1# LaTeX2HTML 2018.3 (Released July 19, 2018) 2# Associate images original text with physical files. 3 4 5$key = q/!;MSF=1.2;LFS=12;AAT/; 6$cached_env_img{$key} = q|<IMG STYLE="vertical-align: 20px" 7 SRC="|."$dir".q|sub_img16.png" 8 ALT="$\!$">|; 9 10$key = q/(C_{ell}^{EB}+C_{ell}^{BE})slash2;MSF=1.2;LFS=12;AAT/; 11$cached_env_img{$key} = q|<IMG STYLE="height: 2.51ex; vertical-align: -0.68ex; " SRC="|."$dir".q|sub_img36.png" 12 ALT="$(C_{\ell}^{EB}+C_{\ell}^{BE})/2$">|; 13 14$key = q/(C_{ell}^{TB}+C_{ell}^{BT})slash2;MSF=1.2;LFS=12;AAT/; 15$cached_env_img{$key} = q|<IMG STYLE="height: 2.51ex; vertical-align: -0.68ex; " SRC="|."$dir".q|sub_img35.png" 16 ALT="$(C_{\ell}^{TB}+C_{\ell}^{BT})/2$">|; 17 18$key = q/(C_{ell}^{TE}+C_{ell}^{ET})slash2;MSF=1.2;LFS=12;AAT/; 19$cached_env_img{$key} = q|<IMG STYLE="height: 2.51ex; vertical-align: -0.68ex; " SRC="|."$dir".q|sub_img34.png" 20 ALT="$(C_{\ell}^{TE}+C_{\ell}^{ET})/2$">|; 21 22$key = q/(theta,phi);MSF=1.2;LFS=12;AAT/; 23$cached_env_img{$key} = q|<IMG STYLE="height: 2.33ex; vertical-align: -0.68ex; " SRC="|."$dir".q|sub_img81.png" 24 ALT="$(\theta,\phi) $">|; 25 26$key = q/-1.637510^{30};MSF=1.2;LFS=12;AAT/; 27$cached_env_img{$key} = q|<IMG STYLE="height: 1.92ex; vertical-align: -0.29ex; " SRC="|."$dir".q|sub_img132.png" 28 ALT="$-1.6375 10^{30}$">|; 29 30$key = q/0<|s|le100;MSF=1.2;LFS=12;AAT/; 31$cached_env_img{$key} = q|<IMG STYLE="height: 2.33ex; vertical-align: -0.68ex; " SRC="|."$dir".q|sub_img11.png" 32 ALT="$0 < \vert s\vert \le 100$">|; 33 34$key = q/0dots;MSF=1.2;LFS=12;AAT/; 35$cached_env_img{$key} = q|<IMG STYLE="height: 1.57ex; vertical-align: -0.10ex; " SRC="|."$dir".q|sub_img178.png" 36 ALT="$0\dots$">|; 37 38$key = q/10^circ;MSF=1.2;LFS=12;AAT/; 39$cached_env_img{$key} = q|<IMG STYLE="height: 1.63ex; vertical-align: -0.10ex; " SRC="|."$dir".q|sub_img137.png" 40 ALT="$10^\circ$">|; 41 42$key = q/180slashpi;MSF=1.2;LFS=12;AAT/; 43$cached_env_img{$key} = q|<IMG STYLE="height: 2.33ex; vertical-align: -0.68ex; " SRC="|."$dir".q|sub_img131.png" 44 ALT="$180/\pi$">|; 45 46$key = q/1slashsqrt{4pi};MSF=1.2;LFS=12;AAT/; 47$cached_env_img{$key} = q|<IMG STYLE="height: 2.56ex; vertical-align: -0.68ex; " SRC="|."$dir".q|sub_img129.png" 48 ALT="$1/\sqrt{4\pi}$">|; 49 50$key = q/2^{128}-1approx3.410^{38};MSF=1.2;LFS=12;AAT/; 51$cached_env_img{$key} = q|<IMG STYLE="height: 1.92ex; vertical-align: -0.29ex; " SRC="|."$dir".q|sub_img168.png" 52 ALT="$2^{128} - 1 \approx 3.4 10^{38}$">|; 53 54$key = q/2^{31}-1simeq2.110^9;MSF=1.2;LFS=12;AAT/; 55$cached_env_img{$key} = q|<IMG STYLE="height: 1.92ex; vertical-align: -0.29ex; " SRC="|."$dir".q|sub_img118.png" 56 ALT="$2^{31}-1 \simeq 2.1 10^9$">|; 57 58$key = q/2^{63}-1simeq9.210^{18};MSF=1.2;LFS=12;AAT/; 59$cached_env_img{$key} = q|<IMG STYLE="height: 1.92ex; vertical-align: -0.29ex; " SRC="|."$dir".q|sub_img119.png" 60 ALT="$2^{63}-1 \simeq 9.2 10^{18}$">|; 61 62$key = q/2pi;MSF=1.2;LFS=12;AAT/; 63$cached_env_img{$key} = q|<IMG STYLE="height: 1.57ex; vertical-align: -0.10ex; " SRC="|."$dir".q|sub_img125.png" 64 ALT="$2\pi$">|; 65 66$key = q/3pislash4;MSF=1.2;LFS=12;AAT/; 67$cached_env_img{$key} = q|<IMG STYLE="height: 2.33ex; vertical-align: -0.68ex; " SRC="|."$dir".q|sub_img167.png" 68 ALT="$3\pi/4$">|; 69 70$key = q/4pi;MSF=1.2;LFS=12;AAT/; 71$cached_env_img{$key} = q|<IMG STYLE="height: 1.57ex; vertical-align: -0.10ex; " SRC="|."$dir".q|sub_img126.png" 72 ALT="$4\pi$">|; 73 74$key = q/=-1.637510^{30};MSF=1.2;LFS=12;AAT/; 75$cached_env_img{$key} = q|<IMG STYLE="height: 1.92ex; vertical-align: -0.29ex; " SRC="|."$dir".q|sub_img146.png" 76 ALT="$= -1.6375 10^{30}$">|; 77 78$key = q/B(ell);MSF=1.2;LFS=12;AAT/; 79$cached_env_img{$key} = q|<IMG STYLE="height: 2.33ex; vertical-align: -0.68ex; " SRC="|."$dir".q|sub_img8.png" 80 ALT="$B(\ell)$">|; 81 82$key = q/B_ell;MSF=1.2;LFS=12;AAT/; 83$cached_env_img{$key} = q|<IMG STYLE="height: 2.04ex; vertical-align: -0.45ex; " SRC="|."$dir".q|sub_img78.png" 84 ALT="$B_\ell$">|; 85 86$key = q/C(ell);MSF=1.2;LFS=12;AAT/; 87$cached_env_img{$key} = q|<IMG STYLE="height: 2.33ex; vertical-align: -0.68ex; " SRC="|."$dir".q|sub_img7.png" 88 ALT="$C(\ell)$">|; 89 90$key = q/C_ell;MSF=1.2;LFS=12;AAT/; 91$cached_env_img{$key} = q|<IMG STYLE="height: 2.04ex; vertical-align: -0.45ex; " SRC="|."$dir".q|sub_img105.png" 92 ALT="$C_\ell$">|; 93 94$key = q/C_ell^B;MSF=1.2;LFS=12;AAT/; 95$cached_env_img{$key} = q|<IMG STYLE="height: 2.51ex; vertical-align: -0.67ex; " SRC="|."$dir".q|sub_img196.png" 96 ALT="$C_\ell^B$">|; 97 98$key = q/C_ell^E;MSF=1.2;LFS=12;AAT/; 99$cached_env_img{$key} = q|<IMG STYLE="height: 2.51ex; vertical-align: -0.67ex; " SRC="|."$dir".q|sub_img195.png" 100 ALT="$C_\ell^E$">|; 101 102$key = q/C_ell^T;MSF=1.2;LFS=12;AAT/; 103$cached_env_img{$key} = q|<IMG STYLE="height: 2.51ex; vertical-align: -0.67ex; " SRC="|."$dir".q|sub_img194.png" 104 ALT="$C_\ell^T$">|; 105 106$key = q/C_ell^{BB};MSF=1.2;LFS=12;AAT/; 107$cached_env_img{$key} = q|<IMG STYLE="height: 2.51ex; vertical-align: -0.67ex; " SRC="|."$dir".q|sub_img108.png" 108 ALT="$C_\ell^{BB}$">|; 109 110$key = q/C_ell^{BE};MSF=1.2;LFS=12;AAT/; 111$cached_env_img{$key} = q|<IMG STYLE="height: 2.51ex; vertical-align: -0.67ex; " SRC="|."$dir".q|sub_img114.png" 112 ALT="$C_\ell^{BE}$">|; 113 114$key = q/C_ell^{BT};MSF=1.2;LFS=12;AAT/; 115$cached_env_img{$key} = q|<IMG STYLE="height: 2.51ex; vertical-align: -0.67ex; " SRC="|."$dir".q|sub_img113.png" 116 ALT="$C_\ell^{BT}$">|; 117 118$key = q/C_ell^{EB};MSF=1.2;LFS=12;AAT/; 119$cached_env_img{$key} = q|<IMG STYLE="height: 2.51ex; vertical-align: -0.67ex; " SRC="|."