From 24a5630b8b6ba37bd922c778e60cbf449e7a6db6 Mon Sep 17 00:00:00 2001
From: Juan Marín Noguera <juan@mnpi.eu>
Date: Tue, 22 Apr 2025 19:50:20 +0200
Subject: Fix some line overflows

---
 3.3.3.lyx | 75 +++++++++++++++++++++++++++++++++++++--------------------------
 3.4.2.lyx |  3 ++-
 2 files changed, 46 insertions(+), 32 deletions(-)

diff --git a/3.3.3.lyx b/3.3.3.lyx
index 71fc21f..d03c773 100644
--- a/3.3.3.lyx
+++ b/3.3.3.lyx
@@ -298,9 +298,10 @@ status open
 In this case,
  the law reduces to
 \begin_inset Formula 
-\[
-\sigma(h,k,0)+\sigma(k,h,0)=12\sum_{0\leq j<k}\left(\left(\frac{j}{k}\right)\right)\left(\left(\frac{hj}{k}\right)\right)+12\sum_{0\leq j<h}\left(\left(\frac{j}{h}\right)\right)\left(\left(\frac{kj}{h}\right)\right)=\frac{h}{k}+\frac{k}{h}+\frac{1}{hk}-3.
-\]
+\begin{multline*}
+\sigma(h,k,0)+\sigma(k,h,0)=\\
+=12\sum_{0\leq j<k}\left(\left(\frac{j}{k}\right)\right)\left(\left(\frac{hj}{k}\right)\right)+12\sum_{0\leq j<h}\left(\left(\frac{j}{h}\right)\right)\left(\left(\frac{kj}{h}\right)\right)=\frac{h}{k}+\frac{k}{h}+\frac{1}{hk}-3.
+\end{multline*}
 
 \end_inset
 
@@ -355,10 +356,13 @@ for
 
 so substituting above,
 \begin_inset Formula 
-\begin{align*}
-S & \coloneqq kh(h-1)=\sum_{j=1}^{k-1}\left(\frac{hj}{k}-\left(\left(\frac{hj}{k}\right)\right)-\frac{1}{2}\right)\left(\frac{hj}{k}-\left(\left(\frac{hj}{k}\right)\right)+\frac{1}{2}\right)+2\sum_{j=1}^{h-1}j\left(\frac{kj}{h}-\left(\left(\frac{kj}{h}\right)\right)+\frac{1}{2}\right)\\
- & =\frac{h^{2}}{k^{2}}\sum_{j=1}^{k-1}j^{2}-\frac{2h}{k}\sum_{j=1}^{k-1}j\left(\left(\frac{hj}{k}\right)\right)+\sum_{j=1}^{k-1}\left(\left(\frac{hj}{k}\right)\right)^{2}-\frac{k-1}{4}+\frac{2k}{h}\sum_{j=1}^{h-1}j^{2}-2\sum_{j=1}^{h-1}j\left(\left(\frac{kj}{h}\right)\right)+\frac{h^{2}-h}{2}.
-\end{align*}
+\begin{multline*}
+S\coloneqq kh(h-1)=\\
+=\sum_{j=1}^{k-1}\left(\frac{hj}{k}-\left(\left(\frac{hj}{k}\right)\right)-\frac{1}{2}\right)\left(\frac{hj}{k}-\left(\left(\frac{hj}{k}\right)\right)+\frac{1}{2}\right)+\\
++2\sum_{j=1}^{h-1}j\left(\frac{kj}{h}-\left(\left(\frac{kj}{h}\right)\right)+\frac{1}{2}\right)=\\
+=\frac{h^{2}}{k^{2}}\sum_{j=1}^{k-1}j^{2}-\frac{2h}{k}\sum_{j=1}^{k-1}j\left(\left(\frac{hj}{k}\right)\right)+\sum_{j=1}^{k-1}\left(\left(\frac{hj}{k}\right)\right)^{2}-\frac{k-1}{4}+\frac{2k}{h}\sum_{j=1}^{h-1}j^{2}-\\
+-2\sum_{j=1}^{h-1}j\left(\left(\frac{kj}{h}\right)\right)+\frac{h^{2}-h}{2}.
+\end{multline*}
 
