1# LaTeX2HTML 2018.3 (Released July 19, 2018) 2# Associate images original text with physical files. 3 4 5$key = q/0^circ;MSF=1.2;LFS=12;AAT/; 6$cached_env_img{$key} = q|<IMG STYLE="height: 1.63ex; vertical-align: -0.10ex; " SRC="|."$dir".q|fac_img50.png" 7 ALT="$0^\circ$">|; 8 9$key = q/2cdotN_{mathrm{side}}<ell_{mathrm{max}}<3cdotN_{mathrm{side}}-1;MSF=1.2;LFS=12;AAT/; 10$cached_env_img{$key} = q|<IMG STYLE="height: 2.04ex; vertical-align: -0.45ex; " SRC="|."$dir".q|fac_img48.png" 11 ALT="$2 \cdot N_{\mathrm{side}}< \ell_{\mathrm{max}}< 3 \cdot N_{\mathrm{side}}- 1$">|; 12 13$key = q/3theta_{mathrm{pix}};MSF=1.2;LFS=12;AAT/; 14$cached_env_img{$key} = q|<IMG STYLE="height: 2.33ex; vertical-align: -0.75ex; " SRC="|."$dir".q|fac_img57.png" 15 ALT="$3\theta_{\mathrm{pix}}$">|; 16 17$key = q/4cdot;MSF=1.2;LFS=12;AAT/; 18$cached_env_img{$key} = q|<IMG STYLE="height: 1.52ex; vertical-align: -0.10ex; " SRC="|."$dir".q|fac_img67.png" 19 ALT="$4\cdot$">|; 20 21$key = q/8N_lambdaN_p;MSF=1.2;LFS=12;AAT/; 22$cached_env_img{$key} = q|<IMG STYLE="height: 2.33ex; vertical-align: -0.75ex; " SRC="|."$dir".q|fac_img61.png" 23 ALT="$8 N_\lambda N_p$">|; 24 25$key = q/B(ell);MSF=1.2;LFS=12;AAT/; 26$cached_env_img{$key} = q|<IMG STYLE="height: 2.33ex; vertical-align: -0.68ex; " SRC="|."$dir".q|fac_img21.png" 27 ALT="$B(\ell)$">|; 28 29$key = q/B(ell)=1;MSF=1.2;LFS=12;AAT/; 30$cached_env_img{$key} = q|<IMG STYLE="height: 2.33ex; vertical-align: -0.68ex; " SRC="|."$dir".q|fac_img23.png" 31 ALT="$B(\ell) = 1$">|; 32 33$key = q/C(ell);MSF=1.2;LFS=12;AAT/; 34$cached_env_img{$key} = q|<IMG STYLE="height: 2.33ex; vertical-align: -0.68ex; " SRC="|."$dir".q|fac_img100.png" 35 ALT="$C(\ell)$">|; 36 37$key = q/C(ell)B(ell)^2w_{mathrm{pix}}(ell)^2;MSF=1.2;LFS=12;AAT/; 38$cached_env_img{$key} = q|<IMG STYLE="height: 2.56ex; vertical-align: -0.75ex; " SRC="|."$dir".q|fac_img78.png" 39 ALT="$C(\ell) B(\ell)^2 40w_{\mathrm{pix}}(\ell)^2$">|; 41 42$key = q/C^B_{ell};MSF=1.2;LFS=12;AAT/; 43$cached_env_img{$key} = q|<IMG STYLE="height: 2.51ex; vertical-align: -0.67ex; " SRC="|."$dir".q|fac_img86.png" 44 ALT="$C^B_{\ell}$">|; 45 46$key = q/C^E_{ell};MSF=1.2;LFS=12;AAT/; 47$cached_env_img{$key} = q|<IMG STYLE="height: 2.51ex; vertical-align: -0.67ex; " SRC="|."$dir".q|fac_img85.png" 48 ALT="$C^E_{\ell}$">|; 49 50$key = q/C^T_{ell};MSF=1.2;LFS=12;AAT/; 51$cached_env_img{$key} = q|<IMG STYLE="height: 2.51ex; vertical-align: -0.67ex; " SRC="|."$dir".q|fac_img84.png" 52 ALT="$C^T_{\ell}$">|; 53 54$key = q/C^{BB}_ell;MSF=1.2;LFS=12;AAT/; 55$cached_env_img{$key} = q|<IMG STYLE="height: 2.51ex; vertical-align: -0.67ex; " SRC="|."$dir".q|fac_img29.png" 56 ALT="$C^{BB}_\ell$">|; 57 58$key = q/C^{BE}_ell;MSF=1.2;LFS=12;AAT/; 59$cached_env_img{$key} = q|<IMG STYLE="height: 2.51ex; vertical-align: -0.67ex; " SRC="|."$dir".q|fac_img35.png" 60 ALT="$C^{BE}_\ell$">|; 61 62$key = q/C^{BT}_ell;MSF=1.2;LFS=12;AAT/; 63$cached_env_img{$key} = q|<IMG STYLE="height: 2.51ex; vertical-align: -0.67ex; " SRC="|."$dir".q|fac_img34.png" 64 ALT="$C^{BT}_\ell$">|; 65 66$key = q/C^{EB}_ell;MSF=1.2;LFS=12;AAT/; 67$cached_env_img{$key} = q|<IMG STYLE="height: 2.51ex; vertical-align: -0.67ex; " SRC="|."