On the Girth of Graphs Critical with Respect to Edge-Colourings

A graph G with maximum valency r is called critical if r + 1colours are needed for an edgecolouring, but every proper subgraphrequires at most r. In this note we consider the minimum orderf(r, g) of a critical graph of maximum valency r and girth g.We show that f(r, 3) = r+1 or r+2 according as r is even orodd, f(r, 4) = 2r+1,f(3, 5) = 9 and f(3, 6) = 15.     

On small graphs critical with respect to edge colourings

A simple graph is said to be of class 1 or of class 2 according as its chromatic index equals the maximum degree or is one greater. A graph of class 2 is called critical if all its proper subgraphs have smaller chromatic index. It has been conjectured by Beineke and Wilson and by Jakobsen that all critical graphs have odd order. In this paper we verify the truth of this conjecture for all graphs of order less than 12 and all graphs of order 12 and maximum degree 3. We also determine all critical graphs through order 7.     

A theoretical characterization of the structure formation enthalpy,and fluzional behaviour of B2H6 and AlBH6

Summary A comprehensive study of the binary association complexes B₂H₆ and AlBH₆ has been performed by ab initio molecular orbital theory. Reliable formation enthalpies can be computed only be extended basis sets and a reasonably complete account of correlation. The greater stability towards neutral dissociation of AlBH₆ with respect to B₂H₆ obtained at the Hartree-Fock level employing the 6-21G^* basis set ( 10 kcal/mol) is reduced to only 2 kcal/mol when the basis set is sufficiently saturated and correlation energy properly included. The value of the activation energy for hydrogen scrambling in AlBH₆ is much less sensitive to the method used, although correlation still plays a significant rôle reducing the potential energy barrier from 11.4 to 7.7 kcal/mol.     

Observation of inclusive η production in ee annihilation at CM energies of 34 GeV

The inclusive production of η mesons at CM energies around 34 GeV has been studied for e⁺e⁻ annihilation into hadrons. The average number of η-mesons per event is found to be 0.72 ± 0.10 (stat.) ± 0.18 (syst.). The abundance of η's is studied as a function of event shape parameters and it is compared to the corresponding π⁰ rates.     

Nuclear activation as a current detector for ion beams produced by a high intensity laser

This work aims to demonstrate that nuclear activation of an irradiated metal foil can determine the number of accelerated particles in the initial irradiation beam. Even if this is not a novel technique it can be really useful to find out whether other commonly used techniques fail. Tests of the method have been carried out using the proton beam accelerated by the cyclotron of the University of Birmingham. Different metal targets (Cu, Cd and Ti) were irradiated and a HpGe detector was used to count the decay photons emitted by the radioactive isotopes produced. Agreement between proton intensities determined by foil activation and ionisation chamber measurements were ～16%.     

Measurement of the p to s wave branching ratio of 187Re beta decay from beta environmental fine structure

The mixed occurrence of s-wave and p-wave contributions in a first forbidden unique Gamow-Teller beta decay has been investigated for the first time by measuring the beta environmental fine structure (BEFS) in a 187Re crystalline compound. The experiment has been carried out with an array of eight AgReO4 thermal detectors operating at a temperature of approximately 100 mK. A fit of the observed BEFS spectrum indicates the p-wave electron emission as the dominant channel. The complete understanding of the BEFS distortion of the 187Re beta decay spectrum is crucial for future experiments aiming at the precise calorimetric measurement of the antineutrino mass.     

Neutrino physics with cryogenic detectors

The recent results on neutrino oscillations and the consequent need to measure the value of the neutrino mass are briefly discussed. The operating principle of cryogenic detectors working at low temperatures, where the small heat capacity allows one to record and measure the temperature increase due to the tiny energy lost by a particle in form of heat is described. An application of these detectors is the measurement, or at least an upper constraint, of the neutrino mass in β decay. This approach is complementary and can, in the future, be competitive with experiments based on the spectrometric measurement of the electron energy. The search for neutrinoless double beta decay could reach a better sensitivity on the mass if a neutrino is a Majorana particle. A large cryogenic detector, named CUORICINO, on neutrinoless double beta decay (DBD) of ¹³⁰Te already yields the best constraint on the absolute value of the Majorana neutrino mass. A much larger detector, named CUORE, for Cryogenic Underground Observatory for Rare Events, is currently under construction. With its active mass of 750 kg of natural TeO₂ it aims to reach the sensitivity in the determination of the Majorana neutrino mass suggested by the results of neutrino oscillation under the inverse hierarchy hypothesis. The problem is closely connected with what I call “the second mystery of Ettore Majorana” who suggested a particle that would violate the lepton number.     

Enhancement of two-photon absorption cross-section and singlet-oxygen generation in porphyrins upon beta-functionalization with donor-acceptor substituents

The microwave-enhanced synthesis, comparative singlet oxygen sensitization efficiency, and nonlinear optical characterization of a new beta-functionalized porphyrin and its copper complex are described. We show that the introduction of a donor-acceptor push-pull conjugated fragment in the beta position strongly perturbs the porphyrin electronic structure leading to a remarkable one- and two-photon NIR absorption enhancement.     

Personalized Metabolic Profile by Synergic Use of NMR and HRMS

A new strategy that takes advantage of the synergism between NMR and UHPLC–HRMS yields accurate concentrations of a high number of compounds in biofluids to delineate a personalized metabolic profile (SYNHMET). Metabolite identification and quantification by this method result in a higher accuracy compared to the use of the two techniques separately, even in urine, one of the most challenging biofluids to characterize due to its complexity and variability. We quantified a total of 165 metabolites in the urine of healthy subjects, patients with chronic cystitis, and patients with bladder cancer, with a minimum number of missing values. This result was achieved without the use of analytical standards and calibration curves. A patient’s personalized profile can be mapped out from the final dataset’s concentrations by comparing them with known normal ranges. This detailed picture has potential applications in clinical practice to monitor a patient’s health status and disease progression.     

The Representation Polyhedron of a Semiorder

Let a finite semiorder, or unit interval order, be given. When suitably defined, its numerical representations are the solutions of a system of linear inequalities. They thus form a convex polyhedron. We show that the facets of the representation polyhedron correspond to the noses and hollows of the semiorder. Our main result is to prove that the system defining the polyhedron is totally dual integral. As a consequence, the coordinates of the vertices and the components of the extreme rays of the polyhedron are all integral multiples of a common value. Total dual integrality is in turn derived from a particular property of the oriented cycles in the directed graph of noses and hollows of a strictly upper diagonal step tableau. Our approach delivers also a new proof for the existence of the minimal representation of a semiorder, a concept originally discovered by Pirlot (Theory Decis 28:109–141, 1990). Finding combinatorial interpretations of the vertices and extreme rays of the representation polyhedron is left for future work.