The proof of Yoneda's lemma is indicated by the following commutative diagram:

3 ) The theory of convergence via filters, due to Henri Cartan and developed by Bourbaki in 1937, leads to the following criterion: assuming the ultrafilter lemma, a space is compact if and only if each ultrafilter on the space converges.

In mathematics, and more specifically in homological algebra, the splitting lemma states that in any abelian category, the following statements for short exact sequence are equivalent.

He began by adopting the following lemma: " All sound, even continuous musical variation, is conceived as an assemblage of a large number of elementary sounds adequately disposed in time.

In the following subsections we discuss versions of Itō's lemma for different types of stochastic processes.

In its simplest form, Itō's lemma states the following: for an Itō drift-diffusion process

For each g in G let X < sup > g </ sup > denote the set of elements in X that are fixed by g. Burnside's lemma asserts the following formula for the number of orbits, denoted | X / G |:

An important property of the exponential map is the following lemma of Gauss ( yet another Gauss's lemma ): given any tangent vector v in the domain of definition of exp < sub > p </ sub >, and another vector w based at the tip of v ( hence w is actually in the double-tangent space T < sub > v </ sub >( T < sub > p </ sub > M )) and orthogonal to v, remains orthogonal to v when pushed forward via the exponential map.

The following lemma shows a simple yet useful upper bound for the spectral radius of a matrix:

Schur's lemma is frequently applied in the following particular case.

Schur's lemma, in the special case of a single representation, says the following.

The following year he played a student who raised objections during Jill Clayburgh's proof of the snake lemma in the film It's My Turn.

Taking in above expression outside the integral sign and expanding the integrand with the help of the Hadamard lemma one can obtain the following useful expression for the derivative of matrix exponent:

A useful result pertaining to subadditive sequences is the following lemma due to Michael Fekete.

To prove Fitting's lemma, we take an endomorphism f of M and consider the following two sequences of submodules.

The proof of is based on the following lemma due to Cauchy:

We first have to show that the ascending Kleene chain of f exists in L. To show that, we prove the following lemma:

The second part of Zeckendorf's theorem ( uniqueness ) requires the following lemma:

Their utility lies in the following lemma:

In mathematics, particularly in set theory, Fodor's lemma states the following:

The following is Nakayama's lemma, as stated in:

The following corollary is also known as Nakayama's lemma, and it is in this form that it most often appears.

The following result manifests Nakayama's lemma in terms of generators

When I try to work out my reasons for feeling that this passage is of critical significance, I come up with the following ideas, which I shall express very briefly here and revert to in a later essay.

Being somewhat delicate in health, at the age of sixteen he was sent to Southern Europe, for which he at once developed a passion, so that he spent nearly all of the following ten years abroad, at first in Italy, then in Greece, Egypt, Asia Minor, and Palestine.

That fall he submitted to Professor Baker the first acts and outlines of the following acts of several plays, six of them, according to some of his associates, and he also worked on a play that he first called Niggertown, the material for which he had collected during the summer at home.

The great increase in the amount of archaeological activity, and therefore information, in the years immediately preceeding and following the Second World War has brought to light data which has changed the complection of the Saxon Shore dispute.

Both abolition of war and new techniques of production, particularly robot factories, greatly increase the world's wealth, a situation described in the following passage, which has the true utopian ring: `` Everything was so cheap that the necessities of life were free, provided as a public service by the community, as roads, water, street lighting and drainage had once been.

Dr. A. V. Astin, NBS Director, opened the 5-day session with introductory remarks, following which a total of twenty-six papers were given throughout the week by NBS scientists, from both the Washington and Boulder Laboratories.

Special events and their dates which will make interesting shooting in the Midwest area, include the following:

`` It is as though '', I said on the historic three-hour, coast-to-coast radio broadcast which I bought ( following Father Coughlin and pre-empting the Eddie Cantor, Manhattan Merry-go-round and Major Bowes shows ) `` That Man in the White House, like some despot of yore, insisted on reading my diary, raiding my larder and ransacking my lingerie!!

The mixture was then extracted with alkali and with water following which the carbon tetrachloride was distilled on a Vigreux column, a 25% center cut being retained which was then degassed under vacuum in the presence of Af.

This chlorine-carbon tetrachloride solution was illuminated for a day following which the flask was resealed onto a vacuum system and the excess chlorine distilled off.

The desired amounts of inactive chlorine and radioactive chlorine were likewise condensed in these cells on the vacuum line following which they were frozen down and the manifold as a whole was sealed off.

In the following paper it is shown that in a certain definite sense, exactly an odd number of squares can be inscribed in every such curve which does not contain an infinite number of inscribed squares.

While the interpretations that have been given are inferences only, they gain support from such comments as the following, which was made by one of the Kohnstamm-negative subjects who did not, on the first trial, perceive the tilt illusion.

-- Syllables are linguistic units centering in peaks which are usually vocalic but, as has been noted, are consonantal under certain circumstances, and which may or may not be combined with preceding and/or following consonants or combinations of consonants.

The Lincoln Mills decision authorizes a whole new body of federal `` common law '' which, as Mr. Justice Frankfurter pointed out in dissent, leads to one of the following `` incongruities '': `` ( ( 1 ) conflict in federal and state court interpretations of collective bargaining agreements ; ;

The parent of a group filing consolidated returns might be treated as the same corporation following a reorganization defined in section 368(a)(1), but as a different corporation for this purpose after a tax-free acquisition by another corporation which had not, for example, elected to file consolidated returns with its own subsidiaries.

The general basis on which these common costs are assigned to differently measured units of service will be illustrated by the following highly simplified problem of an electric-utility cost analysis.

The storms of the past had died away, and the great upheaval which was to mark the following century had not yet begun to disturb men's minds.

The epic poems, the consolidation of the Greek pantheon, the rise of firm political units, the self-awareness which could permit painted and sculptured representations of men -- all these had to await the progress of following decades.

Since I have already discussed his moral position, that discussion is incorporated by reference into the following pages, which will focus on the empirical and analytic side of Fromm's treatment.

In view of these shortcomings in both the amount and the interpretation of survey-type findings on public opinion, and considering the criticisms which can be brought against Fromm's philosophical anthropology, such a passage as the following cannot be taken seriously.

The following items may be specified in actual or symbolic form in the operands of those instructions which refer to the particular items: channel, unit, combined channel and unit, combined arm and file, unit record synchronizers, inquiry synchronizers, and alteration switches.

Conversely, experimenters would consider as impressive such statements as the following, which, if they turned out to be hits, are so unusual as to be really significant: