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Generalizing wavelets by adding desired redundancy and flexibility, framelets (i.e., wavelet frames) are of interest and importance in many applications such as image processing and numerical algorithms. Several key properties of framelets are high vanishing moments for sparse multiscale representation, fast framelet transforms for numerical efficiency, and redundancy for robustness. However, it is a challenging problem to study and construct multivariate nonseparable framelets, mainly due to their intrinsic connections to factorization and syzygy modules of multivariate polynomial matrices. Moreover, all the known multivariate tight framelets derived from spline refinable scalar functions have only one vanishing moment, and framelets derived from refinable vector functions are barely studied yet in the literature. In this paper, we circumvent the above difficulties through the approach of quasi-tight framelets, which behave almost identically to tight framelets. Employing the popular oblique extension principle (OEP), from an arbitrary compactly supported M-refinable vector function ϕ with multiplicity greater than one, we prove that we can always derive from ϕ a compactly supported multivariate quasi-tight framelet such that: (i) all the framelet generators have the highest possible order of vanishing moments; (ii) its associated fast framelet transform has the highest balancing order and is compact. For a refinable scalar function ϕ (i.e., its multiplicity is one), the above item (ii) often cannot be achieved intrinsically but we show that we can always construct a compactly supported OEP-based multivariate quasi-tight framelet derived from ϕ satisfying item (i). We point out that constructing OEP-based quasi-tight framelets is closely related to the generalized spectral factorization of Hermitian trigonometric polynomial matrices. Our proof is critically built on a newly developed result on the normal form of a matrix-valued filter, which is of interest and importance in itself for greatly facilitating the study of refinable vector functions and multiwavelets/multiframelets. This paper provides a comprehensive investigation on OEP-based multivariate quasi-tight multiframelets and their associated framelet transforms with high balancing orders. This deepens our theoretical understanding of multivariate quasi-tight multiframelets and their associated fast multiframelet transforms.

In the study of the constant in Ahlfors' second fundamental theorem involving a set E_{q} of q points, branch values of covering surfaces outside E_{q} bring a lot of troubles. To avoid this situation, for a given surface S, it is useful to construct a new surface So such that L(So) =H(S), and all branch values of So are contained in E_{q}. The goal of this paper is to prove the existence of such So, which generalizes Lemma 9.1 and Theorem 10.1 in Zhang G.Y.: The precise bound for the area-length ratio in Ahifors' theory of covering surfaces. Invent math 191:197-253(2013), This paper is a preparation for the further research on Ahlfors' theorey of covering surfaces

Let $A$ be a (left and right) Noetherian ring that is semiperfect. Let $e$ be an idempotent of $A$ and consider the ring $\Gamma:=(1-e)A(1-e)$ and the semi-simple right $A$-module $S_e : = eA/e{\rm rad}A$. In this paper, we investigate the relationship between the global dimensions of $A$ and $\Gamma$, by using the homological properties of $S_e$. More precisely, we consider the Yoneda ring $Y(e):={\rm Ext}^*_A(S_e,S_e)$ of $e$. We prove that if $Y(e)$ is artinian of finite global dimension, then $A$ has finite global dimension if and only if so is $\Gamma$. We also investigate the situation where both $A,\Gamma$ have finite global dimension. When $A$ is Koszul and finite dimensional, this implies that $Y(e)$ has finite global dimension. We end the paper with a reduction technique to compute the Cartan determiant of artin algebras. We prove that if $Y(e)$ has finite global dimension, then the Cartan determinants of $A$ and $\Gamma$ coincide. This provides a new way to approach the long-standing Cartan determinant conjecture., Comment: 14 pages

In this paper, we find all positive squarefree integers d such that the Pell equation X2-dY2 = +-1 has at least two positive integer solutions (X,Y) and (X',Y') such that both X and X' have Zeckendorf representations with at most two terms. This paper has been accepted for publication in SCIENCE CHINA Mathematics.

