A short note on the joint entropy of n/2-wise independence

In this note, we prove a tight lower bound on the joint entropy of $n$ unbiased Bernoulli random variables which are $n/2$-wise independent. For general $k$-wise independence, we give new lower bounds by adapting Navon and Samorodnitsky's Fourier proof of the `LP bound' on error correcting codes. This counts as partial progress on a problem asked by Gavinsky and Pudl\'ak.Comment: 6 pages, some errors fixe

Monte Carlo studies on neutron interactions in radiobiologial experiments

Monte Carlo method was used to study the characteristics of neutron interactions with cells underneath a water medium layer with varying thickness. The following results were obtained. (1) The fractions of neutron interaction with ¹H, ¹²C, ¹⁴N and ¹⁶O nuclei in the cell layer were studied. The fraction with ¹H increased with increasing medium thickness, while decreased for ¹²C, ¹⁴N and ¹⁶O nuclei. The bulges in the interaction fractions with ¹²C, ¹⁴N and ¹⁶O nuclei were explained by the resonance spikes in the interaction cross-section data. The interaction fraction decreased in the order: ¹H > ¹⁶O > ¹²C > ¹⁴N. (2) In general, as the medium thickness increased, the number of “interacting neutrons” which exited the medium and then further interacted with the cell layer increased. (3) The area under the angular distributions for “interacting neutrons” decreased with increasing incident neutron energy. Such results would be useful for deciphering the reasons behind discrepancies among existing results in the literature

Monte Carlo studies on neutron interactions in radiobiological experiments - Fig 7

<p>Interaction fractions of neutrons with (a) ¹H, (b) ¹²C, (c) ¹⁴N and (d) ¹⁶O within cell layer for 100 keV incident neutrons.</p

Monte Carlo studies on neutron interactions in radiobiological experiments - Fig 10

<p>Penetration fractions of neutrons from the cell-medium system for neutrons with incident energies of (a) 100 keV, (b) 2 MeV and (c) 10 MeV.</p

Three-dimensional neutron tracking module and geometry of interaction with water-cell system.

<p>Three-dimensional neutron tracking module and geometry of interaction with water-cell system.</p

Monte Carlo studies on neutron interactions in radiobiological experiments - Fig 12

<p>Angular distributions of neutrons exiting the medium layer that interact in the cell layer for incident neutron energies of (a) 100 keV, (b) 2 MeV and (c) 10 MeV.</p

Schematic diagram showing the neutron irradiation setup.

<p>Schematic diagram showing the neutron irradiation setup.</p

Mean free path for neutrons in the water medium as a function of the neutron energy.

<p>Mean free path for neutrons in the water medium as a function of the neutron energy.</p

Flowchart for the algorithm in the neutron collision estimator in the NRUneutron code.

<p>Flowchart for the algorithm in the neutron collision estimator in the NRUneutron code.</p