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Formally, this is achieved by mapping the input x to a two dimensional vector u ( x )

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## Some Related Sentences

Formally and is

__Formally__organized vocational programs supported

**by**federal funds allow high school students

**to**gain experience in

**a**field of work which

__is__likely

**to**lead

**to**

**a**full-time job on graduation.

__Formally__

**,**

**the**set of all context-free languages

__is__identical

**to**

**the**set of languages accepted

**by**pushdown automata

**(**PDA ).

More rigorously

**,****the**divergence of**a****vector**field F at**a**point p__is__defined as**the**limit of**the**net flow of F across**the**smooth boundary of**a**three**dimensional**region V divided**by****the**volume of V as V shrinks**to**p.__Formally__**,**__Formally__

**,**

**the**base

__is__known as Naval Support Facility Diego Garcia

**(**

**the**US activity

**)**or Permanent Joint Operating Base

**(**PJOB

**)**Diego Garcia

**(**

**the**UK's term ).

__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__

**,**oxidation state

__is__

**the**hypothetical charge that an atom would have if all bonds

**to**atoms of different elements were 100 % ionic.

__Formally__

**,**an inner product space

__is__

**a**

**vector**space V over

**the**field together with an inner product

**,**i. e., with

**a**map

__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__**,**when working over**the**reals**,**as here**,****this**__is__accomplished**by**considering**the**limit as ε → 0 ; but**the**" infinitesimal " language generalizes directly**to**Lie groups over general rings.__Formally__

**,**

**a**profinite group

__is__

**a**Hausdorff

**,**compact

**,**and totally disconnected topological group: that

__is__

**,**

**a**topological group that

__is__also

**a**Stone space.

__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.

__Formally__

**,**

**a**frame

__is__defined

**to**be

**a**lattice L in which finite meets distribute over arbitrary joins

**,**i. e. every

**(**even infinite

**)**subset

Formally and by

*

__Formally__appoints**the**Prime Minister of Finland selected__by__**the**Parliament**,**and formally appoints**the**rest of**the**cabinet**(**Council of State**)**as proposed__by__**the**Prime Minister__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.

__Formally__

**,**these failed when they were rejected

__by__

**the**Church of England's General Synod in 1972 ; conversations and co-operation continued

**,**however

**,**leading in 2003

**to**

**the**signing of

**a**covenant between

**the**

**two**churches.

__Formally__

**the**" head of state " can also personally be

**the**" head of government "

**(**ex officio or

__by__ad hoc cumulation such as an absolute monarch nominating himself

**)**but otherwise has formal precedence over

**the**Head of Government and other ministers

**,**whether he

**is**their actual political superior

**(**absolute monarch

**,**executive president

**)**or rather theoretical or ceremonial in character.

__Formally__

**,**in

**the**finite-dimensional case

**,**if

**the**linear map

**is**represented as

**a**multiplication

__by__

**a**matrix A and

**the**translation as

**the**addition of

**a**

**vector**

**,**an affine map acting on

**a**

**vector**can be represented as

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__named " United Press Associations " for incorporation and legal purposes

**,**but publicly known and identified as United Press or UP

**,**it was created

__by__

**the**1907 uniting of three smaller news syndicates

__by__

**the**Midwest newspaper publisher E. W. Scripps.

__Formally__

**,**most of these people were convicted

__by__NKVD troikas (" triplets ")– special courts martial.

__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__

**,**

**a**decision problem

**is**P-complete

**(**complete for

**the**complexity class P

**)**if it

**is**in P and that every problem in P can be reduced

**to**it

__by__using an appropriate reduction.

__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__

**,**

**a**transductive support

**vector**machine

**is**defined

__by__

**the**following primal optimization problem:

__Formally__

**,**DFS

**is**an uninformed search that progresses

__by__expanding

**the**first child node of

**the**search tree that appears and thus going deeper and deeper until

**a**goal node

**is**found

**,**or until it hits

**a**node that has no children.

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 and mapping

__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__

**,**

**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__

**,**given

**a**graph G

**,**

**a**vertex labeling

**is**

**a**function

__mapping__vertices of G

**to**

**a**set of labels.

__Formally__

**,**

**a**signed graph Σ

**is**

**a**pair

**(**G

**,**σ

**)**that consists of

**a**graph G =

**(**V

**,**E

**)**and

**a**sign

__mapping__or signature σ from E

**to**

**the**sign group

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