Symplectic geometry of semisimple orbits

Let G be a complex semisimple group, T G a maximal torus and B a Borel subgroup of G containing T. Let Ω be the Kostant-Kirillov holomorphic symplectic structure on the adjoint orbit O = Ad(G)c G/Z(c), where c Lie(T), and Z(c) is the centralizer of c in G. We prove that the real symplectic form Re Ω (respectively, Im Ω) on O is exact if and only if all the eigenvalues ad(c) are real (respectively, purely imaginary). Furthermore, each of these real symplectic manifolds is symplectomorphic to the cotangent bundle of the partial flag manifold G/Z(cc)B, equipped with the Liouville symplectic form. The latter result is generalized to hyperbolic adjoint orbits in a real semisimple Lie algebra.     

Vanishing of Chern Classes of the De Rham Bundles for Some Families of Moduli Spaces

Given a family of nonsingular complex projective surfaces, there is a corresponding family of Hilbert schemes of zero dimensional subschemes. We prove that the Chern classes, with values in the rational Chow groups, of the de Rham bundles for such a family of Hilbert schemes vanish. A similar result is proved for any relative moduli space of rank one sheaves with trivial integral first Chern class over any family of complex projective surfaces.     

Design of hybrid regrouping PSO–GA based sub-optimal networked control system with random packet losses

In this paper, a new approach has been presented to design sub-optimal state feedback regulators over networked control systems with random packet losses. The optimal regulator gains, producing guaranteed stability are designed with the nominal discrete time model of a plant using Lyapunov technique which produces a few set of bilinear matrix inequalities (BMIs). In order to reduce the computational complexity of the BMIs, a genetic algorithm (GA) based approach coupled with the standard interior point methods for LMIs has been adopted. A regrouping particle swarm optimization based method is then employed to optimally choose the weighting matrices for the state feedback regulator design that gets passed through the GA based stability checking criteria i.e. the BMIs. This hybrid optimization methodology put forward in this paper not only reduces the computational difficulty of the feasibility checking condition for optimum stabilizing gain selection but also minimizes other time domain performance criteria like expected value of the set-point tracking error with optimum weight selection based LQR design for the nominal system.     

The topology of parabolic character varieties of free groups

Let $G$ be a complex affine algebraic reductive group, and let $K\,\subset \, G$ be a maximal compact subgroup. Fix h $\,:=\,(h_{1}\,,\ldots \,,h_{m})\,\in \, K^{m}$ . For $n\, \ge \, 0$ , let $\mathsf X _{\mathbf{{h}},n}^{G}$ (respectively, $\mathsf X _{\mathbf{{h}},n}^{K}$ ) be the space of equivalence classes of representations of the free group on $m+n$ generators in $G$ (respectively, $K$ ) such that for each $1\le i\le m$ , the image of the $i$ -th free generator is conjugate to $h_{i}$ . These spaces are parabolic analogues of character varieties of free groups. We prove that $\mathsf X _{\mathbf{{h}},n}^{K}$ is a strong deformation retraction of $\mathsf X _{\mathbf{{h}},n}^{G}$ . In particular, $\mathsf X _{\mathbf{{h}},n}^{G}$ and $\mathsf X _{\mathbf{{h}},n}^{K}$ are homotopy equivalent. We also describe explicit examples relating $\mathsf X _{\mathbf{{h}},n}^{G}$ to relative character varieties.     

Stable bundles and extension of structure group

Let EG be a polystable principal G-bundle over a compact connected Kähler manifold, where G is a complex reductive group, and a homomorphism to another complex reductive group. We give a sufficient condition under which the principal H-bundle obtained by extending the structure group of EG using ρ is stable.     