Let A,K be positive integers and u=-2,-1,1 or 2. The main contribution of the paper is a proof that each of the D(u^2)-triples {K,A^2K+2uA,(A+1)^2K+2u(A+1)} has unique extension to a D(u^2)-quadruple., This paper has 20 pages and has been accepted for publication in SCIENCE CHINA Mathematics

In this paper, we consider a nonlinear integral eigenvalue problem, which is a reformulation of the transmission eigenvalue problem arising in the inverse scattering theory. The boundary element method is employed for discretization, which leads to a generalized matrix eigenvalue problem. We propose a novel method based on the spectral projection. The method probes a given region on the complex plane using contour integrals and decides if the region contains eigenvalue(s) or not. It is particularly suitable to test if zero is an eigenvalue of the generalized eigenvalue problem, which in turn implies that the associated wavenumber is a transmission eigenvalue. Effectiveness and efficiency of the new method are demonstrated by numerical examples., Comment: The paper has been accepted for publication in SCIENCE CHINA Mathematics

This is the second paper in a series following Tian and Xu (2015), on the construction of a mathematical theory of the gauged linear σ-model (GLSM). In this paper, assuming the existence of virtual moduli cycles and their certain properties, we define the correlation function of GLSM for a fixed smooth rigidified r-spin curve.

In this paper, we introduce the notions of differential graded (DG) Poisson algebra and DG Poisson module. Let $A$ be any DG Poisson algebra. We construct the universal enveloping algebra of $A$ explicitly, which is denoted by $A^{ue}$. We show that $A^{ue}$ has a natural DG algebra structure and it satisfies certain universal property. As a consequence of the universal property, it is proved that the category of DG Poisson modules over $A$ is isomorphic to the category of DG modules over $A^{ue}$. Furthermore, we prove that the notion of universal enveloping algebra $A^{ue}$ is well-behaved under opposite algebra and tensor product of DG Poisson algebras. Practical examples of DG Poisson algebras are given throughout the paper including those arising from differential geometry and homological algebra., Comment: Accepted by Science China Mathematics

In the previous paper \cite{L-Z}, for a characteristic problem with not necessarily small initial data given on a complete null cone decaying like that in the work \cite{Ch-K} of the stability of Minkowski spacetime by Christodoulou and Klainerman, we proved the local existence in retarded time, which means the solution to the vacuum Einstein equations exists in a uniform future neighborhood, while the global existence in retarded time is the weak cosmic censorship conjecture. In this paper, we prove that the local existence in retarded time still holds when the data is assumed to decay slower, like that in Bieri's work \cite{Bie} on the extension to the stability of Minkowski spacetime. Such decay guarantees the existence of the limit of the Hawking mass on the initial null cone, when approaching to infinity, in an optimal way., Comment: 30 pages. arXiv admin note: text overlap with arXiv:1406.0048

In this paper, we first show that for a Banach space X there is a fully order-reversing mapping T from conv(X) (the cone of all extended real-valued lower semicontinuous proper convex functions defined on X) onto itself if and only if X is reflexive and linearly isomorphic to its dual X*. Then we further prove the following generalized Artstein-Avidan-Milman representation theorem: For every fully order-reversing mapping T: conv(X) → conv(X) there exist a linear isomorphism U: X → X*, x 0 * , φ0 ∈ X*, α > 0 and r0 ∈ ℝ so that $$(Tf)(x) = \alpha ({\cal F}f)(Ux + x_0^ *) + \left\langle {{\varphi _0},x} \right\rangle + {r_0},\;\;\;\;\;\forall x \in X,$$ where ${\cal F}$ : conv(X) → conv(X*) is the Fenchel transform. Hence, these resolve two open questions. We also show several representation theorems of fully order-preserving mappings defined on certain cones of convex functions. For example, for every fully order-preserving mapping S: semn(X) → semn(X) there is a linear isomorphism U: X → X so that $$(Sf)(x) = f(Ux),\;\;\;\;\;\forall f \in {\rm{semn}}(X),\;\;\;\;\;x \in X,$$ where semn(X) is the cone of all lower semicontinuous seminorms on X.

