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Then X can be read as " X is possibly true " ( or false, as the case may be ).
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Then and X
) Then X < sub > i </ sub > is the value ( or realization ) produced by a given run of the process at time i. Suppose that the process is further known to have defined values for mean μ < sub > i </ sub > and variance σ < sub > i </ sub >< sup > 2 </ sup > for all times i. Then the definition of the autocorrelation between times s and t is
Then X is compact if and only if X is a complete lattice ( i. e. all subsets have suprema and infima ).
Then the quotient space X /~ can be naturally identified with a torus: take a square piece of paper, bend and glue together the upper and lower edge to form a cylinder, then bend the resulting cylinder so as to glue together its two open ends, resulting in a torus.
Then a presheaf on X is a contravariant functor from O ( X ) to the category of sets, and a sheaf is a presheaf which satisfies the gluing axiom.
Then the joint distribution of X and Y is completely determined by our channel and by our choice of, the marginal distribution of messages we choose to send over the channel.
* Let the index set I of an inverse system ( X < sub > i </ sub >, f < sub > ij </ sub >) have a greatest element m. Then the natural projection π < sub > m </ sub >: X → X < sub > m </ sub > is an isomorphism.
Then the emf, E < sub > X </ sub >, of the same cell containing the solution of unknown pH is measured.
Then the Zariski tangent space at a point p ∈ X is the collection of K-derivations D: O < sub > X, p </ sub >→ K, where K is the ground field and O < sub > X, p </ sub > is the stalk of O < sub > X </ sub > at p.
Then and can
Then, all but blind, he said there was nothing in Back to Methuselah --, -- `` G.B.S. ought to have known that '', -- and `` I look at my bookshelves despairingly, knowing that I can have nothing more to do with them ''.
Then during washing, the greasy soil rolls back at the edges so that emulsified droplets can disengage themselves from the sorbed oil mass, with the aid of mechanical action, and enter the aqueous phase.
Then every linear operator T in V can be written as the sum of a diagonalizable operator D and a nilpotent operator N which commute.
Then the enthusiasm and energy of all elements can be channeled to produce cumulative progress toward a common objective.
Then, after I'm back, another fifty so you can put some mileage on yourself and have a solid alibi somewhere while I take care of your seat cover boy ''.
Then we can express the variables as the sum of the ( time averaged ) mean field () that varies in space and a small fluctuating field () that varies in space and time.
Then nodules of blue earth have to be removed and an opaque crust must be cleaned off, which can be done in revolving barrels containing sand and water.
Then, in 1828, Friedrich Wöhler published a paper on the synthesis of urea, proving that organic compounds can be created artificially.
Then the reservists have to serve up to three weeks a year and can be called up to serve two weeks during a non-military crisis.
Then, according to Adkins ( 1975 ), it can be shown that one can write a further thermodynamic version of the above calorimetric rules:
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.
Then each point p of the line can be specified by its distance from O, taken with a + or − sign depending on which half-line contains p.
Then the Cartesian product set D < sub > 1 </ sub > D < sub > 2 </ sub > can be made into a directed set by defining ( n < sub > 1 </ sub >, n < sub > 2 </ sub >) ≤ ( m < sub > 1 </ sub >, m < sub > 2 </ sub >) if and only if n < sub > 1 </ sub > ≤ m < sub > 1 </ sub > and n < sub > 2 </ sub > ≤ m < sub > 2 </ sub >.
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