From 12f15e19985c8a66780abf5d6cd5b03c5dab96ef Mon Sep 17 00:00:00 2001
From: =?UTF-8?q?Gi=C3=B9=20Marcer?= <giuliamarcer@yahoo.it>
Date: Tue, 12 May 2020 22:38:14 +0200
Subject: [PATCH] ex-4: revised and typo-fixed

---
 ex-4/main.c          |   3 +
 ex-4/plot.py         |   2 +-
 notes/images/dip.pdf | Bin 16088 -> 15805 bytes
 notes/images/fit.pdf | Bin 17733 -> 17371 bytes
 notes/sections/4.md  | 175 +++++++++++++++++++------------------------
 5 files changed, 80 insertions(+), 100 deletions(-)

diff --git a/ex-4/main.c b/ex-4/main.c
index 364bc1c..ab7c2ed 100644
--- a/ex-4/main.c
+++ b/ex-4/main.c
@@ -41,6 +41,9 @@ int main(int argc, char **argv)
   size_t n = 50;
   double p_max = 10;
   size_t go = 0;
+
+  // Get eventual CLI arguments.
+  //
   int res = parser(&N, &n, &p_max, argc, argv, &go);
   if (go == 0 && res == 1)
     {
diff --git a/ex-4/plot.py b/ex-4/plot.py
index 9f9ecea..05ab11a 100755
--- a/ex-4/plot.py
+++ b/ex-4/plot.py
@@ -15,7 +15,7 @@ edges = np.linspace(0, bins*step, bins+1)
 plt.rcParams['font.size'] = 17
 
 plt.hist(edges[:-1], edges, weights=counts,
-         color='#eacf00', edgecolor='#3d3d3c')
+         histtype='stepfilled', color='#e3c5ca', edgecolor='#92182b')
 plt.title('Vertical component distribution', loc='right')
 plt.xlabel(r'$p_h$')
 plt.ylabel(r'$\left<|p_v|\right>$')
diff --git a/notes/images/dip.pdf b/notes/images/dip.pdf
index 2a6ba888cd45fbea7734342c744b3f0cf75f70c5..3dec79605440aeb912e77f4450d8acecb5d942cc 100644
GIT binary patch
delta 2845
zcmZuw2{e@b9?tg3e1r&TEMv)5yze{ScNWo=EFpZV%UH@fVoZxA48AG5Fqw$4eJM#)
z$CS?!<Fh2P3@Y0#X|p>w%M^*LEBBmxyEFaIdEfIt&-<L;?|Giz?;VM4jAq*kiGs$4
z_$&vZP<m;e-7z#yremI`7p5hiP(Mgh`RBu<C(eu<53!ke>y>_k<0VpKw7RfbyL=<h
zztnV`-Q6RdK^r(5*g1OC?@T&2;>$Wj&udy-$-;y(`ncogYV&QE5x}tNjFCtxw=TTl
zuF;E6QN-xP<UfFg7!9&}Tf%A0fF5QjTb`#7Lw_wKF-BC#<Z4O{5Sj`mj_W=2U;nJ_
z7Sr>gdiBL<L6wV#<tq>LwP|OC(OYfi`mA%uSciI;Sy7yprQf~(HX_xWVQ?t>rdF%e
zAamza&cRdmCfqVO`h|mBqMrprqPF2lpQ}R-I!7t2CH0<-OqZ$s$zbpKa^2*1j$BA-
z-y;2v;)T{OyB7TgVU^3jPT4J01noURJaWS@fQ8XgOY8DuAOY5SYoiwWQ}Q=MF5Qgj
z?OY*9?x`v}J|*E}(V{fNLsOj#A}2t138uw#h_vIbNJnRtQ1Y)%er<ASDYp~%K~4tS
zMs00WXD-$uui3Pv=s>h9r(2}Zo)cPcs7|j_GCwckfPntbRtAn_Wha<>s+LTo<b5&?
zcM3~MRn>Mc7Oo`nK-W}O+*kC5vR#sLcA0Q|l4!w7R0~BcTyx?5{*x|VJXOV9s-B+Z
z;H5ghqR2SzT48XK)^FDn%iKi+lDG|ps8L<{N;BQ<x8*%Eg#yBlS({^XZEBb@ztbP|
zz55_~{XQa#XKUR`9hl1Y-sLjH)AB<`B#?QY52m1E7c+I=wYT0p!{kw!w!!tCg|7E1
zL@?k(BM0qY>S!1S;Fw#fl1-`E&H}ZU*pa!>N+eM|%SWPn-pvFp8Y(X68+%co$_IUn
z&wK(c!R1C63}&LGy7gi}TCoI$o_fyX0@JUpG3jGH%@UuJcJ;1w82;q*RhDI}1om|6
zjzMwaD5g)$@PsdmyEmg`bP5pmaP%m$25daa&HCS^sC*I5y@jqmf|?!no~1C)ffW?>
z@%z$w@JADmH=x!R&xzeAoueY2AF&>N<%1XPYIf{yydD1-FLs<HO&4>dCuAp3BxjeV
z_o34LJmqYwcZ9eJ(a<W^`c5o2U{yH$*j1sGI{<(gi~@9vFASslKaAyEtgmphk)1Rj
zALu&AqWtFF5!Gloov`~&RP%!{R6f4Nj&0bnEiEGIqFjsq;Nz;>$ITswZ<sY}IS=H$
zjX+GsgJ->;%QJdY;e(p=?ZJm<5*r)Pwx#IMJLp6sK>lq5C;SSP0aE4>@~h7~p4!Lz
z4(-E|x$k|44g~h(Co@RAb&w(p`&HSfR4IJ(9a`#QQL6@1{#?hA_g6AV-NU{^v(1^@
z*=|;KJxso@${^ik`wr=^u&NneJqXoGt*1T*q`dBDRU1<1J#H%q`Mv?VYKY1p&G{q;
z52;)s&DD3@3+MUPyw*&&c@>^cdd1y*w59$LyaJP{E;ur^7Eh)s63NsE6Eb!90GWz2
z^J_5ONcr<vJV+^%wOIDgKk2AAx%N@3eR_9yk!n%Tj+{cm@He!Hwt097(O^=GN1??A
z_Kw}4H`+`tVcS8@80I3WW9YWQPVvyyK;rqPylxZTa!5spbmZfp7OM~?lKAJ5sk*T@
zM&k?4z1@;*H#QUPeK?>QXB8CjC{xppU=+IXo#>SD?3s+ot|}uP!wJqk-d(Xhoh|pK
zhaqOg(dF^+(!$)v*WYOntzdlVrPMn85A;hyYs7(>p7MB{#>?;yitcRc5!7T*qTc6h
zIWcE;QNMNH*rHW)C$7j>!}Nni+;c52hux;;=b5iuPnR+_oUJY1KOvuA3n;$Tz$4M{
zb-H0iu5$L})|r6@)1sYc%O7#a`ypFZ)pvWn+8Nt1EL+Fh?8PK;s%YZlxc*D@M{jk%
z<`uUyBrZ?8gnIAL&dEq58a!{y^|DTAb?8r5k!{m`FzwBG{CBHJ&FO{9Y~~?FCj(o#
zff3ML0#?rNwTvN^CG^G}fXw`3a`y(bGoC&>>f@tu_|#J1CO__z4Z2b@lwcBOFL|{n
zQcd`ws%^vN#S01+%4TKCHm9}e2x?)(>ilS%OpIGvRk3?7E&DG<j<SY3N$1<pwic`S
z2eG7%%(YXpD^&~!8hM*^dCAHz^PNMb1GRQm^p_)^)HjM0LUd^C%HkC@HM6mUMH_WZ
z6Q7U*oOH@euC&I7FX?ec(<%RA9_gZ8O-tp@&s}#{2$z*!=QP&;PTzP*_b_d=N3S)d
z+p8_pok%KQ%*9F=8<H>DgRjr^jL$|_Qi%KYzjB94WoQo#jYqt$37^<rqe_ggf1x&f
zrLWdStttO*@?dT9+T2<IZ)SOBb#eK+T3@a7in~h#YeVEOeU}oH{~5o~un>Q5>i64O
zpCA+nssrj<7n~#v{uf2E-f8GmiK8Zsc0m9H05Awa5CB8Q1^}c306JOgggrvoEG6Q>
z&9^Jjl5Pxd7W#hF&#0reE}$_;fCMlI3kzUatg#>l|AdD__;{O*kS$&o+61OUpul%5
z0`S)cuvmg14?qMkV+6+W_k$p;AO=I&?>zbs<^hNR@izt`Jh(;tfd@j!PcRTC5GEG?
z6M;B@@Pok3{+q`bFEEMCxk30Jh+8}u#tLEx!Y>+t35bAW1o{aE)A9Tx18@Kg@HfU`
zae^3(<zE#5hu{P;9R5egTSo98_#c7v|Lq=t#|ox^M+6<?aRdP|1b`sWgs{2&{6+}I
z|8H(9e*#``aRJDfKN)ay3;Cl40l|@g0Ku4F3<yF334$ODnhF#Iae%-gfp|Rs+`-MC
U7tBgAvqB*VjMCP&bhJYK8y(6$yZ`_I

