It was all right to put a bunch of ranchers onto horses, to call them Night Riders, to set out to attack the largest mining combination the country had ever seen if all they wanted was adventure.

They recognized that slavery was a moral issue and not merely an economic interest, and that to recognize it explicitly in their Constitution would be in explosive contradiction to the concept of sovereignty they had set forth in the Declaration of 1776 that `` all men are created equal, that they are endowed by their Creator with certain unalienable rights, that among them are life, liberty and the pursuit of happiness.

We of the liberal-led world got all set for peace and rehabilitation.

The Axioms required to make the theoretical machinery operate are set out tersely and powerfully, so that all permissible operations within the theory can be traced rigorously back to these axioms, rules, and primitive notions.

For by now the original cause of the quarrel, Philip's seizure of Gascony, was only one strand in the spider web of French interests that overlay all western Europe and that had been so well and closely spun that the lightest movement could set it trembling from one end to the other.

He could produce carefully constructed orations, set and formal speeches, artfully and prayerfully made by writing and rewriting with all the aid his tutor and others could provide, and then delivered verbatim from memory.

In entering this union we will be surrendering most, if not all, of our economic autonomy to international bodies such as the Atlantic Institute ( recently set up ) or the O.E.C.D., I.M.F. and others.

The extent to which we can persuade the less developed countries to appraise their own resources, to set targets toward which they should be working, to establish in the light of this forward perspective the most urgent priorities for their immediate attention, and to do the other things which they must do to help themselves, all on a realistic long-term basis, will depend importantly on the incentives we place before them.

( For each State, make all computations set forth in items 1 to 8 above, and then add the results obtained for each State in item 8.

Before the Juniors entered the ring the Steward announced that after all Juniors had moved their dogs around the ring and set them up, they could relax with their dogs.

Above all, don't set up extravagant fringe benefits just to buy employee good will.

The huge market for changeable signs has spurred a universal demand for individual plastic letters, in all shapes and sizes -- and a number of companies are set up to supply them.

You can save all that shimming time if you set your windows in one, two, three order -- first the stud on one side, then the window, then the stud on the other side.

It was all set up so there would be no dust anywhere and so that their children would color in the coloring room, paint in the painting room, play with blocks in the block house, and do all the other things in the proper rooms at exactly the right time.

After all, if you were going to set up a workshop you had to have the proper equipment and that was that.

The set of all binomial distributions is called the family of binomial distributions, but in general discussions this expression is often shortened to `` the binomial distribution '', or even `` the binomial '' when the context is clear.

A unified set of regulations, applicable to all areas, was issued in 1929, and a complementary series of demographic inquiries in selected areas was instituted at the same time.

Unfortunately, however, and for reasons to be discussed in the following chapter, no rate relationships can be made completely nondiscriminatory as long as all or some of the rates must be set above marginal costs in order to yield adequate revenues.

J. Wheaton Smith, editor of the Warren Telegraph stated that `` the ends of justice must be satisfied, a solitary example must be set, in order that all those misnamed philantropists, who, actuated by a blind zeal, dare to instigate riot, treason, and murder, may heed it and shape their future course accordingly ''.

The set of all decisions is called the operating policy or, more simply, the policy.

The set of vectors Af constitutes the operating policy or, more briefly, the policy, and a policy is admissible if all the Af belong to S.

Houghton said 6 p.m. Friday had been set for a canvass of all 480 homes in the subdivision, which is located northeast of Dequindre and 14 Mile Road East.

The statement also points to a classic paradox: The more men turn toward God, who is not only in himself the paradigm of all unity but also the only ground on which human unity can ultimately be established, the more men splinter into groups and set themselves apart from one another.

The set of all automorphisms of an object forms a group, called the automorphism group.

If the automorphisms of an object X form a set ( instead of a proper class ), then they form a group under composition of morphisms.

The composition of two automorphisms is again an automorphism, and with this operation the set of all automorphisms of a group < var > G </ var >, denoted by Aut (< var > G </ var >), forms itself a group, the automorphism group of < var > G </ var >.

In other words, an automorphism of E / F is an isomorphism α from E to E such that α ( x ) = x for each x in F. The set of all automorphisms of E / F forms a group with the operation of function composition.

If V is a vector space over the field F, the general linear group of V, written GL ( V ) or Aut ( V ), is the group of all automorphisms of V, i. e. the set of all bijective linear transformations V → V, together with functional composition as group operation.

The set of automorphisms of a hypergraph H (= ( X, E )) is a group under composition, called the automorphism group of the hypergraph and written Aut ( H ).

Just as O ( n ) is the group of automorphisms which keep the quadratic polynomials x < sup > 2 </ sup > + y < sup > 2 </ sup > + ... invariant, F < sub > 4 </ sub > is the group of automorphisms of the following set of 3 polynomials in 27 variables.

F < sub > 4 </ sub > is the only exceptional lie group which gives the automorphisms of a set of real commutative polynomials.

In the case of transformation laws in quantum mechanics, the requisite automorphisms are unitary ( or antiunitary ) linear transformations of the Hilbert space V. Under Galilean relativity or special relativity, the mathematics of frames of reference is particularly simple, and in fact restricts considerably the set of physically meaningful observables.

An important example of this is when elements of G act by conjugation, so that the set of operators consists of the inner automorphisms.

Note that automatically G acts by automorphisms ( bijections ) on the set.

into the group of automorphisms of the object X in the category C. The pair ( X, ρ ) defines a homogeneous space provided ρ ( G ) is a transitive group of symmetries of the underlying set of X.

In a polynomial ring in some finite set of indeterminates, there is an action by ring automorphisms of the symmetric group on ( the indices of ) the indeterminates ( simultaneously substituting each of them for another according to the permutation used ).

For example, all order automorphisms of a power set S = 2 < sup > R </ sup > are induced by permutations of R. The papers cited above treat only sets S of functions on R < sup > n </ sup > satisfying some condition of convexity and prove that all order automorphisms are induced by linear or affine transformations of R < sup > n </ sup >.

Over an algebraically closed field, this is the only form ; however, over other fields, there are often many other forms, or “ twists ” of E < sub > 8 </ sub >, which are classified in the general framework of Galois cohomology ( over a perfect field k ) by the set H < sup > 1 </ sup >( k, Aut ( E < sub > 8 </ sub >)) which, because the Dynkin diagram of E < sub > 8 </ sub > ( see below ) has no automorphisms, coincides with H < sup > 1 </ sup >( k, E < sub > 8 </ sub >).

( The set of morphisms from the symmetric group S < sub > 3 </ sub > of order three to itself,, has ten elements: an element z whose product on either side with every element of E is z ( the homomorphism sending every element to the identity ), three elements such that their product on one fixed side is always itself ( the projections onto the three subgroups of order two ), and six automorphisms.

While in the first two cases the surface X admits infinitely many conformal automorphisms ( in fact, the conformal automorphism group is a complex Lie group of dimension three for a sphere and of dimension one for a torus ), a hyperbolic Riemann surface only admits a discrete set of automorphisms.

An invariant set of an operation T is also said to be stable under T. For example, the normal subgroups that are so important in group theory are those subgroups that are stable under the inner automorphisms of the ambient group.

Yet another equivalent definition of a rack is that it is a set where each element acts on the left and right as automorphisms of the rack, with the left action being the inverse of the right one.

The inner automorphisms of this group correspond to permuting the six elements from which we defined the morphemes and synthemes ; these permutations act on the Tutte – Coxeter graph by permuting the vertices on each side of its bipartition while keeping each of the two sides fixed as a set.