mathematic

I was wondering what the rules are for a series to converge or not. In my calculus class our professor said that if it is bounded and monotonic then it will converge, but what are all the rules or characteristics a sequence must have?

The rule you gave is for a sequence to converge, not for a series. Also a sequence may converge if it is not monotonic.

hmm, I see what you mean about my first question, you'd be looking at a completely different thing if you were in a position to substitute variables, not a differential equation...

 

I understand what the total derivative does, when variables depend on other variables, but - and I'm proposing a totally improvised idea, I haven't tried to do anything with it mathematically yet - why can't you differentiate something with respect to all variables, even when those variables don't rely on each other (just a thought, there's probably some topological reason why )

 

Something like

 

[math] z = f(w, x, y) [/math]

 

[math] \frac{dz}{dwdxdy} = ...? [/math]

 

looks pretty stupid, but I don't see why it couldn't work...

[math] \frac{d^3z}{dwdxdy} = [/math] is a well defined concept (using ∂ rather than d).

I think the problem below is an nCr problem [n!/(r!*(n-r)!)], but I'm not quite sure how to approach it. Any help would be appreciated.

 

There are 20 numbers, 1-20.

Someone would randomly pick 5 numbers.

 

I need to create sets of 10 numbers so that at least one set contains the 5 numbers that the other person pick. What's the minimum number of sets I'd need to create?

What is the relation between the sets you create. One extreme - each set of 10 is independent of any other. Other extreme, two sets of ten each, i.e. no number in both sets.

Answers - case 1 - there is no minimum, since you could be unlucky enough to create sets without choosing any of the 5. case 2 - 2 sets will use all the numbers, so the 5 chosen will all show up in one or the other.

Clumping is a bad term for what happens to dark matter. All that we do know is that the density of dark matter will vary in different regions of space, but as mentioned already there is no mechanism for dark matter particles to stick together.

I would also like to know why.

Perhaps whoever gave the -1 (or anyone else for that matter) would explain why the explanation is faulty.

 

 

 

Are you saying that switching does, or does not, improve your chances of winning the car?

 

 

 

By switching your probability of getting the car is 2/3.

Pu239 (the most common isotope) has a half life of 24110 years, so it would not be found on earth, unless recently created. The longest lasting is Pu244 with a half life of 82,000,000 years, making it hard to find since the earth is around 4.5 billion years old.

Hi,

I am trying to prove that a limit exists at a point using the epsilon delta definition in the complex plane, but I can't seem to reach a conclusion.

Here's what I have been trying to get at:

 

[LATEX]\lim_{z\to z_o} z^2+c = {z_o}^2 +c[/LATEX]

 

[LATEX] |z^2+c-{z_o}^2-c|<\epsilon \ whenever\ 0<|z-z_o|<\delta[/LATEX]

 

[LATEX]LH=|z^2-{z_o}^2|=|z-z_o||z+z_o|[/LATEX]

 

[LATEX]=|z-z_o||\overline{z+z_o}|[/LATEX]

 

[LATEX]=|z-z_o||\bar{z}+\bar{z_o}|[/LATEX]

 

[LATEX]=|z\bar{z} +z\bar{z_o} -{z_o}\bar{z} -z_o\bar{z_o}|[/LATEX]

 

 

 

[LATEX]=| |z|^2 -|z_o|^2 +2Im(zz_o) |[/LATEX]

 

[LATEX]\leq||z|^2 -|z_o|^2 +2|z||z_o|| \ (because\ Im(z)\leq|z|)[/LATEX]

 

But I can't get any further. I did this much thinking I could factor it to the square of delta, but that didn't work out because of the positive 2zzo term.If anyone can help me out here, it would be great. Thanks.

You are overly complicating the problem. |z+z0| < (2+δ)|z0| for |z-z0| < δ. You should be able to work out the δ, ε relationship - remember z0 is fixed.

For question 1 - your observation is correct, although I wouldn't call it a differential equation. You are simply taking a derivative.

For question 2 - if x and y are not connected in any way, then total derivative has no meaning. You just have the partials.

ok.I'll take a stab at it and say,that if the photon is massless it has zero momentum,and therefore does not effect the atoms velocity or direction of travel,only its internal structure.yes/no?

http://en.wikipedia.org/wiki/Photon

Photon has momentum.

So, based on the motion of stars in galaxies and galaxy cluster motion etc., there is roughly 5 times more "dark matter" than ordinary matter in the universe. No one knows what this dark matter is. But, presumedly, it is in our solar system. So how does it affect the orbits of our planets around the Sun and moons around the planets, not to mention comets etc. ?