In this paper, we deal with the existence and concentration of normalized solutions to the supercritical nonlinear Schrodinger equation $$\left\{ {\matrix{ { - {\rm{\Delta }}u + V\left( x \right)u = {\mu _q}u + a{{\left| u \right|}^q}u\,\,\,{\rm{in}}\,{\mathbb{R}^2},} \hfill \cr {\int_{{\mathbb{R}^2}} {{{\left| u \right|}^2}dx = 1,} } \hfill \cr } } \right.$$ where μq is the Lagrange multiplier. We show that for q > 2 close to 2, the problem admits two solutions: one is the local minimal solution uq and the other one is the mountain pass solution υq. Furthermore, we study the limiting behavior of uq and υq when q → 2+. Particularly, we describe precisely the blow-up formation of the excited state υq.

We classify, up to some lattice-theoretic equivalence, all possible configurations of rational double points that can appear on a surface whose minimal resolution is a complex Enriques surface., 29 pages. This paper has been accepted for publication in SCIENCE CHINA Mathematics

In this paper, we study isoparametric hypersurfaces in Finsler space forms by investigating focal points, tubes and parallel hypersurfaces of submanifolds. We prove that the focal submanifolds of isoparametric hypersurfaces are anisotropic-minimal and obtain the Cartan-type formula in a Finsler space form with vanishing reversible torsion, from which we give some classifications on the number of distinct principal curvatures or their multiplicities., Comment: 22 pages, comments are welcome!

Hilbert’s Tenth Problem (HTP) asked for an algorithm to test whether an arbitrary polynomial Diophantine equation with integer coefficients has solutions over the ring ℤ of integers. This was finally solved by Matiyasevich negatively in 1970. In this paper we obtain some further results on HTP over ℤ. We prove that there is no algorithm to determine for any P(z1,…,z9) ∈ ℤ[z1,…,z9] whether the equation P(z1,…,z9) = 0 has integral solutions with z9 ⩾ 0. Consequently, there is no algorithm to test whether an arbitrary polynomial Diophantine equation P(z1,…,z11) = 0 (with integer coefficients) in 11 unknowns has integral solutions, which provides the best record on the original HTP over ℤ. We also prove that there is no algorithm to test for any P(z1,…,z17) ∈ ℤ[z1,…,z17] whether P(z 1 2 , …, z 17 2 ) = 0 has integral solutions, and that there is a polynomial Q(z1, …, z20) ∈ ℤ[z1,…,z20] such that {Q(z 1 2 , …, z 20 2 ): z1, …, z20 ∈ ℤ} ∩ {0, 1, 2, …} coincides with the set of all primes.

This paper is concerned with the well-posedness and large-time behavior of a two-dimensional PDE-ODE hybrid chemotaxis system describing the initiation of tumor angiogenesis. We first transform the system via a Cole-Hopf type transformation into a parabolic-hyperbolic system and then show that the solution of the transformed system converges to a constant equilibrium state as time tends to infinity. Finally we reverse the Cole-Hopf transformation and obtain the relevant results for the pre-transformed PDE-ODE hybrid system. In contrast to the existing related results, where continuous initial data is imposed, we are able to prove the asymptotic stability for discontinuous initial data with large oscillations. The key ingredient in our proof is the use of the so-called “effective viscous flux”, which enables us to obtain the desired energy estimates and regularity. The technique of the “effective viscous flux” turns out to be a very useful tool to study chemotaxis systems with initial data having low regularity and was rarely (if not) used in the literature for chemotaxis models.

A general analysis framework is presented in this paper for many different types of finite element methods (including various discontinuous Galerkin methods). For the second-order elliptic equation −div(α∇u) = f, this framework employs four different discretization variables, uh, ph, ŭh and $${\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}}\over p} _h}$$ , where uh and ph are for approximation of u and p = −α∇u inside each element, and ŭh and $${\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}}\over p} _h}$$ are for approximation of residual of u and p · n on the boundary of each element. The resulting 4-field discretization is proved to satisfy two types of inf-sup conditions that are uniform with respect to all discretization and penalization parameters. As a result, many existing finite element and discontinuous Galerkin methods can be analyzed using this general framework by making appropriate choices of discretization spaces and penalization parameters.

