Dirac spectral flow on contact three manifolds II: Thurston--Winkelnkemper contact forms

Given an open book decomposition $(\Sigma,\tau)$ of a three manifold $Y$, Thurston and Winkelnkemper [TW] construct a specific contact form $a$ on $Y$. Given a spin-c Dirac operator $D$ on $Y$, the contact form naturally associates a one parameter family of Dirac operators D_r = D - \frac{ir}{2}\cl(a) for $r\geq0$. When $r>>1$, we prove that the spectrum of D_r = D_0 - \frac{ir}{2}\cl(a) within $[-(r^{1/2})/2, (r^{1/2})/2]$ are almost uniformly distributed. With the result in Part I, it implies that the subleading order term of the spectral flow from $D_0$ to $D_r$ is of order $r (\log r)^{\frac{9}{2}}$. Besides the interests of the spectral flow, the method of this paper provide a tool to analyze the Dirac operator on an open book decomposition.Comment: 48 pages. All comments welcom

Dirac spectral flow on contact three manifolds I: eigensection estimates and spectral asymmetry

Let $Y$ be a compact, oriented 3-manifold with a contact form $a$ and a metric $ds^2$. Suppose that $F\to Y$ is a principal bundle with structure group $U(2) = SU(2)\times_{\pm1}S^1$ such that $F/S^1$ is the principal SO(3) bundle of orthonormal frames for $TY$. A unitary connection $A_0$ on the Hermitian line bundle $F\times_{\det U(2)}\mathbb{C}$ determines a self-adjoint Dirac operator $D_0$ on the $\mathbb{C}^2$-bundle $F\times_{U(2)}\mathbb{C}^2$. The contact form $a$ can be used to perturb the connection $A_0$ by $A_0-ira$. This associates a one parameter family of Dirac operators $D_r$ for $r\geq0$. When $r>>1$, we establish a sharp sup-norm estimate on the eigensections of $D_r$ with small eigenvalues. The sup-norm estimate can be applied to study the asymptotic behavior of the spectral flow from $D_0$ to $D_r$. In particular, it implies that the subleading order term of the spectral flow is strictly smaller than the order of $r^{\frac{3}{2}}$. We also relate the $\eta$-invariant of $D_r$ to certain spectral asymmetry function involving only the small eigenvalues of $D_r$.Comment: 48 pages. All comments welcom

A New Condition for Blow-up Solutions to Discrete Semilinear Heat Equations on Networks

The purpose of this paper is to introduce a new condition $$\hbox{(C)$\hspace{1cm} \alpha \int_{0}^{u}f(s)ds \leq uf(u)+\beta u^{2}+\gamma,\,\,u>0$}$$ for some $\alpha, \beta, \gamma>0$ with $0<\beta\leq\frac{\left(\alpha-2\right)\lambda_{0}}{2}$, where $\lambda_{0}$ is the first eigenvalue of discrete Laplacian $\Delta_{\omega}$, with which we obtain blow-up solutions to discrete semilinear heat equations \begin{equation*} \begin{cases} u_{t}\left(x,t\right)=\Delta_{\omega}u\left(x,t\right)+f(u(x,t)), & \left(x,t\right)\in S\times\left(0,+\infty\right),\\ u\left(x,t\right)=0, & \left(x,t\right)\in\partial S\times\left[0,+\infty\right),\\ u\left(x,0\right)=u_{0}\geq0(nontrivial), & x\in\overline{S} \end{cases} \end{equation*} on a discrete network $S$. In fact, it will be seen that the condition (C) improves the conditions known so far.Comment: 19 page

A strong stability condition on minimal submanifolds and its implications

We identify a strong stability condition on minimal submanifolds that implies uniqueness and dynamical stability properties. In particular, we prove a uniqueness theorem and a C^1 dynamical stability theorem of the mean curvature flow for minimal submanifolds that satisfy this condition. The latter theorem states that the mean curvature flow of any other submanifold in a C^1 neighborhood of such a minimal submanifold exists for all time, and converges exponentially to the minimal one. This extends our previous uniqueness and stability theorem [arXiv:1605.03645] which applies only to calibrated submanifolds of special holonomy ambient manifolds.Comment: 50 pages; Appendix C to gives the details for the C^2 convergence; to appear in J. Reine Angew. Math. (Crelle's Journal

Luttinger surgery and Kodaira dimension

In this note we show that the Lagrangian Luttinger surgery preserves the symplectic Kodaira dimension. Some constraints on Lagrangian tori in symplectic four manifolds with non-positive Kodaira dimension are also derived.Comment: 23 pages; to appear in Asian Journal of Mathematic

A New Condition for the Concavity Method of Blow-up Solutions to p-Laplacian Parabolic Equations

In this paper, we consider an initial-boundary value problem of the p-Laplacian parabolic equations \begin{equation} \begin{cases} u_{t}\left(x,t\right)=\mbox{div}(|\nabla u\left(x,t\right)|^{p-2}\nabla u(x,t))+f(u(x,t)), & \left(x,t\right)\in \Omega\times\left(0,+\infty\right), \newline u\left(x,t\right)=0, & \left(x,t\right)\in\partial \Omega\times\left[0,+\infty\right), \newline u\left(x,0\right)=u_{0}\geq0, & x\in\overline{\Omega}, \end{cases} \end{equation} where $p\geq2$ and $\Omega$ is a bounded domain of $\mathbb{R}^{N}$ $(N\geq1)$ with smooth boundary $\partial\Omega$. The main contribution of this work is to introduce a new condition \mbox{$(C_{p})$$\hspace{1cm} \alpha \int_{0}^{u}f(s)ds \leq uf(u)+\beta u^{p}+\gamma,\,\,u>0$} for some $\alpha, \beta, \gamma>0$ with $0<\beta\leq\frac{\left(\alpha-p\right)\lambda_{1, p}}{p}$, where $\lambda_{1, p}$ is the first eigenvalue of p-Laplacian $\Delta_{p}$, and we use the concavity method to obtain the blow-up solutions to the above equations. In fact, it will be seen that the condition $(C_{p})$ improves the conditions ever known so far.Comment: 15 pages. arXiv admin note: text overlap with arXiv:1706.0349

