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+\usepackage{../../../../preamble_tikz} \begin{document} @@ -18,6 +18,5 @@ % Set intersection label \node at (-0.6,0) {$A\setminus B$}; - \end{tikzpicture} \end{document} \ No newline at end of file diff --git a/Mathematics/1st/Fundamentals_of_mathematics/Images/intersection.pdf b/Mathematics/1st/Fundamentals_of_mathematics/Images/intersection.pdf index 3f8b0bad32dfe817c7d9b44bcea65ace78f1817c..77c198eec10a8c7021f85ae7040df4b113327721 100644 GIT binary patch delta 3398 zcmai0XH=7kwhe?Ds!{}LLBJtmLcWj?5D3zHmkvW$YLKRYN`fdzxpWApd+*1aweG)j&f4poA7}5gPhSdMO%h%5IKLtk3H5jNhA1mT zWDPy=w}L&PXe=`A96gNi%}T|83CTXYC!;XOvG_GAwTcR9pld6|j23smw5E$;x0`o2 zNr}7x%r1|3AS%UrFo|n6hfm0 zI_4;4r$A59?sBHEXTF8#n)d9#pR@Vsz`y4#dbXjZ>c>6-8Qrva7x4Y$W!ay?h+;_rH z+R?3cev+H`8emvoqgQ8W8-lQ0vZ)m}b*gwaaLN8*DFe=Rbm6qERAjsyFJ(xsj<;>T zW=Mf&haei)i;@Y9scN^<0XecwN#udOWALb}ZzyR*JaAaz)gi3vsmv<4#dinp%>P^B zgv#9zN@OTw%3LG-gHC*6|JV?sDsmKtOU&rJ9ef_lTrpo6G_Nq8Ax*Y*XsY<34*e3_ 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b/Mathematics/1st/Fundamentals_of_mathematics/Images/intersection.tex index 490b63a..bcdb73a 100644 --- a/Mathematics/1st/Fundamentals_of_mathematics/Images/intersection.tex +++ b/Mathematics/1st/Fundamentals_of_mathematics/Images/intersection.tex @@ -1,5 +1,5 @@ \documentclass{standalone} -\usepackage{../../../../preamble_formulas} +\usepackage{../../../../preamble_tikz} \begin{document} diff --git a/Mathematics/1st/Fundamentals_of_mathematics/Images/union.pdf b/Mathematics/1st/Fundamentals_of_mathematics/Images/union.pdf index bfa58eadb02f61d52bee58496723b6123c45e854..fad675f87f67512076b1c0c43ce0934ec096465c 100644 GIT binary patch delta 3347 zcmZ{mXFMBx*T<9Cv1{*8)Qlt~Vz)M-W{sdmsoJ|m5lXqJ3l*!j*0`uun-W@z8nvrR z>z@Qa8j}s^+B1pSMuxNYUepf>uu^D4Tn1B$>&3#Lb5hH{Uul^Jg)3`*Q{V4Y|%2 ztovu{_g6|l=d5uGzLcR>Y62w3`V6V6w<*B&W@ONer&o4;f&>oY?bKS6m6{D#^m8pc zge8ct9&!tmJ+9KwIL66S%TDM+#dtz@F+r84t=ic%916xQ8GY<;%T=rQn0Z8vhP)5h;BbJ4#$T^E#eX4N3f~pJfy$C3mIQRq4Y_<`ok7 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80fa521..e6aea75 100644 --- a/Mathematics/1st/Linear_algebra/Images/first_isomorphism.tex +++ b/Mathematics/1st/Linear_algebra/Images/first_isomorphism.tex @@ -1,16 +1,17 @@ \documentclass{standalone} -\usepackage{../../../../preamble_formulas} +\usepackage{../../../../preamble_tikz} +\usepackage{../../../../preamble_math} \begin{document} \begin{tikzpicture} - %%% NODES - \path (0,0) node(x) {$U$} - (2,0) node(y) {$\im f$} (0,-2) node(z) {$\displaystyle\quot{U}{\ker f}$}; - %%% LINES - \draw[very thin,->] (x) --node[above,midway] {$f$} (y); - \draw[very thin,->] (z) --node[right,pos=0.45] {$\tilde{f}$} (y); - \draw[very thin,->] (x) --node[left,midway] {$\pi$} (z); + %%% NODES + \path (0,0) node(x) {$U$} + (2,0) node(y) {$\im f$} (0,-2) node(z) {$\displaystyle\quot{U}{\ker f}$}; + %%% LINES + \draw[very thin,->] (x) --node[above,midway] {$f$} (y); + \draw[very thin,->] (z) --node[right,pos=0.45] {$\tilde{f}$} (y); + \draw[very thin,->] (x) --node[left,midway] {$\pi$} (z); 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b/Mathematics/1st/Real-valued_functions/Images/concavity.tex @@ -1,11 +1,11 @@ \documentclass{standalone} -\usepackage{../../../../preamble_formulas} +\usepackage{../../../../preamble_tikz} \begin{document} \begin{tikzpicture} - \begin{axis}[xmin=-1,xmax=1,ymin=0,ymax=2, - axis equal image] - \addplot [thin,samples=200,smooth,blue] {2-2*x*x}; - \end{axis} + \begin{axis}[xmin=-1,xmax=1,ymin=0,ymax=2, + axis equal image] + \addplot [thin,samples=200,smooth,blue] {2-2*x*x}; + \end{axis} \end{tikzpicture} \end{document} \ No newline at end of file diff --git a/Mathematics/1st/Real-valued_functions/Images/convexity.pdf b/Mathematics/1st/Real-valued_functions/Images/convexity.pdf index 7ecb8faef48abaa1d38c25b7fa87b2ef8263057f..e22d5dfdddf7fbd09a95bb53bcb7b8ab9b83a85e 100644 GIT binary patch delta 10090 zcmaiaWl$a6wk_@!+#Lc0+qi5rxVt-SG`JIB<1WEr<1WE9xCM6!?hqunJG^}7o^$KI zdbjFzRd=nPb9D9kF=qE%YmCSX=b2cL}B|U^RK4>C~S(R zZZ0-|5rDslzxjWze-UaRS4VdjbC4?~_uq&b3M!id$id3Zni9ZGnMCQvNI8c&slqV1 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b/Mathematics/1st/Real-valued_functions/Images/discontinuity2.tex @@ -1,14 +1,14 @@ \documentclass{standalone} -\usepackage{../../../../preamble_formulas} +\usepackage{../../../../preamble_tikz} \begin{document} \begin{tikzpicture} - \begin{axis}[xmin=0,xmax=1,ymin=0,ymax=1, - axis equal image] - \addplot [thin,samples=200,smooth,blue,domain=0:0.5] {x*x}; - \addplot [thin,samples=200,smooth,blue,domain=0.5:1] {0.75-(x-1)^2}; - \draw [blue,fill=white] (axis cs:0.5,0.25) circle [radius=1.5pt]; - \draw [blue,fill=blue] (axis cs:0.5,0.5) circle [radius=1.5pt]; - \end{axis} + \begin{axis}[xmin=0,xmax=1,ymin=0,ymax=1, + axis equal image] + \addplot [thin,samples=200,smooth,blue,domain=0:0.5] {x*x}; + \addplot [thin,samples=200,smooth,blue,domain=0.5:1] {0.75-(x-1)^2}; + \draw [blue,fill=white] (axis cs:0.5,0.25) circle [radius=1.5pt]; + \draw [blue,fill=blue] (axis cs:0.5,0.5) circle [radius=1.5pt]; + \end{axis} \end{tikzpicture} \end{document} \ No newline at end of file diff --git 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a/Mathematics/1st/Real-valued_functions/Images/discontinuity4.tex +++ b/Mathematics/1st/Real-valued_functions/Images/discontinuity4.tex @@ -1,17 +1,17 @@ \documentclass{standalone} -\usepackage{../../../../preamble_formulas} +\usepackage{../../../../preamble_tikz} \begin{document} \begin{tikzpicture} - \begin{axis}[xmin=0,xmax=1,ymin=0,ymax=1, - axis equal image] - \addplot [thin,samples=200,smooth,blue,domain=0.2:1] {0.25 * sin(deg(1/x))+0.5}; - \addplot [thin,samples=2000,smooth,blue,domain=0.1:0.2] {0.25 * sin(deg(1/x))+0.5}; - \addplot [thin,samples=1000,smooth,blue,domain=0.07:0.1] {0.25 * sin(deg(1/x))+0.5}; - \addplot [thin,samples=1000,smooth,blue,domain=0.05:0.7] {0.25 * sin(deg(1/x))+0.5}; - \addplot [thin,samples=1000,smooth,blue,domain=0.03:0.5] {0.25 * sin(deg(1/x))+0.5}; - \addplot [thin,samples=2000,smooth,blue,domain=0.019:0.3] {0.25 * sin(deg(1/x))+0.5}; - \draw [rounded corners=0.1pt,thin,blue,fill=blue] (0.0025,0.75) rectangle (0.019,0.25); - \end{axis} + \begin{axis}[xmin=0,xmax=1,ymin=0,ymax=1, + axis equal image] + \addplot [thin,samples=200,smooth,blue,domain=0.2:1] {0.25 * sin(deg(1/x))+0.5}; + \addplot [thin,samples=2000,smooth,blue,domain=0.1:0.2] {0.25 * sin(deg(1/x))+0.5}; + \addplot [thin,samples=1000,smooth,blue,domain=0.07:0.1] {0.25 * sin(deg(1/x))+0.5}; + \addplot [thin,samples=1000,smooth,blue,domain=0.05:0.7] {0.25 * sin(deg(1/x))+0.5}; + \addplot [thin,samples=1000,smooth,blue,domain=0.03:0.5] {0.25 * sin(deg(1/x))+0.5}; + \addplot [thin,samples=2000,smooth,blue,domain=0.019:0.3] {0.25 * sin(deg(1/x))+0.5}; + \draw [rounded corners=0.1pt,thin,blue,fill=blue] (0.0025,0.75) rectangle (0.019,0.25); + \end{axis} \end{tikzpicture} \end{document} \ No newline at end of file diff --git a/Mathematics/1st/Real-valued_functions/Images/lower_sum.pdf b/Mathematics/1st/Real-valued_functions/Images/lower_sum.pdf index f27e921e9f0e09409f35e0a79ec3a10870b1bad1..479a35ff984bd1cb61c6688f13165519037309f3 100644 GIT binary patch delta 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b/Mathematics/1st/Real-valued_functions/Images/upper_sum.tex @@ -1,5 +1,5 @@ \documentclass{standalone} -\usepackage{../../../../preamble_formulas} +\usepackage{../../../../preamble_tikz} \begin{document} \begin{tikzpicture}[ diff --git a/Mathematics/2nd/Algebraic_structures/Images/circles.pdf b/Mathematics/2nd/Algebraic_structures/Images/circles.pdf index 5318443f4e7b1082f3c27398b92c2fad2288cc14..1e9c31141656361ca8c7c74b43b091cfc8f2faf3 100644 GIT binary patch delta 10410 zcmai)WmH^Gw)Sy?LvV-SPD2BY2Pe3@ySrNhK?1>oH%@SZYw!TUU4py2yI%e?bKg66 zX6|}FoVBXz?2=V$KXrb4*SUIyea(Q2o5bg(V5e|2u|XCPKxUIOcd&G`qTt{F@gx|- z(Sw(D23$xaFxw899_UF6i~WPJG1s7i-8_X(vVDSr zB{`M9Vy14ZsAc6|r<=v$9JNSCl9wvnJXo!_LpMh65rRcliegTqbCig>~!c z#;Gs{Q?FVl+=<2^gdcTj+y}Z^t7k5kK1mUqL%w<4SJp#18Zm(p|)COG@S9WHFGkBWh&pPw&V^3jbN%f z86Dy}(wvQzfj^p+aW*5&>qP?8bHvxjbyR^;?$Kpsva1fi4L%QLTohS}tD{5_i;i*#IanbFJMEEhvjyYb_E52Mn8D(=+lqK)klbmm~2l*r18@=C^ zkAsgPf~1x#XMHGm7-U#K5`4+P`567QM^sK3L&gvx_(Z@rLE)9-)o16V;>(+-jpyg? 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(0,-2) node(z) {$Q(R)$}; - %%% LINES - \draw[very thin,->] (x) --node[above,midway] {$\phi$} (y); - \draw[very thin,->] (z) --node[right,pos=0.45] {$\psi$} (y); - \draw[very thin,right hook->] (x) --node[left,midway] {$i$} (z); + %%% NODES + \path (0,0) node(x) {$R$} + (2,0) node(y) {$K$} (0,-2) node(z) {$Q(R)$}; + %%% LINES + \draw[very thin,->] (x) --node[above,midway] {$\phi$} (y); + \draw[very thin,->] (z) --node[right,pos=0.45] {$\psi$} (y); + \draw[very thin,right hook->] (x) --node[left,midway] {$i$} (z); \end{tikzpicture} \end{document} \ No newline at end of file diff --git a/Mathematics/2nd/Discrete_mathematics/Images/bipartite.pdf b/Mathematics/2nd/Discrete_mathematics/Images/bipartite.pdf index 06744c8f25748984c53deafb143edbbdd48de3b6..263378dac9cb34e60fc2b1bb2b941e1066e92886 100644 GIT binary patch delta 410 zcmeAbY!ICAg3)qfm*!+4#;~;cpY_+SzI!w0vAdCnzP$h2Q?sA$?W!(GFn?Ne=Yf~8 z-pNW?n+Vae?YETE-p=)0<@NbAtC)<+hV6}WXFIRSFTX{D0s`aPIXBZ_JT{b^slw%BE)*5=L=2Q>=u}{ALxlS+t 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z$lRxJMT5VxYP-*!+NpgUtEv}Tmzuc8Txro8O?b9(Jc!M^@>oaF{%&gL#*)nwjEl?2 z19A^KVWoCyt*%_D>ee90@DzksniUVQ1IWQ8vMTEC2`o6IcQsCe~QoK|B#>ZDehQCt47!utpdhXhE_7@dOe`1dU9A z|KGCuip)QY9u$U1>+^TEw!@mtcaWYtXb3r+>;_j-%#k9q`e2`6g0sk?7a{{n7`!A28LS*OLkkR-OCzHCEp~WBI2b zmzV4{*s@P0^X#Yj%8H!npvt(KT~7(i>6_si|M?oVe_0`yN8|7!RxJ&RC1TJ}EiJ1f H*3iEJRVxJ< diff --git a/Mathematics/2nd/Linear_geometry/Images/disk.tex b/Mathematics/2nd/Linear_geometry/Images/disk.tex index 662552b..a349250 100644 --- a/Mathematics/2nd/Linear_geometry/Images/disk.tex +++ b/Mathematics/2nd/Linear_geometry/Images/disk.tex @@ -1,35 +1,43 @@ \documentclass{standalone} -\usepackage{../../../../preamble_formulas} +\usepackage{../../../../preamble_tikz} \begin{document} \begin{tikzpicture}[scale=2] - \tkzDefPoint(0,0){O} - \tkzDefPoint(0,1){P} - \tkzDefPoint(-1,0){M} - \tkzDefPoint(1,0){N} - \tkzDefPoint(-0.73,0.683){A} - \tkzDefPoint(0.929,0.38){B} - \tkzDefPoint(0.978,-0.209){C} - \tkzDefPoint(-0.939,0.344){D} - \tkzDefPoint(-0.223,0.975){E} - \tkzDefPoint(-0.589,-0.808){F} - \tkzDefPoint(0.395,0.919){G} - \tkzDefPoint(0.596,-0.803){H} - \tkzDefPoint(-0.972,-0.234){I} - \tkzDefPoint(-0.768,-0.641){J} - \tkzDefPoint(-0.053,-0.999){K} - %%% LINES - \tkzDrawCircle(O,A) - \tkzClipCircle(O,A) % only print hyperbolic lines inside the unit circle. - \tkzDrawCircle[orthogonal through=A and B](O,P) - \tkzDrawCircle[orthogonal through=C and B](O,P) - \tkzDrawCircle[orthogonal through=C and D](O,P) - \tkzDrawCircle[orthogonal through=E and F](O,P) - \tkzDrawCircle[orthogonal through=G and H](O,P) - \tkzDrawCircle[orthogonal through=I and J](O,P) - \tkzDrawCircle[orthogonal through=I and K](O,P) - \draw[semithick] (-1,0) -- (1,0); + \tkzDefPoint(0,0){O} + \tkzDefPoint(0,1){P} + \tkzDefPoint(-1,0){M} + \tkzDefPoint(1,0){N} + \tkzDefPoint(-0.73,0.683){A} + \tkzDefPoint(0.929,0.38){B} + \tkzDefPoint(0.978,-0.209){C} + \tkzDefPoint(-0.939,0.344){D} + \tkzDefPoint(-0.223,0.975){E} + \tkzDefPoint(-0.589,-0.808){F} + \tkzDefPoint(0.395,0.919){G} + \tkzDefPoint(0.596,-0.803){H} + \tkzDefPoint(-0.972,-0.234){I} + \tkzDefPoint(-0.768,-0.641){J} + \tkzDefPoint(-0.053,-0.999){K} + %%% LINES + \tkzDrawCircle[thick, black](O,A) + \tkzClipCircle(O,A) % only print hyperbolic lines inside the unit circle. + \tkzDefCircle[orthogonal through=A and B](O,P) \tkzGetPoints{a}{b} + \tkzDrawCircle[semithick,black](a,b) + \tkzDefCircle[orthogonal through=C and B](O,P) \tkzGetPoints{c}{d} + \tkzDrawCircle[semithick,black](c,d) + \tkzDefCircle[orthogonal through=C and D](O,P) \tkzGetPoints{e}{f} + \tkzDrawCircle[semithick,black](e,f) + \tkzDefCircle[orthogonal through=E and F](O,P) \tkzGetPoints{g}{h} + \tkzDrawCircle[semithick,black](g,h) + \tkzDefCircle[orthogonal through=G and H](O,P) \tkzGetPoints{i}{j} + \tkzDrawCircle[semithick,black](i,j) + \tkzDefCircle[orthogonal through=I and J](O,P) \tkzGetPoints{k}{l} + \tkzDrawCircle[semithick,black](k,l) + \tkzDefCircle[orthogonal through=I and K](O,P) \tkzGetPoints{m}{n} + \tkzDrawCircle[semithick,black](m,n) + + \draw[semithick,black] (M) -- (N); \end{tikzpicture} \end{document} \ No newline at end of file diff --git a/Mathematics/2nd/Linear_geometry/Images/fano.pdf b/Mathematics/2nd/Linear_geometry/Images/fano.pdf index 481433db369bbff03d5cbf907148602ef18b6545..94c9f9b1f4b3544eb46c351d5ad369be880cf4fb 100644 GIT binary patch delta 1582 zcmai!dpOez7{`&+ex~7s67?JAp8bCNZCIq(>~YDFQn{p9+iKC~Y)dX>akR9trs?KZ zY3t^xtk4A&S=5P2QIxL2NK5JRxE{`P&UwxsJz|j4tWkQkr!YY^7 z@h#0*-S$o&6C3QxZ$Iw)>785NP0^fcn2Avq(eJOjw11AvTy&llEBn#m?yMkLVKuP{ zo27`zJe2A1=IG~PwfirCtPE2ZA|+j}&b)LXOLfn*yTY4MZ?5eP4F0WsgCc!=7vFj8 zYG;3H_IbLuINY>94v0sN#%l(Qcp9fSTdBjGd;AN|0(^c=c^)g@8!%Q z!49`0ebkTXky4fY5cdNv3B58}c`j&-5L{|!=RDdR+az&Di{8I(G-n%Zs-#s&L}i)} zOM1W32ki^gmYXCFyd5n(`vkk~E7Jlqup6QH@iP#LJJ+w2_h5Oqzpxf9Vrp5}z9V#1 zlx-A6p~;CFG)-5ASJ%cTY{sn-?=L@MtDTplkd5>lh=0?ihKypH?@-EAq?7Mc@b!m` z@_esX$6bk=ZA94~l(mzS@{-lirO#P<&qN;9SzqEk`l4#Yy6n@zj*8Pof??b8J@>f@ 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(1.5,0) arc (0:180:1.25); - \draw[very thin] (1.5,0) arc (0:180:0.25); - \draw[very thin] (-0.25,0) arc (0:180:0.75); + %%% LINES + \draw[very thin] (-2,0) -- (2,0); + \draw[very thin] (0,0) -- (0,2); + \draw[very thin] (1.5,0) -- (1.5,2); + \draw[very thin] (-1.1,0) -- (-1.1,2); + %%% CIRCLES + \draw[very thin] (1.5,0) arc (0:180:1.25); + \draw[very thin] (1.5,0) arc (0:180:0.25); + \draw[very thin] (-0.25,0) arc (0:180:0.75); \end{tikzpicture} \end{document} \ No newline at end of file diff --git a/Mathematics/2nd/Linear_geometry/Images/klein.pdf b/Mathematics/2nd/Linear_geometry/Images/klein.pdf index acca33303e8e702448d2b4f7d2cf2058ebaf1d83..e6dd95c55115a78010d91d0d5e60fc3a94da3fcb 100644 GIT binary patch delta 1112 zcmew$bDw9zB{53{1BLvgEG`=xE`6WWy!4U`1tU|l$%0Ip^{tb9^NuL+w0^g2)o*wz zpL2#YE%qx%ztMJ&0}73nzkPn!{%%^f z^PoUpOVr^9(!V~P?x>aD876tUqF`za zj}Q|>b8v(#opLhokbywk`^v6Lradh=LXQqqt%y>5!CP1$JlSoLnb(}IeYNRlq6GH3 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\draw[very thin] (-0.394,0.919) -- (1,0); - \draw[very thin] (-0.87,0.493) -- (0.922,0.388); - \draw[very thin] (-0.991,0.13) -- (0.851,0.525); - \draw[very thin] (-0.968,-0.25) -- (0.777,0.63); - \draw[very thin] (0.664,0.748) -- (0.09,-0.996); + %%% LINES + \draw[very thin] (0,1) -- (0,-1); + \draw[very thin] (-0.394,0.919) -- (1,0); + \draw[very thin] (-0.87,0.493) -- (0.922,0.388); + \draw[very thin] (-0.991,0.13) -- (0.851,0.525); + \draw[very thin] (-0.968,-0.25) -- (0.777,0.63); + \draw[very thin] (0.664,0.748) -- (0.09,-0.996); - %%% CIRCLE - \draw[very thin] (0,0) circle (1); + %%% CIRCLE + \draw[very thin] (0,0) circle (1); \end{tikzpicture} \end{document} \ No newline at end of file diff --git a/Mathematics/2nd/Linear_geometry/Images/pappus.pdf b/Mathematics/2nd/Linear_geometry/Images/pappus.pdf index 47a00be96090f90e4c9cd00c2fe9f5c207e1765f..949b04d5774fd966951b4f278873c0d311736bee 100644 GIT binary patch delta 1547 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zA@NRl%NGHOq~LhPP!C$dt7iRXzN$#YiJH5Txn^0W_R6^EtHzLHS%;Bt_0NU!FRaJM z(yRMt%_g1?EwHB8(4W8CS+=LWzg6>1_ia>eZ+B|VD~&2Agl&z#b z*~I@10}u`|T{1u~V+fAKeRLY%kGWO#OMm9x1fMm25E3BEC%Y9K+<)91L*8ij#(<@_cA@Ls$rKY+K^nWi|Je>cplBMxlS%4sIH495jkes8r zg*Q--6UYNLzyzt=_*%RoOpq2(4-A9=|Laq8a&iZ9^S|1w{YSTS0&)xdUkyp1o-h}; z04Fy;w*;SnwA6nWKZHk=o0D4tA`KQ7`1pE>0{{Ow|6j^$nEzEnxWUPIZYg(mp_2GS2>41s-8?GYJd7R--aGD2&)^YAsBQs0tpwKk zC!pv$eQpP7U5pd%ESjX;G|5%=23DRX^ru1|7qQbhf<0_+&JUrB996ycXS6){{hKgPPhO7 diff --git a/Mathematics/2nd/Numerical_methods/Images/cobweb1.tex b/Mathematics/2nd/Numerical_methods/Images/cobweb1.tex index 2d6c1a6..fb1da28 100644 --- a/Mathematics/2nd/Numerical_methods/Images/cobweb1.tex +++ b/Mathematics/2nd/Numerical_methods/Images/cobweb1.tex @@ -1,5 +1,5 @@ \documentclass{standalone} -\usepackage{../../../../preamble_formulas} +\usepackage{../../../../preamble_tikz} \begin{document} \pgfkeys{/pgf/declare function={f(\x)=cos(deg(\x));}} % Here you can declare multivariable functions like this: @@ -11,23 +11,23 @@ % speech(\t) = 1 + source(\t)*filter(\t); % } \begin{tikzpicture} - \begin{axis}[xmin = 0, xmax = 1.2,ymin = 0, ymax = 1.2, axis equal image] - \addplot [thin,samples=200,smooth,blue] {f(x)}; - \addplot [thin,samples=200,smooth,black] {x}; - \pgfplotsextra{ - \def\x{0.125} - \def\y{0} - \draw [very thin,red] (axis cs:\x,\y) \foreach \i in {0,...,16}{ - \pgfextra{ - \pgfkeys{/pgf/fpu=true, /pgf/fpu/output format=fixed} - \pgfmathparse{f(\x)}\xdef\y{\pgfmathresult} - } - -- (axis cs:\x,\y) - \pgfextra{\xdef\x{\y}} - -- (axis cs:{\x},{\y}) - };} - \draw [thin,color_green2,dashed] (axis cs:0.739,0) -- node[left,pos=0.05]{$\alpha$} (axis cs:0.739,0.739); - \legend{$g(x)$} - \end{axis} + \begin{axis}[xmin = 0, xmax = 1.2,ymin = 0, ymax = 1.2, axis equal image] + \addplot [thin,samples=200,smooth,blue] {f(x)}; + \addplot [thin,samples=200,smooth,black] {x}; + \pgfplotsextra{ + \def\x{0.125} + \def\y{0} + \draw [very thin,red] (axis cs:\x,\y) \foreach \i in {0,...,16}{ + 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-1,5 +1,5 @@ \documentclass{standalone} -\usepackage{../../../../preamble_formulas} +\usepackage{../../../../preamble_tikz} \begin{document} \pgfkeys{/pgf/declare function={f(\x)=1.65*\x-0.3;}} % Here you can declare multivariable functions to like this: @@ -11,37 +11,37 @@ % speech(\t) = 1 + source(\t)*filter(\t); % } \begin{tikzpicture} - \begin{axis}[xmin=0,xmax=1.2,ymin=0,ymax=1.2,axis equal image] - \addplot [thin,samples=200,smooth,blue] {f(x)}; - \addplot [thin,samples=200,smooth,black] {x}; - %%%%% Red lines 1 - \pgfplotsextra{ - \def\x{0.44} - \def\y{0} - \draw [very thin,red] (axis cs:\x,\y) \foreach \i in {0,...,6}{ - \pgfextra{ - \pgfkeys{/pgf/fpu=true, /pgf/fpu/output format=fixed} - \pgfmathparse{f(\x)}\xdef\y{\pgfmathresult} - } - -- (axis cs:\x,\y) - \pgfextra{\xdef\x{\y}} - -- (axis cs:{\x},{\y}) - };} - %%%%% Red lines 2 - \pgfplotsextra{ - \def\x{0.48} - \def\y{0} - \draw [very thin,red] (axis cs:\x,\y) \foreach \i in {0,...,7}{ - \pgfextra{ - \pgfkeys{/pgf/fpu=true, /pgf/fpu/output format=fixed} - \pgfmathparse{f(\x)}\xdef\y{\pgfmathresult} - } - -- (axis cs:\x,\y) - \pgfextra{\xdef\x{\y}} - -- (axis cs:{\x},{\y}) - };} - \draw [thin,color_green2,dashed] (axis cs:0.4615,0) -- node[left,pos=0.1]{$\alpha$} (axis cs:0.4615,0.4615); - \legend{$g(x)$} - \end{axis} + \begin{axis}[xmin=0,xmax=1.2,ymin=0,ymax=1.2,axis equal image] + \addplot [thin,samples=200,smooth,blue] {f(x)}; + \addplot [thin,samples=200,smooth,black] {x}; + %%%%% Red lines 1 + \pgfplotsextra{ + \def\x{0.44} + \def\y{0} + \draw [very thin,red] (axis cs:\x,\y) \foreach \i in {0,...,6}{ + \pgfextra{ + \pgfkeys{/pgf/fpu=true, /pgf/fpu/output format=fixed} + \pgfmathparse{f(\x)}\xdef\y{\pgfmathresult} + } + -- (axis cs:\x,\y) + \pgfextra{\xdef\x{\y}} + -- (axis cs:{\x},{\y}) + };} + %%%%% Red lines 2 + \pgfplotsextra{ + \def\x{0.48} + \def\y{0} + \draw [very thin,red] (axis cs:\x,\y) \foreach \i in {0,...,7}{ + \pgfextra{ + \pgfkeys{/pgf/fpu=true, /pgf/fpu/output format=fixed} + \pgfmathparse{f(\x)}\xdef\y{\pgfmathresult} + } + -- (axis cs:\x,\y) + \pgfextra{\xdef\x{\y}} + -- (axis cs:{\x},{\y}) + };} + \draw [thin,color_green2,dashed] (axis cs:0.4615,0) -- node[left,pos=0.1]{$\alpha$} (axis cs:0.4615,0.4615); + \legend{$g(x)$} + \end{axis} \end{tikzpicture} \end{document} \ No newline at end of file diff --git a/Mathematics/2nd/Numerical_methods/Images/cobweb4.pdf b/Mathematics/2nd/Numerical_methods/Images/cobweb4.pdf index 55c2741173ab7e6c3082361beaebedb0c55257fc..1815436345220bb219046fc6d42c2c9e3f4ff9a9 100644 GIT binary patch delta 15349 zcmZ{LQ*>of*JW%c72Bxb#z(ec`5Nh5YXAK4`H}GTm~sC`5^!