Hyperplane projections of the unit ball of ℓ p n

Let B _p ⁿ ={x∈\R ⁿ ;\; \sum _i=1 ⁿ |x _i | ^p ≤ 1} , 1≤ p\le+∈fty . We study the extreme values of the volume of the orthogonal projection of B _p ⁿ onto hyperplanes H\subset \R ⁿ . For a fixed H , we prove that the ratio vol(P _H B _p ⁿ )/ vol(B _p ^n-1 ) is non-decreasing in p∈[1,+∈fty] . Received May 21, 2001, and in revised form September 2, 2001. Online publication December 17, 2001.     

Embedding the diamond graph inLpand dimension reduction inL1

We show that any embedding of the level k diamond graph of Newman and Rabinovich [NR] into L_p, 1 < p 2, requires distortion at least . An immediate corollary is that there exist arbitrarily large n-point sets such that any D-embedding of X into requires . This gives a simple proof of a recent result of Brinkman and Charikar [BrC] which settles the long standing question of whether there is an L₁ analogue of the Johnson-Lindenstrauss dimension reduction lemma [JL].     

Solution of Shannon's problem on the monotonicity of entropy

It is shown that if are independent and identically distributed square-integrable random variables, then the entropy of the normalized sum

is an increasing function of .

The result also has a version for non-identically distributed random variables or random vectors.

     

Proton transport through water-filled carbon nanotubes

Proton transfer along 1D chains of water molecules inside carbon nanotubes is studied by simulations. Ab initio molecular dynamics and an empirical valence bond model yield similar structures and time scales. The proton mobility along 1D water chains exceeds that in bulk water by a factor of 40, but is reduced if orientational defects are present. Excess protons interact with hydrogen-bonding defects through long-range electrostatics, resulting in coupled motion of protons and defects.     

Triple-hybrid sub-micron particles: Ag@polymer@silica

Journal of Sol-Gel Science and Technology - Methods for the synthesis of triple-hybrid (metal, organic polymer, ceramic component) submicron particles of silver@(polymer@silica) spheres were...     

Ultrametric subsets with large Hausdorff dimension

It is shown that for every ε∈(0,1), every compact metric space (X,d) has a compact subset S?X that embeds into an ultrametric space with distortion O(1/ε), and $$\dim_H(S)\geqslant (1-\varepsilon)\dim_H(X),$$ where dim _H (?) denotes Hausdorff dimension. The above O(1/ε) distortion estimate is shown to be sharp via a construction based on sequences of expander graphs.     

Solution of the Propeller Conjecture in \mathbb R ^3

It is shown that every measurable partition $\{A_1,\ldots , A_k\}$ { A 1 , … , A k } of $\mathbb R ^3$ R 3 satisfies 1 $$\begin{aligned} \sum _{i=1}^k\big \Vert \int _{A_i} x\mathrm{{e}}^{-\frac{1}{2}\Vert x\Vert _2^2}\mathrm{{d}}x\big \Vert _2^2\leqslant 9\pi ^2. \end{aligned}$$ ∑ i = 1 k ‖ ∫ A i x e - 1 2 ‖ x ‖ 2 2 d x ‖ 2 2 ? 9 π 2 . Let $\{P_1,P_2,P_3\}$ { P 1 , P 2 , P 3 } be the partition of $\mathbb R ^2$ R 2 into $120^{\circ }$ 120 ° sectors centered at the origin. The bound (1) is sharp, with equality holding if $A_i=P_i\times \mathbb R $ A i = P i × R for $i\in \{1,2,3\}$ i ∈ { 1 , 2 , 3 } and $A_i=\emptyset $ A i = ? for $i\in \{4,\ldots ,k\}$ i ∈ { 4 , … , k } . This settles positively the $3$ 3 -dimensional Propeller Conjecture of Khot and Naor [(Mathematika 55(1-2):129–165, 2009 (FOCS 2008)]. The proof of (1) reduces the problem to a finite set of numerical inequalities which are then verified with full rigor in a computer-assisted fashion. The main consequence (and motivation) of (1) is complexity-theoretic: the unique games hardness threshold of the kernel clustering problem with $4\times 4$ 4 × 4 centered and spherical hypothesis matrix equals $\frac{2\pi }{3}$ 2 π 3 .     

SEMILLAC: A new hybrid atomic model of hot dense plasmas

SEMILLAC is a fast, yet highly accurate method to calculate ionic population distributions in plasmas at a given electron temperature and density. SEMILLAC solves rate equations for non-relativistic configurations population distributions. It considers electron collisional, radiative and autoionizing atomic processes. The code is designed to be highly versatile so it can be used for modeling a wide range of laboratory plasmas. The population distributions can be calculated for steady state or time dependent conditions, with or without the presence of a radiation field. SEMILLAC is designed to be used as a tool for population distributions calculations and spectroscopic modeling of plasmas. Our aim is to get high accuracy while keeping the code fast enough to be used for standard PC calculations. At the heart of our method, average transitions energies and rate coefficients are calculated for a restricted set of simple non-relativistic ionic configurations using the HULLAC code. We then use this basic set to calculate energies and rates coefficients of more complex, multiply excited configurations.     

Euclidean quotients of finite metric spaces

This paper is devoted to the study of quotients of finite metric spaces. The basic type of question we ask is: Given a finite metric space M and α?1, what is the largest quotient of (a subset of) M which well embeds into Hilbert space. We obtain asymptotically tight bounds for these questions, and prove that they exhibit phase transitions. We also study the analogous problem for embeddings into ?_p, and the particular case of the hypercube.     

Polymer-Bromine Interactions. I. Interactions with Polyacrylonitrile

The present paper deals with the interactions of bromine with poly-acrylonitrile (PAN). Kinetics and equilibria of the sorption of Br₂ on PAN were studied at a concentration range of 0.01–0.1 mol/L and a temperature range of 25–40°C. Two kinds of sorption were found: a “reversible” sorption removable by water and an “irreversible” sorption removable by aqueous ammonia solutions. The irreversibly sorbed bromine is presumably linked by charge transfer to the nitrile groups of the PAN, as evidenced by UV spectra. The irreversible sorption follows the reversible sorption and is slower. Partition coefficients obtained from the linear Freundlich isotherms increased with temperature and, at 40°C, the values obtained were 97, 65, and 32 L/kg for the total, irreversible, and reversible sorptions, respectively. At 25°C the chemical potential, enthalpy, and change in entropy for the irreversible sorption were ?2.0 kcal/mol, 9.4 kcal/mol, and 38 cal·mol^?1·K^?1. Effects of a 6-day Br₂ treatment and ammonia rinse were: decrease in dry T _g from 74.5 to 61°C and in water from 38 to 35°C; no significant decrease in M _W ; decrease in tensile strength measured after the bromine stage, and improvement after ammonia stage; increased swollen dimensions from 57% in water to 75%; and stabilization of swollen dimensions upon drying. The results support the existence of two phases in the less ordered regions of PAN.