Smoothing property of the heat kernel on the one-dimensional torus
Let $G=G(x,t)$ be the heat kernel on the one-dimensional torus $\mathbb{T}^1,$ with $x \in \mathbb{T}^1$ and $t \in (0,T].$$G$ is given by \begin{equation}G(x,t) = (4 \pi t)^{-1/2} \sum_{k \in...
View ArticleCompactness of unit sphere in 1-norm; Gamelin and Greene
In Introduction to Topology by Gamelin and Greene, I'm working an exercise to show the equivalence of norms in $\mathbb R^n$. This exercise succeeds another exercise where various equivalent...
View ArticleWhat is the meaning of the derivative of a complex function.
For the derivative of the complex function, when it is analytic, then it satisfies Cauchy-Riemann equation. So in this case we have $f'(z_{0})=u_{x}(x_{0},y_{0})+iv_{x}(x_{0},y_{0})$. But if we...
View ArticleExercise 2.34 in the book A first course in Sobolev spaces - Leoni
Let $u:[a,b] \rightarrow \mathbb R^N$i) Let $g:[a,b] \rightarrow \mathbb R^N$ and take one $x_0 \in [a,b]$. Prove that$$ \left\| \int_a^b g(x)dx\right\| \geq \int_a^b \|g(x) \| dx -2 \int_a^b \| g(x)...
View ArticleWhat is the name of this theorem and where can I find its detailed proof?
Here is the theorem I want to prove:I guess long ago when I studied analysis this theorem was given in some textbook with its proof. I think it was Kolomogorov book, or Royden Real analysis fourth...
View ArticleProve or disprove $\sum\limits_{\mathrm{cyc}}\frac{n-1+x_{2}x_{3}\cdots...
Let $x_{i}>0 (i=1,2,\cdots,n)$. Prove or disprove that$$\dfrac{n-1+x_{2}x_{3}\cdots x_{n}}{1+(n-1)x_{1}}+\dfrac{n-1+x_{1}x_{3}\cdots x_{n}}{1+(n-1)x_{2}}+\cdots+\dfrac{n-1+x_{1}x_{2}\cdots...
View ArticleCan we evaluate the generalized Ahmed integral...
Does anyone have any idea on how to evaluate the following generalized Ahmed integral?$$I(t):=\int_{0}^{1}\frac{\arctan(t\sqrt{2+x^2})}{\sqrt{2+x^2}(1+x^2)}\,dx$$Here is my...
View ArticleOperator strong $(p,p)$ implies adjoint strong $(p',p')$
Let $T$ be a linear and continuous operator $T: L^{2}(\mathbb{R}^{d}) \to L^{2}(\mathbb{R}^{d})$. I wonder to what extent we can conclude if $T$ is strong $(p,p)$, we will obtain that $T'$, the adjoint...
View ArticleClik here to view.
Extending the Euler-Mascheroni Constant to higher dimensions $\gamma_n$.
Denote an $n$-dimensional gamma constant as $\gamma_n$.$\large \textbf{Background}$This article gives a nice $2D$ graphical representation of Euler's constant.The area of the blue dots represents the...
View ArticleShow that $\int_0^1 e^{-\frac{1}{x}}dx > \frac{1}{4}e^{-\frac{4}{3}}$
I wanted to prove that some integral is bounded by a number. After some steps I got that I need to show this: $\int_0^1 e^{-\frac{1}{x}}dx > \frac{1}{4}e^{-\frac{4}{3}}$. But I can't find a good...
View ArticleCoordinate-free proof of isomorphism between derivations and tangent space
It is important in differential geometry that the space of derivations at a point $p\in V$ (where $V$ is a finite-dimensional real vector space) is isomorphic to the space of partial derivatives (or...
View ArticleDoes an increasing sequence of reals converge if the difference of...
If $a_n$ is a sequence such that$$a_1 \leq a_2 \leq a_3 \leq \dotsb$$ and has the property that $a_{n+1}-a_n \to 0$, then can we conclude that $a_n$ is convergent?I know that without the condition that...
View ArticleIntegral of small function over small interval
Let $X=[0,1]$. For all $\varepsilon>0$, let $A_\varepsilon\subset X$ satisfy $\mathrm{Leb}(A_\epsilon)=O(\varepsilon)$. Further, suppose there exists a family of functions $f_\varepsilon \in L^1(X)$...
View ArticleHow to deal with $\forall M > 0$ in this limit?
For the limit $$\lim_{x\to +\infty} x^2+3 = +\infty$$ by definition we have that $\forall M > 0, \exists N > 0$ such that $\forall x > N$ we have $f(x) > M$.But in this case then $\forall x...
View ArticleHow to calculate arc-length reparameterization of a regular curve.
I am self-studying differential geometry, so please forgive my question if it is lacking in some way. I have a regular curve $\gamma : (0,\pi) \to \mathbb{R}^2$ defined by $$\gamma(t) = \left ( \sin...
View ArticleWhat is the Haar measure on the unit sphere?
I try to understand the proof of Lemma 4.2. in the paper 'The Euler equations as a differential inclusion' by De Lellis and Székelyhidi. In the proof they use the Haar measure on the unit sphere...
View ArticleIs it true that $\sum a_n$ converges $\implies \exists k\in\mathbb{N}$ such...
If $\sum a_n$ converges, then $(a_n)\rightarrow 0$ and $a_n<\dfrac{1}{n}$ for infinitely many $n$.But can we show that $\exists k\in\mathbb{N}$ such that $\forall n\ge k, ~a_n<\dfrac{1}{n}$ ?If...
View ArticleFinding the Maximum and Minimum Distance by Lagrange's Method of Multipliers
Q. Find the maximum and minimum distance of a point from origin such that the point lies in the curve $3x^2+4xy+6y^2=140$I am unable to solve these three equations simultaneously for...
View ArticleCan changing Gradient Descent step size/learing rate from constant 1/L to...
If instead of the classical $1/L$ constant step size we have adaptive step sizes chosen with exact line search or Armijo (let's say) can this alter the Big-O complexity of the convergence rate?In...
View ArticleHow to evaluate $\lim_{(x,y)\to(0,0)}\frac{\sqrt{x^2y^2-1}-1}{x^2+y^2}$
I need to show if this limit exist or not:$$\lim_{(x,y)\to(0,0)}\frac{\sqrt{x^2y^2-1}-1}{x^2+y^2}$$I tried to solve using polar coordinates but It doenst help me much. Please if someone can help me
View Article