1# LaTeX2HTML 2018.3 (Released July 19, 2018) 2# Associate images original text with physical files. 3 4 5$key = q/(2ell+1)C_ell;MSF=1.2;LFS=12;AAT/; 6$cached_env_img{$key} = q|<IMG STYLE="height: 2.33ex; vertical-align: -0.68ex; " SRC="|."$dir".q|intro_img41.png" 7 ALT="$(2\ell+1)C_\ell$">|; 8 9$key = q/(textbf{e}_{1},textbf{e}_{2})=(textbf{e}_theta,textbf{e}_phi);MSF=1.2;LFS=12;AAT/; 10$cached_env_img{$key} = q|<IMG STYLE="height: 2.45ex; vertical-align: -0.75ex; " SRC="|."$dir".q|intro_img68.png" 11 ALT="$(\textbf{e}_{1}, \textbf{e}_{2})=(\textbf{e}_\theta , \textbf{e}_\phi )$">|; 12 13$key = q/,_{pm2}Y_l^m;MSF=1.2;LFS=12;AAT/; 14$cached_env_img{$key} = q|<IMG STYLE="height: 2.27ex; vertical-align: -0.67ex; " SRC="|."$dir".q|intro_img64.png" 15 ALT="$ _{\pm2}Y_l^m$">|; 16 17$key = q/-ell_{mathrm{max}};MSF=1.2;LFS=12;AAT/; 18$cached_env_img{$key} = q|<IMG STYLE="height: 2.04ex; vertical-align: -0.45ex; " SRC="|."$dir".q|intro_img19.png" 19 ALT="$-\ell_{\mathrm{max}}$">|; 20 21$key = q/-sqrt{2};MSF=1.2;LFS=12;AAT/; 22$cached_env_img{$key} = q|<IMG STYLE="height: 2.39ex; vertical-align: -0.43ex; " SRC="|."$dir".q|intro_img128.png" 23 ALT="$-\sqrt{2}$">|; 24 25$key = q/1slash(4pi);MSF=1.2;LFS=12;AAT/; 26$cached_env_img{$key} = q|<IMG STYLE="height: 2.33ex; vertical-align: -0.68ex; " SRC="|."$dir".q|intro_img38.png" 27 ALT="$1/(4\pi)$">|; 28 29$key = q/1slashOmega_{mathrm{pix}};MSF=1.2;LFS=12;AAT/; 30$cached_env_img{$key} = q|<IMG STYLE="height: 2.45ex; vertical-align: -0.75ex; " SRC="|."$dir".q|intro_img156.png" 31 ALT="$1/\Omega_{\mathrm{pix}}$">|; 32 33$key = q/2,times,2;MSF=1.2;LFS=12;AAT/; 34$cached_env_img{$key} = q|<IMG STYLE="height: 1.57ex; vertical-align: -0.29ex; " SRC="|."$dir".q|intro_img54.png" 35 ALT="$2 \times 2$">|; 36 37$key = q/2C_{ell}^{mathrm{CURL}};MSF=1.2;LFS=12;AAT/; 38$cached_env_img{$key} = q|<IMG STYLE="height: 2.56ex; vertical-align: -0.67ex; " SRC="|."$dir".q|intro_img125.png" 39 ALT="$2C_{\ell}^{\mathrm{CURL}} $">|; 40 41$key = q/2C_{ell}^{mathrm{C}};MSF=1.2;LFS=12;AAT/; 42$cached_env_img{$key} = q|<IMG STYLE="height: 2.56ex; vertical-align: -0.67ex; " SRC="|."$dir".q|intro_img124.png" 43 ALT="$2C_{\ell}^{\mathrm{C}} $">|; 44 45$key = q/2C_{ell}^{mathrm{GRAD}};MSF=1.2;LFS=12;AAT/; 46$cached_env_img{$key} = q|<IMG STYLE="height: 2.56ex; vertical-align: -0.67ex; " SRC="|."$dir".q|intro_img121.png" 47 ALT="$2C_{\ell}^{\mathrm{GRAD}} $">|; 48 49$key = q/2C_{ell}^{mathrm{G}};MSF=1.2;LFS=12;AAT/; 50$cached_env_img{$key} = q|<IMG STYLE="height: 2.56ex; vertical-align: -0.67ex; " SRC="|."$dir".q|intro_img120.png" 51 ALT="$2C_{\ell}^{\mathrm{G}} $">|; 52 53$key = q/2^{128}-1approx3.410^{38};MSF=1.2;LFS=12;AAT/; 54$cached_env_img{$key} = q|<IMG STYLE="height: 1.92ex; vertical-align: -0.29ex; " SRC="|."$dir".q|intro_img179.png" 55 ALT="$2^{128}-1 \approx 3.4 10^{38}$">|; 56 57$key = q/2ell+1;MSF=1.2;LFS=12;AAT/; 58$cached_env_img{$key} = q|<IMG STYLE="height: 1.81ex; vertical-align: -0.28ex; " SRC="|."$dir".q|intro_img42.png" 59 ALT="$2\ell+1$">|; 60 61$key = q/4p,,4p+1,,4p+2,,4p+3;MSF=1.2;LFS=12;AAT/; 62$cached_env_img{$key} = q|<IMG STYLE="height: 2.04ex; vertical-align: -0.55ex; " SRC="|."$dir".q|intro_img9.png" 63 ALT="$4p, 4p+1, 4p+2, 4p+3$">|; 64 65$key = q/4pi;MSF=1.2;LFS=12;AAT/; 66$cached_env_img{$key} = q|<IMG STYLE="height: 1.57ex; vertical-align: -0.10ex; " SRC="|."$dir".q|intro_img40.png" 67 ALT="$4\pi$">|; 68 69$key = q/4u,,4u+1,,4u+2,,4u+3;MSF=1.2;LFS=12;AAT/; 70$cached_env_img{$key} = q|<IMG STYLE="height: 2.04ex; vertical-align: -0.55ex; " SRC="|."$dir".q|intro_img10.png" 71 ALT="$4u, 4u+1, 4u+2, 4u+3$">|; 72 73$key = q/7^circ;MSF=1.2;LFS=12;AAT/; 74$cached_env_img{$key} = q|<IMG STYLE="height: 1.63ex; vertical-align: -0.10ex; " SRC="|."$dir".q|intro_img2.png" 75 ALT="$7^\circ$">|; 76 77$key = q/C_X(ell);MSF=1.2;LFS=12;AAT/; 78$cached_env_img{$key} = q|<IMG STYLE="height: 2.33ex; vertical-align: -0.68ex; " SRC="|."$dir".q|intro_img49.png" 79 ALT="$C_X(\ell)$">|; 80 81$key = q/C_ell;MSF=1.2;LFS=12;AAT/; 82$cached_env_img{$key} = q|<IMG STYLE="height: 2.04ex; vertical-align: -0.45ex; " SRC="|."$dir".q|intro_img34.png" 83 ALT="$C_\ell$">|; 84 85$key = q/C_{Cell};MSF=1.2;LFS=12;AAT/; 86$cached_env_img{$key} = q|<IMG STYLE="height: 2.04ex; vertical-align: -0.45ex; " SRC="|."$dir".q|intro_img110.png" 87 ALT="$C_{C\ell}$">|; 88 89$key = q/C_{X,ell};MSF=1.2;LFS=12;AAT/; 90$cached_env_img{$key} = q|<IMG STYLE="height: 2.27ex; vertical-align: -0.75ex; " SRC="|."$dir".q|intro_img46.png" 91 ALT="$C_{X,\ell}$">|; 92 93$key = q/C_{ell}^{mathrm{CURL}};MSF=1.2;LFS=12;AAT/; 94$cached_env_img{$key} = q|<IMG STYLE="height: 2.56ex; vertical-align: -0.67ex; " SRC="|."$dir".q|intro_img122.png" 95 ALT="$C_{\ell}^{\mathrm{CURL}} $">|; 96 97$key = q/C_{ell}^{mathrm{GRAD}};MSF=1.2;LFS=12;AAT/; 98$cached_env_img{$key} = q|<IMG STYLE="height: 2.56ex; vertical-align: -0.67ex; " SRC="|."$dir".q|intro_img118.png" 99 ALT="$C_{\ell}^{\mathrm{GRAD}} $">|; 100 101$key = q/C_{ell}^{mathrm{T-GRAD}}rule[.3cm]{0cm}{.2cm};MSF=1.2;LFS=12;AAT/; 102$cached_env_img{$key} = q|<IMG STYLE="height: 2.68ex; vertical-align: -0.68ex; " SRC="|."$dir".q|intro_img126.png" 103 ALT="$C_{\ell}^{\mathrm{T-GRAD}}\rule[.3cm]{0cm}{.2cm}$">|; 104 105$key = q/C_{ell}^{mathrm{TEMP}};MSF=1.2;LFS=12;AAT/; 106$cached_env_img{$key} = q|<IMG STYLE="height: 2.51ex; vertical-align: -0.67ex; " SRC="|."$dir".q|intro_img115.png" 107 ALT="$C_{\ell}^{\mathrm{TEMP}} $">|; 108 109$key = q/C_{ell}^{mathrm{TG}};MSF=1.2;LFS=12;AAT/; 110$cached_env_img{$key} = q|<IMG STYLE="height: 2.56ex; vertical-align: -0.67ex; " SRC="|."