|zRPXOMG&*PRjAoBxj{Onx7@6YCd zccwp5?_>bNiN_&db{tvhbz+)tDa3gD8G2?WYz4}?OttB;ciov;eo7D zU@v7M&okO~T0o-ZD6etNO-bjXy}SV>mpiP5rdRqb!(aB_6@A9QvEHQQ{ZSCUwbIHA z^%3{}b>EW*^Z85YewvxHqs2R)VoQ+hUh0Lx{;hRWi6MWW8|owTif487>w_fVr4uN< zS4Q=)4ibVVP;vU9I7{9u^fB5p_M3Rx?#(YT^GXEu3lmlKwtE!J?}>Cdhvw^(S;82+xw$Vvc9IY(8Krc zts^7u6^?@snE<`{r(3eCqBi0?J?5E42yr3lXkamIv|rWdOwDU5E&oB*O*@}&=4}X{ 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a/Mathematics/3rd/Differential_equations/Images/phase_space.tex b/Mathematics/3rd/Differential_equations/Images/phase_space.tex index 8c23bd0..e1638b9 100644 --- a/Mathematics/3rd/Differential_equations/Images/phase_space.tex +++ b/Mathematics/3rd/Differential_equations/Images/phase_space.tex @@ -1,5 +1,5 @@ \documentclass{standalone} -\usepackage{../../../../preamble_formulas} +\usepackage{../../../../preamble_tikz} \begin{document} @@ -20,7 +20,6 @@ samples y=10, y dir=reverse, every axis y label/.append style={at=(ticklabel* cs:0)}] - ] % blue cone \addplot3 [draw=color_blue2,samples=200,samples y=1,domain = 0:17] ({x},{\r*cos(deg(x))},{\r*sin(deg(x))}); diff --git a/Mathematics/3rd/Differential_equations/Images/rpsector.pdf b/Mathematics/3rd/Differential_equations/Images/rpsector.pdf index ccc918327d90bcd7014a4c2c186cb46524f52e84..4529a3832a037e260527c56866f3d83db058eae1 100644 GIT binary patch delta 2285 zcmai0c{J3G8a6E^LmGwgY3QfO&a7r+vNp=THI@d=$k;Q?;!9zKPf=MKm8G&3*~gyZ 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-\usepackage{../../../../preamble_formulas} +\usepackage{../../../../preamble_tikz} \begin{document} \begin{tikzpicture}[decorated curves red/.style={ diff --git a/Mathematics/3rd/Differential_equations/Images/semistable_cycle2.pdf b/Mathematics/3rd/Differential_equations/Images/semistable_cycle2.pdf index 378a36dfaa5390150e5b5addf9f32d61f7d8bd44..f2dfb26726e32c327a6b99c5acbbdfd9e15bddb4 100644 GIT binary patch delta 28398 zcmV(zK<2-J;sK@L0g%!dIUq0~Z(?c+JUj|7Ol59obZ8(lH#0Idk&GvQ#Jx+dCCzfB z*XJnC1Wv-=j}cUxQbPkMK-DP*((U;pi2{>P79^A#de{hs;r|NJ2S zSN-q%ga2>+W8W12|M#DN|LecyMgRKmKJyQsk2{GQ`R?OC`QrTCnSbj2+3U~Q3`1zOrOHWt+?7K_lp?!`1@BjAWfB4`2%fHO_XO_KxdOkaU@B8rgIm`L^ z?=D(%EnDYOP%yEK!n*-Oe+sdfYHPb;h3l+|wUul?Ec)+d_wi~Ucp?ac48xlMM~ z?&E#kJU`>lXyxX&&D(l6+1;~0*`9XY{4?9#`4e;OF@Dyca>jdzx>q&P8fE2W5u^TS$`P=Td|J*;L^52k^Wp{u7 z+;w|?UG1aIpK5E|+WS6xSnc{^E9EO-Ctx>w2_{@^-;kZt?a``z%c*(YKBqQhySIKq zxQ?yJPwju&86`hGw>=6%`<5ph%WkcXKmFtv+E3V8n&|d_R(A+nwI`{QRck3L=T`fy 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b/Mathematics/3rd/Differential_equations/Images/singularities_dim2.tex index 26deade..8b3b2f0 100644 --- a/Mathematics/3rd/Differential_equations/Images/singularities_dim2.tex +++ b/Mathematics/3rd/Differential_equations/Images/singularities_dim2.tex @@ -1,5 +1,7 @@ \documentclass{standalone} -\usepackage{../../../../preamble_formulas} +\usepackage{../../../../preamble_tikz} +\usepackage{../../../../preamble_math} + \begin{document} \begin{tikzpicture}[every node/.style={scale=0.7}] diff --git a/Mathematics/3rd/Differential_equations/Images/unstable_cycle.pdf b/Mathematics/3rd/Differential_equations/Images/unstable_cycle.pdf index a1f43b1120834b926cc39e8778cefe37b2aaccdf..ea0ddbc3d626f712176d69c36cbd11b23a2074dd 100644 GIT binary patch delta 28162 zcmV(vKs$WA z1^?k5|MVsQ$Irj~U;1_B&%V1-vhf6w2}fA@X(?>Wo) z`EM>-b1hr+TI!rBJG(TKt=UV;R;l&??N2MK+?3UB@2~yY^VTPt_Kp2dukFn5vbjxm z*6!ooZl0g!}s$Tnjea{enU{?9aVdsggd==pkIQxbWQW3vR-(ioK9#OVZX^A#7{e zD(<~TLGd0(uF-_Z6Vs?0=Z)Yz!y 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a/Mathematics/3rd/Differential_geometry/Images/theorem_immersions.tex b/Mathematics/3rd/Differential_geometry/Images/theorem_immersions.tex index 2fb7e5a..f482afd 100644 --- a/Mathematics/3rd/Differential_geometry/Images/theorem_immersions.tex +++ b/Mathematics/3rd/Differential_geometry/Images/theorem_immersions.tex @@ -1,5 +1,6 @@ \documentclass{standalone} -\usepackage{../../../../preamble_formulas} +\usepackage{../../../../preamble_tikz} +\usepackage{../../../../preamble_math} \begin{document} diff --git a/Mathematics/3rd/Differential_geometry/Images/theorem_submersions.pdf b/Mathematics/3rd/Differential_geometry/Images/theorem_submersions.pdf index 7bafdf60e2e1fe34498ffb989bd6be262fa806c4..10e612faa3e8a22230e51598d05e0f781bbea235 100644 GIT binary patch delta 7964 zcmd6sWmFv9x~&@suECuq5G+{Z+PDQvaDuzLYXiYOG=vZ=1cv|#?i$=BxVyXS>3rYb z=k9%P_PFQg`O#IYRvbXuE}8w42RF@JF>XJIiel%GyOHa6Z*s9h^ z*0v2rz}bv18@#==aj}A8v)dLA@@5YlwP3}_yn>A9m$k=(kjV_6;0Z?I?kqg-j~~{? 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a/Mathematics/3rd/Topology/Images/triangularization.tex +++ b/Mathematics/3rd/Topology/Images/triangularization.tex @@ -1,5 +1,6 @@ \documentclass[border=0.05cm]{standalone} -\usepackage{../../../../preamble_formulas} +\usepackage{../../../../preamble_tikz} +\usepackage{../../../../preamble_math} \def\sepArrow{0.56} \def\sepArrowDouble{0.615} diff --git a/Mathematics/4th/Dynamical_systems/Images/blowdown.pdf b/Mathematics/4th/Dynamical_systems/Images/blowdown.pdf index 170c5deaacefb303b5763b2a9d23bb6350a57c34..f1899651250f73a493f0dee8063d9b3d93f07d7a 100644 GIT binary patch delta 6222 zcmaiYXEfXm_pRtguhESXMrZVzVDu<6(Q9;~_ozPv(c9=Pm?z2@J%|=vbdOGyC_yqo zgy;$Xy#MvCyY9MQ?zeN!-e;XJ`73p5zLL zJh9iFVz-xL?`OO1cyQ$1_x_K)O202{#}{YXKHe+i1E;_nj-Ah;Ihtel zwvU0k^ABI=y^pV>Zw1|VNbdmm)ZwW6OnNLyWCZZ-wjYUQGA*iV3;xB?;JWe0BCMWs zCGQmUduBK9KK%FT%*#{ZdiN4zw9VmiUSv*>C)U%&x|Oo!VRqH;KQ`;Ve!CaVzF{f8 zQ`VQ?^*CPzt@3-`#Pup~zF*Ri+G@7g?Y~sJ`+2vqb~bSz8XmfCk#K)Z9)8nv6yA6v 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b/Mathematics/4th/Dynamical_systems/Images/nilpot-j.tex index a9fc3bf..8347769 100644 --- a/Mathematics/4th/Dynamical_systems/Images/nilpot-j.tex +++ b/Mathematics/4th/Dynamical_systems/Images/nilpot-j.tex @@ -1,5 +1,5 @@ \documentclass{standalone} -\usepackage{../../../../preamble_formulas} +\usepackage{../../../../preamble_tikz} \begin{document} \begin{tikzpicture}[decorated curves/.style={ diff --git a/Mathematics/4th/Dynamical_systems/Images/nilpot-k.pdf b/Mathematics/4th/Dynamical_systems/Images/nilpot-k.pdf index 9c07f7ed42df08767b40f752782b9bfcdcaebce9..5f8f19ccf8d95346d60db96cc62cf7466f228bb9 100644 GIT binary patch delta 7913 zcmaiZWl)?!mo@G(5F7#n8Qf*?1h-(p-5G+r+XDm(7Muik9SH93?jC{#2~KbbkmcRl z@B6h?`{#7uzPC@`>b_mo)u&8qMk0QeR0zxkb~3lc5*5YbRDwBvbhiczaB=e|(a_ZM~m#mZcYjM5>wnwpx0`=4AT)j7v>Y>`t0Uwb{b*75sSMy0A5nr?4Eii!- z3c68_EEU8w3gFqs29|7idH zH?8abH9%9t@VI@+p2B$jxWySbMtE`oq+T00YehYkK^3(6OjM=p37V{1%6y)G!k*5i zRDt`)E3XbWZ%g^e$L?L>HJ+lI^NYfqo8)x;VD5Kswtd$NMc-^9i!pDHQfm~cWgEhz 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a/Mathematics/4th/Dynamical_systems/Images/nilpot-l.tex b/Mathematics/4th/Dynamical_systems/Images/nilpot-l.tex index 89a80d0..026e098 100644 --- a/Mathematics/4th/Dynamical_systems/Images/nilpot-l.tex +++ b/Mathematics/4th/Dynamical_systems/Images/nilpot-l.tex @@ -1,5 +1,5 @@ \documentclass{standalone} -\usepackage{../../../../preamble_formulas} +\usepackage{../../../../preamble_tikz} \begin{document} \begin{tikzpicture}[decorated curves/.style={ diff --git a/Mathematics/4th/Dynamical_systems/Images/nilpot-m.pdf b/Mathematics/4th/Dynamical_systems/Images/nilpot-m.pdf index 719f583ba6198f2c29bda469b997d425f7c0ae20..cf70672aeff010c5a7231c58453a430fead9b906 100644 GIT binary patch delta 7681 zcmaiZRZyH!vn?)zy9Em}Xc%0AyUXD2PH+$5!`+AA!QFxecL^RGf&_ObNFd0~f9>U* zs`Jva_Ui8Hs$IKk*Xoqol1Q2*5u}08xL7!VgoQyI%GS=do^~|+5N@vISSFy$NFl*$ zDS7QT;VZ)J)xP;EMGc0*4Ln*?6AkiB<<|`x(N5Y1`jjB18+C;rQPeK3{QUg)BbQHS z=s!#YK|NO&ucCjhI1-=k)wU)*<_}(}vUN5J&}sRmLK1{57_=i 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b/Mathematics/4th/Dynamical_systems/Images/saddle-node_neg.tex @@ -1,5 +1,5 @@ \documentclass{standalone} -\usepackage{../../../../preamble_formulas} +\usepackage{../../../../preamble_tikz} \begin{document} \begin{tikzpicture}[decorated curves/.style={ diff --git a/Mathematics/4th/Dynamical_systems/Images/saddle-node_pos.pdf b/Mathematics/4th/Dynamical_systems/Images/saddle-node_pos.pdf index e222f40578929c0be1805531eb1db3a6af2b8a70..174477829c15ebc56e9acb5b407a1c6c01c973fa 100644 GIT binary patch delta 4841 zcmai2bx_ogw?$B5=@RKB78Y1ycj;Ifl=#x!u{68Hf+A(mR0N6Zp)u|_4=_>= zA(Ijh1(tHg>tNdSnwuvg=VTb6l_Vj?BvI1P#}G$HJ~-jTN`Vj2wf)7#hmVzxWO1ac zx*DdM?i!%5cv;+g^(*zy<@w3k;1y^6KIm%t@+Wur zXO^(*$@}4hCw$%nlB=_v53?Z~kq*roORszB07AUYkX7ABH$CjpNEe%ShjVU+uDy2$ zB@h;NHuGbAWoK&9-l8CX@y}pA?U}IBaAo;q_>iqvPvw_yNg@92T`kzf3m2ng_wVRXF|AfA)wr=eDMhN=kUkHr>5!fgOf!$nYF{+u%_H- zP2f6Tf0&`3-sax*dY9t1l25o>lJ)GTR=Jg(vw$BK${NR^(Y>g8-CTU5M+>es%kme4JH~pHg-Yr` 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b/Physics/Basic/Classical_mechanics/Images/areal_vel.tex index 96fd1a3..f3ed224 100644 --- a/Physics/Basic/Classical_mechanics/Images/areal_vel.tex +++ b/Physics/Basic/Classical_mechanics/Images/areal_vel.tex @@ -1,5 +1,5 @@ \documentclass{standalone} -\usepackage{../../../../preamble_formulas} +\usepackage{../../../../preamble_tikz} \def\a{2} \def\b{1.5} \begin{document} diff --git a/Physics/Basic/Classical_mechanics/Images/conics.pdf b/Physics/Basic/Classical_mechanics/Images/conics.pdf index 2666bdcfa06b787951b187b18346038cc0e12be0..fea32ef71f4ab98135372f20b1d330bdd5238003 100644 GIT binary patch delta 9489 zcmaiaWl$VYvnB5C4#Axn+}#=6A-KD{3~qrS1Hl~<+$FdNcXxN!KnU>i?Y`aGAFpbw z?vL(Ur@H_2J>6aBu1-V$nuJc8BNU(lP`Q}fqX-M5aDc3wZ9HwMc=-X`5HmPN2p^&i z82kuc#(dj;mgU&*`JLbM=EnQSKk)uJcw(e%$P}<~!~%IK8IqXXDCm z-6Yl~#vm_}Krpv{@n8hemg%Ow;{3!Y}GIz71<2YyEKwqa#wayIk%X) zZ(IVeBiFFy+&4b3)9I5_Zig^pGqCC-)m~GafIGYYj-QX`|1$s1|6oKM$6@Jb9uBk3-_o7dMKMSgN%ELU3tEW+ett-2=|S%`Sc`{2 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a/Physics/Basic/Structure_of_matter_and_thermodynamics/Images/carnot.tex b/Physics/Basic/Structure_of_matter_and_thermodynamics/Images/carnot.tex index 4aa452b..39e803a 100644 --- a/Physics/Basic/Structure_of_matter_and_thermodynamics/Images/carnot.tex +++ b/Physics/Basic/Structure_of_matter_and_thermodynamics/Images/carnot.tex @@ -1,5 +1,6 @@ \documentclass{standalone} -\usepackage{../../../../preamble_formulas} +\usepackage{../../../../preamble_tikz} +\usepackage{../../../../preamble_math} \begin{document} \begin{tikzpicture}[scale=1, every node/.style={scale=0.5}] diff --git a/Physics/Basic/Structure_of_matter_and_thermodynamics/Images/decay_chain.pdf b/Physics/Basic/Structure_of_matter_and_thermodynamics/Images/decay_chain.pdf index 5bae981a77fb7adf3390e2a8cbeea800f7e8dc54..22eb06df2aa55173c19faccea7944affe2414307 100644 GIT binary patch delta 2301 zcmai0c{J3E9!Ci?WW+03uAfl2G4q=-8)GzRELk!}W6h8ydBe?&Woj6tQd-m?TOx$$ z-kMN@WEZAnsj)^>DmAz46t3>M@1A$gedoOYzMt=UzUOnk%XwdcTWariJb(bg{f|Ki 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b/Physics/Basic/Structure_of_matter_and_thermodynamics/Images/fermi.tex @@ -1,5 +1,6 @@ \documentclass{standalone} -\usepackage{../../../../preamble_formulas} +\usepackage{../../../../preamble_tikz} +\usepackage{../../../../preamble_math} \begin{document} \begin{tikzpicture} diff --git a/Physics/Basic/Structure_of_matter_and_thermodynamics/Images/heat_engine.pdf b/Physics/Basic/Structure_of_matter_and_thermodynamics/Images/heat_engine.pdf index c6eb4e0bcdabfb684cc30d7c5bd71efc621b3b32..c73ceff3eaba45b05ad988d2775149e92e860296 100644 GIT binary patch delta 5486 zcmeI0c{CLMyT_$$gODXc4Ms#UGng5Su@AD0o$Rvj%ODh!T}5F+!l1~WovabrvXgzw z$dgU64N1h|Jim;3=}rGTjf$-fy2O6d&qWVvl5HRL>X0{eQcu63b>rmS6w@AfaSnWvy8033hMil_ODyzd^hL2_vDB`?5UiMVyeJ!&6q*@?{ zO^7wSPHPwnYb&~Po8pU+iGj_NI{Mjmu0A&jv|n3h<9iEdN1Uv&{8|w8R81@T2dVsm z%;TkRMYA51Vm8ioGm3Y$2Y06XrJmeY<^X?=_3PkIhfObs8#zV>qJ9A0$(0XX?LTZ4mi&)~lGIg$4SU5vnys7|0`&(r&RuTkZ$;vWwfd 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Options are anchor=center,south west,south east,north west,north east,north,south,west... - ] - \shade[shading=rainbow,shading angle=90,opacity=0.3] (axis cs:400e-9,0) rectangle (axis cs:700e-9,5e13); - \node[font=\tiny, black!50] at (axis cs: 300e-9,4.78e13) {\textbf{UV}}; - \node[font=\tiny, black!50] at (axis cs: 550e-9,4.78e13) {\textbf{VISIBLE}}; - \node[font=\tiny, black!50] at (axis cs: 1850e-9,4.78e13) {\textbf{INFRARED}}; + \def\c{299792458} + \def\h{6.626070*10^(-34)} + \def\kb{1.380649*10^(-23)} + \begin{axis}[ + width=12.5cm, + height=10cm, + xmin = 1e-7, + xmax = 3e-6, + %xtick distance = 0.001, + ymin = -1e10, + ymax = 5e13, + % axis equal image, + ylabel={Spectral radiance [$\text{W}\cdot\text{m}^{-3}$]}, + xlabel={Wavelength [m]}, + legend cell align = {left}, % + legend style={at={(axis cs:2.94e-6,4.5e13)},anchor=north east} % position of the legend box and anchor is the point on the box to be fitted exactly at the point of cs:<>,<>. Options are anchor=center,south west,south east,north west,north east,north,south,west... + ] + \shade[shading=rainbow,shading angle=90,opacity=0.3] (axis cs:400e-9,0) rectangle (axis cs:700e-9,5e13); + \node[font=\tiny, black!50] at (axis cs: 300e-9,4.78e13) {\textbf{UV}}; + \node[font=\tiny, black!50] at (axis cs: 550e-9,4.78e13) {\textbf{VISIBLE}}; + \node[font=\tiny, black!50] at (axis cs: 1850e-9,4.78e13) {\textbf{INFRARED}}; - % Planck - \addplot [thin,samples=200,smooth,blue,domain=1e-7:3e-6] {2*pi*\h*\c*\c/(x^5*(e^(\h*\c/(x*\kb*5000))-1))}; % 5000 K - \addplot [thin,samples=200,smooth,color_green2,domain=1e-7:3e-6] {2*pi*\h*\c*\c/(x^5*(e^(\h*\c/(x*\kb*4000))-1))}; % 4000 K - \addplot [thin,samples=200,smooth,red,domain=1e-7:3e-6] {2*pi*\h*\c*\c/(x^5*(e^(\h*\c/(x*\kb*3000))-1))}; % 3000 K - % Rayleigh-Jeans - \addplot [thin,samples=200,smooth,black,domain=1.25e-6:3e-6] {2*pi*\c*\kb*5000/x^4}; % 5000 K - \legend{5000 K - Planck,4000 K - Planck,3000 K - Planck,5000 K - Rayleigh-Jeans} - \end{axis} + % Planck + \addplot [thin,samples=200,smooth,blue,domain=1e-7:3e-6] {2*pi*\h*\c*\c/(x^5*(e^(\h*\c/(x*\kb*5000))-1))}; % 5000 K + \addplot [thin,samples=200,smooth,color_green2,domain=1e-7:3e-6] {2*pi*\h*\c*\c/(x^5*(e^(\h*\c/(x*\kb*4000))-1))}; % 4000 K + \addplot [thin,samples=200,smooth,red,domain=1e-7:3e-6] {2*pi*\h*\c*\c/(x^5*(e^(\h*\c/(x*\kb*3000))-1))}; % 3000 K + % Rayleigh-Jeans + \addplot [thin,samples=200,smooth,black,domain=1.25e-6:3e-6] {2*pi*\c*\kb*5000/x^4}; % 5000 K + \legend{5000 K - Planck,4000 K - Planck,3000 K - Planck,5000 K - Rayleigh-Jeans} + \end{axis} \end{tikzpicture} \end{document} \ No newline at end of file diff --git a/main_math.idx b/main_math.idx index 9112fa2..cb3b813 100644 --- a/main_math.idx +++ b/main_math.idx @@ -1004,6 +1004,7 @@ \indexentry{Newton's divided differences\\method|hyperpage}{93} \indexentry{equally-spaced|hyperpage}{93} \indexentry{Hermite interpolation problem|hyperpage}{94} +\indexentry{Runge's phenomenon|hyperpage}{94} \indexentry{Spline|hyperpage}{94} \indexentry{spline|hyperpage}{94} \indexentry{cubic splines|hyperpage}{94} @@ -2945,10 +2946,6 @@ \indexentry{Riesz representation theorem|hyperpage}{312} \indexentry{adjoint operator|hyperpage}{312} \indexentry{self-adjoint|hyperpage}{312} -\indexentry{continuous|hyperpage}{313} -\indexentry{coercive|hyperpage}{313} -\indexentry{symmetric|hyperpage}{313} -\indexentry{Lax-Milgram theorem|hyperpage}{313} \indexentry{orthogonal system|hyperpage}{313} \indexentry{orthonormal system|hyperpage}{313} \indexentry{complete|hyperpage}{313} @@ -2961,7 +2958,7 @@ \indexentry{Fourier transform|hyperpage}{313} \indexentry{Riesz-Fischer theorem|hyperpage}{313} \indexentry{Parseval identity|hyperpage}{313} -\indexentry{Spectral theorem|hyperpage}{314} +\indexentry{Spectral theorem|hyperpage}{313} \indexentry{Hilbert-Schmidt spectral representation theorem|hyperpage}{314} \indexentry{Fredholm alternative|hyperpage}{314} \indexentry{Law of total probability|hyperpage}{315} @@ -3087,245 +3084,282 @@ \indexentry{rotation number|hyperpage}{338} \indexentry{semi-conjugate|hyperpage}{338} \indexentry{conjugate|hyperpage}{338} -\indexentry{measure on $\mathcal {C}(\ensuremath {\mathbb {T}}^1)$|hyperpage}{338} -\indexentry{probability measure|hyperpage}{338} -\indexentry{pushforward measure|hyperpage}{339} +\indexentry{measure on $\mathcal {C}(\ensuremath {\mathbb {T}}^1)$|hyperpage}{339} +\indexentry{probability measure|hyperpage}{339} +\indexentry{push-forward measure|hyperpage}{339} \indexentry{invariant|hyperpage}{339} \indexentry{$F$-invariant|hyperpage}{339} -\indexentry{$\sigma $-algebra|hyperpage}{340} -\indexentry{$\sigma $-algebra|hyperpage}{340} -\indexentry{measurable space|hyperpage}{340} -\indexentry{$\sigma $-algebra generated by $\mathcal {F}$|hyperpage}{340} -\indexentry{measurable|hyperpage}{340} -\indexentry{Measure|hyperpage}{340} -\indexentry{measure|hyperpage}{340} -\indexentry{measure space|hyperpage}{340} -\indexentry{integral of $f$ with respect to $\mu $|hyperpage}{340} -\indexentry{integral of $f$ with respect to $\mu $|hyperpage}{340} -\indexentry{Monotone convergence theorem|hyperpage}{340} -\indexentry{Fatou's lemma|hyperpage}{340} -\indexentry{Dominated convergence theorem|hyperpage}{340} -\indexentry{Product measure|hyperpage}{340} -\indexentry{product measure|hyperpage}{340} -\indexentry{$\sigma $-finite|hyperpage}{340} -\indexentry{Fubini|hyperpage}{340} -\indexentry{density|hyperpage}{341} -\indexentry{absolutely continuous|hyperpage}{341} -\indexentry{Radon-Nikodym|hyperpage}{341} -\indexentry{probability space|hyperpage}{341} -\indexentry{events|hyperpage}{341} -\indexentry{Random variable|hyperpage}{341} -\indexentry{$E$-valued random variable|hyperpage}{341} -\indexentry{Expectation|hyperpage}{341} -\indexentry{expectation of $X$|hyperpage}{341} -\indexentry{law of $X$|hyperpage}{341} -\indexentry{$\sigma $-algebra generated by $X$|hyperpage}{341} -\indexentry{Jensen's inequality|hyperpage}{341} -\indexentry{conditional expectation of $X$ given $\mathcal {G}$|hyperpage}{341} -\indexentry{Tower property|hyperpage}{342} -\indexentry{conditional expectation of $X$ given $Y$|hyperpage}{342} -\indexentry{probability kernel|hyperpage}{342} -\indexentry{conditional law of $X$ given $Y$|hyperpage}{342} -\indexentry{filtration|hyperpage}{342} -\indexentry{filtered probability space|hyperpage}{342} -\indexentry{adapted|hyperpage}{342} -\indexentry{martingale|hyperpage}{342} -\indexentry{submartingale|hyperpage}{342} -\indexentry{supermartingale|hyperpage}{342} -\indexentry{stopping time|hyperpage}{342} -\indexentry{stopped process|hyperpage}{342} -\indexentry{uniformly integrable|hyperpage}{342} -\indexentry{converges weakly|hyperpage}{343} -\indexentry{converges strongly|hyperpage}{343} -\indexentry{converges weakly-*|hyperpage}{343} -\indexentry{Banach-Alaoglu theorem|hyperpage}{343} -\indexentry{strongly lower-semicontinuous|hyperpage}{343} -\indexentry{weakly lower-semicontinuity|hyperpage}{343} -\indexentry{weak-* lower-semicontinuity|hyperpage}{343} -\indexentry{support plane|hyperpage}{343} -\indexentry{Sobolev spaces|hyperpage}{344} -\indexentry{Sobolev spaces|hyperpage}{344} -\indexentry{Poincaré's inequality|hyperpage}{344} -\indexentry{Poincaré-Wirtinger's inequality|hyperpage}{344} -\indexentry{embedded|hyperpage}{345} -\indexentry{compactly embedded|hyperpage}{345} -\indexentry{Gagliardo, Nirengerg and Sobolev's inequality|hyperpage}{345} -\indexentry{Hölder continuity|hyperpage}{345} -\indexentry{$\mathcal {C}^{k,\theta }$-Hölder continuous|hyperpage}{345} -\indexentry{Morrey's embedding|hyperpage}{345} -\indexentry{extension|hyperpage}{345} -\indexentry{extension operator|hyperpage}{345} -\indexentry{first order reflection|hyperpage}{346} -\indexentry{Reillich-Kondrachov's compactness theorem|hyperpage}{346} -\indexentry{trace operator|hyperpage}{346} -\indexentry{Gradient