Mashups offer new functionality by combining, aggregating and transforming resources and services available on the Web. This chapter deals with mathematical mashups and focuses on those that process mathematical knowledge rather than, e.g., huge amounts of numeric data, as it is the structure of knowledge that distinguishes mathematics from other application domains. The resources that mathematical mashups process primarily include formulae and the services they offer involve computation. In knowledge-rich mashups, the formulae are not hard-coded into the implementation but represented as explicit data structures, and often also presented to the user. These structures are different from the (re)presentations mashups usually process. To allow for automated processing, formulae need to be represented neither as plain text nor as images, but in a symbolic way. The representation of choice is, for compatibility and scalability reasons discussed in this chapter, usually not JSON or RDF, but semantic XML markup. Besides tables or graphs, mathematical mashups may also require formulae to be presented to the end user. The highest de- gree of interaction with formulae is offered by MathML – in those browsers that fully support it. After introducing typical education and engineering use cases that benefit from mathematical mashups, this chapter reviews the conceptual and technical foundations for representing and pre- senting mathematical formulae, discussing MathML as well as alternatives. We continue with a review of mathematical web services and collections of mathematical knowledge that provide suitable building blocks for mathematical mashups. We then present the Planetarysystem, a math- enabled social semantic web portal that provides an environment for executable papers, and the SAlly framework that mashes up user interfaces of software applications with mathematical web services. Both environments mash up assistive services by hooking them into document structures, which have been annotated with terms from a mathematical background ontology. We conclude with an outlook towards contributing collections of mathematical knowledge to the Web of Data, and outline how such linked open datasets can drive further mathematical mashups in the near future. (C) Springer-Verlag Berlin Heidelberg 2013 |

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