delta 3091
zcmZuw2{=^y8|JELZn(uQSsF{DD@^Cidd`(4r6g2nnxbUMPBEmAb1SJ3GL>y?xyVw8
z6b5BTC8?W9V;MWqGKolq{|t4n|4hGmo_Wsmp7(t3_rBlvs|tQ~vdBz)5ddHbbW3p>
z!cNwsb!=U^mG%u4=9f<C+E!zn?^W_R(d#Ptm0x<6Tau2<Ve7b*w=rFXpsbyL%)@(1
zMQ1WxqQ&eOdee9H#oVfs-KrC(L(a8!jnxd<kMhnX4*tsY(Tv$ynd|I(YII`eGwC=_
z9z8wW$$K&0K0avsc6_PQLG!w|5l_{iLL|beUs<_6WYOfh^pTYA*vW<ilp^J^<`co!
z<vHNczM`7X8x0RO&nBS<$z2~;wEb?jn=T9YZO&%p;JyxzF0n+;+>+1!jdpovbN97?
zoT;41Li93j4Ee<twenb}-qIi|^&G{^QF#vg4n%Q0uOIL1bfng#aGPe=@dj*WPfa4F
z7QNg`TG*uDezR%?-jz$S8oC9gO%36)hq(J{neqhZvaZQXyP~l7u>G1!S7i@%D!kX!
z^i3VDf9~O}@VjqPf*;43^-uhfPQBXX%tns2OJ5Z(`}G#<=zohSXZpo4=n7+3!2Cuv
z)}2S^2P6^#uL80o#@p5M5YT?LM8qPk+#hfZ$NfA24r=#ZKA>Y_?S^xl5*tjxJi+U2
zYTe^mjT^mRZxdJNy_FOqHuiMIAm-20V)B%+D?L$fOs`tSFK0;tJe@##Y~VraOON+l
z9Mf);5tbB1o%z@lyd$CHuHD1-BsP${xO~vYjJ~{45<qx54%kN+-A9xg;TKkD;z~Vz
z@Q$D7BqhpUmn6Qh++ysXdM&KQne|b;Kxs47&`mI|x499{R=M?euY0oeX33)W$~`F&
zax#Jv+l|XtcCStjp&I;k!gVQ42mT|!EsqUs%8Cn^bz*6%X|K6gN$ao}k>f7q@EeiO
zLa<?VUwxbUSs&w%aBXWD#j75;*_J43)>oQ7YIx>S_d)!L-j0OG3i*>o#C(T6E47Q`
zUXV1D=F8WgK7$hYG4bB7?mkG;m>-&Q&bhqGSA00Dx4X~U=SPAde4)<Glfr7*wn}}w
z+X}}IV%!uBCIar-z0^^g!tyv04O=H)lvBA^f$y-mpT9Znz?GQvGH3tpU2Plr`K2fs
zO36{K)IGnN7tTIrXIeZ_R$g>eY{+tXn=ux%!ZueiyN(*NUbm+3mH&75(%6W}GAp8a
zB4g;u+2p8sbZl3phh-UsXx`I%GCo=ZQNC9`c7_)Bg_hX1o?X<IS}di8vcT$~?=efl
zPFP0k=VAF%#}vS~`wxe6o#*ms?3CB9@M!VMw_zRiGCzoM^WcxHk<8O|upBYdkIJ=@
z&Ud0>Gs2!Z*L|Fek9JH8-@cv=X!Wm+zcub!GCLoX9mnJoQvWhH+`Y=GR*F)#nlqpF
zx!0xg?Ts;8%&LH-X}`{{x0O8-RC4%H%TK8)m({ZEh~@ypK}xR2f~3Pwixw70X^@Y+
zu~*vmOYw$^Nzzo&_ubS>GcNs<=kUwEsGiXBFW&gMRUYs$ONfoU{wj!F^hM%NS0pbm
zK=5X+{ipY{)VYE!GWyNF7U#+N9vMvw>v-QipTg?CV1{{DDBk`fN>yy^UfbJ>l)_Nj
z0|l<jcef4wYggXdlkZSFqCS<-9#(K%jQ4aoH}PoDXHO3b>+aLG&(Rp~vErV&TZn8p
z9`7VrEk=D-ca^O7)@6d^J1oi%qMjehOOWd^!b)Wg+_XrRZnUos@RV|8cuY`T68YQ3
znEWU8B$t2p#S8Sd9pB*(`U%Q}g)&TcbY!!7#hH9GbJ^0UjK?+w;g`}4w7*8qkW`kW
zF=NbV`n*aHr__`WV<yOS+SpD>h4A+!42kHaQ^W5Dz70zD!%guu4tmEfS7it3uAy06
zkAHKX6*A=zG*FYcfj7w8kb`;83Fhboua{ZH1HymxV|Fx_U0}6Cu7ym+2^-~s)3;@3
z92@tmc=f&}J>}R-OY@oQy$Pm{_N-FhUOt3sEz0TrJ4j_Y!rsU(b(dRrq;LG4Py5QW
z2>R~bn6$|zqa?`RLke-+X;?Gic5#2p*>t>v^>(c%d-arW89eMGIQbnsc5+Mt{R{I;
z<{Bq;!kHjX+p3EJp-nQG6){Yo$O0YP*=nzg=Xo-I@0<*53M&)OKHsOcEaNDu#KnIJ
zyd-Ggz>8b5lIswLipks9R<Uq>xv-5cu4%Yn%T$?DwNpmEU9DP4M%uyHT@R^kFZIqd
zgE+`$d(iR0olWJTSvf}h53U<fcRkZs>)gUPXEbyXT3m`9rT$fLiZWrBP6gVXhQm!~
z0L_qk(>kX*^V^EhV%*zAquXvSEl<QVZ|oF&GSVN1pjy*dt#dk6k9PbKbh#iQ!$Z|K
zTZz49ykY|575Gq`LZjBBE0$-U4dUhFbm^}$SUOex3UAm*yuoR_@wLNU7rz;f_#?Wu
zPQ4#4z*t+d61h14vEMQGo3bBLxGK%7ut%yZtAl47<I}nfLxQFUdak~hJTdcLqavV2
z0~s`43*}vFqr1vjO-P;~kACZ$$*65=Ij=HPQB^TB-1{!%Q!ZMsZD=+-t-WDMpz9i?
zI!d#F*Igf42Tyz3$Ho)9oPaj-5psj(p0C59H+#M$NLXbk#%r_;jyHbl8*0y=oE&2}
zbfVGdn`nb(w2$G*Q08=VeDfVNI@;o0=Z0~^YHnsQ>adS9&EM~^gSX)6h@%Gz1XTf5
z!2=E;0RMs{dWNc2$!n?p4YD9i1^|%%6C7|qy$4bf2kFz0A^%?=HgP2r0M)s7sy?cy
zc@lvD@v92X0*r_wh+rffUKAtzghwLMhhYnZ2oQz%03L#cYXEo>B+A1RMKA(PAPLvQ
z5pkjz1mXldWPu?7j{}8;K|Bb|5f|`49Q+fEKoSiN{8@uAw4g!0KK~Fwk;DNa5hO1l
z&hsFgC<YUSn+8ZYqKIP>@h2FBkcCGBknwm}SeQ&AieeB>cqITbOcKE$fV|N0oDqls
z%r}Vqw@&~>5={YuMIFO@8Q}(DJR~9l69G{k`TxyH^4sCx1OmN#qdX!U1;`IlI9m`9
zodF2q@WSdKfA+%aAP4~>^*|W@zBuatT`e3)79KQ6fB=B*t-loo6Ce}@W3qEA>VFf8
B@oE47

