If $A$ has strictly positive reach, does the set $\{ x \in A \colon...
Let $A \subseteq \mathbb{R}^n$ with $\text{reach}(A) > 0$ (see https://en.wikipedia.org/wiki/Reach_(mathematics) ).Define for any $\epsilon>0$, the "removal of $\epsilon$-thick...
View Articledifference between Sobolev space and bounded variation space
I learned Sobolev spaces & bounded variation spaces. I read this sentence:Sobolev spaces include functions such that $\int |f^{(1)}(x)|^2 dx < \infty$, but do not include functions that are...
View ArticleA problem on finding the limit of the sum
$$u_{n} = \frac{1}{1\cdot n} + \frac{1}{2\cdot(n-1)} + \frac{1}{3\cdot(n-2)} + \dots + \frac{1}{n\cdot1}.$$Show that, $\lim_{n\rightarrow\infty} u_n = 0$.The only approach I can see is either finding...
View ArticleHow to differentiate this functions?
For a single valued function it is easy to differentiate $f(x+g(t))$ and $f(xg(t))$ for some independent variable $t$ but in vector valued function this is not quite easy to differtiate thatlet $f :...
View ArticleAbout $\int_0^1\bigg(\frac{x^a-x^b}{\log(x)}\bigg)^2...
For $a>-\frac{1}{2}, b>-\frac{1}{2}$, we have$$\int_0^1\bigg(\frac{x^a-x^b}{\log(x)}\bigg)^2 \text{d}x=\log\bigg(\frac{(2a+1)^{2a+1}(2b+1)^{2b+1}}{(a+b+1)^{2(a+b+1)}}\bigg)$$For example, choosing...
View ArticleIntersection of dense sets in a Banach space
Suppose $(X, ||.||_X)$ be a Banach space. Let $Y, Z$ be two dense subsets.We know that $Y \cap Z$ may not be dense in $X$ (rationals and irrationals on real line) even if $Y \cap Z \neq \emptyset$...
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Prove the compactness theorem for Radon measures by using Banach-Alaoglu theorem
I was reading the proof of the compactness theorem for Radon measures (Theorem 1.5.15) from Leon Simon's book: Geometric Measure TheoryI was confused by the highlighted part. I hadn't learned the...
View ArticleDilation invariant $ \mathcal{H}^k $ measurable set.
Assume that $ E\subset\mathbb{R}^n $ such that $ \mathcal{H}^k(E)>0 $ and for any compact set $ K\subset\mathbb{R}^n $, $ \mathcal{H}^k(E\cap K)<+\infty $, where $ \mathcal{H}^k $ denotes the $ k...
View ArticleUnsure whether this series converges or diverges
I am having difficulty with the following question regarding the convergence of a seriesSuppose, $(\sum a_n) $is an absolutely convergent series, and $(\sum b_n)$ is a conditionally convergent series....
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$M \subset \mathbb{R}^3$ is a rotation of circle $\{ x^2 + z^2 = 4x-3, 1
$M \subset \mathbb{R}^3$ is a surface created by rotation of set$\{ x^2 + z^2 = 4x-3, \ 1<x<2, \ -1<z<1 \}$ by $\pi$ around axis OZ.Find $\int_M \max(x,y,z) d \lambda_2$.I know that $x^2 +...
View ArticleHow can we show that this integral is nonnegative?
Let$c_0>0$ and $\ell\in[0,1]$;$(E,\mathcal E,\lambda)$ be a measure space;$\mu$ be a probability measure on $(E,\mathcal E)$;$p:E\to[0,\infty)$ be $\mathcal E$-measurable with $$p_\lambda:=\int...
View ArticleCan oscillations occur in moment sequence of a random variable?
If $X$ is a nonnnegative, nondegenerate random variable, the Lp norm $m(p) = E[X^p]^{1/p}$ is a strictly increasing function of $p\geq 1$. I am curious about whether it is ever possible that...
View ArticleConvergent solutions to Laplace's equation in the plane, in boundary value...
In my PDE module, the general solution to Laplace's equation $\nabla^2 T$ in the plane (in polar coordinates) was shown to be $$T(r,\theta)=A_0+B_0\log...
View ArticleI want to know some other technique to solve this question about limits.
Let $f(x) = x^(1/3)$ for $x ∈ (0, ∞)$, and $θ(h)$ be a function such that,$f(3 + h) − f(3) = h\times f(3 +θ(h)\times h)$ ,for all $h ∈ (−1, 1)$. Then Lim θ(h) as h-> 0 is equal to what???I tried...
View ArticleHessian Matrix test for critical points
Let $\Omega\subseteq \mathbb{R}^n$ be an open set and let be $f:\Omega \to \mathbb{R}$ a real values multivariable function. Suppose that $f\in \mathcal{C}^2(\Omega)$.I can define the gradient of...
View ArticleConvex hull of cartesian product in general vector spaces
Is it true that $\text{conv}(X)\times\text{conv}(Y)\subset \text{conv}(X\times Y)$, where $X,Y$ are subsets of a (not-necessarily-finite-dimensional) vector space? If the answer is “no”, what if we are...
View ArticleQuestion about application of Montel's Theorem
Lemma Suppose that $f_n : \mathbb{D} \to \mathbb{D}$ is holomorphic. If $z \in \mathbb{D}$ and $|f_n(z)| \to 1$, then $|f_n(w)| \to 1$ for every $w \in \mathbb{D}$.Proof: Suppose that conclusion fails....
View ArticleYet another definition of uniform integrability. Is it equivalent to the...
I came across this posting where a definition of uniform integrability (found in Tao, T., Introduction Measure Theory. AMS, GTM vol 126, 2011) is given as follows:Definition T: Suppose...
View ArticleA one-sided discrete intermediate value theorem
In this post the author states a discrete version of the Intermediate Value Theorem as follows:For integers $ a < b $, let $ f $ be a function from the integers in $ [a, b] $ to $ \mathbb{Z} $ that...
View ArticleMultiple Integral Problem with Dirac Delta Constraint: Seeking Guidance
I am working on a challenging multiple integral problem and would appreciate any assistance. The integral is as follows:$$\int_{-\infty}^{+\infty} \int_{-\infty}^{+\infty} \ldots...
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