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Some fixes

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Antti H S Laaksonen 2017-02-26 13:10:29 +02:00
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34
chapter21.tex
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@ -205,7 +205,9 @@ so the result of the function is $[2,2,2,3]$.
\index{sieve of Eratosthenes} \index{sieve of Eratosthenes}
The \key{sieve of Eratosthenes}\footnote{Eratosthenes (c. 276 BC -- c. 194 BC) was a Greek mathematician.} is a preprocessing The \key{sieve of Eratosthenes}
%\footnote{Eratosthenes (c. 276 BC -- c. 194 BC) was a Greek mathematician.}
is a preprocessing
algorithm that builds an array using which we algorithm that builds an array using which we
can efficiently check if a given number between $2 \ldots n$ can efficiently check if a given number between $2 \ldots n$
is prime and, if it is not, find one prime factor of the number. is prime and, if it is not, find one prime factor of the number.
@ -355,8 +357,8 @@ For example,
Numbers $a$ and $b$ are \key{coprime} Numbers $a$ and $b$ are \key{coprime}
if $\textrm{gcd}(a,b)=1$. if $\textrm{gcd}(a,b)=1$.
\key{Euler's totient function} $\varphi(n)$\footnote{Euler \key{Euler's totient function} $\varphi(n)$
presented this function in 1763.} %\footnote{Euler presented this function in 1763.}
gives the number of coprime numbers to $n$ gives the number of coprime numbers to $n$
between $1$ and $n$. between $1$ and $n$.
For example, $\varphi(12)=4$, For example, $\varphi(12)=4$,
@ -434,12 +436,16 @@ int modpow(int x, int n, int m) {
\index{Fermat's theorem} \index{Fermat's theorem}
\index{Euler's theorem} \index{Euler's theorem}
\key{Fermat's theorem}\footnote{Fermat discovered this theorem in 1640.} states that \key{Fermat's theorem}
%\footnote{Fermat discovered this theorem in 1640.}
states that
\[x^{m-1} \bmod m = 1\] \[x^{m-1} \bmod m = 1\]
when $m$ is prime and $x$ and $m$ are coprime. when $m$ is prime and $x$ and $m$ are coprime.
This also yields This also yields
\[x^k \bmod m = x^{k \bmod (m-1)} \bmod m.\] \[x^k \bmod m = x^{k \bmod (m-1)} \bmod m.\]
More generally, \key{Euler's theorem}\footnote{Euler published this theorem in 1763.} states that More generally, \key{Euler's theorem}
%\footnote{Euler published this theorem in 1763.}
states that
\[x^{\varphi(m)} \bmod m = 1\] \[x^{\varphi(m)} \bmod m = 1\]
when $x$ and $m$ are coprime. when $x$ and $m$ are coprime.
Fermat's theorem follows from Euler's theorem, Fermat's theorem follows from Euler's theorem,
@ -519,8 +525,9 @@ cout << x*x << "\n"; // 2537071545
\index{Diophantine equation} \index{Diophantine equation}
A \key{Diophantine equation}\footnote{Diophantus of Alexandria was a Greek A \key{Diophantine equation}
mathematician who lived in the 3th century.} is an equation of the form %\footnote{Diophantus of Alexandria was a Greek mathematician who lived in the 3th century.}
is an equation of the form
\[ ax + by = c, \] \[ ax + by = c, \]
where $a$, $b$ and $c$ are constants where $a$, $b$ and $c$ are constants
and we should find the values of $x$ and $y$. and we should find the values of $x$ and $y$.
@ -640,7 +647,9 @@ are solutions.
\index{Lagrange's theorem} \index{Lagrange's theorem}
\key{Lagrange's theorem}\footnote{J.-L. Lagrange (1736--1813) was an Italian mathematician.} states that every positive integer \key{Lagrange's theorem}
%\footnote{J.-L. Lagrange (1736--1813) was an Italian mathematician.}
states that every positive integer
can be represented as a sum of four squares, i.e., can be represented as a sum of four squares, i.e.,
$a^2+b^2+c^2+d^2$. $a^2+b^2+c^2+d^2$.
For example, the number 123 can be represented For example, the number 123 can be represented
@ -651,8 +660,9 @@ as the sum $8^2+5^2+5^2+3^2$.
