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

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

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

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

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

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

Demonstration of an inductively coupled ring trap for cold atoms

We report the first demonstration of an inductively coupled magnetic ring trap for cold atoms. A uniform, ac magnetic field is used to induce current in a copper ring, which creates an opposing magnetic field that is time-averaged to produce a smooth cylindrically symmetric ring trap of radius 5 mm. We use a laser-cooled atomic sample to characterize the loading efficiency and adiabaticity of the magnetic potential, achieving a vacuum-limited lifetime in the trap. This technique is suitable for creating scalable toroidal waveguides for applications in matter-wave interferometry, offering long interaction times and large enclosed areas

Guessing Games on Triangle-free Graphs

9 pages, submitted to Electronic Journal of Combinatoric9 pages, submitted to Electronic Journal of CombinatoricThe guessing game introduced by Riis is a variant of the "guessing your own hats" game and can be played on any simple directed graph G on n vertices. For each digraph G, it is proved that there exists a unique guessing number gn(G) associated to the guessing game played on G. When we consider the directed edge to be bidirected, in other words, the graph G is undirected, Christofides and Markstr om introduced a method to bound the value of the guessing number from below using the fractional clique number Kf(G). In particular they showed gn(G) >= |V(G)| - Kf(G). Moreover, it is pointed out that equality holds in this bound if the underlying undirected graph G falls into one of the following categories: perfect graphs, cycle graphs or their complement. In this paper, we show that there are triangle-free graphs that have guessing numbers which do not meet the fractional clique cover bound. In particular, the famous triangle-free Higman-Sims graph has guessing number at least 77 and at most 78, while the bound given by fractional clique cover is 50