Upper-Branch Instability of Flow in Pipes of Large Aspect Ratio with Three-Dimensional Nonlinear Viscous Critical Layers

We consider the upper-branch neutral stability of flow in pipesof large aspect ratio, basically extending the work of F. T.Smith to the nonlinear regime. The inclusion of weak nonlinearityleads to an eigenproblem whose solution depends on the propertiesof three-dimensional nonlinear critical layers. Two specialcases are considered. The first is for very small amplitude perturbations, where R is a Reynolds numberbased on the height of the tube and which is assumed large.Then a fully analytical solution of the three-dimensional criticallayers is possible, from which the linear results of Smith maybe deduced. The second case studied is that of flow in a rectangularpipe, where a solution of the nonlinear critical layer problemcan be obtained. Further analysis of neutral modes in this lattercase suggests the possible existence, inter alia, of neutralmodes for finite aspect ratio tubes. These modes depend on thescaled amplitude and have O(1) wavespeeds.     

On the dynamics of cosmological particle production

The time evolution of the particle number density and spectrum of massive scalar particles, coupled conformally to a classical Friedman-Robertson-Walker gravitational field is followed numerically. Not only for a pure radiation cosmos, but also for one with an inflationary interlude, the density of particles produced remains constant in time. This results in a constant equivalent temperature scale up to the Compton time of the massive particles, and opens the possibility that during a sufficiently long inflation the energy density of the particles produced can exceed that of the radiation background.     

Crystal structures of dimethylsuccinate and dimethyloxalate: Carbonyl group orientation for C13 chemical shielding tensor studies

The crystal structures of dimethylsuccinate (DMS) and dimethyloxalate (DMO) have been determined to facilitate the determination of the C-13 chemical shielding tensors of the carbonyl carbon in esters. Crystals of DMS are monoclinic, space groupC2/c,Z=4,a=13.154(4),b=6.156(1),c=9.363(4)Å,=98.53(3)°. The structure was solved by direct methods and refined by leastsquares procedures to giveR=0.071 for 932 observed data. Crystals of DMO are monoclinic space group,P2₁/n,Z=2, witha=3.891(1),b=11.879(2),c=6.213(2) Å,=103.32(2)°. The structure is the same (within experimental error) as that reported by Dougill and Jeffrey (1953) and refined to giveR=0.074 for 395 observed data.     

Weighted Voronoi tessellation with a poisson field of centroids

One considers a weighted Voronoi tessellation generated by a system of n Poisson fields of centroids. One investigates the composition and the boundary field of this structure. It is proved that the boundary field intensity between grains of types i and j (1ijn) is equal to where p_i is the volume part, _i>0 is the weight (for the comparison of distances) of the i-th component, c is a scaling factor. One obtains formulas for the boundary field intensities in plane and linear sections of the structure: . Estimators for the parameters p_i and _i from the observation of the linear section are proposed.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 153, pp. 160–172, 1986.     

Spectrum of the algebra of pseudodifferential operators in a halfspace

One considers C*-algebras, generated by pseudodifferential operators in a halfspace. The operators act in functional spaces with weighted norms. One describes the set of all classes of equivalent irreducible representations of algebras and reveals the topology on the spectrum. One investigates, in particular, the dependence of the spectrum on the choice of the weighted functional space.Translated from Problemy Matematicheskogo Analiza, No. 10, pp. 160–180, 1986.     

Crystal and molecular structure of 1R,3S,5S,7R,10R-3,6,6,10,14-Pentamethyltricyclo[10.3.0.05,7]pentadeca-11,14-diene-1, 10-dihydroxy-2,13-dione (Japodagrol), C20H28O4

A new cytotoxic macrocyclic diterpenoid named Japodagrol, C₂₀H₂₈O₄, has been isolated from the ornamental plantJatropha podagrica, family Euphorbiaceae, grown in Nigeria. The structure of the title compound, C₂₀H₂₈O₄, was solved by X-ray analysis.M _r =332.44, monoclinic space groupC2,a=23.285(4),b=6.5105(12),c=12.505(3) Å,=98.504(17),V=1874.8 ų,Z=4,D _c =1.178 Mg/m³. CuK radiation (graphite crystal monochromator, =1.54184 A), (CuK)=6.13 cm^–1,F(000)=720,T=290 K. Final conventionalR factor=0.035,R _w =0.038 for 3349 observed reflections. The structure was solved with the programDirdif. The 5-membered ring is closed to a half-chair form. The compound contains inter- and intramolecular hydrogen bonds.     

Singular series of genus 2 for sums of squares

Let f₁, ..., f_h be representatives of classes in the genus of a positive quadratic form f, r_i(k) the number of representations of k by the form f_i, (k) the value of the singular Hardy-Littlewood series, E_i the number of integer automorphisms of f_i. We derive an expression for in terms of the values of singular series of genus 1 and 2. This expression is evaluated for the simplest model f(x)=x ₁ ² +...+x _m ² , m1 (mod 8), k is a prime in-integer, k=1(mod 8).Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 151, pp. 26–39, 1986.     

Influence of crystal-field on the thermal conductivity of PrNi5

The thermal conductivity of PrNi₅ in the temperature range 0·5 to 20 K is investigated. The analysis of crystalline electric field effect on the experimentally obtained temperature dependence of the thermal conductivity indicates that beside the main electronic component also other contributions (phonons, magnetic excitons) can take place in the heat transport in this intermetallic compound.The authors would like to thank Dr. G. Eska for lending the sample and Dr. . Jáno for his interest in the work and stimulating discussions.     

Effects of small external magnetic fields on the hyperfine field of iron in iron metal above the curie temperature: A new method for studying critical phenomena

By applying magnetic fields up to 2 kG, the equation of state and the critical dynamics of a spin system near its critical point can be studied with the Mössbauer effect. As an example, the equation of state near the Curie temperature of iron is investigated yielding the critical exponent =4.84±0.15.     

Humulenoids from plants of the family Apiaceae spatial structure of α-apienes

A conformational analysis has been made by the method of molecular mechanics of the -apienes — natural sesquiterpenes isolated from plants of the Apiaceae family. The results of the mathematical method with the inclusion of those x-ray structural studies have shown that in the molecules under consideration it is possible to expect the predominant realization of four conformations of the 11-membered ring:₁₄A ₈ ⁷ ,¹⁴A ₇ ⁸ ,₁₄A ₇ ⁸ , and¹⁴A ₈ ⁷ . These four conformations of unsubstituted -apiene and its natural derivatives have been characterized energetically, and the barriers to their interconversions have been evaluated.Institute of Chemistry of Plant Substances, Uzbekistan Academy of Sciences, Tashkent. Translated from Khimiya Prirodnykh Soedinenii, No. 2, pp. 197–202, March–April, 1992.