Parity check matrices and product representations of squares

Let \(N_{\mathbb{F}} \)(n,k,r) denote the maximum number of columns in an n-row matrix with entries in a finite field \(\mathbb{F}\) in which each column has at most r nonzero entries and every k columns are linearly independent over \(\mathbb{F}\). We obtain near-optimal upper bounds for \(N_{\mathbb{F}} \)(n,k,r) in the case k > r. Namely, we show that \(N_\mathbb{F} (n,k,r) \ll n^{\frac{r}{2} + \frac{{cr}}{k}} \) where \(c \approx \frac{4}{3}\) for large k. Our method is based on a novel reduction of the problem to the extremal problem for cycles in graphs, and yields a fast algorithm for finding short linear dependencies. We present additional applications of this method to a problem on hypergraphs and a problem in combinatorial number theory.     

Antichains in partially ordered sets of singular cofinality

In their paper from 1981, Milner and Sauer conjectured that for any poset , if , then P must contain an antichain of size κ. We prove that for λ > cf(λ) = κ, if there exists a cardinal μ < λ such that cov(λ, μ, κ, 2) = λ, then any poset of cofinality λ contains λ ^κ antichains of size κ. The hypothesis of our theorem is very weak and is a consequence of many well-known axioms such as GCH, SSH and PFA. The consistency of the negation of this hypothesis is unknown.      

Detection of microscopic anisotropy in gray matter and in a novel tissue phantom using double Pulsed Gradient Spin Echo MR

A double Pulsed Gradient Spin Echo (d-PGSE) MR experiment was used to measure and assess the degree of local diffusion anisotropy in brain gray matter, and in a novel "gray matter" phantom that consists of randomly oriented tubes filled with water. In both samples, isotropic diffusion was observed at a macroscopic scale while anisotropic diffusion was observed at a microscopic scale, however, the nature of the resulting echo attenuation profiles were qualitatively different. Gray matter, which contains multiple cell types and fibers, exhibits a more complicated echo attenuation profile than the phantom. Since microscopic anisotropy was observed in both samples in the low q regime comparable to that achievable in clinical scanner, it may offer a new potential contrast mechanism for characterizing gray matter microstructure in medical and biological applications.     

The surface measure and cone measure on the sphere of

We prove a concentration inequality for the norm on the sphere for . This inequality, which generalizes results of Schechtman and Zinn (2000), is used to study the distance between the cone measure and surface measure on the sphere of . In particular, we obtain a significant strengthening of the inequality derived by Naor and Romik (2003), and calculate the precise dependence of the constants that appeared there on .

     

Water Structure Recovery in Chaotropic Anion Recognition: High‐Affinity Binding of Dodecaborate Clusters to γ‐Cyclodextrin

Dodecaborate anions of the type B₁₂X₁₂^2? and B₁₂X₁₁Y^2? (X=H, Cl, Br, I and Y=OH, SH, NH₃⁺, NR₃⁺) form strong (K_a up to 10⁶ L mol^?1, for B₁₂Br₁₂^2?) inclusion complexes with γ‐cyclodextrin (γ‐CD). The micromolar affinities reached are the highest known for this native CD. The complexation exhibits highly negative enthalpies (up to ?25 kcal mol^?1) and entropies (TΔS up to ?18.4 kcal mol^?1, both for B₁₂I₁₂^2?), which position these guests at the bottom end of the well‐known enthalpy‐entropy correlation for CDs. The high driving force can be traced back to a chaotropic effect, according to which chaotropic anions have an intrinsic affinity to hydrophobic cavities in aqueous solution. In line with this argument, salting‐in effects revealed dodecaborates as superchaotropic dianions.     

On guessing generalized clubs at the successors of regulars

König, Larson and Yoshinobu initiated the study of principles for guessing generalized clubs, and introduced a construction of a higher Souslin tree from the strong guessing principle.Complementary to the author’s work on the validity of diamond and non-saturation at the successor of singulars, we deal here with a successor of regulars. It is established that even the non-strong guessing principle entails non-saturation, and that, assuming the necessary cardinal arithmetic configuration, entails a diamond-type principle which suffices for the construction of a higher Souslin tree.We also establish the consistency of GCH with the failure of the weakest form of generalized club guessing. This, in particular, settles a question from the original paper.     

High throughput production of single core double emulsions in a parallelized microfluidic device

Double emulsions are useful templates for microcapsules and complex particles, but no method yet exists for making double emulsions with both high uniformity and high throughput. We present a parallel numbering-up design for microfluidic double emulsion devices, which combines the excellent control of microfluidics with throughput suitable for mass production. We demonstrate the design with devices incorporating up to 15 dropmaker units in a two-dimensional or three-dimensional array, producing single-core double emulsion drops at rates over 1 kg day(-1) and with diameter variation less than 6%. This design provides a route to integrating hundreds of dropmakers or more in a single chip, facilitating industrial-scale production rates of many tons per year.     

Discretization and affine approximation in high dimensions

Lower estimates are obtained for the macroscopic scale of affine approximability of vector-valued Lipschitz functions on finite-dimensional normed spaces, completing the work of Bates, Johnson, Lindenstrauss, Preiss and Schechtman. This yields a new approach to Bourgain’s discretization theorem for superreflexive targets.     

Plant cystoliths: a complex functional biocomposite of four distinct silica and amorphous calcium carbonate phases

Plant cystoliths are mineralized objects that are formed by specialized cells in the leaves of certain plants. The main mineral component of cystoliths by volume is amorphous calcium carbonate (ACC) and the minor component is silica. We show that the silica stalk is formed first and is essential for ACC formation. Furthermore, the cystolith is shown to be composed of four distinct mineral phases with different chemical properties: an almost pure silica phase grades into a Mg-rich silica phase. This Mg-rich silica is overlaid by a relatively stable ACC phase. A bulky and less stable ACC phase encapsulates the first ACC phase. This architecture poses interesting questions about the role of Mg in the silica phase and suggests a strategy for ACC stabilization that takes advantage of a precise regulation of the mineral-growth microenvironment.