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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.

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## Some Related Sentences

Formally and outcomes

__Formally__established in 1922

**,**over

**the**decades Woodcraft Rangers has modified Seton ’ s original emphasis on outdoor life

**to**incorporate activities

**that**meet

**the**needs of

**an**increasingly urban population

**,**

**but**

**the**goal of changing behavior and encouraging positive

__outcomes__through interaction and education remains central

**to**its mission today

**.**

Formally and Y

__Formally__

**,**cl ( S ) denotes

**the**smallest subset

__Y__of M

**that**contains S such

**that**for

**each**reaction ( A

**,**B )

__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__

**,**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__

**,**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__it can be seen just

**as**

**an**ordinary function from X

**to**

**the**power set of

__Y__

**,**written

**as**φ: X → 2

**<**sup

**>**

__Y__

**</**sup >.

Formally and <

__Formally__

**,**

**the**theorem

**is**stated

**as**follows: There exist unique integers q and r such

**that**a = qd + r and 0 ≤ r

__<__| d |,

**where**| d | denotes

**the**absolute value of d

**.**

__Formally__

**,**a unique factorization domain

**is**defined

**to**be

**an**integral domain R in which every non-zero and non-unit x of R can be written

**as**a product ( including

**an**empty product ) of irreducible elements

**p**

__<__

**sub**

**>**

**i**

**</**

**sub**

**>**of R and a unit u:

__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__

**,**

**the**ith row

**,**jth column element of A

__<__sup

**>**T

**</**sup

**>**

**is**

**the**jth row

**,**ith column element of A:

__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__

**,**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__

**,**a ringed space ( X

**,**O

__<__

**sub**

**>**X

**</**

**sub**>)

**is**a topological space X together with a sheaf of rings O

__<__

**sub**

**>**X

**</**

**sub**

**>**on X

**.**

__Formally__

**,**if we write F

__<__

**sub**

**>**Δ

**</**

**sub**>( x )

**to**mean

**the**f-polynomial of Δ

**,**then

**the**h-polynomial of Δ

**is**

__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__

**,**a Lie superalgebra

**is**a ( nonassociative ) Z

__<__

**sub**

**>**2

**</**

**sub**>- graded algebra

**,**or superalgebra

**,**over a commutative ring ( typically R or C ) whose product

**,**called

**the**Lie superbracket or supercommutator

**,**satisfies

**the**two conditions ( analogs of

**the**usual Lie algebra axioms

**,**with grading ):

Formally and >

Formally and i

__Formally__

**,**

**an**inner product space

**is**a vector space V over

**the**field together with

**an**inner product

**,**

__i__

**.**e., with a map

__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__

**,**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__

**,**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__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

**.**

a

**<****sub****>**__i__**</****sub**>, a**<****sub****>**__i__+ 1**</****sub**>,..., which**is**a suffix of w**.**__Formally__**,****the**satisfaction relation between a word and**an**LTL formula**is**defined**as**follows:__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__

**,**

**the**person who directly draws

**the**funds ("

**the**payee ") instructs his or her bank

**to**collect (

__i__

**.**e., debit )

**an**amount directly from another's ("

**the**payer's ") bank account designated

**by**

**the**payer and pay those funds into a bank account designated

**by**

**the**payee

**.**

Formally and </

__Formally__

**,**for a countable set of events A

**<**

**sub**

**>**1

__</__

**sub**>, A

**<**

**sub**

**>**2

__</__

**sub**>, A

**<**

**sub**

**>**3

__</__

**sub**>, ..., we have

__Formally__

**,**a deterministic Büchi automaton

**is**a tuple A = ( Q

**,**Σ

**,**δ

**,**q

**<**

**sub**

**>**0

__</__

**sub**>, F )

**that**consists of

**the**following components:

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__

**,**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__

**,**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__this approximation

**is**founded on

**the**variational principle

**,**valid for Hamiltonians

**that**

__are__bounded from below

**.**

__Formally__

**,**

**the**sets of free and bound names of a process in π – calculus

__are__defined inductively

**as**follows

**.**

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