For any Liouville number $\alpha $, all of the following are transcendental numbers: ${e}^\alpha $, $\log _{e}\alpha $, $\sin \alpha $, $\cos \alpha $, $\tan \alpha $, $\sinh \alpha $, $\cosh \alpha $, $\tanh \alpha $, $\arcsin \alpha $ and the inverse functions evaluated at $\alpha $ of the listed trigonometric and hyperbolic functions, noting that wherever multiple values are involved, every such value is transcendental. This remains true if ‘Liouville number’ is replaced by ‘U-number’, where U is one of Mahler’s classes of transcendental numbers.

In this paper, we consider some dividend problems in the perturbed compound Poisson model under a constant barrier dividend strategy. We approximate the expected present value of dividend payments before ruin and the expected discounted penalty function based on the COS method, and construct some nonparametric estimators by using a random sample on claim number and individual claim sizes. Under a large sample size setting, we perform an error analysis of the estimators. We also provide some simulation results to verify the effectiveness of this estimation method when the sample size is finite.

The aim of this study was to investigate if there is an adverse effect of multiple controlled ovarian stimulation (COS) on the maturity of oocytes (MI and MII), fertilization rate, embryo developmental qualities and clinical pregnancy rates in donation cycles. In total, 65 patients undergoing oocyte donation cycles multiple times were included in this study. Patients were grouped as group A that consisted of donors with ≤2 stimulation cycles while B consisted of donors with ≥3 stimulation cycles; and group C included donors who had ≤15 oocytes, while group D had donors with ≥16 oocytes. Numbers of oocytes obtained, MI and MII oocytes, fertilization, embryo quality and clinical pregnancy outcomes were compared. Significant statistical differences were observed in total number of oocytes obtained, maturity of oocytes (MI and MII), fertilization rate, embryo qualities and clinical pregnancy outcomes of donors in groups A–D. Donors with ≤2 ovarian stimulation cycles had lower numbers of immature oocytes than donors with three or more stimulation cycles. However, donors with ≥3 stimulation cycles had higher numbers of mature oocytes, zygotes, with better day 3 embryo qualities and higher clinical pregnancy rates than donors with ≤2 stimulation cycles. Repeated COS does not seem to have any adverse effect on ovarian response to higher dose of artificial gonadotropin, as quality of oocytes collected and their embryological developmental potential were not affected by the number of successive stimulation cycles. The effect of multiple COS on the health of the oocyte donor needs to be assessed for future purpose.

In this paper, the density of the time to ruin is studied in the context of the classical compound Poisson risk model. Both one-dimensional and two-dimensional Fourier-cosine series expansions are used to approximate the density of the time to ruin, and the approximation errors are also obtained. Some numerical examples are also presented to show that the proposed method is very efficient.