Newbetuts

Why is the optimal solution to $ \min\limits_{S \in \mathbb S^{n}_{+}} -\log(\det(S))+\text{trace}(\Sigma S)$ equal to $\Sigma^{-1}$

If $\phi \in C^1_c(\mathbb R)$ then $ \lim_n \int_\mathbb R \frac{\sin(nx)}{x}\phi(x)\,dx = \pi\phi(0)$.

How to convince a high school student that differentials don't work like fractions in general?

A set $A \subseteq \mathbb{R}$ is closed if and only if every convergent sequence in $\mathbb{R}$ completely contained in $A$ has its limit in $A$

Why is the outer measure of the set of irrational numbers in the interval [0,1] equal to 1?

Solve $f (x + y) + f (y + z) + f (z + x) \ge 3f (x + 2y + 3z)$

$g(x)=\sup \{f(y): y\in B(x)\}$ is lsc on $R^{n}$ where $B(x)$ is a open ball with fixed radius $r$

Calculate the maximum value

Given a real function $g$ satisfying certain conditions, can we construct a convex $h$ with $h \le g$?

Finite Series $\sum_{k=1}^{n-1}\frac1{1-\cos(\frac{2k\pi}{n})}$

Second Countability of Euclidean Spaces

$\dim\{f:(-1,1)\to \mathbb{R}\mid f^{(n)}(0)=0 ~\forall~ n~\geq0\}=\infty$ if $f\in C^\infty[-1,1]$

Proof that $\frac{1 + \sqrt{5}}{2}$ is irrational.

What is the norm measuring in function spaces

convergence of $\sum \limits_{n=1}^{\infty }\bigl\{ \frac {1\cdot3 \cdots (2n-1)} {2\cdot 4\cdots (2n)}\cdot \frac {4n+3} {2n+2}\bigr\} ^{2}$

Canonical metric on product of two complete metric spaces

Deriving an explicit form for the nested sine integral $\int_{-\infty}^t \sin\left(A\sin(\omega t)-A\sin(\omega s)\right)e^{s-t}ds$ [closed]

New posts in real-analysis

Why is the optimal solution to $ \min\limits_{S \in \mathbb S^{n}_{+}} -\log(\det(S))+\text{trace}(\Sigma S)$ equal to $\Sigma^{-1}$

How to control the tail behavior of a class of series

Is $f(x)$ Riemann integrable on $[0,2]?$ Yes/No

$\int_0^\infty ne^{-nx}\sin\left(\frac1{x}\right)\;dx\to ?$ as $n\to\infty$

If $\phi \in C^1_c(\mathbb R)$ then $ \lim_n \int_\mathbb R \frac{\sin(nx)}{x}\phi(x)\,dx = \pi\phi(0)$.

How to convince a high school student that differentials don't work like fractions in general?

A set $A \subseteq \mathbb{R}$ is closed if and only if every convergent sequence in $\mathbb{R}$ completely contained in $A$ has its limit in $A$

Why is the outer measure of the set of irrational numbers in the interval [0,1] equal to 1?

Solve $f (x + y) + f (y + z) + f (z + x) \ge 3f (x + 2y + 3z)$

$g(x)=\sup \{f(y): y\in B(x)\}$ is lsc on $R^{n}$ where $B(x)$ is a open ball with fixed radius $r$

Calculate the maximum value

Given a real function $g$ satisfying certain conditions, can we construct a convex $h$ with $h \le g$?

Finite Series $\sum_{k=1}^{n-1}\frac1{1-\cos(\frac{2k\pi}{n})}$

Second Countability of Euclidean Spaces

$\dim\{f:(-1,1)\to \mathbb{R}\mid f^{(n)}(0)=0 ~\forall~ n~\geq0\}=\infty$ if $f\in C^\infty[-1,1]$

Proof that $\frac{1 + \sqrt{5}}{2}$ is irrational.

What is the norm measuring in function spaces

convergence of $\sum \limits_{n=1}^{\infty }\bigl\{ \frac {1\cdot3 \cdots (2n-1)} {2\cdot 4\cdots (2n)}\cdot \frac {4n+3} {2n+2}\bigr\} ^{2}$

Canonical metric on product of two complete metric spaces

Deriving an explicit form for the nested sine integral $\int_{-\infty}^t \sin\left(A\sin(\omega t)-A\sin(\omega s)\right)e^{s-t}ds$ [closed]