# version

1. ### Clever version of separation of variables

I came across a very interesting method to "separate" variables in an eigenvalue equation. But I'm not quite sure it works as advertised. Here's the sitch. We have, as usual: H \Psi = E \Psi, where H is dependent on both radial and angular variables, so we expect \Psi to contain both radial...
2. ### Using the Local Version of Cauchy's Theorem

I have been stumped on these integrals... My professor did no examples and I am having trouble visualizing what I'm supposed to be doing. Any help is greatly appreciated! By means of the local Cauchy theorem, find: \int_{|z|=1} \sqrt{9-z^2} ~dz. I immediately thought to use trig substitution...
3. ### Gradient of Multiple Linear Regression – Vectorized Version of ALS Update Formulas

Based on the attached image, how do we differentiate/get the gradient of the cost function \min\limits_{x\geq0}||y-\bar{A}x||^2_2? Refer to the attached image for the exact update rule function for both Matrices.
4. ### Constructing a matrix version of the transformation algorithm.

Algorithms like the transformation algorithm: (x, y) --> (x/k + p, ay + d) are not generally used in mathematics. Instead, we use matrices. (and then the question gives some general information about matrices, such as how to identify the, how to add and multiply them, etc. This is just basic...
5. ### Hahn-Banach theorem, geometric version (question about proofstep)

Hello, my question is about a proofstep in the proof of the second geometric version of the Hahn-Banach theorem. The theorem is: Let X be a normed, real or complex vectorspace. Let A,B non-empty, convex subsets of X such that the intersection of A and B is empty. Let A be closed and B compact...
6. ### Help with understanding the proof of Lucas' formula (strong version of induction)

Hi. I don't understand the proof of Lucas' formula. I would be very happy is someone could explain it step by step. Especially the last to lines of the proof. Here is a link to the proof. - Gøran
7. ### Prove that a sequence has a given value (discrete version of calculus proof)

This is from a discrete math textbook, but obviously calculus-based, so I'm asking it here. Let s_1,\ldots ,s_n be a sequence satisfying (a) s_1 is a positive integer and s_n is a negative integer. (b) For all i, 1\leq i<n, s_{i+1}=s_i+1 or s_{i+1}=s_i-1. Prove that there exists i, 1<i<n, such...
8. ### Proving a different version of L'Hopital's Rule

Suppose f: (a,b) \rightarrow \mathbb{R}, g: (a,b) \rightarrow \mathbb{R} are differentiable and that g(x) and g'(x) are always non-zero. Suppose that lim_{x \rightarrow b^-}f(x)=0=lim_{x \rightarrow b^-}g(x) and that lim_{x \rightarrow b^-}\frac{f'(x)}{g'(x)}=\infty {Equation *} then lim_{x...
9. ### Cauchy Theorem Homological version

Hi, I'm having trouble understanding a certain part of the proof for the homological version of the Cauchy theorem. I'm going through a proof that given in this link: http://www.math.uiuc.edu/~r-ash/CV/CV3.pdf starting on page 9. At the end of page 10, it states that g(w,z)=...
10. ### New version of Fooplot

Great new graphing calculator in your web browser. Fooplot ( www.fooplot.com ) just released a new version of their online graphing calculator with some new features (and it works in IE again now). And it lets you save images: Also, there's this other new calculator out there called Desmos (...
11. ### Solving equations with indeces (version 2)

Hello everyone, I have come undone on solving another equation with indices in: \frac{3^{5x+2}}{9^{1-x}}=\frac{27^{4+3x}}{729} I've a feeling it involves simplifying the numerator and denominator of the second half so that I can multiply with the first half, but I'm doing something...
12. ### NOT version of the Inclusion-Exclusion principle

It may sound ridiculous but I can't find this anywhere on the web. Thanks in advance
13. ### Models for Cooling, 9.4 in the Stewart Calculus book, version 6.

1.#7 Page 618: A peach pie is removed from the oven at 5:00 PM. At that time it is piping hot, 100 C. At 5:10 PM its temperature is 80 C; at 5:20 PM it is 65 C. What is the temperature of the room? The answer is 20 degrees Celsius but I'm not sure how to get there from this step (0...
14. ### Relation between a series and its alternating version.

Hello! Here is something that I've been wondering for quite a while. Consider the following two infinite series: S = \sum_{k\ge0}\frac{1}{(ak+b)(ck+d)}, ~~ T = \sum_{k\ge0}\frac{(-1)^k}{(ak+b)(ck+d)} Assuming that all issues of convergence are sorted, what's the relation (if any) between the...
15. ### how to prove this version of L'Hopital's rule

Please prove the following version of L'Hopital's rule. Suppose that f,g : (a,b) \rightarrow \Re are differentiable with g(x) and g'(x) never equal to zero. suppose also that: lim_{x \rightarrow b-} f(x) =0 lim_{x \rightarrow b-} g(x) =0 lim_{x \rightarrow b-} \frac{f'(x)}{g'(x)} =...
16. ### DSolve Mathematica version 7

When I use DSolve with Mathematica version 7 for an IVP, it doesn't generate the answer. However, my professor has the same version and it works just fine for him (I sent him the my file and it works fine on his computer). I have attached the Mathematica file. Is there a preference setting...
17. ### New computer website version

I've just been notified that this website has been upgraded with many new features: Computer Forums
18. ### Riemann Sums, Sparknotes version.

Ok, I am aware of how to do (not fluently) summations, but Riemann sums throw me off so much. So here is my best description: n \sumk^2 i=1 This turns to : n \sumf(i) i=1 Where f(i)=i^2. Ok, I know n is the number of "rectangles" which will approach infinity. Your counter is i, which replaces...
19. ### finite field version of the Bollobás theorem

The proof of Bollobas's theorem on subspaces uses the fact that we can factor out by a subspace in general position. This requires that the Field is infinite. How can I prove the finite field version of the theorem?
20. ### A weaker version of Arzela Ascoli Thm

I'll summarize this question as follows: Suppose (fn) is a an equicontinuous sequence of functions in C0[M, R]. Suppose p is a point in M such that fn(p) is a bounded sequence of real numbers. What is the condition on M so that (fn) is uniformly bounded (that is, sup {lfn(x)l} <= C for some C}...