For example, the division example above is surjective ( or onto ) because every rational number may be expressed as a quotient of an integer and a natural number.

If F is also surjective, then the Banach space X is called reflexive.

If f is also surjective and therefore bijective ( since f is already defined to be injective ), then S is called countably infinite.

* An equivalence relation ~ on X is the equivalence kernel of its surjective projection π: X → X /~.

The canonical map ker: X ^ X → Con X, relates the monoid X ^ X of all functions on X and Con X. ker is surjective but not injective.

Given two groups G and H and a group homomorphism f: G → H, let K be a normal subgroup in G and φ the natural surjective homomorphism G → G / K ( where G / K is a quotient group ).

* An epimorphism ( sometimes called a cover ) is a surjective homomorphism.

For instance, since a function is bijective if and only if it is both injective and surjective, a module homomorphism is an isomorphism if and only if it is both a monomorphism and an epimorphism.

However, the same is not true for epimorphisms ; for instance, the inclusion of Z as a ( unitary ) subring of Q is not surjective, but an epimorphic ring homomorphism.

Thus, a function with a codomain is invertible if and only if it is both injective ( one-to-one ) and surjective ( onto ).

Because the exponential map is surjective on some neighbourhood N of e, it is common to call elements of the Lie algebra infinitesimal generators of the group G. The subgroup of G generated by N is the identity component of G.

For example, the exponential map of SL < sub > 2 </ sub >( R ) is not surjective.

Also, exponential map is not surjective nor injective for infinite-dimensional ( see below ) Lie groups modelled on C < sup >∞</ sup > Fréchet space, even from arbitrary small neighborhood of 0 to corresponding neighborhood of 1.

* As an easy corollary of the Nikolov-Segal result above, any surjective discrete group homomorphism φ: G → H between profinite groups G and H is continuous as long as G is topologically finitely-generated.

And all the time, she had the heat of hatred in her, like charcoal that is burning on its under side, but not visibly.

There was a measure of protection in its concrete walls and ceiling, but the engineers who hastily installed it were well aware that concrete is not much better than prayer, if as efficacious, when a direct hit comes along.

The long-settled areas of states like Virginia and South Carolina developed the ante-bellum culture to its richest flowering, and there the memory is more precious, and the consciousness of loss the greater.

As it is, they consider that the North is now reaping the fruits of excess egalitarianism, that in spite of its high standard of living the `` American way '' has been proved inferior to the English and Scandinavian ways, although they disapprove of the socialistic features of the latter.

Reduced to its simplest terms, it is an assumption of a collective duty to compensate for the inability of individuals to cope with the rigors of the era.

It is one of the ironic quirks of history that the viability and usefulness of nationalism and the territorial state are rapidly dissipating at precisely the time that the nation-state attained its highest number ( approximately 100 ).

While sovereignty has roots in antiquity, in its present usage it is essentially modern.

There is unceasing pressure, but its sources are immediate.

Others are confined to vast reservations, and not only does the Australian government justifiably not wish them to be viewed as exhibits in a zoo, but on their reservations they are extremely fugitive, shunning camps, coming together only for corroborees at which their strange culture comes to its highest pitch -- which is very low indeed.

But the South is, and has been for the past century, engaged in a wide-sweeping urbanization which, oddly enough, is not reflected in its literature.

All but the most rabid of Confederate flag wavers admit that the Old Southern tradition is defunct in actuality and sigh that its passing was accompanied by the disappearance of many genteel and aristocratic traditions of the reputedly languid ante-bellum way of life.

The answers derived by these means may determine not only the temporal organization of the dance but also its spatial design, special slips designating the location on the stage where the movement is to be performed.

The other variables include the dancer who is to perform the movement and the length of time he is to take in its performance.

Most of these, with horrible exceptions, were conceived as is a ship, not as an attempt to quell the ocean of mankind, nor to deny its force, but as a means to survive and enjoy it.

Hemingway's fiction is supported by a `` moral '' backbone and in its search for ultimate meaning hints at a religious dimension.

In addition, they have been converted to Zen Buddhism, with its glorification of all that is `` natural '' and mysteriously alive, the sense that everything in the world is flowing.

Jazz is good not only because it promotes wholeness but because of its decided sexual effect.

This confession serves to make clear in part what is behind this sexual revolution: the craving for sensation for its own sake, the need for change, for new experiences.

What one actually remembers is its greenness.

the mill-pond is quiet, its surface dark and shadowed, and there does not seem to be much water in it.

But the highroad, according to the description of its traffic, belongs to life as it is lived in unawareness of death, while the way to the churchyard belongs to some other sort of life: a suffering form, an existence wholly comprised in the awareness of death.

For every group G there is a natural group homomorphism G → Aut ( G ) whose image is the group Inn ( G ) of inner automorphisms and whose kernel is the center of G. Thus, if G has trivial center it can be embedded into its own automorphism group.

He is also credited with the first port of the GNU C compiler to the INTEL 386 architecture, used in the original implementation of the Linux kernel.

Every character is automatically continuous from A to C, since the kernel of a character is a maximal ideal, which is closed.

As an algebra, a unital commutative Banach algebra is semisimple ( i. e., its Jacobson radical is zero ) if and only if its Gelfand representation has trivial kernel.

nuclei ; from Latin or, meaning kernel ) is a membrane-enclosed organelle found in eukaryotic cells.

When a transition between user mode and kernel mode is required in an operating system, a context switch is not necessary ; a mode transition is not by itself a context switch.

In mathematics, it is closely related to the Poisson kernel, which is the fundamental solution for the Laplace equation in the upper half-plane.

Only the kernel deals with files, but it handles all user-space interaction with ( virtual ) files in a manner that is transparent to the user-space programs.

The strategy is based on a coupling of special microprocessor hardware features, often involving the memory management unit, to a special correctly implemented operating system kernel.

Debian GNU / Linux, which includes the GNU OS tools and Linux kernel, is a popular and influential Linux distribution.

The advantage of loadable device drivers is that they can be loaded only when necessary and then unloaded, thus saving kernel memory.

The primary benefit of running a driver in user mode is improved stability, since a poorly written user mode device driver cannot crash the system by overwriting kernel memory.

The equivalence kernel of a function f is the equivalence relation ~ defined by.

The equivalence kernel of an injection is the identity relation.

Less formally, the equivalence relation ker on X, takes each function f: X → X to its kernel ker f. Likewise, ker ( ker ) is an equivalence relation on X ^ X.

This equivalence relation is known as the kernel of.