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Here is the problem I am working on:
If $M$ is a finitely generated torsion module over a PID $R$ such that there exists $m \in M$ with $ann(m) = (r)$, where $R \ni r = order(M)$, what can you say about the invariant factors?
What exactly is $order(M)$?
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My background is mostly in (applied) math with healthy doses of physics and computer science. Are there any good introductions to protein folding and its challenges for someone with that kind of quantitative background but very little knowledge of biology, chemistry, etcetera?
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I would like to compute the following Jacobian with respect to $x$:
$$\frac{\partial trace[A(x)A(x)^{T}]}{\partial x}$$
where $A(x) \in \mathbb{R}^{n\times m}$, $x \in \mathbb{R}^{m\times 1}$. The result should be a vector in $\mathbb{R}^{m\times 1}$. So far I have something like:
$$\frac{\partia...
Let $g(z)$ be a continuously differentiable function on the complex plane that is zero outside of som compact set. Show that $$g(x)=-\frac{1}{\pi}\iint_\mathbb{C}\frac{\partial g}{\partial \overline{z}}\frac{1}{z-w}dx\,dy,\quad w\in \mathbb{C}.$$ Remark. If we integrate this formally by parts...
In functional analysis, a branch of mathematics, the strong operator topology, often abbreviated SOT, is the locally convex topology on the set of bounded operators on a Hilbert space H induced by the seminorms of the form
T
↦
‖
T
x
‖
{\displaystyle T\mapsto \|Tx\|}
, as x varies in H.
Equivalently, it is the coarsest topology such that, for each fixed x in H, the evaluation map
(
T
,
x
)
↦
T
x
{\displaystyle (T,x)\mapsto...
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