Exponential Iterated Integrals and the Relative Solvable Completion of the Fundamental Group of a Manifold

We develop a class of integrals on a manifold M called exponential iterated integrals, an extension of K. T. Chen's iterated integrals. It is shown that the matrix entries of any upper triangular representation of the fundamental group of M can be expressed via these new integrals. The ring of exponential iterated integrals contains the coordinate rings for a class of universal representations, called relative solvable completions, of the fundamental group. We consider exponential iterated integrals in the particular case of fibered knot complements, where the fundamental group always has a faithful relative solvable completion.Comment: 30 pages. New version is revised throughout, and includes a stronger result in section 7 and more extensive discussion of example

Social action on social media

This paper examines a new way of detecting and measuring social action, especially that which takes place below the radar. Abstract People try to help others in a wide number of ways. Taken together this is social action - the heart of civil society, and the foundation of a healthy one. However, some social action is hard to spot. It may be unregistered, be carried out with little or no income, or have little formal governance. This paper examines a new way of detecting and measuring social action – especially that which takes place below the radar. It uses a new methodology developed by CASM to use social media to spot, collect and measure social action that normally is carried out below the radar. It uses natural language processing algorithms to analyse, and sort large quantities of Tweets related to two key events: the flooding of 2014, and the launch of the Step up to Serve Campaign. This paper finds: Disasters, accidents and catastrophes are likely to create a explosions of Tweets too large to manually read. Some people will use Twitter to either offer or ask for help. This will often be specific to the disaster, spontaneous, and by people operating outside of any organization or charity. Twitter is a significant new forum which people will use in response to events to try to help each other. And it recommends: An Ebay for social action on social media’: Connecting social action supply with demand: When social action information is found, it could be centralized onto a real-time online platform, information exchange or brokerage hub, clearly related to a specific event and segmented either being offered

Graphical Methods in Device-Independent Quantum Cryptography

We introduce a framework for graphical security proofs in device-independent quantum cryptography using the methods of categorical quantum mechanics. We are optimistic that this approach will make some of the highly complex proofs in quantum cryptography more accessible, facilitate the discovery of new proofs, and enable automated proof verification. As an example of our framework, we reprove a previous result from device-independent quantum cryptography: any linear randomness expansion protocol can be converted into an unbounded randomness expansion protocol. We give a graphical proof of this result, and implement part of it in the Globular proof assistant.Comment: Publishable version. Diagrams have been polished, minor revisions to the text, and an appendix added with supplementary proof