 \end_inset
 
@@ -372,9 +376,10 @@ Now,
 
 so
 \begin_inset Formula 
-\[
-\sigma(h,k,0)=12\sum_{j=0}^{k-1}\left(\left(\frac{j}{k}\right)\right)\left(\left(\frac{hj}{k}\right)\right)=12\sum_{j=1}^{k-1}\left(\frac{j}{k}-\frac{1}{2}\right)\left(\left(\frac{hj}{k}\right)\right)=12\sum_{j=1}^{k-1}\frac{j}{k}\left(\left(\frac{hj}{k}\right)\right).
-\]
+\begin{multline*}
+\sigma(h,k,0)=12\sum_{j=0}^{k-1}\left(\left(\frac{j}{k}\right)\right)\left(\left(\frac{hj}{k}\right)\right)=12\sum_{j=1}^{k-1}\left(\frac{j}{k}-\frac{1}{2}\right)\left(\left(\frac{hj}{k}\right)\right)=\\
+=12\sum_{j=1}^{k-1}\frac{j}{k}\left(\left(\frac{hj}{k}\right)\right).
+\end{multline*}
 
 \end_inset
 
@@ -389,7 +394,8 @@ In addition,
 and in particular
 \begin_inset Formula 
 \begin{align*}
-\sum_{j=1}^{k-1}\left(\left(\frac{hj}{k}\right)\right)^{2} & =\sum_{j=1}^{k-1}\left(\left(\frac{j}{k}\right)\right)^{2}=\sum_{j=1}^{k-1}\frac{j^{2}}{k^{2}}-\sum_{j=1}^{k-1}\frac{j}{k}+\frac{k-1}{4}=\frac{k}{3}-\frac{1}{2}+\frac{1}{6k}-\frac{k-1}{2}+\frac{k-1}{4}\\
+\sum_{j=1}^{k-1}\left(\left(\frac{hj}{k}\right)\right)^{2} & =\sum_{j=1}^{k-1}\left(\left(\frac{j}{k}\right)\right)^{2}\\
+ & =\sum_{j=1}^{k-1}\frac{j^{2}}{k^{2}}-\sum_{j=1}^{k-1}\frac{j}{k}+\frac{k-1}{4}=\frac{k}{3}-\frac{1}{2}+\frac{1}{6k}-\frac{k-1}{2}+\frac{k-1}{4}\\
  & =\frac{k}{3}-\frac{1}{2}+\frac{1}{6k}-\frac{k-1}{4},
 \end{align*}
 
@@ -398,8 +404,9 @@ and in particular
 so finally
 \begin_inset Formula 
 \begin{align*}
-S & =\frac{kh^{2}}{3}-\frac{h^{2}}{2}+\frac{h^{2}}{6k}-\frac{h}{6}\sigma(h,k,0)+\frac{k}{3}-\frac{1}{2}+\frac{1}{6k}-\frac{k-1}{2}+\frac{2kh^{2}}{3}-kh+\frac{k}{3}-\frac{h}{6}\sigma(k,h,0)+\frac{h^{2}}{2}-\frac{h}{2}\\
- & =kh(h-1)-\frac{h}{2}+\frac{h^{2}}{6k}+\frac{k}{6}+\frac{1}{6k}-\frac{h}{6}\sigma(h,k,0)-\frac{h}{6}\sigma(k,h,0).
+S= & \frac{kh^{2}}{3}-\frac{h^{2}}{2}+\frac{h^{2}}{6k}-\frac{h}{6}\sigma(h,k,0)+\frac{k}{3}-\frac{1}{2}+\frac{1}{6k}-\frac{k-1}{2}+\frac{2kh^{2}}{3}-kh+\\
+ & +\frac{k}{3}-\frac{h}{6}\sigma(k,h,0)+\frac{h^{2}}{2}-\frac{h}{2}\\
+= & kh(h-1)-\frac{h}{2}+\frac{h^{2}}{6k}+\frac{k}{6}+\frac{1}{6k}-\frac{h}{6}\sigma(h,k,0)-\frac{h}{6}\sigma(k,h,0).
 \end{align*}
 
