) 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 reflexive if and only if each X < sub > j </ sub > is reflexive.

Then X is separable if and only if X ′ is separable.

Then all elements of X equivalent to each other are also elements of the same equivalence class.

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 the expectation of this random variable X is defined as

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.

Then ƒ is invertible if there exists a function g with domain Y and range X, with the property:

* 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 for a specific value x of X, the function L ( θ | x )

Then the observation that X

Then the emf, E < sub > X </ sub >, of the same cell containing the solution of unknown pH is measured.

Then X has cardinality at most and cardinality at most if it is first countable.

Then A is dense in C ( X, R ) if and only if it separates points.

Then ρ will be the finest completely regular topology on X which is coarser than τ.

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 he would get to his feet, as though rising in honor of his own remarkable powers, and say almost invariably, `` Gentlemen, this is an amazing story!!

Then, Jesus indicated that God's forgiveness is unlimited.

Certainly, the meaning is clearer to one who is not familiar with Biblical teachings, in the New English Bible which reads: `` Then Jesus arrived at Jordan from Galilee, and he came to John to be baptized by him.

Then it added: `` It is not possible to determine how extensive these ill effects will be -- nor how many people will be affected ''.

Then the words fell into a pattern: `` Mollie the Mutton is scratching her nose, Scratching her nose in the rain.

Then he thought of Aaron Blaustein standing in his rich house saying: `` God is tired of taking the blame.

Then it is marked on the inside where it comes in contact with the transom, frames, keelson and all the battens.

Then it is replaced and fastened.

Then the chines are rounded off and the bottom is rough-sanded in preparation.

Then, a group of eggs is deposited in a cavity in the beebread loaf and the egg compartment is closed.

Then there is a diagonalizable operator D on V and a nilpotent operator N in V such that ( A ) Af, ( b ) Af.

Then in 2 we show that any line involution with the properties that ( A ) It has no complex of invariant lines, and ( B ) Its singular lines form a complex consisting exclusively of the lines which meet a twisted curve, is necessarily of the type discussed in 1.

Then, too, the utmost clinical flexibility is necessary in judiciously combining carefully timed family-oriented home visits, single and group office interviews, and appropriate telephone follow-up calls, if the worker is to be genuinely accessible and if the predicted unhealthy outcome is to be actually averted in accordance with the principles of preventive intervention.

Then the editorial added prophetically: `` how far they may reach in Asia is yet undetermined, but they fall far short of our dreams of the war conferences ''.

Then she catapults into `` everything and everybody '', putting particular violence on `` everybody '', indicating to the linguist that this is a spot to flag -- that is, it is not congruent to the patient's general style of speech up to this point.

Then comes the time when the last wire is removed and Susie walks out a healthier and more attractive girl than when she first went to the orthodontist.

Then, with the new affluence, there is actually a sallying forth into the wide, wide world beyond the precincts of New York.

Then, if the middle number is activated to its greatest potential in terms of this square, through multiplying it by the highest number, 9 ( which is the square of the base number ), the result is 45 ; ;

Then, with a shrug of pretended indifference, she took a compact from her purse and went through the motions of fixing her make-up.

Then is a compact topological space ; this follows from the Tychonoff theorem.

Then by Arzelà – Ascoli theorem the space K is compact.

Then, depending on whether a laptop or compact or extended keyboard type is used, the shortcut is ++ 5 or + 5 ( numeric keypad ) or ++ i ( laptop ).

Let X be a normed topological vector space over F, compatible with the absolute value in F. Then in X *, the topological dual space X of continuous F-valued linear functionals on X, all norm-closed balls are compact in the weak -* topology.

Conversely, the band's transition from lengthy, complex songs to more compact, simplistic, radio-friendly material was not welcomed by critics ; Rolling Stone's review of ... And Then There Were Three ... read: "... this contemptible opus is but the palest shadow of the group's earlier accomplishments.

Then the Alexandroff extension of X is a certain compact space X * together with an open embedding c: X → X * such that the complement of X in X * consists of a single point, typically denoted ∞.

Let K be a closed subset of a compact set T in R < sup > n </ sup > and let C < sub > K </ sub > be an open cover of K. Then

Curds are mentioned in the Middle Irish parodic tale The Vision of Mac Conglinne, the relevant portion of which reads: Stately, pleasantly it sat, A compact house and strong. Then I went in: The door of it was dry meat, The threshold was bare bread, cheese-curds the sides. Smooth pillars of old cheese, And sappy bacon propsAlternate ranged ; Fine beams of mellow cream, White rafters-real curds, Kept up the house.

Then osteoblasts deposit compact bone within the resorption pit.

Let X be a compact complex manifold of complex dimension n. Then X is an orientable smooth manifold of real dimension 2n, so its cohomology groups lie in degrees zero through 2n.

* Let ρ be a unitary representation of a compact group G on a complex Hilbert space H. Then H splits into an orthogonal direct sum of irreducible finite-dimensional unitary representations of G.

Then, the design of the compact kitchen unit “ UNIREX ” was designed in which one single large appliance had more functions, from the conservation to the cooking phase.

Suppose that H is a locally compact Hausdorff group with a compact subgroup K. Then H acts on the quotient space X = H / K.

Let be the set of smooth functions with compact support on the real line Then, the map

Then T is also compact.

Then it is well known that it possesses a unique, up-to-scale left-( or right -) rotation invariant ring ( borel regular in the case of second countable ) measure ( left and right probability measure in the case of compact ), the Haar measure.

Given a compact subset K of X and an open subset U of Y, let V ( K, U ) denote the set of all functions such that Then the collection of all such V ( K, U ) is a subbase for the compact-open topology on C ( X, Y ).

Then the spectrum for the Laplace-Beltrami operator on is discrete and real, since the Laplace operator is self adjoint with compact resolvent ; that is

Then passing to the second person the compact would be set out in the second line by the offering of the nuptial gifts as a guarantee.

Then, for every ε > 0, there exists a compact E ⊂ such that f restricted to E is continuous and

Then the fundamental class, denoted, of a compact n-dimensional manifold M is a generator of.

Then the set of all tensor-preserving, self-conjugate natural transformations of F is a closed set of End ( F ), which is in fact a ( compact ) group whenever G is a ( compact ) group.