├── .DS_Store ├── ._wordcount_selection.tex ├── .gitattributes ├── .vscode ├── ltex.dictionary.en-US.txt ├── ltex.disabledRules.en-US.txt └── ltex.hiddenFalsePositives.en-US.txt ├── Images ├── R^2.png ├── R^3.png ├── agm3D.png ├── agm3dsect.png ├── agmx=0.png ├── agmx=5.png ├── background.pdf ├── branchingtree.png ├── chap1.png ├── chap2.png ├── colvec.png ├── complexplane.png ├── complexvector.png ├── condprop.jpg ├── conjugate.png ├── converttab.png ├── creodocs_logo.pdf ├── directedvec.png ├── func.jpg ├── function&relation.png ├── intfunc.png ├── locus1.png ├── mtxvis.png ├── orange1.jpg ├── orange2.jpg ├── orange3.jpg ├── paralaw.png ├── polar2.png ├── polarcord.png ├── quadratic.png ├── rootdemoivre.png ├── time complexity.png ├── triangle2.png ├── triangleablt0.png ├── v2bintree.png ├── venn.png └── x+y.png ├── LegrandOrangeBook.cls ├── License ├── README.md ├── Visualization ├── ._wordcount_selection.tex ├── Graph.nb ├── visualization.ipynb └── visualization.nb ├── content ├── Part2 Discrete math │ ├── ._wordcount_selection.tex │ ├── 1.Boolean Algebra.tex │ ├── 2.Number Theory.tex │ ├── 3.Relation and Order.tex │ ├── 4.Graph Theory.tex │ ├── 5.Basics of Abstract Algebra.tex │ └── 6.Introductory Topology and Category Theory.tex ├── Part3 Single-variable Calculus │ ├── Chapter1.tex │ ├── Chapter2.tex │ ├── Chapter3.tex │ ├── Chapter4.tex │ ├── Chapter5.tex │ ├── Chapter6.tex │ └── Chapter7.tex ├── Part4 Multi-variable Calculus │ └── Chapter1.tex ├── Part5 Linear Algebra │ └── Chapter1.tex ├── Part6 Probability and Combinatorics │ ├── ._wordcount_selection.tex │ ├── 1.Counting and Probability Axioms.tex │ ├── 2.Conditional Probability and Independence.tex │ ├── 3.Discrete Distribution and Random Variable.tex │ ├── 4.Continuous Distribution.tex │ ├── 5.Joint Cumulative Distribution.tex │ ├── 6.Limit Theory in Probability.tex │ └── 7.Stochastic Process.tex ├── Part7 Statistics │ └── Chapter1.tex ├── Part8 Basic Information Theory │ └── Chapter1.tex └── part1 Intro │ ├── ._wordcount_selection.tex │ ├── 1.proof and proposition.tex │ ├── 2.Set, Sequence, Function, and Summation.tex │ ├── 3.Algorithm and Number System.tex │ ├── 4.Inequality.tex │ └── 5.Complex Number.tex ├── indexstyle.ist ├── main.aux ├── main.bbl ├── main.bbl-SAVE-ERROR ├── main.bcf ├── main.blg ├── main.dvi ├── main.fdb_latexmk ├── main.fls ├── main.idx ├── main.ilg ├── main.ind ├── main.lof ├── main.log ├── main.lot ├── main.out.ps ├── main.pdf ├── main.ptc ├── main.run.xml ├── main.synctex.gz ├── main.tex ├── main.toc ├── numbers.svg └── sample.bib /.DS_Store: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/Eryc123Y/Math-for-Computer-Science/2f6872b17ff96eafd1f0aea454e25d18c6cd473c/.DS_Store -------------------------------------------------------------------------------- /._wordcount_selection.tex: -------------------------------------------------------------------------------- 1 | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -------------------------------------------------------------------------------- /.gitattributes: -------------------------------------------------------------------------------- 1 | # Auto detect text files and perform LF normalization 2 | * text=auto 3 | -------------------------------------------------------------------------------- /.vscode/ltex.dictionary.en-US.txt: -------------------------------------------------------------------------------- 1 | surjective 2 | Xingyu 3 | Xingyu 4 | cyclicity 5 | Surjective 6 | codomain 7 | nonnegative 8 | Mathematica 9 | Eq 10 | textbfSolution 11 | Moivre 12 | nD 13 | addants 14 | summand 15 | Monotonicity 16 | Normals 17 | Extrema 18 | Rolle 19 | sublist 20 | Bezout 21 | coprime 22 | Diophantus 23 | subproblem 24 | gcd 25 | congruences 26 | solutionStyle 27 | LinearSearch 28 | moduli 29 | XNOR 30 | Minterms 31 | maxterms 32 | subfunctions 33 | minterm 34 | minterms 35 | maxterm 36 | surjections 37 | surjection 38 | Extensionality 39 | Dom 40 | poset 41 | irreflective 42 | -------------------------------------------------------------------------------- /.vscode/ltex.disabledRules.en-US.txt: -------------------------------------------------------------------------------- 1 | UPPERCASE_SENTENCE_START 2 | THE_SUPERLATIVE 3 | EN_UNPAIRED_BRACKETS 4 | SENT_START_CONJUNCTIVE_LINKING_ADVERB_COMMA 5 | COMMA_PARENTHESIS_WHITESPACE 6 | ENGLISH_WORD_REPEAT_RULE 7 | ENGLISH_WORD_REPEAT_BEGINNING_RULE 8 | IF_IS 9 | IN_A_X_MANNER 10 | ON_ADDITION 11 | MORFOLOGIK_RULE_EN_US 12 | LETS_LET 13 | -------------------------------------------------------------------------------- /.vscode/ltex.hiddenFalsePositives.en-US.txt: -------------------------------------------------------------------------------- 1 | {"rule":"NOW","sentence":"^\\QThere is no any overlap between graphs except the intersection \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q.\\E$"} 2 | 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For 435 | the avoidance of doubt, this paragraph does not form part of the 436 | public licenses. 437 | 438 | Creative Commons may be contacted at creativecommons.org. 439 | -------------------------------------------------------------------------------- /README.md: -------------------------------------------------------------------------------- 1 | # Math for CS 2 | 3 | ## Introduction 4 | Welcome to "Math for CS" – an open-source mathematics textbook (note?) tailored for Computer Science students. This project serves as a comprehensive resource for those seeking to deepen their understanding of mathematical concepts crucial in the field of CS. It's also a personal journey, chronicling my learning path as I explore various topics in computer science mathematics. 5 | 6 | ## Project Status 7 | This textbook (or probably note?) is a work in progress and will be updated irregularly. Expect new topics and interesting exercises to be added as my study journey progresses (I'm currently a Y1 student for CS). 8 | 9 | ## About the Book 10 | "Math for CS" aims to cover a wide range of mathematic topics relevant to computer science, including but not limited to: 11 | - Extention of high school and middle school Knowledge (done, but there may be revision in future) 12 | - Discrete Math (To be finished by May) 13 | - Continuous Math (complete Calculus) (To be finished by the end of the year) 14 | - Probability, combinatorics, and statistics. (Working on the discrete part) 15 | - Linear Algebra(with advanced contents) 16 | - Information Theory(basics) 17 | - Important Pure Math knowledge including abstract algebra, category theory and topology(scattered in important contents, no current plan to make it a independent chapter, but maybe I will write some independent oroject on them). 18 | Each topic is accompanied by exercises and examples to reinforce learning and understanding. 19 | All suggestions or corrections are welcome. 20 | 21 | ## Using the Book 22 | The book is written in \LaTeX, ensuring high-quality typesetting, especially for mathematical expressions. To use this book: 23 | You can: 24 | - Clone the repository to your local machine and compile to PDF. 25 | - Download "main.pdf" dirrectly after new commits. 26 | 27 | ## How to Contribute 28 | All contributions are welcome! Whether it's fixing typos, improving explanations, or adding new content, your help is appreciated. 29 | 30 | ## License 31 | This work is released under the CC BY-NC-SA 4.0 DEED license. 32 | 33 | ## Contact 34 | For questions and feedback, feel free to post it under issue. 35 | -------------------------------------------------------------------------------- /Visualization/._wordcount_selection.tex: -------------------------------------------------------------------------------- 1 | import matplotlib.pyplot as plt 2 | from mpl_toolkits.mplot3d import Axes3D 3 | from itertools import product 4 | 5 | # Define the vertices of a unit cube 6 | vertices = list(product([0, 1], repeat=3)) 7 | 8 | # Create the figure and 3D axis 9 | fig = plt.figure(figsize=(18, 5)) 10 | 11 | # Add subplots for B^2, B^3 and an illustration for B^4 12 | # B^2 13 | ax1 = fig.add_subplot(131) 14 | ax1.set_title('2D Boolean Domain $\\mathbb{B}^2$') 15 | ax1.set_xticks([0, 1]) 16 | ax1.set_yticks([0, 1]) 17 | for s, t in product([0, 1], repeat=2): 18 | ax1.text(s, t, f'({s}{t})', ha='center', va='center') 19 | ax1.set_xlim(-0.5, 1.5) 20 | ax1.set_ylim(-0.5, 1.5) 21 | ax1.grid(True) 22 | 23 | # B^3 24 | ax2 = fig.add_subplot(132, projection='3d') 25 | ax2.set_title('3D Boolean Domain $\\mathbb{B}^3$') 26 | for (s, t, u) in vertices: 27 | ax2.text(s, t, u, f'({s}{t}{u})', ha='center', va='center') 28 | ax2.set_xlim(-0.5, 1.5) 29 | ax2.set_ylim(-0.5, 1.5) 30 | ax2.set_zlim(-0.5, 1.5) 31 | ax2.set_xticks([0, 1]) 32 | ax2.set_yticks([0, 1]) 33 | ax2.set_zticks([0, 1]) 34 | ax2.view_init(30, 30) 35 | 36 | # B^4 visualization would require a 4D space which we cannot represent in a 3D plot. 37 | # Instead, we'll create an illustration with placeholder texts. 38 | ax3 = fig.add_subplot(133) 39 | ax3.text(0.5, 0.5, '4D Boolean Domain $\\mathbb{B}^4$\n(Tesseract)', ha='center', va='center', fontsize=12) 40 | ax3.set_xticks([]) 41 | ax3.set_yticks([]) 42 | for spine in ax3.spines.values(): 43 | spine.set_visible(False) 44 | 45 | # Show the plots 46 | plt.tight_layout() 47 | plt.show() 48 | -------------------------------------------------------------------------------- /Visualization/Graph.nb: -------------------------------------------------------------------------------- 1 | (* Content-type: application/vnd.wolfram.mathematica *) 2 | 3 | (*** Wolfram Notebook File ***) 4 | (* http://www.wolfram.com/nb *) 5 | 6 | (* CreatedBy='Mathematica 14.0' *) 7 | 8 | (*CacheID: 234*) 9 | (* Internal cache information: 10 | NotebookFileLineBreakTest 11 | NotebookFileLineBreakTest 12 | NotebookDataPosition[ 158, 7] 13 | NotebookDataLength[ 27016, 668] 14 | NotebookOptionsPosition[ 24440, 616] 15 | NotebookOutlinePosition[ 24862, 632] 16 | CellTagsIndexPosition[ 24819, 629] 17 | WindowFrame->Normal*) 18 | 19 | (* Beginning of Notebook Content *) 20 | Notebook[{ 21 | 22 | Cell[CellGroupData[{ 23 | Cell[BoxData[ 24 | RowBox[{ 25 | RowBox[{"g", "=", 26 | RowBox[{"Graph", "[", 27 | RowBox[{ 28 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673 | } 674 | ] 675 | *) 676 | 677 | -------------------------------------------------------------------------------- /content/Part2 Discrete math/._wordcount_selection.tex: -------------------------------------------------------------------------------- 1 | Basic Relation -------------------------------------------------------------------------------- /content/Part2 Discrete math/4.Graph Theory.tex: -------------------------------------------------------------------------------- 1 | \chapterimage{orange2.jpg} 2 | \chapterspaceabove{6.75cm} 3 | \chapterspacebelow{7.25cm} 4 | \chapter{Graph Theory} 5 | \begin{remark} 6 | This chapter involves the knowledge of combinatorics, which is in chapter \ref{counting}. 