Hell, they were fightin' each other so hard they had no time for anyone else.

They were silent for a little while, each looking glum.

They were headed straight for each other on a collision course.

After that, it requires several minutes of concentrated work, including six separate and deliberate actions by a minimum of three men sitting at three separate stations in a bomber, each with another man beside him to help, for an armed bomb to be released.

`` We the people of the Confederate States, each state acting in its sovereign and independent character, in order to form a permanent federal government, establish justice, insure domestic tranquility, and secure the blessings of liberty to ourselves and our posterity -- invoking the favor and the guidance of Almighty God -- do ordain and establish this Constitution for the Confederate States of America ''.

Whitehead contends that the human way of understanding existence as a unity of interlocking and interdependent processes which constitute each other and which cause each other to be and not to be is possible only because the basic form of such an understanding, for all its vagueness and tendency to mistake the detail, is initially given in the way man feels the world.

Perry Mason and Hamilton Burger, Nero Wolfe and Inspector Cramer spend more time fighting each other than they do in looking for the criminal.

In spite of the dreams of the host for oneness in the group, the Af incoming messages for each guest overload his receiving system beyond comprehension if N exceeds about six.

This arrangement was for Copernicus literally monstrous: `` With ( the Ptolemaists ) it is as though an artist were to gather the hands, feet, head and other members for his images from divers models, each part excellently drawn, but not related to a single body ; ;

Such a list must naturally be selective, and the treatment of each man is brief, for I am interested only in their general ideas on the moral measure of literature.

The brother and sister seemed to be a sort of mutual-aid society, a little fortress of kindness for each other in a hard world.

In the central contest, that for Mayor, they may have found some pertinent points in what each faction has said about the other.

The answer is different for each autistic child, but for most there is an answer.

If the railroads, for example, regularly slaughtered 25,000 passengers each year, the high priests of the cult would have cause to tremble for their personal safety, for such a holocaust would excite demands for the hanging of every railroad president in the United States.

Two things have happened in recent months to bring the Council into perspective: each provides a basis for renewed hope and joy.

I know of no other solution than the one N.C. proposes -- to do what we can for each sufferer as he confronts us, hoping that this will spread beyond him to others at some time and some place.

We hoped that its practitioners and teachers might be put on some sort of reserve list and called back for refresher courses each year or so.

Warren and White each carried, in addition to their own weapons and ammo, a box of ammo for the ROK's machine gun.

The original Nernst heat theorem makes the weaker and less controversial claim that the entropy change for any isothermal process approaches zero as T → 0:

A homomorphism between two Boolean algebras A and B is a function f: A → B such that for all a, b in A:

The systematic name for sucrose, O-α-D-glucopyranosyl -( 1 → 2 )- D-fructofuranoside, indicates four things:

The systematic name for lactose is O-β-D-galactopyranosyl -( 1 → 4 )- D-glucopyranose.

This is because we arrange things such that for every integer, there is a distinct odd integer: ... − 2 → − 3, − 1 → − 1, 0 → 1, 1 → 3, 2 → 5, ...; or, more generally, n → 2n + 1.

* Consider the set K of all functions ƒ: → satisfying the Lipschitz condition | ƒ ( x ) − ƒ ( y )| ≤ | x − y | for all x, y ∈.

The free ions recombine to create new chemical compounds ( for example, breaking atmospheric oxygen into single oxygen → 2O, which then recombine creating ozone ).

Some definitions also add that for any production rule of the form u → v of a context-sensitive grammar, it shall be true that | u |≤| v |.

A metric space X is complete if and only if every decreasing sequence of non-empty closed subsets of X, with diameters tending to 0, has a non-empty intersection: if F < sub > n </ sub > is closed and non-empty, for every n, and diam ( F < sub > n </ sub >) → 0, then there is a point x ∈ X common to all sets F < sub > n </ sub >.

Given a subset X of a manifold M and a subset Y of a manifold N, a function f: X → Y is said to be smooth if for all p in X there is a neighborhood of p and a smooth function g: U → N such that the restrictions agree ( note that g is an extension of f ).

for some natural number n. Moreover, since, the commutator subgroup is normal in G. For any homomorphism f: G → H,

Namely φ is universal for homomorphisms from G to an abelian group H: for any abelian group H and homomorphism of groups f: G → H there exists a unique homomorphism F: G < sup > ab </ sup > → H such that.

As usual for objects defined by universal mapping properties, this shows the uniqueness of the abelianization G < sup > ab </ sup > up to canonical isomorphism, whereas the explicit construction G → G / shows existence.

The flow now defines a map, the Poincaré map F: S → S, for points starting in S and returning to S. Not all these points will take the same amount of time to come back, but the times will be close to the time it takes x < sub > 0 </ sub >.

A dynamical system is the tuple, with a manifold ( locally a Banach space or Euclidean space ), the domain for time ( non-negative reals, the integers, ...) and f an evolution rule t → f < sup > t </ sup > ( with ) such that f < sup > t </ sup > is a diffeomorphism of the manifold to itself.

A map τ: X → X is said to be Σ-measurable if and only if, for every σ ∈ Σ, one has.

Arcturus is notable for its high proper motion, larger than any first magnitude star in the stellar neighborhood other than α Centauri.

for an arbitrary real or complex number α ( the order of the Bessel function ); the most common and important cases are for α an integer or half-integer.

Although α and − α produce the same differential equation, it is conventional to define different Bessel functions for these two orders ( e. g., so that the Bessel functions are mostly smooth functions of α ).

Bessel functions of the first kind, denoted as J < sub > α </ sub >( x ), are solutions of Bessel's differential equation that are finite at the origin ( x = 0 ) for integer α, and diverge as x approaches zero for negative non-integer α.

Plot of Bessel function of the first kind, J < sub > α </ sub >( x ), for integer orders α = 0, 1, 2.

On the other hand, for integer order α, the following relationship is valid ( note that the Gamma function becomes infinite for negative integer arguments ):

Plot of Bessel function of the second kind, Y < sub > α </ sub >( x ), for integer orders α = 0, 1, 2.

When α is an integer, moreover, as was similarly the case for the functions of the first kind, the following relationship is valid:

For integer order α = n, J < sub > n </ sub > is often defined via a Laurent series for a generating function:

This orthogonality relation can then be used to extract the coefficients in the Fourier – Bessel series, where a function is expanded in the basis of the functions J < sub > α </ sub >( x u < sub > α, m </ sub >) for fixed α and varying m.