The Stiefel and Grassmann manifolds have been studied extensively and play a prominent role in many applications. Their geodesic structure is well understood; in particular, minimal geodesics and the cut locus in the Grassmann manifold admit classical descriptions. The oriented Grassmann manifold is less frequently treated explicitly. In this note we review the relevant Stiefel and Grassmann results for Hilbert spaces, including the infinite-dimensional case, give direct proofs adapted to our notation, and derive an explicit minimal-geodesic criterion and a cut-locus characterization for the oriented Grassmann manifold.
We show that {\mathbf{B}}_{\mathrm{dR}}^{+} is the universal thickening of {\mathbf{C}}_{p} . More generally, we show that, if S is a reduced affinoid algebra, {\mathcal{O}}{\mathbb{B}}_{\mathrm{dR}}^{+}({\bar S}) is the universal S -thickening of the completion of {\bar S} . Nous montrons que {\mathbf{B}}_{\mathrm{dR}}^{+} est l’épaississement universel de {\mathbf{C}}_{p} . Plus généralement, nous montrons que, si S est une algèbre affinoïde réduite, {\mathcal{O}}{\mathbb{B}}_{\mathrm{dR}}^{+}({\bar S}) est le S -épaississement universel du complété de {\bar S} .
In this paper we investigate the optimal problem for the L_{p} mixed chord integral. The existence of the L_{p} chord integral–Petty bodies is established, and the L_{p} geominimal chord integral is proposed and its properties, such as invariance under orthogonal matrices, homogeneity, isoperimetric-type inequalities and cyclic-type inequalities, which are provided as well.
Let A(n) denote the largest absolute value of the coefficients of the n-th cyclotomic polynomial Phi(n)(x). In this paper, for odd primes 7 < q < r with r equivalent to +/- 3 (mod 7q), we show that A(7qr) = 1 if q equivalent to 1 (mod 21), 2 if q equivalent to 2, 10, 11, 19, 20 (mod 21), 3 if q equivalent to 16 (mod 21), 2 or 3 if q equivalent to 4, 5, 8, 13,17 (mod 21).
In the theory of shape optimization, the rearrangements of sets are a key concept, because they allow us to keep some properties of the original set while improving other aspects. This paper is devoted to the proof of an isoperimetric property of the double spherical cap rearrangement of planar sets. In particular, we prove that, under the assumption of disconnection of non-trivial spherical slices, the rearranged set has a lower perimeter than the original one. In the general case, the symmetrized set does not decrease the perimeter, but we show that the "excess" is bounded above by 2H(1)(F/, where P denotes the set of radii such that the spherical slice is a non-trivial arc of a circle. Additionally, the higher-dimensional case is briefly discussed; in particular, an explicit counterexample is given, thus explaining why an analogous result cannot hold. The main reason for this is that, in dimension N = 3 or higher, the union of two spherical caps of equal size does not minimize the (N-2/-dimensional measure of the boundary.
