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June 1

Packing pac-men...

I was wondering what percentage of the plane could be packed for each possible partial circular disk (identical in size) running from n=0 to an entire circle (2Pi). At 2Pi, the answer is the standard circle packing, .9069, and as the measure approaches 0, the covering approaches 1. (The slices alternate direction and pack into columns with squiggly sides that get straighter as the arc gets smaller. I'm wondering whether at some point below 2Pi, say about 1.8Pi, the percentage covering is less than that of the full circle as the amount uncovered in each circle goes up, without a neighboring circle being able to cover much of it. Not sure of the minimum there, but half circles (Pi) would be able to pack at least as well as circles...) Ideas?Naraht (talk) 03:47, 1 June 2016 (UTC)

Interesting question. Simplifying assumption: the best way to pack two pac-men is by placing the top of one mouth flat against the bottom of the other mouth. Assume that the radius is 1, and let $\alpha$ be the radius of the missing wedge (the mouth). Now, if I'm visualizing my triangles correctly, when you pack two pac-men together like this, the distance between centers is $2\cos(\alpha )$.

So two full circles occupy a 2 by 4 rectangle and have total area 2pi, while two pac-men occupy a 2 by $(2+2\cos(\alpha ))$ rectangle and have total area $2\pi -\alpha$. The two circles have a better ratio of filled area to rectangle area; to see this, compare the two expressions, cross multiply, replace cosine with its second-order approximation and then simplify, and this becomes the claim that $\pi \alpha ^{2}/2<\alpha$, which is true for sufficiently small $\alpha$.

Of course, this isn't the end of the story, since in an optimal packing, these rectangles will overlap for different pairs. But that effect should be smaller for pac-men than for full circles. So my handwavey claim is that for sufficiently small positive $\alpha$, the packing is worse than for full circles.--2406:E006:45D:1:146D:8FF:3DC1:AE2C (talk) 04:47, 1 June 2016 (UTC)

Sort of surprised this didn't get more comments. With considered thought, I'm not even sure that the function plane coverage = f(disk_percentage) is even continuous, there may be places where things get bizarre. For example, a partial disk that is exactly one third of a disk will have a coverage equal to the full disk, but what happens with something just a bit bigger.Naraht (talk) 17:12, 3 June 2016 (UTC)

"For example, ..." I don't think this is right. I believe that the other construction you mentioned (glue sectors together along their radii, alternating whether curves are "up" or "down", then layer stacks like this together) is close to 94% density when sectors are 1/3 of a circle. On the other hand, I agree that it is not clear that the function is continuous. --JBL (talk) 20:38, 3 June 2016 (UTC)

More concretely, the density of the packing of sectors where you make strips of alternate up-down sectors sharing their radii, then layer these strips in the densest possible way, is ${\frac {t}{\sin t\cdot ({\sqrt {4-\sin ^{2}t}}-\cos t)}}$, where t is half the central angle of the sector. (Here the sector is at most half a disk.) At t = π/2 we get the same as the dense packing of circles, and for slightly smaller t we actually get worse than this, but for every t < 1.33257 we get something denser from this packing (and the density is monotonically increasing to 1). --JBL (talk) 21:46, 3 June 2016 (UTC)

Fractional Game Theory

I can't imagine it, but mathematics is full of surprises. Is there such in the literature to date? I know it won't be here. Number of players a non-integer or fuzzy intermediate value.Julzes (talk) 10:41, 1 June 2016 (UTC)

In combinatorial game theory games can have non-integer or fuzzy values, but the number of players is an integer (usually 2). Gandalf61 (talk) 14:01, 1 June 2016 (UTC)

In MMORPGs, players vary by level of activity. Some are online several times a day, while others are on once a week, or less. That might qualify. StuRat (talk) 15:33, 1 June 2016 (UTC)

As far as I know, no one has developed a theory of MMORPGs, or if they have Blizzard hasn't told anyone. But it seems to me that game theory with a large but undetermined number of players is a description of economics. --RDBury (talk) 01:48, 2 June 2016 (UTC)

June 3

Function question

Hello, this should be a relatively simple question. Recently I've learned about multi-variable functions, e.g. $f\colon \mathbb {R} ^{n}\to \mathbb {R}$; and of course I am familiar with single-variable functions, e.g. $f\colon \mathbb {R} \to \mathbb {R}$. I realized that in both cases, no matter how many real numbers are input, the function only outputs a single real number. This got me wondering, can the following functions (or if not, perhaps equations instead) exist?

