Fluctuation–Dissipation Relation for Systems with Spatially Varying Friction

When a particle diffuses in a medium with spatially dependent friction coefficient \(\alpha (r)\) at constant temperature \(T\) , it drifts toward the low friction end of the system even in the absence of any real physical force \(f\) . This phenomenon, which has been previously studied in the context of non-inertial Brownian dynamics, is termed “spurious drift”, although the drift is real and stems from an inertial effect taking place at the short temporal scales. Here, we study the diffusion of particles in inhomogeneous media within the framework of the inertial Langevin equation. We demonstrate that the quantity which characterizes the dynamics with non-uniform \(\alpha (r)\) is not the displacement of the particle \(\Delta r=r-r^0\) (where \(r^0\) is the initial position), but rather \(\Delta A(r)=A(r)-A(r^0)\) , where \(A(r)\) is the primitive function of \(\alpha (r)\) . We derive expressions relating the mean and variance of \(\Delta A\) to \(f\) , \(T\) , and the duration of the dynamics \(\Delta t\) . For a constant friction coefficient \(\alpha (r)=\alpha \) , these expressions reduce to the well known forms of the force-drift and fluctuation–dissipation relations. We introduce a very accurate method for Langevin dynamics simulations in systems with spatially varying \(\alpha (r)\) , and use the method to validate the newly derived expressions.     

In quest of reversibility of Friedel-Crafts acyl rearrangements in the pyrene series

Friedel-Crafts acyl rearrangements in PPA of diacetylpyrenes (80–120 °C), dibenzoylpyrenes (80–200 °C), and bis(4-flurobenzoyl)pyrenes (80–120 °C) and Scholl reactions in AlCl₃/NaCl of dibenzoylpyrenes (140–200 °C) have been studied. The substrates were 1-AcPY, 2-AcPY, 1,3-Ac₂PY, 1,6-Ac₂PY, 1,8-Ac₂PY, 2,7-Ac₂PY, 1-BzPY, 1,6-Bz₂PY, 1,8-Bz₂PY, 1-4FBzPY, 1,6-4FBz₂PY, 1,8-4FBz₂PY. The mixtures of pyrene, 1-AcPY, 2-AcPY, 1,3-Ac₂PY, 1,6-Ac₂PY, 1,8-Ac₂PY, and 2,7-Ac₂PY were separated by HPLC. The following reversible intermolecular isomerizations were established: 1,6-Ac₂PY ? 1,8-Ac₂PY, 1,6-Bz₂PY ? 1,8-Bz₂PY, and 1,6-4'FBz₂PY ? 1,8-4'FBz₂PY, albeit not in high yields. The results substantiate Gore’s 1955 proposition that “The Friedel–Crafts acylation reaction of reactive aromatic hydrocarbons is a reversible process.” The isomerizations reported here differ from the few previously reported completely reversible intramolecular Friedel-Crafts acyl rearrangements. At ≥ 140 °C, in PPA and in AlCl₃/NaCl, 1,6-Bz₂PY and 1,8-Bz₂PY underwent a highly regioselective double Scholl reaction to give pyranthrone (3) and deacylations to 1-BzPy (and pyrene), followed by mono-Scholl reactions to give 8H-dibenzo[def,qr]chrysen-8-one (1), and 11H-indeno[2,1-a]pyren-11-one (2). The formation of 3 and not the expected tribenzo[a,ghi,o]perylene-7,16-dione (4) from 1,8-Bz₂PY indicates that 1,8-Bz₂PY has first undergone isomerization to 1,6-Bz₂PY. The present study confirms the linkage between Friedel-Crafts acyl rearrangements and the Scholl reaction.

     

On the convergence of circle packings to the Riemann map

Rodin and Sullivan (1987) proved Thurston’s conjecture that a scheme based on the Circle Packing Theorem converges to the Riemann mapping, thereby providing a refreshing geometric view of Riemann’s Mapping Theorem. We now present a new proof of the Rodin–Sullivan theorem. This proof is based on the argument principle, and has the following virtues. 1. It applies to more general packings. The Rodin–Sullivan paper deals with packings based on the hexagonal combinatorics. Later, quantitative estimates were found, which also worked for bounded valence packings. Here, the bounded valence assumption is unnecessary and irrelevant. 2. Our method is rather elementary, and accessible to non-experts. In particular, quasiconformal maps are not needed. Consequently, this gives an independent proof of Riemann’s Conformal Mapping Theorem. (The Rodin–Sullivan proof uses results that rely on Riemann’s Mapping Theorem.) 3. Our approach gives the convergence of the first and second derivatives, without significant additional difficulties. While previous work has established the convergence of the first two derivatives for bounded valence packings, now the bounded valence assumption is unnecessary. Oblatum 15-V-1995 & 13-XI-1995     

