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A polyuble of a Manin triple can be regarded as the ``$n$-th power'' of it, which plays an important rule in the study of Poisson geometry, mathematical physics and Lie theory. In this paper, we first construct an isomorphism between the $mn$-ble and the $n$-ubles of $m$-uble by colored graph and point out it is unique. Then, we construct a class of Poisson homogeneous spaces and obtain a class of Poisson homeomorphisms between them based on the first main result. Last, we apply first two main results to multi-flag varieties as well as multi-double flag varieties and construct a class of global Poisson isomorphisms between them as well as their $T$-leaves.

Let $G$ be a connected complex semi-simple Lie group and ${\mathcal {B}}$ its flag variety. For every positive integer $n$, we introduce a Poisson groupoid over ${{\mathcal {B}}}^n$, called the $n$th total configuration Poisson groupoid of flags of $G$, which contains a family of Poisson sub-groupoids whose total spaces are generalized double Bruhat cells and bases generalized Schubert cells in ${\mathcal {B}}^n$. Certain symplectic leaves of these Poisson sub-groupoids are then shown to be symplectic groupoids over generalized Schubert cells. We also give explicit descriptions of symplectic leaves in three series of Poisson varieties associated to $G$.

Poisson spaces provide the natural setting for Hamiltonian mechanics and compactifications of symplectic manifolds. In this paper, we give a brief survey on a class of Poisson manifolds related to Lie theory which have their origins in quantum groups.

Let $G$ be a connected and simply connected complex semisimple Lie group. For a collection of homogeneous $G$-spaces $G/Q$, we construct a finite atlas ${\mathcal{A}}_{\rm BS}(G/Q)$ on $G/Q$, called the Bott-Samelson atlas, and we prove that all of its coordinate functions are positive with respect to the Lusztig positive structure on $G/Q$. We also show that the standard Poisson structure $\pi_{G/Q}$ on $G/Q$ is presented, in each of the coordinate charts of ${\mathcal{A}}_{\rm BS}(G/Q)$, as a symmetric Poisson CGL extension (or a certain localization thereof) in the sense of Goodearl-Yakimov, making $(G/Q, \pi_{G/Q}, {\mathcal{A}}_{\rm BS}(G/Q))$ into a Poisson-Ore variety. Examples of $G/Q$ include $G$ itself, $G/T$, $G/B$, and $G/N$, where $T \subset G$ is a maximal torus, $B \subset G$ a Borel subgroup, and $N$ the uniradical of $B$.

Classification standards such as the Mammal Species of the World (MSW) aim to unify name usages at the global scale, but may nevertheless experience significant levels of taxonomic change from one edition to the next. This circumstance challenges the biodiversity and phylogenetic data communities to develop more granular identifiers to track taxonomic congruence and incongruence in ways that both humans and machines can process, i.e., to logically represent taxonomic provenance across multiple classification hierarchies. Here we show that reasoning over taxonomic provenance is feasible for two classifications of primates corresponding to the second and third MSW editions. Our approach entails three main components: (1) individuation of name usages as taxonomic concepts, (2) articulation of concepts via human-asserted Region Connection Calculus (RCC-5) relationships, and (3) the use of an Answer Set Programming toolkit to infer and visualize logically consistent alignments of these taxonomic input constraints. Our use case entails the Primates sec. Groves (1993; MSW2 - 317 taxonomic concepts; 233 at the species level) and Primates sec. Groves (2005; MSW3 - 483 taxonomic concepts; 376 at the species level). Using 402 concept-to-concept input articulations, the reasoning process yields a single, consistent alignment, and infers 153,111 Maximally Informative Relations that constitute a comprehensive provenance resolution map for every concept pair in the Primates sec. MSW2/MSW3. The entire alignment and various partitions facilitate quantitative analyses of name/meaning dissociation, revealing that approximately one in three paired name usages across treatments is not reliable - in the sense of the same name identifying congruent taxonomic meanings. We conclude with an optimistic outlook for logic-based provenance tools in next-generation biodiversity and phylogeny data platforms., Partial manuscript; figures down-sampled

We introduce Euler/X, a toolkit for logic-based taxonomy integration. Given two taxonomies and a set of alignment constraints between them, Euler/X provides tools for detecting, explaining, and reconciling inconsistencies; finding all possible merges between (consistent) taxonomies; and visualizing merge results. Euler/X employs a number of different underlying reasoning systems, including first-order reasoners (Prover9 and Mace4), answer set programming (DLV and Potassco), and RCC reasoners (PyRCC8). We demonstrate the features of Euler/X and provide experimental results showing its feasibility on various synthetic and real-world examples., 8 pages, 14 figures, WFLP 2013

Publisher Summary This chapter introduces a laser-heated pedestal growth technology for single-crystal fiber fabrication. A phase conjugate fiber holographic construction architecture is also presented that has the advantage of minimizing the intramodal noise excitation. Angular and wavelength selectivities for photorefractive (PR) fiber hologram storage using coupled wave theory are analyzed. Because the reflection-type wavelength-multiplexed holograms offer a higher and uniform selectivity over a wider range of the construction angles, the focus is on this type of fiber holograms. Cross-talk noise between the wavelength-multiplexed channel is evaluated, and it is found that narrowing the laser linewidth lowers the cross-talk between the holographic channels. Successive recordings partially erase the previous recorded holographic gratings; therefore, the recording-erasure dynamics of the fiber hologram is evaluated. The storage dynamics of the PR fiber hologram is also demonstrated experimentally.