Showing papers in "Science China-mathematics in 2019"
TL;DR: In this article, a monotone finite volume method for the time fractional Fokker-Planck equations was developed and theoretically proved its unconditional stability, and the convergence rate of this method was shown to be of order 1 in the space and if the space grid becomes suffciently fine, it can be improved to order 2.
Abstract: We develop a monotone finite volume method for the time fractional Fokker-Planck equations and theoretically prove its unconditional stability. We show that the convergence rate of this method is of order 1 in the space and if the space grid becomes suffciently fine, the convergence rate can be improved to order 2. Numerical results are given to support our theoretical findings. One characteristic of our method is that it has monotone property such that it keeps the nonnegativity of some physical variables such as density, concentration, etc.
51 citations
TL;DR: A unified model, known as MgNet, is developed that simultaneously recovers some convolutional neural networks for image classification and multigrid methods for solving discretized partial differential equations (PDEs) and modified CNN models (with fewer weights and hyperparameters) are developed.
Abstract: We develop a unified model, known as MgNet, that simultaneously recovers someconvolutional neural networks (CNN) for image classification and multigrid (MG)methods for solving discretized partial differential equations (PDEs). This model isbased on close connections that we have observed and uncovered between the CNNand MG methodologies. For example, pooling operation and feature extraction in CNNcorrespond directly to restriction operation and iterative smoothers in MG, respectively.As the solution space is often the dual of the data space in PDEs, the analogous conceptof feature space and data space (which are dual to each other) is introduced in CNN.With such connections and new concept in the unified model, the function of variousconvolution operations and pooling used in CNN can be better understood.As a result, modified CNN models (with fewer weights and hyperparameters)are developed that exhibit competitive and sometimes better performance incomparison with existing CNN models when applied to both CIFAR-10 and CIFAR-100 data sets.
46 citations
TL;DR: In this article, a unified study for the design and analysis of different finite element methods (FEMs), including conforming and nonconforming FEMs, mixed FEM, hybrid FEM and discontinuous Galerkin (DG) methods, is presented.
Abstract: A unified study is presented in this paper for the design and analysis of different finite element methods (FEMs), including conforming and nonconforming FEMs, mixed FEMs, hybrid FEMs, discontinuous Galerkin (DG) methods, hybrid discontinuous Galerkin (HDG) methods and weak Galerkin (WG) methods. Both HDG and WG are shown to admit inf-sup conditions that hold uniformly with respect to both mesh and penalization parameters. In addition, by taking the limit of the stabilization parameters, a WG method is shown to converge to a mixed method whereas an HDG method is shown to converge to a primal method. Furthermore, a special class of DG methods, known as the mixed DG methods, is presented to fill a gap revealed in the unified framework.
45 citations
TL;DR: In this article, the optimal enhanced dissipation rate for the 2D linearized Navier-Stokes equations around the bar state called the Kolmogorov flow was solved by developing the hypocoercivity method introduced by Villani.
Abstract: In this paper, we solve Beck and Wayne’s conjecture on the optimal enhanced dissipation rate for the 2-D linearized Navier-Stokes equations around the bar state called the Kolmogorov flow by developing the hypocoercivity method introduced by Villani (2009).
38 citations
TL;DR: In this paper, it was shown that every surjective isometry between the unit spheres of two compact C*-algebras A and B, without assuming any restriction on the rank of their direct summands, extends to a real linear isometry from A into B.
Abstract: Let f : S(E) → S(B) be a surjective isometry between the unit spheres of two weakly compact JB*-triples not containing direct summands of rank smaller than or equal to 3. Suppose E has rank greater than or equal to 5. Applying techniques developed in JB*-triple theory, we prove that f admits an extension to a surjective real linear isometry T : E → B. Among the consequences, we show that every surjective isometry between the unit spheres of two compact C*-algebras A and B, without assuming any restriction on the rank of their direct summands (and in particular when A = K(H) and B = K(H′)), extends to a surjective real linear isometry from A into B. These results provide new examples of infinite-dimensional Banach spaces where Tingley’s problem admits a positive answer.
32 citations
TL;DR: In this paper, it was shown that the parameters mσ(G) and mσq(G), respectively, make possible to bound the σ-nilpotent length of a finite soluble group G in the case when mσ (G) = |П(G)|.