diff --git a/notes/images/fit.pdf b/notes/images/fit.pdf
index 7299f99f0bec43e862a86f6d8ca4e7d2706d73e8..9aaab98ac273c4e6b1bdd5fb8a6efaf0dff65578 100644
GIT binary patch
delta 4392
zcmZux2UJs8*A+oQEYyG?QboGtr3dMK6p$`Ogn)q3ks?UR0HFks7NjF0A{{9bl-?1f
zs<a8ch;%|PBmQyzZ_Ruk%w6lgch|oAp0oG4=e^c&DpDv_Nh1|DJLNZmemf|5cUx>+
z_kPT*w0L%mGVX~wuGYniPXNdJm6s|GaYR1q>0b0V(5!d3S#v6%!5BR{HJaDFv%^Xp
zN}U+rnD1TTA<6CUwo5ql?rk2-?k{|UJ$)>=K~5w^3qBay@1O8H*mplz*xI^Xw-p=Q
ztHq~^+%U{9C@sgTt+fXK;T&MT`un}OO&}<(zKohU?@Zb@EK+6PM?PEJEBEuN_~GYM
zDf0vJM}IK`kLq@jt^Y%b1cwGZkLrc&9Zu5D6ukG4aAwYlN^UP!=^?zbfuYgx8#8h3
zg_)yI)u?B6<;nu<Ew}8Uw&*9(^$kzm8aG>B(N<1AzJcDt3ZPyut7?c`m{J$t=lT=X
zwr|(jKQHWi;eO+pOWLw~Fq61ktC&6|?kUFT+{}0Sp*@kMRpWl&$(Qe>ar%fEc&TdV
z?IuLIvYPtPckhUdJa0_#joe*KagW0dtv=2xjaQP!<Z&!LigR7=r;4m;x)KtiFIW*)
zafUm25qObu8=F_j-JsWyrmOMrPe#eh##-Y`X7Wi!@-FA4-?J(8R^=cZy<u!urz@(A
zMt<A<`f8Dk7l6GFpo@EuVCVXwOa;|eIwU1!$vA$NUgBap&rFa#D4A(bJ!;9u-5+N=
zF?PAgpo91Do39oXWwX&ktFz2ld~7Q1YC6v}#zNKl8g|5@E!kvSP;y-ux}6#J;qhwm
z+0nP$Gs9jDHkv{l)IAqd9=3aZf7lxR;ZD#<AB&V+Tj<QPg$WLS@3LU#rc#>Q?T+!i
z?7d~L0jiI9_?syY%f_#2VW4fU=`ZqGr7D>AK}E%xP%p&}w30}Zn=9ycVjBbbf>S8l
z_*p&mF$QcPP)%n}JHm{knZuY^2RsS=67S6)a>ehWcd*ch)yVsx<_-9h@&M|z+XwnS
z2@<72bk#{0xhA<^h`A=U40pr#b>!M~PQOX4T_G69{{UY>HIT15@I_ru8;++XFJWND
zG3ynbFc<H{ThSYb)c2l<U#y1K2(0sRqFVIYw`#EO&PrEpf~)!)n_a||M{u=)m8S&B
z$ts~<+fk<o)#J*!gZyvlTyO-}cL5Er8xzh>Uy_hqSH>v>H|}6`_&iN%P)&p@!S6*y
z-i1G1=Zd(%Fheie-!3~5tYJC3R@dPDRG43DT_<g=;w4$B#zOgFt9VU>E)+dRl|nDt
zq11-OjeS_1T6UqK)j=ievC7TR6f-~V-{p(aR(e}|S!+^^iR!hIZ!h<*%;2zpCEW9H
z{LEv%MmW*l=Gh9TxmC4bQR$fC(6H0Ky~A#<pyyN`<^2;*-0bA6wk`mbZBB5Kygi<+
z_|~@Vph{Y9On>4e%?wiJ6emewnKH4voFr%<+*1o06%bKq{cL5VkCRkh2q)S)+dK5q
z4-LO?EH0IaFtcPIAyYP1DI1&i%S2TpKA(ZowjDc)41>?i)wy+p!}kY@N^4E-S^0pf
z+IAK_DI1%dqyn;_=iQmNF}3z3Z97ga%!icUz~Ywg;Yu+}d0zX5A-0GR(H1xHVgbv&
z6vsFgxYGFobGW%R5V5jQ`I_C)hs?6XZ@A_<nOl+IKEU{CXS)3h`;jCmqLPG^)bd=V
z9&~XFF6zOTTq2{krIrp0+@`nD8PJAq5K&4<t)_e(DfZm#emdc8?|#~*b8*Q4TG<Mc
z!dM&i0jpIjJHLs-Vt<{-+})_)4s1zSOaDS)N>^5YXfm94f1LX55EkUVmgZthS5MZg
zmmwPoj2DMLId5>4U`R|vI^^9DqJrpYeJ{9E|7@PRZqL9cE%2!?W4yYB4|B)4khiYp
z5&90>y~Grjj0~P9^|YM&>E?aSLDBq6vzocO5+6N%`mpRV6BV;PsRGuD+~)X0D)wQ4
zP#XR09@bzao4ahTv@at^lBDccch|ls8Bc9N^A&+M$Rf)A`=kngT?n^paCOj)617X=
zIHSi&vm(}yrFpj2BZOanJT3lenRaE^iWY7Oi(3`Q0Ys)I70$IWq}R9m_ng`Jh!$lw
zjpSIf&%>rOJl#{U!@H&XdiPx28U|v_wDMSuqWg-i;;PsAKJ>Dnne$qVPvIe=gXIjE
zFZ4ah4j&4h%htBuo82jtJaW9Gn8_q!{)S=y=ck#Yk^xN_YVj7s2~*!S2bor<A;dGp
zs_fW!Vx{ztg8>rH(azznT+QL$?Cuk1%fmO$=JnY29J!zTkpw9!IRaCT8iK=(Fwe&S
zGZC*kdSvGcThZCtGVhgm-_GNFzwTgl<Syw`v>PHINu3=btFkClcJ|D;)~SV{mkkCk