7 | You may read through that chapter from basic counting method to random variables. 8 | If you have learned what have been discussed so far, those content will not cause 9 | too many troubles. Do at least read through \textbf{Permutation and Combination} 10 | before dive into this chapter. 11 | \end{remark} 12 | -------------------------------------------------------------------------------- /content/Part2 Discrete math/5.Basics of Abstract Algebra.tex: -------------------------------------------------------------------------------- 1 | \chapterimage{orange2.jpg} 2 | \chapterspaceabove{6.75cm} 3 | \chapterspacebelow{7.25cm} 4 | \chapter{Basics of Abstract Algebra} 5 | \lipsum 6 | \section{Fundamentals of Algebraic Structures} 7 | \subsection{Groups} 8 | \subsection{Rings} 9 | \subsection{Fields} 10 | \subsection{Exercises} 11 | \section{Operations on Algebraic Structures} 12 | \subsection{Homomorphisms} 13 | \subsection{Isomorphisms} 14 | \subsection{Exercises} 15 | \section{Applications of Algebraic Structures} 16 | \subsection{Cryptography} 17 | \subsection{Coding Theory} 18 | \subsection{Exercises} 19 | -------------------------------------------------------------------------------- /content/Part2 Discrete math/6.Introductory Topology and Category Theory.tex: -------------------------------------------------------------------------------- 1 | \chapterimage{orange2.jpg} 2 | \chapterspaceabove{6.75cm} 3 | \chapterspacebelow{7.25cm} 4 | \chapter{Introductory Topology and Category Theory} 5 | \lipsum 6 | 7 | 8 | \section{Basic Topology} 9 | \lipsum 10 | \subsection{Introduction to Topological Spaces} 11 | \subsubsection{Open and Closed Sets} 12 | \subsubsection{Basis for a Topology} 13 | \subsection{Continuity and Limits} 14 | \subsubsection{Continuous Functions} 15 | \subsubsection{Limit Points and Convergence} 16 | \subsection{Compactness and Connectedness} 17 | \subsubsection{Compact Spaces} 18 | \subsubsection{Connected Spaces} 19 | \subsection{Applications of Topology} 20 | \subsubsection{Topology in Computer Science} 21 | \subsubsection{Topology in Physics} 22 | 23 | \section{Category Theory Fundamentals} 24 | \lipsum 25 | \subsection{Introduction to Categories} 26 | \subsubsection{Objects and Morphisms} 27 | \subsubsection{Examples of Categories} 28 | \subsection{Functors and Natural Transformations} 29 | \subsubsection{Definition of Functors} 30 | \subsubsection{Natural Transformations between Functors} 31 | \subsection{Limits and Colimits} 32 | \subsubsection{Universal Properties} 33 | \subsubsection{Construction of Limits and Colimits} 34 | \subsection{Applications of Category Theory} 35 | \subsubsection{Category Theory in Programming Languages} 36 | \subsubsection{Category Theory in Logic and Set Theory} -------------------------------------------------------------------------------- /content/Part3 Single-variable Calculus/Chapter1.tex: -------------------------------------------------------------------------------- 1 | \chapterimage{orange2.jpg} 2 | \chapterspaceabove{6.75cm} 3 | \chapterspacebelow{7.25cm} 4 | \chapter{Function Monotonicity, Parity and Periodicity} 5 | 6 | \section{Monotonicity of Function} 7 | 8 | 9 | \section{Parity of Function} 10 | 11 | 12 | \section{Periodicity of Function} -------------------------------------------------------------------------------- /content/Part3 Single-variable Calculus/Chapter2.tex: -------------------------------------------------------------------------------- 1 | \chapterimage{orange2.jpg} 2 | \chapterspaceabove{6.75cm} 3 | \chapterspacebelow{7.25cm} 4 | \chapter{Abstract and Piece-wise Function} 5 | 6 | \section{Abstract Function} 7 | 8 | \section{Piece-wise Function} -------------------------------------------------------------------------------- /content/Part3 Single-variable Calculus/Chapter3.tex: -------------------------------------------------------------------------------- 1 | \chapterimage{orange2.jpg} 2 | \chapterspaceabove{6.75cm} 3 | \chapterspacebelow{7.25cm} 4 | \chapter{Limit and Continuity} 5 | \section{Limit of Sequence} 6 | 7 | 8 | \section{limit of Function} 9 | 10 | 11 | \section{Continuity} 12 | 13 | 14 | \section{Application of Limit} -------------------------------------------------------------------------------- /content/Part3 Single-variable Calculus/Chapter4.tex: -------------------------------------------------------------------------------- 1 | \chapterimage{orange2.jpg} 2 | \chapterspaceabove{6.75cm} 3 | \chapterspacebelow{7.25cm} 4 | \chapter{Differential Calculus} 5 | 6 | \section{Derivative Basics} 7 | \subsection{Definition of Derivative} 8 | \subsection{Geometric Meaning of Derivative} 9 | \subsection{Physical Meaning of Derivative} 10 | 11 | \section{Basic Derivative Rules} 12 | \subsection{Derivatives of Elementary Functions} 13 | \subsection{Product Rule, Quotient Rule, Chain Rule} 14 | 15 | \section{Higher-order Derivatives} 16 | \subsection{Second-order Derivatives and Applications} 17 | \subsection{Calculation and Significance of Higher-order Derivatives} 18 | 19 | \section{Derivatives of Abnormal Function} 20 | \subsection{Derivatives of Implicit Functions} 21 | \subsection{Derivatives of Parametric Equations} 22 | 23 | \section{Differentiation} 24 | 25 | \section{Related Rates} 26 | \subsection{Relationships between Rates of Change of Different Quantities} 27 | 28 | \section{Taylor Series} 29 | \subsection{Taylor Expansion of Functions} 30 | \subsection{Application of Taylor Series} 31 | 32 | 33 | \section{Applications of Differential Calculus} 34 | \subsection{Tangents and Normals of Curves} 35 | % \subsection{Analysis of Velocity and Acceleration in Motion} 36 | \subsection{Mean Value Theorem in Differential Calculus} 37 | \subsubsection{Fermat's Theorem} 38 | \subsubsection{Rolle's Theorem} 39 | \subsubsection{Lagrange's Mean Value Theorem} 40 | \subsubsection{Cauchy's Mean Value Theorem} 41 | \subsubsection{Indeterminate Form Limit} 42 | \subsection{Extrema Problems and Optimization} -------------------------------------------------------------------------------- /content/Part3 Single-variable Calculus/Chapter5.tex: -------------------------------------------------------------------------------- 1 | \chapterimage{orange2.jpg} 2 | \chapterspaceabove{6.75cm} 3 | \chapterspacebelow{7.25cm} 4 | \chapter{integral calculus} 5 | 6 | \section{Fundamentals of Integration} 7 | \subsection{Definition of the Integral} 8 | \subsection{Properties of Integrals} 9 | \subsection{The Fundamental Theorem of Calculus} 10 | 11 | \section{Techniques of Integration} 12 | \subsection{Basic Integration Formulas} 13 | \subsection{Integration by Substitution} 14 | \subsection{Integration by Parts} 15 | \subsection{Trigonometric Integrals} 16 | \subsection{Partial Fractions} 17 | 18 | \section{Applications of Integration} 19 | \subsection{Area Under Curves} 20 | \subsection{Volumes of Solids of Revolution} 21 | \subsection{Arc Length and Surface Area} 22 | \subsection{Center of Mass and Moments} 23 | 24 | \section{Improper Integrals} 25 | \subsection{Convergence and Divergence of Improper Integrals} 26 | \subsection{Applications of Improper Integrals} 27 | 28 | \section{Numerical Integration Methods} 29 | \subsection{The Trapezoidal Rule} 30 | \subsection{Simpson's Rule} -------------------------------------------------------------------------------- /content/Part3 Single-variable Calculus/Chapter6.tex: -------------------------------------------------------------------------------- 1 | \chapterimage{orange2.jpg} 2 | \chapterspaceabove{6.75cm} 3 | \chapterspacebelow{7.25cm} 4 | \chapter{Differential Equation} -------------------------------------------------------------------------------- /content/Part3 Single-variable Calculus/Chapter7.tex: -------------------------------------------------------------------------------- 1 | \chapterimage{orange2.jpg} 2 | \chapterspaceabove{6.75cm} 3 | \chapterspacebelow{7.25cm} 4 | \chapter{Infinite Series} -------------------------------------------------------------------------------- /content/Part4 Multi-variable Calculus/Chapter1.tex: -------------------------------------------------------------------------------- 1 | \chapterimage{orange2.jpg} 2 | \chapterspaceabove{6.75cm} 3 | \chapterspacebelow{7.25cm} 4 | \chapter{integral calculus} 5 | 6 | 7 | \chapterimage{orange2.jpg} 8 | \chapterspaceabove{6.75cm} 9 | \chapterspacebelow{7.25cm} 10 | \chapter{Introduction to Multivariable Functions} 11 | \subsection{Concepts of Multivariable Functions} 12 | \subsection{Graphs and Contour Plots} 13 | 14 | 15 | \chapterimage{orange2.jpg} 16 | \chapterspaceabove{6.75cm} 17 | \chapterspacebelow{7.25cm} 18 | \chapter{Partial Derivatives} 19 | \subsection{Definition and Interpretation} 20 | \subsection{Higher-Order Partial Derivatives} 21 | \subsection{Chain Rule in Multiple Variables} 22 | 23 | 24 | \chapterimage{orange2.jpg} 25 | \chapterspaceabove{6.75cm} 26 | \chapterspacebelow{7.25cm} 27 | \chapter{Multiple Integrals} 28 | \subsection{Double Integrals} 29 | \subsubsection{Iterated Integrals} 30 | \subsubsection{Double Integrals over General Regions} 31 | \subsection{Triple Integrals} 32 | \subsubsection{Cylindrical and Spherical Coordinates} 33 | 34 | 35 | \chapterimage{orange2.jpg} 36 | \chapterspaceabove{6.75cm} 37 | \chapterspacebelow{7.25cm} 38 | \chapter{Vector Calculus} 39 | \section{Vector Fields} 40 | \section{Gradient, Divergence, and Curl} 41 | \section{Line and Surface Integrals} -------------------------------------------------------------------------------- /content/Part5 Linear Algebra/Chapter1.tex: -------------------------------------------------------------------------------- 1 | \chapterimage{orange2.jpg} 2 | \chapterspaceabove{6.75cm} 3 | \chapterspacebelow{7.25cm} 4 | \chapter{Vectors Space and the Geometry of Space} 5 | 6 | 7 | 8 | 9 | 10 | 11 | \chapterimage{orange2.jpg} 12 | \chapterspaceabove{6.75cm} 13 | \chapterspacebelow{7.25cm} 14 | \chapter{Matrices and Systems of Equations} 15 | 16 | 17 | \chapterimage{orange2.jpg} 18 | \chapterspaceabove{6.75cm} 19 | \chapterspacebelow{7.25cm} 20 | \chapter{Determinant of Matrix} 21 | 22 | 23 | \chapterimage{orange2.jpg} 24 | \chapterspaceabove{6.75cm} 25 | \chapterspacebelow{7.25cm} 26 | \chapter{Orthogonality} 27 | 28 | 29 | \chapterimage{orange2.jpg} 30 | \chapterspaceabove{6.75cm} 31 | \chapterspacebelow{7.25cm} 32 | \chapter{Linear Transformations} 33 | 34 | \chapterimage{orange2.jpg} 35 | \chapterspaceabove{6.75cm} 36 | \chapterspacebelow{7.25cm} 37 | \chapter{Eigenvalues and Eigenvector} 38 | 39 | 40 | \chapterimage{orange2.jpg} 41 | \chapterspaceabove{6.75cm} 42 | \chapterspacebelow{7.25cm} 43 | \chapter{Singular Value Decomposition} 44 | 45 | 46 | \chapterimage{orange2.jpg} 47 | \chapterspaceabove{6.75cm} 48 | \chapterspacebelow{7.25cm} 49 | \chapter{Complex Vector and Matrices} 50 | 51 | 52 | 53 | \chapterimage{orange2.jpg} 54 | \chapterspaceabove{6.75cm} 55 | \chapterspacebelow{7.25cm} 56 | \chapter{Matrix Differential Calculus} -------------------------------------------------------------------------------- /content/Part6 Probability and Combinatorics/._wordcount_selection.tex: -------------------------------------------------------------------------------- 1 | \begin{align*} 2 | p_m = \frac{(m+1)^2-m}{\binom{4m+2}{2}} = \frac{m^2+m+1}{8m^2+6m+1}>\frac{1}{8}. 