$dir".q|fac_img32.png" 68 ALT="$C^{EB}_\ell$">|; 69 70$key = q/C^{EE}_ell;MSF=1.2;LFS=12;AAT/; 71$cached_env_img{$key} = q|<IMG STYLE="height: 2.51ex; vertical-align: -0.67ex; " SRC="|."$dir".q|fac_img28.png" 72 ALT="$C^{EE}_\ell$">|; 73 74$key = q/C^{ET}_ell;MSF=1.2;LFS=12;AAT/; 75$cached_env_img{$key} = q|<IMG STYLE="height: 2.51ex; vertical-align: -0.67ex; " SRC="|."$dir".q|fac_img33.png" 76 ALT="$C^{ET}_\ell$">|; 77 78$key = q/C^{EtimesB}_{ell}ne0;MSF=1.2;LFS=12;AAT/; 79$cached_env_img{$key} = q|<IMG STYLE="height: 2.62ex; vertical-align: -0.68ex; " SRC="|."$dir".q|fac_img96.png" 80 ALT="$C^{E\times B}_{\ell} \ne 0$">|; 81 82$key = q/C^{TB}_ell;MSF=1.2;LFS=12;AAT/; 83$cached_env_img{$key} = q|<IMG STYLE="height: 2.51ex; vertical-align: -0.67ex; " SRC="|."$dir".q|fac_img31.png" 84 ALT="$C^{TB}_\ell$">|; 85 86$key = q/C^{TE}_ell;MSF=1.2;LFS=12;AAT/; 87$cached_env_img{$key} = q|<IMG STYLE="height: 2.51ex; vertical-align: -0.67ex; " SRC="|."$dir".q|fac_img30.png" 88 ALT="$C^{TE}_\ell$">|; 89 90$key = q/C^{TT}_ell;MSF=1.2;LFS=12;AAT/; 91$cached_env_img{$key} = q|<IMG STYLE="height: 2.51ex; vertical-align: -0.67ex; " SRC="|."$dir".q|fac_img27.png" 92 ALT="$C^{TT}_\ell$">|; 93 94$key = q/C^{TtimesB}_{ell}ne0;MSF=1.2;LFS=12;AAT/; 95$cached_env_img{$key} = q|<IMG STYLE="height: 2.62ex; vertical-align: -0.68ex; " SRC="|."$dir".q|fac_img95.png" 96 ALT="$C^{T\times B}_{\ell} \ne 0$">|; 97 98$key = q/C^{TtimesE}_{ell};MSF=1.2;LFS=12;AAT/; 99$cached_env_img{$key} = q|<IMG STYLE="height: 2.62ex; vertical-align: -0.68ex; " SRC="|."$dir".q|fac_img87.png" 100 ALT="$C^{T\times E}_{\ell}$">|; 101 102$key = q/C_ell;MSF=1.2;LFS=12;AAT/; 103$cached_env_img{$key} = q|<IMG STYLE="height: 2.04ex; vertical-align: -0.45ex; " SRC="|."$dir".q|fac_img24.png" 104 ALT="$C_\ell$">|; 105 106$key = q/C_{ell};MSF=1.2;LFS=12;AAT/; 107$cached_env_img{$key} = q|<IMG STYLE="height: 2.04ex; vertical-align: -0.45ex; " SRC="|."$dir".q|fac_img88.png" 108 ALT="$C_{\ell}$">|; 109 110$key = q/DeltaI,DeltaQ;MSF=1.2;LFS=12;AAT/; 111$cached_env_img{$key} = q|<IMG STYLE="height: 2.16ex; vertical-align: -0.55ex; " SRC="|."$dir".q|fac_img107.png" 112 ALT="$\Delta I, \Delta Q$">|; 113 114$key = q/DeltaU;MSF=1.2;LFS=12;AAT/; 115$cached_env_img{$key} = q|<IMG STYLE="height: 1.75ex; vertical-align: -0.10ex; " SRC="|."$dir".q|fac_img108.png" 116 ALT="$\Delta U$">|; 117 118$key = q/Deltaequivleft(frac{partial^2}{partialtheta^2}+cotthetafrac{partial}{partialtheta}+frac{partial^2}{sin^2thetapartialvarphi^2}right);MSF=1.2;LFS=12;AAT/; 119$cached_env_img{$key} = q|<IMG STYLE="height: 3.56ex; vertical-align: -1.26ex; " SRC="|."$dir".q|fac_img109.png" 120 ALT="$\Delta \equiv \left( 121\frac{\partial^2}{\partial\theta^2} 122+ \cot\theta\frac{\partial}{\partial\theta} + 123\frac{\partial^2}{\sin^2\theta\partial\varphi^2} 124\right)$">|; 125 126$key = q/IGerror_der|S720,533|R0,2/; 127$cached_env_img{$key} = q|<IMG STYLE="" 128 SRC="|."$dir".q|IGimg1.png" 129 ALT="Image error_der">|; 130 131$key = q/Lambda;MSF=1.2;LFS=12;AAT/; 132$cached_env_img{$key} = q|<IMG STYLE="height: 1.69ex; vertical-align: -0.10ex; " SRC="|."