Acta Mathematica Sinica-English Series最新文献

In this article, we study the hyperbolic dynamics of geodesic flows on Riemannian (not necessarily compact) manifolds with no conjugate points. By hyperbolic dynamics, we focus on the Anosov Closing Lemma, the local product structure, and the transitivity of the geodesic flows on the set of rank 1 non-wandering set Ω₁ under the conditions of bounded asymptote and uniform visibility. As an application, we further discuss on some generic properties of the set of invariant probability measures.

Federated learning (FL) has becoming a prevailing paradigm which enables small-scale devices to collaboratively learn a shared model efficiently and trains a machine learning model without exchanging data. However, though the original data never leave the local machines in federated learning, possible privacy leakage still exists. To make strong privacy guarantee, in this paper, we incorporate the notion of differential privacy (DP) to study the federated averaging (FedAvg) algorithm. In particular, by adding calibrated gaussian noise, we propose a set of differentially private federated averaging algorithms (DP-FedAvg) under the full and partial participation schemes. We provide tight analysis of the privacy bound by using advanced composition and privacy amplification techniques. We also analyze the convergence bound of DP-FedAvg without having the assumptions: (i) the data are the independent identically distribution (IID), and (ii) all the devices are active. It turns out that the convergence rate is consistent with the one without DP guarantee. The effectiveness of our algorithms is demonstrated by synthetic and real datasets.

Let G be a complex, connected reductive Lie group which is the complexification of a compact Lie group K. Let M be a ℚ-Fano G-compactification. In this paper, we first prove a uniqueness result of K × K-invariant (singular) Kähler–Einstein metrics on M. Then we show the existence of (singular) Kähler–Einstein metric implies properness of the reduced Ding functional. This gives a refinement of the properness conjecture on group compactifications.

This paper discusses variable selection for interval-censored failure time data, a general type of failure time data that commonly arise in many areas such as clinical trials and follow-up studies. Although some methods have been developed in the literature for the problem, most of the existing procedures apply only to specific models. In this paper, we consider the data arising from a general class of partly linear additive generalized odds rate models and propose a penalized variable selection approach through maximizing a derived penalized likelihood function. In the method, the Bernsetin polynomials are employed to approximate both the unknown baseline hazard functions and the nonlinear covariate effects functions, and for the implementation of the method, a coordinate descent algorithm is developed. Also the asymptotic properties of the proposed estimators, including the oracle property, are established. An extensive simulation study is conducted to assess the finite-sample performance of the proposed estimators and indicates that it works well in practice. Finally, the proposed method is applied to a set of real data on Alzheimer’s disease.

Testing the association of two high-dimensional random vectors is of fundamental importance in the statistical theory and applications. In this paper, we propose a new test statistic based on the Frobenius norm and subtracting bias technique, which is generally applicable to high-dimensional data without restricting the distributional Assumptions. The limiting null distribution of the proposed test is shown to be a random variable combining a finite chi-squared-type mixture with a normal approximation. Our proposed test method can also be a normal approximation or a finite chi-squared-type mixtures under additional regularity conditions. To make the test statistic applicable, we introduce a wild bootstrap method and demonstrate its validity. The finite-sample performance of the proposed test via Monte Carlo simulations reveals that it performs better at controlling the empirical size than some existing tests, even when the normal approximation is invalid. Real data analysis is devoted to illustrating the proposed test.

Consider the Kirchhoff equation with Hartree type nonlinearity

where a, b > 0, λ, μ ∈ ℝ, 2 < q < 6, 0 < α < 3, and I_α is the Riesz potential integral operator of order α. Solutions with prescribed mass ({|u|_{{L^2}({{mathbb{R}^3}})}} = c > 0), also known as normalized solutions, are of particular interest in the current paper. Under various assumptions on μ, c and q, we establish the existence, nonexistence and asymptotic behavior of normalized solutions for the above elliptic equation.

Hamel functions and Funk functions of a spray are generalization of locally projectively flat Finsler metrics and Funk metrics respectively. In this paper, we study sprays on Hamel or Funk functions model. We use the Funk metric to construct a family of sprays and obtain some of their curvature properties and metrizability conditions. We prove that there exist local Funk functions on a R-flat spray manifold. On certain projectively flat Berwald spray manifolds, we construct a multitude of nonzero Funk functions. We introduce a new class of sprays called Hamel or Funk sprays associated to given sprays and Hamel or Funk functions, and then obtain some special properties of a Hamel or Funk spray of scalar curvature, especially on its metrizability and the special form of its Riemann curvature.

This note concerns the rapid uniform convergence of Cesàro weighted Birkhoff averages via irrational rotations on tori under certain conditions, involving arbitrary polynomial and exponential convergence rates. We discuss both finite dimensional and infinite dimensional cases, and give Diophantine rotations as examples. These provide the universality of rapid convergence for Cesàro weighted type, which is quite different from L^p (p > 1) convergence for the unweighted one. We also show a certain optimality about our convergence rate. Besides, we introduce a multimodal weighted approach to adapt to the data sparsity, which still preserves exponential convergence.

We investigate the incidence algebras arising from one-branch extensions of “rectangles”. There are four different ways to form such extensions, and all four kinds of incidence algebras turn out to be derived equivalent. We provide realizations for all of them as endomorphism algebra of tilting modules or tilting complexes over a Nakayama algebra. Meanwhile, an unexpected derived equivalence between Nakayama algebras N(2r − 1, r) and N(2r − 1, r + 1) has been found. As an application, we obtain the explicit formulas of the Coxeter polynomials for a large family of Nakayama algebras, i.e., the Nakayama algebras N(n, r) with ({n over 2} < r < n).

This paper develops an Itô-type fractional pathwise integration theory for fractional Brownian motion with Hurst parameters (H in ({1 over 3}, {1 over 2}]), using the Lyons’ rough path framework. This approach is designed to fill gaps in conventional stochastic calculus models that fail to account for temporal persistence prevalent in dynamic systems such as those found in economics, finance, and engineering. The pathwise-defined method not only meets the zero expectation criterion but also addresses the challenges of integrating non-semimartingale processes, which traditional Itô calculus cannot handle. We apply this theory to fractional Black–Scholes models and high-dimensional fractional Ornstein–Uhlenbeck processes, illustrating the advantages of this approach. Additionally, the paper discusses the generalization of Itô integrals to rough differential equations (RDE) driven by fBM, emphasizing the necessity of integrand-specific adaptations in the Itô rough path lift for stochastic modeling.