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Eigenvalues of an integral operator

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.

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Full-rank Hadamard product given a certain structure

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 […]

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Dense subset of a metric space has a collar neighborhood

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 $\phi(\partial D)=D \times \{0\}$?

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Group law in universal central extension of Thompson’s group T

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 […]

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Structure of non-big divisors in an abelian variety

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 $A$?

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How can I see the relation between shtukas and the Langlands conjecture?

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} / […]

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Reference: Sparre-Andersen theorem for random walks

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…

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Does having the derivative in the limit suffice to solve the function at the limit?

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) = […]

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Explicit lifting characterization of complete lattices among posets?

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 […]

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Why can certain matrices be solved in this manner?

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