From 4f670b750af5c11e1eac16d9cd8556455f89f46a Mon Sep 17 00:00:00 2001
From: Juan Marín Noguera <juan@mnpi.eu>
Date: Fri, 16 May 2025 22:18:44 +0200
Subject: Changed layout for more manageable volumes

---
 3.2.1.lyx | 320 --------------------------------------------------------------
 1 file changed, 320 deletions(-)
 delete mode 100644 3.2.1.lyx

(limited to '3.2.1.lyx')

diff --git a/3.2.1.lyx b/3.2.1.lyx
deleted file mode 100644
index f78e3ee..0000000
--- a/3.2.1.lyx
+++ /dev/null
@@ -1,320 +0,0 @@
-#LyX 2.4 created this file. For more info see https://www.lyx.org/
-\lyxformat 620
-\begin_document
-\begin_header
-\save_transient_properties true
-\origin unavailable
-\textclass book
-\begin_preamble
-\input defs
-\end_preamble
-\use_default_options true
-\maintain_unincluded_children no
-\language english
-\language_package default
-\inputencoding utf8
-\fontencoding auto
-\font_roman "default" "default"
-\font_sans "default" "default"
-\font_typewriter "default" "default"
-\font_math "auto" "auto"
-\font_default_family default
-\use_non_tex_fonts false
-\font_sc false
-\font_roman_osf false
-\font_sans_osf false
-\font_typewriter_osf false
-\font_sf_scale 100 100
-\font_tt_scale 100 100
-\use_microtype false
-\use_dash_ligatures true
-\graphics default
-\default_output_format default
-\output_sync 0
-\bibtex_command default
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-\spacing single
-\use_hyperref false
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-\use_package amsmath 1
-\use_package amssymb 1
-\use_package cancel 1
-\use_package esint 1
-\use_package mathdots 1
-\use_package mathtools 1
-\use_package mhchem 1
-\use_package stackrel 1
-\use_package stmaryrd 1
-\use_package undertilde 1
-\cite_engine basic
-\cite_engine_type default
-\biblio_style plain
-\use_bibtopic false
-\use_indices false
-\paperorientation portrait
-\suppress_date false
-\justification true
-\use_refstyle 1
-\use_formatted_ref 0
-\use_minted 0
-\use_lineno 0
-\index Index
-\shortcut idx
-\color #008000
-\end_index
-\secnumdepth 3
-\tocdepth 3
-\paragraph_separation indent
-\paragraph_indentation default
-\is_math_indent 0
-\math_numbering_side default
-\quotes_style english
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-\end_header
-
-\begin_body
-
-\begin_layout Standard
-\begin_inset ERT
-status open
-
-\begin_layout Plain Layout
-
-
-\backslash
-exerc1[10]
-\end_layout
-
-\end_inset
-
-Example (3) shows a situation in which 
-\begin_inset Formula $X_{4}=X_{0}$
-\end_inset
-
-,
- so the sequence begins again from the beginning.
- Give an example of a linear congruential sequence with 
-\begin_inset Formula $m=10$
-\end_inset
-
- for which 
-\begin_inset Formula $X_{0}$
-\end_inset
-
- never appears again in the sequence.
-\end_layout
-
-\begin_layout Standard
-\begin_inset ERT
-status open
-
-\begin_layout Plain Layout
-
-
-\backslash
-answer 
-\end_layout
-
-\end_inset
-
-This happens if 
-\begin_inset Formula $a=c=0$
-\end_inset
-
- and 
-\begin_inset Formula $0<X_{0}<m$
-\end_inset
-
-;
- then the sequence starts with 
-\begin_inset Formula $X_{0}\neq0$
-\end_inset
-
- and then it's constant at 0.
-\end_layout
-
-\begin_layout Standard
-\begin_inset ERT
-status open
-
-\begin_layout Plain Layout
-
-
-\backslash
-rexerc2[M20]
-\end_layout
-
-\end_inset
-
-Show that if 
-\begin_inset Formula $a$
-\end_inset
-
- and 
-\begin_inset Formula $m$
-\end_inset
-
- are relatively prime,
- the number 
-\begin_inset Formula $X_{0}$
-\end_inset
-
- will always appear in the period.
-\end_layout
-
-\begin_layout Standard
-\begin_inset ERT
-status open
-
-\begin_layout Plain Layout
-
-
-\backslash
-answer 
-\end_layout
-
-\end_inset
-
-For 
-\begin_inset Formula $a\geq2$
-\end_inset
-
-,
- by Euler's theorem,
- 
-\begin_inset Formula $a^{\varphi(m)}\equiv1\mod m$
-\end_inset
-
-,
- so 
-\begin_inset Formula $X_{\varphi(m)}=X_{0}$
-\end_inset
-
- by Equation (6).
- For 
-\begin_inset Formula $a=1$
-\end_inset
-
-,
- 
-\begin_inset Formula $X_{m}=(X_{0}+mc)\bmod m=X_{0}$
-\end_inset
-
-,
- and for 
-\begin_inset Formula $a=0$
-\end_inset
-
-,
- 
-\begin_inset Formula $a$
-\end_inset
-
- and 
-\begin_inset Formula $m$
-\end_inset
-
- are not relatively prime.
-\end_layout
-
-\begin_layout Standard
-\begin_inset ERT
-status open
-
-\begin_layout Plain Layout
-
-
-\backslash
-exerc3[M10]
-\end_layout
-
-\end_inset
-
-If 
-\begin_inset Formula $a$
-\end_inset
-
- and 
-\begin_inset Formula $m$
-\end_inset
-
- are not relatively prime,
- explain why the sequence will be somewhat handicapped and probably not very random;
- hence we will generally want the multiplier 
-\begin_inset Formula $a$
-\end_inset
-
- to be relatively prime to the modulus 
-\begin_inset Formula $m$
-\end_inset
-
-.
-\end_layout
-
-\begin_layout Standard
-\begin_inset ERT
-status open
-
-\begin_layout Plain Layout
-
-
-\backslash
-answer 
-\end_layout
-
-\end_inset
-
-Let 
-\begin_inset Formula $p$
-\end_inset
-
- be a common prime between 
-\begin_inset Formula $a$
-\end_inset
-
- and 
-\begin_inset Formula $c$
-\end_inset
-
-,
-\begin_inset Formula 
-\[
-X_{n}=(a^{n}X_{0}+(a^{n-1}+a^{n-2}+\dots+1)c)\bmod m,
-\]
-
-\end_inset
-
-so 
-\begin_inset Formula 
-\[
-X_{n}\bmod p=c\bmod p
-\]
-
-\end_inset
-
- for any 
-\begin_inset Formula $n\geq1$
-\end_inset
-
-.
-\end_layout
-
-\end_body
-\end_document
-- 
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