Let V a representation of G, and form the vector bundle V = Q ×< sub > G </ sub > V over M. Then the principal G-connection α on Q induces a covariant derivative on V, which is a first order linear differential operator

A principal G-connection on a principal G-bundle P over a smooth manifold M is a particular type of connection which is compatible with the action of the group G.

Let π: P → M be a smooth principal G-bundle over a smooth manifold M. Then a principal G-connection on P is a differential 1-form on P with values in the Lie algebra of G which is G-equivariant and reproduces the Lie algebra generators of the fundamental vector fields on P.

Sometimes the term principal G-connection refers to the pair ( P, ω ) and ω itself is called the connection form or connection 1-form of the principal connection.

A principal G-connection ω on P determines an Ehresmann connection on P in the following way.

Conversely, an Ehresmann connection H ⊂ TP ( or v: TP → V ) on P defines a principal G-connection ω if and only if it is G-equivariant in the sense that.

The curvature form of a principal G-connection ω is the-valued 2-form Ω defined by

Fibrinogen, the principal protein of vertebrate blood clotting, is a hexamer containing two sets of three different chains ( α, β, and γ ), linked to each other by disulfide bonds.

Using Log we obtain the principal value of z < sup > α </ sup >, i. e.,

It induces a principal connection α on the associated principal G-bundle P ×< sub > H </ sub > G. This is a Cartan connection.

The principal connection α on Q can be recovered from the form η by taking Q to be the associated bundle P ×< sub > H </ sub > G. Conversely, the form η can be recovered from α by pulling back along the inclusion P ⊂ Q.

Since α is a principal connection, it induces a connection on any associated bundle to Q.

Suppose that M is a Cartan geometry modelled on G / H, and let ( Q, α ) be the principal G-bundle with connection, and ( P, η ) the corresponding reduction to H with η equal to the pullback of α.

When I is a principal ideal αO < sub > K </ sub > then N ( I ) is equal to the absolute value of the norm to Q of α, for α an algebraic integer.

If | α | < δ, the branches are called principal, because they equal their real analogons on the positive real axis.

The norm of a principal ideal generated by α is the ideal generated by the field norm of α.

As a solution to the Hamilton-Jacobi equation, the principal function contains N + 1 undetermined constants, the first N of them denoted as α < sub > 1 </ sub >, α < sub > 2 </ sub > ... α < sub > N </ sub >, and the last one coming from the integration of.

The promenade, for example, continues to take place on the Chahar Bagh, a mile-long garden of plane and poplar trees that now serves as the city's principal street.

Potemkin's Army of Ekaterinoslav, totaling, it was claimed, 40,000 regular troops and 6,000 irregulars of the Cossack Corps, had invested Islam's principal stronghold on the north shore of the Black Sea, the fortress town of Oczakov, and was preparing to test the Turk by land and sea.

Two of the principal addresses were delivered by prominent Protestants, and when the speaker was a Catholic, one `` discussant '' on the dais tended to be of another religious persuasion.

We had a couple of schools in this country, the principal one being on the Marshall Field estate out in Lloyd's Neck.

Subject to the limitations hereinafter provided, the Secretary of the Treasury is authorized and directed to pay, as prescribed by Section 8 of this Title, an amount not exceeding the principal of each award, plus accrued interests on such awards as bear interest, certified pursuant to Section 5 of this Title, in accordance with the award.

to make payments of $1,000 on the principal of each award of more than $1,000 in principal amount, certified pursuant to Section 5 of this Title ; ;

after completing the payments prescribed by paragraphs ( 2 ) and ( 3 ) of this subsection, to make payments, from time to time in ratable proportions, on account of the unpaid principal of all awards in the principal amount of more than $1,000, according to the proportions which the unpaid principal of such awards bear to the total amount in the fund available for distribution at the time such payments are made ; ;

after payment has been made of the principal amounts of all such awards, to make pro rata payments on account of accrued interest on such awards as bear interest.

The extent of such interference -- which may be so slight as to be undetectable at any point where either of the stations renders a usable signal, or may be so great as to virtually destroy the service areas of both stations -- depends on many factors, among the principal ones being the distance between the stations, their respective radiated power, and, of particular significance here, the time of day.

Barco, his state of apprehension gone, never to return, had assumed a matter-of-factness which remained his principal attitude from that time on.

In their very first collages, Braque and Picasso draw or paint over and on the affixed paper or cloth, so that certain of the principal features of their subjects as depicted seem to thrust out into real, bas-relief space -- or to be about to do so -- while the rest of the subject remains imbedded in, or flat upon, the surface.

If it is owned, taxes must be paid, and if the place is not free of mortgage, there will be interest and payments on the principal to take care of.

The Ainus were a primitive people, already living on the island before the principal ancestors of the Japanese came from Southern Asia.

Ships from the West Coast rotated on six-month tours of duty with the Seventh Fleet, and Yokosuka was the Seventh Fleet's principal port for maintenance, upkeep and shore liberty.

Mr. Willis bought Zenith Plastic Products, a skeleton corporation of sorts which had undergone many vicissitudes and whose principal assets were a couple of electronics plants on Long Island engaged in working out government contracts, and installed Freddy in an executive position.

There are three principal families of abugidas, depending on whether vowels are indicated by modifying consonants by diacritics, distortion, or orientation.

Sultan Ahmed III had become unpopular by reason of the excessive pomp and costly luxury in which he and his principal officers indulged ; on September 20, 1730, a mutinous riot of seventeen janissaries, led by the Albanian Patrona Halil, was aided by the citizens as well as the military until it swelled into an insurrection in front of which the Sultan was forced to give up the throne.

Most notably, Ajax is not wounded in any of the battles described in the Iliad, and he is the only principal character on either side who does not receive personal assistance from any of the gods who take part in the battles.

His principal theological works are a commentary in three volumes on the Books of the Sentences of Peter Lombard ( Magister Sententiarum ), and the Summa Theologiae in two volumes.

Opposition to the union seemed to subside somewhat for a time upon the publication of Tsar Nicholas II's congratulations to the king on his engagement and of his acceptance to act as the principal witness at the wedding.

However we are told by Longinus that Ammonius wrote nothing, and if Ammonius was the principal influence on Plotinus, then it is unlikely that Ammonius would have been a Christian.

This dance took place at Q ' umarkaj and involved a gathering of all the principal lineages subject to the K ' iche ' Kingdom of Q ' umarkaj, and as is described in the K ' iche ' chronicle Título de Totonicapán, they were expected to bring tribute, slaves and sacrifices.

* USA Today Q & A with Jefferson County High principal Molly Howard

* USA's top principal could teach CEOs a thing or two, USA Today Q & A with principal of Jefferson County High

A principal objection to the 2SH is that it requires a hypothetical document, Q, the existence of which is not attested in any way, either by existing fragments ( and a great many fragments of early Christian documents do exist ) or by early Church tradition.

The class number of Q (√ d ) is the number of equivalence classes of ideals of the ring of integers of Q (√ d ), where two ideals I and J are equivalent if and only if there exist principal ideals ( a ) and ( b ) such that ( a ) I = ( b ) J.

Thus, the ring of integers of Q (√ d ) is a principal ideal domain ( and hence a unique factorization domain ) if and only if the class number of Q (√ d ) is equal to 1.

* a principal G-bundle Q over M

* a principal H-subbundle P of Q ( i. e., a reduction of structure group )

Let a be a root of P, and Q < sub > a, t </ sub > the product of P by the principal part of the Laurent series of f at a.

Four views of the professional school counselor and principal relationship: A Q methodology study.

* The unique square root of a in Q < sub > n </ sub > is called the principal square root of a modulo n