Math Help Forum - Other Advanced Topics

Show that f is differentiable and bounded

katielaw — Fri, 20 Nov 2009 21:50:04 GMT

I'm sorry, I posed this in the wrong forum. Mods please delete.

rearranged series

friday616 — Fri, 20 Nov 2009 20:51:03 GMT

Let the summation from k=1 to infinity of a_k be a nonabsolutely convergent series. Show that there is a rearrangement of this series so that the sequence of a partial sums of the rearranged series converges to infinity.

Also, show that there is a rearrangement of this series so that the sequence of partial sums of the rearranged series is bounded but does not converge.

Thanks!

Any help appreciated with y�=sin x sketch

shadz — Thu, 19 Nov 2009 22:39:16 GMT

Any help appreciated with a tricky question of the form - using your knowledge of the graph of y=sin x, sketch, with reasons, the graph of y�=sin x for 0 is larger than or equal to x is larger than or equal to 3 pie.
(Rofl)

Algorithm

liberty — Thu, 19 Nov 2009 09:58:44 GMT

Let A be a 1- dimensional array with integers and "min" , "max", "number"
three integer numbers. find an algorithm that answers, if the value "number" is in the array e.g a [i] = number for some i, 1<= min <= i <= max <= n, and the complexity of the algorithms must be O(lgn)

I need some help on that! Thanks in advance!

Need help with random walk algorithm

Marril — Thu, 19 Nov 2009 09:20:07 GMT

I have a problem considering a two-dimensional grid.
The grid size is (L*D) where L and D are known. Each grid site has either 1 or 0 as its value. The total amount of 1's is known and equal to N. The rest of the (L*D - N) sites are 0's.
At the beggining the N 1's are spread randomly over the grid.

I wish to create an algorithm which in every steps randomly selects one of the (1) spots, and moves it (randomly) to one of its four nearest neighbours. This move is only allowed if the neighbour isn't occupied (0) and the site is inside the grid, as to preserve the total number of 1's.

Descriptive example:

0 0 0 0 0
0 0 1 0 1
0 0 1 0 0

At the nex step, the bold 1 can move up, left and right, but not down (since there's a 1 in that spot). The italics 1 can move up, down and left, but not right (since it is the end of the grid).

Currently I use this type of algorithm:
1. Randomly choose a 1 in the grid.
2. Randomly choose a direction (out of 4 allowed).
3. If possible, move the 1 to the new spot and replace the old spot by 0. If not possible, go to step 1.

As it appears, this algorithm has a severe flaw: The 1's start out nearly evenly distributed. As I perform increasingly many iterations of the above algorithms, the 1's seem to clump up near the edges of the grid, and the middle stays pretty much empty.

This is a very bad behavior, since I require an algorithm where the 1's stay evenly distributed no matter how many moves are performed. This is to simulate sort of a simple classical gas in a 2D container.

I suspect that this phenomenon arises from the following cause: 1's at the perimeters of the grid have on average a lower probability to move than those in the middle, since they have less spots open for them to move to.

How can I change this algorithm as to still move only one (1) every step, and maintain the even distribution of 1's in the grid?

If it helps simplify the matter, you can consider a 1 dimesional case of this question, because the other dimension in my problem has quasi-periodic boundary conditions (so 1's don't clump up at the top or bottom of the grid).

The problem in Algorithms

turboman — Thu, 19 Nov 2009 04:02:25 GMT

Please help me to work on the following question:

Consider the following sequence of steps:

1. count:=0
2. answer:=5
3. while count<=9 do
3.1 count:=count+2
3.2 answer:=answer+count
4. output answer

b. Construct a trace table to show the value of all variables.
c. What is the value of answer after 3 iterations of the for-do loop?
d. Is the sequence of steps an algorithm? Giv a reason for your answer.

Please help to solve the above problem.

Cheers,

Complex numbers

arze — Thu, 19 Nov 2009 03:37:44 GMT

I don't know where to post this, so I'll put it here.
If and and a, b, x, and y are real, prove that

How do i go about doing this? Other than that
and
Thanks

The problem in Functions.

turboman — Thu, 19 Nov 2009 02:50:08 GMT

Dear All,

The following is my question:

A function f:{1,2,3,4,} -> {0,1,2,3} is defined by the rule: f(n)=(n+6) mod 4
a. Draw an arrow diagram for this function.

