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140 DANIEL J. MADDEN Then the continued fraction expansion of √ d is m, vA0, −→ N , −→ N , −→ N , . . . , −→ N , Bk−1, vA1, −→ N , −→ N , −→ N , . . . , −→ N , Bk−2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vAk−1, −→ N , −→ N , −→ N , . . . , −→ N , , B0, m, B0, ← N, ← N, ← N, . . . , ← N, vAk−1, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B0, ← N, ← N, ← N, . . . , ← N, vA0, 2m where each N appears n times. In particular, if we begin with 1 1 N = , 2 3 then U = Tn−2(2) − Tn(2) − Tn−1(2) + Tn+1(2) 6 V = Tn+1(2) − Tn−1(2) 6 W = Tn+2(2) − Tn(2) − Tn+1(2) + Tn−1(2) 6 r = 4 + Tn+2(2) − 2T2n(2) − 2T2n+1(2) . 12 After this we use the values set in Proposition 2: and This leads to √d = Ai = ri − 1 W Bi = 2V Ai m = V Ak d = d(u, v, w, δ; l, k) = m 2 + 2r k . Ak; A0, 1, 2, 1, 2, . . . . . . . . . , 1, 2, Bk−1, A1, 1, 2, 1, 2, . . . . . . . . . , 1, 2, Bk−2, A2, 1, 2, 1, 2, . . . . . . . . . , 1, 2, Bk−3, . . . . . . . . . . . . . . . . . . . . . . . . . . .
CONSTRUCTING FAMILIES OF LONG CONTINUED FRACTIONS 141 Ak−2, 1, 2, 1, 2, . . . . . . . . . , 1, 2, B1, Ak−1, 1, 2, 1, 2, . . . . . . . . . , 1, 2, B1, Ak, B0 B1 B2 2, 1, 2, . . . . . . . . . , 2, 1, Ak−1, 2, 1, 2, . . . . . . . . . , 2, 1, Ak−2, 2, 1, 2, . . . . . . . . . , 2, 1, Ak−3, . . . . . . . . . . . . . . . . . . . . . . . . . . . Bk−2, 2, 1, 2, . . . . . . . . . , 2, 1, A1, Bk−1, 2, 1, 2, . . . . . . . . . , 2, 1, A0, 2Ak where in each line the (1,2) pair appears n times. After the zeros are removed the repeating part has length 2k(2n + 2) − 2. Section 7. The next technique for finding sequences to start the process involves factoring 2v1 ± 1. To begin, factor 2v2 + ɛ = ww1. Write w v = [a1; a2, . . . , an]. We can arrange it so that (−1) n = ɛ. Now 0 1 0 1 0 1 u v · · · = . 1 a1 1 a2 1 an x w Since uw − xv = ɛ, and ww1 − 2v 2 = ɛ. So x ≡ 2v (mod w). This leads us to Proposition 2. In practice, this method of producing sequences a1, a2, . . . , an can be used to produce very explicit, yet complicated, families. <strong>For</strong> example, let us start with the prime 3; if v ≡ 2 (mod 3), then 2v 2 + 1 ≡ 0 mod 3. (ɛ = 1.) Let v = 2 + 3n. Choose Then w = 2v2 + 1 3 = 6n 2 + 8n + 3. w v = 6n2 + 8n + 3 = 2n + 1 + 3n + 2 6n2 + 8n + 3 − 6n2 − 7n − 2 3n + 2 3n + 2 n + 1 = 2n + 1 + = 2 + n n + 1 n + 1 3n + 2
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Pacific Journal of Mathematics Volu
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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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82 GILLES CARRON à l’infini s’
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84 GILLES CARRON isométriques au d
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86 GILLES CARRON Comme D est ellipt
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88 GILLES CARRON Réciproquement si
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90 GILLES CARRON où H est la proje
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SOME CHARACTERIZATION, UNIQUENESS A
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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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