• $f\colon \mathbb {R} ^{n}\to \mathbb {R} ^{n}$
• $f\colon \mathbb {R} ^{n}\to \mathbb {R} ^{n+1}$
• $f\colon \mathbb {R} ^{n}\to \mathbb {R} ^{n-1}$

Thanks! 74.15.5.167 (talk) 02:29, 3 June 2016 (UTC)

Sure. You can have a function from any set to any set, except that the set on the right can't be the empty set, unless the set on the left is also the empty set. (If you allow partial functions, you don't even have that restriction.) --Trovatore (talk) 02:36, 3 June 2016 (UTC)

For an example of a function from Rⁿ to Rⁿ, see Matrix difference equation#Non-homogeneous first-order matrix difference equations and the steady state. For an example from R^n×n to R^n×n, see Matrix difference equation#Nonlinear matrix difference equations: Riccati equations. Loraof (talk) 03:05, 3 June 2016 (UTC)

Vector fields in Euclidean space are a common example of functions from $\mathbb {R} ^{n}$ to itself, for instance. I'd think that you'll study the calculus of such functions once you study the calculus of real-valued functions on multivariate domains.--Jasper Deng (talk) 09:24, 3 June 2016 (UTC)

Point to clarify, regarding "single number". For a function F:R^n->R^n, the input and the output are both single elements of their respective sets. So if F(a,b,c)=(x,y,z), it is still sending a single input to a single output, even though we can characterize the output as an ordered n-tuple that sort of looks like a list of many numbers. For functions that sort of drop dimensions look at Projection_(set_theory) and projection_(mathematics). For A:R^n->R^m, consider $A\in \mathbb {R} ^{m\cdot n},\;x\in \mathbb {R} ^{n}$. Then using matrix multiplication, Ax=b is a mapping that can increase or decrease the dimension. SemanticMantis (talk) 14:42, 3 June 2016 (UTC)

A paper map of the Earth surface is a function from (some subset of) $\mathbb {R} ^{2}$, representing some range of (φ,λ), into (some other subset of) $\mathbb {R} ^{2}$, corresponding to (x,y) coordinates on a sheet of paper.
You can paint a picture of a landscape, which is a kind of a projection from $\mathbb {R} ^{3}$ to $\mathbb {R} ^{2}$.
You can make a 3D model of the ocean bottom's depth, thus creating a real representation of $\mathbb {R} ^{2}\to \mathbb {R}$ function.
You can describe a projectile trajectory in time as $\mathbb {R} \ni t\mapsto (x,y,z)\in \mathbb {R} ^{3}$
Etc, etc, etc... --CiaPan (talk) 16:48, 3 June 2016 (UTC)

Symbol: X-like, but with a bar on top?

I believe once to have seen a math symbol looking like an X, the upper part being closed with a bar. Is there such a symbol, and if yes, what is it used for? --KnightMove (talk) 06:28, 3 June 2016 (UTC)

In probability theory ${\overline {X}}$ usually means a mean value of a random variable $X$. --CiaPan (talk) 06:42, 3 June 2016 (UTC)

But that bar isn't closing the top of the X.

Looking over the "Mathematical Symbols" sections of the Unicode code charts, I can find symbols looking like an x with a bar closing any of the other three sides, but not the top! The three I found are charted on these two pages and identified as:

• 22C9 (⋉) left normal factor semidirect product
• 22CA (⋊) right normal factor semidirect product
• 2A32 (⨲) semidirect product with bottom closed

Note that the words "with bottom closed" seem to be describing the symbol rather than explaining its meaning. --69.159.60.83 (talk) 06:50, 3 June 2016 (UTC)

According to The Comprehensive LaTeX Symbol List, three different LaTeX symbol packages have the top-closed product as \utimes, suggesting that someone has used it for something, but I don't know who or what. -- BenRG (talk) 08:36, 3 June 2016 (UTC)

(ec) In a set theory it is a complement of a set, and in Boolean algebra it is sometimes used for negation. --CiaPan (talk) 06:42, 3 June 2016 (UTC)

See also

• List of mathematical symbols, section 'Letter modifiers' and
• Overline, section 'Math and science'.

CiaPan (talk) 06:53, 3 June 2016 (UTC)

For clarity, KnightMove is asking about somthing like $\ltimes$ rotated 90° clockwise. Looks a bit like a folding table. —  crh ₂₃  (Talk) 09:54, 3 June 2016 (UTC)

I have to concede that my wording was ambiguous, so thanks for the clarifying. --KnightMove (talk) 05:58, 4 June 2016 (UTC)

June 4

How numerate is the general population?

Can you spot the error in this notice which was posted in my local library this week (answer below)?

Spring Bank Holiday
All Hackney libraries will be
closed on Monday 30th May 2016
Normal opening hours will resume
from Tuesday, 1st June 2016

What percentage of the population, if asked to state the number of days in a given month, could come up with the right answer without consulting a calendar? What percentage is familiar with the following verse:

Thirty days have September,
April, June and November.
All the rest have thirty - one
Excepting February alone,
which hath but twenty - eight days clear
and twenty - nine in each leap year.
For leap year, coming once in four
gives February one day more.

