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

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

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, 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, 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 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, 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.

* Texts and Commentary: Hegel's Preface to His System in a New Translation With Commentary on Facing Pages, and " Who Thinks Abstractly?

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.

Out of compulsion to say something cheery, Ben Prime blurted, `` Well, we were lucky to be on soft ground when the first floodheads hit.

In truth, we can say that this broke the power of Senator Joseph McCarthy, who was finally exposed in full light to the American people.

Ironically no president we have had would have regretted more than President Eisenhower the possibility to which his own words, in the press conference held at the beginning of August, testified: that unable as he was himself to say his running was best for the country, unconsciously he had placed his party before his nation.

Without saying or seeming to say that in portraying the Sartoris and the Compson families Faulkner's chief concern is social criticism, we can say nevertheless that through those families he dramatizes his comment on the planter dynasties as they have existed since the decades before the Civil War.

As a Humanist, Dr. Huxley interests himself in the possibilities of human development, and one thing we can say about this suggestion, which comes from a leading zoologist, is that, so far as he is concerned, the scientific outlook places no rigid limitation upon the idea of future human evolution.

The unrelieved stranger eventually turned away from the place of his -- shall we dare say his Waterloo??

William Wimsatt and Cleanth Brooks, it seems to me, have a penetrating insight into the way in which this control is effected: `` For if we say poetry is to talk of beauty and love ( and yet not aim at exciting erotic emotion or even an emotion of Platonic esteem ) and if it is to talk of anger and murder ( and yet not aim at arousing anger and indignation ) -- then it may be that the poetic way of dealing with these emotions will not be any kind of intensification, compounding, or magnification, or any direct assault upon the affections at all.

As Sir Charles Oman once said, `` it is no longer fashionable to declare that we can say nothing certain about Old English origins ''.

Now omitting for a moment some recent developments we can say the Saxon Shore hypothesis of Lappenberg and Kemble has undergone virtual eclipse in this century.

the mere fact that he was selected, though as a substitute, to act as interlocutor or moderator for it, or perhaps we should say with Buck as ' father of the act ', is in itself a difficult phase of his development to grasp.

If we make it established custom that whenever butchery on the highways grows excessive, say beyond 25,000 per annum, then somebody is going to hang, it follows that the more eminent the victim, the more impressive the lesson.

We sympathize with Mr. Kennedy, but we feel bound to say that his budget review doesn't please us either, although for very different reasons.

Then we pull out under our mortar and artillery cover, but nobody pulls out until I say so.

This is why I say we just can't go ahead and disarm the Germans and pull down our own defenses.

Indeed, if pressed, we would say what the late Robert Henri, American painter, said to a pupil, `` Anything will do for a subject: it's what you do with it that counts ''.

What can we say of the probabilities of the different possible numbers of bull's-eyes??

or we may say that his attitude was commendable, showing his independence of mind, in his refusal to adjust to the opinions of others.

When an evaluative situation is set up, and no concern is with the details that lead to an over-all estimate, we say that roleplaying is used for evaluation.

In that event, we can correctly say that we have received an authoritative interpretation of the matter, and one which we can follow statewide with confidence that the policy will not be overthrown in other Superior Courts.

The theory claims to show by analysis that when we say, `` That is good '', we do not mean to assert a character of the subject of which we are thinking.