Acta Mathematica Sinica-English Series最新文献

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.

We prove that there exists an open and dense subset ({cal U}) in the space of C² expanding self-maps of the circle ({mathbb T}) such that the Lyapunov minimizing measures of any (T in {cal U}) are uniquely supported on a periodic orbit. This answers a conjecture of Jenkinson-Morris in the C² topology.

In this paper, we prove the Wulff–Gage isoperimetric inequality for origin-symmetric convex bodies and the uniqueness of the log-Minkowski problem in ℝ². Then we give a new proof of the log-Minkowski inequality of curvature entropy for origin-symmetric convex bodies with C² boundaries.

The case–cohort design has been widely used to reduce the cost of covariate measurements in large cohort studies. In this paper, we study the high-dimensional accelerated failure time (AFT) model under the case–cohort design. Based on ℓ₀-regularization and a newly defined weight function, we propose a weighted least squares procedure for variable selection and parameter estimation. Computationally, we develop a support detection and root finding (SDAR) algorithm, where the support is first determined based on the primal and dual information, then the estimator is obtained by solving the weighted least squares problem restricted to the estimated support. We show the proposed algorithm is essentially one Newton-type algorithm, thus it is more efficient and stable compared with other regularized methods. Theoretically, we establish a sharp error bound for the solution sequences generated from the proposed method. Furthermore, we propose an adaptive version of the proposed SDAR algorithm, which determines the support size of the estimated coefficient in a data-driven manner. Extensive simulation studies demonstrate the superior performance of the proposed procedures, especially for the computational efficiency. As an illustration, we apply the proposed method to a malignant breast tumor gene expression data.

The present paper is devoted to studying local derivations on the N = 2 super-BMS₃ algebra based on such researches for the super Virasoro algebra and the Lie algebra W(2, 2). We prove that every local derivation on the N = 2 super-BMS₃ algebra is a derivation. It contributes to determine all local derivations on the general truncated super Virasoro algebras.