Abstract: Let σ = {σi | i ∈ I} be some partition of the set of all primes P. A set H of subgroups of G is said to be a complete Hallσ-set of G if every member ≠ 1 of H is a Hall σi-subgroup of G, for some i ∈ I, and H contains exactly one Hall σi-subgroup of G for every σi ∈ σ(G). A subgroup H of G is said to be: σ-permutable or σ-quasinormal in G if G possesses a complete Hall σ-set H such that HAx = AxH for all A ∈ H and x ∈ G: σ-subnormal in G if there is a subgroup chain A = A0 ≤ A1 ≤ · · · ≤ At = G such that either $${A_{i - 1}}\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{ \triangleleft } {A_i}$$
or Ai=(Ai-1)Ai is a finite σi-group for some σi ∈ σ for all i = 1;:::; t. If Mn 1, some (n−1)-maximal subgroup is not σ-subnormal (not σ-quasinormal, respectively) in G, we write mσ(G) = n (mσq(G) = n, respectively). In this paper, we show that the parameters mσ(G) and mσq(G) make possible to bound the σ-nilpotent length lσ(G) (see below the definitions of the terms employed), the rank r(G) and the number |П(G)| of all distinct primes dividing the order |G| of a finite soluble group G. We also give the conditions under which a finite group is σ-soluble or σ-nilpotent, and describe the structure of a finite soluble group G in the case when mσ(G) = |П(G)|. Some known results are generalized.
27 citations
TL;DR: In this article, the authors derived five basic identities for Sheffer polynomials by using generalized Pascal functional and Wronskian matrices, and applied twelve basic identities, seven from previous results, to degenerate Bernoulli polynomial and Korobov polynomorphism of the first kind and got some new identities.
Abstract: In this paper, we derive five basic identities for Sheffer polynomials by using generalized Pascal functional and Wronskian matrices. Then we apply twelve basic identities for Sheffer polynomials, seven from previous results, to degenerate Bernoulli polynomials and Korobov polynomials of the first kind and get some new identities. In addition, letting $\lambda~\rightarrow~0$ in such identities gives usthose for Bernoulli polynomials and Bernoulli polynomials of the second kind.
25 citations
TL;DR: In this paper, a new kind of approximations of the identity with exponential decay (for short, exp-ATI) is introduced, motivated by the breakthrough work of Auscher and Hytonen (2013) on orthonormal bases of regular wavelets on spaces of homogeneous type.
Abstract: Assume that (X, d, μ) is a space of homogeneous type in the sense of Coifman and Weiss (1971, 1977). In this article, motivated by the breakthrough work of Auscher and Hytonen (2013) on orthonormal bases of regular wavelets on spaces of homogeneous type, we introduce a new kind of approximations of the identity with exponential decay (for short, exp-ATI). Via such an exp-ATI, motivated by another creative idea of Han et al. (2018) to merge the aforementioned orthonormal bases of regular wavelets into the frame of the existed distributional theory on spaces of homogeneous type, we establish the homogeneous continuous/discrete Calderon reproducing formulae on (X, d, μ), as well as their inhomogeneous counterparts. The novelty of this article exists in that d is only assumed to be a quasi-metric and the underlying measure μ a doubling measure, not necessary to satisfy the reverse doubling condition. It is well known that Calderon reproducing formulae are the cornerstone to develop analysis and, especially, harmonic analysis on spaces of homogeneous type.
23 citations
TL;DR: In this paper, the martingale Musielak-Orlicz Hardy spaces were introduced via the maximal function, the quadratic variation and the conditional quadrastic variation of martingales.
