Planar semi-martingales

The concept of a semi-martingale is extended to processes with index set in the plane. The definitions of planar semi-martingales are similar to those of two parameter bounded variation. Necessary and sufficient conditions for a Doob-Meyer decomposition are obtained, and a maximal inequality and almost everywhere convergence theorem is given for planar semi-martingales in LlogL.     

Products of roots of the identity

It is proved that every invertible bounded linear operator on a complex infinite-dimensional Hilbert space is a product of five -th roots of the identity for every 2$">. For invertible normal operators four factors suffice in general.

     

Products of Michael spaces and completely metrizable spaces

For disjoint subsets of the Michael space has the topology obtained by isolating the points in and letting the points in retain the neighborhoods inherited from . We study normality of the product of Michael spaces with complete metric spaces. There is a ZFC example of a Lindelöf Michael space , of minimal weight , with Lindelöf but with not normal. ( denotes the countable product of a discrete space of cardinality .) If denotes , the normality of implies the normality of for any complete metric space (of arbitrary weight). However, the statement `` normal implies normal' is axiom sensitive.

     

The Löwenheim—Skolem—Mal'tsev Theorem for ℍ\mathbb{F}-Structures

We deal with the problem asking whether hereditarily finite superstructures have elementary extensions of the form . In so doing, we settle the question whether a theory for some hereditarily finite superstructure have models of arbitrarily large cardinality. A Hanf number is shown to exist, and we provide an exact bound for the countable case.     

Σ-Subsets of Natural Numbers

It is shown that the class of all possible families of -subsets of finite ordinals in admissible sets coincides with a class of all non-empty families closed under e-reducibility and . The construction presented has the property of being minimal under effective definability. Also, we describe the smallest (w.r.t. inclusion) classes of families of subsets of natural numbers, computable in hereditarily finite superstructures. A new series of examples is constructed in which admissible sets lack in universal -function. Furthermore, we show that some principles of classical computability theory (such as the existence of an infinite non-trivial enumerable subset, existence of an infinite computable subset, reduction principle, uniformization principle) are always satisfied for the classes of all -subsets of finite ordinals in admissible sets.     

On the Ershov Upper Semilattice

We find some links between -reducibility and T-reducibility. We prove that (1) if a quasirigid model is strongly -definable in a hereditarily finite admissible set over a locally constructivizable B-system, then it is constructivizable; (2) every abelian p-group and every Ershov algebra is locally constructivizable; (3) if an antisymmetric connected model is -definable in a hereditarily finite admissible set over a countable Ershov algebra then it is constructivizable.     

Nonconforming elements in least-squares mixed finite element methods

In this paper we analyze the finite element discretization for the first-order system least squares mixed model for the second-order elliptic problem by means of using nonconforming and conforming elements to approximate displacement and stress, respectively. Moreover, on arbitrary regular quadrilaterals, we propose new variants of both the rotated nonconforming element and the lowest-order Raviart-Thomas element.

     

On individual stability of semigroups

Let be a -semigroup with generator on a Banach space . Let be a fixed element. We prove the following individual stability results.

(i) Suppose is an ordered Banach space with weakly normal closed cone and assume there exists such that for all . If the local resolvent admits a bounded analytic extension to the right half-plane 0\}$">, then for all and we have

(ii) Suppose is a rearrangement invariant Banach function space over with order continuous norm. If is an element such that defines an element of , then for all and we have

     

Calculation of Out‐of‐Plane Vibrations of a Porphin Molecule

Using the DFT/B3LYP method with the base set 631G^**, we carried out calculation of the frequencies of the normal vibrations of porphin and of its five isotopic types. Scaling of force constants for outofplane vibrations has been performed in independent natural coordinates. The symmetry coordinates are introduced and a force field for outof plane vibrations of a porphin molecule in independent coordinates of symmetry is obtained. A new correlation of the frequencies of vibrations in the type of the symmetry B _1u for the isotopic type of the d ₂ porphin molecule is suggested on the basis of discrete analysis of the distribution of a potential energy.     

Normal Liquid Helium-3 in the Static Fluctuation Approximation

In this paper normal liquid helium-3 is studied for the first time within the framework of the so-called static fluctuation approximation. This is based on the replacement of the square of the local-field operator with its mean value. A closed set of nonlinear integral equations is derived for neutral many-fermionic systems. This set is solved numerically by an iteration method for a realistic interhelium potential. The thermodynamic properties are then obtained for normal liquid helium-3. The quadratic-fluctuation approximation is found to be valid for this system in the low-temperature limit (0.25 K). Our results are presented in a set of figures. The role of the interaction is emphasized, and the functional dependence on the temperature of key thermodynamic quantities is derived for normal liquid helium-3.