Randy

What are the applications of topology? What sort of problems does it help solve?

Find all (c, y) pairs, where c is a constant and y is a differentiable function defined in [-a, a] such that the equation:

∫ (from -a to a) (y'g' - cyg) dx = 0

holds true for ALL differentiable functions g defined in [-a, a].

Say we have two independent random variables A and B and we define a third random variable C as C = A + B.

Now say we want to compute P(C = c|A = a). This event occurs iff B = c - a. Thus, P(C = c|A = a) = P(B = c - a).

Similarly, say we want to compute P(A = a|C = c). Again, this event occurs iff B = c - a, and therefore P(A = a|C = c) = P(B = c - a).

Therefore, P(A = a|C = c) = P(C = c|A = a). However, this is definitely not necessarily true. Where did I go wrong?

Here, "largest" is in the sense of Lebesgue measure. Does there even exist a group with nonzero measure? Remember, no rationals except zero are allowed.

A functional F(x, y) = ∫(from 0 to T) f(x(t), y(t), x'(t), y'(t), t)dt

where x and y are functions serving as parameters for F. How do we find x(t) and y(t) simultaneously such that F = 0?

A functional F(y, g) = (∫(from a to b) p(y(x), y'(x), x)dx) * (∫(from a to b) q(y(x), y'(x), x)*g(x)dx) + ∫(from a to b) r(y(x), y'(x), x)*g(x)dx

where p, q and r are known functions and y, g are functions of x which serve as parameters for F. How do we find the value of function y such that F = 0 for all values of function g?

Take the two enantiomers of tartaric acid, for example. Now, the only reason L and D tartaric acid are not super-imposable over each other is because the H and OH groups are interchanged. However, as tartaric acid is an alkane, the carbons can freely and spontaneously rotate about the sigma bond. So, why can't the two central carbons in L tartaric acid rotate 180 degrees about the sigma bond and turn the compound into D tartaric acid?

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

Consider the function f(x) = ∫(from 0 to ∞) sin(xt)/t dt. It can be shown that (refer to the link given below for a proof) that the function is identical to:

f(x) =

π/2, when x > 0

0, when x = 0

-π/2, when x < 0

Therefore the derivative of the function is:

f '(x) = 0, over the domain x ≠ 0

and it is undefined when x = 0. Alternatively, lets try taking the derivative of the function by differentiating under the integral.

f '(x) = d/dx(∫(from 0 to ∞) sin(xt)/t dt)

= ∫(from 0 to ∞) ∂/∂x(sin(xt)/t) dt

= ∫(from 0 to ∞) cos(xt) dt

Here, this integral is undefined for all x as the anti-derivative of the cosine function, i.e., the sine function, is undefined at infinity. Therefore, by this method, the derivative of f(x) is undefined for all x!

What went wrong?

Link for the evaluation of ∫(from 0 to ∞) sin(xt)/t dt:

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

I need a good book which explains the concepts of rotational mechanics; rotational dynamics in particular. It should definitely include rotational mechanics for a translating axis, and id possible a rotating axis too. Most importantly, I need the book to give an intuitive feel of how rotational mechanics works for a translating and rotating axis. I have books which do a pretty good job of this for a fixed axis rotation, mostly explaining it analogous to translatory motion. The problem is, when it comes to the moving axis, or even a translating body with respect to a fixed axis, all books revert to pages and pages of mathematical derivations without saying a word of why the formula comes out to be that way. The math I can grasp; but the physical meaning of the final formula, I can't. In fact, most of those derivations look like you need to know the final formula in order to even carry out the derivations! Its mainly the concepts of torque and angular momentum that I don't get, everything else is more or less okay. Can anybody please suggest any good books which help with the same? Many thanks in advance!

What are the conditions for any vector field to represent the flow field, or the velocity field, of a streamlined liquid flow? The liquid is incompressible and has a constant density; over space and time. I'm not quite sure what the actual definition of "streamlined" is, but I'm using it in the context that it does not change with respect to time; the velocity is a function of position only. One necessary condition I could think of is that the flux of the field over any volume has to be zero, but I'm pretty sure that this condition is in no way sufficient. So, what are the conditions for a vector field to be a streamlined liquid flow field?

