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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

In this paper, a type of accurate a posteriori error estimator is proposed for the Steklov eigenvalue problem based on the complementary approach, which provides an asymptotic exact estimate for the approximate eigenpair. Besides, we design a type of cascadic adaptive finite element method for the Steklov eigenvalue problem based on the proposed a posteriori error estimator. In this new cascadic adaptive scheme, instead of solving the Steklov eigenvalue problem in each adaptive space directly, we only need to do some smoothing steps for linearized boundary value problems on a series of adaptive spaces and solve some Steklov eigenvalue problems on a low dimensional space. Furthermore, the proposed a posteriori error estimator provides the way to refine mesh and control the number of smoothing steps for the cascadic adaptive method. Some numerical examples are presented to validate the efficiency of the algorithm in this paper.

This paper deals with the pulsating waves and entire solutions for a spatially periodic nonlocal dispersal model with a quiescent stage. By the method of super- and subsolutions together with the comparison principle, we establish the existence of pulsating waves with any speed larger than the spreading speed. The construction of the super- and subsolutions depends on the principal eigenvalue theory for a periodic eigenvalue problem with partially degenerate nonlocal dispersal. The joint results of this paper and our recent work (Wang J B, Li W T, Sun J W. Global dynamics and spreading speeds for a partially degenerate system with non-local dispersal in periodic habitats. Proc Roy Soc Edinburgh Sect A, 2018, 148: 849–880) show that the spreading speed coincides with the minimal wave speed of pulsating waves for the considered system. Moreover, combining the rightward and leftward pulsating waves with different speeds and a spatially periodic solution, we prove the existence and qualitative properties of entire solutions other than pulsating waves, which provide some new spreading ways in a heterogeneous habitat.

We begin with the reference measure P0 induced by simple, symmetric nearest neighbor continuous time random walk on Zd starting at 0 with jump rate 2d and then define, for β ⩾ 0, t > 0, the Gibbs probability measure Pβ,t by specifying its density with respect to P0 as $${{d{P_{\beta, t}}} \over {d{P^0}}} = {Z_{\beta, t}}{(0)^{-1}}{{\rm{e}}^{\beta \int_0^t {{\delta _0}({x_s})ds}}},$$ (0.1) where $${Z_{\beta, t}}(0) \equiv {E^0}{\rm{[}}{{\rm{e}}^{\beta \;\int_0^t {{\delta _0}({x_s})ds}}}].$$. This Gibbs probability measure provides a simple model for a homopolymer with an attractive potential at the origin. In a previous paper (Cranston and Molchanov, 2007), we showed that for dimensions d ⩾ 3 there is a phase transition in the behavior of these paths from the diffusive behavior for β below a critical parameter to the positive recurrent behavior for β above this critical value. The critical value was determined by means of the spectral properties of the operator Δ + βδ0, where Δ is the discrete Laplacian on Zd. This corresponds to a transition from a diffusive or stretched-out phase to a globular phase for the polymer. In this paper we give a description of the polymer at the critical value where the phase transition takes place. The behavior at the critical parameter is dimension-dependent.

Nonconforming grids with hanging nodes are frequently used in adaptive finite element computations. In all earlier works on such methods, proper constraints should be enforced on degrees of freedom on edges/faces with hanging nodes to keep continuity, which yield numerical computations much complicated. In 2014, Zhao et al. (2014) presented quadrilateral constraint-free finite element methods on quadrilateral grids with hanging nodes. This paper further develops a hexahedral constraint-free finite element method on hexahedral grids with hanging nodes, which is of greater challenge than the two-dimensional case. Residual-based a posteriori error reliability and efficiency are also established in this paper.

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.

This paper is concerned with the time-step condition of linearized implicit finite difference method for solving the Gross-Pitaevskii equation with an angular momentum rotation term. Unlike the existing studies in the literature, where the cut-off function technique was used to establish the error estimates under some conditions of the time-step size, this paper introduces an induction argument and a ‘lifting’ technique as well as some useful inequalities to build the optimal maximum error estimate without any constraints on the time-step size. The analysis method can be directly extended to the general nonlinear Schrodinger-type equations in twoand three-dimensions and other linear implicit finite difference schemes. As a by-product, this paper defines a new type of energy functional of the grid functions by using a recursive relation to prove that the proposed scheme preserves well the total mass and energy in the discrete sense. Several numerical results are reported to verify the error estimates and conservation laws.

