On submanifolds of Hopf manifolds

The main results we obtain are as follows: an invariant submanifold of a Hopf manifold with semi-flat normal connection is either a complex hypersurface or a totally-umbilical quasi-Einstein submanifold with a flat normal connection. The only totally-umbilical invariant submanifolds of zero scalar curvature of a Hopf manifold are the totally-geodesic flat surfaces. To Professor A. Cossu on the Occasion of his 65th Birthday     

Classification of pairs consisting of a linear and a semilinear map

L. Kronecker has found normal forms for pairs (A, B) of m-by-n matrices over a field F when the admissible transformations are of the type (A, B)→(SAT, SBT), where S and T are invertible matrices over F. For the details about these normal forms we refer to Gantmacher's book on matrices [5, Chapter XII]. See also Dickson's paper [3]. We treat here the following more general problem: Find the normal forms for pairs (A, B) of m-by-n matrices over a division ring D if the admissible transformations are of the type (A, B)→(SAT, SBJ(T)) where J is an automorphism of D. It is surprising that these normal forms (see Theorem 1) are as simple as in the classical case treated by Kronecker. The special case D=C, J=conjugation is essentially equivalent to the recent problem of Dlab and Ringel [4]. This is explained thoroughly in Sec. 6. We conclude with two open problems.     

Irreducible connected Lie subgroups of Gl n (R) are closed

IfG is a connected real Lie group and π:G→Aut (V) a continuous irreducible finite-dimensional real representation then we show that π(G) is closed in Aut(V). A similar result is valid in the complex case.     

Stereochemie und1H-NMR-Spektren einiger vom Pentaerythrit abgeleiteten Spiro-1,3-dioxane

Summary Mono- and trispiro-1,3-dioxanes with mobile, fixed and semimobile structures were obtained by the condensation reaction of pentaerythritol with aldehydes and symmetrical or unsymmetrical ketones. The conformational analysis of the obtained compounds was undertaken by means of¹H-NMR spectroscopy. The spiro-1,3-dioxanes obtained with aldehydes or nonsymmetrical ketones represent fixed (anancomeric) structures. Ketalisation with 4-t-butyl-cyclohexanone leads to a semimobile structure of a trispiro-1,3-dioxane in which the two marginal cyclohexanic rings are fixed, while the two middle ones are mobile, continuously flipping at room temperature. The solvent effect on the NMR spectra is also discussed. Utilisation of benzene-d ₆ makes it possible to interpret some of the complex spectra with superposed signals on the basis of the aromatic solvent induced shift (ASIS).

     

Fluctuation forces in a three-layer medium with rough boundaries. II. Calculations in the second order of perturbation theory

All-Union Scientific-Research Center for the Study of Surface and Vacuum Properties. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 82, No. 3, pp. 360–365, March, 1990.     

Helly-Type Theorems on Common Supporting Lines for Nonoverlapping Families of Convex Bodies in the Plane

Helly-type results are established relating to the existence of a line supporting a family of nonoverlapping convex bodies in the plane.     

Some Shift-Invariant Integral Operators, Univariate Case, Revisited

In recent articles the first author and H. Gonska [e.g., see G. Anastassiou, C. Cottin, and H. Gonska, Global smoothness of approximating functions, Analysis, 11, 43–57 (1991); G. Anastassiou and H. Gonska, On some shift-invariant integral operators, univariate case, Ann. Pol. Math. LXI.3, 225–243 (1995)] studied global smoothness preservation by some univariate and multivariate linear operators over compact domains and ⁿ , n 1. In particular, they studied a very general positive linear integral type operator [e.g., see G. Anastassiou and H. Gonska, On some shift-invariant integral operators, univariate case, Ann. Pol. Math. LXI.3, 225–243 (1995)] over ⁿ that was introduced through a convolution-like integration of another general positive linear operator with a scaling-type function. In this article the authors, among others, extend and generalize [G. Anastassiou and H. Gonska, On some shift-invariant integral operators, univariate case, Ann. Pol. Math. LXI.3, 225–243 (1995)]. Also certain new similar but more general integral operators are introduced and studied. These operators arise in a natural way, and for all these sufficient conditions are given for shift invariance, preservation of higher-order global smoothness and sharpness of the related inequalities, convergence to the unit using the first modulus of continuity, shape preservation, and preservation of continuous probabilistic distribution functions. Several examples of very general specialized operators, old and new, are given that satisfy all the above properties.     

The c‐map,Tits Satake subalgebras and the search for inflaton potentials

In this paper we address the general problem of including inflationary models exhibiting Starobinsky‐like potentials into (symmetric) supergravities. This is done by gauging suitable abelian isometries of the hypermultiplet sector and then truncating the resulting theory to a single scalar field. By using the characteristic properties of the global symmetry groups of the supergravities we are able to make a general statement on the possible α‐attractor models which can obtained upon truncation. We find that in symmetric models group theoretical constraints restrict the allowed values of the parameter α to be . This confirms and generalizes results recently obtained in the literature. Our analysis heavily relies on the mathematical structure of symmetric supergravities, in particular on the so called c‐map connection between Quaternionic Kähler manifolds starting from Special Kähler ones. A general statement on the possible consistent truncations of the gauged models, leading to Starobinsky‐like potentials, requires the essential help of Tits Satake universality classes. The paper is mathematically self‐contained and aims at presenting the involved mathematical structures to a public not only of physicists but also of mathematicians. To this end the main mathematical structures and the general gauging procedure of supergravities is reviewed in some detail.