3 | \end{align*} 4 | 5 | This could be done by many ways, like taking limit, induction or even by investigating the global minimum of the function. 6 | 7 | We use $f(m)$ to denote $\frac{m^2+m+1}{8m^2+6m+1}$ and 8 | \begin{align*} 9 | g(m) = f(m) - \frac{1}{8}. 10 | \end{align*} 11 | \begin{proof} 12 | Now we need to prove that $\forall m\in \Z^{>=1}, g(m) > 0$. 13 | \begin{align} 14 | g(m)&=\frac{8\left(m^2+m+1\right)-\left(8 m^2+6 m+1\right)}{8\left(8 m^2+6 m+1\right)\\ 15 | &=\frac{8 m^2+8 m+8-8 m^2-6 m-1}{8\left(8 m^2+6 m+1\right)}\\ 16 | &=\frac{2 m+7}{8\left(8 m^2+6 m+1\right)}} 17 | \end{align} 18 | 19 | Given $m\in \Z^{>=1}$, $g(m) > 0$ in the specified domain. 20 | 21 | Hence, $p_m > \frac{1}{8}$. This completes the proof. 22 | \end{proof} -------------------------------------------------------------------------------- /content/Part6 Probability and Combinatorics/4.Continuous Distribution.tex: -------------------------------------------------------------------------------- 1 | \chapterimage{orange2.jpg} 2 | \chapterspaceabove{6.75cm} 3 | \chapterspacebelow{7.25cm} 4 | \chapter{Continuous Distribution} 5 | -------------------------------------------------------------------------------- /content/Part6 Probability and Combinatorics/5.Joint Cumulative Distribution.tex: -------------------------------------------------------------------------------- 1 | \chapterimage{orange2.jpg} 2 | \chapterspaceabove{6.75cm} 3 | \chapterspacebelow{7.25cm} 4 | \chapter{Joint Cumulative Distribution} -------------------------------------------------------------------------------- /content/Part6 Probability and Combinatorics/6.Limit Theory in Probability.tex: -------------------------------------------------------------------------------- 1 | \chapterimage{orange2.jpg} 2 | \chapterspaceabove{6.75cm} 3 | \chapterspacebelow{7.25cm} 4 | \chapter{Limit Theory in Probability} -------------------------------------------------------------------------------- /content/Part6 Probability and Combinatorics/7.Stochastic Process.tex: -------------------------------------------------------------------------------- 1 | \chapterimage{orange2.jpg} 2 | \chapterspaceabove{6.75cm} 3 | \chapterspacebelow{7.25cm} 4 | \chapter{Stochastic Process} -------------------------------------------------------------------------------- /content/Part7 Statistics/Chapter1.tex: -------------------------------------------------------------------------------- 1 | \chapterimage{orange2.jpg} 2 | \chapterspaceabove{6.75cm} 3 | \chapterspacebelow{7.25cm} 4 | \chapter{Sampling and Parameters} 5 | 6 | 7 | \chapterimage{orange2.jpg} 8 | \chapterspaceabove{6.75cm} 9 | \chapterspacebelow{7.25cm} 10 | \chapter{Descriptive Statistics} 11 | 12 | 13 | \chapterimage{orange2.jpg} 14 | \chapterspaceabove{6.75cm} 15 | \chapterspacebelow{7.25cm} 16 | \chapter{Graphical Statistics} 17 | 18 | 19 | 20 | \chapterimage{orange2.jpg} 21 | \chapterspaceabove{6.75cm} 22 | \chapterspacebelow{7.25cm} 23 | \chapter{Statistical Inference} 24 | \section{Parameter Estimation} 25 | \section{Interval Estimation} 26 | \section{Hypothesis Testing} 27 | \section{Variance Inference} 28 | \section{Bayesian Inference} 29 | 30 | 31 | \chapterimage{orange2.jpg} 32 | \chapterspaceabove{6.75cm} 33 | \chapterspacebelow{7.25cm} 34 | \chapter{Hypothesis Testing} 35 | 36 | 37 | \chapterimage{orange2.jpg} 38 | \chapterspaceabove{6.75cm} 39 | \chapterspacebelow{7.25cm} 40 | \chapter{Regression and Regressive Analysis} 41 | 42 | 43 | \chapterimage{orange2.jpg} 44 | \chapterspaceabove{6.75cm} 45 | \chapterspacebelow{7.25cm} 46 | \chapter{Basic Multi-variable Statistical analysis} 47 | -------------------------------------------------------------------------------- /content/Part8 Basic Information Theory/Chapter1.tex: -------------------------------------------------------------------------------- 1 | \chapterimage{orange2.jpg} 2 | \chapterspaceabove{6.75cm} 3 | \chapterspacebelow{7.25cm} 4 | \chapter{Measuring of Information} 5 | 6 | 7 | \chapterimage{orange2.jpg} 8 | \chapterspaceabove{6.75cm} 9 | \chapterspacebelow{7.25cm} 10 | \chapter{Information Entropy} 11 | 12 | 13 | \chapterimage{orange2.jpg} 14 | \chapterspaceabove{6.75cm} 15 | \chapterspacebelow{7.25cm} 16 | \chapter{Joined Entropy and Conditional Entropy} 17 | 18 | 19 | \chapterimage{orange2.jpg} 20 | \chapterspaceabove{6.75cm} 21 | \chapterspacebelow{7.25cm} 22 | \chapter{Cross Entropy and Relative Entropy} 23 | 24 | 25 | \chapterimage{orange2.jpg} 26 | \chapterspaceabove{6.75cm} 27 | \chapterspacebelow{7.25cm} 28 | \chapter{Mutual Information} 29 | 30 | 31 | \chapterimage{orange2.jpg} 32 | \chapterspaceabove{6.75cm} 33 | \chapterspacebelow{7.25cm} 34 | \chapter{Differential Entropy} -------------------------------------------------------------------------------- /content/part1 Intro/._wordcount_selection.tex: -------------------------------------------------------------------------------- 1 | \chapterimage{orange2.jpg} 2 | \chapterspaceabove{6.75cm} 3 | \chapterspacebelow{7.25cm} 4 | \chapter{Inequality} 5 | throughout our mathematical journey, equalities are always the very basics of most conclusion, and that is also which we start to learn math for. However, inequalities are not such a know thing as equalities. 6 | Inequalities have many tricky characteristics so that we have to take with care, or things could go wrong. This chapter covers the fundamentals of inequality as a crucial tool for problem-solving in Computer Science. 7 | 8 | \section{Inequality basics} 9 | We all know about inequalities, and the first thing to clarify is the relationship between sizes. How to determine the size relationship between certain numbers? Since the basis for comparing "numbers" with each other corresponds to one, it is stipulated on the number line that the points increase from left to right, and therefore the numbers they represent increase in turn. Listed in ascending order, that is: 10 | 11 | Let \( a, b \) be two real numbers, and the points on the number line are denoted as \( A, B \) respectively. If \( A \) is to the right of \( B \), we say \( a > b \); if \( A \) is to the left of \( B \), we say \( a < b \); if \( A \) coincides with \( B \), we say \( a = b \). 12 | 13 | Thus, for any two real numbers, one and only one of the following three situations must hold: 14 | 15 | \[ 16 | a > b; \quad a = b; \quad a < b. 17 | \] 18 | 19 | The above relationship is also known as the one-dimensional coordinate law. 20 | 21 | \[ 22 | \begin{aligned} 23 | a > b &\Leftrightarrow a - b > 0 \\ 24 | a < b &\Leftrightarrow a - b < 0 \\ 25 | a = b &\Leftrightarrow a - b = 0 26 | \end{aligned} 27 | \] 28 | 29 | Where the symbol “\( \Leftrightarrow \)” (double arrow), read as "if and only if," means that the truth of two propositions depends on each other. That is to say, if one proposition is true, then the other proposition is also true; conversely, if one proposition is false, then the other proposition is also false. 30 | 31 | Based on the derivation of inequalities, in most cases, the above principles are sufficient. The following mathematical laws involve the geometric and algebraic meanings of the sizes of real numbers and the relationship between them. They are the basis for comparing the sizes of two real numbers and for proving inequalities by comparison. Let's review some basic properties of inequalities that are the foundation for our further study. 32 | 33 | \begin{itemize} 34 | \item Symmetry: \( a > b \) if and only if \( b < a \). 35 | \item Transitivity: If \( a > b \) and \( b > c \), then \( a > c \). 36 | \item Addition (Subtraction): If \( a > b \), then \( a + c > b + c \). 37 | \item Multiplication (Division): If \( a > b \) and \( c > 0 \), then \( ac > bc \); if \( a > b \) and \( c < 0 \), then \( ac < bc \). 38 | \item Exponentiation: If \( a > b \), then \( a^n > b^n \), where \( n \) is a positive integer, and \( n \geq 2 \). 39 | \item Root Extraction (Power Root): If \( a > b > 0 \), then \( \sqrt[n]{a} > \sqrt[n]{b} \), where \( n \) is a positive integer, and \( n \geq 2 \). 40 | \item If \( a > b \) and \( c > d \), then \( a + c > b + d \). 41 | \item If \( a > b > 0 \) and \( c > d > 0 \), then \( ac > bd \). 42 | \end{itemize} 43 | \subsection{Exercises} 44 | \begin{exercise} 45 | Explain the following statement. 46 | \begin{enumerate} 47 | \item If \( a > b \), then \( \frac{a}{c} > \frac{b}{c} \); 48 | \item If \( ac < bc \), then \( a < b \); 49 | \item If \( a < b \), then \( \frac{1}{a} > \frac{1}{b} \); 50 | \item If \( ac^2 > bc^2 \), then \( a > b \); 51 | \item If \( a > b \), then \( a^n > b^n \). 52 | \end{enumerate} 53 | \end{exercise} 54 | 55 | \textbf{Solution:} 56 | 57 | \begin{enumerate} 58 | \item If \( c > 0 \), multiplying both sides of \( a > b \) by the positive number \( \frac{1}{c} \) preserves the inequality, hence \( \frac{a}{c} > \frac{b}{c} \). If \( c < 0 \), the direction of the inequality would be reversed, which is not given in the condition, hence we assume \( c > 0 \). 59 | \item Dividing both sides of \( ac < bc \) by \( c \) (assuming \( c \neq 0 \)), we get \( a < b \) because division by a positive number preserves the inequality, and division by a negative number reverses it. 60 | \item Taking the reciprocal of both sides of \( a < b \) reverses the inequality because \( a \) and \( b \) are on opposite sides of the fraction line, hence \( \frac{1}{a} > \frac{1}{b} \) (assuming \( a, b > 0 \) to avoid division by zero). 61 | \item Dividing both sides of \( ac^2 > bc^2 \) by \( c^2 \) (assuming \( c \neq 0 \)) preserves the inequality, hence \( a > b \) because \( c^2 \) is positive regardless of whether \( c \) is positive or negative. 62 | \item Raising both sides of \( a > b \) to a power \( n \) (assuming \( n \) is a positive integer) preserves the inequality because both \( a \) and \( b \) are raised to the same power, hence \( a^n > b^n \). 63 | \end{enumerate} 64 | \begin{exercise} 65 | Given the inequality \( a > b > 0 \), \( c < d < 0 \), \( f < 0 \), show that: 66 | \[ 67 | \frac{f}{a - c} > \frac{f}{b - d}. 