The congruence of its secants is therefore of order Af and class Af.

Therefore, the congruence of its secants, that is the image of a general plane field of lines, is of order Af and class Af.

Finally, the image of a general bundle of lines is a congruence whose order is the order of the congruence of invariant lines, namely Af and whose class is the order of the image congruence of a general plane field of lines, namely Af.

Gauss also made important contributions to number theory with his 1801 book Disquisitiones Arithmeticae ( Latin, Arithmetical Investigations ), which, among things, introduced the symbol ≡ for congruence and used it in a clean presentation of modular arithmetic, contained the first two proofs of the law of quadratic reciprocity, developed the theories of binary and ternary quadratic forms, stated the class number problem for them, and showed that a regular heptadecagon ( 17-sided polygon ) can be constructed with straightedge and compass.

For example, the cyclic group of addition modulo n can be obtained from the integers by identifying elements that differ by a multiple of n and defining a group structure that operates on each such class ( known as a congruence class ) as a single entity.

Like any congruence relation, congruence modulo n is an equivalence relation, and the equivalence class of the integer a, denoted by, is the set.

This set, consisting of the integers congruent to a modulo n, is called the congruence class or residue class or simply residue of the integer a, modulo n. When the modulus n is known from the context, that residue may also be denoted.

for all g < sub > 1 </ sub >, g < sub > 2 </ sub >, h < sub > 1 </ sub >, h < sub > 2 </ sub > ∈ G. For a congruence on a group, the equivalence class containing the identity element is always a normal subgroup, and the other equivalence classes are the cosets of this subgroup.

For a congruence on a ring, the equivalence class containing 0 is always a two-sided ideal, and the two operations on the set of equivalence classes define the corresponding quotient ring.

Equivalently, the primes are evenly distributed ( asymptotically ) among each congruence class modulo d.

( In other words, every congruence class except zero modulo p has a multiplicative inverse.

For an algebra, given a congruence E on, the algebra is called the quotient algebra ( or factor algebra ) of modulo E. There is a natural homomorphism from to mapping every element to its equivalence class.

An important class of congruence subgroups is given by reduction of the ring of entries: in general given a group such as the special linear group SL ( n, Z ) we can reduce the entries to modular arithmetic in Z / NZ for any N > 1, which gives a homomorphism

The natural mapping from the integers to the congruence classes modulo n that takes an integer to its congruence class modulo n respects products.

Therefore, pp-waves belong to the Kundt class ( the class of Lorentzian manifolds admitting a null congruence with vanishing optical scalars ).

Since they constitute a very simple and natural class of Lorentzian manifolds, defined in terms of a null congruence, it is not very surprising that they are also important in other relativistic classical field theories of gravitation.

Some classes of solutions can be invariantly characterized using algebraic symmetries of the Weyl tensor: for example, the class of non-conformally flat null electrovacuum or null dust solutions admitting an expanding but nontwisting null congruence is precisely the class of Robinson / Trautmann spacetimes.

Monogamy is the vice from which the abjectly fearful middle class continue to suffer, whereas the beatnik has the courage to break out of that prison of respectability.

And it is precisely in this poorer economic class that one finds, and has always found, the most racial friction.

Was it supposed, perchance, that A & M ( vocational training, that is ) was quite sufficient for the immigrant class which flooded that part of the New England world in the post-Civil War period, the immigrants having been brought in from Southern Europe, to work in the mills, to make up for the labor shortage caused by migration to the West??

By virtue of his self-reliance, his individualism and his freedom from external restraint, the private eye is a perfect embodiment of the middle class conception of liberty, which amounts to doing what you please and let the devil take the hindmost.

At the same time, because the personal code of the detective coincides with the legal dictates of his society, because he likes to catch criminals, he is in middle class eyes a virtuous man.

Since a civilizational crisis involves also a crisis in private interests and in the ruling class, reaction is normally found among those who feel themselves to be among the ruling class.

The concept of labor as a special class is outmoded, and in the task confronting America as bastion of the free world, labor must learn to put the national interest first if it is itself to survive.

The soldiers themselves cannot stage a successful rebellion, it is assumed: but will their discontent spread to the officer class??

Therefore, under our longstanding allocation rules, on some of these channels no station other than the dominant ( class 1 ) -- A ) station is permitted to operate at night, so that the 1, -- A station can render service, interference free, wherever it can be received.

