Let's discuss Mathematics

[Millman]

I know I'm keeping it simple here.

I understand we don't know all the formulas and ideas for all the science out there. When you think about it simply it makes sense.

Take a normal room say 30 length, width, and height. You could fit littler kids into that room. You could almost fill it up in some cases.

Now take the same room and fill it with the kids grown up 20 years later. It's much more difficult to fit in. Some may have to leave or you expand the room.

That's exactly what's happening to the universe imo. You don't need relativity to prove it when you can tell the mass of all objects varies. It increases and decreases with time.

Further, if at least gravity as we know it and relativity as we know it didn't exist we'd be unable to move.

Are best conclusion is we're one of many outcomes of universes. Like why could any universe even start. The truth is still in mystery.

 

[Atticus]

That's physics, not mathematics.

 

[Souron]

Your post has a your characteristic lack of clarity to it, Millman.

However, it doesn't seem to be related to mathematics. Perhaps the Science questions not worth a thread thread would be more appropriate.

 

[Earthling]

I wouldn't say there is anything necessarily wrong with the post structure- but it does literally look like it was accidentally posted in this thread rather than the science questions thread, because the science questions thread was discussing the expansion of the universe (that it didn't just expand from a center point like the center of a sphere and all that).

 

[Millman]

I will add 'how many scientists does it take to replace a light bulb'.

I will also add 'the adults entering the room didn't shrink over time and neither did the room.'

But how would you know from only a 3D rep your mind's eye provides?

Edit-It's also a matter of detectability.

That we are too massive to see atoms and things without equipment. There may even be some things we can't detect. It's like the Sun trying to detect Earth. Any normal burst should destroy us.

We are on a hunt for the final variables and omniscience. What will we discover?

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[Petek]

Show that any sequence of 10 consecutive positive integers contains at least one number that is relatively prime to the product of all the others.

 

[LulThyme]

Show that any sequence of 10 consecutive positive integers contains at least one number that is relatively prime to the product of all the others.

The following lemma suffices:

Lemma : In any sequence of 10 consecutive positive integers, there is one that is divisible by neither 2, 3, 5 or 7.

Once we have the lemma, it suffices to pick an integer x with the property guaranteed by lemma. If x is not relatively prime to the product of all others, then it is not relatively prime to one of them, let's say y.

In particular, gcd(x,y) is more than 1 but, by Euclid's algorithm, gcd(x,y) divides |x-y| which is at most 9. It follows that one of 2,3,5 or 7 must divide gcd(x,y) and hence x, contradiction.



It remains to prove lemma. Note that it suffices to prove the lemma for integers up to 210=2*3*5*7.

There are different ways to prove this, I simply ran a small program, here are the integers up to 210 satisfying the conclusion of the lemma :


1
11
13
17
19
23
29
31
37
41
43
47
53
59
61
67
71
73
79
83
89
97
101
103
107
109
113
121
127
131
137
139
143
149
151
157
163
167
169
173
179
181
187
191
193
197
199
209

There is no gap greater than 10 (in fact, only 2 gaps of size 10, at the beginning and the end).

 

[Petek]

Nice argument. It took me some time to see why

Note that it suffices to prove the lemma for integers up to 210=2*3*5*7.

holds (The pattern of divisibility by 2, 3, 5, 7 repeats with period 210.)

I proved your lemma directly as follows:

Any sequence of 10 consecutive positive integers contains
* exactly 5 that are divisible by 2.
* at most 4 that are divisible by 3, and of those, at most 2 that are not divisible by 2.
* exactly 2 that are divisible by 5, and of those, exactly one that is not divisible by 2.
* at most two that are divisible by 7, and of those, at most one that is not divisible by 2.

Therefore, any sequence of 10 consecutive positive integers contains at most 5 + 2 + 1 + 1 = 9 integers that are divisible by 2, 3, 5 or 7 and so at least one must be divisible by none of them.

 

[Fifty]

Is it true that R is becoming pretty widely used in academic work in stats? Like, if I learned how to use R and not SPSS or whatever the other commercial packages are, would I be learnin' something useless?

