Acta Mathematica Scientia最新文献

In this paper, we consider entire solutions of higher order homogeneous differential equations with the entire coefficients having the same order, and prove that the entire solutions are of infinite lower order. The properties on the radial distribution, the limit direction of the Julia set and the existence of a Baker wandering domain of the entire solutions are also discussed.

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.

Motivated by recent advances made in the study of dividend control and risk management problems involving the U.S. bankruptcy code, in this paper we follow [44] to revisit the De Finetti dividend control problem under the reorganization process and the regulator’s intervention documented in U.S. Chapter 11 bankruptcy. We do this by further accommodating the fixed transaction costs on dividends to imitate the real-world procedure of dividend payments. Incorporating the fixed transaction costs transforms the targeting optimal dividend problem into an impulse control problem rather than a singular control problem, and hence computations and proofs that are distinct from [44] are needed. To account for the financial stress that is due to the more subtle concept of Chapter 11 bankruptcy, the surplus process after dividends is driven by a piece-wise spectrally negative Lévy process with endogenous regime switching. Some explicit expressions of the expected net present values under a double barrier dividend strategy, new to the literature, are established in terms of scale functions. With the help of these expressions, we are able to characterize the optimal strategy among the set of admissible double barrier dividend strategies. When the tail of the Lévy measure is log-convex, this optimal double barrier dividend strategy is then verified as the optimal dividend strategy, solving our optimal impulse control problem.

Clifford analysis is an important branch of modern analysis; it has a very important theoretical significance and application value, and its conclusions can be applied to the Maxwell equation, Yang-Mill field theory, quantum mechanics and value problems. In this paper, we first give the definition of a quasi-Cauchy type integral in complex Clifford analysis, and get the Plemelj formula for it. Second, we discuss the Hölder continuity for the Cauchy-type integral operators with values in a complex Clifford algebra. Finally, we prove the existence of solutions for a class of linear boundary value problems and give the integral representation for the solution.

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.