 

How is it that we can explain these orbits and motions to such great accuracy using general relativity and only the masses of the Sun, planets, moons, etc? These do not include the effects of dark matter. If there is 5 times as much dark matter as the ordinary matter which makes up our stellar objects, why does it not affect these motions more?

 

Edited to fix my dumb typos.

Although there is a lot more dark matter than ordinary matter, the dark matter is much more spread out, so that within the solar system it has a much lower density.

http://www.universetoday.com/15266/dark-matter-is-denser-in-the-solar-system/

The above may help.

I see,

the only thing I don't get is how 'In relativity, a particle of mass m can reach an arbitrarily high kinetic energy even for velocities <c.'

The additional energy ends up as an increase in mass due to Lorentz transformation.

What units are you using for 3 x 10^8? The speed of light is 3x10^8 meters/sec. - this a speed not an acceleration.

Thank you for a very concise post! When you say "it is a question of which nuclear reactions are possible" you mean if the energy of the cosmic ray is high enough and/or which particles is needed for one specific reaction, right?

Yes (I think - I am not quite sure what you have in mind for your question). What I meant is p + nuclide -> products, depending on whether or not the physics is there.

Hello everyone! Let me begin with that I am not a student in physics, this is just of general interest.

Everytime when I read about cosmic rays and transmutation, the only examples coming up are changes in the Earth's atmosphere. Look at this for example

http://en.wikipedia....heric_chemistry

 

My questions are:

-Are there any limits on which atoms cosmic rays can transmutate?

-Is the earths magnetic field limiting the reaction products of cosmic rays?

-How about in space, are there any limitations there?

1) There are no limits as such. It is a question of which nuclear reactions are possible.

2) The magnetic field affects the direction of incoming rays (which are mostly high energy protons), but not the reactions themselves.

3) In space there are no particular limitations - see 1).

http://en.wikipedia.org/wiki/Tunguska_event

This a recent event like the one described (not nuclear).

Are you discussing this in the context of doing it on a computer?

The use of computers in industry became fairly widespread by the mid fifties. Many large companies had their own, while small companies could rent time at data centers.

How about 12,000 years ago?

12000 years is too short to make much difference. However there is geological evidence that millions of years ago there were concentrations of Uranium that acted like natural reactors.

http://en.wikipedia.org/wiki/Nuclear_force

start here

I don't know what you mean by removing GR effects. The most important effect is ordinary gravity.

Well, that can't be entirely true. As you said, there will always be plenty of space to put a new point, yet all the points in [math]\mathbb{R}^2[/math] can't possibly lie in [math]S[/math], to use the notation I used earlier. This in itself is enough for me to wonder what it would look like. It might just look like a black square, ie. it may appear that all the points in [math]\mathbb{R}^2[/math] are elements of [math]S[/math]. It might not. And DrRockets idea, if correct, would certainly yield an interesting fractal. I haven't gotten around to trying it out yet, I'll post here when I do. But thank you, and everyone else, for your inputs, they're much appreciated.

What I said is entirely true. I never claimed that every point would be in the set - far from it. As long as the set is being created sequentially, there will be only a finite number of points. Looking like a black square seems impossible since you would then have many points lying on a given line. Even the idea of fractal doesn't make too much sense, since the points (unless you describe a different approach to creating the set) form a discrete set in the plane.

In my opinion the survey is poorly designed.

Exactly. But the set can't ever contain all points in the plane. Thus, my question; what does it look like?

 

 

EDIT: Ah, DrRocket replied while I was writing a reply. Thanks for the idea.

 

EDIT2: That won't work; even after the first subdivision, every vertex will be colinear with two other vertices, for example the points {0}, {0, 1, 2}, {1, 2} in the following image:

The set won't look like anything in particular - just a random set of points in the plane.

I thought as much, but I was thinking of a different type of set, although I'm not even sure if it'll make sense. Imagine this process. Two points are given. You chose a third (at random) that's not colinear with the two and repeat the process. In general, given n points, you chose the (n+1)st so that no three of them are colinear. If repeated indefinitely, what set will I get? Can the process even be repeated indefinitely?

The process can continue indefinitely without any particular pattern and there is nothing to make it stop. Given n points which are not collinear, there are n(n-1)/2 lines determined by these points. There is plenty of empty space to put in an additional point.

Looks good!