Duality in the projective plane is a special case of duality for projective spaces, transformations of PG ( n, K ) ( also denoted by KP < sup > n </ sup >) with K a field, that interchange objects of dimension r with objects of dimension n-1-r ( = codimension r + 1 ).

: From the point of view of the definition, this is an expression of the Poincaré Duality isomorphism where is the quotient of the field of fractions of by, considered as a-module, and where is the conjugate-module to ie: as an abelian group it is identical to but the covering transformation acts by.

* Duality: Every statement, theorem, or definition in category theory has a dual which is essentially obtained by " reversing all the arrows ".

** Duality ( order theory ), a concept regarding binary relations

This formulation of Poincaré Duality has become quite popular as it provides a means to define Poincaré Duality for any generalized homology theories provided one has a Thom Isomorphism for that homology theory.

* Duality ( order theory )

* Duality principle ( optimization theory )

* Duality principle in functional analysis, used in the large sieve method of analytic number theory

A treatment of this theory, Residues and Duality ( 1966 ) by Robin Hartshorne, became an accessible reference.

In Chinese creation myth | Daoist creation myth, " The Way gave birth to unity, Unity gave birth to duality, Duality gave birth to trinity, Trinity gave birth to the myriad creatures.

Duality with pluralism is considered a logical fallacy.

Duality is also sometimes called reciprocity or polarity.

Duality is most commonly defined in terms of polar reciprocation about a concentric sphere.

Duality ( opposition ) that creates the energy ( or libido ) of the psyche, which Jung asserts expresses itself only through symbols: " It is the energy that manifests itself in the life process and is perceived subjectively as striving and desire.

Originally shamanistic practices that have over time may once have been monotheistic later once since deification of an Earth Mother to the Sky father as a Duality, also including ancestor worship and an addition for a " sun deity ", polytheizing the religion and more deities were incorporated after-wards.

* E. H. Carlson, Wave – Particle Duality: Light on Project PHYSNET

Third-Wave Feminism and the Need to Reweave the Nature / Culture Duality, NWSA Journal 16 ( 3 ): 154-179.

Xiuhtecuhtli is a manifestation of Ometecuhtli, the Lord of Duality, and according to the Florentine Codex Xiuhtecuhtli was considered to be mother and father of the Gods, who dwelled in the turquoise enclosure in the center of earth.

Miguel Leon-Portilla interprets the name " Ometeotl " as " Lord of the Duality " and argues that Ometeotl was the supreme creator deity of the Aztecs, and that the Aztecs envisioned this deity as a mystical entity with a dual nature akin to the European concept of the trinity.

Duality and unity are both significant.

Duality implies distinctive cultural development while unity implies common purpose and community.

* Berthold-Georg Englert, Marlan O. Scully & Herbert Walther, Quantum Optical Tests of Complementarity, Nature, Vol 351, pp 111 – 116 ( 9 May 1991 ) and ( same authors ) The Duality in Matter and Light Scientific American, pg 56 – 61, ( December 1994 ).

Gerrard teamed with Pieter Bourke ( Snog, Soma ) to issue Duality in April 1998.

This paradox is known as the Wave – particle Duality Paradox.

Duality in the Fano plane: Each point corresponds to a line and vice versa.

** The interview was also sampled in Konkhra's track " Religion is a Whore " and The Faceless's " Planetary Duality.

Duality, as such, is the assertion that truth is invariant under this operation on statements.

According to Giddens there is a " Duality of structure " by which social practice, which is the principal unit of investigation, has both a structural and an agency-component.

* B. R. Greene and M. R. Plesser, " Duality in Calabi-Yau Moduli Space ".

Field automorphisms are important to the theory of field extensions, in particular Galois extensions.

Acoustic theory is the field relating to mathematical description of sound waves.

A unified interpretation of antiparticles is now available in quantum field theory, which solves both these problems.

This is an example of renormalization in quantum field theory — the field theory being necessary because the number of particles changes from one to two and back again.

In quantum field theory, this process is allowed only as an intermediate quantum state for times short enough that the violation of energy conservation can be accommodated by the uncertainty principle.

These processes are important in the vacuum state and renormalization of a quantum field theory.

This section draws upon the ideas, language and notation of canonical quantization of a quantum field theory.

This technique is the most widespread method of computing amplitudes in quantum field theory today.

* Quantum field theory and the list of particles

His research program vastly extended the scope of the field, incorporating major elements of commutative algebra, homological algebra, sheaf theory, and category theory into its foundations.

Then, following the programme he outlined in his talk at the 1958 International Congress of Mathematicians, he introduced the theory of schemes, developing it in detail in his Éléments de géométrie algébrique ( EGA ) and providing the new more flexible and general foundations for algebraic geometry that has been adopted in the field since that time.

His theory of schemes has become established as the best universal foundation for this major field, because of its great expressive power as well as technical depth.

The Gaussian theory, however, is only true so long as the angles made by all rays with the optical axis ( the symmetrical axis of the system ) are infinitely small, i. e. with infinitesimal objects, images and lenses ; in practice these conditions are not realized, and the images projected by uncorrected systems are, in general, ill defined and often completely blurred, if the aperture or field of view exceeds certain limits.

Until his return from Mexico in 1930 Whorf had been entirely an autodidact in linguistic theory and field methodology, yet he had already made a name for himself in Middle American linguistics.

Since a leg theory attack means the batsman is more likely to hit the ball on the leg side, additional fielders on that side of the field can be effective in preventing runs and taking catches.

* BCS theory correctly predicts the Meissner effect, i. e. the expulsion of a magnetic field from the superconductor and the variation of the penetration depth ( the extent of the screening currents flowing below the metal's surface ) with temperature.

BCS theory relates the value of the critical field at zero temperature to the value of the transition temperature and the density of states at the Fermi energy.

The analogy was completed when Hawking, in 1974, showed that quantum field theory predicts that black holes should radiate like a black body with a temperature proportional to the surface gravity of the black hole.

Unital Banach algebras over the complex field provide a general setting to study spectral theory.

Today, when mathematicians talk about " set theory " as a field, they usually mean axiomatic set theory.