A $\Gamma$-convergence result for doubling metric measures and associated perimeters

Abstract. In this paper we study the notion of perimeter associated with doubling metric measures or strongly weights. We prove that the metric perimeter in the sense of L. Ambrosio and M. Miranda jr. coincides with the metric Minkowski content and can be obtained also as a -limit of Modica-Mortola type degenerate integral functionals. Received: 27 August 2001 / Accepted: 29 November 2001 / Published online: 10 June 2002 Investigation supported by University of Bologna, funds for selected research topics and by GNAMPA of INdAM, Italy. The authors are very grateful to Luigi Ambrosio and Francesco Serra Cassano for making their preprints available to them, for listening with patience and for many unvaluable suggestions.     

Effects of vitrification suppression on structure morphology, conductivity and dielectric properties of vanadium phosphate glasses

Vitrification suppression in the (V₂O₅)_1−x (P₂O₅)_x glasses where x=0.10, 0.15, 0.20, and 0.25 was controlled by changing the rate of quenching glasses. The structure variations occurring in the glasses were detected by differential thermal analysis and optical microscope. The results implied the separation and growth of V₂O₅ orthorhombic microcrystal in the samples with x=0.10 and 0.15 whereas other samples did not illustrate remarkable changes in their microstructure. However, in temperature range between 300 and 473 K a semiconducting behavior for all samples appears during the study of electrical conductivity-temperature dependence. A decrease in conductivity values accompanied with some variations in activation energies by reducing quenching rate was observed. The conductivity results suggested that the conduction occurs by the phonon assisted hopping of a small polaron between V⁴⁺ and V⁵⁺ states at relatively higher temperature range above θ_D/2. Whereas at relatively low temperatures the conduction may occur by electron jumping between filled and empty states at Fermi level in the disordered matrix besides polaronic conduction. Reasonable values for the density of localized states, carrier concentration and carrier mobility were estimated and discussed. Also, dielectric constant and dielectric loss were studied as a function of frequency at different temperatures confirming the structure variations in the glass system.     

On the origin of irreproducible behaviour of Na1−xLixNbO3 solid solutions

The irreproducibility of electrical properties of Na_0.88Li_0.12NbO₃ solid solution on thermal cycling reported by M.A.L. Nobre and S. Lanfredi is explained by gradual decomposition of the supersaturated solid solution below ca. 800 °C.     

Optical and electric properties of Nb-doped anatase TiO2 single crystal

Nb-doped anatase TiO₂ single crystal has been grown by chemical vapour transport method. Raman spectra shows that the obtained crystal with Nb of 0.08 wt% has typical anatase structure. An absorption band was observed at around 2.2 eV, which seems to be due to the d-d transition in the conduction band. The electron paramagnetic resonance and electric resistivity measurements show that the doped niobium makes quite shallow donor level whose orbital is d_xy-like centered at the titanium position of anatase.     

Defect clusters in titanium-substituted spinel-related lithium ferrite

The cation distribution in spinel-related titanium-substituted lithium ferrite, Li_0.5+0.5xFe_2.5−1.5xTi_xO₄ has been explored using interatomic potential and ab initio calculations. The results suggest that the cation distribution with Ti⁴⁺ substituting for Fe³⁺ on octahedral B sites and excess Li⁺ substituting for Fe³⁺ on tetrahedral A sites is stabilised by the formation of clusters of two octahedrally coordinated Ti⁴⁺ ions and one tetrahedrally coordinated Li⁺ ion linked through a common oxygen.     

Two different synthetic routes involving the reaction of dodecylamine or nicotinamide with crystalline lamellar vanadylphosphate

The crystalline lamellar compound, VOPO₄·2H₂O, was employed as host to react with a long monoamine aliphatic chain, dodecylamine, and with an aromatic derivative, nicotinamide, using two distinct synthetic approaches: in the solid state and from aqueous solution. The resulting compounds were characterized by elemental analysis, infrared spectroscopy, X-ray diffractometry, thermogravimetry and SEM microscopy. From X-ray diffraction patterns, the calculated increase of the interlayer distance was 2.05 nm for the dodecylamine, containing product, which is in agreement with the intercalation process, and was observed with both reactions routes. On the contrary, the guest nicotinamide molecule was not inserted into the lamellar cavity. The reaction in the solid state caused a small modification of the phosphate microstructure, in comparison with the solution procedure. The aliphatic amine molecules are oriented in a bilayer inside the host cavity by forming an angle of 67° with the inorganic lamella. The thermal decomposition for both series of synthesized compounds did not demonstrate any difference in behavior.     

EPR investigations of oxidation effects in (V1−xMox)2−δO3

Electron paramagnetic resonance (EPR) investigations has been carried out on the new family of molybdenum doped vanadium sesquioxides (V_1−xMo_x)_2−δO₃. The oxidation effects were monitored from the rate of paramagnetic V⁴⁺ created when the sample is exposed to the air. The effects of the oxidation time, sample temperature, and annealing at 1000 °C under a diluted hydrogen atmosphere on the EPR signal features are analyzed. The V⁴⁺ concentration in the oxidized samples is determined and the relaxation effects driven by the conduction electrons are pointed out from the thermal behaviour of the EPR line features. EPR spectra of all the oxidized samples also reveal a small ferromagnetic contribution strongly correlated with the V⁴⁺ content.     

Laplacians and the Cheeger Inequality for Directed Graphs

We consider Laplacians for directed graphs and examine their eigenvalues. We introduce a notion of a circulation in a directed graph and its connection with the Rayleigh quotient. We then define a Cheeger constant and establish the Cheeger inequality for directed graphs. These relations can be used to deal with various problems that often arise in the study of non-reversible Markov chains including bounding the rate of convergence and deriving comparison theorems.Received September 8, 2004     

NDMAP: the best material to study the Haldane's prediction

As predicted by Haldane, spin, S=1 one-dimensional (1D) Heisenberg antiferromagnet (HAF) has an energy gap between the singlet ground state and first excited triplet. On application of magnetic field, the triplet state Zeeman splits and the energy of one of the triplet state becomes zero at a critical field, H_c. Above H_c the system recovers magnetism. Then, we expect that a quasi-1D HAF will show a magnetic long-range ordering (LRO) at low temperatures due to the inter-chain coupling. This field-induced LRO has not been observed before due to complication of the crystal structure in the materials studied so far and/or technical difficulty.From a heat capacity measurement on a single crystal of an S=1 quasi-Q1D HAF, Ni(C₅H₁₄N₂)₂N₃(PF₆), we found an anomaly at a temperature in finite fields indicating a field-induced phase transition. A magnetic LRO is confirmed by a neutron diffraction measurement on the same sample. The temperature versus magnetic field phase diagram of this compound is constructed and discussed.