Symplectic Structures of Moduli Space¶of Higgs Bundles over a Curve and Hilbert Scheme¶of Points on the Canonical Bundle

The moduli space of triples of the form (E,θ,s) are considered, where (E,θ) is a Higgs bundle on a fixed Riemann surface X, and s is a nonzero holomorphic section of E. Such a moduli space admits a natural map to the moduli space of Higgs bundles simply by forgetting s. If (Y,L) is the spectral data for the Higgs bundle (E,θ), then s defines a section of the line bundle L over Y. The divisor of this section gives a point of a Hilbert scheme, parametrizing 0-dimensional subschemes of the total space of the canonical bundle K _X , since Y is a curve on K _X . The main result says that the pullback of the symplectic form on the moduli space of Higgs bundles to the moduli space of triples coincides with the pullback of the natural symplectic form on the Hilbert scheme using the map that sends any triple (E,θ,s) to the divisor of the corresponding section of the line bundle on the spectral curve. Received: 15 January 2000 / Accepted: 25 March 2001     

A Multifunctional Subphthalocyanine Nanosphere for Targeting,Labeling, and Killing of Antibiotic‐Resistant Bacteria

Developing a material that can combat antibiotic‐resistant bacteria, a major global health threat, is an urgent requirement. To tackle this challenge, we synthesized a multifunctional subphthalocyanine (SubPc) polymer nanosphere that has the ability to target, label, and photoinactivate antibiotic‐resistant bacteria in a single treatment with more than 99 % efficiency, even with a dose as low as 4.2 J cm^?2 and a loading concentration of 10 nM . The positively charged nanosphere shell composed of covalently linked SubPc units can increase the local concentration of photosensitizers at therapeutic sites. The nanosphere shows superior performance compared to corresponding monomers presumably because of their enhanced water dispersibility, higher efficiency of singlet‐oxygen generation, and phototoxicity. In addition, this material is useful in fluorescence labeling of living cells and shows promise in photoacoustic imaging of bacteria in vivo.     

Ordering poly(trimethylsilyl styrene‐block‐D,L‐lactide) block copolymers in thin films by solvent annealing using a mixture of domain‐selective solvents

Controlling the morphology, domain orientation, and domain size of block copolymer (BCP) thin films is desirable for many applications in nanotechnology. These properties can be tuned during solvent annealing by varying the solvent choice and degree of swelling which affect the effective miscibility and volume fraction of the BCP domains. In this work, we demonstrate with a bulk lamellae‐forming BCP, poly(4‐trimethylsilylstyrene‐block‐D ,L ‐lactide) (PTMSS‐b‐PLA), that varying the composition of a mixture of solvent vapors containing cyclohexane (PTMSS‐selective) and acetone (PLA‐selective), enables formation of perpendicularly oriented lamellae with sub‐20‐nm pitch lines. The BCP domain periodicity was also observed to increase by 30%, compared to bulk, following solvent annealing. Furthermore, solvent annealing alone is shown to induce a transition from a disordered to an ordered BCP. We rationalize our observations by hypothesizing that the use of a combination of domain selective solvent mixtures serves to increase the effective repulsion between the blocks of the copolymer. We furnish results from self‐consistent field theory calculations to support the proposed mechanism. © 2013 Wiley Periodicals, Inc. J. Polym. Sci. Part B: Polym. Phys. 2014 , 52, 36–45     

B(C6F5)3‐Catalyzed Transfer Hydrogenation of Imines and Related Heteroarenes Using Cyclohexa‐1,4‐dienes as a Dihydrogen Source

The strong boron Lewis acid tris(pentafluorophenyl)borane, B(C₆F₅)₃, is shown to abstract a hydride from suitably donor‐substituted cyclohexa‐1,4‐dienes, eventually releasing dihydrogen. This process is coupled with the FLP‐type (FLP=frustrated Lewis pair) hydrogenation of imines and nitrogen‐containing heteroarenes that are catalyzed by the same Lewis acid. The net reaction is a B(C₆F₅)₃‐catalyzed, i.e., transition‐metal‐free, transfer hydrogenation using easy‐to‐access cyclohexa‐1,4‐dienes as reducing agents. Competing reaction pathways with or without the involvement of free dihydrogen are discussed.