$dir".q|sub_img111.png" 120 ALT="$C_\ell^{EB}$">|; 121 122$key = q/C_ell^{EE};MSF=1.2;LFS=12;AAT/; 123$cached_env_img{$key} = q|<IMG STYLE="height: 2.51ex; vertical-align: -0.67ex; " SRC="|."$dir".q|sub_img107.png" 124 ALT="$C_\ell^{EE}$">|; 125 126$key = q/C_ell^{ET};MSF=1.2;LFS=12;AAT/; 127$cached_env_img{$key} = q|<IMG STYLE="height: 2.51ex; vertical-align: -0.67ex; " SRC="|."$dir".q|sub_img112.png" 128 ALT="$C_\ell^{ET}$">|; 129 130$key = q/C_ell^{TB};MSF=1.2;LFS=12;AAT/; 131$cached_env_img{$key} = q|<IMG STYLE="height: 2.51ex; vertical-align: -0.67ex; " SRC="|."$dir".q|sub_img110.png" 132 ALT="$C_\ell^{TB}$">|; 133 134$key = q/C_ell^{TE};MSF=1.2;LFS=12;AAT/; 135$cached_env_img{$key} = q|<IMG STYLE="height: 2.51ex; vertical-align: -0.67ex; " SRC="|."$dir".q|sub_img109.png" 136 ALT="$C_\ell^{TE}$">|; 137 138$key = q/C_ell^{TT};MSF=1.2;LFS=12;AAT/; 139$cached_env_img{$key} = q|<IMG STYLE="height: 2.51ex; vertical-align: -0.67ex; " SRC="|."$dir".q|sub_img106.png" 140 ALT="$C_\ell^{TT}$">|; 141 142$key = q/C_ell^{TtimesE};MSF=1.2;LFS=12;AAT/; 143$cached_env_img{$key} = q|<IMG STYLE="height: 2.62ex; vertical-align: -0.68ex; " SRC="|."$dir".q|sub_img197.png" 144 ALT="$C_\ell^{T\times E}$">|; 145 146$key = q/C_{ell}^{BB};MSF=1.2;LFS=12;AAT/; 147$cached_env_img{$key} = q|<IMG STYLE="height: 2.51ex; vertical-align: -0.67ex; " SRC="|."$dir".q|sub_img24.png" 148 ALT="$C_{\ell}^{BB}$">|; 149 150$key = q/C_{ell}^{BE};MSF=1.2;LFS=12;AAT/; 151$cached_env_img{$key} = q|<IMG STYLE="height: 2.51ex; vertical-align: -0.67ex; " SRC="|."$dir".q|sub_img32.png" 152 ALT="$C_{\ell}^{BE}$">|; 153 154$key = q/C_{ell}^{BT};MSF=1.2;LFS=12;AAT/; 155$cached_env_img{$key} = q|<IMG STYLE="height: 2.51ex; vertical-align: -0.67ex; " SRC="|."$dir".q|sub_img31.png" 156 ALT="$C_{\ell}^{BT}$">|; 157 158$key = q/C_{ell}^{EB};MSF=1.2;LFS=12;AAT/; 159$cached_env_img{$key} = q|<IMG STYLE="height: 2.51ex; vertical-align: -0.67ex; " SRC="|."$dir".q|sub_img28.png" 160 ALT="$C_{\ell}^{EB}$">|; 161 162$key = q/C_{ell}^{EE};MSF=1.2;LFS=12;AAT/; 163$cached_env_img{$key} = q|<IMG STYLE="height: 2.51ex; vertical-align: -0.67ex; " SRC="|."$dir".q|sub_img23.png" 164 ALT="$C_{\ell}^{EE}$">|; 165 166$key = q/C_{ell}^{ET};MSF=1.2;LFS=12;AAT/; 167$cached_env_img{$key} = q|<IMG STYLE="height: 2.51ex; vertical-align: -0.67ex; " SRC="|."$dir".q|sub_img30.png" 168 ALT="$C_{\ell}^{ET}$">|; 169 170$key = q/C_{ell}^{TB};MSF=1.2;LFS=12;AAT/; 171$cached_env_img{$key} = q|<IMG STYLE="height: 2.51ex; vertical-align: -0.67ex; " SRC="|."$dir".q|sub_img27.png" 172 ALT="$C_{\ell}^{TB}$">|; 173 174$key = q/C_{ell}^{TE};MSF=1.2;LFS=12;AAT/; 175$cached_env_img{$key} = q|<IMG STYLE="height: 2.51ex; vertical-align: -0.67ex; " SRC="|."$dir".q|sub_img25.png" 176 ALT="$C_{\ell}^{TE}$">|; 177 178$key = q/C_{ell}^{TT};MSF=1.2;LFS=12;AAT/; 179$cached_env_img{$key} = q|<IMG STYLE="height: 2.51ex; vertical-align: -0.67ex; " SRC="|."$dir".q|sub_img22.png" 180 ALT="$C_{\ell}^{TT}$">|; 181 182$key = q/N_{mathrm{side}}=sqrt{N_{mathrm{pix}}slash12};MSF=1.2;LFS=12;AAT/; 183$cached_env_img{$key} = q|<IMG STYLE="height: 3.56ex; vertical-align: -1.21ex; " SRC="|."$dir".q|sub_img150.png" 184 ALT="$N_{\mathrm{side}}=\sqrt{N_{\mathrm{pix}}/12}$">|; 185 186$key = q/P_{ellm}!!;MSF=1.2;LFS=12;AAT/; 187$cached_env_img{$key} = q|<IMG STYLE="height: 2.04ex; vertical-align: -0.45ex; " SRC="|."$dir".q|sub_img169.png" 188 ALT="$P_{\ell m}\!\!$">|; 189 190$key = q/P_{ellm}(theta);MSF=1.2;LFS=12;AAT/; 191$cached_env_img{$key} = q|<IMG STYLE="height: 2.33ex; vertical-align: -0.68ex; " SRC="|."$dir".q|sub_img41.png" 192 ALT="$P_{\ell m}(\theta)$">|; 193 194$key = q/P_{ellm};MSF=1.2;LFS=12;AAT/; 195$cached_env_img{$key} = q|<IMG STYLE="height: 2.04ex; vertical-align: -0.45ex; " SRC="|."$dir".q|sub_img10.png" 196 ALT="$P_{\ell m}$">|; 197 198$key = q/P_{ellm}^mathrm{P};MSF=1.2;LFS=12;AAT/; 199$cached_env_img{$key} = q|<IMG STYLE="height: 2.51ex; vertical-align: -0.67ex; " SRC="|."$dir".q|sub_img148.png" 200 ALT="$P_{\ell m}^\mathrm{P}$">|; 201 202$key = q/P_{ellm}^mathrm{T};MSF=1.2;LFS=12;AAT/; 203$cached_env_img{$key} = q|<IMG STYLE="height: 2.51ex; vertical-align: -0.67ex; " SRC="|."$dir".q|sub_img147.png" 204 ALT="$P_{\ell m}^\mathrm{T}$">|; 205 206$key = q/T_{mathrm{CPU}}propto1+2times;MSF=1.2;LFS=12;AAT/; 207$cached_env_img{$key} = q|<IMG STYLE="height: 1.98ex; vertical-align: -0.45ex; " SRC="|."$dir".q|sub_img145.png" 208 ALT="$T_{\mathrm{CPU}} \propto 1 + 2 \times $">|; 209 210$key = q/XneY;MSF=1.2;LFS=12;AAT/; 211$cached_env_img{$key} = q|<IMG STYLE="height: 2.21ex; vertical-align: -0.55ex; " SRC="|."$dir".q|sub_img19.png" 212 ALT="$X \ne Y$">|; 213 214$key = q/Y_{ellm}(theta,phi)=lambda_{ellm}(theta)e^{imphi};MSF=1.2;LFS=12;AAT/; 215$cached_env_img{$key} = q|<IMG STYLE="height: 2.51ex; vertical-align: -0.68ex; " SRC="|."$dir".q|sub_img159.png" 216 ALT="$Y_{\ell m}(\theta,\phi) = \lambda_{\ell m}(\theta) e^{i m \phi}$">|; 217 218$key = q/Y_{ellm};MSF=1.2;LFS=12;AAT/; 219$cached_env_img{$key} = q|<IMG STYLE="height: 2.04ex; vertical-align: -0.45ex; " SRC="|."$dir".q|sub_img44.png" 220 ALT="$Y_{\ell m}$">|; 221 222$key = q/[0,2pi];MSF=1.2;LFS=12;AAT/; 223$cached_env_img{$key} = q|<IMG STYLE="height: 2.33ex; vertical-align: -0.68ex; " SRC="|."$dir".q|sub_img86.png" 224 ALT="$[0, 2\pi] $">|; 225 226$key = q/[0,pi];MSF=1.2;LFS=12;AAT/; 227$cached_env_img{$key} = q|<IMG STYLE="height: 2.33ex; vertical-align: -0.68ex; " SRC="|."