Anisotropy mapping in rat brains using Intermolecular Multiple Quantum Coherence Effects

This document reports an unconventional and rapidly developing approach to magnetic resonance imaging (MRI) using intermolecular multiple-quantum coherences (iMQCs). Rat brain images are acquired using iMQCs. We detect iMQCs between spins that are 10 {\mu}m to 500 {\mu}m apart. The interaction between spins is dependent on different directions. We can choose the directions on physical Z, Y and X axis by choosing correlation gradients along those directions. As an important application, iMQCs can be used for anisotropy mapping. In the rat brains, we investigate tissue microstructure. We simulated images expected from rat brains without microstructure. We compare those with experimental results to prove that the dipolar field from the overall shape only has small contributions to the experimental iMQC signal. Because of the underlying low signal to noise ratio (SNR) in iMQCs, this anisotropy mapping method still has comparatively large potentials to grow. The ultimate goal of my project is to develop creative and effective methods of tissue microstructure anisotropy mapping. Recently we found that combining phase data of iMQCs images with phase data of modified-crazed images is very promising to construct microstructure maps. Some information and initial results are shown in this document

Minimum codegree threshold for Hamilton l-cycles in k-uniform hypergraphs

For $1\le \ell<k/2$, we show that for sufficiently large $n$, every $k$-uniform hypergraph on $n$ vertices with minimum codegree at least $\frac n{2 (k-\ell)}$ contains a Hamilton $\ell$-cycle. This codegree condition is best possible and improves on work of H\`an and Schacht who proved an asymptotic result.Comment: 22 pages, 0 figure. Accepted for publication in JCTA. arXiv admin note: text overlap with arXiv:1307.369

Blow up for some semilinear wave equations in multi-space dimensions

In this paper, we discuss a new nonlinear phenomenon. We find that in $n\geq 2$ space dimensions, there exists two indexes $p$ and $q$ such that the cauchy problems for the nonlinear wave equations {equation} \label{0.1} \Box u(t,x) = |u(t,x)|^{q}, \ \ x\in R^{n}, {equation} and {equation} \label{0.2} \Box u(t,x) = |u_{t}(t,x)|^{p}, \ \ x\in R^{n} {equation} both have global existence for small initial data, while for the combined nonlinearity, the solutions to the Cauchy problem for the nonlinear wave equation {equation} \label{0.3} \Box u(t,x) = | u_{t}(t,x)|^{p} + |u(t,x)|^{q}, \ \ x\in R^{n}, {equation} with small initial data will blow up in finite time. In the two dimensional case, we also find that if $q=4$, the Cauchy problem for the equation \eqref{0.1} has global existence, and the Cauchy problem for the equation {equation} \label{0.4} \Box u(t,x) = u (t,x)u_{t}(t,x)^{2}, \ \ x\in R^{2} {equation} has almost global existence, that is, the life span is at least $\exp (c\varepsilon^{-2})$ for initial data of size $\varepsilon$. However, in the combined nonlinearity case, the Cauchy problem for the equation {equation} \label{0.5} \Box u(t,x) = u(t,x) u_{t}(t,x)^{2} + u(t,x)^{4}, \ \ x\in R^{2} {equation} has a life span which is of the order of $\varepsilon^{-18}$ for the initial data of size $\varepsilon$, this is considerably shorter in magnitude than that of the first two equations. This solves an open optimality problem for general theory of fully nonlinear wave equations (see \cite{Katayama}).Comment: 13 page

Forbidding Hamilton cycles in uniform hypergraphs

For $1\le d\le \ell< k$, we give a new lower bound for the minimum $d$-degree threshold that guarantees a Hamilton $\ell$-cycle in $k$-uniform hypergraphs. When $k\ge 4$ and $d< \ell=k-1$, this bound is larger than the conjectured minimum $d$-degree threshold for perfect matchings and thus disproves a well-known conjecture of R\"odl and Ruci\'nski. Our (simple) construction generalizes a construction of Katona and Kierstead and the space barrier for Hamilton cycles.Comment: 6 pages, 0 figur

Life-Span of Solutions to Critical Semilinear Wave Equations

The final open part of the famous Strauss conjecture on semilinear wave equations of the form \Box u=|u|^{p}, i.e., blow-up theorem for the critical case in high dimensions was solved by Yordanov and Zhang, or Zhou independently. But the estimate for the lifespan, the maximal existence time, of solutions was not clarified in both papers. Recently, Takamura and Wakasa have obtained the sharp upper bound of the lifespan of the solution to the critical semilinear wave equations, and their method is based on the method in Yordanov and Zhang. In this paper, we give a much simple proof of the result of Takamura and Wakasa by using the method in Y. Zhou for space dimensions n\geq 2. Simultaneously, this estimate of the life span also proves the last open optimality problem of the general theory for fully nonlinear wave equations with small initial data in the case n=4 and quadratic nonlinearity(One can see Li and Chen for references on the whole history).Comment: 12 pages, no figure

