Rényi entropies are compared to generalized log-Fisher information and variational entropies in the context of translation, scale and concentration invariance. It is proved that the Rényi entropies occupy a special place amongst these entropies. It is also shown that Shannon entropy is centrally positioned amidst the Rényi entropies.

In this paper we study the existence of maximizers for two families of interpolation inequalities, namely a generalized Gagliardo–Nirenberg inequality and a new inequality involving the Riesz energy. Two basic tools in our argument are a generalization of Lieb’s Translation Lemma and a Riesz energy version of the Brézis–Lieb lemma.

We review the construction known as the Nahm transform in a generalized context, which includes all the examples of this construction already described in the literature. The Nahm transform for translation invariant instantons on R is presented in an uniform manner. We also analyze two new examples, the first of which being the first example involving a four-manifold that is not hyperk”ahler.

We construct generalized polygons (‘parking garages’) in which the billiard flow satisfies the Veech dichotomy, although the associated translation surface obtained from the Zemlyakov-Katok unfolding is not a lattice surface. We also explain the difficulties in constructing a genuine polygon with these properties.

Unitary representations of the fundamental group of a Kähl-er manifold correspond to polystable vector bundles (with vanishing Chern classes). Semisimple linear representations correspond to poly-stable Higgs bundles. In this paper we find the objects corresponding to affine representations: the linear part gives a Higgs bundle and the translation part corresponds to an element of a generalized...

In this paper, using a generalized Jacobi-Dunkl translation operator, we prove a generalization of Titchmarsh’s theorem for functions in the k-JacobiDunkl-Lipschitz class defined by the finite differences of order k ∈ N∗ and Sobolev spaces associated with the Jacobi-Dunkl operator.

We analyze the emergence of a minimal length for a large class of generalized commutation relations, preserving commutation of the position operators and translation invariance as well as rotation invariance (in dimension higher than one). We show that the construction of the maximally localized states based on squeezed states generally fails. Rather, one must resort to a constrained variationa...

In this paper we compute transitive cardinal coefficients of ideals on generalized Cantor spaces. In particular, we observe that there exists a null set A ⊆ 21 such that for every null set B ⊆ 21 we can find x ∈ 21 such that the set A ∪ (A + x) cannot be covered by any translation of the set B.

The present paper proposes a novel method for 3D model content-based search based on the 3D Generalized Radon Transform. A set of descriptor vectors is extracted, which represent significant shape characteristics and completely invariant in terms of translation, scaling and rotation. Experimental results show that the proposed method can be used for 3D model search and retrieval in a highly eff...

Approximately five years ago Thas observed a remarkable coincidence relating certain generalized quadrangles constructed in [Ka3] to flocks of cones and translation planes [Th] (cf. [Ka2]). While there is a rapidly growing literature concerning this connection, there as yet has been no explanation for it. This note contains an observation providing at least some kind of explanation. It also con...