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1. Property of exponentials and sine functions?

That is correct, however, I made a typo in my DE. It should be x''-4x'+5x=0. The idea is that you can find the characteristic equation from the DE, hence the roots, and then the general solution. So you may work backwards as well. Try it again.
2. Property of exponentials and sine functions?

No, that is a bit complicated. If you have the differential equation x''-4x'+x=0, how would you find the general solution? Just go through a few steps, then should have the solution to your problem as well. If not, I will give you another hint.
3. Property of exponentials and sine functions?

3 and 4 do not matter. Work backwards. If the solution in this case involves sines and cosines then we can conclude that the roots to the auxiliary equation are complex. Find the roots and then the polynomial. You will then have your differential equation and hence the values of r and k.
4. Logarithmic singularities are locally square integrable

Homework Statement I will like to show that the function f:\mathbb{R}^2\rightarrow \mathbb{R} defined by f(x)=\ln\bigg(1+\dfrac{\mu}{|x-x_0|^2}\bigg),\quad\mu>0 is in L^2(\mathbb{R}^2). Homework Equations A function is in L^2(\mathbb{R}^2) if its norm its finite, i.e...
5. Where to get started with Numerical Solutions to PDEs?

I have established existence and uniqueness of solutions to a nonlinear elliptic system of PDE's and would like to see how the solutions look like numerically. I have some programming background on MatLab and C++. I also understand basic ODE numerical schemes. But I am not so sure, if I need to...
6. Showing that a function is in $L^2(\mathbb{R}^2)$

Homework Statement Suppose f\in L^2(\mathbb{R}^2). Is f+c\in L^2(\mathbb{R}^2) where c is a constant? Homework Equations f\in L^2(\mathbb{R}^2) if ||f||_2<∞. The Attempt at a Solution I think the answer is no because ∫_{\mathbb{R}^2}{c^2}dx=∞. However, I am still unsure. Any guidance...
7. Finiteness of an integral given an $L^2$ function

Homework Statement Let \Omega be a torus and g\in L^2(\Omega) be a scalar value function. Is \int_{\Omega}{e^g}dx<\infty? Homework Equations The Attempt at a Solution Not sure where to start. However, if g\in W^{1,2}(\Omega) then I can show that the answer is yes by applying...
8. Convexity of a functional using the Hessian

Homework Statement Consider the functional I:W^{1,2}(\Omega)\times W^{1,2}(\Omega)\rightarrow \mathbb{R} such that I(f_1,f_2)=\int_{\Omega}{\dfrac{1}{2}|\nabla f_1|^2+\dfrac{1}{2}|\nabla f_1|^2+e^{f_1+f_2}-f_1-f_2}dx. I would like to show that the functional is strictly convex by using the...
9. Show that matrix A is not invertible by finding non trivial solutions

I am not sure what you mean by elemination failing. However, elimination give many rows of zero and hence infinitely many solutions for Bx=0. Consider doing elimination on C for a 2x2 matrix with the same columns and see what are your solutions. If you have learn the determinant, then the...
10. Show that matrix A is not invertible by finding non trivial solutions

Bx=0 and Cx=0 are homogeneous system of equations. So there are only two posibilities for their solutions. 1. Either the solutions are trivial and unique or 2. Infinitely many solutions. Ask yourself if B is invertible? If it is, then multiply both sides of Bx=0 by its inverse to obtain only...
11. Property of a limit of functions of average value zero in L^2 space

Got it. Thanks. Using Holder's inequality and noting that the size of \Omega is finite the results follows since f_k\rightarrow f in L^2(\Omega).
12. Solving Differential Equations

Assuming K and L are constants. How about dividing by (L-N) on both sides from dN/dt = k(L-N) instead of distributing the K? This equation is separable.
13. Property of a limit of functions of average value zero in L^2 space

Homework Statement Let f_k\rightarrow f in L^2(\Omega) where |\Omega| is finite. If \int_{\Omega}{f_k(x)}dx=0 for all k=1,2,3,\ldots, then \int_{\Omega}{f(x)}dx=0. Homework Equations The Attempt at a Solution I started by playing around with Holder's inequality and constructing...
14. Closed subspace of a Sobolev Space

