where is the Gelfand representation of x defined as follows: is the continuous function from Δ ( A ) to C given by The spectrum of in the formula above, is the spectrum as element of the algebra C ( Δ ( A )) of complex continuous functions on the compact space Δ ( A ).

Further studies on animals that were traditionally assumed to be cold-blooded have shown that most creatures incorporate different variations of the three terms defined above, along with their counterparts ( ectothermy, poikilothermy and bradymetabolism ), thus creating a broad spectrum of body temperature types.

It is impossible to represent spectrum indigo exactly on a computer screen, because true spectrum indigo is outside the color triangle or gamut of the RGB color space defined by the monitor primaries.

However, the International Prototype Metre remained the standard until 1960, when the eleventh CGPM defined the metre in the new International System of Units ( SI ) as equal to 1, 650, 763. 73 wavelengths of the orange-red emission line in the electromagnetic spectrum of the krypton-86 atom in a vacuum.

Max Eastman defined the spectrum of satire in terms of " degrees of biting ," as ranging from satire proper at the hot-end, and " kidding " at the violet-end ; Eastman adopted the term kidding to denote what is just satirical in form, but is not really firing at the target.

Many species can see light with frequencies outside the " visible spectrum ," which is defined in terms of human vision.

In contrast to the TCSEC's precisely defined hierarchy of six evaluation classes, the more recently introduced Common Criteria ( CC )— which derive from a blend of more or less technically mature standards from various NATO countries — provide a more tenuous spectrum of seven " evaluation classes " that intermix features and assurances in an arguably non-hierarchical manner and lack the philosophic precision and mathematical stricture of the TCSEC.

Tourette's is defined as part of a spectrum of tic disorders, which includes transient and chronic tics.

The term ethnic cleansing has been defined as a spectrum, or continuum by some historians.

The pulsating stars swell and shrink regularly by stellar radius, magnitude and spectrum, most often with a defined period, sometimes semiregularly with an average period and amplitude, or a pseudoperiod.

The microwave spectrum is usually defined as electromagnetic energy ranging from approximately 1 GHz to 100 GHz in frequency, but older usage includes lower frequencies.

Its official declared policy is now defined as ' full spectrum dominance '.

* Some forms of spread spectrum transmission ( and most forms of ultra-wideband ) are mathematically defined as being devoid of carrier waves.

Note that historical opinions fall on a spectrum, rather than into tightly defined camps.

of the Lyapunov spectrum it is possible to obtain the so-called Kaplan – Yorke dimension, that is defined as follows:

Since the first ionization energy of hydrogen and oxygen are both 14 eV, the spectrum of ionizing radiation is commonly defined to start at approximately 10 eV ( equivalent to a far ultraviolet wavelength of 124 nanometers ).

The power spectrum is defined as

The peripheral spectrum of an operator is defined as the set of points in its spectrum which have modulus equal to its spectral radius.

In other words, the spectrum of such an operator, which was defined as a generalization of the concept of eigenvalues, consists in this case only of the usual eigenvalues, and possibly 0.

The spectrum of lissencephaly is only now becoming more defined as neuroimaging and genetics has provided more insights into migration disorders.

If X is an algebraic variety carrying the Zariski topology, we can define a locally ringed space by taking O < sub > X </ sub >( U ) to be the ring of rational functions defined on the Zariski-open set U which do not blow up ( become infinite ) within U. The important generalization of this example is that of the spectrum of any commutative ring ; these spectra are also locally ringed spaces.

A collection of new technologies are taking advantage of unlicensed spectrum including Wi-Fi, Ultra Wideband, spread spectrum, software defined radio, cognitive radio, and mesh networks.

If A is commutative then the center of A is equal to A, so that a commutative R-algebra can be defined simply as a homomorphism of commutative rings.

Although most often used for matrices whose entries are real or complex numbers, the definition of the determinant only involves addition, subtraction and multiplication, and so it can be defined for square matrices with entries taken from any commutative ring.

The notion of greatest common divisor can more generally be defined for elements of an arbitrary commutative ring, although in general there need not exist one for every pair of elements.

