Multivariate integration of functions depending explicitly on the minimum and the maximum of the variables

By using some basic calculus of multiple integration, we provide an alternative expression of the integral $$\int_{]a,b[^n} f(\mathbf{x},\min x_i,\max x_i) d\mathbf{x},$$ in which the minimum and the maximum are replaced with two single variables. We demonstrate the usefulness of that expression in the computation of orness and andness average values of certain aggregation functions. By generalizing our result to Riemann-Stieltjes integrals, we also provide a method for the calculation of certain expected values and distribution functions.Comment: 15 page

LIFE AND WORK OF DOMENICO CANDELORO: AN APPRECIATION

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q-deformations of two-dimensional Yang-Mills theory: Classification, categorification and refinement

We characterise the quantum group gauge symmetries underlying q-deformations of two-dimensional Yang-Mills theory by studying their relationships with the matrix models that appear in Chern-Simons theory and six-dimensional N=2 gauge theories, together with their refinements and supersymmetric extensions. We develop uniqueness results for quantum deformations and refinements of gauge theories in two dimensions, and describe several potential analytic and geometric realisations of them. We reconstruct standard q-deformed Yang-Mills amplitudes via gluing rules in the representation category of the quantum group associated to the gauge group, whose numerical invariants are the usual characters in the Grothendieck group of the category. We apply this formalism to compute refinements of q-deformed amplitudes in terms of generalised characters, and relate them to refined Chern-Simons matrix models and generalized unitary matrix integrals in the quantum beta-ensemble which compute refined topological string amplitudes. We also describe applications of our results to gauge theories in five and seven dimensions, and to the dual superconformal field theories in four dimensions which descend from the N=(2,0) six-dimensional superconformal theory.Comment: 71 pages; v2: references added; final version to be published in Nuclear Physics