Abstractly, we can say that D is a linear transformation from some vector space V to another one, W. We know that D ( c ) = 0 for any constant function c. We can by general theory ( mean value theorem ) identify the subspace C of V, consisting of all constant functions as the whole kernel of D. Then by linear algebra we can establish that D < sup >− 1 </ sup > is a well-defined linear transformation that is bijective on Im D and takes values in V / C.

Abstractly, if one starts with a complex vector space W and takes the complexification of the underlying real space, one obtains a space isomorphic to the direct sum of W and its conjugate:

Abstractly, an array reference is simply a procedure of two arguments: an array and a subscript vector, which could be expressed as ; but many languages provide special syntax like ; similarly an array element update is abstractly something like, but many languages provide syntax like.

Abstractly, the probability model for a classifier is a conditional model

Abstractly, this is equivalent to stabilising a flag: upper triangular matrices are precisely those that preserve the standard flag, which is given by the standard ordered basis and the resulting flag < math > 0 <

Abstractly, this circle is a cyclic group of order twelve, and may be identified with the residue classes modulo twelve.

Abstractly, it can be defined as the integer triples ( a, b, c ), associated to 3 < sup > a </ sup > 5 < sup > b </ sup > 7 < sup > c </ sup >, where the distance measure is not the usual Euclidean distance but rather the Euclidean distance deriving from the vector space norm

Abstractly we can say that this is a projective line in the space of all conics, on which we take

Abstractly, these are the left adjoint and right adjoint, respectively, to the inclusion functors of in D. In addition, the truncation functors fit into a triangle, and this is in fact the unique triangle satisfying the third axiom above:

Abstractly it is assumed that the economy consists only of capitalist production and that the capitalist economy is equal to capitalist society.

Abstractly, the method is as follows:

Abstractly, a hierarchy can be modelled mathematically as a rooted tree: the root of the tree forms the top level, and the children of a given vertex are at the same level, below their common parent.

Abstractly programmable toolpath guidance began with mechanical solutions, such as in musical box cams and Jacquard looms.

Abstractly, we can identify the chain rule as a 1-cocycle.

Abstractly, Weyl groups are finite Coxeter groups, and are important examples of these.

Abstractly, 1 / 8-schisma tuning may be considered the analog, among schismatic tunings, of 1 / 4-comma meantone among meantone tunings, as it also has pure interval ratios of 2: 1 and 5: 4, though with much more accurate interval ratios of 3: 2 and 6: 5 ( less than a quarter of a cent off from just intonation ) than its meantone counterpart.

Abstractly, a parametric equation defines a relation as a set of equations.

Abstractly, the matrix exponential gives the connection between a matrix Lie algebra and the corresponding Lie group.

What exactly are we trying to prove??

And to top it all I am often sentimental on purpose, trying to prove to myself that I am not afraid of sentiment.

They were, I felt, people invariably trying to prove not who, but what they were, and trying to determine what, not who, others were.

In the piece, Swift adopts the " technique of a political arithmetician " to show the utter ridiculousness of trying to prove any proposal with dispassionate statistics.

But instead of trying to force the opportunity to form a Liberal government, Asquith decided instead to allow Labour the chance of office in the belief that they would prove incompetent and this would set the stage for a revival of Liberal fortunes at Labour's expense.

Denmark is trying to prove that the North Pole is geographically connected to Greenland.

In 1958 Bing proved a weak version of the Poincaré conjecture: if every simple closed curve of a compact 3-manifold is contained in a 3-ball, then the manifold is homeomorphic to the 3-sphere .< ref > Bing also described some of the pitfalls in trying to prove the Poincaré conjecture.

The ascetic Priscillian of Avila was martyred by the Catholic usurper Magnus Maximus in 385, who was trying to prove his correct religious credentials against heretics, before the Visigothic period, and the persecution continued in subsequent generations as " Priscillianist " heretics were rooted out.

He spent much of his later life trying to prove, by close textual reading, that Christ had not been crucified but stoned to death.

It is only in the later strips that Fatty has been made fun of for his weight, to which he usually just reacts in trying to prove the others wrong.

The play opens on 10 April 1809, in a garden front room of a country house in Derbyshire with tutor Septimus Hodge trying to distract his 13 year-old pupil Thomasina from her enquiries as to the meaning of a " carnal embrace " by challenging her to prove Fermat's Last Theorem so he can focus on reading the poem ' The Couch of Eros ', a piece written by another character, Mr. Ezra Chater.

Unlike many commentators on Euclid before and after him ( including of course Saccheri ), Khayyám was not trying to prove the parallel postulate as such but to derive it from an equivalent postulate he formulated from " the principles of the Philosopher " ( Aristotle ):

Among the most fervent supporters of the artificial-canal hypothesis was the American astronomer Percival Lowell, who spent much of his life trying to prove the existence of intelligent life on the red planet.

Falling victim to a love spell in the episode " Him ", she displays dangerous behavior such as attacking people and trying to commit suicide to prove her " love " for classmate RJ Brooks.

The Rosenbergs ' two sons, Robert Meeropol and Michael Meeropol, spent years trying to prove the innocence of their parents.

However Davies refused to elaborate on what he called the " suicide theory of enigmatic death ," as he feared that " some of the old guard would be furious if they thought I was trying to prove it.

However, in trying to prove that he was the one who killed LeChuck, Guybrush makes the mistake of showing Largo that he has LeChuck's beard.

Quentin spends much of his time trying to prove his father wrong, but is unable to.

He confronts her trying to prove his masculine authority, but soon breaks at Gloria's strictness.

By assuming what it is trying to prove, that point undermines the entire argument.

Adams noted that " with Gold, I was trying to prove something to myself.

To some extent these philosophers were all trying to meet the challenge laid down by Epicureanism, and the success of Epicurus was in developing a system of philosophy which would prove to be more comprehensive and sophisticated than its rivals.

They ultimately concocted an unusual strategy of trying to prove that, in addition to illegally handling information, Lee was trying to injure the United States by denying it the exclusivity of the nuclear information.