Network Communication with operators in Dedekind Finite and Stably Finite Rings

Messages in communication networks often are considered as "discrete" taking values in some finite alphabet (e.g. a finite field). However, if we want to consider for example communication based on analogue signals, we will have to consider messages that might be functions selected from an infinite function space. In this paper, we extend linear network coding over finite/discrete alphabets/message space to the infinite/continuous case. The key to our approach is to view the space of operators that acts linearly on a space of signals as a module over a ring. It turns out that modules over many rings $R$ leads to unrealistic network models where communication channels have unlimited capacity. We show that a natural condition to avoid this is equivalent to the ring $R$ being Dedekind finite (or Neumann finite) i.e. each element in $R$ has a left inverse if and only if it has a right inverse. We then consider a strengthened capacity condition and show that this requirement precisely corresponds to the class of (faithful) modules over stably finite rings (or weakly finite). The introduced framework makes it possible to compare the performance of digital and analogue techniques. It turns out that within our model, digital and analogue communication outperforms each other in different situations. More specifically we construct: 1) A communications network where digital communication outperforms analogue communication. 2) A communication network where analogue communication outperforms digital communication. The performance of a communication network is in the finite case usually measured in terms band width (or capacity). We show this notion also remains valid for finite dimensional matrix rings which make it possible (in principle) to establish gain of digital versus analogue (analogue versus digital) communications

Planar tautologies hard for resolution.

We prove exponential lower bounds on the resolution proofs of some tautologies, based on rectangular grid graphs. More specifically, we show a 2/sup /spl Omega/(n)/ lower bound for any resolution proof of the mutilated chessboard problem on a 2n/spl times/2n chessboard as well as for the Tseitin tautology (G. Tseitin, 1968) based on the n/spl times/n rectangular grid graph. The former result answers a 35 year old conjecture by J. McCarthy (1964)

Tree resolution proofs of the weak pigeon-hole principle.

We prove that any optimal tree resolution proof of PHPn m is of size 2&thetas;(n log n), independently from m, even if it is infinity. So far, only a 2Ω(n) lower bound has been known in the general case. We also show that any, not necessarily optimal, regular tree resolution proof PHPn m is bounded by 2O(n log m). To the best of our knowledge, this is the first time the worst case proof complexity has been considered. Finally, we discuss possible connections of our result to Riis' (1999) complexity gap theorem for tree resolution

Bose-Einstein condensates in `giant' toroidal magnetic traps

The experimental realisation of gaseous Bose-Einstein condensation (BEC) in 1995 sparked considerable interest in this intriguing quantum fluid. Here we report on progress towards the development of an 87Rb BEC experiment in a large (~10cm diameter) toroidal storage ring. A BEC will be formed at a localised region within the toroidal magnetic trap, from whence it can be launched around the torus. The benefits of the system are many-fold, as it should readily enable detailed investigations of persistent currents, Josephson effects, phase fluctuations and high-precision Sagnac or gravitational interferometry.Comment: 5 pages, 3 figures (Figs. 1 and 2 now work

Efficient Charge Separation in 2D Janus van der Waals Structures with Build-in Electric Fields and Intrinsic p-n Doping

Janus MoSSe monolayers were recently synthesised by replacing S by Se on one side of MoS$_2$ (or vice versa for MoSe$_2$). Due to the different electronegativity of S and Se these structures carry a finite out-of-plane dipole moment. As we show here by means of density functional theory (DFT) calculations, this intrinsic dipole leads to the formation of built-in electric fields when the monolayers are stacked to form $N$-layer structures. For sufficiently thin structures ($N<4$) the dipoles add up and shift the vacuum level on the two sides of the film by $\sim N \cdot 0.7$ eV. However, for thicker films charge transfer occurs between the outermost layers forming atomically thin n- and p-doped electron gasses at the two surfaces. The doping concentration can be tuned between about $5\cdot 10^{12}$ e/cm$^{2}$ and $2\cdot 10^{13}$ e/cm$^{2}$ by varying the film thickness. The surface charges counteract the static dipoles leading to saturation of the vacuum level shift at around 2.2 eV for $N>4$. Based on band structure calculations and the Mott-Wannier exciton model, we compute the energies of intra- and interlayer excitons as a function of film thickness suggesting that the Janus multilayer films are ideally suited for achieving ultrafast charge separation over atomic length scales without chemical doping or applied electric fields. Finally, we explore a number of other potentially synthesisable 2D Janus structures with different band gaps and internal dipole moments. Our results open new opportunities for ultrathin opto-electronic components such as tunnel diodes, photo-detectors, or solar cells

Laser cooling with a single laser beam and a planar diffractor

A planar triplet of diffraction gratings is used to transform a single laser beam into a four-beam tetrahedral magneto-optical trap. This `flat' pyramid diffractor geometry is ideal for future microfabrication. We demonstrate the technique by trapping and subsequently sub-Doppler cooling 87Rb atoms to 30microKelvin.Comment: 3 pages, 4 figure

A large magnetic storage ring for Bose-Einstein condensates

Cold atomic clouds and Bose-Einstein condensates have been stored in a 10cm diameter vertically-oriented magnetic ring. An azimuthal magnetic field enables low-loss propagation of atomic clouds over a total distance of 2m, with a heating rate of less than 50nK/s. The vertical geometry was used to split an atomic cloud into two counter-rotating clouds which were recombined after one revolution. The system will be ideal for studying condensate collisions and ultimately Sagnac interferometry.Comment: 4 pages, 5 figure

Spatial interference from well-separated condensates

We use magnetic levitation and a variable-separation dual optical plug to obtain clear spatial interference between two condensates axially separated by up to 0.25 mm -- the largest separation observed with this kind of interferometer. Clear planar fringes are observed using standard (i.e. non-tomographic) resonant absorption imaging. The effect of a weak inverted parabola potential on fringe separation is observed and agrees well with theory.Comment: 4 pages, 5 figures - modified to take into account referees' improvement