Convergence of fundamental solutions of linear parabolic equations under Cheeger-Gromov convergence

In this note we show the convergence of the fundamental solutions of the parabolic equations assuming the Cheeger-Gromov convergence of the underlying manifolds and the uniform $L^1$-bound of the solutions. We also prove a local integral estimate of fundamental solutions.Comment: 19 pages

Special Lagrangian Tori on a Borcea-Voisin Threefold

We show the existence of special Lagrangian tori on one family of Borcea-Voisin threefolds. We also construct a family of special Lagrangian submanifolds on the total space of the canonical line bundle of projective spaces.Comment: 6 page

Maxima and minima of independent and non-identically distributed bivariate Gaussian triangular arrays

In this paper, joint limit distributions of maxima and minima on independent and non-identically distributed bivariate Gaussian triangular arrays is derived as the correlation coefficient of $i$th vector of given $n$th row is the function of $i/n$. Furthermore, second-order expansions of joint distributions of maxima and minima are established if the correlation function satisfies some regular conditions.Comment: 16 page

Monochromatic 4-term arithmetic progressions in 2-colorings of Zn\mathbb Z_n

This paper is motivated by a recent result of Wolf \cite{wolf} on the minimum number of monochromatic 4-term arithmetic progressions(4-APs, for short) in $\Z_p$, where $p$ is a prime number. Wolf proved that there is a 2-coloring of $\Z_p$ with 0.000386% fewer monochromatic 4-APs than random 2-colorings; the proof is probabilistic and non-constructive. In this paper, we present an explicit and simple construction of a 2-coloring with 9.3% fewer monochromatic 4-APs than random 2-colorings. This problem leads us to consider the minimum number of monochromatic 4-APs in $\Z_n$ for general $n$. We obtain both lower bound and upper bound on the minimum number of monochromatic 4-APs in all 2-colorings of $\Z_n$. Wolf proved that any 2-coloring of $\Z_p$ has at least $(1/16+o(1))p^2$ monochromatic 4-APs. We improve this lower bound into $(7/96+o(1))p^2$. Our results on $\Z_n$ naturally apply to the similar problem on $[n]$ (i.e., $\{1,2,..., n\}$). In 2008, Parillo, Robertson, and Saracino \cite{prs} constructed a 2-coloring of $[n]$ with 14.6% fewer monochromatic 3-APs than random 2-colorings. In 2010, Butler, Costello, and Graham \cite{BCG} extended their methods and used an extensive computer search to construct a 2-coloring of $[n]$ with 17.35% fewer monochromatic 4-APs (and 26.8% fewer monochromatic 5-APs) than random 2-colorings. Our construction gives a 2-coloring of $[n]$ with 33.33% fewer monochromatic 4-APs (and 57.89% fewer monochromatic 5-APs) than random 2-colorings.Comment: 23 pages, 4 figure

Principal stratification analysis using principal scores

Practitioners are interested in not only the average causal effect of the treatment on the outcome but also the underlying causal mechanism in the presence of an intermediate variable between the treatment and outcome. However, in many cases we cannot randomize the intermediate variable, resulting in sample selection problems even in randomized experiments. Therefore, we view randomized experiments with intermediate variables as semi-observational studies. In parallel with the analysis of observational studies, we provide a theoretical foundation for conducting objective causal inference with an intermediate variable under the principal stratification framework, with principal strata defined as the joint potential values of the intermediate variable. Our strategy constructs weighted samples based on principal scores, defined as the conditional probabilities of the latent principal strata given covariates, without access to any outcome data. This principal stratification analysis yields robust causal inference without relying on any model assumptions on the outcome distributions. We also propose approaches to conducting sensitivity analysis for violations of the ignorability and monotonicity assumptions, the very crucial but untestable identification assumptions in our theory. When the assumptions required by the classical instrumental variable analysis cannot be justified by background knowledge or cannot be made because of scientific questions of interest, our strategy serves as a useful alternative tool to deal with intermediate variables. We illustrate our methodologies by using two real data examples, and find scientifically meaningful conclusions

A comparative study of fracture in Al: quantum mechanical vs. empirical atomistic description

