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-rw-r--r--Master/texmf-dist/doc/pdftex/tests/38-interword-space-at-font-switch/Makefile59
-rw-r--r--Master/texmf-dist/doc/pdftex/tests/38-interword-space-at-font-switch/a.bib71
-rw-r--r--Master/texmf-dist/doc/pdftex/tests/38-interword-space-at-font-switch/a2.tex697
-rw-r--r--Master/texmf-dist/doc/pdftex/tests/38-interword-space-at-font-switch/a3.bbl37
-rw-r--r--Master/texmf-dist/doc/pdftex/tests/38-interword-space-at-font-switch/a3.tex303
-rwxr-xr-xMaster/texmf-dist/doc/pdftex/tests/38-interword-space-at-font-switch/compare_pdfs.rb40
-rw-r--r--Master/texmf-dist/doc/pdftex/tests/38-interword-space-at-font-switch/f.tex19
-rw-r--r--Master/texmf-dist/doc/pdftex/tests/38-interword-space-at-font-switch/f2.tex3
-rw-r--r--Master/texmf-dist/doc/pdftex/tests/38-interword-space-at-font-switch/f3.tex13
-rw-r--r--Master/texmf-dist/doc/pdftex/tests/38-interword-space-at-font-switch/from-latex-tagged-pdf/arXiv-2401.05361v1.tex7
-rw-r--r--Master/texmf-dist/doc/pdftex/tests/38-interword-space-at-font-switch/from-latex-tagged-pdf/glosuba.tex2
-rw-r--r--Master/texmf-dist/doc/pdftex/tests/38-interword-space-at-font-switch/sample2e.tex4
-rw-r--r--Master/texmf-dist/doc/pdftex/tests/38-interword-space-at-font-switch/texlive-en.tex1
13 files changed, 1209 insertions, 47 deletions
diff --git a/Master/texmf-dist/doc/pdftex/tests/38-interword-space-at-font-switch/Makefile b/Master/texmf-dist/doc/pdftex/tests/38-interword-space-at-font-switch/Makefile
index 39265f51806..28bfb4f41e2 100644
--- a/Master/texmf-dist/doc/pdftex/tests/38-interword-space-at-font-switch/Makefile
+++ b/Master/texmf-dist/doc/pdftex/tests/38-interword-space-at-font-switch/Makefile
@@ -7,6 +7,10 @@ pdftex24 = /opt/texlive/2020/bin/local/pdftex-tl24
pdftex25 = /opt/texlive/2020/bin/local/pdftex2
pdflatex24 = $(pdftex24) -fmt pdflatex-24
pdflatex25 = $(pdftex25) -fmt pdflatex-25
+cmp_vimdiff = vimdiff
+cmp_visual = ./compare_pdfs.rb
+
+cmp_vimdiff = true
test:
# $(pdftex21) -ini f.tex && mv f.pdf f-21.pdf && ./postpdf.sh f-21.pdf
@@ -14,8 +18,8 @@ test:
# $(pdftex23) -ini f.tex && mv f.pdf f-23.pdf && ./postpdf.sh f-23.pdf
$(pdftex24) -ini f.tex && mv f.pdf f-24.pdf && ./postpdf.sh f-24.pdf
$(pdftex25) -ini f.tex && mv f.pdf f-25.pdf && ./postpdf.sh f-25.pdf
- ./compare_pdfs.rb f-2[45].pdf --res=300
- vimdiff -R f-2[45]--pp.pdf
+ $(cmp_visual) f-2[45].pdf --res=300
+ $(cmp_vimdiff) -R f-2[45]--pp.pdf
test2:
# $(pdftex21) -fmt pdflatex-21 f2.tex && mv f2.pdf f2-21.pdf && ./postpdf.sh f2-21.pdf
@@ -23,8 +27,17 @@ test2:
# $(pdftex23) -fmt pdflatex-23 f2.tex && mv f2.pdf f2-23.pdf && ./postpdf.sh f2-23.pdf
$(pdflatex24) f2.tex && mv f2.pdf f2-24.pdf && ./postpdf.sh f2-24.pdf
$(pdflatex25) f2.tex && mv f2.pdf f2-25.pdf && ./postpdf.sh f2-25.pdf
- # ./compare_pdfs.rb f2-2[45].pdf --res=300
- vimdiff -R f2-2[45]--pp.pdf
+ $(cmp_visual) f2-2[45].pdf --res=300
+ $(cmp_vimdiff) -R f2-2[45]--pp.pdf
+
+test3:
+ # $(pdftex21) -fmt pdflatex-21 f3.tex && mv f3.pdf f3-21.pdf && ./postpdf.sh f3-21.pdf
+ # $(pdftex22) -fmt pdflatex-22 f3.tex && mv f3.pdf f3-22.pdf && ./postpdf.sh f3-22.pdf
+ # $(pdftex23) -fmt pdflatex-23 f3.tex && mv f3.pdf f3-23.pdf && ./postpdf.sh f3-23.pdf
+ $(pdflatex24) f3.tex && mv f3.pdf f3-24.pdf && ./postpdf.sh f3-24.pdf
+ $(pdflatex25) f3.tex && mv f3.pdf f3-25.pdf && ./postpdf.sh f3-25.pdf
+ $(cmp_visual) f3-2[45].pdf --res=300
+ $(cmp_vimdiff) -R f3-2[45]--pp.pdf
sample2e:
make fmt
@@ -33,7 +46,8 @@ sample2e:
# $(pdftex23) -fmt pdflatex-23 sample2e.tex && mv sample2e.pdf sample2e-23.pdf && ./postpdf.sh sample2e-23.pdf
$(pdflatex24) sample2e.tex && mv sample2e.pdf sample2e-24.pdf && ./postpdf.sh sample2e-24.pdf
$(pdflatex25) sample2e.tex && mv sample2e.pdf sample2e-25.pdf && ./postpdf.sh sample2e-25.pdf
- vimdiff -f -R sample2e-2[45]--pp.pdf
+ $(cmp_visual) sample2e-2[45].pdf --res=300
+ $(cmp_vimdiff) -f -R sample2e-2[45]--pp.pdf
a1:
$(pdflatex24) a1.tex
@@ -44,8 +58,32 @@ a1:
$(pdflatex25) a1.tex
$(pdflatex25) a1.tex
mv a1.pdf a1-25.pdf && ./postpdf.sh a1-25.pdf
- ./compare_pdfs.rb a1-24.pdf a1-25.pdf --fuzz=10 --threshold=10 --res=300
- vimdiff -R a1-24--pp.pdf a1-25--pp.pdf
+ $(cmp_visual) a1-2[45].pdf --res=300
+ $(cmp_vimdiff) -R a1-2[45]--pp.pdf
+
+a2:
+ $(pdflatex24) a2.tex
+ $(pdflatex24) a2.tex
+ $(pdflatex24) a2.tex
+ mv a2.pdf a2-24.pdf && ./postpdf.sh a2-24.pdf
+ $(pdflatex25) a2.tex
+ $(pdflatex25) a2.tex
+ $(pdflatex25) a2.tex
+ mv a2.pdf a2-25.pdf && ./postpdf.sh a2-25.pdf
+ $(cmp_visual) a2-2[45].pdf --res=300 --res=300
+ $(cmp_vimdiff) -R a2-2[45]--pp.pdf
+
+a3:
+ $(pdflatex24) a3.tex
+ $(pdflatex24) a3.tex
+ $(pdflatex24) a3.tex
+ mv a3.pdf a3-24.pdf && ./postpdf.sh a3-24.pdf
+ $(pdflatex25) a3.tex
+ $(pdflatex25) a3.tex
+ $(pdflatex25) a3.tex
+ mv a3.pdf a3-25.pdf && ./postpdf.sh a3-25.pdf
+ $(cmp_visual) a3-2[45].pdf --res=300 --res=300
+ $(cmp_vimdiff) -R a3-2[45]--pp.pdf
texlive-en:
$(pdflatex24) texlive-en.tex
@@ -56,8 +94,8 @@ texlive-en:
# $(pdflatex25) texlive-en.tex
# $(pdflatex25) texlive-en.tex
mv texlive-en.pdf texlive-en-25.pdf && ./postpdf.sh texlive-en-25.pdf
- ./compare_pdfs.rb texlive-en-24.pdf texlive-en-25.pdf --fuzz=10 --threshold=10 --res=150
- vimdiff -R texlive-en-24--pp.pdf texlive-en-25--pp.pdf
+ $(cmp_visual) texlive-en-2[45].pdf --res=300
+ $(cmp_vimdiff) -R texlive-en-2[45]--pp.pdf
fmt:
# $(pdftex21) -ini -etex -env pdflatex.ini && mv pdflatex.fmt pdflatex-21.fmt
@@ -66,8 +104,5 @@ fmt:
$(pdftex24) -ini -etex -env pdflatex.ini && mv pdflatex.fmt pdflatex-24.fmt
$(pdftex25) -ini -etex -env pdflatex.ini && mv pdflatex.fmt pdflatex-25.fmt
-cmp:
- ./compare_pdfs.rb sample2e-2[45].pdf --fuzz=10 --threshold=10 --res=300
-
clean:
$(clean)
diff --git a/Master/texmf-dist/doc/pdftex/tests/38-interword-space-at-font-switch/a.bib b/Master/texmf-dist/doc/pdftex/tests/38-interword-space-at-font-switch/a.bib
new file mode 100644
index 00000000000..d8dd74cb97d
--- /dev/null
+++ b/Master/texmf-dist/doc/pdftex/tests/38-interword-space-at-font-switch/a.bib
@@ -0,0 +1,71 @@
+@book{bishop,
+ AUTHOR = {C.M. Bishop},
+ TITLE = {Pattern Recognition and Machine Learning},
+ PUBLISHER = {Springer},
+ %ADDRESS = {Chicago},
+ YEAR = {2006},
+ %EDITION = {16th}
+}
+
+@inproceedings{foster2019complexity,
+ title={The complexity of making the gradient small in stochastic convex optimization},
+ author={Foster, Dylan J and Sekhari, Ayush and Shamir, Ohad and Srebro, Nathan and Sridharan, Karthik and Woodworth, Blake},
+ booktitle={Conference on Learning Theory},
+ pages={1319--1345},
+ year={2019},
+ organization={PMLR}
+}
+
+@inproceedings{zhang2020complexity,
+ title={Complexity of finding stationary points of nonconvex nonsmooth functions},
+ author={Zhang, Jingzhao and Lin, Hongzhou and Jegelka, Stefanie and Sra, Suvrit and Jadbabaie, Ali},
+ booktitle={International Conference on Machine Learning},
+ pages={11173--11182},
+ year={2020},
+ organization={PMLR}
+}
+
+@book{chembook,
+ AUTHOR = {T.F. Edgar and D.M. Himmelblau},
+ TITLE = {Optimization of Chemical Processes},
+ PUBLISHER = {McGraw Hill},
+ %ADDRESS = {Chicago},
+ YEAR = {1988},
+ %EDITION = {16th}
+}
+
+@book{globook,
+ AUTHOR = {R. Horst and H. Tuy},
+ TITLE = {Global Optimization: Deterministic Approaches},
+ PUBLISHER = {Springer-Verlag},
+ %ADDRESS = {Chicago},
+ YEAR = {1996},
+ %EDITION = {16th}
+}
+
+@book{sorbook,
+ AUTHOR = {R.I. Soare},
+ TITLE = {Turing Computability: Theory and Applications},
+ PUBLISHER = {Springer-Verlag},
+ %ADDRESS = {Chicago},
+ YEAR = {2016},
+ %EDITION = {16th}
+}
+
+@book{nest,
+ AUTHOR = {Y. Nesterov},
+ TITLE = {Introductory Lectures on Convex Optimization A Basic Course},
+ PUBLISHER = { Kluwer Academic Publishers},
+ %ADDRESS = {Chicago},
+ YEAR = {2003},
+ %EDITION = {16th}
+}
+
+@book{pour,
+ AUTHOR = {M. Pour-El and J. Richards},
+ TITLE = {Computability in analysis and physics},
+ PUBLISHER = {Springer},
+ ADDRESS = { Heidelberg},
+ YEAR = {1989},
+ %EDITION = {16th}
+}
diff --git a/Master/texmf-dist/doc/pdftex/tests/38-interword-space-at-font-switch/a2.tex b/Master/texmf-dist/doc/pdftex/tests/38-interword-space-at-font-switch/a2.tex
new file mode 100644
index 00000000000..007d825dd9d
--- /dev/null
+++ b/Master/texmf-dist/doc/pdftex/tests/38-interword-space-at-font-switch/a2.tex
@@ -0,0 +1,697 @@
+\documentclass[11pt,a4paper]{article}
+\usepackage[latin1]{inputenc}
+\usepackage{amsmath}
+\usepackage{amsthm}
+\usepackage{amsfonts}
+\usepackage{amsfonts,amsthm,latexsym,amsmath,amssymb,amscd,epsfig,psfrag,enumerate}
+\usepackage{graphics,graphicx, bezier, float, color, hyperref}
+\usepackage{amssymb,url}
+\usepackage{multienum}
+\usepackage[table]{xcolor}
+\usepackage{multicol,multirow}
+\usepackage{graphicx}
+\usepackage{fancyvrb}
+\usepackage{parskip}
+\usepackage[toc,page]{appendix}
+\sloppy
+\setlength{\parindent}{0pt}
+\setlength\parskip{0.1in}
+\usepackage[top=2.7cm, bottom=2.7cm, left=1.5cm, right=1.5cm]{geometry}
+\usepackage{xcolor}
+
+\usepackage{blkarray}
+\newtheorem{theorem}{Theorem}[section]
+\newtheorem{lemma}{Lemma}[section]
+\newtheorem{cor}{Corollary}[section]
+\newtheorem{remark}{Remark}[section]
+\usepackage[none]{hyphenat}[section]
+\newtheorem{definition}{Definition}[section]
+\newcommand\numberthis{\addtocounter{equation}{1}\tag{\theequation}}
+\numberwithin{equation}{section}
+\numberwithin{table}{section}
+\numberwithin{figure}{section}
+
+\title{Lucas numbers that are palindromic concatenations of
+ two distinct repdigits}
+\author{Herbert Batte$^{1,*} $}
+\date{}
+
+\usepackage{microtype}
+\begin{document}
+\pdfcompresslevel=0
+\pdfobjcompresslevel=0
+\pdfinterwordspaceon
+
+\maketitle
+\abstract{ Let $ \{L_n\}_{n\geq 0} $ be the sequence of Lucas numbers. In this paper, we determine all Lucas numbers that are palindromic concatenations of two distinct repdigits. }
+
+{\bf Keywords and phrases}: Lucas numbers; linear forms in logarithms; Repdigits; Baker--Davenport reduction method.
