Let RPⁿ be a bumpy and irreversible Finsler n-dimensional real projective space with reversibility λ and flag curvature K satisfying (λ/1+λ)² < K ≤ 1 when n is odd, and K ≥ 0 when n is even. We show that if there exist exactly 2[n+1/2] prime closed geodesics on such RPⁿ, then all of them are non-contractible, which coincides with the Katok's examples.

In this paper, we use a unified framework to study Poisson stable (including stationary, periodic, quasi-periodic, almost periodic, almost automorphic, Birkhoff recurrent, almost recurrent in the sense of Bebutov, Levitan almost periodic, pseudo-periodic, pseudo-recurrent and Poisson stable) solutions for semilinear stochastic differential equations driven by infinite dimensional Lévy noise with large jumps. Under suitable conditions on drift, diffusion and jump coefficients, we prove that there exist solutions which inherit the Poisson stability of coefficients. Further we show that these solutions are globally asymptotically stable in square-mean sense. Finally, we illustrate our theoretical results by several examples.

In this paper, we introduce a new notion of integrability for billiard tables, namely, integrability away from the boundary. One key feature of our notion is that the integrable region could be disjoint from the boundary with a positive distance. We prove that if a strictly convex billiard table, whose boundary is a small perturbation of an ellipse with small eccentricity, is integrable in this sense, then its boundary must be itself an ellipse.

In this paper we introduce two metrics: the max metric d_{n, q} and the mean metric d_{n, q}. We give an equivalent characterization of rigid measure preserving systems by the two metrics. It turns out that an invariant measure μ on a topological dynamical system (X, T) has bounded complexity with respect to d_{n, q} if and only if μ has bounded complexity with respect to d_{n, q} if and only if (X, B_X, μ, T) is rigid. We also obtain computation formulas of the measure-theoretic entropy of an ergodic measure preserving system (resp. the topological entropy of a topological dynamical system) by the two metrics d_{n, q} and d_{n, q}.

In this paper we review and systematize the index method in closed geodesic problem. As we know, the closed geodesic problem on compact Riemannian or Finsler manifold is a famous problem, and has far from been resolved. In recent years, the Maslov-type index theory for symplectic path has been applied to studying the closed geodesic problem, and has induced a great number of results on the multiplicity and stability of closed geodesics. We will systematically introduce these progresses in this review.

This paper is devoted to the study of the solitary wave solutions for the delayed coupled Higgs field equation
\begin{document}$\left\{\begin{array}{ll} u_{tt}-u_{xx}-\alpha u+\beta f*uu^2-2uv-\tau u(u^2)_x=0,\\ \displaystyle v_{tt}+v_{xx}-\beta(u^2)_{xx}=0.\nonumber \end{array} \right.$ \end{document}
We first establish the existence of solitary wave solutions for the corresponding equation without delay and perturbation by using the Hamiltonian system method. Then we consider the persistence of solitary wave solutions of the delayed coupled Higgs field equation by using the method of dynamical system, especially the geometric singular perturbation theory, invariant manifold theory and Fredholm theory. According to the relationship between solitary wave and homoclinic orbit, the coupled Higgs field equation is transformed into the ordinary differential equations with fast variables by using the variable substitution. It is proved that the equations with perturbation also possess homoclinic orbit, and thus we obtain the existence of solitary wave solutions of the delayed coupled Higgs field equation.

In this survey we will present the symbolic extension theory in topological dynamics, which was built over the past twenty years.

The well-known Hartman–Grobman Theorem says that a C¹ hyperbolic diffeomorphism F can be locally linearized by a homeomorphism Φ. For parameterized systems F_θ, known results show that the corresponding homeomorphisms Φ^θ exist uniquely in a functional space equipped with the supremum norm and depend continuously on the parameter θ. In this paper, we further extend the results to Hölder dependence of Φ^θ on θ by Pugh's strategy, but introducing a kind of special Hölder norm instead of the usual supremum norm in the proof to control the linear parts of F_θ. This requires a new Hölder linearization result for every F_θ.

In this paper, we investigate the orbital stability of the peaked solitons (peakons) for the modified Camassa–Holm equation with cubic nonlinearity. We consider a minimization problem with an appropriately chosen constraint, from which we establish the orbital stability of the peakons under H¹ ∩ W^{1, 4} norm.

This paper is devoted to studying the asymptotic behavior of the solution to nonlocal Fisher-KPP type reaction diffusion equations in heterogeneous media. The kernel K is assumed to depend on the media. First, we give an estimate of the upper and lower spreading speeds by generalized principal eigenvalues. Second, we prove the existence of spreading speeds in the case where the media is periodic or almost periodic by showing that the upper and lower generalized principal eigenvalues are equal.