The present paper is concerned with a class of quasi-linear elliptic degenerate equations. The degenerate operator arises from analysis of manifolds with singularities. The variational methods are applied here to verify the existence of infinitely many solutions for the problem.

Consider a compact symplectic sub-orbifold groupoid $\sf S$ of a compact symplectic orbifold groupoid $(\mathsf X,\omega)$. Let $\mathsf X_{\mathfrak a}$ be the weight-$\mathfrak a$ blowup of $\sf X$ along $\sf S$, and $\mathsf D_{\mathfrak a}=\mathsf{PN}_{\mathfrak a}$ be the exceptional divisor, where $\sf N$ is the normal bundle of $\sf S$ in $\sf X$. In this paper we show that the absolute orbifold Gromov--Witten theory of $\mathsf X_{\mathfrak a}$ can be effectively and uniquely reconstructed from the absolute orbifold Gromov--Witten theories of $\sf X$, $\sf S$ and $\mathsf D_{\mathfrak a}$, the natural restriction homomorphism $H^*_{\text{CR}}({\sf X})\rightarrow H^*_{\text{CR}}({\sf S})$ and the first Chern class of the tautological line bundle over $\mathsf D_{\mathfrak a}$. To achieve this we first prove similar results for the relative orbifold Gromov--Witten theories of $(\mathsf X_{\mathfrak a}|\mathsf D_{\mathfrak a})$ and $(\mathsf N_{\mathfrak a}|\mathsf D_{\mathfrak a})$. As applications of these results, we prove an orbifold version of a conjecture of Maulik--Pandharipande on the Gromov--Witten theory of blowups along complete intersections, a conjecture on the Gromov--Witten theory of root constructions and a conjecture on Leray--Hirsch result for orbifold Gromov--Witten theory of Tseng--You., Comment: 44pages; some typos fixed; comments are welcome; accepted for publication in SCIENCE CHINA Mathematics

This article is concerned with the well-posedness as well as long-term dynamics of a wide class of non-autonomous, non-local, fractional, stochastic FitzHugh-Nagumo systems driven by nonlinear noise defined on the entire spaceℝRN. The well-posedness is proved for the systems with polynomial drift terms of arbitrary order as well as locally Lipschitz nonlinear diffusion terms by utilizing the pathwise and mean square uniform estimates. The mean random dynamical system generated by the solution operators is proved to possess a unique weak pullback mean random attractor in a Bochner space. The existence of invariant measures is also established for the autonomous systems with globally Lipschitz continuous diffusion terms. The idea of uniform tail-estimates of the solutions in the appropriate spaces is employed to derive the tightness of a family of probability distributions of the solutions in order to overcome the non-compactness of the standard Sobolev embeddings on ℝN as well as the lack of smoothing effect on one component of the solutions. The results of this paper are new even when the fractional Laplacian is replaced by the standard Laplacian.

In this paper, we study transitive partially hyperbolic diffeomorphisms with one-dimensional topologically neutral center, meaning that the length of the iterate of small center segments remains small. Such systems are dynamically coherent. We show that there exists a continuous metric along the center foliation which is invariant under the dynamics. As an application, we classify the transitive partially hyperbolic diffeomorphisms on 3-manifolds with topologically neutral center., 29 pages, 2 figures

In this paper, we deal with a singular quasilinear critical elliptic equation of Lichnerowicz type involving the p-Laplacian operator. With the help of the subcritical approach from variational method, we obtain the non-existence, existence, and multiplicity results under some given assumptions., 32 pages,2 figures. It has been accepted for publication in SCIENCE CHINA Mathematics

