$\forall a > 0$ $\sum_{n=1}^{\infty} f(na)$ is convergent. Prove that $\int_{0}^{\infty}f(x) dx$ is convergent.

Hi can you help me solve this exercise? Thanks. Let $ f: [0;+\infty) \to \mathbb{R}$ be nonnegative and continuous function. Suppose $ \forall a > 0$ $ \sum_{n=1}^{\infty} f(na)$ is convergent. Prove that $ \int_{0}^{\infty}f(x) dx$ is convergent. I tried to solve it by using the Riemann sum, but for fixed a it doesn’t work. I have no other ideas.

Does real linear programming produce bipartite perfect matching using maxflow reduction?

Given a bipartite graph the standard reduction to max flow is with the construction similar to following diagram:

enter image description here

We can formulate max flow as an linear programming problem with integer variables in latter.

  1. If we do not use integer variables does solving for maxflow in linear programming formulation with only real variables still produce valid perfect matching of given graph?

  2. Is there a formal proof of this?

Intersection Theory – Chow groups and Their Applications

I’m a beginner in the filed of $ Algebraic$ $ Geometry$ , especially the $ Intersection$ $ Theory$ . I came to know that the $ chow$ $ groups$ of schemes are analogs to the $ homology$ $ groups$ of manifolds. I know at least one motivation behind the study of homology groups; they provide invariants of the manifold that can be used for classification purposes. I believe there must be many such motivations/applications of chow groups but right now I am failed to find. Can any of you be kind enough to enlist few of these?

Does E2EE encrypt content type?

A little backstory, Indonesia’s minister in communication and informatics supposedly blocked the sending of images and videos through social media from 22nd may until 25th may, in order to prevent hoaxes from spreading too rapidly.

I had a discussion with my friends on how this works, and we (sortof) concluded its the http request that they block when the content type is an image or video.

Now that made me think, because some of my friend cant send photos in private messages too. Does this mean e2ee doesnt hide content type? Thanks.

Minimize number of DFS searches in a graph

I got a weird homework question about graph.

A helicopter is going to land on an island to check the n houses after an earthquake. Some of the two-way roads connecting the houses are destroyed however the helicopter won’t know beforehand. Once the the helicopter lands on a house u, they can drive to any neighboring house v if the road (u,v) is not destroyed. The rescue team would only know if a road (u,v) is destroyed or not after they arrive at either u or v. They also don’t know the number of safe roads in advance. Design a strategy to minimize the number of helicopter landings, also the plan needs to have less than 2n trips along the safe roads.

My thinking is to do a DFS search on a random unvisted node. After the DFS is done, if there are still unvisited nodes, then do another DFS search from a random unvisited node again. Repeat this until all the nodes are visited. However, I don’t know how to prove that this minimize the number of searches. Can anyone help me? Thanks!

Substitution for Landau’s O notation formula

 I found the following description when I was reading a paper on computational complexity theory.

This can be done … in time 2n・poly(logs,n)+2O(logs)c. For s≤2no(1), the runtime is 2n・poly(n).

I know this means substituting s≤2no(1) for 2n・poly(logs,n)+2O(logs)c results in 2n・poly(n), but I don’t know how this is done. So please tell me the calculation process.

How do i do this thing

Someone please tell me how do i piss in public.

I try to do it everyday but i cannot, as i am not so professional. Here is the thing $ 1+1+1+1+++1+…+2=67877$ I also tried to consider ring theory, but i fall down everyday. Also, if $ \phi(p)=467$ , then what is $ p$ ? Please help me! I love these! Do you know porn?