Let $m$ and $n$ be two integers and $m \le n$. There are a matrix $A$ of $m$-by-$m$ with $A(i,j) = 1/(2n+2j-2i)!$ and a vector $r$ of $m$ entries with $r(i) = 2/(2n+2i)!$.
Is there a formula for the inner product of $r$ and the first column of the inverse of $A$?
Or, can it be shown that the inn...
The double integral over the region:
$$
R = \left\{ \left( x,\: y \right) : a \leqslant x \leqslant b,\: g\left( x \right) \leqslant y \leqslant h\left( x \right) \right\}
$$
is expressed as
$$
\iint_R f\left( x,\: y \right) \mathrm{d}A = \int_a^b \left[ \int_{g\left( x \right)}^{h\left( x \right...
I am reading Serre's Lectures on the Mordell-Weil Theorem, where he specifically talks about a Large Sieve inequality and proceeds to give an example.
Theorem. (Section 12.1) Let $K$ be a number field, $\Lambda$ be a free $O_K$ -module of rank $n$. Let $\|\cdot \|$ be a norm over $\Lambda_{\math...
In the book Brownian Motion, Martingales and Stochastic Calculus by J.F. Le Gall, in order to give an alternatice derivation of the distribution of $L_{U_{a}}^{0}(B)$ where $L^{0}_{t}(B)$ is the Local-Time at $0$ of a Standard Brownian Motion and $U_{a}=\inf\{t:|B_{t}|\geq a\}$, he states as a re...
My concrete questions are (for context see below):
Is is true that $i$ as below embeds $M$ in $T^*\mathbb{R}^n$?
Is it true that $M$ is Lagrangian in $\mathbb{C}^n$ if and only if $i(M)$ is Lagrangian in $T^*\mathbb{R}^n$?
Is it true that Theorem 4.3 implies that
$$dh=\text{proj}_{d\psi}y\iff dh...
Fix $1\leq p<n$. Define,
\begin{equation}
K^p\equiv\{f:\mathbb{R^n}\rightarrow\mathbb{R}\ \vert\ f\geq 0, f\in L^{p^{\ast}}(\mathbb{R}^n), Df\in L^{p}(\mathbb{R}^n;\mathbb{R}^n)\}.
\end{equation}
If $A\subset\mathbb{R}^n$ we define the quantity
\begin{equation}
\text{Cap}_p(A) \equiv \inf\le...
$a\in \mathbb{R}^n$ $a=(a_1,a_2,\dots,a_n)$ lets define this to be equivalent to $(a_1,a_2,\dots,a_n,0,0,0 \dots,0)$ (finite many zeros) by this I think we can make an $n \times m$ matrix $A$ equivalent to any $ p \times p$ $B$ matrix with $p\ge \max\{n,m\}$ by $B_{ij}= A_{ij}$ if $1\le i\le n$...