Minimum vertex degree threshold for C43C_4^3-tiling

We prove that the vertex degree threshold for tiling \C_4^3 (the 3-uniform hypergraph with four vertices and two triples) in a 3-uniform hypergraph on $n\in 4\mathbb N$ vertices is $\binom{n-1}2 - \binom{\frac34 n}2+\frac38n+c$, where $c=1$ if $n\in 8\mathbb N$ and $c=-\frac12$ otherwise. This result is best possible, and is one of the first results on vertex degree conditions for hypergraph tiling.Comment: 16 pages, 0 figure. arXiv admin note: text overlap with arXiv:0903.2867 by other author

Blow up of Solutions to Semilinear Wave Equations with variable coefficients and boundary

This paper is devoted to studying the following two initial-boundary value problems for semilinear wave equations with variable coefficients on exterior domain with subcritical exponent in $n$ space dimensions: u_{tt}-partial_{i}(a_{ij}(x)\partial_{j}u)=|u|^{p}, (x,t)\in \Omega^{c}\times(0,+\infty), n\geq 3 and u_{tt}-\partial_{i}(a_{ij}(x)\partial_{j}u)=|u_{t}|^{p}, (x,t)\in \Omega^{c}\times (0,+\infty), n\geq 1, where $a_{ij}(x)=\delta_{ij}, when |x|\geq R. The exponents$p$satisfies$ 1<p<p_{1}(n)$in (0.1), and$p \leq p_{2}(n)$in (0.2), where$p_{1}(n)$ is the larger root of the quadratic equation (n-1)p^{2}-(n+1)p-2=0, and p_{2}(n)=\frac{2}{n-1}+1, respectively. It is well-known that the numbers p_{1}(n) and p_{2}(n) are the critical exponents. We will establish two blowup results for the above two initial-boundary value problems, it is proved that there can be no global solutions no matter how small the initial data are, and also we give the lifespan estimate of solutions for above problems

On multipartite Hajnal-Szemer\'edi theorems

Let $G$ be a $k$-partite graph with $n$ vertices in parts such that each vertex is adjacent to at least $\delta^*(G)$ vertices in each of the other parts. Magyar and Martin \cite{MaMa} proved that for $k=3$, if $\delta^*(G)\ge 2/3n$ and $n$ is sufficiently large, then $G$ contains a $K_3$-factor (a spanning subgraph consisting of $n$ vertex-disjoint copies of $K_3$) except that $G$ is one particular graph. Martin and Szemer\'edi \cite{MaSz} proved that $G$ contains a $K_4$-factor when $\delta^*(G)\ge 3/4n$ and $n$ is sufficiently large. Both results were proved by the Regularity Lemma. In this paper we give a proof of these two results by the absorbing method. Our absorbing lemma actually works for all $k\ge 3$.Comment: 15 pages, no figur

Minimum vertex degree threshold for loose Hamilton cycles in 3-uniform hypergraphs

We show that for sufficiently large $n$, every 3-uniform hypergraph on $n$ vertices with minimum vertex degree at least $\binom{n-1}2 - \binom{\lfloor\frac34 n\rfloor}2 + c$, where $c=2$ if $n\in 4\mathbb{N}$ and $c=1$ if $n\in 2\mathbb{N}\setminus 4\mathbb{N}$, contains a loose Hamilton cycle. This degree condition is best possible and improves on the work of Bu\ss, H\`an and Schacht who proved the corresponding asymptotical result.Comment: 23 pages, 1 figure, Accepted for publication in JCT

Dipolar spinor Bose-Einstein condensates

Under many circumstances, the only important two-body interaction between atoms in ultracold dilute atomic vapors is the short-ranged isotropic s-wave collision. Recent studies have shown, however, that situations may arise where the dipolar interaction between atomic magnetic or electric dipole moments can play a significant role. The long-range anisotropic nature of the dipolar interaction greatly enriches the static and dynamic properties of ultracold atoms. In the case of dipolar spinor condensates, the interplay between the dipolar interaction and the spin exchange interaction may lead to nontrivial spin textures. Here we pay particular attention to the spin vortex state that is analogous to the magnetic vortex found in thin magnetic films.Comment: 12 pages, 10 figures. A review on our recent work on dipolar spinor BECs, to appear in the book "Electromagnetic, Magnetostatic and Exchange Interaction Vortices in Confined Magnetic Structures