Refined intersection products and limiting linear subspaces of hypersurfaces

Let $X$ be a hypersurface of degree $d$ in $\Bbb P^n$ and $F_X$ be the scheme of $\Bbb P^r$'s contained in $X$. If $X$ is generic, then $F_X$ will have the expected dimension (or empty) and its class in the Chow ring of $G(r+1,n+1)$ is given by the top Chern class of the vector bundle $S^dU^*$, where $U$ is the universal subbundle on the Grassmannian $G(r+1,n+1)$. When we deform a generic $X$ into a degenerate $X_0$, the dimension of $F_X$ can jump. In this case, there is a subscheme $F_{lim}$ of $F_{X_0}$ with the expected dimension which consists of limiting $\Bbb P^r$'s in $X_0$ with respect to a general deformation. The simplest example is the well-known case of $27$ lines in a generic cubic surface. If we degenerate the surface into the union of a plane and a quadric, then there are infinitely many lines in the union. Which $27$ lines are the limiting ones and how many of them are in the plane and how many of them are in the quadric? The goal of this paper is to study $F_{lim}$ in general.Comment: AmS-Tex, 26 page

Residual Intersections and Some Applications

We give a new residual intersection decomposition for the refined intersection products of Fulton-MacPherson. Our formula refines the celebrated residual intersection formula of Fulton, Kleiman, Laksov, and MacPherson. The new decomposition is more likely to be compatible with the canonical decomposition of the intersection products and each term in the decomposition thus has simple geometric meaning. Our study is motivated by its applications to some geometric problems. In particular, we use the decomposition to find the distribution of limiting linear subspaces in degenerations of hypersurfaces. A family of identities for characteristic classes of vector bundles is also obtained as another consequence. This paper will appear in Duke Math. Jour.Comment: 23 pages, Ams-Tex Version 2.

Chern classes and degenerations of hypersurfaces and their lines

We study limiting lines on degenerations of generic hypersurfaces in $P^n$.Comment: 18 pages, AmS-Tex V2.

Global performance of multireference density functional theory for low-lying states in sdsd-shell nuclei

We present a comprehensive study of low-lying states in even-even Ne, Mg, Si, S, Ar isotopes with the multireference density functional theory (MR-DFT) based on a relativistic point-coupling energy density functional (EDF). Beyond mean-field (BMF) effects are taken into account by configuration mixing of both particle-number and angular-momentum projected axially deformed states with generator coordinate method (GCM). Global performance of the MR-DFT for the properties of both ground state and of the first $2^+, 4^+$ states is examined, in comparison with previous studies based on nonrelativistic EDFs and available data. Our results indicate that an EDF parameterized at the BMF level is demanded to achieve a quantitative description

Mixing Time of Random Walk on Poisson Geometry Small World

This paper focuses on the problem of modeling for small world effect on complex networks. Let's consider the supercritical Poisson continuous percolation on $d$-dimensional torus $T^d_n$ with volume $n^d$. By adding "long edges (short cuts)" randomly to the largest percolation cluster, we obtain a random graph $\mathscr G_n$. In the present paper, we first prove that the diameter of $\mathscr G_n$ grows at most polynomially fast in $\ln n$ and we call it the Poisson Geometry Small World. Secondly, we prove that the random walk on $\mathscr G_n$ possesses the rapid mixing property, namely, the random walk mixes in time at most polynomially large in $\ln n$.Comment: 23 page

Existence of Positive Solutions for a class of Quasilinear Schr\"{o}dinger Equations of Choquard type

In this paper, we study the following quasilinear Schr\"{o}dinger equation of Choquard type $$-\triangle u+V(x)u-\triangle (u^{2})u=(I_\alpha *|u|^p)|u|^{p-2}u, \ \ x \in \mathbb{R}^{N},$$ where $N\geq 3$,\ $0<\alpha<N$, $\frac{2(N+\alpha)}{N}\leq p<\frac{2(N+\alpha)}{N-2}$ and $I_\alpha$ is a Riesz potential. Under appropriate assumptions on $V(x)$, we establish the existence of positive solutions

On The Modified Newman-Watts Small World and Its Random Walk

It is well known that adding "long edges (shortcuts)" to a regularly constructed graph will make the resulted model a small world. Recently, \cite{W} indicated that, among all long edges, those edges with length proportional to the diameter of the regularly constructed graph may play the key role. In this paper, we modify the original Newman-Watts small world by adding only long special edges to the $d$-dimensional lattice torus (with size $n^d$) according to \cite{W}, and show that the diameter of the modified model and the mixing time of random walk on it grow polynomially fast in $\ln n$.Comment: 17 pages. arXiv admin note: substantial text overlap with arXiv:1703.0825

On a Lower Bound for the Time Constant of First-Passage Percolation

We consider the Bernoulli first-passage percolation on $\mathbb Z^d (d\ge 2)$. That is, the edge passage time is taken independently to be 1 with probability $1-p$ and 0 otherwise. Let ${\mu(p)}$ be the time constant. We prove in this paper that $$\mu(p_1)-\mu({p_2})\ge \frac{\mu(p_2)}{1-p_2}(p_2-p_1)$$ for all $0\leq p_1<p_2< 1$ by using Russo's formula.Comment: 7 page

Uniqueness of the critical probability for percolation in the two dimensional Sierpinski carpet lattice

We prove that the critical probability for the Sierpinski carpet lattice in two dimensions is uniquely determined. The transition is sharp. This extends the Kumagai's result to the original Sierpinski carpet lattice.Comment: 22pages; typos added(erased pre-finel comments

On The Time Constant for Last Passage Percolation on Complete Graph

This paper focuses on the time constant for last passage percolation on complete graph. Let $G_n=([n],E_n)$ be the complete graph on vertex set $[n]=\{1,2,\ldots,n\}$, and i.i.d. sequence $\{X_e:e\in E_n\}$ be the passage times of edges. Denote by $W_n$ the largest passage time among all self-avoiding paths from 1 to $n$. First, it is proved that $W_n/n$ converges to constant $\mu$, where $\mu$ is called the time constant and coincides with the essential supremum of $X_e$. Second, when $\mu<\infty$, it is proved that the deviation probability $P(W_n/n\leq \mu-x)$ decays as fast as $e^{-\Theta(n^2)}$, and as a corollary, an upper bound for the variance of $W_n$ is obtained. Finally, when $\mu=\infty$, lower and upper bounds for $W_n/n$ are given.Comment: 12 pages, 1 figur