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Formally, a Menger sponge can be defined as follows:

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## Some Related Sentences

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 defined

__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 ).

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__

**,**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 ).

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__

**,**

**a**transductive support vector machine is

__defined__by the following primal optimization problem

**:**

__Formally__

**,**bending modulus is

__defined__

**as**the energy required to deform

**a**membrane from its intrinsic curvature to some other curvature.

__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 index of H in G is

__defined__

**as**the number of cosets of H in G. ( The number of left cosets of H in G is always equal to the number of right cosets.

__Formally__

**,**it is

**a**norm

__defined__on the space of bounded linear operators between two given normed vector spaces.

__Formally__

**,**

**a**cardinal κ is

__defined__to

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

**:**< sup > 2 </ sup > →

__Formally__

**,**the sets of free and bound names of

**a**process in π – calculus are

__defined__inductively

**as**

**follows**.

__Formally__

**,**an absolute coequalizer of

**a**pair in

**a**category C is

**a**coequalizer

**as**

__defined__above but with the added property that given any functor F ( Q ) together with F ( q ) is the coequalizer of F ( f ) and F ( g ) in the category D. Split coequalizers are examples of absolute coequalizers.

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