Science China Mathematics

This is an expository article on the recent progress on the arithmetic fundamental lemma conjecture, based largely on Zhang (2019). Beside stating the local conjecture, we will present three global intersection problems along with some constructions of algebraic cycles.

Xem xét một nhóm nhóm con-đĩa symplectic S của một nhóm nhóm-đĩa symplectic (X, ω) có kích thước hữu hạn. Đặt Xa là phép thổi phồng trọng số-a của X dọc theo S, và Da = PNa là divisor ngoại lệ, trong đó N là bụng bình thường của S trong X. Trong bài viết này, chúng tôi chỉ ra rằng lý thuyết Gromov-Witten đĩa tuyệt đối của Xα có thể được tái cấu trúc một cách hiệu quả và độc nhất từ lý thuyết Gromov-Witten đĩa tuyệt đối của X, S và Dα, homomorphism hạn chế tự nhiên H*CR(X) → H*CR(S) và lớp Chern đầu tiên của gói line tautological trên Dα. Để đạt được điều này, chúng tôi trước tiên chứng minh các kết quả tương tự cho các lý thuyết Gromov-Witten đĩa tương đối của (Xα | Dα) và (Nα | Dα). Làm ứng dụng cho các kết quả này, chúng tôi chứng minh một phiên bản đĩa của giả thuyết của Maulik và Pandharipande (Topology, 2006) về lý thuyết Gromov-Witten của các phép thổi phồng dọc theo các giao diện hoàn chỉnh, một giả thuyết về lý thuyết Gromov-Witten của các cấu trúc gốc và một giả thuyết về kết quả Leray-Hirsch cho lý thuyết Gromov-Witten đĩa của Tseng và You (J Pure Appl Algebra, 2016).

Giorgi conjectured in 1979 that if a sequence of functionals converges in the sense of Г-convergence to a limiting functional, then the corresponding gradient flows will converge as well after changing timescale appropriately. It is shown that this conjecture holds true for a rather wide kind of functionals.

We obtain an equation among invariants obtained from the Alexander module of an amphicheiral link. For special cases, it deduces necessary conditions on the Alexander polynomial. By using the present results and some known results, we show that the Alexander polynomial of an algebraically split component-preservingly (±)-amphicheiral link with even components is zero, and we determine prime amphicheiral links with at least 2 components and up to 9 crossings.

Let X be an RD-space. In this paper, the authors establish the boundedness of the commutator T b f = bT f − T(bf) on L p , p ∈ (1,∞), where T is a Calderón-Zygmund operator related to the admissible function ρ and b ∈ BMO θ (X) ⊇ BMO(X). Moreover, they prove that T b is bounded from the Hardy space H 1 (X) into the weak Lebesgue space L weak 1 (X). This can be used to deal with the Schrödinger operators and Schrödinger type operators on the Euclidean space ℝ n and the sub-Laplace Schrödinger operators on the stratified Lie group $\mathbb{G}$ .

In this paper we define some weak martingale Hardy spaces and three kinds of weak atoms. They are the counterparts of martingale Hardy spaces and atoms in the classical martingale H p-theory. And then three atomic decomposition theorems for martingales in weak martingale Hardy spaces are proved. With the help of the weak atomic decompositions of martingale, a sufficient condition for a sublinear operator defined on the weak martingale Hardy spaces to be bounded is given. Using the sufficient condition, we obtain a series of martingale inequalities with respect to the weak L p-norm, the inequalities of weak (p,p)-type and some continuous imbedding relationships between various weak martingale Hardy spaces. These inequalities are the weak versions of the basic inequalities in the classical martingale H p-theory.

The solar radio spectra of the gyro-synchrotron radiation without absorption from mildly relativistic non-thermal electron have been presented using the model with the spherical symmetric nonhomogeneity. These spectra have the following features: (i) Twists and turns of the spectra, which appear at the joint between the medium and high frequencies, (ii) Flux density drop at the lower frequencies. Even if there are no absorption in the radio sources, flux density drop at the lower frequencies is still present because of magnetic nonhomogeneity. (iii) These theoretical spectra belong to C type. This provides us a new interpretation for the observed solar burst C type spectra.

Some inequalities for moments of partial sums of aB -valued strong mixing field are established and their applications to the weak and strong laws of large numbers and the complete convergences are discussed.

In this paper, we prove the local existence, uniqueness and stability of a supersonic shock for the supersonic isothermal incoming flow past a curved cone. Major difficulties include constructing an appropriate solution and treating the Neumann boundary conditions and local stability condition.

We show large deviation expansions for sums of independent and bounded from above random variables. Our moderate deviation expansions are similar to those of Cramér (1938), Bahadur and Ranga Rao (1960), and Sakhanenko (1991). In particular, our results extend Talagrand’s inequality from bounded random variables to random variables having finite (2 + δ)-th moments, where δ ∈ (0; 1]. As a consequence, we obtain an improvement of Hoeffding’s inequality. Applications to linear regression, self-normalized large deviations and t-statistic are also discussed.