$dir".q|sub_img85.png" 228 ALT="$[0, \pi] $">|; 229 230$key = q/_{0}a^+_{ellm}=-a^T_{ellm};MSF=1.2;LFS=12;AAT/; 231$cached_env_img{$key} = q|<IMG STYLE="height: 2.51ex; vertical-align: -0.68ex; " SRC="|."$dir".q|sub_img67.png" 232 ALT="$ _{0}a^+_{\ell m} = -a^T_{\ell m}$">|; 233 234$key = q/_{0}a^-_{ellm}=0;MSF=1.2;LFS=12;AAT/; 235$cached_env_img{$key} = q|<IMG STYLE="height: 2.21ex; vertical-align: -0.68ex; " SRC="|."$dir".q|sub_img68.png" 236 ALT="$ _{0}a^-_{\ell m} = 0$">|; 237 238$key = q/a=sum_i|x_i-m|slashn;MSF=1.2;LFS=12;AAT/; 239$cached_env_img{$key} = q|<IMG STYLE="height: 2.33ex; vertical-align: -0.68ex; " SRC="|."$dir".q|sub_img92.png" 240 ALT="$a= \sum_i\vert x_i-m\vert/n$">|; 241 242$key = q/a^B_{ellm};MSF=1.2;LFS=12;AAT/; 243$cached_env_img{$key} = q|<IMG STYLE="height: 2.51ex; vertical-align: -0.67ex; " SRC="|."$dir".q|sub_img76.png" 244 ALT="$a^B_{\ell m}$">|; 245 246$key = q/a^E_{ellm};MSF=1.2;LFS=12;AAT/; 247$cached_env_img{$key} = q|<IMG STYLE="height: 2.51ex; vertical-align: -0.67ex; " SRC="|."$dir".q|sub_img75.png" 248 ALT="$a^E_{\ell m}$">|; 249 250$key = q/a^T_{ellm};MSF=1.2;LFS=12;AAT/; 251$cached_env_img{$key} = q|<IMG STYLE="height: 2.51ex; vertical-align: -0.67ex; " SRC="|."$dir".q|sub_img74.png" 252 ALT="$a^T_{\ell m}$">|; 253 254$key = q/a_{ellm};MSF=1.2;LFS=12;AAT/; 255$cached_env_img{$key} = q|<IMG STYLE="height: 1.46ex; vertical-align: -0.45ex; " SRC="|."$dir".q|sub_img6.png" 256 ALT="$a_{\ell m}$">|; 257 258$key = q/a_{ellm}^B;MSF=1.2;LFS=12;AAT/; 259$cached_env_img{$key} = q|<IMG STYLE="height: 2.51ex; vertical-align: -0.67ex; " SRC="|."$dir".q|sub_img40.png" 260 ALT="$a_{\ell m}^B$">|; 261 262$key = q/a_{ellm}^E;MSF=1.2;LFS=12;AAT/; 263$cached_env_img{$key} = q|<IMG STYLE="height: 2.51ex; vertical-align: -0.67ex; " SRC="|."$dir".q|sub_img39.png" 264 ALT="$a_{\ell m}^E$">|; 265 266$key = q/a_{ellm}^T;MSF=1.2;LFS=12;AAT/; 267$cached_env_img{$key} = q|<IMG STYLE="height: 2.51ex; vertical-align: -0.67ex; " SRC="|."$dir".q|sub_img38.png" 268 ALT="$a_{\ell m}^T$">|; 269 270$key = q/a_{ellm}^mathrm{(pix)};MSF=1.2;LFS=12;AAT/; 271$cached_env_img{$key} = q|<IMG STYLE="height: 2.97ex; vertical-align: -0.68ex; " SRC="|."$dir".q|sub_img155.png" 272 ALT="$a_{\ell m}^\mathrm{(pix)}$">|; 273 274$key = q/a_{ellm}^mathrm{(pix)}=a_{ellm}w_{mathrm{pix}}(ell);MSF=1.2;LFS=12;AAT/; 275$cached_env_img{$key} = q|<IMG STYLE="height: 3.03ex; vertical-align: -0.75ex; " SRC="|."$dir".q|sub_img156.png" 276 ALT="$a_{\ell m}^\mathrm{(pix)}= a_{\ell m}w_{\mathrm{pix}}(\ell)$">|; 277 278$key = q/a_{ellm}longrightarrowa_{ellm}b(ell);MSF=1.2;LFS=12;AAT/; 279$cached_env_img{$key} = q|<IMG STYLE="height: 2.33ex; vertical-align: -0.68ex; " SRC="|."$dir".q|sub_img77.png" 280 ALT="$a_{\ell m} 281\longrightarrow a_{\ell m} b(\ell) $">|; 282 283$key = q/backslash;MSF=1.2;LFS=12;AAT/; 284$cached_env_img{$key} = q|<IMG STYLE="height: 2.33ex; vertical-align: -0.68ex; " SRC="|."$dir".q|sub_img17.png" 285 ALT="$\backslash$">|; 286 287$key = q/bar{x}equivsum_{p=1}^{N}frac{x(p)}{N};MSF=1.2;LFS=12;AAT/; 288$cached_env_img{$key} = q|<IMG STYLE="height: 3.15ex; vertical-align: -0.97ex; " SRC="|."$dir".q|sub_img142.png" 289 ALT="$\bar{x} \equiv \sum_{p=1}^{N} \frac{x(p)}{N}$">|; 290 291$key = q/cal{P};MSF=1.2;LFS=12;AAT/; 292$cached_env_img{$key} = q|<IMG STYLE="height: 1.75ex; vertical-align: -0.10ex; " SRC="|."$dir".q|sub_img175.png" 293 ALT="$\cal{P}$">|; 294 295$key = q/cos^{-1};MSF=1.2;LFS=12;AAT/; 296$cached_env_img{$key} = q|<IMG STYLE="height: 1.92ex; vertical-align: -0.10ex; " SRC="|."$dir".q|sub_img87.png" 297 ALT="$\cos^{-1}$">|; 298 299$key = q/dgeq4;MSF=1.2;LFS=12;AAT/; 300$cached_env_img{$key} = q|<IMG STYLE="height: 1.98ex; vertical-align: -0.41ex; " SRC="|."$dir".q|sub_img33.png" 301 ALT="$d\geq4$">|; 302 303$key = q/dgeq6;MSF=1.2;LFS=12;AAT/; 304$cached_env_img{$key} = q|<IMG STYLE="height: 1.98ex; vertical-align: -0.41ex; " SRC="|."$dir".q|sub_img26.png" 305 ALT="$d\geq 6$">|; 306 307$key = q/dgeq9;MSF=1.2;LFS=12;AAT/; 308$cached_env_img{$key} = q|<IMG STYLE="height: 1.98ex; vertical-align: -0.41ex; " SRC="|."$dir".q|sub_img29.png" 309 ALT="$d\geq 9$">|; 310 311$key = q/displaystyle(-1)^m{_{|s|}}a^{+*}_{ellm},;MSF=1.2;LFS=12;AAT/; 312$cached_env_img{$key} = q|<IMG STYLE="height: 2.68ex; vertical-align: -0.85ex; " SRC="|."$dir".q|sub_img64.png" 313 ALT="$\displaystyle (-1)^m {_{\vert s\vert}}a^{+*}_{\ell m},$">|; 314 315$key = q/displaystyle(-1)^m{_{|s|}}a^{-*}_{ellm}.;MSF=1.2;LFS=12;AAT/; 316$cached_env_img{$key} = q|<IMG STYLE="height: 2.56ex; vertical-align: -0.85ex; " SRC="|."$dir".q|sub_img66.png" 317 ALT="$\displaystyle (-1)^m {_{\vert s\vert}}a^{-*}_{\ell m}.$">|; 318 319$key = q/displaystyle-({_{|s|}}a_{ellm}+(-1)^s{_{-|s|}}a_{ellm})slash2;MSF=1.2;LFS=12;AAT/; 320$cached_env_img{$key} = q|<IMG STYLE="height: 2.56ex; vertical-align: -0.85ex; " SRC="|."$dir".q|sub_img59.png" 321 ALT="$\displaystyle - ( {_{\vert s\vert}}a_{\ell m} + (-1)^s {_{-\vert s\vert}}a_{\ell m} )/2$">|; 322 323$key = q/displaystyle-({_{|s|}}a_{ellm}-(-1)^s{_{-|s|}}a_{ellm})slash(2i);MSF=1.2;LFS=12;AAT/; 324$cached_env_img{$key} = q|<IMG STYLE="height: 2.56ex; vertical-align: -0.85ex; " SRC="|."$dir".q|sub_img61.png" 325 ALT="$\displaystyle - ( {_{\vert s\vert}}a_{\ell m} - (-1)^s {_{-\vert s\vert}}a_{\ell m} )/(2i)$">|; 326 327$key = q/displaystyle0<phileqpislash2,;MSF=1.2;LFS=12;AAT/; 328$cached_env_img{$key} = q|<IMG STYLE="height: 2.33ex; vertical-align: -0.68ex; " SRC="|."