$dir".q|fac_img106.png" 133 ALT="$\Lambda$">|; 134 135$key = q/N_lambda<2^{31}=2147483648;MSF=1.2;LFS=12;AAT/; 136$cached_env_img{$key} = q|<IMG STYLE="height: 2.27ex; vertical-align: -0.45ex; " SRC="|."$dir".q|fac_img63.png" 137 ALT="$N_\lambda < 2^{31} = 2147483648$">|; 138 139$key = q/N_lambda={{tt{nsmax}({{tt{nlmax}+1)({{tt{nlmax}+2);MSF=1.2;LFS=12;AAT/; 140$cached_env_img{$key} = q|<IMG STYLE="height: 2.33ex; vertical-align: -0.68ex; " SRC="|."$dir".q|fac_img62.png" 141 ALT="$N_\lambda = {\tt nsmax} ({\tt nlmax}+1)({\tt nlmax}+2)$">|; 142 143$key = q/N_{mathrm{side}}ge2;MSF=1.2;LFS=12;AAT/; 144$cached_env_img{$key} = q|<IMG STYLE="height: 2.04ex; vertical-align: -0.45ex; " SRC="|."$dir".q|fac_img53.png" 145 ALT="$N_{\mathrm{side}}\ge 2$">|; 146 147$key = q/P(ell);MSF=1.2;LFS=12;AAT/; 148$cached_env_img{$key} = q|<IMG STYLE="height: 2.33ex; vertical-align: -0.68ex; " SRC="|."$dir".q|fac_img20.png" 149 ALT="$P(\ell)$">|; 150 151$key = q/P(ell)=1;MSF=1.2;LFS=12;AAT/; 152$cached_env_img{$key} = q|<IMG STYLE="height: 2.33ex; vertical-align: -0.68ex; " SRC="|."$dir".q|fac_img22.png" 153 ALT="$P(\ell) = 1541$">|; 155 156$key = q/P_{ellm}(theta);MSF=1.2;LFS=12;AAT/; 157$cached_env_img{$key} = q|<IMG STYLE="height: 2.33ex; vertical-align: -0.68ex; " SRC="|."$dir".q|fac_img52.png" 158 ALT="$P_{\ell m}(\theta)$">|; 159 160$key = q/P_{ellm};MSF=1.2;LFS=12;AAT/; 161$cached_env_img{$key} = q|<IMG STYLE="height: 2.04ex; vertical-align: -0.45ex; " SRC="|."$dir".q|fac_img51.png" 162 ALT="$P_{\ell m}$">|; 163 164$key = q/P_{lm}(theta);MSF=1.2;LFS=12;AAT/; 165$cached_env_img{$key} = q|<IMG STYLE="height: 2.33ex; vertical-align: -0.68ex; " SRC="|."$dir".q|fac_img59.png" 166 ALT="$P_{lm}(\theta)$">|; 167 168$key = q/X,Yin{T,E,B};MSF=1.2;LFS=12;AAT/; 169$cached_env_img{$key} = q|<IMG STYLE="height: 2.33ex; vertical-align: -0.68ex; " SRC="|."$dir".q|fac_img12.png" 170 ALT="$X,Y \in \{T,E,B\}$">|; 171 172$key = q/Y_{ellm};MSF=1.2;LFS=12;AAT/; 173$cached_env_img{$key} = q|<IMG STYLE="height: 2.04ex; vertical-align: -0.45ex; " SRC="|."$dir".q|fac_img40.png" 174 ALT="$Y_{\ell m}$">|; 175 176$key = q/[1,{{tt{na}];MSF=1.2;LFS=12;AAT/; 177$cached_env_img{$key} = q|<IMG STYLE="height: 2.33ex; vertical-align: -0.68ex; " SRC="|."$dir".q|fac_img71.png" 178 ALT="$[1, {\tt na}]$">|; 179 180$key = q/a_{ellm};MSF=1.2;LFS=12;AAT/; 181$cached_env_img{$key} = q|<IMG STYLE="height: 1.46ex; vertical-align: -0.45ex; " SRC="|."$dir".q|fac_img14.png" 182 ALT="$a_{\ell m}$">|; 183 184$key = q/alpha;MSF=1.2;LFS=12;AAT/; 185$cached_env_img{$key} = q|<IMG STYLE="height: 1.11ex; vertical-align: -0.10ex; " SRC="|."$dir".q|fac_img68.png" 186 ALT="$\alpha$">|; 187 188$key = q/alpha_0=left(1-sum_i^n|alpha_i|^2right)^{1slash2};MSF=1.2;LFS=12;AAT/; 189$cached_env_img{$key} = q|<IMG STYLE="height: 3.03ex; vertical-align: -0.68ex; " SRC="|."$dir".q|fac_img76.png" 190 ALT="$\alpha_0 = 191\left(1 - \sum_i^n \vert\alpha_i\vert^2\right)^{1/2}$">|; 192 193$key = q/alpha_i;MSF=1.2;LFS=12;AAT/; 194$cached_env_img{$key} = q|<IMG STYLE="height: 1.46ex; vertical-align: -0.45ex; " SRC="|."