$dir".q|intro_img129.png" 111 ALT="$C_{\ell}^{\mathrm{TG}} $">|; 112 113$key = q/C_{ell}^{mathrm{T}};MSF=1.2;LFS=12;AAT/; 114$cached_env_img{$key} = q|<IMG STYLE="height: 2.51ex; vertical-align: -0.67ex; " SRC="|."$dir".q|intro_img117.png" 115 ALT="$C_{\ell}^{\mathrm{T}} $">|; 116 117$key = q/C_{ell}^{mathrm{pix}};MSF=1.2;LFS=12;AAT/; 118$cached_env_img{$key} = q|<IMG STYLE="height: 2.80ex; vertical-align: -0.68ex; " SRC="|."$dir".q|intro_img165.png" 119 ALT="$C_{\ell}^{\mathrm{pix}}$">|; 120 121$key = q/C_{ell}^{mathrm{unpix}};MSF=1.2;LFS=12;AAT/; 122$cached_env_img{$key} = q|<IMG STYLE="height: 2.80ex; vertical-align: -0.68ex; " SRC="|."$dir".q|intro_img166.png" 123 ALT="$C_{\ell}^{\mathrm{unpix}}$">|; 124 125$key = q/C_{mathrm{B},ell};MSF=1.2;LFS=12;AAT/; 126$cached_env_img{$key} = q|<IMG STYLE="height: 2.27ex; vertical-align: -0.75ex; " SRC="|."$dir".q|intro_img123.png" 127 ALT="$C_{\mathrm{B},\ell} $">|; 128 129$key = q/C_{mathrm{C},ell};MSF=1.2;LFS=12;AAT/; 130$cached_env_img{$key} = q|<IMG STYLE="height: 2.27ex; vertical-align: -0.75ex; " SRC="|."$dir".q|intro_img127.png" 131 ALT="$C_{\mathrm{C},\ell} $">|; 132 133$key = q/C_{mathrm{E},ell};MSF=1.2;LFS=12;AAT/; 134$cached_env_img{$key} = q|<IMG STYLE="height: 2.27ex; vertical-align: -0.75ex; " SRC="|."$dir".q|intro_img119.png" 135 ALT="$C_{\mathrm{E},\ell} $">|; 136 137$key = q/C_{mathrm{T},ell};MSF=1.2;LFS=12;AAT/; 138$cached_env_img{$key} = q|<IMG STYLE="height: 2.27ex; vertical-align: -0.75ex; " SRC="|."$dir".q|intro_img116.png" 139 ALT="$C_{\mathrm{T},\ell} $">|; 140 141$key = q/DeltaT;MSF=1.2;LFS=12;AAT/; 142$cached_env_img{$key} = q|<IMG STYLE="height: 1.69ex; vertical-align: -0.10ex; " SRC="|."$dir".q|intro_img44.png" 143 ALT="$\Delta T$">|; 144 145$key = q/DeltaTslashT;MSF=1.2;LFS=12;AAT/; 146$cached_env_img{$key} = q|<IMG STYLE="height: 2.33ex; vertical-align: -0.68ex; " SRC="|."$dir".q|intro_img43.png" 147 ALT="$\Delta T/T $">|; 148 149$key = q/Deltawslashw<1.710^{-3};MSF=1.2;LFS=12;AAT/; 150$cached_env_img{$key} = q|<IMG STYLE="height: 2.45ex; vertical-align: -0.68ex; " SRC="|."$dir".q|intro_img176.png" 151 ALT="$\Delta w/w < 1.7 10^{-3}$">|; 152 153$key = q/Deltawslashw<710^{-4};MSF=1.2;LFS=12;AAT/; 154$cached_env_img{$key} = q|<IMG STYLE="height: 2.45ex; vertical-align: -0.68ex; " SRC="|."$dir".q|intro_img174.png" 155 ALT="$\Delta w/w < 7 10^{-4}$">|; 156 157$key = q/F_{2,lm}(theta)=0;MSF=1.2;LFS=12;AAT/; 158$cached_env_img{$key} = q|<IMG STYLE="height: 2.39ex; vertical-align: -0.75ex; " SRC="|."$dir".q|intro_img89.png" 159 ALT="$F_{2,lm}(\theta)=0$">|; 160 161$key = q/N_{mathrm{pix}}sim10^7;MSF=1.2;LFS=12;AAT/; 162$cached_env_img{$key} = q|<IMG STYLE="height: 2.56ex; vertical-align: -0.75ex; " SRC="|."$dir".q|intro_img4.png" 163 ALT="$N_{\mathrm{pix}}\sim 10^7$">|; 164 165$key = q/N_{mathrm{side}}le128;MSF=1.2;LFS=12;AAT/; 166$cached_env_img{$key} = q|<IMG STYLE="height: 2.04ex; vertical-align: -0.45ex; " SRC="|."$dir".q|intro_img173.png" 167 ALT="$N_{\mathrm{side}}\le 128$">|; 168 169$key = q/Omega_{mathrm{pix}};MSF=1.2;LFS=12;AAT/; 170$cached_env_img{$key} = q|<IMG STYLE="height: 2.33ex; vertical-align: -0.75ex; " SRC="|."$dir".q|intro_img154.png" 171 ALT="$\Omega_{\mathrm{pix}}$">|; 172 173$key = q/P_ell(cosbeta)rightarrow1;MSF=1.2;LFS=12;AAT/; 174$cached_env_img{$key} = q|<IMG STYLE="height: 2.33ex; vertical-align: -0.68ex; " SRC="|."$dir".q|intro_img103.png" 175 ALT="$P_\ell(\cos\beta) \rightarrow 1$">|; 176 177$key = q/P_ell;MSF=1.2;LFS=12;AAT/; 178$cached_env_img{$key} = q|<IMG STYLE="height: 2.04ex; vertical-align: -0.45ex; " SRC="|."$dir".q|intro_img149.png" 179 ALT="$P_\ell$">|; 180 181$key = q/P_ell^2(cosbeta)rightarrowsin^2betafrac{(ell+2)!}{8(ell-2)!};MSF=1.2;LFS=12;AAT/; 182$cached_env_img{$key} = q|<IMG STYLE="height: 3.50ex; vertical-align: -1.28ex; " SRC="|."$dir".q|intro_img104.png" 183 ALT="$P_\ell^2(\cos 184\beta) \rightarrow \sin^2 \beta \frac{(\ell+2)!}{8 (\ell-2)!}$">|; 185 186$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="|."$dir".q|intro_img147.png" 188 ALT="$P_{\ell m}$">|; 189 190$key = q/QpmiU;MSF=1.2;LFS=12;AAT/; 191$cached_env_img{$key} = q|<IMG STYLE="height: 2.10ex; vertical-align: -0.55ex; " SRC="|."$dir".q|intro_img62.png" 192 ALT="$Q\pm iU$">|; 193 194$key = q/X_{1,lm}(textbf{n})=(;_2Y_{lm}+;_{-2}Y_{lm})slash2;MSF=1.2;LFS=12;AAT/; 195$cached_env_img{$key} = q|<IMG STYLE="height: 2.39ex; vertical-align: -0.75ex; " SRC="|."$dir".q|intro_img78.png" 196 ALT="$X_{1,lm}(\textbf{n})=(\;_2Y_{lm}+\;_{-2}Y_{lm})/2$">|; 197 198$key = q/X_{2,lm}(textbf{n})=(;_2Y_{lm}-;_{-2}Y_{lm})slash2;MSF=1.2;LFS=12;AAT/; 199$cached_env_img{$key} = q|<IMG STYLE="height: 2.39ex; vertical-align: -0.75ex; " SRC="|."$dir".q|intro_img79.png" 200 ALT="$X_{2,lm}(\textbf{n})=(\;_2Y_{lm}-\;_{-2}Y_{lm})/ 2$">|; 201 202$key = q/Y_{ellm};MSF=1.2;LFS=12;AAT/; 203$cached_env_img{$key} = q|<IMG STYLE="height: 2.04ex; vertical-align: -0.45ex; " SRC="|."$dir".q|intro_img11.png" 204 ALT="$Y_{\ell m}$">|; 205 206$key = q/_{pm2}a_{lm};MSF=1.2;LFS=12;AAT/; 207$cached_env_img{$key} = q|<IMG STYLE="height: 1.46ex; vertical-align: -0.61ex; " SRC="|."$dir".q|intro_img71.png" 208 ALT="$_{\pm 2}a_{lm}$">|; 209 210$key = q/a_{21}^{GRAD}=-a_{2-1}^{GRAD}=1;MSF=1.2;LFS=12;AAT/; 211$cached_env_img{$key} = q|<IMG STYLE="height: 2.56ex; vertical-align: -0.83ex; " SRC="|."$dir".q|intro_img53.png" 212 ALT="$a_{21}^{GRAD}= - a_{2-1}^{GRAD}=1$">|; 213 214$key = q/a_{21}^{TEMP}=-a_{2-1}^{TEMP}=1;MSF=1.2;LFS=12;AAT/; 215$cached_env_img{$key} = q|<IMG STYLE="height: 2.51ex; vertical-align: -0.83ex; " SRC="|."