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September 5

The Sorgenfrey plane

Ok I know how the usual proof of non-normality goes, but I've been thinking about something that smells fishy to me. Take the line y=-x. The proof goes by showing the set of all rational points and irrationals can't be separated by open sets using Baire's category theorem. But what if we do this: for every rational p/q, cover the point on the line with [p/q,p/q+1/(q^q)) X [-p/q,-p/q+1(q^q)), where X denotes cartesian product. Notice how this half open square is VERY small. It certainly looks plausible that every irrational has a half open square (with the irrational at the bottom left corner) that's disjoint from the union of the rational squares, because every irrational is surrounded by a cloud of rationals with very large denominator. I've been trying to show this is not possible but it seems to have something to do with approximation of irrationals by rationals, continued fractions maybe (which I don't know much about). What am I doing wrong? Breath of the Dying (talk) 09:00, 5 September 2009 (UTC)[reply]

Eh, only quadratic irrationals like square roots can be approximated by continued fractions. (At least if you mean the conventional type with only 1's as numerators.) Professor M. Fiendish, Esq. 09:17, 5 September 2009 (UTC)[reply]

That is not true. Algebraist 12:16, 5 September 2009 (UTC)[reply]

You were probably thinking about the fact that quadratic irrationals have a repeating continued fraction expansion. -- Meni Rosenfeld (talk) 17:15, 5 September 2009 (UTC)[reply]

That particular scheme will fail for irrational points that can be well-approximated by rationals. For example, setting

c=\sum _{j=1}^{\infty }10^{-j!}

, any neighbourhood of (c,-c) intersects your neighbourhood of the rational points. Algebraist 12:22, 5 September 2009 (UTC)[reply]

As to the first question, as you recalled: in this topology, the closure of any neighborhood of the closed set

\{(r,-r):r\in \mathbb {Q} \}

, meets the anti-diagonal in uncountably many points. If this is OK for you, I really don't see why it is not OK for you that in particular, the nbd you wrote has this property.--pma (talk) 17:58, 5 September 2009 (UTC)[reply]

Great retrosnub icosidodecahedron

I keep seeing conflicting vertex figures for this one.

Some sources say the vertfig is 3.³/₂.3.⁵/₃.3.
Other sources say it is ³/₂.³/₂.³/₂.⁵/₃.³/₂.
And still others say it is ^{(3.3.3.3.⁵/₂)}/₂.

Which is right?

On Wikipedia, the vertfig shown is ^{(3.3.3.3.⁵/₂)}/₂, but application of the Wythoff symbol |³/₂ ⁵/₃ 2 gives 3.³/₂.3.⁵/₃.3. No idea where ³/₂.³/₂.³/₂.⁵/₃.³/₂ came from. Professor M. Fiendish, Esq. 09:04, 5 September 2009 (UTC)[reply]

Just going by the vertex figure image there, which I'm assuming is correct, it looks to be a pentagram with the faces being four triangles and one pentagram so I guess the problem is how do you represent that with the notation? Just listing the 5 faces would be ambiguous with the case where they're arranged in a pentagon, so wikipedia handles that with the "/2" at the end to represent 5/2 instead of 5. It looks like another way is to specify the orientation of the faces relative to the vertex. 3/2 is a triangle but connecting the vertices in the opposite order, and similarly 5/3 is an inverted pentagram. I'm not really sure on how the orientations dictate the shape of the vertex figure, but it could be that those other two versions you mentioned both represent the same thing. Vertex configuration seems to explain. Rckrone (talk) 18:04, 5 September 2009 (UTC)[reply]

Wouldn't ³/₂.³/₂.³/₂.⁵/₃.³/₂ (all numbers <2) be indistinguishable from 3.3.3.3.⁵/₂? That is, if all the faces are backward, isn't it the same as if all are forward? —Tamfang (talk) 07:08, 10 September 2009 (UTC)[reply]

I agree with you there. Unless it turns the polyhedron inside-out or something :-). I think the first has a problem because 3.3/2 would have 2 Δs over each other. Professor M. Fiendish, Esq. 03:18, 13 September 2009 (UTC)[reply]

Uniform Continuity of sin(x³)/(x+1)

Hi again guys, sorry about the influx of analysis from me on here, I'm working through a bundle of questions and saving the ones which really stump me for here! I'm trying to work out whether ${\frac {\sin(x^{3})}{x+1}}$ is uniformly continuous over $[0,\infty )$ and I have no idea how to begin. The fact sin(x³) has a derivative tending to infinity leads me to think it might not be, but then the function decreases in magnitude to 0 too which may affect things, and I'm unsure even what answer to expect, let alone how to proceed.

Any help would be greatly appreciated!

