3,137 research outputs found
Circular law for random discrete matrices of given row sum
Let be a random matrix of size and let
be the eigenvalues of . The empirical spectral
distribution of 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 are i.i.d. copies of a random variable with mean zero and variance
, then the empirical spectral distribution of the normalized matrix
of converges almost surely to the uniform
distribution \mu_\cir over the unit disk as tends to infinity.
In this paper we show that the empirical spectral distribution of the
normalized matrix of , a random matrix whose rows are independent random
vectors of given row-sum with some fixed integer satisfying
, also obeys the circular law. The key ingredient is a new
polynomial estimate on the least singular value of
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 be an by random matrix whose entries are independent real
random variables with mean zero, variance one and with subexponential tail. We
show that the logarithm of 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
Improving the Gilbert-Varshamov bound for -ary codes
Given positive integers , and , denote by the maximum size of a -ary code of length and minimum distance . The famous Gilbert-Varshamov bound asserts that , where is the volume of a -ary sphere of radius . Extending a recent work of Jiang and Vardy on binary codes, we show that for any positive constant less than there is a positive constant such that for . This confirms a conjecture by Jiang and Vardy
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
. 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%)
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