Searcharxiv⌕ Search

arXiv subjects

Hao Wu

Publications and source records attributed to Hao Wu.

At least 919 records · Page 51Linked to original sources
arXiv2014

A Family of Transverse Link Homologies

We define a homology $\mathcal{H}_N$ for closed braids by applying Khovanov and Rozansky's matrix factorization construction with potential $ax^{N+1}$. Up to a grading shift, $\mathcal{H}_0$ is the HOMFLYPT homology defined in arXiv:math/0505056. We demonstrate that, for $N \geq 1$, $\mathcal{H}_N$ is a $\mathbb{Z}_2\oplus\mathbb{Z}^{\oplus 3}$-graded $\mathbb{Q}[a]$-module that is invariant under transverse Markov moves, but not under negative stabilization/de-stabilization. Thus, for $N\geq 1$, this homology is an invariant for transverse links in the standard contact $S^3$, but not for smooth links. We also discuss the decategorification of $\mathcal{H}_N$ and the relation between $\mathcal{H}_N$ and the $\mathfrak{sl}(N)$ Khovanov-Rozansky homology defined in arXiv:math/0401268.

math.GT↗
arXiv2014

On the Transverse Khovanov-Rozansky Homologies: Graded Module Structure and Stabilization

In arXiv:1308.3152, the author proved that the Khovanov-Rozansky homology $\mathcal{H}_N$ with potential $ax^{N+1}$ is an invariant for transverse links in the standard contact $3$-sphere. In the current paper, we study the $\mathbb{Z}_2 \oplus \mathbb{Z}^{\oplus 3}$-graded $\mathbb{Q}[a]$-module structure of $\mathcal{H}_N$, which leads to better understanding of the effect of stabilization on $\mathcal{H}_N$. As an application, we compute $\mathcal{H}_N$ for all transverse unknots.

math.GT↗
arXiv2013

Robust exponential attractors for the modified phase-field crystal equation

We consider the modified phase-field crystal (MPFC) equation that has recently been proposed by P. Stefanovic et al. This is a variant of the phase-field crystal (PFC) equation, introduced by K.-R. Elder et al., which is characterized by the presence of an inertial term $βϕ_{tt}$. Here $ϕ$ is the phase function standing for the number density of atoms and $β\geq 0$ is a relaxation time. The associated dynamical system for the MPFC equation with respect to the parameter $β$ is analyzed. More precisely, we establish the existence of a family of exponential attractors $\mathcal{M}_β$ that are Hölder continuous with respect to $β$.

math.AP↗
arXiv2013

Majorana Excitations, Spin- and Mass Currents on the Surface of Topological Superfluid $^3$He-B

The B-phase of superfluid $^3$He is a 3D time-reversal invariant (TRI) topological superfluid with an isotropic energy gap separating the ground-state and bulk continuum states. We report calculations of surface spectrum, spin- and mass current densities originating from the Andreev surface states for confined $^3$He-B. The surface states are Majorana Fermions with their spins polarized transverse to their direction of propagation along the surface. The negative energy states give rise to a ground-state helical spin current confined on the surface. The spectral functions reveal the subtle role of the spin-polarized surface states in relation to the ground-state spin current. By contrast, these states do not contribute to the $T=0$ mass current. Superfluid flow through a channel of confined $^3$He-B is characterized by the flow field, $p_s$. The flow field breaks 2D rotational symmetry and time reversal (T). However, the Bogoliubov-Nambu Hamiltonian remains invariant under $T$ combined with a 180 degree rotation about the normal to the film. As a result the B-phase in the presence of a superflow remains a topological phase with a gapless spectrum of Majorana modes on the surface. Thermal excitation of the Doppler shifted Majorana branches leads to a power law suppression of the superfluid mass current, providing a direct signature of the Majorana branches of surface excitations in the fully gapped 3D topological superfluid, $^3$He-B. Results are reported for the superfluid fraction (mass current) and helical spin current for confined $^3$He-B, including the temperature dependences, as well as dependences on confinement, interactions between quasiparticles and pressure.

cond-mat.supr-con↗
arXiv2013

SU(3) deconfining phase transition with finite volume corrections due to a confined exterior

Using the geometry of a double-layered torus we investigate the deconfining phase transition of pure SU(3) lattice gauge theory by Markov chain Monte Carlo simulations. In one layer, called "outside", the temperature is set below the deconfining temperature and in the other, called "inside", it is iterated to a pseudo-transition temperature. Lattice sizes are chosen in a range suggested by the physical volumes achieved in relativistic heavy ion collisions and both temperatures are kept close enough to stay in the SU(3) scaling region. Properties of the transition are studied as function of the volume for three outside temperatures. When compared with infinite volume extrapolations, small volume corrections of the deconfining temperature and width become competitive with those found by including quarks. Effective finite size scaling exponents of the specific and Polyakov loop susceptibilities are also calculated.

