Recent content by Physics_wiz

Yes, I see now how I wrote doesn't make sense. I was trying to use this fact to solve the problem in my last post of the "Expressing multi-variable functions" Thread, but I guess I can't use this fact to check for whether a function exists or not.

I remember before reading bits and pieces about how if we have a function of two variables, say f = f(x,y), then it must be true that d/dx(df/dy) = d/dy(df/dx), where the "d"'s are partials. Can anyone guide me to what this theorem is called or to its implications? Also, does it work in...

Anyone? Do the functions N and M exist so I can write c = N(d,e,M(a,b))? By the way, N and M can be anything...I just wanted to know if they exist.

Ok, here's where the original question came from...maybe this helps. Say I have a function F(a,b,c) = G(d,e). Assume the Implicit Function Theorem conditions are satisfied. So, I can solve for c as follows: c = H(G(d,e),a,b). Now, in this case, can I write c as c = N(d,e,M(a,b))? Why or why not?

I read a little about cardinality...I think I understand. Can anyone direct me to where I can look to answer my original question? (i.e. what topics names I can look up in calculus or analysis)

I don't think I agree with that. Let f(x_1,x_2,x_3,x_4)=x_1x_3+x_4. Now, what function g(x_1,x_2,u(x_3,x_4)) is equal to f(x_1,x_2,x_3,x_4)

I have a simple question, let's say I have a function f = f(h(a,b), c, d). Can I express this as f = g(a, b, u(c,d))? Are the two expressions equivalent or is one different/more general than the other?

Does anyone know a formula for the inverse of a sum of two upper triangular matrices?

ok I got it I think. If I set |x-a|<|a|, then |x|-|a|<|a| or |x/a|<2. So the original |f(x)-f(a)|<3|x-a|. So, if delta = epsilon/3 then the definition given in the first post will be satisfied. Side note, does that imply uniform continuity since delta depends only on epsilon?

Homework Statement Show that f(x) = x/(1+x^2) is continuous on R Homework Equations f is continuous at a if for any epsilon > 0, there exists a number delta > 0 such that if |x-a|<delta, then |f(x)-f(a)|<epsilon. The Attempt at a Solution |f(x) - f(a)| = |x/(1+x^2) - a/(1+a^2)|...

Say I have a Gaussian random number generator that generates random numbers with an unknown mean x. I get a few random numbers from the generator and I want to estimate x. The estimate will, of course, be the average of the numbers (y), but how confident can I be that x is within a value, a, of y?

Yes, you're right in your interpretation. However, the equality between the middle part and the last part is the most crucial step for my purposes, so I can't take it out.

Say I have a function g = g(x,y). Now, I have another function defined as: f(x,y) = \partial / \partial x [g(x,y)]. Is the following true: f(a,b) = (\partial / \partial x [g(x,y)])_{x = a, y = b} = \partial / \partial a [g(a,b)] a, b, x, y are all variables. Is this true in general for...

Yes, I understand. However, I am trying to find a general formula for upper triangular matrices. Something along the lines of the inverse formula for 2x2 matrices. I remember my linear algebra teacher telling us that formulas like that exist for higher dimension matrices.

Does anyone know a formula to find the inverse of an upper triangular matrix of dimension n (with a reference preferred)?