Pre—Separation Axioms in Fuzzifying Topology

1　 IntroductionYing[5,6 ] introduced and elementally developed so called fuzzifying topology with the semanticmethod of continuous valued L ogic.Shen[7] introduced and studied T0 -,T1-,T2 (Hausdorff) -,T3(regularity) -,T4 (normality) -separation axioms in fuzzifying topology.In [3 ]the concepts of thefamily of fuzzifying pre-open sets,fuzzifying pre-neighbourhood structure of a point and fuzzifyingpre-closure are introduced and studied.It is worth to mention that pre-separation axioms are …     

Approximation Properties AP s and p-Nuclear Operators (the Case 0 < s ≤ 1)

We investigate Banach spaces possessing (or not possessing) the approximation properties AP _s, 0 < s ≤ 1, in connection with the following known question in the geometric theory of operators: under which conditions on Banach spaces X and Y and on positive numbers r and p does the p-nuclearity of the second adjoint of a continuous operator T from X to Y imply the p-nuclearity of T? Actually, we give necessary and sufficient conditions under which this question is answered affirmatively. In addition, the corresponding counterexamples are obtained in the maximally strong form. For instance, it is shown (and this statement is a significant strengthening of the previous results of that sort) that there exists a pair of separable Banach spaces Z and W such that the spaces Z ^** and W have Schauder bases, while for every p, 1 ≤ p < 2, there is a non-p-nuclear operator from W to Z with a p-nuclear second adjoint. Earlier, in similar examples, the corresponding spaces did not possess even the Grothendieck approximation property. The technique developed in this paper does not allow us to treat the case p > 2. That case will be studied in a forthcoming paper of the author. Bibliography: 11 titles.     

Rovibrational analysis of the absorption spectrum of H2O around 1.02 μm by ICLAS-VECSEL

The very weak water vapor absorption spectrum has been investigated between 9520 and 10 010 cm⁻¹ by intracavity laser absorption spectroscopy based on a vertical external cavity surface emitting laser. The rovibrational assignment performed on the basis of the ab initio calculations of Schwenke and Partridge has allowed for determining 156 new energy levels belonging to a total of 13 vibrational states. The assignment process is detailed. The results are compared with the available databases and discussed in regard of recent investigations by Fourier transform spectroscopy.     

A Theorem on Multiplicators in Nonhomogeneous Hölder Spaces and Some of Its Applications

In Sec. 2, sufficient conditions of the boundedness of convolution operators with kernel m in a nonhomogeneous Hölder space are given in terms of the Fourier-image of m. After that, the results of Sec. 2 are used to prove the solvability in a Hölder space of the Cauchy problem for linear systems of the hydrodynamic type. Bibliography: 6 titles.     

On Necessary and Sufficient Conditions for the Asymptotic Stability of Impulsive Systems

We prove that the sufficient conditions for the asymptotic stability of impulsive systems obtained by Gurgula and Perestyuk are also necessary conditions.     

On Quasiclassical Field Theories Invariant with Respect to a Lie Group

Mathematical Notes -     

Minimal, rigid foliations by curves on ℂℙ n

We prove the existence of minimal and rigid singular holomorphic foliations by curves on the projective space ℂℙ ⁿ for every dimension n≥2 and every degree d≥2. Precisely, we construct a foliation ℱ which is induced by a homogeneous vector field of degree d, has a finite singular set and all the regular leaves are dense in the whole of ℂℙ ⁿ . Moreover, ℱ satisfies many additional properties expected from chaotic dynamics and is rigid in the following sense: if ℱ is conjugate to another holomorphic foliation by a homeomorphism sufficiently close to the identity, then these foliations are also conjugate by a projective transformation. Finally, all these properties are persistent for small perturbations of ℱ.?This is done by considering pseudo-groups generated on the unit ball 𝔹 ⁿ ⊆ℂ ⁿ by small perturbations of elements in Diff(ℂ ⁿ ,0). Under open conditions on the generators, we prove the existence of many pseudo-flows in their closure (for the C ⁰-topology) acting transitively on the ball. Dynamical features as minimality, ergodicity, positive entropy and rigidity may easily be derived from this approach. Finally, some of these pseudo-groups are realized in the transverse dynamics of polynomial vector fields in ℂℙ ⁿ . Received March 7, 2002 / final version received November 26, 2002?Published online February 7, 2003 Most of this work has been carried out during a visit of the first author to IMPA/RJ and a visit of the second author to the University of Lille 1. We would like to thank these institutes for hospitality and express our gratitude to CNPq-Brazil and CNRS-France for the financial support which made these visits possible. We are also indebted to Paulo Sad, Marcel Nicolau and the referee whose comments helped us to improve on the preliminary version. Finally, the second author has partially conducted this research for the Clay Mathematics Institute.     

Reduction of nonholonomic variation problems to isoperimetric ones and connections in principal bundles

Translated from Matematicheskie Zametki, Vol. 49, No. 5, pp. 37–44, May, 1991.