$dir".q|sub_img185.png" 329 ALT="$\displaystyle 0< \phi \leq \pi/2,$">|; 330 331$key = q/displaystyle<comment_mark>13282slash3>zgeq0,;MSF=1.2;LFS=12;AAT/; 332$cached_env_img{$key} = q|<IMG STYLE="height: 2.33ex; vertical-align: -0.68ex; " SRC="|."$dir".q|sub_img186.png" 333 ALT="$\displaystyle %[Nside*(Nside+2)/4] 3342/3 > z \geq 0,$">|; 335 336$key = q/displaystyle<comment_mark>14082slash3>zgeq0,;MSF=1.2;LFS=12;AAT/; 337$cached_env_img{$key} = q|<IMG STYLE="height: 2.33ex; vertical-align: -0.68ex; " SRC="|."$dir".q|sub_img188.png" 338 ALT="$\displaystyle %[Nside*(Nside+2)/4] 3392/3 > z \geq 0,$">|; 340 341$key = q/displaystyleA_{ij}equivsum_{pincal{P}}s_i(p)w(p)s_j(p),;MSF=1.2;LFS=12;AAT/; 342$cached_env_img{$key} = q|<IMG STYLE="height: 4.66ex; vertical-align: -2.94ex; " SRC="|."$dir".q|sub_img174.png" 343 ALT="$\displaystyle A_{ij} \equiv \sum_{p \in \cal{P}} s_i(p) w(p) s_j(p),$">|; 344 345$key = q/displaystyleC_{12}^{XY}(ell)=frac{1}{2ell+1}sum_{m=-ell}^{ell}a_{1,ellm}^Xa_{2,ellm}^{Y*},;MSF=1.2;LFS=12;AAT/; 346$cached_env_img{$key} = q|<IMG STYLE="height: 6.06ex; vertical-align: -2.82ex; " SRC="|."$dir".q|sub_img18.png" 347 ALT="$\displaystyle C_{12}^{XY}(\ell) = \frac{1}{2 \ell +1} 348\sum_{m=-\ell}^{\ell} a_{1,\ell m}^X 349a_{2,\ell m}^{Y*},$">|; 350 351$key = q/displaystyleC_{{12}}^{{XY}}(ell)equivfrac{C_{12}^{XY}(ell)+C_{12}^{YX}(ell)}{2}=frac{C_{12}^{XY}(ell)+C_{21}^{XY}(ell)}{2};MSF=1.2;LFS=12;AAT/; 352$cached_env_img{$key} = q|<IMG STYLE="height: 5.01ex; vertical-align: -1.69ex; " SRC="|."$dir".q|sub_img20.png" 353 ALT="$\displaystyle C_{\{12\}}^{\{XY\}}(\ell) \equiv \frac{C_{12}^{XY}(\ell)+C_{12}^{YX}(\ell)}{2} = \frac{C_{12}^{XY}(\ell)+C_{21}^{XY}(\ell)}{2}$">|; 354 355$key = q/displaystyleb_iequivsum_{pincal{P}}s_i(p)w(p)m(p),;MSF=1.2;LFS=12;AAT/; 356$cached_env_img{$key} = q|<IMG STYLE="height: 4.66ex; vertical-align: -2.94ex; " SRC="|."$dir".q|sub_img173.png" 357 ALT="$\displaystyle b_i \equiv \sum_{p \in \cal{P}} s_i(p) w(p) m(p),$">|; 358 359$key = q/displaystylem'(p)=m(p)-sum_{i=0}^{d^2-1}f_is_i(p).;MSF=1.2;LFS=12;AAT/; 360$cached_env_img{$key} = q|<IMG STYLE="height: 6.24ex; vertical-align: -2.60ex; " SRC="|."$dir".q|sub_img177.png" 361 ALT="$\displaystyle m'(p) = m(p) - \sum_{i=0}^{d^2-1} f_i s_i(p).$">|; 362 363$key = q/displaystylephi=0,quad{{rm{or}quadphi=frac{pi}{4N_{mathrm{side}}}.<comment_mark>1329;MSF=1.2;LFS=12;AAT/; 364$cached_env_img{$key} = q|<IMG STYLE="height: 4.60ex; vertical-align: -2.04ex; " SRC="|."$dir".q|sub_img187.png" 365 ALT="$\displaystyle \phi=0, \quad{\rm or}\quad \phi=\frac{\pi}{4N_{\mathrm{side}}}. %\nonumber %[Nside] 366$">|; 367 368$key = q/displaystylephi=0,quad{{rm{or}quadphi=frac{pi}{4N_{mathrm{side}}}.<comment_mark>1409;MSF=1.2;LFS=12;AAT/; 369$cached_env_img{$key} = q|<IMG STYLE="height: 4.60ex; vertical-align: -2.04ex; " SRC="|."$dir".q|sub_img189.png" 370 ALT="$\displaystyle \phi=0, \quad{\rm or}\quad \phi=\frac{\pi}{4N_{\mathrm{side}}}. %\nonumber %[Nside] 371$">|; 372 373$key = q/displaystylesum_{j=0}^{d^2-1}A_{ij}f_j=b_i;MSF=1.2;LFS=12;AAT/; 374$cached_env_img{$key} = q|<IMG STYLE="height: 6.53ex; vertical-align: -2.91ex; " SRC="|."$dir".q|sub_img172.png" 375 ALT="$\displaystyle \sum_{j=0}^{d^2-1} A_{ij} f_j = b_i$">|; 376 377$key = q/displaystyletextbf{a}^{(n)}=textbf{a}^{(n-1)}+textbf{A}.left(textbf{w}.textbf{m}-textbf{S}.textbf{a}^{(n-1)}right),;MSF=1.2;LFS=12;AAT/; 378$cached_env_img{$key} = q|<IMG STYLE="height: 3.56ex; vertical-align: -1.26ex; " SRC="|."$dir".q|sub_img138.png" 379 ALT="$\displaystyle \textbf{a}^{(n)} = \textbf{a}^{(n-1)} + \textbf{A}. \left( \textbf{w}.\textbf{m}- \textbf{S}.\textbf{a}^{(n-1)} \right),$">|; 380 381$key = q/displaystylez=cos(theta)ge2slash3,;MSF=1.2;LFS=12;AAT/; 382$cached_env_img{$key} = q|<IMG STYLE="height: 2.33ex; vertical-align: -0.68ex; " SRC="|."$dir".q|sub_img183.png" 383 ALT="$\displaystyle z=\cos(\theta)\ge 2/3,$">|; 384 385$key = q/displaystyle{_s}S(p)=sum_{ellm}{_s}a_{ellm}{_s}Y_{ellm}(p);MSF=1.2;LFS=12;AAT/; 386$cached_env_img{$key} = q|<IMG STYLE="height: 4.37ex; vertical-align: -2.66ex; " SRC="|."$dir".q|sub_img53.png" 387 ALT="$\displaystyle {_s}S(p) = \sum_{\ell m} {_s}a_{\ell m} {_s}Y_{\ell m}(p)$">|; 388 389$key = q/displaystyle{_{|s|}}a^{+}_{ell-m};MSF=1.2;LFS=12;AAT/; 390$cached_env_img{$key} = q|<IMG STYLE="height: 2.68ex; vertical-align: -0.85ex; " SRC="|."$dir".q|sub_img63.png" 391 ALT="$\displaystyle {_{\vert s\vert}}a^{+}_{\ell-m}$">|; 392 393$key = q/displaystyle{_{|s|}}a^{+}_{ellm};MSF=1.2;LFS=12;AAT/; 394$cached_env_img{$key} = q|<IMG STYLE="height: 2.68ex; vertical-align: -0.85ex; " SRC="|."$dir".q|sub_img57.png" 395 ALT="$\displaystyle {_{\vert s\vert}}a^{+}_{\ell m}$">|; 396 397$key = q/displaystyle{_{|s|}}a^{-}_{ell-m};MSF=1.2;LFS=12;AAT/; 398$cached_env_img{$key} = q|<IMG STYLE="height: 2.21ex; vertical-align: -0.85ex; " SRC="|."$dir".q|sub_img65.png" 399 ALT="$\displaystyle {_{\vert s\vert}}a^{-}_{\ell-m}$">|; 400 401$key = q/displaystyle{_{|s|}}a^{-}_{ellm};MSF=1.2;LFS=12;AAT/; 402$cached_env_img{$key} = q|<IMG STYLE="height: 2.21ex; vertical-align: -0.85ex; " SRC="|."$dir".q|sub_img60.png" 403 ALT="$\displaystyle {_{\vert s\vert}}a^{-}_{\ell m}$">|; 404 405$key = q/dots;MSF=1.2;LFS=12;AAT/; 406$cached_env_img{$key} = q|<IMG STYLE="height: 0.23ex; vertical-align: -0.10ex; " SRC="|."$dir".q|sub_img1.png" 407 ALT="$\dots $">|; 408 409$key = q/ell;MSF=1.2;LFS=12;AAT/; 410$cached_env_img{$key} = q|<IMG STYLE="height: 1.69ex; vertical-align: -0.10ex; " SRC="|."