In this paper, we obtain the necessary and sufficient condition of the pre-compact sets in the variable exponent Lebesgue spaces, which is also called the Riesz-Kolmogorov theorem. The main novelty appearing in this approach is the constructive approximation which does not rely on the boundedness of the Hardy-Littlewood maximal operator in the considered spaces such that we do not need the log-Holder continuous conditions on the variable exponent. As applications, we establish the boundedness of Riemann-Liouville integral operators and prove the compactness of truncated Riemann-Liouville integral operators in the variable exponent Lebesgue spaces. Moreover, applying the Riesz-Kolmogorov theorem established in this paper, we obtain the existence and the uniqueness of solutions to a Cauchy type problem for fractional differential equations in variable exponent Lebesgue spaces.

This paper aims to introduce some new ideas into the study of submodules in Hilbert spaces of analytic functions. The effort is laid out in the Hardy space over the bidisk H2(D2). A closed subspace M in H2(D2) is called a submodule if z i M ⊂ M (i = 1, 2). An associated integral operator (defect operator) C M captures much information about M. Using a Kreĭn space indefinite metric on the range of C M , this paper gives a representation of M. Then it studies the group (called Lorentz group) of isometric self-maps of M with respect to the indefinite metric, and in finite rank case shows that the Lorentz group is a complete invariant for congruence relation. Furthermore, the Lorentz group contains an interesting abelian subgroup (called little Lorentz group) which turns out to be a finer invariant for M.

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

This paper is devoted to the existence of the traveling waves of the equations describing a diffusive susceptible-exposed-infected-recovered (SEIR) model. The existence of traveling waves depends on the basic reproduction rate and the minimal wave speed. We obtain a more precise estimation of the minimal wave speed of the epidemic model, which is of great practical value in the control of serious epidemics. The approach in this paper is to use the Schauder fixed point theorem and the Laplace transform. We also give some numerical results on the minimal wave speed.

This paper concerns with the existence of solutions for the following fractional Kirchhoff problem with critical nonlinearity: $${\left( {\int {\int {_{{\mathbb{R}^{2N}}}\frac{{{{\left| {u\left( x \right) - u\left( y \right)} \right|}^2}}}{{{{\left| {x - y} \right|}^{N + 2s}}}}dxdy} } } \right)^{\theta - 1}}{\left( { - \Delta } \right)^s}u = \lambda h\left( x \right){u^{p - 1}} + {u^{2_s^* - 1}} in {\mathbb{R}^N},$$ where (−Δ) s is the fractional Laplacian operator with 0 < s < 1, 2 s * = 2N/(N − 2s), N > 2s, p ∈ (1, 2 s *), θ ∈ [1, 2 s */2), h is a nonnegative function and λ a real positive parameter. Using the Ekeland variational principle and the mountain pass theorem, we obtain the existence and multiplicity of solutions for the above problem for suitable parameter λ > 0. Furthermore, under some appropriate assumptions, our result can be extended to the setting of a class of nonlocal integro-differential equations. The remarkable feature of this paper is the fact that the coefficient of fractional Laplace operator could be zero at zero, which implies that the above Kirchhoff problem is degenerate. Hence our results are new even in the Laplacian case.

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

Reflected Ornstein-Uhlenbeck process is a process that returns continuously and immediately to the interior of the state space when it attains a certain boundary. It is an extended model of the traditional Ornstein-Uhlenbeck process being extensively used in finance as a one-factor short-term interest rate model. In this paper, under certain constraints, we are concerned with the problem of estimating the unknown parameter in the reflected Ornstein-Uhlenbeck processes with the general drift coefficient. The methodology of estimation is built upon the maximum likelihood approach and the method of stochastic integration. The strong consistency and asymptotic normality of estimator are derived. As a by-product of the use, we also establish Girsanovs theorem of our model in this paper.

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.