A comparative study of fracture in Al is carried out by using quantum mechanical and empirical atomistic description of atomic interaction at crack tip. The former is accomplished with the density functional theory (DFT) based Quasicontinuum method (QCDFT) and the latter with the original Quasicontinuum method (EAM-QC). Aside from quantitative differences, the two descriptions also yield qualitatively distinctive fracture behavior. While EAM-QC predicts a straight crack front and a micro-twinning at the crack tip, QCDFT finds a more rounded crack profile and the absence of twinning. Although many dislocations are emitted from the crack tip in EAM-QC, they all glide on a single slip plane. In contrast, only two dislocations are nucleated under the maximum load applied in QCDFT, and they glide on two adjacent slip planes. The electron charge density develops sharp corners at the crack tip in EAM-QC, while it is smoother in QCDFT. The physics underlying these differences is discussed.Comment: 24 pages,9 figure

High-order Phase Transition in Random Hypergrpahs

In this paper, we study the high-order phase transition in random $r$-uniform hypergraphs. For a positive integer $n$ and a real $p\in [0,1]$, let $H:=H^r(n,p)$ be the random $r$-uniform hypergraph with vertex set $[n]$, where each $r$-set is selected as an edge with probability $p$ independently randomly. For $1\leq s \leq r-1$ and two $s$-sets $S$ and $S'$, we say $S$ is connected to $S'$ if there is a sequence of alternating $s$-sets and edges $S_0,F_1,S_1,F_2, \ldots, F_k, S_k$ such that $S_0,S_1,\ldots, S_k$ are $s$-sets, $S_0=S$, $S_k=S'$, $F_1,F_2,\ldots, F_k$ are edges of $H$, and $S_{i-1}\cup S_i\subseteq F_i$ for each $1\leq i\leq k$. This is an equivalence relation over the family of all $s$-sets ${[n]\choose s}$ and results in a partition: ${V\choose s}=\cup_i C_i$. Each $C_i$ is called an { $s$-th-order} connected component and a component $C_i$ is {\em giant} if $|C_i|=\Theta(n^s)$. We prove that the sharp threshold of the existence of the $s$-th-order giant connected components in $H^r(n,p)$ is $\frac{1}{\big({r\choose s}-1\big){n\choose r-s}}$. Let $c={n\choose r-s}p$. If $c$ is a constant and $c<\tfrac{1}{\binom{r}{s}-1}$, then with high probability, all $s$-th-order connected components have size $O(\ln n)$. If $c$ is a constant and $c > \tfrac{1}{\binom{r}{s}-1}$, then with high probability, $H^r(n,p)$ has a unique giant connected $s$-th-order component and its size is $(z+o(1)){n\choose s}$, where $$z=1-\sum_{j=0}^\infty \frac{\left({r\choose s}j -j+1 \right)^{j-1}}{j!}c^je^{-c\left({r\choose s}j -j+1\right)}.$$Comment: We revised the paper substantially based on the referees' reports and rewrote Section

On the asymptotic scalar curvature ratio of complete Type I-like ancient solutions to the Ricci flow on non-compact 3-manifolds

The main result of this paper is: Given any constant C, there is $(\epsilon,k,L)$ such that if a complete, orientable, noncompact odd-dimensional manifold with bounded positive sectional curvature contains a $(\epsilon,k,L)$-neck, then the asymptotic scalar curvature ratio is bigger or equal to C. As a application we proved that the asymptotic scalar curvature ratio of a complete noncompact ancient Type I-like solution to the Ricci flow with bounded positive sectional curvature on an orientable 3-manifold, is infinity.Comment: 28 pages, no figure

New proofs of Perelman's theorem on shrinking Breathers in Ricci flow

We give two new proofs of Perelman's theorem that shrinking breathers of Ricci flow on closed manifolds are gradient Ricci solitons, using the fact that the singularity models of type I solutions are shrinking gradient Ricci solitons and the fact that non-collapsed type I ancient solutions have rescaled limits being shrinking gradient Ricci solitons.Comment: 6 page

Some elementary consequences of Perelman's canonical neighborhood theorem

In this purely expository note, we recall a few known properties of 3-dimensional singularity models. These properties are direct consequences of Perelman's canonical neighborhood theorem for 3-dimensional Ricci flow and compactness theorem for 3-dimensional kappa-solutions.Comment: 3 page