Does it hold that: $\{\frac{1}n:\:n\in\mathbb{N}\}\cong\mathbb{N}$ (each set...
I think this is true and all that's needed to prove it is to let $f:{\{\frac{1}n:\:n\in\mathbb{N}\}\to\mathbb{N}},$$\frac{1}n\to\ n$Obviously then, $f$ is surjective and injective and continuity of $f$...
View ArticleMatrix of the derivative of a differentiable function $f$ at the origin with...
Let $f(x,y)=(u(x,y),v(x,y)):\Bbb R^2\rightarrow\Bbb R^2$ be a differentiable function.Let $A$ denote the matrix of derivative of $f$ at the origin with respect to the standard basis of $\Bbb...
View ArticleHow to show $ \|fg\|_{L^2}\leq...
Let $ f,g\in C_c^1(\mathbb{R}^2) $, show that$$\|fg\|_{L^2}\leq \|\partial_{x_1}f\|_{L^2}\|g\|_{L^2}+\|f\|_{L^2}\|\partial_{x_2}g\|_{L^2}.$$Here is my try. Without loss of generality, we can assume...
View ArticleHow to get a variable in a closed form given an equation
I need to express a variable in a closed form given an equation, $k^4 = 2^k.$ Taking the logarithm from both sides I get $k^{\frac{1}{k}} = 2^{\frac{1}{4}}$ which is not helping much. I tried other...
View ArticleSingular extremal of a constrained variational problem
Consider the following constrained variational problem: $$\min_{u \in H^{1}(I)} \{\mathcal{F}(u) : u(\pm 1) = 1, \mathcal{G}(u) = 1/3 \},$$ where $I = [-1, 1] \subseteq \mathbb{R}, H^1 (I) := H^{1,...
View ArticleProving the nested interval property (proof verification)
The text gives the following definition: "a set $\{I_n\mid n \in \mathbb N\}$ of nonempty closed bounded intervals is called nested intervals (clumsy, I know, I am translating from the German here),...
View ArticleHow to compute integral $\int_0^\infty |\sin x|e^{-x}dx$
When I plug $$\int_0^\infty |\sin x|e^{-x}dx$$into symbolab, it tells me the first step is to find the equivalent expression for the integrand at $0\leq x \leq \infty$. It says this expression is $\sin...
View ArticleAny way to prove this inequality involving trigamma functions.
While solving a problem I succeeded to reduce it to the following inequality:$$\forall \{a,b,z\in\mathbb R_+,\ a\ne b\}:\quad...
View ArticleThe definition of the integral of a function f with respect to a convolution...
While reading the book "Measure Theory and Probability Theory" by Krishna B. Athreya and Soumendra N. Lahiri, I encountered a problem(Proposition 5.5.4). What does (μ∗λ)(dx) represent and what is its...
View ArticleProve that $g_n = \sup_{j\geq n}|f_j-f|$ is $\mathscr{A}$-measurable.
My QuestionLet $(X,\mathscr{A},\mu)$ be a measure space, and let $f$ and $f_1,f_2,\dots$ be real-valued $\mathscr{A}$-measurable functions on $X$. Suppose $\mu$ is finite and $\{f_n\}$ converges to $f$...
View ArticleRudin's RCA, Theorem $7.16$: The Fundamental Theorem of Calculus.
There is the equality: $$ f(x) - f(a) = \int_a^x f'(t)dt \ \ (a \leq x \leq b). \tag{1}$$There is assumption by Rudin:Suppose $f$ is continuous on $[a,b], f$ is differentiable at almost every point of...
View ArticleCan we apply calculus to emotions? What is the derivative of sadness?
Can we apply concepts of Calculus to things such as emotions, philosophy of history?
View Articleif the intersection of a decreasing sequence of closed sets is non empty,...
in the converse of Cantor's intersection theorem we have the intersection of a decreasing sequence of closed sets with diameters tending to zero is nonempty, then the metric space is complete.What if...
View ArticleIf $f_n\to f$ a.e. and $\limsup_{n\to +\infty} \|f_n\|\le K,$ does it imply...
Let $(H, \|\cdot\|)$ denote a Hilbert space which is continuously embedded in $L^2(\mathbb R^n)$. Let $\{f_n\}$ be a sequence such that$$ f_n\to f \text{ a.e. in } \mathbb R^n,$$and$$\limsup_{n\to...
View ArticleLim inf of function and lim inf of sequence [closed]
I just want to know if $\liminf\limits_{x\rightarrow a} f(x) = \liminf\limits_{k\rightarrow \infty} f(x_k)$, when $x_k \rightarrow a$; and why. All I could show is $\liminf\limits_{x\rightarrow a} f(x)...
View ArticleProve that $\lim_{n\rightarrow \infty} \frac{3n+5}{2n+7} = \frac{3}{2}$...
Prove that $\lim_{n\rightarrow \infty} \frac{3n+5}{2n+7} = \frac{3}{2}$Proof:Let $a_n = \frac{3n+5}{2n+7}$. Then, $\left | a_n-\frac{3}{2} \right |=\frac{11}{2(2n+7)}<\frac{3}{n}$. Given $\epsilon...
View ArticleShow that $M = \{ (r(t)\cos(s),r(t)\sin(s),t)\mid t \in I, 0 \leq s \leq 2\pi...
I want to show that the set $M = \{(r(t)\cos(s), r(t)\sin(s), t) \mid t \in I, 0 \leq s \leq 2\pi)\}$ is a manifold. Where $I \subset \mathbb{R}$ is an open interval and $r: I \to (0,\infty)$ is a...
View ArticleHow do we find the decimal (or base) expansion of a real number upto first n...
Suppose we have a sequence ${x_n}$ that converges to some limit, say $\pi$. How do we compute out the decimal expansion of $\pi$ from that sequence? How could the epsilon definition help us here? What...
View ArticleOn the convergence of $\sum_{n\geq0}a_n\cos(nx)$
Let $(a_n)_{n\in\mathbb{N}}$ be a sequence of real numbers such that$\sum_{n\geq0}a_n$ converges absolutely. Can we say that$\sum_{n\geq0}a_n\cos(nx)$ converges to a continuous function?I know that...
View ArticleA question abount Young's inequality
I'm reading this answerI report it for completeness:Let $C$ be the graph of $v = f(u)$ over the interval $[0,f^{-1}(b)]$. If $f(a) > b$, then $f^{-1}(b) < a$, in which case\begin{align}\int_0^a...
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