On the logarithmic probability that a random integral ideal is A\mathscr A-free

This extends a theorem of Davenport and Erd\"os on sequences of rational integers to sequences of integral ideals in arbitrary number fields $K$. More precisely, we introduce a logarithmic density for sets of integral ideals in $K$ and provide a formula for the logarithmic density of the set of so-called $\mathscr A$-free ideals, i.e. integral ideals that are not multiples of any ideal from a fixed set $\mathscr A$.Comment: 9 pages, to appear in S. Ferenczi, J. Ku{\l}aga-Przymus and M. Lema\'nczyk (eds.), Chowla's conjecture: from the Liouville function to the M\"obius function, Lecture Notes in Math., Springe

Direct ink writing of custom UV curable rubbers with radiation absorbing particles and its challenges

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Tensile Fracture of Welded Polymer Interfaces: Miscibility, Entanglements and Crazing

Large-scale molecular simulations are performed to investigate tensile failure of polymer interfaces as a function of welding time $t$. Changes in the tensile stress, mode of failure and interfacial fracture energy $G_I$ are correlated to changes in the interfacial entanglements as determined from Primitive Path Analysis. Bulk polymers fail through craze formation, followed by craze breakdown through chain scission. At small $t$ welded interfaces are not strong enough to support craze formation and fail at small strains through chain pullout at the interface. Once chains have formed an average of about one entanglement across the interface, a stable craze is formed throughout the sample. The failure stress of the craze rises with welding time and the mode of craze breakdown changes from chain pullout to chain scission as the interface approaches bulk strength. The interfacial fracture energy $G_I$ is calculated by coupling the simulation results to a continuum fracture mechanics model. As in experiment, $G_I$ increases as $t^{1/2}$ before saturating at the average bulk fracture energy $G_b$. As in previous simulations of shear strength, saturation coincides with the recovery of the bulk entanglement density. Before saturation, $G_I$ is proportional to the areal density of interfacial entanglements. Immiscibiltiy limits interdiffusion and thus suppresses entanglements at the interface. Even small degrees of immisciblity reduce interfacial entanglements enough that failure occurs by chain pullout and $G_I \ll G_b$

Effect of body deformability on microswimming

In this work we consider the following question: given a mechanical microswimming mechanism, does increased deformability of the swimmer body hinder or promote the motility of the swimmer? To answer this we study a microswimmer model composed of deformable beads connected with springs. We determine the velocity of the swimmer analytically, starting from the forces driving the motion and assuming that the oscillations in the effective radii of the beads are known and are much smaller than the radii themselves. We find that to the lowest order, only the driving frequency mode of the surface oscillations contributes to the swimming velocity, and that this velocity may both rise and fall with the deformability of the beads depending on the spring constant. To test these results, we run immersed boundary lattice Boltzmann simulations of the swimmer, and show that they reproduce both the velocity-promoting and velocity-hindering effects of bead deformability correctly in the predicted parameter ranges. Our results mean that for a general swimmer, its elasticity determines whether passive deformations of the swimmer body, induced by the fluid flow, aid or oppose the motion

The role of dipole interactions in hyperthermia heating colloidal clusters of densely-packed superparamagnetic nanoparticles

This work aims to investigate the influence of inter-particle dipole interactions on hyperthermia heating colloidal clusters of densely-packed Fe3O4 nanoparticles at low field intensity. Emulsion droplet solvent evaporation method was used to assemble oleic acid modified Fe3O4 particles into compact clusters which were stabilized by surfactant in water. Both experimental and simulation works were conducted to study their heating performance at different cluster’s sizes. The dipole interactions improve the heating only when the clusters are small enough to bring an enhancement in clusters’ shape anisotropy. The shape anisotropy is reduced at greater clusters’ sizes, since the shapes of the clusters become more and more spherical. Consequently, the dipole interactions change to impair the heating efficiency at larger sizes. When the clusters are totally isotropic in shape, the heating efficiency is lower than that of non-interacting particles despite the cluster’s size, although the efficiency increases by a little bit at a particular size most likely due to the dipole couplings. In these situations, one has to use particles with higher magnetic anisotropy and/or saturation magnetization to improve the heating

Nature's lessons in design: nanomachines to scaffold, remodel and shape membrane compartments.

Compartmentalisation of cellular processes is fundamental to regulation of metabolism in Eukaryotic organisms and is primarily provided by membrane-bound organelles. These organelles are dynamic structures whose membrane barriers are continually shaped, remodelled and scaffolded by a rich variety of highly sophisticated protein complexes. Towards the goal of bottom-up assembly of compartmentalised protocells in synthetic biology, we believe it will be important to harness and reconstitute the membrane shaping and sculpting characteristics of natural cells. We review different in vitro membrane models and how biophysical investigations of minimal systems combined with appropriate theoretical modelling have been used to gain new insights into the intricate mechanisms of these membrane nanomachines, paying particular attention to proteins involved in membrane fusion, fission and cytoskeletal scaffolding processes. We argue that minimal machineries need to be developed and optimised for employment in artificial protocell systems rather than the complex environs of a living organism. Thus, well-characterised minimal components might be predictably combined into functional, compartmentalised protocellular materials that can be engineered for wide-ranging applications