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Formally the algorithm can be described as:

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## Some Related Sentences

Formally and algorithm

__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 and can

__Formally__, arbitrage transactions have negative skew – prices

__can__get a small amount closer ( but often no closer than 0 ), while they

__can__get very far apart.

__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**

__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__, 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**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**case where only a subset of parameters is defined is still a composite hypothesis ; nonetheless,

**the**term point hypothesis is often applied in such cases, particularly where

**the**hypothesis test

__can__

**be**structured in such a way that

**the**distribution of

**the**test statistic (

**the**distribution under

**the**null hypothesis ) does not depend on

**the**parameters whose values have not been specified under

**the**point null hypothesis.

__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__,

**the**poem

__can__

**be**understood

**as**

**the**combination of two sonnets, though

**the**spacing of

**the**stanzas is irregular.

__Formally__, Mbenga – Mbuti Pygmy music consists of at most only four parts, and

__can__

**be**

**described**

**as**an, " ostinato with variations ," or similar to a passacaglia, in that it is cyclical.

__Formally__,

**the**complexity class MA is

**the**set of decision problems that

__can__

**be**decided in polynomial time by an Arthur – Merlin protocol where Merlin's only move precedes any computation by Arthur.

__Formally__, a system is said to

**be**observable if, for any possible sequence of state and control vectors,

**the**current state

__can__

**be**determined in finite time using only

**the**outputs ( this definition is slanted towards

**the**state space representation ).

__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__,

**the**issue is that interfertile " able to interbreed " is not a transitive relation – if A

__can__breed with B, and B

__can__breed with C, it does not follow that A

__can__breed with C – and thus does not define an equivalence relation.

Formally and be

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__, 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 frame is defined to

__be__a lattice L in which finite meets distribute over arbitrary joins, i. e. every ( even infinite ) subset

__Formally__,

**the**convex hull may

__be__defined

**as**

**the**intersection of all convex sets containing X or

**as**

**the**set of all convex combinations of points in X.

__Formally__, a detailed balanced process is

**the**one on which

**the**probability to

__be__in

**the**state x and transit to x ' must

__be__

**the**same

**as**to

__be__in

**the**state x ' and transit to x.

If R is a ring, let R denote

**the**ring of polynomials in**the**indeterminate X over R. Hilbert proved that if R is " not too large ", in**the**sense that if R is Noetherian,**the**same must__be__true for R.__Formally__,__Formally__, a government would lay out its reasons for going to war,

**as**well

**as**its intentions in prosecuting it and

**the**steps that might

__be__taken to avert it.

Formally and described

__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**Guild of Fools and Joculators and College of Clowns (

__described__in Wyrd Sisters and appears in Men at Arms and Making Money )

__Formally__

__described__in Stephen Gilligan's article

**:**The relational self

**:**

**the**expanding of love beyond desire ( 1996 ), and expanded on in his book

**:**The Courage to Love ( 1997 ), it has become a major new approach in psychotherapy and healing.

__Formally__, an Hamiltonian system is a dynamical system completely

__described__by

**the**scalar function,

**the**Hamiltonian.

__Formally__

__described__, it alternates slow sections in a modal F with faster sections, " Neue Kraft fühlend " ( with renewed strength ), in D. The slow sections each have two elements, ( 1 ) a passage reminiscent of

**the**opening of

**the**first movement in which

**the**instruments overlap each other with a brief motive ; ( 2 ) a chorale,

**the**actual song.

__Formally__, this is

__described__in algebraic notation like this

**:**( 19 + 1 ) + ( 15 − 1 ) = x, but even a young student might use this technique without calling it algebra.

0.110 seconds.