A body in a damped oscillator eventually stops at the origin. Can we the body to the positive side by giving a positive driving force? Unfortunately, due to the oscillatory motion of the body, it is not true in general. In this paper, we give a sufficient condition on the driving force guaranteeing the asymptotic positivity of the position of the body, which means the negative part of the position vanishes in time. Also the result will be extended to a wider class of differential equations including the damped oscillator.

We prove the unimodality of some coloured  -Eulerian polynomials, which involve the flag excedances, the major index and the fixed points on coloured permutation groups, via two recurrence formulas. In particular, we confirm a recent conjecture of Mongelli about the unimodality of the flag excedances over type B derangements. Furthermore, we find the coloured version of Gessel’s hook factorisation, which enables us to interpret these two recurrences combinatorially. We also provide a combinatorial proof of a symmetric and unimodal expansion for the coloured derangement polynomial, which was first established by Shin and Zeng using continued fractions.

A new approach is used to describe the large time behavior of the nonlocal differential equation initially studied in T.-N. Nguyen (On the ω-limit set of a nonlocal differential equation: application of rearrangement theory. Differ. Integr. Equ. arXiv:1601.06491, 2016). Our approach is based upon the existence of infinitely many Lyapunov functionals and allows us to extend the analysis performed in T.-N. Nguyen (On the ωlimit set of a nonlocal differential equation: application of rearrangement theory. Differ. Integr. Equ. arXiv:1601.06491, 2016).

Generalizing the notion of the degree of a finite-to-one factor code from a shift of finite type, the class degree of a possibly infinite-to-one factor code extends many important properties of degree. In this paper, introducing relative class degree, we study how class degrees change as two factor codes are composed, in comparison with degrees.

A hex tree is an ordered tree of which each vertex has updegree 0, 1, or 2, and an edge from a vertex of updegree 1 is either left, median, or right. We present a refined enumeration of symmetric hex trees via a generalized binomial transform. It turns out that the refinement has a natural combinatorial interpretation by means of supertrees. We describe a bijection between symmetric hex trees and a certain class of supertrees. Some algebraic properties of the polynomials obtained in this procedure are also studied.

 The r-ary number sequences given by (b (r) n )n≥0 = 1 (r − 1)n + 1 rn n  are analogs of the sequence of the Catalan numbers 1 n+1 2n n
. Their history goes back at least to Lambert [8] in 1758 and they are of considerable interest in sequential testing. Usually, the sequences are considered separately and the generalizations can go in several directions. Here we link the various r first by introducing a new combinatorial structure related to GR trees and then algebraically as well. This GR transition generalizes to give r-ary analogs of many sequences of combinatorial interest. It also lets us find infinite numbers of combinatorially defined sequences that lie between the Catalan numbers and the Ternary numbers, or more generally, between b (r) n and b (r+1) n 

Let Y$Y$ be a smooth hypersurface of degree d$d$ in a projective space Pn${？}^n$and take a point y$y$ in Pn\Y${？}^n\backslash Y$. Let Hilb[m]y(Y/Pn−1)${\mathrm{\text{Hilb}}}_y^{[m]}(Y/{？}^{n-1})$ be the relative Hilbert scheme parametrizing zero-dimensional subscheme, of length m$m$, of fibers of the projection morphism Y→Pn−1$Y\to {？}^{n-1}$ from y$y$. In this paper we present an embedding of the relative Hilbert scheme Hilb[m]y(Y/Pn−1)${\mathrm{\text{Hilb}}}_y^{[m]}(Y/{？}^{n-1})$into a weighted projective space and describe its defining ideal for general y$y$. We also study line bundles on the relative Hilbert scheme Hilb[m]y(Y/Pn−1)${\mathrm{\text{Hilb}}}_y^{[m]}(Y/{？}^{n-1})$ for n≥4$n\ge 4$ and general y

$y$..

Let $Z\subset P^N$ be a Fano manifold whose Picard group is generated by the hyperplane section class. Assume that $Z$ is covered by lines and $i\left(Z\right)\ge 3$. Let $？:X^Z\to Z$ be a double cover, branched along a smooth hypersurface section of degree $2m$,$1\le m\le i\left(Z\right)-2$. We describe the defining ideal of the variety of minimal rationaltangents at a general point. As an application, we show that if $Z\subset P^N$ is defined by quadratic equations and $2\le m\le i\left(Z\right)-2$, then the morphism $？$ satisfies the Cartan–Fubini type rigidity property.

We consider ordered trees with a distinguished vertex which we call a mutator. There are many situations where this model arises. An ordered tree could represent a river network, supply lines, an employee organization chart, a phylogenetic tree, or a family tree. The mutator could be a dam or a source of pollution, a break in a supply line, a corrupt employee, and so on. We look at such things as the number of affected vertices, distance from the root, and the effect of various succession rules. Among the types of trees these results apply to are ordered trees defined by a uniform updegree requirement. We find some profiles of trees with a mutation in terms of Riordan matrices. The asymptotic formulas for the ratios of vertices of the new type to all vertices are also derived by using singularity analysis of generating functions.

We study elliptic equations with measurable nonlinearities in nonsmooth domains. We establish an optimal global $W^{1,p}$ estimate under the condition that the associated nonlinearity is allowed to be merely measurable in one variable but has a sufficiently small BMO semi-norm in the other variables, while the underlying domain is sufficiently flat in the Reifenberg sense that the boundary of the domain is locally trapped between two narrow strips.