Thanks a lot, Spamalert101 (talk) 14:32, 5 September 2009 (UTC)[reply]

It's pointwise continuous, and tends to a limit as x tends to infinity, so it's uniformly continuous. This is a variant of the Heine–Cantor theorem. This is very similar to your last question: in both cases we have a function with good local behaviour that is badly behaved at infinity, so multiplying it by a function that decays to 0 at infinity gives good global behaviour. Algebraist 14:54, 5 September 2009 (UTC)[reply]

And to make it quantitative, you may try and prove that your function is Hölder continuous of exponent 1/2: for instance,

\scriptstyle |f(x)-f(y)|\leq 5{\sqrt {|x-y|}}

holds true for all

x

and

y

in

\scriptstyle [0,\infty )

. --pma (talk) 20:50, 5 September 2009 (UTC)[reply]

The restriction to any bounded interval [0, a] is a continuous function on a compact set and so is uniformly continuous, and the restriction to [a, ∞] only takes values near 0, so it can't vary by more than ε if a is big enough. Michael Hardy (talk) 20:58, 5 September 2009 (UTC)[reply]

Properties of an isosceles triangle

Let ABC be triangle with AB=AC=6 cm . If the circumradius of the triangle is 5 cm, find the length of BC . —Preceding unsigned comment added by 117.98.100.98 (talk) 16:44, 5 September 2009 (UTC)[reply]

This looks a lot like a homework problem. I will give you this hint, though: if you draw the triangle with its circumscribed circle, you will notice that the line between the vertex A of the triangle and the center of the circle is perpendicular to the side BC of the triangle. When you get your answer and punch it into your calculator (which you shouldn't do until the last step), the 3rd and 4th digits after the decimal should be 6 and 9 respectively, assuming I did my math correctly.

Note that my approach was thrown together from very faint recollections from my geometry class many years ago, and may not be the way your teacher is expecting you to solve it. --COVIZAPIBETEFOKY (talk) 18:12, 5 September 2009 (UTC)[reply]

"...the 3rd and 4th digits after the decimal should be 6 and 9 respectively, assuming I did my math correctly." Of course, I found a mistake in my work. The answer after correcting my work still includes both of those dirty digits, though; just not in the same place. --COVIZAPIBETEFOKY (talk) 18:20, 5 September 2009 (UTC)[reply]

A hint: Use Heron's formula. Bo Jacoby (talk) 18:22, 5 September 2009 (UTC).[reply]

There certainly may be something I'm missing here, but I'm pretty sure this leads nowhere. The area by Heron's formula and the area given by (1/2)base*height are exactly the same expressions, so setting those two equal certainly doesn't really help at all. Was there something else you had in mind? --COVIZAPIBETEFOKY (talk) 18:36, 5 September 2009 (UTC)[reply]

You can use law of sines and the area formula Δ=ab*sinC/2 to relate the circum-diameter to the area. Anyway, Circumscribed circle lists a formula that can be applied directly to this problem. Rckrone (talk) 18:54, 5 September 2009 (UTC)[reply]

Here is a simple approach to the problem, based on properties of triangles you should have already seen in school.

Calling the center of the circumcircle O, note that the line AO bisects BC at a right angle, at say point D (see isosceles triangle).
Note that triangle AOB is also an isosceles triangle, and the angle bisector of angle AOB bisects the line AB at right angles, at point E. Can you determine the lengths of sides AO, OB, AE, EB and OE ?
Now what can you say about triangles AOE and ABD ? How do you use that to calculate length of side DB and hence, BC ?

Hint: Look up similar triangles and pythagoras theorem. Also, no calulator is required to solve this problem! 98.220.252.228 (talk) 18:46, 5 September 2009 (UTC)[reply]

Triangle ABO has sides 5,5,6. s=(5+5+6)/2=8. Area according to Heron is √(8·3·3·2)=12. This is = (1/2)base*height = (1/2)·5·(|BC|/2). Solve to get |BC|=48/5=9.6 . This, COVIZAPIBETEFOKY, is what I had in mind. Bo Jacoby (talk) 20:54, 5 September 2009 (UTC).[reply]

I'm guessing that by "isoscale" you mean "isosceles". Remember that if angles α and β are opposite sides a and b respectively, then

{\frac {a}{\sin \alpha }}={\frac {b}{\sin \beta }}={\text{circumdiameter}}

(see law of sines). So you have

{\frac {6}{\sin \beta }}=10,

and hence

\sin \beta ={\frac {3}{5}}.