In this paper, we obtain a necessary and sufficient condition for a $U(n)$-invariant complex Finsler metric $F$ on domains in $\mathbb{C}^n$ to be strongly convex, which also makes it possible to investigate relationship between real and complex Finsler geometry via concrete and computable examples. We prove a rigid theorem which states that a $U(n)$-invariant strongly convex complex Finsler metric $F$ is a real Berwald metric if and only if $F$ comes from a $U(n)$-invariant Hermitian metric. We give a characterization of $U(n)$-invariant weakly complex Berwald metrics with vanishing holomorphic sectional curvature and obtain an explicit formula for holomorphic curvature of $U(n)$-invariant strongly pseudoconvex complex Finsler metric. Finally, we prove that the real geodesics of some $U(n)$-invariant complex Finsler metric restricted on the unit sphere $\pmb{S}^{2n-1}\subset\mathbb{C}^n$ share a specific property as that of the complex Wrona metric on $\mathbb{C}^n$.cc, 22 pages, has been accepted for publication in SCIENCE CHINA Mathematics

In this paper we put forward a viral propagation model with a nonlinear infection rate and free boundaries and investigate the dynamical properties. This model is composed of two ordinary differential equations and one partial differential equation, in which the spatial range of the first equation is the whole space ℝ, and the last two equations have free boundaries. As a new mathematical model, we prove the existence, uniqueness and uniform estimates of the global solution, and provide the criteria for spreading and vanishing, and the long time behavior of the solution components u, v and w. Comparing this model with the corresponding ordinary differential systems, the basic reproduction number $${{\cal R}_0}$$ plays a different role. We find that when $${{\cal R}_0} \le 1$$, the virus cannot spread successfully; when $${{\cal R}_0} > 1$$, the successful spread of virus depends on the initial value and varying parameters.

We give the extension formulae on almost complex manifolds and give decompositions of the extension formulae. As applications, we study $(n,0)$-forms, the $(n,0)$-Dolbeault cohomology group and $(n,q)$-forms on almost complex manifolds., a new subsection about $(n,q)$-forms added, some typos revised, no change in mathematical content. This paper has been accepted for publication in SCIENCE CHINA Mathematics

In this paper, numerical analysis is carried out for a class of history-dependent variational-hemivariational inequalities arising in contact problems. Three different numerical treatments for temporal discretization are proposed to approximate the continuous model. Fixed-point iteration algorithms are employed to implement the implicit scheme and the convergence is proved with a convergence rate independent of the time step-size and mesh grid-size. A special temporal discretization is introduced for the history-dependent operator, leading to numerical schemes for which the unique solvability and error bounds for the temporally discrete systems can be proved without any restriction on the time step-size. As for spatial approximation, the finite element method is applied and an optimal order error estimate for the linear element solutions is provided under appropriate regularity assumptions. Numerical examples are presented to illustrate the theoretical results., 33 pages,6 figures

In this paper some new results concerning the Cp classes introduced by Muckenhoupt (1981) and later extended by Sawyer (1983), are provided. In particular, we extend the result to the full expected range p < 0, to the weak norm, to other operators and to their vector-valued extensions. Some of those results rely upon sparse domination, which in the vector-valued case are provided as well. We will also provide sharp weighted estimates for vector-valued extensions relying on those sparse domination results.

In this paper, we compute sub-Riemannian limits of Gaussian curvature for a Euclidean $C^2$-smooth surface in the affine group and the group of rigid motions of the Minkowski plane away from characteristic points and signed geodesic curvature for Euclidean $C^2$-smooth curves on surfaces. We get Gauss-Bonnet theorems in the affine group and the group of rigid motions of the Minkowski plane., It has been accepted for publication in SCIENCE CHINA Mathematics

In this paper, we calculate the norm of the string Fourier transform on subfactor planar algebras and characterize the extremizers of the inequalities for parameters 0 < p,q ⩽ ∞. Furthermore, we establish Renyi entropic uncertainty principles for subfactor planar algebras.