$dir".q|fac_img70.png" 195 ALT="$\alpha_i$">|; 196 197$key = q/alpha_i=(0,0,0.2);MSF=1.2;LFS=12;AAT/; 198$cached_env_img{$key} = q|<IMG STYLE="height: 2.33ex; vertical-align: -0.68ex; " SRC="|."$dir".q|fac_img81.png" 199 ALT="$\alpha_i = (0, 0, 0.2)$">|; 200 201$key = q/b_B(ell);MSF=1.2;LFS=12;AAT/; 202$cached_env_img{$key} = q|<IMG STYLE="height: 2.33ex; vertical-align: -0.68ex; " SRC="|."$dir".q|fac_img3.png" 203 ALT="$b_B(\ell)$">|; 204 205$key = q/b_E(ell);MSF=1.2;LFS=12;AAT/; 206$cached_env_img{$key} = q|<IMG STYLE="height: 2.33ex; vertical-align: -0.68ex; " SRC="|."$dir".q|fac_img2.png" 207 ALT="$b_E(\ell)$">|; 208 209$key = q/b_T(ell);MSF=1.2;LFS=12;AAT/; 210$cached_env_img{$key} = q|<IMG STYLE="height: 2.33ex; vertical-align: -0.68ex; " SRC="|."$dir".q|fac_img1.png" 211 ALT="$b_T(\ell)$">|; 212 213$key = q/backslash;MSF=1.2;LFS=12;AAT/; 214$cached_env_img{$key} = q|<IMG STYLE="height: 2.33ex; vertical-align: -0.68ex; " SRC="|."$dir".q|fac_img54.png" 215 ALT="$\backslash$">|; 216 217$key = q/bullet;MSF=1.2;LFS=12;AAT/; 218$cached_env_img{$key} = q|<IMG STYLE="height: 0.93ex; vertical-align: -0.10ex; " SRC="|."$dir".q|fac_img38.png" 219 ALT="$\bullet$">|; 220 221$key = q/deltaC_ellproptoepsiloncdotC_ell;MSF=1.2;LFS=12;AAT/; 222$cached_env_img{$key} = q|<IMG STYLE="height: 2.10ex; vertical-align: -0.45ex; " SRC="|."$dir".q|fac_img46.png" 223 ALT="$\delta C_\ell \propto \epsilon \cdot C_\ell$">|; 224 225$key = q/displaystyleC^{XY}_{ellm};MSF=1.2;LFS=12;AAT/; 226$cached_env_img{$key} = q|<IMG STYLE="height: 2.62ex; vertical-align: -0.67ex; " SRC="|."$dir".q|fac_img10.png" 227 ALT="$\displaystyle C^{XY}_{\ell m}$">|; 228 229$key = q/displaystyleC^{XY}_{ellm}b_{X}(ell)b_{Y}(ell),;MSF=1.2;LFS=12;AAT/; 230$cached_env_img{$key} = q|<IMG STYLE="height: 2.62ex; vertical-align: -0.68ex; " SRC="|."$dir".q|fac_img11.png" 231 ALT="$\displaystyle C^{XY}_{\ell m} b_{X}(\ell)b_{Y}(\ell),$">|; 232 233$key = q/displaystylea^{X}_{ellm};MSF=1.2;LFS=12;AAT/; 234$cached_env_img{$key} = q|<IMG STYLE="height: 2.62ex; vertical-align: -0.67ex; " SRC="|."$dir".q|fac_img7.png" 235 ALT="$\displaystyle a^{X}_{\ell m}$">|; 236 237$key = q/displaystylea^{X}_{ellm}b_{X}(ell),;MSF=1.2;LFS=12;AAT/; 238$cached_env_img{$key} = q|<IMG STYLE="height: 2.62ex; vertical-align: -0.68ex; " SRC="|."$dir".q|fac_img9.png" 239 ALT="$\displaystyle a^{X}_{\ell m} b_{X}(\ell),$">|; 240 241$key = q/displaystylea_{ellm}^mathrm{IN}frac{B^mathrm{OUT}(ell)P^mathrm{OUT}(ell)}{B^mathrm{IN}(ell)P^mathrm{IN}(ell)},;MSF=1.2;LFS=12;AAT/; 242$cached_env_img{$key} = q|<IMG STYLE="height: 5.65ex; vertical-align: -2.27ex; " SRC="|."$dir".q|fac_img19.png" 243 ALT="$\displaystyle a_{\ell m}^\mathrm{IN} 244\frac{B^\mathrm{OUT}(\ell) P^\mathrm{OUT}(\ell)}{B^\mathrm{IN}(\ell) 245P^\mathrm{IN}(\ell)},$">|; 246 247$key = q/displaystylea_{ellm}^mathrm{OUT};MSF=1.2;LFS=12;AAT/; 248$cached_env_img{$key} = q|<IMG STYLE="height: 2.68ex; vertical-align: -0.67ex; " SRC="|."