Effective electrostatic interactions in mixtures of charged colloids

We present a theory of effective electrostatic interactions in polydisperse suspensions of charged macroions, generalizing to mixtures a theory previously developed for monodisperse suspensions. Combining linear response theory with a random phase approximation for microion correlations, we coarse-grain the microion degrees of freedom to derive general expressions for effective macroion-macroion pair potentials and a one-body volume energy. For model mixtures of charged hard-sphere colloids, we give explicit analytical expressions. The resulting effective pair potentials have the same general form as predicted by linearized Poisson-Boltzmann theory, but consistently incorporate dependence on macroion density and excluded volume via the Debye screening constant. The volume energy, which depends on the average macroion density, contributes to the free energy and so can influence thermodynamic properties of deionized suspensions. To validate the theory, we compute radial distribution functions of binary mixtures of oppositely charged colloidal macroions from molecular dynamics simulations of the coarse-grained model (with implicit microions), taking effective pair potentials as input. Our results agree closely with corresponding results from more computationally intensive Monte Carlo simulations of the primitive model (with explicit microions). Simulations of a mixture with large size and charge asymmetries indicate that charged nanoparticles can enhance electrostatic screening of charged colloids. The theory presented here lays a foundation for future large-scale modeling of complex mixtures of charged colloids, nanoparticles, and polyelectrolytes.Comment: 11 pages, 5 figure

Tunable Band gap of Iron-Doped Lanthanum-Modified Bismuth Titanate Synthesized by the Thermal Decomposition of a Secondary Phase

The photoelectric properties of complex oxides have prompted interest in materials with a tunable band gap, because the absorption The substitution of iron atoms in La-modified bismuth titanate (BLT) can lead to dramatic improvements in the band gap, however, the substitution of iron atoms in BLT without forming a BiFeO3 secondary phase is quite challenging. Therefore, a series of Fe-doped BLT (Fe-BLT) samples were characterized using a solid reaction at various calcination temperatures (300~900{\deg}C) to remove the secondary phase. The structural and optical properties were analyzed by X-ray diffraction and ultraviolet-visible absorption spectroscopy. This paper reports a new route to synthesize a pure Fe-BLT phase with a reduced optical band gap by high temperature calcination due to the thermal decomposition of BiFeO3 during high temperature calcination. This simple route to reduce the second phase can be adapted to other complex oxides for use in emerging oxide optoelectronic devices.Comment: 15 pages, 4 figure

A New Condition for the Concavity Method of Blow-up Solutions to Semilinear Heat Equations

In this paper, we consider the semilinear heat equations under Dirichlet boundary condition u_{t}\left(x,t\right)=\Delta u\left(x,t\right)+f(u(x,t)), & \left(x,t\right)\in \Omega\times\left(0,+\infty\right), u\left(x,t\right)=0, & \left(x,t\right)\in\partial \Omega\times\left[0,+\infty\right), u\left(x,0\right)=u_{0}\geq0, & x\in\overline{\Omega}, where $\Omega$ is a bounded domain of $\mathbb{R}^{N}$ $(N\geq1)$ with smooth boundary $\partial\Omega$. The main contribution of our work is to introduce a new condition $$(C) \alpha \int_{0}^{u}f(s)ds \leq uf(u)+\beta u^{2}+\gamma,\,\,u>0$$ for some $\alpha, \beta, \gamma>0$ with $0<\beta\leq\frac{\left(\alpha-2\right)\lambda_{0}}{2}$, where $\lambda_{0}$ is the first eigenvalue of Laplacian $\Delta$, and we use the concavity method to obtain the blow-up solutions to the semilinear heat equations. In fact, it will be seen that the condition (C) improves the conditions known so far.Comment: 7 page

Dark matter halo assembly bias: environmental dependence in the non-Markovian excursion set theory

In the standard excursion set model for the growth of structure, the statistical properties of halos are governed by the halo mass and are independent of the larger scale environment in which the halos reside. Numerical simulations, however, have found the spatial distributions of halos to depend not only on their mass but also on the details of their assembly history and environment. Here we present a theoretical framework for incorporating this "assembly bias" into the excursion set model. Our derivations are based on modifications of the path integral approach of Maggiore & Riotto (2010) that models halo formation as a non-Markovian random walk process. The perturbed density field is assumed to evolve stochastically with the smoothing scale and exhibits correlated walks in the presence of a density barrier. We write down conditional probabilities for multiple barrier crossings, and derive from them analytic expressions for descendant and progenitor halo mass functions and halo merger rates as a function of both halo mass and the linear overdensity of the larger-scale environment of the halo. Our results predict a higher halo merger rate and higher progenitor halo mass function in regions of higher overdensity, consistent with the behavior seen in N-body simulations.Comment: 13 pages, 1 figure, ApJ in pres