hep-lat↗
arXiv2013

Non-isothermal viscous Cahn--Hilliard equation with inertial term and dynamic boundary conditions

We consider a non-isothermal modified Cahn--Hilliard equation which was previously analyzed by M. Grasselli et al. Such an equation is characterized by an inertial term and a viscous term and it is coupled with a hyperbolic heat equation. The resulting system was studied in the case of no-flux boundary conditions. Here we analyze the case in which the order parameter is subject to a dynamic boundary condition. This assumption requires a more refined strategy to extend the previous results to the present case. More precisely, we first prove the well-posedness for solutions with bounded energy as well as for weak solutions. Then we establish the existence of a global attractor. Finally, we prove the convergence of any given weak solution to a single equilibrium by using a suitable Lojasiewicz--Simon inequality.

math.AP↗
arXiv2013

Mechanical properties and microdomain separation of fluid membranes with anchored polymers

The entropic effects of anchored polymers on biomembranes are studied using simulations of a meshless membrane model combined with anchored linear polymer chains. The bending rigidity and spontaneous curvature are investigated for anchored ideal and excluded-volume polymer chains. Our results are in good agreement with the previous theoretical predictions. It is found that the polymer reduces the line tension of membrane edges, as well as the interfacial line tension between membrane domains, leading to microdomain formation. Instead of the mixing of two phases as observed in typical binary fluids, densely anchored polymers stabilize small domains. A mean field theory is proposed for the edge line tension reduced by anchored ideal chains, which reproduces our simulation results well.

cond-mat.soft↗
arXiv2013

Projected and Hidden Markov Models for calculating kinetics and metastable states of complex molecules

Markov state models (MSMs) have been successful in computing metastable states, slow relaxation timescales and associated structural changes, and stationary or kinetic experimental observables of complex molecules from large amounts of molecular dynamics simulation data. However, MSMs approximate the true dynamics by assuming a Markov chain on a clusters discretization of the state space. This approximation is difficult to make for high-dimensional biomolecular systems, and the quality and reproducibility of MSMs has therefore been limited. Here, we discard the assumption that dynamics are Markovian on the discrete clusters. Instead, we only assume that the full phase- space molecular dynamics is Markovian, and a projection of this full dynamics is observed on the discrete states, leading to the concept of Projected Markov Models (PMMs). Robust estimation methods for PMMs are not yet available, but we derive a practically feasible approximation via Hidden Markov Models (HMMs). It is shown how various molecular observables of interest that are often computed from MSMs can be computed from HMMs / PMMs. The new framework is applicable to both, simulation and single-molecule experimental data. We demonstrate its versatility by applications to educative model systems, an 1 ms Anton MD simulation of the BPTI protein, and an optical tweezer force probe trajectory of an RNA hairpin.

physics.chem-ph↗
arXiv2013

Long-time dynamics of the nonhomogeneous incompressible flow of nematic liquid crystals

We study the long-time behavior of global strong solutions to a hydrodynamic system for nonhomogeneous incompressible nematic liquid crystal flows driven by two types of external forces in a smooth bounded domain in $\mathbb{R}^2$. For arbitrary large regular initial data with the initial density being away from vacuum, we prove the decay of the velocity field for both cases. Furthermore, for the case with asymptotically autonomous external force, we can prove the convergence of the density function and the director vector as time goes to infinity. Estimates on convergence rate are also provided.

math.AP↗
arXiv2013

Global solution to the three-dimensional compressible flow of liquid crystals

The Cauchy problem for the three-dimensional compressible flow of nematic liquid crystals is considered. Existence and uniqueness of the global strong solution are established in critical Besov spaces provided that the initial datum is close to an equilibrium state $(1,{\bf 0}, \hat{\d})$ with a constant vector $\hat{\d}\in S^2$. The global existence result is proved via the local well-posedness and uniform estimates for proper linearized systems with convective terms.

math.AP↗
arXiv2013

Well-posedness and longtime behavior for the modified phase-field crystal equation

We consider a modification of the so-called phase-field crystal (PFC) equation introduced by K.R. Elder et al. This variant has recently been proposed by P. Stefanovic et al. to distinguish between elastic relaxation and diffusion time scales. It consists of adding an inertial term (i.e. a second-order time derivative) into the PFC equation. The mathematical analysis of the resulting equation is more challenging with respect to the PFC equation, even at the well-posedness level. Moreover, its solutions do not regularize in finite time as in the case of PFC equation. Here we analyze the modified PFC (MPFC) equation endowed with periodic boundary conditions. We first prove the existence and uniqueness of a solution with initial data in a bounded energy space. This solution satisfies some uniform dissipative estimates which allow us to study the global longtime behavior of the corresponding dynamical system. In particular, we establish the existence of an exponential attractor. Then we demonstrate that any trajectory originating from the bounded energy phase space does converge to a unique equilibrium. This is done by means of a suitable version of the Łojasiewicz-Simon inequality. A convergence rate estimate is also given.

