Acta Mathematica Scientia最新文献

This paper is devoted to considering the quasiperiodicity of complex differential polynomials, complex difference polynomials and complex delay-differential polynomials of certain types, and to studying the similarities and differences of quasiperiodicity compared to the corresponding properties of periodicity.

Though atomic decomposition is a very useful tool for studying the boundedness on Hardy spaces for some sublinear operators, untill now, the boundedness of operators on weighted Hardy spaces in a multi-parameter setting has been established only by almost orthogonality estimates. In this paper, we mainly establish the boundedness on weighted multi-parameter local Hardy spaces via atomic decomposition.

We find the exact forms of meromorphic solutions of the nonlinear differential equations

where q, Q are nonzero polynomials, Q ≡ Const., and p₁, p₂, α₁, α₂ are nonzero constants with α₁ ≠ α₂. Compared with previous results on the equation p(z)f³ + q(z)f″ = − sin α(z) with polynomial coefficients, our results show that the coefficient of the term f^(k) perturbed by multiplying an exponential function will affect the structure of its solutions.

Let X = {X(t) ∈ ℝ^d, t ∈ℝ^N} be a centered space-time anisotropic Gaussian field with indices H = (H₁, ⋯, H_N) ∈ (0, 1)^N, where the components X_i (i = 1, ⋯, d) of X are independent, and the canonical metric (sqrt {{{mathbb{E}({X_i}(t) - {X_i}(s))}^2}} ,(i = 1, cdots ,d)) is commensurate with ({gamma ^{{alpha _i}}}(sumlimits_{j = 1}^N {|{t_j} - {s_j}{|^{{H_j}}})} ) for s = (s₁, ⋯, s_N), t = (t₁, ⋯, t_N) ∈ ℝ^N, α_i ∈ (0, 1], and with the continuous function γ(·) satisfying certain conditions. First, the upper and lower bounds of the hitting probabilities of X can be derived from the corresponding generalized Hausdorff measure and capacity, which are based on the kernel functions depending explicitly on γ (·). Furthermore, the multiple intersections of the sample paths of two independent centered space-time anisotropic Gaussian fields with different distributions are considered. Our results extend the corresponding results for anisotropic Gaussian fields to a large class of space-time anisotropic Gaussian fields.

In this paper, we study the one-dimensional motion of viscous gas near a vacuum, with the gas connecting to a vacuum state with a jump in density. The interface behavior, the pointwise decay rates of the density function and the expanding rates of the interface are obtained with the viscosity coefficient μ(ρ) = ρ^α for any 0 < α < 1; this includes the time-weighted boundedness from below and above. The smoothness of the solution is discussed. Moreover, we construct a class of self-similar classical solutions which exhibit some interesting properties, such as optimal estimates. The present paper extends the results in [Luo T, Xin Z P, Yang T. SIAM J Math Anal, 2000, 31(6): 1175–1191] to the jump boundary conditions case with density-dependent viscosity.

In this paper, we establish two transformation formulas for nonterminating basic hypergeometric series by using Carlitz’s inversions formulas and Jackson’s transformation formula. In terms of application, by specializing certain parameters in the two transformations, four Rogers-Ramanujan type identities associated with moduli 20 are obtained.

We consider the singular Riemann problem for the rectilinear isentropic compressible Euler equations with discontinuous flux, more specifically, for pressureless flow on the left and polytropic flow on the right separated by a discontinuity x = x(t). We prove that this problem admits global Radon measure solutions for all kinds of initial data. The over-compressing condition on the discontinuity x = x(t) is not enough to ensure the uniqueness of the solution. However, there is a unique piecewise smooth solution if one proposes a slip condition on the right-side of the curve x = x(t) + 0, in addition to the full adhesion condition on its left-side. As an application, we study a free piston problem with the piston in a tube surrounded initially by uniform pressureless flow and a polytropic gas. In particular, we obtain the existence of a piecewise smooth solution for the motion of the piston between a vacuum and a polytropic gas. This indicates that the singular Riemann problem looks like a control problem in the sense that one could adjust the condition on the discontinuity of the flux to obtain the desired flow field.

In this paper, we define a new class of control functions through aggregate special functions. These class of control functions help us to stabilize and approximate a tri-additive ψ-functional inequality to get a better estimation for permuting tri-homomorphisms and permuting tri-derivations in unital C*-algebras and Banach algebras by the vector-valued alternative fixed point theorem.

This is a survey of local and global classification results concerning Dupin hypersurfaces in Sⁿ (or Rⁿ) that have been obtained in the context of Lie sphere geometry. The emphasis is on results that relate Dupin hypersurfaces to isoparametric hypersurfaces in spheres. Along with these classification results, many important concepts from Lie sphere geometry, such as curvature spheres, Lie curvatures, and Legendre lifts of submanifolds of Sⁿ (or Rⁿ), are described in detail. The paper also contains several important constructions of Dupin hypersurfaces with certain special properties.

In this note, we prove a logarithmic Sobolev inequality which holds for compact submanifolds without a boundary in manifolds with asymptotically nonnegative sectional curvature. Like the Michale-Simon Sobolev inequality, this inequality contains a term involving the mean curvature.