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3.0

Jun 29, 2018
06/18

by
Hao Wu

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We derive the alternating arm exponents of critical Ising model. We obtain six different patterns of alternating boundary arm exponents which correspond to the boundary conditions $(\ominus\oplus)$, $(\ominus\text{free})$ and $(\text{free}\text{free})$, and the alternating interior arm exponents.

Topics: Probability, Mathematics

Source: http://arxiv.org/abs/1605.00985

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Sep 23, 2013
09/13

by
Hao Wu

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Motivated by the works of Krasner [arXiv:0801.4018] and Lobb [arXiv:1103.1412], we simplify the Khovanov-Rozansky chain complexes of open 2-braids. As an application, we show that, for a knot containing a "long" 2-braid, the sl(N) Rasmussen invariant of this knot depends linearly on the length of this 2-braid. We refine this result for (2,2k+1) cable knots and, as a simple corollary, compute the sl(2) Rasmussen invariants of (2,2k+1) cables of slice and amphicheiral knots.

Source: http://arxiv.org/abs/1111.3669v2

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Jul 20, 2013
07/13

by
Hao Wu

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This paper consists of three parts. First, we generalize the Jaeger Formula to express the Kauffman-Vogel graph polynomial as a state sum of the Murakami-Ohtsuki-Yamada graph polynomial. Then, we demonstrate that reversing the orientation and the color of a MOY graph along a simple circuit does not change the sl(N) Murakami-Ohtsuki-Yamada polynomial or the sl(N) homology of this MOY graph. In fact, reversing the orientation and the color of a component of a colored link only changes the sl(N)...

Source: http://arxiv.org/abs/1107.5333v2

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Sep 21, 2013
09/13

by
Hao Wu

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We classify up to isotopy the tight contact structures on small Seifert spaces with $e_0\neq0,-1,-2$. (The first version contains on the $e_0

Source: http://arxiv.org/abs/math/0402167v2

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Jul 20, 2013
07/13

by
Hao Wu

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We consider a nonlinear plate equation with thermal memory effects due to non-Fourier heat flux laws. First we prove the existence and uniqueness of global solutions as well as the existence of a global attractor. Then we use a suitable Lojasiewicz--Simon type inequality to show the convergence of global solutions to single steady states as time goes to infinity under the assumption that the nonlinear term $f$ is real analytic. Moreover, we provide an estimate on the convergence rate.

Source: http://arxiv.org/abs/0804.1806v2

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Sep 18, 2013
09/13

by
Hao Wu

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In this paper we consider the Cahn-Hilliard equation endowed with Wentzell boundary condition which is a model of phase separation in a binary mixture contained in a bounded domain with permeable wall. Under the assumption that the nonlinearity is analytic with respect to unknown dependent function, we prove the convergence of a global solution to an equilibrium as time goes to infinity by means of a suitable \L ojasiewicz-Simon type inequality with boundary term. Estimates of convergence rate...

Source: http://arxiv.org/abs/0705.3362v1

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Sep 18, 2013
09/13

by
Hao Wu

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We apply Heegaard-Floer homology theory to establish generalized slicing Bennequin inequalities closely related to a recent result of T. Mrowka and Y. Rollin proved using Seiberg-Witten monopoles.

Source: http://arxiv.org/abs/math/0505279v2

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Jul 20, 2013
07/13

by
Hao Wu

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We describe all random sets that satisfy the radial conformal restriction property, therefore providing the analogue in the radial case of results of Lawler, Schramm and Werner in the chordal case.

Source: http://arxiv.org/abs/1304.5712v1

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3.0

Jun 30, 2018
06/18

by
Hao Wu

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We investigate a non-isothermal diffuse-interface model that describes the dynamics of two-phase incompressible flows with thermo-induced Marangoni effect. The governing PDE system consists of the Navier--Stokes equations coupled with convective phase-field and energy transport equations, in which the surface tension, fluid viscosity and thermal diffusivity are temperature dependent functions. First, we establish the existence and uniqueness of local strong solutions when the spatial dimension...

Topics: Mathematics, Analysis of PDEs

Source: http://arxiv.org/abs/1408.5186

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3.0

Jun 30, 2018
06/18

by
Hao Wu

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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.

Topics: Symplectic Geometry, Mathematics, Geometric Topology

Source: http://arxiv.org/abs/1403.6083

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Jun 29, 2018
06/18

by
Hao Wu

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We derive the arm exponents of SLE$_{\kappa}$ for $\kappa\in (4,8)$ and explain how to combine them with the convergence of the interface to obtain the arm exponents of critical FK-Ising model. We obtain six different patterns of boundary arm exponents and three different patterns of interior arm exponents of critical FK-Ising model.