I know how to do it but I do not know how to draw it in here.

b. Is f one-to-one?

Yes, it is.

c. Is f onto?

Yes, it is.

d. Does f^-1 exist? Explain your answer.

I have a problem in this part. Please explain the answer to me.

Cheers,

Saying an expression

Bingk — Tue, 17 Nov 2009 23:26:54 GMT

I've been studying a paper, and I recently tried discussing it with someone in person, and that made me realize I need some help with talking in math :)

How would you say the following in an efficient way (i.e. without having to explain it all the time, specially the index of summation)?

where is the Legendre symbol, n is a positive odd integer and p is a prime such that p>n+1.

I'm not sure how I would say the summation part (the index) because it's not a straightforward from i=1 to p-1. It's a combination of unique (not equal) i's arranged in order, where 0
Also, I'm not sure how to say the Legendre part ... would it be "the Legendre of i_1 over 3"?

Thanks, any help would be much appreciated :)

Projection of a sphere onto a line

Peem — Tue, 17 Nov 2009 14:51:06 GMT

I need to define a function, let's say b(x), that defines the projection of a 3D sphere of radius r centred on the origin - onto a line.

I'm a bit lost as to how I should start this to be perfectly honest. Involved algebra, integration of x^2 + y^2 + z^2 = r^2, something else entirely?

There is a bunch of stuff involving Fourier transforms and such thereafter for multiple spheres but if anyone might be able to give me a pointer for this...

complex numbers: |z+1| + |z-1| = 3

lampak — Mon, 16 Nov 2009 19:01:02 GMT

How can I solve an equation like this:

I've tried expanding z as

and computing the equation

but this way it quickly gets terribly complicated. Plus I have two variables in one expression.

Any other ideas?

Complex Analysis: cauchy's theorem

rickgoz — Mon, 16 Nov 2009 11:34:02 GMT

Hello,

I have a problem with the following exercise:

Use one of the consequences of Cauchy's Theorem to evaluate

(i) when B:z=e^(it) (0<=t<=pi)

Can I said that B is a contour in C. The region C is simply-connected and Cauchy's theorem can be used.

So =0

Is it right?

Extended essay on Mathematics

Kareir — Fri, 13 Nov 2009 23:22:20 GMT

Hey all,

First post on this forum, so.. hi (Hi)

Anyway, to get to the point.. I live in Ireland, and i'm doing a program called the IB. It more or less covers the last two years of American High School, i think.. One of the requirements is that we write an extended (4000 word) essay on an academic topic. I've been considering writing it on Maths, but not quite sure where to start. It has to be a fairly specific title, and as I'm hoping to study engineering in college, i was thinking of a title along the lines of "Applications of Differential Calculus in Engineering".

I was wondering whether you guys would know a good place to start on this topic. We cover diff. Calculus fairly well, although it'll probably be next year before we get to the "Applications of Diff. Calc." section. If you have any suggestions on the best place to start looking, or even a better, more specific title, then it'd be welcome..

Oh, also, apologies if this isn't the correct topic for this board, or even the forum. And while I'm in 'high school', i figured this is a slightly more university-scale question, though i could be wrong.

Thanks a lot,
_Kar.

a number sequence problem

enyllebeam — Fri, 13 Nov 2009 01:32:38 GMT

hi guys, im a psychologist bowing down to the mathematics...I have unfortunately encountered a problem whilst doing my research. It involves coming up with a umber sequence based on the following constraints:

basically it's like this:

96 individual objects in total
48 are presented ONCE only
48 are presented THREE times (objects 25-48, 73-96)

the problem: come up with a sequence/pattern that goes something like....

25 X X 25 X X X X X X 25 and 25 X X X X X X 25 X X 25 (ideally half the repeated objects will have the first pattern and the other half will have the reverse)

so the objects after 3 or after 7 instances

in total, there should be 192 objects presented to the participant.

bonus points if you can break it into blocks of 3 or 4 or whatever just so i can give the participant a break when i test them for my experiment.

the main thing is counterbalancing as this is part of a hierarchy of counterbalanced conditions.............

I apologise that I can't lay out the problem in a maths equation, unfortunately my maths abilities only extend to A-levels and this is quite beyond me.

any help would be appreciated!

algebraic geometry

esisk — Thu, 12 Nov 2009 00:24:18 GMT

Studying for prelims:

Why do x_1, x_2 not generate the sections of O_(A^2) over A^2
thank you