Index formulas for generalized Wiener-Hopf operators and boson-fermion correspondence in 2N dimensions

The kernels of operators associated with special chiral gauge transformations (kinks) in the 2N-dimensional Dirac theory are explicitly determined. The result is used to obtain index formulas for Fredholm operators corresponding to continuous chiral gauge transformations. Moreover, the Fock space quadratic forms corresponding to the kinks are proved to converge to the Dirac field as the kink size goes to zero. It is also shown that forN 1, 2(mod 4) the Majorana field can be reached in a similar fashion.Work supported by the Netherlands Organisation for the Advancement of Research (NWO)     

Disorder arising from the location of enantiomers in a twist-boat structure: 2,6-trans-2,4,6-tris(trichloromethyl)-1-oxa-3,5-dithian

Crystals of the title compound are orthorhombic, Pca2₁;a=27.586(4),b=10.509(3),c=11.080(2) Å,V=3212(1) ų,Z=8. The structure was solved by direct methods, from data collected at room temperature on an Enraf-Nonius CAD4 diffractometer, and refined by full-matrix least squares to a finalR value of 0.055 using 2043 reflections. One of the two independent molecules in the asymmetric unit is disordered. The disorder is manifested mainly between the oxygen atom and one sulfur atom and is caused by opposite enantiomers occupying corresponding positions in different unit cells. The endocyclic torsion angles in both independent molecules are very similar but the disposition of the heteroatoms on the twist-boat framework is different in the two molecules. Site occupancy of the two models on this framework in the disordered molecule is in the ratio 2179 with the smaller proportion being very similar to the ordered molecule and the remainder similar to that found in the normal molecule oftrans-4-chloro-2,4,6-tris(trichloromethyl)-1-oxa-3,5-dithian (Irving and Irving, 1988).     

Effect of the flow regime in an underexpanded jet issuing into a subsonic counterstream on the pressure distribution on a body

The pressure distribution on a cone with a half-angle =75°, from which a single central underexpanded jet issues into a subsonic counterstream, has been experimentally investigated. The effect of the flow regime in the jet on the pressure distribution is demonstrated. Generalized relations for the pressure on the body are obtained for various jet-flow momentum ratios J and flow Mach numbers M = 0.35–0.9; the Mach number M_a at the exit of the conical nozzle with half-angle _a=10° was equal to 2.9. The working medium of the jet and the flow was air with stagnation temperatures T_0a = T₀ 260–265°K. The ratio of the nozzle outlet radius to the radius of the maximum cross section of the cone _a/R_M=0.1.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 182–185, January-February, 1988.     

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.     

On 2-spannedness for the adjunction mapping

LetL be a line bundle on a smooth connected projective manifold X of dimension n. We extend to any dimension the definition of k-spannedness forL; this is a notion of k-th order embedding which was recently given in the case of curves and surfaces. Then, by a reduction to the surfaces case, we prove that the adjoint bundle K_x+(n–1)L is 2-spanned ifL is (at least) 3-spanned.     

Intra- versus intermolecular hydrogen bonds in salicylamide derivatives

The crystal structures, solid-state infrared patterns, and thermal properties of two polymorphs of 4-nitrosalicylanilide are presented. In both polymorphs, intramolecular hydrogen bonds are found between the phenol oxygen and the amide proton, and intermolecular hydrogen bonds are found between the amide carbonyl oxygen and the phenol proton. These hydrogen bond patterns are compared to those found in other known salicylamide derivatives and an analysis is given of the factors contributing to preferences for intra- or intermolecular hydrogen bonds in these structures. Crystal data: polymorph, orthorhombic,Pbca,a=11.003(4),b=27.959(7),c=7.622(5) Å,Z=4,V=2345(3) ų, andR=0.038 (1351 reflections); polymorph, monoclinic,P2₁/a,a=28.36(1),b=11.64(1),c=7.293(8) Å,=90.68(6)°,Z=8,V=2408 ų, andR=0.043 (2425 reflections).     

On a characteristic of random matrices connected with unconditional convergence almost everywhere

В статье рассматрива ются множестваM _N , 1≦N<∞ всех систем функций Φ={?(x)} _j ^=1/N , заданных на [0,1] с где (ε _{i, j} ) _{i, j} ^=1/N — матрица с э лементами ± 1. Изучаетс я поведение наМ _N функц ии $$\alpha _N (\Phi ) = \mathop {\sup }\limits_\sigma \mathop {\sup }\limits_{\sum a_j^2 = 1} (\int\limits_0^1 {\mathop {\sup }\limits_{1 \leqq k \leqq N} (\sum\limits_{j = 1}^k {a_j \varphi _{\sigma (j)} (x)} } )^2 dx)^{1/2} $$ гдеσ: {1, ...,N}?{1, ...,N}. Дока зьгаается, что сущест вуют абсолютные постоянн ыеc ₃,c ₄>0,y ₀>1, такие, что для любог оN=1,2, ... иy>y ₀ $$\mu _N (\{ \Phi \in {\rm M}_{\rm N} :\alpha _N (\Phi )/\left\| \Phi \right\| > y\} ) \leqq c_3 \exp [ - \exp (c_4 y)N]$$ гдеμ _N — мера наM _N с µ _N ({Ф}) = 2^?N2 дл я любой системыΦ∈M     

Distribution density of the norm of a stable vector

Let B be a Banach space,X be a stable B -valued random vector with exponentd(0,2), and P(·) be the distribution density of the norm of X. In this paper we study the question of the boundedness of P. In particular, we construct examples of a space B with a symmetric stable vector X with exponentd(1,2) with unbounded P and prove that if X is a nondegenerate strictly stable vector with exponentd(0,1), then P is bounded.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 158, pp. 105–114, 1987.The author is grateful to Yu. A. Davydov, V. I. Paulauskas, V. Yu. Bentkus, and D. Pap for stimulating discussions of the subject of this paper. When the paper was finished the author learned that similar results are found in [9].     

Superbranching processes and projections of random Cantor sets

We study sequences (X ₀, X ₁, ...) of random variables, taking values in the positive integers, which grow faster than branching processes in the sense that , for m, n0, where the X _n (m, i) are distributed as X _n and have certain properties of independence. We prove that, under appropriate conditions, X _n ^1/n almost surely and in L ¹, where =sup E(X _n )^1/n . Our principal application of this result is to study the Lebesgue measure and (Hausdorff) dimension of certain projections of sets in a class of random Cantor sets, being those obtained by repeated random subdivisions of the M-adic subcubes of [0, 1] ^d . We establish a necessary and sufficient condition for the Lebesgue measure of a projection of such a random set to be non-zero, and determine the box dimension of this projection.Work done partly whilst visiting Cornell University with the aid of a Fulbright travel grant