descent algorithm|hyperpage}{347} -\indexentry{gradient descent algorithm|hyperpage}{347} -\indexentry{Armijo-type rule|hyperpage}{347} -\indexentry{Frank-Wolfe-type method|hyperpage}{347} -\indexentry{$L$-co-coercive|hyperpage}{347} -\indexentry{firmly non-expansive|hyperpage}{347} -\indexentry{Baillon-Haddad|hyperpage}{347} -\indexentry{strongly convex|hyperpage}{347} -\indexentry{$\gamma $-convex|hyperpage}{347} -\indexentry{Newton method|hyperpage}{348} -\indexentry{Newton method|hyperpage}{348} -\indexentry{multistep method|hyperpage}{348} -\indexentry{Heavy ball method|hyperpage}{348} -\indexentry{Conjugate gradient|hyperpage}{348} -\indexentry{conjugate gradient method|hyperpage}{348} -\indexentry{Nesterov's accelerated gradient method|hyperpage}{348} -\indexentry{Nesterov's method|hyperpage}{348} -\indexentry{subgradient descent method|hyperpage}{348} -\indexentry{implicit descent|hyperpage}{348} -\indexentry{nonexpansive|hyperpage}{348} -\indexentry{avergaed|hyperpage}{348} -\indexentry{Opial's lemma|hyperpage}{348} -\indexentry{Krasnoselskii-Mann's convergence theorem|hyperpage}{349} -\indexentry{linear second-order PDE|hyperpage}{350} -\indexentry{non-divergence form|hyperpage}{350} -\indexentry{divergence form|hyperpage}{350} -\indexentry{Dirichlet boundary condition|hyperpage}{350} -\indexentry{homogeneous|hyperpage}{350} -\indexentry{Neumann boundary condition|hyperpage}{350} -\indexentry{homogeneous|hyperpage}{350} -\indexentry{uniformly elliptic|hyperpage}{350} -\indexentry{Abstract Fredholm alternative|hyperpage}{350} -\indexentry{weak formulation|hyperpage}{350} -\indexentry{formal adjoint|hyperpage}{350} -\indexentry{homogeneous adjoint problem|hyperpage}{350} -\indexentry{resolvent set|hyperpage}{351} -\indexentry{spectrum|hyperpage}{351} -\indexentry{Inner regularity|hyperpage}{351} -\indexentry{Regularity up to the boundary|hyperpage}{351} -\indexentry{Weak maximum principle|hyperpage}{351} -\indexentry{Weak maximum principle|hyperpage}{351} -\indexentry{strong solution|hyperpage}{352} -\indexentry{classical solution|hyperpage}{352} -\indexentry{Weak maximum principle|hyperpage}{352} -\indexentry{Hopf's lemma|hyperpage}{352} -\indexentry{interior ball condition|hyperpage}{352} -\indexentry{Strong maximum principle|hyperpage}{352} -\indexentry{Continuation method|hyperpage}{352} -\indexentry{Brower fixed point|hyperpage}{353} -\indexentry{Schauder fixed point|hyperpage}{353} -\indexentry{Schaefer fixed point|hyperpage}{353} -\indexentry{Montecarlo estimator|hyperpage}{354} -\indexentry{seed|hyperpage}{354} -\indexentry{Mersenne Twister algorithm|hyperpage}{354} -\indexentry{Acceptance-rejection method|hyperpage}{354} -\indexentry{Box-Muller method|hyperpage}{355} -\indexentry{Polar method|hyperpage}{355} -\indexentry{antithetic method|hyperpage}{355} -\indexentry{control variate|hyperpage}{355} -\indexentry{multiple control variate estimator|hyperpage}{356} -\indexentry{equivalent|hyperpage}{356} -\indexentry{importance sampling estimator|hyperpage}{356} -\indexentry{Euler method|hyperpage}{357} -\indexentry{continuous Euler scheme|hyperpage}{357} -\indexentry{Strong error of the Euler scheme|hyperpage}{357} -\indexentry{Weak error of the Euler scheme|hyperpage}{357} -\indexentry{Romberg Extrapolation|hyperpage}{357} -\indexentry{Romberg Extrapolation|hyperpage}{357} -\indexentry{Dirichlet boundary conditions|hyperpage}{359} -\indexentry{Neumann boundary conditions|hyperpage}{359} -\indexentry{Robin boundary conditions|hyperpage}{359} -\indexentry{conforming Galerkin method|hyperpage}{359} -\indexentry{Céa's lemma|hyperpage}{359} -\indexentry{finite element|hyperpage}{359} -\indexentry{basis functions|hyperpage}{360} -\indexentry{local interpolant|hyperpage}{360} -\indexentry{subdivision|hyperpage}{360} -\indexentry{global interpolant|hyperpage}{360} -\indexentry{triangulation|hyperpage}{360} -\indexentry{affinely equivalent|hyperpage}{360} -\indexentry{Bramble-Hilbert lemma|hyperpage}{360} -\indexentry{diameter|hyperpage}{360} -\indexentry{insphere diameter|hyperpage}{360} -\indexentry{condition number|hyperpage}{360} -\indexentry{Local interpolation error|hyperpage}{361} -\indexentry{regular|hyperpage}{361} -\indexentry{Global interpolation error|hyperpage}{361} -\indexentry{spectral methods|hyperpage}{361} -\indexentry{trial functions|hyperpage}{361} -\indexentry{globally smooth|hyperpage}{361} -\indexentry{Galerkin methods|hyperpage}{361} -\indexentry{Collocation methods|hyperpage}{361} -\indexentry{collocation points|hyperpage}{361} -\indexentry{$\tau $ methods|hyperpage}{361} -\indexentry{interpolant|hyperpage}{361} -\indexentry{Runge phenomenon|hyperpage}{362} -\indexentry{Chebyshev polynomials|hyperpage}{362} -\indexentry{interpolant|hyperpage}{362} -\indexentry{jointly Gaussian|hyperpage}{363} -\indexentry{Brownian motion|hyperpage}{363} -\indexentry{Strong law of large numbers for Brownian motion|hyperpage}{363} -\indexentry{Markov property for Brownian motion|hyperpage}{363} -\indexentry{natural filtration|hyperpage}{363} -\indexentry{Martingale|hyperpage}{363} -\indexentry{martingale|hyperpage}{363} -\indexentry{adapted|hyperpage}{363} -\indexentry{sub-martingale|hyperpage}{363} -\indexentry{super-martingale|hyperpage}{363} -\indexentry{hitting time|hyperpage}{364} -\indexentry{Doob's optional sampling theorem|hyperpage}{364} -\indexentry{stopped process|hyperpage}{364} -\indexentry{Orthogonality of martingales|hyperpage}{364} -\indexentry{Doob's maximal inequality|hyperpage}{364} -\indexentry{absoulte variation|hyperpage}{364} -\indexentry{finite variation|hyperpage}{364} -\indexentry{Quadratic variation|hyperpage}{364} -\indexentry{mesh|hyperpage}{364} -\indexentry{continuous local martingale|hyperpage}{365} -\indexentry{localizing sequence|hyperpage}{365} -\indexentry{Doob's optional sampling theorem for local martingales|hyperpage}{365} -\indexentry{Levy's characterization of Brownian motion|hyperpage}{366} -\indexentry{isometry|hyperpage}{366} -\indexentry{partial isometry|hyperpage}{366} -\indexentry{Wiener integral|hyperpage}{366} -\indexentry{Wiener isometry|hyperpage}{366} -\indexentry{Wiener integral|hyperpage}{366} -\indexentry{Wiener integral|hyperpage}{366} -\indexentry{Chasles relation|hyperpage}{366} -\indexentry{progressive|hyperpage}{367} -\indexentry{Itô integral|hyperpage}{368} -\indexentry{Itô isometry|hyperpage}{368} -\indexentry{Itô integral|hyperpage}{368} -\indexentry{generalized Itô integral|hyperpage}{368} -\indexentry{Stochastic dominated convergence theorem|hyperpage}{368} -\indexentry{Itô process|hyperpage}{369} -\indexentry{martingale term|hyperpage}{369} -\indexentry{drift term|hyperpage}{369} -\indexentry{stochastic differential|hyperpage}{369} -\indexentry{quadratic variation|hyperpage}{369} -\indexentry{Stochastic integration by parts|hyperpage}{370} -\indexentry{Itô term|hyperpage}{370} -\indexentry{Itô's formula|hyperpage}{370} -\indexentry{Doléans-Dade exponential|hyperpage}{370} -\indexentry{Novikov's condition|hyperpage}{370} -\indexentry{Giranov's theorem|hyperpage}{371} -\indexentry{drift|hyperpage}{371} -\indexentry{diffusion|hyperpage}{371} -\indexentry{stochastic differential equation|hyperpage}{371} -\indexentry{SDE|hyperpage}{371} -\indexentry{solution of the SDE|hyperpage}{371} -\indexentry{Gronwall's lemma|hyperpage}{371} -\indexentry{Existence and uniqueness of solutions of SDEs|hyperpage}{371} -\indexentry{Langevin equation|hyperpage}{371} -\indexentry{Geometric Brownian motion|hyperpage}{372} -\indexentry{Black-Scholes process|hyperpage}{372} -\indexentry{homogeneous SDE|hyperpage}{372} -\indexentry{diffusions|hyperpage}{372} -\indexentry{Invariance under time shift|hyperpage}{372} -\indexentry{Generator|hyperpage}{372} -\indexentry{generator|hyperpage}{372} -\indexentry{Kolmogorov's equation|hyperpage}{372} -\indexentry{Feynman-Kac's formula|hyperpage}{373} -\indexentry{Itô's formula|hyperpage}{374} -\indexentry{Flow property|hyperpage}{374} -\indexentry{Dynamic programming principle|hyperpage}{374} -\indexentry{Fokker-Planck equation|hyperpage}{374} -\indexentry{control parameter|hyperpage}{375} -\indexentry{Finite horizon problem|hyperpage}{375} -\indexentry{running cost|hyperpage}{375} -\indexentry{terminal cost|hyperpage}{375} -\indexentry{Infinte horizon problem|hyperpage}{375} +\indexentry{orbit|hyperpage}{339} +\indexentry{positive orbit|hyperpage}{339} +\indexentry{negative orbit|hyperpage}{339} +\indexentry{positively invariant|hyperpage}{340} +\indexentry{negatively invariant|hyperpage}{340} +\indexentry{invariant|hyperpage}{340} +\indexentry{positively recurrent points|hyperpage}{340} +\indexentry{negatively recurrent points|hyperpage}{340} +\indexentry{wandering point|hyperpage}{340} +\indexentry{wandering domain|hyperpage}{340} +\indexentry{non-wandering|hyperpage}{340} +\indexentry{minimal|hyperpage}{340} +\indexentry{minimal|hyperpage}{340} +\indexentry{measure of $U$|hyperpage}{340} +\indexentry{measure of $A$|hyperpage}{340} +\indexentry{support of $\mu $|hyperpage}{340} +\indexentry{Cantor set|hyperpage}{341} +\indexentry{uniquely ergodic|hyperpage}{341} +\indexentry{bounded variation|hyperpage}{342} +\indexentry{Denjoy-Koksma inequality|hyperpage}{342} +\indexentry{Denjoy theorem|hyperpage}{343} +\indexentry{$\sigma $-algebra|hyperpage}{344} +\indexentry{$\sigma $-algebra|hyperpage}{344} +\indexentry{measurable space|hyperpage}{344} +\indexentry{$\sigma $-algebra generated by $\mathcal {F}$|hyperpage}{344} +\indexentry{measurable|hyperpage}{344} +\indexentry{Measure|hyperpage}{344} +\indexentry{measure|hyperpage}{344} +\indexentry{measure space|hyperpage}{344} +\indexentry{integral of $f$ with respect to $\mu $|hyperpage}{344} +\indexentry{integral of $f$ with respect to $\mu $|hyperpage}{344} +\indexentry{Monotone convergence theorem|hyperpage}{344} +\indexentry{Fatou's lemma|hyperpage}{344} +\indexentry{Dominated convergence theorem|hyperpage}{344} +\indexentry{Product measure|hyperpage}{344} +\indexentry{product measure|hyperpage}{344} +\indexentry{$\sigma $-finite|hyperpage}{344} +\indexentry{Fubini|hyperpage}{344} +\indexentry{density|hyperpage}{345} +\indexentry{absolutely continuous|hyperpage}{345} +\indexentry{Radon-Nikodym|hyperpage}{345} +\indexentry{probability space|hyperpage}{345} +\indexentry{events|hyperpage}{345} +\indexentry{Random variable|hyperpage}{345} +\indexentry{$E$-valued random variable|hyperpage}{345} +\indexentry{Expectation|hyperpage}{345} +\indexentry{expectation of $X$|hyperpage}{345} +\indexentry{law of $X$|hyperpage}{345} +\indexentry{$\sigma $-algebra generated by $X$|hyperpage}{345} +\indexentry{Jensen's inequality|hyperpage}{345} +\indexentry{conditional expectation of $X$ given $\mathcal {G}$|hyperpage}{345} +\indexentry{Tower property|hyperpage}{346} +\indexentry{conditional expectation of $X$ given $Y$|hyperpage}{346} +\indexentry{probability kernel|hyperpage}{346} +\indexentry{conditional law of $X$ given $Y$|hyperpage}{346} +\indexentry{filtration|hyperpage}{346} +\indexentry{filtered probability space|hyperpage}{346} +\indexentry{adapted|hyperpage}{346} +\indexentry{martingale|hyperpage}{346} +\indexentry{submartingale|hyperpage}{346} +\indexentry{supermartingale|hyperpage}{346} +\indexentry{stopping time|hyperpage}{346} +\indexentry{stopped process|hyperpage}{346} +\indexentry{uniformly integrable|hyperpage}{346} +\indexentry{converges weakly|hyperpage}{347} +\indexentry{converges strongly|hyperpage}{347} +\indexentry{converges weakly-*|hyperpage}{347} +\indexentry{Banach-Alaoglu theorem|hyperpage}{347} +\indexentry{strongly lower-semicontinuous|hyperpage}{347} +\indexentry{weakly lower-semicontinuity|hyperpage}{347} +\indexentry{weak-* lower-semicontinuity|hyperpage}{347} +\indexentry{support plane|hyperpage}{347} +\indexentry{Sobolev spaces|hyperpage}{348} +\indexentry{Sobolev spaces|hyperpage}{348} +\indexentry{Poincaré's inequality|hyperpage}{348} +\indexentry{Poincaré-Wirtinger's inequality|hyperpage}{348} +\indexentry{embedded|hyperpage}{349} +\indexentry{compactly embedded|hyperpage}{349} +\indexentry{Gagliardo, Nirengerg and Sobolev's inequality|hyperpage}{349} +\indexentry{Hölder continuity|hyperpage}{349} +\indexentry{$\mathcal {C}^{k,\theta }$-Hölder continuous|hyperpage}{349} +\indexentry{Morrey's embedding|hyperpage}{349} +\indexentry{extension|hyperpage}{349} +\indexentry{extension operator|hyperpage}{349} +\indexentry{first order reflection|hyperpage}{350} +\indexentry{Reillich-Kondrachov's compactness theorem|hyperpage}{350} +\indexentry{trace operator|hyperpage}{350} +\indexentry{Gradient descent algorithm|hyperpage}{351} +\indexentry{gradient descent algorithm|hyperpage}{351} +\indexentry{Armijo-type rule|hyperpage}{351} +\indexentry{Frank-Wolfe-type method|hyperpage}{351} +\indexentry{$L$-co-coercive|hyperpage}{351} +\indexentry{firmly non-expansive|hyperpage}{351} +\indexentry{Baillon-Haddad|hyperpage}{351} +\indexentry{strongly convex|hyperpage}{351} +\indexentry{$\gamma $-convex|hyperpage}{351} +\indexentry{Newton method|hyperpage}{352} +\indexentry{Newton method|hyperpage}{352} +\indexentry{multistep method|hyperpage}{352} +\indexentry{Heavy ball method|hyperpage}{352} +\indexentry{Conjugate gradient|hyperpage}{352} +\indexentry{conjugate gradient method|hyperpage}{352} +\indexentry{Nesterov's accelerated gradient method|hyperpage}{352} +\indexentry{Nesterov's method|hyperpage}{352} +\indexentry{subgradient descent method|hyperpage}{352} +\indexentry{implicit descent|hyperpage}{352} +\indexentry{nonexpansive|hyperpage}{352} +\indexentry{avergaed|hyperpage}{352} +\indexentry{Opial's lemma|hyperpage}{352} +\indexentry{Krasnoselskii-Mann's convergence theorem|hyperpage}{353} +\indexentry{linear second-order PDE|hyperpage}{354} +\indexentry{non-divergence form|hyperpage}{354} +\indexentry{divergence form|hyperpage}{354} +\indexentry{Dirichlet boundary condition|hyperpage}{354} +\indexentry{homogeneous|hyperpage}{354} +\indexentry{Neumann boundary condition|hyperpage}{354} +\indexentry{homogeneous|hyperpage}{354} +\indexentry{uniformly elliptic|hyperpage}{354} +\indexentry{weak formulation|hyperpage}{354} +\indexentry{variational formulation|hyperpage}{354} +\indexentry{weak solution|hyperpage}{354} +\indexentry{strong solution|hyperpage}{354} +\indexentry{classical solution|hyperpage}{354} +\indexentry{continuous|hyperpage}{354} +\indexentry{coercive|hyperpage}{354} +\indexentry{symmetric|hyperpage}{354} +\indexentry{Lax-Milgram theorem|hyperpage}{355} +\indexentry{Abstract Fredholm alternative|hyperpage}{356} +\indexentry{formal adjoint|hyperpage}{356} +\indexentry{homogeneous adjoint problem|hyperpage}{356} +\indexentry{resolvent set|hyperpage}{356} +\indexentry{spectrum|hyperpage}{356} +\indexentry{Inner regularity|hyperpage}{357} +\indexentry{Regularity up to the boundary|hyperpage}{357} +\indexentry{Weak maximum principle|hyperpage}{357} +\indexentry{Weak maximum principle|hyperpage}{358} +\indexentry{Weak maximum principle|hyperpage}{358} +\indexentry{Hopf's lemma|hyperpage}{358} +\indexentry{interior ball condition|hyperpage}{358} +\indexentry{Strong maximum principle|hyperpage}{358} +\indexentry{Continuation method|hyperpage}{359} +\indexentry{Brower fixed point|hyperpage}{359} +\indexentry{Schauder fixed point|hyperpage}{359} +\indexentry{Schaefer fixed point|hyperpage}{359} +\indexentry{Montecarlo estimator|hyperpage}{360} +\indexentry{seed|hyperpage}{360} +\indexentry{Mersenne Twister algorithm|hyperpage}{360} +\indexentry{Acceptance-rejection method|hyperpage}{360} +\indexentry{Box-Muller method|hyperpage}{361} +\indexentry{Polar method|hyperpage}{361} +\indexentry{antithetic method|hyperpage}{361} +\indexentry{control variate|hyperpage}{361} +\indexentry{multiple control variate estimator|hyperpage}{362} +\indexentry{equivalent|hyperpage}{362} +\indexentry{importance sampling estimator|hyperpage}{362} +\indexentry{Euler method|hyperpage}{363} +\indexentry{continuous Euler scheme|hyperpage}{363} +\indexentry{Strong error of the Euler scheme|hyperpage}{363} +\indexentry{Weak error of the Euler scheme|hyperpage}{363} +\indexentry{Romberg Extrapolation|hyperpage}{363} +\indexentry{Romberg Extrapolation|hyperpage}{363} +\indexentry{finite difference estimator|hyperpage}{364} +\indexentry{Black-Scholes model|hyperpage}{364} +\indexentry{tangent process|hyperpage}{364} +\indexentry{price of an American option|hyperpage}{365} +\indexentry{risk-free interest rate|hyperpage}{365} +\indexentry{discretization method|hyperpage}{365} +\indexentry{naive approach|hyperpage}{365} +\indexentry{Tsitsiklis-Van Roy method|hyperpage}{365} +\indexentry{Tsitsiklis-Van Roy method|hyperpage}{365} +\indexentry{Longstaff-Schwartz method|hyperpage}{366} +\indexentry{Longstaff-Schwartz method|hyperpage}{366} +\indexentry{Rogers's lemma|hyperpage}{366} +\indexentry{Dirichlet boundary conditions|hyperpage}{367} +\indexentry{Neumann boundary conditions|hyperpage}{367} +\indexentry{Robin boundary conditions|hyperpage}{367} +\indexentry{conforming Galerkin method|hyperpage}{367} +\indexentry{Céa's lemma|hyperpage}{367} +\indexentry{finite element|hyperpage}{367} +\indexentry{basis functions|hyperpage}{368} +\indexentry{local interpolant|hyperpage}{368} +\indexentry{subdivision|hyperpage}{368} +\indexentry{global interpolant|hyperpage}{368} +\indexentry{triangulation|hyperpage}{368} +\indexentry{affinely equivalent|hyperpage}{368} +\indexentry{Bramble-Hilbert lemma|hyperpage}{368} +\indexentry{diameter|hyperpage}{368} +\indexentry{insphere diameter|hyperpage}{368} +\indexentry{condition number|hyperpage}{368} +\indexentry{Local interpolation error|hyperpage}{369} +\indexentry{regular|hyperpage}{369} +\indexentry{Global interpolation error|hyperpage}{369} +\indexentry{spectral methods|hyperpage}{369} +\indexentry{trial functions|hyperpage}{369} +\indexentry{globally smooth|hyperpage}{369} +\indexentry{Galerkin methods|hyperpage}{369} +\indexentry{Collocation methods|hyperpage}{369} +\indexentry{collocation points|hyperpage}{369} +\indexentry{$\tau $ methods|hyperpage}{369} +\indexentry{interpolant|hyperpage}{369} +\indexentry{interpolant|hyperpage}{370} +\indexentry{jointly Gaussian|hyperpage}{371} +\indexentry{Brownian motion|hyperpage}{371} +\indexentry{Strong law of large numbers for Brownian motion|hyperpage}{371} +\indexentry{Markov property for Brownian motion|hyperpage}{371} +\indexentry{natural filtration|hyperpage}{371} +\indexentry{Martingale|hyperpage}{371} +\indexentry{martingale|hyperpage}{371} +\indexentry{adapted|hyperpage}{371} +\indexentry{sub-martingale|hyperpage}{371} +\indexentry{super-martingale|hyperpage}{371} +\indexentry{hitting time|hyperpage}{372} +\indexentry{Doob's optional sampling theorem|hyperpage}{372} +\indexentry{stopped process|hyperpage}{372} +\indexentry{Orthogonality of martingales|hyperpage}{372} +\indexentry{Doob's maximal inequality|hyperpage}{372} +\indexentry{absoulte variation|hyperpage}{372} +\indexentry{finite variation|hyperpage}{372} +\indexentry{Quadratic variation|hyperpage}{372} +\indexentry{mesh|hyperpage}{372} +\indexentry{continuous local martingale|hyperpage}{373} +\indexentry{localizing sequence|hyperpage}{373} +\indexentry{Doob's optional sampling theorem for local martingales|hyperpage}{373} +\indexentry{Levy's characterization of Brownian motion|hyperpage}{374} +\indexentry{isometry|hyperpage}{374} +\indexentry{partial isometry|hyperpage}{374} +\indexentry{Wiener integral|hyperpage}{374} +\indexentry{Wiener isometry|hyperpage}{374} +\indexentry{Wiener integral|hyperpage}{374} +\indexentry{Wiener integral|hyperpage}{374} +\indexentry{Chasles relation|hyperpage}{374} +\indexentry{progressive|hyperpage}{375} +\indexentry{Itô