The perfect matching polytope and solid bricks

The perfect matching polytope of a graph G is the convex hull of the set of incidence vectors of perfect matchings of G. Edmonds (J. Res. Nat. Bur. Standards Sect. B 69B 1965 125) showed that a vector x in Q^E belongs to the perfect matching polytope of G if and only if it satisfies the inequalities: (i) x0 (non-negativity), (ii) x(∂(v))=1, for all vV (degree constraints) and (iii) x(∂(S))1, for all odd subsets S of V (odd set constraints). In this paper, we characterize graphs whose perfect matching polytopes are determined by non-negativity and the degree constraints. We also present a proof of a recent theorem of Reed and Wakabayashi.     

The Manin—Drinfeld theorem and Ramanujan sums

The Manin—Drinfeld theorem asserts the finiteness of the cuspidal divisor class group of a modular curve corresponding to a congruence subgroup. The purpose of the note is to draw attention to the connection between this theorem and Ramanujan sums, and to the question of what happens for non-congruence subgroups. Research partially supported by an NSERC grant. Alfred P Sloan Fellow. Research partially supported by the NSF.     

On the number of real quadratic fields with class number divisible by 3

We find a lower bound for the number of real quadratic fields whose class groups have an element of order . More precisely, we establish that the number of real quadratic fields whose absolute discriminant is and whose class group has an element of order is improving the existing best known bound of R. Murty.

     

Decay of126m Sb and the energy levels of126Te

The decay characteristics of 19 min^126m Sb have been studied using Ge(Li) and scintillation spectrometers. The sources of^126m Sb were obtained by the (n, p) reaction on enriched¹²⁶Te. The gamma rays of energies 414.4±0.3, 666.7±0.5 and 696.1±0.5 keV were found decaying with a half-life of 19 min. The study of beta spectra employing an anthracene crystal yielded two beta groups with end-point energies of 1.9±0.15 and 2.5±0.15 MeV. The results of singles and coincidence measurements have been incorporated into a decay scheme with excited states of¹²⁶Te at 666.7, 1,362.8 and 1,777.2 keV.     

Linear flow induced in fluid particle suspension by an infinite differentially rotating disk

The steady, axisymmetric laminar flow of a homogeneous incompressible fluid with suspended particles occupying the half-infinite space over a differentially rotating rigid plane boundary is analyzed in this paper. The effect of suspended particles is described by two parametersf and τ. The mass concentration parameterf is a measure of the concentration of suspended dust particles. The interaction parameter τ is a measure of the rate at which the velocity of dust particles adjusts to changes in the fluid velocity and depends upon the size of the individual particles. Due to Ekman suction, the particle density remains no longer a constant in the boundary layer but varies with the axial coordinate ξ. Flow characteristics and density variations are studied as functions off, τ and ξ. Possible limiting cases for τ≪1 and τ≫1 which correspond to the case of fine dust and coarse dust respectively are derived and discussed.     

On the number of groups of a given order

Letting G(n) denote the number of nonisomorphic groups of order n, it is shown that for square-free n, G(n) ≤ ?(n) and G(n) ≤ (log n)^c on a set of positive density. Letting F_k(x) denote the number of n ≤ x for which G(n) = k, it is shown that $\text{F}{}_2\text{(x) =}\text{O(x(}\text{log}{}_4\text{x)}\text{(}\text{log}{}_3\text{x)}{}^2\text{)}$, where log_rx denotes the r-fold iterated logarithm.     

Elemental analysis of geological samples by cyclic activation using a 14 MeV neutron generator and applying flux corrections

The solvent extraction of the rare earth elements, yttrium and scandium with trioctylphosphine oxide in the presence of strong complexing and various salting-out agents has been studied. As a result, a technique was developed for the extractionchromatography recovery of the rare earth elements and yttrium with simultaneous purification practically from all the elements, including scandium, usually present in the mineral samples.