[en] Normal surfaces over the complex plane are considered that are logarithmic (i.e., all its singularities are of the quotient type) and of which all reduced homology groups with rational coefficients vanish. It is proved that all such planes are rational. 16 refs

[en] After showing that Rosen's bimetric theory of gravity is a harmonic map, the geometry of the ten-dimensional harmonic mapping space (HMS), and of its nine-dimensional symmetric submanifolds, which are the leaves of the codimension one foliation of the HMS, is detailed. Both structures are global affinely symmetric spaces. For each, the metric, connections, and Riemann, Ricci, and scalar curvatures are given. The Killing vectors in each case are also worked out and related to the ''conserved quantities'' naturally associated with the harmonic mapping character of the theory. The structure of the Rosen HMS is very much like that determined by the DeWitt metric on the six-dimensional Wheeler superspace of all positive definite three-dimensional metrics. It is clear that a slight modification of the Rosen HMS metric will yield the corresponding metric on the space of all four-dimensional metrics of Lorentz signature. Finally, interesting avenues of further research are indicated, particularly with respect to the structure and comparison of Lagrangian-based gravitational theories which are similar to Einstein's general relativity

[en] A criterion for the linear compactness of submodules of certain linearly topologized modules of continuous mappings is established.

[en] The concepts of neutrosophic normal soft group, neutrosophic soft cosets, neutrosophic soft homomorphism are introduced and illustrated by suitable examples in this paper. Several related properties and structural characteristics are investigated. Some of their basic theorems are also established.

[en] In this paper, we show that a Lip n-microbundle contains a Lip n-bundle and that any two such are isomorphic. (author). 5 refs

[en] In this paper, at first the properties of isolated zero points of complex polynomial mappings are discussed using the integral expression of a degree of C¹ mapping, Sard's Theorems and homotopy. Then we prove that for almost every complex polynomial mapping P:Cⁿ→Cⁿ, the zero set H^-1(0) of the homotopy H(z,t)=tP(z)+(1-t)Q(z) consists of s=Π_j=1ⁿs_j disjoint differential curves, and the zero set φ_δj^-1(0) of its piecewise linear approximation φ_δj consists of some broken lines which do not meet ν-dimensional simplexes (0≤ν≤2n-1), where Q(z)=(Q₁(z),...,Q_n(z)), Q_j(z)=z_j^sj-b_j^sj, b_j≠0,j=1,...,n. When (lim_j→∞δ_j=0), these broken lines tend to corresponding differential curves in H^-1(0). (author). 4 refs

[en] Let U be a continuous representation of a (connected) locally compact group G in a separated locally convex space E. It is shown that the study of U is equivalent to the study of a family Usub(i) of continuous representations of G in Frechet spaces Fsub(i). If U is equicontinuous, the Fsub(i) are Banach spaces, and the Usub(i) are realized by isomeric operators. When U is topologically irreducible, it is Naemark equivalent to a Frechet (or isomeric Banach, in the equicontinuous case) continuous representation. Similar results hold for semi-groups. (Auth.)

[en] We show that the entropy of the relation inverse to a rational map of the Riemann sphere or to a piecewise monotone map of the interval is zero. We give also an example of map the inverse image of which has entropy bigger than the entropy of the map itself, answering a question of J. PALIS. (author). 8 refs., 4 figs

[en] Li and Yorke originally introduced the notion of chaos for continuous self-map of the interval I = (0,1). In the present paper we show that an interval self-map with positive topological entropy has a chaoticity more complicated than the chaoticity in the sense of Li and Yorke. The main result is that if f:I → I is continuous and has a periodic point with odd period > 1 then there exists a closed subset K of I invariant with respect to f such that the periodic points are dense in K, the periods of periodic points in K form an infinite set and f|K is topologically mixing. (author). 9 refs

[en] The aim of the present work is to follow the D-functors and modify it to be able to correct the errors in Cogosvili's paper 'on some properties of D-functions generated by cohomology'. Moreover, we state and prove the sufficient conditions which guarantee the exactness of the D-functor. (author). 14 refs