zsW$#>;VGGLOOILkbjwi_r%c_>1PuG#5TR*C(Vm5}t8rn{c1uKB4(o!lf{Uw;=tvtS
zIs{f6pP#Ioar*aR_Cm9_Ew8qgWg-;}Kt54};n}}?1RcbW@lg?-GrM%o9a`X~`CmD<
zW)}J>3@Y=BF%M5tyy>QAG2=(?uC|@M!Cz@PZ}}r@sA0~1X4<E>(Zyd_i7!is>*8mY
z?0ei%d}?b}dTn3*BAa!AI|r=o4X~|N?g8vABC94<QT}Kj^Xyd7K09?pNm3HRudDVo
z+8zJ;Ze1ksvRsj7RtwP;rhN&GF)&q)jr=(vF^AviIQ0T^UNq368ISwJ?uxMl587U|
z$e2DHCYAwr=Svh~UxWr<T$I$&X>gJT;M;Pz8B2X%&PCq!9*-{hn484;j@ZfWZ`i=d
zK9{=DzDK&}=|$REzm?E-c75+`zRm5KiAaJbMh>s@cN@lYF<kxx@#f$}fEucmFpr9P
zlEkKNGHo(3sAyN-w^bx2^9FUXFgL;@wsUk}j?Z^$q8_>AA4M{J3jI2yzmTf7i%?SN
zx?H$u_*^P$_rbIpx>yL>glnI?*@|5yKtffZ2naU}$R|5?22GmO&dI-Lw)tE(a3>5S
zf3#>(X01oPaL+H;>05I|T2sD+b8fL|U~hkkx2D^@rj(Ryy(?MqisLYHF{2?2(ljsK
z-eZfu@7n@2G6{v3i%=a2W#^0(&Gn&c6~J1oqXPnMS_89o^PGb|HEc;WSkPAon$MGS
z+)~3^nkbv6+FE7gfSdq!vRMVm_CkeT;tqQDW11L8v*{iS8`|~;^_}$c%kOh5;(W^d
zZ+=6l$s4F#rxQ--yM<(W01wd-=j^X_Nnmz(^2%q$%ln1dk%=OV>-G$vy5pygo3~`u
zKdxi8u}LiPg?SBJ36qR2cx<=OgGYWwypkK!eL)`kznT3=GV_s6Ga-;TG*&d;Ng;Jh
ztTM``hDTr;%Uo<uE@4y4Q_s|H84?GT-!x6dS{!NIy1htpM^1TdUYmG@P+M_ZR<vjd
z=mMWVI%nM~odFN5$_&YB{<so{&m5Az^23PDGAvv(-(auQ_<U95wGr%~p=R2PtAgG1
zG6+`vY*6q=HjNa6tdQk7u9OZ&IKfd*CCC?UZ-;}G-iD@&s%$z=Jsz0WD^EWAM65<M
z!<N@K?NG?9o*~szAzH@bpvl!KoV%aZckLF4{d_13m8IMr1!g4C4YbQU!y6sbJZH0D
zQA87c3ft|SRg4e<eZ`vBG+L!T%Jn0KXVhx0b}Z7Zsp~o_&oF(ZuALv9>zg%qv@4x1
z)RzqE<CPQQHZ6G+cU`2)C}6Ze;f&+7R43`3s<`eHF!m<)PXVZd=4*j)c)Oh!3%%rq
z|HCV(bZ}0sQq^nyb9@!tLI%i@^Ts*d@{>6jNk@9&nBa8>91nKhA>jAVVNke{<#}o!
zQ>}|Zz{8CMaVZv*gBvqX66g!Ap)@s8XhgmFO}AIM3(aWeZfW-$FV*Y7aEWY3y5ob>
zP-21vXmc_7;8CKc=V+h5;89o*zGi2z*znv)iR$&sCPmPnZARy4lpxyT{E!|8+!_-q
zaFwOQ;~-6f4vEzY7(sf3@~(6A^6?C^@P>sx&#`BFKz)HMglD~%gSy~%H-dEa^wo4f
zn^7CQRN|yApKpA4w=r|*u-Bz-fjgES?*enRS-XBP88+ayZ%A}>XV5~u6!QBwE-?$r
zt>0_LXnvNO&8pL8jqPrX9Qu=<95?jc4xD>2C@!2Iy8aUTGK*<wg;&ijo@@E8Mj)i>
z1Glr{%Y!p)bL@F%W|)|7aw*PiXv%PzTb!*HqGGkX{*2Fb#wwn`Q9N}j)~KFNz~{AU
z#-aS$H9XL(ks_@7?de9AZULxamNuc79~8)G{Vad%K}9CVZ$eT&G~C3@F3=5o+jkwl
zpF&Dw-eaI7ESExT=A@%3L38%A6UHI-yu+S;twDRx_l3hxt4U=CBOH2a%Tk&@3msj4
z?!Fpc9Dl1fCcVC-U-)z@v1A8cIdOBiW2tUF>i+x1`Pw=OVZJ13l7{-j29MN0qNg^n
zUwM~Dw$^*US(5Z=7g=A2-CZ2ZS!qY|$oY9~zMYV(g?s;8CHDFHA25WRs>aY;J3Vl7
ze{79${Iznnw4(<AUNG;k1r!P({{{)bRpD}X#&41!2p9qe!vP2c0!BelVql067%W71
zB!7hpN(h#hqdZ&XRcyfkk{A4I@H+9*|Fr<%NI2>Qh5}R49pj-Xr%qsyf8oI(1Xl$O
zEEI`?96LY}h&&zy3_~MM@L&k&2@C;&K##|RqQECG7#M}7@URrEf4c<3VCdtC0T>kd
zi}>F>00x}Iph)P+s=@wEAQJvhykGe#13`)S5BMJpiH04=kO25327w+I4MxJDCmbVD
zCovQjfFHjbFbV-6jweQ;p(ijj7=64az-S2cBnE~5m*f98hen|O5=im?Px;a46IB2x
z4SU=%0EWVkivhq0#PK8m7>PQ;1Hk`d?yvd-0PJLk0;pdf)35QjKmhoc?uMMa5&!}R
wPKbe^PE-(pf@M+1?+$=M5XXA~fWqJ>s(=6^k%Vn!ReA^%pcfQWxuZ(|f1X}^Hvj+t

