Complex biological products, such as those used to treat various forms of cancer, are typically produced by mammalian cells in bioreactors. The most important class of such biological medicines is proteins. These typically bind to sugars (glycans) in a process known as glycosylation, creating glycoproteins, which are more stable and effective medicines. The glycans are large polymers that are formed by a long sequence of enzyme catalysed reactions. This sequence is not always completed, thus leading to a heterogeneous glycoprotein distribution. A better comprehension of this distribution could lead to more efficient production of high quality drugs. To understand how the manufacturing process can affect the extent of glycosylation of protein, a non-linear ODE model of glycoprotein production is developed which describes the bioreactor configuration as well as the protein production and glycosylation reactions within the cell. The entire system evolves eventually to a stable steady state. The earlier evolution is critical however, as the amount of product produced and its quality varies over time. The model is considered as two coupled systems: the bioreactor submodel and the glycosylation submodel. To investigate the early time evolution within the bioreactor submodel, analytical and numerical properties are derived using matched asymptotic expansions and a finite difference scheme for a range of initial conditions. This leads to qualitatively different regimes for aglycosylated protein production, which affect the glycosylation submodel. The discrete glycoprotein distribution is approximated as continuous and written as a first-order PDE, with good agreement between the discrete and continuous models. The PDE is found to admit shocks, but the existence of these shocks is dependent on the early time evolution within the bioreactor submodel and leads to higher levels of glycosylation at early time. This suggests that changing the bioreactor configuration can lead to higher quality product at certain times.

In this article, we consider the spatial homogenisation of a multi-phase model for avascular tumour growth and response to chemotherapeutic treatment. The key contribution of this work is the derivation of a system of homogenised partial differential equations describing macroscopic tumour growth, coupled to transport of drug and nutrient, that explicitly incorporates details of the structure and dynamics of the tumour at the microscale. In order to derive these equations, we employ an asymptotic homogenisation of a microscopic description under the assumption of strong interphase drag, periodic microstructure, and strong separation of scales. The resulting macroscale model comprises a Darcy flow coupled to a system of reaction–advection partial differential equations. The coupled growth, response, and transport dynamics on the tissue scale are investigated via numerical experiments for simple academic test cases of microstructural information and tissue geometry, in which we observe drug- and nutrient-regulated growth and response consistent with the anticipated dynamics of the macroscale system.

Euler noted the relation $6^{3}\,=\,3^{3}+4^{3}+5^{3}$ and asked for other instances of cubes that are sums of consecutive cubes. Similar problems have been studied by Cunningham, Catalan, Gennochi, Lucas, Pagliani, Cassels, Uchiyama, Stroeker and Zhongfeng Zhang. In particular, Stroeker determined all squares that can be written as a sum of at most 50 consecutive cubes. We generalize Stroeker’s work by determining all perfect powers that are sums of at most 50 consecutive cubes. Our methods include descent, linear forms in two logarithms and Frey–Hellegouarch curves.

Let $\unicode[STIX]{x1D703}$ be an arithmetic function and let ${\mathcal{B}}$ be the set of positive integers $n=p_{1}^{\unicode[STIX]{x1D6FC}_{1}}\cdots p_{k}^{\unicode[STIX]{x1D6FC}_{k}}$ which satisfy $p_{j+1}\leqslant \unicode[STIX]{x1D703}(p_{1}^{\unicode[STIX]{x1D6FC}_{1}}\cdots p_{j}^{\unicode[STIX]{x1D6FC}_{j}})$ for $0\leqslant j<k$ . We show that ${\mathcal{B}}$ has a natural density, provide a criterion to determine whether this density is positive, and give various estimates for the counting function of ${\mathcal{B}}$ . When $\unicode[STIX]{x1D703}(n)/n$ is non-decreasing, the set ${\mathcal{B}}$ coincides with the set of integers $n$ whose divisors $1=d_{1}<d_{2}<\cdots <d_{\unicode[STIX]{x1D70F}(n)}=n$ satisfy $d_{j+1}\leqslant \unicode[STIX]{x1D703}(d_{j})$ for $1\leqslant j<\unicode[STIX]{x1D70F}(n)$ .

Assuming the Riemann Hypothesis, Soundararajan [Ann. of Math. (2) 170 (2009), 981–993] showed that $\int _{0}^{T}|\unicode[STIX]{x1D701}(1/2+\text{i}t)|^{2k}\ll T(\log T)^{k^{2}+\unicode[STIX]{x1D716}}$ . His method was used by Chandee [Q. J. Math.62 (2011), 545–572] to obtain upper bounds for shifted moments of the Riemann Zeta function. Building on these ideas of Chandee and Soundararajan, we obtain, conditionally, upper bounds for shifted moments of Dirichlet $L$ -functions which allow us to derive upper bounds for moments of theta functions.