Cluster algebras in scattering amplitudes with special 2D kinematics

We study the cluster algebra of the kinematic configuration space $Conf_n(\mathbb{P}^3)$ of a n-particle scattering amplitude restricted to the special 2D kinematics. We found that the n-points two loop MHV remainder function found in special 2D kinematics depend on a selection of \XX-coordinates that are part of a special structure of the cluster algebra related to snake triangulations of polygons. This structure forms a necklace of hypercubes beads in the corresponding Stasheff polytope. Furthermore in $n = 12$, the cluster algebra and the selection of \XX-coordinates in special 2D kinematics replicates the cluster algebra and the selection of \XX-coordinates of $n=6$ two loop MHV amplitude in 4D kinematics.Comment: 22 page

The mass function of GX 339-4 from spectroscopic observations of its donor star

We obtained 16 VLT/X-shooter observations of GX 339-4 in quiescence in the period May - September 2016 and detected absorption lines from the donor star in its NIR spectrum. This allows us to measure the radial velocity curve and projected rotational velocity of the donor for the first time. We confirm the 1.76 day orbital period and we find that $K_2$ = $219 \pm 3$ km s$^{-1}$, $\gamma = 26 \pm 2$ km s$^{-1}$ and $v \sin i = 64 \pm 8$ km s$^{-1}$. From these values we compute a mass function $f(M) =1.91 \pm 0.08~M_{\odot}$, a factor $\sim 3$ lower than previously reported, and a mass ratio $q = 0.18 \pm 0.05$. We confirm the donor is a K-type star and estimate that it contributes $\sim 45-50\%$ of the light in the $J$- and H-band. We constrain the binary inclination to $37^\circ < i < 78^\circ$ and the black hole mass to $2.3~M_{\odot} < M_\mathrm{BH} < 9.5~M_{\odot}$. GX 339-4 may therefore be the first black hole to fall in the 'mass-gap' of $2-5~M_{\odot}$.Comment: 11 pages, 7 figures, accepted for publication in Ap

Time-resolved photometry of the nova remnants DM Gem, CP Lac, GI Mon, V400 Per, CT Ser and XX Tau

We present the first results of a photometric survey of poorly studied nova remnants in the Northern Hemisphere. The main results are as follows: DM Gem shows a modulation at 0.123 d (probably linked to the orbit) and rapid variations at ∼22 min. A moderate resolution spectrum taken at the time of the photometric observations shows intense He II λ4686 and Bowen emission, characteristic of an intermediate polar or a SW Sex star. Variability at 0.127 d and intense flickering (or quasi-periodic oscillations) are the main features of the light curve of CP Lac. A 0.1-mag dip lasting for ∼45 min is observed in GI Mon, which could be an eclipse. A clear modulation (probably related to the orbital motion) either at 0.179 d or 0.152 d is observed in the B-band light curve of V400 Per. The results for CT Ser point to an orbital period close to 0.16 d. Intense flickering is also characteristic of this old nova. Finally, XX Tau shows a possible periodic signal near 0.14 d and displays fast variability at ∼24 min. Its brightness seems to be modulated at ∼5 d. We relate this long periodicity to the motion of an eccentric/tilted accretion disc in the binary

Finite size scaling of the bayesian perceptron

We study numerically the properties of the bayesian perceptron through a gradient descent on the optimal cost function. The theoretical distribution of stabilities is deduced. It predicts that the optimal generalizer lies close to the boundary of the space of (error-free) solutions. The numerical simulations are in good agreement with the theoretical distribution. The extrapolation of the generalization error to infinite input space size agrees with the theoretical results. Finite size corrections are negative and exhibit two different scaling regimes, depending on the training set size. The variance of the generalization error vanishes for $N \rightarrow \infty$ confirming the property of self-averaging.Comment: RevTeX, 7 pages, 7 figures, submitted to Phys. Rev.