delta 4781
zcmZux2{@E%8*WkdCQf80WS7|&OJrZhl6}uEgfbyx`68tVhe4w(SsGy^`%Xn66WNoH
zZHR-Al%#*uIsN}{&O6sN*LOYd{XNffFEgVt`=7_|fBbYmFqnZM2()ut|NfQB*Yq%J
zpqqmPuQ<nvMDZx-F{sp0hRF8^#GkbGCJx^jfhl3eSVfMdL?XGB)?bfVD99#S>G<0^
zX7#Ow-Rd%k*s<TK+m4%1mGcUr^sJlBjE;FFaC%dxd+sD6^mxMJH?t!YGb5B$wr<}W
z515=bdx^o?0N$+fpe%?sdtp55$8{Bvs%YY&nXoRd$DTX&?@kxh2tZr9E<|R}zN(lN
z?kb+`^&Gi-1hyfeG6eYo?fW`vKlR}i4*i|^ZWpDny7%Rd!7H;*q}hAzD-k@66UwTY
zrb{^qL%5bTtjeJzna;C;Z{|;E@mO9jtqJ8B%Fb?f+yRUmONxQW%@0F2Wmg1suD~)*
z>c{r4HoFw+66;dn{!5px319u1aFR0DmQ*3U!kViYHzO6|%AP2Eb>qh5n6zYkuQ<%X
zMhWYG)BUq3YUty9dk?;cW4!J<#D#B!)`$00)g9zqiHhIHgW!p>=(@JoT|7-_s+HVv
zWyM6tQQ(EBkm|m5K?9-{bKo;Ifpyum58?bLH|ol6wu7F8wR2K^84Bfx7l`M|lx^p|
zs;!aSt6%mLSFE3f&6qGbtWrvOTB6i)hVA0sy=Qc4c4fxZr|Mcfz)-E&0Mh2b<Ho9+
zt-F2Jmd#D`m1hGiaH^r?6d`@y@cei!<(*XkHc=5Ba>|)<tjQzLx)35R4e6#`MC_v~
zrGY&^)>D=(id*#!bLQare)3blIAD(13Opi>i9M+LlV9u}jq{iN*8V)t>IOyFWsIs6
z$(6?>IvY~#e9)ivk%I*$$0S9`L8Z2W+G{MnNBTBj2>ngChm*-{6(|~XcE@Hk6lqRa
zw@vOc2nkMpqmh!_=p9IG<DmZ4^lF!yu<E*s0nu|=)F5<l0enlEP20xJha={fHF447
zE*rEy$r7AR-F?3Wve!QE<w2WEatpmxNLzmOrvaET!gzsI$&uC?W?xPG7+~Et{ZqGf
zqng*^j#Y}-KDClIo0{J$i#B@7@0O=|dyGi^esaGwXjB@MitK3WYu>leQ>9<Yb=jXa
z5!lW^ZHrJE@uYW+Zh=HaTwc=^S5}bF)%66?O5W;mv(?x^=W?Ux<t5^wxGUTHD8X{h
zrvL<;I^bBQh-EP~6qg+Ny^TJeFE8}@0$6@jFNj)umGOP!G~=A3ItPi*s0@QGkCsC`
zs4TnwcSNwKa2Gm#EMoM8oG-R6%#Dj4*^N|JgPChC$33=E3qcY_E^Yg&mcY*HPrJ)s
zjW1LcwZe(6UVz5i-wB+qcCa$du%ziOz~^f{w)T$(_V-0D;Y$|4Iagr&32%Pun_&zs
zNL*vhEtY%Oul(%=AI;W9b`A6<kG|kYmqiSh63bD$;&DFI{C?{d^s`WtKMAMO6vwTw
z)3re!jvGTlsNEa`v>+!O4fS;_RV1@f&z(hd#h+0Q+c&P15L}RDDxz?_)KbtA&+edF
z5U)au>(BL(EWJLKF{#>Od0q4Taepi6C^3r;EVU7|^q%C_9~%1SSEZ3dl5(t2B&TpI
za>{S~PPRW&5|S`XJ|YWH8axegLnT@QzalTGGe>DLs!+rxs?88%+@Zlg*neGQBiee4
zXm8{NtiLWAb?1*m|8#K0NTcRQz9JVcaV8e6Q~ZW9YS%g~thy+#f$ksV==ykf&{<ii
zP_$x<r!|shqAQb+Db9y2OL$Q>?Xm$HVGb;mApZVq2#L?}FikZhA2CQNg};4>bLB{!
zfVKQ-TREqYniv`<dMa@P5Q;E<;!GUCKZWH0Ticy$^Rn2nJCdpa4x|yOOxddO^Jjde
zhi5xM1;$eoX_5+e6tXkOV~oaCA%BH0&F~(yX>@W|8Vu?;YWAIbDrg<kR0;@o5=KBj
z0v^aj-wT?l&SV^B7yR5DWHMUo_VVn#k?MZG(GZhaMcV5h&6OG3%_=%cnU_Yp$JhOu
zR>POr-j%kO)t?!uBrFR~S}ou1D&X)>k*?pU$|rda;vKTaS9QZ5NOl~R5XCR%m9EuA
z8b|bcW^nr5k9qe=ZE?-(NH{=wv$kPNcCq^5X4OA6>=`h6JR*4+S>M@vEnm0bqKlNP
zE+uaCd1>p5iXKq=%bfWamd(ibk3sNc9vPJa>EzY>IE{_T8^pYJzCWbr-oEDgW}?9^
zi1z3XtPT;%2?qD&tz}*kKeo&t@P?7^CFTaF>#bu>opU#YJfcNd`<C7T)T$iW&p9{n
zpp+~6S5xXp(ZNS$I5(Uk@3SqBUV6j$_Uy<z9CvQ%od~~qXLVbL`GhY)%Q+x~MmfP$
z=7DD77cbb!TAsd-U(F8|_QdUHISH=8u}${l87u?J`GOL-S(6OxvbWn0MC&<u*gk$?
zvTeYz8RIY(IK}`fD@Gmn2i{uB3^g>*hlXutpZb2iD^DS-X?v$(qUHO};HWVGY=~|u
zw68t)alr#(yal)4YI^I<9t5Q?YBcX;KR@HR;}=+IUo4g9n2znbeXGkWxMZicBUjJ0
z5)09lIXvGktdMp38+CQAtG{M(G=wPaVQvgcuS0KXv98>|dRy=8eLyza@Xm(2vI@f>