$dir".q|sub_img21.png" 411 ALT="$\ell$">|; 412 413$key = q/ell<46340;MSF=1.2;LFS=12;AAT/; 414$cached_env_img{$key} = q|<IMG STYLE="height: 1.69ex; vertical-align: -0.16ex; " SRC="|."$dir".q|sub_img12.png" 415 ALT="$\ell < 46340 $">|; 416 417$key = q/ell=128;MSF=1.2;LFS=12;AAT/; 418$cached_env_img{$key} = q|<IMG STYLE="height: 1.69ex; vertical-align: -0.10ex; " SRC="|."$dir".q|sub_img37.png" 419 ALT="$\ell = 128$">|; 420 421$key = q/ell=;MSF=1.2;LFS=12;AAT/; 422$cached_env_img{$key} = q|<IMG STYLE="height: 1.69ex; vertical-align: -0.10ex; " SRC="|."$dir".q|sub_img115.png" 423 ALT="$\ell=$">|; 424 425$key = q/ell_{mathrm{max}};MSF=1.2;LFS=12;AAT/; 426$cached_env_img{$key} = q|<IMG STYLE="height: 2.04ex; vertical-align: -0.45ex; " SRC="|."$dir".q|sub_img144.png" 427 ALT="$\ell_{\mathrm{max}}$">|; 428 429$key = q/ell_{mathrm{max}}=64;MSF=1.2;LFS=12;AAT/; 430$cached_env_img{$key} = q|<IMG STYLE="height: 2.04ex; vertical-align: -0.45ex; " SRC="|."$dir".q|sub_img170.png" 431 ALT="$\ell_{\mathrm{max}}=64$">|; 432 433$key = q/ell_{mathrm{max}}=m_{mathrm{max}}=128;MSF=1.2;LFS=12;AAT/; 434$cached_env_img{$key} = q|<IMG STYLE="height: 2.04ex; vertical-align: -0.45ex; " SRC="|."$dir".q|sub_img80.png" 435 ALT="$\ell_{\mathrm{max}}=m_{\mathrm{max}}= 128$">|; 436 437$key = q/ell_{mathrm{max}}=m_{mathrm{max}}=64;MSF=1.2;LFS=12;AAT/; 438$cached_env_img{$key} = q|<IMG STYLE="height: 2.04ex; vertical-align: -0.45ex; " SRC="|."$dir".q|sub_img181.png" 439 ALT="$\ell_{\mathrm{max}}= m_{\mathrm{max}}= 64$">|; 440 441$key = q/ell_{mathrm{max}}^3;MSF=1.2;LFS=12;AAT/; 442$cached_env_img{$key} = q|<IMG STYLE="height: 2.51ex; vertical-align: -0.67ex; " SRC="|."$dir".q|sub_img180.png" 443 ALT="$\ell_{\mathrm{max}}^3$">|; 444 445$key = q/ell_{mathrm{max}}leq4N_{mathrm{side}};MSF=1.2;LFS=12;AAT/; 446$cached_env_img{$key} = q|<IMG STYLE="height: 2.04ex; vertical-align: -0.45ex; " SRC="|."$dir".q|sub_img157.png" 447 ALT="$\ell_{\mathrm{max}}\leq 4N_{\mathrm{side}}$">|; 448 449$key = q/ellge|m|,ellge|s|;MSF=1.2;LFS=12;AAT/; 450$cached_env_img{$key} = q|<IMG STYLE="height: 2.33ex; vertical-align: -0.68ex; " SRC="|."$dir".q|sub_img54.png" 451 ALT="$\ell \ge \vert m\vert, \ell \ge \vert s\vert$">|; 452 453$key = q/ellin{;MSF=1.2;LFS=12;AAT/; 454$cached_env_img{$key} = q|<IMG STYLE="height: 2.33ex; vertical-align: -0.68ex; " SRC="|."$dir".q|sub_img79.png" 455 ALT="$\ell \in \{$">|; 456 457$key = q/ellle4N_{mathrm{side}};MSF=1.2;LFS=12;AAT/; 458$cached_env_img{$key} = q|<IMG STYLE="height: 2.04ex; vertical-align: -0.45ex; " SRC="|."$dir".q|sub_img104.png" 459 ALT="$\ell\le4 N_{\mathrm{side}}$">|; 460 461$key = q/ellleq1024;MSF=1.2;LFS=12;AAT/; 462$cached_env_img{$key} = q|<IMG STYLE="height: 1.98ex; vertical-align: -0.41ex; " SRC="|."$dir".q|sub_img116.png" 463 ALT="$\ell 464\leq 1024$">|; 465 466$key = q/ellleqell_{mathrm{max}};MSF=1.2;LFS=12;AAT/; 467$cached_env_img{$key} = q|<IMG STYLE="height: 2.04ex; vertical-align: -0.45ex; " SRC="|."$dir".q|sub_img179.png" 468 ALT="$\ell \leq \ell_{\mathrm{max}}$">|; 469 470$key = q/gammasimeq0.577ldots;MSF=1.2;LFS=12;AAT/; 471$cached_env_img{$key} = q|<IMG STYLE="height: 2.10ex; vertical-align: -0.55ex; " SRC="|."$dir".q|sub_img128.png" 472 ALT="$\gamma \simeq 0.577\ldots$">|; 473 474$key = q/ge4;MSF=1.2;LFS=12;AAT/; 475$cached_env_img{$key} = q|<IMG STYLE="height: 1.92ex; vertical-align: -0.41ex; " SRC="|."$dir".q|sub_img190.png" 476 ALT="$\ge 4$">|; 477 478$key = q/htmlref{convert_nest2ring}{sub:convert_nest2ring};MSF=1.2;LFS=12;AAT/; 479$cached_env_img{$key} = q|<IMG STYLE="height: 2.04ex; vertical-align: -0.55ex; " SRC="|."$dir".q|sub_img98.png" 480 ALT="$\htmlref{convert\_nest2ring}{sub:convert_nest2ring}$">|; 481 482$key = q/htmlref{convert_ring2nest}{sub:convert_ring2nest};MSF=1.2;LFS=12;AAT/; 483$cached_env_img{$key} = q|<IMG STYLE="height: 2.04ex; vertical-align: -0.55ex; " SRC="|."$dir".q|sub_img99.png" 484 ALT="$\htmlref{convert\_ring2nest}{sub:convert_ring2nest}$">|; 485 486$key = q/index=ell^2+ell+m+1;MSF=1.2;LFS=12;AAT/; 487$cached_env_img{$key} = q|<IMG STYLE="height: 2.04ex; vertical-align: -0.28ex; " SRC="|."$dir".q|sub_img73.png" 488 ALT="$index=\ell^2+\ell+m+1$">|; 489 490$key = q/k=sum(x_i-m)^4slash(nsigma^4)-3;MSF=1.2;LFS=12;AAT/; 491$cached_env_img{$key} = q|<IMG STYLE="height: 2.45ex; vertical-align: -0.68ex; " SRC="|."$dir".q|sub_img96.png" 492 ALT="$k = \sum(x_i-m)^4 / (n\sigma^4) - 3$">|; 493 494$key = q/lambda_{00}(theta_1);MSF=1.2;LFS=12;AAT/; 495$cached_env_img{$key} = q|<IMG STYLE="height: 2.33ex; vertical-align: -0.68ex; " SRC="|."$dir".q|sub_img160.png" 496 ALT="$\lambda_{00}(\theta_1)$">|; 497 498$key = q/lambda_{00}(theta_2);MSF=1.2;LFS=12;AAT/; 499$cached_env_img{$key} = q|<IMG STYLE="height: 2.33ex; vertical-align: -0.68ex; " SRC="|."$dir".q|sub_img165.png" 500 ALT="$\lambda_{00}(\theta_2)$">|; 501 502$key = q/lambda_{10}(theta_1);MSF=1.2;LFS=12;AAT/; 503$cached_env_img{$key} = q|<IMG STYLE="height: 2.33ex; vertical-align: -0.68ex; " SRC="|."$dir".q|sub_img161.png" 504 ALT="$\lambda_{10}(\theta_1)$">|; 505 506$key = q/lambda_{11}(theta_1);MSF=1.2;LFS=12;AAT/; 507$cached_env_img{$key} = q|<IMG STYLE="height: 2.33ex; vertical-align: -0.68ex; " SRC="|."$dir".q|sub_img163.png" 508 ALT="$\lambda_{11}(\theta_1)$">|; 509 510$key = q/lambda_{20}(theta_1);MSF=1.2;LFS=12;AAT/; 511$cached_env_img{$key} = q|<IMG STYLE="height: 2.33ex; vertical-align: -0.68ex; " SRC="|."$dir".q|sub_img162.png" 512 ALT="$\lambda_{20}(\theta_1)$">|; 513 514$key = q/lambda_{21}(theta_1);MSF=1.2;LFS=12;AAT/; 515$cached_env_img{$key} = q|<IMG STYLE="height: 2.33ex; vertical-align: -0.68ex; " SRC="|."