Topics: Probability, Mathematical Physics, Mathematics

Source: http://arxiv.org/abs/1604.06639

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46

Sep 18, 2013
09/13

by
Hao Wu

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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...

Source: http://arxiv.org/abs/1211.6732v3

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Sep 20, 2013
09/13

by
Hao Wu

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We construct an equivariant colored sl(N)-homology for links, which generalizes both the colored sl(N)-homology defined by the author and the equivariant sl(N)-homology defined by Krasner. The construction is a straightforward generalization of that of the colored sl(N)-homology. The proof of invariance is based on a simple observation which allows us to translate the proof of the invariance of the colored sl(N)-homology into the new setting. As an application, we prove that deformations over C...

Source: http://arxiv.org/abs/1002.2662v2

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56

Sep 20, 2013
09/13

by
Hao Wu

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We prove the quantum filtration on the Khovanov-Rozansky link cohomology H_p with a general degree (n+1) monic potential polynomial p(x) is invariant under Reidemeister moves, and construct a spectral sequence converging to H_p that is invariant under Reidemeister moves, whose E_1 term is isomorphic to the Khovanov-Rozansky sl(n)-cohomology H_n. Then we define a generalization of the Rasmussen invariant, and study some of its properties. We also discuss relations between upper bounds of the...

Source: http://arxiv.org/abs/math/0612406v2

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Sep 21, 2013
09/13

by
Hao Wu

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This paper is concerned with the asymptotic behavior of the solution to the semilinear parabolic equation with dynamical boundary condition. Our main goal is to prove the convergence of a global solution to an equilibrium as time goes to infinity by means of a suitable \L ojasiewicz--Simon type inequality.

Source: http://arxiv.org/abs/1204.6018v1

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Sep 21, 2013
09/13

by
Hao Wu

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We establish some inequalities about the Khovanov-Rozansky cohomologies of braids. These give new upper bounds of the self-linking numbers of transversal links in standard contact $S^3$ which is sharper than the well known bound given by the HOMFLY polynomial. We also introduce a sequence of transversal link invariants, and discuss some of their properties.

Source: http://arxiv.org/abs/math/0508064v3

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3.0

Jun 30, 2018
06/18

by
Hao Wu

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In arXiv:math/0508510, Rasmussen observed that the Khovanov-Rozansky homology of a link is a finitely generated module over the polynomial ring generated by the components of this link. In the current paper, we study the module structure of the middle HOMFLYPT homology, especially the Betti numbers of this module. For each link, these Betti numbers are supported on a finite subset of $\mathbb{Z}^4$. One can easily recover from these Betti numbers the Poincar\'e polynomial of the middle HOMFLYPT...

Topics: Geometric Topology, Mathematics

Source: http://arxiv.org/abs/1703.07257

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Sep 23, 2013
09/13

by
Hao Wu

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We study a simplified system of the original Ericksen--Leslie equations for the flow of nematic liquid crystals. This is a coupled non-parabolic dissipative dynamic system. We show the convergence of global classical solutions to single steady states as time goes to infinity (uniqueness of asymptotic limit) by using the \L ojasiewicz--Simon approach. Moreover, we provide an estimate on the convergence rate. Finally, we discuss some possible extensions of the results to certain generalized...

Source: http://arxiv.org/abs/0904.0390v1

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Jun 28, 2018
06/18

by
Hao Wu

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It has been quite a long time since AI researchers in the field of computer science stop talking about simulating human intelligence or trying to explain how brain works. Recently, represented by deep learning techniques, the field of machine learning is experiencing unprecedented prosperity and some applications with near human-level performance bring researchers confidence to imply that their approaches are the promising candidate for understanding the mechanism of human brain. However apart...

Topics: Artificial Intelligence, Computing Research Repository

Source: http://arxiv.org/abs/1509.08891

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Jun 28, 2018
06/18

by
Hao Wu

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We determine the cycle packing number of a directed graph using elementary projective algebraic geometry. Our idea is rooted in the Khovanov-Rozansky theory. In fact, using the Khovanov-Rozansky homology of a graph, we also obtain algebraic methods of detecting directed and undirected cycles containing a particular vertex or edge.