For a monotone twist map, under certain non-degenerate condition, we showed the existence of infinitely many homoclinic and heteroclinic orbits between two periodic neighboring minimal orbits with the same rotation number, which indicates chaotic dynamics. Our results also apply to geodesics of smooth Riemannian metrics on the two-dimension torus.

Let X = G/Γ be a homogeneous space with ambient group G containing the group H = (SO(n, 1))^k and x∈ X be such that Hx is dense in X. Given an analytic curve φ: I=[a, b] → H, we will show that if φ satisfies certain geometric condition, then for a typical diagonal subgroup A ={a(t): t ∈ R} ? H the translates {a(t)φ(I)x: t >0} of the curve φ(I)x will tend to be equidistributed in X as t → +∈∞. The proof is based on Ratner's theorem and linearization technique.

We give a brief survey on the dynamics of vector fields with singularities. The aim of this survey is not to list all results in this field, but only to introduce some results from several viewpoints and some technics.

This paper is concerned with the derivative nonlinear Schr¨ odinger equation with periodic boundary conditions. We obtain complete Birkhoff normal form of order six. As an application, the long time stability for solutions of small amplitude is proved.

In this paper, we consider the brake orbits of a reversible even Hamiltonian system near an equilibrium. Let the Hamiltonian system (HS) ?=JH'(x) satisfies H(0)=0, H'(0)=0, reversible and even conditions H(Nx)=H(x) and H(-x)=H(x) for all x∈ R²ⁿ. Suppose the quadratic form Q(x)=1/2〈H"(0)x, x〉 is non-degenerate. Fix τ₀>0 and assume that R²ⁿ=E⊕F decomposes into linear subspaces E and F which are invariant under the flow associated to the linear system ?=JH"(0)x and such that each solution of the above linear system in E is τ₀-periodic whereas no solution in F \ {0} is τ₀-periodic. Write σ(τ₀)=σ_Q(τ₀) for the signature of Q_E. If σ(τ₀)≠0, we prove that either there exists a sequence of brake orbits x_k→ 0 with τ_k-periodic on the hypersurface H^-1(0) where τ_k→ τ₀; or for each λ close to 0 with λ σ(τ₀)>0 the hypersurface H^-1(λ) contains at least 1/2σ(τ₀) distinct brake orbits of the Hamiltonian system (HS) near 0 with periods near τ₀. Such result for periodic solutions was proved by Bartsch in 1997.

In this paper, we construct random horseshoes of Anosov systems driven by an equicontinuous system based on an ergodic measure with positive entropy.

In this paper, we define mean index for non-periodic orbits in Hamiltonian systems and study its properties. In general, the mean index is an interval in R which is uniformly continuous on the systems. We show that the index interval is a point for a quasi-periodic orbit. The mean index can be considered as a generalization of rotation number defined by Johnson and Moser in the study of almost periodic Schr¨ odinger operators. Motivated by their works, we study the relation of Fredholm property of the linear operator and the mean index at the end of the paper.

We study the homology of the dual de Rham complex as functors on the category of abelian groups. We give a description of homology of the dual de Rham complex up to degree 7 for free abelian groups and present a corrected version of the proof of Jean’s computations of the zeroth homology group.

This is a set of lecture notes for the first author’s lectures on the difference equations in 2019 at the Institute of Advanced Study for Mathematics at Zhejiang University. We focus on explicit computations and examples. The convergence of local solutions is discussed.

We give an overview of five rationalization theories for spaces (Bousfield-Kan’s ?-completion; Sullivan’s rationalization; Bousfield’s homology rationalization; Casacuberta-Peschke’s Ω-rationalization; Gómez-Tato-Halperin-Tanré’s π₁-fiberwise rationalization) that extend the classical rationalization of simply connected spaces. We also give an overview of the corresponding rationalization theories for groups (?-completion; H?-localization; Baumslag rationalization) that extend the classical Malcev completion.

We provide a construction of the moduli space of stable coherent sheaves in the world of non-archimedean geometry, where we use the notion of Berkovich non-archimedean analytic spaces. The motivation for our construction is Tony Yue Yu’s non-archimedean enumerative geometry in Gromov—Witten theory. The construction of the moduli space of stable sheaves using Berkovich analytic spaces will give rise to the non-archimedean version of Donaldson—Thomas invariants. In this paper we give the moduli construction over a non-archimedean field K. We use the machinery of formal schemes, that is, we define and construct the formal moduli stack of (semi)-stable coherent sheaves over a discrete valuation ring R, and taking generic fiber we get the non-archimedean analytic moduli space of semistable coherent sheaves over the fractional non-archimedean field K. We generalize Joyce’s d-critical scheme structure in [37] or Kiem—Li’s virtual critical manifolds in [38] to the world of formal schemes, and Berkovich non-archimedean analytic spaces. As an application, we provide a proof for the motivic localization formula for a d-critical non-archimedean K-analytic space using global motive of vanishing cycles and motivic integration on oriented formal d-critical schemes. This generalizes Maulik’s motivic localization formula for the motivic Donaldson—Thomas invariants.

