Absolute, Relative, and Tate Cohomology of Modules of Finite Gorenstein Dimension

We study finitely generated modules M over a ring R, noetherianon both sides. If M has finite Gorenstein dimension G-dim_RMin the sense of Auslander and Bridger, then it determines twoother cohomology theories besides the one given by the absolutecohomology functors . Relative cohomology functors are defined for all non-negative integers n; they treat the modules of Gorensteindimension 0 as projectives and vanish for n > G-dim_RM. Tatecohomology functors are defined for all integers n; all groups vanish if M or N has finite projective dimension. Comparisonmorphisms and link these functors. We give a self-contained treatmentof modules of finite G-dimension, establish basic propertiesof relative and Tate cohomology, and embed the comparison morphismsinto a canonical long exact sequence . We show that these results provide efficient tools for computingold and new numerical invariants of modules over commutativelocal rings. 2000 Mathematical Subject Classification: 16E05, 13H10, 18G25.     

Nonlinear forced vibrations of a cylindrical shell with two internal resonances

Forced vibrations of cylindrical shells described by a system of three ordinary differential equations are studied. There are two internal resonances. Standing and traveling waves in the shells are described by a system of six modulation equations derived using the multiple-scales method. These waves are analyzed for stability __________ Translated from Prikladnaya Mekhanika, Vol. 42, No. 2, pp. 51–58, February 2006.     

Damping of Free Elastic Vibrations in Linear Systems

The possibility of using a Mises truss as an absorber of free elastic vibrations in a linear elastic system is examined. The nonlinear normal mode method is used to analyze nonlinear vibrations. A local nonlinear normal mode is shown to be favorable for vibration damping__________Translated from Prikladnaya Mekhanika, Vol. 41, No. 2, pp. 110–117, February 2005.     

Stable cohomology over local rings

For a commutative noetherian ring R with residue field k stable cohomology modules have been defined for each n∈Z, but their meaning has remained elusive. It is proved that the k-rank of any characterizes important properties of R, such as being regular, complete intersection, or Gorenstein. These numerical characterizations are based on results concerning the structure of Z-graded k-algebra carried by stable cohomology. It is shown that in many cases it is determined by absolute cohomology through a canonical homomorphism of algebras . Some techniques developed in the paper are applicable to the study of stable cohomology functors over general associative rings.     

Analysis of forced vibrations by nonlinear modes

The combination of Rausher method and nonlinear modes is suggested to analyze the forced vibrations of nonlinear discrete systems. The basis of the Rausher method is iterative procedure. In this case, the analysis of a nonautonomous dynamical system reduces to the multiple solutions of the autonomous ones. As an example, the forced vibrations of shallow arch close to equilibrium position are considered in this paper. The results of the analysis are shown on the frequency response.     

Ab initio calculations of endo-and exohedral C60 fullerene complexes with Li+ ion and the endohedral C60 fullerene complex with Li2 dimer

The results of ab initio Hartree-Fock calculations of endo-and exohedral C₆₀ fullerene complexes with the Li⁺ ion and Li₂ dimer are presented. The coordination of the Li⁺ ion and the Li₂ dimer in the endohedral complexes and the coordination of Li⁺ ion in the exohedral complex of C₆₀ fullerene are determined by the geometry optimization using the 3–21G basis set. In the endohedral Li⁺C₆₀ complex, the Li⁺ ion is displaced from the center of the C₆₀ cage to the centers of carbon hexa-and pentagons by 0.12 nm. In the Li₂ dimer encapsulated inside the C₆₀ cage, the distance between the lithium atoms is 0.02 nm longer than that in the free molecule. The calculated total and partial one-electron densities of states of C₆₀ fullerene are in good agreement with the experimental photoelectron and X-ray emission spectra. Analysis of one-electron density of states of the endohedral Li⁺@C₆₀ complex indicates an ionic bonding between the Li atoms and the C₆₀ fullerene. In the Li⁺C₆₀ and Li⁺@C₆₀ complexes, there is a strong electrostatic interaction between the Li⁺ ion and the fullerene.     

Thermo-dynamical contours of electronic-vibrational spectra simulated using the statistical quantum�Cmechanical methods

Three polycyclic organic molecules in various solvents focused on thermo-dynamical aspects were theoretically investigated using the recently developed statistical quantum mechanical/classical molecular dynamics method for simulating electronic-vibrational spectra. The absorption bands of estradiol, benzene, and cyanoanthracene have been simulated, and most notably, the increase in the spectral intensity for the lowest excited state transition as the temperature is increased observed experimentally is well reproduced. In addition, this method has been extended to treat luminescent processes also, and it is seen that the experimental emission spectrum of cyanoanthracene is also well described. The method still needs further refinement, but results to date, including those presented in this work, document clearly that our model is one which is able to treat the many complex effects that the environment have on electronic absorption and emission spectra.     

Homology of perfect complexes

It is proved that the sum of the Loewy lengths of the homology modules of a finite free complex F over a local ring R is bounded below by a number depending only on R. This result uncovers, in the structure of modules of finite projective dimension, obstructions to realizing R as a closed fiber of some flat local homomorphism. Other applications include, as special cases, uniform proofs of known results on free actions of elementary abelian groups and of tori on finite CW complexes. The arguments use numerical invariants of objects in general triangulated categories, introduced here and called levels. They allow one to track, through changes of triangulated categories, homological invariants like projective dimension, as well as structural invariants like Loewy length. An intermediate result sharpens, with a new proof, the New Intersection Theorem for commutative algebras over fields. Under additional hypotheses on the ring R stronger estimates are proved for Loewy lengths of modules of finite projective dimension.     

Kinetics of relaxation and crystallization of sodium metaphosphate glass

Kinetics of structural relaxation and crystallization of NaPO₃ glassforming melt is studied by means of thermal analyses. It is demonstrated that activation energy depends strongly on fictive temperature T_f. The dependence of the onset temperature T_o of the glass transition interval on the heating rate q⁺ is investigated for samples that were previously cooled down at a rate q⁻ of about 850 K/min. The dimensionless fragility F is a measure of the dependence of the activation energy for spatial rearrangement on the changes of structure. According to the present results F is large for NaPO₃ (i.e. phosphates are ‘fragile’ substances).