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1. 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...
2. 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...
3. 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...
4. 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...
5. 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...
6. 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...
7. 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...
8. 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...
9. 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.
10. 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...
11. 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...
12. 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...
13. 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...
14. 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...
15. 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...
16. Integral using Lebesgue Measure

Homework Statement Find the integral of the function f(x)=3 when x is rational and 2 when x is irrational on the interval [0,1]. Homework Equations The Attempt at a Solution So I partition [0,1] into two disjoint sets A and B. A=[0,1] \cap Q and B=[0,1] \cap Q^{c}. Now the...
17. Differential Topology: 1-dimensional manifold

Homework Statement Given S1={(x,y) in R2: x2+y2=1}. Show that S1 is a 1-dimensional manifold. Homework Equations The Attempt at a Solution Let f1:(-1,1)->S1 s.t. f1(x)=(x,(1-x2)1/2). This mapping is a diffeomorphism from (-1,1) onto the top half of the circle S1. I was...
18. Parametrization vs. coordinate system

I am reading Differential Topology by Guillemin and Pollack. Definition: X in RN is a k-dimensional manifold if it is locally diffeomorphic to Rk. Suppose U is an open subset of Rk and V is a neighborhood of a point x in X. A diffeomorphism f:U->V is called a parametrization of the...
19. Abstract algebra: Rings and Ideals

Homework Statement The problem is to show that a subset A of a ring S is an ideal where A has certain properties. S is a ring described as a cartisian product of two other rings (i.e., S=(RxZ,+,*)). I have already proved that A is a subring of S and proved one direction of the definition of an...
20. Connected vs. Path Connected Sets

In general, if S is a connected set, can I conclude that S must be path connected? Definition 1: S is connected if it is not disconnected. A set S is disconnected if it can be written as the union of two mutually separated sets, where mutually separated sets are two nonempty sets that do not...
21. Partial derivatives

Homework Statement A mapping f from an open subset S of Rn into Rm is called smooth if it has continuous partial derivatives of all orders. However, when the domain S is not open one cannot usually speak of partial derivatives. Why? Homework Equations The Attempt at a Solution In the 1...
22. Simple harmonic motion proof

Homework Statement Prove that there exists a number A>0 and \phi such that acos(ct)+bsin(ct)=Acos(ct-\phi). Homework Equations a,b,c are predermined constants where c>0. From this equation I can justify conclusions regarding the amplitude, frequency, and so forth of a simple harmonic...
23. Factor Groups

Homework Statement Let G be a group, and let H be a normal subgroup of G. Must show that every subgroup K' of the factor group G/H has the form K'=K/H, where K is a subgroup of G that contains H. Homework Equations I dont see how to get started. The Attempt at a Solution I wrote...
24. Classifying finite groups

Homework Statement Prove that the group of order 175 is abelian. Homework Equations The Attempt at a Solution |G|=175=527. Using the Sylow theorems it can be determined that G has only one Sylow 2-subgroup of order 25 called it H and only one Sylow 7-subgroups called it K. Thus...
25. Sylow 2-subgroups of S4

Homework Statement I want to find the Sylow 2-subgroups of the permutation group S4 Homework Equations I dont understand why is my application of Sylow's third theorem wrong. The Attempt at a Solution The order of S4 is 24=233. Thus, there are Sylow 2-subgroups and Sylow...