\index{Zeckendorf's theorem} \index{Zeckendorf's theorem}
\index{Fibonacci number} \index{Fibonacci number}
\key{Zeckendorf's theorem}\footnote{E. Zeckendorf published the theorem in 1972 \cite{zec72}; \key{Zeckendorf's theorem}
however, this was not a new result.} states that every %\footnote{E. Zeckendorf published the theorem in 1972 \cite{zec72}; however, this was not a new result.}
states that every
positive integer has a unique representation positive integer has a unique representation
as a sum of Fibonacci numbers such that as a sum of Fibonacci numbers such that
no two numbers are equal or consecutive no two numbers are equal or consecutive
@ -693,7 +703,9 @@ produces the smallest Pythagorean triple
\index{Wilson's theorem} \index{Wilson's theorem}
\key{Wilson's theorem}\footnote{J. Wilson (1741--1793) was an English mathematician.} states that a number $n$ \key{Wilson's theorem}
%\footnote{J. Wilson (1741--1793) was an English mathematician.}
states that a number $n$
is prime exactly when is prime exactly when
\[(n-1)! \bmod n = n-1.\] \[(n-1)! \bmod n = n-1.\]
For example, the number 11 is prime, because For example, the number 11 is prime, because

23
chapter22.tex
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@ -342,8 +342,9 @@ corresponds to the binomial coefficient formula.
\index{Catalan number} \index{Catalan number}
The \key{Catalan number}\footnote{E. C. Catalan (1814--1894) The \key{Catalan number}
was a Belgian mathematician.} $C_n$ equals the %\footnote{E. C. Catalan (1814--1894) was a Belgian mathematician.}
$C_n$ equals the
number of valid number of valid
parenthesis expressions that consist of parenthesis expressions that consist of
$n$ left parentheses and $n$ right parentheses. $n$ left parentheses and $n$ right parentheses.
@ -679,8 +680,9 @@ elements should be changed.
\index{Burnside's lemma} \index{Burnside's lemma}
\key{Burnside's lemma}\footnote{Actually, Burnside did not discover this lemma; \key{Burnside's lemma}
he only mentioned it in his book \cite{bur97}.} can be used to count %\footnote{Actually, Burnside did not discover this lemma; he only mentioned it in his book \cite{bur97}.}
can be used to count
the number of combinations so that the number of combinations so that
only one representative is counted only one representative is counted
for each group of symmetric combinations. for each group of symmetric combinations.
@ -766,10 +768,10 @@ with 3 colors is
\index{Cayley's formula} \index{Cayley's formula}
\key{Cayley's formula}\footnote{While the formula \key{Cayley's formula}
is named after A. Cayley, % \footnote{While the formula is named after A. Cayley,
who studied it in 1889, % who studied it in 1889, it was discovered earlier by C. W. Borchardt in 1860.}
it was discovered earlier by C. W. Borchardt in 1860.} states that states that
there are $n^{n-2}$ labeled trees there are $n^{n-2}$ labeled trees
that contain $n$ nodes. that contain $n$ nodes.
The nodes are labeled $1,2,\ldots,n$, The nodes are labeled $1,2,\ldots,n$,
@ -832,8 +834,9 @@ be derived using Prüfer codes.
\index{Prüfer code} \index{Prüfer code}
A \key{Prüfer code}\footnote{In 1918, H. Prüfer proved A \key{Prüfer code}
Cayley's theorem using Prüfer codes \cite{pru18}.} is a sequence of %\footnote{In 1918, H. Prüfer proved Cayley's theorem using Prüfer codes \cite{pru18}.}
is a sequence of
$n-2$ numbers that describes a labeled tree. $n-2$ numbers that describes a labeled tree.
The code is constructed by following a process The code is constructed by following a process
that removes $n-2$ leaves from the tree. that removes $n-2$ leaves from the tree.

14
chapter29.tex
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@ -14,7 +14,7 @@ and our task is to calculate its area.
For example, a possible input for the problem is as follows: For example, a possible input for the problem is as follows:
\begin{center} \begin{center}
\begin{tikzpicture}[scale=0.44] \begin{tikzpicture}[scale=0.45]
\draw[fill] (6,2) circle [radius=0.1]; \draw[fill] (6,2) circle [radius=0.1];
\draw[fill] (5,6) circle [radius=0.1]; \draw[fill] (5,6) circle [radius=0.1];
@ -27,7 +27,7 @@ One way to approach the problem is to divide
the quadrilateral into two triangles by a straight the quadrilateral into two triangles by a straight
line between two opposite vertices: line between two opposite vertices:
\begin{center} \begin{center}
\begin{tikzpicture}[scale=0.44] \begin{tikzpicture}[scale=0.45]
\draw[fill] (6,2) circle [radius=0.1]; \draw[fill] (6,2) circle [radius=0.1];
\draw[fill] (5,6) circle [radius=0.1]; \draw[fill] (5,6) circle [radius=0.1];
@ -41,8 +41,8 @@ line between two opposite vertices:
After this, it suffices to sum the areas After this, it suffices to sum the areas
of the triangles. of the triangles.
The area of a triangle can be calculated, The area of a triangle can be calculated,
for example, using \key{Heron's formula}\footnote{Heron of Alexandria for example, using \key{Heron's formula}
(c. 10--70) was a Greek mathematician.} %\footnote{Heron of Alexandria (c. 10--70) was a Greek mathematician.}
\[ \sqrt{s (s-a) (s-b) (s-c)},\] \[ \sqrt{s (s-a) (s-b) (s-c)},\]
where $a$, $b$ and $c$ are the lengths where $a$, $b$ and $c$ are the lengths
of the triangle's sides and of the triangle's sides and
@ -57,7 +57,7 @@ two arbitrary opposite vertices.
