Showing papers in "Science China-mathematics in 2018"

The Brezis-Nirenberg type critical problem for the nonlinear Choquard equation

Abstract: We establish some existence results for the Brezis-Nirenberg type problem of the nonlinear Choquard equation $$ - \Delta u = \left( {\int_\Omega {\frac{{{{\left| {u\left( y \right)} \right|}^{2_\mu ^*}}}}{{{{\left| {x - y} \right|}^\mu }}}dy} } \right){\left| u \right|^{2_\mu ^* - 2}}u + \lambda uin\Omega ,$$ , where Ω is a bounded domain of R N with Lipschitz boundary, λ is a real parameter, N ≥ 3, $$2_\mu ^* = \left( {2N - \mu } \right)/\left( {N - 2} \right)$$ is the critical exponent in the sense of the Hardy-Littlewood-Sobolev inequality.

Exponential convergence of the deep neural network approximation for analytic functions

Abstract: We prove that for analytic functions in low dimension, the convergence rate of the deep neural network approximation is exponential.

Tensor absolute value equations

Abstract: This paper is concerned with solving some structured multi-linear systems, which are called tensor absolute value equations This kind of absolute value equations is closely related to tensor complementarity problems and is a generalization of the well-known absolute value equations in the matrix case We prove that tensor absolute value equations are equivalent to some special structured tensor complementary problems Some sufficient conditions are given to guarantee the existence of solutions for tensor absolute value equations We also propose a Levenberg-Marquardt-type algorithm for solving some given tensor absolute value equations and preliminary numerical results are reported to indicate the efficiency of the proposed algorithm

Purely Hom-Lie bialgebras

Abstract: In this paper, we first show that there is a Hom-Lie algebra structure on the set of $(\sigma,\sigma)$-derivations of an associative algebra. Then we construct the dual representation of a representation of a Hom-Lie algebra. We introduce the notions of a Manin triple for Hom-Lie algebras and a purely Hom-Lie bialgebra. Using the coadjoint representation, we show that there is a one-to-one correspondence between Manin triples for Hom-Lie algebras and purely Hom-Lie bialgebras. Finally, we study coboundary purely Hom-Lie bialgebras and construct solutions of the classical Hom-Yang-Baxter equations in some special Hom-Lie algebras using Hom-$\mathcal~O$-operators.

Reflected solutions of backward stochastic differential equations driven by ${\boldsymbol~G}$-Brownian motion

Abstract: In this paper, we study the reflected solutions of one-dimensional backward stochastic differential equations driven by $G$-Brownian motion. The reflection keeps the solution above a given stochastic process. In order to derive the uniqueness of reflected $G$-BSDEs, we apply a “martingale condition" instead of the Skorohod condition. Similar to the classical case, we prove the existence by approximation via penalization. We then give some applications including a generalized Feynman-Kac formula of an obstacle problem for fully nonlinear partial differential equation and option pricing of American types under volatility uncertainty.

Stochastic Hamiltonian flows with singular coefficients

Abstract: In this paper, we study the following stochastic Hamiltonian system in ℝ2d (a second order stochastic differential equation): $$d{\dot X_t} = b({X_t},{\dot X_t})dt + \sigma ({X_t},{\dot X_t})d{W_t},({X_0},{\dot X_0}) = (x,v) \in \mathbb{R}^{2d},$$ where b(x; v) : ℝ2d → ℝd and σ(x; v): ℝ2d → ℝd ⊗ ℝd are two Borel measurable functions. We show that if σ is bounded and uniformly non-degenerate, and b ∈ H 2/3,0 and ∇σ ∈ Lp for some p > 2(2d+1), where H , is the Bessel potential space with differentiability indices α in x and β in v, then the above stochastic equation admits a unique strong solution so that (x, v) ↦ Zt(x, v) := (Xt, Ẋt)(x, v) forms a stochastic homeomorphism flow, and (x, v) ↦ Zt(x, v) is weakly differentiable with ess.supx, v E(supt∈[0, T] |∇Zt(x, v)|q) < ∞ for all q ⩾ 1 and T ⩾ 0. Moreover, we also show the uniqueness of probability measure-valued solutions for kinetic Fokker-Planck equations with rough coefficients by showing the well-posedness of the associated martingale problem and using the superposition principle established by Figalli (2008) and Trevisan (2016).