math.AP↗
arXiv2013

A colored sl(N)-homology for links in S^3

Fix an integer N>1. To each diagram of a link colored by 1,...,N, we associate a chain complex of graded matrix factorizations. We prove that the homotopy type of this chain complex is invariant under Reidemeister moves. When every component of the link is colored by 1, this chain complex is isomorphic to the chain complex defined by Khovanov and Rozansky in arXiv:math/0401268. The homology of this chain complex decategorifies to the Reshetikhin-Turaev sl(N) polynomial of links colored by exterior powers of the defining representation.

math.GT↗
arXiv2013

Effects of anchored flexible polymers on mechanical properties of model biomembranes

We have studied biomembranes with grafted polymer chains using a coarse-grained membrane simulation, where a meshless membrane model is combined with polymer chains. We focus on the polymer-induced entropic effects on mechanical properties of membranes. The spontaneous curvature and bending rigidity of the membranes increase with increasing polymer density. Our simulation results agree with the previous theoretical predictions.

physics.bio-ph↗
arXiv2013

Global weak solution and blow-up criterion of the general Ericksen-Leslie system for nematic liquid crystal flows

In this paper we investigate the three dimensional general Ericksen-Leslie (E--L) system with Ginzburg-Landau type approximation modeling nematic liquid crystal flows. First, by overcoming the difficulties from lack of maximum principle for the director equation and high order nonlinearities for the stress tensor, we prove existence of global-in-time weak solutions under physically meaningful boundary conditions and suitable assumptions on the Leslie coefficients, which ensures that the total energy of the E--L system is dissipated. Moreover, for the E--L system with periodic boundary conditions, we prove the local well-posedness of classical solutions under the so-called Parodi's relation and establish a blow-up criterion in terms of the temporal integral of both the maximum norm of the curl of the velocity field and the maximum norm of the gradient of the liquid crystal director field.

math.AP↗
arXiv2013

Intersections of SLE Paths: the double and cut point dimension of SLE

We compute the almost-sure Hausdorff dimension of the double points of chordal SLE_kappa for kappa > 4, confirming a prediction of Duplantier-Saleur (1989) for the contours of the FK model. We also compute the dimension of the cut points of chordal SLE_kappa for kappa > 4 as well as analogous dimensions for the radial and whole-plane SLE_kappa(rho) processes for kappa > 0. We derive these facts as consequences of a more general result in which we compute the dimension of the intersection of two flow lines of the formal vector field e^{ih/chi}, where h is a Gaussian free field and chi > 0, of different angles with each other and with the domain boundary.

math.PR↗
arXiv2013

Equivariant Khovanov-Rozansky Homology and Lee-Gornik Spectral Sequence

Lobb observed in [arXiv:1103.1412] that each equivariant sl(N) Khovanov-Rozansky homology over C[a] admits a standard decomposition of a simple form. In the present paper, we derive a formula for the corresponding Lee-Gornik spectral sequence in terms of this decomposition. Based on this formula, we give a simple alternative definition of the Lee-Gornik spectral sequence using exact couples. We also demonstrate that an equivariant sl(N) Khovanov-Rozansky homology over C[a] can be recovered from the corresponding Lee-Gornik spectral sequence via this formula. Therefore, these two algebraic invariants are equivalent and contain the same information about the link. As a byproduct of the exact couple construction, we generalize Lee's endomorphism on the rational Khovanov homology to a natural exterior algebra action on the sl(N) Khovanov-Rozansky homology. A numerical link invariant called torsion width comes up naturally in our work. It determines when the corresponding Lee-Gornik spectral sequence collapses and is bounded from above by the homological thickness of the sl(N) Khovanov-Rozansky homology. We use the torsion width to explain why the Lee spectral sequences of certain H-thick links collapse so fast.

math.GT↗
arXiv2013

Long-time behavior for a hydrodynamic model on nematic liquid crystal flows with asymptotic stabilizing boundary condition and external force

In this paper, we consider a simplified Ericksen-Leslie model for the nematic liquid crystal flow. The evolution system consists of the Navier-Stokes equations coupled with a convective Ginzburg-Landau type equation for the averaged molecular orientation. We suppose that the Navier-Stokes equations are characterized by a no-slip boundary condition and a time-dependent external force g(t), while the equation for the molecular director is subject to a time-dependent Dirichlet boundary condition h(t). We show that, in 2D, each global weak solution converges to a single stationary state when h(t) and g(t) converge to a time-independent boundary datum h_\infty and 0, respectively. Estimates on the convergence rate are also obtained. In the 3D case, we prove that global weak solutions are eventually strong so that results similar to the 2D case can be proven. We also show the existence of global strong solutions, provided that either the viscosity is large enough or the initial datum is close to a given equilibrium.

math.AP↗