Enumeration of sequences of given specification according to rises,falls and maxima

Letn = (a₁.a₂ … a_N) denote a sequence of integers a_i={1.2.…n}. A rise is a part a_i.a_i+1 with a_i <a_i+1: a fall is a pair with a_ia_i+1: a level is a pair with a_i = a_i+1. A maximum is a triple a_i-1.a_ia_i+1 with a_i-1?a_i.a_i?a_i+1. If e_i is the number of a_j?n witha_j = i, then [e₁…e_n] is called the specification of n. In addition, a conventional rise is counted to the left of a₁ and a conventional fall to the right of a_N: ifa₁?a₂, then a₁ is counted as a conventional maximum, similarly if a_N-1 ? a_N thena_N is a conventional maximum. Simon Newcomb's problem is to find the number of sequences n with given specification and r rises; the refined problem of determining the number of sequences of given specification with r rises and s falls has also been solved recently. The present paper is concerned with the problem of finding the number of sequences of given specification with r rises, s falls. λ levels and λ maxima. A generating function for this enumerant is obtained as the quotient of two continuants. In certain special cases this result simplifies considerably.     

A chiral nitronate dianion from (=d)-glyceraldehyde. Enantiospecific syntheses of 2,3-dideoxy-3-nitro furanosides and pyranosides

A new enantiospecific protocol for the synthesis of 2,3-dideoxy-3- nitro and 3-amino sugars is reported. Chiral nitronate dianion 3, derived from (+)-(=d)-glyceraldehyde, is allylated and the resulting adduct 7 is transformed to either the methyl furanoside 9 or the methyl pyranoside 10 depending on the reaction sequence.     

Collective contributions to E1, ΔK=1 matrix elements in odd-odd deformed nuclei

The half-life of the 184.3 keV state in¹⁶⁵Dy has been remeasured asT _1/2=1.0±0.1 ns in order to deduce a reliable value for the transition matrix element of the depopulatingE l,ΔK=1 transition. Experimental E1,ΔK=1 transition matrix elements in odd-odd deformed nuclei are found to be systematically larger than similar values in adjacent odd-A isotones. This effect is supposed to be associated with higher-order vibrational admixtures coupled via RPC andp— n interaction mixings.     

The size of Bose-Einstein effects in the statistical bootstrap model

We have determined the shift in Hagedorn temperature due to Bose statistics and calculated explicitly the cluster expansions of fireball level densities. This enabled us to investigate Bose effects in one-particle spectra and integrated correlations. Apart from the low secondary energy region of the one-particle distributions, the nonasymptotic behaviour of these quantities is little affected.     

The energy dependence of single particle spectra in e+e− annihilation

We present explicit predictions of the statistical bootstrap model for inclusive single particle spectra in e⁺e^? annihilation. The distribution (ω/σ_tot·E) (dσ/p² dp) is found to become, for E ? 3 GeV, a function only of the secondary energy ω, independent of the incident e⁺d^? energy E.     

Isomeric E1 and M1 transitions in 177Ta and in neighbouring odd-A Lu and Ta isotopes

Delayed γ-ray spectra as well as time distributions have been measured applying the r.f.-γ method and the method of delayed γ-γ coincidences. Using the reaction ¹⁷⁵Lu(α, 2n)¹⁷⁷Ta, the half-lives of the levels at 70.6 and 73.6 keV in ¹⁷⁷Ta have been determined as $\text{T}{}_{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{= 80±10}\text{ns}$ and $\text{T}{}_{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{= 370±50}\text{ns}$, respectively. In further experiments, upper limits for the lifetimes of the $\text{9}\text{2}{}^{\mathrm{?}}\text{[514]}$ and $\text{5}\text{2}{}^{\mathrm{+}}\text{[402]}$ states in ¹⁷¹Lu and ¹⁷³Lu were estimated. Comparisons of the experimental half-lives to Weisskopf and Nilsson model predictions are performed taking into account pairing correlations. For the calculations, a modified Nilsson Hamiltonian has been used. The hindrance factors obtained are summarized together with those deduced for corresponding transitions in neighbouring odd-proton nuclei. From the experimental results in ¹⁷⁷Ta, conclusions have been drawn on the possible value of the constant k₂ occurring in the magnetic dipole operator.     

Some observations on the NOE effect in 1H NMR spectroscopy

The NOE enhancement factor in ¹H NMR spectroscopy is shown to be a function of dipole-dipole interaction of the solute. The NOE increases with increasing concentration if a monomer-dimer equilibrium exists in solution.