8,755 research outputs found
Dirac spectral flow on contact three manifolds II: Thurston--Winkelnkemper contact forms
Given an open book decomposition of a three manifold ,
Thurston and Winkelnkemper [TW] construct a specific contact form on .
Given a spin-c Dirac operator on , the contact form naturally associates
a one parameter family of Dirac operators D_r = D - \frac{ir}{2}\cl(a) for
. When , we prove that the spectrum of D_r = D_0 -
\frac{ir}{2}\cl(a) within are almost uniformly
distributed. With the result in Part I, it implies that the subleading order
term of the spectral flow from to is of order . 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 be a compact, oriented 3-manifold with a contact form and a
metric . Suppose that is a principal bundle with structure group
such that is the principal SO(3) bundle
of orthonormal frames for . A unitary connection on the Hermitian
line bundle determines a self-adjoint Dirac
operator on the -bundle .
The contact form can be used to perturb the connection by
. This associates a one parameter family of Dirac operators for
. When , we establish a sharp sup-norm estimate on the
eigensections of with small eigenvalues. The sup-norm estimate can be
applied to study the asymptotic behavior of the spectral flow from to
. In particular, it implies that the subleading order term of the spectral
flow is strictly smaller than the order of . We also relate
the -invariant of to certain spectral asymmetry function involving
only the small eigenvalues of .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 for some with
, where is
the first eigenvalue of discrete Laplacian , 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 . 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 and is a bounded domain of
with smooth boundary . 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 with
, where is the first eigenvalue of p-Laplacian , 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 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 is a
bounded domain of with smooth boundary
. The main contribution of our work is to introduce a new
condition for some with
, where is
the first eigenvalue of Laplacian , 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
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