z)NxpwCnaRm63ZTPt{~99=c)IUi8-#@WP%qVa`4+@Pxy_7!gKO7<EOlWLRaG$hoUUC
z?KPPPn%zpZ=k1&Vb#J@f7C!;%3}T2Y#1X@}vy3?3uc}ythRtEnIaJXiU$Ohx%LPf-
z;D!fS>*AYxT-rBHWC5bGs13K!QD{){(5Z1%MnXVFabbd>M)G7>p_zepX(R>lmwT3O
zLekAV`K`mFH!;Sc(jr49Ynp&Oe!2r6ja|(@RPy00e-lHf=4kDiXNL|KQ_JUdc&px2
zYj;aC#ri^1J5pJ6v;VrDjD5*_a7E%vW%6oE(5Ecj0V4LjdNshKo{GwF_6d(Q(t}$Y
zEGb&@5-2Y`!%s>c5;x*slvUQ7D`b}SbV!nSxZX4sZN2h&jq|YZWu2j6Wrm5k^WtkB
zIt<g%H*h#lY@88WS->f;aOT()>Dw0K3R>j`iPhXsq+F4?x)Tpx_j?j<Kloy<(^|m!
zg^A7DrZl@&cD4a9S%(-umA_z|?_`(L;j|v5#q`R%A}JXfo2vJ4N@}QYiZ`1SbD7I=
zv&TK5Hp|<S>R6nW^;)b4ciV;PdGVPPHTwF^6G25A1x)eoIOH*q806mUs3>9TZLXJy
zIy36S%eb>ouIcU6OS}|oFyg<nk#UKSdK%1PYq_-6{hB=#@Jb+cQcK7&xH{6UghIjY
z60WhzDGdYbL5;6%_W2{JX{0+uX6!{u=aUgHsVQT|CbG{OygtkC1^A5F$Jb4~+vWw;
z3(9ykuy!st&lrpMY4yIVnDv(O`OFz2A}ar3qOmA@qj$5<Q0UXMIv;q3kIXSG6~%Yj
zwf!Mfm!+8jXfEK91p9%|)Uy|cj|W#=Nh9g3o>5zkc8p0^yR4TU>8@FMj={k2^W~yY
ze$6^^1GR*70iAW2gIulatgcnu+WoBCtKeWi-dq3j9S6f(Cl7q~4SJ`p;)|7?vB*Qm
zyD8P^4x@*|^+lDBK|8>;Cn)8~eXTNIF3Gu|BdQf8fCwY*<(Z?cA=;J7rdorx!)`_B
z5OEu2F|VX%hi@cwZ==LSk<Rv8Yf!5R53>l4q92xAUQVv`=u0P0-gYJhKc|52lKD-p
zaC>yv-n;zh*|491O=`MAwQo~8M$FjU2ifQ9dG&5WY+6Sa<*s4Pnz;AHc`x=N51Y*c
zaQ1*=U|w2+eCyoZMY!PkWlMXFsuz8}#wpDf)}0~7DT#~gaX|o9X+;Y!z&vb~@!q2^
z&nq@e*t_5ULW3_+64HWRIn9l~u;e`^#w!`m#E$+T!DyFvMSN`-54CNm_L~C#h|*|7
zbq#2>I8N7(YN^&K5`qUz!#%ph*gzi~+|H{3c`aLtvGGpspP|z1d~*Y?vPg%LkwCtd
z`AgeIGV`*J47En(2ju!XT&s^<|I+U9<*tOFVHvk|=6RPV#oDS~{Mowl{M~Nx`nj2s
zh4GD~g-1J43zez(wSG9sft96G{Zd(U_Ar+!i+(|OrQ5SKHi4x8Dj0n&t5e!CQ(E9N
za4{N`&NVC^IgL^mFEJo(IWqT>(T9cdH$xotxh^CVnvAUB8NQd&DQ_Bx$KvoZ;*TOm
z1{0O?CLXHZy;j)xB5;1O{h_l1v8~tEZUfb=e}d?JutdxJg+=^D;p<{v0Myo#Tr3o8
zXLO=MVT&Zj99A+XF8ruK%>=|h5STduq+#KJXnseaZH(T;tA)=Zk=;*mL-nst;w8B3
zWBv>?FDfbG;wxM7*A@yjEv_)UDMN_b_{t+Y`Rt{ELQ*r8BZ?(;u>8Z3XwwYMN0V-b
zZi1hCkH4;vC=c|KTo`AdY7L(s?zrz#YZOa)O5he8Yp#_$^ns5k9vZG%m}^%m3ETxg
z+<_b%JpMqkON>eBfI8@oLj8?DWYeE+>Fqy>5OX#038V12-B?_fO11tW6}2UX`?P_{
z57u-^v(PisTEMGpJ*@Wl!|79Grv+L4tgCHa#h43D1DAXzFo8!|-YiwNj-1{5XxZ*~
z#-Yg@J#vdKJS^AG+tqJH4@5++YAFJbkD4t(1wgq4{@)jfMO%UHSgSTa_btB>r*ArH
zR|cs5G4E79)@al&Tmvr6;9t!)2P;c^13j|%4<0HjU)|O9qLKpBnOO}3#&WgnvhNc*
zSF)PEE<NvG0UWcMo4?I&6nmCN*hO4f{ak}m*!lj=U%|1Z9t&?d{r$LU9mrma#of;h
zcR9e_*?0HJ-^H5+0ucd=>^{JdXvFU#;VHjV1p|nx0D@ppU@(OC**y?j!jIGZ`ym8=
zRaM%zovIc9LnB0f-ii2$u>1lcAy6cJ4T+(vL6K<s8k!ylgMjzoXoHYH=Msd~-~b$g
zpaUQfu)WQ}2pEhWM{BjWITC@Psj_>1lLEsbDEb;4fuiAn{|BNGdj~_{DEKb$e{m2v
z;$LbIFrAtR^uH{Kgv0*ZZ+AZJAW?KyfKd=6^uNGgINC1sH3Vkw(qI$@Ove}v{g)bm
zfb8`QjK-khdk14+VEP&yxi>3d3=&LVLqY$C@lPVONFshNNaO$8CkPlyrvgn>I>rz%
z3c7b02m}nH8v=nK=y9<BBllB(2n0o+Tqp#!R~ZQOUs4aHw*&%3qv@7`LFp8Pz~HCh
ld%c6ekTAL+!O+mXzC)03ID!zTq0NH8z*tV6)H2p)`47$?8EOCk

