 # cosine

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## Posts posted by cosine

### Statement vs. Hypothesis

Yeah, there's another meaning of Hypothesis when you are doing statistics.

### Linear congruential generator

Are you sure your formula is correct? It would seem to me that the numbers generated by this would be from 0 to 31.

Mr. Skeptic brings up a good point... Mary, are your 1000 numbers an ordered sequence?

### Find the total mass of a wire with a certain density ?

Density (rho) can be a function of volume, too, not necessarily scalar. For that matter, if the object is denser in the center and gets less dense as it moves away from the center, then the rho is not scalar, it's a function of space (x,y,z).

Oh I see, k in the p equation is the 'really k' from later on! Sorry CalleighMay! Um... are you still working on this problem?

### Deriving continuous Brownian Motion from discrete Brownian motion.

It can be intuitively difficult to imagine such a function because when we draw something it tends to be smooth (a.e. -for technicality's sake). But continuous Brownian Motion is an example of a function that is continuous everywhere but nowhere differentiable.

(Note: Brownian Motion is also referred to as a Weiner Process.)

The Wiener process plays an important role both in pure and applied mathematics. In pure mathematics, the Wiener process gave rise to the study of continuous time martingales. It is a key process in terms of which more complicated stochastic processes can be described. As such, it plays a vital role in stochastic calculus, diffusion processes and even potential theory. It is the driving process of Schramm-Loewner evolution. In applied mathematics, the Wiener process is used to represent the integral of a Gaussian white noise process, and so is useful as a model of noise in electronics engineering, instruments errors in filtering theory and unknown forces in control theory.

The Wiener process has applications throughout the mathematical sciences. In physics it is used to study Brownian motion, the diffusion of minute particles suspended in fluid, and other types of diffusion via the Fokker-Planck and Langevin equations. It also forms the basis for the rigorous path integral formulation of quantum mechanics (by the Feynman-Kac formula, a solution to the Schrödinger equation can be represented in terms of the Wiener process) and the study of eternal inflation in physical cosmology. It is also prominent in the mathematical theory of finance, in particular the Black–Scholes option pricing model.

Merged post follows:

Consecutive posts merged
it might help if he considers the bean machine (no, i'm not kidding http://en.wikipedia.org/wiki/Bean_machine). i think that might be closer to what he's looking for.

hey insane_alien, I know what you mean by the bean machine, that is the sort of image I have in mind! Actually, what I want to see is that as the balls and spaces get smaller and more plentiful, the end result of the bean machine approximates a bell curve. I just want to see that in math language (aka by passing to the limit).

### riemman summs

I see why you might say that, but no it is not like that. You can only take Reimann integrals over continuous domains, but because your theta goes from (-pi, pi), you're not allowed to jump from pi to -pi!

If you could do the reimann sum as you were saying, then you would necessarily have:

$\int^{-\pi+\epsilon}_{\pi-\epsilon} d\theta = -\int^{\pi-\epsilon}_{-\pi+\epsilon} d\theta$

However, all is not lost because the region you highlighted is just:

$\int^{-\pi+\epsilon}_{-\pi} d\theta + \int^{\pi}_{\pi-\epsilon} d\theta$

### Linear congruential generator

You can actually do this naïvely with brute force. there are 32 choices for a and 32 choices for c, so you only have 934 possibilities. So take any two numbers next two each other in the sequence and use it as a check for all the possible values of a and c. Store the ones that work for this pair, and then try these out on another pair in the sequence. Do it until needed. This should be O(n).

### Deriving continuous Brownian Motion from discrete Brownian motion.

Hey guys, long time no post! In the past few years I wound up getting a BA in math and then accidently wound up going to grad school. I'm giving an informal presentation in a few days where I will introduce Brownian motion to some colleagues (no laxitive jokes, please!).

Anyway, for the presentation I want to pass from discrete Brownian motion to continuous Brownian motion, but I need some help connecting the dots! I'll put what I have so far here and then maybe you can point the way. Thanks!

