To understand the concept, it is useful to study several examples first, of which there are many: all free objects, direct product and direct sum, free group, free lattice, Grothendieck group, product topology, Stone – Čech compactification, tensor product, inverse limit and direct limit, kernel and cokernel, pullback, pushout and equalizer.

The concept of excitons was first proposed by Yakov Frenkel in 1931, when he described the excitation of atoms in a lattice of insulators.

This concept is arguably the " most complete " notion of a meet-semilattice that is not yet a lattice ( in fact, only the top element may be missing ).

The concept describes the lack of an electron at a position where one could exist in an atom or atomic lattice.

The design concept for utilizing quartz crystals as a filtering component was first established by Walter Cady in 1922, but it was largely Walter Mason's work in the late 1920s and early 1930s that devised methods for incorporating crystals into LC lattice filter networks which set the groundwork for much of the progress in telephone communications.

This type of definition traces back, in one line, to the lattice concept of the 1930s.

The family of these concepts obeys the mathematical axioms defining a lattice, and is called more formally a concept lattice.

In his article Restructuring Lattice Theory ( 1982 ) initiating formal concept analysis as a mathematical discipline, Rudolf Wille starts from a discontent with the current lattice theory and pure mathematics in general: The production of theoretical results-often achieved by " elaborate mental gymnastics "-were impressive, but the connections between neighbouring domains, even parts of a theory were getting weaker.

A related concept is that of the irreducible Brillouin zone, which is the first Brillouin zone reduced by all of the symmetries in the point group of the lattice.

Although the concept of a two-dimensional lattice is quite simple, there is a considerable amount of specialized notation and language concerning the lattice that occurs in mathematical literature.

During his work on the propagation of electron waves in a crystal lattice, he introduced the concept of Brillouin zones in 1930.

Cast iron railing along Sadovaya Street constitutes one of the essential components in the overall concept, perhaps it even served a prototype for the Summer Garden lattice in the late 1770s.

The types of structural unit that exist can be related to the concept of the coincidence site lattice, in which repeated units are formed from points where the two misoriented lattices happen to coincide.

Nevertheless, it may be helpful to cite one example -- that of employment -- for, as will be shown below, it cuts across both facets of the new concept.

National responsibility for individual welfare is a concept not limited to the United States or even to the Western nations.

This and other fears of the solar system have disappeared gradually, first, with the Ptolemaic system and its built-in concept of periodicity and then, more firmly, with the Newtonian innovation of an universal force that could account quantitatively for both terrestial and celestial motions.

In seeking for such meaning and purpose, Albert Schweitzer seized upon the concept of the `` sacredness of life ''.

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.

A realistic balancing of the need for new forms of international organization on the one hand, and our capacity to achieve them on the other, must be approached through the concept of `` community ''.

It is ironic that Washington is having to struggle so for a concept that for six years it bypassed as unreasonable.

There was only one way to accomplish this: by design, by drawing diagrams and sketches in which he probed the remotest corner of his mind for creative ideas to carry his concept.

That we had the wit and wisdom to adopt Mr. Lowell's concept and make it the base for our processes of selection is one reason why our selections have been, it may be said truly, pretty uniformly good.

It is this sort of experience that makes the concept of high fidelity of real musical significance for the home music listener.

This is interesting for it combines both the thermodynamic concept of a minimum Gibbs function for equilibrium and minimum mechanical potential energy for equilibrium.

Necessity for an over-all concept

This is stated to emphasize the necessity for an over-all concept of submarine defense, one which would provide positions of relative importance to ASW elements based on projected potentialities.

Therefore, a broad concept of over-all submarine defense is needed for co-ordination of the Navy's efforts, for a logical presentation to the public, for industry's guidance, and as a basis for a program to the Congress.

That which follows will be a discussion of principles and possible content for an over-all concept of antisubmarine warfare.

In an over-all ASW concept, dependence on and effort expended for such systems should be limited to those with proven capabilities.

In discussing the ways this crisis is germane to consderations for the prevention of illegitimacy, we shall again superimpose Erikson's concept on our data.

The tremendous emphasis on the 5 in the Lo Shu square -- for purely mathematical reasons -- and the fact that this number so neatly symbolized the heart and center of the universe, could well explain why the Old Chinese seem to have so revered the number 5, and why they put so much stress on the concept of Centrality.

He was closer to understanding it in English now, although it could never have the inevitability of the Martian concept it stood for.

Abstraction in mathematics is the process of extracting the underlying essence of a mathematical concept, removing any dependence on real world objects with which it might originally have been connected, and generalizing it so that it has wider applications or matching among other abstract descriptions of equivalent phenomena.

The concept of an abelian group is one of the first concepts encountered in undergraduate abstract algebra, with many other basic objects, such as a module and a vector space, being its refinements.

Some philosophers deny that the concept of " being " has any meaning at all, since we only define an object's existence by its relation to other objects, and actions it undertakes.

However it is important to note that the objects of a category need not be sets nor the arrows functions ; any way of formalising a mathematical concept such that it meets the basic conditions on the behaviour of objects and arrows is a valid category, and all the results of category theory will apply to it.

These categories surely have some objects that are " special " in a certain way, such as the empty set or the product of two topologies, yet in the definition of a category, objects are considered to be atomic, i. e., we do not know whether an object A is a set, a topology, or any other abstract concept – hence, the challenge is to define special objects without referring to the internal structure of those objects.

An empirical or an a posteriori concept is a general representation ( Vorstellung ) or non-specific thought of that which is common to several specific perceived objects ( Logic, I, 1., § 1, Note 1 )

An enumerative definition of a concept or term is an extensional definition that gives an explicit and exhaustive listing of all the objects that fall under the concept or term in question.

The concept of dimension is not restricted to physical objects.

From Weierstrass he derives the idea that we generate the concept of number by counting a certain collection of objects.

These are called " universal names "; their meaning is a " concept " and refers to a series of objects ( the extension of the concept ).

Melanie Klein extended Freud's concept of fantasy to cover the developing child's relationship to a world of internal objects.

Physicists began to understand the concept of a black hole, and to identify quasars as one of these objects ' astrophysical manifestations.

When the mind makes a generalization, it extracts the essence of a concept based on its analysis of similarities from many discrete objects.

The concept of momentum plays a fundamental role in explaining the behavior of variable-mass objects such as a rocket ejecting fuel or a star accreting gas.

The authors call this concept a ‘ conduit metaphor .’ By this they meant that a speaker can put ideas or objects into words or containers, and then send them along a channel, or conduit, to a listener who takes that idea or object out of the container and makes meaning of it.

* For the general treatment of the concept of a product, see product ( category theory ), which describes how to combine two objects of some kind to create an object, possibly of a different kind.

One attempt to overcome these limitations is to replace ordinary quantum field theory, which is based on the classical concept of a point particle, with a quantum theory of one-dimensional extended objects: string theory.

The concept of surface finds application in physics, engineering, computer graphics, and many other disciplines, primarily in representing the surfaces of physical objects.

Alan Kay has commented that despite the attention given to objects, messaging is the most important concept in Smalltalk: " The big idea is ' messaging ' -- that is what the kernel of Smalltalk / Squeak is all about ( and it's something that was never quite completed in our Xerox PARC phase ).