The classical critical Trudinger-Moser inequality in ℝ2 under the constraint $\int_{\mathbb{R}{^2}} {\left( {{{\left| {\nabla u} \right|}^2} + {{\left| u \right|}^2}} \right)dx\;} \leqslant \;1$ was established through the technique of blow-up analysis or the rearrangement-free argument: for any τ > 0, it holds that $$\mathop {\sup }\limits_{\matrix{{u \in {H^1}\left( {{\mathbb{R}^2}} \right)} \cr {\int_{{\mathbb{R}^2}} {\left( {\tau {{\left| u \right|}^2} + {{\left| {{\nabla u}} \right|}^2}} \right)dx} \leqslant 1} \cr } } \int_{\mathbb{R}{^2}} {\left( {{\rm{e}^{4\pi }}^{{{\left| u \right|}^2}} - 1} \right)dx} \;\leqslant \;C\left( \tau \right) < + \infty,$$ and 4π is sharp. However, if we consider the less restrictive constraint $\int_{\mathbb{R}{^2}} {\left( {{{\left| {\nabla u} \right|}^2} + V\left( x \right){u^2}} \right)dx\;} \leqslant \;1$ , where V(x) is nonnegative and vanishes on an open set in ℝ2, it is unknown whether the sharp constant of the Trudinger-Moser inequality is still 4π. The loss of a positive lower bound of the potential V(x) makes this problem become fairly nontrivial. The main purpose of this paper is two-fold. We will first establish the Trudinger-Moser inequality $$\mathop {\sup }\limits_{u \in {H^1}\left( {\mathbb{R}{^2}} \right),\;\int_{\mathbb{R}{^2}} {\left( {{{\left| {\nabla u} \right|}^2} + V\left( x \right){u^2}} \right)dx} \leqslant 1} \int_{\mathbb{R}{^2}} {\left( {{\rm{e}^{4\pi {u^2}}} - 1} \right)dx} \;\leqslant \;C\left( V \right) < \infty,$$ when V is nonnegative and vanishes on an open set in ℝ2. As an application, we also prove the existence of ground state solutions to the following Schrodinger equations with critical exponential growth (0.1) $$ - {\rm{\Delta }} u + V\left( x \right)u = f\left( u \right)\;\;\;\;{\rm{in}}\;\mathbb{R}{^2},$$ where V(x) ⩾ 0 and vanishes on an open set of ℝ2 and f has critical exponential growth. Having the positive constant lower bound for the potential V(x) (e.g., the Rabinowitz type potential) has been the standard assumption when one deals with the existence of solutions to the above Schroodinger equations when the nonlinear term has the exponential growth. Our existence result seems to be the first one without this standard assumption.

In this paper we derive a relative volume comparison of Ricci flow under a certain local curvature condition. It is a refinement of Perelman’s no local collapsing theorem in Perelman (2002).

In this paper, we study the mean curvature flow with oblique derivative boundary conditions. We prove the longtime existence by choosing a suitable auxiliary function. Also, we prove the asymptotic behavior of the mean curvature flow with zero oblique derivative boundary data which is a generalization of Huisken’s original result about prescribed perpendicular contact angle.

In this paper, we propose a new correlation, called stable correlation, to measure the dependence between two random vectors. The new correlation is well defined without the moment condition and is zero if and only if the two random vectors are independent. We also study its other theoretical properties. Based on the new correlation, we further propose a robust model-free feature screening procedure for ultrahigh dimensional data and establish its sure screening property and rank consistency property without imposing the subexponential or sub-Gaussian tail condition, which is commonly required in the literature of feature screening. We also examine the finite sample performance of the proposed robust feature screening procedure via Monte Carlo simulation studies and illustrate the proposed procedure by a real data example.

The testing covariance equality is of importance in many areas of statistical analysis, such as microarray analysis and quality control. Conventional tests for the finite-dimensional covariance do not apply to high-dimensional data in general, and tests for the high-dimensional covariance in the literature usually depend on some special structure of the matrix and whether the dimension diverges. In this paper, we propose a jack-knife empirical likelihood method to test the equality of covariance matrices. The asymptotic distribution of the new test is regardless of the divergent or fixed dimension. Simulation studies show that the new test has a very stable size with respect to the dimension and it is also more powerful than the test proposed by Schott (2007) and studied by Srivastava and Yanagihara (2010). Furthermore, we illustrate the method using a breast cancer dataset.

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.

