Molecular phylogenetics, the study of reconstructing evolutionary trees, is a well-established field of scientific endeavor. However, in certain circumstances evolution is not completely tree-like. For example, a comparison of gene trees representing a set of present-day species and reconstructed for different genetic loci often reveals conflicting tree topologies. These discrepancies are not a...

Let X be a finite set, N be a reticulation-visible network on X, and T be a rooted binary phylogenetic tree. We show that there is a polynomial-time algorithm for deciding whether or not N displays T . Furthermore, for all |X| ≥ 1, we show that N has at most 8|X| − 7 vertices in total and at most 3|X| − 3 reticulation vertices, and that these upper bounds are sharp.

‎‎‎This paper presents a remarkable formula for spectral distance of a given block normal matrix $G_{D_0} = begin{pmatrix}‎ ‎A & B \‎ ‎C & D_0‎ ‎end{pmatrix} $ to set of block normal matrix $G_{D}$ (as same as $G_{D_0}$ except block $D$ which is replaced by block $D_0$)‎, ‎in which $A in mathbb{C}^{ntimes n}$ is invertible‎, ‎$ B in mathbb{C}^{ntimes m}‎, ‎C in mathbb{C}^{mti...

A graph class $\mathcal{F}$ has the induced Erd\H{o}s-P\'osa property if there exists a function $f$ such that for every graph $G$ and every positive integer $k$, $G$ contains either $k$ pairwise vertex-disjoint induced subgraphs that belong to $\mathcal{F}$, or a vertex set of size at most $f(k)$ hitting all induced copies of graphs in $\mathcal{F}$. Kim and Kwon (SODA'18) showed that for a cy...

Knot polynomials colored with symmetric representations of $SL_q(N)$ satisfy difference equations as functions representation parameter, which look like quantization classical ${\cal A}$-polynomials. However, they are quite difficult to derive and investigate. Much simpler should be the for coefficients differential expansion nicknamed quantum C}$-polynomials. It turns out that, each knot, one ...

Let $ { mathcal{R}} L$ be the ring of real-valued continuous functions on a frame $L$ as the pointfree  version of $C(X)$, the ring of all real-valued continuous functions on a topological space $X$. Since $C_c(X)$ is the largest subring of $C(X)$ whose elements have countable image, this motivates us to present the pointfree  version of $C_c(X).$The main aim of this paper is to present t...