We classify and analyze the orbits of the Kepler problem on surfaces of constant curvature (both positive and negative, ${{\mathbb{S}}^{2}}$ and ${{\mathbb{H}}^{2}}$, respectively) as functions of the angular momentum and the energy. Hill's regions are characterized, and the problem of time-collision is studied. We also regularize the problem in Cartesian and intrinsic coordinates, depending on the constant angular momentum, and we describe the orbits of the regularized vector field. The phase portraits both for ${{\mathbb{S}}^{2}}$ and ${{\mathbb{H}}^{2}}$ are pointed out.

Jean-Louis Loday introduced a class of symmetric operads generated by one bilinear operation subject to one relation making each left-normed product of three elements equal to a linear combination of right-normed products: $({{a}_{1}}{{a}_{2}}){{a}_{3}}\,=\,\sum{{{_{\sigma }}_{\in {{s}_{3}}}}\,}{{x}_{\sigma }}\,{{a}_{\sigma \,(1)}}({{a}_{\sigma (2)\,}}{{a}_{\sigma (3)}})$. Such an operad is called a parametrized one-relation operad. For a particular choice of parameters $\{{{x}_{\sigma }}\}$, this operad is said to be regular if each of its components is the regular representation of the symmetric group; equivalently, the corresponding free algebra on a vector space $V$ is, as a graded vector space, isomorphic to the tensor algebra of $V$. We classify, over an algebraically closed field of characteristic zero, all regular parametrized one-relation operads. In fact, we prove that each such operad is isomorphic to one of the following five operads: the left-nilpotent operad defined by the relation $(({{a}_{1}}{{a}_{2}}){{a}_{3}})\,=\,0$, the associative operad, the Leibniz operad, the dual Leibniz (Zinbiel) operad, and the Poisson operad. Our computational methods combine linear algebra over polynomial rings, representation theory of the symmetric group, and Gröbner bases for determinantal ideals and their radicals.

We provide a simple, general argument to obtain improvements of concentration-type inequalities starting from improvements of their corresponding isoperimetric-type inequalities. We apply this argument to obtain robust improvements of the Brunn-Minkowski inequality (for Minkowski sums between generic sets and convex sets) and of the Gaussian concentration inequality. The former inequality is then used to obtain a robust improvement of the Riesz rearrangement inequality under certain natural conditions. These conditions are compatible with the applications to a finite-range nonlocal isoperimetric problem arising in statistical mechanics.

Building on our previous work, we study the non-relative homology of quantum group convolution algebras. Our main result establishes the equivalence of amenability of a locally compact quantum group $\mathbb{G}$ and 1-injectivity of ${{L}^{\infty }}\left( \widehat{\mathbb{G}} \right)$ as an operator ${{L}^{1}}\left( \widehat{\mathbb{G}} \right)$-module. In particular, a locally compact group $G$ is amenable if and only if its group von Neumann algebra $\text{VN}\left( G \right)$ is 1-injective as an operator module over the Fourier algebra $A\left( G \right)$. As an application, we provide a decomposability result for completely bounded ${{L}^{1}}\left( \widehat{\mathbb{G}} \right)$-module maps on ${{L}^{\infty }}\left( \widehat{\mathbb{G}} \right)$, and give a simplified proof that amenable discrete quantum groups have co-amenable compact duals, which avoids the use of modular theory and the Powers-Størmer inequality, suggesting that our homological techniques may yield a new approach to the open problem of duality between amenability and co-amenability.

In this paper, we prove that on a compact, $n$-dimensional Alexandrov space with curvature at least −1, the Wasserstein barycenter of Borel probability measures ${{\mu }_{1}},\ldots ,{{\mu }_{m}}$ is absolutely continuous with respect to the $n$-dimensional Hausdorff measure if one of them is.

In this paper, we characterize the closures of convex hulls of unitary orbits of self-adjoint operators in unital, separable, simple ${{\text{C}}^{*}}$ -algebras with non-trivial tracial simplex, real rank zero, stable rank one, and strict comparison of projections with respect to tracial states. In addition, an upper bound for the number of unitary conjugates in a convex combination needed to approximate a self-adjoint are obtained.

For a lattice $L$ of ${{\mathbb{R}}^{n}}$, a sphere $S\left( c,r \right)$ of center $c$ and radius $r$ is called empty if for any $v\,\in \,L$ we have $\parallel v\,-\,c\parallel \,\,\ge \,r$. Then the set $S\left( c,r \right)\,\cap \,L$ is the vertex set of a $Delaunay\,polytope\,P\,=\,\text{conv}\left( S\left( c,r \right)\cap L \right)$. A Delaunay polytope is called perfect if any affine transformation $\phi $ such that $\phi \left( P \right)$ is a Delaunay polytope is necessarily an isometry of the space composed with an homothety.

Perfect Delaunay polytopes are remarkable structures that exist only if $n\,=\,1$ or $n\,\ge \,6$, and they have shown up recently in covering maxima studies. Here we give a general algorithm for their enumeration that relies on the Erdahl cone. We apply this algorithm in dimension seven, which allows us to find that there are only two perfect Delaunay polytopes: 321, which is a Delaunay polytope in the root lattice ${{\text{E}}_{7}}$, and the Erdahl Rybnikov polytope.

We then use this classification in order to get the list of all types of Delaunay simplices in dimension seven and found that there are eleven types.

Let $\text{P}\,\text{=}\,\text{M}\,\text{N}$ be a Levi decomposition of a maximal parabolic subgroup of a connected reductive group $\text{G}$ over a $p$-adic field $F$. Assume that there exists ${{w}_{0}}\,\in \,G\left( F \right)$ that normalizes $\text{M}$ and conjugates $\text{p}$ to an opposite parabolic subgroup. When $\text{N}$ has a Zariski dense $\text{Int}\,\text{M}$-orbit, $\text{F}$. Shahidi and $\text{X}$. Yu described a certain distribution $D$ on $\text{M}\left( F \right)$, such that, for irreducible unitary supercuspidal representations $\pi $ of $\text{M}\left( F \right)$ with $\pi \,\cong \,\pi \,\circ \,\text{Int}\,{{w}_{0}},\,\text{Ind}_{\text{P}\left( F \right)}^{\text{G}\left( F \right)}\,\pi $ is irreducible if and only if $D\left( f \right)\,\ne \,0$ for some pseudocoefficient $f$ of $\pi $. Since this irreducibility is conjecturally related to $\pi $ arising via transfer from certain twisted endoscopic groups of $\text{M}$, it is of interest to realize $D$ as endoscopic transfer from a simpler distribution on a twisted endoscopic group $\text{H}$ of $\text{M}$. This has been done in many situations where $\text{N}$ is abelian. Here we handle the standard examples in cases where $\text{N}$ is nonabelian but admit a Zariski dense $\text{Int}\,\text{M}$-orbit.