Acta Mathematica Sinica-English Series最新文献

In this paper, we establish Schrödinger maximal estimates associated with the finite type phase

where m ≥ 4 is an even number. Following [12], we prove an L² fractal restriction estimate associated with the surface

as the main result, which also gives results on the average Fourier decay of fractal measures associated with these surfaces. The key ingredients of the proof include the rescaling technique from [16], Bourgain–Demeter’s ℓ² decoupling inequality, the reduction of dimension arguments from [17] and induction on scales. We notice that our Theorem 1.1 has some similarities with the results in [8]. However, their results do not cover ours. Their arguments depend on the positive definiteness of the Hessian matrix of the phase function, while our phase functions are degenerate.

In this paper, an infinite dimensional KAM theorem with unbounded perturbations and double normal frequencies is established under qualitative non-degenerate conditions. This is an extension of the degenerate KAM theorem with bounded perturbations by Bambusi, Berti, Magistrelli, and us. As applications, for derivative nonlinear Schrödinger equation with periodic boundary conditions, quasi-periodic solutions around constant solutions are obtained.

In this paper, we consider the distributed inference for heterogeneous linear models with massive datasets. Noting that heterogeneity may exist not only in the expectations of the subpopulations, but also in their variances, we propose the heteroscedasticity-adaptive distributed aggregation (HADA) estimation, which is shown to be communication-efficient and asymptotically optimal, regardless of homoscedasticity or heteroscedasticity. Furthermore, a distributed test for parameter heterogeneity across subpopulations is constructed based on the HADA estimator. The finite-sample performance of the proposed methods is evaluated using simulation studies and the NYC flight data.

A cover of a graph G is a graph H with vertex set V (H) = ∪_v∈V(G) L_v, where L_v = {v} × [s], and the edge set M = ∪_uv∈E(G) M_uv, where M_uv is a matching between L_u and L_v. A vertex set T ⊆ V (H) is a transversal of H if ∣T ∩ L_v∣ = 1 for each v ∈ V(G). Let f be a nonnegative integer valued function on the vertex-set of H. If for any nonempty subgraph Γ of H[T], there exists a vertex x ∈ V (H) such that d(x) < f(x), then T is called a strictly f-degenerate transversal. In this paper, we give a sufficient condition for the existence of strictly f-degenerate transversal for planar graphs without chorded 6-cycles. As a consequence, every planar graph without subgraphs isomorphic to the configurations is DP-4-colorable.

In this paper, we introduce a new concept of expansiveness, similar to the separating property. Specifically, we consider a compact Riemannian manifold M without boundary and a C¹ vector field X on M, which generates a flow φ_t on M. We say that X is rescaling separating on a compact invariant set Λ of X if there is a constant δ > 0 such that, for any x, y ∈ Λ, if d(φ_t(x), φ_t(y)) ≤ δ∥X (φ_t(x))∥ for all t ∈ ℝ, then y ∈ Orb(x). We prove that if X is rescaling separating on Λ and every singularity of X in Λ is hyperbolic, then any C¹ vector field Y, whose flow commutes with φ_t on Λ, must be collinear to X on Λ. As applications of this result, we show that the centralizer of a rescaling separating C¹ vector field without nonhyperbolic singularity is quasi-trivial. We also proved that there is an open and dense set ({cal U} subset {{cal X}^{1}}(M)) such that for any star vector field (X in {cal U}), the centralizer of X is collinear to X on the chain recurrent set of X.

A scattered operator is a bounded linear operator with at most countable spectrum. In this paper, we prove that for any elementary operator on ({cal B}({cal H})), not only for finite length but also for infinite length, if the range of the elementary operator is contained in scattered operators, then the corresponding sum of multipliers is a compact operator. We also prove that for some special classes of elementary operators, such as the elementary operators of length 2, higher order inner derivations and generalized inner derivation, if the range of the elementary operator is contained in the set of scattered operators, then the range is contained in the set of power compact operators. At the same time, the multipliers of the corresponding elementary operators are characterized.

In the spirit of Morse homology initiated by Witten and Floer, we construct two ∞-categories ({cal A}) and ({cal B}). The weak one ({cal A}) comes out of the Morse–Smale pairs and their higher homotopies, and the strict one ({cal B}) concerns the chain complexes of the Morse functions. Based on the boundary structures of the compactified moduli space of gradient flow lines of Morse functions with parameters, we build up a weak ∞-functor ({cal F}:{cal A} rightarrow {cal B}). Higher algebraic structures behind Morse homology are revealed with the perspective of defects in topological quantum field theory.

In this paper, we first give characterizations of weighted Besov spaces with variable exponents via Peetre’s maximal functions. Then we obtain decomposition characterizations of these spaces by atom, molecule and wavelet. As an application, we obtain the boundedness of the pseudo-differential operators on these spaces.

Let (mathbb{B}) be a unit ball in ℝ², (W_{0}^{1,2}(mathbb{B})) be the standard Sobolev space. For any ϵ > 0, de Figueiredo, do Ó, dos Santons, Yang and Zhu proved the existence of extremals of a Trudinger-Moser inequality in the unit ball. Precisely,

can be attained by some radially symmetric function (u_{epsilon}in W_{0}^{1,2}(mathbb{B})) with (int_{mathbb{B}}vertnabla u_{epsilon}vert^{2}dx=1). In this note, we concern the compactness of the function family {u_ϵ}_ϵ>0 and prove that up to a subsequence u_ϵ converges to some function u₀ in (C^{1}(overline{mathbb{B}})) as ϵ → 0. Furthermore, u₀ is an extremal function of the supremum

Let us explain the result in geometry. Denote (omega_{0}=dx_{1}^{2}+dx_{2}^{2}) be the standard Euclidean metric. Define a conical metric (omega_{epsilon}=vert xvert^{2epsilon}omega_{0}) for (xinmathbb{B}). Then the extremal family {u_ϵ}_ϵ>0 of the following Trudinger-Moser functionals

under the constraint (uin W_{0}^{1,2}(mathbb{B})) and (int_{mathbb{B}}vertnabla_{omega_{epsilon}u}vert^{2}dv_{omega_{epsilon}}leq 1) is compact as ϵ → 0.

Let L/F be a finite Galois extension of number fields of degree n and let p be a prime which does not divide n. We shall study the p^j-rank of (K_{2i}(mathcal{O}_{L})) via its Galois module structure following the approaches of Iwasawa and Komatsu–Nakano. Along the way, we generalize previous observations of Browkin, Wu and Zhou on K₂-groups to higher even K-groups. We also give examples to illustrate our results. Finally, we apply our discussion to refine a result of Kitajima pertaining to the p-rank of even K-groups in the cyclotomic ℤ_l-extension, where l ≠ p.