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124 DANIEL J. MADDEN times. Further we will see how to produce surds with periods of ever increasing complexity. We will describe a method that takes the period of one example and inserts it inside a more complicated recurring pattern. <strong>For</strong> example, starting with a sequence a of length k2 taken from one continued fraction, we can construct another continued fraction that involves a recurring pattern bi,a, ci that appears k1 times in the repeating part. The process is recursive, and it can be used to produce ever more elaborate sequences of partial quotients. Finally, our method will allow us to choose the partial quotients to be of any size so our fractions can be chosen to limit the occurrence of small partial quotients. While there is no end to the number and complexity of families our methods can produce, they become increasingly difficult to write down explicitly in terms of the controlling parameters. The notation that follows gets rather involved, and we will be forced to set some conventions just to make things readable. To start we will no longer overline the repeating quotients in the expansion a quadratic surd, and agree that √ d = [a0; a1, a2, . . . . . . . . . , an] means that the quotients a1, a2, . . . . . . . . . , an repeat. We will still use [a0; a1, a2, . . . . . . . . . , an] for the finite continued fraction in hopes that the context will allow the correct interpretation. Suppose we have a matrix 0 1 0 1 0 1 N = · · · . 1 a1 1 a2 1 an We will use the shorthand notation N = {a1, a2, . . . . . . . . . , an} to express such a product. We will write −→ N to denote the sequence a1, a2, . . . . . . . . . , an. We will denote the reverse sequence as ← N which could just as well be written −→ N T . as Any such matrix can be viewed as a fractional linear functions, and we will denote this action as s t sx + t [x] = u v ux + v . Of course, the composition of functions corresponds to matrix multiplication. To set out the method used in the proofs, we will begin with an example. This example will illustrate the steps used in this paper to evaluate a continued fraction, and allow readers to reconcile the layout of these steps with their own style. The example we will use is in one of the sequences we will construct later, √ 31. In this case, the expansion claimed is √ 31 = [5; 1, 1, 3, 5, 3, 1, 1, 10].
CONSTRUCTING FAMILIES OF LONG CONTINUED FRACTIONS 125 We can compute the convergents of the first cycle using the familiar PQ chart: 5 1 1 3 5 3 1 1 10 Q 1 0 1 1 2 7 37 118 155 273 2885 P 0 1 5 6 11 39 206 657 863 1520 16063 <strong>For</strong> our purposes it is better to change the last quotient in this chart from 10 to 5, then the last two columns look like 1 5 273 1520 1520 8463 This calculation can be expressed in our matrix notation as 273 1520 M = {5; 1, 1, 3, 5, 3, 1, 1, 5} = . 1520 8463 Because the continued fraction repeats, the value of this fraction satisfies the quadratic equation 273x + 1520 1 M[x] = = 1520x + 8463 x . This reduce to 273x 2 = 8463 or x 2 = 31. Further we can obtain the smallest solutions to Pell’s equation from these last two columns 1520 2 − 273 2 · 31 = 1. In general, if √ d = [a0; a1, a2, . . . . . . . . . , 2a0], then the corresponding matrix M = {a0, a1, a2, . . . . . . . . . , a0} must have the form s t M = t sd where s 2 d − t 2 = ±1. Then √ d is the solution to the quadratic equation: M[x] = 1 x .
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PACIFIC JOURNAL OF MATHEMATICS Vol.
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EIGENVALUES ASYMPTOTICS 3 (i) If pm
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EIGENVALUES ASYMPTOTICS 5 Corollary
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Thus, by the Itô formula, we have
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Therefore, K1(t) ≡ t 2−d/2 ≤
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EIGENVALUES ASYMPTOTICS 11 The chan
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EIGENVALUES ASYMPTOTICS 13 Here Res
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IMAGINARY QUADRATIC FIELDS 17 Table
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IMAGINARY QUADRATIC FIELDS 19 and t
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IMAGINARY QUADRATIC FIELDS 21 Propo
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〈σ 〈σ〉 〈σ, τ〉 2 〈1
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IMAGINARY QUADRATIC FIELDS 25 Then
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IMAGINARY QUADRATIC FIELDS 27 were
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IMAGINARY QUADRATIC FIELDS 29 Thus
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IMAGINARY QUADRATIC FIELDS 31 [3] ,
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34 CARINA BOYALLIAN spherical serie
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36 CARINA BOYALLIAN Here I G MAN (
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38 CARINA BOYALLIAN where k ≤ [ n
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40 CARINA BOYALLIAN The case G = Sp
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42 CARINA BOYALLIAN we can, since i
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44 CARINA BOYALLIAN and this formul
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46 CARINA BOYALLIAN Here, J(ν) int
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48 CARINA BOYALLIAN [V] D. Vogan, R
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50 ROBERT F. BROWN AND HELGA SCHIRM
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Page 78 and 79: 76 ROBERT F. BROWN AND HELGA SCHIRM
Page 84 and 85: 82 GILLES CARRON à l’infini s’
Page 86 and 87: 84 GILLES CARRON isométriques au d
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Page 92 and 93: 90 GILLES CARRON où H est la proje
Page 94 and 95: 92 GILLES CARRON 2.a. Exemples repo
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Page 100 and 101: 98 GILLES CARRON Proposition 2.7. N
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Page 104 and 105: 102 GILLES CARRON compact. De même
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Page 132 and 133: 128 DANIEL J. MADDEN Now the whole
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Page 136 and 137: 132 DANIEL J. MADDEN We can simplif
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Page 140 and 141: 136 DANIEL J. MADDEN and d = d(u, v
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Page 144 and 145: 140 DANIEL J. MADDEN Then the conti
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SOME CHARACTERIZATION, UNIQUENESS A
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and SOME CHARACTERIZATION, UNIQUENE
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SYMPLECTIC SUBMANIFOLDS FROM SURFAC
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208 PAULO TIRAO Theorem. The affine
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210 PAULO TIRAO It follows from the
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212 PAULO TIRAO K ρ1 (0,−1,1) =
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214 PAULO TIRAO Theorem 2.7. Let ρ
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216 PAULO TIRAO Notation: By A = [
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218 PAULO TIRAO Now is not difficul
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220 PAULO TIRAO Regard H n (Z2 ⊕
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222 PAULO TIRAO is an isomorphism.
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224 PAULO TIRAO Lemma 4.2. Let Λ1
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226 PAULO TIRAO (c)1 B1 B2 B3 B1 0
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228 PAULO TIRAO ⎛ ⎜ B2 = ⎜
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230 PAULO TIRAO being those in whic
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232 PAULO TIRAO Hence, if ρ = m1χ
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(1.6) HÖLDER REGULARITY FOR ∂ 23
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HÖLDER REGULARITY FOR ∂ 239 can
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HÖLDER REGULARITY FOR ∂ 241 wher
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HÖLDER REGULARITY FOR ∂ 243 Theo
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Note (4.7) bD∩U ′ i dzA j J H
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HÖLDER REGULARITY FOR ∂ 247 p. 1
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similarly, we can prove HÖLDER REG
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HÖLDER REGULARITY FOR ∂ 251 For
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Now if n1 ≥ 2, l ∈ S1, then (4.
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HÖLDER REGULARITY FOR ∂ 255 Refe
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