 

[GoodGame]

Is it true that R is becoming pretty widely used in academic work in stats? Like, if I learned how to use R and not SPSS or whatever the other commercial packages are, would I be learnin' something useless?

I've taken some higher undergrad and low level grad stats lately, and what I heard in class was that SPSS was preferred though Minitab was acceptable. I have no clue what "R" is. In professional use in the past (scientific) a range of graphing/statistical packages has been acceptable (MS Excel, Graphpad Prism, Kaleidoscope). SPSS is almost a programming language for stats is preferred for the customization in display and algorithm that it allows, which was the impression I got.

I think your best bet would be to look at the journals you would actually cite in your own work, and see what their author submission guidelines are, which should include information on acceptable statistical analysis.

 

[Dirichlet]

Is it true that R is becoming pretty widely used in academic work in stats? Like, if I learned how to use R and not SPSS or whatever the other commercial packages are, would I be learnin' something useless?

Go for R. It is widely used, especially if you look at the people who actually do research on statistics, not research with statistics. SPSS is IMO second choice, anyway. Use SAS or R. R if you can, SAS if you must.

 

[Atticus]

Wow! There's a prestigious mathematician in the thread now!

 

[Fifty]

Wow! There's a prestigious mathematician in the thread now!

Its so nice being noticed!


In other news, I don't understand how the empty set axiom and the axiom of pairing in ZF are mutually consistent (could be that the bookywooky I read them in misstated them). ENLIGHTEN ME!

 

[Mise]

If you learn R it shouldn't be hard to learn SPSS or SAS later if you need it. By that I mean, if you're capable of learning one of those packages, you're quite capable of learning another if it becomes necessary. But even if you could only ever learn one of them, R is a fine choice.

 

[Atticus]

In other news, I don't understand how the empty set axiom and the axiom of pairing in ZF are mutually consistent (could be that the bookywooky I read them in misstated them). ENLIGHTEN ME!

Tell us what's wrong with them first, that's the easier procedure. My first thought is that you read/the book printed some quantification wrong.

 

[Fifty]

Tell us what's wrong with them first, that's the easier procedure. My first thought is that you read/the book printed some quantification wrong.

NEVERMIND it was a misread by me in the statement of the empty set axiom (I thought it said that the empty set was an ELEMENT of every set, rather than a SUBSET of every set).

My issue was that if the empty set was an element of every set, then the axiom of pairing can't be true. Let a = {1} and b = {2}, then by the axiom of pairing there is a set C with only {1} and {2} as elements. But if the empty set is an element in every set, then C also has the empty set as an element, so it can't just have {1} and {2} as elements.

 

[pboily]

Wow! There's a prestigious mathematician in the thread now!

Its so nice being noticed!

[wiki=http://en.wikipedia.org/wiki/Johann_Peter_Gustav_Lejeune_Dirichlet]Johan Peter Gustav Lejeune Fifty[/wiki]

 

[PlutonianEmpire]

A Boeing 747-400 is at the Earth's rotational Equator, flying in a straight line heading due east following the equator, at typical cruising speed, Mach 0.84.

The plane has just crossed the terminator into night, with the sun having just set, and it is spring equinox. With the plane's speed and the planet's rotation in mind, how long is it before the plane crosses the terminator again and the sun rises, assuming the plane doesn't deviate from its course following the equator and has enough fuel. How many hours?

 

[sanabas]

Plane's flying at ~1029 km/h.

Terminator's moving at ~532 km/h.

So plane is moving at ~497 km/h relative to the terminator, and needs to cover ~6378 km. So ~ 12 hours, 50 minutes. Plus another 2 minutes because flying 10 km up makes the diameter 10 pi km longer.

 

[PlutonianEmpire]

Umm, are you sure? The Earth's rotation is 24 hours, so at the equator, nights are only 12 hours, and since the plane is at the equator, and is going WITH the Earth's rotation, not against it, I'd think it would cross the terminator again in less than 12, 13 hours.