Is $W_0^{1,p}$ weakly closed?
Is $W_0^{1,p}(\Omega)$ weakly closed? $W_0^{1,p}(\Omega)$ is the closure of $C_0^{\infty}(\Omega)$ with respect to the norm of $W_0^{1,p}(\Omega)$, and I've been trying to figure out if it is true that...
View ArticleWhy is the Lebesgue integral of a two variable measurable function over one...
Suppose $f\geq0$ is a measurable function $f:X_1 \times X_2 \rightarrow [0,\infty]$ according to the product measure $\mu_1 \otimes \mu_2$ defined by the $\sigma$-finite measures $\mu_1 :\Sigma_1...
View ArticleLimit of $\sin^n(n)$
Lately I thought about these limits:$$\lim_{n\to\infty}\sin^n(n),\quad\lim_{n\to\infty}\cos^n(n)$$Intuitively, I would say that they do not exist, but I wasn't able to prove it rigorously. Any help...
View ArticleProve divergence of an alternating sequence
Prove that the divergence of the following sequence.$$s_n=\frac{(-1)^nn}{2n-1}$$The following is the sample answerNote that $\exists N\in\mathbb{N}\, s.t.\,\forall k\geq...
View ArticleFubini's theorem in Stochastic Integral
The short rate under the Merton model is$r_t = r_s + \theta (t - s) + \sigma \int_s^t d w_u$Now we integrate above to get bond price from s to T$\int_s^T r_t \, dt = \int_s^T r_s \, dt + \theta...
View ArticleProve $\sum\limits_{i=1}^{N} m_i \ c_i \ \ln...
I am interested in solving a rather peculiar inequality that, like some of my previous posts, is physics-related because it comes from a physical condition that must be satisfied by a mathematical...
View ArticleIs this space of progressively measurable processes a Banach space
Suppose $(\Omega,\mathcal F,\mathbb P)$ is a complete probability space, on which a filtration is defined. Let $P$ be the set of progressively measurable processes $X$ such that $\Vert...
View ArticleLet $g(x) = \lim_{r\to 0} {\left((x+1)^{r+1} -...
Task:The function $g(x) = \lim_{r\to 0} {\left((x+1)^{r+1} - x^{r+1}\right)^{\frac{1}{r}}}$ is defined for any positive real $x$. Calculate $\lim_{x\to\infty}{\frac{g(x)}{x}}$.My attempt:My idea was...
View ArticleProve that $\lim_{n\to \infty}\int_0^{n^{-\beta}}nf(x)dx=0$, for all $f\in...
Prove that $$\lim_{n\to \infty}\int_0^{n^{-\beta}}nf(x)dx=0,\forall f\in L^q[0,1], $$ where $\beta=\frac s{s-1}$, $1<s<q$.I tried to use Hölder's inequality, but it doesn't work.Could you please...
View ArticleUniform convergence of sequence of functions with Cauchy sequences
I have that for a given sequence of functions f_n:[0,1]→R s.t.For every Cauchy sequence from the domain f_n(x_n) -> 0 when n->inf, prove that f_n is uniformly convergent, can somebody help me...
View ArticleShow that if $\sum_{|k|=0}^\infty c_k \delta_k$ is tempered distribution then...
This question is the anti-theorem of the following exercise and the solution of original exercise is here.Exercise 2.3.8 from Classical Fourier Analysis by Loukas GrafakosSuppose...
View ArticleConvergence of a Specific Recursive Sequence
I'm trying to prove that the following recursive sequence converges to 1:$$x_1=0$$$$x_{n+1}=x_n-2\frac{e^{x_n}-ex_n}{e^{x_n}-e}$$The sequence has no fixed points, but...
View ArticleEquivalences for $G$-regularity and $F$-regularity in measure theory
I am doing some exercises from my real analysis class and got stuck trying to prove the following equivalences:Let $ A \subseteq \mathbb{R} $. Prove that the following statements are equivalent:a) $ A...
View ArticleHow to solve this question only in the sense of Riemann integral? [closed]
Sovle the infinite limit of Riemann-integral$\displaystyle\lim_{n\to\infty}\int_0^1\displaystyle\frac{\mathrm{d}x}{1+\left(1+\frac{x}{n}\right)^n}$
View ArticleBaby Rudin 1.6(c)
Working the exercice in my copy of the french translation. I looked everywhere and no one seems to be asking this question.The problem, in french at least, says $B(x)$ is a subset or rational numbers...
View ArticleValues of quadratic function and its discriminant
The question might be trivial, but I confused in one step which I tried to mention.Let $f:\mathbb{R}\rightarrow \mathbb{R}$ be $f(x)=ax^2+bx+c$ where $a\neq 0$.Suppose that $f(x)\ge 0$ for all...
View ArticleIs there a shorter expression for infinite multiplication? [duplicate]
I'm trying find an alternate (shorter?) expression for product of $\,\dfrac{x^2-1}{x^2}\,$ over integers greater than $\,1\,$, i.e.$\,\dfrac{3}{4}\cdot\dfrac{8}{9}\cdot\dfrac{15}{16}\cdots\quad$ An...
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Upper and lower bounds on sine integral?
Define the sine integral $Si(x)$ by$$\def\Si{\operatorname{Si}}\Si(x) = \int^x_0 \frac{\sin t}{t} \, dt$$I want to establish upper and lower bounds on $\frac{1}{\pi} \Si(\pi x)$. The answer here proves...
View ArticleHow to prove that $A = \{x \in \mathbb{R}^n: 1 \leq \|x\| \leq 2\}$ is...
I'm trying to understand if $A = \{x \in \mathbb{R}^n: 1 \leq \|x\| \leq 2\}$ is connected. For reference, this problem is from Jerold Marsden and Michael Hoffman's Classical Elementary Analysis (2nd...
View ArticleProve that $\frac{\ln(1+ax)}{\ln(1+bx)}$ is increasing for $a,b>0$ and $a
Hello I want to preove that the fuction f defined in $(0,+\infty)$ by $f(x)=\frac{\ln(1+ax)}{\ln(1+bx)}$ is increasing for every a,b>0 and a<bI tried to calculate the derivative but I don't know...
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