integral|hyperpage}{376} +\indexentry{Itô isometry|hyperpage}{376} +\indexentry{Itô integral|hyperpage}{376} +\indexentry{generalized Itô integral|hyperpage}{376} +\indexentry{Stochastic dominated convergence theorem|hyperpage}{376} +\indexentry{Itô process|hyperpage}{377} +\indexentry{martingale term|hyperpage}{377} +\indexentry{drift term|hyperpage}{377} +\indexentry{stochastic differential|hyperpage}{377} +\indexentry{quadratic variation|hyperpage}{377} +\indexentry{Stochastic integration by parts|hyperpage}{378} +\indexentry{Itô term|hyperpage}{378} +\indexentry{Itô's formula|hyperpage}{378} +\indexentry{Doléans-Dade exponential|hyperpage}{378} +\indexentry{Novikov's condition|hyperpage}{378} +\indexentry{Giranov's theorem|hyperpage}{379} +\indexentry{drift|hyperpage}{379} +\indexentry{diffusion|hyperpage}{379} +\indexentry{stochastic differential equation|hyperpage}{379} +\indexentry{SDE|hyperpage}{379} +\indexentry{solution of the SDE|hyperpage}{379} +\indexentry{Gronwall's lemma|hyperpage}{379} +\indexentry{Existence and uniqueness of solutions of SDEs|hyperpage}{379} +\indexentry{Langevin equation|hyperpage}{379} +\indexentry{Geometric Brownian motion|hyperpage}{380} +\indexentry{Black-Scholes process|hyperpage}{380} +\indexentry{homogeneous SDE|hyperpage}{380} +\indexentry{diffusions|hyperpage}{380} +\indexentry{Invariance under time shift|hyperpage}{380} +\indexentry{Generator|hyperpage}{380} +\indexentry{generator|hyperpage}{380} +\indexentry{Kolmogorov's equation|hyperpage}{380} +\indexentry{Feynman-Kac's formula|hyperpage}{381} +\indexentry{Itô's formula|hyperpage}{382} +\indexentry{Flow property|hyperpage}{382} +\indexentry{Dynamic programming principle|hyperpage}{382} +\indexentry{Fokker-Planck equation|hyperpage}{382} +\indexentry{control parameter|hyperpage}{383} +\indexentry{Finite horizon problem|hyperpage}{383} +\indexentry{running cost|hyperpage}{383} +\indexentry{terminal cost|hyperpage}{383} +\indexentry{Infinte horizon problem|hyperpage}{383} diff --git a/main_math.ilg b/main_math.ilg index 887c781..0f87c9c 100644 --- a/main_math.ilg +++ b/main_math.ilg @@ -1,6 +1,6 @@ This is makeindex, version 2.17 [TeX Live 2023] (kpathsea + Thai support). -Scanning input file main_math.idx.......done (3331 entries accepted, 0 rejected). -Sorting entries...............................done (41166 comparisons). -Generating output file main_math.ind.......done (2842 lines written, 0 warnings). +Scanning input file main_math.idx.......done (3365 entries accepted, 0 rejected). +Sorting entries................................done (42220 comparisons). +Generating output file main_math.ind.......done (2868 lines written, 0 warnings). Output written in main_math.ind. Transcript written in main_math.ilg. diff --git a/main_math.ind b/main_math.ind index 19a8386..eeb866e 100644 --- a/main_math.ind +++ b/main_math.ind @@ -1,13 +1,13 @@ \begin{theindex} \item $(G,\cdot )$-space, \hyperpage{211} - \item $E$-valued random variable, \hyperpage{341} + \item $E$-valued random variable, \hyperpage{345} \item $F$-distribution with degrees of freedom $d_1$ and $d_2$, \hyperpage{252} \item $F$-invariant, \hyperpage{339} \item $F$-linearly independent, \hyperpage{170} \item $K$-field morphism, \hyperpage{165} - \item $L$-co-coercive, \hyperpage{347} + \item $L$-co-coercive, \hyperpage{351} \item $N$-th partial sum of $Sf$, \hyperpage{83} \item $N$-th partial sum of the series, \hyperpage{76}, \hyperpage{105} @@ -25,24 +25,24 @@ \item $\alpha $-limit set, \hyperpage{131} \item $\ensuremath {\mathbb {C}}$-differentiable, \hyperpage{109} \item $\ensuremath {\mathbb {R}}$-differentiable, \hyperpage{110} - \item $\gamma $-convex, \hyperpage{347} + \item $\gamma $-convex, \hyperpage{351} \item $\mathcal {C}^0$-close, \hyperpage{336} \item $\mathcal {C}^1$-close, \hyperpage{336} - \item $\mathcal {C}^{k,\theta }$-Hölder continuous, \hyperpage{345} + \item $\mathcal {C}^{k,\theta }$-Hölder continuous, \hyperpage{349} \item $\omega $-limit point, \hyperpage{131} \item $\omega $-limit set, \hyperpage{131} \item $\sigma $-additivity, \hyperpage{173}, \hyperpage{291} \item $\sigma $-algebra, \hyperpage{172}, \hyperpage{291}, - \hyperpage{340} + \hyperpage{344} \item $\sigma $-algebra generated, \hyperpage{172} - \item $\sigma $-algebra generated by $X$, \hyperpage{341} + \item $\sigma $-algebra generated by $X$, \hyperpage{345} \item $\sigma $-algebra generated by $\boldsymbol {\mathrm {X}}$, \hyperpage{321} - \item $\sigma $-algebra generated by $\mathcal {F}$, \hyperpage{340} + \item $\sigma $-algebra generated by $\mathcal {F}$, \hyperpage{344} \item $\sigma $-algebra of all Lebesgue measurable sets in $\ensuremath {\mathbb {R}}^n$, \hyperpage{175} - \item $\sigma $-finite, \hyperpage{340} - \item $\tau $ methods, \hyperpage{361} + \item $\sigma $-finite, \hyperpage{344} + \item $\tau $ methods, \hyperpage{369} \item $d$-dimensional standard Brownian motion, \hyperpage{332} \item $i$-th pivot, \hyperpage{14} \item $k$-linear map, \hyperpage{155} @@ -93,13 +93,13 @@ \item Absolute geometry, \hyperpage{64} \item absolute value, \hyperpage{25} \item absolutely continuous, \hyperpage{178}, \hyperpage{180}, - \hyperpage{341} + \hyperpage{345} \item absolutely convergent, \hyperpage{77}, \hyperpage{106}, \hyperpage{117}, \hyperpage{300} \item absolutely stable, \hyperpage{261} - \item absoulte variation, \hyperpage{364} - \item Abstract Fredholm alternative, \hyperpage{350} - \item Acceptance-rejection method, \hyperpage{354} + \item absoulte variation, \hyperpage{372} + \item Abstract Fredholm alternative, \hyperpage{356} + \item Acceptance-rejection method, \hyperpage{360} \item acceptation region, \hyperpage{199} \item ACCP, \hyperpage{46} \item accumulation point, \hyperpage{26}, \hyperpage{54} @@ -107,7 +107,7 @@ \item Adams method, \hyperpage{262} \item Adams-Bashforth method, \hyperpage{262} \item Adams-Moulton method, \hyperpage{262} - \item adapted, \hyperpage{342}, \hyperpage{363} + \item adapted, \hyperpage{346}, \hyperpage{371} \item adherence, \hyperpage{54} \item adherent point, \hyperpage{54}, \hyperpage{207} \item adjacency matrix, \hyperpage{50} @@ -122,7 +122,7 @@ \item affine plane, \hyperpage{66} \item affine space, \hyperpage{69} \item affine subvariety, \hyperpage{70} - \item affinely equivalent, \hyperpage{360} + \item affinely equivalent, \hyperpage{368} \item affinely independents, \hyperpage{69} \item affinity, \hyperpage{70} \item AIC, \hyperpage{257} @@ -155,7 +155,7 @@ \item angle-preserving, \hyperpage{149} \item anisotropic, \hyperpage{74} \item annihilator, \hyperpage{19} - \item antithetic method, \hyperpage{355} + \item antithetic method, \hyperpage{361} \item aperiodic, \hyperpage{320} \item approximation of identity, \hyperpage{79}, \hyperpage{241} \item approximations of the identity, \hyperpage{235} @@ -167,7 +167,7 @@ \item Archimedean property, \hyperpage{25} \item argument, \hyperpage{104} \item Argument principle, \hyperpage{118} - \item Armijo-type rule, \hyperpage{347} + \item Armijo-type rule, \hyperpage{351} \item Arnold family, \hyperpage{337} \item Artin's lemma, \hyperpage{169} \item Arzelà-Ascoli theorem, \hyperpage{121}, \hyperpage{125}, @@ -192,7 +192,7 @@ \item autonomous, \hyperpage{122} \item average, \hyperpage{289} \item average response, \hyperpage{253} - \item avergaed, \hyperpage{348} + \item avergaed, \hyperpage{352} \item Axiom of Archimedes, \hyperpage{63} \item Axiom of choice, \hyperpage{213} \item Axiom of completeness, \hyperpage{63} @@ -205,13 +205,13 @@ \item backward Euler method, \hyperpage{258} \item Backward-time central-space, \hyperpage{268}, \hyperpage{274} - \item Baillon-Haddad, \hyperpage{347} + \item Baillon-Haddad, \hyperpage{351} \item Baire's theorem, \hyperpage{309} \item balance equation, \hyperpage{329} \item ball, \hyperpage{205} \item Banach fixed-point theorem, \hyperpage{125} \item Banach space, \hyperpage{301} - \item Banach-Alaoglu theorem, \hyperpage{343} + \item Banach-Alaoglu theorem, \hyperpage{347} \item Banach-Steinhaus theorem, \hyperpage{310} \item bandlimited, \hyperpage{239} \item Barrow's law, \hyperpage{298} @@ -221,7 +221,7 @@ \item basic feasible solutions, \hyperpage{51} \item basin, \hyperpage{131} \item basis, \hyperpage{16}, \hyperpage{206} - \item basis functions, \hyperpage{360} + \item basis functions, \hyperpage{368} \item Bautin's theorem, \hyperpage{226} \item Bayes estimate, \hyperpage{203} \item Bayes' formula, \hyperpage{174} @@ -254,7 +254,8 @@ \item bipartite, \hyperpage{50} \item birth and death process, \hyperpage{329} \item Bisection method, \hyperpage{90} - \item Black-Scholes process, \hyperpage{372} + \item Black-Scholes model, \hyperpage{364} + \item Black-Scholes process, \hyperpage{380} \item block matrix, \hyperpage{15} \item blow-down, \hyperpage{225} \item Blow-up in cartesian coordinates, \hyperpage{226} @@ -283,14 +284,15 @@ \item bounded from above, \hyperpage{25, 26} \item bounded from below, \hyperpage{25, 26} \item bounded set, \hyperpage{54} + \item bounded variation, \hyperpage{342} \item box topology, \hyperpage{209} \item Box-Cox transformation, \hyperpage{257} - \item Box-Muller method, \hyperpage{355} - \item Bramble-Hilbert lemma, \hyperpage{360} + \item Box-Muller method, \hyperpage{361} + \item Bramble-Hilbert lemma, \hyperpage{368} \item bridge, \hyperpage{50} \item Brouwer's fixed-point theorem, \hyperpage{215} - \item Brower fixed point, \hyperpage{353} - \item Brownian motion, \hyperpage{330}, \hyperpage{363} + \item Brower fixed point, \hyperpage{359} + \item Brownian motion, \hyperpage{330}, \hyperpage{371} \item Broyden's method, \hyperpage{263} \item Broyden-Fletcher-Goldfarb-Shanno method, \hyperpage{264} \item BTCS, \hyperpage{268} @@ -306,7 +308,7 @@ \hyperpage{51} \item canonical form of a linear programming to minimize, \hyperpage{51} - \item Cantor set, \hyperpage{209} + \item Cantor set, \hyperpage{209}, \hyperpage{341} \item Cantor's theorem, \hyperpage{9} \item Cantor-Bernstein theorem, \hyperpage{9} \item Cardano-Vieta's formulas, \hyperpage{114} @@ -375,9 +377,9 @@ \item Characteristic function, \hyperpage{191} \item characteristic function, \hyperpage{9}, \hyperpage{191} \item characteristic polynomial, \hyperpage{20}, \hyperpage{49} - \item Chasles relation, \hyperpage{366} + \item Chasles relation, \hyperpage{374} \item Chebyshev method, \hyperpage{90} - \item Chebyshev polynomials, \hyperpage{96}, \hyperpage{362} + \item Chebyshev polynomials, \hyperpage{96} \item Chebyshev's inequality, \hyperpage{186}, \hyperpage{297} \item chi-squared distribution with $n$ degrees of freedom, \hyperpage{197} @@ -390,7 +392,7 @@ \item class $\mathcal {C}^n$ at a point $a\in \ensuremath {\mathbb {R}}$, \hyperpage{31} \item class $\mathcal {C}^p$, \hyperpage{31} - \item classical solution, \hyperpage{352} + \item classical solution, \hyperpage{354} \item Classification of affine quadrics, \hyperpage{75} \item Classification of compact connected surfaces, \hyperpage{219} \item Classification of connected 1-manifolds, \hyperpage{217} @@ -427,13 +429,13 @@ \item codimension, \hyperpage{223} \item coefficient of determination, \hyperpage{252} \item coefficients, \hyperpage{11} - \item coercive, \hyperpage{313} + \item coercive, \hyperpage{354} \item cofactor, \hyperpage{136} \item cofactor matrix, \hyperpage{15} \item Cofinite topology, \hyperpage{206} \item collineation, \hyperpage{66} - \item Collocation methods, \hyperpage{361} - \item collocation points, \hyperpage{361} + \item Collocation methods, \hyperpage{369} + \item collocation points, \hyperpage{369} \item column rank, \hyperpage{16} \item Combinations with repetition, \hyperpage{10} \item Combinations without repetition, \hyperpage{10} @@ -444,7 +446,7 @@ \item Compact space, \hyperpage{213} \item compact subset, \hyperpage{213} \item compact support, \hyperpage{79} - \item compactly embedded, \hyperpage{345} + \item compactly embedded, \hyperpage{349} \item compactness, \hyperpage{213} \item Comparison test, \hyperpage{76}, \hyperpage{80}, \hyperpage{183} @@ -483,13 +485,13 @@ \item computational efficiency, \hyperpage{91} \item concave, \hyperpage{30} \item Condensation test, \hyperpage{76} - \item condition number, \hyperpage{99}, \hyperpage{360} + \item condition number, \hyperpage{99}, \hyperpage{368} \item conditional consistency, \hyperpage{269} \item conditional expectation, \hyperpage{186} \item conditional expectation of $X$ given $\mathcal {G}$, - \hyperpage{341} - \item conditional expectation of $X$ given $Y$, \hyperpage{342} - \item conditional law of $X$ given $Y$, \hyperpage{342} + \hyperpage{345} + \item conditional expectation of $X$ given $Y$, \hyperpage{346} + \item conditional law of $X$ given $Y$, \hyperpage{346} \item conditional probability, \hyperpage{174} \item conditional probability density function, \hyperpage{182} \item conditional probability mass function, \hyperpage{182} @@ -505,7 +507,7 @@ \item configuration, \hyperpage{67} \item conformal, \hyperpage{119}, \hyperpage{149} \item conformal representation, \hyperpage{119} - \item conforming Galerkin method, \hyperpage{359} + \item conforming Galerkin method, \hyperpage{367} \item Congruence axioms, \hyperpage{63} \item congruence relation, \hyperpage{63} \item conic, \hyperpage{73} @@ -514,9 +516,9 @@ \item conjugate, \hyperpage{104}, \hyperpage{132}, \hyperpage{202}, \hyperpage{338} \item conjugate dynamical systems, \hyperpage{131} - \item Conjugate gradient, \hyperpage{348} + \item Conjugate gradient, \hyperpage{352} \item Conjugate gradient method, \hyperpage{264} - \item conjugate gradient method, \hyperpage{348} + \item conjugate gradient method, \hyperpage{352} \item conjugate Poisson kernel, \hyperpage{248} \item conjugation action, \hyperpage{41} \item connected, \hyperpage{50}, \hyperpage{54}, \hyperpage{214} @@ -538,20 +540,20 @@ \item Construction of a non-SAS geometry, \hyperpage{66} \item contact, \hyperpage{144} \item contact of order $\geq n$ at $a$, \hyperpage{31} - \item Continuation method, \hyperpage{352} + \item Continuation method, \hyperpage{359} \item Continuity axioms, \hyperpage{63} \item Continuity correction, \hyperpage{192} \item continuity correction, \hyperpage{192} \item Continuity from above, \hyperpage{173} \item Continuity from below, \hyperpage{173} \item continuous, \hyperpage{28}, \hyperpage{55}, \hyperpage{106}, - \hyperpage{130}, \hyperpage{313} + \hyperpage{130}, \hyperpage{354} \item continuous at $x_0$, \hyperpage{28} \item Continuous equation, \hyperpage{277} \item continuous equation, \hyperpage{277} - \item continuous Euler scheme, \hyperpage{357} + \item continuous Euler scheme, \hyperpage{363} \item Continuous function, \hyperpage{208} - \item continuous local martingale, \hyperpage{365} + \item continuous local martingale, \hyperpage{373} \item Continuous memorylessness property, \hyperpage{178} \item Continuous uniform distribution, \hyperpage{178} \item continuous uniform distribution, \hyperpage{178} @@ -559,8 +561,8 @@ \item contractible, \hyperpage{139} \item contraction, \hyperpage{55}, \hyperpage{90} \item contrast matrix, \hyperpage{201} - \item control parameter, \hyperpage{375} - \item control variate, \hyperpage{355} + \item control parameter, \hyperpage{383} + \item control variate, \hyperpage{361} \item converge in mean, \hyperpage{298} \item converge in norm $L^p$, \hyperpage{87} \item convergent, \hyperpage{26}, \hyperpage{54}, \hyperpage{76}, @@ -574,10 +576,10 @@ \item converges in probability, \hyperpage{187} \item converges in the $p$-th mean, \hyperpage{189} \item converges pointwise, \hyperpage{77, 78}, \hyperpage{106, 107} - \item converges strongly, \hyperpage{343} + \item converges strongly, \hyperpage{347} \item converges uniformly, \hyperpage{77, 78}, \hyperpage{106, 107} - \item converges weakly, \hyperpage{343} - \item converges weakly-*, \hyperpage{343} + \item converges weakly, \hyperpage{347} + \item converges weakly-*, \hyperpage{347} \item convex, \hyperpage{30}, \hyperpage{115} \item convex functional, \hyperpage{308} \item convolution, \hyperpage{79}, \hyperpage{240}, \hyperpage{244} @@ -627,7 +629,7 @@ \item cyclotomic, \hyperpage{170} \item càd, \hyperpage{326} \item càdlàg, \hyperpage{176} - \item Céa's lemma, \hyperpage{359} + \item Céa's lemma, \hyperpage{367} \indexspace @@ -653,8 +655,10 @@ \item Dehomogenization, \hyperpage{74} \item dehomogenization, \hyperpage{70} \item Delta method, \hyperpage{196} + \item Denjoy theorem, \hyperpage{343} + \item Denjoy-Koksma inequality, \hyperpage{342} \item dense, \hyperpage{207} - \item density, \hyperpage{278}, \hyperpage{341} + \item density, \hyperpage{278}, \hyperpage{345} \item density function, \hyperpage{178} \item Dependence on $\boldsymbol {\mathrm {\lambda }}$, \hyperpage{129} @@ -682,7 +686,7 @@ \item diagonal, \hyperpage{20} \item diagonalizable, \hyperpage{20} \item Diagonalization theorem, \hyperpage{20} - \item diameter, \hyperpage{360} + \item diameter, \hyperpage{368} \item dicyclic group, \hyperpage{42} \item diffeomorphism, \hyperpage{57} \item Differentiability criterion, \hyperpage{56} @@ -698,11 +702,11 @@ \item differential operator over distributions, \hyperpage{245} \item differential system, \hyperpage{122} \item differentiation operator, \hyperpage{285} - \item diffusion, \hyperpage{371} + \item diffusion, \hyperpage{379} \item diffusion coefficient, \hyperpage{278} \item Diffusion equation, \hyperpage{278} \item diffusion flux, \hyperpage{278} - \item diffusions, \hyperpage{372} + \item diffusions, \hyperpage{380} \item diffusivity, \hyperpage{278} \item digital topology, \hyperpage{206} \item dihedral group, \hyperpage{42} @@ -718,8 +722,8 @@ \item directional derivative, \hyperpage{55} \item director subspace, \hyperpage{70} \item Dirichlet, \hyperpage{274} - \item Dirichlet boundary condition, \hyperpage{350} - \item Dirichlet boundary conditions, \hyperpage{359} + \item Dirichlet boundary condition, \hyperpage{354} + \item Dirichlet boundary conditions, \hyperpage{367} \item Dirichlet kernel, \hyperpage{84}, \hyperpage{233} \item Dirichlet problem, \hyperpage{140}, \hyperpage{288} \item Dirichlet problem in the disc, \hyperpage{288} @@ -741,6 +745,7 @@ \item Discrete topology, \hyperpage{206} \item Discrete uniform distribution, \hyperpage{177} \item discrete uniform distribution, \hyperpage{177} + \item discretization method, \hyperpage{365} \item discriminant, \hyperpage{169} \item displacement current, \hyperpage{278} \item distance, \hyperpage{53}, \hyperpage{82}, \hyperpage{205}, @@ -755,7 +760,7 @@ \hyperpage{241} \item distributional derivative, \hyperpage{242}, \hyperpage{285} \item divergence, \hyperpage{61} - \item divergence form, \hyperpage{350} + \item divergence form, \hyperpage{354} \item Divergence theorem, \hyperpage{160} \item Divergence theorem on $\ensuremath {\mathbb {R}}^2$, \hyperpage{62} @@ -763,22 +768,22 @@ \hyperpage{62}, \hyperpage{160} \item divergent, \hyperpage{26}, \hyperpage{76}, \hyperpage{105, 106} \item divided difference, \hyperpage{93} - \item Doléans-Dade exponential, \hyperpage{370} + \item Doléans-Dade exponential, \hyperpage{378} \item domain, \hyperpage{115}, \hyperpage{148} \item dominant eigenvalue, \hyperpage{100} \item dominant eigenvector, \hyperpage{100} \item Dominated convergence theorem, \hyperpage{184}, \hyperpage{189}, - \hyperpage{297}, \hyperpage{340} - \item Doob's maximal inequality, \hyperpage{364} - \item Doob's optional sampling theorem, \hyperpage{364} + \hyperpage{297}, \hyperpage{344} + \item Doob's maximal inequality, \hyperpage{372} + \item Doob's optional sampling theorem, \hyperpage{372} \item