Renormalization : A number theoretical model

We analyse the Dirichlet convolution ring of arithmetic number theoretic functions. It turns out to fail to be a Hopf algebra on the diagonal, due to the lack of complete multiplicativity of the product and coproduct. A related Hopf algebra can be established, which however overcounts the diagonal. We argue that the mechanism of renormalization in quantum field theory is modelled after the same principle. Singularities hence arise as a (now continuously indexed) overcounting on the diagonals. Renormalization is given by the map from the auxiliary Hopf algebra to the weaker multiplicative structure, called Hopf gebra, rescaling the diagonals.Comment: 15 pages, extended version of talks delivered at SLC55 Bertinoro,Sep 2005, and the Bob Delbourgo QFT Fest in Hobart, Dec 200

Improving the Representation and Conversion of Mathematical Formulae by Considering their Textual Context

Mathematical formulae represent complex semantic information in a concise form. Especially in Science, Technology, Engineering, and Mathematics, mathematical formulae are crucial to communicate information, e.g., in scientific papers, and to perform computations using computer algebra systems. Enabling computers to access the information encoded in mathematical formulae requires machine-readable formats that can represent both the presentation and content, i.e., the semantics, of formulae. Exchanging such information between systems additionally requires conversion methods for mathematical representation formats. We analyze how the semantic enrichment of formulae improves the format conversion process and show that considering the textual context of formulae reduces the error rate of such conversions. Our main contributions are: (1) providing an openly available benchmark dataset for the mathematical format conversion task consisting of a newly created test collection, an extensive, manually curated gold standard and task-specific evaluation metrics; (2) performing a quantitative evaluation of state-of-the-art tools for mathematical format conversions; (3) presenting a new approach that considers the textual context of formulae to reduce the error rate for mathematical format conversions. Our benchmark dataset facilitates future research on mathematical format conversions as well as research on many problems in mathematical information retrieval. Because we annotated and linked all components of formulae, e.g., identifiers, operators and other entities, to Wikidata entries, the gold standard can, for instance, be used to train methods for formula concept discovery and recognition. Such methods can then be applied to improve mathematical information retrieval systems, e.g., for semantic formula search, recommendation of mathematical content, or detection of mathematical plagiarism.Comment: 10 pages, 4 figure

Coefficients of Wronskian Hermite polynomials

We study Wronskians of Hermite polynomials labeled by partitions and use the combinatorial concepts of cores and quotients to derive explicit expressions for their coefficients. These coefficients can be expressed in terms of the characters of irreducible representations of the symmetric group, and also in terms of hook lengths. Further, we derive the asymptotic behavior of the Wronskian Hermite polynomials when the length of the core tends to infinity, while fixing the quotient. Via this combinatorial setting, we obtain in a natural way the generalization of the correspondence between Hermite and Laguerre polynomials to Wronskian Hermite polynomials and Wronskians involving Laguerre polynomials. Lastly, we generalize most of our results to polynomials that have zeros on the p‐sta

Put to the Test: For a New Sociology of Testing

In an age defined by computational innovation, testing seems to have become ubiquitous, and tests are routinely deployed as a form of governance, a marketing device, an instrument for political intervention, and an everyday practice to evaluate the self. This essay argues that something more radical is happening here than simply attempts to move tests from the laboratory into social settings. The challenge that a new sociology of testing must address is that ubiquitous testing changes the relations between science, engineering and sociology: Engineering is today in the very stuff of where society happens. It is not that the tests of 21st Century engineering occur within a social context but that it is the very fabric of the social that is being put to the test. To understand how testing and the social relate today, we must investigate how testing operates on social life, through the modification of its settings. One way to clarify the difference is to say that the new forms of testing can be captured neither within the logic of the field test nor of the controlled experiment. Whereas tests once happened inside social environments, today’s tests directly and deliberately modify the social environment

RWPRI and (rm2T)1(rm 2T)_1 flag-transitive linear spaces

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RWPRI and (2T)1 Flag-transitive Linear Spaces

Abstract. The classification of finite flag-transitive linear spaces is almost complete. For the thick case, this result was announced by Buekenhout, Delandtsheer, Doyen, Kleidman, Liebeck and Saxl, and in the thin case (where the lines have 2 points), it amounts to the classification of 2-transitive groups, which is generally considered to follow from the classification of finite simple groups. These two classifications actually leave an open case, which is the so-called 1-dimensional case. In this paper, we work with two additional assumptions. These two conditions, namely (2T)1 and RWPri, are taken from another field of study in Incidence Geometry and allow us to obtain a complete classification, which we present at the end of this paper. In particular, for the 1-dimensional case, we show that the only (2T)1 flag-transitive linear spaces are AG(2, 2) and AG(2, 4), with AΓL(1, 4) and AΓL(1, 16) as respective automorphism groups. 1

Studies on -adrenergic activation of hepatic glucose output. The role of mitochondrial calcium release in -adrenergic activation of phosphorylase in perfused rat liver.

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