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- https://math.libretexts.org/Bookshelves/Combinatorics_and_Discrete_Mathematics/Combinatorics_and_Graph_Theory_(Guichard)/01%3A_Fundamentals/1.02%3A_ExamplesSuppose we have a chess board, and a collection of tiles, like dominoes, each of which is the size of two squares on the chess board. First we need to be clear on the rules: the board is covered if th...Suppose we have a chess board, and a collection of tiles, like dominoes, each of which is the size of two squares on the chess board. First we need to be clear on the rules: the board is covered if the dominoes are laid down so that each covers exactly two squares of the board; no dominoes overlap; and every square is covered. To make the problem more interesting, we allow the board to be rectangular of any size, and we allow some squares to be removed from the board.
- https://math.libretexts.org/Bookshelves/Calculus/Calculus_(Guichard)/17%3A_Differential_EquationsDifferential Equations are the language in which the laws of nature are expressed. Understanding properties of solutions of differential equations is fundamental to much of contemporary science and en...Differential Equations are the language in which the laws of nature are expressed. Understanding properties of solutions of differential equations is fundamental to much of contemporary science and engineering. Ordinary differential equations (ODE's) deal with functions of one variable, which can often be thought of as time.
- https://math.libretexts.org/Bookshelves/Calculus/Calculus_(Guichard)/03%3A_Rules_for_Finding_DerivativesIt is tedious to compute a limit every time we need to know the derivative of a function. Fortunately, we can develop a small collection of examples and rules that allow us to compute the derivative o...It is tedious to compute a limit every time we need to know the derivative of a function. Fortunately, we can develop a small collection of examples and rules that allow us to compute the derivative of almost any function we are likely to encounter. Many functions involve quantities raised to a constant power, such as polynomials and more complicated combinations . So we start by examining powers of a single variable; this gives us a building block for more complicated examples.
- https://math.libretexts.org/Under_Construction/Development_Details/Media_Repo/Appendix_B%3A_Answers_and_Hints_for_Selected_Exercises
- https://math.libretexts.org/Bookshelves/Combinatorics_and_Discrete_Mathematics/Combinatorics_and_Graph_Theory_(Guichard)/00%3A_Front_Matter/04%3A_LicensingA detailed breakdown of this resource's licensing can be found in Back Matter/Detailed Licensing.
- https://math.libretexts.org/Bookshelves/Calculus/Exercises_(Calculus)/Exercises%3A_Calculus_(Guichard)/zz%3A_Back_Matter/10%3A_Index
- https://math.libretexts.org/Bookshelves/Combinatorics_and_Discrete_Mathematics/Combinatorics_and_Graph_Theory_(Guichard)/05%3A_Graph_Theory/5.06%3A_Optimal_Spanning_TreesFor example, if a graph represents a network of roads, the weight of an edge might be the length of the road between its two endpoints, or the amount of time required to travel from one endpoint to th...For example, if a graph represents a network of roads, the weight of an edge might be the length of the road between its two endpoints, or the amount of time required to travel from one endpoint to the other, or the cost to bury cable along the road from one end to the other.
- https://math.libretexts.org/Bookshelves/Combinatorics_and_Discrete_Mathematics/Combinatorics_and_Graph_Theory_(Guichard)/04%3A_Systems_of_Distinct_Representatives/4.E%3A_Systems_of_Distinct_Representatives_(Exercises)An \(n\times n\) Latin square \(A\) is symmetric if it is symmetric around the main diagonal, that is, \(A_{i,j}=A_{j,i}\) for all \(i\) and \(j\). The transpose \(A^\top\) of a Latin square \(A\) is ...An \(n\times n\) Latin square \(A\) is symmetric if it is symmetric around the main diagonal, that is, \(A_{i,j}=A_{j,i}\) for all \(i\) and \(j\). The transpose \(A^\top\) of a Latin square \(A\) is the reflection of \(A\) across the main diagonal, so that \(A_{i,j}^\top=A_{j,i}\). Show directly that that the size of a minimum vertex cover in \(G\) is the minimum value of \(n-k+|\bigcup_{j=1}^k A_{i_j}|\), as mentioned above.
- https://math.libretexts.org/Bookshelves/Calculus/Exercises_(Calculus)/Exercises%3A_Calculus_(Guichard)/01%3A_Analytic_GeometryThese are homework exercises to accompany David Guichard's "General Calculus" Textmap.
- https://math.libretexts.org/Bookshelves/Combinatorics_and_Discrete_Mathematics/Combinatorics_and_Graph_Theory_(Guichard)/00%3A_Front_Matter
- https://math.libretexts.org/Bookshelves/Combinatorics_and_Discrete_Mathematics/Combinatorics_and_Graph_Theory_(Guichard)/05%3A_Graph_Theory/5.11%3A_Directed_GraphsThus, the entire sum \(S\) has value \[\sum_{e\in E_s^+} f(e)-\sum_{e\in E_s^-}f(e).\nonumber\] On the other hand, we can write the sum \(S\) as \[ \sum_{v\in U}\sum_{e\in E_v^+}f(e)- \sum_{v\in U}\su...Thus, the entire sum \(S\) has value \[\sum_{e\in E_s^+} f(e)-\sum_{e\in E_s^-}f(e).\nonumber\] On the other hand, we can write the sum \(S\) as \[ \sum_{v\in U}\sum_{e\in E_v^+}f(e)- \sum_{v\in U}\sum_{e\in E_v^-}f(e). \nonumber\] Every arc \(e=(x,y)\) with both \(x\) and \(y\) in \(U\) appears in both sums, that is, in \[\sum_{v\in U}\sum_{e\in E_v^+}f(e),\nonumber\] when \(v=x\), and in \[\sum_{v\in U}\sum_{e\in E_v^-}f(e),\nonumber\] when \(v=y\), and so the flow in such arcs contributes \(…