Abstract: In this article, we introduce the martingale Musielak-Orlicz Hardy spaces $$H_\varphi ^*\left({\rm{\Omega}} \right)$$, Pϕ(Ω), $$H_\varphi ^S\left({\rm{\Omega}} \right)$$, Qϕ(Ω) and $$H_\varphi ^s\left({\rm{\Omega}} \right)$$, respectively, via the maximal function, the quadratic variation and the conditional quadratic variation of martingales. We then establish the atomic characterizations of $$H_\varphi ^s\left({\rm{\Omega}} \right)$$, Pϕ(Ω) and Qϕ(Ω). As applications, we obtain the dual space of $$H_\varphi ^s\left({\rm{\Omega}} \right)$$ and several martingale inequalities which further clarify the relations among $$H_\varphi ^*\left({\rm{\Omega}} \right)$$, Pϕ(Ω), $$H_\varphi ^S\left({\rm{\Omega}} \right)$$, Qϕ(Ω) and $$H_\varphi ^s\left({\rm{\Omega}} \right)$$. Especially, as special cases, the results on atomic characterizations of $$H_\varphi ^s\left({\rm{\Omega}} \right)$$, Pϕ(Ω) and Qϕ(Ω) as well as on the dual space of $$H_\varphi ^s\left({\rm{\Omega}} \right)$$ in the weighted case are also new.
22 citations
TL;DR: In this paper, the authors proved the C 1, 1-regularity of the envelope of a C1, 1 function on a compact Hermitian manifold and presented the examples to show this regularity is sharp.
Abstract: In this paper, we prove the C1,1-regularity of the plurisubharmonic envelope of a C1,1 function on a compact Hermitian manifold. We also present the examples to show this regularity is sharp.
22 citations
TL;DR: The traditional approaches or measurements on the associations among the observed variables, such as correlation coefficient, mutual information and conditional mutual information (CMI), are reviewed, and recently developed theories and methods are summarized.
Abstract: Detecting direct associations or inferring networks based on the observed data is an important issue in many fields, including biology, physics, engineering and social studies. In this work, we focus on the information theoretic approaches in the network reconstruction or the direct association detection, in particular, for biological networks. We not only review the traditional approaches or measurements on the associations among the observed variables, such as correlation coefficient, mutual information and conditional mutual information (CMI), but also summarize recently developed theories and methods. The new theoretic works include: information geometry to give a unified framework in detecting causality/association, the partial independence to alleviate the singularity of CMI, and multiscale analysis of CMI to avoid the underestimation issue of CMI. The new methods include part mutual information (PMI) and partial associations (PA), which improve the old measurements in avoiding both overestimation and underestimation. All those theories and methods make important contributions as major advances in the development of network inference.
TL;DR: In this paper, the intersection of the non-klt (non Kawamata log terminal) locus of (X, Δ) with any fiber Xs has at most two connected components.
Abstract: Let (X, Δ) be a log pair over S, such that-(KX + Δ) is nef over S. It is conjectured that the intersection of the non-klt (non Kawamata log terminal) locus of (X, Δ) with any fiber Xs has at most two connected components. We prove this conjecture in dimension no larger than 4 and in arbitrary dimension assuming the termination of klt flips.
TL;DR: In this article, it was shown that the Hellys selection principle can be extended to continuous group actions on median pretrees (for example, dendrons or linearly ordered sets).
Abstract: We show that group actions on many treelike compact spaces are not too complicated dynamically.We first observe that an old argument of Seidler (1990) implies that every action of a topological group $G$on a regular continuum is null and therefore also tame. As every local dendron is regular, one concludes that every action of $G$ on a local dendron is null. We then use a more direct method to show that every continuous group action of $G$ on a dendron is Rosenthal representable, hence also tame. Similar results are obtained for median pretrees.As a related result, we show that Hellys selection principle can be extended to bounded monotone sequencesdefined on median pretrees (for example, dendrons or linearly ordered sets). Finally, we point out some applications of these results to continuous group actions on dendrites.
TL;DR: In this article, the authors state some refinements of conjectures of Gan-Gross-Prasad and Kudla concerning the central derivatives of L-series and special cycles on Shimura varieties.
Abstract: In this note, we state some refinements of conjectures of Gan-Gross-Prasad and Kudla concerning the central derivatives of L-series and special cycles on Shimura varieties. The analogues of our formulation for special values of L-series are written in terms of invariant linear forms on autormorphic representations defined by integrations of matrix coefficients.
TL;DR: In this paper, the Poisson stability of motions for monotone non-autonomous dynamical systems and solutions for some classes of nonautonomous evolution equations (ODEs, FDEs and parabolic PDEs) is studied.
