On the density of the odd values of the partition function, II: An infinite conjectural framework

We continue our study of a basic but seemingly intractable problem in integer partition theory, namely the conjecture that $p(n)$ is odd exactly $50\%$ of the time. Here, we greatly extend on our previous paper by providing a doubly-indexed, infinite framework of conjectural identities modulo 2, and show how to, in principle, prove each such identity. However, our conjecture remains open in full generality. A striking consequence is that, under suitable existence conditions, if any $t$-multipartition function is odd with positive density and $t\not \equiv 0$ (mod 3), then $p(n)$ is also odd with positive density. These are all facts that appear virtually impossible to show unconditionally today. Our arguments employ a combination of algebraic and analytic methods, including certain technical tools recently developed by Radu in his study of the parity of the Fourier coefficients of modular forms.Comment: 14 pages. To appear in the J. of Number Theor

On the density of the odd values of the partition function

The purpose of this note is to introduce a new approach to the study of one of the most basic and seemingly intractable problems in partition theory, namely the conjecture that the partition function $p(n)$ is equidistributed modulo 2. Our main result will relate the densities, say $\delta_t$, of the odd values of the $t$-multipartition functions $p_t(n)$, for several integers $t$. In particular, we will show that if $\delta_t>0$ for some $t\in \{5,7,11,13,17,19,23,25\}$, then (assuming it exists) $\delta_1>0$; that is, $p(n)$ itself is odd with positive density. Notice that, currently, the best unconditional result does not even imply that $p(n)$ is odd for $\sqrt{x}$ values of $n\le x$. In general, we conjecture that $\delta_t=1/2$ for all $t$ odd, i.e., that similarly to the case of $p(n)$, all multipartition functions are in fact equidistributed modulo 2. Our arguments will employ a number of algebraic and analytic methods, ranging from an investigation modulo 2 of some classical Ramanujan identities and several other eta product results, to a unified approach that studies the parity of the Fourier coefficients of a broad class of modular form identities recently introduced by Radu.Comment: Several changes with respect to the 2015 version. 18 pages. To appear in the Annals of Combinatoric

Ultra high level liquid argon purification using electron emission from a tip array

Abstract An array of 148 sharp tips was used to inject large quantities of free electrons in liquid argon and to purify it at a level · 0.09 ppb in few hours. We also evaluated the gettering capability of the stainless steel surface and measured the I–V characteristic of the array

Irradiation of carbon nanotubes with carbon projectiles: A molecular dynamics study

The irradiation of carbon based nanostructures with ions and electrons has been shown to be an appropriate tool to tailor their properties. The defects induced in the nanostructures during irradiation are able to modify their mechanical and electronic properties. Here we simulate the irradiation of carbon nanotubes with carbon ions using a molecular dynamics code. We use the Tersoff potential joined smoothly to the Universal Ziegler-Biersack-Littmark potential at short distances. We study the number of defects produced after irradiation with a single carbon ion finding a saturation with its energy at ∼ 3 keV. We observe, after continuum irradiation with low energy ions, the formation of bumps in the irradiated region. For larger energy ions we find that the diameter of the nanotube shrinks as shown in previous works.This work has been financially supported by Project FIS2010-17225 by the Spanish Ministerio de Ciencia e Innovación

Simulation of swift boron clusters traversing amorphous carbon foils

We use a simulation code to study the interaction of swift boron clusters (Bn+, n=2–6, 14) with amorphous carbon foils. We analyze different aspects of this interaction, such as the evolution of the cluster structure inside the target, the energy and angle distributions at the detector or the stopping power ratio. Our simulation code follows in detail the motion of the cluster fragments through the target and in the vacuum until reaching a detector, taking into account the following interactions: (i) wake force, (ii) Coulomb repulsion among cluster fragments, (iii) stopping force, and (iv) elastic scattering with the target nuclei. Electron capture and loss by each fragment is also included in the code, affecting the above-mentioned interactions. The clusters size grows inside the foil due mainly to the Coulomb explosion but this increase is less pronounced in the plane transversal to the beam direction because of the alignment effect of the wake forces. We obtain an enhancement of the stopping power ratio that increases with the projectile energy and with the number of molecular constituents. Our results agree very well with the available experimental data for the thicker foils (≳10 μg∕cm2) and are compatible (within the experimental error bars) for the thinner foils.This work has been financially supported by the Spanish Ministerio de Educación y Ciencia (Contract Nos. BFM2003-04457-C02-01 and BFM2003-04457-C02-02). S.H.A. thanks the Fundación CajaMurcia for financial support and C.D.D. thanks the Spanish Ministerio de Educación y Ciencia for support under the Ramón y Cajal Program

