Circular law for random discrete matrices of given row sum

Let $M_n$ be a random matrix of size $n\times n$ and let $\lambda_1,...,\lambda_n$ be the eigenvalues of $M_n$. The empirical spectral distribution $\mu_{M_n}$ of $M_n$ is defined as \mu_{M_n}(s,t)=\frac{1}{n}# \{k\le n, \Re(\lambda_k)\le s; \Im(\lambda_k)\le t\}. The circular law theorem in random matrix theory asserts that if the entries of $M_n$ are i.i.d. copies of a random variable with mean zero and variance $\sigma^2$, then the empirical spectral distribution of the normalized matrix $\frac{1}{\sigma\sqrt{n}}M_n$ of $M_n$ converges almost surely to the uniform distribution \mu_\cir over the unit disk as $n$ tends to infinity. In this paper we show that the empirical spectral distribution of the normalized matrix of $M_n$, a random matrix whose rows are independent random $(-1,1)$ vectors of given row-sum $s$ with some fixed integer $s$ satisfying $|s|\le (1-o(1))n$, also obeys the circular law. The key ingredient is a new polynomial estimate on the least singular value of $M_n$

Non-abelian Littlewood-Offord inequalities

In 1943, Littlewood and Offord proved the first anti-concentration result for sums of independent random variables. Their result has since then been strengthened and generalized by generations of researchers, with applications in several areas of mathematics. In this paper, we present the first non-abelian analogue of Littlewood-Offord result, a sharp anti-concentration inequality for products of independent random variables.Comment: 14 pages Second version. Dependence of the upper bound on the matrix size in the main results has been remove

Random matrices: Law of the determinant

Let $A_n$ be an $n$ by $n$ random matrix whose entries are independent real random variables with mean zero, variance one and with subexponential tail. We show that the logarithm of $|\det A_n|$ satisfies a central limit theorem. More precisely, \begin{eqnarray*}\sup_{x\in {\mathbf {R}}}\biggl|{\mathbf {P}}\biggl(\frac{\log(|\det A_n|)-({1}/{2})\log (n-1)!}{\sqrt{({1}/{2})\log n}}\le x\biggr)-{\mathbf {P}}\bigl(\mathbf {N}(0,1)\le x\bigr)\biggr|\\\qquad\le\log^{-{1}/{3}+o(1)}n.\end{eqnarray*}Comment: Published in at http://dx.doi.org/10.1214/12-AOP791 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

Helminth eggs die-off and nutrients : human excreta storage experiment

Are the current practices of handling human excreta for agricultural purposes by farmers in Vietnam good enough?This study set up an excreta storage experiment to research how to inactivate Ascaris lumbricoides eggs and stillmaintain the nutrient value of human excreta

Vanishing Twist near Focus-Focus Points

We show that near a focus-focus point in a Liouville integrable Hamiltonian system with two degrees of freedom lines of locally constant rotation number in the image of the energy-momentum map are spirals determined by the eigenvalue of the equilibrium. From this representation of the rotation number we derive that the twist condition for the isoenergetic KAM condition vanishes on a curve in the image of the energy-momentum map that is transversal to the line of constant energy. In contrast to this we also show that the frequency map is non-degenerate for every point in a neighborhood of a focus-focus point.Comment: 13 page

. In vitro propagation of the new orchid Dendrobium trankimianum T. Yukawa

Dendrobium trankimianum T. Yukawa is a beautiful, endemic orchid of Vietnam, a new species with a first - published description in 2004. It is very rare and expected to be added to the IUCN Red List status - CR. In vitro studies of orchid D. trankimianum T. Yukawa were conducted in order to conserve and increase the genetic pool of this precious wild orchid species. The results showed that full-strength MS medium supplemented with 2.0 mg/L BA and 0.5 mg/L NAA (10.24 PLBs/explant; 90.11% explants formed PLBs) or full-strength MS medium supplemented with 1.5 mg/L TDZ and 0.5 mg/L NAA (14.11 PLBs/explant; 92.06% explants formed PLBs) were the most suitable for protocorm formation. For subculture, suitable growth of shoots were obtained on full-strength MS medium supplemented 1.5 mg/L BA (22.35 shoots/explant; shoots length of 1.96 cm) and full-strength MS medium supplemented with 60 g ripe banana per liter (25.11 shoots/explant; shoots length of 2.12 cm). The shoots in vitro were transferred to half-strength MS supplemented with different concentrations of IAA, IBA and NAA to investigate root formation. The best rooting occurred at 0,5 mg/L NAA (7.91 roots/shoot; root length of 4.01 cm; 98.51% root formation). The plantlets with uniform growth were planted on different substrate: Eco clean soil, Coconut fiber, Fern fiber, 50% Rice husk in combination with 50% Eco clean soil for research the most suitable substrate. After 60 days of transplantion and acclimatization, the result showed that Fern fiber was suitable substrate for plantlet growth in a nursery garden (8.0 roots/ explant; root length of 5.5 cm; survival rate of 93.29%)