Sobolev regularity and an enhanced Jensen inequality

We derive a new criterion for a real-valued function $u$ to be in the Sobolev space $W^{1,2}(\R^n)$. This criterion consists of comparing the value of a functional $\int f(u)$ with the values of the same functional applied to convolutions of $u$ with a Dirac sequence. The difference of these values converges to zero as the convolutions approach $u$, and we prove that the rate of convergence to zero is connected to regularity: $u\in W^{1,2}$ if and only if the convergence is sufficiently fast. We finally apply our criterium to a minimization problem with constraints, where regularity of minimizers cannot be deduced from the Euler-Lagrange equation.Comment: 10 page

When is the inverse of an invertible convex function itself convex?

We provide a sufficient condition for an invertible (locally strongly) convex vector-valued function on $\mathbb{R}^N$ to have a (locally strongly) convex inverse. We show under suitable conditions that if the gradient of each component of the inverse has negative entries, then this inverse is (locally strongly) convex if the original is

A consistent treatment of link and writhe for open rods, and their relation to end rotation

We combine and extend the work of Alexander & Antman \cite{alexander.82} and Fuller \cite{fuller.71,fuller.78} to give a framework within which precise definitions can be given of topological and geometrical quantities characterising the contortion of open rods undergoing large deformations under end loading. We use these definitions to examine the extension of known results for closed rods to open rods. In particular, we formulate the analogue of the celebrated formula $Lk=Tw+Wr$ (link equals twist plus writhe) for open rods and propose an end rotation, through which the applied end moment does work, in the form of an integral over the length of the rod. The results serve to promote the variational analysis of boundary-value problems for rods undergoing large deformations.Comment: 17 pages, 4 figure

Bounding ζ(s)\zeta(s) in the critical strip

Assuming the Riemann Hypothesis, we make use of the recently discovered \cite{CLV} extremal majorants and minorants of prescribed exponential type for the function $\log\left(\tfrac{4 + x^2}{(\alpha-1/2)^2 + x^2}\right)$ to find upper and lower bounds with explicit constants for $\log|\zeta(\alpha + it)|$ in the critical strip, extending the work of Chandee and Soundararajan \cite{CS}

Rare enemies and rare friends: adaptations that make other adaptations maladaptive

We show that certain adaptations can make other adaptations maladaptive. For example, one line of defence against an enemy can make an otherwise valuable, but subsequent line of defence detrimental. This can occur through indirect rare enemy effects

The adaptiveness of defence strategies against cuckoo parasitism

Most bird species of the Eurasian Cuckoo, 'Cuculuscanorus', often display egg-discrimination behaviour butchick-rejection behaviour has never been reported.In this paper, we analyse ahost-cuckoo association in which both population dynamics andevolutionary dynamics are explored in a discrete-time model.We introduce four host types, each with their own defence behaviour, displayingeither egg or chick rejection, neither or both. We also introducefitness functions for each of these host types.Although we can characterise the long term behaviour in many cases by a simpleheuristic argument which is in accordance with common views in ecology, thereare a number of other phenomena that are not explained within thisframework: we describe stable oscillatory behaviour and coexistence oftwo defensive host types. We analyse the scenariosin which chick rejection may establish itself and give a first explanationas to why this defence trait has never been recorded in nature.We find that chick rejectors generally are at an intrinsicdisadvantage with respect to a host type that rejects eggs.Hosts benefit more from rejecting cuckoo eggs than cuckoo chicks, and ourmodel suggests that this is chiefly responsible for the absence of chickrejection. Moreover, even though it seems that chick rejection must beuseful as an extra defence, it is shown that hosts with both defencestrategies are less likely to establish themselves in competitionwith egg-rejectors than hosts which reject chicks only.These results provide insight in the extent to whichadaptations may be perfected by natural selection

Reconstruction of sea surface temperatures from the oxygen isotope composition of fossil planktic foraminifera.

Knowledge of the historic surface temperature of sea water is of importance for the calibration of climate models. The oxygen isotope composition of the shells of several species of planktic foraminifera can be used as a measure for this sea surface temperature. In this paper we investigate how mathematical models can contribute to the process of extracting information about the temperature at which the foraminifera lived from measurement of the oxygen isotope composition of their shells. A simple model is proposed which captures both the average and the variability of the temperature. Preliminary findings suggest that this model forms a solid basis for future research

Understanding start-up problems in yeast glycolysis

Yeast glycolysis has been the focus of research for decades, yet a number of dynamical aspects of yeast glycolysis remain poorly understood at present. If nutrients are scarce, yeast will provide its catabolic and energetic needs with other pathways, but the enzymes catalysing upper glycolytic fluxes are still expressed. We conjecture that this overexpression facilitates the rapid transition to glycolysis in case of a sudden increase in nutrient concentration. However, if starved yeast is presented with abundant glucose, it can enter into an imbalanced state where glycolytic intermediates keep accumulating, leading to arrested growth and cell death. The bistability between regularly functioning and imbalanced phenotypes has been shown to depend on redox balance. We shed new light on these phenomena with a mathematical analysis of an ordinary differential equation model, including NADH to account for the redox balance. In order to gain qualitative insight, most of the analysis is parameter-free, i.e., without assigning a numerical value to any of the parameters. The model has a subtle bifurcation at the switch between an inviable equilibrium state and stable flux through glycolysis. This switch occurs if the ratio between the flux through upper glycolysis and ATP consumption rate of the cell exceeds a fixed threshold. If the enzymes of upper glycolysis would be barely expressed, our model predicts that there will be no glycolytic flux, even if external glucose would be at growth-permissable levels. The existence of the imbalanced state can be found for certain parameter conditions independent of the mentioned bifurcation. The parameter-free analysis proved too complex to directly gain insight into the imbalanced states, but the starting point of a branch of imbalanced states can be shown to exist in detail. Moreover, the analysis offers the key ingredients necessary for successful numerical continuation, which highlight the existence of this bistability and the influence of the redox balance

Some studies on the deformation of the membrane in an RF MEMS switch

Radio Frequency (RF) switches of Micro Electro Mechanical Systems (MEMS) are appealing to the mobile industry because of their energy efficiency and ability to accommodate more frequency bands. However, the electromechanical coupling of the electrical circuit to the mechanical components in RF MEMS switches is not fully understood. In this paper, we consider the problem of mechanical deformation of electrodes in RF MEMS switch due to the electrostatic forces caused by the difference in voltage between the electrodes. It is known from previous studies of this problem, that the solution exhibits multiple deformation states for a given electrostatic force. Subsequently, the capacity of the switch that depends on the deformation of electrodes displays a hysteresis behaviour against the voltage in the switch. We investigate the present problem along two lines of attack. First, we solve for the deformation states of electrodes using numerical methods such as finite difference and shooting methods. Subsequently, a relationship between capacity and voltage of the RF MEMS switch is constructed. The solutions obtained are exemplified using the continuation and bifurcation package AUTO. Second, we focus on the analytical methods for a simplified version of the problem and on the stability analysis for the solutions of deformation states. The stability analysis shows that there exists a continuous path of equilibrium deformation states between the open and closed state