Hyperbolic-parabolic deformations of rational maps

Abstract: We develop a Thurston-like theory to characterize geometrically finite rational maps, and then apply it to study pinching and plumbing deformations of rational maps. We show that under certain conditions the pinching path converges uniformly and the quasiconformal conjugacy converges uniformly to a semi-conjugacy from the original map to the limit. Conversely, every geometrically finite rational map with parabolic points is the landing point of a pinching path for any prescribed plumbing combinatorics.

Convergence of ground state solutions for nonlinear Schrödinger equations on graphs

Abstract: We consider the nonlinear Schrodinger equation -Δu + (λa(x) + 1)u = |u|p-1u on a locally finite graph G = (V,E). We prove via the Nehari method that if a(x) satisfies certain assumptions, for any λ > 1, the equation admits a ground state solution uλ. Moreover, as λ → 1, the solution uλ converges to a solution of the Dirichlet problem -Δu+u = |u|p-1u which is defined on the potential well Ω. We also provide a numerical experiment which solves the equation on a finite graph to illustrate our results.

Boundedness of Hausdorff operators on Lebesgue spaces and Hardy spaces

Abstract: In this paper, we study the boundedness of the Hausdorff operator Hϕ on the real line ℝ. First, we start with an easy case by establishing the boundedness of the Hausdorff operator on the Lebesgue space Lp(ℝ) and the Hardy space H1(ℝ). The key idea is to reformulate Hϕ as a Calderon-Zygmund convolution operator, from which its boundedness is proved by verifying the Hormander condition of the convolution kernel. Secondly, to prove the boundedness on the Hardy space Hp(ℝ) with 0 < p < 1; we rewrite the Hausdorff operator as a singular integral operator with the non-convolution kernel. This novel reformulation, in combination with the Taibleson-Weiss molecular characterization of Hp(ℝ) spaces, enables us to obtain the desired results. Those results significantly extend the known boundedness of the Hausdorff operator on H1(ℝ).

Holographic software for quantum networks

Abstract: We introduce a pictorial approach to quantum information, called holographic software. Our software captures both algebraic and topological aspects of quantum networks. It yields a bi-directional dictionary to translate between a topological approach and an algebraic approach. Using our software, we give a topological simulation for quantum networks. The string Fourier transform (SFT) is our basic tool to transform product states into states with maximal entanglement entropy. We obtain a pictorial interpretation of Fourier transformation, of measurements, and of local transformations, including the n-qudit Pauli matrices and their representation by Jordan-Wigner transformations. We use our software to discover interesting new protocols for multipartite communication. In summary, we build a bridge linking the theory of planar para algebras with quantum information.

W-entropy formulas on super Ricci flows and Langevin deformation on Wasserstein space over Riemannian manifolds

Abstract: In this survey paper, we give an overview of our recent works on the study of the W-entropy for the heat equation associated with the Witten Laplacian on super-Ricci flows and the Langevin deformation on the Wasserstein space over Riemannian manifolds. Inspired by Perelman’s seminal work on the entropy formula for the Ricci flow, we prove the W-entropy formula for the heat equation associated with the Witten Laplacian on n-dimensional complete Riemannian manifolds with the CD(K,m)-condition, and the W-entropy formula for the heat equation associated with the time-dependent Witten Laplacian on n-dimensional compact manifolds equipped with a (K,m)-super Ricci flow, where K ∈ R and m ∈ [n,∞]. Furthermore, we prove an analogue of the W-entropy formula for the geodesic flow on the Wasserstein space over Riemannian manifolds. Our result improves an important result due to Lott and Villani (2009) on the displacement convexity of the Boltzmann-Shannon entropy on Riemannian manifolds with non-negative Ricci curvature. To better understand the similarity between above two W-entropy formulas, we introduce the Langevin deformation of geometric flows on the tangent bundle over the Wasserstein space and prove an extension of the W-entropy formula for the Langevin deformation. We also make a discussion on the W-entropy for the Ricci flow from the point of view of statistical mechanics and probability theory. Finally, to make this survey more helpful for the further development of the study of the W-entropy, we give a list of problems and comments on possible progresses for future study on the topic discussed in this survey.

