Joint detection of a given number of reference fragments in a quasi-periodic sequence and its partition into segments containing series of identical fragments

The problem of joint a posteriori detection of reference fragments in a quasi-periodic sequence and its partition into segments containing series of recurring fragments from the reference tuple is solved. It is assumed that (i) an ordered reference tuple of sequences to be detected is given, (ii) the number of desired fragments is known, (iii) the index of the sequence term corresponding to the beginning of a fragment is a deterministic (not random) value, and (iv) a sequence distorted by an additive uncorrelated Gaussian noise is available for observation. It is established that the problem consists in testing a set of hypotheses about the mean of a random Gaussian vector. The cardinality of the set grows exponentially as the vector dimension (i.e., the sequence length) increases. An efficient a posteriori algorithm producing a maximum-likelihood optimal solution to the problem is substantiated. Time and space complexity bounds related to the parameters of the problem are derived. The results of numerical simulation are presented.     

Energy spectrum of <Emphasis Type="Italic">p</Emphasis> electrons in a plane crystal lattice and in a strong magnetic field

Harper equations are derived for a _px, _py electronic system. Analysis is carried out for extreme points of the quasi-continuous spectrum in the cases when the number of magnetic flux quanta through a unit cell is a rational number and calculations are made for square and triangular lattices as well as for a honeycomb lattice with two nonequivalent atoms. The possibility of application of the results for explaining the fractional Hall effect is considered.     

Bundles, Handcuffs, and Local Freedom

We answer a question of J. Anderson's by producing infinitely many commensurability classes of fibered hyperbolic 3-manifolds whose fundamental groups contain subgroups that are locally free and not free. These manifolds are obtained by performing 0–surgery on a collection of knots with the same properties.     

Stereochemical aspects of the formation of toxisterols A

Mechanisms for the formation of toxisterol A are suggested. The connection between the conformation of the initial previtamin and the stereochemistry of the reaction products is discussed.Lensoviet Leningrad Technological Institute. Translated from Khimiya Prirodnykh Soedinenii, No. 2, pp. 225–229, March–April, 1989.     

Arylation of 2,5-dimethyl-4-phenylpyridine

The position of the pyridine ring is arylated by reaction of 2,5-dimethyl-4-phenylpyridine with phenyllithium and p-tolyllithium. Destructive oxidation of the diaryl-substituted pyridines primarily affects the pyridine ring.Translated from Khimiya Geterotsiklicheskikh Soedinenii, No. 10, pp. 1359–1361, October, 1970.     

Five-membered-ring-substituted 1-methyl-1H-indeno[1,2-b]pyridines

1-Methyl-1H-indeno[1,2-b]pyridine and 1-methyl-1H-5-(, -dicarbomethoxyvinyl)-(formyl, acetyl)indeno[3,2-b]pyridines were obtained by treatment of N-methyl-4-azafluorenium iodide, as well as mixtures of it with acetylenedicarboxylic ester, dimethylformamide (DMF), and phosphorus oxychloride or acetic anhydride, with bases. 4-Azafluoronenone was used to synthesize 9-(p-methoxyphenyl)-4-azafluoren-9-ol, which was reduced to 9-(p-methoxyphenyl)-4-azafluorene, and 1-methyl-1H-5-(p-methoxyphenyl)indeno[3,2-b]pyridine was obtained from the methiodide of the latter.Translated from Khimiya Geterotsiklicheskikh Soedinenii, No. 10, pp. 1382–1386, October, 1981.