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Some fixes

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Antti H S Laaksonen 2017-02-26 13:10:29 +02:00
parent 8a936ed246
commit 25c948597e
4 changed files with 60 additions and 45 deletions
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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}
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
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.
@ -355,8 +357,8 @@ For example,
Numbers $a$ and $b$ are \key{coprime}
if $\textrm{gcd}(a,b)=1$.
\key{Euler's totient function} $\varphi(n)$\footnote{Euler
presented this function in 1763.}
\key{Euler's totient function} $\varphi(n)$
%\footnote{Euler presented this function in 1763.}
gives the number of coprime numbers to $n$
between $1$ and $n$.
For example, $\varphi(12)=4$,
@ -434,12 +436,16 @@ int modpow(int x, int n, int m) {
\index{Fermat'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
\[x^{m-1} \bmod m = 1\]
when $m$ is prime and $x$ and $m$ are coprime.
This also yields
\[x^k \bmod m = x^{k \bmod (m-1)} \bmod m.\]
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
\[x^{\varphi(m)} \bmod m = 1\]
when $x$ and $m$ are coprime.
Fermat's theorem follows from Euler's theorem,
@ -519,8 +525,9 @@ cout << x*x << "\n"; // 2537071545
\index{Diophantine equation}
A \key{Diophantine equation}\footnote{Diophantus of Alexandria was a Greek
mathematician who lived in the 3th century.} is an equation of the form
A \key{Diophantine equation}
%\footnote{Diophantus of Alexandria was a Greek mathematician who lived in the 3th century.}
is an equation of the form
\[ ax + by = c, \]
where $a$, $b$ and $c$ are constants
and we should find the values of $x$ and $y$.
@ -640,7 +647,9 @@ are solutions.
\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.,
$a^2+b^2+c^2+d^2$.
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{Fibonacci number}
\key{Zeckendorf's theorem}\footnote{E. Zeckendorf published the theorem in 1972 \cite{zec72};
however, this was not a new result.} states that every
\key{Zeckendorf's theorem}
%\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
as a sum of Fibonacci numbers such that
no two numbers are equal or consecutive
@ -693,7 +703,9 @@ produces the smallest Pythagorean triple
\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
\[(n-1)! \bmod n = n-1.\]
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}
The \key{Catalan number}\footnote{E. C. Catalan (1814--1894)
was a Belgian mathematician.} $C_n$ equals the
The \key{Catalan number}
%\footnote{E. C. Catalan (1814--1894) was a Belgian mathematician.}
$C_n$ equals the
number of valid
parenthesis expressions that consist of
$n$ left parentheses and $n$ right parentheses.
@ -679,8 +680,9 @@ elements should be changed.
\index{Burnside's lemma}
\key{Burnside's lemma}\footnote{Actually, Burnside did not discover this lemma;
he only mentioned it in his book \cite{bur97}.} can be used to count
\key{Burnside's lemma}
%\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
only one representative is counted
for each group of symmetric combinations.
@ -766,10 +768,10 @@ with 3 colors is
\index{Cayley's formula}
\key{Cayley's formula}\footnote{While the formula
is named after A. Cayley,
who studied it in 1889,
it was discovered earlier by C. W. Borchardt in 1860.} states that
\key{Cayley's formula}
% \footnote{While the formula is named after A. Cayley,
% who studied it in 1889, it was discovered earlier by C. W. Borchardt in 1860.}
states that
there are $n^{n-2}$ labeled trees
that contain $n$ nodes.
The nodes are labeled $1,2,\ldots,n$,
@ -832,8 +834,9 @@ be derived using Prüfer codes.
\index{Prüfer code}
A \key{Prüfer code}\footnote{In 1918, H. Prüfer proved
Cayley's theorem using Prüfer codes \cite{pru18}.} is a sequence of
A \key{Prüfer code}
%\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.
The code is constructed by following a process
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:
\begin{center}
\begin{tikzpicture}[scale=0.44]
\begin{tikzpicture}[scale=0.45]
\draw[fill] (6,2) 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
line between two opposite vertices:
\begin{center}
\begin{tikzpicture}[scale=0.44]
\begin{tikzpicture}[scale=0.45]
\draw[fill] (6,2) 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
of the triangles.
The area of a triangle can be calculated,
for example, using \key{Heron's formula}\footnote{Heron of Alexandria
(c. 10--70) was a Greek mathematician.}
for example, using \key{Heron's formula}
%\footnote{Heron of Alexandria (c. 10--70) was a Greek mathematician.}
\[ \sqrt{s (s-a) (s-b) (s-c)},\]
where $a$, $b$ and $c$ are the lengths
of the triangle's sides and
@ -57,7 +57,7 @@ two arbitrary opposite vertices.
For example, in the following situation,
the division line is outside the quadrilateral:
\begin{center}
\begin{tikzpicture}[scale=0.44]
\begin{tikzpicture}[scale=0.45]
\draw[fill] (6,2) 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}
However, another way to draw the line works:
\begin{center}
\begin{tikzpicture}[scale=0.44]
\begin{tikzpicture}[scale=0.45]
\draw[fill] (6,2) 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}
\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
of the polygon have integer coordinates.
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.
\emph{Annals of Mathematics}, 3(1/4):35--39, 1901.
\bibitem{bur97}
W. Burnside.
\emph{Theory of Groups of Finite Order},
Cambridge University Press, 1897.
% \bibitem{bur97}
% W. Burnside.
% \emph{Theory of Groups of Finite Order},
% Cambridge University Press, 1897.
\bibitem{cod15}
Codeforces: On ''Mo's algorithm'',
@ -241,21 +241,21 @@
Where to use and how not to use polynomial string hashing.
\emph{Olympiads in Informatics}, 7(1):90--100, 2013.
\bibitem{pic99}
G. Pick.
Geometrisches zur Zahlenlehre.
\emph{Sitzungsberichte des deutschen naturwissenschaftlich-medicinischen Vereines
für Böhmen "Lotos" in Prag. (Neue Folge)}, 19:311--319, 1899.
% \bibitem{pic99}
% G. Pick.
% Geometrisches zur Zahlenlehre.
% \emph{Sitzungsberichte des deutschen naturwissenschaftlich-medicinischen Vereines
% für Böhmen "Lotos" in Prag. (Neue Folge)}, 19:311--319, 1899.
\bibitem{pri57}
R. C. Prim.
Shortest connection networks and some generalizations.
\emph{Bell System Technical Journal}, 36(6):1389--1401, 1957.
\bibitem{pru18}
H. Prüfer.
Neuer Beweis eines Satzes über Permutationen.
\emph{Arch. Math. Phys}, 27:742--744, 1918.
% \bibitem{pru18}
% H. Prüfer.
% Neuer Beweis eines Satzes über Permutationen.
% \emph{Arch. Math. Phys}, 27:742--744, 1918.
\bibitem{q27}
27-Queens Puzzle: Massively Parallel Enumeration and Solution Counting.
@ -306,9 +306,9 @@
\emph{Des Rösselsprunges einfachste und allgemeinste Lösung}.
Schmalkalden, 1823.
\bibitem{zec72}
E. Zeckendorf.
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.
% \bibitem{zec72}
% E. Zeckendorf.
% 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.
\end{thebibliography}