Question about existence and uniqueness of lifting of a curve
Theorem: Suppose that $p : X \to Y$ is a covering map.If $\gamma:[0,1] \to Y$ is continuous, $x_0 \in X$ and $p(x_0) = \gamma(0)$ then there exists a unique continuous function $\tilde{\gamma} : [0,1]...
View ArticleFind $f(x)$ assuming that $f(\sin x)+f(\cos x)=2x-\frac{\pi}{2}$
If $f(x)$ is a real valued function such that $$f(\sin x)+f(\cos x)=2x-\frac{\pi}{2}$$Find $f(x)$.I did $x\to\arcsin x$ and then $x\to \arccos x$ and I obtained $2\arcsin x=2\arccos x$ or...
View ArticleIf the restriction of a function is continuous at a point c then the function...
Let $I$ and $J$ be intervals such that $J \subseteq I$ and let $c \in J$. If $f|_J$ is continuous at $c$, then $f$ is continuous at $c$. This is true only if J is an open interval.Why do we need the...
View ArticleWhy $\infty=\sum_{i=1}^\infty...
I was wondering why $\sum_{i=1}^\infty \frac{1}{n+i}$ diverges but $\lim_{n\rightarrow\infty}\sum_{i=1}^n \frac{1}{n+i}=\log2$. While assuming integral as limit of series, we find out...
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How does Big-O complexity of the number of iterations relate to the...
I have a presentation in less than a day and my brain seems to be stuck.I'm trying to see what is the rough convergence rate of my algorithm on the Rosenbrock function and the data don't make much...
View Articlecalculate the triple integral $e^x$ over the unit sphere [closed]
Calculate the following triple integral over the unit ball.$$∭_{𝑥^2+𝑦^2+𝑧^2≤1}𝑒^xd𝑥d𝑦d𝑧$$
View Article$ \sup \{|x_n| \mid n\in \mathbb N \}= \lim_{n\to \infty} (\lim_{p\to...
I have to prove that for a bounded sequence $ {x_n}$$$ \sup \{|x_n| \mid n\in \mathbb N \}= \lim_{n\to \infty} (\lim_{p\to \infty}(\sum_{k=1}^n |x_k|^p)^{1/p}) $$I know that$$ \sup \{|x_n| \mid n\in...
View Article$f$ convex $\Rightarrow$ $L(f)\geq f$
Assume that $L:C[a,b]\to C[a,b]$ is a sequence of linear operator acting on the set $C[a,b]$ of continuous functions over [a,b]. Besides, $L(1)=1$ and $L(x)=x$. I recently found that from this is...
View ArticleFunctions between complete metric space and metric space
Let $(X,d_x)$ be a complete metric space and $(Y,d_y)$ just a metric space. We know that $f:X \rightarrow Y$ is a bijection. Must $Y$ be complete?Case 1/ $f$ is continuous and $f^{-1}$ is uniformly...
View ArticleHow to find closed subsets of $A,B$ such that $\lambda A+(1-\lambda)B$ is not...
How to find closed subsets of $A,B$ such that $\lambda A+(1-\lambda)B$ is not closed for a fixed $0<\lambda<1$.As is well-known that if $A,B$ are bounded closed subsets of $\Bbb R^n$, then for...
View ArticleGiven a non-empty open set $\Omega \subset \mathbb R^n$, can one guarantee...
Context. Let $1 \leqslant p < \infty$ and $0 \leqslant \lambda \leqslant n$ be fixed elements. Moreover, suppose that $\Omega \subset \mathbb R^n$ is a non-empty open set. Throughout this post I...
View ArticleResources for making error analysis of semi-analytical methods for solving...
I want resources explaining how to perform an error analysis of semi-analytical methods for solving differential equations with details.What I found in papers is very hard to grasp. I'm looking for...
View ArticleProving that convergence of norms and convergence a.e. implies strong...
I have in my notes the following theoremTheorem$(Y,\mathcal{F},\mu)$$\sigma-$finite measure space, $p\geqslant 1$, $\{f_n\}\subset L^p(Y)$ sequence of functions, $f\in L^p(Y)$ such that...
View ArticleBarycentre of a ball
I saw the following definition for the barycentre of a set $\Omega \subseteq \mathbb{R}^n$:$$\mathrm{bc}^\Omega=\frac{1}{\mathrm{vol}(\Omega)}\int_\Omega x dx \in\mathbb{R}^n.$$So I wanted to compute...
View ArticleOn the contrary, a theorem of Rudin's real and complex analysis
A theorem of Rodin's real and complex analysissuppose $\mu$ and $\nu_1$,$\nu_2$ are measures on a $\sigma$_algebra m, and $\mu$ is positive:a)if $\nu$ is concentrated on A, so is...
View ArticleThe operator norm regarding to the difference between a mollified function...
Let $\rho_\epsilon$ be a mollifier that has support in $B(0,\epsilon)$, define the operator$$T_\epsilon (f)=\rho_\epsilon*f-f,$$for every $f\in L^2(\mathbb R^d)$, can we prove or disprove that the...
View Articlecomplex measure,total variation
let m be a $\sigma$_algebra , and$\mu$ is a complex measure on X,if $\left|\mu\right|$(X)=$\mu$(X)show that $\mu$(E)=$\mu$(E) for each E member of $\sigma$_ algebra m.I tried to prove, but...
View Article$L_p$ norm estimate of a sum
Let $R>0$, and $(B_k)_{k \in \Bbb{N}}$ a collection of disjoint balls of radius R and let $f$ be a measurable function on $\mathbb{R}^n$ of the form$$f = \sum_{k=1}^{\infty} a_k X_{B_k}$$for some...
View ArticleClosed form for the pdf of two IG variables
ContextI would like to estimate the distribution of the difference of two inverse gaussian variables. The convolution doesn't lead to any special functions according to Mathematica. Then, I would like...
View ArticleGradient of softmax function with the inner product argument
Suppose that $x_i$ and $y_j$ are vectors in $\mathbb{R}^d$, where $i,j\in\{1,2,\dots,N\}$. Let the loss function be defined as $$\mathcal{L} = -\frac{1}{N}\ln\left(\frac{\exp({\frac{x_i.y_i}{\tau\lVert...
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