Dynamics of Finger Formation in Laplacian Growth Without Surface Tension

We study the dynamics of “finger” formation in Laplacian growth without surface tension in a channel geometry (the Saffman–Taylor problem). We present a pedagogical derivation of the dynamics of the conformal map from a strip in the complex plane to the physical channel. In doing so we pay attention to the boundary conditions (no flux rather than periodic) and derive a field equation of motion for the conformal map. We first consider an explicit analytic class of conformal maps that form a basis for solutions in infinitely long channels, characterized by meromorphic derivatives. The great bulk of these solutions can lose conformality due to finite time singularities. By considerations of the nature of the analyticity of these solutions, we show that those solutions which are free of such singularities inevitably result in a single asymptotic “finger” whose width is determined by initial conditions. This is in contradiction with the experimental results that indicate selection of a finger of width 1/2. In the last part of this paper we show that such a solution might be determined by the boundary conditions of a finite body of fluid, e.g. finiteness can lead to pattern selection.     

Asymptotic theory of flow separation from wings of low aspect ratio

Numerous methods have been developed to calculate the aerodynamic characteristics of wings of low aspect ratio in the case when there is flow separation from the wing edges. Among the methods based on direct solution of the three-dimensional Euler equations there are the method of discrete vortices [1, 2] and the panel method [3]. In addition, numerical and asymptotic methods [4, 5] based on the theory of slender bodies [6] are used. One of the most important shortcomings of this theory is the dependence of the flow pattern at a given section of the wing on only the upstream flow. The obtained solutions therefore contain no information about the influence of the trailing edge of the wing, on which, as is well known, the Chaplygin-Zhukovskii condition is satisfied. The aim of the present paper is to construct an asymptotic theory of higher approximation and a corresponding numerical method for calculating flow separation from wings of low aspect ratio in which this shortcoming is absent.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 141–147, July–August, 1982.     

Optically induced rotation of an exciton spin in a semiconductor quantum dot

We demonstrate control over the spin state of a semiconductor quantum dot exciton using a polarized picosecond laser pulse slightly detuned from a biexciton resonance. The control pulse follows an earlier pulse, which generates an exciton and initializes its spin state as a coherent superposition of its two nondegenerate eigenstates. The control pulse preferentially couples one component of the exciton state to the biexciton state, thereby rotating the exciton's spin direction. We detect the rotation by measuring the polarization of the exciton spectral line as a function of the time difference between the two pulses. We show experimentally and theoretically how the angle of rotation depends on the detuning of the second pulse from the biexciton resonance.     

A Construction of Almost Steiner Systems

Let n, k, and t be integers satisfying . A Steiner system with parameters t, k, and n is a k‐uniform hypergraph on n vertices in which every set of t distinct vertices is contained in exactly one edge. An outstanding problem in Design Theory is to determine whether a nontrivial Steiner system exists for . In this note we prove that for every and sufficiently large n, there exists an almost Steiner system with parameters t, k, and n; that is, there exists a k‐uniform hypergraph on n vertices such that every set of t distinct vertices is covered by either one or two edges.     

Crossing numbers of sequences of graphs II: Planar tiles

We describe a method of creating an infinite family of crossing‐critical graphs from a single small planar map, the tile, by gluing together many copies of the tile together in a circular fashion. This method yields all known infinite families of k‐crossing‐critical graphs. Furthermore, the method yields new infinite families, which extend from (4,6) to (3.5,6) the interval of rationals r for which there is, for some k, an infinite sequence of k‐crossing‐critical graphs all having average degree r. © 2003 Wiley Periodicals, Inc. J Graph Theory 42: 332–341, 2003     

Gas permeability and n‐butane solubility of poly(1‐trimethylgermyl‐1‐propyne)

