A Note on the Use of Sum in the Logic of Proofs

The Logic of Proofs~LP, introduced by Artemov, encodes the same reasoning as the modal logic~S4 using proofs explicitly present in the language. In particular, Artemov showed that three operations on proofs (application~$\cdot$, positive introspection~!, and sum~+) are sufficient to mimic provability concealed in S4~modality. While the first two operations go back to G{\"o}del, the exact role of~+ remained somewhat unclear. In particular, it was not known whether the other two operations are sufficient by themselves. We provide a positive answer to this question under a very weak restriction on the axiomatization of LP.

Transverse-momentum spectra of electroweak bosons near threshold at NNLO

We obtain the next-to-next-to-leading order corrections to transverse-momentum spectra of W, Z and Higgs bosons near the partonic threshold. In the threshold limit, the electroweak boson recoils against a low-mass jet and all radiation is either soft, or collinear to the jet or the beam directions. We extract the virtual corrections from known results for the relevant two-loop four-point amplitudes and combine them with the soft and collinear two-loop functions as defined in Soft-Collinear Effective Theory. We have implemented these results in a public code PeTeR and present numerical results for the threshold resummed cross section of W and Z bosons at next-to-next-to-next-to-leading logarithmic accuracy, matched to next-to-leading fixed-order perturbation theory. The two-loop corrections lead to a moderate increase in the cross section and reduce the scale uncertainty by about a factor of two. The corrections are significantly larger for Higgs production.

Human intervertebral disc stiffness correlates better with the Otsu threshold computed from axial T2 map of its posterior annulus fibrosus than with clinical classifications

Degeneration of the intervertebral disc, sometimes associated with low back pain and abnormal spinal motions, represents a major health issue with high costs. A non-invasive degeneration assessment via qualitative or quantitative MRI (magnetic resonance imaging) is possible, yet, no relation between mechanical properties and T2 maps of the intervertebral disc (IVD) has been considered, albeit T2 relaxation time values quantify the degree of degeneration. Therefore, MRI scans and mechanical tests were performed on 14 human lumbar intervertebral segments freed from posterior elements and all soft tissues excluding the IVD. Degeneration was evaluated in each specimen using morphological criteria, qualitative T2 weighted images and quantitative axial T2 map data and stiffness was calculated from the load-deflection curves of in vitro compression, torsion, lateral bending and flexion/extension tests. In addition to mean T2, the OTSU threshold of T2 (TOTSU), a robust and automatic histogram-based method that computes the optimal threshold maximizing the distinction of two classes of values, was calculated for anterior, posterior, left and right regions of each annulus fibrosus (AF). While mean T2 and degeneration schemes were not related to the IVDs' mechanical properties, TOTSU computed in the posterior AF correlated significantly with those classifications as well as with all stiffness values. TOTSU should therefore be included in future degeneration grading schemes.