$dir".q|fac_img18.png" 249 ALT="$\displaystyle a_{\ell m}^\mathrm{OUT}$">|; 250 251$key = q/displaystylea_{ellm}longrightarrowa_{ellm}left[C(l)right]^{1slash2}B(ell)w_{{rm{pix}(ell);MSF=1.2;LFS=12;AAT/; 252$cached_env_img{$key} = q|<IMG STYLE="height: 2.97ex; vertical-align: -0.75ex; " SRC="|."$dir".q|fac_img77.png" 253 ALT="$\displaystyle a_{\ell m} \longrightarrow a_{\ell m} \left[C(l)\right]^{1/2} B(\ell) w_{\rm pix}(\ell)$">|; 254 255$key = q/displaystylerho_{{rm{SHO}(t)=left|psi_nright|^2=e^{-t^2slash2sigma_0^2}left|sum_0}^{n}alpha_iC_iH_ileft(frac{t}{sqrt{2}sigma_0}right)right|^2;MSF=1.2;LFS=12;AAT/; 256$cached_env_img{$key} = q|<IMG STYLE="height: 6.29ex; vertical-align: -2.60ex; " SRC="|."$dir".q|fac_img72.png" 257 ALT="$\displaystyle \rho_{\rm SHO}(t) = \left\vert \psi_n\right\vert^2 = e^{-t^2/2\si... 258...{i=0}^{n} \alpha_i 259C_i H_i \left(\frac{t}{\sqrt{2}\sigma_0}\right)\right\vert^2$">|; 260 261$key = q/ell(ell+1)C(ell)slash2pi;MSF=1.2;LFS=12;AAT/; 262$cached_env_img{$key} = q|<IMG STYLE="height: 2.33ex; vertical-align: -0.68ex; " SRC="|."$dir".q|fac_img102.png" 263 ALT="$\ell(\ell+1)C(\ell)/2\pi$">|; 264 265$key = q/ell;MSF=1.2;LFS=12;AAT/; 266$cached_env_img{$key} = q|<IMG STYLE="height: 1.69ex; vertical-align: -0.10ex; " SRC="|."$dir".q|fac_img5.png" 267 ALT="$\ell$">|; 268 269$key = q/ell=256;MSF=1.2;LFS=12;AAT/; 270$cached_env_img{$key} = q|<IMG STYLE="height: 1.69ex; vertical-align: -0.10ex; " SRC="|."$dir".q|fac_img79.png" 271 ALT="$\ell=256$">|; 272 273$key = q/ell=64;MSF=1.2;LFS=12;AAT/; 274$cached_env_img{$key} = q|<IMG STYLE="height: 1.69ex; vertical-align: -0.10ex; " SRC="|."$dir".q|fac_img97.png" 275 ALT="$\ell=64$">|; 276 277$key = q/ell>2cdotN_{mathrm{side}};MSF=1.2;LFS=12;AAT/; 278$cached_env_img{$key} = q|<IMG STYLE="height: 2.04ex; vertical-align: -0.45ex; " SRC="|."$dir".q|fac_img44.png" 279 ALT="$\ell > 2\cdot N_{\mathrm{side}}$">|; 280 281$key = q/ell>2cdotnsmax;MSF=1.2;LFS=12;AAT/; 282$cached_env_img{$key} = q|<IMG STYLE="height: 1.69ex; vertical-align: -0.16ex; " SRC="|."$dir".q|fac_img83.png" 283 ALT="$\ell > 2\cdot nsmax$">|; 284 285$key = q/ell>3cdotN_{mathrm{side}}-1;MSF=1.2;LFS=12;AAT/; 286$cached_env_img{$key} = q|<IMG STYLE="height: 2.04ex; vertical-align: -0.45ex; " SRC="|."$dir".q|fac_img49.png" 287 ALT="$\ell > 3 \cdot N_{\mathrm{side}}-1$">|; 288 289$key = q/ell_{mathrm{max}}+1;MSF=1.2;LFS=12;AAT/; 290$cached_env_img{$key} = q|<IMG STYLE="height: 2.04ex; vertical-align: -0.45ex; " SRC="|."$dir".q|fac_img4.png" 291 ALT="$\ell_{\mathrm{max}}+1$">|; 292 293$key = q/ell_{mathrm{max}};MSF=1.2;LFS=12;AAT/; 294$cached_env_img{$key} = q|<IMG STYLE="height: 2.04ex; vertical-align: -0.45ex; " SRC="|."$dir".q|fac_img25.png" 295 ALT="$\ell_{\mathrm{max}}$">|; 296 297$key = q/ell_{mathrm{max}}=3cdotN_{mathrm{side}}-1;MSF=1.2;LFS=12;AAT/; 298$cached_env_img{$key} = q|<IMG STYLE="height: 2.04ex; vertical-align: -0.45ex; " SRC="|."$dir".q|fac_img45.png" 299 ALT="$\ell_{\mathrm{max}}= 3 \cdot N_{\mathrm{side}}-1$">|; 300 301$key = q/ell_{mathrm{max}}approx2cdotN_{mathrm{side}};MSF=1.2;LFS=12;AAT/; 302$cached_env_img{$key} = q|<IMG STYLE="height: 2.04ex; vertical-align: -0.45ex; " SRC="|."