Acta Mathematica Scientia最新文献

In this paper, conjugate k-holomorphic functions and generalized k-holomorphic functions are defined in the two-dimensional complex space, and the corresponding Riemann boundary value problems and the inverse problems are discussed on generalized bicylinders. By the characteristics of the corresponding functions and boundary properties of the Cauchy type singular integral operators with conjugate k-holomorphic kernels, the general solutions and special solutions of the corresponding boundary value problems are studied in a detailed fashion, and the integral expressions of the solutions are obtained.

We consider the singular Dirichlet problem for the Monge-Ampère type equation ({rm det} D^2 u=b(x)g(-u)(1+|nabla u|^2)^{q/2}, u<0, x in Omega, u|_{partial Omega}=0), where Ω is a strictly convex and bounded smooth domain in ℝⁿ, q ∈ [0, n +1), g ∈ C^∞ (0, ∞) is positive and strictly decreasing in (0, ∞) with (limlimits_{srightarrow 0^+}g(s)=infty), and b ∈ C^∞ (Ω) is positive in Ω. We obtain the existence, nonexistence and global asymptotic behavior of the convex solution to such a problem for more general b and g. Our approach is based on the Karamata regular variation theory and the construction of suitable sub-and super-solutions.

Infinite matrix theory is an important branch of function analysis. Every linear operator on a complex separable infinite dimensional Hilbert space corresponds to an infinite matrix with respect a orthonormal base of the space, but not every infinite matrix corresponds to an operator. The classical Schur test provides an elegant and useful criterion for the boundedness of linear operators, which is considered a respectable mathematical accomplishment. In this paper, we prove the compact version of the Schur test. Moreover, we provide the Schur test for the Schatten class S₂. It is worth noting that our main results can be applicable to the general matrix without limitation on non-negative numbers. We finally provide the Schur test for compact operators from l_p into l_q.

This paper addresses the estimation problem of an unknown drift parameter matrix for a fractional Ornstein-Uhlenbeck process in a multi-dimensional setting. To tackle this problem, we propose a novel approach based on rough path theory that allows us to construct pathwise rough path estimators from both continuous and discrete observations of a single path. Our approach is particularly suitable for high-frequency data. To formulate the parameter estimators, we introduce a theory of pathwise Itô integrals with respect to fractional Brownian motion. By establishing the regularity of fractional Ornstein-Uhlenbeck processes and analyzing the long-term behavior of the associated Lévy area processes, we demonstrate that our estimators are strongly consistent and pathwise stable. Our findings offer a new perspective on estimating the drift parameter matrix for fractional Ornstein-Uhlenbeck processes in multi-dimensional settings, and may have practical implications for fields including finance, economics, and engineering.

In this paper, by characterizing Carleson measures, we investigate a class of bounded Toeplitz operator between weighted Bergman spaces with Békollé weights over the half-plane for all index choices.

This paper is devoted to the Cauchy problem for the generalized damped Boussinesq equation with a nonlinear source term in the natural energy space. With the help of linear time-space estimates, we establish the local existence and uniqueness of solutions by means of the contraction mapping principle. The global existence and blow-up of the solutions at both subcritical and critical initial energy levels are obtained. Moreover, we construct the sufficient conditions of finite time blow-up of the solutions with arbitrary positive initial energy.

For any s ∈ (0, 1), let the nonlocal Sobolev space X^s(ℝ^N) be the linear space of Lebesgue measure functions from ℝ^N to ℝ such that any function u in X^s(ℝ^N) belongs to L²(ℝ^N) and the function

is in L²(ℝ^N, ℝ^N). First, we show, for a coercive function V(x), the subspace

of X^s(ℝ^N) is embedded compactly into L^p(ℝ^N) for (pin[2,2_s^*)), where (2_s^*) is the fractional Sobolev critical exponent. In terms of applications, the existence of a least energy sign-changing solution and infinitely many sign-changing solutions of the nonlocal Schrödinger equation

are obtained, where (-{cal{L}_K}) is an integro-differential operator and V is coercive at infinity.

A cautious projection BFGS method is proposed for solving nonconvex unconstrained optimization problems. The global convergence of this method as well as a stronger general convergence result can be proven without a gradient Lipschitz continuity assumption, which is more in line with the actual problems than the existing modified BFGS methods and the traditional BFGS method. Under some additional conditions, the method presented has a superlinear convergence rate, which can be regarded as an extension and supplement of BFGS-type methods with the projection technique. Finally, the effectiveness and application prospects of the proposed method are verified by numerical experiments.

For a general normed vector space, a special optimal value function called a maximal time function is considered. This covers the farthest distance function as a special case, and has a close relationship with the smallest enclosing ball problem. Some properties of the maximal time function are proven, including the convexity, the lower semicontinuity, and the exact characterizations of its subdifferential formulas.

In this paper, we reconstruct strongly-decaying block sparse signals by the block generalized orthogonal matching pursuit (BgOMP) algorithm in the l₂-bounded noise case. Under some restraints on the minimum magnitude of the nonzero elements of the strongly-decaying block sparse signal, if the sensing matrix satisfies the the block restricted isometry property (block-RIP), then arbitrary strongly-decaying block sparse signals can be accurately and steadily reconstructed by the BgOMP algorithm in iterations. Furthermore, we conjecture that this condition is sharp.