In this paper we study some extremal problems for the family $S_g^0(\mathbb{B}_X)$ of normalized univalent mappings with $g$-parametric representation on the unit ball $\mathbb{B}_X$ of an $n$-dimensional JB$^*$-triple $X$ with $r\geq 2$, where $r$ is the rank of $X$ and $g$ is a convex (univalent) function on the unit disc $\mathbb{U}$, which satisfies some natural assumptions. We obtain sharp coefficient bounds for the family $S_g^0(\mathbb{B}_X)$, and examples of bounded support points for various subsets of $S_g^0(\mathbb{B}_X)$. Our results are generalizations to bounded symmetric domains of known recent results related to support points for families of univalent mappings on the Euclidean unit ball $\mathbb{B}^n$ and the unit polydisc $\mathbb{U}^n$ in $\mathbb{C}^n$. Certain questions will be also mentioned. Finally, we point out sharp coefficient bounds and bounded support points for the family $S_g^0(\mathbb{B}^n)$ and for special compact subsets of $S_g^0(\mathbb{B}^n)$, in the case $n\geq 2$., SCIENCE CHINA Mathematics

Let $\fg$ be an affine Kac-Moody algebra, and $\mu$ a diagram automorphism of $\fg$. In this paper, we give an explicit realization for the universal central extension $\wh\fg[\mu]$ of the twisted loop algebra of $\fg$ related to $\mu$, which provides a Moody-Rao-Yokonuma presentation for the algebra $\wh\fg[\mu]$ when $\mu$ is non-transitive, and the presentation is indeed related to the quantization of toroidal Lie algebras.

In this paper, new Levin methods are presented for calculating oscillatory integrals with algebraic and/or logarithmic singularities. To avoid singularity, the technique of singularity separation is applied and then the singular ODE occurring in classic Levin methods is converted into two kinds of non-singular ODEs. The solutions of one can be obtained explicitly, while those of the other can be solved efficiently by collocation methods. The proposed methods can attach arbitrarily high asymptotic orders and also enjoy superalgebraic convergence with respect to the number of collocation points. Several numerical experiments are presented to validate the efficiency of the proposed methods., Accepted for publication in SCIENCE CHINA Mathematics

Using a relative version of Auslander’s formula, we give a functorial approach to show that the bounded derived category of every Artin algebra admits a categorical resolution. This, in particular, implies that the bounded derived categories of Artin algebras of finite global dimension determine bounded derived categories of all Artin algebras. Hence, this paper can be considered as a typical application of functor categories, introduced in representation theory by Auslander (1971), to categorical resolutions.

In this paper, we consider the {\it tensor absolute value equations} (TAVEs), which is a newly introduced problem in the context of multilinear systems. Although the system of TAVEs is an interesting generalization of matrix {\it absolute value equations} (AVEs), the well-developed theory and algorithms for AVEs are not directly applicable to TAVEs due to the nonlinearity (or multilinearity) of the problem under consideration. Therefore, we first study the solutions existence of some classes of TAVEs with the help of degree theory, in addition to showing, by fixed point theory, that the system of TAVEs has at least one solution under some checkable conditions. Then, we give a bound of solutions of TAVEs for some special cases. To find a solution to TAVEs, we employ the generalized Newton method and report some preliminary results., Comment: 26 pages and 4 tables

In this paper, we give a description for steady Ricci solitons with a linear decay of sectional curvature. In particular, we classify all 3-dimensional steady Ricci solitons and 4-dimensional $\kappa$-noncollpased steady Ricci solitons with nonnegative sectional curvature under the linear curvature decay.

The sparse linear programming (SLP) is a linear programming problem equipped with a sparsity constraint, which is nonconvex, discontinuous and generally NP-hard to solve due to the combinatorial property involved. In this paper, by rewriting the sparsity constraint into a disjunctive form, we present an explicit formula of Lagrangian dual problem for the SLP, in terms of an unconstrained piecewise-linear convex programming problem which admits a strong duality. Furthermore, we show a saddle point theorem based on the strong duality and give two first-order necessary and sufficient optimality conditions for the saddle point problem without any constraint qualification for SLP. Additionally, as an extension, we show that these main results are also valid for the case when the nonnegative box constraint $[0,u]$ in the SLP is replaced by $[-u,u]$ for any positive vector $u$.