This paper continues to derive the globally hyperbolic moment model of arbitrary order for the three-dimensional special relativistic Boltzmann equation with the Anderson-Witting collision. The method is the model reduction by the operator projection. Finding an orthogonal basis of the weighted polynomial space is crucial and built on infinite families of the complicate relativistic Grad type orthogonal polynomials depending on a parameter and the real spherical harmonics instead of the irreducible tensors. We study the properties of those functions carefully, including their recurrence relations, their derivatives with respect to the independent variable and parameter, and the zeros of the orthogonal polynomials. Our moment model is proved to be globally hyperbolic and linearly stable. Moreover, the Lorentz-covariance and the quasi-one-dimensional case, the non-relativistic and ultra-relativistic limits are also studied.

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

Unique expansions in non-integer bases have been investigated in many papers during the last thirty years. They are often conveniently generated by labeled directed graphs. We give a precise description of the set of sequences generated by these graphs. This provides a geometric explanation of many former abstract results in this domain. Our results are illustrated by many examples.

In this paper, a topological classification of piecewise monotone functions (abbreviated as PM functions) with nonmonotonicity height equal to 1 which are strictly increasing on their characteristic intervals and have finitely many fixed points is presented.

In this paper, we apply local discontinuous Galerkin methods to the pattern formation dynamical model in polymerizing action flocks. Optimal error estimates for the density and filament polarization in different norms are established. We use a semi-implicit spectral deferred correction time method for time discretization, which allows a relative large time step and avoids computation of a Jacobian matrix. Numerical experiments are presented to verify the theoretical analysis and to show the capability for simulations of action wave formation.

For an element A in a unital C*-algebra ℬ, the operator-valued 1-form ωa(z) = (z − A)−1dz is analytic on the resolvent set ρ(A), which plays an important role in the functional calculus of A. This paper defines a class of Hermitian metrics on ρ(A) through the coupling of the operator-valued (1, 1)-form ΩA = −ω*A Λ ωa with tracial and vector states. Its main goal is to study the connection between A and the properties of the metric concerning curvature, arc length, completeness and singularity. A particular example is when A is quasi-nilpotent, in which case the metric lives on the punctured complex plane ℂ {0}. The notion of the power set is defined to gauge the “blow-up” rate of the metric at 0, and examples are given to indicate a likely link with A’s hyper-invariant subspaces.

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.

In this paper, we consider the stability, semi-stability and canonical metric structures on transverse Higgs bundles over a class of foliation manifolds, also a transversal Bogomolov inequality is obtained.

In this paper we prove the strong unique continuation property for a class of fourth order elliptic equations involving strongly singular potentials. Our argument is to establish some Hardy-Rellich type inequalities with boundary terms and introduce an Almgren’s type frequency function to show some doubling conditions for the solutions to the above-mentioned equations.

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.

Fractional operators are widely used in mathematical models describing abnormal and nonlocal phenomena. Although there are extensive numerical methods for solving the corresponding model problems, theoretical analysis such as the regularity result, or the relationship between the left-side and right-side fractional operators is seldom mentioned. Instead of considering the fractional derivative spaces, this paper starts from discussing the image spaces of Riemann-Liouville fractional integrals of LP(Ω) functions, since the fractional derivative operators that are often used are all pseudo-differential. Then the high regularity situation—the image spaces of Riemann-Liouville fractional integral operators on the Wm,p(Ω) space is considered. Equivalent characterizations of the defined spaces, as well as those of the intersection of the left-side and right-side spaces are given. The behavior of the functions in the defined spaces at both the nearby boundary point/points and the points in the domain is demonstrated in a clear way. Besides, tempered fractional operators are shown to be reciprocal to the corresponding Riemann-Liouville fractional operators, which is expected to contribute some theoretical support for relevant numerical methods. Last, we also provide some instructions on how to take advantage of the introduced spaces when numerically solving fractional equations.

The generalized conjugate phase retrieval problem aims to reconstruct a complex signal x ∈ ℂn from quadratic measurements x* A1x,…, x* Amx, where A1,…, Am ∈ ℝn×n are real symmetric matrices. The equivalent formulation for generalized conjugate phase retrieval along with the minimal measurement number required for accurate retrieval (up to a global phase factor as well as conjugacy) are derived in this paper. We present a set of nine vectors in ℝ4 and prove that it is conjugate phase retrievable on ℂ4. This result implies the measurement number bound 4n − 6 is not optimal for some n, which confirms a conjecture in the article by Evans and Lai (2019).

