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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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## 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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