Topics: Combinatorics, Commutative Algebra, Mathematics

Source: http://arxiv.org/abs/1508.07337

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47

Sep 18, 2013
09/13

by
Hao Wu

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We generalize the Morton-Franks-Williams inequality to the colored $\mathfrak{sl}(N)$ link homology defined in arXiv:0907.0695, which gives infinitely many new bounds for the braid index and the self linking number. A key ingredient of our proof is a composition product for the general MOY graph polynomial, which generalizes that of Wagner arXiv:math/0702230v1.

Source: http://arxiv.org/abs/1102.0586v1

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62

Sep 21, 2013
09/13

by
Hao Wu

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We generalize the works of Lee [arXiv:math/0210213v3] and Gornik [arXiv:math/0402266v2] to construct a basis for generic deformations of the colored sl(N)-homology defined in [arXiv:1002.2662v1]. As applications, we construct non-degenerate pairings and co-pairings which lead to dualities of generic deformations of the colored sl(N)-homology. We also define and study colored sl(N)-Rasmussen invariants. Among other things, we observe that these invariants vanish on amphicheiral knots and discuss...

Source: http://arxiv.org/abs/1011.2254v2

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4.0

Jun 30, 2018
06/18

by
Hao Wu

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We review some of the results related to conformal restriction: the chordal case and the radial case. We describe Brownian intersection exponents, conformal restriction property and SLE, and study their properties.

Topics: Probability, Mathematics

Source: http://arxiv.org/abs/1409.1898

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5.0

Jun 29, 2018
06/18

by
Hao Wu

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We prove that the degree of the Hilbert polynomial of the HOMFLYPT homology of a closed braid $B$ is $l-1$, where $l$ is the number of components of $B$. This controls the growth of the HOMFLYPT homology with respect to its polynomial grading. The Hilbert polynomial also reveals a link polynomial hidden in the HOMFLYPT polynomial.

Topics: Geometric Topology, Mathematics

Source: http://arxiv.org/abs/1604.05222

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5.0

Jun 29, 2018
06/18

by
Hao Wu

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We consider the planar Ising model in rectangle $(\Omega; x^L, x^R, y^R, y^L)$ with alternating boundary condition: $\ominus$ along $(x^Lx^R)$ and $(y^Ry^L)$, $\xi^R\in\{\oplus, \text{free}\}$ along $(x^Ry^R)$, and $\xi^L\in\{\oplus, \text{free}\}$ along $(y^Lx^L)$. We prove that the interface of critical Ising model with these boundary conditions converges to the so-called hypergeometric SLE$_3$. The method developed in this paper does not require constructing new holomorphic observable and...

Topics: Probability, Mathematics

Source: http://arxiv.org/abs/1610.06113

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3.0

Jun 30, 2018
06/18

by
Hao Wu

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This article pertains to the classification of pairs of simple random curves with conformal Markov property and symmetry. Such pairs correspond to scaling limits of pairs of interfaces in critical lattice model with alternating boundary conditions in topological rectangles. Conformal Markov property and symmetry single out a two-parameter family of random curves: Hypergeometric SLE, denoted by hSLE$_{\kappa}(\nu)$ for $\kappa\in (0,4]$ and $\nu

Topics: Probability, Mathematics

Source: http://arxiv.org/abs/1703.02022

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56

Sep 23, 2013
09/13

by
Hao Wu

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The Reshetikhin-Turaev sl(N) polynomial of links colored by wedge powers of the defining representation has been categorified via several different approaches. Here, we give a concise introduction to the categorification using matrix factorizations, which is a direct generalization of the Khovanov-Rozansky homology. Full details of the construction are given in [arXiv:0907.0695]. We also briefly review deformations and applications of this categorification given in [arXiv:1002.2662,...

Source: http://arxiv.org/abs/1110.2076v2

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62

Sep 22, 2013
09/13

by
Hao Wu

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We use the Ozsv\'ath-Szab\'o contact invariants to distinguish between tight contact structures obtained by Legendrian surgeries on stabilized Legendrian links in tight contact 3-manifolds. We also discuss the implication of our result on the tight contact structures on the Brieskon homology spheres $-\Sigma(2,3,6n-1)$.

Source: http://arxiv.org/abs/math/0501074v7

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57

Sep 22, 2013
09/13

by
Hao Wu

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We discuss the relations between the $e_0$ invariants of a small Seifert space and the twisting numbers of Legendrian vertical circles in it.

Source: http://arxiv.org/abs/math/0310034v2

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25

Jun 27, 2018
06/18

by
Hao Wu

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Nowadays, represented by Deep Learning techniques, the field of machine learning is experiencing unprecedented prosperity and its influence is demonstrated in academia, industry and civil society. "Intelligent" has become a label which could not be neglected for most applications; celebrities and scientists also warned that the development of full artificial intelligence may spell the end of the human race. It seems that the answer to building a computer system that could...