Doob's optional sampling theorem for local martingales, - \hyperpage{365} + \hyperpage{373} \item dot product, \hyperpage{53} \item double, \hyperpage{90} \item Double dual space, \hyperpage{19} \item double dual space, \hyperpage{19} - \item drift, \hyperpage{371} - \item drift term, \hyperpage{369} + \item drift, \hyperpage{379} + \item drift term, \hyperpage{377} \item Du Bois-Reymond's test, \hyperpage{107} \item Du-Fort-Frankel scheme, \hyperpage{274} \item dual basis, \hyperpage{19} @@ -790,7 +795,7 @@ \item Duhamel principle, \hyperpage{287} \item dummy variable, \hyperpage{253} \item dyadic partition of order $n$, \hyperpage{183} - \item Dynamic programming principle, \hyperpage{374} + \item Dynamic programming principle, \hyperpage{382} \item dynamical system, \hyperpage{130} \indexspace @@ -811,7 +816,7 @@ \item elliptic equation, \hyperpage{277} \item elliptic point, \hyperpage{150} \item elliptic sector, \hyperpage{133} - \item embedded, \hyperpage{345} + \item embedded, \hyperpage{349} \item embedded methods, \hyperpage{260} \item empirical distribution, \hyperpage{202} \item empty set, \hyperpage{6} @@ -835,7 +840,7 @@ \item equivalence relation, \hyperpage{8} \item equivalent, \hyperpage{13}, \hyperpage{17}, \hyperpage{23}, \hyperpage{73}, \hyperpage{130}, \hyperpage{132}, - \hyperpage{306}, \hyperpage{356} + \hyperpage{306}, \hyperpage{362} \item equivalent dynamical systems, \hyperpage{131} \item Ergotic theorem, \hyperpage{324, 325} \item error of type I, \hyperpage{199} @@ -857,7 +862,7 @@ \item Euclidean vector space, \hyperpage{24} \item Euclidian division, \hyperpage{11} \item Euler characteristic, \hyperpage{218} - \item Euler method, \hyperpage{258}, \hyperpage{357} + \item Euler method, \hyperpage{258}, \hyperpage{363} \item Euler theorem, \hyperpage{50} \item Euler's formula, \hyperpage{108}, \hyperpage{150} \item Euler's theorem, \hyperpage{11} @@ -870,12 +875,12 @@ \item even extension, \hyperpage{84} \item even periodic extension, \hyperpage{283} \item event, \hyperpage{173} - \item events, \hyperpage{341} + \item events, \hyperpage{345} \item evolute, \hyperpage{144} \item exact, \hyperpage{156} \item Examples of Euclidean motions, \hyperpage{73} \item Excluded point topology, \hyperpage{206} - \item Existence and uniqueness of solutions of SDEs, \hyperpage{371} + \item Existence and uniqueness of solutions of SDEs, \hyperpage{379} \item Existence of a lift, \hyperpage{337} \item Existence of orthogonal polynomials, \hyperpage{96} \item Existence of the rotation number, \hyperpage{338} @@ -883,10 +888,10 @@ \item exists and it is finite, \hyperpage{298} \item expansion map, \hyperpage{335} \item Expansive fixed point theorem, \hyperpage{229} - \item Expectation, \hyperpage{183}, \hyperpage{341} + \item Expectation, \hyperpage{183}, \hyperpage{345} \item expectation, \hyperpage{178}, \hyperpage{182--184}, \hyperpage{249} - \item expectation of $X$, \hyperpage{341} + \item expectation of $X$, \hyperpage{345} \item explicit, \hyperpage{258} \item explicit Euler method, \hyperpage{258} \item explicit form, \hyperpage{122} @@ -901,8 +906,8 @@ \item exponential series, \hyperpage{48} \item extended complex plane, \hyperpage{105} \item extended real numbers, \hyperpage{105} - \item extension, \hyperpage{126}, \hyperpage{308}, \hyperpage{345} - \item extension operator, \hyperpage{345} + \item extension, \hyperpage{126}, \hyperpage{308}, \hyperpage{349} + \item extension operator, \hyperpage{349} \item exterior, \hyperpage{54} \item exterior angle, \hyperpage{160} \item Exterior angle theorem, \hyperpage{64} @@ -919,7 +924,7 @@ \item Fano configuration, \hyperpage{67} \item fast Fourier transform, \hyperpage{240} \item Fatou's lemma, \hyperpage{184}, \hyperpage{297}, - \hyperpage{340} + \hyperpage{344} \item feasible region, \hyperpage{51} \item feasible solution, \hyperpage{51} \item Fejér kernel, \hyperpage{85}, \hyperpage{234} @@ -929,7 +934,7 @@ \item FEM, \hyperpage{275} \item Fermat's little theorem, \hyperpage{11} \item Fermat's principle, \hyperpage{278} - \item Feynman-Kac's formula, \hyperpage{373} + \item Feynman-Kac's formula, \hyperpage{381} \item FFT, \hyperpage{240} \item Fick's law, \hyperpage{278} \item Fick's law of diffusion, \hyperpage{278} @@ -940,37 +945,38 @@ \item field of complex numbers, \hyperpage{104} \item field of fractions, \hyperpage{46}, \hyperpage{163} \item field of rational functions, \hyperpage{163} - \item filtered probability space, \hyperpage{342} - \item filtration, \hyperpage{321}, \hyperpage{342} + \item filtered probability space, \hyperpage{346} + \item filtration, \hyperpage{321}, \hyperpage{346} \item filtration space, \hyperpage{321} \item finer, \hyperpage{205} \item finer than, \hyperpage{32}, \hyperpage{58} \item finite, \hyperpage{25}, \hyperpage{49}, \hyperpage{164}, \hyperpage{213}, \hyperpage{294} \item finite $k$-th moment, \hyperpage{185} + \item finite difference estimator, \hyperpage{364} \item finite difference method, \hyperpage{265} \item finite difference scheme, \hyperpage{268} - \item finite element, \hyperpage{359} + \item finite element, \hyperpage{367} \item finite element method, \hyperpage{275} \item finite expectation, \hyperpage{182--184} \item Finite field, \hyperpage{166} \item finite field, \hyperpage{166} - \item Finite horizon problem, \hyperpage{375} + \item Finite horizon problem, \hyperpage{383} \item finite moment of order $k$, \hyperpage{185} \item Finite subadditivity, \hyperpage{173} - \item finite variation, \hyperpage{364} + \item finite variation, \hyperpage{372} \item Finite-dimensional distributions, \hyperpage{333} \item finite-dimensional distributions, \hyperpage{333} \item finite-rank operator, \hyperpage{307} \item finitely generated, \hyperpage{165} - \item firmly non-expansive, \hyperpage{347} + \item firmly non-expansive, \hyperpage{351} \item first and second characteristic polynomials, \hyperpage{261} \item First Borel-Cantelli lemma, \hyperpage{188} \item first fundamental form, \hyperpage{148} \item first integral, \hyperpage{134}, \hyperpage{154} \item First isomorphism theorem, \hyperpage{18}, \hyperpage{39}, \hyperpage{44} - \item first order reflection, \hyperpage{346} + \item first order reflection, \hyperpage{350} \item First Sylow theorem, \hyperpage{41} \item Fisher information, \hyperpage{195} \item Fisher's theorem, \hyperpage{197} @@ -986,10 +992,10 @@ \item flow, \hyperpage{128}, \hyperpage{132} \item Flow box theorem, \hyperpage{132} \item flow of the linear ODE, \hyperpage{126} - \item Flow property, \hyperpage{374} + \item Flow property, \hyperpage{382} \item flux, \hyperpage{62}, \hyperpage{159} - \item Fokker-Planck equation, \hyperpage{374} - \item formal adjoint, \hyperpage{350} + \item Fokker-Planck equation, \hyperpage{382} + \item formal adjoint, \hyperpage{356} \item Formal derivative, \hyperpage{167} \item formal derivative, \hyperpage{167} \item Forward and backward difference formula of order 1, @@ -1004,7 +1010,7 @@ \hyperpage{244}, \hyperpage{313} \item Fourier transform operator, \hyperpage{232} \item Fourier's law, \hyperpage{278} - \item Frank-Wolfe-type method, \hyperpage{347} + \item Frank-Wolfe-type method, \hyperpage{351} \item Fredholm alternative, \hyperpage{314} \item Fredholm operator with kernel $K$, \hyperpage{307} \item free variables, \hyperpage{14} @@ -1016,7 +1022,7 @@ \item FTBS, \hyperpage{268} \item FTCS, \hyperpage{268} \item FTFS, \hyperpage{268} - \item Fubini, \hyperpage{340} + \item Fubini, \hyperpage{344} \item Fubini's theorem, \hyperpage{59}, \hyperpage{175}, \hyperpage{300} \item Fubini's theorem for elementary regions, \hyperpage{59} @@ -1041,9 +1047,9 @@ \indexspace - \item Gagliardo, Nirengerg and Sobolev's inequality, \hyperpage{345} + \item Gagliardo, Nirengerg and Sobolev's inequality, \hyperpage{349} \item Galerkin approximation, \hyperpage{275} - \item Galerkin methods, \hyperpage{361} + \item Galerkin methods, \hyperpage{369} \item Galois, \hyperpage{168} \item Galois extension, \hyperpage{168} \item Galois group, \hyperpage{166} @@ -1080,42 +1086,42 @@ \item generalized eigenvector, \hyperpage{128} \item generalized heat kernel, \hyperpage{286} \item Generalized Hölder's inequality, \hyperpage{237} - \item generalized Itô integral, \hyperpage{368} + \item generalized Itô integral, \hyperpage{376} \item generalized solution, \hyperpage{245} \item generating set, \hyperpage{16} - \item Generator, \hyperpage{372} - \item generator, \hyperpage{372} + \item Generator, \hyperpage{380} + \item generator, \hyperpage{380} \item generator tree, \hyperpage{50} \item generatrix, \hyperpage{147} \item genus $g$ orientable surface, \hyperpage{217} \item genus $h$ non-orientable surface, \hyperpage{217} \item geodesic, \hyperpage{153} \item geodesic curvature, \hyperpage{153} - \item Geometric Brownian motion, \hyperpage{372} + \item Geometric Brownian motion, \hyperpage{380} \item Geometric distribution, \hyperpage{177} \item geometric distribution, \hyperpage{177} \item geometric multiplicity, \hyperpage{20} \item Gershgorin circle theorem, \hyperpage{92} - \item Giranov's theorem, \hyperpage{371} + \item Giranov's theorem, \hyperpage{379} \item Glide orthogonal reflections, \hyperpage{73} \item glide reflection, \hyperpage{72} \item glide vector, \hyperpage{73} \item Global Gau\ss -Bonnet theorem, \hyperpage{160} - \item global interpolant, \hyperpage{360} - \item Global interpolation error, \hyperpage{361} + \item global interpolant, \hyperpage{368} + \item Global interpolation error, \hyperpage{369} \item global maximum, \hyperpage{57} \item global minimum, \hyperpage{57} \item global stable manifold, \hyperpage{222} \item global truncation error, \hyperpage{258} \item global unstable manifold, \hyperpage{222} - \item globally smooth, \hyperpage{361} + \item globally smooth, \hyperpage{369} \item Goodness of fit, \hyperpage{200} \item Goursat's theorem, \hyperpage{112} \item Gra\ss mann formula, \hyperpage{16}, \hyperpage{67} \item gradient, \hyperpage{56} \item Gradient descent, \hyperpage{264} - \item Gradient descent algorithm, \hyperpage{347} - \item gradient descent algorithm, \hyperpage{347} + \item Gradient descent algorithm, \hyperpage{351} + \item gradient descent algorithm, \hyperpage{351} \item gradient vector field, \hyperpage{61} \item Gram-Schmidt process, \hyperpage{23}, \hyperpage{313} \item graph, \hyperpage{49}, \hyperpage{55}, \hyperpage{227} @@ -1127,7 +1133,7 @@ \item Green's formula, \hyperpage{159} \item Green's theorem, \hyperpage{62} \item grid, \hyperpage{268} - \item Gronwall's lemma, \hyperpage{371} + \item Gronwall's lemma, \hyperpage{379} \item Group, \hyperpage{37} \item group, \hyperpage{37} \item Group morphism, \hyperpage{38} @@ -1156,7 +1162,7 @@ \item heat flux, \hyperpage{278} \item heat kernel, \hyperpage{285} \item Heaviside step function, \hyperpage{242}, \hyperpage{285} - \item Heavy ball method, \hyperpage{348} + \item Heavy ball method, \hyperpage{352} \item Heine's theorem, \hyperpage{55} \item Heine-Borel theorem, \hyperpage{214} \item Hermite interpolation problem, \hyperpage{94} @@ -1177,7 +1183,7 @@ \item Hilbert-Schmidt operator with kernel $K$, \hyperpage{307} \item Hilbert-Schmidt spectral representation theorem, \hyperpage{314} - \item hitting time, \hyperpage{364} + \item hitting time, \hyperpage{372} \item holding times, \hyperpage{327} \item holes, \hyperpage{216} \item holomorphic, \hyperpage{109} @@ -1188,14 +1194,14 @@ \item homoclinic bifurcation, \hyperpage{226} \item homoclinic orbit, \hyperpage{133} \item homogeneous, \hyperpage{49}, \hyperpage{122}, \hyperpage{126}, - \hyperpage{350} - \item homogeneous adjoint problem, \hyperpage{350} + \hyperpage{354} + \item homogeneous adjoint problem, \hyperpage{356} \item homogeneous coordinates, \hyperpage{67} \item Homogeneous distribution, \hyperpage{245} \item homogeneous of degree $k$, \hyperpage{221} \item homogeneous of degree $r\in \ensuremath {\mathbb {R}}$, \hyperpage{245} - \item homogeneous SDE, \hyperpage{372} + \item homogeneous SDE, \hyperpage{380} \item Homogenization, \hyperpage{74} \item homogenization, \hyperpage{70} \item homography, \hyperpage{67} @@ -1205,7 +1211,7 @@ \item homotopic, \hyperpage{139} \item homotopy, \hyperpage{139} \item Hopf bifurcation theorem, \hyperpage{226} - \item Hopf's lemma, \hyperpage{352} + \item Hopf's lemma, \hyperpage{358} \item Hopf-bifurcation, \hyperpage{226} \item Hurwitz's theorem, \hyperpage{119} \item hyperbolic, \hyperpage{268}, \hyperpage{273} @@ -1226,7 +1232,7 @@ \item hypothesis test, \hyperpage{199} \item Hölder condition, \hyperpage{247} \item Hölder conjugates, \hyperpage{302} - \item Hölder continuity, \hyperpage{345} + \item Hölder continuity, \hyperpage{349} \item Hölder's inequality, \hyperpage{302, 303} \indexspace @@ -1243,12 +1249,12 @@ \item Immersion, \hyperpage{146} \item immersion, \hyperpage{146} \item implicit, \hyperpage{258} - \item implicit descent, \hyperpage{348} + \item implicit descent, \hyperpage{352} \item implicit Euler method, \hyperpage{258} \item implicit form, \hyperpage{122} \item Implicit function theorem, \hyperpage{57} \item Implicit scheme in finite differences, \hyperpage{286} - \item importance sampling estimator, \hyperpage{356} + \item importance sampling estimator, \hyperpage{362} \item improper, \hyperpage{37}, \hyperpage{203} \item improper integral, \hyperpage{79} \item in perspective with respect to a line, \hyperpage{68} @@ -1285,7 +1291,7 @@ \item infinite product topology, \hyperpage{209} \item infinitesimal generator, \hyperpage{328} \item infinitesimal transition scheme, \hyperpage{328} - \item Infinte horizon problem, \hyperpage{375} + \item Infinte horizon problem, \hyperpage{383} \item inflection point, \hyperpage{31} \item Information, \hyperpage{256} \item information of an event, \hyperpage{256} @@ -1297,8 +1303,8 @@ \item initial values, \hyperpage{48} \item injective, \hyperpage{7} \item inner product, \hyperpage{24}, \hyperpage{82}, \hyperpage{310} - \item Inner regularity, \hyperpage{351} - \item insphere diameter, \hyperpage{360} + \item Inner regularity, \hyperpage{357} + \item insphere diameter, \hyperpage{368} \item integrable, \hyperpage{32}, \hyperpage{182, 183} \item integrable function, \hyperpage{59} \item integrable function over $E$, \hyperpage{297} @@ -1314,7 +1320,7 @@ \item integral of $f$ over a measurable set $E\subseteq \ensuremath {\mathbb {R}}^n$, \hyperpage{296} \item integral of $f$ over the region $R$, \hyperpage{148} - \item integral of $f$ with respect to $\mu $, \hyperpage{340} + \item integral of $f$ with respect to $\mu $, \hyperpage{344} \item integral of $s$ over $\ensuremath {\mathbb {R}}^n$, \hyperpage{295} \item integral of $s$ over a measurable set $E$, \hyperpage{295} @@ -1326,14 +1332,14 @@ \item inter-arrival times, \hyperpage{327} \item Interior, \hyperpage{206} \item interior, \hyperpage{54}, \hyperpage{206} - \item interior ball condition, \hyperpage{352} + \item interior ball condition, \hyperpage{358} \item interior point, \hyperpage{54}, \hyperpage{157}, \hyperpage{207} \item interior product, \hyperpage{156} \item Intermediate value theorem, \hyperpage{28}, \hyperpage{55}, \hyperpage{94}, \hyperpage{215} \item internally studentized residuals, \hyperpage{254} - \item interpolant, \hyperpage{361, 362} + \item interpolant, \hyperpage{369, 370} \item Interpolation inequality, \hyperpage{303} \item interpolation problem, \hyperpage{93} \item intersection, \hyperpage{6} @@ -1342,8 +1348,8 @@ \item Intervals for $\mu $ and $\sigma ^2$, \hyperpage{198} \item invariance level, \hyperpage{71} \item Invariance of the MLE, \hyperpage{195} - \item Invariance under time shift, \hyperpage{372} - \item invariant, \hyperpage{131}, \hyperpage{339} + \item Invariance under time shift, \hyperpage{380} + \item invariant, \hyperpage{131}, \hyperpage{339, 340} \item invariant algebraic curve, \hyperpage{135} \item invariant subspace, \hyperpage{21} \item invariant under an affinity, \hyperpage{71} @@ -1373,7 +1379,7 @@ \item isolated singularity, \hyperpage{116}, \hyperpage{160} \item isometric, \hyperpage{23} \item isometry, \hyperpage{23}, \hyperpage{74}, \hyperpage{148}, - \hyperpage{366} + \hyperpage{374} \item isomorphic, \hyperpage{17}, \hyperpage{38}, \hyperpage{50}, \hyperpage{306} \item isomorphism, \hyperpage{38} @@ -1383,11 +1389,11 @@ \item iteration matrix, \hyperpage{99} \item itinerary, \hyperpage{230} \item Itinerary lemma, \hyperpage{229} - \item Itô integral, \hyperpage{368} - \item Itô isometry, \hyperpage{368} - \item Itô process, \hyperpage{369} - \item Itô term, \hyperpage{370} - \item Itô's formula, \hyperpage{370}, \hyperpage{374} + \item Itô integral, \hyperpage{376} + \item Itô isometry, \hyperpage{376} + \item Itô process, \hyperpage{377} + \item Itô term, \hyperpage{378} + \item Itô's formula, \hyperpage{378}, \hyperpage{382} \item ivp, \hyperpage{122} \indexspace @@ -1397,13 +1403,13 @@ \item Jacobian determinant, \hyperpage{55} \item Jacobian matrix, \hyperpage{55} \item Jeffrey's prior, \hyperpage{203} - \item Jensen's inequality, \hyperpage{236}, \hyperpage{341} + \item Jensen's inequality, \hyperpage{236}, \hyperpage{345} \item joint pdf, \hyperpage{180} \item joint pmf, \hyperpage{180} \item joint probability density function, \hyperpage{180} \item Joint probability mass function, \hyperpage{180} \item joint probability mass function, \hyperpage{180} - \item jointly Gaussian, \hyperpage{363} + \item jointly Gaussian, \hyperpage{371} \item Jordan arc, \hyperpage{61} \item Jordan block, \hyperpage{21} \item Jordan closed curve, \hyperpage{61} @@ -1425,10 +1431,10 @@ \item Kolmogorov system, \hyperpage{221} \item Kolmogorov's backward equation, \hyperpage{328} \item Kolmogorov's continuity theorem, \hyperpage{331} - \item Kolmogorov's equation, \hyperpage{372} + \item Kolmogorov's equation, \hyperpage{380} \item Kolmogorov's forward equation, \hyperpage{328} \item Kolmogorov's strong law of large numbers, \hyperpage{191} - \item Krasnoselskii-Mann's convergence theorem, \hyperpage{349} + \item Krasnoselskii-Mann's convergence theorem, \hyperpage{353} \item Kronecker delta, \hyperpage{19} \item Kronecker's lemma, \hyperpage{164} \item Kronecker's theorem, \hyperpage{167} @@ -1445,19 +1451,19 @@ \item Lagrange's theorem, \hyperpage{39} \item Laguerre polynomials, \hyperpage{96} \item Lamé coefficients, \hyperpage{277} - \item Langevin equation, \hyperpage{371} + \item Langevin equation, \hyperpage{379} \item Laplace equation, \hyperpage{140}, \hyperpage{288} \item Laplacian, \hyperpage{61} \item lattice of subgroups, \hyperpage{169} \item Laurent series, \hyperpage{117} \item Laurent series theorem, \hyperpage{117} - \item law of $X$, \hyperpage{341} + \item law of $X$, \hyperpage{345} \item Law of the iterated logarithm, \hyperpage{333} \item Law of total expectation, \hyperpage{186, 187}, \hyperpage{315} \item Law of total probability, \hyperpage{174}, \hyperpage{315} \item Lax theorem, \hyperpage{261} \item Lax-Friedrichs scheme, \hyperpage{268} - \item Lax-Milgram theorem, \hyperpage{313} + \item Lax-Milgram theorem, \hyperpage{355} \item Lax-Richtmyer equivalence theorem, \hyperpage{273} \item Lax-Wendroff, \hyperpage{271} \item Lax-Wendroff scheme, \hyperpage{271} @@ -1487,7 +1493,7 @@ \item level set, \hyperpage{55} \item leverage, \hyperpage{253} \item levorotation, \hyperpage{145} - \item Levy's characterization of Brownian motion, \hyperpage{366} + \item Levy's characterization of Brownian motion, \hyperpage{374} \item lexicographic degree, \hyperpage{163} \item lexicographic order, \hyperpage{163} \item lift, \hyperpage{337} @@ -1523,7 +1529,7 @@ \item