On Products and Diagonals of Mappings in Generalized Topological Spaces

Based on the theory of products of generalized topologies,we introduce the product mappings and the diagonal mappings in generalized topological spaces in this paper.We investigate some basic properties(especially,the continuity,openness and closedness)of the product mappings and the diagonal mappings in generalized topological spaces.Some applications are given to answer two questions raised in[3].

Biharmonic product maps between doubly warped product manifolds

The biharmonicity of the product map Φ2=φ×ψ and the two generalized projections φ-and ψ-are analyzed. Some results are obtained, that is, Φ2 is a proper biharmonic map if and only if b is a non-constant solution of -1/f2 Jφ（dφ（grad（lnb）））＋n/2 grad｜dφ（grad（lnb））｜2=0 and f is a non-constant solution of -1/b2Jψ（dψ（grad（lnf）））＋m/2grad｜dψ（grad（lnf））｜2=0, and Φ2=φ×ψ is a proper biharmonic map if and only if φ-and ψ-are proper biharmonic maps.

f-Harmonic maps of doubly warped product manifolds

In this paper,we study f-harmonicity of some special maps from or into a doubly warped product manifold.First we recall some properties of doubly twisted product manifolds.After showing that the inclusion maps from Riemannian manifolds M and N into the doubly warped product manifold M ×（μ,λ） N can not be proper f-harmonic maps,we use projection maps and product maps to construct nontrivial f-harmonic maps.Thus we obtain some similar results given in [21],such as the conditions for f-harmonicity of projection maps and some characterizations for non-trivial f-harmonicity of the special product maps.Furthermore,we investigate non-trivial f-harmonicity of the product of two harmonic maps.

Sensitivity of the Product System of Set-Valued Non-Autonomous Discrete Dynamical Systems

This paper is concerned with the sensitivity of set-valued discrete systems. Firstly, this paper obtained the equivalence between or and the product system in sensitivity, infinite sensitivity, F-sensitivity, (F1, F2)-sensitivity. Then, the relation between (X, f1,∞) or (Y, g1,∞) and in ergodic sensitivity is obtained. Where is the set-valued dynamical system induced by a non-autonomous discrete dynamical system (X, f1,∞).

Product portfolio map:a visual tool for supporting product variant discovery and structuring

A fundamental criterion for reusing and continuously improving knowledge in product development is ensuring that the knowledge is explicit and visual.This paper is based on the situation of an engineer-to-order（ETO） manufacturing company,where historically grown product variety and related knowledge are diffuse（tacit）.Consequently,several resources are used in（re）developing derivatives of previous products rather than innovating new ones.To establish a more competitive configure-to-order（CTO） product strategy,product knowledge needs to be revealed,systemized,and structured,and thus made explicit.Hence,product-specific knowledge and product variants have been analyzed and subsequently mapped at architectural,functional,and physical levels in one unified map and tested in the form of a proof-of-concept（POC）demonstrator with the introduced SME company.The result is a product portfolio map that forms a base for defining a systemized,transparent,unified product variant overview,which can be used as a basis for implementing a cross-variant product architecture and supporting knowledge-based approaches.

Hand-feel soil texture observations to evaluate the accuracy of digital soil maps for local prediction of soil particle size distribution:A case study in Central France

Digital maps of soil properties are now widely available.End-users now can access several digital soil mapping(DSM)products of soil properties,produced using different models,calibration/training data,and covariates at various spatial scales from global to local.Therefore,there is an urgent need to provide easy-to-understand tools to communicate map uncertainty and help end-users assess the reliability of DSM products for use at local scales.In this study,we used a large amount of hand-feel soil texture(HFST)data to assess the performance of various published DSM products on the prediction of soil particle size distribution in Central France.We tested four DSM products for soil texture prediction developed at various scales(global,continental,national,and regional)by comparing their predictions with approximately 3200 HFST observations realized on a 1:50000 soil survey conducted after release of these DSM products.We used both visual comparisons and quantitative indicators to match the DSM predictions and HFST observations.The comparison between the low-cost HFST observations and DSM predictions clearly showed the applicability of various DSM products,with the prediction accuracy increasing from global to regional predictions.This simple evaluation can determine which products can be used at the local scale and if more accurate DSM products are required.