68 | \] 69 | \end{exercise} 70 | \begin{proof} 71 | Since \( a > b > 0 \) and \( c < d < 0 \), then \( a - c > b - d \) because subtracting a smaller negative number is the same as adding a larger positive number. Given that \( f < 0 \), when dividing by a larger positive number, the result is smaller because a negative number divided by a positive number yields a negative result, and the further away the divisor is from zero, the smaller the quotient. 72 | 73 | Therefore: 74 | \[ 75 | \frac{f}{a - c} > \frac{f}{b - d}. 76 | \] 77 | \end{proof} 78 | 79 | \section{Solving Quadratic Inequality} 80 | Recall that, when we solve Quadratic equations, we use discriminant to solve equations, which is also applicable to inequalities. 81 | For a quadratic inequality of the form $ax^2 + bx + c > 0$ or $ax^2 + bx + c < 0$ (where $a > 0$), the solution set can be determined by the discriminant $\Delta = b^2 - 4ac$: 82 | \begin{enumerate} 83 | \item If $\Delta > 0$, the quadratic equation $ax^2 + bx + c = 0$ has two distinct real roots $x_1$ and $x_2$, and $x_1 < x_2$. The solution set for $y = ax^2 + bx + c$ being greater than zero (when $y = 0$) is for values of $x$ either less than $x_1$ or greater than $x_2$, and the solution set for $ax^2 + bx + c < 0$ is $\{ x \mid x_1 < x < x_2 \}$. 84 | \item If $\Delta = 0$, then $ax^2 + bx + c = 0$ has one real root, specifically $x_1 = x_2 = -\frac{b}{2a}$. The solution set for $y = ax^2 + bx + c$ being greater than zero is all $x$ except $x \neq -\frac{b}{2a}$, and there is no solution set where $ax^2 + bx + c < 0$. 85 | \item If $\Delta < 0$, then $ax^2 + bx + c = 0$ has no real roots, and the parabola $y = ax^2 + bx + c$ does not intersect the x-axis. The solution set for $ax^2 + bx + c > 0$ is all real numbers, and there is no solution set where $ax^2 + bx + c < 0$. 86 | \end{enumerate} 87 | \begin{figure}[ht!] 88 | \centering 89 | \includegraphics[width=0.8\textwidth]{quadratic.png} 90 | \caption{Quadratic function graphs based on the discriminant.} 91 | \end{figure} -------------------------------------------------------------------------------- /indexstyle.ist: -------------------------------------------------------------------------------- 1 | delim_0 "\\dotfill\ " % Filler between section heading and page number 2 | delim_1 "\\dotfill\ " % Filler between subsection heading and page number 3 | headings_flag 1 4 | heading_prefix "\\nopagebreak\n\\vspace*{0.1cm}\\tikz\\node at (0pt,0pt) [draw=none, fill=ocre!50, line width=1pt, inner sep=5pt]{\\parbox{0.993\\linewidth-2\\fboxsep-2\\fboxrule-2pt}{\\centering\\large\\sffamily\\bfseries\\textcolor{white}{" heading_suffix "}}};\\vspace*{0.2cm}\\nopagebreak\n" % Styling of index 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and $F_2$ after Grouping}}{191}{figure.6.2}% 29 | \contentsline {figure}{\numberline {6.3}{\ignorespaces Visualization of Boolean Domain}}{193}{figure.6.3}% 30 | \addvspace {10\p@ } 31 | \contentsline {figure}{\numberline {7.1}{\ignorespaces Binary, Octal and Hexadecimal Representation}}{210}{figure.7.1}% 32 | \addvspace {10\p@ } 33 | \contentsline {figure}{\numberline {8.1}{\ignorespaces Visual Comparison of a Relation and a Function}}{240}{figure.8.1}% 34 | \contentsline {figure}{\numberline {8.2}{\ignorespaces $R=\{(a,b)\mid a$ divides $b\}$}}{240}{figure.8.2}% 35 | \contentsline {figure}{\numberline {8.3}{\ignorespaces $R_1$ and $R_2$ in Digraph}}{253}{figure.8.3}% 36 | \addvspace {10\p@ } 37 | \addvspace {10\p@ } 38 | \addvspace {10\p@ } 39 | \addvspace {10\p@ } 40 | \addvspace {10\p@ } 41 | \addvspace {10\p@ } 42 | \addvspace {10\p@ } 43 | \addvspace {10\p@ } 44 | \addvspace {10\p@ } 45 | \addvspace {10\p@ } 46 | \addvspace {10\p@ } 47 | \addvspace {10\p@ } 48 | \addvspace {10\p@ } 49 | \addvspace {10\p@ } 50 | \addvspace {10\p@ } 51 | \addvspace {10\p@ } 52 | \addvspace {10\p@ } 53 | \addvspace {10\p@ } 54 | \addvspace {10\p@ } 55 | \addvspace {10\p@ } 56 | \addvspace {10\p@ } 57 | \addvspace {10\p@ } 58 | \addvspace {10\p@ } 59 | \addvspace {10\p@ } 60 | \addvspace {10\p@ } 61 | \addvspace {10\p@ } 62 | \addvspace {10\p@ } 63 | \addvspace {10\p@ } 64 | \addvspace {10\p@ } 65 | \addvspace {10\p@ } 66 | \addvspace {10\p@ } 67 | \addvspace {10\p@ } 68 | \addvspace {10\p@ } 69 | \addvspace {10\p@ } 70 | \addvspace {10\p@ } 71 | \addvspace {10\p@ } 72 | \addvspace {10\p@ } 73 | \addvspace {10\p@ } 74 | \addvspace {10\p@ } 75 | \addvspace {10\p@ } 76 | \addvspace {10\p@ } 77 | \addvspace {10\p@ } 78 | \addvspace {10\p@ } 79 | \addvspace {10\p@ } 80 | \contentsfinish 81 | -------------------------------------------------------------------------------- /main.lot: -------------------------------------------------------------------------------- 1 | \addvspace {10\p@ } 2 | \addvspace {10\p@ } 3 | \contentsline {table}{\numberline {2.1}{\ignorespaces Basic Elementary Functions}}{51}{table.2.1}% 4 | \addvspace {10\p@ } 5 | \contentsline {table}{\numberline {3.1}{\ignorespaces Execution of RPM}}{90}{table.3.1}% 6 | \contentsline {table}{\numberline {3.2}{\ignorespaces Common time complexities in Big O notation}}{92}{table.3.2}% 7 | \addvspace {10\p@ } 8 | \addvspace {10\p@ } 9 | \addvspace {10\p@ } 10 | \contentsline {table}{\numberline {6.1}{\ignorespaces Common Algebraic Laws}}{169}{table.6.1}% 11 | \contentsline {table}{\numberline {6.2}{\ignorespaces Common Boolean Operators Truth Tables}}{171}{table.6.2}% 12 | \contentsline {table}{\numberline {6.3}{\ignorespaces Addition and Multiplication Rule of 0 and 1}}{172}{table.6.3}% 13 | \contentsline {table}{\numberline {6.4}{\ignorespaces Truth table for \((A \land \neg B) \lor (\neg A \land B)\)}}{172}{table.6.4}% 14 | \contentsline {table}{\numberline {6.5}{\ignorespaces Basic Boolean Identities}}{172}{table.6.5}% 15 | \contentsline {table}{\numberline {6.6}{\ignorespaces Truth values of material conditional, biconditional, and XOR for all possible inputs.}}{174}{table.6.6}% 16 | \contentsline {table}{\numberline {6.7}{\ignorespaces Truth table for the expression \( x \lor ((\lnot y) \land (\lnot z)) \)}}{177}{table.6.7}% 17 | \contentsline {table}{\numberline {6.8}{\ignorespaces De Morgan's law for quantifiers}}{197}{table.6.8}% 18 | \addvspace {10\p@ } 19 | \addvspace {10\p@ } 20 | \addvspace {10\p@ } 21 | \addvspace {10\p@ } 22 | \addvspace {10\p@ } 23 | \addvspace {10\p@ } 24 | \addvspace {10\p@ } 25 | \addvspace {10\p@ } 26 | \addvspace {10\p@ } 27 | \addvspace {10\p@ } 28 | \addvspace {10\p@ } 29 | \addvspace {10\p@ } 30 | \addvspace {10\p@ } 31 | \addvspace {10\p@ } 32 | \addvspace {10\p@ } 33 | \addvspace {10\p@ } 34 | \addvspace {10\p@ } 35 | \addvspace {10\p@ } 36 | \addvspace {10\p@ } 37 | \addvspace {10\p@ } 38 | \addvspace {10\p@ } 39 | \addvspace {10\p@ } 40 | \addvspace {10\p@ } 41 | \addvspace {10\p@ } 42 | \addvspace {10\p@ } 43 | \addvspace {10\p@ } 44 | \addvspace {10\p@ } 45 | \contentsline {table}{\numberline {33.1}{\ignorespaces $(i,j)$-divisible sequences $(m\leq 3)$}}{375}{table.33.1}% 46 | \contentsline {table}{\numberline {33.2}{\ignorespaces $(i,j)$-divisible sequences $(m\leq 5)$}}{375}{table.33.2}% 47 | \addvspace {10\p@ } 48 | \addvspace {10\p@ } 49 | \addvspace {10\p@ } 50 | \addvspace {10\p@ } 51 | \addvspace {10\p@ } 52 | \addvspace {10\p@ } 53 | \addvspace {10\p@ } 54 | \addvspace {10\p@ } 55 | \addvspace {10\p@ } 56 | \addvspace {10\p@ } 57 | \addvspace {10\p@ } 58 | \addvspace {10\p@ } 59 | \addvspace {10\p@ } 60 | \addvspace {10\p@ } 61 | \addvspace {10\p@ } 62 | \addvspace {10\p@ } 63 | \addvspace {10\p@ } 64 | \addvspace {10\p@ } 65 | \addvspace {10\p@ } 66 | \contentsfinish 67 | -------------------------------------------------------------------------------- /main.out.ps: 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141 | {\upshape} 142 | {\parindent} 143 | {\bfseries} 144 | {:} 145 | {3pt} 146 | {} 147 | 148 | 149 | \theoremstyle{solutionStyle} 150 | \newtheorem*{solution}{Solution} 151 | 152 | % Define a new theorem style called 'axiom_style' 153 | \newtheoremstyle{axiom_style}% name of the style to be used 154 | {3pt}% Space above 155 | {3pt}% Space below 156 | {\normalfont}% Body font 157 | {}% Indent amount 158 | {\bfseries}% Theorem head font 159 | {.}% Punctuation after theorem head 160 | {.5em}% Space after theorem head 161 | {}% Theorem head spec (can be left empty, meaning 'normal') 162 | 163 | % Apply the theorem style 164 | \theoremstyle{axiom_style} 165 | 166 | % Define the 'axiom' theorem with chapter numbering 167 | \newtheorem{axiom}{Axiom}[chapter] 168 | %\addbibresource{sample.bib} % Bibliography file 169 | 170 | \definecolor{ocre}{RGB}{243, 102, 25} % Define the color used for highlighting throughout the book 171 | 172 | \chapterimage{orange1.jpg} % Chapter heading image 173 | \chapterspaceabove{6.5cm} % Default whitespace from the top of the page to the chapter title on chapter pages 174 | \chapterspacebelow{6.75cm} % Default amount of vertical whitespace from the top margin to the start of the text on chapter pages 175 | 176 | %---------------------------------------------------------------------------------------- 177 | 178 | \begin{document} 179 | 180 | %---------------------------------------------------------------------------------------- 181 | % TITLE PAGE 182 | %---------------------------------------------------------------------------------------- 183 | 184 | \titlepage % Output the title page 185 | {\includegraphics[width=\paperwidth]{background.pdf}} % Code to output the background image, which should be the same dimensions as the paper to fill the page entirely; leave empty for no background image 186 | { % Title(s) and author(s) 187 | \centering\sffamily % Font styling 188 | {\Huge\bfseries Math for Computer Science\par} % Book title 189 | \vspace{16pt} % Vertical whitespace 190 | {\LARGE A Journey Through Rigorous Mathematical Foundations} % Subtitle 191 | \vspace{24pt} % Vertical whitespace 192 | {\huge\bfseries \\ Eric Yang Xingyu\par} % Author name 193 | } 194 | 195 | %---------------------------------------------------------------------------------------- 196 | % COPYRIGHT PAGE 197 | %---------------------------------------------------------------------------------------- 198 | 199 | \thispagestyle{empty} % Suppress headers and footers on this page 200 | 201 | ~\vfill % Push the text down to the bottom of the page 202 | 203 | \noindent Copyright \copyright\ 2024 Eric Yang Xingyu\\ % Copyright notice 204 | 205 | \noindent \textsc{Published by Publisher}\\ % Publisher 206 | This \LaTeX \space template is from 207 | \noindent \textsc{\href{https://www.latextemplates.com/template/legrand-orange-book}{book-website.com}}\\ % URL 208 | 209 | \noindent Licensed under the Creative Commons Attribution-NonCommercial 4.0 License (the ``License''). You may not use this file except in compliance with the License. You may obtain a copy of the License at \url{https://creativecommons.org/licenses/by-nc-sa/4.0}. Unless required by applicable law or agreed to in writing, software distributed under the License is distributed on an \textsc{``as is'' basis, without warranties or conditions of any kind}, either express or implied. See the License for the specific language governing permissions and limitations under the License.