For example, in the following situation, For example, in the following situation,
the division line is outside the quadrilateral: the division line is outside the quadrilateral:
\begin{center} \begin{center}
\begin{tikzpicture}[scale=0.44] \begin{tikzpicture}[scale=0.45]
\draw[fill] (6,2) circle [radius=0.1]; \draw[fill] (6,2) circle [radius=0.1];
\draw[fill] (3,2) circle [radius=0.1]; \draw[fill] (3,2) circle [radius=0.1];
@ -70,7 +70,7 @@ the division line is outside the quadrilateral:
\end{center} \end{center}
However, another way to draw the line works: However, another way to draw the line works:
\begin{center} \begin{center}
\begin{tikzpicture}[scale=0.44] \begin{tikzpicture}[scale=0.45]
\draw[fill] (6,2) circle [radius=0.1]; \draw[fill] (6,2) circle [radius=0.1];
\draw[fill] (3,2) circle [radius=0.1]; \draw[fill] (3,2) circle [radius=0.1];
@ -573,7 +573,7 @@ along the boundary of the polygon.
\index{Pick's theorem} \index{Pick's theorem}
\key{Pick's theorem} \cite{pic99} provides another way to calculate \key{Pick's theorem} provides another way to calculate
the area of a polygon provided that all vertices the area of a polygon provided that all vertices
of the polygon have integer coordinates. of the polygon have integer coordinates.
According to Pick's theorem, the area of the polygon is According to Pick's theorem, the area of the polygon is

34
list.tex
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@ -45,10 +45,10 @@
Nim, a game with a complete mathematical theory. Nim, a game with a complete mathematical theory.
\emph{Annals of Mathematics}, 3(1/4):35--39, 1901. \emph{Annals of Mathematics}, 3(1/4):35--39, 1901.
\bibitem{bur97} % \bibitem{bur97}
W. Burnside. % W. Burnside.
\emph{Theory of Groups of Finite Order}, % \emph{Theory of Groups of Finite Order},
Cambridge University Press, 1897. % Cambridge University Press, 1897.
\bibitem{cod15} \bibitem{cod15}
Codeforces: On ''Mo's algorithm'', Codeforces: On ''Mo's algorithm'',
@ -241,21 +241,21 @@
Where to use and how not to use polynomial string hashing. Where to use and how not to use polynomial string hashing.
\emph{Olympiads in Informatics}, 7(1):90--100, 2013. \emph{Olympiads in Informatics}, 7(1):90--100, 2013.
\bibitem{pic99} % \bibitem{pic99}
G. Pick. % G. Pick.
Geometrisches zur Zahlenlehre. % Geometrisches zur Zahlenlehre.
\emph{Sitzungsberichte des deutschen naturwissenschaftlich-medicinischen Vereines % \emph{Sitzungsberichte des deutschen naturwissenschaftlich-medicinischen Vereines
für Böhmen "Lotos" in Prag. (Neue Folge)}, 19:311--319, 1899. % für Böhmen "Lotos" in Prag. (Neue Folge)}, 19:311--319, 1899.
\bibitem{pri57} \bibitem{pri57}
R. C. Prim. R. C. Prim.
Shortest connection networks and some generalizations. Shortest connection networks and some generalizations.
\emph{Bell System Technical Journal}, 36(6):1389--1401, 1957. \emph{Bell System Technical Journal}, 36(6):1389--1401, 1957.
\bibitem{pru18} % \bibitem{pru18}
H. Prüfer. % H. Prüfer.
Neuer Beweis eines Satzes über Permutationen. % Neuer Beweis eines Satzes über Permutationen.
\emph{Arch. Math. Phys}, 27:742--744, 1918. % \emph{Arch. Math. Phys}, 27:742--744, 1918.
\bibitem{q27} \bibitem{q27}
27-Queens Puzzle: Massively Parallel Enumeration and Solution Counting. 27-Queens Puzzle: Massively Parallel Enumeration and Solution Counting.
@ -306,9 +306,9 @@
\emph{Des Rösselsprunges einfachste und allgemeinste Lösung}. \emph{Des Rösselsprunges einfachste und allgemeinste Lösung}.
Schmalkalden, 1823. Schmalkalden, 1823.
\bibitem{zec72} % \bibitem{zec72}
E. Zeckendorf. % E. Zeckendorf.
Représentation des nombres naturels par une somme de nombres de Fibonacci ou de nombres de Lucas. % Représentation des nombres naturels par une somme de nombres de Fibonacci ou de nombres de Lucas.
\emph{Bull. Soc. Roy. Sci. Liege}, 41:179--182, 1972. % \emph{Bull. Soc. Roy. Sci. Liege}, 41:179--182, 1972.
\end{thebibliography} \end{thebibliography}