Abstract: In this paper, we study the Poisson stability (in particular, stationarity, periodicity, quasi-periodicity, Bohr almost periodicity, almost automorphy, recurrence in the sense of Birkhoff, Levitan almost periodicity, pseudo periodicity, almost recurrence in the sense of Bebutov, pseudo recurrence, Poisson stability) of motions for monotone nonautonomous dynamical systems and of solutions for some classes of monotone nonautonomous evolution equations (ODEs, FDEs and parabolic PDEs). As a byproduct, some of our results indicate that all the trajectories of monotone systems converge to the above mentioned Poisson stable trajectories under some suitable conditions, which is interesting in its own right for monotone dynamics.
TL;DR: In this article, a generalized Trudinger-Moser inequality on a compact Riemannian surface with conical singularity was established using the method of blow-up analysis, and the extremal function for the above inequality exists.
Abstract: In this paper, using the method of blow-up analysis, we establish a generalized Trudinger-Moser inequality on a compact Riemannian surface with conical singularity. Precisely, let (Σ,D) be such a surface with divisor $$D=\Sigma_{i=1}^m\beta_{i}p_{i}$$
, where βi > −1 and pi ∈ Σ for i = 1, …, m, and g be a metric representing D. Denote b0 = 4π(1 + min1⩽i⩽mβi). Suppose ψ : Σ → ℝ is a continuous function with ∫Σψdvg ≠ 0 and define $$\lambda _1^{**} (\sum ,g) = \mathop {\inf }\limits_{u \in H^1 (\sum ,g),\smallint _\sum \psi udv_g = 0,\smallint _\sum u^2 dv_g = 1} \int_\sum {\left| {
abla _g u} \right|^2 dv_g .}$$
Then for any
$$\alpha\in[0,\lambda_1^{**}(\Sigma, g))$$
, we have $$\mathop {\sup }\limits_{u \in H^1 (\sum ,g),\smallint _\sum \psi u = 0,\smallint _\sum \left| {
abla _g u} \right|^2 dv_g - \alpha \smallint _\sum u^2 dv_g \leqslant 1} \int_\sum {e^{b_0 u^2 } dv_g < + \infty .}$$
When b > b0, the integrals $$\int_\sum {e^{bu^2 } dv_g }$$
are still finite, but the supremum is infinity. Moreover, we prove that the extremal function for the above inequality exists.
TL;DR: In this paper, the authors discuss some recent progress in the study of non-Kahler manifolds, in particular the Hermitian geometry of flat canonical connections and Kahler-like connections, and discuss a number of conjectures and open questions in this direction.
Abstract: In this paper, we discuss some recent progress in the study of non-Kahler manifolds, in particular the Hermitian geometry of flat canonical connections and Kahler-like connections. We also discuss a number of conjectures and open questions in this direction.
TL;DR: In this paper, the authors considered infinite depth gravity water waves in two space dimensions and showed that the cubic nonlinear Schrodinger equation (NLSE) is well approximated by the natural cubic time scale.
Abstract: This article is concerned with infinite depth gravity water waves in two space dimensions. We consider this system expressed in position-velocity potential holomorphic coordinates. Our goal is to study this problem with small wave packet data, and to show that this is well approximated by the cubic nonlinear Schrodinger equation (NLS) on the natural cubic time scale.
TL;DR: In this article, the Yau gradient estimate for harmonic maps into a metric space (X, dX) with curvature bounded above by a constant κ (κ ⩾ 0) in the sense of Alexandrov is shown.
Abstract: In this paper, we show the Yau’s gradient estimate for harmonic maps into a metric space (X, dX) with curvature bounded above by a constant κ (κ ⩾ 0) in the sense of Alexandrov. As a direct application, it gives some Liouville theorems for such harmonic maps. This extends the works of Cheng (1980) and Choi (1982) to harmonic maps into singular spaces.
TL;DR: In this paper, the authors considered three-dimensional incompressible Navier-Stokes equations with different viscous coefficients in the vertical and horizontal variables and proved the global well-posedness of this system.
Abstract: We consider three-dimensional incompressible Navier-Stokes equations (NS) with different viscous coefficients in the vertical and horizontal variables. In particular, when one of these viscous coefficients is large enough compared with the initial data, we prove the global well-posedness of this system. In fact, we obtain the existence of a global strong solution to (NS) when the initial data verifies an anisotropic smallness condition which takes into account the different roles of the horizontal and vertical viscosity.