$dir".q|intro_img52.png" 216 ALT="$a_{21}^{TEMP}= - a_{2-1}^{TEMP}=1$">|; 217 218$key = q/a_{ellm};MSF=1.2;LFS=12;AAT/; 219$cached_env_img{$key} = q|<IMG STYLE="height: 1.46ex; vertical-align: -0.45ex; " SRC="|."$dir".q|intro_img26.png" 220 ALT="$a_{\ell m}$">|; 221 222$key = q/beta;MSF=1.2;LFS=12;AAT/; 223$cached_env_img{$key} = q|<IMG STYLE="height: 2.10ex; vertical-align: -0.55ex; " SRC="|."$dir".q|intro_img90.png" 224 ALT="$\beta$">|; 225 226$key = q/betarightarrow0;MSF=1.2;LFS=12;AAT/; 227$cached_env_img{$key} = q|<IMG STYLE="height: 2.10ex; vertical-align: -0.55ex; " SRC="|."$dir".q|intro_img102.png" 228 ALT="$\beta \rightarrow 0$">|; 229 230$key = q/costheta=a+bslash(pislash2-phi)^2;MSF=1.2;LFS=12;AAT/; 231$cached_env_img{$key} = q|<IMG STYLE="height: 2.45ex; vertical-align: -0.68ex; " SRC="|."$dir".q|intro_img7.png" 232 ALT="$\cos \theta = a + b / (\pi/2 - \phi) ^2 $">|; 233 234$key = q/costheta=a+bslashphi^2;MSF=1.2;LFS=12;AAT/; 235$cached_env_img{$key} = q|<IMG STYLE="height: 2.45ex; vertical-align: -0.68ex; " SRC="|."$dir".q|intro_img6.png" 236 ALT="$\cos \theta = a + b / \phi^2 $">|; 237 238$key = q/costheta=apmbcdotphi;MSF=1.2;LFS=12;AAT/; 239$cached_env_img{$key} = q|<IMG STYLE="height: 2.10ex; vertical-align: -0.55ex; " SRC="|."$dir".q|intro_img5.png" 240 ALT="$\cos \theta = a \pm b \cdot \phi $">|; 241 242$key = q/delta_{i,j};MSF=1.2;LFS=12;AAT/; 243$cached_env_img{$key} = q|<IMG STYLE="height: 2.39ex; vertical-align: -0.75ex; " SRC="|."$dir".q|intro_img75.png" 244 ALT="$\delta_{i,j}$">|; 245 246$key = q/displaystyle(-1)^m(1-x^2)^{mslash2}frac{d^m}{dx^m}P_{ell}(x),;MSF=1.2;LFS=12;AAT/; 247$cached_env_img{$key} = q|<IMG STYLE="height: 4.84ex; vertical-align: -1.69ex; " SRC="|."$dir".q|intro_img151.png" 248 ALT="$\displaystyle (-1)^m (1-x^2)^{m/2} \frac{d^m}{dx^m} P_{\ell}(x),$">|; 249 250$key = q/displaystyle(-1)^mlambda_{ell|m|},quadtextrm{for~}m<0,;MSF=1.2;LFS=12;AAT/; 251$cached_env_img{$key} = q|<IMG STYLE="height: 2.56ex; vertical-align: -0.85ex; " SRC="|."$dir".q|intro_img144.png" 252 ALT="$\displaystyle (-1)^m \lambda_{\ell \vert m\vert}, \quad\textrm{for } 253m < 0,$">|; 254 255$key = q/displaystyle(1-x^2)frac{d^2}{dx^2}P_{ellm}-2xfrac{d}{dx}P_{ellm}+left(ell(ell+1)-frac{m^2}{1-x^2}right)P_{ellm};MSF=1.2;LFS=12;AAT/; 256$cached_env_img{$key} = q|<IMG STYLE="height: 5.83ex; vertical-align: -2.42ex; " SRC="|."$dir".q|intro_img148.png" 257 ALT="$\displaystyle (1-x^2)\frac{d^2}{dx^2}P_{\ell m} - 2x \frac{d}{dx}P_{\ell m} 258+ \left(\ell(\ell+1) - \frac{m^2}{1-x^2}\right) P_{\ell m}$">|; 259 260$key = q/displaystyle-sin2psiQ+cos2psiU;MSF=1.2;LFS=12;AAT/; 261$cached_env_img{$key} = q|<IMG STYLE="height: 2.10ex; vertical-align: -0.55ex; " SRC="|."$dir".q|intro_img61.png" 262 ALT="$\displaystyle -\sin 2\psi Q + \cos 2\psi U$">|; 263 264$key = q/displaystyle-sum_l{2l+1over4pi}C_{Cl}F_{1,l0}(beta);MSF=1.2;LFS=12;AAT/; 265$cached_env_img{$key} = q|<IMG STYLE="height: 5.65ex; vertical-align: -2.66ex; " SRC="|."$dir".q|intro_img100.png" 266 ALT="$\displaystyle - \sum_l {2l+1 \over 4 \pi} C_{Cl} F_{1,l0}(\beta)$">|; 267 268$key = q/displaystyle-sum_{lm}a_{B,lm}X_{1,lm}-ia_{E,lm}X_{2,lm};MSF=1.2;LFS=12;AAT/; 269$cached_env_img{$key} = q|<IMG STYLE="height: 4.37ex; vertical-align: -2.66ex; " SRC="|."$dir".q|intro_img77.png" 270 ALT="$\displaystyle -\sum_{lm} a_{B,lm} X_{1,lm}-i a_{E,lm} X_{2,lm}$">|; 271 272$key = q/displaystyle-sum_{lm}a_{E,lm}X_{1,lm}+ia_{B,lm}X_{2,lm};MSF=1.2;LFS=12;AAT/; 273$cached_env_img{$key} = q|<IMG STYLE="height: 4.37ex; vertical-align: -2.66ex; " SRC="|."$dir".q|intro_img76.png" 274 ALT="$\displaystyle -\sum_{lm} a_{E,lm} X_{1,lm} 275+i a_{B,lm}X_{2,lm}$">|; 276 277$key = q/displaystyle0,quadtextrm{for},|m|>ell.;MSF=1.2;LFS=12;AAT/; 278$cached_env_img{$key} = q|<IMG STYLE="height: 2.33ex; vertical-align: -0.68ex; " SRC="|."$dir".q|intro_img145.png" 279 ALT="$\displaystyle 0, \quad\textrm{for} \vert m\vert > \ell.$">|; 280 281$key = q/displaystyle2^{mathrm{floor}left(log_2(uslash4)slash2right)},;MSF=1.2;LFS=12;AAT/; 282$cached_env_img{$key} = q|<IMG STYLE="height: 2.62ex; vertical-align: -0.55ex; " SRC="|."$dir".q|intro_img8.png" 283 ALT="$\displaystyle 2^{\mathrm{floor}\left( \log_2(u/4)/2 \right)},$">|; 284 285$key = q/displaystyle2sqrt{(l-2)!(l-m)!over(l+2)!(l+m)!}.;MSF=1.2;LFS=12;AAT/; 286$cached_env_img{$key} = q|<IMG STYLE="height: 7.05ex; vertical-align: -2.81ex; " SRC="|."$dir".q|intro_img88.png" 287 ALT="$\displaystyle 2 \sqrt{(l-2)!(l-m)! \over (l+2)!(l+m)!}.$">|; 288 289$key = q/displaystyleC_{ell}^mathrm{pix};MSF=1.2;LFS=12;AAT/; 290$cached_env_img{$key} = q|<IMG STYLE="height: 2.80ex; vertical-align: -0.68ex; " SRC="|."$dir".q|intro_img167.png" 291 ALT="$\displaystyle C_{\ell}^\mathrm{pix}$">|; 292 293$key = q/displaystyleD_{X,ell};MSF=1.2;LFS=12;AAT/; 294$cached_env_img{$key} = q|<IMG STYLE="height: 2.27ex; vertical-align: -0.75ex; " SRC="|."$dir".q|intro_img47.png" 295 ALT="$\displaystyle D_{X,\ell}$">|; 296 297$key = q/displaystyleF_{1,lm}(theta);MSF=1.2;LFS=12;AAT/; 298$cached_env_img{$key} = q|<IMG STYLE="height: 2.39ex; vertical-align: -0.75ex; " SRC="|."$dir".q|intro_img83.png" 299 ALT="$\displaystyle F_{1,lm}(\theta)$">|; 300 301$key = q/displaystyleF_{2,lm}(theta);MSF=1.2;LFS=12;AAT/; 302$cached_env_img{$key} = q|<IMG STYLE="height: 2.39ex; vertical-align: -0.75ex; " SRC="|."$dir".q|intro_img85.png" 303 ALT="$\displaystyle F_{2,lm}(\theta)$">|; 304 305$key = q/displaystyleM_{ellm};MSF=1.2;LFS=12;AAT/; 306$cached_env_img{$key} = q|<IMG STYLE="height: 2.04ex; vertical-align: -0.45ex; " SRC="|."