Cocalc Chapter 3 Tutorial Ipynb

Often when looking at a network, we want to find the most "important" nodes, for some definition of important. The most basic measure of centrality is the degree, or number of links attached to a node. Let's look at the Enron executive email graph: Edge lists are a simple, plain text format for storing graphs. Since this simple file format doesn't contain information about data types, all node names are assumed to be strings by default. When the node names are given by integers, as they are in this example, we should specify the nodetype=int keyword argument to avoid confusion with the node names.

We're going to make use of Python's built-in max function in order to find the node with the highest degree. First, let us recall what the max function does. At its most basic, the max function returns the "greatest" item in a sequence: Let us draw the curve given by the implicit equation C={z∈C:∣cos⁡(x4)∣=1}.\mathcal{C} = \left\{ z \in \mathbb{C} : |\cos(x^4)| = 1\right\}.C={z∈C:∣cos(x4)∣=1}. Example. (Random walk) Starting from the origin O, a particle moves a distance ℓ\ellℓ every ttt seconds, in a random direction, independently of the preceding moves.

Let us draw an example of particle trajectory. The red line goes from the initial to the final position. Example. (Uniformly distributed sequences) Given a real sequence (un)n∈N∗(u_n)_{n\in \mathbb{N}^*}(un​)n∈N∗​, we construct the polygonal line whose successive vertices are the points in the complex plane zN=∑n≤Ne2iπun.z_N = \sum_{n\leq N} e^{2i \pi u_n}.zN​=n≤N∑​e2iπun​. un=n2u_n = n\sqrt{2}un​=n2​ and N=200N=200N=200 un=nln⁡(n)2u_n = n \ln(n) \sqrt{2}un​=nln(n)2​ and N=10000N=10000N=10000

CoCalc is a cloud-based service that provides infrastructure and services that are useful for running courses based on Jupyter Notebooks. It is used for teaching by Universities around the world. Basic access, without internet access, is free; however, it is well-worth paying for premium access if you are going to support a class full of students or need internet access. Paying users get more resources and support - vital if you are to rely on it for paying students. Read more about how how to pay for a course in the CoCalc Wiki. All material moved to the more comprehensive CoCalc Manual

For a list of authors see the contributors section. Ready to explore the elegant mathematical theory behind linear programming optimization? Next: Linear Programming with SageMath in CoCalc - Chapter 4 Discover duality theory and sensitivity analysis - the theoretical foundations of optimization. This notebook contains Chapter 3 from the main Linear Programming with SageMath in CoCalc notebook. For the complete course, please refer to the main notebook: Linear Programming with SageMath in CoCalc.ipynb

This notebook contains Chapter 3 from the main Combinatorics Fundamentals with Python in CoCalc notebook. For the complete course, please refer to the main notebook: Combinatorics Fundamentals with Python in CoCalc.ipynb A combination is a selection of objects where order does not matter. Combinations of n objects taken r at a time: C(n,r)=(nr)=n!r!(n−r)!C(n,r) = \binom{n}{r} = \frac{n!}{r!(n-r)!}C(n,r)=(rn​)=r!(n−r)!n!​ (nr)=(nn−r)\binom{n}{r} = \binom{n}{n-r}(rn​)=(n−rn​) (Symmetry) The physical interpretation of determinants is to compute the area enclosed by vectors.

For instance, consider the matrix A=[abcd] A = \left[\begin{matrix} a & b \\ c & d \end{matrix}\right] A=[ac​bd​] The determinant represents the area of the parallelogram formed by the vectors [ac]and[bd] \left[\begin{matrix} a \\... Here we demonstrate with a matrix [2003] \left[\begin{matrix} 2 & 0\cr 0 & 3 \end{matrix}\right] [20​03​] It's also easy to understand the area formed by these two vectors are actually a rectangle. If the matrix is not diagonal, the area will take the form of a parallelogram. Similarly, in 3D space, the determinant is the volume of a parallelepiped. First let's draw a parallelogram. However, determinant can be a negative number, it means we flip the area like flipping a piece of paper.

What if two vectors are linearly dependent? The area between vectors will be zero. To create a vector in Sage, use the vector command. Exercise: Create the vector x=(1,2,…,100)x = (1, 2, \ldots, 100)x=(1,2,…,100). Exercise: Create the vector y=(12,22,…,1002)y = (1^2, 2^2, \ldots, 100^2)y=(12,22,…,1002). Exercise: Find the dot product of x and y.

[The above problems are essentially the first problem on Exercise Set 1 of William Stein's Math 480b.] To create a vector in Sage, use the vector command. Exercise: Create the vector x=(1,2,…,100)x = (1, 2, \ldots, 100)x=(1,2,…,100). Exercise: Create the vector y=(12,22,…,1002)y = (1^2, 2^2, \ldots, 100^2)y=(12,22,…,1002). Exercise: Find the dot product of x and y. [The above problems are essentially the first problem on Exercise Set 1 of William Stein's Math 480b.]

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