$dir".q|fac_img39.png" 303 ALT="$\ell_{\mathrm{max}}\approx 2\cdot N_{\mathrm{side}}$">|; 304 305$key = q/ell_{mathrm{max}}le1.5N_{mathrm{side}};MSF=1.2;LFS=12;AAT/; 306$cached_env_img{$key} = q|<IMG STYLE="height: 2.04ex; vertical-align: -0.45ex; " SRC="|."$dir".q|fac_img43.png" 307 ALT="$\ell_{\mathrm{max}}\le 1.5 N_{\mathrm{side}}$">|; 308 309$key = q/ell_{max};MSF=1.2;LFS=12;AAT/; 310$cached_env_img{$key} = q|<IMG STYLE="height: 2.04ex; vertical-align: -0.45ex; " SRC="|."$dir".q|fac_img60.png" 311 ALT="$\ell_{max}$">|; 312 313$key = q/epsilon<0.1;MSF=1.2;LFS=12;AAT/; 314$cached_env_img{$key} = q|<IMG STYLE="height: 1.57ex; vertical-align: -0.16ex; " SRC="|."$dir".q|fac_img47.png" 315 ALT="$\epsilon <0.1$">|; 316 317$key = q/frac{4pi}{N_{mathrm{pix}}}sum_pw(p)Y_{ellm}(p)=sqrt{4pi}delta_{ell0}delta_{m0};MSF=1.2;LFS=12;AAT/; 318$cached_env_img{$key} = q|<IMG STYLE="height: 3.26ex; vertical-align: -1.34ex; " SRC="|."$dir".q|fac_img41.png" 319 ALT="$\frac{4\pi}{N_{\mathrm{pix}}} \sum_p w(p) Y_{\ell m}(p) = \sqrt{4 \pi} \delta_{\ell 0}\delta_{m 0}$">|; 320 321$key = q/hyperref{iseed}{}{}{fac:sky_ng_sim:iseed}=1;MSF=1.2;LFS=12;AAT/; 322$cached_env_img{$key} = q|<IMG STYLE="height: 1.52ex; vertical-align: -0.10ex; " SRC="|."$dir".q|fac_img82.png" 323 ALT="$\hyperref{iseed}{}{}{fac:sky_ng_sim:iseed}=1$">|; 324 325$key = q/hyperref{iseed}{}{}{fac:synfast:iseed};MSF=1.2;LFS=12;AAT/; 326$cached_env_img{$key} = q|<IMG STYLE="vertical-align: 20px" 327 SRC="|."$dir".q|fac_img99.png" 328 ALT="$\hyperref{iseed}{}{}{fac:synfast:iseed}$">|; 329 330$key = q/hyperref{iseed}{}{}{fac:synfast:iseed}=-1;MSF=1.2;LFS=12;AAT/; 331$cached_env_img{$key} = q|<IMG STYLE="height: 1.57ex; vertical-align: -0.29ex; " SRC="|."$dir".q|fac_img98.png" 332 ALT="$\hyperref{iseed}{}{}{fac:synfast:iseed}=-1$">|; 333 334$key = q/ige1;MSF=1.2;LFS=12;AAT/; 335$cached_env_img{$key} = q|<IMG STYLE="height: 1.92ex; vertical-align: -0.41ex; " SRC="|."$dir".q|fac_img75.png" 336 ALT="$i\ge 1$">|; 337 338$key = q/ldots;MSF=1.2;LFS=12;AAT/; 339$cached_env_img{$key} = q|<IMG STYLE="height: 0.23ex; vertical-align: -0.10ex; " SRC="|."$dir".q|fac_img65.png" 340 ALT="$\ldots$">|; 341 342$key = q/le1446;MSF=1.2;LFS=12;AAT/; 343$cached_env_img{$key} = q|<IMG STYLE="height: 1.92ex; vertical-align: -0.41ex; " SRC="|."$dir".q|fac_img64.png" 344 ALT="$\le 1446$">|; 345 346$key = q/le3;MSF=1.2;LFS=12;AAT/; 347$cached_env_img{$key} = q|<IMG STYLE="height: 1.92ex; vertical-align: -0.41ex; " SRC="|."$dir".q|fac_img69.png" 348 ALT="$\le 3$">|; 349 350$key = q/left({partialT}slash{partialtheta},{partialQ}slash{partialtheta},{partialU}slashaltheta};{partialT}slash{partialphi}slashsintheta,ldotsright);MSF=1.2;LFS=12;AAT/; 351$cached_env_img{$key} = q|<IMG STYLE="height: 2.33ex; vertical-align: -0.68ex; " SRC="|."$dir".q|fac_img92.png" 352 ALT="$\left({\partial T}/{\partial \theta}, {\partial Q}/{\partial \theta}, {\partial 353U}/{\partial \theta}; {\partial T}/{\partial \phi}/\sin\theta, \ldots \right) 354$">|; 355 356$key = q/left({partialT}slash{partialtheta},{partialT}slash{partialphi}slashsinthetaright);MSF=1.2;LFS=12;AAT/; 357$cached_env_img{$key} = q|<IMG STYLE="height: 2.33ex; vertical-align: -0.68ex; " SRC="|."