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Indeed and following
Indeed, following, suppose ƒ is a complex function defined in an open set Ω ⊂ C. Then, writing for every z ∈ Ω, one can also regard Ω as an open subset of R < sup > 2 </ sup >, and ƒ as a function of two real variables x and y, which maps Ω ⊂ R < sup > 2 </ sup > to C. We consider the Cauchy – Riemann equations at z = 0 assuming ƒ ( z ) = 0, just for notational simplicity – the proof is identical in general case.
Indeed we have the following result.
Indeed, two roads, which diverged at Othoca ( modern Santa Giusta ) connected Caralis to Turris, the more important keeping inland and the other following the west coast.
Indeed, there are non-Bayesian updating rules that also avoid Dutch books ( as discussed in the literature on " probability kinematics " following the publication of Richard C. Jeffrey's rule, which applies Bayes ' rule to the case where the evidence itself is assigned a probability ).
Observe first the following: if a is a limit point of S, then any finite collection C of open sets, such that each open set U ∈ C is disjoint from some neighborhood V < sub > U </ sub > of a, fails to be a cover of S. Indeed, the intersection of the finite family of sets V < sub > U </ sub > is a neighborhood W of a in R < sup > n </ sup >.
Indeed, in England the word " dollar " was in use for the Thaler for 200 years before the issue of the United States dollar, and until the half crown ceased to be used following decimalisation in 1971, the term " half a dollar " could be heard for " half a crown ".
Indeed, following the 1960s, it increasingly became home to a number of squatters, artists, bohemians and also political radicals.
" Indeed, the act was passed by Parliament on 21 May, the same day as a royal proclamation against seditious writings was issued, and more libel cases would be brought by the government in the following two years than had been in all the preceding years of the eighteenth century.
Indeed, enjoyment of the natural surroundings of the walk is the primary motivating factor for many people following the route.
Indeed, more guilds are known in Thyatira than any other contemporary city in the Roman province of Asia ( inscriptions mention the following: wool-workers, linen-workers, makers of outer garments, dyers, leather-workers, tanners, potters, bakers, slave-dealers, and bronze-smiths ).
Indeed, the concept of self-esteem is approached since then in humanistic psychology as an inalienable right for every person, summarized in the following sentence:
" Indeed, Russian historian Oleg Sokolov noted that Wellington had committed a serious strategic error by following the French into northern Portugal, and that this decision could have had grievous consequences for the Anglo-Portuguese.
Indeed, in 1969 a " Birding Glossary " appeared in Birding Volume 1, No. 2, which stated the following:
Indeed, Against Apion, the writing of Josephus in 95 CE, treated the text of the Hebrew Bible as a closed canon to which "... no one has ventured either to add, or to remove, or to alter a syllable ..." However, we know that for a long time following this date that the divine inspiration of Esther, the Song of Songs, and Ecclesiastes was often under scrutiny.
Indeed, the geometrization of space that would characterize Cubism can already be observed in his works as early as 1905, following the lead of Seurat and Cézanne.
Indeed, the transitional government that ruled the country from the cease-fire agreement ( signed in Lusaka on 7 September 1974 ) to independence ( set for 25 June of the following year ) acted in a very conciliatory fashion.
Indeed, the Wall Street Crash of 1929 and the following Great Depression, which affected France from 1931, caused much anxiety and disturbance, as in other countries.
Indeed, special relativity largely comes down to studying a kind of noneuclidean trigonometry in Minkowski spacetime, as suggested by the following table :< ref group = A >
Indeed, these locally focused routers were often built along existing roads and steam railways with which they competed, sometimes following the latter from terminal to terminal.
Indeed, in the early years following the Scottish Reformation there was actually internal sectarian tension between Church of Scotland Presbyterians and ' High Church ' Anglicans, whom they regarded as having retained too many attitudes and practices from the pre-Reformation Catholic era.
Indeed, following the success of The Deluge, Goehr was commissioned a new cantata, Sutter's Gold for choir, baritone and orchestra.
Indeed, the addition of a couple of key players could have seen Omagh challenge for the following seasons premiership.
Indeed, as a result of the war and the lessons that were learned about the potential of airpower in the region, following the war, the Chief of the Air Staff, Sir Hugh Trenchard, proposed controlling the frontier by air power alone.
Indeed, following the construction of the spectral decomposition for a bounded, normal, not self-adjoint, operator T, one sees that to verify that T

Indeed and theorem
i. e., z is a vector orthogonal to the vector v ( Indeed, z is the projection of u onto the plane orthogonal to v .) We can thus apply the Pythagorean theorem to
Indeed, it is not hard to see that it is equivalent to the Boolean prime ideal theorem ( BPIT ), a well-known intermediate point between the axioms of Zermelo-Fraenkel set theory ( ZF ) and the ZF theory augmented by the axiom of choice ( ZFC ).
Indeed, if ( X ) holds for such a, multiplying both sides by a yields the original form of the theorem,
Indeed, according to the Kochen-Specker theorem, in the quantum mechanical formalism it is impossible that, for a given quantum state, each one of these measurable properties ( observables ) has a determinate ( sharp ) value.
Indeed, it is needed to prove both the mean value theorem and the existence of Taylor series.
An important consequence of the domain invariance theorem is that R < sup > n </ sup > cannot be homeomorphic to R < sup > m </ sup > if m ≠ n. Indeed, no non-empty open subset of R < sup > n </ sup > can be homeomorphic to any open subset of R < sup > m </ sup > in this case.
) Indeed, a triumph of nineteenth century geometry was the proof of the uniformization theorem, the analogous topological classification of smooth two-manifolds, where Hamilton showed that the Ricci flow does indeed evolve a negatively curved two-manifold into a two-dimensional multi-holed torus which is locally isometric to the hyperbolic plane.
Indeed, this restatement of Dickson's lemma was used by Paul Gordan in 1899 as part of a proof of Hilbert's basis theorem.
This is a special case of the Chebotarev density theorem for the Nth cyclotomic field K. Indeed, the Galois group of K / Q is abelian and can be canonically identified with the group of invertible residue classes mod N. The splitting invariant of a prime p not dividing N is simply its residue class because the number of distinct primes into which p splits is φ ( N )/ m, where m is multiplicative order of p modulo N ; hence by the Chebotarev density theorem, primes are asymptotically uniformly distributed among different residue classes coprime to N.
Indeed, by the Stone – Weierstrass theorem, any continuous function on the unit interval can be expressed as a limit of polynomials, so any class of functions containing the polynomials and closed under limits will necessarily include all continuous functions.
Indeed, whereas the time hierarchy theorem has seen little remarkable improvement since its inception, the nondeterministic space hierarchy theorem has seen at least one important improvement by Viliam Geffert in his 2003 paper " Space hierarchy theorem revised ".
Indeed, a statement cannot rightly be called a " theorem " if a counterexample disproving it exists.
Indeed, the theorem is often stated only for non-compact manifolds.
Indeed, for any a and b in the algebraic closure of k one has by the binomial theorem
Indeed, from the statement of that theorem, we have where:
Indeed, according to Donsker's theorem, the discrete random walk would, in the scaling limit, approach the true Brownian motion.
Indeed, the bounded convergence theorem ( a corollary of the dominated convergence theorem ) of real analysis states that if a sequence of functions on a set of finite measure is uniformly bounded and converges pointwise, then passage of the limit under the integral is valid.

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