+
+{\bf 2020 Mathematics Subject Classification}: 11B39, 11D61, 11J86
+
+\thanks{$ ^{*} $ Corresponding author}
+
+\section{Introduction}\label{intro}
+\subsection{Background}
+\label{sec:1.1}
+Consider the Lucas number sequence $\{L_n\}_{n\ge 0}$, which starts with $L_0=2$, $L_1=1$, and follows the pattern $L_{n+2}=L_{n+1}+L_{n}$ for all $n \geq 0$. The initial numbers in this sequence are
+$$
+2,\;1,\;3,\;4,\;7,\;11,\;18,\;29,\;47,\;76,\;123,\;199,\ldots.
+$$
+A \textit{repdigit} in base 10 is a positive number $N$ made up of a single repeating digit. Specifically, $N$ is written as
+\[
+N = \underbrace{\overline{d\cdots d}}_{\ell \text{ times}} = d \left( \frac{10^\ell - 1}{9} \right),
+\]
+with positive integers \(d\) and \( \ell \), where \(0 \leq d \leq 9\) and \( \ell \geq 1\). Our study adds to the extensive research on the Diophantine characteristics of certain sequences defined by recurrence relations. Particularly, we explore how the terms of these sequences can be expressed as sequences within themselves or as combinations thereof. The work of Luca and Banks \cite{banks}, despite its broad scope, yielded some limited results regarding the count of such sequence terms. The case of Fibonacci numbers composed of two repdigits was addressed in \cite{ala}, with the largest identified as \(F_{14} = 377\).
+
+Recent work has also looked into the relationship between linear recurrence numbers and repdigits. For example, all repdigits formed by the addition of two Padovan numbers were identified in \cite{gar}. This was expanded upon by the Ddamulira in \cite{dda}, who also looked at Padovan numbers that are the concatenation of two different repdigits, finding the largest to be \(P_{21} = 200\) in \cite{dda2}.
+
+Further contributions to this field of research have been made by Bedná\v rik and Trojovská \cite{bed}, Boussayoud et al. \cite{bou}, Bravo and Luca \cite{bravo}, and others \cite{ddam, erd, raya, tro, troj, qu}. The specific findings in \cite{qu} were revisited in \cite{er}, which confirmed that the only Lucas numbers that can be formed by combining two repdigits are
+$$ 11,~18, ~29, ~47,~76, ~199, ~322.$$
+This finding, also reported in \cite{qu}, was derived through different methods. An interesting follow-up to this work \cite{er} would be to identify Lucas numbers that are \textit{palindromic}. In this context, a number is a \textit{palindrome} if it reads the same backwards as forwards. To begin exploring this, we currently examine a more constrained Diophantine equation:
+\begin{align}\label{eq1.1l}
+ L_n = \overline{\underbrace{d_1 \ldots d_1}_{\ell \text{ times}}\underbrace{d_2 \ldots d_2}_{m \text{ times}}\underbrace{d_1 \ldots d_1}_{\ell \text{ times}}},
+\end{align}
+where \( d_1, d_2 \in \{0, 1, 2, \ldots, 9\}, \) with \( d_1 > 0 \) and \(d_1\ne d_2\). Similar work has been done in \cite{chal} proving that $151$ and $616$ are the only Padovan numbers that are palindromic concatenations of two distinct repdigits.
+\newpage
+Here, we present the following result.
+\subsection{Main Results}\label{sec:1.2l}
+\begin{theorem}\label{thm1.1l}
+ There is no Lucas number which is a palindromic concatenation of two distinct repdigits.
+\end{theorem}
+
+\section{Methods}
+\subsection{Preliminaries}
+Here, we start with the well-known Binet formula for the sequence of Lucas numbers. It is given by
+\begin{align}\label{eq2.1l}
+ L_n = \alpha^n +\beta^n,~~~\text{where}~~\alpha=\dfrac{1+\sqrt{5}}{2}, ~~\beta=\dfrac{1-\sqrt{5}}{2}.
+\end{align}
+Note that $\beta=-\alpha^{-1}$ and $|\beta|<1$. It was shown in \cite{BRL} that
+\begin{align}\label{eq2.2l}
+ \alpha^{n-1} \le L_n \le2\alpha^n, \quad \text{holds for all} \quad n\ge0.
+\end{align}
+We go back and rewrite relation \eqref{eq1.1l} as
+\begin{align}\label{eq2.3l}
+ L_n &= \overline{\underbrace{d_1 \ldots d_1}_{\ell \text{ times}}\underbrace{d_2 \ldots d_2}_{m \text{ times}}\underbrace{d_1 \ldots d_1}_{\ell \text{ times}}}\nonumber\\
+ &=\overline{\underbrace{d_1 \ldots d_1}_{\ell \text{ times}}\cdot 10^{\ell+m}+\underbrace{d_2 \ldots d_2}_{m \text{ times}}\cdot 10^{\ell}+\underbrace{d_1 \ldots d_1}_{\ell \text{ times}}}\nonumber\\
+ &=\dfrac{1}{9}\left(d_1\cdot 10^{2\ell+m}-(d_1-d_2)\cdot 10^{\ell+m} +(d_1-d_2)\cdot 10^{\ell}-d_1 \right),
+\end{align}
+where \( d_1, d_2 \in \{0, 1, 2, \ldots, 9\}\), \( d_1 > 0 \) and \(d_1\ne d_2\).
+
+By equation \eqref{eq2.2l}, assume for a moment that $n>1000$, then equation \eqref{eq2.2l} together with \eqref{eq2.3l} imply that
+\begin{align*}
+ 2\alpha^n\ge L_n >10^{2\ell +m-1},
+\end{align*}
+and taking logarithms both sides yields $(2\ell+m-1)\log 10<n\log \alpha+\log 2$, which simplifies as
+\begin{align*}
+ (2\ell+m)\log 10<n\log \alpha+3<n,
+\end{align*}
+so that
+\begin{align}\label{eq2.4l}
+ 2\ell+m<n,
+\end{align}
+holds for all $n>1000$.
+
+\subsection{Linear forms in logarithms}
+We use three times Baker--type lower bounds for nonzero linear forms in three logarithms of algebraic numbers. There are many such bounds mentioned in the literature like that of Baker and W{\"u}stholz from \cite{BW} or Matveev from \cite{matl}. Before we can formulate such inequalities we need the notion of height of an algebraic number recalled below.
+
+
+\begin{definition}\label{def2.1l}
+ Let $ \gamma $ be an algebraic number of degree $ d $ with minimal primitive polynomial over the integers $$ a_{0}x^{d}+a_{1}x^{d-1}+\cdots+a_{d}=a_{0}\prod_{i=1}^{d}(x-\gamma^{(i)}), $$ where the leading coefficient $ a_{0} $ is positive. Then, the logarithmic height of $ \gamma$ is given by $$ h(\gamma):= \dfrac{1}{d}\Big(\log a_{0}+\sum_{i=1}^{d}\log \max\{|\gamma^{(i)}|,1\} \Big). $$
+\end{definition}
+ In particular, if $ \gamma$ is a rational number represented as $\gamma:=p/q$ with coprime integers $p$ and $ q\ge 1$, then $ h(\gamma ) = \log \max\{|p|, q\} $.
+The following properties of the logarithmic height function $ h(\cdot) $ will be used in the rest of the paper without further reference:
+\begin{equation}\nonumber
+ \begin{aligned}
+ h(\gamma_{1}\pm\gamma_{2}) &\leq h(\gamma_{1})+h(\gamma_{2})+\log 2;\\
+ h(\gamma_{1}\gamma_{2}^{\pm 1} ) &\leq h(\gamma_{1})+h(\gamma_{2});\\
+ h(\gamma^{s}) &= |s|h(\gamma) \quad {\text{\rm valid for}}\quad s\in \mathbb{Z}.
+ \end{aligned}
+\end{equation}
+
+A linear form in logarithms is an expression
+\begin{equation}
+ \label{eq:Lambdal}
+ \Lambda:=b_1\log \gamma_1+\cdots+b_t\log \gamma_t,
+\end{equation}
+where for us $\gamma_1,\ldots,\gamma_t$ are positive real algebraic numbers and $b_1,\ldots,b_t$ are nonzero integers. We assume, $\Lambda\ne 0$. We need lower bounds
+for $|\Lambda|$. We write ${\mathbb K}:={\mathbb Q}(\gamma_1,\ldots,\gamma_t)$ and $D$ for the degree of ${\mathbb K}$.
+We start with the general form due to Matveev \cite{matl}.
+
+\begin{theorem}[Matveev, \cite{matl}]
+ \label{thm:Matl}
+ Put $\Gamma:=\gamma_1^{b_1}\cdots \gamma_t^{b_t}-1=e^{\Lambda}-1$. Assume $\Gamma\ne 0$. Then
+ $$
+ \log |\Gamma|>-1.4\cdot 30^{t+3}\cdot t^{4.5} \cdot D^2 (1+\log D)(1+\log B)A_1\cdots A_t,
+ $$
+ where $B\ge \max\{|b_1|,\ldots,|b_t|\}$ and $A_i\ge \max\{Dh(\gamma_i),|\log \gamma_i|,0.16\}$ for $i=1,\ldots,t$.
+\end{theorem}
+
+
+\subsection{Reduction methods}
+Typically, the estimates from Matveev's theorem are excessively large to be practical in computations. To refine these estimates, we employ a modified approach based on the Baker--Davenport reduction method. Our adaptation follows the method introduced by Dujella and Pethö (\cite{duj}, Lemma 5a). When considering a real number \( r \), we use \( \| r \| \) to represent the smallest distance between \( r \) and any integer, which is formally written as \( \min\{|r - n| : n \in \mathbb{Z}\} \).
+\begin{lemma}[Dujella \& Pethö, \cite{duj}]\label{dujl}
+ Let \( \tau \neq 0 \), and \( A, B, \mu \) be real numbers with \( A > 0 \) and \( B > 1 \). Let \( M > 1 \) be a positive integer and suppose that \( p/q \) is a convergent of the continued fraction expansion of \( \tau \) with \( q > 6M \). Let
+ \[
+ \varepsilon := \| \mu q \| - M \| \tau q \|.
+ \]
+ If \( \varepsilon > 0 \), then there is no solution of the inequality
+ \[
+ 0 < |m\tau - n + \mu| < AB^{-k}
+ \]
+ in positive integers \( m, n, k \) with
+ \[
+ \frac{\log(Aq/\varepsilon)}{\log B} \leq k \quad \text{and} \quad m \leq M.