$dir".q|fac_img89.png" 358 ALT="$\left({\partial T}/{\partial \theta}, {\partial T}/{\partial \phi}/\sin\theta \right) 359$">|; 360 361$key = q/left({partial^2T}slash{partialtheta^2},ldots;{partial^2T}slash{partialthetapartialphi}slashsintheta,ldots;right.;MSF=1.2;LFS=12;AAT/; 362$cached_env_img{$key} = q|<IMG STYLE="height: 2.45ex; vertical-align: -0.68ex; " SRC="|."$dir".q|fac_img93.png" 363 ALT="$\left({\partial^2 T}/{\partial \theta^2},\ldots; {\partial^2 T}/{\partial 364\theta\partial\phi}/\sin\theta,\ldots ;\right. $">|; 365 366$key = q/left({partial^2T}slash{partialtheta^2},{partial^2T}slash{partialthetapartialphi}slashsintheta,right.;MSF=1.2;LFS=12;AAT/; 367$cached_env_img{$key} = q|<IMG STYLE="height: 2.45ex; vertical-align: -0.68ex; " SRC="|."$dir".q|fac_img90.png" 368 ALT="$\left({\partial^2 T}/{\partial \theta^2}, {\partial^2 T}/{\partial 369\theta\partial\phi}/\sin\theta,\right. $">|; 370 371$key = q/left.{partial^2T}slash{partialphi^2}slashsin^2thetaldotsright);MSF=1.2;LFS=12;AAT/; 372$cached_env_img{$key} = q|<IMG STYLE="height: 2.45ex; vertical-align: -0.68ex; " SRC="|."$dir".q|fac_img94.png" 373 ALT="$\left.{\partial^2 T}/{\partial \phi^2}/\sin^2\theta \ldots \right) $">|; 374 375$key = q/left.{partial^2T}slash{partialphi^2}slashsin^2thetaright);MSF=1.2;LFS=12;AAT/; 376$cached_env_img{$key} = q|<IMG STYLE="height: 2.45ex; vertical-align: -0.68ex; " SRC="|."$dir".q|fac_img91.png" 377 ALT="$\left.{\partial^2 T}/{\partial \phi^2}/\sin^2\theta \right) $">|; 378 379$key = q/left|psi_0right|^2,;MSF=1.2;LFS=12;AAT/; 380$cached_env_img{$key} = q|<IMG STYLE="height: 2.74ex; vertical-align: -0.68ex; " SRC="|."$dir".q|fac_img74.png" 381 ALT="$\left\vert\psi_0\right\vert^2,$">|; 382 383$key = q/mu;MSF=1.2;LFS=12;AAT/; 384$cached_env_img{$key} = q|<IMG STYLE="height: 1.57ex; vertical-align: -0.55ex; " SRC="|."$dir".q|fac_img56.png" 385 ALT="$\mu$">|; 386 387$key = q/nge100;MSF=1.2;LFS=12;AAT/; 388$cached_env_img{$key} = q|<IMG STYLE="height: 1.92ex; vertical-align: -0.41ex; " SRC="|."$dir".q|fac_img13.png" 389 ALT="$n\ge 100$">|; 390 391$key = q/partial(Q,U)slashpartialtheta;MSF=1.2;LFS=12;AAT/; 392$cached_env_img{$key} = q|<IMG STYLE="height: 2.33ex; vertical-align: -0.68ex; " SRC="|."$dir".q|fac_img103.png" 393 ALT="$\partial (Q,U)/\partial\theta$">|; 394 395$key = q/partialXslashpartialtheta;MSF=1.2;LFS=12;AAT/; 396$cached_env_img{$key} = q|<IMG STYLE="height: 2.33ex; vertical-align: -0.68ex; " SRC="|."$dir".q|fac_img15.png" 397 ALT="$\partial X/\partial\theta$">|; 398 399$key = q/partial^2(Q,U)slash(partialthetapartialvarphisintheta);MSF=1.2;LFS=12;AAT/; 400$cached_env_img{$key} = q|<IMG STYLE="height: 2.45ex; vertical-align: -0.68ex; " SRC="|."$dir".q|fac_img105.png" 401 ALT="$\partial^2 (Q,U)/(\partial\theta\partial\varphi\sin\theta)$">|; 402 403$key = q/partial^2(Q,U)slashpartialtheta^2;MSF=1.2;LFS=12;AAT/; 404$cached_env_img{$key} = q|<IMG STYLE="height: 2.45ex; vertical-align: -0.68ex; " SRC="|."