linear programming to minimize, \hyperpage{51} \item linear recurrence relation of order $k$, \hyperpage{48} \item linear saddles, \hyperpage{227} - \item linear second-order PDE, \hyperpage{350} + \item linear second-order PDE, \hyperpage{354} \item linear span, \hyperpage{16} \item Linearity, \hyperpage{76} \item linearization, \hyperpage{221} @@ -1552,8 +1558,8 @@ \item local equivalence, \hyperpage{132} \item local extremum, \hyperpage{30}, \hyperpage{57} \item Local Gau\ss -Bonnet theorem, \hyperpage{160} - \item local interpolant, \hyperpage{360} - \item Local interpolation error, \hyperpage{361} + \item local interpolant, \hyperpage{368} + \item Local interpolation error, \hyperpage{369} \item local isometry, \hyperpage{148} \item local maximum, \hyperpage{30}, \hyperpage{57} \item local minimum, \hyperpage{30}, \hyperpage{57} @@ -1562,7 +1568,7 @@ \item local transversal section, \hyperpage{132} \item local truncation error, \hyperpage{262} \item local truncation errors, \hyperpage{258} - \item localizing sequence, \hyperpage{365} + \item localizing sequence, \hyperpage{373} \item locally, \hyperpage{214} \item locally bounded, \hyperpage{121}, \hyperpage{304} \item locally compact, \hyperpage{214} @@ -1580,6 +1586,7 @@ \item Logarithmic test, \hyperpage{76} \item logistic map, \hyperpage{228} \item logit, \hyperpage{257} + \item Longstaff-Schwartz method, \hyperpage{366} \item loop, \hyperpage{49}, \hyperpage{215} \item Lorenz system, \hyperpage{228} \item loss function, \hyperpage{203} @@ -1615,11 +1622,11 @@ \item marginal probability mass functions, \hyperpage{180} \item Markov chain, \hyperpage{318} \item Markov property, \hyperpage{318} - \item Markov property for Brownian motion, \hyperpage{363} + \item Markov property for Brownian motion, \hyperpage{371} \item Markov's inequality, \hyperpage{185} - \item Martingale, \hyperpage{363} - \item martingale, \hyperpage{342}, \hyperpage{363} - \item martingale term, \hyperpage{369} + \item Martingale, \hyperpage{371} + \item martingale, \hyperpage{346}, \hyperpage{371} + \item martingale term, \hyperpage{377} \item material derivative operator, \hyperpage{277} \item Matrix, \hyperpage{13} \item matrix, \hyperpage{13}, \hyperpage{18} @@ -1648,19 +1655,21 @@ \item Mean value theorem for vector-valued functions, \hyperpage{56} \item mean vector, \hyperpage{181}, \hyperpage{196} \item measurable, \hyperpage{175}, \hyperpage{293, 294}, - \hyperpage{340} - \item measurable space, \hyperpage{340} - \item Measure, \hyperpage{291}, \hyperpage{340} - \item measure, \hyperpage{174}, \hyperpage{291}, \hyperpage{340} + \hyperpage{344} + \item measurable space, \hyperpage{344} + \item Measure, \hyperpage{291}, \hyperpage{344} + \item measure, \hyperpage{174}, \hyperpage{291}, \hyperpage{344} + \item measure of $A$, \hyperpage{340} + \item measure of $U$, \hyperpage{340} \item measure on $\mathcal {C}(\ensuremath {\mathbb {T}}^1)$, - \hyperpage{338} - \item measure space, \hyperpage{175}, \hyperpage{340} + \hyperpage{339} + \item measure space, \hyperpage{175}, \hyperpage{344} \item Melnikov's method, \hyperpage{227} \item memoryless, \hyperpage{177, 178} \item meromorphic, \hyperpage{118} - \item Mersenne Twister algorithm, \hyperpage{354} + \item Mersenne Twister algorithm, \hyperpage{360} \item Mesh, \hyperpage{275} - \item mesh, \hyperpage{275}, \hyperpage{364} + \item mesh, \hyperpage{275}, \hyperpage{372} \item mesh-points, \hyperpage{258} \item Method of characteristics, \hyperpage{279} \item Method of moments, \hyperpage{194} @@ -1668,7 +1677,7 @@ \item metric space, \hyperpage{53}, \hyperpage{205}, \hyperpage{300} \item metrizable, \hyperpage{187}, \hyperpage{212} \item Meusnier's theorem, \hyperpage{150} - \item minimal, \hyperpage{221} + \item minimal, \hyperpage{221}, \hyperpage{340} \item minimal element, \hyperpage{9} \item minimal polynomial, \hyperpage{20} \item minimal surface, \hyperpage{149} @@ -1698,16 +1707,16 @@ \item moment-generating function, \hyperpage{186} \item monic, \hyperpage{11} \item Monotone convergence theorem, \hyperpage{184}, \hyperpage{296}, - \hyperpage{340} + \hyperpage{344} \item monotonic, \hyperpage{26}, \hyperpage{28} \item monotonically decreasing, \hyperpage{26} \item monotonically increasing, \hyperpage{26} - \item Montecarlo estimator, \hyperpage{354} + \item Montecarlo estimator, \hyperpage{360} \item Montel's theorem, \hyperpage{121} \item more efficient than, \hyperpage{194} \item Morera's theorem, \hyperpage{114} \item morphism of field extensions, \hyperpage{165} - \item Morrey's embedding, \hyperpage{345} + \item Morrey's embedding, \hyperpage{349} \item MSE, \hyperpage{194} \item multi-index notation, \hyperpage{57} \item multicollinearity, \hyperpage{255} @@ -1716,14 +1725,14 @@ \item Multinomial distrbution, \hyperpage{180} \item multinomial distribution, \hyperpage{180} \item multiple, \hyperpage{10} - \item multiple control variate estimator, \hyperpage{356} + \item multiple control variate estimator, \hyperpage{362} \item multiple edges, \hyperpage{49} \item Multiple shooting method, \hyperpage{264} \item multiple shooting method, \hyperpage{264} \item multiplicative group, \hyperpage{43} \item multiplicity, \hyperpage{114} \item multistep, \hyperpage{268} - \item multistep method, \hyperpage{348} + \item multistep method, \hyperpage{352} \item multivalued function, \hyperpage{104} \item multivariate cdf, \hyperpage{181} \item multivariate cumulative distribution function, \hyperpage{181} @@ -1741,33 +1750,36 @@ \indexspace + \item naive approach, \hyperpage{365} \item Nart-Vila theorem, \hyperpage{171} \item Natural cubic spline, \hyperpage{94} - \item natural filtration, \hyperpage{363} + \item natural filtration, \hyperpage{371} \item Navier-Cauchy equation, \hyperpage{277} \item negation, \hyperpage{8} \item negative, \hyperpage{141} \item negative basis, \hyperpage{142} \item Negative binomial distribution, \hyperpage{178} \item negative binomial distribution, \hyperpage{178} + \item negative orbit, \hyperpage{339} \item negative part, \hyperpage{77} \item negative semi-orbit, \hyperpage{130} \item negative-definite, \hyperpage{23} \item negative-semidefinite, \hyperpage{23} - \item negatively invariant, \hyperpage{131} + \item negatively invariant, \hyperpage{131}, \hyperpage{340} + \item negatively recurrent points, \hyperpage{340} \item negatively rotated, \hyperpage{227} \item negatively stable, \hyperpage{131} \item negatively-oriented, \hyperpage{142} \item neighbourhood, \hyperpage{25}, \hyperpage{54}, \hyperpage{207} - \item Nesterov's accelerated gradient method, \hyperpage{348} - \item Nesterov's method, \hyperpage{348} + \item Nesterov's accelerated gradient method, \hyperpage{352} + \item Nesterov's method, \hyperpage{352} \item Neumann, \hyperpage{274} - \item Neumann boundary condition, \hyperpage{350} - \item Neumann boundary conditions, \hyperpage{359} + \item Neumann boundary condition, \hyperpage{354} + \item Neumann boundary conditions, \hyperpage{367} \item Neumann series, \hyperpage{308} \item Neville's algorithm, \hyperpage{93} \item Newton method, \hyperpage{262}, \hyperpage{264}, - \hyperpage{348} + \hyperpage{352} \item Newton's divided differences\\method, \hyperpage{93} \item Newton-Raphson method, \hyperpage{90} \item Newton-Raphson modified method, \hyperpage{90} @@ -1786,13 +1798,14 @@ \item non-degenerate, \hyperpage{74} \item non-degenerated, \hyperpage{227} \item non-Desarguesian planes, \hyperpage{68} - \item non-divergence form, \hyperpage{350} + \item non-divergence form, \hyperpage{354} \item Non-parametric bootstrap, \hyperpage{202} \item Non-Paschian geometry, \hyperpage{65} \item Non-SAS geometry, \hyperpage{66} \item non-singular, \hyperpage{131} + \item non-wandering, \hyperpage{340} \item noncommutative ring, \hyperpage{42} - \item nonexpansive, \hyperpage{348} + \item nonexpansive, \hyperpage{352} \item nonsingular, \hyperpage{22} \item norm, \hyperpage{24}, \hyperpage{53}, \hyperpage{94}, \hyperpage{300}, \hyperpage{305} @@ -1817,7 +1830,7 @@ \item normed vector space, \hyperpage{53}, \hyperpage{300} \item not integrable, \hyperpage{182} \item not orientation-preserving, \hyperpage{217} - \item Novikov's condition, \hyperpage{370} + \item Novikov's condition, \hyperpage{378} \item null hypothesis, \hyperpage{199} \item null recurrent, \hyperpage{324} \item null set, \hyperpage{174}, \hyperpage{293} @@ -1853,9 +1866,10 @@ \item Open mapping theorem, \hyperpage{116}, \hyperpage{309} \item open sets, \hyperpage{205} \item operator, \hyperpage{124}, \hyperpage{305} - \item Opial's lemma, \hyperpage{348} + \item Opial's lemma, \hyperpage{352} \item opposite orientations, \hyperpage{142}, \hyperpage{217} - \item orbit, \hyperpage{8}, \hyperpage{40}, \hyperpage{130} + \item orbit, \hyperpage{8}, \hyperpage{40}, \hyperpage{130}, + \hyperpage{339} \item Orbit linear structure, \hyperpage{8} \item Orbit structure, \hyperpage{8} \item Orbit-stabilizer theorem, \hyperpage{40} @@ -1896,7 +1910,7 @@ \item orthogonal system, \hyperpage{313} \item orthogonal with respect to the weight $\omega (x)$, \hyperpage{96} - \item Orthogonality of martingales, \hyperpage{364} + \item Orthogonality of martingales, \hyperpage{372} \item orthonormal, \hyperpage{22}, \hyperpage{82} \item orthonormal system, \hyperpage{82}, \hyperpage{313} \item orthonormalization, \hyperpage{313} @@ -1935,7 +1949,7 @@ \item partial differential equation, \hyperpage{139} \item Partial fraction decomposition theorem, \hyperpage{117} \item partial inverse Fourier transform, \hyperpage{233} - \item partial isometry, \hyperpage{366} + \item partial isometry, \hyperpage{374} \item partial order relation, \hyperpage{9} \item Partial pivoting, \hyperpage{102} \item partially ordered set, \hyperpage{9} @@ -1996,12 +2010,12 @@ \item Poincaré index formula, \hyperpage{139} \item Poincaré inequality, \hyperpage{289} \item Poincaré map, \hyperpage{136} - \item Poincaré's inequality, \hyperpage{344} + \item Poincaré's inequality, \hyperpage{348} \item Poincaré's method, \hyperpage{226} \item Poincaré-Bendixson theorem, \hyperpage{136} \item Poincaré-Hopf theorem, \hyperpage{139}, \hyperpage{161} \item Poincaré-Hopf theorem on $S^2$, \hyperpage{139} - \item Poincaré-Wirtinger's inequality, \hyperpage{344} + \item Poincaré-Wirtinger's inequality, \hyperpage{348} \item points, \hyperpage{63}, \hyperpage{205} \item points of the quadric, \hyperpage{73} \item pointwise bounded, \hyperpage{125}, \hyperpage{304} @@ -2013,7 +2027,7 @@ \item Poisson summation formula, \hyperpage{238}, \hyperpage{240} \item Polar form, \hyperpage{108} \item polar form, \hyperpage{108} - \item Polar method, \hyperpage{355} + \item Polar method, \hyperpage{361} \item Polarization identity, \hyperpage{310} \item Pole, \hyperpage{116} \item pole, \hyperpage{116} @@ -2032,13 +2046,15 @@ \item poset, \hyperpage{9} \item positive, \hyperpage{141}, \hyperpage{158} \item positive basis, \hyperpage{142} + \item positive orbit, \hyperpage{339} \item positive part, \hyperpage{77} \item positive recurrent, \hyperpage{324} \item positive semi-orbit, \hyperpage{130} \item positive-definite, \hyperpage{23} \item positive-semidefinite, \hyperpage{23} - \item positively invariant, \hyperpage{131} + \item positively invariant, \hyperpage{131}, \hyperpage{340} \item positively oriented, \hyperpage{142} + \item positively recurrent points, \hyperpage{340} \item positively rotated, \hyperpage{227} \item positively stable, \hyperpage{131} \item posterior, \hyperpage{202} @@ -2060,6 +2076,7 @@ \item preditions, \hyperpage{250} \item preimage, \hyperpage{7} \item preserves orientation, \hyperpage{337} + \item price of an American option, \hyperpage{365} \item primal, \hyperpage{51} \item prime, \hyperpage{10}, \hyperpage{12}, \hyperpage{44} \item Prime number theorem, \hyperpage{10} @@ -2080,18 +2097,18 @@ \item prior distribution, \hyperpage{202} \item probability, \hyperpage{173} \item probability density function, \hyperpage{178} - \item probability kernel, \hyperpage{342} + \item probability kernel, \hyperpage{346} \item Probability mass function, \hyperpage{176} \item probability mass function, \hyperpage{176} - \item probability measure, \hyperpage{338} - \item probability space, \hyperpage{173}, \hyperpage{341} + \item probability measure, \hyperpage{339} + \item probability space, \hyperpage{173}, \hyperpage{345} \item probability-generating function, \hyperpage{316} \item product, \hyperpage{11}, \hyperpage{13}, \hyperpage{106} - \item Product measure, \hyperpage{340} - \item product measure, \hyperpage{340} + \item Product measure, \hyperpage{344} + \item product measure, \hyperpage{344} \item product topology, \hyperpage{209} \item products of group subsets, \hyperpage{40} - \item progressive, \hyperpage{367} + \item progressive, \hyperpage{375} \item projection, \hyperpage{71} \item Projection theorem, \hyperpage{311} \item Projections, \hyperpage{71} @@ -2118,7 +2135,7 @@ \item pure death process, \hyperpage{329} \item purely inseparable, \hyperpage{168} \item purely transcendental, \hyperpage{165} - \item pushforward measure, \hyperpage{339} + \item push-forward measure, \hyperpage{339} \item Pythagorean, \hyperpage{64} \item Pythagorean plane, \hyperpage{64} \item Pythagorean theorem, \hyperpage{72}, \hyperpage{310} @@ -2133,8 +2150,8 @@ \item Quadratic loss function, \hyperpage{203} \item quadratic polynomial, \hyperpage{162} \item quadratic space, \hyperpage{74} - \item Quadratic variation, \hyperpage{364} - \item quadratic variation, \hyperpage{369} + \item Quadratic variation, \hyperpage{372} + \item quadratic variation, \hyperpage{377} \item quadrature formula, \hyperpage{95} \item quadrature formula with weight $\omega (x)$, \hyperpage{97} \item quadric, \hyperpage{73} @@ -2162,10 +2179,10 @@ \item radius of curvature, \hyperpage{144} \item radix-2 decimation-in-time (DIT) FFT, \hyperpage{240} \item Radix-2 DIT Cooley-Tukey FFT algorithm, \hyperpage{240} - \item Radon-Nikodym, \hyperpage{341} + \item Radon-Nikodym, \hyperpage{345} \item Radó theorem, \hyperpage{218} \item random sample, \hyperpage{193} - \item Random variable, \hyperpage{341} + \item Random variable, \hyperpage{345} \item random variable, \hyperpage{175} \item random vector, \hyperpage{179} \item Random walk, \hyperpage{319} @@ -2209,13 +2226,13 @@ \item Regula falsi method, \hyperpage{90} \item regular, \hyperpage{49}, \hyperpage{58}, \hyperpage{131}, \hyperpage{141}, \hyperpage{195, 196}, \hyperpage{212}, - \hyperpage{328}, \hyperpage{361} + \hyperpage{328}, \hyperpage{369} \item regular distributions, \hyperpage{241} \item regular domain, \hyperpage{148} \item regular region, \hyperpage{160} \item regular surface, \hyperpage{147} - \item Regularity up to the boundary, \hyperpage{351} - \item Reillich-Kondrachov's compactness theorem, \hyperpage{346} + \item Regularity up to the boundary, \hyperpage{357} + \item Reillich-Kondrachov's compactness theorem, \hyperpage{350} \item Related samples with unknown variances, \hyperpage{199} \item relative condition numbers, \hyperpage{90} \item relative error, \hyperpage{89} @@ -2233,7 +2250,7 @@ \item residuals, \hyperpage{250} \item residue, \hyperpage{116} \item Residues theorem, \hyperpage{117} - \item resolvent set, \hyperpage{351} + \item resolvent set, \hyperpage{356} \item response, \hyperpage{253} \item restrictions, \hyperpage{51} \item Reuter criterion, \hyperpage{330} @@ -2261,11 +2278,13 @@ \item ring, \hyperpage{42} \item Ring morphism, \hyperpage{43} \item ring morphism, \hyperpage{43} + \item risk-free interest rate, \hyperpage{365} \item RK methods, \hyperpage{260} \item Robin, \hyperpage{274} - \item Robin boundary conditions, \hyperpage{359} + \item Robin boundary conditions, \hyperpage{367} + \item Rogers's lemma, \hyperpage{366} \item Rolle's theorem, \hyperpage{30} - \item Romberg Extrapolation, \hyperpage{357} + \item Romberg Extrapolation, \hyperpage{363} \item Romberg method, \hyperpage{96} \item root, \hyperpage{12} \item root condition, \hyperpage{262} @@ -2281,9 +2300,9 @@ \item row rank, \hyperpage{16} \item Ruffini's rule, \hyperpage{12} \item ruled surface, \hyperpage{151} - \item Runge phenomenon, \hyperpage{362} + \item Runge's phenomenon, \hyperpage{94} \item Runge-Kutta-Fehlberg method, \hyperpage{260} - \item running cost, \hyperpage{375} + \item running cost, \hyperpage{383} \indexspace @@ -2306,8 +2325,8 @@ \item SAS criterion, \hyperpage{63, 64} \item scalars, \hyperpage{15} \item scale, \hyperpage{179} - \item Schaefer fixed point, \hyperpage{353} - \item Schauder fixed point, \hyperpage{353} + \item Schaefer fixed point, \hyperpage{359} + \item Schauder fixed point, \hyperpage{359} \item Schrödinger equation, \hyperpage{279} \item Schwartz space, \hyperpage{243} \item Schwarz lemma, \hyperpage{120} @@ -2317,7 +2336,7 @@ \item score function, \hyperpage{195} \item Score test, \hyperpage{202} \item score test, \hyperpage{202} - \item SDE, \hyperpage{371} + \item SDE, \hyperpage{379} \item Secant method, \hyperpage{90} \item secant-like method, \hyperpage{263} \item Second Borel-Cantelli lemma, \hyperpage{189} @@ -2328,7 +2347,7 @@ \item Second Sylow theorem, \hyperpage{41} \item section, \hyperpage{299} \item sectorial decomposition, \hyperpage{133} - \item seed, \hyperpage{354} + \item seed, \hyperpage{360} \item segment, \hyperpage{72} \item Segmented regression, \hyperpage{253} \item self-adjoint, \hyperpage{312} @@ -2415,11 +2434,11 @@ \item Slutsky's theorem, \hyperpage{190} \item small change, \hyperpage{279} \item Sobolev space, \hyperpage{289} - \item Sobolev spaces, \hyperpage{344} + \item Sobolev spaces, \hyperpage{348} \item solution, \hyperpage{90}, \hyperpage{139} \item solution of a system of equations, \hyperpage{13} \item solution of the ODE, \hyperpage{122} - \item solution of the SDE, \hyperpage{371} + \item solution of the SDE, \hyperpage{379} \item solvable, \hyperpage{41}, \hyperpage{170} \item solvable by radicals, \hyperpage{170} \item SOR, \hyperpage{100} @@ -2427,11 +2446,11 @@ \item space of rapidly decreasing functions, \hyperpage{243} \item special orthogonal group, \hyperpage{145} \item specific heat capacity, \hyperpage{278} - \item spectral methods, \hyperpage{361} + \item spectral methods, \hyperpage{369} \item spectral radius, \hyperpage{98} - \item Spectral theorem, \hyperpage{24}, \hyperpage{314} + \item Spectral theorem, \hyperpage{24}, \hyperpage{313} \item spectral values, \hyperpage{309} - \item spectrum, \hyperpage{97}, \hyperpage{309}, \hyperpage{351} + \item spectrum, \hyperpage{97}, \hyperpage{309}, \hyperpage{356} \item sphere, \hyperpage{54} \item Spline, \hyperpage{94} \item spline, \hyperpage{94} @@ -2478,10 +2497,10 @@ \item stiff equations, \hyperpage{261} \item stiffness matrix, \hyperpage{276} \item Stirling's formula, \hyperpage{81} - \item stochastic differential, \hyperpage{369} - \item stochastic differential equation, \hyperpage{371} - \item Stochastic dominated convergence theorem, \hyperpage{368} - \item Stochastic integration by parts, \hyperpage{370} + \item stochastic differential, \hyperpage{377} + \item stochastic differential equation, \hyperpage{379} + \item Stochastic dominated convergence theorem, \hyperpage{376} + \item Stochastic integration by parts, \hyperpage{378} \item Stochastic matrix, \hyperpage{318} \item stochastic matrix, \hyperpage{318} \item Stochastic process, \hyperpage{316} @@ -2491,8 +2510,8 @@ \item Stokes' theorem, \hyperpage{62} \item Stolz-Cesàro theorem, \hyperpage{27} \item Stone-Weierstra\ss \ theorem, \hyperpage{304} - \item stopped process, \hyperpage{342}, \hyperpage{364} - \item stopping