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Formal and ring
* Formal derivative, an operation on elements of a polynomial ring which mimics the form of the derivative from calculus
Formal group laws over a ring R are often constructed by writing down their logarithm as a power series with coefficients in R ⊗ Q, and then proving that the coefficients of the corresponding formal group over R ⊗ Q actually lie in R. When working in positive characteristic, one typically replaces R with a mixed characteristic ring that has a surjection to R, such as the ring W ( R ) of Witt vectors, and reduces to R at the end.
* Formal derivative, in mathematics, an operation on elements of a polynomial ring which mimics the form of the derivative from calculus

Formal and construction
Formal agreement was reached on 16 January 1845 between the L & SWR, the GWR and the Southampton & Dorchester, agreeing exclusive areas of influence for future railway construction as between the parties.
Formal construction began in 1927 during the Gerardo Machado administration.

Formal and algebraic
Formal methods are best described as the application of a fairly broad variety of theoretical computer science fundamentals, in particular logic calculi, formal languages, automata theory, and program semantics, but also type systems and algebraic data types to problems in software and hardware specification and verification.
Formal groups are intermediate between Lie groups ( or algebraic groups ) and Lie algebras.

spectrum and ring
* The spectrum of any commutative ring with the Zariski topology ( that is, the set of all prime ideals ) is compact, but never Hausdorff ( except in trivial cases ).
* the Zariski topology on an algebraic variety or on the spectrum of a ring, used in algebraic geometry,
Prime ideals are the points of algebro-geometric objects, via the notion of the spectrum of a ring.
In abstract algebra and algebraic geometry, the spectrum of a commutative ring R, denoted by Spec ( R ), is the set of all proper prime ideals of R. It is commonly augmented with the Zariski topology and with a structure sheaf, turning it into a locally ringed space.
It follows readily from the definition of the spectrum of a ring, the space of prime ideals of equipped with the Zariski topology, that the Krull dimension of is precisely equal to the irreducible dimension of its spectrum.
* In the category of schemes, Spec ( Z ) the prime spectrum of the ring of integers is a terminal object.
The empty scheme ( equal to the prime spectrum of the trivial ring ) is an initial object.
As a rule of thumb, any sort of construction that takes as input a fairly general object ( often of an algebraic, or topological-algebraic nature ) and outputs a compact space is likely to use Tychonoff: e. g., the Gelfand space of maximal ideals of a commutative C * algebra, the Stone space of maximal ideals of a Boolean algebra, and the Berkovich spectrum of a commutative Banach ring.
Modern algebraic geometry takes the spectrum of a ring ( the set of proper prime ideals ) as its starting point.
Thus, V ( S ) is " the same as " the maximal ideals containing S. Grothendieck's innovation in defining Spec was to replace maximal ideals with all prime ideals ; in this formulation it is natural to simply generalize this observation to the definition of a closed set in the spectrum of a ring.
* Spec k, the spectrum of the polynomial ring over a field k, which is also denoted, the affine line: the polynomial ring is known to be a principal ideal domain and the irreducible polynomials are the prime elements of k. If k is algebraically closed, for example the field of complex numbers, a non-constant polynomial is irreducible if and only if it is linear, of the form t − a, for some element a of k. So, the spectrum consists of one closed point for every element a of k and a generic point, corresponding to the zero ideal.
Just as in classical algebraic geometry, any spectrum or projective spectrum is compact, and if the ring in question is Noetherian then the space is a Noetherian space.
In the NMR spectrum of a dimethyl derivative, two nonequivalent signals are found for the two methyl groups indicating that the molecular conformation of this cation not perpendicular ( as in A ) but is bisected ( as in B ) with the empty p-orbital and the cyclopropyl ring system in the same plane:
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 scheme is a locally ringed space such that every point has a neighbourhood, which, as a locally ringed space, is isomorphic to a spectrum of a ring.
If the same signal is sent to both inputs of a ring modulator, the resultant harmonic spectrum is the original frequency domain doubled ( if f < sub > 1 </ sub >
An equivalent but streamlined construction is given by the Proj construction, which is an analog of the spectrum of a ring, denoted " Spec ", which defines an affine scheme.
Let P be a finitely generated projective module over a commutative ring R and X be the spectrum of R. The rank of P at a prime ideal in X is the rank of the free-module.

spectrum and construction
Depending on the client's needs and the jurisdiction's requirements, the spectrum of the architect's services may be extensive ( detailed document preparation and construction review ) or less inclusive ( such as allowing a contractor to exercise considerable design-build functions ).
Boroughs have the power to advise the mayor with nonbinding opinions on a large spectrum of topics ( environment, construction, public health, local markets ); in addition they are supplied with an autonomous founding in order to finance local activities.
It conducts a broad spectrum of inter-disciplinary scientific research in three main areas: particle and high energy physics ; photon science ; and the development, construction and operation of particle accelerators.
Color cameras require a more complex construction to differentiate wavelength and color has less meaning outside of the normal visible spectrum because the differing wavelengths do not map uniformly into the system of color vision used by humans.
This set can be identified with the spectrum of the C *- algebra associated to G by the group C *- algebra construction.
The work of technologists is usually focused on the portion of the technological spectrum closest to product improvement, manufacturing, construction, and engineering operational functions.
Kirwan stated in an interview that the band was " formed to be political ", with the socialist lyrics attracting one half of the political spectrum, and the songs of the day-to-day life in America attracting traditionally right-leaning " cops, firemen and construction workers.
The arrangement allowed Plough to have an inexpensive FM presence in the years before FM came to dominate the radio spectrum, and created enough interest amongst the student body that the Georgia Board of Regents obtained a license and construction permit for its own station at the university, FM 88. 5 WRAS.
There is a corresponding construction using real vector bundles instead of complex vector bundles, which gives an 8-periodic spectrum.
This will contribute to the construction of an experimental fast spectrum installation ( MYRRHA ), allowing a. o.
Method construction approaches can be organized in a flexibility spectrum ranging from ' low ' to ' high '.
Lying at the ' low ' end of this spectrum are rigid methods, whereas at the ' high ' end there are modular method construction.
A series of interviews conducted in the late 1970s revealed a wide spectrum of users, including construction workers, a " trendy East Side NYC couple " at a " chic NYC nightclub ", a Los Angeles businesswoman " in the middle of a particularly hectic public-relations job " ( who confided to the reporter that " I could really use a popper now "), and frenetic disco dancers amid " flashing strobe lights and the pulsating beat of music in discos across the country.
The subsidiary has a spectrum of financial products and services for corporate, construction equipments etc.

spectrum and algebraic
In other words, there exists a faithfully flat map X → S such that any point in X has a quasi-compact open neighborhood U whose image is an open affine subscheme of S, such that base change to U yields a finite product of copies of GL < sub > 1, U </ sub > = G < sub > m </ sub >/ U. One particularly important case is when S is the spectrum of a field K, making a torus over S an algebraic group whose extension to some finite separable extension L is a finite product of copies of G < sub > m </ sub >/ L.
In algebraic topology, a branch of mathematics, a spectrum is an object representing a generalized cohomology theory.
An example of such a sheaf is that associated in algebraic geometry to an R-module M, R being any commutative ring, on the spectrum of a ring Spec ( R ).
* Topological modular forms, an E-infinity ring spectrum used in algebraic topology
Especially, it studies the spectrum of the adjacency matrix, or the Laplacian matrix of a graph ( this part of algebraic graph theory is also called spectral graph theory ).

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