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June 27
Arccos(1.1)
I was recently looking at a problem which involved taking the inverse cosine of a calculated number which, in one case, turned out to be greater than 1. A quick examination revealed that the way I'd sketched out the problem made no physical sense but it occurred to me that this might just mean "has no solution in the real numbers".
Is there a definition for arccos(1.1) which lies outside the reals?
2A01:E34:EF5E:4640:BD32:A0DC:FFF7:43C0 (talk) 11:38, 27 June 2022 (UTC)
- Using we have that
- So it makes some sense to use
- although there is no obvious reason to prefer this choice over --Lambiam 12:33, 27 June 2022 (UTC)
- You should use, of course, whichever corresponds to arccos on the usual domain, which will depend on which branch of the complex logarithm you choose.
- The result, depending on which branch of the function you use, is approximately ±0.44356825 i + n * 2π, where n is an integer. IpseCustos (talk) 14:50, 27 June 2022 (UTC)
- Thanks a lot both of you. 37.166.79.23 (talk) 18:23, 27 June 2022 (UTC)
What's a Mobius strip with a 1-dimensional centerline called?
Or the roll equivalent of a great circle? A great circle of course being the geodesic where you pitch at constant rate in 3D Euclidean space but don't yaw or roll and the "straight Mobius" or "zero radius helix" being the geodesic of rolling at constant rate with no pitch or yaw. Sagittarian Milky Way (talk) 23:09, 27 June 2022 (UTC)
- The Möbius strip is a 2-dimensional metrizable manifold, but from the way the question is phrased it seems that you think of the strip as being embedded in 3-dimensional Euclidean space, inheriting its metric. These embeddings are not unique; some give rise to an Euclidean metric on the strip, while others don't. Are there Möbius strips whose centre line is not 1-dimensional? It seems to me that the centre line of anything, being a line, is 1-dimensional. --Lambiam 11:22, 28 June 2022 (UTC)
- Unglue the ends and pull the rectangle taut without untwisting it. Sagittarian Milky Way (talk) 12:08, 28 June 2022 (UTC)
- And then what? Is that different from twisting one end of a plain strip a half turn while holding it taut? Topologically, it is a disk, twisted or not. --Lambiam 17:42, 28 June 2022 (UTC)
- Same thing, it becomes not a spiral, not a typical helix but a another kind of twister with zero diameter if it was infinitely thin. Sagittarian Milky Way (talk) 22:18, 28 June 2022 (UTC)
- It is no longer flat (in the inherited metric), so the twisting also causes a deformation. We can consider the spiraling surface parametrized by two parameters, and embedded in by in which and are constants. If this degenerates into simply a straight line, indistinguishable from --Lambiam 22:52, 28 June 2022 (UTC)
- What does geometry call it if it's not infinitely thin? If a line segment on the z=seconds since t=0 plane is spinning on the z-axis and the midpoint is always x=0 y=0 z=time then the centerline of the surface that touched the segment would be the z-axis thus flat despite the surface not being flat. Sagittarian Milky Way (talk) 01:26, 29 June 2022 (UTC)
- SMW, it's really hard to figure out what your actual question is. Are you asking for the name of this surface? I don't know that it has a standard name. --Trovatore (talk) 01:39, 29 June 2022 (UTC)
- Yes what do you call this surface (doesn't have to be in that location or orientation) Sagittarian Milky Way (talk) 02:12, 29 June 2022 (UTC)
- SMW, it's really hard to figure out what your actual question is. Are you asking for the name of this surface? I don't know that it has a standard name. --Trovatore (talk) 01:39, 29 June 2022 (UTC)
- What does geometry call it if it's not infinitely thin? If a line segment on the z=seconds since t=0 plane is spinning on the z-axis and the midpoint is always x=0 y=0 z=time then the centerline of the surface that touched the segment would be the z-axis thus flat despite the surface not being flat. Sagittarian Milky Way (talk) 01:26, 29 June 2022 (UTC)
- It is no longer flat (in the inherited metric), so the twisting also causes a deformation. We can consider the spiraling surface parametrized by two parameters, and embedded in by in which and are constants. If this degenerates into simply a straight line, indistinguishable from --Lambiam 22:52, 28 June 2022 (UTC)
- Same thing, it becomes not a spiral, not a typical helix but a another kind of twister with zero diameter if it was infinitely thin. Sagittarian Milky Way (talk) 22:18, 28 June 2022 (UTC)
- And then what? Is that different from twisting one end of a plain strip a half turn while holding it taut? Topologically, it is a disk, twisted or not. --Lambiam 17:42, 28 June 2022 (UTC)
- Unglue the ends and pull the rectangle taut without untwisting it. Sagittarian Milky Way (talk) 12:08, 28 June 2022 (UTC)
- Helicoid. More precisely, a helicoid stretches out from the z-axis to infinity, as when we allow the parameter above to range over --Lambiam 07:35, 29 June 2022 (UTC)
- If is constrained, you get an (idealized) Archimedes screw. --Lambiam 09:03, 29 June 2022 (UTC)
- Fusillioid? Sagittarian Milky Way (talk) 11:47, 29 June 2022 (UTC)
June 28
Rubik’s Cube and Wang Tiles
Hi! I wonder if anyone has ever written about the relationship between a Rubik’s Cube and a set of Wang tiles.
Thanks
Duomillia (talk) 03:20, 28 June 2022 (UTC)
- People write about imagined relationships between celebrities, but not so much when it comes to mathematical entities. Is there a relationship? I don't see one. --Lambiam 08:00, 28 June 2022 (UTC)
Easy Graph Question
A grade 10th question surprised me- it asked to plot the graph for 15x-30y+1=0 and 3x-(24/4)y+(1/5)=0 and find if the lines are parallel, coincident, or intersecting. How do you plot this graph with reasonable coordinates? --ExclusiveEditor Notify Me! 13:18, 28 June 2022 (UTC)
- Put them in slope-intercept form. See https://en.wikipedia.org/wiki/Linear_equation#Slope%E2%80%93intercept_form_or_Gradient-intercept_form --Modocc (talk) 13:36, 28 June 2022 (UTC)
@Modocc:The problem is that this problem is for junior high school students who are not taught slope-intercept form but just taught methods like elimination, substitution, and then putting some value to x or y and then finding the other variable's value. Also by this method(taught to students), we find coordinates like 1/30 and 1/15 which are not easily plottable on a graph. So thinking on this level we want a simple answer, or if there is an error in the question itself. --ExclusiveEditor Notify Me! 16:14, 28 June 2022 (UTC)
- To find the answer, it does not really matter what the coordinates are. Take x in the range [−5, 5], so y remains within [-3, 3]. I don't see a problem. --Lambiam 17:56, 28 June 2022 (UTC)
If the slopes of the two lines are unequal (compare ratios), then they intersect. Otherwise if they have a point in common (solve equations), they are the same line. Otherwise, they are parallel. 2601:648:8202:350:0:0:0:FD2B (talk) 06:20, 2 July 2022 (UTC)
- It is also easy to see that the two equations of the question have the same solution set: multiply both sides of the second one by 5 and simplify, and it turns into the first equation. But the question asks specifically to plot the graphs; given the limited size of paper sheets, this requires, with the most plausible procedure, the student to select the range of the abscissa; that of the ordinate then follows. I took the question to mean, "how is a student supposed to select this range?". For this specific question it does not matter; if the question had been about the two equations 47x + y + 2389 = 0 and 48x + y + 2438 = 0, not only the range of the abscissa but also the scales (aspect ratio) become important. --Lambiam 10:37, 2 July 2022 (UTC)
July 1
Balance between betting odds in sports betting (and similar games)
In the case of non-fixed odds betting like sports betting, horse racing, political market, etc, assuming the game is not rigged, and has many players that have equal access to loads of information:
would the game have a tendency towards a balance? That is, chances of winning 10x are approx. 1/10?
