Testing the reliability of forecasting systems

The problem of statistically evaluating forecasting systems is revisited. The forecaster claims the forecasts to exhibit a certain nominal statistical behaviour; for instance, the forecasts provide the expected value (or certain quantiles) of the verification, conditional on the information available at forecast time. Forecasting systems that indeed exhibit the nominal behaviour are referred to as reliable. Statistical tests for reliability are presented (based on an archive of verification–forecast pairs). As noted previously, devising such tests is encumbered by the fact that the dependence structure of the verification–forecast pairs is not known in general. Ignoring this dependence though might lead to incorrect tests and too-frequent rejection of forecasting systems that are actually reliable. On the other hand, reliability typically implies that the forecast provides information about the dependence structure, and using this in conjunction with judicious choices of the test statistic, rigorous results on the asymptotic distribution of the test statistic are obtained. These results are used to test for reliability under minimal additional assumptions on the statistical properties of the verification–forecast pairs. Applications to environmental forecasts are discussed. A python implementation of the discussed methods is available online

A reason for fusion rules to be even

We show that certain tensor product multiplicities in semisimple braided sovereign tensor categories must be even. The quantity governing this behavior is the Frobenius-Schur indicator. The result applies in particular to the representation categories of large classes of groups, Lie algebras, Hopf algebras and vertex algebras.Comment: 6 pages, LaTe

Nonnegatively curved homogeneous metrics obtained by scaling fibers of submersions

We consider invariant Riemannian metrics on compact homogeneous spaces G/H where an intermediate subgroup K between G and H exists, so that the homogeneous space G/H is the total space of a Riemannian submersion. We study the question as to whether enlarging the fibers of the submersion by a constant scaling factor retains the nonnegative curvature in the case that the deformation starts at a normal homogeneous metric. We classify triples of groups (H,K,G) where nonnegative curvature is maintained for small deformations, using a criterion proved by Schwachh\"ofer and Tapp. We obtain a complete classification in case the subgroup H has full rank and an almost complete classification in the case of regular subgroups.Comment: 23 pages; minor revisions, to appear in Geometriae Dedicat

When and where do ECMWF seasonal forecast systems exhibit anomalously low signal‐to‐noise ratio?

Seasonal predictions of wintertime climate in the Northern Hemisphere midlatitudes, while showing clear correlation skill, suffer from anomalously low signal‐to‐noise ratio. The low signal‐to‐noise ratio means that forecasts need to be made with large ensemble sizes and require significant post‐processing to correct the forecast distribution. In this study, a recently introduced statistical model of seasonal climate predictability is adapted so that it can be used to examine the signal‐to‐noise ratio in two versions of the ECMWF seasonal forecast system. Three novel features of the low signal‐to‐noise ratio are revealed. The low signal‐to‐noise ratio is present only for forecasts initialized on 1 November and not for forecasts initialized on 1 December. The low signal‐to‐noise ratio is predominantly a feature of the lower and middle troposphere and is not present in the stratosphere. The low signal‐to‐noise ratio is linked to low signal amplitude of the forecast systems in early winter. Future studies attempting to examine the signal‐to‐noise ratio should focus on the extent to which this early winter variability is predictable

On the total curvatures of a tame function

Given a definable function f, enough differentiable, we study the continuity of the total curvature function t --> K(t), total curvature of the level {f=t}, and the total absolute curvature function t-->|K| (t), total absolute curvature of the level {f=t}. We show they admits at most finitely many discontinuities

On all possible static spherically symmetric EYM solitons and black holes

We prove local existence and uniqueness of static spherically symmetric solutions of the Einstein-Yang-Mills equations for any action of the rotation group (or SU(2)) by automorphisms of a principal bundle over space-time whose structure group is a compact semisimple Lie group G. These actions are characterized by a vector in the Cartan subalgebra of g and are called regular if the vector lies in the interior of a Weyl chamber. In the irregular cases (the majority for larger gauge groups) the boundary value problem that results for possible asymptotically flat soliton or black hole solutions is more complicated than in the previously discussed regular cases. In particular, there is no longer a gauge choice possible in general so that the Yang-Mills potential can be given by just real-valued functions. We prove the local existence of regular solutions near the singularities of the system at the center, the black hole horizon, and at infinity, establish the parameters that characterize these local solutions, and discuss the set of possible actions and the numerical methods necessary to search for global solutions. That some special global solutions exist is easily derived from the fact that su(2) is a subalgebra of any compact semisimple Lie algebra. But the set of less trivial global solutions remains to be explored.Comment: 26 pages, 2 figures, LaTeX, misprints corrected, 1 reference adde

Smash products for secondary homotopy groups

We construct a smash product operation on secondary homotopy groups yielding the structure of a lax symmetric monoidal functor. Applications on cup-one products, Toda brackets and Whitehead products are considered. In particular we prove a formula for the crossed effect of the cup-one product operation on unstable homotopy groups of spheres which was claimed by Barratt-Jones-Mahowald.Comment: We give a clearer description of the tensor product of symmetric sequences of quadratic pair module