$dir".q|intro_img131.png" 307 ALT="$\displaystyle M_{\ell m}$">|; 308 309$key = q/displaystyleN_{lm}(theta);MSF=1.2;LFS=12;AAT/; 310$cached_env_img{$key} = q|<IMG STYLE="height: 2.33ex; vertical-align: -0.68ex; " SRC="|."$dir".q|intro_img87.png" 311 ALT="$\displaystyle N_{lm}(\theta)$">|; 312 313$key = q/displaystyleN_{lm}left[-left({l-m^2oversin^2theta}+{1over2}l(l-1)right)P_l^m(costheta)+(l+m){costhetaoversin^2theta}P_{l-1}^m(costheta)right];MSF=1.2;LFS=12;AAT/; 314$cached_env_img{$key} = q|<IMG STYLE="height: 5.83ex; vertical-align: -2.42ex; " SRC="|."$dir".q|intro_img84.png" 315 ALT="$\displaystyle N_{lm} 316\left[ -\left({l-m^2 \over \sin^2\theta} 317+{1 \over 2}l(l-... 318... \theta) 319+(l+m) {\cos \theta \over \sin^2 \theta} 320P_{l-1}^m(\cos\theta)\right]$">|; 321 322$key = q/displaystyleN_{lm}{moversin^2theta}[-(l-1)costhetaP_l^m(costheta)+(l+m)P_{l-1}^m(costheta)],;MSF=1.2;LFS=12;AAT/; 323$cached_env_img{$key} = q|<IMG STYLE="height: 4.25ex; vertical-align: -1.69ex; " SRC="|."$dir".q|intro_img86.png" 324 ALT="$\displaystyle N_{lm}{m \over 325\sin^2 \theta} 326[ -(l-1)\cos \theta P_l^m(\cos \theta)+(l+m) P_{l-1}^m(\cos\theta)],$">|; 327 328$key = q/displaystyleP_{ellm};MSF=1.2;LFS=12;AAT/; 329$cached_env_img{$key} = q|<IMG STYLE="height: 2.04ex; vertical-align: -0.45ex; " SRC="|."$dir".q|intro_img150.png" 330 ALT="$\displaystyle P_{\ell m}$">|; 331 332$key = q/displaystyleP_{ell}(x);MSF=1.2;LFS=12;AAT/; 333$cached_env_img{$key} = q|<IMG STYLE="height: 2.33ex; vertical-align: -0.68ex; " SRC="|."$dir".q|intro_img152.png" 334 ALT="$\displaystyle P_{\ell}(x)$">|; 335 336$key = q/displaystyleQ^{prime};MSF=1.2;LFS=12;AAT/; 337$cached_env_img{$key} = q|<IMG STYLE="height: 2.33ex; vertical-align: -0.55ex; " SRC="|."$dir".q|intro_img58.png" 338 ALT="$\displaystyle Q^{\prime}$">|; 339 340$key = q/displaystyleU^{prime};MSF=1.2;LFS=12;AAT/; 341$cached_env_img{$key} = q|<IMG STYLE="height: 1.87ex; vertical-align: -0.10ex; " SRC="|."$dir".q|intro_img60.png" 342 ALT="$\displaystyle U^{\prime}$">|; 343 344$key = q/displaystyleY_{ellm}(theta,phi);MSF=1.2;LFS=12;AAT/; 345$cached_env_img{$key} = q|<IMG STYLE="height: 2.33ex; vertical-align: -0.68ex; " SRC="|."$dir".q|intro_img139.png" 346 ALT="$\displaystyle Y_{\ell m}(\theta,\phi)$">|; 347 348$key = q/displaystylecos2psiQ+sin2psiU;MSF=1.2;LFS=12;AAT/; 349$cached_env_img{$key} = q|<IMG STYLE="height: 2.10ex; vertical-align: -0.55ex; " SRC="|."$dir".q|intro_img59.png" 350 ALT="$\displaystyle \cos 2\psi Q + \sin 2\psi U$">|; 351 352$key = q/displaystyledelta_{m,m^prime}C_{Xl}quadlanglea_{T,lm}^{*}a_{E,lm}rangle=delta_{m,m^prime}C_{Cl},;MSF=1.2;LFS=12;AAT/; 353$cached_env_img{$key} = q|<IMG STYLE="height: 2.62ex; vertical-align: -0.97ex; " SRC="|."$dir".q|intro_img73.png" 354 ALT="$\displaystyle \delta_{m,m^\prime}C_{Xl} 355\quad 356\langle a_{T,lm}^{*}a_{E,lm}\rangle=\delta_{m,m^\prime}C_{Cl},$">|; 357 358$key = q/displaystylef({gamma});MSF=1.2;LFS=12;AAT/; 359$cached_env_img{$key} = q|<IMG STYLE="height: 2.33ex; vertical-align: -0.68ex; " SRC="|."$dir".q|intro_img12.png" 360 ALT="$\displaystyle f({ \gamma})$">|; 361 362$key = q/displaystylefrac{1}{2^ellell!}frac{d^ell}{dx^ell}(x^2-1)^ell.;MSF=1.2;LFS=12;AAT/; 363$cached_env_img{$key} = q|<IMG STYLE="height: 5.13ex; vertical-align: -1.69ex; " SRC="|."$dir".q|intro_img153.png" 364 ALT="$\displaystyle \frac{1}{2^\ell \ell!}\frac{d^\ell}{dx^\ell} (x^2-1)^\ell.$">|; 365 366$key = q/displaystylefrac{1}{2ell+1}sum_{m}verthat{a}_{ellm}vert^2.;MSF=1.2;LFS=12;AAT/; 367$cached_env_img{$key} = q|<IMG STYLE="height: 5.30ex; vertical-align: -2.42ex; " SRC="|."$dir".q|intro_img32.png" 368 ALT="$\displaystyle \frac{1}{2\ell +1}\sum_{m} \vert\hat{a}_{\ell m}\vert^2.$">|; 369 370$key = q/displaystylefrac{4pi}{N_{mathrm{pix}}}sum_{p=0}^{N_{mathrm{pix}}-1}Y^ast_{ellm}(gamma_p)f(gamma_p),;MSF=1.2;LFS=12;AAT/; 371$cached_env_img{$key} = q|<IMG STYLE="height: 6.47ex; vertical-align: -2.88ex; " SRC="|."$dir".q|intro_img28.png" 372 ALT="$\displaystyle \frac{4\pi}{N_{\mathrm{pix}}}\sum_{p=0}^{N_{\mathrm{pix}}-1} 373Y^\ast_{\ell m}(\gamma_p) f(\gamma_p),$">|; 374 375$key = q/displaystylefrac{ell(ell+1)}{(2pi)T_{CMB}^2}C_{X,ell},;MSF=1.2;LFS=12;AAT/; 376$cached_env_img{$key} = q|<IMG STYLE="height: 5.54ex; vertical-align: -2.27ex; " SRC="|."$dir".q|intro_img48.png" 377 ALT="$\displaystyle \frac{\ell(\ell+1)}{(2\pi)T_{CMB}^2}C_{X,\ell},$">|; 378 379$key = q/displaystylehat{C}_ell;MSF=1.2;LFS=12;AAT/; 380$cached_env_img{$key} = q|<IMG STYLE="height: 2.62ex; vertical-align: -0.45ex; " SRC="|."$dir".q|intro_img31.png" 381 ALT="$\displaystyle \hat{C}_\ell$">|; 382 383$key = q/displaystylehat{a}_{ellm};MSF=1.2;LFS=12;AAT/; 384$cached_env_img{$key} = q|<IMG STYLE="height: 2.04ex; vertical-align: -0.45ex; " SRC="|."$dir".q|intro_img27.png" 385 ALT="$\displaystyle \hat{a}_{\ell m}$">|; 386 387$key = q/displaystyleintd{u{w_p({u{)Y_{ellm}({u{),;MSF=1.2;LFS=12;AAT/; 388$cached_env_img{$key} = q|<IMG STYLE="height: 4.31ex; vertical-align: -1.67ex; " SRC="|."$dir".q|intro_img160.png" 389 ALT="$\displaystyle \int d\u w_p(\u ) Y_{\ell m}(\u ),$">|; 390 391$key = q/displaystyleintd{u{w_p({u{)f({u{);MSF=1.2;LFS=12;AAT/; 392$cached_env_img{$key} = q|<IMG STYLE="height: 4.31ex; vertical-align: -1.67ex; " SRC="|."$dir".q|intro_img155.png" 393 ALT="$\displaystyle \int d\u w_p(\u )f(\u )$">|; 394 395$key = q/displaystylelambda_{ellm}(costheta)e^{{i}mphi};MSF=1.2;LFS=12;AAT/; 396$cached_env_img{$key} = q|<IMG STYLE="height: 2.62ex; vertical-align: -0.68ex; " SRC="|."