**

**the**space

**of**

__complex__lines through

**the**origin

**of**

**an**

**(**

**n**+

**1**)- dimensional

__complex__vector space

**.**

__Formally__

**,**

**the**novel

**is**notable because

**of**its lack

**of**paragraphing

**,**a digressive style

**,**

**the**blending

**of**fact and fiction

**,**very long and

__complex__sentences

**(**one sentence

**is**about 9 pages long

**)**as well as

**the**inclusion

**of**a set

**of**mysterious and evocative photographs

**,**scattered throughout

**the**book

**,**which enhance

**the**melancholy message

**of**

**the**text

**.**

Formally and variables

__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__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__

**,**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__

**,**dependence refers to any situation in which random

__variables__do not satisfy a mathematical condition

**of**probabilistic independence

**.**

__Formally__

**,**a constraint satisfaction problem

**is**defined as a triple

**,**where

**is**a set

**of**

__variables__

**,**

**is**a domain

**of**values

**,**and

**is**a set

**of**constraints

**.**

__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__

**,**

**the**algorithm's performance will be a random variable determined by

**the**random bits ; thus either

**the**running time

**,**

**or**

**the**output

**(**

**or**both

**)**

**are**random

__variables__

**.**

__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__

**,**propositional models can be represented by sets

**of**propositional

__variables__; namely

**,**each model

**is**represented by

**the**set

**of**propositional

__variables__it assigns to true

**.**

__Formally__

**,**

**the**extension

**of**circumscription that incorporate varying and fixed

__variables__

**is**as follows

**,**where

**is**

**the**set

**of**

__variables__to minimize

**,**

**the**fixed

__variables__

**,**and

**the**varying

__variables__

**are**those not in:

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