$dir".q|sub_img164.png" 516 ALT="$\lambda_{21}(\theta_1)$">|; 517 518$key = q/lambda_{ellm};MSF=1.2;LFS=12;AAT/; 519$cached_env_img{$key} = q|<IMG STYLE="height: 2.04ex; vertical-align: -0.45ex; " SRC="|."$dir".q|sub_img158.png" 520 ALT="$\lambda_{\ell m}$">|; 521 522$key = q/ldots;MSF=1.2;LFS=12;AAT/; 523$cached_env_img{$key} = q|<IMG STYLE="height: 0.23ex; vertical-align: -0.10ex; " SRC="|."$dir".q|sub_img9.png" 524 ALT="$\ldots$">|; 525 526$key = q/le4N_{mathrm{side}};MSF=1.2;LFS=12;AAT/; 527$cached_env_img{$key} = q|<IMG STYLE="height: 2.04ex; vertical-align: -0.45ex; " SRC="|."$dir".q|sub_img134.png" 528 ALT="$\le 4N_{\mathrm{side}}$">|; 529 530$key = q/le;MSF=1.2;LFS=12;AAT/; 531$cached_env_img{$key} = q|<IMG STYLE="height: 1.87ex; vertical-align: -0.41ex; " SRC="|."$dir".q|sub_img117.png" 532 ALT="$\le$">|; 533 534$key = q/left(textbf{w}.textbf{m}-textbf{S}.textbf{a}^{(n-1)}right);MSF=1.2;LFS=12;AAT/; 535$cached_env_img{$key} = q|<IMG STYLE="height: 3.56ex; vertical-align: -1.26ex; " SRC="|."$dir".q|sub_img140.png" 536 ALT="$\left(\textbf{w}.\textbf{m}- \textbf{S}.\textbf{a}^{(n-1)}\right)$">|; 537 538$key = q/left(textbf{w}.textbf{m}right);MSF=1.2;LFS=12;AAT/; 539$cached_env_img{$key} = q|<IMG STYLE="height: 2.33ex; vertical-align: -0.68ex; " SRC="|."$dir".q|sub_img139.png" 540 ALT="$\left(\textbf{w}.\textbf{m}\right)$">|; 541 542$key = q/left({partialT}slash{partialtheta},{partialQ}slash{partialtheta},{partialU}slash{partialtheta};right.;MSF=1.2;LFS=12;AAT/; 543$cached_env_img{$key} = q|<IMG STYLE="height: 2.33ex; vertical-align: -0.68ex; " SRC="|."$dir".q|sub_img47.png" 544 ALT="$\left({\partial T}/{\partial 545\theta}, {\partial Q}/{\partial \theta}, {\partial U}/{\partial \theta};\right. $">|; 546 547$key = q/left({partialT}slash{partialtheta},{partialT}slash{partialphi}slashsinthetaright);MSF=1.2;LFS=12;AAT/; 548$cached_env_img{$key} = q|<IMG STYLE="height: 2.33ex; vertical-align: -0.68ex; " SRC="|."$dir".q|sub_img46.png" 549 ALT="$\left({\partial T}/{\partial \theta}, {\partial T}/{\partial \phi}/\sin\theta \right) 550$">|; 551 552$key = q/left({partial^2T}slash{partialtheta^2},{partial^2Q}slash{partialtheta^2},{partial^2Q}slash{partialtheta^2},ldotsright);MSF=1.2;LFS=12;AAT/; 553$cached_env_img{$key} = q|<IMG STYLE="height: 2.45ex; vertical-align: -0.68ex; " SRC="|."$dir".q|sub_img51.png" 554 ALT="$\left({\partial^2 T}/{\partial \theta^2}, {\partial^2 Q}/{\partial \theta^2}, 555{\partial^2 Q}/{\partial \theta^2}, \ldots \right) $">|; 556 557$key = q/left({partial^2T}slash{partialtheta^2},{partial^2T}slash{partialthetapartialphi}slashsintheta,right.;MSF=1.2;LFS=12;AAT/; 558$cached_env_img{$key} = q|<IMG STYLE="height: 2.45ex; vertical-align: -0.68ex; " SRC="|."$dir".q|sub_img49.png" 559 ALT="$\left({\partial^2 T}/{\partial \theta^2}, {\partial^2 T}/{\partial 560\theta\partial\phi}/\sin\theta,\right. $">|; 561 562$key = q/left.{partialT}slash{partialphi}slashsintheta,ldotsright);MSF=1.2;LFS=12;AAT/; 563$cached_env_img{$key} = q|<IMG STYLE="height: 2.33ex; vertical-align: -0.68ex; " SRC="|."$dir".q|sub_img48.png" 564 ALT="$\left.{\partial T}/{\partial \phi}/\sin\theta, \ldots \right) 565$">|; 566 567$key = q/left.{partial^2T}slash{partialphi^2}slashsin^2thetaright);MSF=1.2;LFS=12;AAT/; 568$cached_env_img{$key} = q|<IMG STYLE="height: 2.45ex; vertical-align: -0.68ex; " SRC="|."$dir".q|sub_img50.png" 569 ALT="$\left.{\partial^2 T}/{\partial \phi^2}/\sin^2\theta \right) $">|; 570 571$key = q/lezle;MSF=1.2;LFS=12;AAT/; 572$cached_env_img{$key} = q|<IMG STYLE="height: 1.87ex; vertical-align: -0.41ex; " SRC="|."$dir".q|sub_img43.png" 573 ALT="$\le z \le$">|; 574 575$key = q/m=sum_ix_islashn;MSF=1.2;LFS=12;AAT/; 576$cached_env_img{$key} = q|<IMG STYLE="height: 2.33ex; vertical-align: -0.68ex; " SRC="|."$dir".q|sub_img91.png" 577 ALT="$m= \sum_i x_i / n$">|; 578 579$key = q/mathbf{phi};MSF=1.2;LFS=12;AAT/; 580$cached_env_img{$key} = q|<IMG STYLE="height: 2.10ex; vertical-align: -0.55ex; " SRC="|."$dir".q|sub_img4.png" 581 ALT="$\mathbf{\phi}$">|; 582 583$key = q/mathbf{theta};MSF=1.2;LFS=12;AAT/; 584$cached_env_img{$key} = q|<IMG STYLE="height: 1.69ex; vertical-align: -0.10ex; " SRC="|."$dir".q|sub_img2.png" 585 ALT="$\mathbf{\theta}$">|; 586 587$key = q/mge0;MSF=1.2;LFS=12;AAT/; 588$cached_env_img{$key} = q|<IMG STYLE="height: 1.92ex; vertical-align: -0.41ex; " SRC="|."$dir".q|sub_img62.png" 589 ALT="$m\ge 0$">|; 590 591$key = q/partialXslashpartialtheta;MSF=1.2;LFS=12;AAT/; 592$cached_env_img{$key} = q|<IMG STYLE="height: 2.33ex; vertical-align: -0.68ex; " SRC="|."$dir".q|sub_img13.png" 593 ALT="$\partial X/\partial\theta$">|; 594 595$key = q/partial^2Xslash(partialthetapartialphisintheta);MSF=1.2;LFS=12;AAT/; 596$cached_env_img{$key} = q|<IMG STYLE="height: 2.45ex; vertical-align: -0.68ex; " SRC="|."$dir".q|sub_img14.png" 597 ALT="$\partial^2 X/(\partial\theta\partial\phi\sin\theta)$">|; 598 599$key = q/partial^2Xslashpartialtheta^2;MSF=1.2;LFS=12;AAT/; 600$cached_env_img{$key} = q|<IMG STYLE="height: 2.45ex; vertical-align: -0.68ex; " SRC="|."$dir".q|sub_img15.png" 601 ALT="$\partial^2 X/\partial \theta^2$">|; 602 603$key = q/phi;MSF=1.2;LFS=12;AAT/; 604$cached_env_img{$key} = q|<IMG STYLE="height: 2.10ex; vertical-align: -0.55ex; " SRC="|."$dir".q|sub_img133.png" 605 ALT="$\phi$">|; 606 607$key = q/phi=0;MSF=1.2;LFS=12;AAT/; 608$cached_env_img{$key} = q|<IMG STYLE="height: 2.10ex; vertical-align: -0.55ex; " SRC="|."