The Cosmic Time in Terms of the Redshift

In cosmology one labels the time t since the Big Bang in terms of the redshift of light emitted at t, as we see it now. In this Note we derive a formula that relates t to z which is valid for all redshifts. One can go back in time as far as one wishes, but not to the Big Bang at which the redshift tends to infinity.     

On the Helical Crystals of Cholesterol Monohydrate

We review in this short perspective the history of cholesterol crystals and crystal structures. We address in particular the helical crystals that form in vitro and in pathology from environments rich in bile acids or from phospholipid membranes. We review the known mechanisms leading to crystals with chiral morphology, from screw-dislocation mediated growth to mechanisms involving asymmetric mechanical strain. We propose a mechanism for cholesterol helical crystal development based on the monoclinic cholesterol monohydrate crystal structure. We suggest that curvature arises in few layers thick crystals due to the tension induced between the hydrophobic layer and the ice-like H-bonded lattice of the water molecules with the cholesterol hydroxy groups. Helicity would ensue through a combination of the curvature and the fast growth of a thin ribbon in one crystal direction.     

Theory of laser enhancement and suppression of cold reactions: the fermion-boson 6Li+7Li2<-->(variant Planck's over 2pi omega0) 6Li7Li+7Li radiative collision

We present a nonperturbative time-dependent quantum mechanical theory of the laser catalysis and control of a bifurcating A+BC<-->(variant Planck's over 2pi omega(0))ABC*(v)<-->(variant Planck's over 2pi omega(0) )AB+C reaction, with ABC*(v) denoting an intermediate, electronically excited, complex of ABC in the vth vibrational state. We apply this theory to the low collision energy fermion-boson light-induced exchange reaction, (6)Li((2)S)+(7)Li(2)((3)Sigma(u)(+))<-->(variant Planck's over 2pi omega(0))((6)Li(7)Li(7)Li)*<-->(variant Planck's over 2pi omega(0))(6)Li(7)Li((3)Sigma(+))+(7)Li((2)S). We show that at very low collision energies and energetically narrow (approximately 0.01 cm(-1)) initial reactant wave packets, it is possible to tune the yield of the exchange reaction from 0 to near-unity (yield >or=99%) values. Controllability is somewhat reduced at collisions involving energetically wider (approximately 1 cm(-1)) initial reactant wave packets. At these energetic bandwidths, the radiative reactive control, although still impressive, is limited to the 0%-76% reactive-probabilities range.     

Thermally induced fluid reversed hexagonal (H(II)) mesophase

In the present study we characterized the microstructures of the Lc and HII phases in a glycerol monooleate (GMO)/tricaprylin (TAG)/water mixture as a function of temperature. We studied the factors that govern the formation of a low-viscosity HII phase at relatively elevated temperatures (>35 degrees C). This phase has very valuable physical characteristics and properties. The techniques used were differential scanning calorimetry (DSC), wide- and small-angle X-ray scattering (WAXS and SAXS, respectively), NMR (self-diffusion and (2)H NMR), and Fourier transform infrared (FTIR) spectroscopies. The reverse hexagonal phase exhibited relatively rapid flow of water in the inner channels within the densely packed cylindrical aggregates of GMO with TAG molecules located in the interstices. The existence of two water diffusion peaks reflects the existence of both mobile water and hydration water at the GMO-water interface (hydrogen exchange between the GMO hydroxyls and water molecules). Above 35 degrees C, the sample became fluid yet hexagonal symmetry was maintained. The fluidity of the HII phase is explained by a significant reduction in the domain size and also perhaps cylinder length. This phenomenon was characterized by higher mobility of the GMO, lower mobility of the water, and a significant dehydration process.     

Reduced limits for nonlinear equations with measures

We consider equations (E) −Δu+g(u)=μ in smooth bounded domains Ω⊂RN, where g is a continuous nondecreasing function and μ is a finite measure in Ω. Given a bounded sequence of measures (μk), assume that for each k?1 there exists a solution uk of (E) with datum μk and zero boundary data. We show that if uk→u# in L¹(Ω), then u# is a solution of (E) relative to some finite measure μ#. We call μ# the reduced limit of (μk). This reduced limit has the remarkable property that it does not depend on the boundary data, but only on (μk) and on g. For power nonlinearities g(t)=|t|^q−1t, ∀t∈R, we show that if (μk) is nonnegative and bounded in W−2,q(Ω), then μ and μ# are absolutely continuous with respect to each other; we then produce an example where μ#≠μ.     

Efficiently computing the permanent and Hafnian of some banded Toeplitz matrices

We present a new efficient method for computing the permanent and Hafnian of certain banded Toeplitz matrices. The method covers non-trivial cases for which previous known methods do not apply. The main idea is to use the elements of the first row and column, which determine the entire Toeplitz matrix, to construct a digraph in which certain paths correspond to permutations that the permanent and Hafnian count. Since counting paths can be done efficiently, the permanent and Hafnian for those matrices is easily obtainable.     

The Odd-Distance Plane Graph

The vertices of the odd-distance graph are the points of the plane ℝ². Two points are connected by an edge if their Euclidean distance is an odd integer. We prove that the chromatic number of this graph is at least five. We also prove that the odd-distance graph in ℝ² is countably choosable, while such a graph in ℝ³ is not. The research of J. Maňuch was supported in part by MITACS (Mathematics of Information Technology and Complex Systems). The research of M. Rosenfeld was supported in part by the Chancellor Research Grant and the Institute of Technology, UWT. The research of S. Shelah was supported by the United States-Israel Binational Science Foundation (Grant no. 2002323), and by NSF grant No. NSF-DMS 0600940. No. 923 on Shelah’s publication list. The research of L. Stacho was supported in part by NSERC (Natural Science and Engineering Research Council of Canada) grant.     

Synthesis of styrene–acrylamide copolymer by surfactant‐free sonicated dynamic interfacial polymerization

This paper summarizes a study on emulsifier‐free ultrasonically assisted in situ dynamic interfacial emulsion copolymerization process of acrylamide and styrene. The resulting emulsions are stable and uniform for several months. Thermogravimetric analysis (TGA) curves and reaction conversion measurements have provided an important knowledge regarding the emulsifier‐free polymerization method. Solvent extractions (water, methanol, and xylene) have shown that the polymerization product is essentially a styrene–acrylamide copolymer. The copolymer produced is a block copolymer, PS‐b‐PAM, where each block contains small amounts of the other comonomer. The produced emulsions are film forming at room temperature in spite of the very high block T_gs, owing to a unique water plasticization effect of the polyacrylamide blocks. Some films prepared from the PS‐b‐PAM have resulted in clear and transparent films. The presented interfacial dynamic polymerization process is fast, reaching 81% conversion within 2 hr of sonication at 4°C (low temperature owing to molecular weight and kinetic considerations), and produces very stable PS‐b‐PAM emulsions. TGA was extensively used as an analytical tool for determination of the reaction parameters and composition of the acrylamide–styrene copolymers. Copyright © 2011 John Wiley & Sons, Ltd.