$dir".q|intro_img140.png" 397 ALT="$\displaystyle \lambda_{\ell m}(\cos\theta) e^{{i} 398m\phi}$">|; 399 400$key = q/displaystylelambda_{ellm}(x);MSF=1.2;LFS=12;AAT/; 401$cached_env_img{$key} = q|<IMG STYLE="height: 2.33ex; vertical-align: -0.68ex; " SRC="|."$dir".q|intro_img141.png" 402 ALT="$\displaystyle \lambda_{\ell m}(x)$">|; 403 404$key = q/displaystylelambda_{ellm};MSF=1.2;LFS=12;AAT/; 405$cached_env_img{$key} = q|<IMG STYLE="height: 2.04ex; vertical-align: -0.45ex; " SRC="|."$dir".q|intro_img143.png" 406 ALT="$\displaystyle \lambda_{\ell m}$">|; 407 408$key = q/displaystylelangleQQrangle+langleUUrangle;MSF=1.2;LFS=12;AAT/; 409$cached_env_img{$key} = q|<IMG STYLE="height: 2.33ex; vertical-align: -0.68ex; " SRC="|."$dir".q|intro_img107.png" 410 ALT="$\displaystyle \langle QQ \rangle + \langle UU\rangle$">|; 411 412$key = q/displaystylelangleQ_{r}(1)Q_{r}(2)rangle;MSF=1.2;LFS=12;AAT/; 413$cached_env_img{$key} = q|<IMG STYLE="height: 2.33ex; vertical-align: -0.68ex; " SRC="|."$dir".q|intro_img95.png" 414 ALT="$\displaystyle \langle Q_{r}(1)Q_{r}(2) \rangle$">|; 415 416$key = q/displaystylelangleT(1)Q_{r}(2)rangle;MSF=1.2;LFS=12;AAT/; 417$cached_env_img{$key} = q|<IMG STYLE="height: 2.33ex; vertical-align: -0.68ex; " SRC="|."$dir".q|intro_img99.png" 418 ALT="$\displaystyle \langle T(1)Q_{r}(2) 419\rangle$">|; 420 421$key = q/displaystylelangleT(1)U_{r}(2)rangle;MSF=1.2;LFS=12;AAT/; 422$cached_env_img{$key} = q|<IMG STYLE="height: 2.33ex; vertical-align: -0.68ex; " SRC="|."$dir".q|intro_img101.png" 423 ALT="$\displaystyle \langle T(1)U_{r}(2) \rangle$">|; 424 425$key = q/displaystylelangleTQrangle=langleTUrangle;MSF=1.2;LFS=12;AAT/; 426$cached_env_img{$key} = q|<IMG STYLE="height: 2.33ex; vertical-align: -0.68ex; " SRC="|."$dir".q|intro_img109.png" 427 ALT="$\displaystyle \langle TQ\rangle = \langle TU\rangle$">|; 428 429$key = q/displaystylelangleTTrangle;MSF=1.2;LFS=12;AAT/; 430$cached_env_img{$key} = q|<IMG STYLE="height: 2.33ex; vertical-align: -0.68ex; " SRC="|."$dir".q|intro_img105.png" 431 ALT="$\displaystyle \langle TT \rangle$">|; 432 433$key = q/displaystylelangleT_1T_2rangle;MSF=1.2;LFS=12;AAT/; 434$cached_env_img{$key} = q|<IMG STYLE="height: 2.33ex; vertical-align: -0.68ex; " SRC="|."$dir".q|intro_img93.png" 435 ALT="$\displaystyle \langle T_1T_2 \rangle$">|; 436 437$key = q/displaystylelangleU_{r}(1)U_{r}(2)rangle;MSF=1.2;LFS=12;AAT/; 438$cached_env_img{$key} = q|<IMG STYLE="height: 2.33ex; vertical-align: -0.68ex; " SRC="|."$dir".q|intro_img97.png" 439 ALT="$\displaystyle \langle U_{r}(1)U_{r}(2) \rangle$">|; 440 441$key = q/displaystylelanglea_{X,lm}^{*}a_{X,lm^prime}rangle;MSF=1.2;LFS=12;AAT/; 442$cached_env_img{$key} = q|<IMG STYLE="height: 2.62ex; vertical-align: -0.97ex; " SRC="|."$dir".q|intro_img72.png" 443 ALT="$\displaystyle \langle a_{X,lm}^{*} 444a_{X,lm^\prime}\rangle$">|; 445 446$key = q/displaystyleleft(array{{cc}X_{1,ellm}<col_mark>iX_{2,ellm}<row_mark>-iX_{2,ellm}<col_mark>X_{1,ellm}array{right);MSF=1.2;LFS=12;AAT/; 447$cached_env_img{$key} = q|<IMG STYLE="height: 5.83ex; vertical-align: -2.42ex; " SRC="|."$dir".q|intro_img132.png" 448 ALT="$\displaystyle \left( 449\begin{array}{cc} X_{1,\ell m} & i X_{2,\ell m} \\\\ 450-i X_{2,\ell m} & X_{1,\ell m} 451\end{array} \right)$">|; 452 453$key = q/displaystyleleft(array{{c}Q'<row_mark>U'array{right)=left(array{{cc}cos2psi<col_ight)left(array{{c}Q<row_mark>Uarray{right),<comment_mark>138;MSF=1.2;LFS=12;AAT/; 454$cached_env_img{$key} = q|<IMG STYLE="height: 5.83ex; vertical-align: -2.42ex; " SRC="|."$dir".q|intro_img111.png" 455 ALT="$\displaystyle \left(\begin{array}{c} 456Q'\ U' 457\end{array}\right) = 458\left(\be... 459...array} \right) 460\left(\begin{array}{c} 461Q\ U 462\end{array} \right),%\nonumber 463$">|; 464 465$key = q/displaystyleleft(array{{c}a_{E,ellm}'<row_mark>a_{B,ellm}'array{right)=left(arra}a_{E,ellm}<row_mark>a_{B,ellm}array{right).<comment_mark>139;MSF=1.2;LFS=12;AAT/; 466$cached_env_img{$key} = q|<IMG STYLE="height: 5.83ex; vertical-align: -2.42ex; " SRC="|."$dir".q|intro_img112.png" 467 ALT="$\displaystyle \left(\begin{array}{c} 468a_{E,\ell m}'\ a_{B,\ell m}' 469\end{arra... 470...\begin{array}{c} 471a_{E,\ell m}\ a_{B,\ell m} 472\end{array} \right).%\nonumber 473$">|; 474 475$key = q/displaystyleleft(frac{1}{N_{mathrm{pix}}}sum_{p=0}^{N_{mathrm{pix}}-1}w^2_{ell}(p)right)^{1slash2}.;MSF=1.2;LFS=12;AAT/; 476$cached_env_img{$key} = q|<IMG STYLE="height: 7.58ex; vertical-align: -3.00ex; " SRC="|."$dir".q|intro_img171.png" 477 ALT="$\displaystyle \left(\frac{1}{N_{\mathrm{pix}}}\sum_{p=0}^{N_{\mathrm{pix}}-1} w^2_{\ell}(p)\right)^{1/2}.$">|; 478 479$key = q/displaystyleleft(frac{4pi}{2ell+1}sum_{m=-ell}^{ell}left|w_{ellm}(p)right|^2right)^{1slash2},;MSF=1.2;LFS=12;AAT/; 480$cached_env_img{$key} = q|<IMG STYLE="height: 7.58ex; vertical-align: -3.00ex; " SRC="|."$dir".q|intro_img164.png" 481 ALT="$\displaystyle \left(\frac{4 \pi}{2\ell+1}\sum_{m=-\ell}^{\ell} \left\vert w_{\ell m}(p)\right\vert^2\right)^{1/2},$">|; 482 483$key = q/displaystylephantom{1.1}{left(array{{c}Qrule[.3cm]{0cm}{.2cm}rule[-.3cm]{0cm}{.2ark>Urule[.3cm]{0cm}{.2cm}rule[-.3cm]{0cm}{.2cm}array{right)};MSF=1.2;LFS=12;AAT/; 484$cached_env_img{$key} = q|<IMG STYLE="height: 8.22ex; vertical-align: -3.92ex; " SRC="|."$dir".q|intro_img137.png" 485 ALT="$\displaystyle \phantom{1.1}{ 486\left( 487\begin{array}{c} Q \rule[.3cm]{0cm}{.2cm}\... 488...}{.2cm}\ U \rule[.3cm]{0cm}{.2cm}\rule[-.3cm]{0cm}{.2cm}\end{array}\right) 489}$">|; 490 491$key = q/displaystylephantom{1.2}{left(array{{c}Qrule[.3cm]{0cm}{.2cm}rule[-.3cm]{0cm}{.2ark>Urule[.3cm]{0cm}{.2cm}rule[-.3cm]{0cm}{.2cm}array{right)};MSF=1.2;LFS=12;AAT/; 492$cached_env_img{$key} = q|<IMG STYLE="height: 8.22ex; vertical-align: -3.92ex; " SRC="|."