$dir".q|sub_img135.png" 609 ALT="$\phi=0$">|; 610 611$key = q/phiin[0,2pi[;MSF=1.2;LFS=12;AAT/; 612$cached_env_img{$key} = q|<IMG STYLE="height: 2.33ex; vertical-align: -0.68ex; " SRC="|."$dir".q|sub_img5.png" 613 ALT="$\phi\in[0,2\pi[$">|; 614 615$key = q/pi;MSF=1.2;LFS=12;AAT/; 616$cached_env_img{$key} = q|<IMG STYLE="height: 1.11ex; vertical-align: -0.10ex; " SRC="|."$dir".q|sub_img3.png" 617 ALT="$\pi$">|; 618 619$key = q/pisimeq3.14159ldots;MSF=1.2;LFS=12;AAT/; 620$cached_env_img{$key} = q|<IMG STYLE="height: 1.57ex; vertical-align: -0.10ex; " SRC="|."$dir".q|sub_img124.png" 621 ALT="$\pi \simeq 3.14159\ldots$">|; 622 623$key = q/pislash180;MSF=1.2;LFS=12;AAT/; 624$cached_env_img{$key} = q|<IMG STYLE="height: 2.33ex; vertical-align: -0.68ex; " SRC="|."$dir".q|sub_img130.png" 625 ALT="$\pi/180$">|; 626 627$key = q/pislash2;MSF=1.2;LFS=12;AAT/; 628$cached_env_img{$key} = q|<IMG STYLE="height: 2.33ex; vertical-align: -0.68ex; " SRC="|."$dir".q|sub_img123.png" 629 ALT="$\pi/2$">|; 630 631$key = q/pislash4;MSF=1.2;LFS=12;AAT/; 632$cached_env_img{$key} = q|<IMG STYLE="height: 2.33ex; vertical-align: -0.68ex; " SRC="|."$dir".q|sub_img122.png" 633 ALT="$\pi/4$">|; 634 635$key = q/pislash5;MSF=1.2;LFS=12;AAT/; 636$cached_env_img{$key} = q|<IMG STYLE="height: 2.33ex; vertical-align: -0.68ex; " SRC="|."$dir".q|sub_img166.png" 637 ALT="$\pi/5$">|; 638 639$key = q/pm;MSF=1.2;LFS=12;AAT/; 640$cached_env_img{$key} = q|<IMG STYLE="height: 1.46ex; vertical-align: -0.29ex; " SRC="|."$dir".q|sub_img136.png" 641 ALT="$\pm$">|; 642 643$key = q/propto1024;MSF=1.2;LFS=12;AAT/; 644$cached_env_img{$key} = q|<IMG STYLE="height: 1.57ex; vertical-align: -0.10ex; " SRC="|."$dir".q|sub_img198.png" 645 ALT="$\propto 1024$">|; 646 647$key = q/propto1slashsqrt{{{rm{n_obs}};MSF=1.2;LFS=12;AAT/; 648$cached_env_img{$key} = q|<IMG STYLE="height: 2.51ex; vertical-align: -0.68ex; " SRC="|."$dir".q|sub_img171.png" 649 ALT="$\propto 1/\sqrt{{\rm n\_obs}}$">|; 650 651$key = q/psi,theta,varphi;MSF=1.2;LFS=12;AAT/; 652$cached_env_img{$key} = q|<IMG STYLE="height: 2.16ex; vertical-align: -0.55ex; " SRC="|."$dir".q|sub_img100.png" 653 ALT="$\psi, \theta, \varphi 654$">|; 655 656$key = q/psi;MSF=1.2;LFS=12;AAT/; 657$cached_env_img{$key} = q|<IMG STYLE="height: 2.10ex; vertical-align: -0.55ex; " SRC="|."$dir".q|sub_img101.png" 658 ALT="$\psi$">|; 659 660$key = q/psi=pislash3,theta=0.5,varphi=0;MSF=1.2;LFS=12;AAT/; 661$cached_env_img{$key} = q|<IMG STYLE="height: 2.33ex; vertical-align: -0.68ex; " SRC="|."$dir".q|sub_img182.png" 662 ALT="$\psi=\pi/3, \theta=0.5, \varphi=0 $">|; 663 664$key = q/s=sum(x_i-m)^3slash(nsigma^3);MSF=1.2;LFS=12;AAT/; 665$cached_env_img{$key} = q|<IMG STYLE="height: 2.45ex; vertical-align: -0.68ex; " SRC="|."$dir".q|sub_img95.png" 666 ALT="$s = \sum(x_i-m)^3 / (n\sigma^3)$">|; 667 668$key = q/s_1(p)=x,s_2(p)=y,s_3(p)=z;MSF=1.2;LFS=12;AAT/; 669$cached_env_img{$key} = q|<IMG STYLE="height: 2.33ex; vertical-align: -0.68ex; " SRC="|."$dir".q|sub_img176.png" 670 ALT="$s_1(p)=x, s_2(p)=y, s_3(p)=z$">|; 671 672$key = q/sigma;MSF=1.2;LFS=12;AAT/; 673$cached_env_img{$key} = q|<IMG STYLE="height: 1.11ex; vertical-align: -0.10ex; " SRC="|."$dir".q|sub_img94.png" 674 ALT="$\sigma$">|; 675 676$key = q/sigma^2=sum(x_i-m)^2slash(n-1);MSF=1.2;LFS=12;AAT/; 677$cached_env_img{$key} = q|<IMG STYLE="height: 2.45ex; vertical-align: -0.68ex; " SRC="|."$dir".q|sub_img93.png" 678 ALT="$\sigma^2 = \sum(x_i-m)^2/ (n-1)$">|; 679 680$key = q/sigmaequivsqrt{sum_{p=1}^{N}frac{(x(p)-bar{x})^2}{N-1}};MSF=1.2;LFS=12;AAT/; 681$cached_env_img{$key} = q|<IMG STYLE="height: 3.56ex; vertical-align: -1.10ex; " SRC="|."$dir".q|sub_img141.png" 682 ALT="$\sigma \equiv \sqrt{\sum_{p=1}^{N}\frac{(x(p)-\bar{x})^2}{N-1}}$">|; 683 684$key = q/simeq1.8010^{308};MSF=1.2;LFS=12;AAT/; 685$cached_env_img{$key} = q|<IMG STYLE="height: 1.92ex; vertical-align: -0.10ex; " SRC="|."$dir".q|sub_img121.png" 686 ALT="$\simeq 1.80 10^{308}$">|; 687 688$key = q/simeq3.4010^{38};MSF=1.2;LFS=12;AAT/; 689$cached_env_img{$key} = q|<IMG STYLE="height: 1.92ex; vertical-align: -0.10ex; " SRC="|."$dir".q|sub_img120.png" 690 ALT="$\simeq 3.40 10^{38}$">|; 691 692$key = q/sqrt{2};MSF=1.2;LFS=12;AAT/; 693$cached_env_img{$key} = q|<IMG STYLE="height: 2.39ex; vertical-align: -0.43ex; " SRC="|."$dir".q|sub_img127.png" 694 ALT="$\sqrt{2}$">|; 695 696$key = q/textbf{w}^2netextbf{w};MSF=1.2;LFS=12;AAT/; 697$cached_env_img{$key} = q|<IMG STYLE="height: 2.45ex; vertical-align: -0.55ex; " SRC="|."$dir".q|sub_img143.png" 698 ALT="$\textbf{w}^2 \ne \textbf{w}$">|; 699 700$key = q/textstyle;MSF=1.2;LFS=12;AAT/; 701$cached_env_img{$key} = q|<IMG STYLE="vertical-align: 20px" 702 SRC="|."$dir".q|sub_img184.png" 703 ALT="$\textstyle $">|; 704 705$key = q/textstylemyequal;MSF=1.2;LFS=12;AAT/; 706$cached_env_img{$key} = q|<IMG STYLE="height: 0.58ex; vertical-align: -0.10ex; " SRC="|."$dir".q|sub_img58.png" 707 ALT="$\textstyle \myequal$">|; 708 709$key = q/textstyleparbox{2.2cm}{Will~return:parbox[t]{2cm}{tt{abbbbbbbbC103}}};MSF=1.2;LFS=12;AAT/; 710$cached_env_img{$key} = q|<IMG STYLE="height: 10.08ex; vertical-align: -4.53ex; " SRC="|."$dir".q|sub_img97.png" 711 ALT="$\textstyle \parbox{2.2cm}{ 712Will return: 713\parbox[t]{2cm}{\tt{a\ bbbbbbbb\ C 10 3}}}$">|; 714 715$key = q/textstyleparbox{8.2cm}{Will~print~out:hfillparbox[t]{8cm}{tt--------------------------------------------------}onabi-pro(ordualcore)computer};MSF=1.2;LFS=12;AAT/; 716$cached_env_img{$key} = q|<IMG STYLE="height: 15.33ex; vertical-align: -7.19ex; " SRC="|."