Abstract. A lattice L is constructed with the property that every interval has finite height, but there exists no strictly order-preserving map from L to Z. A 1979 problem of Erné (posed at the 1981 Banff Conference on Ordered Sets) is thus solved. It is also shown that if a poset P has no uncountable antichains, then it admits a strictly order-preserving map into Z if and only if every interva...

This paper aims to describe the geometrical structure and explicit expressions of family of finitely parametrized probability densities over smooth manifold M. The geometry of family of probability densities on M are inherited from probability densities on Euclidean spaces {Uα} via bundle morphisms, induced by an orientation-preserving diffeomorphisms ρα : Uα → M. Current literature inherits de...

The Kohonen Self Organizing Map (SOM), is a topology preserving map that maps data from higher dimensions onto a (typically) two dimensional grid of lattice points[3]. The aim of Self-Organization is to generate a topology preserving mapping, where the neighborhood relations in the input space are preserved as well as possible, in the neighborhood relations of the units of the map[2]. One of th...

We present a new method of analysis of measure-preserving dynamical systems, based on frequency analysis and ergodic theory, which extends our earlier work [1]. Our method employs the novel concept of harmonic time average [2], and is realized as a computational algorithms for visualization of periodic and quasi-periodic sets or arbitrary periodicity in the phase space. Besides identifying all ...

This paper studies the creation of schematic maps from traditional vector-based, carto-graphic information. An algorithm is proposed to modify positions of lines in the original input map with the goal of producing as output a schematic map that meets certain geometric and aesthetic criteria. Special emphasis is placed here on preserving topological structure of features during this transformat...

We propose a new approach to analyzing dynamical systems that combine hyperbolic and non-hyperbolic (“center”) behavior, e.g. partially hyperbolic diffeomorphisms. A number of applications illustrate its power. We find that any ergodic automorphism of the 4-torus with two eigenvalues in the unit circle is stably Bernoulli among symplectic maps. Indeed, any nearby symplectic map has no zero Lyap...

We introduce the notions of strongly zero-product (strongly Jordan zero-product) preserving maps on normed algebras. These notions are generalization of the concepts of zero-product and Jordan zero-product preserving maps. Also for a non-zero vector space V and for a non-zero linear functional f on V, we equip V with a multiplication, converting V into an associative algebra, denoted by Vf . We...

Abstract We say that a smooth normed space X has property (SL) , if every mapping $$f:X \rightarrow X$$ f : X → preserving the semi-inner product on is linear. It well known Hilbert and same true for finite-dimensional space. In this pa...