This is about the fourth order Runge-Kutta method for numerically approximating the solutions of differential equations (RK4). In the lower order Runge-Kutta methods (RK2 and RK3), we basically take the average of the increment and the next predicted increments to get the actual increment we're going to add. So, in RK2, the increment will be (k1 + k2)/2 and in RK3 the increment is (k1 + k2 + k3)/3; where k1 is the increment from Euler's method and k2 and k3 are the predicted increments we get using Euler's method after we add the first increment. However, when we come to RK4, the expression becomes (k1 + 2k2 + 2k3 + k4)/6, essentially giving k2 and k3 a higher weight compared to k1 and k4. Why is this done? Won't just (k1 + k2 + k3 + k4)/4 work? And if at all we need to give more weightage to any of the terms, shouldn't it be for the first term, as its the only one which is directly evaluated? Thanks in advance!

In the poem "The Rime Of The Ancient Mariner" by Samuel Taylor Coleridge, the Mariner shoots down the albatross and later feels guilty and repents. But, why did he shoot it down in the first place?

We have a 10x10 square grid (or any other size, for that matter), with the bottom-right square marked "Start" and the top-left square marked "Finish". Every other square has either a 0 or a 1. From any square, we can move to any square adjacent to it in all four directions. The objective is to get from Start to Finish such that the sum of all the squares we have traveled through is maximum. We know the distribution of the 0s and 1s in the grid. How do we design an algorithm to achieve this purpose? Could somebody please sketch the working of an algorithm; in brief?

Can somebody please suggest a few books on advanced physics? I need a few books which are comprehensive and cover every aspect of a particular topic. I'm tired of seeing "The proof is beyond the scope of this textbook" over and over again, -_-. I preferably want a general physics book, but if you can name a book which is comprehensive in a particular topic like thermodynamics, classical mechanics, etc; that's fine too. I would also prefer it if the book is mathematically inclined. Many thanks in advance!

I read somewhere that there exist convergent functions that are not uniformly convergent. But I haven't managed to think of a single convergent function that isn't uniformly convergent, and I have no idea what such a function would be like. Please help!

The operation of division is well defined with an algorithmic approach for all integer denominators. It is also defined for rational numbers : a/(p/q) = aq/p. It is even defined for all surds : a/b^(1/q) = ab^(1 - 1/q)/b. Although I'm not sure, i suppose it can subsequently be defined for all algebraic numbers also. But trouble arises when it comes to creating an algorithmic approach for dividing by a transcendental denominator. How do we evaluate something like 3/π? One way would be to define it as follows:

3/π = lim (n ->∞) 3 * 10^n/[π * 10^(n)]

where [x] is the floor function. Alternatively:

3/π = 3 * π^(-1) = 3 * e^(-ln(π))

= 3 * lim (n -> ∞) Σ(from 0 to n) (-ln(π))^n/n!

by the Maclaurin series expansion of e^x. However, these are more of approximations than actual algorithms to evaluate 3/π digit by digit. So, is there any algorithmic approach to divide by a transcendental number?

Prove or disprove the following:

lim x ->a (f(x)) = lim x ->g(a) (f(g^(-1)(x)))

Read : The limit as x tends to a of f(x) is equal to the limit as x tends to g(a) of f(g^(-1)(x)), where g^(-1)(x) is the inverse function of g(x).

Any number can be represented in the following form:

√(n +√(n + √(n + ....

Now, lets try to represent both 0 and 1 in this form by finding the corresponding n.

0:

0 = √(n +√(n + √(n + ....

0^2 = n +√(n + √(n + ....

0 - n = √(n +√(n + √(n + ...

which is exactly what we started with; so the right side of the equation is again 0:

0 - n = 0

n = 0

So, 0 = √(0 +√(0 + √(0 + .... (equation 1)

1:

1 = √(n +√(n + √(n + ....

1^2 = n +√(n + √(n + ....

1 - n = √(n +√(n + √(n + ....

which is exactly what we started with. So, the right side is equal to 1.

1 - n = 1

n = 0

So, 1 = √(0 +√(0 + √(0 + .... (equation 2)

Now, the right sides of equations 1 and 2 are the same. So, we can equate their left sides to get:

1 = 0

What is wrong with this proof?

Isn't 1 + 1 a simpler computation than 2 + 2? So why don't they say "easy as 1 + 1"?

By this, I mean that their plurals are homonyms but the words themselves are not homonyms.

Homonyms = Two words which are spelled the same but have different meanings. An example is 'right', which can refer to the direction or mean 'correct'.