In fact, one important aspect of their very religion is the annihilation of men ''.

In fact it has caused us to give serious thought to moving our residence south, because it is not easy for the most objective Southerner to sit calmly by when his host is telling a roomful of people that the only way to deal with Southerners who oppose integration is to send in troops and shoot the bastards down.

The fact is due mainly to international wars, both hot and cold.

While the pattern is uneven, some having gained more than others, nationalism has in fact served the Western peoples well.

In point of fact, this is a beige box with a bright red door, about one and a half feet square and hung from the wall about six feet from the door to Wisman's right.

The fact is that the Southern Confederacy differed from the earlier one almost as much as the Federal Constitution did.

To my knowledge, Lincoln remains the only Head of State and Commander-in-Chief who, while fighting a fearful war whose issue was in doubt, proved man enough to say this publicly -- to give his foe the benefit of the fact that in all human truth there is some error, and in all our error, some truth.

In fact the accumulation of the hardware of destruction is day by day increasing our fear of each other.

The new fact the initiates of this cult have to learn is that they must move toward simplicity.

The magic circle is, in fact, a symbol of and preparation for the metaphysical orgasm ''.

Operating as a one man police force in fact if not in name, he is at once more independent and more dedicated than the police themselves.

`` I may possibly be a greater risk than is the normal person of my age '', the President had said on February 29th of the election year, ignoring the fact that no one of his age had ever lived out another term.

In the incessant struggle with recalcitrant political fact he learns to focus the essence of a problem in the significant detail, and to articulate the distinctions which clarify the detail as significant, with what is sometimes astounding rapidity.

we accord it its place there, and in Lawrence's treatment we are given the innocent fantasy of a child, in fact, the form in which oedipal love is expressed in childhood.

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.

How much they esteemed him is shown by the fact that their underground committee selected him as one of the few who would be helped to escape.

That is not to deny that he has been aware of traditions, of course, that he is steeped in them, in fact, or that he has dealt with them, in his books.

There is evidence to suggest, in fact, that many authors of the humorous sketches were prompted to write them -- or to make them as indelicate as they are -- by way of protesting against the artificial refinements which had come to dominate the polite letters of the South.

It seems quite obvious that all the really difficult tasks of human beings arise from the fact that man is not one, but many.

If our sincerity is granted, and it is granted, the discrepancy can only be explained by the fact that we have come to believe hearsay and legend about ourselves in preference to an understanding gained by earnest self-examination.

Perhaps the mere fact that by plucking on the nerves nature can awaken in the most ordinary of us, temporarily anyway, the sleeping poet, and in poets can discover their immortality, is the most remarkable of all the remarkable phenomena to which we can attest??

In fact, when A is a commutative unital C *- algebra, the Gelfand representation is then an isometric *- isomorphism between A and C ( Δ ( A )).

In fact, all C *- algebras that are finite dimensional as vector spaces are of this form, up to isomorphism.

So the logarithm function is in fact a group isomorphism from the group ( R < sup >+</ sup >,< big >×</ big >) to the group ( R ,< big >+</ big >).

In fact, the parallel of f is an isomorphism for every morphism f if and only if the pre-abelian category is an abelian category.

In fact, this definition applies more generally to the situation where the category is a concrete category whose underlying set functor is conservative, meaning that if the underlying map of sets is a bijection, then the original morphism is an isomorphism.

If one requires G and Q to be abelian groups, then the set of isomorphism classes of extensions of Q by a given ( abelian ) group N is in fact a group, which is isomorphic to

The first Stiefel – Whitney class classifies smooth real line bundles ; in particular, the collection of ( equivalence classes of ) real line bundles are in correspondence with elements of the first cohomology with Z / 2Z coefficients ; this correspondence is in fact an isomorphism of abelian groups ( the group operations being tensor product of line bundles and the usual addition on cohomology ).

One can, in fact, define the tensor algebra T ( V ) as the unique algebra satisfying this property ( specifically, it is unique up to a unique isomorphism ), but one must still prove that an object satisfying this property exists.

* the fact that for commutative C *- algebras, this representation is an isometric isomorphism.

In fact there is an isomorphism proved of the type

* h-principles are largely about showing that higher homotopy groups agree ( rather, that the inclusion is an isomorphism on these groups ) – showing that, once an inclusion has been shown to be 1-connected, it is in fact n-connected, possibly for all n ;

In fact, up to isomorphism, it is the unique 1-dimensional compact, connected Lie group.

This is true in the first cases mentioned, where L is a free K-module, or K contains the field of rational numbers, using the construction outlined here ( in fact, the result is a coalgebra isomorphism, and not merely a K-module isomorphism, equipping both S ( L ) and U ( L ) with their natural coalgebra structures such that for v ∈ L ).

In fact, it is an isomorphism, and its inverse is simply

Up to field isomorphism, this fact characterizes the field of surcomplex numbers within any fixed set theory.

The importance of skeletons comes from the fact that they are ( up to isomorphism of categories ), canonical representatives of the equivalence classes of categories under the equivalence relation of equivalence of categories.

* Planar graphs ( In fact, planar graph isomorphism is in log space, a class contained in P .)

In fact, it was later shown that graph isomorphism is low for ZPP < sup > NP </ sup >.

contains the graph isomorphism problem, and in fact this problem is low for.

The utility of this isomorphism comes from the fact that su < sub > 2 </ sub > is the complexification of the rotation algebra, and so its irreducible representations correspond to the well-known representations of the spatial rotation group ; for each j in ½Z, one has the ( 2j + 1 )- dimensional spin-j representation spanned by the spherical harmonics with j as the highest weight.

His investigation, also in 1826, of the two crystalline modifications of sulfur threw much light on the fact that the two minerals calc-spar and aragonite have the same composition but different crystalline forms, a property which Mitscherlich called isomorphism.

In fact from the Kummer – Vandiver conjecture and the norm residue isomorphism theorem follow a full conjectural calculation of the K-groups for all values of n ; see Quillen – Lichtenbaum conjecture for details.

The use of the word " the " ( as in " the left Kan extension ") is justified by the fact that, as with all universal constructions, if the object defined exists, then it is unique up to unique isomorphism.