$dir".q|intro_img133.png" 493 ALT="$\displaystyle \phantom{1.2}{ 494\left( 495\begin{array}{c} Q \rule[.3cm]{0cm}{.2cm}\... 496...}{.2cm}\ U \rule[.3cm]{0cm}{.2cm}\rule[-.3cm]{0cm}{.2cm}\end{array}\right) 497}$">|; 498 499$key = q/displaystylesqrt{2l+1over4pi};_{s_2}Y_{l-s_1}(beta,psi_1)e^{-is_2psi_2};MSF=1.2;LFS=12;AAT/; 500$cached_env_img{$key} = q|<IMG STYLE="height: 5.83ex; vertical-align: -2.00ex; " SRC="|."$dir".q|intro_img92.png" 501 ALT="$\displaystyle \sqrt{2l+1 \over 4 \pi} 502\;_{s_2} Y_{l-s_1}(\beta,\psi_1)e^{-is_2\psi_2}$">|; 503 504$key = q/displaystylesqrt{frac{2ell+1}{4pi}frac{(ell-m)!}{(ell+m)!}}P_{ellm}(x),quadtextrm{for~}mge0;MSF=1.2;LFS=12;AAT/; 505$cached_env_img{$key} = q|<IMG STYLE="height: 7.05ex; vertical-align: -2.81ex; " SRC="|."$dir".q|intro_img142.png" 506 ALT="$\displaystyle \sqrt{ \frac{2\ell+1}{4\pi} 507\frac{(\ell-m)!}{(\ell+m)!} } P_{\ell m}(x), \quad\textrm{for } 508m\ge 0$">|; 509 510$key = q/displaystylesum_ell{2ell+1over4pi}C_{Tell};MSF=1.2;LFS=12;AAT/; 511$cached_env_img{$key} = q|<IMG STYLE="height: 5.65ex; vertical-align: -2.66ex; " SRC="|."$dir".q|intro_img106.png" 512 ALT="$\displaystyle \sum_\ell {2\ell+1 \over 4 \pi} 513C_{T\ell}$">|; 514 515$key = q/displaystylesum_l{2ell+1over4pi}left(C_{Eell}+C_{Bell}right);MSF=1.2;LFS=12;AAT/; 516$cached_env_img{$key} = q|<IMG STYLE="height: 5.65ex; vertical-align: -2.66ex; " SRC="|."$dir".q|intro_img108.png" 517 ALT="$\displaystyle \sum_l {2\ell+1 \over 4 \pi} \left(C_{E\ell} 518+C_{B\ell}\right)$">|; 519 520$key = q/displaystylesum_l{2l+1over4pi}C_{Tl}P_l(cosbeta);MSF=1.2;LFS=12;AAT/; 521$cached_env_img{$key} = q|<IMG STYLE="height: 5.65ex; vertical-align: -2.66ex; " SRC="|."$dir".q|intro_img94.png" 522 ALT="$\displaystyle \sum_l {2l+1 \over 4 \pi} 523C_{Tl} P_l(\cos \beta)$">|; 524 525$key = q/displaystylesum_l{2l+1over4pi}[C_{Bl}F_{1,l2}(beta)-C_{El}F_{2,l2}(beta)];MSF=1.2;LFS=12;AAT/; 526$cached_env_img{$key} = q|<IMG STYLE="height: 5.65ex; vertical-align: -2.66ex; " SRC="|."$dir".q|intro_img98.png" 527 ALT="$\displaystyle \sum_l {2l+1 \over 4 \pi} 528[C_{Bl} F_{1,l2}(\beta)-C_{El} F_{2,l2}(\beta) ]$">|; 529 530$key = q/displaystylesum_l{2l+1over4pi}[C_{El}F_{1,l2}(beta)-C_{Bl}F_{2,l2}(beta)];MSF=1.2;LFS=12;AAT/; 531$cached_env_img{$key} = q|<IMG STYLE="height: 5.65ex; vertical-align: -2.66ex; " SRC="|."$dir".q|intro_img96.png" 532 ALT="$\displaystyle \sum_l {2l+1 \over 4 \pi} [C_{El} 533F_{1,l2}(\beta)-C_{Bl} F_{2,l2}(\beta)]$">|; 534 535$key = q/displaystylesum_m;_{s_1}Y_{lm}^*(textbf{n}_1);_{s_2}Y_{lm}(textbf{n}_2);MSF=1.2;LFS=12;AAT/; 536$cached_env_img{$key} = q|<IMG STYLE="height: 4.14ex; vertical-align: -2.42ex; " SRC="|."$dir".q|intro_img91.png" 537 ALT="$\displaystyle \sum_m \;_{s_1} Y_{lm}^*(\textbf{n}_1) 538\;_{s_2} Y_{lm}(\textbf{n}_2)$">|; 539 540$key = q/displaystylesum_{ell=0}^{ell_{mathrm{max}}}sum_{m}a_{ellm}Y_{ellm}(gamma),;MSF=1.2;LFS=12;AAT/; 541$cached_env_img{$key} = q|<IMG STYLE="height: 6.06ex; vertical-align: -2.66ex; " SRC="|."$dir".q|intro_img14.png" 542 ALT="$\displaystyle \sum_{\ell =0}^{\ell_{\mathrm{max}}}\sum_{m}a_{\ell m}Y_{\ell m}(\gamma),$">|; 543 544$key = q/displaystylesum_{ell=0}^{ell_{mathrm{max}}}sum_{m}a_{ellm}w_{ellm}(p),;MSF=1.2;LFS=12;AAT/; 545$cached_env_img{$key} = q|<IMG STYLE="height: 6.06ex; vertical-align: -2.66ex; " SRC="|."$dir".q|intro_img158.png" 546 ALT="$\displaystyle \sum_{\ell =0}^{\ell_{\mathrm{max}}}\sum_{m}a_{\ell m}w_{\ell m}(p),$">|; 547 548$key = q/displaystylesum_{ellm}M_{ellm}{left(array{{c}-a_{ellm}^{mathrm{GRAD}}rule[.3cm]{RL}}rule[.3cm]{0cm}{.2cm}rule[-.3cm]{0cm}{.2cm}array{right)}.;MSF=1.2;LFS=12;AAT/; 549$cached_env_img{$key} = q|<IMG STYLE="height: 8.22ex; vertical-align: -3.92ex; " SRC="|."$dir".q|intro_img134.png" 550 ALT="$\displaystyle \sum_{\ell m} M_{\ell m} { 551\left( 552\begin{array}{c} -a_{\ell m}^{... 553...thrm{CURL}} \rule[.3cm]{0cm}{.2cm}\rule[-.3cm]{0cm}{.2cm}\end{array}\right) 554}.$">|; 555 556$key = q/displaystylesum_{ellm}M_{ellm}{left(array{{c}-sqrt{2}a_{ellm}^{mathrm{GRAD}}ruleRL}}rule[.3cm]{0cm}{.2cm}rule[-.3cm]{0cm}{.2cm}array{right)}.;MSF=1.2;LFS=12;AAT/; 557$cached_env_img{$key} = q|<IMG STYLE="height: 8.22ex; vertical-align: -3.92ex; " SRC="|."$dir".q|intro_img138.png" 558 ALT="$\displaystyle \sum_{\ell m} M_{\ell m} { 559\left( 560\begin{array}{c} -\sqrt{2}a_{\... 561...thrm{CURL}} \rule[.3cm]{0cm}{.2cm}\rule[-.3cm]{0cm}{.2cm}\end{array}\right) 562}.$">|; 563 564$key = q/displaystylesum_{ellm}M_{ellm}{left(array{{c}sqrt{2}a_{mathrm{E},ellm}rule[.3cm]llm}rule[.3cm]{0cm}{.2cm}rule[-.3cm]{0cm}{.2cm}array{right)},;MSF=1.2;LFS=12;AAT/; 565$cached_env_img{$key} = q|<IMG STYLE="height: 8.22ex; vertical-align: -3.92ex; " SRC="|."$dir".q|intro_img136.png" 566 ALT="$\displaystyle \sum_{\ell m} M_{\ell m} { 567\left( 568\begin{array}{c} \sqrt{2}a_{\m... 569...{B},\ell m} \rule[.3cm]{0cm}{.2cm}\rule[-.3cm]{0cm}{.2cm}\end{array}\right) 570},$">|; 571 572$key = q/displaystylesum_{lm}a_{-2,lm};_{-2}Y_{lm}(textbf{n}).;MSF=1.2;LFS=12;AAT/; 573$cached_env_img{$key} = q|<IMG STYLE="height: 4.37ex; vertical-align: -2.66ex; " SRC="|."$dir".q|intro_img67.png" 574 ALT="$\displaystyle \sum_{lm} 575a_{-2,lm}\;_{-2}Y_{lm}(\textbf{n}).$">|; 576 577$key = q/displaystylesum_{lm}a_{2,lm};_2Y_{lm}(textbf{n});MSF=1.2;LFS=12;AAT/; 578$cached_env_img{$key} = q|<IMG STYLE="height: 4.37ex; vertical-align: -2.66ex; " SRC="|."