+ \]
+\end{lemma}
+
+
+Finally, we present an analytic argument which is Lemma 7 in \cite{guz}.
+\begin{lemma}[Lemma 7 in \cite{guz}]\label{guzl} If $ s \geq 1 $, $T > (4s^2)^s$ and $T > \displaystyle \frac{z}{(\log z)^s}$, then $$z < 2^s T (\log T)^s.$$
+\end{lemma}
+SageMath 9.5 is used to perform all the computations in this work.
+
+\section{Proof of Theorem \ref{thm1.1l}}
+\subsection{The low range $n\le 1000$}
+Using a basic SageMath script, we investigated all possible solutions to the Diophantine equation \eqref{eq1.1l}. The parameters $d_1, d_2$ were taken from the set $\{0, 1, 2, \ldots, 9\}$ with the conditions that $d_1 > 0$ and $d_1 \neq d_2$, and we restricted our search to $1 \leq \ell, m \leq n \leq 1000$. This search did not result in any solutions, see Appendix 1. From this point, we will only consider cases where $n > 1000$.
+
+\subsection{The case $n> 1000$}
+In this subsection, we proceed to examine \eqref{eq2.3l} in three different ways. We first prove the following result.
+\begin{lemma}\label{lem3.1l}
+ Let $\ell$, $m$ and $n>1000$ be solutions to the Diophantine equation \eqref{eq1.1l}, then
+ $$\ell<5\cdot 10^{12}\log n.$$
+\end{lemma}
+\begin{proof}
+We go back to \eqref{eq2.3l} and rewrite it using \eqref{eq2.1l} as
+\begin{align*}
+ L_n
+ &=\dfrac{1}{9}\left(d_1\cdot 10^{2\ell+m}-(d_1-d_2)\cdot 10^{\ell+m} +(d_1-d_2)\cdot 10^{\ell}-d_1 \right),\\
+ 9(\alpha^n+\beta^n)
+ &=d_1\cdot 10^{2\ell+m}-(d_1-d_2)\cdot 10^{\ell+m} +(d_1-d_2)\cdot 10^{\ell}-d_1, \\
+ 9\alpha^n-d_1\cdot 10^{2\ell+m} &=-9\beta^n-(d_1-d_2)\cdot 10^{\ell+m} +(d_1-d_2)\cdot 10^{\ell}-d_1.
+\end{align*}
+Therefore, we have that
+\begin{align*}
+ \left|9\alpha^n-d_1\cdot 10^{2\ell+m}\right| &=\left|-9\beta^n-(d_1-d_2)\cdot 10^{\ell+m} +(d_1-d_2)\cdot 10^{\ell}-d_1\right|\\
+ &\le 9\alpha^{-n}+27\cdot 10^{\ell+m},\quad\text{since}\quad \beta=-\alpha^{-1},\\
+ &<28\cdot 10^{\ell+m},
+\end{align*}
+where in the last inequality we have used the fact that $n>1000$. Now, dividing both sides by $d_1\cdot 10^{2\ell+m}$, we get
+\begin{align}\label{eq3.1l}
+ \left|\dfrac{9}{d_1}\cdot\alpha^n\cdot 10^{-2\ell-m}-1\right|
+ &<28\cdot 10^{-\ell}.
+\end{align}
+Let
+$$
+\Gamma=\dfrac{9}{d_1}\cdot\alpha^n\cdot 10^{-2\ell-m}-1=e^{\Lambda}-1.
+$$
+Notice that $\Lambda\ne 0$, otherwise we would have
+\begin{align*}
+ \alpha^n=\dfrac{d_1\cdot 10^{2\ell+m}}{9},
+\end{align*}
+which is impossible since the left--hand side is irrational and the right--hand side is rational. The algebraic number field containing the following $\gamma_i$'s is $\mathbb{K} := \mathbb{Q}(\sqrt{5})$. We have $D = 2$, $t :=3$,
+\begin{equation}\nonumber
+ \begin{aligned}
+ \gamma_{1}&:=9/d_1,\quad\gamma_{2}:=\alpha,\quad\gamma_{3}:=10,\\
+ b_{1}&:=1,\quad \quad~~ b_{2}:=n,\quad b_{3}:=-2\ell-m.
+ \end{aligned}
+\end{equation}
+Since $h(\gamma_{1})=h(9/d_1)\le \log 9 <2.2$, $h(\gamma_{2})=h(\alpha)=0.5\log \alpha <0.25$ and $h(\gamma_{3})=h(10)= \log 10 <2.31$, we take $A_1:=4.4$, $A_2:=0.5$ and $A_3:=4.62$. Next, $B \geq \max\{|b_i|:i=1,2,3\}$. By equation \eqref{eq2.4l}, $2\ell+m<n$, so we take $B:=n$. Now, by Theorem \ref{thm:Matl},
+\begin{align}\label{eq3.2l}
+ \log |\Gamma| &> -1.4\cdot 30^{6} \cdot 3^{4.5}\cdot 2^2 (1+\log 2)(1+\log n)\cdot 4.4\cdot 0.5\cdot 4.62\nonumber\\
+ &> -9.9\cdot 10^{12}(1+\log n).
+\end{align}
+Comparing \eqref{eq3.1l} and \eqref{eq3.2l}, we get
+\begin{align*}
+ \ell\log 10-\log 28&<9.9\cdot 10^{12}(1+\log n),\\
+ \ell&<4.3\cdot 10^{12}(1+\log n)+1.5\\
+ &=4.3\cdot 10^{12}\log n\left(\dfrac{1}{\log n}+1+\dfrac{1.5}{4.3\cdot 10^{12}\log n}\right),
+\end{align*}
+which leads to $\ell<5\cdot 10^{12}\log n$, for all $n>1000$.
+\end{proof}
+Next, we prove the following.
+\begin{lemma}\label{lem3.2l}
+ Let $\ell$, $m$ and $n>1000$ be solutions to the Diophantine equation \eqref{eq1.1l}, then
+ $$m<3\cdot 10^{25}(\log n)^2.$$
+\end{lemma}
+\begin{proof}
+Again, we go back to \eqref{eq2.3l} and rewrite it using \eqref{eq2.1l} as
+\begin{align*}
+ L_n
+ &=\dfrac{1}{9}\left(d_1\cdot 10^{2\ell+m}-(d_1-d_2)\cdot 10^{\ell+m} +(d_1-d_2)\cdot 10^{\ell}-d_1 \right),\\
+ 9(\alpha^n+\beta^n)
+ &=d_1\cdot 10^{2\ell+m}-(d_1-d_2)\cdot 10^{\ell+m} +(d_1-d_2)\cdot 10^{\ell}-d_1, \\
+ 9\alpha^n-d_1\cdot 10^{2\ell+m} +(d_1-d_2)\cdot 10^{\ell+m}&=-9\beta^n +(d_1-d_2)\cdot 10^{\ell}-d_1.
+\end{align*}
+Therefore, we have that
+\begin{align*}
+ \left|9\alpha^n-(d_1\cdot 10^{\ell}-(d_1-d_2))\cdot 10^{\ell+m}\right| &=\left|-9\beta^n +(d_1-d_2)\cdot 10^{\ell}-d_1\right|\\
+ &\le 9\alpha^{-n}+18\cdot 10^{\ell},\quad\text{since}\quad \beta=-\alpha^{-1},\\
+ &<19\cdot 10^{\ell},
+\end{align*}
+where in the last inequality we have used the fact that $n>1000$.
+
+Now, dividing both sides by $(d_1\cdot 10^{\ell}-(d_1-d_2))\cdot 10^{\ell+m}$, we get
+\begin{align}\label{eq3.3l}
+ \left|\dfrac{9}{(d_1\cdot 10^{\ell}-(d_1-d_2))}\cdot\alpha^n\cdot 10^{-\ell-m}-1\right|
+ &<\dfrac{19}{(d_1\cdot 10^{\ell}-(d_1-d_2))}\cdot 10^{-m}<19\cdot 10^{-m}.
+\end{align}
+Let
+$$
+\Gamma_1=\dfrac{9}{(d_1\cdot 10^{\ell}-(d_1-d_2))}\cdot\alpha^n\cdot 10^{-\ell-m}-1=e^{\Lambda_1}-1.
+$$
+Notice that $\Lambda_1\ne 0$, otherwise we would have
+\begin{align*}
+ \alpha^n=\dfrac{(d_1\cdot 10^{\ell}-(d_1-d_2))\cdot 10^{\ell+m}}{9},
+\end{align*}
+which is impossible since $ \alpha^n$ is irrational and the right--hand side is rational. The algebraic number field containing the following $\gamma_i$'s is $\mathbb{K} := \mathbb{Q}(\sqrt{5})$. Again, we have $D = 2$, $t :=3$,
+\begin{equation}\nonumber
+ \begin{aligned}
+ \gamma_{1}&:=\dfrac{9}{(d_1\cdot 10^{\ell}-(d_1-d_2))},\quad\gamma_{2}:=\alpha,\quad\gamma_{3}:=10,\\
+ b_{1}&:=1,\quad \quad\quad\quad\quad\quad\quad\quad\quad\quad b_{2}:=n,~~~ b_{3}:=-\ell-m.
+ \end{aligned}
+\end{equation}
+In order to determine what $A_1$ will be, we need to find the find the maximum of the quantities $h(\gamma_{1} )$ and $|\log \gamma_1 |$. We note that
+\begin{align*}
+ h(\gamma_{1})&=h\left(\dfrac{9}{d_1\cdot 10^{\ell}-(d_1-d_2)}\right) \le h(9)+h(d_1)+\ell h(10)+h(d_1-d_2)+\log 2 \le 4\log 9+\ell \log 10\\
+ &<4\log 9+(5\cdot 10^{12}\log n)\log 10<1.2\cdot 10^{13}\log n,
+\end{align*}
+where we used Lemma \ref{lem3.1l} and the fact that $n>1000$. On the other note,
+\begin{align*}
+ |\log \gamma_1 |&=\left|\log\left(\dfrac{9}{d_1\cdot 10^{\ell}-(d_1-d_2)}\right)\right| \le \log 9+\left|\log (d_1\cdot 10^{\ell}-(d_1-d_2))\right|\\
+ &\le \log 9+\log \left(d_1\cdot 10^{\ell}\right)+\left|\log \left(1-\dfrac{(d_1-d_2)}{d_1\cdot 10^{\ell}}\right)\right|\\
+ &\leq \log 9 + \log d_1+\ell \log 10 + \left| \frac{|d_1 - d_2|}{d_1 \cdot 10^\ell} + \frac{1}{2} \left( \frac{|d_1 - d_2|}{d_1 \cdot 10^\ell} \right)^2 + \cdots \right| \\
+ &\leq 2\log 9+\ell \log 10 + \frac{1}{10^\ell} + \frac{1}{2 \cdot 10^{2\ell}} + \cdots \\
+ &< 2\log 9+ (5 \cdot 10^{12} \log n)\log 10 + \frac{1}{10^\ell - 1} < 1.16 \cdot 10^{13} \log n,
+\end{align*}
+where still we used Lemma \ref{lem3.1l} and the fact that $n>1000$. Since $Dh(\gamma_{1})>|\log \gamma_1 |$, we can take $A_1:=2.4 \cdot 10^{13} \log n$. As before in the proof of Lemma \ref{lem3.1l}, we can still take $A_2:=0.5$, $A_3:=4.62$ and $B:=n$. Now, by Theorem \ref{thm:Matl},
+\begin{align}\label{eq3.4l}
+ \log |\Gamma| &> -1.4\cdot 30^{6} \cdot 3^{4.5}\cdot 2^2 (1+\log 2)(1+\log n)\cdot 0.5\cdot 4.62\cdot 2.4 \cdot 10^{13} \log n\nonumber\\
+ &> -6.2\cdot 10^{25}(\log n)^2.
+\end{align}
+Comparing \eqref{eq3.3l} and \eqref{eq3.4l}, we get
+\begin{align*}
+ m\log 10-\log 19&<6.2\cdot 10^{25}(\log n)^2,\\
+ m&<2.7\cdot 10^{25}(\log n)^2+1.3
+ =2.7\cdot 10^{25}(\log n)^2\left(1+\dfrac{1.3}{2.7\cdot 10^{25}(\log n)^2}\right),
+\end{align*}
+which leads to $m<3\cdot 10^{25}(\log n)^2$, for all $n>1000$.
+\end{proof}
+Lastly in this subsection, we prove the following.