I am having a hard time making this conclusion. How about if the sequence is bounded?
15. Impact of PhD granting instititution on career

Thank you all for your feedback.
16. Impact of PhD granting instititution on career

This is what I thought. I have an excellent dissertation adviser. Well know in his field, but before I asked him of opinion I wanted to be more informed so I am seeking all inputs.
17. Impact of PhD granting instititution on career

Does the institution where an individual receives their PhD degree in mathematics have a significant impact on their career? The question evolves from a merger among universities: If a university is under a merger with another well established university with a much much better reputation in...
18. Closed subspace of a Sobolev Space

Homework Statement I am considering the space \tilde{W}^{1,2}(\Omega) to be the class of functions in W^{1,2}(\Omega) satisfying the property that its average value on \Omega is 0. I would like to show that \tilde{W}^{1,2}(\Omega) is a closed subspace of W^{1,2}(\Omega). Homework...
19. Divergence Theorem on a surface without boundary

Reading through Spivak's Calculus on Manifolds and some basic books in Analysis I notice that the divergence theorem is derived for surfaces or manifolds with boundary. I am trying to understand the case where I can apply the divergence theorem on a surface without boundary.
20. Calculating weak limits

Homework Statement Prove that the sequence \{sin(kx)\} converges weakly to 0 in L^2(0,1). Homework Equations A sequence of elements \{f_k\} in a Banach space X is to converge weakly to an element x\in X if L(f_k)→L(f) as k→∞ for each L in the dual of X. The Attempt at a Solution...
21. Programs Quality of your PHD thesis

Hello everyone, I feel that I can complete a thesis within a year but its quality would not be as great if I spend two years in it. However, I would like to complete my thesis so that I can begin during research on my own and enjoy the privileges of having a PHD whatever they might be. The...
22. Limit and diffirentiability of a function

So, I have made some progress by rewriting the problem and using L'hopitals rule. But I am still off by a factor of \dfrac{p}{2}. rewriting: |f+tg|^2=f\bar{f}+tf\bar{g}+t\bar{f}g+t^2g\bar{g} When I apply L'Hopitals rule I get \dfrac{p}{2}|f|^{p-2}(\bar{f}g+f\bar{g})
23. Limit and diffirentiability of a function

Homework Statement For complex numbers f and g, and for 1<p<\infty we have \lim_{t\rightarrow 0}\dfrac{|f+tg|^p-|f|^p}{t}=|f|^{p-2}(\bar{f}g+f\bar{g}); i.e., |f+tg|^p is differentiable. I would like to show that the above statement is true. Homework Equations The Attempt at a...
24. Weakly convergent sequences are bounded

Thanks for your comments. I was able to resolve my issue with this problem.
25. Weakly convergent sequences are bounded

Homework Statement I would like to show that a weakly convergent sequence is necessarily bounded. The Attempt at a Solution I would like to conclude that if I consider a sequence {Jx_k} in X''. Then for each x' in X' we have that \sup|Jx_k(x')| over all k is finite. I am not sure why...

thank you
27. Countable VS finite

Homework Statement Are there countably many rational numbers in the interval (0,1) or are there finitely many? Homework Equations The Attempt at a Solution I am confused. There are countably many rational numbers in the interval (0,1). Does this mean I can list them all in such a...
28. Measure on the real line

Ok, I have an idea on how to prove this now. Any finite cover of A must also cover the interval (0,1). Thus the sum of the lengths of the intervals that cover A must be greater than 1 since they also cover the interval (0,1). However, I still am having difficulty writing down an argument to show...
29. Measure on the real line

Homework Statement Let A be the set of all rational numbers between 0 and 1. Show that for any "finite" collection of intervals I_n that cover A the following inequality holds: \sum I_n \geq 1 . Homework Equations We are using the definition of the outer measure here. Where the outer...
30. Integral using Lebesgue Measure

Thanks for the hint. So 1=len([0,1])=len(A)+len(B)=len(B).