$dir".q|intro_img66.png" 579 ALT="$\displaystyle \sum_{lm} 580a_{2,lm}\;_2Y_{lm}(\textbf{n})$">|; 581 582$key = q/displaystylesum_{lm}a_{T,lm}Y_{lm}(textbf{n});MSF=1.2;LFS=12;AAT/; 583$cached_env_img{$key} = q|<IMG STYLE="height: 4.37ex; vertical-align: -2.66ex; " SRC="|."$dir".q|intro_img65.png" 584 ALT="$\displaystyle \sum_{lm} a_{T,lm} Y_{lm}(\textbf{n})$">|; 585 586$key = q/displaystylew^2_{ell}C_{ell}^mathrm{unpix};MSF=1.2;LFS=12;AAT/; 587$cached_env_img{$key} = q|<IMG STYLE="height: 2.80ex; vertical-align: -0.68ex; " SRC="|."$dir".q|intro_img168.png" 588 ALT="$\displaystyle w^2_{\ell} C_{\ell}^\mathrm{unpix}$">|; 589 590$key = q/displaystylew_ell(p)Y_{ellm}(p);MSF=1.2;LFS=12;AAT/; 591$cached_env_img{$key} = q|<IMG STYLE="height: 2.33ex; vertical-align: -0.68ex; " SRC="|."$dir".q|intro_img162.png" 592 ALT="$\displaystyle w_\ell(p) Y_{\ell m}(p)$">|; 593 594$key = q/displaystylew_{ellm}(p);MSF=1.2;LFS=12;AAT/; 595$cached_env_img{$key} = q|<IMG STYLE="height: 2.33ex; vertical-align: -0.68ex; " SRC="|."$dir".q|intro_img159.png" 596 ALT="$\displaystyle w_{\ell m}(p)$">|; 597 598$key = q/displaystylew_{ell}(p);MSF=1.2;LFS=12;AAT/; 599$cached_env_img{$key} = q|<IMG STYLE="height: 2.33ex; vertical-align: -0.68ex; " SRC="|."$dir".q|intro_img163.png" 600 ALT="$\displaystyle w_{\ell}(p)$">|; 601 602$key = q/displaystylew_{ell};MSF=1.2;LFS=12;AAT/; 603$cached_env_img{$key} = q|<IMG STYLE="height: 1.46ex; vertical-align: -0.45ex; " SRC="|."$dir".q|intro_img170.png" 604 ALT="$\displaystyle w_{\ell}$">|; 605 606$key = q/displaystyle{left(array{{c}Qrule[.3cm]{0cm}{.2cm}rule[-.3cm]{0cm}{.2cm}<row_mark>-Urule[.3cm]{0cm}{.2cm}rule[-.3cm]{0cm}{.2cm}array{right)};MSF=1.2;LFS=12;AAT/; 607$cached_env_img{$key} = q|<IMG STYLE="height: 8.22ex; vertical-align: -3.92ex; " SRC="|."$dir".q|intro_img135.png" 608 ALT="$\displaystyle { 609\left( 610\begin{array}{c} Q \rule[.3cm]{0cm}{.2cm}\rule[-.3cm]{0cm}{.2cm}\ -U \rule[.3cm]{0cm}{.2cm}\rule[-.3cm]{0cm}{.2cm}\end{array}\right) 611}$">|; 612 613$key = q/ell;MSF=1.2;LFS=12;AAT/; 614$cached_env_img{$key} = q|<IMG STYLE="height: 1.69ex; vertical-align: -0.10ex; " SRC="|."$dir".q|intro_img36.png" 615 ALT="$\ell$">|; 616 617$key = q/ell=2N_{mathrm{side}};MSF=1.2;LFS=12;AAT/; 618$cached_env_img{$key} = q|<IMG STYLE="height: 2.04ex; vertical-align: -0.45ex; " SRC="|."$dir".q|intro_img175.png" 619 ALT="$\ell=2N_{\mathrm{side}}$">|; 620 621$key = q/ell=4N_{mathrm{side}};MSF=1.2;LFS=12;AAT/; 622$cached_env_img{$key} = q|<IMG STYLE="height: 2.04ex; vertical-align: -0.45ex; " SRC="|."$dir".q|intro_img177.png" 623 ALT="$\ell=4N_{\mathrm{side}}$">|; 624 625$key = q/ell>ell_{mathrm{max}};MSF=1.2;LFS=12;AAT/; 626$cached_env_img{$key} = q|<IMG STYLE="height: 2.04ex; vertical-align: -0.45ex; " SRC="|."$dir".q|intro_img18.png" 627 ALT="$\ell>\ell_{\mathrm{max}}$">|; 628 629$key = q/ell_{mathrm{max}};MSF=1.2;LFS=12;AAT/; 630$cached_env_img{$key} = q|<IMG STYLE="height: 2.04ex; vertical-align: -0.45ex; " SRC="|."$dir".q|intro_img20.png" 631 ALT="$\ell_{\mathrm{max}}$">|; 632 633$key = q/ellle4N_{mathrm{side}};MSF=1.2;LFS=12;AAT/; 634$cached_env_img{$key} = q|<IMG STYLE="height: 2.04ex; vertical-align: -0.45ex; " SRC="|."$dir".q|intro_img172.png" 635 ALT="$\ell\le 4N_{\mathrm{side}}$">|; 636 637$key = q/f(gamma);MSF=1.2;LFS=12;AAT/; 638$cached_env_img{$key} = q|<IMG STYLE="height: 2.33ex; vertical-align: -0.68ex; " SRC="|."$dir".q|intro_img39.png" 639 ALT="$f(\gamma)$">|; 640 641$key = q/f(gamma)slashsqrt{4pi};MSF=1.2;LFS=12;AAT/; 642$cached_env_img{$key} = q|<IMG STYLE="height: 2.56ex; vertical-align: -0.68ex; " SRC="|."$dir".q|intro_img37.png" 643 ALT="$f(\gamma)/\sqrt{4 \pi}$">|; 644 645$key = q/f(gamma_p);MSF=1.2;LFS=12;AAT/; 646$cached_env_img{$key} = q|<IMG STYLE="height: 2.45ex; vertical-align: -0.75ex; " SRC="|."$dir".q|intro_img35.png" 647 ALT="$f(\gamma_p)$">|; 648 649$key = q/f(gamma_p)longrightarrowf(gamma_p)w(gamma_p);MSF=1.2;LFS=12;AAT/; 650$cached_env_img{$key} = q|<IMG STYLE="height: 2.45ex; vertical-align: -0.75ex; " SRC="|."$dir".q|intro_img33.png" 651 ALT="$f(\gamma_p) \longrightarrow f(\gamma_p) w(\gamma_p)$">|; 652 653$key = q/f({gamma});MSF=1.2;LFS=12;AAT/; 654$cached_env_img{$key} = q|<IMG STYLE="height: 2.33ex; vertical-align: -0.68ex; " SRC="|."$dir".q|intro_img23.png" 655 ALT="$f({\gamma})$">|; 656 657$key = q/frac{ell(ell+1)}{2pi}C_X(ell);MSF=1.2;LFS=12;AAT/; 658$cached_env_img{$key} = q|<IMG STYLE="height: 3.09ex; vertical-align: -0.90ex; " SRC="|."$dir".q|intro_img50.png" 659 ALT="$\frac{\ell(\ell+1)}{2\pi}C_X(\ell)$">|; 660 661$key = q/gamma_{p};MSF=1.2;LFS=12;AAT/; 662$cached_env_img{$key} = q|<IMG STYLE="height: 1.75ex; vertical-align: -0.75ex; " SRC="|."$dir".q|intro_img24.png" 663 ALT="$\gamma_{p}$">|; 664 665$key = q/ge;MSF=1.2;LFS=12;AAT/; 666$cached_env_img{$key} = q|<IMG STYLE="height: 1.87ex; vertical-align: -0.41ex; " SRC="|."$dir".q|intro_img113.png" 667 ALT="$\ge$">|; 668 669$key = q/hat{C}_ell;MSF=1.2;LFS=12;AAT/; 670$cached_env_img{$key} = q|<IMG STYLE="height: 2.62ex; vertical-align: -0.45ex; " SRC="|."$dir".q|intro_img30.png" 671 ALT="$\hat{C}_\ell$">|; 672 673$key = q/hat{a}_{ellm};MSF=1.2;LFS=12;AAT/; 674$cached_env_img{$key} = q|<IMG STYLE="height: 2.04ex; vertical-align: -0.45ex; " SRC="|."$dir".q|intro_img29.png" 675 ALT="$\hat{a}_{\ell m}$">|; 676 677$key = q/intd{u{w_p({u{)=1;MSF=1.2;LFS=12;AAT/; 678$cached_env_img{$key} = q|<IMG STYLE="height: 2.45ex; vertical-align: -0.75ex; " SRC="|."$dir".q|intro_img157.png" 679 ALT="$\int d\u w_p(\u ) = 1$">|; 680 681$key = q/langlecdotsrangle;MSF=1.2;LFS=12;AAT/; 682$cached_env_img{$key} = q|<IMG STYLE="height: 2.33ex; vertical-align: -0.68ex; " SRC="|."