In the case of the complex numbers, C, there is a unique nontrivial automorphism that sends R into R: complex conjugation, but there are infinitely ( uncountably ) many " wild " automorphisms ( assuming the axiom of choice ).

Since complex scalar fields admit two different kinds of annihilation operators, which are related by conjugation, such fields describe charged bosons.

then performs a complex conjugation, and then a matrix transpose, one ends up with the ket

* Complex conjugation, the involution multiplying the imaginary part of a complex number by − 1

* Conformal conjugation in complex dynamics

* Conjugate element ( field theory ), a generalization of complex conjugation

The set of constructible numbers can be completely characterized in the language of field theory: the constructible numbers form the quadratic closure of the rational numbers: the smallest field extension of which is closed under square root and complex conjugation.

where the star denotes complex conjugation.

The results on the existence of a solution to the conjugation equation depend on the eigenvalues of J and the degree of smoothness required from h. As J does not need to have any special symmetries, its eigenvalues will typically be complex numbers.

* Gal ( C / R ) has two elements, the identity automorphism and the complex conjugation automorphism.

Indo-European languages usually inflect verbs for several grammatical categories in complex paradigms, although some, like English, have simplified verb conjugation to a large extent.

A Klein surface is, as for Riemann surfaces, a surface with an atlas allowing the transition functions to be composed using complex conjugation.

Conjugative plasmids contain tra genes, which perform the complex process of conjugation, the transfer of plasmids to another bacterium ( Fig.

* If the base field is C, then for all complex numbers λ, where denotes the complex conjugation of λ.

As a real algebra, Cℓ < sup > 0 </ sup >< sub > 2, 0 </ sub >( R ) is isomorphic to field of complex numbers C. As a result, it admits a conjugation operation ( analogous to complex conjugation ), sometimes called the reverse of a Clifford element, defined by

If and are any complex numbers, the inner product is the standard inner product and the bar notation is used for complex conjugation then the inequality may be restated in more explicitly as

For quantum systems which are invariant under time reversal the Hamiltonian can be made real and symmetric, so that the action of time-reversal on the wave-function is just complex conjugation.

The complex field has only two isomorphisms that leave the real line fixed, the identity and conjugation.

ANOVA is difficult to teach, particularly for complex experiments, with split-plot designs being notorious.

More complex assistive technology devices have been developed over time, and as a result, sports for people with disabilities " have changed from being a clinical therapeutic tool to an increasingly competition-oriented activity ".

Nearest and next-nearest neighbors form a distorted octahedral complex, with the three atoms in the same double-layer being slightly closer than the three atoms in the next.

The nearest and next-nearest neighbors form a distorted octahedral complex, with the three atoms in the same double-layer being slightly closer than the three atoms in the next.

Basic theories, such as arithmetic, real analysis and complex analysis are often introduced non-axiomatically, but implicitly or explicitly there is generally an assumption that the axioms being used are the axioms of Zermelo – Fraenkel set theory with choice, abbreviated ZFC, or some very similar system of axiomatic set theory like Von Neumann – Bernays – Gödel set theory, a conservative extension of ZFC.

Egyptians also believed that being mummified and put in a sarcophagus ( an ancient Egyptian " coffin " carved with complex symbols and designs, as well as pictures and hieroglyphs ) was the only way to have an afterlife.

A further group of spherical organelles are distributed throughout the cell rather than being localized at the apical complex and are known as the dense granules.

The 13th century Iranian writer al-Kashani describes a more complex process whereby tutiya was mixed with raisins and gently roasted before being added to the surface of the molten metal.

3D beading is also associated with the stigma of being " too complex " for most beaders to manage, although this sentiment is largely due to the apparent complexity of many oriental beading diagrams.

As of 2011, the BIOS is being replaced by the more complex Extensible Firmware Interface ( EFI ) in many new machines, but BIOS remains in widespread use.

Its purpose is to unburden the application software programmer from the often complex details of the particular computer being used, including such accessories as communications devices, printers, device readers, displays and keyboards, and also to partition the computer's resources such as memory and processor time in a safe and stable manner.

They offer the advantage of re-engaging without additional user input, but being spring-loaded are prone to both spring fatigue and their more complex mechanisms becoming balky from dirt, ice, or other contamination.

Its origins are complex and may date to the Crusades, later being influenced by French gamblers.

The puzzling phenomenon of two individuals being exposed to the same evidence and being able to reach different conclusions, has been frequently explained ( particularly by Daniel Kahneman ) by reference to a ' bounded rationality ' - that is most judgments are made by fast acting heuristics ( system 1 ) that work well in every day situations, but are not amenable to decision making about complex subjects such as climate change.

By the 1990s, many of these bands and artists had disbanded, were no longer performing, or were being carried by independent labels because their music tended to be more lyrically complex ( and often more controversial ) than mainstream Christian pop.

Holomorphy is the property of a complex function of being differentiable at every point of an open and connected subset of ( this is called a domain in ).

The three Greek Orthodox chapels of St. James the Just, St. John the Baptist and of the Forty Martyrs of Sebaste, south of the rotunda and on the west side of the front courtyard originally formed the baptistery complex of the Constantinean church, the southern most chapel being the vestibule, the middle chapel being the actual baptistery and the north chapel being the chamber in which the patriarch chrismated the newly baptized before leading them into the rotunda north of this complex.

Some consider it an advantage of the Kimball method, that the data warehouse ends up being " segmented " into a number of logically self-contained ( up to and including The Bus ) and consistent data marts, rather than a big and often complex centralized model.

Then in 2 we show that any line involution with the properties that ( A ) It has no complex of invariant lines, and ( B ) Its singular lines form a complex consisting exclusively of the lines which meet a twisted curve, is necessarily of the type discussed in 1.

Since the complex of singular lines is of order K and since there is no complex of invariant lines, it follows from the formula Af that the order of the involution is Af.

A Banach *- algebra A is a Banach algebra over the field of complex numbers, together with a map *: A → A called involution which has the following properties:

:, involution ( i. e., the conjugate of the conjugate of a complex number z is again that number )

* More generally, the conjugation involution in any Cayley – Dickson algebra such as the complex numbers, quaternions and octonions, if a blind eye is turned on the nonassociativity of the latter.

* The involution is pointwise complex conjugation.

The involution also acts on the complex structure ( 1, 1 )- form J

* Let be the complexification of a real semisimple Lie algebra, then complex conjugation on is an involution on.

In Theosophy, Anthroposophy and Rosicrucianism, involution and evolution are part of a complex sequence of cosmic cycles, called Round.