Compositions of random transpositions

LetY=(y1,y2, ...),y1≥y2≥..., be the list of sizes of the cycles in the composition ofcn transpositions on the set {1, 2, ...,n}. We prove that ifc>1/2 is constant andn → ∞, the distribution off(c)Y/n converges toPD(1), the Poisson-Dirichlet distribution with parameter 1, where the functionf is known explicitly. A new proof is presented of the theorem by Diaconis, Mayer-Wolf, Zeitouni and Zerner stating that thePD(1) measure is the unique invariant measure for the uniform coagulation-fragmentation process. In loving memory of my parents, Hanna and Mickey Schramm     

Synthesis of lsopropyl-1-thio-Β-D-glucopyranoside (IPTGlc), an inducer ofaspergillus niger b1 Β-glucosidase production

Production of Β-glucosidase inAspergillus niger Bl is subjected to catabolic repression by glucose.Aspergillus niger Bl grown on bran as a carbon source secreted Β-glucosidase. The maximum level of the enzyme was reached after 7 d of fermentation. Addition of 1% glucose to the medium suppressed Β-glucosidase production to undetectable levels. In this study, the organic synthesis of a potential inducer of Β-glucosidase production by A.niger Bl’s reported. Isopropyl-1-thio-Β-D-glucopyranoside (IPTGlc) was synthesized using a two-step organic synthesis protocol. The H-NMR data agreed with those reported previously for the galactoside analog. When IPTGlc was added 24 h after inoculation at a final concentration of 0.4 mM, similar levels of Β-glucosidase were reached 3 to 4 d earlier as compared to fermentation without IPTGlc induction. In practice, this may translate to a more efficient method of producing Β-glucosidase from this fungus.     

Daubechies wavelets as a basis set for density functional pseudopotential calculations

Daubechies wavelets are a powerful systematic basis set for electronic structure calculations because they are orthogonal and localized both in real and Fourier space. We describe in detail how this basis set can be used to obtain a highly efficient and accurate method for density functional electronic structure calculations. An implementation of this method is available in the ABINIT free software package. This code shows high systematic convergence properties, very good performances, and an excellent efficiency for parallel calculations.     

Colorful Strips

We study the following geometric hypergraph coloring problem: given a planar point set and an integer k, we wish to color the points with k colors so that any axis-aligned strip containing sufficiently many points contains all colors. We show that if the strip contains at least 2k − 1 points, such a coloring can always be found. In dimension d, we show that the same holds provided the strip contains at least k(4 ln k + ln d) points. We also consider the dual problem of coloring a given set of axis-aligned strips so that any sufficiently covered point in the plane is covered by k colors. We show that in dimension d the required coverage is at most d(k − 1) + 1. This complements recent impossibility results on decomposition of strip coverings with arbitrary orientations. From the computational point of view, we show that deciding whether a three-dimensional point set can be 2-colored so that any strip containing at least three points contains both colors is NP-complete. This shows a big contrast with the planar case, for which this decision problem is easy.     

Wiring of redox enzymes on three dimensional self-assembled molecular scaffold

The integration of biological molecules and nanoscale components provides a fertile basis for the construction of hybrid materials of synergic properties and functions. Stable protein 1 (SP1), a highly stable ring shaped protein, was recently used to display different functional domains, to bind nanoparticles (NPs), and to spontaneously form two and three-dimensional structures. Here we show an approach to wire redox enzymes on this self-assembled protein-nanoparticle hybrid. Those hybrids are genetically engineered SP1s, displaying glucose oxidase (GOx) enzymes tethered to the protein inner pore. Moreover, the Au-NP-protein hybrids self-assembled to multiple enzymatic layers on the surface. By wiring the redox enzymes to the electrode, we present an active structure for the bioelectrocatalytic oxidation of glucose. This system demonstrates for the first time a three-dimensional assembly of multiple catalytic modules on a protein scaffold with an efficient electrical wiring of the enzyme units on an electrode surface, thus implementing a hybrid electrically active unit for nanobioelectronic applications.