Here we go:

Here is a diagram of the simplest Brownian motion:

      1
/
0
\
-1

Where each possible outcome (either 1 or -1) has a probability of 1/2. Let's say this happens over the discrete time period /\s. Now let's look at two time steps:

          2
/
1
/    \
0         0
\    /
-1
|  \
|   -2
|    |
[u]/\[/u]s, 2[u]/\[/u]s

Now we can define p(t, x), a probability density function, where t is the time, and x is the position on the vertical axis. There is a counting arguement where you define n, the number of time steps that have passed, as t//\s, that allows you to define this function explicitly. I won't go into the arguement here, but instead just show that:

p(t, x) = {n}CHOOSE{(n+x)/2} / 2^n

Ok, so this part has been straight forward. What I want to do is find the limit of p(t, x) as /\s -> 0. Actually, it will be that p(t, x) is a normal distribution in x, where t is the squareroot of the variance, and 0 is the mean. Any ideas on how to show this though by letting /\s -> 0? Thanks again for the help guys!

### Find the total mass of a wire with a certain density ?

Find the total mass of the wire with density p.

And it gives:

r(t)=2 cos ti + 2 sin tj + 3tk

and p(x,y,z)=k+z

(the p is a different looking p, most likely represents something else, something that sounds like roe maybe? lol. and k is really k below)

and: (k>0), 0<=t<=2pi

your p is very strange as it is the sum of a vector and a scalar value. i j and k are vectors. how they work is like this:

you could write the point (1, 2, 3) as 1i + 2j + 3k

also this p is weird because density is a scalar value, not a vector value, so there should be no i's j's or k's that don't cancel out in its formula.

when you have this figured out the straight forward way to look at this would be to use cylindrical co-ordinates. the cross-section of the object at any given z is a circle of radius 2. thats because cos t i + sin t j is the parametrization of a circle in a single variable.

maybe when you come back with more details on p we can help you out more

### What is good training for holding political office?

Philosophy should be a good course for presidential candidates.

I seriously think that people who have something to gain by winning the presidency should not be president.

### solving this SDE

Where's the F?

### URGENT: Programming factorials

I need help how do I create a program that calculates factorials? Step 1. Write a research proposal that requires the calculation of factorials.

Step 2. Budget for a program that already calculates factorials.

Step 3. Get research grant.

Step 4. Purchase and install program.

4 lines of code, easy!

### Method for simple Maths

As an aside, I believe this method is part of a larger curriculum called Vedic mathematics, which claims to have roots in the Hindu Vedas, and has some following in Indian education.

### Mathematics a Social Discipline?

But none of that applies specifically to maths, does it?

I think you're meaning that that doesn't only apply to mathematics, in which case I totally agree. I'm just trying to dispell the illusions that most mathematicians lock themselves up alone in offices and out pops formulae

### Mathematics a Social Discipline?

Well I would argue that it is a social discipline because you the social mathematician is infinitely better than the unsocial one. First off, explaining your thoughts is a very social skill. I appeal to authority now when I agree with the Feynmann/Einstein quote (depending on your source) that "If you can't explain it to a 6 year old [or an interested undergraduate], then you don't understand it enough yourself." And secondly the development of mathematics is much richer and quicker when you're bouncing ideas with a partner instead of only on your own path. I think of it much like this, If you consider a person as a vector, His ideas can span the space with the basis of one vector, aka a 1-dimensional space. Then add another person as a vector into the thought process, and the thoughts spanned by the two are 2-dimensional instead of 1!

### Mathematics a Social Discipline?

saying "Please" and "Thank you" is, Maths is Not however.

I beg to differ!

Mathematician A: Let's assume Fermat's Last Theorem to be true...

Mathematician B: No.

Mathematician B: No.

Mathematician A: Pretty please? It will simplify the proof...

Mathematician B: Well, okay then.

Mathematician A: Well if we assume Fermat's Last Theorem, then my proof of it follows from our assumption! Haha I win!

Mathematician C: B got pwned!

Mathematician A: Thanks C.

### my friends theory

This one is a little tougher.

The imaginary number, $i=\sqrt{-1}$

Now, let $-1=-1$

We can also write this as

$\frac{-1}{1}=\frac{1}{-1}$

Take the square root,

$\frac{\sqrt{-1}}{\sqrt{1}}=\frac{\sqrt{1}}{\sqrt{-1}}$

$\frac{i}{1}=\frac{1}{i}$

Since by definiton, $i^2=-1$ then

$1=-1$

I'm stumped, what is the trick behind this one?