Topics: Artificial Intelligence, Computing Research Repository

Source: http://arxiv.org/abs/1505.04813

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63

Sep 23, 2013
09/13

by
Hao Wu

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We establish a relation between several easy-to-calculate numerical invariants of generic knots in $\mathbb{H}^2 \times S^1$, and use it to give a new method of computing the Thurston-Bennequin number of Legendrian knots in the tight contact $\mathbb{R}^3$.

Source: http://arxiv.org/abs/math/0312224v2

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Sep 18, 2013
09/13

by
Hao Wu

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We use the spanning tree model for Khovanov homology to study Legendrian links. This leads to an alternative proof for Ng's Khovanov bound for the Thurston-Bennequin number and to both a necessary and a sufficient condition for this bound to be sharp.

Source: http://arxiv.org/abs/math/0605630v4

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56

Sep 19, 2013
09/13

by
Hao Wu

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We review Bennequin type inequalities established using various versions of the Khovanov-Rozansky cohomology. Then we give a new proof of a Bennequin type inequality established by the author, and derive new Bennequin type inequalities for knots using Gornik's version of the Khovanov-Rozansky cohomology, which generalize those established by Shumakovitch, Plamenevskaya and Kawamura using the Rasmussen invariant.

Source: http://arxiv.org/abs/math/0703210v2

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Jun 25, 2018
06/18

by
Hao Wu

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When studying a metastable dynamical system, a prime concern is how to decompose the phase space into a set of metastable states. Unfortunately, the metastable state decomposition based on simulation or experimental data is still a challenge. The most popular and simplest approach is geometric clustering which is developed based on the classical clustering technique. However, the prerequisites of this approach are: (1) data are obtained from simulations or experiments which are in global...

Topics: Learning, Computing Research Repository, Numerical Analysis, Computing Research Repository, Systems...

Source: http://arxiv.org/abs/1501.00125

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5.0

Jun 21, 2021
06/21

by
Hao Wu (whowechina)

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JGAurora Z603S does not support TPU printing. If you insist to print TPU, filament will bend and come out before it goes into the hot end. So you need something to keep it straight before it is pushed into the hot end. I designed this thing by TinkerCad. It is a permanent add-on to the original Z603S extruder, the small hole is for the filament and the big hole is for the spring, you'll figure out how it works. This addon works perfectly on my Z603S, compatible for PLA and TPU. JGAurora may...

Topics: stl, thingiverse, 3D Printer Accessories

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42

Sep 23, 2013
09/13

by
Maurizio Grasselli; Hao Wu

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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...

Source: http://arxiv.org/abs/1111.1285v3

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Jun 28, 2018
06/18

by
Ellen Powell; Hao Wu

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We study the level lines of a Gaussian free field in a planar domain with general boundary data $F$. We show that the level lines exist as continuous curves under the assumption that $F$ is regulated (i.e., admits left and right limits at every point), and satisfies certain inequalities. Moreover, these level lines are a.s. determined by the field. This allows us to define and study a generalization of the SLE$_4(\underline{\rho})$ process, now with a continuum of force points. A crucial...

Topics: Mathematics, Probability

Source: http://arxiv.org/abs/1509.02462

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Sep 23, 2013
09/13

by
Hao Wu; Frank Noé

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When studying high-dimensional dynamical systems such as macromolecules, quantum systems and polymers, a prime concern is the identification of the most probable states and their stationary probabilities or free energies. Often, these systems have metastable regions or phases, prohibiting to estimate the stationary probabilities by direct simulation. Efficient sampling methods such as umbrella sampling, metadynamics and conformational flooding have developed that perform a number of simulations...

Source: http://arxiv.org/abs/1212.6711v1

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3.0

Jun 29, 2018
06/18

by
Hao Wu; Dapeng Zhan

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We derive boundary arm exponents for SLE. Combining with the convergence of critical lattice models to SLE, these exponents would give the alternating half-plane arm exponents for the corresponding lattice models.

Topics: Probability, Mathematics

Source: http://arxiv.org/abs/1606.05998

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Sep 21, 2013
09/13

by
Antonio Segatti; Hao Wu

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We consider a hydrodynamic system that models the Smectic-A liquid crystal flow. The model consists of the Navier-Stokes equation for the fluid velocity coupled with a fourth-order equation for the layer variable $\vp$, endowed with periodic boundary conditions. We analyze the long-time behavior of the solutions within the theory of infinite-dimensional dissipative dynamical systems. We first prove that in 2D, the problem possesses a global attractor $\mathcal{A}$ in certain phase space. Then...

