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.

* Cartesian morphism, formalisation of pull-back operation in category theory

Because the group operation associates, parentheses have only one necessary use in group theory: to set the scope of the inverse operation.

This set together with the group operation of the elliptic group theory form an Abelian group, with the point at infinity as identity element.

A category of this sort can be viewed as augmented with a unary operation, called inverse by analogy with group theory.

In the mathematical field of representation theory, group representations describe abstract groups in terms of linear transformations of vector spaces ; in particular, they can be used to represent group elements as matrices so that the group operation can be represented by matrix multiplication.

Even though the set may be the same, the same function might be a homomorphism, say, in group theory ( sets with a single operation ) but not in ring theory ( sets with two related operations ), because it fails to preserve the additional operation that ring theory considers.

* induced representation, in representation theory: an operation for obtaining a representation of an object from one of its subobjects.

* The group of smooth maps from a manifold to a finite dimensional Lie group is an example of a gauge group ( with operation of pointwise multiplication ), and is used in quantum field theory and Donaldson theory.

A monoid for which the operation is commutative for some, but not all elements is a trace monoid ; trace monoids commonly occur in the theory of concurrent computation.

In set theory also we have the negation operation of the complement which obeys this property: a set A and a set ( A < sup > C </ sup >)< sup > C </ sup > ( where A < sup > C </ sup > represents the complement of A ) are the same.

The simplest theory of operation for a pulley system assumes that the pulleys and lines are weightless, and that there is no energy loss due to friction.

In group theory, given a group G under a binary operation *, a subset H of G is called a subgroup of G if H also forms a group under the operation *.

At the close of the 19th century, the reductionism of atomic theory began to advance into the atom itself ; determining, through physics, the nature of the atom and the operation of chemical reactions.

The theory of operation is that the blood on the buried bean will draw away the wart.

But in Boole's original system, + was a partial operation: in the language of set theory it would correspond only to disjoint union of subsets.

The JFET was predicted by Julius Lilienfeld in 1925 and by the mid-1930s its theory of operation was sufficiently well known to justify a patent.

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.

Thales searched for a simple material-form directly perceptible by the senses, behind the appearances of things, and his theory is also related to the older animism.

Aberrations occur because the simple paraxial theory is not a completely accurate model of the effect of an optical system on light, rather than due to flaws in the optical elements.

The physical model behind cosmic inflation is extremely simple, however it has not yet been confirmed by particle physics, and there are difficult problems reconciling inflation and quantum field theory.

simple: Category theory

In contrast simple molecular orbital theory predicts that the hydrogen molecule dissociates into a linear superposition of hydrogen atoms and positive and negative hydrogen ions, a completely unphysical result.

However, this political restriction is less confining than it may first appear in that the Marxist historical framework is surprisingly flexible, and a rather simple matter to modify an alternative historical theory to use language that at least does not challenge the Marxist interpretation of history.

By the late nineteenth century, however, a handful of experimental anomalies remained that could not be explained by the simple wave theory.

To illustrate this idea, we can formulate a very simple hidden variable theory for the above thought experiment.

Another recent theory ( see EPR paradox above ) resulting from the analysis of an EPR communication set up, has the simple device based on removing the effective retarded time terms in the Lorentz transform to yield a preferred absolute reference frame.

There are many distinct FFT algorithms involving a wide range of mathematics, from simple complex-number arithmetic to group theory and number theory ; this article gives an overview of the available techniques and some of their general properties, while the specific algorithms are described in subsidiary articles linked below.

simple: Game theory

Known as a top-down formation scenario, this theory is quite simple yet no longer widely accepted because observations of the early universe strongly suggest that objects grow from bottom-up ( i. e. smaller objects merging to form larger ones ).

In 1907, beginning with a simple thought experiment involving an observer in free fall, he embarked on what would be an eight-year search for a relativistic theory of gravity.

simple: Graph theory

simple: Grand unification theory

In simple chemistry, as per valence bond theory, the carbon atom must follow the " 4-hydrogen rule ", which states that the maximum number of atoms available to bond with carbon is equal to the number of electrons that are attracted into the outer shell of carbon. In terms of shells, carbon consists of an incomplete outer shell, which comprises 4 electrons, and thus has 4 electrons available for covalent or dative bonding.

The theory explains how simple chunks of information form the building blocks of schemata, which are more complex structures.

The hydrogen atom has special significance in quantum mechanics and quantum field theory as a simple two-body problem physical system which has yielded many simple analytical solutions in closed-form.

simple: Information theory

His theory is that Muslims have an aggressive personality with relatively closed, simple minds, and were less amenable to reason.

The theory was simple and had been first proposed by Frisius.

Bradley explained this effect in the context of Newton's corpuscular theory of light, by showing that the aberration angle was given by simple vector addition of the Earth's orbital velocity and the velocity of the corpuscles of light, just as vertically falling raindrops strike a moving object at an angle.

A particularly rich theory has been developed for finite groups, which culminated with the monumental classification of finite simple groups announced in 1983.