diff --git a/notes/sections/4.md b/notes/sections/4.md
index f86c4c7..ed8f274 100644
--- a/notes/sections/4.md
+++ b/notes/sections/4.md
@@ -3,14 +3,14 @@
 ## Kinematic dip PDF derivation
 
 Consider a great number of non-interacting particles, each of which with a
-random momentum $\vec{p}$ with module between 0 and $p_{\text{max}}$ randomly
+random momentum $\vec{P}$ with module between 0 and $P_{\text{max}}$ randomly
 angled with respect to a coordinate system {$\hat{x}$, $\hat{y}$, $\hat{z}$}.
 Once the polar angle $\theta$ is defined, the momentum vertical and horizontal
-components of a particle, which will be referred as $\vec{p_v}$ and $\vec{p_h}$
+components of a particle, which will be referred as $\vec{P_v}$ and $\vec{P_h}$
 respectively, are the ones shown in @fig:components.  
 If $\theta$ is evenly distributed on the sphere and the same holds for the
-module $p$, which distribution will the average value of $|p_v|$ follow as a
-function of $p_h$?
+module $P$, which distribution will the average value of $|P_v|$ follow as a
+function of $P_h$?
 
 \begin{figure}
 \hypertarget{fig:components}{%
@@ -30,9 +30,9 @@ function of $p_h$?
   \draw [thick, dashed, cyclamen]   (5,7.2)   -- (3.8,6);
   \draw [ultra thick, ->, pink]     (5,2)     -- (5,7.2);
   \draw [ultra thick, ->, pink]     (5,2)     -- (3.8,0.8);
-  \node at (4.8,1.1) {$\vec{p_h}$};
-  \node at (5.5,6.6) {$\vec{p_v}$};
-  \node at (3.3,5.5) {$\vec{p}$};
+  \node at (4.8,1.1) {$\vec{P_h}$};
+  \node at (5.5,6.6) {$\vec{P_v}$};
+  \node at (3.3,5.5) {$\vec{P}$};
   % Angle
   \draw [thick, cyclamen] (4.4,4) to [out=80,in=210] (5,5);
   \node at (4.7,4.2) {$\theta$};
@@ -41,11 +41,11 @@ function of $p_h$?
 }
 \end{figure}
 
-Since the aim is to compute $\langle |p_v| \rangle (p_h)$, the conditional
-distribution probability of $p_v$ given a fixed value of $p_h = x$ must first
-be determined. It can be computed as the ratio between the probablity of
-getting a fixed value of $P_v$ given $x$ over the total probability of $x$:
-
+Since the aim is to compute $\langle |P_v| \rangle (P_h)$, the conditional
+distribution probability of $P_v$ given a fixed value of $P_h = x$ must first
+be determined. It can be computed as the ratio between the probability of
+getting a fixed value of $P_v$ given $x$ over the total probability of
+getting that $x$:
 $$
   f (P_v | P_h = x) = \frac{f_{P_h , P_v} (x, P_v)}
   {\int_{\{ P_v \}} d P_v f_{P_h , P_v} (x, P_v)}
@@ -53,35 +53,31 @@ $$
 $$
 
 where $f_{P_h , P_v}$ is the joint pdf of the two variables $P_v$ and $P_h$ and
-the integral runs over all the possible values of $P_v$ given $P_h$.
-
-This joint pdf can simly be computed from the joint pdf of $\theta$ and $p$ with
-a change of variables. For the pdf of $\theta$ $f_{\theta} (\theta)$, the same
-considerations done in @sec:3 lead to:
-
+the integral $I$ runs over all the possible values of $P_v$ given a certain
+$P_h$.  
+$f_{P_h , P_v}$ can simply be computed from the joint pdf of $\theta$ and $P$
+with a change of variables. For the pdf of $\theta$ $f_{\theta} (\theta)$, the
+same considerations done in @sec:3 lead to:
 $$
   f_{\theta} (\theta) = \frac{1}{2} \sin{\theta} \chi_{[0, \pi]} (\theta)
 $$
 
-whereas, being $p$ evenly distributed:
-
+whereas, being $P$ evenly distributed:
 $$
-  f_p (p) = \chi_{[0, p_{\text{max}}]} (p)
+  f_P (P) = \chi_{[0, P_{\text{max}}]} (P)
 $$
 
 where $\chi_{[a, b]} (y)$ is the normalized characteristic function which value
-is $1/N$ between $a$ and $b$, where $N$ is the normalization term, and 0
+is $1/N$ between $a$ and $b$ (where $N$ is the normalization term) and 0
 elsewhere. Being a couple of independent variables, their joint pdf is simply
 given by the product of their pdfs:
-
 $$
-  f_{\theta , p} (\theta, p) = f_{\theta} (\theta) f_p (p)
+  f_{\theta , P} (\theta, P) = f_{\theta} (\theta) f_P (P)
   = \frac{1}{2} \sin{\theta} \chi_{[0, \pi]} (\theta)
-    \chi_{[0, p_{\text{max}}]} (p)
+    \chi_{[0, P_{\text{max}}]} (P)
 $$
 
 Given the new variables:
-
 $$
   \begin{cases}
     P_v = P \cos(\theta) \\
@@ -89,12 +85,11 @@ $$
   \end{cases}
 $$
 
-with $\theta \in [0, \pi]$, the previous ones are written as:
-
+with $\theta \in [0, \pi]$, the previous ones can be written as:
 $$
   \begin{cases}
     P = \sqrt{P_v^2 + P_h^2} \\
-    \theta = \text{atan}^{\star} ( P_h/P_v ) :=
+    \theta = \text{atan}_2 ( P_h/P_v ) :=
       \begin{cases}
         \text{atan} ( P_h/P_v )       &\incase P_v > 0 \\
         \text{atan} ( P_h/P_v ) + \pi &\incase P_v < 0
@@ -103,24 +98,21 @@ $$
 $$
 
 which can be shown having Jacobian:
-
 $$
   J = \frac{1}{\sqrt{P_v^2 + P_h^2}}
 $$
 
 Hence:
-
 $$  
   f_{P_h , P_v} (P_h, P_v) =
-    \frac{1}{2} \sin[ \text{atan}^{\star} ( P_h/P_v )]
-    \chi_{[0, \pi]} (\text{atan}^{\star} ( P_h/P_v )) \cdot \\
+    \frac{1}{2} \sin[ \text{atan}_2 ( P_h/P_v )]
+    \chi_{[0, \pi]} (\text{atan}_2 ( P_h/P_v )) \cdot \\
     \frac{\chi_{[0, p_{\text{max}}]} \left( \sqrt{P_v^2 + P_h^2} \right)}
     {\sqrt{P_v^2 + P_h^2}}
 $$
 
 from which, the integral $I$ can now be computed. The edges of the integral
 are fixed by the fact that the total momentum can not exceed $P_{\text{max}}$:
-
 $$
   I = \int
   \limits_{- \sqrt{P_{\text{max}}^2 - P_h}}^{\sqrt{P_{\text{max}}^2 - P_h}}
@@ -128,39 +120,32 @@ $$
 $$
 
 after a bit of maths, using the identity:
-
 $$
-  \sin[ \text{atan}^{\star} ( P_h/P_v )] = \frac{P_h}{\sqrt{P_h^2 + P_v^2}}
+  \sin[ \text{atan}_2 ( P_h/P_v )] = \frac{P_h}{\sqrt{P_h^2 + P_v^2}}
 $$
 
-and the fact that both the characteristic functions are automatically satisfied
-within the integral limits, the following result arises:
-
+and the fact that both the characteristic functions play no role within the
+integral limits, the following result arises:
 $$
   I = 2 \, \text{atan} \left( \sqrt{\frac{P_{\text{max}}^2}{x^2} - 1} \right)
 $$
 
 from which:
-
 $$
   f (P_v | P_h = x) = \frac{x}{P_v^2 + x^2} \cdot
   \frac{1}{2 \, \text{atan}
            \left( \sqrt{\frac{P_{\text{max}}^2}{x^2} - 1} \right)}
 $$
 
