In the past two decades, a rigorous solution for the shape of human red blood cell (RBC) with a negative spontaneous curvature c0 has been derived with the Helfrich model under the condition that both the osmotic pressure Δp and tensile stress λ are equal to zero. By fitting the experimentally observed shape of RBC, c0R0 has been predicted to be –1.62, theminus golden ratio, where R0 is the radius of a sphere which has the same surface area as RBC. In this paper, we verify this prediction by comparing experimental data with an analytical equation describing the relation between volume and surface area. Furthermore, it is also found ρmax /ρB ≈ 1.6 with ρmax the maximal radius and ρB the characteristic radius of RBC, showing an approximate beautiful golden cross section of RBC. On the basis of a complete numerical calculation, we propose a mechanism behind the beauty of the minus golden ratio that c0R0 results from the balance between the minimization of the surface area and the requirement of adequate deformability of RBC to allow it passing through the spleen, the so called “physical fitness test”.

We prove that the Fibonacci morphism is an automorphism of infinite order of free Burnside groups for all odd $n\geq 665$ and even $n = 16k \geq 8000$.

A multivariate distribution for describing the life-lengths of the components of a system which operates in an environment that is different from the test bench environment has been proposed by Lindley and Singpurwalla (1986). In this paper, the properties of the reliability function of such a system are studied and comparisons made with the reliability function obtained under the assumption of independence. It is interesting to note that the reliability function of parallel redundant systems whose components share a common unknown environment cannot be characterized by any of the well-known classes of distributions that have been proposed in the mathematical theory of reliability. This observation suggests the need for defining a new class of failure distributions. A formula for making Bayesian inferences for the reliability function is also given.

One of the earliest population models to be studied gives rise to the Fibonacci sequence and has a history dating back more than 750 years. A stochastic version of the model is discussed in this paper, its basic defining property being E(Xn | Xn−1, · ··, X0) = Xn−1 + Xn−2 a.s. The process {Xn} mimics many of the standard properties of the Fibonacci sequence. In particular, under mild additional conditions, a.s. as n → where α is the ‘golden ratio'