time, \hyperpage{321}, \hyperpage{342} + \item stopped process, \hyperpage{346}, \hyperpage{372} + \item stopping time, \hyperpage{321}, \hyperpage{346} \item strict Lyapunov function, \hyperpage{137} \item strictly concave, \hyperpage{30} \item strictly convex, \hyperpage{30} @@ -2501,16 +2520,16 @@ \item strictly diagonally dominant by rows, \hyperpage{99} \item strictly increasing, \hyperpage{26}, \hyperpage{28} \item Strong duality theorem, \hyperpage{52} - \item Strong error of the Euler scheme, \hyperpage{357} + \item Strong error of the Euler scheme, \hyperpage{363} \item Strong law of large numbers, \hyperpage{191} \item Strong law of large numbers for Brownian motion, - \hyperpage{363} + \hyperpage{371} \item Strong Markov property, \hyperpage{321, 322} - \item Strong maximum principle, \hyperpage{352} - \item strong solution, \hyperpage{352} + \item Strong maximum principle, \hyperpage{358} + \item strong solution, \hyperpage{354} \item strongly consistent estimator, \hyperpage{194} - \item strongly convex, \hyperpage{347} - \item strongly lower-semicontinuous, \hyperpage{343} + \item strongly convex, \hyperpage{351} + \item strongly lower-semicontinuous, \hyperpage{347} \item Structal stability, \hyperpage{336} \item Strum's sequence, \hyperpage{92} \item Student's $t$-distribution with $n$ degrees of freedom, @@ -2518,22 +2537,22 @@ \item Sturm's sequence, \hyperpage{92} \item Sturm's theorem, \hyperpage{92} \item Sturm-Picone comparison theorem, \hyperpage{283} - \item sub-martingale, \hyperpage{363} + \item sub-martingale, \hyperpage{371} \item sub-multiplicativity, \hyperpage{97} \item subadditive, \hyperpage{338} \item subalgebra, \hyperpage{303} \item subbasis, \hyperpage{206} \item subcover, \hyperpage{213} - \item subdivision, \hyperpage{360} + \item subdivision, \hyperpage{368} \item subfield generated, \hyperpage{163} - \item subgradient descent method, \hyperpage{348} + \item subgradient descent method, \hyperpage{352} \item Subgroup, \hyperpage{37} \item subgroup, \hyperpage{37} \item subgroup generated, \hyperpage{37} \item sublinear, \hyperpage{306} \item submanifold, \hyperpage{146} \item submanifold with boundary, \hyperpage{157} - \item submartingale, \hyperpage{342} + \item submartingale, \hyperpage{346} \item Submersion, \hyperpage{146} \item submersion, \hyperpage{146} \item subordinated matrix norm, \hyperpage{97} @@ -2556,12 +2575,13 @@ \hyperpage{107} \item sum of the series in an uniform sense, \hyperpage{78}, \hyperpage{107} - \item super-martingale, \hyperpage{363} + \item super-martingale, \hyperpage{371} \item superadditive, \hyperpage{338} - \item supermartingale, \hyperpage{342} + \item supermartingale, \hyperpage{346} \item support, \hyperpage{8}, \hyperpage{79}, \hyperpage{146}, \hyperpage{176}, \hyperpage{180}, \hyperpage{242} - \item support plane, \hyperpage{343} + \item support of $\mu $, \hyperpage{340} + \item support plane, \hyperpage{347} \item support points, \hyperpage{93} \item supremum, \hyperpage{25} \item Supremum axiom, \hyperpage{25} @@ -2576,7 +2596,7 @@ \item Sylow $p$-subgroup, \hyperpage{41} \item Sylvester's criterion, \hyperpage{58} \item Sylvester's law of inertia, \hyperpage{23} - \item symmetric, \hyperpage{22}, \hyperpage{62}, \hyperpage{313} + \item symmetric, \hyperpage{22}, \hyperpage{62}, \hyperpage{354} \item Symmetric difference formula of order 1, \hyperpage{95} \item Symmetric difference formula of order 2, \hyperpage{95} \item symmetric group, \hyperpage{8}, \hyperpage{38} @@ -2597,6 +2617,7 @@ \item tangent line to the graph at the point $(x_0,f(x_0))$, \hyperpage{29} \item tangent plane, \hyperpage{147} + \item tangent process, \hyperpage{364} \item tangent space, \hyperpage{147}, \hyperpage{156, 157} \item tangent vector, \hyperpage{141}, \hyperpage{147}, \hyperpage{156} @@ -2609,7 +2630,7 @@ \item Taylor's theorem, \hyperpage{57} \item tempered distribution, \hyperpage{244} \item tends to the boundary, \hyperpage{126} - \item terminal cost, \hyperpage{375} + \item terminal cost, \hyperpage{383} \item terminal point, \hyperpage{215} \item test, \hyperpage{200} \item test functions, \hyperpage{240} @@ -2658,9 +2679,9 @@ \item totally isotropic, \hyperpage{74} \item Tower formula, \hyperpage{165} \item tower of fields, \hyperpage{165} - \item Tower property, \hyperpage{342} + \item Tower property, \hyperpage{346} \item trace, \hyperpage{141}, \hyperpage{289} - \item trace operator, \hyperpage{346} + \item trace operator, \hyperpage{350} \item Trace theorem, \hyperpage{289} \item Traffic flow equation, \hyperpage{280} \item trail, \hyperpage{49} @@ -2683,14 +2704,14 @@ \item Trapezoidal rule, \hyperpage{95} \item traversable, \hyperpage{50} \item tree, \hyperpage{50} - \item trial functions, \hyperpage{361} + \item trial functions, \hyperpage{369} \item triangle, \hyperpage{160} \item Triangular inequality, \hyperpage{25}, \hyperpage{72} \item triangular inequality, \hyperpage{53}, \hyperpage{205}, \hyperpage{300} \item triangular system, \hyperpage{97} \item triangularization, \hyperpage{218} - \item triangulation, \hyperpage{360} + \item triangulation, \hyperpage{368} \item triple, \hyperpage{90} \item trivial $\sigma $-algebra, \hyperpage{172} \item Trivial topology, \hyperpage{206} @@ -2698,6 +2719,7 @@ \item truncated, \hyperpage{266} \item truncated Hilbert transform, \hyperpage{247} \item truncated singular value decomposition, \hyperpage{266} + \item Tsitsiklis-Van Roy method, \hyperpage{365} \item TSVD, \hyperpage{266} \item two times differentiable, \hyperpage{31} \item two times differentiable at $a$, \hyperpage{31} @@ -2722,12 +2744,13 @@ \item uniformly bounded, \hyperpage{125}, \hyperpage{304} \item uniformly continuous, \hyperpage{33}, \hyperpage{55} \item uniformly convergent, \hyperpage{117} - \item uniformly elliptic, \hyperpage{350} + \item uniformly elliptic, \hyperpage{354} \item uniformly equicontinuous, \hyperpage{125}, \hyperpage{304} - \item uniformly integrable, \hyperpage{342} + \item uniformly integrable, \hyperpage{346} \item uniformly most powerful, \hyperpage{200} \item union, \hyperpage{6} \item unique factorization domain, \hyperpage{45} + \item uniquely ergodic, \hyperpage{341} \item Uniqueness of Dirichlet problem, \hyperpage{288} \item Uniqueness of the heat equation, \hyperpage{287} \item Uniqueness of the heat equation on the unbounded domains, @@ -2768,7 +2791,7 @@ \item variance inflation factor, \hyperpage{255} \item Variation of constants, \hyperpage{123} \item variational equations, \hyperpage{129} - \item variational formulation, \hyperpage{275} + \item variational formulation, \hyperpage{275}, \hyperpage{354} \item Variations without repetition, \hyperpage{10} \item vector field, \hyperpage{60}, \hyperpage{152}, \hyperpage{154}, \hyperpage{156} @@ -2791,18 +2814,21 @@ \item Wald test, \hyperpage{201} \item Wald theorem, \hyperpage{315} \item walk, \hyperpage{49} + \item wandering domain, \hyperpage{340} + \item wandering point, \hyperpage{340} \item Waring's method, \hyperpage{164} \item Wave equation, \hyperpage{140}, \hyperpage{277} \item wave equation, \hyperpage{140} \item wave function, \hyperpage{279} \item Weak duality theorem, \hyperpage{51} - \item Weak error of the Euler scheme, \hyperpage{357} - \item weak formulation, \hyperpage{275}, \hyperpage{350} + \item Weak error of the Euler scheme, \hyperpage{363} + \item weak formulation, \hyperpage{275}, \hyperpage{354} \item Weak law of large numbers, \hyperpage{190} - \item Weak maximum principle, \hyperpage{351, 352} - \item weak-* lower-semicontinuity, \hyperpage{343} + \item Weak maximum principle, \hyperpage{357, 358} + \item weak solution, \hyperpage{354} + \item weak-* lower-semicontinuity, \hyperpage{347} \item weakly consistent estimator, \hyperpage{194} - \item weakly lower-semicontinuity, \hyperpage{343} + \item weakly lower-semicontinuity, \hyperpage{347} \item Weierstra\ss ' theorem, \hyperpage{55}, \hyperpage{114}, \hyperpage{214} \item Weierstra\ss \ approximation theorem, \hyperpage{79} @@ -2815,8 +2841,8 @@ \item well-ordered set, \hyperpage{9} \item well-posed, \hyperpage{258} \item well-posed in the Hadamard sense, \hyperpage{258} - \item Wiener integral, \hyperpage{366} - \item Wiener isometry, \hyperpage{366} + \item Wiener integral, \hyperpage{374} + \item Wiener isometry, \hyperpage{374} \item Wiener process, \hyperpage{330} \item winding number, \hyperpage{113} \item Wintner lemma, \hyperpage{126} diff --git a/main_math.tex b/main_math.tex index 99f1fe4..e2f76bf 100644 --- a/main_math.tex +++ b/main_math.tex @@ -1,5 +1,6 @@ \documentclass[10pt,a4paper]{report} -\usepackage{preamble_formulas} +\usepackage{preamble_general} +\usepackage{preamble_math} \usepackage{preamble_style} \title{Summaries of the Mathematics degree at UAB} diff --git a/main_physics.idx b/main_physics.idx index 1d79a89..eae497b 100644 --- a/main_physics.idx +++ b/main_physics.idx @@ -497,7 +497,16 @@ \indexentry{kinematic viscosity|hyperpage}{44} \indexentry{characteristic length|hyperpage}{44} \indexentry{characteristic velocity|hyperpage}{44} +\indexentry{convective term|hyperpage}{44} +\indexentry{inertia term|hyperpage}{44} +\indexentry{diffusive term|hyperpage}{44} +\indexentry{dissipation term|hyperpage}{44} \indexentry{characteristic length|hyperpage}{44} \indexentry{characteristic velocity|hyperpage}{44} \indexentry{Reynolds number|hyperpage}{44} \indexentry{similar|hyperpage}{44} +\indexentry{Helmholtz-Hodge decomposition theorem|hyperpage}{44} +\indexentry{Stokes' equations|hyperpage}{45} +\indexentry{equation of state|hyperpage}{45} +\indexentry{speed of sound|hyperpage}{45} +\indexentry{Mach number|hyperpage}{45} diff --git a/main_physics.ilg b/main_physics.ilg index 9e4190e..54ae92e 100644 --- a/main_physics.ilg +++ b/main_physics.ilg @@ -1,6 +1,6 @@ This is makeindex, version 2.17 [TeX Live 2023] (kpathsea + Thai support). -Scanning input file main_physics.idx....done (503 entries accepted, 0 rejected). -Sorting entries.......done (4815 comparisons). -Generating output file main_physics.ind....done (555 lines written, 0 warnings). +Scanning input file main_physics.idx....done (512 entries accepted, 0 rejected). +Sorting entries.......done (4973 comparisons). +Generating output file main_physics.ind....done (564 lines written, 0 warnings). Output written in main_physics.ind. Transcript written in main_physics.ilg. diff --git a/main_physics.ind b/main_physics.ind index aa4848d..73e368d 100644 --- a/main_physics.ind +++ b/main_physics.ind @@ -108,6 +108,7 @@ \item control roads, \hyperpage{26} \item Convection, \hyperpage{29} \item convection, \hyperpage{29} + \item convective term, \hyperpage{44} \item Coulomb's law, \hyperpage{4} \item crystalline, \hyperpage{28} \item Curvature, \hyperpage{11} @@ -124,10 +125,12 @@ \item Density, \hyperpage{16} \item diamagnetic, \hyperpage{9} \item Diffusion equation, \hyperpage{29} + \item diffusive term, \hyperpage{44} \item discharge of a fluid, \hyperpage{17} \item displacement current, \hyperpage{10} \item Dissipated power, \hyperpage{17} \item dissipated power, \hyperpage{17} + \item dissipation term, \hyperpage{44} \item Distribution of molecular velocities, \hyperpage{19} \item down, \hyperpage{27} \item Drag forces, \hyperpage{18} @@ -161,6 +164,7 @@ \item Entropy, \hyperpage{33} \item entropy, \hyperpage{33} \item equation of movement, \hyperpage{11} + \item equation of state, \hyperpage{45} \item Equipartition theorem, \hyperpage{20} \item Equipartition theorem on diatomic molecules, \hyperpage{20} \item Examples of potential energies, \hyperpage{14} @@ -224,6 +228,7 @@ \item Heisenberg's uncertainty principle, \hyperpage{21} \item Helmholtz free energy, \hyperpage{33} \item Helmholtz's theorem, \hyperpage{43} + \item Helmholtz-Hodge decomposition theorem, \hyperpage{44} \item Hermitian, \hyperpage{36} \item Hilbert space, \hyperpage{35} \item homogeneous, \hyperpage{40} @@ -236,6 +241,7 @@ \item ideal fluid, \hyperpage{39} \item incompressible, \hyperpage{16}, \hyperpage{40} \item inductor, \hyperpage{10} + \item inertia term, \hyperpage{44} \item inertial force, \hyperpage{12} \item Inertial forces, \hyperpage{12} \item inertial frame of reference, \hyperpage{14} @@ -298,6 +304,7 @@ \indexspace + \item Mach number, \hyperpage{45} \item Macroscopic Ohm's law, \hyperpage{6} \item magnetic flux, \hyperpage{9} \item magnetic moment, \hyperpage{7} @@ -453,6 +460,7 @@ \item spectral decomposition, \hyperpage{36} \item spectral radiance, \hyperpage{20} \item speed, \hyperpage{11} + \item speed of sound, \hyperpage{45} \item Spin, \hyperpage{23} \item spin, \hyperpage{23} \item spin quantum number, \hyperpage{23} @@ -466,6 +474,7 @@ \item steady, \hyperpage{41} \item Stefan-Boltzmann law, \hyperpage{20} \item Stern-Gerlach, \hyperpage{37} + \item Stokes' equations, \hyperpage{45} \item Stopping potential, \hyperpage{21} \item strange, \hyperpage{27} \item stream function, \hyperpage{43} diff --git a/main_physics.tex b/main_physics.tex index ad37dd3..77b63aa 100644 --- a/main_physics.tex +++ b/main_physics.tex @@ -1,6 +1,7 @@ \documentclass[10pt,a4paper]{report} -\usepackage{preamble_formulas} +\usepackage{preamble_general} \usepackage{preamble_style} +\usepackage{preamble_math} \title{Summaries of Physics degree at UAB} \author{Víctor Ballester} diff --git a/preamble_general.sty b/preamble_general.sty new file mode 100644 index 0000000..da9d291 --- /dev/null +++ b/preamble_general.sty @@ -0,0 +1,253 @@ +\usepackage{amsthm, amsmath, mathtools, amssymb, physics} % for math and physics commands and symbols. +\usepackage[intlimits]{esint} % for more options on integrals. +\usepackage{standalone} % for compiling individually .tex images. +\usepackage[hypcap=false]{caption} % basically for using the command \captionof{}, which is useful in some environments, others than \begin{figure}. +\usepackage{subcaption} % for subfigures, when many captions come into play. +\usepackage{multicol} % organize the whole document in multiple columns. +\usepackage[USenglish]{babel} % for improving use of English and mainly the use of hyphenation. +\usepackage[version=4]{mhchem} % used for chemistry and nuclear physics. +\usepackage{multirow} % multirow in a table, for example. +\usepackage{array, makecell} % for additional options in tables and tabular. +\usepackage{hyperref} % link automatically references. +% \usepackage{hycolor} % implements options of colors for hyperref (it ables the user to use colors from xcolor package) +\usepackage[capitalise,nameinlink]{cleveref} % for clever references. That is, for example, for automatically write "Definition 2.13" instead of only "2.13". +\usepackage{graphicx} % for options like \definecolor{}{}{}, \textcolor{}{},... +\usepackage[usenames,x11names]{xcolor} % to customize RGB colors. +\usepackage{colortbl} % for colors in tables. +\usepackage{tcolorbox} % for coloring boxes. +\setlength{\parindent}{0pt} % Indentation disabled. + +\usepackage{float} % to allow using argument [H] in figure environtment + +\setlength\multlinegap{0pt} % disable the margins on \begin{multline} command. +\allowdisplaybreaks % allows to break an align environment in two pages. + +%% colors - subjects %% +\newcommand{\col}{red}% % DEFAULT +\newcommand{\ana}{DeepSkyBlue2}% % analysis +\newcommand{\alg}{Green3}% % algebra +\newcommand{\geo}{Purple2}% % geometry and topology +\newcommand{\apl}{Red2}% % applied mathematics +\newcommand{\pro}{Orange2}% % probability and statistics +\newcommand{\phy}{SlateBlue2}% % physics + +\ExplSyntaxOn +\NewExpandableDocumentCommand{\refcol}{m}{ + \str_case_e:nnF {\FirstWord{#1}}{ + {FM}{\alg} % fundamentals of mathematics + {LA}{\alg} % linear algebra + {RVF}{\ana} % real valued functions + {AS}{\alg} % algebraic structures + {DM}{\alg} % discrete mathematics + {FSV}{\ana} % functions of several variables + {LG}{\geo} % linear geometry + {MA}{\ana} % mathematical analysis + {NM}{\apl} % numerical methods + {CA}{\ana} % complex analysis + {DE}{\apl} % differential equations + {DG}{\geo} % differential geometry + {GT}{\alg} % Galois theory + {P}{\pro} % probability + {S}{\pro} % statistics + {TOP}{\geo} % topology + {AT}{\geo} % algebraic topology + {A}{\alg} % arithmetic + {CAL}{\alg} % commutative algebra + {DS}{\apl} % dynamical systems + {HA}{\ana} % harmonic analysis + {LM}{\pro} % linear models + {NC}{\apl} % numeric calculus + {NIPDE}{\apl} % numeric integration of pde + {PDE}{\apl} % partial differential equations + {RFA}{\ana} % real and functional analysis + {SP}{\pro} % stochastic processes + {ADS}{\apl} % advanced dynamical systems + {AP}{\pro} % advanced probability + {ATFAPDE}{\ana} % advanced topics in functional analysis and pdes + {IEPDE}{\ana} % introduction to evolution pdes + {INLEPDE}{\ana} % introduction to non linear elliptic pdes + {LTLD}{\pro} % limit theorems and large deviations + {SC}{\pro} % stochastic calculus + {INLP}{\phy} % Instabilities and nonlinear phenomena + {MM}{\pro} % Montecarlo methods + {CO}{\apl} % Continuous optimization + {NMPDE}{\apl} % Numerical methods for pdes + {SCO}{\pro} % Stochastic control + {JP}{\pro} % Jump processes + }{\col}% +} +\ExplSyntaxOff +%%%%%%%%%%%%%%%%%%%%%%% + +\newcommand\FirstWord[1]{\@removedots#1:\@nil}% +\newcommand\@removedots{}% +\def\@removedots#1:#2\@nil{#1}% + +\newcommand\changecolor[1]{ + \renewcommand{\col}{\refcol{#1}} + \hypersetup{linkcolor = \col} +} + +%% for references (hyperref) +\hypersetup{ + colorlinks = true, + linkcolor = Blue4, % color of internal links (sections, pages, etc.) + filecolor = cyan, % color for URLs which open local files + citecolor = green, % color for bibliographical citations in text. + urlcolor = cyan, % color of URL links (mail, web) +} +%%%%%%%%%%%%%%%%%%%% + +%% Color box for important theorems +\def\sep{1mm} +\newtcolorbox{important}{colback=\col!5, colframe=\col, left=\sep, right=\sep, top=\sep, bottom=\sep} +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +%%%%%%% Enumeration - commands %%%%%%% +\usepackage{enumitem} % for the command [leftmargin=1.15cm] in customized-label enumerations +\renewcommand{\labelenumi}{\arabic{enumi}.} +\renewcommand{\theenumi}{\thetheorem-\arabic{enumi}} + +\renewcommand{\labelenumii}{\roman{enumii})} +\renewcommand{\theenumii}{\theenumi.\roman{enumii})} + +\renewcommand{\labelenumiii}{\alph{enumiii})} +\renewcommand{\theenumiii}{\theenumii.\alph{enumiii})} + + +% for doing things like: +% 1. .............. +% 2. .............. +% 3, 4. ........... +% 5. .............. +\newcommand{\doubleitem}{% + \begingroup + \stepcounter{enumi}% + \edef\tmp{\arabic{enumi}, }% + \stepcounter{enumi} + \edef\tmp{\endgroup\noexpand\item[\tmp\labelenumi]}% + \tmp} + +%% for proving two-way implications +% Usage: +% \begin{itemizeiff} +% First implication ... +% \item Second implication ... +% \end{itemizeiff} +\newlist{itemizeiff}{enumerate}{2} +\setlist[itemizeiff]{ + first=\item[{$\implies\!\!$)}], + label={$\impliedby\!\!$)}, + leftmargin=1.2cm, +} + +\newcommand*\ifftoright{\item[$\implies$:]} +\newcommand*\ifftoleft{\item[$\ \impliedby$:]} +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +%%%%% Theorems, definitions,... environments %%%%% +\newtheoremstyle{math_theorems} +{\topsep} % ABOVESPACE +{\topsep} % BELOWSPACE +{} % BODYFONT +{0pt} % INDENT (empty value is the same as 0pt) +{\bfseries} % HEADFONT +{.