\\ % License information, replace this with your own license (if any) 210 | 211 | \noindent \textit{First amendment, March 2024} % Printing/edition date 212 | 213 | %---------------------------------------------------------------------------------------- 214 | % TABLE OF CONTENTS 215 | %---------------------------------------------------------------------------------------- 216 | 217 | \pagestyle{empty} % Disable headers and footers for the following pages 218 | 219 | \tableofcontents % Output the table of contents 220 | 221 | \listoffigures % Output the list of figures, comment or remove this command if not required 222 | 223 | \listoftables % Output the list of tables, comment or remove this command if not required 224 | 225 | \pagestyle{fancy} % Enable default headers and footers again 226 | 227 | \cleardoublepage % Start the following content on a new page 228 | 229 | %---------------------------------------------------------------------------------------- 230 | % PART 231 | %---------------------------------------------------------------------------------------- 232 | 233 | \part{Introductory Topics} 234 | \input{content/part1 Intro/1.proof and proposition.tex} 235 | \input{content/part1 Intro/2.Set, Sequence, Function, and Summation.tex} 236 | \input{content/part1 Intro/3.Algorithm and Number System.tex} 237 | \input{content/part1 Intro/4.Inequality.tex} 238 | \input{content/part1 Intro/5.Complex Number.tex} 239 | 240 | \part{Further Discrete Mathematics and Theories} 241 | \input{content/Part2 Discrete math/1.Boolean Algebra.tex} 242 | \input{content/Part2 Discrete math/2.Number Theory.tex} 243 | \input{content/Part2 Discrete math/3.Relation and Order.tex} 244 | \input{content/Part2 Discrete math/4.Graph Theory.tex} 245 | \input{content/Part2 Discrete math/5.Basics of Abstract Algebra.tex} 246 | \input{content/Part2 Discrete math/6.Introductory Topology and Category Theory.tex} 247 | %------------------------------------------------ 248 | \part{Single-variable Calculus} 249 | \input{content/Part3 Single-variable Calculus/Chapter1.tex} 250 | \input{content/Part3 Single-variable Calculus/Chapter2.tex} 251 | \input{content/Part3 Single-variable Calculus/Chapter3.tex} 252 | \input{content/Part3 Single-variable Calculus/Chapter4.tex} 253 | \input{content/Part3 Single-variable Calculus/Chapter5.tex} 254 | \input{content/Part3 Single-variable Calculus/Chapter6.tex} 255 | \input{content/Part3 Single-variable Calculus/Chapter7.tex} 256 | %------------------------------------------------ 257 | \part{Multi-variable and Vector Calculus} 258 | \input{content/Part4 Multi-variable Calculus/Chapter1.tex} 259 | %------------------------------------------------ 260 | \part{Linear Algebra} 261 | \input{content/Part5 Linear Algebra/Chapter1.tex} 262 | %------------------------------------------------ 263 | \part{Probability and Combinatorics} 264 | \input{content/Part6 Probability and Combinatorics/1.Counting and Probability Axioms.tex} 265 | \input{content/Part6 Probability and Combinatorics/2.Conditional Probability and Independence.tex} 266 | \input{content/Part6 Probability and Combinatorics/3.Discrete Distribution and Random Variable.tex} 267 | \input{content/Part6 Probability and Combinatorics/4.Continuous Distribution.tex} 268 | \input{content/Part6 Probability and Combinatorics/5.Joint Cumulative Distribution.tex} 269 | \input{content/Part6 Probability and Combinatorics/6.Limit Theory in Probability.tex} 270 | \input{content/Part6 Probability and Combinatorics/7.Stochastic Process.tex} 271 | %------------------------------------------------ 272 | \part{Statistics} 273 | \input{content/Part7 Statistics/Chapter1.tex} 274 | %---------------------------------------------------------------------------------------- 275 | \part{Information Theory} 276 | \input{content/Part8 Basic Information Theory/Chapter1.tex} 277 | 278 | 279 | 280 | %\stopcontents[part] % Manually stop the 'part' table of contents here so the previous Part page table of contents doesn't list the following chapters 281 | 282 | 283 | \end{document} 284 | -------------------------------------------------------------------------------- /main.toc: -------------------------------------------------------------------------------- 1 | \contentsline {part}{\numberline {I}Introductory Topics}{17}{part.1}% 2 | \contentsline {chapter}{\numberline {1}Mathematical Proof Strategy}{19}{chapter.1}% 3 | \contentsline {section}{\numberline {1.1}Propositions and Axiomatic System}{19}{section.1.1}% 4 | \contentsline {section}{\numberline {1.2}Direct Proof}{23}{section.1.2}% 5 | \contentsline {subsection}{\numberline {1.2.1}Exercises}{24}{subsection.1.2.1}% 6 | \contentsline {section}{\numberline {1.3}Proof by Cases}{25}{section.1.3}% 7 | \contentsline {subsection}{\numberline {1.3.1}Exercises}{26}{subsection.1.3.1}% 8 | \contentsline {section}{\numberline {1.4}Indirect Proof}{28}{section.1.4}% 9 | \contentsline {subsection}{\numberline {1.4.1}Proof by Contradiction}{28}{subsection.1.4.1}% 10 | \contentsline {subsection}{\numberline {1.4.2}Proof by Contrapositive}{32}{subsection.1.4.2}% 11 | \contentsline {subsection}{\numberline {1.4.3}Exercises}{32}{subsection.1.4.3}% 12 | \contentsline {section}{\numberline {1.5}Mathematical Induction}{33}{section.1.5}% 13 | \contentsline {subsection}{\numberline {1.5.1}Framework of MI}{34}{subsection.1.5.1}% 14 | \contentsline {subsection}{\numberline {1.5.2}Strong Mathematical Induction}{35}{subsection.1.5.2}% 15 | \contentsline {subsection}{\numberline {1.5.3}Exercises}{36}{subsection.1.5.3}% 16 | \contentsline {chapter}{\numberline {2}Set, Sequence, Function, and Summation}{43}{chapter.2}% 17 | \contentsline {section}{\numberline {2.1}Set}{43}{section.2.1}% 18 | \contentsline {section}{\numberline {2.2}Properties of Sets with Proofs}{44}{section.2.2}% 19 | \contentsline {subsection}{\numberline {2.2.1}Exercises}{47}{subsection.2.2.1}% 20 | \contentsline {section}{\numberline {2.3}Function: a perspective from Set Theory}{49}{section.2.3}% 21 | \contentsline {subsection}{\numberline {2.3.1}Function and Operation on Function}{49}{subsection.2.3.1}% 22 | \contentsline {subsection}{\numberline {2.3.2}Elementary Functions and More on Cartesian Product}{50}{subsection.2.3.2}% 23 | \contentsline {subsection}{\numberline {2.3.3}Partial and Total Function}{55}{subsection.2.3.3}% 24 | \contentsline {subsection}{\numberline {2.3.4}Injective, Surjective, and Bijective Function}{56}{subsection.2.3.4}% 25 | \contentsline {subsection}{\numberline {2.3.5}Exercises}{57}{subsection.2.3.5}% 26 | \contentsline {section}{\numberline {2.4}Summation}{63}{section.2.4}% 27 | \contentsline {subsection}{\numberline {2.4.1}Sigma Notation}{64}{subsection.2.4.1}% 28 | \contentsline {subsection}{\numberline {2.4.2}Properties and Techniques of Sigma Notation}{64}{subsection.2.4.2}% 29 | \contentsline {subsubsection}{\numberline {2.4.2.1}Manipulation of Sigma Notation}{65}{subsubsection.2.4.2.1}% 30 | \contentsline {subsubsection}{\numberline {2.4.2.2}Multiple Sums}{66}{subsubsection.2.4.2.2}% 31 | \contentsline {subsection}{\numberline {2.4.3}Exercises}{67}{subsection.2.4.3}% 32 | \contentsline {section}{\numberline {2.5}Sequence}{72}{section.2.5}% 33 | \contentsline {subsection}{\numberline {2.5.1}Introduction}{72}{subsection.2.5.1}% 34 | \contentsline {subsection}{\numberline {2.5.2}Special Sequences}{73}{subsection.2.5.2}% 35 | \contentsline {subsubsection}{\numberline {2.5.2.1}Algorithmic Sequence}{73}{subsubsection.2.5.2.1}% 36 | \contentsline {subsubsection}{\numberline {2.5.2.2}Geometric Sequence}{75}{subsubsection.2.5.2.2}% 37 | \contentsline {subsubsection}{\numberline {2.5.2.3}characteristic Sequence}{76}{subsubsection.2.5.2.3}% 38 | \contentsline {subsection}{\numberline {2.5.3}Exercises}{76}{subsection.2.5.3}% 39 | \contentsline {chapter}{\numberline {3}Algorithm and Number System}{81}{chapter.3}% 40 | \contentsline {section}{\numberline {3.1}Numbers}{81}{section.3.1}% 41 | \contentsline {subsection}{\numberline {3.1.1}Typology of Numbers}{81}{subsection.3.1.1}% 42 | \contentsline {subsection}{\numberline {3.1.2}The Real Number System}{83}{subsection.3.1.2}% 43 | \contentsline {subsection}{\numberline {3.1.3}Floor, Ceiling, and Remainder}{85}{subsection.3.1.3}% 44 | \contentsline {subsubsection}{\numberline {3.1.3.1}Properties of Integer Function}{85}{subsubsection.3.1.3.1}% 45 | \contentsline {subsubsection}{\numberline {3.1.3.2}Remainder and Integer Function}{86}{subsubsection.3.1.3.2}% 46 | \contentsline {subsection}{\numberline {3.1.4}exercises}{87}{subsection.3.1.4}% 47 | \contentsline {section}{\numberline {3.2}Algorithm and Algorithm Analysis}{89}{section.3.2}% 48 | \contentsline {subsection}{\numberline {3.2.1}Algorithm}{89}{subsection.3.2.1}% 49 | \contentsline {subsubsection}{\numberline {3.2.1.1}What is an Algorithm?}{89}{subsubsection.3.2.1.1}% 50 | \contentsline {subsubsection}{\numberline {3.2.1.2}Pseudocode}{90}{subsubsection.3.2.1.2}% 51 | \contentsline {subsection}{\numberline {3.2.2}Algorithm Analysis}{91}{subsection.3.2.2}% 52 | \contentsline {subsubsection}{\numberline {3.2.2.1}Time Complexity}{92}{subsubsection.3.2.2.1}% 53 | \contentsline {subsubsection}{\numberline {3.2.2.2}Space Complexity}{93}{subsubsection.3.2.2.2}% 54 | \contentsline {subsection}{\numberline {3.2.3}Exercises}{93}{subsection.3.2.3}% 55 | \contentsline {chapter}{\numberline {4}Inequality}{97}{chapter.4}% 56 | \contentsline {section}{\numberline {4.1}Inequality basics}{97}{section.4.1}% 57 | \contentsline {subsection}{\numberline {4.1.1}Exercises}{98}{subsection.4.1.1}% 58 | \contentsline {section}{\numberline {4.2}Solving Quadratic Inequality}{99}{section.4.2}% 59 | \contentsline {section}{\numberline {4.3}important Inequalities}{100}{section.4.3}% 60 | \contentsline {subsection}{\numberline {4.3.1}The Triangle Inequality}{100}{subsection.4.3.1}% 61 | \contentsline {subsection}{\numberline {4.3.2}The Arithmetic-Geometric Mean Inequality}{102}{subsection.4.3.2}% 62 | \contentsline {subsection}{\numberline {4.3.3}Exercises}{106}{subsection.4.3.3}% 63 | \contentsline {subsection}{\numberline {4.3.4}Cauchy-Schwarz Inequality}{111}{subsection.4.3.4}% 64 | \contentsline {subsection}{\numberline {4.3.5}Rearrangement Inequality}{113}{subsection.4.3.5}% 65 | \contentsline {subsection}{\numberline {4.3.6}Exercises}{114}{subsection.4.3.6}% 66 | \contentsline {chapter}{\numberline {5}Complex Number}{125}{chapter.5}% 67 | \contentsline {section}{\numberline {5.1}Algebra of Complex Number}{126}{section.5.1}% 68 | \contentsline {subsection}{\numberline {5.1.1}Exercises}{127}{subsection.5.1.1}% 69 | \contentsline {section}{\numberline {5.2}Point representation of Complex Number}{129}{section.5.2}% 70 | \contentsline {subsection}{\numberline {5.2.1}Exercises}{132}{subsection.5.2.1}% 71 | \contentsline {section}{\numberline {5.3}Vector and Polar Form}{135}{section.5.3}% 72 | \contentsline {subsection}{\numberline {5.3.1}Vector Form of Complex Number}{135}{subsection.5.3.1}% 73 | \contentsline {subsection}{\numberline {5.3.2}Polar Form of Complex Number}{137}{subsection.5.3.2}% 74 | \contentsline {subsubsection}{\numberline {5.3.2.1}The Polar Coordinate}{137}{subsubsection.5.3.2.1}% 75 | \contentsline {subsubsection}{\numberline {5.3.2.2}Polar Expression of Complex Number}{137}{subsubsection.5.3.2.2}% 76 | \contentsline {subsection}{\numberline {5.3.3}Exercises}{140}{subsection.5.3.3}% 77 | \contentsline {section}{\numberline {5.4}Exponential Form}{144}{section.5.4}% 78 | \contentsline {subsection}{\numberline {5.4.1}De Moivre's Theorem}{146}{subsection.5.4.1}% 79 | \contentsline {subsection}{\numberline {5.4.2}Exercises}{147}{subsection.5.4.2}% 80 | \contentsline {section}{\numberline {5.5}Finding Complex Roots}{150}{section.5.5}% 81 | \contentsline {subsection}{\numberline {5.5.1}Solving Quadratic Equations Over the Complex Numbers}{150}{subsection.5.5.1}% 82 | \contentsline {subsection}{\numberline {5.5.2}Solving Polynomial Equations on Complex Number}{153}{subsection.5.5.2}% 83 | \contentsline {subsection}{\numberline {5.5.3}Solving Equation with De Moivre's Theorem}{157}{subsection.5.5.3}% 84 | \contentsline {subsection}{\numberline {5.5.4}Exercises}{160}{subsection.5.5.4}% 85 | \contentsline {part}{\numberline {II}Further Discrete Mathematics and Theories}{167}{part.2}% 86 | \contentsline {chapter}{\numberline {6}Boolean Algebra and Further Logic}{169}{chapter.6}% 87 | \contentsline {section}{\numberline {6.1}Boolean Expression and Truth Table}{169}{section.6.1}% 88 | \contentsline {subsection}{\numberline {6.1.1}Property of Algebra Operation}{169}{subsection.6.1.1}% 89 | \contentsline {subsection}{\numberline {6.1.2}Boolean Expression and Truth Table}{170}{subsection.6.1.2}% 90 | \contentsline {subsection}{\numberline {6.1.3}Boolean Identities}{172}{subsection.6.1.3}% 91 | \contentsline {subsection}{\numberline {6.1.4}Exercises}{175}{subsection.6.1.4}% 92 | \contentsline {section}{\numberline {6.2}Boolean Function}{178}{section.6.2}% 93 | \contentsline {subsection}{\numberline {6.2.1}Representation of Boolean Function}{179}{subsection.6.2.1}% 94 | \contentsline {subsection}{\numberline {6.2.2}Properties of Boolean Function}{181}{subsection.6.2.2}% 95 | \contentsline {subsection}{\numberline {6.2.3}Simplification of Boolean Function}{185}{subsection.6.2.3}% 96 | \contentsline {subsubsection}{\numberline {6.2.3.1}Simplification by Boolean Laws (Algebra)}{185}{subsubsection.6.2.3.1}% 97 | \contentsline {subsubsection}{\numberline {6.2.3.2}Simplification by Karnaugh Maps}{185}{subsubsection.6.2.3.2}% 98 | \contentsline {subsection}{\numberline {6.2.4}Simplification by Quine-McCluskey Method}{188}{subsection.6.2.4}% 99 | \contentsline {subsection}{\numberline {6.2.5}Exercises}{189}{subsection.6.2.5}% 100 | \contentsline {section}{\numberline {6.3}Predicates and Quantifiers}{194}{section.6.3}% 101 | \contentsline {subsection}{\numberline {6.3.1}Predicate}{194}{subsection.6.3.1}% 102 | \contentsline {subsection}{\numberline {6.3.2}Quantifier}{195}{subsection.6.3.2}% 103 | \contentsline {subsection}{\numberline {6.3.3}Nested Quantifier}{197}{subsection.6.3.3}% 104 | \contentsline {section}{\numberline {6.4}Inference and Deduction}{197}{section.6.4}% 105 | \contentsline {chapter}{\numberline {7}Preliminary Number Theory and Cryptography}{199}{chapter.7}% 106 | \contentsline {section}{\numberline {7.1}Divisibility and Modular Arithmetic}{199}{section.7.1}% 107 | \contentsline {subsection}{\numberline {7.1.1}Division and Divisibility}{200}{subsection.7.1.1}% 108 | \contentsline {subsection}{\numberline {7.1.2}Modular Arithmetic}{201}{subsection.7.1.2}% 109 | \contentsline {subsection}{\numberline {7.1.3}Exercises}{204}{subsection.7.1.3}% 110 | \contentsline {section}{\numberline {7.2}Number Representations and Algorithms}{207}{section.7.2}% 111 | \contentsline {subsection}{\numberline {7.2.1}Representations of Numbers and Base Conversion}{207}{subsection.7.2.1}% 112 | \contentsline {subsection}{\numberline {7.2.2}Base Conversion}{208}{subsection.7.2.2}% 113 | \contentsline {subsection}{\numberline {7.2.3}Operation Algorithms of Number}{211}{subsection.7.2.3}% 114 | \contentsline {subsection}{\numberline {7.2.4}Modular Exponentiation Algorithm}{213}{subsection.7.2.4}% 115 | \contentsline {paragraph}{Time Complexity of Algorithm 5}{215}{paragraph*.4}% 116 | \contentsline {subsection}{\numberline {7.2.5}Exercises}{216}{subsection.7.2.5}% 117 | \contentsline {section}{\numberline {7.3}Primes and Greatest Common Divisors}{216}{section.7.3}% 118 | \contentsline {subsection}{\numberline {7.3.1}Primes and Related Algorithms}{216}{subsection.7.3.1}% 119 | \contentsline {subsection}{\numberline {7.3.2}Greatest Common Divisors and Least Common Multiples}{219}{subsection.7.3.2}% 120 | \contentsline {subsection}{\numberline {7.3.3}Exercises}{222}{subsection.7.3.3}% 121 | \contentsline {section}{\numberline {7.4}Solving Congruence}{222}{section.7.4}% 122 | \contentsline {subsection}{\numberline {7.4.1}Linear Congruence}{222}{subsection.7.4.1}% 123 | \contentsline {paragraph}{Solution:}{223}{paragraph*.5}% 124 | \contentsline {subsection}{\numberline {7.4.2}The Chinese Remainder Theorem}{226}{subsection.7.4.2}% 125 | \contentsline {subsection}{\numberline {7.4.3}Fermat's Little Theorem}{230}{subsection.7.4.3}% 126 | \contentsline {chapter}{\numberline {8}Relation}{233}{chapter.8}% 127 | \contentsline {section}{\numberline {8.1}NBG Set Theory and Binary Relation}{233}{section.8.1}% 128 | \contentsline {subsection}{\numberline {8.1.1}Class}{234}{subsection.8.1.1}% 129 | \contentsline {subsection}{\numberline {8.1.2}Binary Relations, Composition and Inverse}{238}{subsection.8.1.2}% 130 | \contentsline {subsection}{\numberline {8.1.3}Mapping, Composition, and Inverse}{241}{subsection.8.1.3}% 131 | \contentsline {subsection}{\numberline {8.1.4}Families of Sets}{243}{subsection.8.1.4}% 132 | \contentsline {subsection}{\numberline {8.1.5}Reflexivity, Symmetry, and Transitivity}{246}{subsection.8.1.5}% 133 | \contentsline {subsection}{\numberline {8.1.6}Exercises}{247}{subsection.8.1.6}% 134 | \contentsline {section}{\numberline {8.2}Representation of Relations}{248}{section.8.2}% 135 | \contentsline {subsection}{\numberline {8.2.1}Representation By Matrix}{248}{subsection.8.2.1}% 136 | \contentsline {subsection}{\numberline {8.2.2}Representation By Digraph}{251}{subsection.8.2.2}% 137 | \contentsline {subsection}{\numberline {8.2.3}Exercises}{253}{subsection.8.2.3}% 138 | \contentsline {section}{\numberline {8.3}Closure of Relations}{255}{section.8.3}% 139 | \contentsline {subsection}{\numberline {8.3.1}Exercises}{255}{subsection.8.3.1}% 140 | \contentsline {section}{\numberline {8.4}Equivalence Relations}{255}{section.8.4}% 141 | \contentsline {subsection}{\numberline {8.4.1}Equivalence}{256}{subsection.8.4.1}% 142 | \contentsline {subsection}{\numberline {8.4.2}Equivalence Classes}{257}{subsection.8.4.2}% 143 | \contentsline {subsection}{\numberline {8.4.3}Exercises}{259}{subsection.8.4.3}% 144 | \contentsline {section}{\numberline {8.5}Order Relations}{262}{section.8.5}% 145 | \contentsline {subsection}{\numberline {8.5.1}Partial, Total, and Well Ordering}{262}{subsection.8.5.1}% 146 | \contentsline {subsection}{\numberline {8.5.2}Lexicographic Order}{264}{subsection.8.5.2}% 147 | \contentsline {subsection}{\numberline {8.5.3}Hasse Diagram}{265}{subsection.8.5.3}% 148 | \contentsline {subsection}{\numberline {8.5.4}Maximal and Minimal Elements}{265}{subsection.8.5.4}% 149 | \contentsline {subsection}{\numberline {8.5.5}Lattices}{265}{subsection.8.5.5}% 150 | \contentsline {subsection}{\numberline {8.5.6}Topological Sorting}{265}{subsection.8.5.6}% 151 | \contentsline {subsection}{\numberline {8.5.7}Exercises}{265}{subsection.8.5.7}% 152 | \contentsline {section}{\numberline {8.6}Special Types of Relations}{267}{section.8.6}% 153 | \contentsline {subsection}{\numberline {8.6.1}Recursive Relations}{267}{subsection.8.6.1}% 154 | \contentsline {subsection}{\numberline {8.6.2}\(n\)-ary Relations}{267}{subsection.8.6.2}% 155 | \contentsline {subsection}{\numberline {8.6.3}Exercises}{267}{subsection.8.6.3}% 156 | \contentsline {chapter}{\numberline {9}Graph Theory}{269}{chapter.9}% 157 | \contentsline {chapter}{\numberline {10}Basics of Abstract Algebra}{271}{chapter.10}% 158 | \contentsline {section}{\numberline {10.1}Fundamentals of Algebraic Structures}{272}{section.10.1}% 159 | \contentsline {subsection}{\numberline {10.1.1}Groups}{272}{subsection.10.1.1}% 160 | \contentsline {subsection}{\numberline {10.1.2}Rings}{272}{subsection.10.1.2}% 161 | \contentsline {subsection}{\numberline {10.1.3}Fields}{272}{subsection.10.1.3}% 162 | \contentsline {subsection}{\numberline {10.1.4}Exercises}{272}{subsection.10.1.4}% 163 | \contentsline {section}{\numberline {10.2}Operations on Algebraic Structures}{272}{section.10.2}% 164 | \contentsline {subsection}{\numberline {10.2.1}Homomorphisms}{272}{subsection.10.2.1}% 165 | \contentsline {subsection}{\numberline {10.2.2}Isomorphisms}{272}{subsection.10.2.2}% 166 | \contentsline {subsection}{\numberline {10.2.3}Exercises}{272}{subsection.10.2.3}% 167 | \contentsline {section}{\numberline {10.3}Applications of Algebraic Structures}{272}{section.10.3}% 168 | \contentsline {subsection}{\numberline {10.3.1}Cryptography}{272}{subsection.10.3.1}% 169 | \contentsline {subsection}{\numberline {10.3.2}Coding Theory}{272}{subsection.10.3.2}% 170 | \contentsline {subsection}{\numberline {10.3.3}Exercises}{272}{subsection.10.3.3}% 171 | \contentsline {chapter}{\numberline {11}Introductory Topology and Category Theory}{273}{chapter.11}% 172 | \contentsline {section}{\numberline {11.1}Basic Topology}{274}{section.11.1}% 173 | \contentsline {subsection}{\numberline {11.1.1}Introduction to Topological Spaces}{276}{subsection.11.1.1}% 174 | \contentsline {subsubsection}{\numberline {11.1.1.1}Open and Closed Sets}{276}{subsubsection.11.1.1.1}% 175 | \contentsline {subsubsection}{\numberline {11.1.1.2}Basis for a Topology}{276}{subsubsection.11.1.1.2}% 176 | \contentsline {subsection}{\numberline {11.1.2}Continuity and Limits}{276}{subsection.11.1.2}% 177 | \contentsline {subsubsection}{\numberline {11.1.2.1}Continuous Functions}{276}{subsubsection.11.1.2.1}% 178 | \contentsline {subsubsection}{\numberline {11.1.2.2}Limit Points and Convergence}{276}{subsubsection.11.1.2.2}% 