In this article, we give a generalization of δ-twisted homology introduced by Jingyan Li, Vladimir Vershinin and Jie Wu, called Δ-twisted homology, which enriches the theory of δ-(co)homology introduced by Alexander Grigor’yan, Yuri Muranov and Shing-Tung Yau. We show that the Mayer—Vietoris sequence theorem holds for Δ-twisted homology. Applying the Δ-twisted ideas to Cartesian products, we introduce the notion of Δ-twisted Cartesian product on simplicial sets, which generalizes the classical work of Barratt, Gugenheim and Moore on twisted Cartesian products of simplicial sets. Under certain hypothesis, we show that the coordinate projection of Δ-twisted Cartesian product admits a fibre bundle structure.

Let M and N be topological spaces, let G be a group, and let τ: G × M → M be a proper free action of G. In this paper, we define a Borsuk—Ulam-type property for homotopy classes of maps from M to N with respect to the pair (G, τ) that generalises the classical antipodal Borsuk—Ulam theorem of maps from the n-sphere Sⁿ to Rⁿ. In the cases where M is a finite pathwise-connected CW-complex, G is a finite, non-trivial Abelian group, τ is a proper free cellular action, and N is either R² or a compact surface without boundary different from S² and RP², we give an algebraic criterion involving braid groups to decide whether a free homotopy class β ∈ [M, N] has the Borsuk—Ulam property. As an application of this criterion, we consider the case where M is a compact surface without boundary equipped with a free action τ of the finite cyclic group Z_n. In terms of the orientability of the orbit space M_τ of M by the action τ, the value of n modulo 4 and a certain algebraic condition involving the first homology group of M_τ, we are able to determine if the single homotopy class of maps from M to R² possesses the Borsuk—Ulam property with respect to (Z_n, τ). Finally, we give some examples of surfaces on which the symmetric group acts, and for these cases, we obtain some partial results regarding the Borsuk—Ulam property for maps whose target is R².

In this article we estimate mutual distances of simple choreographic solutions for the Newtonian three-body problem. Explicit formulas will be proved and our applications include the famous figure-8 orbit.

We establish a blowing down criterion in the context of birational symplectic geometry in dimension 6.

Let K_n denote the complete graph consisting of n vertices, every pair of which forms an edge. We want to know the least possible number of the intersections, when we draw the graph K_n on the plane or on the sphere using continuous arcs for edges. In a paper published in 1960, Richard K. Guy conjectured that the least possible number of the intersections is 1/64(n-1)²(n-3)² if n is odd, or 1/64(n-2)²(n-4)² if n is even. A virgin road V_n is a drawing of a Hamiltonian cycle in K_n consisting of n vertices and n edges such that no other edge-representing arcs cross V_n. A drawing of K_n is called virginal if it contains a virgin road. All drawings considered in this paper will be virginal with respect to a fixed virgin road V_n. We will present a certain drawing of a subgraph of K_n, for each n(≥ 5), which is “characteristic” in the sense that any minimal virginal drawing of K_n containing this subdrawing satisfies Guy’s conjecture.

In this paper we study the cubical stuctures and fundamental groups of real toric spaces. We give an explicit presentation of the fundamental group of the real toric space over a simple polytope. Then using this presentation, we give a description of the existence of non-degenerate colourings on a simple polytope from a homotopy point of view.

With the great advancement of experimental tools, a tremendous amount of biomolecular data has been generated and accumulated in various databases. The high dimensionality, structural complexity, the nonlinearity, and entanglements of biomolecular data, ranging from DNA knots, RNA secondary structures, protein folding configurations, chromosomes, DNA origami, molecular assembly, to others at the macromolecular level, pose a severe challenge in their analysis and characterization. In the past few decades, mathematical concepts, models, algorithms, and tools from algebraic topology, combinatorial topology, computational topology, and topological data analysis, have demonstrated great power and begun to play an essential role in tackling the biomolecular data challenge. In this work, we introduce biomolecular topology, which concerns the topological problems and models originated from the biomolecular systems. More specifically, the biomolecular topology encompasses topological structures, properties and relations that are emerged from biomolecular structures, dynamics, interactions, and functions. We discuss the various types of biomolecular topology from structures (of proteins, DNAs, and RNAs), protein folding, and protein assembly. A brief discussion of databanks (and databases), theoretical models, and computational algorithms, is presented. Further, we systematically review related topological models, including graphs, simplicial complexes, persistent homology, persistent Laplacians, de Rham—Hodge theory, Yau—Hausdorff distance, and the topology-based machine learning models.

