Abstract:
The NEMO-3 detector installed in the Modane Underground Laboratory (LSM, France) is running to search for neutrinoless double beta decay ($\beta\beta0\nu$) with an expected sensitivity for the effective Majorana neutrino mass down to 0.1 eV. New preliminary limits (90% CL), $T_{1/2}(\beta\beta0\nu) > 1.0 \cdot 10^{23}$ y (massive mechanism) and $T_{1/2}(\beta\beta0\nu) > 1.9 \cdot 10^{23}$ y (right-handed currents) were established analyzing $^{100}$Mo data collected during 3800h of measurements in 2003. This is the world's best result for $^{100}$Mo. New preliminary half-lives of $\beta\beta2\nu$-mode for $^{100}$Mo, $^{82}$Se, $^{116}$Cd, $^{150}$Nd are also reported as well as the results of background studies. The future progress and plans of the NEMO-3 project are discussed.

Abstract:
The SuperNEMO project aims to search for neutrinoless double beta decay ($0\nu\beta\beta$) up to a sensitivity of 10$^{26}$ years for the $0\nu\beta\beta$ half-life (down to 50 meV in the effective Majorana neutrino mass), using $\sim$100 kg of source and a `tracko-calo' detector. The current status of the 2007--2009 R&D programme is presented here, focusing on the most challenging aspects: calorimetry, production of sources, low radioactivity measurements, and tracker.

Abstract:
The SuperNEMO project aims to search for neutrinoless double beta decay ($0\nu\beta\beta$) up to a sensitivity of 10$^{26}$ years for the $0\nu\beta\beta$ half-life (down to $\sim$ 50~meV in the effective Majorana neutrino mass), using $\sim$100 kg of source and a `tracko-calo' detector. The current status of the 2006--2010 R\&D programme is discussed here.

Abstract:
The soft gamma repeater SGR 1900+14 was observed in Pushchino observatory since 1988 December using BSA radio telescope operating at 111 MHz. We have detected the pulsed radio emission (Shitov 1999) with the same 5.16 s period that was reported earlier for this object. The timing analysis has shown that this new radio pulsar PSR J1907+0919 associated with SGR 1900+14 has a superstrong magnetic field, which is 8.1 * 10^14 G, thereby confirming that it is a "magnetar". The dispersion measure of PSR J1907+0919 is 281.4(9) pc * cm^(-3) which gives an estimate of the pulsar's distance as about 5.8 kpc.

Abstract:
We provide an example of a 6-by-6 matrix with tropical rank equal to 4 and Kapranov rank equal to 5. This answers a question asked by M. Chan, A. Jensen, and E. Rubei.

Abstract:
In this note, we generalize the technique developed in [13] and prove that every 5xn matrix of tropical rank at most 3 has Kapranov rank at most 3, for the ground field that contains at least 4 elements. For the ground field either F_2 or F_3, we construct an example of a 5x5 matrix with tropical rank 3 and Kapranov rank 4.

Abstract:
We provide a nontrivial upper bound for the nonnegative rank of rank-three matrices, which allows us to prove that [6(n+1)/7] linear inequalities suffice to describe a convex n-gon up to a linear projection.

Abstract:
Let $A$ be a real matrix. The term rank of $A$ is the smallest number $t$ of lines (that is, rows or columns) needed to cover all the nonzero entries of $A$. We prove a conjecture of Li et al. stating that, if the rank of $A$ exceeds $t-3$, there is a rational matrix with the same sign pattern and rank as those of $A$. We point out a connection of the problem discussed with the Kapranov rank function of tropical matrices, and we show that the statement fails to hold in general if the rank of $A$ does not exceed $t-3$.

Abstract:
Let $\chi(A)$ denote the characteristic polynomial of a matrix $A$ over a field; a standard result of linear algebra states that $\chi(A^{-1})$ is the reciprocal polynomial of $\chi(A)$. More formally, the condition $\chi^n(X) \chi^k(X^{-1})=\chi^{n-k}(X)$ holds for any invertible $n\times n$ matrix $X$ over a field, where $\chi^i(X)$ denotes the coefficient of $\lambda^{n-i}$ in the characteristic polynomial $\det(\lambda I-X)$. We confirm a recent conjecture of Niv by proving the tropical analogue of this result.

Abstract:
The tropical arithmetic operations on $\mathbb{R}$ are defined by $a\oplus b=\min\{a,b\}$ and $a\otimes b=a+b$. Let $A$ be a tropical matrix and $k$ a positive integer, the problem of Tropical Matrix Factorization (TMF) asks whether there exist tropical matrices $B\in\mathbb{R}^{m\times k}$ and $C\in\mathbb{R}^{k\times n}$ satisfying $B\otimes C=A$. We show that no algorithm for TMF is likely to work in polynomial time for every fixed $k$, thus resolving a problem proposed by Barvinok in 1993.