Any commutative monoid is endowed with its algebraic preordering ≤, defined by x ≤ y if and only if there exists z such that x + z = y.

Prime numbers give rise to two more general concepts that apply to elements of any commutative ring R, an algebraic structure where addition, subtraction and multiplication are defined: prime elements and irreducible elements.

* In commutative algebra, a commutative ring can be completed at an ideal ( in the topology defined by the powers of the ideal ).

Over a commutative ring R, one must be slightly more careful: a matrix over R is invertible if and only if its determinant is a unit in R, that is, if its determinant is invertible in R. Therefore GL ( n, R ) may be defined as the group of matrices whose determinants are units.

The determinant of a product AB is the product of the determinants of matrices A and B ( not defined when the underlying ring is not commutative ):

Being a consequence of just algebraic expression manipulation, these relations are valid for matrices with entries in any commutative ring ( commutativity must be assumed for determinants to be defined in the first place ).

The set of arithmetic functions forms a commutative ring, the, under pointwise addition ( i. e. f + g is defined by ( f + g )( n )= f ( n ) + g ( n )) and Dirichlet convolution.

In full generality, the exterior algebra can be defined for modules over a commutative ring, and for other structures of interest in abstract algebra.

The radical of an ideal I in a commutative ring R, denoted by Rad ( I ) or, is defined as

The ordered exponential ( also called the path-ordered exponential ) is a mathematical object, defined in non-commutative algebras, which is equivalent to the exponential function of the integral in the commutative algebras.

The determinant is over the ring of n × n matrices whose entries are polynomials in t with coefficients in the commutative algebra of even complex differential forms on M. The curvature form of V is defined as

In a monoidal category, analogs of usual monoids from abstract algebra can be defined using the same commutative diagrams.

Let A be a commutative Banach algebra, defined over the field C of complex numbers.

A ring of symmetric polynomials can be defined over any commutative ring R, and will be denoted Λ < sub > R </ sub >; the basic case is for R = Z.

The commutator of two elements a and b of a ring or an associative algebra is defined by

The anticommutator of two elements a and b of a ring or an associative algebra is defined by

The Dirichlet convolution can be defined for general arithmetic functions, and yields a ring structure, the Dirichlet ring.

A polynomial f in one variable X over a ring R is defined as a formal expression of the form

* The relation defined by iff there exists some injection from x to y. Injection may be replaced by surjection, or any type of structure-preserving function, such as ring homomorphism, or permutation.

The gravity of shepherd moons serves to maintain a sharply defined edge to the ring ; material that drifts closer to the shepherd moon's orbit is either deflected back into the body of the ring, ejected from the system, or accreted onto the moon itself.

An ideal can be used to construct a quotient ring similarly to the way that modular arithmetic can be defined from integer arithmetic, and also similarly to the way that, in group theory, a normal subgroup can be used to construct a quotient group.

Many scientific concepts are of necessity vague, for instance species in biology cannot be precisely defined, owing to unclear cases such as ring species.

The ring takes the place of the category C, and the category of modules over the ring is a category of functors defined on C.

It is also called 100VG-AnyLAN because it was defined to carry both Ethernet and token ring frame types.

* Periods ( ring ) — the ring of numbers which can be expressed as integrals of algebraic differential forms over algebraically defined domains

The switch is programmed to ring a specific extension ( the called phone ) when a defined extension ( the calling phone ) goes off-hook.

In noncommutative ring theory, a maximal right ideal is defined analogously as being a maximal element in the poset of proper right ideals, and similarly, a maximal left ideal is defined to be a maximal element of the poset of proper left ideals.

* An important ideal of the ring called the Jacobson radical can be defined using maximal right ( or maximal left ) ideals.

Also, it is believed that a ring, defined as something other than simply the area given to the wrestlers by spectators, came into being in the 16th century as a result of a tournament organized by the then principal warlord in Japan, Oda Nobunaga.

A formal Laurent series over a ring R is defined in a similar way to a formal power series, except that we also allow finitely many terms of negative degree ( this is different from the classical Laurent series ), that is series of the form