TL;DR: In particular, if RicciN is non-negative, then the first p-eigenvalue is bounded from below by a constant K in terms of the diameter d of a manifold, dimension, K, p and N as mentioned in this paper.
Abstract: In this paper, we give a sharp lower bound for the first (nonzero) p-eigenvalue on a compact Finsler manifold M without boundary or with convex boundary if the weighted Ricci curvature RicciN is bounded from below by a constant K in terms of the diameter d of a manifold, dimension, K, p and N. In particular, if RicciN is non-negative, then the first p-eigenvalue is bounded from below by $$(p - 1){({\textstyle{{{\pi _p}} \over d}})^p}$$, and the equality holds if and only if M is either a circle or a segment.
TL;DR: In this paper, a generalization of group symmetries of modular categories to category symmets in topological phases of matter has been studied, where simple modular categories are defined as the prime ones without any nontrivial normal algebras.
Abstract: Pursuing a generalization of group symmetries of modular categories to category symmetries in topological phases of matter, we study linear Hopf monads. The main goal is a generalization of extension and gauging group symmetries to category symmetries of modular categories, which include also categorical Hopf algebras as special cases. As an application, we propose an analogue of the classification of finite simple groups to modular categories, where we define simple modular categories as the prime ones without any nontrivial normal algebras.
TL;DR: A survey of invariant Einstein metrics on generalized Wallach spaces is given in this article. But invariant equilibria on generalized wallach spaces have not yet been classified except SO(k + l + m)/SO(k) × SO(l + m/SO(m) ×SO(l) × SOM(m).
Abstract: Invariant Einstein metrics on generalized Wallach spaces have been classified except SO(k + l + m)/SO(k) × SO(l) × SO(m). In this paper, we first give a survey on the study of invariant Einstein metrics on generalized Wallach spaces, and prove that there are infinitely many spaces of the type SO(k+l+m)/SO(k) × SO(l) × SO(m) admitting exactly two, three, or four invariant Einstein metrics up to a homothety.
TL;DR: In this paper, the energy form induced by an eigenfunction of a self-adjoint extension of the restriction of the Laplace operator to $$C_c^\infty ({\mathbb{R}^3}\backslash \{ 0\} )$$ was studied.
Abstract: Our concern in this paper is the energy form induced by an eigenfunction of a self-adjoint extension of the restriction of the Laplace operator to $$C_c^\infty ({\mathbb{R}^3}\backslash \{ 0\} )$$). We will prove that this energy form is a regular Dirichlet form with core $$C_c^\infty ({\mathbb{R}^3})$$. The associated diffusion X behaves like a 3-dimensional Brownian motion with a mild radial drift when far from 0, subject to an ever-stronger push toward 0 near that point. In particular, {0} is not a polar set with respect to X. The diffusion X is rotation invariant, and admits a skew-product representation before hitting {0}: its radial part is a diffusion on (0, ∞) and its angular part is a time-changed Brownian motion on the sphere S2. The radial part of X is a “reflected” extension of the radial part of X0 (the part process of X before hitting {0}). Moreover, X is the unique reflecting extension of X0, but X is not a semi-martingale.
TL;DR: In this article, a convex integration approach was used to construct very weak solutions to the 2D Monge-Ampere equation with Holder-continuous first derivatives of exponent β < 1/5.
Abstract: In this short note we revisit the convex integration approach to constructing very weak solutions to the 2D Monge-Ampere equation with Holder-continuous first derivatives of exponent β < 1/5. Our approach is based on combining the approach of Lewicka and Pakzad (2017) with a new diagonalization procedure which avoids the use of conformal coordinates, which was introduced by De Lellis et al. (2018) for the isometric immersion problem.
TL;DR: In this paper, the authors proposed a finite difference scheme for the two-dimensional nonlinear Schrodinger (NLS) equation involving fractional Laplacian, which is based on a weighted and shifted Grunwald-Letnikov difference (WSGD) operator.
