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Page "Quotient algebra" ¶ 8
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Formally and let
Formally, let S and T be finite sets and let F =
If R is a ring, let R denote the ring of polynomials in the indeterminate X over R. Hilbert proved that if R is " not too large ", in the sense that if R is Noetherian, the same must be true for R. Formally,
Formally, let P be an expression in which the variable x is free.
Formally, let
Formally, let be a stochastic process and let represent the cumulative distribution function of the joint distribution of at times.
Formally, let f: < sup > n </ sup > → be the cost function which must be minimized.
Formally, let P and Q be abelian categories, and let
Formally, let A be a real matrix of which we want to compute the eigenvalues, and let A < sub > 0 </ sub >:= A.
Formally, let G be a Coxeter group with reduced root system R and k < sub > v </ sub > a multiplicity function on R ( so k < sub > u </ sub > = k < sub > v </ sub > whenever the reflections σ < sub > u </ sub > and σ < sub > v </ sub > corresponding to the roots u and v are conjugate in G ).
Formally, let p ( x, y ) be a complex polynomial in the complex variables x and y.
Formally, for received words, let denote the Hamming distance between and, that is, the number of positions in which and differ.
Formally, let p and q be two nonzero polynomials, respectively of degree m and n. Thus:
Formally, let P be a poset ( partially ordered set ), and let F be a filter on P ; that is, F is a subset of P such that:
Formally, let denote the pairwise score for against.
Formally, let X be any scheme and S be a sheaf of graded-algebras ( the definition of which is similar to the definition of-modules on a locally ringed space ): that is, a sheaf with a direct sum decomposition

Formally and be
However, shortly after this positive result, Kurt Gödel published On Formally Undecidable Propositions of Principia Mathematica and Related Systems ( 1931 ), showing that in any sufficiently strong axiomatic system there are true statements which cannot be proved in the system.
Formally the algorithm can be described as:
Formally, if M is a set, the identity function f on M is defined to be that function with domain and codomain M which satisfies
Formally, a frame is defined to be a lattice L in which finite meets distribute over arbitrary joins, i. e. every ( even infinite ) subset
Formally a random variable is considered to be a function on the possible outcomes.
Formally the " head of state " can also personally be the " head of government " ( ex officio or by ad hoc cumulation such as an absolute monarch nominating himself ) but otherwise has formal precedence over the Head of Government and other ministers, whether he is their actual political superior ( absolute monarch, executive president ) or rather theoretical or ceremonial in character.
Formally, in the finite-dimensional case, if the linear map is represented as a multiplication by a matrix A and the translation as the addition of a vector, an affine map acting on a vector can be represented as
Formally, the theorem can be stated as follows:
Formally, the convex hull may be defined as the intersection of all convex sets containing X or as the set of all convex combinations of points in X.
Formally, a decision problem is P-complete ( complete for the complexity class P ) if it is in P and that every problem in P can be reduced to it by using an appropriate reduction.
Formally, a detailed balanced process is the one on which the probability to be in the state x and transit to x ' must be the same as to be in the state x ' and transit to x.
Formally, a unique factorization domain is defined to be an integral domain R in which every non-zero and non-unit x of R can be written as a product ( including an empty product ) of irreducible elements p < sub > i </ sub > of R and a unit u:
Formally, powers with positive integer exponents may be defined by the initial condition
Formally, the problem of supervised pattern recognition can be stated as follows: Given an unknown function ( the ground truth ) that maps input instances to output labels, along with training data assumed to represent accurate examples of the mapping, produce a function that approximates as closely as possible the correct mapping.
Formally the self-inductance of a wire loop would be given by the above equation with i
Formally, a government would lay out its reasons for going to war, as well as its intentions in prosecuting it and the steps that might be taken to avert it.
Formally, if is any non-zero polynomial, it must be writable as.
Formally, a Menger sponge can be defined as follows:
Formally, the case where only a subset of parameters is defined is still a composite hypothesis ; nonetheless, the term point hypothesis is often applied in such cases, particularly where the hypothesis test can be structured in such a way that the distribution of the test statistic ( the distribution under the null hypothesis ) does not depend on the parameters whose values have not been specified under the point null hypothesis.

Formally and homomorphism
Formally, an antihomomorphism between X and Y is a homomorphism, where equals Y as a set, but has multiplication reversed: denoting the multiplication on Y as and the multiplication on as, we have.

Formally and .
Formally organized vocational programs supported by federal funds allow high school students to gain experience in a field of work which is likely to lead to a full-time job on graduation.
Formally, arbitrage transactions have negative skew – prices can get a small amount closer ( but often no closer than 0 ), while they can get very far apart.
Formally, their designation is the letter Ž and the number.
Formally, a topological space X is called compact if each of its open covers has a finite subcover.
Formally, they were part of the non-resident sections, but in fact constituted a separate division, largely autonomous in their activities.
Formally, the Congress serves two functions: to approve changes to the Party constitution regarding policy and to elect a Central Committee, about 300 strong.
More rigorously, the divergence of a vector field F at a point p is defined as the limit of the net flow of F across the smooth boundary of a three dimensional region V divided by the volume of V as V shrinks to p. Formally,
Formally, there is a clear distinction: " DFT " refers to a mathematical transformation or function, regardless of how it is computed, whereas " FFT " refers to a specific family of algorithms for computing DFTs.
Formally, one has an implication, not an equivalence, so the solution set may get larger.
Formally, oxidation state is the hypothetical charge that an atom would have if all bonds to atoms of different elements were 100 % ionic.
Formally, a bifunctor is a functor whose domain is a product category.
Formally inaugurated in March 2004, the Global Leadership Foundation works to " promote good governance – democratic institutions, open markets, human rights and the rule of law – and to contribute to the prevention and resolution of conflict through mediation.
Formally given to bishop Philip of Senj, the permission to use the Glagolitic liturgy ( the Roman Rite conducted in Slavic language instead of Latin, not the Byzantine rite ), actually extended to all Croatian lands, mostly along the Adriatic coast.
Formally, the system is said to have memory.
Formally the Prince-Electors elected a King of the Romans, who was elected in Germany but became Holy Roman Emperor only when crowned by the Pope.
Formally, an inner product space is a vector space V over the field together with an inner product, i. e., with a map
Formally neutral and reluctant to become involved with the great powers except as a last resort, Kuwait turned to the United States, the Soviet Union and Britain for naval protection of its tanker fleet after twenty-one ships were attacked in the gulf since late 1986.
* Formally, when working over the reals, as here, this is accomplished by considering the limit as ε → 0 ; but the " infinitesimal " language generalizes directly to Lie groups over general rings.
Massiah: Formally Charged + Deliberate Elicitation.
*, see On Formally Undecidable Propositions of Principia Mathematica and Related Systems for details on English translations.
Formally, these failed when they were rejected by the Church of England's General Synod in 1972 ; conversations and co-operation continued, however, leading in 2003 to the signing of a covenant between the two churches.
Formally introduced in September 1979, Initial samples were released in February 1980, with production chips available over the counter in November.
* Nuclear bunker buster: Formally known as the Robust Nuclear Earth Penetrator ( RNEP ), this program aimed to modify an existing gravity bomb to penetrate into soil and rock in order to destroy underground targets.
Formally, a profinite group is a Hausdorff, compact, and totally disconnected topological group: that is, a topological group that is also a Stone space.

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