Crystals and total positivity on orientable surfaces

We develop a combinatorial model of networks on orientable surfaces, and study weight and homology generating functions of paths and cycles in these networks. Network transformations preserving these generating functions are investigated. We describe in terms of our model the crystal structure and R-matrix of the affine geometric crystal of products of symmetric and dual symmetric powers of type A. Local realizations of the R-matrix and crystal actions are used to construct a double affine geometric crystal on a torus, generalizing the commutation result of Kajiwara et al. (Lett Math Phys, 60(3):211–219, 2002) and an observation of Berenstein and Kazhdan (MSJ Mem, 17:1–9, 2007). We show that our model on a cylinder gives a decomposition and parametrization of the totally non-negative part of the rational unipotent loop group of GL _n .     

Random assignment problems

Analysis of random instances of optimization problems provides valuable insights into the behavior and properties of problem’s solutions, feasible region, and optimal values, especially in large-scale cases. A class of problems that have been studied extensively in the literature using the methods of probabilistic analysis is represented by the assignment problems, and many important problems in operations research and computer science can be formulated as assignment problems. This paper presents an overview of the recent results and developments in the area of probabilistic assignment problems, including the linear and multidimensional assignment problems, quadratic assignment problem, etc.     

Promotion and cyclic sieving via webs

We show that Schützenberger’s promotion on two and three row rectangular Young tableaux can be realized as cyclic rotation of certain planar graphs introduced by Kuperberg. Moreover, following work of the third author, we show that this action admits the cyclic sieving phenomenon.     

On the Formation of N3H3 Isomers in Irradiated Ammonia Bearing Ices: Triazene (H2NNNH) or Triimide (HNHNNH)

The remarkable versatility of triazenes in synthesis, polymer chemistry and pharmacology has led to numerous experimental and theoretical studies. Surprisingly, only very little is known about the most fundamental triazene: the parent molecule with the chemical formula N₃H₃. Here we observe molecular, isolated N₃H₃ in the gas phase after it sublimes from energetically processed ammonia and nitrogen films. Combining theoretical studies with our novel detection scheme of photoionization‐driven reflectron time‐of‐flight mass spectroscopy we can obtain information on the isomers of triazene formed in the films. Using isotopically labeled starting material, we can additionally gain insight in the formation pathways of the isomers of N₃H₃ under investigation and identify the isomers formed as triazene (H₂NNNH) and possibly triimide (HNHNNH).     

Pattern selection during electropolishing due to double-layer effects

We extend our earlier study of nanoscale pattern formation during electropolishing [Nanotechnology 7, 360 (1996); Phys. Rev. B 56, 12 608 (1997)]. The patterns are attributed to preferential adsorption of organic molecules on the convex portion of the electrode due to its enhanced electric field. This local enhancement occurs because of the effect of surface curvature on the double-layer potential drop. By allowing for transport correction to the double-layer potential drop at thermodynamic equilibrium, we estimate this anodic overpotential to be in the realistic mV range and hence verify the Debye-Huckel approximation used in our model. This small anodic overpotential suggests that pattern formation is a generic electropolishing phenomenon whose only requirement is that the polarizability of the organic additive relative to water must lie within a range specified by our theory. We verify this prediction experimentally with a variety of electrolyte solutions. The voltage ranges for specific hexagonal and ridge patterns are well correlated by our model with only a single parameter. (c) 1999 American Institute of Physics.     

Copper(II) bromide and copper(II) acetate complexes of 4,4′‐(p‐phenylene)bipyridazine

4,4′‐(p‐Phenylene)bipyridazine, C₁₄H₁₀N₄, (I), and the coordination compounds catena‐poly[[dibromidocopper(II)]‐μ‐4,4′‐(p‐phenylene)bipyridazine‐κ²N²:N^2′], [CuBr₂(C₁₄H₁₀N₄)]_n, (II), and catena‐poly[[[tetrakis(μ‐acetato‐κ²O:O′)dicopper(II)]‐μ‐4,4′‐(p‐phenylene)bipyridazine‐κ²N¹:N^1′] chloroform disolvate], {[Cu₂(C₂H₃O₂)₄(C₁₄H₁₀N₄)]·2CHCl₃}_n, (III), contain a new extended bitopic ligand. The combination of the p‐phenylene spacer and the electron‐deficient pyridazine rings precludes C—H...π interactions between the lengthy aromatic molecules, which could be suited for the synthesis of open‐framework coordination polymers. In (I), the molecules are situated across a center of inversion and display a set of very weak intermolecular C—H...N hydrogen bonds [3.399 (3) and 3.608 (2) Å]. In (II) and (III), the ligand molecules are situated across a center of inversion and act as N²,N^2′‐bidentate [in (II)] and N¹,N^1′‐bidentate [in (III)] long‐distance bridges between the metal ions, leading to the formation of coordination chains [Cu—N = 2.005 (3) Å in (II) and 2.199 (2) Å in (III)]. In (II), the copper ion lies on a center of inversion and adopts CuN₂Br₄ (4+2)‐coordination involving two long axial Cu—Br bonds [3.2421 (4) Å]. In (III), the copper ion has a tetragonal pyramidal CuO₄N environment. The uncoordinated pyridazine N atom and two acetate O atoms provide a multiple acceptor site for accommodation of a chloroform solvent molecule by trifurcated hydrogen bonding [C—H...O(N) = 3.298 (5)–3.541 (4) Å].     

Modelling of the Effect of the Sheared Poloidal Flow on the Electrostatic Turbulence on the CASTOR Tokamak

Electrostatic drift turbulence of the edge plasma in the CASTOR tokamak is studied numerically by using the Hasegawa-Wakatani equations. The fluctuations of plasma density and potential as well as the corresponding fluctuation-induced particle flux are calculated for regimes with various plasma poloidal flows. Results of numerical simulations are in a reasonable agreement with experimental results.     

Synthesis of 3-Borylated Pyrrolidines by 1,3-Dipolar Cycloaddition of Alkenyl Boronates and Azomethine Ylide

A scalable and efficient process for the preparation of 3-borylated pyrrolidines by 1,3-dipolar cycloaddition of N-benzyl azomethine ylide generated in situ has been developed. The optimized method included the use of LiF in DMSO at 110 °C and was suitable for α-mono-, α,β-di-, and α,β,β-trialkyl-, β,β-(hetera)cycloalkylidene-, CO₂Et-, as well as most β-(het)aryl-substituted alkenyl boropinacolates. The 1,3-dipolar reaction proceeded on a multigram scale providing 3-borylated pyrrolidines with diverse substitution patterns (including fused and spirocyclic ones) in a diastereoselective manner. The Pd(OH)₂-mediated N-debenzylation of pyrrolidine hydrochlorides was successfully performed to give the corresponding bifunctional building blocks on an up to 130 g scale, thus providing a substantial expansion of the synthetic and medicinal chemist's toolbox. Other reactions included the preparation of trifluoroborates, Zweifel-Aggarwal sp³-sp² coupling, and oxidative deborylation as an example of C-heteroatom bond formation.