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.

We use weakly holomorphic modular forms for the Hecke theta group to construct an explicit interpolation formula for Schwartz functions on the real line. The formula expresses the value of a function at any given point in terms of the values of the function and its Fourier transform on the set {0, ± √

We prove that the E8 root lattice and the Leech lattice are universally optimal among point configurations in Euclidean spaces of dimensions 8 and 24, respectively. In other words, they minimize energy for every potential function that is a completely monotonic function of squared distance (for example, inverse power laws or Gaussians), which is a strong form of robustness not previously known for any configuration in more than one dimension. This theorem implies their recently shown optimality as sphere packings, and broadly generalizes it to allow for long-range interactions.The proof uses sharp linear programming bounds for energy. To construct the optimal auxiliary functions used to attain these bounds, we prove a new interpolation theorem, which is of independent interest. It reconstructs a radial Schwartz function f from the values and radial derivatives of f and its Fourier transform f at the radii √ 2n for integers n ≥ 1 in R 8 and n ≥ 2 in R 24 . To prove this theorem, we construct an interpolation basis using integral transforms of quasimodular forms, generalizing Viazovska's work on sphere packing and placing it in the context of a more conceptual theory.

For each N ≥ C d t d we prove the existence of a well-separated spherical t-design in the sphere S d consisting of N points, where C d is a constant depending only on d.