 \end_inset
@@ -517,8 +524,8 @@ The number
 \begin_inset Formula 
 \begin{multline*}
 \sigma(2^{18}+1,2^{35},1)=\frac{(\cancel{2^{18}}+1)+(2^{35}\cancel{-2^{18}}+1)}{2^{35}}+\left((2^{17}-1)-6\cdot0+6\frac{1^{2}}{2^{35}(2^{18}+1)}\right)-\\
--\left(1-6\cdot0+6\frac{1^{2}}{(2^{18}+1)(2^{17}+1)}\right)+\left(1-6\cdot0+6\frac{1^{2}}{(2^{17}+1)2^{17}}\right)-\left(2^{17}-6\cdot1+6\frac{1^{2}}{2^{17}\cdot1}\right)-\\
--3-2+1=\\
+-\left(1-6\cdot0+\frac{6\cdot1^{2}}{(2^{18}+1)(2^{17}+1)}\right)+\left(1-6\cdot0+6\frac{1^{2}}{(2^{17}+1)2^{17}}\right)-\left(2^{17}-6\cdot1+6\frac{1^{2}}{2^{17}}\right)\\
+\\-3-2+1=\\
 =\cancel{1}+\frac{2}{2^{35}}\cancel{+2^{17}}\cancel{-1}+\frac{6}{2^{53}+2^{35}}\cancel{-1}-\frac{6}{2^{35}+2^{18}+2^{17}+1}\cancel{+1}+\frac{6}{2^{34}+2^{17}}\cancel{-2^{17}}+6-\frac{6}{2^{17}}-4=\\
 =2+\frac{1}{2^{34}}+6\left(\frac{1}{2^{53}+2^{35}}-\frac{1}{2^{35}+3\cdot2^{17}+1}+\frac{1}{2^{34}+2^{17}}-\frac{1}{2^{17}}\right)=\\
 =2+\frac{1}{2^{34}}+6\frac{2^{17}+1-2^{35}\cancel{+2^{36}+2^{18}}-2^{53}\cancel{-2^{36}}-2^{35}\cancel{-2^{18}}}{2^{35}(2^{18}+1)(2^{17}+1)}=\\
@@ -530,9 +537,10 @@ The number
 
 Thus,
 \begin_inset Formula 
-\[
-C=\frac{8(2^{17}-1)(2^{16}-1)-3+6\cdot2^{18}(2^{17}-1)}{2^{70}-1}=\frac{91624920407}{393530540239137101141}\cong2.33\cdot10^{-10}.
-\]
+\begin{multline*}
+C=\frac{8(2^{17}-1)(2^{16}-1)-3+6\cdot2^{18}(2^{17}-1)}{2^{70}-1}=\frac{91624920407}{393530540239137101141}\cong\\
+\cong2.33\cdot10^{-10}.
+\end{multline*}
 
 \end_inset
 
@@ -642,7 +650,8 @@ so we may assume
 \begin_inset Formula 
 \begin{multline*}
 S(h,k,c,z)=\sum_{0\leq j<k}\left(\left\lfloor \frac{j}{k}\right\rfloor -\left\lfloor \frac{j-z}{k}\right\rfloor \right)\left(\left(\frac{hj+c}{k}\right)\right)=\\
-=\sum_{0\leq j<k}\left(\cancel{\frac{j}{k}}-\left(\left(\frac{j}{k}\right)\right)\cancel{-\frac{1}{2}}-\frac{\cancel{j}-z}{k}+\left(\left(\frac{j-z}{k}\right)\right)\cancel{+\frac{1}{2}}\right)\left(\left(\frac{hj+c}{k}\right)\right)+\frac{1}{2}\left(\left(\frac{c}{k}\right)\right)-\frac{1}{2}\left(\left(\frac{hz+c}{k}\right)\right)=\\
+=\sum_{0\leq j<k}\left(\cancel{\frac{j}{k}}-\left(\left(\frac{j}{k}\right)\right)\cancel{-\frac{1}{2}}-\frac{\cancel{j}-z}{k}+\left(\left(\frac{j-z}{k}\right)\right)\cancel{+\frac{1}{2}}\right)\left(\left(\frac{hj+c}{k}\right)\right)+\\
++\frac{1}{2}\left(\left(\frac{c}{k}\right)\right)-\frac{1}{2}\left(\left(\frac{hz+c}{k}\right)\right)=\\
 =\sum_{0\leq j<k}\left(\left(\left(\frac{j-z}{k}\right)\right)+\frac{z}{k}-\left(\left(\frac{j}{k}\right)\right)\right)\left(\left(\frac{hj+c}{k}\right)\right)+\frac{1}{2}\left(\left(\frac{c}{k}\right)\right)-\frac{1}{2}\left(\left(\frac{hz+c}{k}\right)\right).
 \end{multline*}
 