} % HEADPUNCT +{5pt plus 1pt minus 1pt} % HEADSPACE +{\color{\col}\thmname{#1}\thmnumber{\@ifnotempty{#1}{ }\@upn{#2}}% + \thmnote{{ \bfseries\color{black}(#3)\index{#3}}}} % CUSTOM-HEAD-SPEC + +\theoremstyle{math_theorems} +\newtheorem{theorem}{Theorem} +\newtheorem{lemma}[theorem]{Lemma} +\newtheorem{proposition}[theorem]{Proposition} +\newtheorem{corollary}[theorem]{Corollary} +\newtheorem{axiom}[theorem]{Axiom} +\newtheorem{definition}[theorem]{Definition} +\newtheorem{law}[theorem]{Law} +\newtheorem{principle}[theorem]{Principle} +\newtheorem{model}[theorem]{Model} + +\newtheoremstyle{math_remarks} +{\topsep} % ABOVESPACE +{\topsep} % BELOWSPACE +{} % BODYFONT +{0pt} % INDENT (empty value is the same as 0pt) +{\itshape} % HEADFONT +{.} % HEADPUNCT +{5pt plus 1pt minus 1pt} % HEADSPACE +{\color{\col}\thmname{#1}\thmnote{{ \color{black}(#3)}}} % CUSTOM-HEAD-SPEC + +\theoremstyle{math_remarks} +\newtheorem{remark}{Remark} + +% \renewcommand*{\qed}{{\color{\col}\qed}} +% \renewcommand{\qedsymbol}{\textcolor{\col}{\qed}} +\newcommand*{\QED}[1][$\square$]{% + \color{\col} + \leavevmode\unskip\penalty9999 \hbox{}\nobreak\hfill + \quad\hbox{#1}% +} + +\makeatletter +\renewenvironment{proof}[1][\proofname]{ + \par% + \pushQED{\QED}% + \normalfont% + \topsep5\p@\@plus\p@\@minus\p@\relax% + \trivlist% + \item\relax% + {% + \color{\col}% + \itshape#1\@addpunct{.}}\hspace\labelsep\ignorespaces% +}{% + % \color{\col!30} + \popQED\endtrivlist\@endpefalse% +} +\makeatother + +\newcommand\explanationname{Sketch of the proof} % could be made language-dependent +\newenvironment{sproof}% +{\renewcommand\proofname\explanationname% + \begin{proof}}% + {\end{proof}} +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +%%%%%% Extra references %%%%%%% +\makeatletter +\if@cref@capitalise % if we pass the option 'capitalise' on cleveref package + % \crefname{theorem}{\bfseries\color{\col} Theorem}{\bfseries\color{\col} Theorems} + % \crefname{lemma}{\bfseries\color{\col} Lemma}{\bfseries\color{\col} Lemmas} + % \crefname{proposition}{\bfseries\color{\col} Proposition}{\bfseries\color{\col} Propositions} + % \crefname{corollary}{\bfseries\color{\col} Corollary}{\bfseries\color{\col} Corollary} + % \crefname{axiom}{\bfseries\color{\col} Axiom}{\bfseries\color{\col} Axioms} + % \crefname{definition}{\bfseries\color{\col} Definition}{\bfseries\color{\col} Definitions} + % \crefname{law}{\bfseries\color{\col} Law}{\bfseries\color{\col} Laws} + % \crefname{principle}{\bfseries\color{\col} Principle}{\bfseries\color{\col} Principles} + % \crefname{enumi}{\textcolor{\col}{Item}}{\textcolor{\col}{Items}} + % \else + % \crefname{theorem}{\bfseries\color{\col} theorem}{\bfseries\color{\col} theorems} + % \crefname{lemma}{\bfseries\color{\col} lemma}{\bfseries\color{\col} lemmas} + % \crefname{proposition}{\bfseries\color{\col} proposition}{\bfseries\color{\col} propositions} + % \crefname{corollary}{\bfseries\color{\col} corollary}{\bfseries\color{\col} corollary} + % \crefname{axiom}{\bfseries\color{\col} axiom}{\bfseries\color{\col} axioms} + % \crefname{definition}{\bfseries\color{\col} definition}{\bfseries\color{\col} definitions} + % \crefname{law}{\bfseries\color{\col} law}{\bfseries\color{\col} laws} + % \crefname{principle}{\bfseries\color{\col} principle}{\bfseries\color{\col} principles} + % \crefname{enumi}{\textcolor{\col}{item}}{\textcolor{\col}{items}} + \crefname{axiom}{Axiom}{Axioms} +\else + \crefname{axiom}{axiom}{axioms} +\fi +\makeatother + +\newcommand{\mref}[1]{\hypersetup{linkcolor = \refcol{#1}}\ref{#1}\hypersetup{linkcolor = \col}} % custom ref +\newcommand{\mcref}[1]{\hypersetup{linkcolor = \refcol{#1}}\cref{#1}\hypersetup{linkcolor = \col}} % custom cref +% \newcommand{\mnameref}[1]{\hypersetup{linkcolor = \refcol{#1}}\nameref{#1}\hypersetup{linkcolor = \col}} % custom nameref +\newcommand{\mnameref}[1]{\hypersetup{linkcolor = \refcol{#1}}\hyperref[{#1}]{\ref{#1} \nameref{#1}}\hypersetup{linkcolor = \col}} % custom nameref +% \newcommand{\mfullref}[1]{\hypersetup{linkcolor = \refcol{#1}}\hyperref[{#1}]{\ref{#1} \nameref{#1}}\hypersetup{linkcolor = \col}} +% \newcommand{\fullref}[1]{\hypersetup{linkcolor = \refcol{#1}}\hyperref[{#1}]{\autoref{#1} \nameref{#1}}\hypersetup{linkcolor = \col}} +% \crefformat{enumerate}{#2\textcolor{\col}{Item} #1#3} +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \ No newline at end of file diff --git a/preamble_formulas.sty b/preamble_math.sty similarity index 50% rename from preamble_formulas.sty rename to preamble_math.sty index 4c6db19..d8a30c5 100644 --- a/preamble_formulas.sty +++ b/preamble_math.sty @@ -1,175 +1,4 @@ \usepackage{amsthm, amsmath, mathtools, amssymb, physics} % for math and physics commands and symbols. -\usepackage[intlimits]{esint} % for more options on integrals. -\usepackage{standalone} % for compiling individually .tex images. -\usepackage[hypcap=false]{caption} % basically for using the command \captionof{}, which is useful in some environments, others than \begin{figure}. -\usepackage{subcaption} % for subfigures, when many captions come into play. -\usepackage{multicol} % organize the whole document in multiple columns. -\usepackage[USenglish]{babel} % for improving use of English and mainly the use of hyphenation. -\usepackage[version=4]{mhchem} % used for chemistry and nuclear physics. -\usepackage{multirow} % multirow in a table, for example. -\usepackage{array, makecell} % for additional options in tables and tabular. -\usepackage{hyperref} % link automatically references. -% \usepackage{hycolor} % implements options of colors for hyperref (it ables the user to use colors from xcolor package) -\usepackage[capitalise,nameinlink]{cleveref} % for clever references. That is, for example, for automatically write "Definition 2.13" instead of only "2.13". -\usepackage{graphicx} % for options like \definecolor{}{}{}, \textcolor{}{},... -\usepackage[usenames,x11names]{xcolor} % to customize RGB colors. -\usepackage{colortbl} % for colors in tables. -\usepackage{tcolorbox} % for coloring boxes. -\setlength{\parindent}{0pt} % Indentation disabled. - -%%%%% tikz images %%%%%% -\usepackage{tikz, pgfplots} % for tikz images. -\usepackage{tkz-euclide} % for getting some useful geometric commands. -\usepackage{tikz-3dplot} % to draw a visual 3d image with tikz (not with pgfplots). -\usepackage[RPvoltages]{circuitikz} % to draw electric circuits with tikz -\usetikzlibrary{arrows,arrows.meta,decorations.pathmorphing,decorations.pathreplacing,decorations.shapes,decorations.markings,patterns,calc,fillbetween,intersections} -\usepgfplotslibrary{fillbetween, polar} -\pgfplotsset{compat=newest} -%%%%%%%%%%%%%%%%%%%%%%%% - -\usepackage{float} % to allow using argument [H] in figure environtment - -\setlength\multlinegap{0pt} % disable the margins on \begin{multline} command. -\allowdisplaybreaks % allows to break an align environment in two pages. - - -%%%%%%%%%%%%%%% Colors %%%%%%%%%%%%%%% -\definecolor{color_blue1}{RGB}{190,190,255} -\definecolor{color_blue2}{RGB}{100,100,255} -\definecolor{color_blue3}{RGB}{0,0,255} -\definecolor{color_blue4}{RGB}{0,0,150} -\definecolor{color_green1}{RGB}{160,240,160} -\definecolor{color_green2}{RGB}{50,200,50} -\definecolor{color_green3}{RGB}{10,150,10} -\definecolor{color_green4}{RGB}{0,80,0} - -%% for creating a rainbow shade -\pgfdeclareverticalshading{rainbow}{100bp}{color(0bp)=(red); - color(25bp)=(red); color(35bp)=(yellow); color(45bp)=(green); - color(55bp)=(cyan); color(65bp)=(blue); color(75bp)=(violet); - color(100bp)=(violet)} -%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - -%% colors - subjects %% -\newcommand{\col}{red}% % DEFAULT -\newcommand{\ana}{DeepSkyBlue2}% % analysis -\newcommand{\alg}{Green3}% % algebra -\newcommand{\geo}{Purple2}% % geometry and topology -\newcommand{\apl}{Red2}% % applied mathematics -\newcommand{\pro}{Orange2}% % probability and statistics -\newcommand{\phy}{SlateBlue2}% % physics - -\ExplSyntaxOn -\NewExpandableDocumentCommand{\refcol}{m}{ - \str_case_e:nnF {\FirstWord{#1}}{ - {FM}{\alg} % fundamentals of mathematics - {LA}{\alg} % linear algebra - {RVF}{\ana} % real valued functions - {AS}{\alg} % algebraic structures - {DM}{\alg} % discrete mathematics - {FSV}{\ana} % functions of several variables - {LG}{\geo} % linear geometry - {MA}{\ana} % mathematical analysis - {NM}{\apl} % numerical methods - {CA}{\ana} % complex analysis - {DE}{\apl} % differential equations - {DG}{\geo} % differential geometry - {GT}{\alg} % Galois theory - {P}{\pro} % probability - {S}{\pro} % statistics - {TOP}{\geo} % topology - {AT}{\geo} % algebraic topology - {A}{\alg} % arithmetic - {CAL}{\alg} % commutative algebra - {DS}{\apl} % dynamical systems - {HA}{\ana} % harmonic analysis - {LM}{\pro} % linear models - {NC}{\apl} % numeric calculus - {NIPDE}{\apl} % numeric integration of pde - {PDE}{\apl} % partial differential equations - {RFA}{\ana} % real and functional analysis - {SP}{\pro} % stochastic processes - {ADS}{\apl} % advanced dynamical systems - {AP}{\pro} % advanced probability - {ATFAPDE}{\ana} % advanced topics in functional analysis and pdes - {IEPDE}{\ana} % introduction to evolution pdes - {INLEPDE}{\ana} % introduction to non linear elliptic pdes - {LTLD}{\pro} % limit theorems and large deviations - {SC}{\pro} % stochastic calculus - {INLP}{\phy} % Instabilities and nonlinear phenomena - {MM}{\pro} % Montecarlo methods - {CO}{\apl} % Continuous optimization - {NMPDE}{\apl} % Numerical methods for pdes - {SCO}{\pro} % Stochastic control - {JP}{\pro} % Jump processes - }{\col}% -} -\ExplSyntaxOff -%%%%%%%%%%%%%%%%%%%%%%% - -\newcommand\FirstWord[1]{\@removedots#1:\@nil}% -\newcommand\@removedots{}% -\def\@removedots#1:#2\@nil{#1}% - -\newcommand\changecolor[1]{ - \renewcommand{\col}{\refcol{#1}} - \hypersetup{linkcolor = \col} -} - -%% for references (hyperref) -\hypersetup{ - colorlinks = true, - linkcolor = Blue4, % color of internal links (sections, pages, etc.) - filecolor = cyan, % color for URLs which open local files - citecolor = green, % color for bibliographical citations in text. - urlcolor = cyan, % color of URL links (mail, web) -} -%%%%%%%%%%%%%%%%%%%% - -%% Color box for important theorems -\def\sep{1mm} -\newtcolorbox{important}{colback=\col!5, colframe=\col, left=\sep, right=\sep, top=\sep, bottom=\sep} -%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - -%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - -%%%%%%% Enumeration - commands %%%%%%% -\usepackage{enumitem} % for the command [leftmargin=1.15cm] in customized-label enumerations -\renewcommand{\labelenumi}{\arabic{enumi}.} -\renewcommand{\theenumi}{\thetheorem-\arabic{enumi}} - -\renewcommand{\labelenumii}{\roman{enumii})} -\renewcommand{\theenumii}{\theenumi.\roman{enumii})} - -\renewcommand{\labelenumiii}{\alph{enumiii})} -\renewcommand{\theenumiii}{\theenumii.\alph{enumiii})} - - -% for doing things like: -% 1. .............. -% 2. .............. -% 3, 4. ........... -% 5. .............. -\newcommand{\doubleitem}{% - \begingroup - \stepcounter{enumi}% - \edef\tmp{\arabic{enumi}, }% - \stepcounter{enumi} - \edef\tmp{\endgroup\noexpand\item[\tmp\labelenumi]}% - \tmp} - - -\newlist{itemizeiff}{enumerate}{2} -\setlist[itemizeiff]{ - first=\item[{$\implies\!\!$)}], - label={$\impliedby\!\!$)}, - leftmargin=1.2cm, -} - -\newcommand*\ifftoright{\item[$\implies$:]} -\newcommand*\ifftoleft{\item[$\ \impliedby$:]} -%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - %%%%%%% Math-commands %%%%%%% %%% GENERAL @@ -444,109 +273,4 @@ \cleaders\hbox{$\mkern-2mu\smash-\mkern-2mu$}\hfill \mkern-7mu\raisebox{-3.66pt}[\p@][\p@]{$\mathord\mathchar"017E$}$ } -%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - -%%%%% Theorems, definitions,... environments %%%%% -\newtheoremstyle{math_theorems} -{\topsep} % ABOVESPACE -{\topsep} % BELOWSPACE -{} % BODYFONT -{0pt} % INDENT (empty value is the same as 0pt) -{\bfseries} % HEADFONT -{.} % HEADPUNCT -{5pt plus 1pt minus 1pt} % HEADSPACE -{\color{\col}\thmname{#1}\thmnumber{\@ifnotempty{#1}{ }\@upn{#2}}% - \thmnote{{ \bfseries\color{black}(#3)\index{#3}}}} % CUSTOM-HEAD-SPEC - -\theoremstyle{math_theorems} -\newtheorem{theorem}{Theorem} -\newtheorem{lemma}[theorem]{Lemma} -\newtheorem{proposition}[theorem]{Proposition} -\newtheorem{corollary}[theorem]{Corollary} -\newtheorem{axiom}[theorem]{Axiom} -\newtheorem{definition}[theorem]{Definition} -\newtheorem{law}[theorem]{Law} -\newtheorem{principle}[theorem]{Principle} -\newtheorem{model}[theorem]{Model} - -\newtheoremstyle{math_remarks} -{\topsep} % ABOVESPACE -{\topsep} % BELOWSPACE -{} % BODYFONT -{0pt} % INDENT (empty value is the same as 0pt) -{\itshape} % HEADFONT -{.} % HEADPUNCT -{5pt plus 1pt minus 1pt} % HEADSPACE -{\color{\col}\thmname{#1}\thmnote{{ \color{black}(#3)}}} % CUSTOM-HEAD-SPEC - -\theoremstyle{math_remarks} -\newtheorem{remark}{Remark} - -% \renewcommand*{\qed}{{\color{\col}\qed}} -% \renewcommand{\qedsymbol}{\textcolor{\col}{\qed}} -\newcommand*{\QED}[1][$\square$]{% - \color{\col} - \leavevmode\unskip\penalty9999 \hbox{}\nobreak\hfill - \quad\hbox{#1}% -} - -\makeatletter -\renewenvironment{proof}[1][\proofname]{ - \par% - \pushQED{\QED}% - \normalfont% - \topsep5\p@\@plus\p@\@minus\p@\relax% - \trivlist% - \item\relax% - {% - \color{\col}% - \itshape#1\@addpunct{.}}\hspace\labelsep\ignorespaces% -}{% - % \color{\col!30} - \popQED\endtrivlist\@endpefalse% -} -\makeatother - -\newcommand\explanationname{Sketch of the proof} % could be made language-dependent -\newenvironment{sproof}% -{\renewcommand\proofname\explanationname% - \begin{proof}}% - {\end{proof}} -%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - -%%%%%% Extra references %%%%%%% -\makeatletter -\if@cref@capitalise % if we pass the option 'capitalise' on cleveref package - % \crefname{theorem}{\bfseries\color{\col} Theorem}{\bfseries\color{\col} Theorems} - % \crefname{lemma}{\bfseries\color{\col} Lemma}{\bfseries\color{\col} Lemmas} - % \crefname{proposition}{\bfseries\color{\col} Proposition}{\bfseries\color{\col} Propositions} - % \crefname{corollary}{\bfseries\color{\col} Corollary}{\bfseries\color{\col} Corollary} - % \crefname{axiom}{\bfseries\color{\col} Axiom}{\bfseries\color{\col} Axioms} - % \crefname{definition}{\bfseries\color{\col} Definition}{\bfseries\color{\col} Definitions} - % \crefname{law}{\bfseries\color{\col} Law}{\bfseries\color{\col} Laws} - % \crefname{principle}{\bfseries\color{\col} Principle}{\bfseries\color{\col} Principles} - % \crefname{enumi}{\textcolor{\col}{Item}}{\textcolor{\col}{Items}} - % \else - % \crefname{theorem}{\bfseries\color{\col} theorem}{\bfseries\color{\col} theorems} - % \crefname{lemma}{\bfseries\color{\col} lemma}{\bfseries\color{\col} lemmas} - % \crefname{proposition}{\bfseries\color{\col} proposition}{\bfseries\color{\col} propositions} - % \crefname{corollary}{\bfseries\color{\col} corollary}{\bfseries\color{\col} corollary} - % \crefname{axiom}{\bfseries\color{\col} axiom}{\bfseries\color{\col} axioms} - % \crefname{definition}{\bfseries\color{\col} definition}{\bfseries\color{\col} definitions} - % \crefname{law}{\bfseries\color{\col} law}{\bfseries\color{\col} laws} - % \crefname{principle}{\bfseries\color{\col} principle}{\bfseries\color{\col} principles} - % \crefname{enumi}{\textcolor{\col}{item}}{\textcolor{\col}{items}} - \crefname{axiom}{Axiom}{Axioms} -\else - \crefname{axiom}{axiom}{axioms} -\fi -\makeatother - -\newcommand{\mref}[1]{\hypersetup{linkcolor = \refcol{#1}}\ref{#1}\hypersetup{linkcolor = \col}} % custom ref -\newcommand{\mcref}[1]{\hypersetup{linkcolor = \refcol{#1}}\cref{#1}\hypersetup{linkcolor = \col}} % custom cref -% \newcommand{\mnameref}[1]{\hypersetup{linkcolor = \refcol{#1}}\nameref{#1}\hypersetup{linkcolor = \col}} % custom nameref -\newcommand{\mnameref}[1]{\hypersetup{linkcolor = \refcol{#1}}\hyperref[{#1}]{\ref{#1} \nameref{#1}}\hypersetup{linkcolor = \col}} % custom nameref -% \newcommand{\mfullref}[1]{\hypersetup{linkcolor = \refcol{#1}}\hyperref[{#1}]{\ref{#1} \nameref{#1}}\hypersetup{linkcolor = \col}} -% \newcommand{\fullref}[1]{\hypersetup{linkcolor = \refcol{#1}}\hyperref[{#1}]{\autoref{#1} \nameref{#1}}\hypersetup{linkcolor = \col}} -% \crefformat{enumerate}{#2\textcolor{\col}{Item} #1#3} -%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \ No newline at end of file +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \ No newline at end of file diff --git a/preamble_tikz.sty b/preamble_tikz.sty new file mode 100644 index 0000000..ae7edb0 --- /dev/null +++ b/preamble_tikz.sty @@ -0,0 +1,27 @@ +%%%%% tikz images %%%%%% +\usepackage{tikz, pgfplots} % for tikz images. +\usepackage{tkz-euclide} % for getting some useful geometric commands. +\usepackage{tikz-3dplot} % to draw a visual 3d image with tikz (not with pgfplots). +\usepackage[RPvoltages]{circuitikz} % to draw electric circuits with tikz +\usetikzlibrary{arrows,arrows.meta,decorations.pathmorphing,decorations.pathreplacing,decorations.shapes,decorations.markings,patterns,calc,fillbetween,intersections} +\usepgfplotslibrary{fillbetween, polar} +\pgfplotsset{compat=newest} +%%%%%%%%%%%%%%%%%%%%%%%% + + +%%%%%%%%%%%%%%% Colors %%%%%%%%%%%%%%% +\definecolor{color_blue1}{RGB}{190,190,255} +\definecolor{color_blue2}{RGB}{100,100,255} +\definecolor{color_blue3}{RGB}{0,0,255} +\definecolor{color_blue4}{RGB}{0,0,150} +\definecolor{color_green1}{RGB}{160,240,160} +\definecolor{color_green2}{RGB}{50,200,50} +\definecolor{color_green3}{RGB}{10,150,10} +\definecolor{color_green4}{RGB}{0,80,0} + +%% for creating a rainbow shade +\pgfdeclareverticalshading{rainbow}{100bp}{color(0bp)=(red); + color(25bp)=(red); color(35bp)=(yellow); color(45bp)=(green); + color(55bp)=(cyan); color(65bp)=(blue); color(75bp)=(violet); + color(100bp)=(violet)} +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

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b/Mathematics/4th/Numerical_integration_of_partial_differential_equations/Images/upwind.tex index 6cb21a4..ec42b53 100644 --- a/Mathematics/4th/Numerical_integration_of_partial_differential_equations/Images/upwind.tex +++ b/Mathematics/4th/Numerical_integration_of_partial_differential_equations/Images/upwind.tex @@ -1,5 +1,5 @@ \documentclass{standalone} -\usepackage{../../../../preamble_formulas} +\usepackage{../../../../preamble_tikz} \def\a{1} \def\lamb{0.8} diff --git a/Mathematics/4th/Partial_differential_equations/Images/traffic_char.pdf b/Mathematics/4th/Partial_differential_equations/Images/traffic_char.pdf index a5b5cdc6c833904247ff7767201429b6da4849a6..f3573878a7a7f655cd37e4f94a44a036c0dbcf95 100644 GIT binary patch delta 20338 zcmV(pK=8k@?gHWL0+7-eIUq0~Z(?c+JUj|7Ol59obZ8(kIWsgllYh1+f6blAu5?Fs zo#%Xt9NL3>9eo4O1lR?5R&oIz$Ql?WNFEdg$S@#1zSp;QoRgV_YZd|kNmlIZR6_G1QQsVwv!M79yq;J zdjt`YPCGB-~Z6;z1FJtzyH=< z`n!{60wWJ`d;HLZ_O4aeUq1K9+N+b80!IOQli&is0{{4vp#y0FCX)*UhJU|cKUPtR 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a/Mathematics/4th/Partial_differential_equations/Images/waves_char_solve.tex +++ b/Mathematics/4th/Partial_differential_equations/Images/waves_char_solve.tex @@ -1,5 +1,6 @@ \documentclass{standalone} -\usepackage{../../../../preamble_formulas} +\usepackage{../../../../preamble_general} +\usepackage{../../../../preamble_tikz} \begin{document} \begin{tikzpicture} diff --git a/Mathematics/4th/Real_and_functional_analysis/Images/young.pdf b/Mathematics/4th/Real_and_functional_analysis/Images/young.pdf index d2515b2b6a7f7a05c53cae5b8a0a5b86b8c97320..16e1a9d2e3048014ea84b9b27d082b5d727d77f3 100644 GIT binary patch delta 53221 zcmV(&K;gfa_XEN81CY`IIg!-Xe}4Y$=kL%5{MVoV%Z{r%|MBr_j5P=SzOZlI{-wH& z361|T8k*ZV`kfTfpjQL_$KU^<)nEUOHvJbP8`aOBdW>}M_(s3BuhyfhhyGuG|HnW5 z{;&V?^OqkD=kLa%r)~eWT5Id~!90Wb(?5*r)9(N6aQ?7x{$ZxpuW`JOe`l(G2m0x; z_*j(xTEqQ&;J^O-_Md-#`)_|(q3xIb@W1}^KmYdg`;Y(h=imO_fB)OR{QLj-AN0Zh z*H8ZZ>wkHE(*2*}Kh4}~(&GP|LABZGrmq$g5u3z&-~9Hq&d>VQeD(SBS4hUXs|R%5 z){Q1}Eu~tm@!R%Y{JeAhSUjUg2O>Dr 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