Indefinite stochastic linear-quadratic optimal control problems with random jumps and related stochastic Riccati equations

Abstract: We discuss the stochastic linear-quadratic (LQ) optimal control problem with Poisson processes under the indefinite case. Based on the wellposedness of the LQ problem, the main idea is expressed by the definition of relax compensator that extends the stochastic Hamiltonian system and stochastic Riccati equation with Poisson processes (SREP) from the positive definite case to the indefinite case. We mainly study the existence and uniqueness of the solution for the stochastic Hamiltonian system and obtain the optimal control with open-loop form. Then, we further investigate the existence and uniqueness of the solution for SREP in some special case and obtain the optimal control in close-loop form.

Efficient contraflow algorithms for quickest evacuation planning

Abstract: The optimization models and algorithms with their implementations on flow over time problems have been an emerging field of research because of largely increasing human-created and natural disasters worldwide. For an optimal use of transportation network to shift affected people and normalize the disastrous situation as quickly and efficiently as possible, contraflow configuration is one of the highly applicable operations research (OR) models. It increases the outbound road capacities by reversing the direction of arcs towards the safe destinations that not only minimize the congestion and increase the flow but also decrease the evacuation time significantly.In this paper, we sketch the state of quickest flow solutions and solve the quickest contraflow problem with constant transit times on arcs proving that the problem can be solved in strongly polynomial time $~O(nm^2~(\log~n)^2)~$, where $~n~$ and $~m~$ are number of nodes and number of arcs, respectively in the network. This contraflow solution has the same computational time bound as that of the best min-cost flow solution. Moreover, we also introduce the contraflow approach with load dependent transit times on arcs and present an efficient algorithm to solve the quickest contraflow problem approximately. Supporting the claim, our computational experiments on Kathmandu road network and on randomly generated instances perform very well matching the theoretical results. For a sufficiently large number of evacuees, about double flow can be shifted with the same evacuation time and about half time is sufficient to push the given flow value with contraflow reconfiguration.

Full characterization of the embedding relations between bm$\alpha$-modulation spaces

Abstract: In this paper, we consider the embedding relations between any two $\alpha$-modulation spaces. Based on an observation that the $\alpha$-modulationspace with smaller $\alpha$ can be regarded as a corresponding decomposition spaceassociated with $\alpha$-covering for larger $\alpha$, we give a complete characterizationof the Fourier multipliers between $\alpha$-modulation spaces with different$\alpha$. Then we establish a full version of optimal embedding relationsbetween $\alpha$-modulation spaces.

A note on the ill-posedness of shear flow for the MHD boundary layer equations

Abstract: For the two-dimensional Magnetohydrodynamics (MHD) boundary layer system, it has been shown that the non-degenerate tangential magnetic field leads to the well-posedness in Sobolevspaces and high Reynolds number limits without any monotonicity condition on the velocity field in our previous worksThis paper aims to show that sufficient degeneracy in thetangential magnetic field at a non-degenerate critical point of the tangential velocity field of shear flow indeed yields instability as for the classical Prandtl equations without magnetic field studied by Gerard-Varet and Dormy (2010) This partially shows the necessity of thenon-degeneracy in the tangential magnetic field for the stability of the boundary layer of MHD in 2D at least in Sobolev spaces

Nonexistence of maximizers for the functional of the centroaffine Minkowski problem

Abstract: The centroaffine Minkowski problem is studied, which is the critical case of the $L_p$-Minkowski problem. It admits a variational structure that plays an important role in studying the existence of solutions. In this paper, we find that there is generally no maximizer of the corresponding functional for the centroaffine Minkowski problem.

Fekete and Szegö problem in one and higher dimensions

Abstract: In this paper, we establish the Fekete and Szegoinequality for a class of holomorphic functions in the unit disk,and then we extend this result to a class of holomorphic mappings on the unit ball in a complex Banach space or on the unit polydisk in $\mathbb{C}^n$.