Edit: Is it order of operations? PEMDAS here in America, BEDMANS in Canada, I don't know other countries' convention names. But anyway before you do the exponent of 1/2 on both sides (aka take the square root) you have to complete the division first! Because you're taking the squareroot of each side in its entirety, you have to perform the operation of each side first.

### can there be a negative infinity?

As a sidenote, I don't understand this bit about limits... If infinity is infinite how can it have a limit? There is noone that can prove that, it's against the definition isn't it? I remember finding limits at infinity in my math unit, but they referred to limits of the function and not to the limits of infinity.. right?

Well there are several types of infinities we talked about. (cf: http://www.scienceforums.net/forum/showpost.php?p=343622&postcount=10)

But the sense in which you are talking about is infinity as a limiting concept. (cf: http://en.wikipedia.org/wiki/Limit_%28mathematics%29)

### Implicit Diff.

indeed, it is B. hehe! awww... how come no one laughed at my joke... nah! it was not even a joke. 'twas just a stupid post.. Oh I see the joke now... lolz

### can there be a negative infinity?

hmm if we take the concept of infinity as a whole, then it exists of course, otherwise we wouldn't know about it would we, it's not imaginery? infinity in itself maybe doesn't exist, but the concept of it does, and that is one of the ways how we can express that concept, isn't it?

Can you restate what you mean?

### Variance of Median

The X =d normally means "X is distributed according to" and the R would be the probability distribution. But, I am unfamilar with what R would mean, too.

Hmm, I looked for examples of probability distributions on wikipedia denoted with an R, I found so far:

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

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

But I haven't any real idea what these are or are commonly used for

### Best Mathematics Books

http://en.wikibooks.org/wiki/Math

here are some open source math texts from wikibooks.

Ah you beat me to it Also I'm sure The Teaching Company has some resources you might like, as I'm thoroughly enjoyed every other product of theirs that I've used. They're audio and video lectures.

Edit: Also along similar lines, http://en.wikiversity.org/wiki/Portal:Mathematics

### Variance of Median

$X =^{d} R(0,\theta)$ and we obtain five independent observations on X: 1.2, 3.7, 2.1, 5.9, and 4.0.

The median $\hat{M}$ is 3.7 and I'm told that $var(\hat{M})=\frac{\theta^2}{28}$. How is this obtained? Do I use the formula $var(\hat{M})=\frac{1}{4nf(m)^2}$?

Could someone please explain what $X =^{d} R(0,\theta)$ means? What is an equals sign superscripted with a d mean? and is R some well-known probability distribution?

### can there be a negative infinity?

I think it should be classified that most of these posts are all dealing with different things referred to as infinity.

Ecoli is talking about the infinity of calculus which technically exists only as a limit.

w=f[z] is talking about the infinities which are transfinite cardinalities.

KLB is talking about To Infinity, and Beyond!

And geoguy and dave are talking about "the point at infinity" which exists in some non-euclidean geometries, most notably (as was said) Reimann geometry, which is a geometry on a sphere.

### Best Mathematics Books

I figured instead of wasting a new thread id add to a older one simular topic. Im a 22 yr old dropout whos deeply wanting to better my math I plan on going back to school soon and Id like to start learning as much as Possible. My deep love for astronomy has triggerd a math bug, and last math Ive taken was Algebra... I believe I failed it due to skipping school.

So anyone know any good algebra books too get someone who hasnt been active in the math world starting? Id love to hear

When you say Algebra... do you mean what is more colloquially called "highschool algebra" or "college algebra" that has to do with variables and polynomials or are you referring to what's more often called "abstract algebra" with groups and rings?

### 0/1

$\frac{0}{1}=0$ I don't know who told you otherwise.

Diving by zero however, is an invalid operation and there simply is no answer.

Excuse me for nitpicking, but there is a technical other consideration.

Any nonzero number divided by zero is undefined (unless working in some non-euclidean planes, then they are "the point at infinity"). As for $\frac{0}{0}$ is sometimes an exception if it is the limit of a function. If it is the limit of a function then it is called an indeterminate form and you need to use calculus to see if it has a value or not. (Cf: L'hopital's rule)

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