A note on harmonic forms on complete manifolds

In this note, we will prove that under certain conditions, the space of polynomial growth harmonic functions and harmonic forms with a fixed growth rate on manifolds which are asymptotically nonnegatively curved is finite dimensional. This is a partial generalization of the works of Li and Colding-Minicozzi. We will also give an explicit estimate for the dimension in case the manifold is a complete surface of finite total curvature. This is a generalization to harmonic forms of the work of Li and the author.

     

Quaternionic geometry of matroids

Building on a recent paper [8], here we argue that the combinatorics of matroids are intimately related to the geometry and topology of toric hyperkähler varieties. We show that just like toric varieties occupy a central role in Stanley’s proof for the necessity of McMullen’s conjecture (or g-inequalities) about the classification of face vectors of simplicial polytopes, the topology of toric hyperkähler varieties leads to new restrictions on face vectors of matroid complexes. Namely in this paper we will give two proofs that the injectivity part of the Hard Lefschetz theorem survives for toric hyperkähler varieties. We explain how this implies the g-inequalities for rationally representable matroids. We show how the geometrical intuition in the first proof, coupled with results of Chari [3], leads to a proof of the g-inequalities for general matroid complexes, which is a recent result of Swartz [20]. The geometrical idea in the second proof will show that a pure O-sequence should satisfy the g-inequalities, thus showing that our result is in fact a consequence of a long-standing conjecture of Stanley.     

All-optical add-drop node for optical packet-switched networks

We have demonstrated all-optical packet add-drop for all-optical packet-switched networks. Intelligent all-optical add-drop of packets is performed, based on all-optical processing of packet headers. The header and payload rates are 5 and 10 Gbits/s, respectively.     

Synthesis of novel 6,11-O-bridged bicyclic ketolides via a palladium-catalyzed bis-allylation

A bridging chemistry process was developed to form an ether bridge between 6-O and 11-O of erythromycin A via a tandem or stepwise palladium-catalyzed bis-pi-allylation. By applying this bridging process, new 6,11-O-bridged bicyclic ketolides (BBKs) were synthesized. These BBKs showed good antibacterial activities against the macrolide-susceptible strains as well as mef-resistant strains and served as a good core for further modifications to study the structure-activity relationship (SAR) and to overcome bacterial resistance. [reaction: see text]     

On Perturbations of Eventually Compact Semigroups Preserving Eventual Compactness

We show, within the context of Miyadera–Voigt perturbations on L^p spaces, that for partial differential equations with delay arising from an immediately compact undelayed equation the delay semigroup is eventually compact. We also indicate how this result can be applied to the spectral analysis and numerical treatment of these equations.     

Dioscorealides and dioscoreanone,novel cytotoxic naphthofuranoxepins,and 1,4-phenanthraquinone from Dioscorea membranacea Pierre

[structure: see text] Commonly used among ingredients in Thai traditional anticancer preparations, the rhizome of Dioscorea membranacea Pierre was found potently cytotoxic and possibly contributed to such a therapeutic effect. Bioassay-guided isolation resulted in two novel cytotoxic naphthofuranoxepins, dioscorealides A (1) and B (2), and a new 1,4-phenanthraquinone, dioscoreanone (3). The structure determination, achieved mainly by means of NMR and CD spectral and X-ray crystallographic analyses, and cytotoxicity are discussed here.     

Fabrication of strong long-period gratings in hydrogen-free fibers with 157-nm F2-laser radiation

Long-period gratings were fabricated in standard telecommunication fiber (Corning SMF-28) by use of what is believed to be record short-wavelength light from a 157-nm F(2) laser. Strong loss peaks were formed without the need for enhancement techniques such as hydrogen loading. The magnitude of the attenuation peak was sensitive to the single-pulse laser fluence, decreasing with increasing pulse fluence as a result of nonuniform 157-nm laser interaction with both the fiber cladding and core. The long-period fiber gratings have good wavelength stability (Dlambda~7 nm) under thermal annealing at 150 degrees C.     

The “Full Müntz Theorem” inL p[0, 1] for 0<p<∞

Denote by span {f ₁,f ₂, …} the collection of all finite linear combinations of the functionsf ₁,f ₂, … over ?. The principal result of the paper is the following. Theorem (Full Müntz Theorem in L_p(A) for p ∈ (0, ∞) and for compact sets A ? [0, 1] with positive lower density at 0). Let A ? [0, 1] be a compact set with positive lower density at 0. Let p ∈ (0, ∞). Suppose (λ _j ) _j=1 ^∞ is a sequence of distinct real numbers greater than ?(1/p). Then span {x ^λ1,x ^λ2,…} is dense in L_p(A) if and only if $\sum\limits_{j = 1}^\infty {\frac{{\lambda _j + \left( {1/p} \right)}}{{\left( {\lambda _j + \left( {1/p} \right)} \right)^2 + 1}} = \infty } $ . Moreover, if $\sum\limits_{j = 1}^\infty {\frac{{\lambda _j + \left( {1/p} \right)}}{{\left( {\lambda _j + \left( {1/p} \right)} \right)^2 + 1}} = \infty } $ , then every function from the L_p(A) closure of {x ^λ1,x ^λ2,…} can be represented as an analytic function on {z ∈ ? \ (?∞,0] : |z| < r_A} restricted to A ∩ (0, r_A) where $r_A : = \sup \left\{ {y \in \mathbb{R}:\backslash ( - \infty ,0]:\left| z \right|< r_A } \right\}$ (m(·) denotes the one-dimensional Lebesgue measure). This improves and extends earlier results of Müntz, Szász, Clarkson, Erdös, P. Borwein, Erdélyi, and Operstein. Related issues about the denseness of {x ^λ1,x ^λ2,…} are also considered.