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By Michel Willem

ISBN-10: 2870851243

ISBN-13: 9782870851241

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Example text

D(k) = (k − d(k))/β · β + d(k) = k We may now formulate a digit serial, right-to-left (least significant first) algorithm for determining the existence and coefficients of a radix polynomial. 2 (DGT Algorithm) Stimulus: A radix β, |β| ≥ 2. A digit set D which is a complete residue system modulo |β|. A radix-β number v ∈ Q|β| . i Response: Either the radix polynomial P ∈ P[β, D], P = m i= di [β] , P = v, or a signal that no such polynomial exists. } The loops L1 and L2 serve to determine , the lower index of the radix polynomial, and insure that the value of r is integral when the while-loop is entered at L3.

Proof Note that D (n) is formed from the values of |β|n distinct (n − 1)th-order polynomials with 0 ∈ D (n) . , P ≡ P (mod |β| ). i n Assume such P and P exist, then n−1 i=0 (di − di )β ≡ 0 (mod |β| ). Let j be the n−1 smallest index such that dj = dj , then i=j (di − di )β i ≡ 0 (mod |β|n ) implies (dj − dj )β j ≡ 0 (mod |β|j +1 ), and hence dj ≡ dj (mod |β|), a contradiction. We then obtain the following corollary of the lemma. 7) for every n ≥ 2. Then j (β n − 1) ∈ for any non-zero j . Proof D (n) is complete for radix β n and also a non-redundant digit set.

2) is satisfied and the digit set {0, 1} is then “non-redundant” for radix β = 2. Obviously, if negative numbers have to be representable, either the radix β has to be negative, or the digit set has to include at least one negative digit value. , |V−2 (v) ∩ P[−2, {0, 1}]| = 1 for all v ∈ Q2 . 1) can be answered affirmatively: any binary number (in Q2 ) can be represented. , in radix string notation: 1 = 11¯ 2 = 11¯ 1¯ 2 = · · · . Despite the redundancy in representations from P[2, {−1, 0, 1}] this turns out to be a very useful system as we shall see in the following chapters.

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Analyse convexe et optimisation by Michel Willem


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