$dir".q|intro_img74.png" 683 ALT="$\langle\cdots \rangle$">|; 684 685$key = q/le;MSF=1.2;LFS=12;AAT/; 686$cached_env_img{$key} = q|<IMG STYLE="height: 1.87ex; vertical-align: -0.41ex; " SRC="|."$dir".q|intro_img114.png" 687 ALT="$\le$">|; 688 689$key = q/m>ell;MSF=1.2;LFS=12;AAT/; 690$cached_env_img{$key} = q|<IMG STYLE="height: 1.69ex; vertical-align: -0.16ex; " SRC="|."$dir".q|intro_img22.png" 691 ALT="$m>\ell$">|; 692 693$key = q/mu;MSF=1.2;LFS=12;AAT/; 694$cached_env_img{$key} = q|<IMG STYLE="height: 1.57ex; vertical-align: -0.55ex; " SRC="|."$dir".q|intro_img51.png" 695 ALT="$\mu$">|; 696 697$key = q/muK;MSF=1.2;LFS=12;AAT/; 698$cached_env_img{$key} = q|<IMG STYLE="height: 2.16ex; vertical-align: -0.55ex; " SRC="|."$dir".q|intro_img45.png" 699 ALT="$\mu K$">|; 700 701$key = q/phiin[0,2pi);MSF=1.2;LFS=12;AAT/; 702$cached_env_img{$key} = q|<IMG STYLE="height: 2.33ex; vertical-align: -0.68ex; " SRC="|."$dir".q|intro_img17.png" 703 ALT="$\phi\in[0,2\pi)$">|; 704 705$key = q/pin[0,N_{mathrm{pix}}-1];MSF=1.2;LFS=12;AAT/; 706$cached_env_img{$key} = q|<IMG STYLE="height: 2.45ex; vertical-align: -0.75ex; " SRC="|."$dir".q|intro_img25.png" 707 ALT="$p\in[0,N_{\mathrm{pix}}-1]$">|; 708 709$key = q/pm2;MSF=1.2;LFS=12;AAT/; 710$cached_env_img{$key} = q|<IMG STYLE="height: 1.57ex; vertical-align: -0.29ex; " SRC="|."$dir".q|intro_img63.png" 711 ALT="$\pm 2$">|; 712 713$key = q/psi;MSF=1.2;LFS=12;AAT/; 714$cached_env_img{$key} = q|<IMG STYLE="height: 2.10ex; vertical-align: -0.55ex; " SRC="|."$dir".q|intro_img55.png" 715 ALT="$\psi$">|; 716 717$key = q/sim10';MSF=1.2;LFS=12;AAT/; 718$cached_env_img{$key} = q|<IMG STYLE="height: 1.75ex; vertical-align: -0.10ex; " SRC="|."$dir".q|intro_img3.png" 719 ALT="$\sim 10'$">|; 720 721$key = q/sim;MSF=1.2;LFS=12;AAT/; 722$cached_env_img{$key} = q|<IMG STYLE="height: 0.58ex; vertical-align: -0.10ex; " SRC="|."$dir".q|intro_img1.png" 723 ALT="$\sim $">|; 724 725$key = q/sqrt{2}C_{ell}^{mathrm{T-GRAD}};MSF=1.2;LFS=12;AAT/; 726$cached_env_img{$key} = q|<IMG STYLE="height: 2.68ex; vertical-align: -0.68ex; " SRC="|."$dir".q|intro_img130.png" 727 ALT="$\sqrt{2}C_{\ell}^{\mathrm{T-GRAD}} $">|; 728 729$key = q/textbf{e}_phi;MSF=1.2;LFS=12;AAT/; 730$cached_env_img{$key} = q|<IMG STYLE="height: 1.75ex; vertical-align: -0.75ex; " SRC="|."$dir".q|intro_img70.png" 731 ALT="$\textbf{e}_\phi $">|; 732 733$key = q/textbf{e}_theta;MSF=1.2;LFS=12;AAT/; 734$cached_env_img{$key} = q|<IMG STYLE="height: 1.46ex; vertical-align: -0.45ex; " SRC="|."$dir".q|intro_img69.png" 735 ALT="$\textbf{e}_\theta $">|; 736 737$key = q/textstyle{myequal{;MSF=1.2;LFS=12;AAT/; 738$cached_env_img{$key} = q|<IMG STYLE="height: 0.58ex; vertical-align: -0.10ex; " SRC="|."$dir".q|intro_img13.png" 739 ALT="$\textstyle \myequal $">|; 740 741$key = q/thetain[0,pi];MSF=1.2;LFS=12;AAT/; 742$cached_env_img{$key} = q|<IMG STYLE="height: 2.33ex; vertical-align: -0.68ex; " SRC="|."$dir".q|intro_img16.png" 743 ALT="$\theta\in[0,\pi]$">|; 744 745$key = q/w_{ellm}(p);MSF=1.2;LFS=12;AAT/; 746$cached_env_img{$key} = q|<IMG STYLE="height: 2.33ex; vertical-align: -0.68ex; " SRC="|."$dir".q|intro_img161.png" 747 ALT="$w_{\ell m}(p)$">|; 748 749$key = q/w_{ell};MSF=1.2;LFS=12;AAT/; 750$cached_env_img{$key} = q|<IMG STYLE="height: 1.46ex; vertical-align: -0.45ex; " SRC="|."$dir".q|intro_img169.png" 751 ALT="$w_{\ell}$">|; 752 753$key = q/xequivcostheta;MSF=1.2;LFS=12;AAT/; 754$cached_env_img{$key} = q|<IMG STYLE="height: 1.69ex; vertical-align: -0.10ex; " SRC="|."$dir".q|intro_img146.png" 755 ALT="$x\equiv\cos\theta$">|; 756 757$key = q/xin]0,1[;MSF=1.2;LFS=12;AAT/; 758$cached_env_img{$key} = q|<IMG STYLE="height: 2.33ex; vertical-align: -0.68ex; " SRC="|."$dir".q|intro_img178.png" 759 ALT="$x\in]0,1[$">|; 760 761$key = q/{F_{(1,2),lm}(theta)};MSF=1.2;LFS=12;AAT/; 762$cached_env_img{$key} = q|<IMG STYLE="height: 2.56ex; vertical-align: -0.85ex; " SRC="|."$dir".q|intro_img82.png" 763 ALT="${F_{(1,2),lm}(\theta)}$">|; 764 765$key = q/{X_{1,lm}(textbf{n})=sqrt{(2l+1)slash4pi}F_{1,lm}(theta)e^{imphi}};MSF=1.2;LFS=12;AAT/; 766$cached_env_img{$key} = q|<IMG STYLE="height: 3.56ex; vertical-align: -1.17ex; " SRC="|."$dir".q|intro_img80.png" 767 ALT="${X_{1,lm}(\textbf{n})=\sqrt{(2l+1) / 4\pi} F_{1,lm}(\theta) e^{im\phi}}$">|; 768 769$key = q/{X_{2,lm}(textbf{n})=sqrt{(2l+1)slash4pi}F_{2,lm}(theta)e^{imphi}};MSF=1.2;LFS=12;AAT/; 770$cached_env_img{$key} = q|<IMG STYLE="height: 3.56ex; vertical-align: -1.17ex; " SRC="|."$dir".q|intro_img81.png" 771 ALT="${X_{2,lm}(\textbf{n})=\sqrt{(2l+1) / 4\pi} F_{2,lm}(\theta) e^{im\phi}}$">|; 772 773$key = q/{ellm};MSF=1.2;LFS=12;AAT/; 774$cached_env_img{$key} = q|<IMG STYLE="height: 1.69ex; vertical-align: -0.10ex; " SRC="|."$dir".q|intro_img21.png" 775 ALT="${\ell m}$">|; 776 777$key = q/{textbf{e}_{1}}^{prime}=cospsi{textbf{e}_{1}}+sinpsi{textbf{e}_{2}};MSF=1.2;LFS=12;AAT/; 778$cached_env_img{$key} = q|<IMG STYLE="height: 2.21ex; vertical-align: -0.55ex; " SRC="|."$dir".q|intro_img56.png" 779 ALT="${\textbf{e}_{1}}^{\prime}=\cos \psi {\textbf{e}_{1}}+\sin\psi {\textbf{e}_{2}} $">|; 780 781$key = q/{textbf{e}_{2}}^{prime}=-sinpsi{textbf{e}_{1}}+cospsi{textbf{e}_{2}};MSF=1.2;LFS=12;AAT/; 782$cached_env_img{$key} = q|<IMG STYLE="height: 2.21ex; vertical-align: -0.55ex; " SRC="|."$dir".q|intro_img57.png" 783 ALT="${\textbf{e}_{2}}^{\prime}=-\sin \psi {\textbf{e}_{1}}+\cos\psi {\textbf{e}_{2}} $">|; 784 785$key = q/{{gamma}};MSF=1.2;LFS=12;AAT/; 786$cached_env_img{$key} = q|<IMG STYLE="height: 1.57ex; vertical-align: -0.55ex; " SRC="|."$dir".q|intro_img15.png" 787 ALT="${{\gamma}}$">|; 788 7891; 790 791