In this paper, we first present a Gearhardt-Pr\"uss type theorem with a sharp bound for m-accretive operators. Then we give two applications: (1) give a simple proof of the result proved by Constantin et al. on relaxation enhancement induced by incompressible flows; (2) show that shear flows with a class of Weierstrass functions obey logarithmically fast dissipation time-scales., Comment: 12 pages

In this paper, we study the integrals of the Ricci curvature over metric balls in a Finsler manifold, which can be viewed as an Lq-norm of the Ricci curvature. By bounding such integrals from above, we obtain several Myers type theorems.

In this paper, we study locally strongly convex affine hyperspheres in the unimodular affine space ℝn+1 which, as Riemannian manifolds, are locally isometric to the Riemannian product of two Riemannian manifolds both possessing constant sectional curvature. As the main result, a complete classification of such affine hyperspheres is established. Moreover, as direct consequences, 3- and 4-dimensional affine hyperspheres with parallel Ricci tensor are also classified.

In this paper, we analyze the asymptotic behaviour of the Hermitian-Yang-Mills flow over a compact non-Kahler manifold $(X,~g)$ with the Hermitian metric $g$ satisfying the Gauduchon and Astheno-Kahler condition.

In this paper, we provide a pathwise spine decomposition for multitype superdiffusions with non-local branching mechanisms under a martingale change of measure. As an application of this decomposition, we obtaina necessary and sufficient condition (called the $L\log~L$ criterion)for the limit of the fundamental martingale to be non-degenerate. This result complements the related results obtained inKyprianou et al. (2012),Kyprianou and Murillo-Salas (2013)and Liu et al. (2009)for superprocesses with purely local branching mechanisms and inKyprianou and Palau (2018) for super Markov chains.

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 $C_c^\infty(\mathbf{R}^3\setminus \{0\})$. We will prove that this energy form is a regular Dirichlet form with core $C_c^\infty(\mathbf{R}^3)$. 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 $(0,\infty)$ and its angular part is a time-changed Brownian motion on the sphere $S^2$. The radial part of $X$ is a "reflected" extension of the radial part of $X^0$ (the part process of $X$ before hitting $\{0\}$). Moreover, $X$ is the unique reflecting extension of $X^0$, but $X$ is not a semi-martingale.

In this paper, we firstly extend Theorem 5.1.1 in \cite {Helein} due to H\'elein to a rescaled branched conformal immersed sequence(c.f. Theorem 1.5). By virtue of this local convergence theorem, we study the blowup behavior of a sequence of branched conformal immersions of closed Riemannian surface in $\mathbb{R}^{n}$ with uniformly bounded areas and Willmore energies. Furthermore, we prove that the integral identity of Gauss curvature is true., Comment: some typos corrected; accepted for publication in SCIENCE CHINA Mathematics

Motivated by the study of coupled K\"ahler-Einstein metrics by Hultgren and Witt Nystr\"om and coupled K\"ahler-Ricci solitons by Hultgren, we study in this paper coupled Sasaki-Einstein metrics and coupled Sasaki-Ricci solitons. We first show an isomorphism between the Lie algebra of all transverse holomorphic vector fields and certain space of coupled basic functions related to coupled twisted Laplacians for basic functions, and obtain extensions of the well-known obstructions to the existence of K\"ahler-Einstein metrics to this coupled case. These results are reduced to coupled K\"ahler-Einstein metrics when the Sasaki structure is regular. Secondly we show the existence of toric coupled Sasaki-Einstein metrics when the basic first Chern class is positive extending the work of Hultgren., Comment: Final version. To appear in Science China Math

This paper begins to study the limiting behavior of a family of Hermitian Yang-Mills (HYM for brevity) metrics on a class of rank two slope stable vector bundles over a product of two elliptic curves with Kahler metrics ωe when e → 0. Here, ωe are flat and have areas e and e-1 on the two elliptic curves, respectively. A family of Hermitian metrics on the vector bundle are explicitly constructed and with respect to them, the HYM metrics are normalized. We then compare the family of normalized HYM metrics with the family of constructed Hermitian metrics by doing estimates. We get the higher order estimates as long as the C0-estimate is provided. We also get the estimate of the lower bound of the C0-norm. If the desired estimate of the upper bound of the C0-norm can be obtained, then it would be shown that these two families of metrics are close to arbitrary order in e in any Ck norms.

