A refined form of the ‘Folk Theorem’ that a smooth action by a compact Lie group can be (canonically) resolved, by iterated blow up, to have unique isotropy type is proved in the context of manifolds with corners. This procedure is shown to capture the simultaneous resolution of all isotropy types in a ‘resolution structure’ consisting of equivariant iterated fibrations of the boundary faces. T...

Let f ∈ Lloc(R) ∩ S ′, S ′ is the Schwartz class of distributions, and ∫ σ(D) f(x)dx = 0 ∀σ ∈ G, (∗) whereD ⊂ Rn, n ≥ 2, is a bounded domain, the closure D̄ of which is C1−diffeomorphic to a closed ball. Then the complement of D̄ is connected and path connected. Here G denotes the group of all rigid motions in Rn. This group consists of all translations and rotations. It is conjectured that if f ...

Because of the hairy ball theorem, the only closed 2-manifold that supports a lattice in its tangent space is T . But, if singular points (i.e. points whose tangent space is not endowed with 2 distinct coordinate directions) are allowed, then it becomes possible to give the tangent space a lattice. Because the lattice is well defined everywhere around the points, the effect of moving around the...

Denote by B the n-dimensional unit ball centred at o. It is known that in every lattice packing of B there is a cylindrical hole of infinite length whenever n ≥ 3. As a counterpart, this note mainly proves the following result: For any fixed , > 0, there exist a periodic point set P (n, ) and a constant c(n, ) such that B + P (n, ) is a packing in R, and the length of the longest segment contai...

− div ( |∇up|p−2∇up ) = f in Ω up = 0 on ∂Ω, where p > 1 and Ω is a bounded open set of R (N ≥ 2) with Lipschitz boundary. We analyze the case where Ω is a ball and the datum f is a non-negative radially decreasing function belonging to the Lorentz space LN,∞(Ω) and the case where the datum f belongs to the dual space W−1,∞(Ω). We are interested in finding the pointwise limit of up as p goes to...

The size of sports fields considerably varies from a few meters for table tennis to hundreds of meters for golf. We first show that this size is mainly fixed by the range of the projectile, that is, by the aerodynamic properties of the ball (mass, surface, drag coefficient) and its maximal velocity in the game. This allows us to propose general classifications for sports played with a ball.

Graphical Abstract “When I was a kid wanted to be an adult. Be careful what you wish for! … The most important thing have learned from my students is that ideas are not necessarily the best ones …” Find out more about Liam Ball in his Introducing Profile.

A deeper understanding of the milling operation of ball mills helps mineral processing engineers to control and optimize them, and therefore, reduce their consuming power. In this work, the milling operation of ball mills is investigated using two methods, i.e. DEM and combined DEM-SPH. First, a pilot scale ball mill with no lifter is simulated by both methods. Then another pilot scale ball mil...