$dir".q|sub_img90.png" 717 ALT="$\textstyle \parbox{8.2cm}{ 718Will print out: \hfill\\\\ 719\parbox[t]{8cm}{\tt 720---... 721... 722------------------- } 723on a bi-pro (or dual core) computer}$">|; 724 725$key = q/theta;MSF=1.2;LFS=12;AAT/; 726$cached_env_img{$key} = q|<IMG STYLE="height: 1.69ex; vertical-align: -0.10ex; " SRC="|."$dir".q|sub_img102.png" 727 ALT="$\theta$">|; 728 729$key = q/varphi;MSF=1.2;LFS=12;AAT/; 730$cached_env_img{$key} = q|<IMG STYLE="height: 1.57ex; vertical-align: -0.55ex; " SRC="|."$dir".q|sub_img103.png" 731 ALT="$\varphi$">|; 732 733$key = q/w_{mathrm{pix}}(ell);MSF=1.2;LFS=12;AAT/; 734$cached_env_img{$key} = q|<IMG STYLE="height: 2.45ex; vertical-align: -0.75ex; " SRC="|."$dir".q|sub_img154.png" 735 ALT="$w_{\mathrm{pix}}(\ell)$">|; 736 737$key = q/x=sinthetacosphi;MSF=1.2;LFS=12;AAT/; 738$cached_env_img{$key} = q|<IMG STYLE="height: 2.10ex; vertical-align: -0.55ex; " SRC="|."$dir".q|sub_img82.png" 739 ALT="$x = \sin\theta\cos\phi $">|; 740 741$key = q/y=sinthetasinphi;MSF=1.2;LFS=12;AAT/; 742$cached_env_img{$key} = q|<IMG STYLE="height: 2.10ex; vertical-align: -0.55ex; " SRC="|."$dir".q|sub_img83.png" 743 ALT="$y=\sin\theta\sin\phi $">|; 744 745$key = q/z=costheta;MSF=1.2;LFS=12;AAT/; 746$cached_env_img{$key} = q|<IMG STYLE="height: 1.69ex; vertical-align: -0.10ex; " SRC="|."$dir".q|sub_img84.png" 747 ALT="$z=\cos\theta $">|; 748 749$key = q/z=sin(mathrm{latitude})=cos(theta).;MSF=1.2;LFS=12;AAT/; 750$cached_env_img{$key} = q|<IMG STYLE="height: 2.33ex; vertical-align: -0.68ex; " SRC="|."$dir".q|sub_img42.png" 751 ALT="$z=\sin(\mathrm{latitude}) = 752\cos(\theta).$">|; 753 754$key = q/z=sin({{rm{latitude})=cos(theta);MSF=1.2;LFS=12;AAT/; 755$cached_env_img{$key} = q|<IMG STYLE="height: 2.33ex; vertical-align: -0.68ex; " SRC="|."$dir".q|sub_img88.png" 756 ALT="$z=\sin({\rm latitude}) = 757\cos(\theta)$">|; 758 759$key = q/{1,ldots,2^{28}=268435456};MSF=1.2;LFS=12;AAT/; 760$cached_env_img{$key} = q|<IMG STYLE="height: 2.45ex; vertical-align: -0.68ex; " SRC="|."$dir".q|sub_img151.png" 761 ALT="$\{1,\ldots,2^{28}=268435456\}$">|; 762 763$key = q/{1,ldots,2^{28}};MSF=1.2;LFS=12;AAT/; 764$cached_env_img{$key} = q|<IMG STYLE="height: 2.45ex; vertical-align: -0.68ex; " SRC="|."$dir".q|sub_img149.png" 765 ALT="$\{1,\ldots,2^{28}\}$">|; 766 767$key = q/{_0}S^+=T;MSF=1.2;LFS=12;AAT/; 768$cached_env_img{$key} = q|<IMG STYLE="height: 2.16ex; vertical-align: -0.45ex; " SRC="|."$dir".q|sub_img69.png" 769 ALT="$ {_0}S^+ = T$">|; 770 771$key = q/{_0}S^-=0.;MSF=1.2;LFS=12;AAT/; 772$cached_env_img{$key} = q|<IMG STYLE="height: 2.04ex; vertical-align: -0.45ex; " SRC="|."$dir".q|sub_img70.png" 773 ALT="$ {_0}S^- = 0.$">|; 774 775$key = q/{_s}Y_{ellm};MSF=1.2;LFS=12;AAT/; 776$cached_env_img{$key} = q|<IMG STYLE="height: 2.04ex; vertical-align: -0.45ex; " SRC="|."$dir".q|sub_img52.png" 777 ALT="$ {_s}Y_{\ell m}$">|; 778 779$key = q/{_s}Y_{ellm}^*=(-1)^{s+m}{_{-s}}Y_{ell-m};MSF=1.2;LFS=12;AAT/; 780$cached_env_img{$key} = q|<IMG STYLE="height: 2.39ex; vertical-align: -0.68ex; " SRC="|."$dir".q|sub_img55.png" 781 ALT="${_s}Y_{\ell m}^* = (-1)^{s+m} {_{-s}}Y_{\ell -m}$">|; 782 783$key = q/{_s}a_{ellm};MSF=1.2;LFS=12;AAT/; 784$cached_env_img{$key} = q|<IMG STYLE="height: 1.46ex; vertical-align: -0.45ex; " SRC="|."$dir".q|sub_img45.png" 785 ALT="${_s}a_{\ell m}$">|; 786 787$key = q/{_s}a_{ellm}^*=(-1)^{s+m}{_{-s}}a_{ell-m};MSF=1.2;LFS=12;AAT/; 788$cached_env_img{$key} = q|<IMG STYLE="height: 2.39ex; vertical-align: -0.68ex; " SRC="|."$dir".q|sub_img56.png" 789 ALT="${_s}a_{\ell m}^* = (-1)^{s+m} {_{-s}}a_{\ell -m}$">|; 790 791$key = q/{_{|s|}}a^+_{ellm};MSF=1.2;LFS=12;AAT/; 792$cached_env_img{$key} = q|<IMG STYLE="height: 2.68ex; vertical-align: -0.85ex; " SRC="|."$dir".q|sub_img71.png" 793 ALT="${_{\vert s\vert}}a^+_{\ell m}$">|; 794 795$key = q/{_{|s|}}a^-_{ellm};MSF=1.2;LFS=12;AAT/; 796$cached_env_img{$key} = q|<IMG STYLE="height: 2.21ex; vertical-align: -0.85ex; " SRC="|."$dir".q|sub_img72.png" 797 ALT="${_{\vert s\vert}}a^-_{\ell m}$">|; 798 799$key = q/{displaymath}N_w=frac{(N_{mathrm{side}}+1)(3N_{mathrm{side}}+1)}{4}simeqfrac{N_{mathrm{pix}}}{16}{displaymath};MSF=1.2;LFS=12;AAT/; 800$cached_env_img{$key} = q|<IMG STYLE="" 801 SRC="|."$dir".q|sub_img152.png" 802 ALT="\begin{displaymath}N_w=\frac{(N_{\mathrm{side}}+1)(3N_{\mathrm{side}}+1)}{4} \simeq \frac{N_{\mathrm{pix}}}{16}\end{displaymath}">|; 803 804$key = q/{displaymath}N_{mathrm{template}}=frac{1+N_{mathrm{side}}(N_{mathrm{side}}+6)}{4}{displaymath};MSF=1.2;LFS=12;AAT/; 805$cached_env_img{$key} = q|<IMG STYLE="" 806 SRC="|."$dir".q|sub_img153.png" 807 ALT="\begin{displaymath}N_{\mathrm{template}}=\frac{1+N_{\mathrm{side}}(N_{\mathrm{side}}+6)}{4}\end{displaymath}">|; 808 809$key = q/{{bf{v}_1;MSF=1.2;LFS=12;AAT/; 810$cached_env_img{$key} = q|<IMG STYLE="height: 1.40ex; vertical-align: -0.45ex; " SRC="|."$dir".q|sub_img191.png" 811 ALT="${\bf v}_1$">|; 812 813$key = q/{{bf{v}_2;MSF=1.2;LFS=12;AAT/; 814$cached_env_img{$key} = q|<IMG STYLE="height: 1.40ex; vertical-align: -0.45ex; " SRC="|."$dir".q|sub_img192.png" 815 ALT="${\bf v}_2$">|; 816 817$key = q/{{bf{v}_3={{bf{v}_1times{{bf{v}_2;MSF=1.2;LFS=12;AAT/; 818$cached_env_img{$key} = q|<IMG STYLE="height: 1.57ex; vertical-align: -0.45ex; " SRC="|."$dir".q|sub_img193.png" 819 ALT="${\bf v}_3 = {\bf v}_1 \times {\bf v}_2$">|; 820 821$key = q/|b|le15^o;MSF=1.2;LFS=12;AAT/; 822$cached_env_img{$key} = q|<IMG STYLE="height: 2.33ex; vertical-align: -0.68ex; " SRC="|."$dir".q|sub_img89.png" 823 ALT="$\vert b\vert \le 15^o$">|; 824 8251; 826 827