In this paper, we present an ontology of mathematical knowledge concepts that covers a wide range of the fields of mathematics and introduces a balanced representation between comprehensive and sensible models. We demonstrate the applications of this representation in information extraction, semantic search, and education. We argue that the ontology can be a core of future integration of math-aware data sets in the Web of Data and, therefore, provide mappings onto relevant datasets, such as DBpedia and ScienceWISE.
To put our research into the context, we summarize the most relevant previous works for representing mathematical knowledge in this section. For a more comprehensive overview of services, ontological models and languages for mathematical knowledge management on the Semantic Web and beyond, we refer the interested reader to C. Lange's survey @cite_6 .
@cite_6
2165738341 1687148191 2078652018 2081752276
Considering data size and privacy concerns in a distributed setting, it is neither desirable nor feasible to translate data from one resource to another in data mining. Rather, it makes more sense to first mine knowledge from one data resource and then translate the discovered knowledge (models) to another for knowledge reuse. Although there have been successful research efforts in knowledge transfer, the knowledge translation problem in the semantically heterogenous scenario has not been addressed adequately. In this paper, we first propose to use Semantic Web ontologies to represent rule-based knowledge to make the knowledge computer "translatable". Instead of an inductive learning approach, we treat knowledge translation as a deductive inference. We elaborate a translation method with both the forward and backward chaining to address the asymmetry of translation. We show the effectiveness of our knowledge translation method in decision tree rules and association rules mined from sports and gene data respectively. In a more general context, this work illustrates the promise of a novel research which leverages ontologies and Semantic Web techniques to extend the knowledge transfer in data mining to the semantically heterogeneous scenario. Mathematics is a ubiquitous foundation of science, technology, and engineering. Specific areas of mathematics, such as numeric and symbolic computation or logics, enjoy considerable software support. Working mathematicians have recently started to adopt Web 2.0 environments, such as blogs and wikis, but these systems lack machine support for knowledge organization and reuse, and they are disconnected from tools such as computer algebra systems or interactive proof assistants. We argue that such scenarios will benefit from Semantic Web technology.Conversely, mathematics is still underrepresented on the Web of [Linked] Data. There are mathematics-related Linked Data, for example statistical government data or scientific publication databases, but their mathematical semantics has not yet been modeled. We argue that the services for the Web of Data will benefit from a deeper representation of mathematical knowledge.Mathematical knowledge comprises structures given in a logical language --formulae, statements e.g. axioms, and theo-ries --, a mixture of rigorous natural language and symbolic notation in documents, application-specific metadata, and discussions about conceptualizations, formalizations, proofs, and counter-examples. Our review of vocabularies for representing these structures covers ontologies for mathematical problems, proofs, interlinked scientific publications, scientific discourse, as well as mathematical metadata vocabularies and domain knowledge from pure and applied mathematics.Many fields of mathematics have not yet been implemented as proper Semantic Web ontologies; however, we show that MathML and OpenMath, the standard XML-based exchange languages for mathematical knowledge, can be fully integrated with RDF representations in order to contribute existing mathematical knowledge to the Web of Data.We conclude with a roadmap for getting the mathematical Web of Data started: what datasets to publish, how to interlink them, and how to take advantage of these new connections. On the Semantic Web, the types of resources and the semantic relationships between resources are defined in an ontology. By using that information, the accuracy of information retrieval can be improved. In this paper, we present effective ranking and search techniques considering the semantic relationships in an ontology. Our technique retrieves top-k resources which are the most relevant to query keywords through the semantic relationships. To do this, we propose a weighting measure for the semantic relationship. Based on this measure, we propose a novel ranking method which considers the number of meaningful semantic relationships between a resource and keywords as well as the coverage and discriminating power of keywords. In order to improve the efficiency of the search, we prune the unnecessary search space using the length and weight thresholds of the semantic relationship path. In addition, we exploit Threshold Algorithm based on an extended inverted index to answer top-k results efficiently. The experimental results using real data sets demonstrate that our retrieval method using the semantic information generates accurate results efficiently compared to the traditional methods. An important challenge in the Knowledge Representation area is on representing and reasoning over temporally changing information. Particularly, a number of authors have been investigating approaches to extend the expressivity beyond what is currently supported by the DL (Description Logics) based languages in order to address this issue, while maintaining compatibility with subclasses of DLs adopted in the Semantic Web. This is mainly due to the increasing popularity of the Semantic Web initiative as well as the role played by DL in that context. In this paper we defend the need of a higher-level foundational framework based on results coming from the discipline of Formal Ontology. We present two complementary proposals for modeling temporally changing information in OWL, based on the most discussed strategy in the literature to address this problem, namely, the use of a perdurantist (or 4D) view of domain entities. Moreover we compare the results with some related work and discuss its limitations and further improvements.