The well-known Eneström-Kakeya theorem states that polynomial $p(z)=\sum_{\nu =0}^n a_\nu z^\nu$, where $0\leq a_0\leq a_1\leq \cdots\leq a_n$, has all of its (complex) zeros in $|z|\leq 1$. Many generalizations this result exist the literature. In paper, we extend one such to quaternionic setting and state possible corollaries.

We characterize the complex differential equations of form $${{dy} \over {dx}} = {a_n}(x){y^n} + {a_{n - 1}}(x){y^{n 1}} \cdots {a_1}(x)y {a_0}(x),$$ where aj(x) are meromorphic functions in variable x for j 0,…,n that admit either a Weierstrass first integral or inverse integrating factor.

Let $a_1, \ldots, a_n \in \mathbb{R}$ satisfy $\sum_i a_i^2 = 1$, and let $\varepsilon_1, \varepsilon_n$ be uniformly random $\pm 1$ signs $X \sum_{i=1}^{n} a_i \varepsilon_i$. It is conjectured that \varepsilon_i$ has $\Pr[X \geq 1] 7/64$. The best lower bound so far $1/20$, due to Oleszkiewicz. In this paper we improve 6/64$.

We introduce the chiral domain of an arrangement cameras $$\mathcal {A} = \{A_1,..., A_m\}$$ which is subset $$\mathbb {P}^3$$ visible in {A}$$ . It generalizes classical definition chirality to include all and offers a unifying framework for studying multiview chirality. give algebraic description allows us define describe version Triggs’ joint image. then use re-derive extend prior results on...

For an element $r$ of a ring $R$, Diophantine $D(r)$ $m$-tuple is $(a_1,a_2,\ldots,a_m)$ elements $R$ such that for all $i,j$ with $i\neq j$, $a_ia_j+r$ perfect square in $R$. In this article, we compute and estimate the measures sets $m$-tuples $\mathbb{Z}_p$ $p$-adic integers, as well its residue field $\mathbb{F}_p$.

In problems of ecological dynamics involving bistability (two disjoint attractors $$A_1$$ and $$A_2$$ ), we present (with various examples) a natural mechanism allowing the transfer from attraction basin to other basin, so reach any initial configuration. The involve an extra species which changes dimension phase space topology basins. disappear in way, that this keeps some relation with apopto...