In this paper we prove that no packing of unit balls in Euclidean space R 8 has density greater than that of the E 8 -lattice packing.We define the density of a packing P as the limit superior ∆ P := lim sup r→∞ ∆ P (r).The number be want to know is the supremum over all possible packing densities ∆ d := sup P⊂R d sphere packing ∆ P , 1 arXiv:1603.04246v2 [math.NT]

Building on Viazovska's recent solution of the sphere packing problem in eight dimensions, we prove that the Leech lattice is the densest packing of congruent spheres in twenty-four dimensions and that it is the unique optimal periodic packing. In particular, we find an optimal auxiliary function for the linear programming bounds, which is an analogue of Viazovska's function for the eight-dimensional case.Comment: 17 page

The precise momentum dependence of the superconducting gap in the iron-arsenide superconductor Ba 1−x K x Fe 2 As 2 ͑BKFA͒ with T c = 32 K was determined from angle-resolved photoemission spectroscopy ͑ARPES͒ via fitting the distribution of the quasiparticle density to a model. The model incorporates finite lifetime and experimental resolution effects, as well as accounts for peculiarities of BKFA electronic structure. We have found that the value of the superconducting gap is practically the same for the inner ⌫ barrel, X pocket, and "blade" pocket, and equals 9 meV, while the gap on the outer ⌫ barrel is estimated to be less than 4 meV, resulting in 2⌬ / k B T c = 6.8 for the large gap and 2⌬ / k B T c Ͻ 3 for the small gap. A large ͑77Ϯ 3 %͒ nonsuperconducting component in the photoemission signal is observed below T c . Details of gap extraction from ARPES data are discussed in Appendixes A and B.

Here we present a calculation of the temperature-dependent London penetration depth, λ(T ), in Ba1−xKxFe2As2 (BKFA) on the basis of the electronic band structure [1, 2] and momentumdependent superconducting gap [3] extracted from angle-resolved photoemission spectroscopy (ARPES) data. The results are compared to the direct measurements of λ(T ) by muon spin rotation (µSR) [4]. The value of λ(T = 0), calculated with no adjustable parameters, equals 270 nm, while the directly measured one is 320 nm; the temperature dependence λ(T ) is also easily reproduced. Such agreement between the two completely different approaches allows us to conclude that ARPES studies of BKFA are bulk-representative. Our review of the available experimental studies of the superconducting gap in the new iron-based superconductors in general allows us to state that all hole-doped of them bear two nearly isotropic gaps with coupling constants 2∆/kBTc = 2.5 ± 1.5 and 7 ± 2.

In this paper we prove the conjecture of Korevaar and Meyers: for each N ≥ c d t d there exists a spherical t-design in the sphere S d consisting of N points, where c d is a constant depending only on d.