Dubins TSP crossing trajectory theorem

In the Dubins TSP (DTSP for short), one needs to visit a set of given points in the plane, and return to the starting point, minimizing the distance of such a trajectory. The difference with the Euclidean TSP (ETSP for short) however is that the visiting entity is curvature constrained by some $ r$ , meaning the vehicle can, at any moment in time, either go straight, or move on the edge of some circle of radius $ r’\geq r$ .

Consider the following theorem, by Le Ny et al. (Theorem 2 in http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.84.982&rep=rep1&type=pdf) which we will call Theorem 1 in this post :

A trajectory based on an ETSP optimal visit order, is not necessarily an optimal trajectory for the DTSP.

(I changed the wording of their theorem to suit this post)

Suppose the next theorem, Theorem 2 :

There exists instances for which the optimal DTSP visit order creates a crossing ETSP trajectory.

My question is : Does Theorem 2 imply Theorem 1 ?
The idea is as follows. Since some instances have the optimal DTSP visit order creating a crossing ETSP trajectory $ T$ , this ETSP trajectory $ T$ is not optimal (due to the triangle inequality). This in turn implies that some optimal ETSP trajectory $ T*$ on these instances (or their corresponding optimal ETSP visit order) cannot be adapted and optimized to obtain the optimal DTSP trajectory. If it could, then $ T = T*$ , which is a contradiction, since $ T$ admits a crossing trajectory.

Crossing Number of $K_5$ on the Tubular Neighborhood of a Trefoil Knot

The crossing number of a simple connected graph $ G$ is the minimum number of edge crossings of $ G$ over all drawing of $ G$ . It’s well known that on the sphere the crossing number of the complete graph $ K_5$ is $ 1$ and on the torus it’s $ 0$ .

What is the crossing number of $ K_5$ on the tubular neighborhood of a trefoil knot?

My intuition is that it’s still $ 0$ , but I’m having trouble showing this.

Avoiding crossing lines in DFD diagrams

I have researched DFDs reasonably extensively, and received some feedback from peers about my first DFD. I have since reworked it to avoid lines crossing by duplicating the ‘Pokemon’ data store.

I have read that it should read left to right. Does this mean I should create another ‘Pokemon Trainer’ entity on the right, or is this not needed?

Scenario: Users (Pokemon Trainers) sign up for an account. They can search for Pokemon from a database. They can ‘catch’ Pokemon which are then stored in their own Pokedex. Users can view their caught Pokemon, along with information about that Pokemon.

First DFD enter image description here

Second DFDPokemon DFD

Crossing US border with music files I’m legally allowed to possess

Apart from directly buying digital music files, there are a few ways in which you could possess such files legally (at least in my country):

  1. Creating them from CDs you legally own, to make listening on the go more convenient. I believe this is legal in the U.S.
  2. Receiving them from a close acquaintance, who have a legal right to possess those files. This is legal in my country, but I’m not so sure it is in the U.S.

While at least the first category (as far as I know) is legal in the U.S. as well, there really is no way for border agents to know, for any given music file, whether I’m legally allowed to have it or not.

If I, a foreign national from a visa waiver program country, were to arrive from overseas at a U.S. airport while carrying a phone, laptop or other storage device containing files in either of these two categories, how likely am I to get in trouble (as in, getting held up or detained, having devices confiscated, etc.) with the CBP?

Updated Border Crossing Status Between Kosovo and Serbia

On my upcoming trip to the Balkans I’m looking to drive from Skopje, North Macedonia, to Tara National Park in Serbia via Pristina, Kosovo. As I plan to exit Kosovo directly into Serbia I would like to also cross into Kosovo from Serbia and, hence, am not planning on taking the most direct route (E 65). I want to cross into Serbia on the A-1, then pass through Presevo on the B35 and cross into Kosovo near Murcibabe. Does anyone know if that border crossing is currently open?

backpacking on ESTA VWP for 75 days a problem at border crossing? [duplicate]

This question already has an answer here:

  • Backpacking 46 days in the US with ESTA 1 answer

I want to go to the US for 2.5 months and I’m a little scared I will have difficulty getting in because it’s for so long. I’m going to backpack and stay with friends, i have two inland flights as well, will spend two weeks in 4 different cities and then i want to go for a hike, maybe rent a car. I’m a student but I don’t have employment because of that. has anyone travelled there for so long and experienced any problems? I’m female, white and from germany.. also don’t know how much money I will have to proof. any experience would be appreciated

Why did the metro bus stop at each railway crossing, despite no warning indicating a train was coming?

I’m travelling in LA and we went to Knott’s Berry Farm for a day out. We took the metro 460 bus, which goes to DisneyLand. During the drive on the way there and back, I noticed a peculiar habit. The bus driver would come to a complete stop at railway crossings and then after a few seconds continue on. Other cars didn’t seem to do this and the driver did this on crossings with warning lights that would indicate a train was coming, but the lights weren’t on.

Why were the bus drivers doing this? I can understand slowing down, to prevent damage and such, but a complete stop seemed odd.

Is Kew Pha Wok an official Thai-Myanmar border crossing?

In 2018, we (a party consisting of Thai and European citizens) visited Kew Pha Wok where we were allowed to cross the border without showing much official documentation or getting stamped. One person in our party left their Thai ID card at a Thai military post on the Thai side and we were allowed to walk to the Myanmar side.

On the Myanmar side, we were greeted by a person in military uniform with Shan State Army logos on it. The person was very friendly, wanted to pose for a picture (below, not allowed on the Thai side) and we were allowed in.

On the other side was a village where we walked around a bit and then we went back. What is the point of this crossing? Is it just for hopping over for a day’s visit or can one actually cross from one country into the other legally with stamps and all?

enter image description here

Is UK transit visa required for flight change at LGW ( Crossing the border)

1) I have a valid schengen visa issued from France and a valid H1B visa from usa. 2) I will be departing from Seattle, USA on May3rd and Landing at LGW (London) via Norwegian Air on 7 AM May 4th ( London time). I will be catching flight to Paris the next day before mid night from LHR

My research says, I don’t require a visa to change flights by crossing the border at LGW and taking another flight next day to France as I have a valid US visa i.e., H1b and travelling from USA with a valid schengen visa to France

I would like to hear from other folks about the same.