On k-symmetries of hyperbolic spaces

We determine all the possible pointwise k-symmetric spaces of negative constant curvature. In general, such spaces are not k-symmetric.In fact we show that, for all $n?3$, $k\ne 2$, $H^n$ is not k-symmetric, i.e., for any set of selected k-symmetries, one for each point of $H^n$, the regularity condition does not hold.     

Cocycle deformations and Galois objects for semisimple Hopf algebras of dimension p3 and pq2

Let p and q be distinct prime numbers. We study the Galois objects and cocycle deformations of the noncommutative, noncocommutative, semisimple Hopf algebras of odd dimension $p^3$ and of dimension $p{q}^2$. We obtain that the $p+1$ non-isomorphic self-dual semisimple Hopf algebras of dimension $p^3$ classified by Masuoka have no non-trivial cocycle deformations, extending his previous results for the 8-dimensional Kac–Paljutkin Hopf algebra. This is done as a consequence of the classification of categorical Morita equivalence classes among semisimple Hopf algebras of odd dimension $p^3$, established by the third-named author in an appendix.     

Weak version of restriction estimates for spheres and paraboloids in finite fields

We study $L^p-L^r$ restriction estimates for algebraic varieties in d-dimensional vector spaces over finite fields. Unlike the Euclidean case, if the dimension d is even, then it is conjectured that the $L^{(2d+2)/(d+3)}-L^2$ Stein–Tomas restriction result can be improved to the $L^{(2d+4)/(d+4)}-L^2$ estimate for both spheres and paraboloids in finite fields. In this paper we show that the conjectured $L^p-L^2$ restriction estimate holds in the specific case when test functions under consideration are restricted to d-coordinate functions or homogeneous functions of degree zero. To deduce our result, we use the connection between the restriction phenomena for our varieties in d dimensions and those for homogeneous varieties in $(d+1)$ dimensions.     

Approximation properties of certain operator-induced norms on Hilbert spaces

We consider a class of operator-induced norms, acting as finite-dimensional surrogates to the $L^2$ norm, and study their approximation properties over Hilbert subspaces of $L^2$. The class includes, as a special case, the usual empirical norm encountered, for example, in the context of nonparametric regression in a reproducing kernel Hilbert space (RKHS). Our results have implications to the analysis of $M$-estimators in models based on finite-dimensional linear approximation of functions, and also to some related packing problems.     

Biharmonic ideal hypersurfaces in Euclidean spaces

Let $x:M\to {\mathbb{E}}^m$ be an isometric immersion from a Riemannian n-manifold into a Euclidean m-space. Denote by Δ and $\overrightarrow{x}$ the Laplace operator and the position vector of M, respectively. Then M is called biharmonic if ${\mathrm{\Delta }}^2\overrightarrow{x}=0$. The following Chen?s Biharmonic Conjecture made in 1991 is well-known and stays open: The only biharmonic submanifolds of Euclidean spaces are the minimal ones. In this paper we prove that the biharmonic conjecture is true for $\delta (2)$-ideal and $\delta (3)$-ideal hypersurfaces of a Euclidean space of arbitrary dimension.     

Norm of the Hilbert matrix on Bergman spaces

It is well known that the Hilbert matrix operator H is a bounded operator from the Bergman space $A^p$ into $A^p$ if and only if $2<p<\infty$. In [5] it was shown that the norm of the Hilbert matrix operator H on the Bergman space $A^p$ is equal to $\frac{\pi }{\sin ?\frac{2\pi }{p}}$, when $4\le p<\infty$, and it was also conjectured that

${6H6}_{A^p\to A^p}=\frac{\pi }{\sin ?\frac{2\pi }{p}},$

when $2<p<4$. In this paper we prove this conjecture.