+\begin{lemma}\label{lem3.3l}
+ Let $\ell$, $m$ and $n>1000$ be solutions to the Diophantine equation \eqref{eq1.1l}, then
+ $$\ell< 5.3\cdot 10^{14},\quad m<3.4\cdot 10^{29}\quad\text{and}\quad n<9\cdot 10^{45}.$$
+\end{lemma}
+
+\begin{proof}
+ Once more, we revisit equation \eqref{eq2.3l} and rewrite it using \eqref{eq2.1l} as
+ \begin{align*}
+ 9\alpha^n-d_1\cdot 10^{2\ell+m} +(d_1-d_2)\cdot 10^{\ell+m}-(d_1-d_2)\cdot 10^{\ell}&=-9\beta^n -d_1.
+ \end{align*}
+ Therefore, we have that
+ \begin{align*}
+ \left|9\alpha^n-(d_1\cdot 10^{\ell+m}-(d_1-d_2)\cdot 10^{m}+(d_1-d_2))\cdot 10^{\ell}\right| &=\left|-9\beta^n -d_1\right|\le 9\alpha^{-n}+9<10,
+ \end{align*}
+ where we have used the fact that $\beta=-\alpha^{-1}$ and $n>1000$. Now, dividing sides both by $9\alpha^n$, we get
+ \begin{align}\label{eq3.5l}
+ \left|\dfrac{(d_1\cdot 10^{\ell+m}-(d_1-d_2)\cdot 10^{m}+(d_1-d_2))}{9}\cdot \alpha^{-n}\cdot 10^{\ell}-1\right|
+ &<\dfrac{10}{9\alpha^{n}}.
+ \end{align}
+ Let
+ $$
+ \Gamma_2=\dfrac{(d_1\cdot 10^{\ell+m}-(d_1-d_2)\cdot 10^{m}+(d_1-d_2))}{9}\cdot \alpha^{-n}\cdot 10^{\ell}-1=e^{\Lambda_2}-1.
+ $$
+ Notice that $\Lambda_2\ne 0$, otherwise we would have
+ \begin{align*}
+ \alpha^n=\dfrac{(d_1\cdot 10^{\ell+m}-(d_1-d_2)\cdot 10^{m}+(d_1-d_2))}{9}\cdot 10^{\ell},
+ \end{align*}
+ which is impossible since $ \alpha^n$ is irrational and the right--hand side is rational. The algebraic number field containing the following $\gamma_i$'s is $\mathbb{K} := \mathbb{Q}(\sqrt{5})$. Again, we have $D = 2$, $t :=3$,
+ \begin{equation}\nonumber
+ \begin{aligned}
+ \gamma_{1}&:=\dfrac{(d_1\cdot 10^{\ell+m}-(d_1-d_2)\cdot 10^{m}+(d_1-d_2))}{9},\quad\gamma_{2}:=\alpha,\quad\gamma_{3}:=10,\\
+ b_{1}&:=1,\quad \quad\quad\quad\quad\quad \quad\quad\quad\quad\quad \quad\quad\quad\quad\quad\quad\quad~~ b_{2}:=-n,~~ b_{3}:=-\ell.
+ \end{aligned}
+ \end{equation}
+Again, in order to determine what $A_1$ will be here, we need to find the find the maximum of the quantities $h(\gamma_{1} )$ and $|\log \gamma_1 |$. We note that
+ \begin{align*}
+ h(\gamma_{1})&=h\left(\dfrac{d_1\cdot 10^{\ell+m}-(d_1-d_2)\cdot 10^{m}+(d_1-d_2)}{9}\right) \\
+ &\le h(9)+h(d_1)+(\ell+m) h(10)+h(d_1-d_2)+mh(10)+h(d_1-d_2)+3\log 2 \\
+ &\le 7\log 9+(\ell+m) \log 10+m \log 10\\
+ &<7\log 9+\left(3.1\cdot 10^{25}(\log n)^2\right)\log 10+\left(3\cdot 10^{25}(\log n)^2\right)\log 10
+ <1.41\cdot 10^{26}(\log n)^2,
+ \end{align*}
+ where we used Lemmas \ref{lem3.1l}, \ref{lem3.2l} and the fact that $n>1000$.
+
+ On the other note,
+ \begin{align*}
+ |\log \gamma_1 |&=\left|\log\left(\dfrac{d_1\cdot 10^{\ell+m}-(d_1-d_2)\cdot 10^{m}+(d_1-d_2)}{9}\right)\right| \\
+ &\le \log 9+\left|\log (d_1\cdot 10^{\ell+m}-(d_1-d_2)\cdot 10^{m}+(d_1-d_2))\right|\\
+ &\le \log 9+\log \left(d_1\cdot 10^{\ell+m}\right)+\left|\log \left(1-\dfrac{(d_1-d_2)(10^m-1)}{d_1\cdot 10^{\ell+m}}\right)\right|\\
+ &\leq \log 9 + \log d_1+(\ell+m) \log 10 + \left| \dfrac{(d_1-d_2)(10^m-1)}{d_1\cdot 10^{\ell+m}} + \frac{1}{2} \left( \dfrac{(d_1-d_2)(10^m-1)}{d_1\cdot 10^{\ell+m}} \right)^2 + \cdots \right| \\
+ &\leq 2\log 9+(\ell+m) \log 10 + \frac{1}{10^\ell} + \frac{1}{2 \cdot 10^{2\ell}} + \cdots \\
+ &< 2\log 9+ \left(3.1\cdot 10^{25}(\log n)^2\right)\log 10 + \frac{1}{10^\ell - 1} < 7.2 \cdot 10^{25} (\log n)^2,
+ \end{align*}
+ where still we used Lemmas \ref{lem3.1l}, \ref{lem3.2l} and the fact that $n>1000$. Since $Dh(\gamma_{1})>|\log \gamma_1 |$, we can take $A_1:=2.82\cdot 10^{26}(\log n)^2$. As before in the proof of Lemma \ref{lem3.1l}, we can still take $A_2:=0.5$, $A_3:=4.62$ and $B:=n$. Now, by Theorem \ref{thm:Matl},
+ \begin{align}\label{eq3.6l}
+ \log |\Gamma| &> -1.4\cdot 30^{6} \cdot 3^{4.5}\cdot 2^2 (1+\log 2)(1+\log n)\cdot 0.5\cdot 4.62\cdot 2.82\cdot 10^{26}(\log n)^2\nonumber\\
+ &> -7.24\cdot 10^{38}(\log n)^3.
+ \end{align}
+ Comparing \eqref{eq3.5l} and \eqref{eq3.6l}, we get
+ \begin{align*}
+ n\log\alpha-\log (10/9)&<7.24\cdot 10^{38}(\log n)^3,\\
+ n&<1.51\cdot 10^{39}(\log n)^3+0.22
+ =1.51\cdot 10^{39}(\log n)^3\left(1+\dfrac{0.22}{1.51\cdot 10^{39}(\log n)^3}\right),
+ \end{align*}
+ which leads to $n<1.52\cdot 10^{39}(\log n)^3$, for all $n>1000$. To proceed from here, let $ s =3\geq 1 $, $T =1.52\cdot 10^{39}> (4s^2)^s=46656$ and $z=n$, then Lemma \ref{guzl} implies that $n < 2^3 \cdot 1.52\cdot 10^{39} (\log 1.52\cdot 10^{39})^3$, or $n<9\cdot 10^{45}$.
+
+Moreover, Lemma \ref{lem3.1l} gives $\ell<5\cdot 10^{12}\log n<5\cdot 10^{12}\log (9\cdot 10^{45})<5.3\cdot 10^{14}$ and Lemma \ref{lem3.2l} gives $m<3\cdot 10^{25}(\log n)^2<3\cdot 10^{25}(\log (9\cdot 10^{45}))^2<3.4\cdot 10^{29}$.
+\end{proof}
+The bounds established in Lemma \ref{lem3.3l} exceed practical computational utility and require reduction. This process is detailed in Subsection \ref{subsecl}.
+
+\subsection{The reduction process}\label{subsecl}
+Here, we apply Lemma \ref{dujl} as follows. First, we return to the inequality \eqref{eq3.1l} and put
+\[
+\Lambda_1 := (2\ell + m) \log 10 - n \log \alpha + \log \left( \frac{d_1}{9} \right).
+\]
+Inequality \eqref{eq3.1l} can be rewritten as
+$|\Gamma_1| = |e^{\Lambda_1} - 1| < 28\cdot10^{-\ell}$. If we assume that \(\ell \geq 2\), then the right--hand side of this inequality is at most \(0.28 < 1/2\). The inequality \(|e^{\Lambda_1} - 1| < x\) for real values of \(x\) and \({\Lambda_1}\) implies that \(|{\Lambda_1}| < 2x\). Thus,
+$|\Lambda_1| < 56\cdot10^{-\ell}$. This implies that
+\[
+\left| (2\ell + m) \log 10 - n \log \alpha + \log \left( \frac{d_1}{9}\right) \right| < 56\cdot10^{-\ell}.
+\]
+Dividing through the above inequality by \(\log \alpha\) gives
+\[
+\left| (2\ell + m) \frac{\log 10}{\log \alpha} - n + \left( \frac{\log(d_1/9)}{\log \alpha} \right) \right| < \frac{56}{\log \alpha}\cdot 10^{-\ell}.
+\]
+So, we apply Lemma \ref{dujl} with the quantities:
+\[
+\tau := \frac{\log 10}{\log \alpha}, \quad \mu(d_1) := \frac{\log(d_1/9)}{\log \alpha}, \quad 1 \leq d_1 \leq 9, \quad A := \frac{56}{\log \alpha}, \quad B := 10.
+\]
+Let \(\tau = [a_0; a_1, a_2, \ldots] = [4; 1, 3, 1, 1, 1, 6, 4, 2, 1, 10, 1, 4, 46, 3, 1, 2, 1, 2, \ldots]\) be the continued fraction expansion of \(\tau\). We set \(M := 10^{46}\) which is an upper bound of \(2\ell + m\). With the help of SageMath in Appendix 2, we find that the convergent
+\[
+\dfrac{p}{q} = \dfrac{p_{98}}{q_{98}} = \dfrac{1645685064668785741047746957258993430046006088389}{343927838259763182336125476035118084206130771252} ,
+\]
+is such that \(q = q_{98} > 6M\). Furthermore, it gives \(\varepsilon > 0.4614141430\), and thus,
+\[
+\ell \leq \frac{\log((56/\log \alpha)q/\varepsilon)}{\log 10} < 52.
+\]
+Therefore, we have that \(\ell < 52\). The case \(\ell < 2\) also holds because \(\ell < 2 < 52\).
+
+Next, for fixed \( d_1, d_2 \in \{0, 1, 2, \ldots, 9\} \), \( d_1 > 0 \), \(d_1 \neq d_2\) and \( 1 \leq \ell < 52 \), we return to inequality \eqref{eq3.3l} and put
+\[
+\Lambda_2 := (\ell + m) \log 10 - n \log \alpha + \log \left( \frac{d_1 \cdot 10^\ell - (d_1 - d_2)}{9} \right).
+\]
+From inequality \eqref{eq3.3l}, we have that
+\(
+|\Gamma_2| = |e^{\Lambda_2} - 1| < 19\cdot10^{-m}.
+\)
+Assume that \( m \geq 2 \), then the right--hand side of this inequality is at most \( 0.19 < 1/2 \). Thus,
+$|\Lambda_2| < 38\cdot10^{-m}$,
+which implies that
+\[
+\left| (\ell + m) \log 10 - n \log \alpha + \log \left( \frac{d_1 \cdot 10^\ell - (d_1 - d_2)}{9} \right) \right| < 38\cdot10^{-m}.
+\]
+Dividing through the above inequality by \( \log \alpha \) gives
+\[
+\left| (\ell + m) \frac{\log 10}{\log \alpha} - n + \frac{\log ((d_1 \cdot 10^\ell - (d_1 - d_2))/9)}{\log \alpha} \right| < \frac{38}{ \log \alpha}\cdot 10^{-m}.
+\]
+Thus, we apply Lemma \ref{dujl} with the quantities:
+\[
+\mu(d_1, d_2) := \frac{\log ((d_1 \cdot 10^\ell - (d_1 - d_2))/9)}{\log \alpha}, \quad A := \frac{38}{\log \alpha}, \quad B := 10.
+\]
+We take the same \( \tau \) as before and its convergent \( p/q = p_{98}/q_{98} \). Since \( \ell + m < 2\ell + m \), we set \( M := 10^{46} \) as an upper bound on \( \ell + m \). With the help of a simple computer program in SageMath (Appendix 3), we get that \( \varepsilon > 0.4906425804 \), and therefore,
+\[
+m \leq \frac{\log((38/\log \alpha)/\varepsilon)}{\log 10} < 54.
+\]
+Hence, we have that \( m < 54 \). The case \( m < 2 \) holds as well since \( m < 2 < 54 \).
