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146 DANIEL J. MADDEN b, 2b(2bn + 1) k−1 , b(2bn + 1), 2b(2bn + 1) k−2 , b(2bn + 1) 2 , . . . , b(2bn + 1) k−1 , b(2bn + 1) 1 , 2b(2bn + 1) k−1 , 2b, b(2bn + 1) k + n, 2b, b(2bn + 1) k−1 , 2b(2bn + 1), b(2bn + 1) k−2 , 2b(2bn + 1) 2 , . . . , 2b(2bn + 1) k−2 , b(2bn + 1) 1 , 2b(2bn + 1) k−1 , b, 2b(2bn + 1) k + 2n If we take n = b, we get b 2 ((2b 2 + 1) k + 1) 2 + 2(2b 2 + 1) k = = b2 (2b2 + 1) 2k + 2(b2 + 1)(2b2 + 1) k + 1 b(2b 2 + 1) k + b; b, 2b(2b 2 + 1) k−1 , b(2b 2 + 1), 2b(2b 2 + 1) k−2 , b(2b 2 + 1) 2 , . . . , b(2b 2 + 1) k−2 , 2b(2b 2 + 1) 1 , b(2b 2 + 1) k−1 , 2b, b(2b 2 + 1) k + b, 2b, b(2b 2 + 1) k−1 , 2b(2b 2 + 1), b(2b 2 + 1) k−2 , 2b(2b 2 + 1) 2 , . . . , 2b(2b 2 + 1) k−2 , b(2b 2 + 1) 1 , 2b(2b 2 + 1) k−1 , b, 2b(2b 2 + 1) k + 2b We can multiply this by b to get a new expansion: = b 4 (2b 2 + 1) 2k + 2b 2 (b 2 + 1)(2b 2 + 1) k + b 2 b 2 (2b 2 + 1) k + b 2 ; 1, 2b 2 (2b 2 + 1) k−1 , (2b 2 + 1), 2b 2 (2b 2 + 1) k−2 , (2b 2 + 1) 2 , . . . , (2b 2 + 1) k−2 , 2b 2 (2b 2 + 1) 1 , (2b 2 + 1) k−1 , 2b 2 , (2b 2 + 1) k + 1, 2b 2 , (2b 2 + 1) k−1 , 2b 2 (2b 2 + 1), (2b 2 + 1) k−2 , 2b 2 (2b 2 + 1) 2 , . . . , 2b 2 (2b 2 + 1) k−2 , (2b 2 + 1) 1 , 2b 2 (2b 2 + 1) k−1 , 1, 2b 2 (2b 2 + 1) k + 2b 2 . References [A] T. Azuhatu, On the fundamental units and class numbers of real quadratic fields, II, Tokyo Journal of Math., 10 (1987), 259-270. [LR] C. Levesque and G. Rind, A few classes of periodic continued fractions, Utilitas Math, 30 (1986), 79-107. . .
CONSTRUCTING FAMILIES OF LONG CONTINUED FRACTIONS 147 [LN] R. Lidl and Niederreiter, Finite Fields, Addison-Wesley Publishing Co. Reading, Massachusetts, 1983. [vdP] A.J. van der Poorten, Explicit formulas for units in certain quadratic number fields, Algorithmic Number Theory (Ithaca, NY, 1994), Lecture Notes in Computer Science, 877, Springer, Berlin, (1994), 194-208. [W] H.C. Williams, A note on the periodic length of the continued fraction length of certain √ D, Utilitas Math., 28 (1985), 201-209. Received October 9, 1998 and revised November 15, 1999. University of Arizona Tucson Arizona 85721 E-mail address: madden@math.arizona.edu
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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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92 GILLES CARRON 2.a. Exemples repo
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94 GILLES CARRON alors si σ ∈ C
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96 GILLES CARRON Ainsi s’il exist
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Page 125: E-mail address: prosk@imath.kiev.ua
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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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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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