Wikipedia:Reference desk/Archives/Mathematics/2023 November 1

Welcome to the Wikipedia Mathematics Reference Desk Archives
The page you are currently viewing is a transcluded archive page. While you can leave answers for any questions shown below, please ask new questions on one of the current reference desk pages.

November 1

Acceptable ambiguity

It's very basic, but I'm curious whether the potential ambiguity in the following sentence is acceptable: If the apex of the pyramid is perpendicular to the center of the square, it is a right square pyramid with four isosceles triangles; otherwise, it is an oblique square pyramid. Is it alright to assume that it will be understood that the perpendicularity is relative to the plane of the base? Remsense聊 05:34, 1 November 2023 (UTC)

In my mathematical English, an apex is a point and can't be perpendicular to anything. I'd say "if the apex lies on a line erected perpendicularly from the center of the square". But I think it would then "be understood that the perpendicularity is relative to the plane of the base". --142.112.221.156 (talk) 05:40, 1 November 2023 (UTC)

142.112.221.156, thank you, yes! I suppose my slightly backwards assumption was precisely because an apex is a point, it would be less ambiguous somehow. — Remsense聊 05:42, 1 November 2023 (UTC)

In less mathematical English one might say, "if the apex of the pyramid is squarely above the centre of the square". While the notion of aboveness is alien to the Platonic concept of a pyramid, the term “base" induces the connotation of a vertical orientation, and the apex being directly "above" the centre is used in definitions in several texts.^[1][2][3]  --Lambiam 09:16, 1 November 2023 (UTC)

Border between finite and infinite

If x = 1, the infinite sum 1/1^x + 1/2^x + 1/3^x + ...... goes to infinity. But if x = 2, the sum is finite. What is the greatest value of x that gives an infinite sum? 24.72.82.173 (talk) 15:04, 1 November 2023 (UTC)

The greatest value is indeed x = 1. For all x > 1, the sum converges. GalacticShoe (talk) 15:21, 1 November 2023 (UTC)

The convergence becomes very slow when $x$ nears $1.$ When $x=1.01,$ you need to sum like $10^{700}$ terms to get the first five decimals of the fractional part of the limit ($100.57794...$) correct.  --Lambiam 17:41, 1 November 2023 (UTC)

It may be worth mentioning that the sum is the Riemann zeta function. It's kind of a thing. --RDBury (talk) 21:19, 1 November 2023 (UTC)

There are a number of ways to prove GalacticShoe's statement. The Cauchy condensation test or integral test for convergence will both work to prove that the sum converges for any x > 1. The proof that it diverges for x = 1 is a well-known gem. Double sharp (talk) 08:18, 2 November 2023 (UTC)

Distance paradox

Some time ago while in park I noticed that for dropping pocket trash and then leaving the park the nearest trash bin (roughly 15 meters away) couldn't necessarily be the best option - there was another trash bin at a greater distance from me, but standing on the exit path from the park (unlike the nearest trash bin which was standing in another direction). That is, if I wanted to drop the trash at the nearest trash bin and leave the park, I would make a loop, adding a total of 30 m to my exit path from the park (15 m of walking towards that nearest bin and then backwards onto exit path). On the other hand, if I drop trash to the far bin along the exit path, I add nothing, but should walk a greater distance towards that bin as compared to the nearest bin. Is any of those two paths more convenient mathematically? 212.180.235.46 (talk) 22:04, 1 November 2023 (UTC)

It depends entirely on what your priority is. You can think of the relevant locations within the park (in this case, such a set of locations would include your current location, the trash bins, and the exits) as forming a graph, with edges weighted by distance. If you simply want to prioritize time getting to the bin as forming the basis of convenience, then the problem is just finding the shortest path from your current location to any of the bins, with no regard for the exits. In this case that would be the 15 meter loop. If you want to prioritize total time taken to throw out your trash and leave, then that becomes the problem of finding the shortest path from your current location to a trash bin and then to an exit. In this case that would be the bin-having exit path you mentioned. GalacticShoe (talk) 00:33, 2 November 2023 (UTC)

I'm thinking this is a generalization of the Travelling salesman problem. Instead of visiting each destination at least once, the destinations are in groups and the idea is to visit a destination in each group at least once. This allows for multiple entrances/exits, trash cans, fountains, food trucks, memorial statues, etc. where you must visit at least one of each, starting and ending with an entrance/exit. It sounds like in this case the park planners have made the calculation unnecessary by putting a trash can at the exit; when in doubt just use that one. --RDBury (talk) 05:19, 2 November 2023 (UTC)

One could also formulate it as finding the shortest cycle on a digraph where: 1. the vertices are the starting point, the bins, and the exits; 2. the edges go from the starting point to all the bins, from all the bins to all the exits, and all the exits back to the starting point; 3. the weighting on the edges to the bins and to the exits are distance-based, while the weighting from the exits back to the start is 0 (rendering it equivalent to finding the shortest path from your starting point to a bin then to an exit.) GalacticShoe (talk) 13:30, 2 November 2023 (UTC)