arXiv subjects
Publications and source records attributed to Kalle Leppälä.
Two conjectures of Allman, Long and Rhodes are proven. The ultrametric exchangeable model captures the symmetries in allele distributions of ultrametric mechanistic models of coalescence and substitution in mathematical phylogeny. The dimension of this model with respect to the species tree leaf count is shown to be greatest for caterpillar trees and least for balanced trees.
We study the $k$:th raw moment of a variable $R$ following the binomial distribution $\text{B}(n, p)$, where $n/k \rightarrow β> 0$. It is known that $\mathbb{E}(R^k)$ is bounded both from below and from above by functions of the form $k^k Ψ^k$. We solve the asymptotically optimal value of $Ψ$ as a function of $p$ and $β$.
We give explicit and asymptotic lower bounds for the quantity $|e^{s/t}-M/N|$ by studying a generalized continued fraction expansion of $e^{s/t}$. In cases $|s|\geq 3$ we improve existing results by extracting a large common factor from the numerators and the denominators of the convergents of that generalized continued fraction.
We study how the asymptotic irrationality exponent of a given generalized continued fraction \[ \K_{n=1}^\infty \frac{a_n}{b_n}\,,\quad a_n, b_n\in \mathbb{Z}^+, \] behaves as a function of growth properties of partial coefficient sequences $(a_n)$ and $(b_n)$.
Let $\mathbb{I}$ denote an imaginary quadratic field or the field $\mathbb{Q}$ of rational numbers and $\mathbb{Z}_{\mathbb{I}}$ its ring of intergers. We shall prove an explicit Baker type lower bound for $\mathbb{Z}_{\mathbb{I}}$-linear form of the numbers \begin{equation}\label{1} 1,\ e^{α_1},...,\ e^{α_m},\quad m\ge 2, \end{equation} where $α_0=0$, $α_1,...,α_m$, are $m+1$ different numbers from the field $\mathbb{I}$. Our work gives gives some improvements to the existing explicit versions of of Baker's work about exponential values at rational points. In particilar, dependences on $m$ are improved.