Jacobian criteria for complete intersections. The graded case

Summary LetP be a positively graded polynomial ring over a fieldk of characteristic zero, letI be a homogeneous ideal ofP, and setR=P/I. The paper investigates the homological properties of someR-modules canonically associated withR, among them the module _R/k of Kähler differentials and the conormal moduleI/I ².It is shown that a subexponential bound on the Betti numbers of any of these modules implies thatI is generated by aP-regular sequence. In particular, the finiteness of the projective dimension of the conormal module impliesR is a complete intersection. Similarly, the finiteness of the projective dimension of the differential module impliesR is a reduced complete intersection. This provides strong converses to some well-known properties of complete intersections, and establishes special cases of conjectures of Vasconcelos.The proofs of these results make extensive use of differential graded homological algebra. The crucial step is to show that any homomorphism of complexes from the minimal cotangent complexL _R/k of André and Quillen into the minimal free resolution of the irrelevant maximal ideal m ofR, which extends the Euler map _R/k , is a split embedding of gradedR-modules.Oblatum 14-IV-1993 & 9-IX-1993Dedicated to Professor Ernst Kunz on his sixtieth birthdayThe first author was partly supported by a grant from NSF. During the peparation of this paper the second author was supported by Purdue University, whose hospitality he wishes to acknowledge     

Thermally modulated differential scanning calorimetry of a glass of composition 45Na2O-40B2O3-10Al2O3-5In2O3

The kinetics of structural relaxation and of glass transition of the 45Na₂O-40B₂O₃-10Al₂O₃-5In₂O₃ glassforming melt is studied by means of standard DSC and of temperature modulated DSC. In this way the dependence of the fictive temperature on cooling rate is determined simultaneously with the determination of the dependence of the dynamic glass transition temperature on modulation frequency. Both sets of data are fitted together in terms of the equation of Ritland-Bartenev. It was found that the activation energy of the structural relaxation exhibits a moderate dependence on temperature with the dimensionless fragility parameter α=3.3 (for strong systems, α is about 1 and increases to about 8 for some very fragile polymeric systems).     

Conditions for metastable crystallization from undercooled melts

The limits of validity of Ostwald's rule of stages are investigated theoretically in the case of crystallization of undercooled melts. The treatment is within the limits of capillary theory. Two basic models are compared: (1) According to the first one (model A), the phase with lower energy of formation of critical nucleus is predominantly formed. In an enantiotropic-type phase diagram there is no region of homogeneous preferential formation of the low temperature phase. If the phase diagram is monotropic-type there is a certain temperature below which the metastable crystalline phase is preferentially formed. (2) The second assumption takes into consideration that the nature of extremely small phases is somewhat undefined. One certainly cannot determine whether, say 3-particle complex, is of phase 1 or of phase 2. Moreover, it is known that properties of extremely small clusters could be different from the corresponding volume phase. The main assumption is that there is a certain crucial size (n-particle complex) at which the nature of the two phases can be distinguished. Complex of the phase, which has lower chemical potential at the crucial size, will be formed first. According to the model, in the case of enantiotropic-type transition there is a critical temperature.     

Effects of strong electron correlations in X-ray and electron spectra of high-T c superconductors

The current state of theoretical and experimental studies on the electronic structure of high-T _c superconductors is analyzed. The agreement between the theory and experimental spectroscopic data is shown to be rather poor in certain cases. The reason is that the X-ray and electronic spectra reveal strong electron correlations. At the same time, no realistic model has been developed up to now in which both one-electron and multielectron mechanisms of the formation of the spectra could be described in a unified way in compounds containing transition and rare-earth elements. In this paper, particular attention is paid to a sudden-perturbation model, by which it has been possible to describe or interpret some X-ray and electronic spectra, including both one-electron and multielectron effects.     

Nonlinear equations of flexural-flexural-torsional oscillations of rotating beamswith arbitrary cross-section

A system of three nonlinear partial differential equations describing the flexural-flexural-torsional vibrations of a rotating slender cantilever beam of arbitrary cross-section is derived using Hamilton’s principle. It is assumed that the center of gravity and the shear center are at different points. The interaction between flexural and torsional vibrations is accounted for in the linear and nonlinear parts of model Published in Prikladnaya Mekhanika, Vol. 44, No. 5, pp. 123–132, May 2008.     

Possible scheme of synthesis-assembling of fullerenes

A new scheme of fullerene formation is proposed on the basis of the similarity between the experimentally detected carbon structures. According to experimental data, the microclusters of C₂ and C₁₀ are synthesized first and then either an intermediate nucleus cluster or an obtainable lower fullerene is assembled from them. A high-symmetry fullerene can be assembled with a high probability from a nucleus cluster with a “good” symmetry. The atomic and electronic structures of molecules such as C₃₆, C₆₀, C₇₀, and C₇₆ are analyzed. For C₃₆, the NMR spectra are calculated and compared with the experimental data.     

Bifurcations at Combination Resonance and Quasiperiodic Vibrations of Flexible Beams

The nonlinear dynamic behavior of flexible beams is described by nonlinear partial differential equations. The beam model accounts for the tension of the neutral axis under vibrations. The Bubnov–Galerkin method is used to derive a system of ordinary differential equations. The system is solved by the multiple-scale method. A system of modulation equations is analyzed