Let $K\in L^2((0,1)\times(0,1))$ and consider the operator defined in $L^2(0,1)$ by $$Lu(x):=u(x)-\int_0^1K(s,x)u(s)ds.$$ What kind of assumption might I impose on $K$ such that $\lambda=1$ will be not an eigenvalue of the operator $L$?. Any ideas?. Thank you.

# Author: Math Dev

Let us assume that we have a full-rank randomly chosen $k\times (m\cdot l)$ matrix, $\boldsymbol{H}$, with $l \leq k \leq (m\cdot l)$ and no specific structure (e.g., a realization of an IID complex Gaussian random matrix), and a full-rank $(m-r)\times m$ matrix, $\boldsymbol{X}$, which has at least one non-zero element per column and its last […]

Let $X$ be a locally-compact metric space and $D\subseteq X$ be dense. Then, does $\partial D \triangleq X -D$ have a collar neighborhood? Ie: an open subset $\partial D\subseteq U\subseteq X$ homeomorphic to $D\times [0,1)$, via $\phi$, for which $\phi(\partial D)=D \times \{0\}$?

I’m having some troubles with universal central extensions and associated cocycles; in particular, I want to understand the group law of the universal central extension of the Thompson’s group $T$. Let us consider the universal central extension $E$ of the Thompson’s group $T$; I know that it is possible to describe $E$ as the set […]

Let $A$ be an abelian variety over $\mathbb{C}$. If $A$ has an effective non-big divisor, then $A$ is not simple. (In a simple abelian variety, every non-zero effective divisor is ample.) What can we say about the structure of effective non-big divisors in $A$?

The following bullet points represent the very peak of my understanding of the resolution of the Langlands program for function fields. Disclaimer: I don’t know what I’m writing about. Drinfeld modules are like the function field analogue of CM elliptic curves. To see this, complexify an elliptic curve $E$ to get a torus $\mathbb{C} / […]

I am looking for a reference (a book e.g.) for the Sparre-Andersen theorem for random walks. I couldn’t find any book by googling…

Suppose that I have a function $f(x, \epsilon)$ and I know that $$ \lim_{\epsilon \to 0} f'(x, \epsilon) = g'(x). $$ Now let $g(x)$ be the function whose derivative appears above. How can I rigorously prove my intuition that $$ \lim_{\epsilon \to 0} f(x, \epsilon) = g(x), $$ given that $\lim_{\epsilon \to 0}f(0, \epsilon) = […]

It’s well-known that the complete lattices are characterized among all posets as the regular-injectives. That is, a poset $L$ is a complete lattice if and only if $L$ has the right lifting property with respect to the class $Emb$ of all embeddings of posets. This remarkable characterization would be even more useful if there were […]

Please follow the link to further understand the solution I created. Matrix Solution 1Click Here