Higgs algebra of curves and loop crystals

We define the Higgs algebra $\mathcal{H}_\P1$ of the projective line, as a convolution algebra of constructible functions on the global nilpotent cone $\underline{\Lambda}_\P1$, a lagrangian substack of the Higgs bundle T^*\Coh_\P1, where \Coh_\P1 is the stack of coherent sheaves on $\P1$. We prove that $\mathcal{H}_\P1$ is isomorphic to (some completion of) $U^+(\hat{sl}_2)$. We use this geometric realization to define a semicanonical basis of $U^+(\hat{sl}_2)$, indexed by irreducible components of $\underline{\Lambda}_\P1$. We also construct a combinatorial data on this set of irreducible components in the spirit of \cite{KS}, which is an affine analog of a crystal. We call it a loop crystal and give some of its properties

Accountability and Intervening Agency: An Asymmetry between Upstream and Downstream Actors

Suppose someone (P1) does something that is wrongful only in virtue of the risk that it will enable another person (P2) to commit a wrongdoing. Suppose further that P1’s conduct does indeed turn out to enable P2’s wrongdoing. The resulting wrong is agentially mediated: P1 is an enabling agent and P2 is an intervening agent. Whereas the literature on intervening agency focuses on whether P2’s status as an intervening agent makes P1’s conduct less bad, I turn this issue on its head by investigating whether P1’s status as an enabling agent makes P2’s conduct more bad. I argue that it does: P2 wrongs not just the victims of ϕ but P1 as well, by acting in a way that wrongfully makes P1 accountable for ϕ. This has serious implications for compensatory and defensive liability in cases of agentially mediated wrongs

On the restriction of the Fourier transform to polynomial curves

We prove a Fourier restriction theorem on curves parametrised by the mapping P(t) = (P1(t),..., Pn(t)), where each of the P1,..., Pn is a real-valued polynomial and t belongs to an interval on which each of the P1,..., Pn "resembles" a monomial

Sum of squared logarithms - An inequality relating positive definite matrices and their matrix logarithm

Let y1, y2, y3, a1, a2, a3 > 0 be such that y1 y2 y3 = a1 a2 a3 and y1 + y2 + y3 >= a1 + a2 + a3, y1 y2 + y2 y3 + y1 y3 >= a1 a2 + a2 a3 + a1 a3. Then the following inequality holds (log y1)^2 + (log y2)^2 + (log y3)^2 >= (log a1)^2 + (log a2)^2 + (log a3)^2. This can also be stated in terms of real positive definite 3x3-matrices P1, P2: If their determinants are equal det P1 = det P2, then tr P1 >= tr P2 and tr Cof P1 >= tr Cof P2 implies norm(log P1) >= norm(log P2), where log is the principal matrix logarithm and norm(P) denotes the Frobenius matrix norm. Applications in matrix analysis and nonlinear elasticity are indicated

Singular normal form for the Painlev\'e equation P1

We show that there exists a rational change of coordinates of Painlev\'e's P1 equation $y''=6y^2+x$ and of the elliptic equation $y''=6y^2$ after which these two equations become analytically equivalent in a region in the complex phase space where $y$ and $y'$ are unbounded. The region of equivalence comprises all singularities of solutions of P1 (i.e. outside the region of equivalence, solutions are analytic). The Painlev\'e property of P1 (that the only movable singularities are poles) follows as a corollary. Conversely, we argue that the Painlev\'e property is crucial in reducing P1, in a singular regime, to an equation integrable by quadratures

Error estimates for Stokes problem with Tresca friction condition

In this work we propose and study a three field mixed formulation for solving the Stokes problem with Tresca-type non-linear boundary conditions. Two Lagrange multipliers are used to enforce div(u)=0 constraint and to regularize the energy functional. The resulting problem is discretised using "P1 bubble/P1-P1" finite elements. Error estimates are derived and several numerical studies are achieved