$dir".q|fac_img104.png" 405 ALT="$\partial^2 (Q,U)/\partial \theta^2$">|; 406 407$key = q/partial^2Xslash(partialthetapartialphisintheta);MSF=1.2;LFS=12;AAT/; 408$cached_env_img{$key} = q|<IMG STYLE="height: 2.45ex; vertical-align: -0.68ex; " SRC="|."$dir".q|fac_img16.png" 409 ALT="$\partial^2 X/(\partial\theta\partial\phi\sin\theta)$">|; 410 411$key = q/partial^2Xslash(partialthetapartialvarphisin(theta));MSF=1.2;LFS=12;AAT/; 412$cached_env_img{$key} = q|<IMG STYLE="height: 2.45ex; vertical-align: -0.68ex; " SRC="|."$dir".q|fac_img101.png" 413 ALT="$\partial^2 X/(\partial\theta\partial\varphi\sin(\theta))$">|; 414 415$key = q/partial^2Xslashpartialtheta^2;MSF=1.2;LFS=12;AAT/; 416$cached_env_img{$key} = q|<IMG STYLE="height: 2.45ex; vertical-align: -0.68ex; " SRC="|."$dir".q|fac_img17.png" 417 ALT="$\partial^2 X/\partial \theta^2$">|; 418 419$key = q/phi;MSF=1.2;LFS=12;AAT/; 420$cached_env_img{$key} = q|<IMG STYLE="height: 2.10ex; vertical-align: -0.55ex; " SRC="|."$dir".q|fac_img37.png" 421 ALT="$\phi$">|; 422 423$key = q/pm;MSF=1.2;LFS=12;AAT/; 424$cached_env_img{$key} = q|<IMG STYLE="height: 1.46ex; vertical-align: -0.29ex; " SRC="|."$dir".q|fac_img55.png" 425 ALT="$\pm$">|; 426 427$key = q/proptoN_{mathrm{pix}}^pproptoN_{mathrm{side}}^{2p};MSF=1.2;LFS=12;AAT/; 428$cached_env_img{$key} = q|<IMG STYLE="height: 2.97ex; vertical-align: -0.91ex; " SRC="|."$dir".q|fac_img66.png" 429 ALT="$\propto N_{\mathrm{pix}}^p 430\propto N_{\mathrm{side}}^{2p}$">|; 431 432$key = q/sigma_0=1;MSF=1.2;LFS=12;AAT/; 433$cached_env_img{$key} = q|<IMG STYLE="height: 1.92ex; vertical-align: -0.45ex; " SRC="|."$dir".q|fac_img80.png" 434 ALT="$\sigma_0=1$">|; 435 436$key = q/sigma_0^2;MSF=1.2;LFS=12;AAT/; 437$cached_env_img{$key} = q|<IMG STYLE="height: 2.51ex; vertical-align: -0.67ex; " SRC="|."$dir".q|fac_img73.png" 438 ALT="$\sigma_0^2$">|; 439 440$key = q/textstylelongrightarrow;MSF=1.2;LFS=12;AAT/; 441$cached_env_img{$key} = q|<IMG STYLE="height: 0.99ex; vertical-align: -0.10ex; " SRC="|."$dir".q|fac_img8.png" 442 ALT="$\textstyle \longrightarrow$">|; 443 444$key = q/theta;MSF=1.2;LFS=12;AAT/; 445$cached_env_img{$key} = q|<IMG STYLE="height: 1.69ex; vertical-align: -0.10ex; " SRC="|."$dir".q|fac_img36.png" 446 ALT="$\theta$">|; 447 448$key = q/theta_{mathrm{pix}};MSF=1.2;LFS=12;AAT/; 449$cached_env_img{$key} = q|<IMG STYLE="height: 2.33ex; vertical-align: -0.75ex; " SRC="|."$dir".q|fac_img58.png" 450 ALT="$\theta_{\mathrm{pix}}$">|; 451 452$key = q/{0,ldots,ell_{mathrm{max}}};MSF=1.2;LFS=12;AAT/; 453$cached_env_img{$key} = q|<IMG STYLE="height: 2.33ex; vertical-align: -0.68ex; " SRC="|."$dir".q|fac_img6.png" 454 ALT="$\{0,\ldots,\ell_{\mathrm{max}}\}$">|; 455 456$key = q/{cal{O}}(N_{mathrm{pix}}^{1slash2}ell_{mathrm{max}}^2);MSF=1.2;LFS=12;AAT/; 457$cached_env_img{$key} = q|<IMG STYLE="height: 3.21ex; vertical-align: -0.91ex; " SRC="|."$dir".q|fac_img26.png" 458 ALT="${\cal {O}}(N_{\mathrm{pix}}^{1/2}\ell_{\mathrm{max}}^2 )$">|; 459 460$key = q/|m|leellle3N_{mathrm{side}};MSF=1.2;LFS=12;AAT/; 461$cached_env_img{$key} = q|<IMG STYLE="height: 2.33ex; vertical-align: -0.68ex; " SRC="|."$dir".q|fac_img42.png" 462 ALT="$\vert m\vert \le \ell \le 3 N_{\mathrm{side}}$">|; 463 4641; 465 466