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., Comment: 31 pages, no figures

For equations of order two with the Dirichlet boundary condition, as the Laplace problem, the Stokes and the Navier-Stokes systems, perforated domains were only studied when the distance between the holes $d_{\varepsilon}$ is equal or much larger than the size of the holes $\varepsilon$. Such a diluted porous medium is interesting because it contains some cases where we have a non-negligible effect on the solution when $(\varepsilon,d_{\varepsilon})\to (0,0)$. Smaller distance was avoided for mathematical reasons and for theses large distances, the geometry of the holes does not affect -- or few -- the asymptotic result. Very recently, it was shown for the 2D-Euler equations that a porous medium is non-negligible only for inter-holes distance much smaller than the size of the holes. For this result, the regularity of holes boundary plays a crucial role, and the permeability criterium depends on the geometry of the lateral boundary. In this paper, we relax slightly the regularity condition, allowing a corner, and we note that a line of irregular obstacles cannot slow down a perfect fluid in any regime such that $\varepsilon \ln d_{\varepsilon} \to 0$., Dedicated to Professor Jean-Yves Chemin on the Occasion of his 60th Birthday. Accepted for publication in SCIENCE CHINA Mathematics

In this paper, we develop a semi-classical analysis on H-type groups. We define semi-classical pseudodifferential 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 consists 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 the finite-dimensioned representations of the group, even though they are negligible with respect to the Plancherel measure., Comment: 29 pages

In this paper, we prove Poincaré and Sobolev inequalities for differential forms in L1(ℝn). The singular integral estimates that it is possible to use for Lp, p > 1, are replaced here with inequalities which go back to Bourgain and Brezis (2007).

In this paper, we introduce and study the conformal mean curvature flow of submanifolds of higher codimension in the Euclidean space $\bbr^n$. This kind of flow is a special case of a general modified mean curvature flow which is of various origination. As the main result, we prove a blow-up theorem concluding that, under the conformal mean curvature flow in $\bbr^n$, the maximum of the square norm of the second fundamental form of any compact submanifold tends to infinity in finite time. Furthermore, by using the idea of Andrews and Baker for studying the mean curvature flow of submanifolds in the Euclidean space, we also derive some more evolution formulas and inequalities which we believe to be useful in our further study of conformal mean curvature flow. Presently, these computations together with our main theorem are applied to provide a direct proof of a convergence theorem concluding that the external conformal forced mean curvature flow of a compact submanifold in $\bbr^n$ with the same pinched condition as Andrews-Baker's will be convergent to a round point in finite time., 47 pages including an appendix; all comments are welcome!

The combinatorial Mandelbrot set is a continuum in the plane, whose boundary can be defined, up to a homeomorphism, as the quotient space of the unit circle by an explicit equivalence relation. This equivalence relation was described by Douady and, in different terms, by Thurston. Thurston used quadratic invariant laminations as a major tool. As has been previously shown by the authors, the combinatorial Mandelbrot set can be interpreted as a quotient of the space of all limit quadratic invariant laminations. The topology in the space of laminations is defined by the Hausdorff distance. In this paper, we describe two similar quotients. In the first case, the identifications are the same but the space is smaller than that taken for the Mandelbrot set. The result (the quotient space) is obtained from the Mandelbrot set by "unpinching" the transitions between adjacent hyperbolic components. In the second case, we do not identify non-renormalizable laminations while identifying renormalizable laminations according to which hyperbolic lamination they can be "unrenormalised" to., Comment: 29 pages, 4 figures