From 25c948597e45fc74ddbfc9c480326f300c747d30 Mon Sep 17 00:00:00 2001 From: Antti H S Laaksonen Date: Sun, 26 Feb 2017 13:10:29 +0200 Subject: [PATCH] Some fixes --- chapter21.tex | 34 +++++++++++++++++++++++----------- chapter22.tex | 23 +++++++++++++---------- chapter29.tex | 14 +++++++------- list.tex | 34 +++++++++++++++++----------------- 4 files changed, 60 insertions(+), 45 deletions(-) diff --git a/chapter21.tex b/chapter21.tex index c49f96f..7f2d221 100644 --- a/chapter21.tex +++ b/chapter21.tex @@ -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 diff --git a/chapter22.tex b/chapter22.tex index bc9ba31..676f857 100644 --- a/chapter22.tex +++ b/chapter22.tex @@ -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. diff --git a/chapter29.tex b/chapter29.tex index 5df5e45..d83f0c8 100644 --- a/chapter29.tex +++ b/chapter29.tex @@ -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 diff --git a/list.tex b/list.tex index be0c952..9cf1518 100644 --- a/list.tex +++ b/list.tex @@ -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} \ No newline at end of file