3-Vertices with fewest 2-neighbors in plane graphs with no long paths of 2-vertices

Let $g(k,t)$ be the minimum integer such that every plane graph with girth g at least $g(k,t)$, minimum degree $\delta =2$ and no $(k+1)$-paths consisting of vertices of degree 2, where $k\ge 1$, has a 3-vertex with at least t neighbors of degree 2, where $1\le t\le 3$.In 2015, Jendrol' and Maceková proved $g(1,1)\le 7$. Later on, Hudák et al. established $g(1,3)=10$, Jendrol', Maceková, Montassier, and Soták proved $g(1,1)\ge 7$, $g(1,2)=8$ and $g(2,2)\ge 11$, and we recently proved that $g(2,2)=11$ and $g(2,3)=14$.Thus $g(k,t)$ is already known for $k=1$ and all t. In this paper, we prove that $g(k,1)=3k+4$, $g(k,2)=3k+5$, and $g(k,3)=3k+8$ whenever $k\ge 2$.     

The sharp conditions of the uniqueness for inverse nodal problems

In this paper we provide the sharp conditions of the uniqueness for inverse nodal Sturm–Liouville problems defined on interval $[0,1]$ with separated boundary conditions. We prove that the potential and boundary parameters can be uniquely determined by a dense nodal subset contained on $[a_1,a_2](?[0,1])$ with $1/2\in (a_1,a_2)$ through two cases of $a_1=0$ and $a_1>0$, where in the latter case the nodal subset also need to be paired. Note that, the dense nodal subset was required to be twin for both cases in the previous works.     

Self-dual codes over F5 and s-extremal unimodular lattices

New s-extremal extremal unimodular lattices in dimensions 38, 40, 42 and 44 are constructed from self-dual codes over ${\mathbb{F}}_5$ by Construction A. In the process of constructing these codes, we obtain a self-dual $[44,22,14]$ code over ${\mathbb{F}}_5$. In addition, the code implies a $[43,22,13]$ code over ${\mathbb{F}}_5$. These codes have larger minimum weights than the previously known $[44,22]$ codes and $[43,22]$ codes, respectively.     

SLE intersecting with random hulls

Lawler, Schramm, and Werner gave in 2003 an explicit formula of the probability that $\mathrm{SLE}(8/3)$ does not intersect a deterministic hull. For general $\mathrm{SLE}(\kappa )$ with $\kappa \ne 8/3$, no such explicit formula has been obtained so far. In this paper, we shall consider a random hull generated by an independent chordal conformal restriction measure and obtain an explicit formula for the probability that $\mathrm{SLE}(\kappa )$ does not intersect this random hull for any $\kappa \in (0,8)$. As a corollary, we will give a new proof of Werner's result on conformal restriction measures.     

The space of lines in cyclic covers of projective space

Take positive integers m, n and d. Let Y be an m-fold cyclic cover of ${\mathbb{P}}^n$ ramified over a general hypersurface $X\subset P^n$ of degree md. In this paper we study the space $F(Y)$ of lines in Y and show that it is smooth of dimension $2(n-1)-d(m-1)$ if $md>2n-3$ and $2(n-1)-d(m-1)\ge 0$. When $2(n-1)=d(m-1)$, our result gives a formula on the number of m-contact order lines of X (see Definition 1.2).     

Isolated maximal d.r.e. degrees

There are very few results about maximal d.r.e. degrees as the construction is very hard to work with other requirements. In this paper we show that there exists an isolated maximal d.r.e. degree. In fact, we introduce a closely related notion called $(m,n)$-cupping degree and show that there exists an isolated $(2,\omega )$-cupping degree, and there exists a proper $(2,1)$-cupping degree. It helps understanding various degree structures in the Ershov Hierarchy.     

Parity considerations for drops in cycles on {1,2,…,n}

In 2019, A. Lazar and M. L. Wachs conjectured that the number of cycles on $[2n]$ with only even-odd drops equals the n-th Genocchi number. In this paper, we restrict our attention to a subset of cycles on $[n]$ such that in all drops in the cycle, the latter entry is odd. We deduce two bivariate generating functions for such a subset of cycles with an extra variable introduced to count the number of odd-odd and even-odd drops, respectively. One of the generating function identities confirms Lazar and Wachs' conjecture, while the other identity implies that the number of cycles on $[2n?1]$ with only odd-odd drops equals the $(n?2)$-th Genocchi median.