@@ -672,9 +681,10 @@ Let's evaluate the sum term by term.
 .
  Finally,
 \begin_inset Formula 
-\[
-\sum_{0\leq j<k}\left(\left(\frac{j-z}{k}\right)\right)\left(\left(\frac{hj+c}{k}\right)\right)=\sum_{-z\leq j<k-z}\left(\left(\frac{j}{k}\right)\right)\left(\left(\frac{hj+c+hz}{k}\right)\right)=\frac{1}{12}\sigma(h,k,c+hz).
-\]
+\begin{multline*}
+\sum_{0\leq j<k}\left(\left(\frac{j-z}{k}\right)\right)\left(\left(\frac{hj+c}{k}\right)\right)=\sum_{-z\leq j<k-z}\left(\left(\frac{j}{k}\right)\right)\left(\left(\frac{hj+c+hz}{k}\right)\right)=\\
+=\frac{1}{12}\sigma(h,k,c+hz).
+\end{multline*}
 
 \end_inset
 
@@ -808,9 +818,10 @@ answer
 
  is
 \begin_inset Formula 
-\[
-\frac{1}{m}\sum_{0\leq x<m}P(x)Q(ax+c)=\frac{1}{m}\sum_{0\leq x<m}\left(\left\lfloor \frac{x-\alpha}{m}\right\rfloor -\left\lfloor \frac{x-\beta}{m}\right\rfloor \right)\left(\left\lfloor \frac{S(x)-\alpha'}{m}\right\rfloor -\left\lfloor \frac{S(x)-\beta'}{m}\right\rfloor \right)
-\]
+\begin{multline*}
+\frac{1}{m}\sum_{0\leq x<m}P(x)Q(ax+c)=\\
+=\frac{1}{m}\sum_{0\leq x<m}\left(\left\lfloor \frac{x-\alpha}{m}\right\rfloor -\left\lfloor \frac{x-\beta}{m}\right\rfloor \right)\left(\left\lfloor \frac{S(x)-\alpha'}{m}\right\rfloor -\left\lfloor \frac{S(x)-\beta'}{m}\right\rfloor \right)
+\end{multline*}
 
 \end_inset
 
@@ -829,7 +840,8 @@ Thus,
 \begin_inset Formula 
 \begin{multline*}
 \sum_{0\leq x<m}\left\lfloor \frac{x-\alpha}{m}\right\rfloor \left\lfloor \frac{S(x)-\alpha'}{m}\right\rfloor =\\
-=\sum_{0\leq x<m}\left(\left(\frac{x}{m}-\frac{1}{2}\right)-\frac{\alpha}{m}-\left(\left(\frac{x-\alpha}{m}\right)\right)\right)\left(\left(\left(\frac{ax+c}{m}\right)\right)-\left(\left(\frac{ax+c-\alpha'}{m}\right)\right)-\frac{\alpha'}{m}\right)+\\
+=\sum_{0\leq x<m}\left(\left(\frac{x}{m}-\frac{1}{2}\right)-\frac{\alpha}{m}-\left(\left(\frac{x-\alpha}{m}\right)\right)\right)\\
+\left(\left(\left(\frac{ax+c}{m}\right)\right)-\left(\left(\frac{ax+c-\alpha'}{m}\right)\right)-\frac{\alpha'}{m}\right)+\\
 +\frac{1}{2}\left(\left\lfloor \frac{S(\alpha)-\alpha'}{m}\right\rfloor +\left\lfloor \frac{S^{-1}(\alpha')-\alpha}{m}\right\rfloor -\left\lfloor \frac{S^{-1}(0)-\alpha}{m}\right\rfloor \right)+\frac{1}{4}([S(\alpha)=\alpha']-[S(\alpha)=0]).
 \end{multline*}
 