It is already known that finitely-generated Fuchsian groups are profinitely rigid among all lattices of connected Lie groups by the result of Bridson, Conder and Reid. Hence the triangle groups are distinguished among themselves by their finite quotients. We focus on the question about quantifying the size of a quotient which separates two triangle groups and give an explicit upper bound.

In this note, we present an L² extension approach to the effectiveness result of strong openness property of multiplier ideal sheaves.

In this article, we first establish an asymptotically sharp Koebe type covering theorem for harmonic K-quasiconformal mappings. Then we use it to obtain an asymptotically Koebe type distortion theorem, a coefficients estimate, a Lipschitz characteristic and a linear measure distortion theorem of harmonic K-quasiconformal mappings. At last, we give some characterizations of the radial John disks with the help of pre-Schwarzian of harmonic mappings.

In this paper we present a necessary and sufficient condition to guarantee that the zeroextended function of the solution for the heat equation in a smaller cylinder is still the solution of the corresponding extension problem in a larger cylinder. We prove the results under the frameworks of classical solutions, strong solutions and weak solutions. Moreover, we generalize these results to uniformly parabolic equations of divergence form.

A generalized stepping stone model with Ξ-resampling mechanism is a two dimensional probability-measure-valued stochastic process whose moment dual is similar to that of the classical stepping stone model except that Kingman’s coalescent is replaced by Ξ-coalescent. We prove the existence of such a process by specifying its moments using the dual function-valued Ξ-coalescent process with geographical labels and migration, and then verifying a multidimensional Hausdorff moment problem. We also characterize the stationary distribution of the generalized stepping stone model and show that it is not reversible if the mutation operator is of uniform jump-type.

In this paper, we prove a Talagrand’s T₂ transportation cost-information inequality for the law of the nonlinear one-dimensional stochastic heat equation driven by a Gaussian noise, which is white in time and which has the covariance of a fractional Brownian motion with Hurst parameter H∈(1/4,1/2) in the space variable, on the continuous path space with respect to the weighted L²-norm.

Let {X_k,i;k≥1,i≥1} be an array of random variables,{X_k;k≥1} be a strictly stationary α-mixing sequence, where X_k=(X_k,1, X_k,2,...). Let {p_n;n≥1} be a sequence of positive integers such that c₁≤ n/p_n≤c₂, where c₁,c₂>0. In this paper, we obtain the asymptotic distributions of the largest entries L_n=max_1≤i<j≤pn|ρ_ij⁽ⁿ⁾| of the sample correlation matrices, where ρ_ij⁽ⁿ⁾ denotes the Pearson correlation coefficient between X⁽ⁱ⁾ and X^(j), X⁽ⁱ⁾=(X_1,i,X_2,i,...). The asymptotic distributions of Ln is derived by using the Chen–Stein Poisson approximation method.

Propensity score is widely used to estimate treatment effects in observational studies. The covariate adjustment using propensity score is the most straightforward method in the literature of causal inference. In this article, we estimate the survival treatment effect with covariate adjustment using propensity score in the semiparametric accelerated failure time model. We establish the asymptotic properties of the proposed estimator by simultaneous estimating equations. We conduct simulation studies to evaluate the finite sample performance of the proposed method. A real data set from the German Breast Cancer Study Group is analyzed to illustrate the proposed method.

The Accelerated Hermitian/skew-Hermitian type Richardson (AHSR) iteration methods are presented for solving non-Hermitian positive definite linear systems with three schemes, by using Anderson mixing. The upper bounds of spectral radii of iteration matrices are studied, and then the convergence theories of the AHSR iteration methods are established. Furthermore, the optimal iteration parameters are provided, which can be computed exactly. In addition, the application to the model convection-diffusion equation is depicted and numerical experiments are conducted to exhibit the effectiveness and confirm the theoretical analysis of the AHSR iteration methods.

On the basis of the spectral analysis of the 4×4 matrix Lax pair, the initial value problem of the coherently-coupled nonlinear Schrödinger system is transformed into a 4×4 matrix Riemann–Hilbert problem. By using the nonlinear steepest decent method, the long-time asymptotics of the solution of the initial value problem for the coherently-coupled nonlinear Schrödinger system is obtained through deforming the Riemann–Hilbert problem into a solvable model one.

Suppose the ground field F is an algebraically closed field characteristic of p>2. In this paper, we investigate the restricted cohomology theory of restricted Lie superalgebras. Algebraic interpretations of low dimensional restricted cohomology of restricted Lie superalgebra are given. We show that there is a family of restricted model filiform Lie superalgebra L^λp,p structures parameterized by elements λ∈F^p. We explicitly describe both the 1-dimensional ordinary and restricted cohomology superspaces of L^λp,p with coefficients in the 1-dimensional trivial module and show that these superspaces are equal. We also describe the 2-dimensional ordinary and restricted cohomology superspaces of L^λp,p with coefficients in the 1-dimensional trivial module and show that these superspaces are unequal.