On the relative homology of cleft extensions of rings and abelian categories

We study the relative homological behaviour of the omnipresent class of cleft extensions of abelian categories. This class of extensions is a natural generalization of the trivial extensions studied in detail by Fossum, Griffith and Reiten and by Palmer and Roos. We apply our results to the relative homology of cleft extensions of rings.     

On set theoretic complete intersections in P k 3

We prove that smooth monomial curves of degree greater than three are not set theoretic complete intersections on a wide class of surfaces, called bihonogeneous surfaces.     

On the average volume of sections of convex bodies

The average section functional as(K) of a star body in Rn is the average volume of its central hyperplane sections: \(as\left( k \right) = \int_{{S^{n - 1}}} {\left| {K \cap {\xi ^ \bot }} \right|} d\sigma \left( \xi \right)\). We study the question whether there exists an absolute constantC > 0 such that for every n, for every centered convex body K in R ⁿ and for every 1 ≤ k ≤ n ? 2,

$$as\left( K \right) \leqslant {C^k}{\left| K \right|^{\frac{k}{n}}}\mathop {\max }\limits_{|E \in G{r_{n - k}}} {\kern 1pt} as\left( {K \cap E} \right)$$

. We observe that the case k = 1 is equivalent to the hyperplane conjecture. We show that this inequality holds true in full generality if one replaces C by CL _K orCd_ovr(K, BP _k ⁿ ), where L _K is the isotropic constant of K and dovr(K, BP _k ⁿ ) is the outer volume ratio distance of K to the class BP _k ⁿ of generalized k-intersection bodies. We also compare as(K) to the average of as(K ∩ E) over all k-codimensional sections of K. We examine separately the dependence of the constants on the dimension when K is in some classical position. Moreover, we study the natural lower dimensional analogue of the average section functional.

     

Magnetohydrodynamics and Plasma Cosmology

We study the linear magnetohydrodynamic (MHD) equations, both in the Newtonian and the general-relativistic limit, as regards a viscous magnetized fluid of finite conductivity and discuss instability criteria. In addition, we explore the excitation of cosmological perturbations in anisotropic spacetimes, in the presence of an ambient magnetic field. Acoustic, electromagnetic (e/m) and fast-magnetosonic modes, propagating normal to the magnetic field, can be excited, resulting in several implications of cosmological significance.     

The Existence of Helical Chirality in a Sterically Congested Naphtho [9]annulenone Preliminary Communication

The existence of the naphtho [9]annulenone 1 in two skeletally enantiomeric forms is demonstrated with the use of ¹H? NMR. spectroscopy in conjunction with a chiral «shift» reagent.     

Optimal systems for equipment maintenance and replacement under Markovian deterioration

Deterioration of equipment is modeled as a multistate discrete time controlled Markov process. The states are classified according to the degree of deterioration. The problem of design of optimal systems for equipment maintenance and replacement is considered when the decision-maker may take, in each stage, one of many available maintenance actions, classified according to their “stochastic effectiveness”; no action and replacement are included as alternatives. It is assumed that the transition probabilities satisfy two conditions which effectively describe a trend for monotonically increasing expected deterioration and rate of deterioration. Under these assumptions it is proved in the paper that the optimal (cost minimizing) decision system in an infinite horizon is of the control limit rule type, rapidly obtained by policy improvement algorithms. A numerical example is presented for a specific practical application; detailed data are available from the authors on request.     

On comparing optimum Alternating Direction Preconditioning and Extrapolated Alternating Direction Implicit schemes

The purpose of this paper is to show the equivalence of Alternating Direction Preconditioning and Extrapolated Alternating Direction Implicit methods in the case of two-level one-parameter optimized schemes as far as convergence rates are concerned, and also to give optimum parameters for the numerical solution of the self-adjoint second-order elliptic partial differential equation and the biharmonic equation.     

An ab initio study of the E 3Πg state of the iodine molecule

The E (3)Π(g) state of the iodine molecule is studied by ab initio multireference methods coupled with effective core potentials and large basis sets. Two potential minima are found, a global featuring an ion-pair character, and a local presenting a purely Rydberg nature. Four avoided crossings along the dissociation coordinate attribute an interesting topology to its potential energy curve, and their effect on the vibrational levels of I(2) is discussed.     

Specializations of Multigradings and the Arithmetical Rank of Lattice Ideals

In this article, we study specializations of multigradings and apply them to the problem of the computation of the arithmetical rank of a lattice ideal I _{L 𝒢}  ? K[x ₁,…, x _n ]. The arithmetical rank of I _{L 𝒢} equals the ?-homogeneous arithmetical rank of I _{L 𝒢} , for an appropriate specialization ? of 𝒢. To the lattice ideal I _{L 𝒢} and every specialization ? of 𝒢, we associate a simplicial complex. We prove that combinatorial invariants of the simplicial complex provide lower bounds for the ?-homogeneous arithmetical rank of I _{L 𝒢} .