On the maps preserving the equality of distance

In this paper, in connection with the known result of Baker and Vogt, we get if preserves equality of distance with dimension X?2 and Y is strictly convex, the range of f contains a segment, then f is affine.     

Note On the equality of star products

We study the problem of finding conditions for two star products of n vectors to be equal when the vectors in each star product are not linearly independent, and the problem of finding conditions for one star product to be zero. We reprove results of M. Marcus and J. Chollet in a very simple way. We present also some open problems.     

On fiber products of elementary maps

The following characterization of fully closed maps is proved: a quotient map between regular spaces is fully closed if and only if it coincides with the fiber product of elementary maps between regular spaces.     

On the local classification of smooth maps induced by Newton's method

The classification of germs of smooth maps f:Rⁿ→Rⁿ induced by the continuous Newton method is addressed in this paper. This classification problem is shown to rely on an equivalence notion located between right and contact equivalences, driving the classification problem from the setting of quasilinear ODEs to the singularity theory framework. One-dimensional problems and regular, n-dimensional cases are easily characterized, and normal forms for them are given. Folded zeros in Rⁿ display a much richer behavior: an invariant and a preliminary normal form are derived for these cases.     

On products of pseudo-open maps and some related matters

As a generalization of closed maps and open maps, Arhangel'skiǐ introduced the notion of pseudo-open maps and showed that such maps are precisely hereditarily quotient. In this paper, we shall investigate the product of two or countably many pseudo-open maps, and refute the plausible conjecture: If each fⁿ is closed, then fⁿ is pseudo-open, or at least quotient. Moreover, as related matters, we shall consider countable products of closed maps.     

On one-relator products induced by generalized triangle groups I

In this paper and a sequel, we study a group which is the quotient of a free product of groups by the normal closure of a single word that is contained in a subgroup which has the form of a free product of two cyclic groups. We use known properties of generalized triangle groups, together with detailed analysis of pictures and of words in free monoids, to prove a number of results such as a Freiheitssatz and the existence of Mayer-Vietoris sequences for such groups under suitable hypotheses. The results generalize those in an earlier article of the second author and Shwartz.     

On one-relator products induced by generalized triangle groups II

This is the second of two papers in which we study a group which is the quotient of a free product of groups by the normal closure of a single word that is contained in a subgroup which has the form of a free product of two cyclic groups. We use known properties of generalized triangle groups, together with detailed analysis of pictures and of words in free monoids, to prove a number of results such as a Freiheitssatz and the existence of Mayer-Vietoris sequences for such groups under suitable hypotheses. The results generalize those in an earlier article of the second author and Shwartz.     

On generalized hankel operators induced by sums and products

Vectorial Hankel operators are studied, in particular the ranges of Hankel operators induced by sums and products of matrix functions defined on the unit circle are determined. The analytical tools involve factorization theorems for operator valued analytic functions and the spectral analysis of operators that intertwine restricted shifts.     

Lower previsions induced by filter maps

We investigate under which conditions a transformation of an imprecise probability model of a certain type (coherent lower previsions, n  -monotone capacities, minitive measures) produces a model of the same type. We give a number of necessary and sufficient conditions, and study in detail a particular class of such transformations, called filter maps. These maps include as particular models multi-valued mappings as well as other models of interest within imprecise probability theory, and can be linked to filters of sets and {0,1}$\{0,1\}$-valued lower probabilities.     

On soft equality

Molodtsov introduced the concept of soft sets, which can be seen as a new mathematical tool for dealing with uncertainty. In this paper, we deal with the algebraic structure of soft sets. The lattice structures of soft sets are constructed. The concept of soft equality is introduced and some related properties are derived. It is proved that soft equality is a congruence relation with respect to some operations and the soft quotient algebra is established.     

Matrices and eigenfunctions induced by Markov maps

The matrix M induced by a Markov map has 1 as the eigenvalue of maximum modulus. If M is also irreducible, then the algebraic and geometric multiplicities of the eigenvalue 1 are also 1. Let g=(g₁,g₂…g_n) be an n-row-vector such that g₁=c> and g_ic, i?2. Then every such g is the unique (upto constant multiples) left eigenvector associated with the eigenvalue 1 of a matrix M induced by a Markov map.     

A note on stability of maps which preserve equality of distance

In this note, we show a generalized stability of maps which preserve equality of distance.     

On the parseval equality

Moscow. Translated from Sibirskii Matematicheskii Zhurnal, Vol. 31, No. 5, pp. 178–181, Spetember–October, 1990.     

On tensor products of completely positive linear maps between pro-C *-algebras

In this paper we define the tensor products of completely positive linear maps between pro-C^*-algebras and discuss about connection between the KSGNS construction associated with the strict completely positive linear maps ρ and θ and the KSGNS construction associated with ρ ⊗ θ. This research was partially supported by CEEX grant -code PR-D11-PT00-48/2005 from The Romanian Ministry of Education and Research and partially by CNCSIS (Romanian National Council for Research in High Education) grant-code A 1065/2006.     

Residual behavior of induced maps

Consider (X,F, μ,T) a Lebesgue probability space and measure preserving invertible map. We call this a dynamical system. For a subsetA ∈F. byT _A:A →A we mean the induced map,T _A(x)=T^rA(x)(x) wherer _A(x)=min{i〉0:T ⁱ(x) ∈A}. Such induced maps can be topologized by the natural metricD(A, A’) = μ(AΔA’) onF mod sets of measure zero. We discuss here ergodic properties ofT _A which are residual in this metric. The first theorem is due to Conze.Theorem 1 (Conze):For T ergodic, T _A is weakly mixing for a residual set of A.Theorem 2:For T ergodic, 0-entropy and loosely Bernoulli, T _A is rank-1, and rigid for a residual set of A.Theorem 3:For T ergodic, positive entropy and loosely Bernoulli, T _A is Bernoulli for a residual set of A.Theorem 4:For T ergodic of positive entropy, T _A is a K-automorphism for a residual set of A. A strengthening of Theorem 1 asserts thatA can be chosen to lie inside a given factor algebra ofT. We also discuss even Kakutani equivalence analogues of Theorems 1–4.