Knutson, Tao, and Woodward [KTW04] formulated a Littlewood–Richardson rule for the cohomology ring of Grassmannians in terms of puzzles. Vakil [Vak06] and Wheeler–Zinn-Justin [WZ16] have found additional triangular puzzle pieces that allow one to express structure constants for K-theory of Grassmannians. Here we introduce two other puzzle pieces of hexagonal shape, each of which gives a Littlew...

Pairwise particle interactions arise in diverse physical systems ranging from insect swarms to self-assembly of nanoparticles. In the presence of long-range attraction and short-range repulsion, such systems can exhibit bound states. We uses linear stability analysis of a ring equilibrium to classify the morphology of patterns in two dimensions. Conditions are identified that assure the wellpos...

We show that the motion of a single hole in the infinite-U Hubbard model with frustrated hopping leads to weak metallic antiferromagnetism of kinetic origin. An intimate relationship is demonstrated between the simplest versions of this problem in one and two dimensions, and two of the most subtle many body problems, namely, the Heisenberg Bethe ring in one dimension and the two-dimensional tri...

Gelidocalamus xunwuensis W.G.Zhang & G.Y.Yang, a new species collected from Xunwu County of Jiangxi Province in China, is described and illustrated. The new species is similar to G. stellatus in the habit, but differs by internodes sparsely hairy with granuliferous warts, culm sheath stiffly hairy, culm sheath blade broadly lanceolate to narrowly triangular, each node with a ring of appressed t...

A least-squares procedure to find the tilts, curvature, astigmatism, coma, and triangular astigmatism by means of measurements of the transverse aberrations using a Hartmann or Shack-Hartmann test is described. The sampling points are distributed in a ring centered on the pupil of the optical system. The properties and characteristics of rings with three, four, five, six, or more sampling point...

A representation for a group or a monoid is called a representation with Borel mold if it can be normalized to a representation in upper triangular matrices whose image of the group or monoid generates the algebra of upper triangular matrices. In [Na2] the moduli of representations with Borel mold has been constructed for each group or monoid. The moduli of representations with Borel mold has s...

The objective of this study is to investigate the fuzzy subhypernear-modules over hypernear-ring by using a triangular norm, which T-fuzzy are generalization submodules hyper near-modules, and some related properties discussed. Idempotent subhypernear-module hypernear-module suggested. relation between level subset presented. Also, image inverse under mapping introduced. direct product i...

In this paper, an atomistic based finite element model is developed to investigate the influence of topological defects on mechanical properties of graphene. The general in-plane stiffness matrix of the hexagonal network structure of graphene is found. Effective elastic modulus of a carbon ring is determined from the equivalence of molecular potential energy related to stretch and angular defor...

For each object in a tensor triangulated category, we construct a natural continuous map from the object’s support—a closed subset of the category’s triangular spectrum—to the Zariski spectrum of a certain commutative ring of endomorphisms. When applied to the unit object this recovers a construction of P. Balmer. These maps provide an iterative approach for understanding the spectrum of a tens...