Inspired by this question. It is known that for the matrix $T_n \in \mathcal{M}_n$ (the space of real-valued $n \times n$ matrices) defined by \begin{equation*} (T_n)_{ij} = \begin{cases} 1 & i \geq j \\ 0 & i < j \end{cases} \end{equation*} and arbitrary $A \in \mathcal{M}_n$, that $\|T_n \circ ...

Motivation: It recently occurred to me that Turing's analysis of the halting problem may be formulated as a fixed-point theorem. Might this intuition from theoretical computer science have informed important developments in the theory of dynamical systems? This question is motivated by discussion...

Let $z \in C$ and consider the following integral equation: $$-\frac{\Gamma(a)}{\Gamma(b)}\frac{1}{ z \dfrac{\mathrm{d}}{\mathrm{d}z} {_{1}F_{1}}(b,b-a;z)}= \int_{0}^{+ \infty}x^{z-1}K(x) \mathrm{d}x$$ I would like to find the kernel $K(x)$ and I would also like to write it as an infinite product...

Let $X$ be a smooth projective surface (over complex numbers). Let $D$ be a divisor on $X$. Then we know that its volume is defined as $$\text{vol}_X(D):= \lim \sup_{m \rightarrow \infty} \frac{h^0(X, \mathcal O_X(mD))}{{m^2}/2}.$$ Suppose that, for a divisor $D$ on $X$, it is known that $\text{...

Let $R$ be a ring and $\text{Mod}\,R$ the category of $R$ modules. For two $R$-modules $X,Y$ one can define $\text{Ext}_R^n(X,Y)$ as follows. We take an injective resolution $0\rightarrow Y\rightarrow I_0 \rightarrow I_1 \rightarrow \dots,$ throw away $Y$ and apply $\text{Hom}_R(X,-)$ to obtain t...