179 | \contentsline {subsection}{\numberline {11.1.3}Compactness and Connectedness}{276}{subsection.11.1.3}% 180 | \contentsline {subsubsection}{\numberline {11.1.3.1}Compact Spaces}{276}{subsubsection.11.1.3.1}% 181 | \contentsline {subsubsection}{\numberline {11.1.3.2}Connected Spaces}{276}{subsubsection.11.1.3.2}% 182 | \contentsline {subsection}{\numberline {11.1.4}Applications of Topology}{276}{subsection.11.1.4}% 183 | \contentsline {subsubsection}{\numberline {11.1.4.1}Topology in Computer Science}{276}{subsubsection.11.1.4.1}% 184 | \contentsline {subsubsection}{\numberline {11.1.4.2}Topology in Physics}{276}{subsubsection.11.1.4.2}% 185 | \contentsline {section}{\numberline {11.2}Category Theory Fundamentals}{276}{section.11.2}% 186 | \contentsline {subsection}{\numberline {11.2.1}Introduction to Categories}{277}{subsection.11.2.1}% 187 | \contentsline {subsubsection}{\numberline {11.2.1.1}Objects and Morphisms}{277}{subsubsection.11.2.1.1}% 188 | \contentsline {subsubsection}{\numberline {11.2.1.2}Examples of Categories}{277}{subsubsection.11.2.1.2}% 189 | \contentsline {subsection}{\numberline {11.2.2}Functors and Natural Transformations}{277}{subsection.11.2.2}% 190 | \contentsline {subsubsection}{\numberline {11.2.2.1}Definition of Functors}{277}{subsubsection.11.2.2.1}% 191 | \contentsline {subsubsection}{\numberline {11.2.2.2}Natural Transformations between Functors}{277}{subsubsection.11.2.2.2}% 192 | \contentsline {subsection}{\numberline {11.2.3}Limits and Colimits}{277}{subsection.11.2.3}% 193 | \contentsline {subsubsection}{\numberline {11.2.3.1}Universal Properties}{277}{subsubsection.11.2.3.1}% 194 | \contentsline {subsubsection}{\numberline {11.2.3.2}Construction of Limits and Colimits}{277}{subsubsection.11.2.3.2}% 195 | \contentsline {subsection}{\numberline {11.2.4}Applications of Category Theory}{277}{subsection.11.2.4}% 196 | \contentsline {subsubsection}{\numberline {11.2.4.1}Category Theory in Programming Languages}{277}{subsubsection.11.2.4.1}% 197 | \contentsline {subsubsection}{\numberline {11.2.4.2}Category Theory in Logic and Set Theory}{277}{subsubsection.11.2.4.2}% 198 | \contentsline {part}{\numberline {III}Single-variable Calculus}{279}{part.3}% 199 | \contentsline {chapter}{\numberline {12}Function Monotonicity, Parity and Periodicity}{281}{chapter.12}% 200 | \contentsline {section}{\numberline {12.1}Monotonicity of Function}{281}{section.12.1}% 201 | \contentsline {section}{\numberline {12.2}Parity of Function}{281}{section.12.2}% 202 | \contentsline {section}{\numberline {12.3}Periodicity of Function}{281}{section.12.3}% 203 | \contentsline {chapter}{\numberline {13}Abstract and Piece-wise Function}{283}{chapter.13}% 204 | \contentsline {section}{\numberline {13.1}Abstract Function}{283}{section.13.1}% 205 | \contentsline {section}{\numberline {13.2}Piece-wise Function}{283}{section.13.2}% 206 | \contentsline {chapter}{\numberline {14}Limit and Continuity}{285}{chapter.14}% 207 | \contentsline {section}{\numberline {14.1}Limit of Sequence}{285}{section.14.1}% 208 | \contentsline {section}{\numberline {14.2}limit of Function}{285}{section.14.2}% 209 | \contentsline {section}{\numberline {14.3}Continuity}{285}{section.14.3}% 210 | \contentsline {section}{\numberline {14.4}Application of Limit}{285}{section.14.4}% 211 | \contentsline {chapter}{\numberline {15}Differential Calculus}{287}{chapter.15}% 212 | \contentsline {section}{\numberline {15.1}Derivative Basics}{288}{section.15.1}% 213 | \contentsline {subsection}{\numberline {15.1.1}Definition of Derivative}{288}{subsection.15.1.1}% 214 | \contentsline {subsection}{\numberline {15.1.2}Geometric Meaning of Derivative}{288}{subsection.15.1.2}% 215 | \contentsline {subsection}{\numberline {15.1.3}Physical Meaning of Derivative}{288}{subsection.15.1.3}% 216 | \contentsline {section}{\numberline {15.2}Basic Derivative Rules}{288}{section.15.2}% 217 | \contentsline {subsection}{\numberline {15.2.1}Derivatives of Elementary Functions}{288}{subsection.15.2.1}% 218 | \contentsline {subsection}{\numberline {15.2.2}Product Rule, Quotient Rule, Chain Rule}{288}{subsection.15.2.2}% 219 | \contentsline {section}{\numberline {15.3}Higher-order Derivatives}{288}{section.15.3}% 220 | \contentsline {subsection}{\numberline {15.3.1}Second-order Derivatives and Applications}{288}{subsection.15.3.1}% 221 | \contentsline {subsection}{\numberline {15.3.2}Calculation and Significance of Higher-order Derivatives}{288}{subsection.15.3.2}% 222 | \contentsline {section}{\numberline {15.4}Derivatives of Abnormal Function}{288}{section.15.4}% 223 | \contentsline {subsection}{\numberline {15.4.1}Derivatives of Implicit Functions}{288}{subsection.15.4.1}% 224 | \contentsline {subsection}{\numberline {15.4.2}Derivatives of Parametric Equations}{288}{subsection.15.4.2}% 225 | \contentsline {section}{\numberline {15.5}Differentiation}{288}{section.15.5}% 226 | \contentsline {section}{\numberline {15.6}Related Rates}{288}{section.15.6}% 227 | \contentsline {subsection}{\numberline {15.6.1}Relationships between Rates of Change of Different Quantities}{288}{subsection.15.6.1}% 228 | \contentsline {section}{\numberline {15.7}Taylor Series}{288}{section.15.7}% 229 | \contentsline {subsection}{\numberline {15.7.1}Taylor Expansion of Functions}{288}{subsection.15.7.1}% 230 | \contentsline {subsection}{\numberline {15.7.2}Application of Taylor Series}{288}{subsection.15.7.2}% 231 | \contentsline {section}{\numberline {15.8}Applications of Differential Calculus}{288}{section.15.8}% 232 | \contentsline {subsection}{\numberline {15.8.1}Tangents and Normals of Curves}{288}{subsection.15.8.1}% 233 | \contentsline {subsection}{\numberline {15.8.2}Mean Value Theorem in Differential Calculus}{288}{subsection.15.8.2}% 234 | \contentsline {subsubsection}{\numberline {15.8.2.1}Fermat's Theorem}{288}{subsubsection.15.8.2.1}% 235 | \contentsline {subsubsection}{\numberline {15.8.2.2}Rolle's Theorem}{288}{subsubsection.15.8.2.2}% 236 | \contentsline {subsubsection}{\numberline {15.8.2.3}Lagrange's Mean Value Theorem}{288}{subsubsection.15.8.2.3}% 237 | \contentsline {subsubsection}{\numberline {15.8.2.4}Cauchy's Mean Value Theorem}{288}{subsubsection.15.8.2.4}% 238 | \contentsline {subsubsection}{\numberline {15.8.2.5}Indeterminate Form Limit}{288}{subsubsection.15.8.2.5}% 239 | \contentsline {subsection}{\numberline {15.8.3}Extrema Problems and Optimization}{288}{subsection.15.8.3}% 240 | \contentsline {chapter}{\numberline {16}integral calculus}{289}{chapter.16}% 241 | \contentsline {section}{\numberline {16.1}Fundamentals of Integration}{290}{section.16.1}% 242 | \contentsline {subsection}{\numberline {16.1.1}Definition of the Integral}{290}{subsection.16.1.1}% 243 | \contentsline {subsection}{\numberline {16.1.2}Properties of Integrals}{290}{subsection.16.1.2}% 244 | \contentsline {subsection}{\numberline {16.1.3}The Fundamental Theorem of Calculus}{290}{subsection.16.1.3}% 245 | \contentsline {section}{\numberline {16.2}Techniques of Integration}{290}{section.16.2}% 246 | \contentsline {subsection}{\numberline {16.2.1}Basic Integration Formulas}{290}{subsection.16.2.1}% 247 | \contentsline {subsection}{\numberline {16.2.2}Integration by Substitution}{290}{subsection.16.2.2}% 248 | \contentsline {subsection}{\numberline {16.2.3}Integration by Parts}{290}{subsection.16.2.3}% 249 | \contentsline {subsection}{\numberline {16.2.4}Trigonometric Integrals}{290}{subsection.16.2.4}% 250 | \contentsline {subsection}{\numberline {16.2.5}Partial Fractions}{290}{subsection.16.2.5}% 251 | \contentsline {section}{\numberline {16.3}Applications of Integration}{290}{section.16.3}% 252 | \contentsline {subsection}{\numberline {16.3.1}Area Under Curves}{290}{subsection.16.3.1}% 253 | \contentsline {subsection}{\numberline {16.3.2}Volumes of Solids of Revolution}{290}{subsection.16.3.2}% 254 | \contentsline {subsection}{\numberline {16.3.3}Arc Length and Surface Area}{290}{subsection.16.3.3}% 255 | \contentsline {subsection}{\numberline {16.3.4}Center of Mass and Moments}{290}{subsection.16.3.4}% 256 | \contentsline {section}{\numberline {16.4}Improper Integrals}{290}{section.16.4}% 257 | \contentsline {subsection}{\numberline {16.4.1}Convergence and Divergence of Improper Integrals}{290}{subsection.16.4.1}% 258 | \contentsline {subsection}{\numberline {16.4.2}Applications of Improper Integrals}{290}{subsection.16.4.2}% 259 | \contentsline {section}{\numberline {16.5}Numerical Integration Methods}{290}{section.16.5}% 260 | \contentsline {subsection}{\numberline {16.5.1}The Trapezoidal Rule}{290}{subsection.16.5.1}% 261 | \contentsline {subsection}{\numberline {16.5.2}Simpson's Rule}{290}{subsection.16.5.2}% 262 | \contentsline {chapter}{\numberline {17}Differential Equation}{291}{chapter.17}% 263 | \contentsline {chapter}{\numberline {18}Infinite Series}{293}{chapter.18}% 264 | \contentsline {part}{\numberline {IV}Multi-variable and Vector Calculus}{295}{part.4}% 265 | \contentsline {chapter}{\numberline {19}integral calculus}{297}{chapter.19}% 266 | \contentsline {chapter}{\numberline {20}Introduction to Multivariable Functions}{299}{chapter.20}% 267 | \contentsline {subsection}{\numberline {20.0.1}Concepts of Multivariable Functions}{299}{subsection.20.0.1}% 268 | \contentsline {subsection}{\numberline {20.0.2}Graphs and Contour Plots}{299}{subsection.20.0.2}% 269 | \contentsline {chapter}{\numberline {21}Partial Derivatives}{301}{chapter.21}% 270 | \contentsline {subsection}{\numberline {21.0.1}Definition and Interpretation}{301}{subsection.21.0.1}% 271 | \contentsline {subsection}{\numberline {21.0.2}Higher-Order Partial Derivatives}{301}{subsection.21.0.2}% 272 | \contentsline {subsection}{\numberline {21.0.3}Chain Rule in Multiple Variables}{301}{subsection.21.0.3}% 273 | \contentsline {chapter}{\numberline {22}Multiple Integrals}{303}{chapter.22}% 274 | \contentsline {subsection}{\numberline {22.0.1}Double Integrals}{303}{subsection.22.0.1}% 275 | \contentsline {subsubsection}{\numberline {22.0.1.1}Iterated Integrals}{303}{subsubsection.22.0.1.1}% 276 | \contentsline {subsubsection}{\numberline {22.0.1.2}Double Integrals over General Regions}{303}{subsubsection.22.0.1.2}% 277 | \contentsline {subsection}{\numberline {22.0.2}Triple Integrals}{303}{subsection.22.0.2}% 278 | \contentsline {subsubsection}{\numberline {22.0.2.1}Cylindrical and Spherical Coordinates}{303}{subsubsection.22.0.2.1}% 279 | \contentsline {chapter}{\numberline {23}Vector Calculus}{305}{chapter.23}% 280 | \contentsline {section}{\numberline {23.1}Vector Fields}{305}{section.23.1}% 281 | \contentsline {section}{\numberline {23.2}Gradient, Divergence, and Curl}{305}{section.23.2}% 282 | \contentsline {section}{\numberline {23.3}Line and Surface Integrals}{305}{section.23.3}% 283 | \contentsline {part}{\numberline {V}Linear Algebra}{307}{part.5}% 284 | \contentsline {chapter}{\numberline {24}Vectors Space and the Geometry of Space}{309}{chapter.24}% 285 | \contentsline {chapter}{\numberline {25}Matrices and Systems of Equations}{311}{chapter.25}% 286 | \contentsline {chapter}{\numberline {26}Determinant of Matrix}{313}{chapter.26}% 287 | \contentsline {chapter}{\numberline {27}Orthogonality}{315}{chapter.27}% 288 | \contentsline {chapter}{\numberline {28}Linear Transformations}{317}{chapter.28}% 289 | \contentsline {chapter}{\numberline {29}Eigenvalues and Eigenvector}{319}{chapter.29}% 290 | \contentsline {chapter}{\numberline {30}Singular Value Decomposition}{321}{chapter.30}% 291 | \contentsline {chapter}{\numberline {31}Complex Vector and Matrices}{323}{chapter.31}% 292 | \contentsline {chapter}{\numberline {32}Matrix Differential Calculus}{325}{chapter.32}% 293 | \contentsline {part}{\numberline {VI}Probability and Combinatorics}{327}{part.6}% 294 | \contentsline {chapter}{\numberline {33}Introduction to Counting and Probability }{329}{chapter.33}% 295 | \contentsline {section}{\numberline {33.1}Counting Principal}{329}{section.33.1}% 296 | \contentsline {subsection}{\numberline {33.1.1}Principal of Counting}{329}{subsection.33.1.1}% 297 | \contentsline {subsection}{\numberline {33.1.2}Pigeonhole Theorem}{333}{subsection.33.1.2}% 298 | \contentsline {subsection}{\numberline {33.1.3}Exercises}{334}{subsection.33.1.3}% 299 | \contentsline {section}{\numberline {33.2}Combination and Permutation with applications}{337}{section.33.2}% 300 | \contentsline {subsection}{\numberline {33.2.1}Permutation}{337}{subsection.33.2.1}% 301 | \contentsline {subsection}{\numberline {33.2.2}Combination}{339}{subsection.33.2.2}% 302 | \contentsline {subsection}{\numberline {33.2.3}Further Interpretation of Counting with Set Theory}{342}{subsection.33.2.3}% 303 | \contentsline {subsection}{\numberline {33.2.4}Binomial and Multinomial Theorem}{344}{subsection.33.2.4}% 304 | \contentsline {subsection}{\numberline {33.2.5}Catalan Number}{348}{subsection.33.2.5}% 305 | \contentsline {subsection}{\numberline {33.2.6}String Number}{348}{subsection.33.2.6}% 306 | \contentsline {subsection}{\numberline {33.2.7}Exercises}{348}{subsection.33.2.7}% 307 | \contentsline {section}{\numberline {33.3}Axioms of Probability}{351}{section.33.3}% 308 | \contentsline {subsection}{\numberline {33.3.1}Sample Space and Events}{352}{subsection.33.3.1}% 309 | \contentsline {subsection}{\numberline {33.3.2}Probability Axioms}{353}{subsection.33.3.2}% 310 | \contentsline {subsection}{\numberline {33.3.3}Exercises}{357}{subsection.33.3.3}% 311 | \contentsline {section}{\numberline {33.4}Finding Probability with Counting}{369}{section.33.4}% 312 | \contentsline {subsection}{\numberline {33.4.1}Some Basic Problems}{369}{subsection.33.4.1}% 313 | \contentsline {subsection}{\numberline {33.4.2}Further Problems}{371}{subsection.33.4.2}% 314 | \contentsline {subsection}{\numberline {33.4.3}Exercises}{373}{subsection.33.4.3}% 315 | \contentsline {chapter}{\numberline {34}Conditional Probability and Independence of Events}{377}{chapter.34}% 316 | \contentsline {section}{\numberline {34.1}Conditional Probability}{378}{section.34.1}% 317 | \contentsline {subsection}{\numberline {34.1.1}Basic Conditional Probability}{378}{subsection.34.1.1}% 318 | \contentsline {subsection}{\numberline {34.1.2}Exercises}{385}{subsection.34.1.2}% 319 | \contentsline {section}{\numberline {34.2}Bayes's Theorem}{385}{section.34.2}% 320 | \contentsline {subsection}{\numberline {34.2.1}Bayes's Theorem and Bayesian Thinking}{385}{subsection.34.2.1}% 321 | \contentsline {subsection}{\numberline {34.2.2}Exercises}{391}{subsection.34.2.2}% 322 | \contentsline {section}{\numberline {34.3}Independence of Events}{391}{section.34.3}% 323 | \contentsline {subsection}{\numberline {34.3.1}Definition of Independence}{391}{subsection.34.3.1}% 324 | \contentsline {subsection}{\numberline {34.3.2}Multiple Independence}{392}{subsection.34.3.2}% 325 | \contentsline {subsection}{\numberline {34.3.3}Exercises}{395}{subsection.34.3.3}% 326 | \contentsline {section}{\numberline {34.4}Further Conditional Probability}{396}{section.34.4}% 327 | \contentsline {subsection}{\numberline {34.4.1}Probability Axiom in Conditional Probability}{396}{subsection.34.4.1}% 328 | \contentsline {subsection}{\numberline {34.4.2}Multi-conditional Probability}{399}{subsection.34.4.2}% 329 | \contentsline {subsection}{\numberline {34.4.3}Exercises}{400}{subsection.34.4.3}% 330 | \contentsline {chapter}{\numberline {35}Random Variable and Discrete Distribution}{403}{chapter.35}% 331 | \contentsline {section}{\numberline {35.1}Random Variable}{403}{section.35.1}% 332 | \contentsline {subsection}{\numberline {35.1.1}Analysis of Random Variables}{405}{subsection.35.1.1}% 333 | \contentsline {subsection}{\numberline {35.1.2}Discrete Random Variables and Discrete Distributions}{406}{subsection.35.1.2}% 334 | \contentsline {subsection}{\numberline {35.1.3}Exercises}{408}{subsection.35.1.3}% 335 | \contentsline {section}{\numberline {35.2}Expectation and Variance}{410}{section.35.2}% 336 | \contentsline {subsection}{\numberline {35.2.1}Expectation of Discrete Random Variable}{411}{subsection.35.2.1}% 337 | \contentsline {subsection}{\numberline {35.2.2}Expectation of Function and Linearity}{412}{subsection.35.2.2}% 338 | \contentsline {subsection}{\numberline {35.2.3}Variance}{416}{subsection.35.2.3}% 339 | \contentsline {subsection}{\numberline {35.2.4}PDF and CDF}{418}{subsection.35.2.4}% 340 | \contentsline {subsection}{\numberline {35.2.5}Composition of Dsicrete Random Variable}{418}{subsection.35.2.5}% 341 | \contentsline {subsection}{\numberline {35.2.6}Exercises}{419}{subsection.35.2.6}% 342 | \contentsline {section}{\numberline {35.3}Common Discrete Distributions}{428}{section.35.3}% 343 | \contentsline {subsection}{\numberline {35.3.1}Bernoulli and Binomial Distribution}{428}{subsection.35.3.1}% 344 | \contentsline {subsection}{\numberline {35.3.2}Poisson Distribution}{432}{subsection.35.3.2}% 345 | \contentsline {subsection}{\numberline {35.3.3}Geometric Distribution}{437}{subsection.35.3.3}% 346 | \contentsline {subsection}{\numberline {35.3.4}Hypergeometric Distribution}{440}{subsection.35.3.4}% 347 | \contentsline {subsection}{\numberline {35.3.5}Exercises}{444}{subsection.35.3.5}% 348 | \contentsline {section}{\numberline {35.4}Other Discrete Distributions}{445}{section.35.4}% 349 | \contentsline {subsection}{\numberline {35.4.1}Discrete Uniform Distribution}{445}{subsection.35.4.1}% 350 | \contentsline {subsection}{\numberline {35.4.2}Negative Binomial Distribution}{445}{subsection.35.4.2}% 351 | \contentsline {subsection}{\numberline {35.4.3}Zeta-Bernoulli Distribution}{445}{subsection.35.4.3}% 352 | \contentsline {subsection}{\numberline {35.4.4}Logarithmic Series Distribution}{445}{subsection.35.4.4}% 353 | \contentsline {subsection}{\numberline {35.4.5}Zipf's Distribution}{445}{subsection.35.4.5}% 354 | \contentsline {subsection}{\numberline {35.4.6}Exercises}{445}{subsection.35.4.6}% 355 | \contentsline {section}{\numberline {35.5}Properties of Random Variable, PDF, and CDF}{445}{section.35.5}% 356 | \contentsline {subsection}{\numberline {35.5.1}Exercises}{445}{subsection.35.5.1}% 357 | \contentsline {chapter}{\numberline {36}Continuous Distribution}{447}{chapter.36}% 358 | \contentsline {chapter}{\numberline {37}Joint Cumulative Distribution}{449}{chapter.37}% 359 | \contentsline {chapter}{\numberline {38}Limit Theory in Probability}{451}{chapter.38}% 360 | \contentsline {chapter}{\numberline {39}Stochastic Process}{453}{chapter.39}% 361 | \contentsline {part}{\numberline {VII}Statistics}{455}{part.7}% 362 | \contentsline {chapter}{\numberline {40}Sampling and Parameters}{457}{chapter.40}% 363 | \contentsline {chapter}{\numberline {41}Descriptive Statistics}{459}{chapter.41}% 364 | \contentsline {chapter}{\numberline {42}Graphical Statistics}{461}{chapter.42}% 365 | \contentsline {chapter}{\numberline {43}Statistical Inference}{463}{chapter.43}% 366 | \contentsline {section}{\numberline {43.1}Parameter Estimation}{463}{section.43.1}% 367 | \contentsline {section}{\numberline {43.2}Interval Estimation}{463}{section.43.2}% 368 | \contentsline {section}{\numberline {43.3}Hypothesis Testing}{463}{section.43.3}% 369 | \contentsline {section}{\numberline {43.4}Variance Inference}{463}{section.43.4}% 370 | \contentsline {section}{\numberline {43.5}Bayesian Inference}{463}{section.43.5}% 371 | \contentsline {chapter}{\numberline {44}Hypothesis Testing}{465}{chapter.44}% 372 | \contentsline {chapter}{\numberline {45}Regression and Regressive Analysis}{467}{chapter.45}% 373 | \contentsline {chapter}{\numberline {46}Basic Multi-variable Statistical analysis}{469}{chapter.46}% 374 | \contentsline {part}{\numberline {VIII}Information Theory}{471}{part.8}% 375 | \contentsline {chapter}{\numberline {47}Measuring of Information}{473}{chapter.47}% 376 | \contentsline {chapter}{\numberline {48}Information Entropy}{475}{chapter.48}% 377 | \contentsline {chapter}{\numberline {49}Joined Entropy and Conditional Entropy}{477}{chapter.49}% 378 | \contentsline {chapter}{\numberline {50}Cross Entropy and Relative Entropy}{479}{chapter.50}% 379 | \contentsline {chapter}{\numberline {51}Mutual Information}{481}{chapter.51}% 380 | \contentsline {chapter}{\numberline {52}Differential Entropy}{483}{chapter.52}% 381 | \contentsfinish 382 | -------------------------------------------------------------------------------- /numbers.svg: -------------------------------------------------------------------------------- 1 | 2 | 3 | 4 | 18 | 20 | 27 | 34 | 41 | 48 | 55 | 62 | 69 | 76 | 83 | 90 | 97 | 104 | 111 | 118 | 125 | 132 | 139 | 146 | 147 | 165 | 167 | 168 | 170 | image/svg+xml 171 | 173 | 174 | 175 | 176 | 177 | 182 | 187 | 192 | 197 | 200 | 204 | 208 | 209 | 213 | 217 | 218 | 223 | 228 | 231 | 235 | 238 | 239 | Rational 250 | 251 | 255 | 259 | 260 | 265 | 270 | 274 | 278 | 279 | 283 | 287 | 288 | Integers 299 | Natural 310 | Real 321 | Irrational 332 | Complex 343 | Imaginary 354 | 355 | 356 | -------------------------------------------------------------------------------- /sample.bib: -------------------------------------------------------------------------------- 1 | @book{Smith:2021qr, 2 | title = {{B}ook {T}itle}, 3 | publisher = {Publisher}, 4 | author = {Smith, J.~M. and Jones, A.~B.}, 5 | year = {2021}, 6 | edition = {7th}, 7 | } 8 | 9 | @article{Smith:2022jd, 10 | author = {Jones, A.~B. and Smith, J.~M.}, 11 | title = {{A}rticle {T}itle}, 12 | journal = {Journal title}, 13 | year = {2022}, 14 | volume = {13}, 15 | pages = {123-456}, 16 | number = {52}, 17 | month = {3}, 18 | publisher = {Publisher}, 19 | doi = {10.1038/s41586-021-03616-x}, 20 | } --------------------------------------------------------------------------------