+
+Lastly, for fixed \( d_1, d_2 \in \{0, 1, 2, \ldots, 9\} \), \( d_1 > 0 \), \(d_1 \neq d_2\), \( 1 \leq \ell <52 \) and \( 1 \leq m < 54 \), we return to inequality \eqref{eq3.5l} and put
+\[
+\Lambda_3 := \ell \log 10 - n \log \alpha + \log \left( \frac{d_1 \cdot 10^{\ell+m} - (d_1 - d_2) \cdot 10^m + (d_1 - d_2)}{9} \right).
+\]
+From inequality \eqref{eq3.5l}, we have that $
+|\Gamma_3| = |e^{\Lambda_3} - 1| < 10/9\alpha^n$. Since \( n > 1000 \), the right--hand side of this inequality is less than \( 1/2 \). Thus, the above inequality implies that
+$
+|\Lambda_3| < 20/9\alpha^n,
+$
+which leads to
+\[
+\left| \ell \log 10 - n \log \alpha + \log \left( \frac{d_1 \cdot 10^{\ell+m} - (d_1 - d_2) \cdot 10^m + (d_1 - d_2)}{9} \right) \right| < \frac{20}{9\alpha^n}.
+\]
+Dividing through the above inequality by \( \log \alpha \) gives,
+\[
+\left| \ell\frac{ \log 10}{\log \alpha} - n + \frac{\log \left( (d_1 \cdot 10^{\ell+m} - (d_1 - d_2) \cdot 10^m + (d_1 - d_2))/9 \right)}{\log \alpha} \right| < \frac{20}{ 9\log \alpha}\cdot \alpha^{-n}.
+\]
+Again, we apply Lemma \ref{dujl} with the quantities:
+\[
+\mu(d_1, d_2) := \frac{\log \left( (d_1 \cdot 10^{\ell+m} - (d_1 - d_2) \cdot 10^m + (d_1 - d_2))/9 \right)}{\log \alpha}, \quad A := \frac{20}{9\log \alpha}, \quad B := \alpha.
+\]
+We can still take the same \( \tau \) and its convergent \( p/q = p_{98}/q_{98} \) as before. Since \( \ell < 2\ell + m \), we choose \( M := 10^{46} \) as an upper bound for \( \ell \). With the help of a simple computer program in SageMath (Appendix 4), we get that \( \varepsilon > 0.4929934686 \), and thus,
+\[
+n \leq \frac{\log((20/9\log \alpha)/\varepsilon)}{\log \alpha} < 269,
+\]
+contradicting our working assumption that \( n > 1000 \), hence Theorem \ref{thm1.1l} holds.\qed
+
+\section*{Conclusion}
+In this work, we showed that there is no Lucas numbers that is a palindromic concatenation of two distinct repdigits. It remains an open problem to determine such palindromes in generalized $k-$Lucas numbers.
+\section*{Acknowledgments}
+The author thanks the Eastern Africa Universities Mathematics Programme (EAUMP) for funding his doctoral studies.
+
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+
+ \bibitem{dda2}
+ Ddamulira, M.: Padovan numbers that are concatenations of two distinct repdigits. Math. Slovaca \textbf{71}(2), 275--284 (2021).
+
+ \bibitem{duj}
+ Dujella, A., Peth\"o, A.: A generalization of a theorem of Baker and Davenport. Q. J. Math. Oxford Ser. 2 \textbf{49}(195), 291--306 (1998).
+
+ \bibitem{er}
+ Erduvan, F., Keskin, R.: Lucas numbers which are concatenations of two repdigits. Boletín de la Sociedad Matemática Mexicana, 27, pp.1-11 (2021).
+
+ \bibitem{erd}
+ Erduvan, F., Keskin, R.: Lucas numbers which are concatenations of three repdigits. Results Math. \textbf{76}(1), 13 (2021).
+
+ \bibitem{gar}
+ García Lomelí, A.C., Hernández Hernández, S.: Repdigits as sums of two Padovan numbers. J. Integer Seq. \textbf{22}(2), Art. 19.2.3 (2019)
+
+ \bibitem{guz}
+ Gúzman Sánchez, S., Luca, F.: Linear combinations of factorials and \( s \)--units in a binary recurrence sequence. Ann. Math. Qué. \textbf{38}(2), 169--188 (2014).
+
+ \bibitem{matl}
+ Matveev, E. M.: An explicit lower bound for a homogeneous rational linear form in the logarithms of algebraic numbers. II. Izvestiya: Mathematics, \textbf{64}(6), 1217 (2000).
+
+ \bibitem{qu}
+ Qu, Y., Zeng, J.: Lucas numbers which are concatenations of two repdigits. Mathematics \textbf{8}(8), 1360 (2020).
+
+ \bibitem{raya}
+ Rayaguru, S.G., Panda, G.K.: Balancing numbers which are concatenations of two repdigits. Bol. Soc. Mat. Mex. \textbf{26}(3), 911--919 (2020).
+
+ \bibitem{tro}
+ Trojovský, P.: Fibonacci numbers with a prescribed block of digits. Mathematics \textbf{8}(4), 639 (2020).
+
+ \bibitem{troj}
+ Trojovský, P.: On repdigits as sums of Fibonacci and Tribonacci numbers. Symmetry \textbf{12}(11), 1774 (2020).
+
+\end{thebibliography}
+
+
+
+\section*{Address}
+$ ^{1} $ Department of Mathematics, School of Physical Sciences, College of Natural Sciences, Makerere University, Kampala, Uganda
+
+Email: \url{hbatte91@gmail.com}
+\pagebreak
+\section*{Appendices}
+\subsection*{Appendix 1}\label{app1}
+\begin{verbatim}
+def generate_lucas_numbers(limit):
+ lucas = [2, 1] # Initial Lucas numbers
+ while len(lucas) < limit:
+ lucas.append(lucas[-1] + lucas[-2])
+ return set(lucas) # Use a set for efficient membership checking
+
+def construct_palindromic_numbers():
+ palindromic_numbers = set()
+ for d1 in range(1, 10): # d1 is a nonzero digit
+ for d2 in range(10): # d2 can be any digit, including zero
+ if d1 != d2:
+ for length in range(1, 100): # Reasonable lengths for repdigits
+ first_part = str(d1) * length
+ second_part = str(d2) * length
+ palindromic_number = int(first_part + second_part + first_part)
+ palindromic_numbers.add(palindromic_number)
+ return palindromic_numbers
+
+# Generate Lucas numbers and palindromic numbers
+lucas_numbers = generate_lucas_numbers(1001)
+palindromic_numbers = construct_palindromic_numbers()
+
+# Find the intersection of the two sets
+palindromic_lucas = lucas_numbers.intersection(palindromic_numbers)
+
+print("Lucas numbers which are palindromic concatenations of two distinct repdigits:")
+print(sorted(palindromic_lucas))
+
+\end{verbatim}
+\subsection*{Appendix 2}\label{app2}
+\begin{verbatim}
+from sage.all import *
+
+# Constants
+a = golden_ratio.n(digits=1000) # Using the golden ratio as an approximation for the
+ # root of x^2 - x - 1
+tau = (log(10) / log(a)).n(digits=1000)
+A = (56 / log(a))
+B = 10
+M = 1 * 10^46
+
+# Continued Fraction and Convergents
+cf_tau = continued_fraction(tau)
+convergents = cf_tau.convergents()
+
+for d1 in range(1, 10): # Iterate through d1 from 1 to 9
+ mu = (log(d1/9) / log(a)).n(digits=1000)
+
+ DD = [] # Initialize empty list for results for each d1
+
+ # Dujella and Pethö Reduction Method
+ for i, convergent in enumerate(convergents):
+ p, q = convergent.numerator(), convergent.denominator()
+ ep = abs(mu * q - round(mu * q)) - M * abs(tau * q - round(tau * q))
+
+ if q > 6 * M and ep > 0:
+ log_expr_a = (log(A * q / ep) / log(B)).n(digits=10)
+ DD.append((i, ep.n(digits=10), log_expr_a))
+ print(f"d1 = {d1}, p_{i}/q_{i} = {p}/{q}")
+ break # Stop after finding the first suitable convergent for this d1
+
+ # Results for each d1
+ if DD:
+ print(f"Results for d1 = {d1}:")
+ print("First few elements of DD:", DD[:1])
+ else:
+ print(f"No suitable convergent found for d1 = {d1}.")
+print("Continued fraction expansion of tau:", cf_tau[:20])
+\end{verbatim}
+
+\subsection*{Appendix 3}\label{app3}
+\begin{verbatim}
+from sage.all import *
+
+# Constants
+a = golden_ratio.n(digits=1000) # Using the golden ratio
+tau = (log(10) / log(a)).n(digits=1000)
+A = (38 / log(a))
+B = 10
+M = 1 * 10^46
+
+# Continued Fraction and Convergents
+cf_tau = continued_fraction(tau)
+convergents = cf_tau.convergents()
+
+# Variables to store maximum values
+max_ep = -Infinity
+max_log_expr_a = -Infinity
+
+for d1 in range(1, 10):
+ for d2 in range(0, 10):
+ if d1 == d2:
+ continue # d1 should not be equal to d2
+ for l in range(1, 52):
+ mu = (log((d1 * 10^l - (d1 - d2)) / 9) / log(a)).n(digits=1000)
+
+ # Dujella and Pethö Reduction Method
+ for i, convergent in enumerate(convergents):
+ p, q = convergent.numerator(), convergent.denominator()
+ ep = abs(mu * q - round(mu * q)) - M * abs(tau * q - round(tau * q))
+
+ if q > 6 * M and ep > 0:
+ log_expr_a = (log(A * q / ep) / log(B)).n(digits=10)
+ if ep > max_ep:
+ max_ep = ep.n(digits=10)
+ if log_expr_a > max_log_expr_a:
+ max_log_expr_a = log_expr_a
+ break # Stop after finding the first suitable convergent
+
+# Print maximum values
+print("Maximum ep across all combinations:", max_ep)
+print("Maximum log_expr_a across all combinations:", max_log_expr_a)
+\end{verbatim}
+
+\subsection*{Appendix 4}\label{app4}
+\begin{verbatim}
+from sage.all import *
+
+# Constants
+a = golden_ratio.n(digits=1000) # Using the golden ratio
+tau = (log(10) / log(a)).n(digits=1000)
+A = (20 / (9 * log(a)))
+B = a
+M = 1 * 10^46
+
+# Continued Fraction and Convergents
+cf_tau = continued_fraction(tau)
+convergents = cf_tau.convergents()
+
+# Variables to store maximum values
+max_ep = -Infinity
+max_log_expr_a = -Infinity
+
+# Iterate through combinations of d1, d2, l, and m
+for d1 in range(1, 10):
+ for d2 in range(0, 10):
+ if d1 == d2:
+ continue # d1 should not be equal to d2
+ for l in range(1, 52):
+ for m in range(1, 54):
+ mu = (log((d1 * 10^(l+m) - (d1 - d2) * 10^m +
+ (d1 - d2)) / 9) / log(a)).n(digits=1000)
+
+ # Dujella and Pethö Reduction Method
+ for convergent in convergents:
+ p, q = convergent.numerator(), convergent.denominator()
+ ep = abs(mu * q - round(mu * q)) - M * abs(tau * q - round(tau * q))
+
+ if q > 6 * M and ep > 0:
+ log_expr_a = (log(A * q / ep) / log(B)).n(digits=10)
+ if ep > max_ep:
+ max_ep = ep.n(digits=10)
+ if log_expr_a > max_log_expr_a:
+ max_log_expr_a = log_expr_a
+ break # Stop after finding the first suitable convergent
+
+# Print maximum values
+print("Maximum ep across all combinations:", max_ep)
+print("Maximum log_expr_a across all combinations:", max_log_expr_a)
+\end{verbatim}
+\end{document}
diff --git a/Master/texmf-dist/doc/pdftex/tests/38-interword-space-at-font-switch/a3.bbl b/Master/texmf-dist/doc/pdftex/tests/38-interword-space-at-font-switch/a3.bbl
new file mode 100644
index 00000000000..2ec95ef93cf
--- /dev/null
+++ b/Master/texmf-dist/doc/pdftex/tests/38-interword-space-at-font-switch/a3.bbl
@@ -0,0 +1,37 @@
+\begin{thebibliography}{1}
+
+\bibitem{foster2019complexity}
+Dylan~J Foster, Ayush Sekhari, Ohad Shamir, Nathan Srebro, Karthik Sridharan,
+ and Blake Woodworth.
+\newblock The complexity of making the gradient small in stochastic convex
+ optimization.
+\newblock In {\em Conference on Learning Theory}, pages 1319--1345. PMLR, 2019.
