In this paper we study classification of homogeneous solutions to the stationary Euler equation with locally finite energy. Written in the form $\bu = \n^\perp \Psi$, $\Psi(r,\th) = r^{\l} \psi(\th)$, for $\l >0$, we show that only trivial solutions exist in the range $0<\l<1/2$, i.e. parallel shear and rotational flows. In other cases many new solutions are exhibited that have hyperbolic, parabolic and elliptic structure of streamlines. In particular, for $\l>9/2$ the number of different non-trivial elliptic solutions is equal to the cardinality of the set $(2,\sqrt{2\l}) \cap \N$. The case $\l = 2/3$ is relevant to Onsager's conjecture. We underline the reasons why no anomalous dissipation of energy occurs for such solutions despite their critical Besov regularity $1/3$.

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This paper studies the Voronoi diagrams on 2-manifold meshes based on geodesic metric (a.k.a. geodesic Voronoi diagrams or GVDs), which have polyline generators. We show that our general setting leads to situations more complicated than conventional 2D Euclidean Voronoi diagrams as well as point-source based GVDs, since a typical bisector contains line segments, hyperbolic segments and parabolic segments. To tackle this challenge, we introduce a new concept, called local Voronoi diagram (LVD), which is a combination of additively weighted Voronoi diagram and line-segment Voronoi diagram on a mesh triangle. We show that when restricting on a single mesh triangle, the GVD is a subset of the LVD and only two types of mesh triangles can contain GVD edges. Based on these results, we propose an efficient algorithm for constructing the GVD with polyline generators. Our algorithm runs in O(nNlogN) time and takes O(nN) space on an n-face mesh with m generators, where N = max{m,n}. Computational results on real-world models demonstrate the efficiency of our algorithm.

Kadison and Kastler introduced a natural metric on the collection of all$C$*-subalgebras of the bounded operators on a separable Hilbert space. They conjectured that sufficiently close algebras are unitarily conjugate. We establish this conjecture when one algebra is separable and nuclear. We also consider one-sided versions of these notions, and we obtain embeddings from certain near inclusions involving separable nuclear$C$*-algebras. At the end of the paper we demonstrate how our methods lead to improved characterisations of some of the types of algebras that are of current interest in the classification programme.

Starting with a commutative ring $R$ and an ideal $I$ , it is possible to define a family of rings $R(I)_{a,b}$ , with $a,b \in R$ , as quotients of the Rees algebra $\oplus_{n \geq0} I^{n}t^{n}$ ; among the rings appearing in this family we find Nagata’s idealization and amalgamated duplication. Many properties of these rings depend only on $R$ and $I$ and not on $a$ , $b$ ; in this paper we show that the Gorenstein and the almost Gorenstein properties are independent of $a$ , $b$ . More precisely, we characterize when the rings in the family are Gorenstein, complete intersection, or almost Gorenstein and we find a formula for the type.