This `` grand division '' permits many costs to be assigned in their entirety to some one class, such as street lighting, or at least to be excluded completely from some important class or classes.

An important class of alcohols are the simple acyclic alcohols, the general formula for which is C < sub > n </ sub > H < sub > 2n + 1 </ sub > OH.

In this fashion, one can prove many important theorems in the class of compact spaces, that do not hold in the context of non-compact ones.

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.

An important class of redox reactions are the electrochemical reactions, where the electrons from the power supply are used as a reducing agent.

It is important to note that historians class the predecessors of Hugh Capet as Robertians, not Capetians.

Perhaps even more important than compensation is that class treatment of claims may be the only way to impose the costs of wrongdoing on the wrongdoer, thus deterring future wrongdoing.

Hilbert spaces are an important class of objects in the area of functional analysis, particularly of the spectral theory of self-adjoint linear operators, that grew up around it during the 20th century.

An important class of Riemannian manifolds is the Riemannian symmetric spaces, whose curvature is not necessarily constant.

An important feature of the scheme is the ability to assign multiple class numbers to a bibliographical item and only use one of them for shelving.

Eiffel's most important contribution to software engineering is design by contract ( DbC ), in which assertions, preconditions, postconditions, and class invariants are employed to help ensure program correctness without sacrificing efficiency.

It is important to compare the class of Euclidean domains with the larger class of principal ideal domains ( PIDs ).

Guatemala City offers all the modern amenities along with important historic sites that a world class city can be expected to have ranging from an IMAX Theater to the Ícaro film festival ( Festival Ícaro ) featuring films produced in Guatemala and Central America.

An important class of idempotent functions is given by projections in a vector space.

The other important contribution to Marx's revision of Hegelianism came from Engels's book, The Condition of the Working Class in England in 1844, which led Marx to conceive of the historical dialectic in terms of class conflict and to see the modern working class as the most progressive force for revolution.

This is the Staudinger synthesis, a facile route to this important class of compounds.

Although this also used the Latin alphabet, it was much less affected by the ban, because Polish was still used by the politically important class of the nobility, and also used predominantly in the biggest towns of Lithuania, and supported by the church.

Another biologically important class of molecules is exemplified by the quinones and hydroquinones, which contain an isoprenoid tail attached to a quinonoid core of non-isoprenoid origin.

* An important class of infinite-dimensional real Lie algebras arises in differential topology.

His letters provide detailed information about an important period in Roman history and offer a vivid picture of the public and private life among the Roman governing class.

It is important to note that unlike subject heading or thesauri where multiple terms can be assigned to the same work, in library classification systems, each work can only be placed in one class.

As the composition of the Earth's crust is dominated by silicon and oxygen, silicate elements are by far the most important class of minerals in terms of rock formation and diversity.

Rogers described the concepts of congruence and incongruence as important ideas in his theory.

In many important cases congruence relations have an alternative representation as substructures of the structure on which they are defined.

Shape of femoral head at the time when Legg-Calve Perthes Disease heals is the most important determinant of risk for degenerative arthritis ; hence, the shape of femoral head and congruence of hip are most useful outcome measures.

If X and Y are algebraic structures of some fixed type ( such as groups, rings, or vector spaces ), and if the function f from X to Y is a homomorphism, then ker f will be a subalgebra of the direct product X × X. Subalgebras of X × X that are also equivalence relations ( called congruence relations ) are important in abstract algebra, because they define the most general notion of quotient algebra.

A Cartan connection provides an identification between the model spaces in the congruence along any curve in S. An important feature of these identifications is that the point of contact of the model space with S always moves with the curve.

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.

Describing the mutual motion of the test particles in a null geodesic congruence in a spacetime such as the Schwarzschild vacuum or FRW dust is a very important problem in general relativity.

For example in a paper reporting on a study involving human subjects, there typically appears a table giving the overall sample size, sample sizes in important subgroups ( e. g., for each treatment or exposure group ), and demographic or clinical characteristics such as the average age, the proportion of subjects of each sex, and the proportion of subjects with related comorbidities.

Other economically important plantain groups include the East African Highland bananas ( Mutika / Lujugira subgroup ) of the AAA Group and the Pacific plantains ( including the Popoulo, Maoli, and Iholena subgroups ) of the AAB Group.

An important condition in the theory is no small subgroups.

The symplectic group comes up in quantum physics as a symmetry on poisson brackets so it is important to understand its subgroups.

There are three important normal subgroups of prime power index, each being the smallest normal subgroup in a certain class:

These groups have important connections to the Sylow subgroups and the transfer homomorphism, as discussed there.

An aggregate pattern is an important statistical concept in many fields that rely on statistics to predict the behavior of large groups, based on the tendencies of subgroups to consistently behave in a certain way.

An important application comes from the result that, if X is a " sufficiently good " topological space, there is a bijection from the collection of all isomorphism classes of connected coverings of X and subgroups of the fundamental group of X.

The most important subgroups of a linear algebraic group, besides its Borel subgroups, are its tori, especially the maximal ones ( similar to the study of maximal tori in Lie groups ).

Discrete normal subgroups play an important role in the theory of covering groups and locally isomorphic groups.

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.

Maximal compact subgroups play an important role in the classification of Lie groups and especially semi-simple Lie groups.

Subgroup series can simplify the study of a group to the study of simpler subgroups and their relations, and several subgroup series can be invariantly defined and are important invariants of groups.

It is especially important to exactly determine the structure of the sample ( and specifically the size of the subgroups ) when subgroup analyses will be performed during the main analysis phase.