+
+\bibitem{globook}
+R.~Horst and H.~Tuy.
+\newblock {\em Global Optimization: Deterministic Approaches}.
+\newblock Springer-Verlag.
+
+\bibitem{nest}
+Y.~Nesterov.
+\newblock {\em Introductory Lectures on Convex Optimization A Basic Course}.
+\newblock Kluwer Academic Publishers.
+
+\bibitem{pour}
+M.~Pour-El and J.~Richards.
+\newblock {\em Computability in analysis and physics}.
+\newblock Springer, Heidelberg, 1989.
+
+\bibitem{sorbook}
+R.I. Soare.
+\newblock {\em Turing Computability: Theory and Applications}.
+\newblock Springer-Verlag.
+
+\bibitem{zhang2020complexity}
+Jingzhao Zhang, Hongzhou Lin, Stefanie Jegelka, Suvrit Sra, and Ali Jadbabaie.
+\newblock Complexity of finding stationary points of nonconvex nonsmooth
+ functions.
+\newblock In {\em International Conference on Machine Learning}, pages
+ 11173--11182. PMLR, 2020.
+
+\end{thebibliography}
diff --git a/Master/texmf-dist/doc/pdftex/tests/38-interword-space-at-font-switch/a3.tex b/Master/texmf-dist/doc/pdftex/tests/38-interword-space-at-font-switch/a3.tex
new file mode 100644
index 00000000000..7e8777e9a14
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+++ b/Master/texmf-dist/doc/pdftex/tests/38-interword-space-at-font-switch/a3.tex
@@ -0,0 +1,303 @@
+\documentclass[a4paper,11pt]{article}
+
+%\usepackage[margin=1in]{geometry}
+\usepackage[utf8]{inputenc}
+\usepackage{amsthm,amsmath,amssymb}
+\usepackage{tikz}
+\usepackage{graphicx,subcaption,caption}
+\usepackage{algorithm,algorithmic}
+\usepackage{pgfplots}
+\usepackage{multirow}
+\usepackage[T1]{fontenc}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+%% If your tex system is less than 2 years old (in 2012) the following
+%% font options are available. If not comment them out.
+\usepackage{tgtermes}
+% otherwise use alternative journal fonts
+%\renewcommand{\rmdefault}{ptm} % system default Times font
+\usepackage{mathptmx}
+%%% additional fonts
+\usepackage[scaled=.92]{helvet}
+%\setoptfont{enc={T1},fam={pop}} % if You have Optima font, uncomment this line
+%%% MATH
+%\usepackage{amsthm,amsmath,amssymb}
+\usepackage{mathrsfs}
+%%% BIBLIOGRAPHY
+%\usepackage{natbib} %% numbers is required.
+%%% LINKS
+\usepackage[colorlinks,citecolor=blue,urlcolor=blue]{hyperref} %%check
+
+\usepackage{enumerate}
+
+%\artstatus{am} %%% leave this alone!! That means you, too!!
+%%%%%%%%theorems%%%%%
+
+%%%% feel free to changes these%%%%%%%
+\newtheorem{theorem}{Theorem}[section]
+\newtheorem{lemma}[theorem]{Lemma}
+\newtheorem{problem}[theorem]{Problem}
+\newtheorem{conjecture}[theorem]{Conjecture}
+\newtheorem{condition}[theorem]{Condition}
+\newtheorem{claim}[theorem]{Claim}
+\newtheorem{question}[theorem]{Question}
+\newtheorem{corollary}[theorem]{Corollary}
+\theoremstyle{definition}
+\newtheorem{definition}[theorem]{Definition}
+\newtheorem{statement}[theorem]{Statement}
+\newtheorem{notation}[theorem]{Notation}
+\theoremstyle{remark}
+\newtheorem{remark}[theorem]{Remark}
+
+
+%\tikzexternalize
+%\newtheorem{theorem}{Theorem}
+%\newtheorem{proposition}{Proposition}
+%\newtheorem{lemma}{Lemma}
+%\newtheorem{assumption}{Assumption}
+%\newtheorem{remark}{Remark}
+%\newtheorem{corollary}{Corollary}
+%\newtheorem{definition}{Definition}
+%\usepackage{hyperref}
+%\providecommand{\keywords}[1]{\textbf{\textit{Keywords---}} #1}
+
+%\firstpageno{1}
+
+\begin{document}
+% \begin{frontmatter}
+ \title{Computability of Approximate Optima of Nonconvex Functions}
+ \author{K. Lakshmanan \\ Department of Computer Science and Engineering, \\ Indian Institute of Technology (BHU) Varanasi, 221005 India. \\ Email: lakshmanank.cse@iitbhu.ac.in}
+ \date{}
+
+ \maketitle
+
+ \begin{abstract}
+ It is known that finding approximate optima of non-convex functions is intractable. We give a simple proof to show that this problem is not even computable. \\
+ \textbf{Keywords:} Computablity of Approximate Optima, Non-convex functions
+ \end{abstract}
+
+ \section{Introduction and Preliminaries}
+ We consider the problem of finding the global minima of a non-convex continuous function $f: C \rightarrow \mathbb{R}$, where $ C \subset \mathbb{R}^d$ is a closed, compact subset. Global minima is the point $x^* \in C$ which satisfies the following property: $f(x^*) \leq f(x)$ for all $x \in C$. The function $f$ attains this minimum at least once by extreme value theorem. Our goal is to find one such point. This problem is well-studied with many books written on the subject, see for example \cite{globook}.
+
+ We note that we consider an oracle setting, where the function values are given by an oracle. This is different from the computablity of optima computable real functions studied for example in \cite{pour}. It is easy to see that a simple grid search will output a sequence of points converging to the global optima. And for a Lipschitz continuous function, it requires an exponential number of oracle calls \cite{nest} if the Lipschitz constant is known. % that the approximations to optima are not computable too. %This is also the case if we are looking for points close to approximate optima.
+
+ Let us define $S = \{x||f(x)-f(x^*)|\leq \epsilon\}$ and term the members of $S$ as $\epsilon$-optima. In optimization literature \cite{foster2019complexity,zhang2020complexity}, it is known that finding approximations to optima is not tractable for non-convex functions. For non-convex functions, $\epsilon$-stationary point which is weaker than $\epsilon$-optima is also known to be not tractable \cite{zhang2020complexity}. We show more in this paper, that this set $S$ is not computable. This is much stronger than saying they it is intractable. We assume for formality finite-precision numbers. This assumption of finite-precision numbers is useful to model real-world systems and we show is not restrictive.
+
+ %and $T = \{y|\exists x \mbox{ s.t., } |x-y| \leq \delta \mbox{ and } |f(x)-f(x^*)|\leq \epsilon \}$.
+ % and that of $T$ as $(\epsilon,\epsilon)$-optima.
+ %Our proof is through simple reductions.
+
+ %\begin{table}[ht]\label{table1}
+ % \centering
+ % \begin{tabular}{|p{4cm}|c c|}
+ % \hline
+ % \multirow{2}{*}{Function Type} & \multicolumn{2}{c}{Function Oracle} \\
+ % & Yes & No \\
+ % \hline
+ % Computable-Real & No & No \\
+ % Computable-Real (Isolated minimum) & \multirow{2}{*}{Yes} & \multirow{2}{*}{Yes}\\
+ % Lipschitz Continuous & Yes & No \\
+ % Differentiable/Smooth & No & No \\
+ % Continuous & No & No \\
+ % \hline
+ % \end{tabular}
+ % \caption{Computable Convergence to Global Minimum }
+ % \label{tab1}
+ %\end{table}
+
+ \subsection{Finite Precision Reals}
+ We now briefly explain what we mean by this.
+ Consider any real number $x \in \mathbb{R}$. Let $r_0$ be the largest integer such that $r_0 \leq x$. Having chosen $r_0,r_1,\ldots,r_{k-1}$ choose largest positive integer $r_k$ such that
+ \[ r_0 + \frac{r_1}{10} + \frac{r_2}{10^2} + \ldots \frac{r_k}{10^k} \leq x. \]
+
+ This is the decimal expansion of the number. We can check that this expansion is unique. We define precision length to be the number $k$.
+ Now for finite precision representaion of a real we need to specify this precision length $k$. And we say for any real $x \in \mathbb{R}$, the numbers $r_0,r_1,\ldots,r_k$ is its finite precision representation. Note that $r_i, 0\leq i \leq k$ can be zero. %We can consider this finite-precision real number as a natural number by representing it as $r_0*10^k+r_1*10^{k-1}+\ldots+r_k$. For $x \in \mathbb{R}^d$,%all finite-precision lengths of its coordinates.
+ For a point $x \in \mathbb{R}^d$, given a precision length $k$ we can have decimal expansions for all it's co-ordinates.
+ Note that though we give binary representations to the Turing machine, for simplicity we assume precisions denote the decimal precisions.
+
+ \begin{remark}\label{gaprem}
+ Suppose $r_1,\ldots,r_k$ is the finite precision representation with length $k$ of some real $x$. Let $\bar x$ be the number with decimal expansion $r_1,\ldots,r_k$ as $x$ and $r_{l}=9$ for $l \geq k+1$. And let $\underline x$ be the number with decimal expansion $r_1,\ldots,r_k$ as $x$ and $r_{l}=0$ for $l \geq k+1$. %We can see that for all reals in $[\underline x,\bar x]$ we have the same finite length representation.
+ And the length of this interval $[\underline x,\bar x]$ is $\epsilon = 10^{-k}$. We then say with precision length $k$ we can represent consecutive numbers with gap greater than or equal to $\epsilon$.
+ \end{remark}
+
+ \subsection{The Problem}
+ We assume there is an oracle for our continuous function $f$. This oracle gives the value $f(x)$ up to any finite-precision for an given finite-precision $x$. The Turing-machine has access to this function oracle. We give also give a value $\epsilon > 0$ as input to the Turing machine. Our main problem is to write any point $x_{o}$ of the finite precision length such that $|f(x_{o})-f(x*)| < \epsilon$ i.e., it should find $\epsilon-$ approximation of the global optima. We show that this problem is not computable.
+
+ Let us assume we have a three-tape Turing machine, one is used for calculations, second is for the giving the finite precision real and the precision length required to the function oracle and third one has the value returned from the oracle \cite{sorbook}. Note that the third tape can also store the previous values. That is suppose we start with $x_0$ and find $x_1,\ldots,x_{k}$ this tape can store all these and also the corresponding function values obtained from the function oracle $f(x_0),\ldots,f(x_k)$ for finding $x_{k+1}$. We now give definition of the standard Turing machine here:
+
+ \begin{definition}
+ Turing machine has a three infinite tapes divided into cells, a reading head which scans one cell of the tape at a time, and a finite set of internal states $Q=\{q_0,q_1,\ldots,q_n\}, \, n \geq 1$. Each cell is either blank or has symbol 1 written on it. In a single step the machine may simultaneously (1) change the from one state to another; (2) change the scanned symbol $s$ to another symbol $s'\in S = \{1,B\}$; (3) move the reading head one cell to the right (R) or left (L). This operation of machine is controlled by a partial map $\Gamma : Q \times S^3 \rightarrow Q \times (S \times \{R,L\})^3$.
+ \end{definition}
+
+ \begin{remark}
+ The map $\Gamma$ viewed as a finite set of quintuples is called a Turing program. The interpretation is that if $(q,s_1,s_2,s_3,q',s'_1,X_1,s'_2,X_2,s'_3,X_3) \in \Gamma$, in state $q$, scanning symbols $s_1, s_2, s_3$ changes state to $q'$ and in the tape $i$ input symbol to $s'_i$ and moves to scan one square to the right if $X_i=R$ (or left if $X_i=L$.) in the tape $i$.
+ \end{remark}
+
+ We consider this problem in the paper.
+
+ \begin{problem}
+ Given a continuous, nonconvex function $f$, is there a Turing machine with access to the function oracle which can find a $\epsilon-$ approximation to the global optima of the function $f$ ?
+ \end{problem}
+
+ \section{Main Theorem}
+
+ %Given a input $x$ of finite precision length $n$, the oracle can return $f(x)$ to any specified precision.
+ Given the objective function $f$, let the set of global minima be denoted by $G^f$. Now consider $\epsilon$-approximation to the global minima. %We consider finite length approximation of the set $G'_{\epsilon}$.
+
+ \begin{lemma}\label{preclem}
+ For all $\epsilon > 0$ there exists a point $x_n^*$ of finite precision length $n$ such that $|f(x^*)-f(x_n^*)| < \epsilon$.