The gas permeability and n‐butane solubility in glassy poly(1‐trimethylgermyl‐1‐propyne) (PTMGP) are reported. As synthesized, the PTMGP product contains two fractions: (1) one that is insoluble in toluene and soluble only in carbon disulfide (the toluene‐insoluble polymer) and (2) one that is soluble in both toluene and carbon disulfide (the toluene‐soluble polymer). In as‐cast films, the gas permeability and n‐butane solubility are higher in films prepared from the toluene‐soluble polymer (particularly in those films cast from toluene) than in films prepared from the toluene‐insoluble polymer and increase to a maximum in both fractions after methanol conditioning. For example, in as‐cast films prepared from carbon disulfide, the oxygen permeability at 35 °C is 330 × 10^?10 cm³ (STP) cm/(cm² s cmHg) for the toluene‐soluble polymer and 73 × 10^?10 cm³ (STP) cm/(cm² s cmHg) for the toluene‐insoluble polymer. After these films are conditioned in methanol, the oxygen permeability increases to 5200 × 10^?10 cm³ (STP) cm/(cm² s cmHg) for the toluene‐soluble polymer and 6200 × 10^?10 cm³ (STP) cm/(cm² s cmHg) for the toluene‐insoluble polymer. The rankings of the fractional free volume and nonequilibrium excess free volume in the various PTMGP films are consistent with the measured gas permeability and n‐butane solubility values. Methanol conditioning increases gas permeability and n‐butane solubility of as‐cast PTMGP films, regardless of the polymer fraction type and casting solvent used, and minimizes the permeability and solubility differences between the various films (i.e., the permeability and solubility values of all conditioned PTMGP films are similar). © 2002 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 40: 2228–2236, 2002     

Asymptotic solution of the axisymmetric problem of ideal fluid flow in the neighborhood of cusped cavities

A denumerable set of axisymmetric solutions of the equations of motion of an ideal fluid in the neighborhood of a cavity closure point, including an infinitely remote point, is obtained. This family of solutions is similar to the well-known family of corresponding two-dimensional solutions (1), (2).Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 84–87, July–August, 1990.     

Crystallographic analysis of metal-ion binding to human ubiquitin

The metal-binding ability of human ubiquitin (hUb) towards a selection of biologically relevant metal ions and complexes has been probed. Different techniques have been used to obtain crystals suitable for crystallographic analysis. In the first type of experiments, crystals of hUb have been soaked in solutions containing copper(II) acetate and two metallodrugs, Zeise salt (K[PtCl(3)(η(2)-C(2)H(4))]·H(2)O) and cisplatin (cis-[PtCl(2)(NH(3))(2)]). The Zeise salt is used in a test for hepatitis, whereas cisplatin is one of the most powerful anticancer drugs in clinical use. The Zeise salt readily reacts with hUb crystals to afford an adduct with three platinum residues per protein molecule, Pt(3)-hUb. In contrast, copper(II) acetate and cisplatin were found to be unreactive for contact times up to one hour and to cause degradation of the hUb crystals for longer times. In the second type of experiments, hUb was cocrystallized with a solution of copper(II) or zinc(II) acetate or cisplatin. Zinc(II) acetate gives, at low metal-to-protein molar ratios (8:1), crystals containing one metal ion per three molecules of protein, Zn-hUb(3) (already reported in previous work), whereas at high metal-to-protein ratios (70:1) gives crystals containing three Zn(II) ions per protein molecule, Zn(3)-hUb. In contrast, once again, copper(II) acetate and cisplatin, even at low metal-to-protein ratios, do not give crystalline material. In the soaking experiment, the Zeise anion leads to simultaneous platination of His68, Met1, and Lys6. Present and previous results of cocrystallization experiments performed with Zn(II) and other Group 12 metal ions allow a comprehensive understanding of the metal-ion binding properties of hUb with His68 as the main anchoring site, followed by Met1 and carboxylic groups of Glu16, Glu18, Glu64, Asp21, and Asp32, to be reached. In the case of platinum, Lys6 can also be a binding site. The amount of bound metal ion, with respect to that of the protein, appears to be a relevant parameter influencing crystal packing.