Abstract: This paper proposes and analyzes an efficient finite difference scheme for the two-dimensional nonlinear Schrodinger (NLS) equation involving fractional Laplacian. The scheme is based on a weighted and shifted Grunwald-Letnikov difference (WSGD) operator for the spatial fractional Laplacian. We prove that the proposed method preserves the mass and energy conservation laws in semi-discrete formulations. By introducing the differentiation matrices, the semi-discrete fractional nonlinear Schrodinger (FNLS) equation can be rewritten as a system of nonlinear ordinary differential equations (ODEs) in matrices formulations. Two kinds of time discretization methods are proposed for the semi-discrete formulation. One is based on the Crank-Nicolson (CN) method which can be proved to preserve the fully discrete mass and energy conservation. The other one is the compact implicit integration factor (cIIF) method which demands much less computational effort. It can be shown that the cIIF scheme can approximate CN scheme with the error O(τ2). Finally numerical results are presented to demonstrate the method’s conservation, accuracy, efficiency and the capability of capturing blow-up.
TL;DR: This paper considers the generalized ADMM, which incorporates an acceleration factor and is more efficient, and proposes using the original ϵ-optimal solution measure, under which it is proved that the G-ADMM converges at a rate of O(1/t).
Abstract: Owing to its efficiency in solving some types of large-scale separable optimization problems with linear constraints, the convergence rate of the alternating direction method of multipliers (ADMM for short) has recently attracted significant attention. In this paper, we consider the generalized ADMM (G-ADMM), which incorporates an acceleration factor and is more efficient. Instead of using a solution measure that depends on a bounded set and cannot be easily estimated, we propose using the original ϵ-optimal solution measure, under which we prove that the G-ADMM converges at a rate of O(1/t). The new bound depends on the penalty parameter and the distance between the initial point and solution set, which is more reasonable than the previous bound.
TL;DR: In this paper, a scaling invariance of the barotropic compressible Navier-Stokes equations was shown for all adiabatic number γ > 1, where κ > 0 does not depend on the initial data.
Abstract: We find a new scaling invariance of the barotropic compressible Navier-Stokes equations. Then it is shown that type-I singularities of solutions with $$\mathop {\lim \sup }\limits_{t
earrow T} |div(t,x)|(T - t) \leqslant \kappa $$
can never happen at time T for all adiabatic number γ > 1. Here κ > 0 does not depend on the initial data. This is achieved by proving the regularity of solutions under $$\rho (t,x) \leqslant \frac{M}{{{{(T - t)}^\kappa }}},M < \infty .$$
This new scaling invariance also motivates us to construct an explicit type-II blowup solution for γ > 1.
TL;DR: In this paper, a semi-classical analysis on H-type groups is presented, which takes into account the finite-dimensional representations of the group, even though they are negligible with respect to the Plancherel measure.
Abstract: In this paper, we develop semi-classical analysis on H-type groups. We define semi-classical pseudodi fferential operators, prove the boundedness of their action on square integrable functions and develop a symbolic calculus. Then, we define the semi-classical measures of bounded families of square integrable functions which consist of a pair formed by a measure defined on the product of the group and its unitary dual, and by a field of trace class positive operators acting on the Hilbert spaces of the representations. We illustrate the theory by analyzing examples, which show in particular that this semi-classical analysis takes into account thefinite-dimensional representations of the group, even though they are negligible with respect to the Plancherel measure.
TL;DR: In this paper, an extended Cesaro operator Tφ with holomorphic symbol φ was defined and characterized for which φ is bounded between the Bloch-type spaces and the little Bloch type spaces.
Abstract: Let $$\mathbb{B}$$
be the unit ball of a complex Banach space X. In this paper, we generalize the Bloch-type spaces and the little Bloch-type spaces to the open unit ball $$\mathbb{B}$$
by using the radial derivative. Next, we define an extended Cesaro operator Tφ with holomorphic symbol φ and characterize those φ for which Tφ is bounded between the Bloch-type spaces and the little Bloch-type spaces. We also characterize those φ for which Tφ is compact between the Bloch-type spaces and the little Bloch-type spaces under some additional assumption on the symbol φ. When $$\mathbb{B}$$
is the open unit ball of a finite dimensional complex Banach space X, this additional assumption is automatically satisfied.