An interpolatory subdivision algorithm for surfaces over arbitrary triangulations

In this paper, an interpolatory subdivision algorithm for surfaces over ar-bitrary triangulations is introduced and its convergence properties over nonuni-form triangulations studied. The so called Butterfly Scheme (interpolatory) is a special case of this algorithm. In our analysis of the algorithm over uniform triangulations, a matrix approach is employed and the idea, of "Cross Differ-ence of Directional Divided Difference" analysis is presented. This method is a generalization of the technique used by Dyn, Gregory and Levin etc. to analyse univariate subdivision algorithms. While for nonuniform data, an extraordi-nary point analysis is introduced and the local subdivision matrix analysis is presented. It is proved that the algorithm produces smooth surfaces over ar-bitrary triangular networks provided the shape parameters are kept within an appropriate range

Estimating Maximally Probable Constrained Relations by Mathematical Programming

Estimating a constrained relation is a fundamental problem in machine learning. Special cases are classification (the problem of estimating a map from a set of to-be-classified elements to a set of labels), clustering (the problem of estimating an equivalence relation on a set) and ranking (the problem of estimating a linear order on a set). We contribute a family of probability measures on the set of all relations between two finite, non-empty sets, which offers a joint abstraction of multi-label classification, correlation clustering and ranking by linear ordering. Estimating (learning) a maximally probable measure, given (a training set of) related and unrelated pairs, is a convex optimization problem. Estimating (inferring) a maximally probable relation, given a measure, is a 01-linear program. It is solved in linear time for maps. It is NP-hard for equivalence relations and linear orders. Practical solutions for all three cases are shown in experiments with real data. Finally, estimating a maximally probable measure and relation jointly is posed as a mixed-integer nonlinear program. This formulation suggests a mathematical programming approach to semi-supervised learning.Comment: 16 page

Effective website for educational institution

This paper examines the current status of a New Zealand educational institution’s website and makes a comparison among 44 samples. Two main sections have been contained in this research, secondary resource collection and observation. The aim of this research is to raise the awareness of this organisation and to attract potential students from the Chinese market by improve the organisation’s official website. This is desk research which has made use of qualitative methods. The results of the secondary resource collection mainly illustrate the general concepts of website design, the key factors of the education website, and Chinese web page preferences. The results from observation of 44 samples have been divided into two aspects, functionality and content. Most educational institution website pages are functional, while their online communication channels are basically limited to social media links; from the content point of view, school websites with Chinese versions generally have poor translations, and existing educational institution website samples can provide the information that the viewer needs through different presentation methods. These differences are explained in the discussion section. In conclusion, it suggested that in the current stage, educational institution websites could gain development by diversifying their communication channels and information presentation