A general truncated regularization framework for contrast-preserving variational signal and image restoration: Motivation and implementation

Abstract: Variational methods have become an important kind of methods in signal and image restoration—a typical inverse problem. One important minimization model consists of the squared $\ell_2$ data fidelity (corresponding to Gaussian noise) and a regularization term constructed by a potential function composed of first order difference operators. It is well known that total variation (TV) regularization, although achieved great successes, suffers from a contrast reduction effect. Using a typical signal, we show that, actually all convex regularizers and most nonconvex regularizers have this effect. With this motivation, we present a general truncated regularization framework. The potential function is a truncation of existing nonsmooth potential functions and thus flat on $(\tau,+\infty)$ for some positive $\tau$. Some analysis in 1D theoretically demonstrate the good contrast-preserving ability of the framework. We also give optimization algorithms with convergence verification in 2D, where global minimizers of each subproblem (either convex or nonconvex) are calculated. Experiments numerically show the advantages of the framework.

Existence of solutions to a class of Kazdan-Warner equations on compact Riemannian surface

Abstract: Let (Σ, g) be a compact Riemannian surface without boundary and λ1(Σ) be the first eigenvalue of the Laplace-Beltrami operator Δ g . Let h be a positive smooth function on Σ. Define a functional $${J_{\alpha ,\beta }}\left( u \right) = \frac{1}{2}\int_\sum {\left( {{{\left| {{ abla _g}u} \right|}^2} - \alpha {u^2}} \right)} d{v_g} - \beta \log \int_\sum {h{e^u}} d{v_g}$$ on a function space H = {u ∈ W1,2(Σ): ∫Σ udv g = 0}. If α 8π, then for any α ∈ R, there holds infu∈HJα,β(u) = −∞. Moreover, we consider the same problem in the case that α is large, where higher order eigenvalues are involved.

Classification of certain qualitative properties of solutions for the quasilinear parabolic equations

Abstract: In this paper, we mainly consider the initial boundary problem for a quasilinear parabolic equation $${u_t} - div\left( {{{\left| { abla u} \right|}^{p - 2}} abla u} \right) = - {\left| u \right|^{\beta - 1}}u + \alpha {\left| u \right|^{q - 2}}u,$$ where p > 1; β > 0, q ≥ 1 and α > 0. By using Gagliardo-Nirenberg type inequality, the energy method and comparison principle, the phenomena of blowup and extinction are classified completely in the different ranges of reaction exponents.

Data-based prediction and causality inference of nonlinear dynamics

Abstract: Natural systems are typically nonlinear and complex, and it is of great interest to be able to reconstruct a system in order to understand its mechanism, which cannot only recover nonlinear behaviors but also predict future dynamics. Due to the advances of modern technology, big data becomes increasingly accessible and consequently the problem of reconstructing systems from measured data or time series plays a central role in many scientific disciplines. In recent decades, nonlinear methods rooted in state space reconstruction have been developed, and they do not assume any model equations but can recover the dynamics purely from the measured time series data. In this review, the development of state space reconstruction techniques will be introduced and the recent advances in systems prediction and causality inference using state space reconstruction will be presented. Particularly, the cutting-edge method to deal with short-term time series data will be focused on. Finally, the advantages as well as the remaining problems in this field are discussed.

On De Giorgi’s conjecture: Recent progress and open problems

Abstract: In 1979, De Giorgi conjectured that the only bounded monotone solutions to the Allen-Cahn equation $$\Delta{u}+u-{u^3}=0\;\text{in}\;\mathbb{R}^N$$ are one-dimensional. This conjecture and its connection with minimal surfaces and Toda systems are the subject of this survey article.

Weak-strong uniqueness for the compressible Navier-Stokes equations with a hard-sphere pressure law

Abstract: We consider the Navier-Stokes equations with a pressure function satisfying a hard-sphere law That means the pressure, as a function of the density, becomes infinite when the density approaches a finite critical value Under some structural constraints imposed on the pressure law, we show a weak-strong uniqueness principle in periodic spatial domains The method is based on a modified relative entropy inequality for the system The main difficulty is that the pressure potential associated with the internal energy of the system is largely dominated by the pressure itself in the area close to the critical density As a result, several terms appearing in the relative energy inequality cannot be controlled by the total energy

On the character of certain tilting modules

Abstract: Let G be a semisimple group over an algebraically closed field of characteristic p > 0. We give a (partly conjectural) closed formula for the character of many indecomposable tilting rational G-modules assuming that p is large.