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

for a proper subdomain D of $${{\overline {\mathbb{R}}} ^n}$$ and for all x,y ∈ D define $${\mu _D}\left({x,\,y} \right) = \mathop {\inf}\limits_{{C_{xy}}} \,\,{\rm{Cap}}\left({D,\,{C_{xy}}} \right),$$ where the infimum is taken over all curves Cxy = γ[0, 1] in D with γ(0) = x and γ(1) = y, and Cap denotes the conformal capacity of condensers. The quantity μD is a metric if and only if the domain D has a boundary of positive conformal capacity. If Cap(∂D) > 0, we call μD the modulus metric of D. Ferrand et al. (1991) have conjectured that isometries for the modulus metric are conformal mappings. Very recently, this conjecture has been proved for n = 2 by Betsakos and Pouliasis (2019). In this paper, we prove that the conjecture is also true in higher dimensions n ⩾ 3.

The Bourgain-Sarnak-Ziegler (BSZ) criterion was initially established for an arbitrary multiplicative function a(n) satisfying |a(n)| ⩽ 1. Recently, Cafferata et al. (2020) showed that the condition |a(n)| ⩽ 1 can be replaced by a(p) 1 to some extent. In this paper, we formulate and prove a further extension of the BSZ criterion, in which the restriction a(p) ⩽ 1 is removed. As applications, we use it together with the analytic theory of automorphic L-functions to prove that there exist some cancellations in the sequence {λπ(n)e(nkα)}n⩽1 on GLm and the Mobius function is disjoint from this sequence.

This paper deals with numerical solutions of nonlinear stiff stochastic differential equations with jump-diffusion and piecewise continuous arguments. By combining compensated split-step methods and balanced methods, a class of compensated split-step balanced (CSSB) methods are suggested for solving the equations. Based on the one-sided Lipschitz condition and local Lipschitz condition, a strong convergence criterion of CSSB methods is derived. It is proved under some suitable conditions that the numerical solutions produced by CSSB methods can preserve the mean-square exponential stability of the corresponding analytical solutions. Several numerical examples are presented to illustrate the obtained theoretical results and the effectiveness of CSSB methods. Moreover, in order to show the computational advantage of CSSB methods, we also give a numerical comparison with the adapted split-step backward Euler methods with or without compensation and tamed explicit methods.

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 investigate the smoothness of invariant manifolds and foliations for random dynamical systems with nonuniform pseudo-hyperbolicity in Hilbert spaces. We discuss on the effect of temperedness and the spectral gaps in the nonuniform pseudo-hyperbolicity so as to prove the existence of invariant manifolds and invariant foliations, which preserve the CN,τ(ω) HOlder smoothness of the random system in the space variable and the measurability of the random system in the sample point. Moreover, we also prove that the stable foliation is CN−1,τ(ω) in the base point.

In this paper, we compare the first order fractional GJMS (see Graham et al. (1992)) operator P1 with the conformal Laplacian P2, on the conformal infinity of a Poincare-Einstein manifold. We derive some inequalities between the Yamabe constants and the first eigenvalues associated with P1 and P2, and prove some rigidity theorems by characterizing the equalities. Similarly, some comparison theorems between P2 and the Paneitz operator P4 or the 6th order GJMS operator P6 are also given.

Recently, iteratively reweighted methods have attracted much interest in compressed sensing, outperforming their unweighted counterparts in most cases. In these methods, decision variables and weights are optimized alternatingly, or decision variables are optimized under heuristically chosen weights. In this paper, we present a novel weighted $l_1$-norm minimization problem for the sparsest solution of underdetermined linear equations. We propose an iteratively weighted thresholding method for this problem, wherein decision variables and weights are optimized simultaneously. Furthermore, we prove that the iteration process will converge eventually. Using the homotopy technique, we enhance the performance of the iteratively weighted thresholding method. Finally, extensive computational experiments show that our method performs better in terms of both running time and recovery accuracy compared with some state-of-the-art methods.