+ \end{lemma}
+
+ \begin{proof}
+ Let $\delta > 0$ be such that $|x-y| < \delta$ implies $|f(x)-f(y)| < \epsilon$. Such an $\delta > 0$ exists for all $\epsilon > 0$ because the function $f$ is continuous. Let $n$ be the precision length required to represent numbers with gap $\epsilon/10$ between consecutive numbers (Remark \ref{gaprem}). Then we see for the global minima $x^*$ (like for all other points) it's finite precision representation $x_n^*$ with precision length $n$ is such that $|x_n^*-x^*|<\epsilon$.
+ \end{proof}
+
+ \begin{definition}
+ Let $G^f_{\epsilon,k}$ be the set of points with given finite precision length $k \geq 1$ where the function value is $\epsilon > 0$ close to the global minima. And $G^f_{\epsilon}$ be the union of all such sets.
+ \end{definition} % We show that this set $G'_{\epsilon}$ is not limit computable.
+ We consider only finite-precision numbers. %As the domain is a compact set, this can be regarded as a subset of natural numbers by multiplying by an appropriate constant. And in particular
+ As there are only finite number of points with precision length $k$, the set $G^f_{\epsilon,k}$ is finite.
+ Since we would like an algorithm to computably converge to a single point, for simplicity we assume the global optima is unique i.e., $G^f$ is a singleton. This is not uncommon in optimization literature as strict convexity gives unique local (global) minima and is assumed for objective functions. Now we state the main theorem.
+
+ \begin{lemma}\label{checklem}
+ There is no algorithm to check if a point $x_k$ is a $\epsilon-$ approximation to the global optima.
+ \end{lemma}
+
+ \begin{proof}
+ We consider an equivalent problem. We define
+ \begin{equation*}
+ h_{x_k}^{\epsilon}(x) := \max \{0,f(x_k)-f(x)-\epsilon\}.
+ \end{equation*}
+ Since our objective function $f$ is continuous, $h_{x_k}^{\epsilon}(\cdot)$ is also continuous.
+ This function is identically zero if and only if $|f(x_k) - f(x)| < \epsilon$, for all $x$. This happens only if $x_k$ is a $\epsilon-$ approximation to the optimum (See Figure \ref{illusfig}). Note that $x_k$ and $x$ are represented with some finite precision. % then $h_{x_k}^{\epsilon}(\cdot)$ is identically zero. %Or to find a finite-precision point $y$ which is $\epsilon-$ approximation to the optimum is same as finding a function $h_y^{\epsilon}(\cdot)$ which is identically zero for some $y$. But this cannot be checked for any particular value of $y$.
+ \begin{figure}[!ht]
+ \centering
+ % \includegraphics{fig1-crop.pdf}
+ \caption{The figure on the right shows a sample objective function. The figure of the left is the function $ \max \{0,f(60)-f(x) \}$. This function is not identically zero as $x=60$ is not the global minima of $f(x)$.}
+ \label{illusfig}
+ \end{figure}
+
+
+ Thus the set $G^f_{\epsilon}$ of all points with finite precision that are $\epsilon$ close to the global minima, is precisely the set of all points $x_k$ where the function $h_{x_k}^{\epsilon}(\cdot)$ is identically zero.
+ That is,
+ \[ x_k \in G^f_{\epsilon} \Leftrightarrow h_{x_k}^{\epsilon}(x) = 0 \mbox{ for all } x \in C \]
+ But this cannot be checked for a particular $x_k$ unless it is checked for all $x$ of any finite precision length. As there are infinitely many such points, there is no Turing machine which can compute (halt) if a function is zero at infinitely many points.
+ So given any point $x_k$, it is not possible to say if it is an $\epsilon-$ approximation to the global optima or not.
+ \end{proof}
+
+ \begin{theorem}\label{mainthm}
+ We assume the objective function we wish to minimize is known by its oracle. There is no algorithm which can compute the $\epsilon$-approximate optima of a continuous, non-convex objective function on a compact domain.
+ \end{theorem}
+
+ \begin{proof}
+ Let $x'_k$ be any point of some finite precision length $n_k$ such that $|f(x^*)-f(x'_k)| < \epsilon$. Such a point exists by Lemma \ref{preclem} i.e., the set $G^f_{\epsilon,n_k}$ is non-empty for $\epsilon > 0$. Suppose that we have an algorithm to find a $\epsilon/2-$ approximation point $x'_k$.% This length $n_k$ can increase with $k$.
+
+ Now for any point $x_k \in C$ we can say it is $\epsilon$ close to optimum if $|f(x_k)-f(x'_k)| < \epsilon/2$ else it is not. Thus we have an algorithm to check if a point is an $\epsilon-$ approximation to the global optima or not. This is a contradiction to Lemma \ref{checklem}. Thus for a $\epsilon > 0$, there is no algorithm to find a $\epsilon-$ approximation to the global optima.
+
+ %we can not compute any $N$ such that for all $n \geq N$ with $\{x_n\}$ having finite precision, $|f(x_n) - f(x^*)| < \epsilon$.
+ %In other words, we see that there is no computable sequence with finite precision length $\{ x_k \}$ converging to the global optimum (Definition \ref{ccdef}). Thus we have shown the theorem.
+ %there is no sequence of computable $\{x_k\}$ with finite-precision length converging to the global minima.
+ \end{proof}
+
+ \begin{corollary}
+ The finite-precision assumption is not restrictive. If there is an algorithm to find a general real number which is $\epsilon-$ approximation, we can take the first $k$ digits which gives $\epsilon-$ approximation to get a finite precision approximation. %If an algorithm outputs real vectors $\{x_k\}$ converging to the global minimum, then by taking the first $k$ digits of each co-ordinate in $x_k$ it can output a sequence of real vectors with finite-precision length converging to the same point. By contra-positive, if there is no algorithm which can output finite-precision points then there can be no algorithm which outputs any sequence of points converging to the global minimum.
+ \end{corollary}
+
+ %\begin{corollary}
+ % The problem of approximating the global minima of a continuous function by an arbitrary $\epsilon > 0$ is not computable.
+ %\end{corollary}
+
+ \begin{corollary}
+ The problem of checking whether local minima $z$ is global is not computable as this also involves checking whether $h_z^{\epsilon}(\cdot)$ is identically zero.
+ \end{corollary}
+
+ \begin{remark}
+ Even in presence of higher order oracles, i.,e oracles which give derivatives of the function, the equivalent problem of checking if $h_z^{\epsilon}(\cdot)$ is identically zero remains. Hence global optima even in presence of these higher-order function oracle is not computable. %(\#\#\# if bound on gradient is not known \~ Lipschitz)
+ \end{remark}
+
+ \begin{remark}
+ As we mentioned before, our result is for algorithms having access to the function oracle. This is different from the setting of computable function and reals studied in computable analysis. \cite{pour}
+ \end{remark}
+
+ \begin{remark}
+ As we can find an ball of some radius $\omega$ where the continuous function $h_z^{\epsilon}(\cdot)$ is non-zero around a local optima. The same proof does not hold for converging to local optima as we can check if $h_z^{\epsilon}(\cdot)$ is identically zero in steps of size less than $\omega$.
+ \end{remark}
+
+ \section{Global Optima Property}
+ In this section, we see a simple property a function satisfies if the global optima is computable. For this let us first define the set $G^f$ of global optima of the function $f:C\rightarrow \mathbb{R}$, $C\subset \mathbb{R}^d$ as
+ \[ G^f := \{x\,| \, \mbox{ for all } y, \, f(x) \leq f(y), \, x \in C \}.\]
+
+ This set $G^f$ is a subset of $C$. As we are interested in finding only one optima, we call $G^f$ computable if atleast one of it's member is. %We now claim that if the global optima is computable then function must satisfy a property of a particular form.
+
+ \begin{definition}
+ A function $f$ satisfies global property if there is a first order (3-ary) predicate $P_{\zeta}(y,x)$ and a $\zeta \in \mathbb{R}$ such that $P_{\zeta}(y,x)$ is True for all $y,x$ in the domain $C$ of the function $f$.
+ \end{definition}
+
+ We say a property $P_{\zeta}(y,x)$ can be computed if $\zeta$ in the definition can be computed.
+
+ \begin{definition}
+ A set of functions $\mathcal{F}$ satisfies a global property if there is a first order predicate $P_{\zeta}(y,x)$ satisfied by all the functions $\mathcal{F}$.% such that the predicate is True for all $y, x$ in the domain $C$ of the function $f$.
+ \end{definition}
+
+ %We represent True by 1 and False by 0. For example if we let
+ %We can see that this property $Q^f_{x^*}(y)$ defines the global minima and is satisfied by all continuous functions on compact domain.
+
+ %\begin{theorem}[Global Optima Property]
+ % There is a algorithm to converge to the global optima of a set of functions $\mathcal{F}$ if and only if there exists a computable global property $P_{\zeta} (y,x)$ that is satisfied by all the functions in $\mathcal{F}$.% for some computable $\zeta \in \mathbb{R}$. %satisfies the property $P_\zeta (y,x)$.% for all $y$ and $x$ in the domain $C$. Note that here the property $P^f_\zeta (y,x)$ need to be satisfied for all $x \in C$ not just the set $G^f$ of global optima.
+ %\end{theorem}
+
+ \begin{remark}
+ %Here we assume we are finding minima. There could be more than one global minima and we need to find one.
+ If the global minima is computable for a function $f$, we can easily define the $\zeta$ to be the norm of a member of $G^f$ which is computable, say $x^*$ and the global property to be
+ \[ P_{\zeta}(y,x) \mbox{ is True for all } x, \, y \mbox{ if there exists a } x \mbox{ s.t. } \parallel x \parallel = \zeta.\]
+ %\begin{table}[h]
+ % \centering
+ % \begin{tabular}{cl}
+ % \multirow{2}{1.4cm}{$P^f_{\zeta}(y,x) =$} &1 if $\parallel x \parallel = \zeta$ \\ %, \mbox{for some } x^* \in G^f, $ \\
+ % &0 o.w.
+ % \end{tabular}
+ %\end{table}
+ It is clear that if global optima is computable then this $P_{\parallel x^* \parallel}(y,x)$ is satisfied by the function $f$ and $\parallel x^* \parallel$ can be computed.
+
+ %Now for the other direction, assume that global minima is not computable. But there is some property $P_{\zeta}(y,x)$ where one can find a computable $\zeta \in \mathbb{R}$ such that the property is satisfied by the function $f$. Now we look at the definition of global minima and define another global property for $f$ as
+
+ %\[P_{\zeta}(y,x) \mbox{ is True for all } x, y, \zeta \mbox{ if there exists a } x \mbox{ s.t. for all } y, f(x) \leq f(y). \]
+
+ %\begin{table}[h]
+ % \centering
+ % \begin{tabular}{cl}
+ % \multirow{2}{1.1cm}{$P^f_{\varepsilon}(y,x)=$} & 1 if $f(x) \leq f(y), \forall y$ \\
+ % &0 o.w.
+ % \end{tabular}
+ %\end{table}
+
+ %And by assumption we can not find a computable $\zeta$, such that $P_{\zeta}(y,x)$ is satisfied by $f$. But this is a contradiction to the existence of global optima $x^*$, as $P_{\zeta}(y,x^*)$ is satisfied by $f$ for all $\zeta$ and this property can be computed if $x^*$ is computable. Thus we have shown the structure of the global property which must be satisfied for the global optima to be computable.
+ %independent of the value of $\varepsilon$
+ \end{remark}
+
+ \begin{remark}
+ Lipschitz continuity is another example of such a global property. Let $P_L(y,x)$ be
+ \[|f(x)-f(y)| \leq L \parallel x - y \parallel, \, \mbox{ for all } x,y \, \in C.\]
+ And here the number $\zeta$ is the Lipschitz constant $L$. Let the set of functions on some compact domain $C$ satisfying the Lipschitz property be denoted by $\mathcal{L}$. It is known that if the Lipschitz constant or an upper bound to it is known then the global optima for this class of functions $\mathcal{L}$ is computable. (For example refer to Theorem 1.1.2 of \cite{nest}). Another example of a global property is bounded derivatives. If a bound on the gradient is known then the global minima can be computed.
+ \end{remark}
+
+ \section{Conclusion}
+ We have proved that there is no algorithm which finds a $\epsilon-$ approximation to the optima of nonconvex function $f$. This result holds even if the function has higher-order derivatives.
+
+ \bibliographystyle{plain}
+ \bibliography{glosub}
+
+ %\begin{thebibliography}{}
+ %\bibitem{bishop} C.M. Bishop., Pattern Recognition and Machine Learning, Springer (2006).
+
+ %\bibitem{chembook} T.F. Edgar and D.M. Himmelblau., Optimization of Chemical Processes, McGraw Hill (1988).