-Finally, putting all the pieces together, the average value of $|P_v|$ can now
-be computed:
-
-\begin{align*}
-  \langle |P_v| \rangle &= \int
+Finally, putting all the pieces together, the average value of $|P_v|$ can be
+computed:
+$$
+  \langle |P_v| \rangle = \int
   \limits_{- \sqrt{P_{\text{max}}^2 - P_h}}^{\sqrt{P_{\text{max}}^2 - P_h}}
-  f (P_v | P_h = x)
-  \\
-  &= [ \dots ]
-  \\
-  &= x \, \frac{\ln \left( \frac{P_{\text{max}}}{x} \right)}
+  f (P_v | P_h = x) = [ \dots ]
+  = x \, \frac{\ln \left( \frac{P_{\text{max}}}{x} \right)}
   {\text{atan} \left( \sqrt{ \frac{P^2_{\text{max}}}{x^2} - 1} \right)}
-\end{align*}
+$$ {#eq:dip}
 
 Namely:
 
@@ -171,80 +156,71 @@ Namely:
 ## Monte Carlo simulation
 
 The same distribution should be found by generating and binning points in a
-proper way.  
-A number of $N = 50'000$ points were generated as a couple of values ($p$,
-$\theta$), with $p$ evenly distrinuted between 0 and $p_{\text{max}}$ and
+proper way. A number of $N = 50'000$ points were generated as a couple of values
+($P$, $\theta$), with $P$ evenly distributed between 0 and $P_{\text{max}}$ and
 $\theta$ given by the same procedure described in @sec:3, namely:
-
 $$
   \theta = \text{acos}(1 - 2x)
 $$
 
-with $x$ uniformely distributed between 0 and 1.  
-The data binning turned out to be a bit challenging. Once $p$ was sorted and
-$p_h$ was computed, the bin in which the latter goes in must be found. If $n$ is
-the number of bins in which the range [0, $p_{\text{max}}$] is willing to be
-divided into, then the width $w$ of each bin is given by:
-
+with $x$ uniformly distributed between 0 and 1.  
+The data binning turned out to be a bit challenging. Once $P$ was sampled and
+$P_h$ was computed, the bin containing the latter's value must be found. If $n$
+is the number of bins in which the range $[0, P_{\text{max}}]$ is divided into,
+then the width $w$ of each bin is given by:
 $$
-  w = \frac{p_{\text{max}}}{n}
+  w = \frac{P_{\text{max}}}{n}
 $$
 
-and the $j^{th}$ bin in which $p_h$ goes in is:
-
+and the $i^{th}$ bin in which $P_h$ goes in is:
 $$
-  j = \text{floor} \left( \frac{p_h}{w} \right)
+  i = \text{floor} \left( \frac{P_h}{w} \right)
 $$
 
-where 'floor' is the function which gives the upper integer lesser than its
+where 'floor' is the function which gives the bigger integer smaller than its
 argument and the bins are counted starting from zero.  
-Then, a vector in which the $j^{\text{th}}$ entry contains the sum $S_j$ of all
-the $|p_v|$s relative to each $p_h$ fallen into the $j^{\text{th}}$ bin and the
-number num$_j$ of entries in the $j^{\text{th}}$ bin was reiteratively updated.
-At the end, the average value of $|p_v|_j$ was computed as $S_j /
-\text{num}_j$.  
-For the sake of clarity, for each sorted couple, it works like this:
+Then, a vector in which the $j^{\text{th}}$ entry contains both the sum $S_j$
+of all the $|P_v|$s relative to each $P_h$ fallen into the $j^{\text{th}}$ bin
+itself and the number num$_j$ of the bin entries was reiteratively updated. At
+the end, the average value of $|P_v|_j$ was computed as $S_j / \text{num}_j$.  
+For the sake of clarity, for each sampled couple the procedure is the
+following. At first $S_j = 0 \wedge \text{num}_j = 0 \, \forall \, j$, then:
 
-  - the couple $[p; \theta]$ is generated;
-  - $p_h$ and $p_v$ are computed;
-  - the $j^{\text{th}}$ bin in which $p_h$ goes in is computed;
-  - num$_j$ is increased by 1;
-  - $S_j$ (which is zero at the beginning of everything) is increased by a factor
-    $|p_v|$.
+  - the couple $(P, \theta)$ is generated,
+  - $P_h$ and $P_v$ are computed,
+  - the $j^{\text{th}}$ bin containing $P_h$ is found,
+  - num$_j$ is increased by 1,
+  - $S_j$ is increased by $|P_v|$.
 
-At the end, $\langle |p_h| \rangle_j$ = $\langle |p_h| \rangle (p_h)$ where:
+For $P_{\text{max}} = 10$ and $n = 50$, the following result was obtained:
 
-$$
-  p_h = j \cdot w + \frac{w}{2} = w \left( 1 + \frac{1}{2} \right)
-$$
+![Sampled points histogram.](images/dip.pdf)
 
-For $p_{\text{max}} = 10$, the following result was obtained:
-
-![Histogram of the obtained distribution.](images/dip.pdf)
-
-In order to check wheter the expected distribution properly metches the
-produced histogram, a chi-squared minimization was applied. Being a simple
+In order to check whether the expected distribution (@eq:dip) properly matches
+the produced histogram, a chi-squared minimization was applied. Being a simple
 one-parameter fit, the $\chi^2$ was computed without a suitable GSL function
-and the error of the so obtained estimation of $p_{\text{max}}$ was given as
-the inverese of the $\chi^3$ second derivative in its minimum, according to the
-Cramér-Rao bound.  
+and the error of the so obtained estimation of $P_{\text{max}}$ was given as
+the inverse of the $\chi^2$ second derivative in its minimum, according to the
+Cramér-Rao bound.
+
 The following results were obtained:
 
 $$
-  p^{\text{oss}}_{\text{max}} = 10.005 \pm 0.018 \with \chi^2 = 0.071
+  P^{\text{oss}}_{\text{max}} = 10.005 \pm 0.018 \with \chi_r^2 = 0.071
 $$
 
-In order to compare $p^{\text{oss}}_{\text{max}}$ with the expected value
-$p_{\text{max}} = 10$, the following compatibility t-test was applied:
+where $\chi_r^2$ is the reduced $\chi^2$, proving a good convergence.  
+In order to compare $P^{\text{oss}}_{\text{max}}$ with the expected value
+$P_{\text{max}} = 10$, the following compatibility t-test was applied:
 
 $$
   p = 1 - \text{erf}\left(\frac{t}{\sqrt{2}}\right)\ \with
-  t = \frac{|p^{\text{oss}}_{\text{max}} - p_{\text{max}}|}
-           {\Delta p_{\text{max}}}
+  t = \frac{|P^{\text{oss}}_{\text{max}} - P_{\text{max}}|}
+           {\Delta P^{\text{oss}}_{\text{max}}}
 $$
 
-where $\Delta p_{\text{max}}$ is the absolute error of $p_{\text{max}}$. At 95%
-confidence level, the values are compatible if $p > 0.05$.
+where $\Delta P^{\text{oss}}_{\text{max}}$ is the $P^{\text{oss}}_{\text{max}}$
+uncertainty. At 95% confidence level, the values are compatible if $p > 0.05$.
 In this case:
 
   - t = 0.295
@@ -254,4 +230,5 @@ which allows to assert that the sampled points actually follow the predicted
 distribution. In @fig:fit, the fit function superimposed on the histogram is
 shown.
 
-![Fitted data.](images/fit.pdf){#fig:fit}
+![Fitted sampled data. $P^{\text{oss}}_{\text{max}} = 10.005
+  \pm 0.018$, $\chi_r^2 = 0.071$.](images/fit.pdf){#fig:fit}