Since two angles are β and one is α and they add up to a half-circle, that gives you α. Then you've got

{\frac {a}{\sin \alpha }}=10

so you can find a. Michael Hardy (talk) 21:11, 5 September 2009 (UTC)[reply]

...and I just noticed the arithmetic is really simple: you don't need numerical approximations from a calculator, since sin β = 3/5. Consequently cos β = √(1 − (3/5)²) and as luck would have it, we've got a Pythagorean triple and so cos β = 4/5 exactly. Therefore sin(β + β) = 2 sin β cos β = 2(3/5)(4/5) = 24/25. Since α + β + β = half-circle, we must have sin α = sin(β + β). So that sine is 24/25. SSo we get a = 48/5 exactly. Michael Hardy (talk) 21:23, 5 September 2009 (UTC)[reply]

How nice that Michael's and mine seemingly different roads lead us to exactly the same result. Bo Jacoby (talk) 06:09, 6 September 2009 (UTC).[reply]

35 should have been 25. -- Meni Rosenfeld (talk) 07:14, 6 September 2009 (UTC)[reply]

In my elementary school times I formed the opinion that almost all rectangular triangles have edges in the ratio (3:4:5). --pma (talk) 21:33, 6 September 2009 (UTC)[reply]

Little did you know that in time, you would learn that this is indeed true - for an appropriate measure on the set of all triangles, of course. :) -- Meni Rosenfeld (talk) 19:44, 7 September 2009 (UTC)[reply]

Dense in C

Resolved

--Shahab (talk) 19:41, 6 September 2009 (UTC)[reply]

I want to show that $\mathbb {Z} [{\frac {i}{2}}]$ is dense in $\mathbb {C}$ , the complex plane. By $\mathbb {Z} [{\frac {i}{2}}]$ I mean the set of all evaluations of polynomials with integer coefficients at ${\frac {i}{2}}$ (where $i=(0,1)$ ). How can I show that any open ball around an arbitrary z contains infinitely many such evaluations?--Shahab (talk) 19:48, 5 September 2009 (UTC)[reply]

Let r be an arbitrary positive real number. We need to show that you can find a polynomial that evaluates to a number within a distance of r from z. Choose integer n such that

{\frac {1}{2^{n}}}<{\frac {r}{\sqrt {2}}}

and n=0 (mod 4). Then choose integers p and q such that

{\frac {p}{2^{n}}}+{\frac {iq}{2^{n+1}}}

is a close to z as possible. That number is

pX^{n}+qX^{n+1}

evaluated at i/2 and is less than r from z. --Tango (talk) 20:34, 5 September 2009 (UTC)[reply]

Thanks. I have a doubt though. How do we choose p and q such that

{\frac {p}{2^{n}}}+{\frac {iq}{2^{n+1}}}

is as close to z as possible? I understand choosing n by the Archimedian property. Similarly what property do we use when choosing p and q?--Shahab (talk) 20:55, 5 September 2009 (UTC)[reply]

To get p, say, we take the integer part of the real part of z, call it a. Then test a2ⁿ+b for b in {0,1,...,2ⁿ-1} and choose the closest. It is a finite number of trials, so you will definitely find a closest. Do the same for q and the job is done. --Tango (talk) 21:01, 5 September 2009 (UTC)[reply]

Nice. You may also say: p and q such that 2p+iq is as close as possible to 2ⁿ⁺¹z (a minimum distance that is certainly not greater than

\scriptstyle {\sqrt {5}}/2

, &c) However as soon as you've observed that Z[i/2] contains all numbers p/2ⁿ+iq/2ⁿ it's done.--pma (talk) 21:12, 5 September 2009 (UTC)[reply]

Yes, that is pretty much equivalent. --Tango (talk) 21:42, 5 September 2009 (UTC)[reply]

Thank you both.--Shahab (talk) 19:41, 6 September 2009 (UTC)[reply]

Divisor of a Riemann surface

Can someone please explain what is meant by a divisor on a compact connected Riemann surface? I cannot find the relevant Wikipedia article (rather unusual). 86.210.207.133 (talk) 23:17, 5 September 2009 (UTC)[reply]

Divisor (algebraic geometry) is the relevant article, I think. --Tango (talk) 23:44, 5 September 2009 (UTC)[reply]

Grrrrr, no mention of Riemann surfaces... 86.210.207.133 (talk) 01:15, 6 September 2009 (UTC)[reply]

Any compact Riemann surface is a projective variety. --PS T 03:57, 6 September 2009 (UTC)[reply]

In this case (because Riemann surfaces have complex dimension 1) a divisor is just a function

D\colon X\to \mathbb {Z}

with discrete support, ie for a compact Riemann surface it's just a finite (formal) linear combination of points on the Riemann surface. --Xedi^Talk 04:54, 7 September 2009 (UTC)[reply]

Function elements

Is there a Wikipedia page concerning function elements?

Mathworld has one... http://mathworld.wolfram.com/FunctionElement.html —Preceding unsigned comment added by 86.210.207.133 (talk) 23:33, 5 September 2009 (UTC)[reply]

Have you tried searching Function elements? Intelligentsium 02:27, 6 September 2009 (UTC)[reply]