When I was little and read that February 1900 had only 28 days I didn't believe it. What percentage of the population knows this, and what was their reaction when they first found out? 151.224.167.104 (talk) 11:36, 4 June 2016 (UTC)

47% - Amazement. — Preceding unsigned comment added by 82.38.219.142 (talk) 12:05, 4 June 2016 (UTC)

This has nothing to do with numeracy or mathematics. Also, you kids get off my lawn. --JBL (talk) 15:21, 4 June 2016 (UTC)

I concur. Besides, not everyone learns it using that mnemonic. (If I forget, I use the piano-keyboard one: if you start on F for January and go up an octave, so that F♯ is February, G is March, etc., the months falling on white keys have 31 days.) Double sharp (talk) 15:39, 4 June 2016 (UTC)

Agreed. I prefer counting on my knuckles.--2406:E006:45D:1:ECD4:E060:2C92:20F4 (talk) 02:44, 5 June 2016 (UTC)

See Thirty days hath September#Knuckle mnemonic.—Wavelength (talk) 15:59, 4 June 2016 (UTC)

Linguistic digression — I think far more people know it as starting "[t]hirty days hath September", but it's nagged at me in the back of my mind for a long time. It's not really grammatical in that version. "Thirty days hath September" is fine by itself, but then there's nothing to do with April, June and November. If you include those three months in the subject, then the subject is plural, so it should be "have", as in your version. My feeling is, if you're going to use archaic grammar, you ought to at least take the trouble to get it right. --Trovatore (talk) 20:23, 5 June 2016 (UTC)

• If you read it as elliptical for "Thirty days hath September, thirty days hath April, thirty days hath June and thirty days hath November", this avoids the grammatical problem. --69.159.60.83 (talk) 18:44, 6 June 2016 (UTC)

Here's an idea for those who are neither musical nor literary. Count the number of months from March, multiply by 6, divide by 5 and discard any remainder. If the answer is odd the month has thirty days, and if it is even the month has 31. All this may sound counter - intuitive, but it's the way the calendar is organised, as the first line of this couplet demonstrates:

Impar luna pari, par fiet in impare mense;
In quo completur mensi lunatio detur.

So,

March - distance = 0, multiply by 6 = 0, divide by 5 =0.

April - distance = 1, multiply by 6 = 6, divide by 5 = 1.

May - distance = 2, multiply by 6 = 12, divide by 5 = 2.

June - distance = 3, multiply by 6 = 18, divide by 5 = 3.

July - distance = 4, multiply by 6 = 24, divide by 5 = 4.

August - distance = 5, multiply by 6 = 30, divide by 5 = 6.

September - distance = 6, multiply by 6 = 36, divide by 5 = 7.

October - distance = 7, multiply by 6 = 42, divide by 5 = 8.

November - distance = 8, multiply by 6 = 48, divide by 5 = 9.

December - distance = 9, multiply by 6 = 54, divide by 5 = 10.

January - distance = 10, multiply by 6 = 60, divide by 5 = 12.

February - distance = 11, multiply by 6 = 66, divide by 5 = 13.

When you get to February, the number "13" reminds you it was considered unlucky and has only 28 days instead of 30. 151.224.133.26 (talk) 16:31, 6 June 2016 (UTC)

June 5

Noetherian ring with non-Artinian total ring of quotients

Is there a commutative Noetherian ring with a non-Artinian total ring of quotients? GeoffreyT2000 (talk) 16:06, 5 June 2016 (UTC)

How about $\mathbb {R} [x,y]/(xy=0)$? It's Noetherian by basic results: fields are Noetherian; if $R$ is Noetherian, then so is $R[x]$; quotients of Noetherian rings are Noetherian. But its ring of quotients is not Artinian, because consider $(x)\supset (x^{2})\supset (x^{3})\supset \dots$.--2406:E006:45D:1:4022:A8FC:6159:3666 (talk) 05:36, 6 June 2016 (UTC)

June 6

Expected distance between point inside a hypercube and a vertex

Let $I_{n}=\int _{0}^{1}\int _{0}^{1}...\int _{0}^{1}{\sqrt {x_{1}^{2}+x_{2}^{2}+...x_{n}^{2}}}d{x_{1}}...d{x_{n}}$. $I_{1}=1/2,I_{2}=({\sqrt {2}}+\sinh ^{-1}{1})/3,I_{3}\approx 0.960591956455053$. What would be the best way to numerically evaluate this integral in higher dimensions? Or, if there's a simple closed form what is it? 24.255.17.182 (talk) 22:01, 6 June 2016 (UTC)