@@ -856,8 +868,8 @@ The terms outside this last sum can be calculated directly.
 and similarly,
 \begin_inset Formula 
 \begin{multline*}
-\sum_{0\leq x<m}\left(\left(\frac{x-\alpha}{m}\right)\right)\left(\left(\frac{ax+c-\alpha'}{m}\right)\right)=\sum_{-\alpha\leq x<m-\alpha}\left(\left(\frac{x}{m}\right)\right)\left(\left(\frac{ax+c-\alpha'+\alpha}{m}\right)\right)=\\
-=\frac{1}{12}\sigma(a,m,c-\alpha'+\alpha).
+\sum_{0\leq x<m}\left(\left(\frac{x-\alpha}{m}\right)\right)\left(\left(\frac{ax+c-\alpha'}{m}\right)\right)=\\
+=\sum_{-\alpha\leq x<m-\alpha}\left(\left(\frac{x}{m}\right)\right)\left(\left(\frac{ax+c-\alpha'+\alpha}{m}\right)\right)=\frac{1}{12}\sigma(a,m,c-\alpha'+\alpha).
 \end{multline*}
 
 \end_inset
@@ -1353,9 +1365,10 @@ If
 ,
  so the probability in this case is
 \begin_inset Formula 
-\[
-\max\left\{ 0,s_{p}^{-1}(\beta')-\max\{\alpha,s_{p}^{-1}(\alpha')\}\right\} +(q-p-1)\frac{\beta'-\alpha'}{a}+\max\left\{ 0,\min\{\beta,s_{q}^{-1}(\beta')\}-s_{q}^{-1}(\alpha')\right\} .
-\]
+\begin{multline*}
+\max\left\{ 0,s_{p}^{-1}(\beta')-\max\{\alpha,s_{p}^{-1}(\alpha')\}\right\} +(q-p-1)\frac{\beta'-\alpha'}{a}+\\
++\max\left\{ 0,\min\{\beta,s_{q}^{-1}(\beta')\}-s_{q}^{-1}(\alpha')\right\} .
+\end{multline*}
 
 \end_inset
 
diff --git a/3.4.2.lyx b/3.4.2.lyx
index 8b8ae25..8d59cc5 100644
--- a/3.4.2.lyx
+++ b/3.4.2.lyx
@@ -448,7 +448,8 @@ so
 \begin_inset Formula 
 \begin{align*}
 \text{mean}(G) & =\dot{G}(1)=n(1+H_{N}-H_{n});\\
-\text{var}(G) & =\ddot{G}(1)+\dot{G}(1)-\dot{G}(1)^{2}=\cancel{n(n-1)}\cancel{+2n\dot{F}(1)}+\ddot{F}(1)\cancel{+n}+\dot{F}(1)\cancel{-n^{2}}-\dot{F}(1)^{2}\cancel{-2n\dot{F}(1)}\\
+\text{var}(G) & =\ddot{G}(1)+\dot{G}(1)-\dot{G}(1)^{2}\\
+ & =\cancel{n(n-1)}\cancel{+2n\dot{F}(1)}+\ddot{F}(1)\cancel{+n}+\dot{F}(1)\cancel{-n^{2}}-\dot{F}(1)^{2}\cancel{-2n\dot{F}(1)}\\
  & =n(H_{N}-H_{n})-n^{2}(H_{N}^{(2)}-H_{n}^{(2)}),\\
 \sigma(G) & =\sqrt{\text{var}(G)}=\sqrt{n(H_{N}-H_{n})-n^{2}(H_{N}^{(2)}-H_{n}^{(2)})}.
 \end{align*}
-- 
cgit v1.2.3