This is more a psychology than a mathematical question, but could betting for the underdog have a consistent positive expected earnings? That is, most people prefer to bet for teams that win more often, even if they get lower betting odds. It just feel like a winning move, even if the math tells us the expected earning are lower. There's also a feeling of reward getting more results right and some punters won't keep track of losses. Bumptump (talk) 18:30, 1 July 2022 (UTC)
- I don't know how bookmakers decide the odds, but they can do it in such a way that they make the same profit percentage regardless of the outcome. Suppose there is a match between A and B, and 60% of bettors bet on A while 40% bet on B. If the bookmaker sets the odds for A to 8/5 and for B to 12/5, they will always make 4%. If a bettor happens to know that actually A and B are a more even match than suggested by the ratio 3 : 2 – specifically, the likelihood of B winning exceeds 5/12 ≈ 42% – their expected outcome when placing a bet on B is positive. If a substantial number of bettors are not rational actors but severely underbet when it comes to underdogs, betting on underdogs may be rational, but whether this is actually so depends on the messy mindset of the mass of gamblers and is not a mathematically resolvable issue. If you have data about a large number of matches with betting odds and actual match outcomes, this can be examined for these data, but note that the results may depend on the type of event, the time period, and the locality. --Lambiam 19:56, 1 July 2022 (UTC)
Look up parimutuel betting. Basically the bookmaker calculates the odds according to the bets received so far, and almost always gets a profit that way. 2601:648:8202:350:0:0:0:FD2B (talk) 06:21, 2 July 2022 (UTC)
- Thanks; the hypothetical example I presented fits with the formulas in Parimutuel betting § Algebraic summary, using n = 2, W1 = 60, W2 = 40, and r = 0.04. The bettors of the question may be placing their bets with betting houses that use different systems, though. In any case, knowing the system used by the house does not help to answer the question whether bettors tend to underbet on underdogs. --Lambiam 10:06, 2 July 2022 (UTC)
July 3
Virtual Graph
Is the graph for cos(xy) = csc(yx)
same as made by Desmos or is it largely different? ExclusiveEditor Notify Me! 13:15, 3 July 2022 (UTC)
- Is there a way to find out what Desmos thinks the graph is without downloading it? I don't have it on my computer. Without getting into the details, it seems likely the graph consists of discrete points, assuming it's non-empty. (It would be the intersection of cos(xy) = 1 and sin(yx) = 1, combined with the intersection of cos(xy) = −1 and sin(yx) = −1.) This could cause problems with a graphing calculator. --RDBury (talk) 14:50, 3 July 2022 (UTC)
- It is not hard to show that the graph is a discrete set. The system xy = 2π, yx = ½π has the solution x ≈ 5.425421011388989, y ≈ 1.086796732660537, so the set is not empty. I think that, more in general, the system xy = 2mπ, yx = (2n + ½)π has a solution for all integers m > 0, n ≥ 0. --Lambiam 15:54, 3 July 2022 (UTC)
- There is an online version of Demos. On entering I get a strange plot of way too many scattered points, plus a notice: ⚠This equation contains fine detail that has not fully been resolved. Learn more. See here. --Lambiam 16:42, 3 July 2022 (UTC)
- On further examination, it seems that the separate graphs of cos(xy) = 1 and csc(yx) = 1 each consist of a set of closely spaced curves; the solution set of the combined system of equations consists of the points where the curves of these two graphs intersect, and there are very many such intersection points. Given a sufficiently high resolution, you can see that the points forming the graph are not randomly scattered, as they appear to be on the Desmos plot, but form a pattern reminiscent of moiré. --Lambiam 19:10, 3 July 2022 (UTC)
Cubic formula
There are formulas from the 16th century for solving the general cubic, that were found with lots of work and ingenuity. Suppose I don't know about them, but want to discover the formula today, using modern "technology". I think the basic idea is that the Galois group for the general cubic is the symmetric group S3, and S3 factors into cyclic subgroups that correspond to square and cube root operations in the field extensions. But I'm missing some intermediate steps about turning that into formulas. The same goes for the quartic, while the quintic is unsolvable because A5 is a simple group with no normal subgroups.
Is there some subtlety to the above? Is this a good "project" to try, if I want to get better at this kind of algebra? I took a class once, but don't remember that much and didn't attempt this at the time (though I wondered about it). Thanks.
Inspiration for this question: [1]. 2601:648:8202:350:0:0:0:FD2B (talk) 22:27, 3 July 2022 (UTC)