Such additional daytime class 2, assignments are appropriate if optimum use is to be made of these frequencies, and the Commission has over the years made a large number of them.

Staley Hanover ( Knight Dream-Sweetmite Hanover ) is a little on the small side but a very compact colt and looks like one to stand training and many future battles with colts in his class.

Consultation with architects, clients, real estate men, fabric houses and furniture companies is essential to the proper development of class problems just as in actual work.

As is well known, detergent actives belong to the chemical class consisting of moderately high molecular weight and highly polar molecules which exhibit the property of forming micelles in solution.

This is particularly true of highly charged ions, especially those ions which fall into the class of polyelectrolytes.

A final class of exceptional lines is identifiable from the following considerations: Since no two generators of Af can intersect, it follows that their image curves can have no free intersections.

Another great danger is that the emerging middle class will feel itself increasingly alienated from the political leaders who still justify their dominance by reference to the struggle for independence or the early phase of nationalism.

Like most of the country, Surigao is officially considered a tropical rainforest climate type, Köppen class Af with an average monthly precipitation of 308. 66 mm ( 12. 5 inches ).

But the practice is likely to be misleading, since it may seem to support a conclusion that, as long as the revenues from any class of service cover the imputed operating expenses plus some return on capital investment, however low, the rates of charge for this service are compensatory.

Ravi Batra argues that " all official economic measures adopted since 1981 ... have devastated the middle class " and that the Occupy Wall Street movement should push for their repeal and thus end the influence of the super wealthy in the political process, which he considers a manifestation of crony capitalism.

Although the Bizerte ( Type PR 48 large patrol craft ) class L ' Audacieux is fitted for SS 12M missiles these are not embarked and its operational status is in some doubt, having not been reported at sea since 1995.

However since Burke there has always been tension between traditional aristocratic conservatism and the wealthy business class.

Notably since US courts are reluctant to take up class actions brought on behalf of injured parties not residing in the US who have suffered damages due to acts or omissions committed outside the US, it may be interesting to combine a US class action and a Dutch collective action to be able come to a settlement that covers plaintiffs worldwide.

The classic view of Domitian is usually negative, since most of the antique sources were related to the Senatorial or aristocratic class, with which Domitian had a notoriously difficult relation.

Since each element of X belongs to a unique cell of any partition of X, and since each cell of the partition is identical to an equivalence class of X by ~, each element of X belongs to a unique equivalence class of X by ~.

These were found also, in its long-term allies Chile, which sold since 2008 Leopard 1 tanks, Leander class frigates.

Class AB is widely considered a good compromise for audio amplifiers, since much of the time the music is quiet enough that the signal stays in the " class A " region, where it is amplified with good fidelity, and by definition if passing out of this region, is large enough that the distortion products typical of class B are relatively small.

The conflict in the countryside from 1952-1954 between the Ladinos and Indians was not a new occurrence, but rather a continuation of a dilemma fueled by a complicated mix of class, regional, political, and ethnic differences since the colonial times.

Custom status codes can be used since, if the user agent encounters a code it does not recognize, it can use the first digit of the code to determine the general class of the response.

The Italian Navy has used six hydrofoils of the Sparviero class since the late 1970s.

Although most felt the Russian Revolution was working class in character, they believed that, since capitalist relations still existed ( because the workers had no say in running the economy ), the Soviet Union ended up as a state capitalist country, with the state replacing the individual capitalist.

The field has since broadened to include every class of materials, including ceramics, polymers, semiconductors, magnetic materials, medical implant materials and biological materials ( materiomics ).

He rejected the theory of class struggle and the Marxist concern with labor, instead claiming, according to Leszek Kołakowski, that since " all questions of material existence have been solved, moral commands and prohibitions are no longer relevant.

It has since come to describe a broader class of content without conventional copyright restrictions.

Any definition which focuses on amusement leaves us unsatisfied since the feelings experienced by a world class chess player are very different from those of a circle of children playing Duck Duck Goose.

Furthermore, since internal class conflict had been overcome, workers could now be duly rewarded for their qualifications and technical skills without contravening Marxism-Leninism.

However, they cannot be used to determine if a language is in a given class, since satisfying the pumping lemma is a necessary, but not sufficient, condition for class membership.

The addition of the C6 fluorine atom has since been demonstrated to not be required for the antibacterial activity of this class ( circa 1997 ).