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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 surjective homomorphism.
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 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 denote
Formally, if we denote the set of stable functions by S ( D ) and the stability radius by r ( f, D ), then:

Formally and for
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 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.
*, see On Formally Undecidable Propositions of Principia Mathematica and Related Systems for details on English translations.
Formally named " United Press Associations " for incorporation and legal purposes, but publicly known and identified as United Press or UP, it was created by the 1907 uniting of three smaller news syndicates by the Midwest newspaper publisher E. W. Scripps.
Formally stated, the FFT is a method for computing the discrete Fourier transform of a sampled signal.
Formally, collective noun forms such as “ a group of people ” are represented by second-order variables, or by first-order variables standing for sets ( which are well-defined objects in mathematics and logic ).
Formally it is precisely in allowing quantification over class variables α, β, etc., that we assume a range of values for these variables to refer to.
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, the sequence of partial sums of some infinite summation converges if for every fixed power of X the coefficient stabilizes: there is a point beyond which all further partial sums have the same coefficient.
Formally, a function ƒ is real analytic on an open set D in the real line if for any x < sub > 0 </ sub > in D one can write
Formally, an elliptic function is a function meromorphic on for which there exist two non-zero complex numbers and with ( in other words, not parallel ), such that and for all.
Formally, if is an open subset of the complex plane, a point of, and is a holomorphic function, then is called a removable singularity for if there exists a holomorphic function which coincides with on.
Formally, we have for the approximation to the full solution A, a series in the small parameter ( here called ), like the following:
In late 1900s the day that of the City was founded is Jakin School is a Private School is Formally Part of Jakin School Board was Grades One through Twelve there is no Football field on this school. In 1966 When Jakin School was Closed were Sent to Blakely-Union Elementary, Junior High, and High School ( which later Became Early County High, Middle, and Elementary School ). while Carver school was built for African-American students grades One through Twelve. although late 1960s when Carver school was closed along with Kestler School was based in Damascus were sent to Washington High & Elementary at Blakely. In addition to small farm agriculture, Jakin's early economic growth resulted from turpentine.
Formally, cl ( S ) denotes the smallest subset Y of M that contains S such that for each reaction ( A, B )
Formally, it is a coalition of U. S. and European CEOs, largely from transnational corporations, launched at the urging of former United States Secretary of Commerce Ron Brown in Seville ( Spain ) in late 1995 at a conference attended by CEOs from more than 100 U. S. and EU companies and by top government representatives led by the Commissioners for Trade and Industry and by the U. S. Secretary of Commerce.
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, the movement is a rondo that acts as the theme for a set of eight variations, capped off by a dramatic coda.
( Formally speaking, this then satisfies the premises of an axiom of well-founded induction, which asserts that these two conditions are sufficient for the proposition to hold for all x.
Formally, the motion merely calls on parliament to thank the monarch or viceroy for the speech via an Address in Reply.
; Electronics: Formally a fixture in radio shacks, now used for assembly and repair of all sorts of electronic equipment including communication, computer, and home entertainment items.
It has been leased from the National Trust since 1974 by the Somerset Wildlife Trust ( Formally Somerset Trust for Nature Conservation ) and is run as a nature reserve and visitor centre.

Formally and against
Formally, a hypothesis is compared against its opposite or null hypothesis (" if I release this ball, it will not fall to the floor ").
Formally declared on October 9, 1970, the Khmer Republic was a right-wing pro-United States military-led government headed by General Lon Nol and Prince Sisowath Sirik Matak that took power in the March 18, 1970 coup against Prince Norodom Sihanouk, then the country's head of state.

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.
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, 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, 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, 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.
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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