Hi chat! I have a question about something in the proof of the formula of the determinant of a block lower triangular matrix in Huffman - Kunze.
Huffman and Kunze first show that if $B$ is obtained from $A$ by adding the rows $c\alpha_j$ to $\alpha_i$ where $i<j$ then $\det A=\det B$. Then they define $D(A,B,C)=\det\begin{bmatrix}
A & B \\ 0 & C
\end{bmatrix}$, if $A$ and $B$ are fixed then $D$ is alternating and $s$-linear as a function of the rows of $C$, so by a previous theorem $$D(A,B,C)=(\det C) D(A,B,I)$$ where $I$ is the identity function with $s\times s$ dimension. Now they claim t…
Let $f:\mathbb{R}^N \to \mathbb{R}$ be a positive definite function. Let $$g(h) = \int_{\mathbb{R}^N}f(x)f(x+h)\mathrm{d}x$$ Due to Bochner's and Parseval's $g$ is a lso a positive definite function.
Define the set $S$ = $\{x_1,x_2,...x_n\}$ containing $n$ distinct points in $\mathbb{R}^N$
Defi...
Today I learned in class the following result, which my professor stated without proof:
Given a Banach space $V$, there exists a compact Hausdorff space $X$ such that $V$ embeds into $C(X)$ as a closed subspace.
Recall that $C(X)$ is the space of all continuous complex valued functions on ...
Let $ p_n $ be the $n$ th prime.
I was confused about the following idea.
$$A = \sum_{n = 1}^{\infty} ( \frac{p_{2n-1}}{p_{2n}} - \frac{p_{2n}}{p_{2n +1}} ) $$
Very confused actually.
Does this even converge ?
Do we need a summability method ?
Does its converge depend strongly on conjectures...
