Many approaches enable teachers to digitalize students' answers and mark them on the computer. However, they are still limited for supporting marking descriptive mathematical answers that can best evaluate learners' understanding. This paper presents clustering of offline handwritten mathematical expressions (HMEs) to help teachers efficiently mark answers in the form of HMEs. In this work, we investigate a method of combining feature types from low-level directional features and multiple levels of recognition: bag-of-symbols, bag-of-relations, and bag-of-positions. Moreover, we propose a marking cost function to measure the marking effort. To show the effectiveness of our method, we used two datasets and another sampled from CROHME 2016 with synthesized patterns to prepare correct answers and incorrect answers for each question. In experiments, we employed the k-means++ algorithm for each level of features and considered their combination to produce better performance. The experiments show that the best combination of all the feature types can reduce the marking cost to about 0.6 by setting the number of answer clusters appropriately compared with the manual one-by-one marking.

In this paper, we study how to automatically classify mathematical expressions written in MathML (Mathematical Markup Language). It is an essential preprocess to resolve analysis problems originated from multi-meaning mathematical symbols. We first define twelve equation classes based on chapter information of mathematics textbooks and then conduct various experiments. Experimental results show an accuracy of 94.75%, by employing the feature combination of tags, operators, strings, and “identifier & operator” bigram.

Synchronous sampling allows alternating current (AC) quantities, such as the root mean square (RMS) values of voltage and power, to be determined with very low uncertainties (on the order of a few parts of 10-6 [1]). In this paper, a mathematical expression for estimating measurement uncertainties in nonideal synchronization with fundamental frequency AC signals is presented. The obtained results were compared with those obtained by measurements with a high-precision instrument used for measuring basic AC values.