+
+ %\bibitem{globook} R. Horst and H. Tuy., Global Optimization: Deterministic Approaches, Springer-Verlag (1996).
+
+ %\bibitem{sorbook} R.I. Soare., Turing Computability: Theory and Applications, Springer-Verlag (2016).
+
+ %\bibitem{nest} Y. Nesterov., Introductory Lectures on Convex Optimization A Basic Course, Kluwer Academic Publishers (2003).
+
+ %\bibitem{pour} M. Pour-El and J. Richards., Computability in analysis and physics, Springer, Heidelberg (1989).
+ %\end{thebibliography}
+
+\end{document}
diff --git a/Master/texmf-dist/doc/pdftex/tests/38-interword-space-at-font-switch/compare_pdfs.rb b/Master/texmf-dist/doc/pdftex/tests/38-interword-space-at-font-switch/compare_pdfs.rb
index cea36599790..41a9cd4479d 100755
--- a/Master/texmf-dist/doc/pdftex/tests/38-interword-space-at-font-switch/compare_pdfs.rb
+++ b/Master/texmf-dist/doc/pdftex/tests/38-interword-space-at-font-switch/compare_pdfs.rb
@@ -40,17 +40,17 @@ def convert_pdf_to_png(pdf_path, output_prefix, resolution=150)
Dir["#{output_prefix}-*.png"].length
end
-def create_diff_visualization(image1, image2, diff_output)
+def create_diff_visualization(image1, image2, diff_output_base)
system("magick '(' \"#{image1}\" -flatten -grayscale Rec709Luminance ')' " \
"'(' \"#{image2}\" -flatten -grayscale Rec709Luminance ')' " \
"'(' -clone 0-1 -compose darken -composite ')' " \
- "-channel RGB -combine \"#{diff_output}\"")
+ "-channel RGB -combine \"#{diff_output_base}\"")
end
-def compare_images(image1, image2, diff_output, fuzz=5)
+def compare_images(image1, image2, diff_output_base, fuzz=5)
# First create both types of difference visualizations
- base_name = File.join(File.dirname(diff_output),
- File.basename(diff_output, ".png"))
+ base_name = File.join(File.dirname(diff_output_base),
+ File.basename(diff_output_base))
compare_output = "#{base_name}-compare.png"
visual_output = "#{base_name}-visual.png"
@@ -77,23 +77,23 @@ def compare_all_pages(dir1, dir2, diff_dir, threshold=100, fuzz=5)
page2 = File.join(dir2, File.basename(page1))
if File.exist?(page2)
- diff_output = File.join(diff_dir, "diff-page-#{page_num}.png")
- diff = compare_images(page1, page2, diff_output)
+ diff_output_base = File.join(diff_dir, "diff-page-#{page_num}-")
+ diff = compare_images(page1, page2, diff_output_base)
if diff == -1
puts "Page #{page_num}: Error comparing pages (possibly different dimensions)"
differences[page_num] = "error"
- elsif diff == 0
- puts "Page #{page_num}: Identical"
- File.delete(diff_output) if File.exist?(diff_output)
- differences[page_num] = 0
elsif diff <= threshold
- puts "Page #{page_num}: Minor differences (#{diff} pixels) - below threshold"
- Dir["#{diff_output}-*.png"].each { |f| File.delete(f) if File.exist?(f) }
+ if diff == 0
+ puts "Page #{page_num}: Identical"
+ else
+ puts "Page #{page_num}: Minor differences (#{diff} pixels) - below threshold"
+ end
+ Dir["#{diff_output_base}*"].each { |file| File.delete(file) if File.exist?(file) }
differences[page_num] = 0
else
- base_name = File.join(File.dirname(diff_output),
- File.basename(diff_output, ".png"))
+ base_name = File.join(File.dirname(diff_output_base),
+ File.basename(diff_output_base))
puts "Page #{page_num}: Significant differences (#{diff} pixels)"
puts "Difference images saved as:"
puts " - Compare method: #{base_name}-compare.png"
@@ -108,11 +108,11 @@ def compare_all_pages(dir1, dir2, diff_dir, threshold=100, fuzz=5)
end
# Check for extra pages in second PDF
- Dir["#{dir2}/*.png"].each_with_index do |page2, idx|
- page_num = idx + 1
- unless File.exist?(File.join(dir1, File.basename(page2)))
- puts "Page #{page_num}: Extra page in second PDF"
- differences[page_num] = "extra"
+ Dir["#{dir2}/*.png"].each do |page2|
+ base_name = File.basename(page2)
+ unless File.exist?(File.join(dir1, base_name))
+ puts "Extra page in second PDF: #{base_name}"
+ differences[base_name] = "extra"
end
end
diff --git a/Master/texmf-dist/doc/pdftex/tests/38-interword-space-at-font-switch/f.tex b/Master/texmf-dist/doc/pdftex/tests/38-interword-space-at-font-switch/f.tex
index 7081f32f5d4..ff22e3af90f 100644
--- a/Master/texmf-dist/doc/pdftex/tests/38-interword-space-at-font-switch/f.tex
+++ b/Master/texmf-dist/doc/pdftex/tests/38-interword-space-at-font-switch/f.tex
@@ -1,23 +1,22 @@
\input plain.tex
\input pdftexconfig.tex
\input pdftexmagfix.tex
+
\pdfcompresslevel=0
\pdfobjcompresslevel=0
-\pdftracingfonts=1
-
\pdfinterwordspaceon
-% \font\f=cmr10 at 10pt
-% \pdffontexpand\f 20 20 10 autoexpand
-% \pdfadjustspacing=2
-% \f
+\font\f=cmr10 at 10pt
+\pdffontexpand\f 20 20 10 autoexpand
+\pdfadjustspacing=2
+\f
text {\bf text} text
-% Lorem ipsum dolor sit amet, consectetuer adipiscing elit
-% Lorem ipsum dolor sit amet, consectetuer adipiscing elit
+Lorem ipsum dolor sit amet, consectetuer adipiscing elit
+Lorem ipsum dolor sit amet, consectetuer adipiscing elit
-% {\bf \TeX{} Web Resources} The web page
-% {\tt https://tug.org/interest.html} has many \TeX{}-
+{\bf \TeX{} Web Resources} The web page
+{\tt https://tug.org/interest.html} has many \TeX{}-
\bye
diff --git a/Master/texmf-dist/doc/pdftex/tests/38-interword-space-at-font-switch/f2.tex b/Master/texmf-dist/doc/pdftex/tests/38-interword-space-at-font-switch/f2.tex
index c014af07620..449f767a27d 100644
--- a/Master/texmf-dist/doc/pdftex/tests/38-interword-space-at-font-switch/f2.tex
+++ b/Master/texmf-dist/doc/pdftex/tests/38-interword-space-at-font-switch/f2.tex
@@ -1,10 +1,11 @@
\documentclass[12pt]{article}
\RequirePackage[a4paper,lines=47,hmargin=2.6cm,vmarginratio=5:4]{geometry}
-\usepackage{microtype}
+\usepackage{microtype}
\begin{document}
\pdfcompresslevel=0
\pdfobjcompresslevel=0
+\pdfinterwordspaceon
\begin{description}
\item[\TeX{} Web Resources] The web page
diff --git a/Master/texmf-dist/doc/pdftex/tests/38-interword-space-at-font-switch/f3.tex b/Master/texmf-dist/doc/pdftex/tests/38-interword-space-at-font-switch/f3.tex
new file mode 100644
index 00000000000..2da6cd6059b
--- /dev/null
+++ b/Master/texmf-dist/doc/pdftex/tests/38-interword-space-at-font-switch/f3.tex
@@ -0,0 +1,13 @@
+\documentclass{article}
+
+\usepackage{microtype}
+\begin{document}
+\pdfcompresslevel=0
+\pdfobjcompresslevel=0
+\pdfinterwordspaceon
+
+a\pdffakespace b
+
+c d \pdffakespace e
+\end{document}
+
diff --git a/Master/texmf-dist/doc/pdftex/tests/38-interword-space-at-font-switch/from-latex-tagged-pdf/arXiv-2401.05361v1.tex b/Master/texmf-dist/doc/pdftex/tests/38-interword-space-at-font-switch/from-latex-tagged-pdf/arXiv-2401.05361v1.tex
index 46284e52b8c..007d825dd9d 100644
--- a/Master/texmf-dist/doc/pdftex/tests/38-interword-space-at-font-switch/from-latex-tagged-pdf/arXiv-2401.05361v1.tex
+++ b/Master/texmf-dist/doc/pdftex/tests/38-interword-space-at-font-switch/from-latex-tagged-pdf/arXiv-2401.05361v1.tex
@@ -36,7 +36,12 @@
\author{Herbert Batte$^{1,*} $}
\date{}
+\usepackage{microtype}
\begin{document}
+\pdfcompresslevel=0
+\pdfobjcompresslevel=0
+\pdfinterwordspaceon
+
\maketitle
\abstract{ Let $ \{L_n\}_{n\geq 0} $ be the sequence of Lucas numbers. In this paper, we determine all Lucas numbers that are palindromic concatenations of two distinct repdigits. }
@@ -689,4 +694,4 @@ for d1 in range(1, 10):
print("Maximum ep across all combinations:", max_ep)
print("Maximum log_expr_a across all combinations:", max_log_expr_a)
\end{verbatim}
-\end{document} \ No newline at end of file
+\end{document}
diff --git a/Master/texmf-dist/doc/pdftex/tests/38-interword-space-at-font-switch/from-latex-tagged-pdf/glosuba.tex b/Master/texmf-dist/doc/pdftex/tests/38-interword-space-at-font-switch/from-latex-tagged-pdf/glosuba.tex
index ac3531e5886..7e8777e9a14 100644
--- a/Master/texmf-dist/doc/pdftex/tests/38-interword-space-at-font-switch/from-latex-tagged-pdf/glosuba.tex
+++ b/Master/texmf-dist/doc/pdftex/tests/38-interword-space-at-font-switch/from-latex-tagged-pdf/glosuba.tex
@@ -171,7 +171,7 @@
This function is identically zero if and only if $|f(x_k) - f(x)| < \epsilon$, for all $x$. This happens only if $x_k$ is a $\epsilon-$ approximation to the optimum (See Figure \ref{illusfig}). Note that $x_k$ and $x$ are represented with some finite precision. % then $h_{x_k}^{\epsilon}(\cdot)$ is identically zero. %Or to find a finite-precision point $y$ which is $\epsilon-$ approximation to the optimum is same as finding a function $h_y^{\epsilon}(\cdot)$ which is identically zero for some $y$. But this cannot be checked for any particular value of $y$.
\begin{figure}[!ht]
\centering
- \includegraphics{fig1-crop.pdf}
+ % \includegraphics{fig1-crop.pdf}
\caption{The figure on the right shows a sample objective function. The figure of the left is the function $ \max \{0,f(60)-f(x) \}$. This function is not identically zero as $x=60$ is not the global minima of $f(x)$.}
\label{illusfig}
\end{figure}
diff --git a/Master/texmf-dist/doc/pdftex/tests/38-interword-space-at-font-switch/sample2e.tex b/Master/texmf-dist/doc/pdftex/tests/38-interword-space-at-font-switch/sample2e.tex
index d30fe91b8a6..9333381c701 100644
--- a/Master/texmf-dist/doc/pdftex/tests/38-interword-space-at-font-switch/sample2e.tex
+++ b/Master/texmf-dist/doc/pdftex/tests/38-interword-space-at-font-switch/sample2e.tex
@@ -4,7 +4,6 @@
% and is used for comments like this one.
\documentclass{article} % Specifies the document class
-\usepackage{microtype}
% The preamble begins here.
\title{An Example Document} % Declares the document's title.
@@ -19,7 +18,8 @@
% This is an alternative definition of
% \ip that is commented out.
-\begin{document} % End of preamble and beginning of text.
+\usepackage{microtype}
+\begin{document}
\pdfcompresslevel=0
\pdfobjcompresslevel=0
\pdfinterwordspaceon
diff --git a/Master/texmf-dist/doc/pdftex/tests/38-interword-space-at-font-switch/texlive-en.tex b/Master/texmf-dist/doc/pdftex/tests/38-interword-space-at-font-switch/texlive-en.tex
index bd352a5ffca..4714d71df33 100644
--- a/Master/texmf-dist/doc/pdftex/tests/38-interword-space-at-font-switch/texlive-en.tex
+++ b/Master/texmf-dist/doc/pdftex/tests/38-interword-space-at-font-switch/texlive-en.tex
@@ -24,6 +24,7 @@
\pdfcompresslevel=0
\pdfobjcompresslevel=0
\pdfinterwordspaceon
+
\maketitle
\begin{multicols}{2}