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Let denote the sequence of convergents to the continued fraction for.
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Let and denote
Let denote the Bézier curve determined by the points P < sub > 0 </ sub >, P < sub > 1 </ sub >, ..., P < sub > n </ sub >.
Let '~' denote an equivalence relation over some nonempty set A, called the universe or underlying set.
Let G denote the set of bijective functions over A that preserve the partition structure of A: ∀ x ∈ A ∀ g ∈ G ( g ( x ) ∈ ).
Let G be a set and let "~" denote an equivalence relation over G. Then we can form a groupoid representing this equivalence relation as follows.
Let n denote a complete set of ( discrete ) quantum numbers for specifying single-particle states ( for example, for the particle in a box problem we can take n to be the quantized wave vector of the wavefunction.
Let ε ( n ) denote the energy of a particle in state n. As the particles do not interact, the total energy of the system is the sum of the single-particle energies.
Let the line of symmetry intersect the parabola at point Q, and denote the focus as point F and its distance from point Q as f. Let the perpendicular to the line of symmetry, through the focus, intersect the parabola at a point T. Then ( 1 ) the distance from F to T is 2f, and ( 2 ) a tangent to the parabola at point T intersects the line of symmetry at a 45 ° angle.
That is, Alice has one half, a, and Bob has the other half, b. Let c denote the qubit Alice wishes to transmit to Bob.
Let H be a Hilbert space, and let H * denote its dual space, consisting of all continuous linear functionals from H into the field R or C. If x is an element of H, then the function φ < sub > x </ sub >, defined by
If V is a real vector space, then we replace V by its complexification V ⊗< sub > R </ sub > C and let g denote the induced bilinear form on V ⊗< sub > R </ sub > C. Let W be a maximal isotropic subspace, i. e. a maximal subspace of V such that g |< sub > W </ sub > = 0.
A possible definition of spoiling based on vote splitting is as follows: Let W denote the candidate who wins the election, and let X and S denote two other candidates.
Let be a sequence of independent and identically distributed variables with distribution function F and let denote the maximum.
Let and sequence
Suppose that in a mathematical language L, it is possible to enumerate all of the defined numbers in L. Let this enumeration be defined by the function G: W → R, where G ( n ) is the real number described by the nth description in the sequence.
Let be a random sample of size n — that is, a sequence of independent and identically distributed random variables drawn from distributions of expected values given by µ and finite variances given by σ < sup > 2 </ sup >.
* The ring of continuous functions from the real numbers to the real numbers is not Noetherian: Let I < sub > n </ sub > be the ideal of all continuous functions f such that f ( x ) = 0 for all x ≥ n. The sequence of ideals I < sub > 0 </ sub >, I < sub > 1 </ sub >, I < sub > 2 </ sub >, etc., is an ascending chain that does not terminate.
Let the samples of be represented by the sequence, and let be represented by the sequence which correspond to the same points in time.
Let the samples of be represented by the sequence, and let be represented by the sequence which correspond to the same points in time.
Such events are followed with Ben's voiceover (" Let me try that again " or " Damn "), after which the sequence starts over to allow the player to retry.
The music used for the final version of the museum sequence is an instrumental cover version of The Smiths ' " Please, Please, Please Let Me Get What I Want ", performed by The Dream Academy.
The film features a notable opening sequence following Manhattan-bound commuters on the Staten Island Ferry accompanied by Carly Simon's song " Let the River Run ", for which she received the Academy Award for Best Song.
Let F be a field and p ( X ) be a polynomial in the polynomial ring F of degree n. The general process for constructing K, the splitting field of p ( X ) over F, is to construct a sequence of fields such that is an extension of containing a new root of p ( X ).
Let ( Ω, F, P ) be a probability space and let F < sub > n </ sub > be a sequence of mutually independent σ-algebras contained in F. Let
The casual " in-the-round " sequence in Elvis Presley's 1968 Comeback Special, and The Beatles informal studio jams documented in the 1970 film Let It Be were both precursors of the " Unplugged " concept, though they were neither conceived nor promoted as such at the time they occurred.
Let I be an interval in the real line R. A function f: I → R is absolutely continuous on I if for every positive number, there is a positive number such that whenever a finite sequence of pairwise disjoint sub-intervals ( x < sub > k </ sub >, y < sub > k </ sub >) of I satisfies
Let the coin tosses be represented by a sequence of independent random variables, each of which is equal to H with probability p, and T with probability Let N be time of appearance of the first H ; in other words,, and If the coin never shows H, we write N is itself a random variable because it depends on the random outcomes of the coin tosses.
Let one round be defined as a sequence of consecutive losses followed by either a win, or bankruptcy of the gambler.
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