David Hilbert’s (1862–1943) so-called Zahlbericht of 1897, which appears here for the first time in English, was the principal textbook on algebraic number theory for a period of at least thirty years after its appearance. Emil Artin, Helmut Hasse, Erich Hecke, Hermann Weyl and many others learned their number theory from this book. Even beyond this immediate impact Hilbert’s Zahlbericht has se...

Let A, B ⊆ R be finite, nonempty subsets, let s ≥ 2 be an integer, and let h1(A,B) denote the minimal number t such that there exist 2t (not necessarily distinct) parallel lines, l1, . . . , lt, l ′ 1, . . . , l ′ t , with A ⊆ ⋃t i=1 li and B ⊆ ⋃t i=1 l ′ i . Suppose h1(A,B) ≥ s. Then we show that: (a) if ||A| − |B|| ≤ s and |A|+ |B| ≥ 4s − 6s+ 3, then |A+B| ≥ (2− 1 s )(|A| + |B|)− 2s+ 1; (b) i...

We prove some concavity properties connected to nonlinear Bernoulli type free boundary problems. In particular, we prove a Brunn-Minkowski inequality and an Urysohn’s type inequality for the Bernoulli Constant and we study the behaviour of the free boundary with respect to the given boundary data. Moreover we prove a uniqueness result regarding the interior non-linear Bernoulli problem.

We introduce f -divergence, a concept from information theory and statistics, for convex bodies in R. We prove that f -divergences are SL(n) invariant valuations and we establish an affine isoperimetric inequality for these quantities. We show that generalized affine surface area and in particular the Lp affine surface area from the Lp Brunn Minkowski theory are special cases of f -divergences.