11 Solutions for active people

Problem Set 11 Solutions

We will remove vertices from a given graph G of treewidth k in a good elimination ordering, and we can assume w.l.o.g. that G is connected. Some cliques in a current graph will be active cliques. For each active clique C and for each subset S of vertices in C, we store information if there exists a vertex cover of the original graph G that chooses exactly vertices in S out of the vertices in C....

Homework 11 Solutions

38. Following the hint, suppose I is a non-zero ideal (since the zero ideal, (0), is clearly principal). Then there exists some a ∈ I with a 6= 0. Then, either a is positive, or −a is positive, and also in I since I is a subring of Z. Thus, I contains positive elements. Now, let S = {x ∈ I |x > 0}. By the Well-Ordering Property of Z, S contains a smallest element, call it c. We claim that, in f...

Math 215 HW #11 Solutions

Math 317 HW #11 Solutions

Proof. Suppose f is differentiable and f ′ is continuous on [a, b]. Then, since [a, b] is compact, f ′ is bounded. In other words, there exists M > 0 such that |f ′(x)| ≤ M for all x ∈ [a, b]. Then, for any x, y ∈ [a, b], the Mean Value Theorem guarantees that there exists c between x and y such that ∣∣∣∣f(x)− f(y) x− y ∣∣∣∣ = |f ′(c)| ≤M. Since the choice of x and y was arbitrary, we see that ...

Cybersecurity Solutions for Active Power Distribution Networks

An active distribution network (ADN) is an electrical-power distribution network that implements a real-time monitoring and control of the electrical resources and the grid. Effective monitoring and control in an ADN is realised by deploying a large number of sensing and actuating intelligent electronic devices (IEDs) and a reliable two-way communication infrastructure that facilitates the tran...