In some systems of axiomatic set theory, relations are extended to classes, which are generalizations of sets.

The chemical nature of catalysts is as diverse as catalysis itself, although some generalizations can be made.

Unlike the gradient and divergence, curl does not generalize as simply to other dimensions ; some generalizations are possible, but only in three dimensions is the geometrically defined curl of a vector field again a vector field.

Although there are some generalizations which can be made for the dozens of other glyphs, which are based on the circle and square, the details are somewhat idiosyncratic and each needs to be memorized.

Finally, it should be noted that some early generalizations of Einstein's gravitational theory, known as classical unified field theories, either introduced a cosmological constant on theoretical grounds or found that it arose naturally from the mathematics.

WMCF advocates ( and includes some examples of ) a cognitive idea analysis of mathematics which analyzes mathematical ideas in terms of the human experiences, metaphors, generalizations, and other cognitive mechanisms giving rise to them.

Most of this article deals with linear combinations in the context of a vector space over a field, with some generalizations given at the end of the article.

" The book's challenges have led some commentators into generalised statements about its content and themes, prompting critic Bernard Benstock to warn against the danger of " boiling down " Finnegans Wake into " insipid pap, and leaving the lazy reader with a predigested mess of generalizations and catchphrases.

Although there are some generalizations of the compactness theorem to non-first-order logics, the compactness theorem itself does not hold in them.

Note, however, that these are generalizations, and some of these may not be true for specific dialects:

" He reveals that Jones was angry at Clampett for making some generalizations in his 1970 interview with Funnyworld that seemingly gave Clampett too much credit.

In trying to rebut this premise, some commentators point to differences between the categories of race and sexual orientation, claiming they are too complex to support any generalizations.

Other concessions were less substantial and merely contained some generalizations regarding the reorganizing and unification of Germany.

Each of these properties may lead to generalizations, in general different from each other ( see the article ' Coherent states in mathematical physics ' for some of these ).

Although it is difficult to make accurate generalizations about reverse imagery, as this was something that varied by emperor, some trends do exist.

In topology, a branch of mathematics, braid theory is an abstract geometric theory studying the everyday braid concept, and some generalizations.

This fallacy may occur when we confuse generalizations (" some ") for categorical statements (" always and everywhere ").

However, some breeders consider these " differences " to be unsound generalizations and that American standards should be updated to reflect the breed's standard in its country of origin, i. e. France, where black has become an acceptable coat colour since 1956 while it is still considered a fault in America.

The languages vary on the environments for the mutations, though some generalizations can be made.

Kruskal himself pioneered some of the generalizations, such as the existence of infinitely many conserved quantities for the sine-Gordon equation.

Francis Edwards Peters says that it is hard to make generalizations about Muhammad's marriages: many of them were political, some compassionate, and some perhaps affairs of the heart.

Sampling surveys, on the other hand, have the goal of obtaining a representative sample of some population of sites or artifacts in order to make generalizations about that population.

Although laws differ widely from one jurisdiction to another, some generalizations are possible:

`` The main bunch is outside, but there are some over there inside the wall ''.

For a brief period each year, the rays of the sun are warm enough to melt some of the snows piled a mile deep at the base of the headwalls, and then the pinnacles glisten in the daytime at high noon, and billions of gallons of water begin their slow seepage under the glaciers and across the rockstrewn hanging valleys on their long, meandering journey to the sea -- running east past the sky-carving massifs of Gurla Mandhata and Kemchenjunga, then turning south and curling down through the jungles of Assam, past the Khasi Hills, and into Bengal, past Sirinjani and Madaripur, until the hard water of the melting snows mingles with the soft drainage of fields and at length fans out to meld with the teeming salt depths of the Bay of Bengal.

to some degree they are being supplanted by a concept of national responsibility.

At least a dozen men, some armed, are never far away from him.

Those three other great activities of the Persians, the bath, the teahouse, and the zur khaneh ( the latter a kind of club in which a leader and a group of men in an octagonal pit move through a rite of calisthenics, dance, chanted poetry, and music ), do not take place in buildings to which entrance tickets are sold, but some of them occupy splendid examples of Persian domestic architecture: long, domed, chalk-white rooms with daises of turquoise tile, their end walls cut through to the orchards and the sky by open arches.

We get some clue from a few remembrances of childhood and from the circumstance that we are probably not much more afraid of people now than man ever was.

We are worried about what people may do with them -- that some crazy fool may `` push the button ''.

This almost trivial example is nevertheless suggestive, for there are some elements in common between the antique fear that the days would get shorter and shorter and our present fear of war.

Even in domains where detailed and predictive understanding is still lacking, but where some explanations are possible, as with lightning and weather and earthquakes, the appropriate kind of human action has been more adequately indicated.

some of them are driven over the borderline of sanity and lose contact with reality.

Underlying these conceptions of mimesis are certain presuppositions concerning the nature of primary human experience which require some exposition before the main argument can proceed.

There are, however, some wonderful chapters at the beginning of the second part, concerning the reactions of the Swedes in adversity.

Not discussed here are some military problems of modern times such as undersea warfare, where the surveillance, sending, transmitting, and receiving are all so inadequate that networks and decision making are not the bottlenecks.

There is probably some significance in the fact that two of the best incest stories I have encountered in recent years are burlesques of the incest myth.

nor is there need to add that among them are some of the most highly individualized and most successful of his characters.

Actually, you could wish for some passion, now and then, but when you look around the world and see the little volcanos of current history which partisan social passions have wrought, you are glad that in these pamphlets there is at least some civilized calm.

We feel uncomfortable at being bossed by a corporation or a union or a television set, but until we have some knowledge about these phenomena and what they are doing to us, we can hardly learn to control them.

The lower-level hypotheses are never ' ad hoc ', never introduced ex post facto just to sweep up within the theory some recalcitrant datum.

and, `` I do think that families are the most beautiful things in all the world '', burst out Jo some five hundred pages later in that popular story of the March family, which had first appeared when Henrietta was eight ; ;

Some are fishing, some driving the team, and many are sitting indolently on the banks of the river.

To some extent predispositions are shaped by exposure to group environments.