On the other hand, the theory of infinite abelian groups is an area of current research.

* Conductor ( class field theory ), a modulus describing the ramification in an abelian extension of local or global fields

The theory for abelian locally compact groups is called Pontryagin duality ; it is considered to be in a satisfactory state, as far as explaining the main features of harmonic analysis goes.

If the group is neither abelian nor compact, no general satisfactory theory is currently known.

In Freiberg Abel did brilliant research in the theory of functions, particularly: elliptic, hyperelliptic, and a new class now known as abelian functions.

Mathematically, QED is an abelian gauge theory with the symmetry group U ( 1 ).

In mathematics, specifically in category theory, a preadditive category is a category that is enriched over the monoidal category of abelian groups.

* A gauge theory is abelian or non-abelian depending on whether its symmetry group is commutative or non-commutative, respectively

In mathematics, more specifically in the field of group theory, a solvable group ( or soluble group ) is a group that can be constructed from abelian groups using extensions.

In mathematics, especially homological algebra and other applications of abelian category theory, the short five lemma is a special case of the five lemma.

In mathematics, especially homological algebra and other applications of abelian category theory, the five lemma is an important and widely used lemma about commutative diagrams.

In the theory of abelian groups, the torsion subgroup A < sub > T </ sub > of an abelian group A is the subgroup of A consisting of all elements that have finite order.

In mathematics, more specifically in the field of group theory, a nilpotent group is a group that is " almost abelian ".

The development of class field theory has provided detailed information about abelian extensions of number fields, function fields of algebraic curves over finite fields, and local fields.

The Kummer theory gives a complete description of the abelian extension case, and the Kronecker – Weber theorem tells us that if K is the field of rational numbers, an extension is abelian if and only if it is a subfield of a field obtained by adjoining a root of unity.

In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups defined from a co-chain complex.

That this and other statements about uncountable abelian groups are provably independent of ZFC shows that the theory of such groups is very sensitive to the assumed underlying set theory.

In mathematics, class field theory is a major branch of algebraic number theory that studies abelian extensions of number fields and function fields of curves over finite fields and arithmetic properties of such abelian extensions.

Thus, while altruistic persons may under some circumstances be outcompeted by less altruistic persons at the individual level, according to group selection theory the opposite may occur at the group level where groups consisting of the more altruistic persons may outcompete groups consisting of the less altruistic persons.

For example, in some groups, the group operation is commutative, and this can be asserted with the introduction of an additional axiom, but without this axiom we can do quite well developing ( the more general ) group theory, and we can even take its negation as an axiom for the study of non-commutative groups.

He also provided an algebraic definition of fundamental groups of schemes and more generally the main structures of a categorical Galois theory.

Instead of focusing merely on the individual objects ( e. g., groups ) possessing a given structure, category theory emphasizes the morphisms – the structure-preserving mappings – between these objects ; by studying these morphisms, we are able to learn more about the structure of the objects.

Using the language of category theory, many areas of mathematical study can be cast into appropriate categories, such as the categories of all sets, groups, topologies, and so on.

The Langlands program is a far-reaching web of these ideas of ' unifying conjectures ' that link different subfields of mathematics, e. g. number theory and representation theory of Lie groups ; some of these conjectures have since been proved.

With the publication of Darwin's theory of evolution in 1859, the concept of a " natural system " of taxonomy gained a theoretical basis, and the idea was born that groups used in a system of classification should represent branches on the evolutionary tree of life.

C *- algebras are now an important tool in the theory of unitary representations of locally compact groups, and are also used in algebraic formulations of quantum mechanics.

The classification of groups of small 2-rank, especially ranks at most 2, makes heavy use of ordinary and modular character theory, which is almost never directly used elsewhere in the classification.

The PDFLP soon gained a reputation as the most intellectual of the Palestinian fedayeen groups, and drew heavily on Marxist-Leninist theory to explain the situation in the Middle East.

This occurs, e. g. in the character theory of finite groups.

This occurs, e. g. in the character theory of finite groups.

According to Husserl, this view of logic and mathematics accounted for the objectivity of a series of mathematical developments of his time, such as n-dimensional manifolds ( both Euclidean and non-Euclidean ), Hermann Grassmann's theory of extensions, William Rowan Hamilton's Hamiltonians, Sophus Lie's theory of transformation groups, and Cantor's set theory.

While many mathematicians before Galois gave consideration to what are now known as groups, it was Galois who was the first to use the word group ( in French groupe ) in a sense close to the technical sense that is understood today, making him among the founders of the branch of algebra known as group theory.

Fundamental groups can be studied using the theory of covering spaces, since a fundamental group coincides with the group of deck transformations of the associated universal covering space.

In category theory, quotient groups are examples of quotient objects, which are dual to subobjects.

Finite fields have applications in many areas of mathematics and computer science, including coding theory, LFSRs, modular representation theory, and the groups of Lie type.

What is the history of criticism but the history of men attempting to make sense of the manifold elements in art that will not allow themselves to be reduced to a single philosophy or a single aesthetic theory??

Accordingly, it is the aim of this essay to advance a new theory of imitation ( which I shall call mimesis in order to distinguish it from earlier theories of imitation ) and a new theory of invention ( which I shall call symbol for reasons to be stated hereafter ).

The central concern of Erich Auerbach's impressive volume called Mimesis is to describe the shift from a classic theory of imitation ( based upon a recognition of levels of truth ) to a Christian theory of imitation in which the levels are dissolved.

And it is clearly argued by Lord Percy of Newcastle, in his remarkable long essay, The Heresy Of Democracy, and in a more general way by Voegelin, in his New Science Of Politics, that this same Rousseauan idea, descending through European democracy, is the source of Marx's theory of the dictatorship of the proletariat.

Thus, in no ordinary sense of ' simplicity ' is the Ptolemaic theory simpler than the Copernican.

Analogously, anyone who argues that Einstein's theory of gravitation is simpler than Newton's, must say rather more to explain how it is that the latter is mastered by student-physicists, while the former can be managed ( with difficulty ) only by accomplished experts.

In a sense, Einstein's theory is simpler than Newton's, and there is a corresponding sense in which Copernicus' theory is simpler than Ptolemy's.

The strongest appeal of the Copernican formulation consisted in just this: ideally, the justification for dealing with special problems in particular ways is completely set out in the basic ' rules ' of the theory.

My reply is that I associate myself with all those who affirm that Gentile-Jewish relations should contribute to the theory and practice of human dignity.

Finally we may note that the idea appears in educational theory where its influence is at present widespread.

The point is that an ethical critic, with an assist from Freud, can seize on this theory to argue that tragedy provides us with a harmless outlet for our hostile urges.

In his study Samuel Johnson, Joseph Wood Krutch takes this line when he says that what Aristotle really means by his theory of catharsis is that our evil passions may be so purged by the dramatic ritual that it is `` less likely that we shall indulge them through our own acts ''.

But beginning, for all practical purposes, with Frederick Seebohm's English Village Community scholars have had to reckon with a theory involving institutional and agrarian continuity between Roman and Anglo-Saxon times which is completely at odds with the reigning concept of the Anglo-Saxon invasions.

that theory was and is sound.