AMAT 583

Test #2

Spring 2021

拓扑数据分析代写 The procedures and rules are exactly the same as in the previous test. The total number of points is 30.Your solutions need to be written

The procedures and rules are exactly the same as in the previous test. The total number of points is 30. Your solutions need to be written out carefully either by hand or typed up if you know LaTeX. You can do that on paper and then scan or on your electronic  device such as a tablet. Either way your submission needs to be in PDF format by email to bgoldfarb@albany.edu. Please make sure you write your name at the top of your paper. The deadline for the electronic submission is midnight on Monday, March 29. 拓扑数据分析代写

A quick note for those of you who work together. 拓扑数据分析代写

Most of these problems come from homework assignments. If you solved them together with someone, you need to sit down independently and write up solutions without consulting with others. The essence of the solution can be the same but the narrative should be different.

Question 1 (6 points). Give an example of a set P in R² and a value of the parameter t for which C_t(P) and R_t(P) are not the same complex.拓扑数据分析代写

Question 2 (6 points). Prove that the Hamming distance gives a metric on binary se- quences of fixed length. You need to verify all of the axioms for metric spaces in this case.

Question 3 (6 points). Prove that the vertices V(G) in a graph G is a metric space where the distance is measured by lengths of shortest paths between the vertices.

You need to verify all of the axioms for metric spaces in this case. 拓扑数据分析代写

Question 4 (6 points). Consider the following subset P of the 2-dimensional plane:

{(0, 1), (0, 2), (0, 6), (0, 7), (1, 2), (1, 6), (5, 2), (5, 3)}.拓扑数据分析代写

Construct the clustering barcode for this set P with respect to the max distance.

Question 5 (6 points). Find all real-valued solutions to the following system consisting of just one linear equation

2x₁ + x₂ − 2x₃ + x₄ + 5x₅ + x₆ = 9.

Make sure to identify free variables and write the answer in terms of a particular solu- tion plus a linear combination of homogeneous solutions.