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

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

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

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

Abstractly, in trying to prove a proposition P, one assumes that it is false, and that therefore there is at least one counterexample.

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.

There's only one way they can get out now and that's through the Gap -- if we ride hard we can take them ''.

`` That is, if we can be sure this is Colcord's money '' --

`` I know what we can do '', I said.

I think I have a way so we can carry on without his suspecting us ''.

and now I think we can use the knowledge they passed on to us.

`` We can get it if we dig '', he said patiently.

`` I suppose '', he muttered, `` I can sell the outfit for enough to send you home to your folks, once we find a settlement ''.

Once we send out the whole pie, they can put their pieces into it.

It is well then that in this hour both of `` national peril '' and of `` national opportunity '' we can take counsel with the men who made the nation.

We have staved off a war and, since our behavior has involved all these elements, we can only keep adding to our ritual without daring to abandon any part of it, since we have not the slightest notion which parts are effective.

That, I thought, is at least one thing I can find out when we meet.

But now we can keep it out no longer, because we have come into a time when `` it invades our experience at every moment.

How can we make sense of this division??

However, for convenience we will stick to the idea that information can be classified according to Table 1.

This is an unsolved problem which probably has never been seriously investigated, although one frequently hears the comment that we have insufficient specialists of the kind who can compete with the Germans or Swiss, for example, in precision machinery and mathematics, or the Finns in geochemistry.

We hear equally fervent concern over the belief that we have not enough generalists who can see the over-all picture and combine our national skills and knowledge for useful purposes.

But, for practical purposes, we have people who can be considered as such.

What I am here to do is to report on the gyrations of the struggle -- a struggle that amounts to self-redefinition -- to see if we can predict its future course.

One of the obvious conclusions we can make on the basis of the last election, I suppose, is that we, the majority, were dissatisfied with Eisenhower conservatism.

Kant's goal in his critical philosophy was to identify what claims we are and are not justified in making, and the antinomies are a particularly illustrative example of his larger project.

This function is undefined at the point π, since tan ( π / 2 ) is undefined ; we will identify this point with our point ∞.

As cultures, we identify ourselves through the collective dreams we call creation myths, or cosmogonies.

This means that the similarity of our thoughts and the continuity of them in this similarity do not mean that we can identify ourselves as a self but that our thoughts are similar.

It assumes that we can identify the DTD with the relative URI reference " example. dtd "; the " people_list " after "! DOCTYPE " tells us that the root tags, or the first element defined in the DTD, is called " people_list ":

In order to better understand the world of appearances and objects, phenomenology attempts to identify the invariant features of how objects are perceived and pushes attributions of reality into their role as an attribution about the things we perceive ( or an assumption underlying how we perceive objects ).

Both in relativistic and non-relativistic quantum physics, we can identify the group velocity of a particle's wave function with the particle velocity.

The integration and application of what we know about the brain was strengthened in 2000 when the American Federation of Teachers stated: " It is vital that we identify what science tells us about how people learn in order to improve the education curriculum.

* Some scholars hold that the Mishnah and Talmud have been extensively shaped by later editorial redaction, but that it contains sources which we can identify and describe with some level of reliability.

So we can identify the " class of all sets " with the predicate x = x or any equivalent predicate.

In other words, we may be " temporarily unconscious " of whether an image is white, black or yellow and concentrate our attention on the fact that it is a man and on just those attributes necessary to identify it as a man ( but not as any particular one ).

Everybody understands what we mean when we talk about playing a game, and we can even clearly identify and correct inaccurate uses of the word, all without reference to any definition that consists of necessary and sufficient conditions for the application of the concept of a game.

Since we've assumed that we can infallibly identify programs for computing squares, we can determine whether t, which depends on a and i, is such a program, and that for every a and i ; thus we have obtained a program that decides whether program a halts on input i. Note that our halting-decision algorithm never executes t, but only passes its description to the squaring-identification program, which by assumption always terminates ; since the construction of the description of t can also be done in a way that always terminates, the halting-decision cannot fail to halt either.

Such a relation closely corresponds to what is usually called the extension of a predicate in first-order logic except that here we identify the places in the predicate with attribute names.

A counterpoint to the above is that while these considerations harmonize belief in God with our inability to identify his reasons for permitting evil, there remains a question as to why we have not been given a clear and unambiguous assurance by God that he has good reasons for allowing evil, which would be within our ability to understand.

Schwalbe believes that these distinctions are important, because society wants to identify and categorize people as soon as we see them.

The operator on the left represents the particle energy reduced by its rest energy, which is just the classical energy, so we recover Pauli's theory if we identify his 2-spinor with the top components of the Dirac spinor in the non-relativistic approximation.