(η5-Cyclo­penta­dien­yl)[(1,2,3,4,4a,12a-η)-naphtho­[2,3-b][1,4]benzodioxine]iron(II) hexa­fluoridophosphate

At 296 (2) K, both complexed rings in the iron(II) complex cation of the title salt, [Fe(C5H5)(C16H10O2)]PF6, are almost parallel [dihedral angle between planes = 2.4 (3)°]. The quaternary C atoms of the complexed arene ring are located at the longest distance from the Fe atom, with Fe—C distances of 2.112 (4) and 2.105 (3) Å, which are slightly longer than the average Fe—C distance for this ring (2.083 Å). The Fe ion is located 1.660 (1) and 1.543 (1) Å, respectively, from the cyclo­penta­dienyl and the complexed arene ring

Collision cascade effects near an edge dislocation dipole in alpha-Fe: Induced dislocation mobility and enhanced defect clustering

Collision cascades near a 1/2⟨111⟩{110} edge dipole in alpha-iron have been studied using molecular dynamics simulations for a recoil energy of 20 keV and two temperatures, 20 K and 300 K. These simulations show that the collision cascade induces the migration of the dislocations through glide along its slip plane. The motion of the dislocations starts at the peak of the collision cascade and expands a time scale much longer than the cascade duration, until restoring the equilibrium distance of the dipole, regardless of the damage produced by the cascade. At the initial stages, kinks are formed at the dislocation that enhance glide. When defects reach the dislocations, jogs are produced. We show that the initial dislocation motion is triggered by the shock wave of the collision cascade. The cascade morphology is also strongly influenced by the presence of the dislocations, having an elongated form at the peak of the displacement, which demonstrates the strong interaction of the dislocations with the cascade even at the early stages. Finally, we show that larger vacancy clusters are formed in the presence of dislocations compared to isolated cascades and that these clusters are larger for 300 K compared to 20 K.This work was partly supported by the Generalitat Valenciana through PROMETEO2017/139. This work has been carried out within the framework of the EUROfusion Consortium and has received funding from the Euratom research and training programme 2014–2018 and 2019–2020 under grant agreement No 633053. The views and opinions expressed herein do not necessarily reflect those of the European Commission. EM gratefully acknowledges support from the U.S. DOE, Office of Science, Office of Fusion Energy Sciences, and Office of Advanced Scientific Computing Research through the Scientific Discovery through Advanced Computing (SciDAC) project on Plasma-Surface Interactions (award no. DE-SC0008875)

A novel experimental approach for the detection of the dynamic Casimir effect

The Casimir effect is a well-known macroscopic consequence of quantum vacuum fluctuations, but whereas the static effect (Casimir force) has long been observed experimentally, the dynamic Casimir effect is up to now undetected. From an experimental viewpoint a possible detection would imply the vibration of a mirror at gigahertz frequencies. Mechanical motions at such frequencies turn out to be technically unfeasible. Here we present a different experimental scheme where mechanical motions are avoided, and the results of laboratory tests showing that the scheme is practically feasible. We think that at present this approach gives the only possibility of detecting this phenomenon.Comment: Submitted to the Physical Review Letters. RevTeX. 4 pages, 2 figure

Resonance frequency shift in a cavity with a thin conducting film near a conducting wall

We show that a very thin conducting film (whose thickness can be much smaller than the skin depth), placed nearby a wall of an electromagnetic cavity, can produce the same shift of the resonance frequency as a bulk conducting slab, provided the displacement of the film from the wall is much bigger than the skin depth. We derive a simple analytical formula for the frequency shift and compare it with exact numerical calculations and experimental data

Surface and subsurface contributions to the build-up of forces on bed particles

In nature and in many industrial applications, the boundary of a channel flow is made of solid particles which form a porous wall, so that there is a mutual influence between the free flow and the subsurface flow developing inside the pores. While the influence of the porous wall on the free flow has been well studied, less well characterized is the subsurface flow, due to the practical difficulties in gathering information in the small spaces given by the pores. It is also not clear whether the subsurface flow can host turbulent events able to contribute significantly to the build-up of forces on the particles, potentially leading to their dislodgement. Through large eddy simulations, we investigate the interface between a free flow and a bed composed of spherical particles in a cubic arrangement. The communication between surface and subsurface flow is in this case enhanced, with relatively strong turbulent events happening also inside the pores. After comparing the simulation results with a previous experimental work from a similar setting, the forces experienced by the boundary particles are analysed. While it remains true that the lift forces are largely dependent on the structure of the free flow, turbulence inside the pores can also give a significant contribution. Pressure inside the pores is weakly correlated to the pressure in the free flow, and strong peaks above and below a particle can happen independently. Ignoring the porous layer below the particle from the computations leads then in this case to an underestimation of the lift forces