Source: http://arxiv.org/abs/1011.0358v1

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Sep 24, 2013
09/13

by
Wendelin Werner; Hao Wu

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We study some conformally invariant dynamic ways to construct the Conformal Loop Ensembles with simple loops introduced in earlier papers by Sheffield, and by Sheffield and Werner. One outcome is a conformally invariant way to measure a distance of a CLE(4) loop to the boundary "within" the CLE(4), when one identifies all points of each loop.

Source: http://arxiv.org/abs/1112.1211v2

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Jun 30, 2018
06/18

by
Xianpeng Hu; Hao Wu

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We consider the Cauchy problem for incompressible viscoelastic fluids in the whole space $\mathbb{R}^d$ ($d=2,3$). By introducing a new decomposition via Helmholtz's projections, we first provide an alternative proof on the existence of global smooth solutions near equilibrium. Then under additional assumptions that the initial data belong to $L^1$ and their Fourier modes do not degenerate at low frequencies, we obtain the optimal $L^2$ decay rates for the global smooth solutions and their...

Topics: Mathematics, Analysis of PDEs

Source: http://arxiv.org/abs/1411.0518

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40

Sep 21, 2013
09/13

by
Hao Wu; Jie Jiang

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In this paper, we study the Cauchy problem of a time-dependent drift-diffusion-Poisson system for semiconductors. Existence and uniqueness of global weak solutions are proven for the system with a higher-order nonlinear recombination-generation rate R. We also show that the global weak solution will converge to a unique equilibrium as time tends to infinity.

Source: http://arxiv.org/abs/1108.5844v2

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Sep 23, 2013
09/13

by
Jason Miller; Hao Wu

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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...

Source: http://arxiv.org/abs/1303.4725v1

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6.0

Jun 29, 2018
06/18

by
Gábor Pete; Hao Wu

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We construct an aggregation process of chordal SLE(\kappa) excursions in the unit disk, starting from the boundary, growing towards all inner points simultaneously, invariant under all conformal self-maps of the disk. We prove that this conformal growth process of excursions, abbreviated as CGE(\kappa), exists iff \kappa\in [0,4), and that it does not create additional fractalness: the Hausdorff dimension of the closure of all the SLE(\kappa) arcs attached is 1+\kappa/8 almost surely. We...

Topics: Probability, Mathematical Physics, Mathematics

Source: http://arxiv.org/abs/1601.05713

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59

Sep 21, 2013
09/13

by
Hao Wu; Xiang Xu

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In this paper we study the well-posedness and long-time dynamics of a diffuse-interface model for the mixture of two viscous incompressible Newtonian fluids with thermo-induced Marangoni effects. The governing system consists of modified Navier--Stokes equations coupled with phase-field and energy transport equations. We first derive an energy inequality that illustrates the dissipative nature of the system under the assumption that the initial temperature variation is properly small. Then we...

Source: http://arxiv.org/abs/1204.6013v2

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63

Sep 23, 2013
09/13

by
Xianpeng Hu; Hao Wu

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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...

Source: http://arxiv.org/abs/1202.4512v2

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3.0

Jun 30, 2018
06/18

by
Eveliina Peltola; Hao Wu

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This article pertains to the classification of multiple Schramm-Loewner evolutions (SLE). We construct the pure partition functions of multiple SLE$_\kappa$ with $\kappa\in (0,4]$ and relate them to certain extremal multiple SLE measures, thus verifying a conjecture from [BBK05, KP16]. We prove that the two approaches to construct multiple SLEs --- the global, configurational construction of [KL07, Law09a] and the local, growth process construction of [BBK05, Dub07, Gra07, KP16] --- agree. The...

Topics: Probability, Mathematical Physics, Mathematics

Source: http://arxiv.org/abs/1703.00898

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Sep 23, 2013
09/13

by
Hao Wu; Xiang Xu

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In this paper, we study a hydrodynamic system modeling the deformation of vesicle membranes in incompressible viscous fluids. The system consists of the Navier-Stokes equations coupled with a fourth order phase-field equation. In the three dimensional case, we prove the existence/uniqueness of local strong solutions for arbitrary initial data as well as global strong solutions under the large viscosity assumption. We also establish some regularity criteria in terms of the velocity for local...

Source: http://arxiv.org/abs/1202.4869v2

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Jul 20, 2013
07/13

by
Hao Wu; Hiroshi Noguchi

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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.

Source: http://arxiv.org/abs/1304.0651v1