Gorenstein projective modules and Frobenius extensions

Abstract: We prove that for a Frobenius extension, if a module over the extension ring is Gorenstein projective, then its underlying module over the base ring is Gorenstein projective; the converse holds if the Frobenius extension is either left-Gorenstein or separable (e.g., the integral group ring extension $\mathbb{Z}\subset~\mathbb{Z}G$).Moreover, for the Frobenius extension $R\subset~A=R[x]/(x^2)$, we show that: a graded $A$-module is Gorenstein projective in $\mathrm{GrMod}(A)$, if and only if its ungraded $A$-module is Gorenstein projective, if and only if its underlying $R$-module is Gorenstein projective. It immediately follows that an $R$-complex is Gorenstein projective if and only if all its items are Gorenstein projective $R$-modules.

Residual-based a posteriori error estimates for symmetric conforming mixed finite elements for linear elasticity problems

Abstract: A posteriori error estimators for the symmetric mixed finite element methods for linearelasticity problems with Dirichlet and mixed boundary conditions are proposed. Reliability and efficiency of the estimators are proved. Numerical examples are presentedto verify the theoretical results.

The non-cutoff Vlasov-Maxwell-Boltzmann system with weak angular singularity

Abstract: We establish the global existence of small-amplitude solutions near a global Maxwellian to the Cauchy problem of the Vlasov-Maxwell-Boltzmann system for non-cutoff soft potentials with weak angular singularity. This extends the work of Duan et al. (2013), in which the case of strong angular singularity is considered, to the case of weak angular singularity.

Affine-periodic solutions by averaging methods

Abstract: This paper concerns the existence of affine-periodic solutions for perturbed affine-periodic systems.This kind of affine-periodic solutions has the form of $x(t+T)\equiv~Qx(t)$ with some nonsingular matrix $Q$, which may be quasi-periodic when $Q$ is an orthogonal matrix.It can be even unbounded but $\frac{x(t)}{|x(t)|}$ is quasi-periodic, like a helical line, for example $x(t)={\rm~e}^{at}(\cos~\omega~t,\sin~\omega~t)$, when $Q$ is not an orthogonal matrix.The averaging method of higher order for finding affine-periodic solutions is given by topological degree.

Penalized profile least squares-based statistical inference for varying coefficient partially linear errors-in-variables models

Abstract: The purpose of this paper is two fold. First, we investigate estimation for varying coefficient partially linear models in which covariates in the nonparametric part are measured with errors. As there would be some spurious covariates in the linear part, a penalized profile least squares estimation is suggested with the assistance from smoothly clipped absolute deviation penalty. However, the estimator is often biased due to the existence of measurement errors, a bias correction is proposed such that the estimation consistency with the oracle property is proved. Second, based on the estimator, a test statistic is constructed to check a linear hypothesis of the parameters and its asymptotic properties are studied. We prove that the existence of measurement errors causes intractability of the limiting null distribution that requires a Monte Carlo approximation and the absence of the errors can lead to a chi-square limit. Furthermore, confidence regions of the parameter of interest can also be constructed. Simulation studies and a real data example are conducted to examine the performance of our estimators and test statistic.

The existence and multiplicity of solutions of a fractional Schrödinger-Poisson system with critical growth

Abstract: In this paper, we study the existence and multiplicity of solutions for the following fractional Schrodinger-Poisson system: (0.1) $$\left\{ \begin{gathered} {\varepsilon ^{2s}}{\left( { - \Delta } \right)^s}u + V\left( x \right)u + \phi u = {\left| u \right|^{2_s^* - 2}}u + f\left( u \right)in{\mathbb{R}^3}, \hfill \\ {\varepsilon ^{2s}}{\left( { - \Delta } \right)^s}\phi = {u^2}in{\mathbb{R}^3}, \hfill \\ \end{gathered} \right.$$ where $$\frac{3}{4} < s < 1, 2_s^*:=\frac{6}{3-2s}$$ the fractional critical exponent for 3-dimension, the potential V: ℝ3 → ℝ is continuous and has global minima, and f is continuous and supercubic but subcritical at infinity. We prove the existence and multiplicity of solutions for System (0.1) via variational methods.