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

Characteristics of protons exiting from a polyethylene converter irradiated by neutrons with energies between 1 keV and 10 MeV

<p>Figs. 1 to 4, and Figs. 6 to 10</p

Electron ejection fractions during a photoelectric interaction from various atomic subshells for (a) 10 keV, (b) 100 keV and (c) 1 MeV incident photon energies, for a range of water medium thicknesses.

<p>Electron ejection fractions during a photoelectric interaction from various atomic subshells for (a) 10 keV, (b) 100 keV and (c) 1 MeV incident photon energies, for a range of water medium thicknesses.</p

Proton range in PE as a function of the proton energy (obtained using SRIM).

<p>Proton range in PE as a function of the proton energy (obtained using SRIM).</p

Distribution of protons as a function of the proton energy with the thickness of the PE layer as a parameter (in cm).

<p>(a) For incident neutron energy of 0.5 MeV; (b) for incident neutron energy of 1 MeV; (c) for incident neutron energy of 5 MeV; (d) for incident neutron energy of 10 MeV.</p

Schematic diagrams of photon interaction mechanisms considered in the NRUphoton code.

<p>Schematic diagrams of photon interaction mechanisms considered in the NRUphoton code.</p

Cumulative distribution functions (CDFs) for incoherent and coherent scattering.

<p>Cumulative distribution functions (CDFs) for incoherent and coherent scattering.</p

Geometry models and consideration of neutrons beam direction.

<p>(a) Geometry of interactions in the PE layer. (b) Schematic diagram showing the neutron beam direction.</p

Electron hits on the exposed cell array with 10⁴ cells for 100 keV incident photons for administered doses of (a) 10 mGy, (b) 20 mGy, (c) 50 mGy, (d) 100 mGy and (e) 500 mGy.

<p>Electron hits on the exposed cell array with 10⁴ cells for 100 keV incident photons for administered doses of (a) 10 mGy, (b) 20 mGy, (c) 50 mGy, (d) 100 mGy and (e) 500 mGy.</p

Dose delivered per incident photon in the IOC from photons with different incident energies from the source.

<p>Dose delivered per incident photon in the IOC from photons with different incident energies from the source.</p