Morse-Palais lemma for nonsmooth functionals on normed spaces

Using elementary differential calculus we get a version of the Morse-Palais lemma. Since we do not use powerful tools in functional analysis such as the implicit theorem or flows and deformations in Banach spaces, our result does not require the -smoothness of functions nor the completeness of spaces. Therefore it is stronger than the classical one but its proof is very simple.

     

Proper groupoids and momentum maps: Linearization, affinity, and convexity

We show that proper Lie groupoids are locally linearizable. As a consequence, the orbit space of a proper Lie groupoid is a smooth orbispace (a Hausdorff space which locally looks like the quotient of a vector space by a linear compact Lie group action). In the case of proper (quasi-)symplectic groupoids, the orbit space admits a natural integral affine structure, which makes it into an affine orbifold with locally convex polyhedral boundary, and the local structure near each boundary point is isomorphic to that of a Weyl chamber of a compact Lie group. We then apply these results to the study of momentum maps of Hamiltonian actions of proper (quasi-)symplectic groupoids, and show that these momentum maps preserve natural transverse affine structures with local convexity properties. Many convexity theorems in the literature can be recovered from this last statement and some elementary results about affine maps.     

Anisotropic wet etching on birefringent calcite crystal

In this paper, a mechanism for wet etching on birefringent calcite crystal is discussed. By altering the etchant type, its concentration and the stirring speed, the etching mechanism can be confined within the surface reaction regime, resulting in anisotropic etching. Acetic acid can be used to form V shapes defined by the {101̄1} planes with an intersecting angle of 51 degree. V-groove and blazed grating, with a grating period as small as 2 μm, were successfully fabricated. Such gratings can be used as transmission and reflection devices in the visible and infrared ranges. In addition, a transmission polarization dependent device can also be made by covering the calcite with index matching material. PACS 42.70.-a; 42.79.Dj     

Tetrahydrofuran Activation Assisted Synthesis of Nanosized Lithium Niobate and Lithium Tantalate

Black solid precursors obtained from reactions between MCl₅ (M = Nb, Ta) and alkyllithiums, n‐butyllithium (ⁿBuLi) and ethyllithium (EtLi), in tetrahydrofuran (THF) were heat treated under vacuum at 673–973 K to form nano‐sized particles (20–100 nm in diameter) of lithium niobate (LiNbO₃) and lithium tantalate (LiTaO₃). Stoichiometry of the reactants is critical and affects the phases of the products. Based on the volatile byproducts detected, a reaction pathway involving the activation of THF by alkyllithiums is proposed to be important for the formation of LiNbO₃ and LiTaO₃ precursors.     

Sydnones IV. Synthesis of Sydnones Analogous to the Photochromic Compounds

N-(6-methoxy-3-pyridyl)-sydnone (I), N-(4-nitrobenzyl)sydnone (II-1), and N-(2-nitrobenzyl)sydnone (II-2) have been prepared. Condensation of 2, 4-dinitrochlorohenzene with N-(carbethoxymethyl)sydnone (II-4) produced N-[(2, 4-dinitrophenyl) carbethoxymethyl]-sydnone (II-5). Under normal conditions none of them are photochromic.     

Bilayer lipid membranes formed from thermotropic liquid crystals

This paper reports a new type of bilayer lipid membrane (BLM) with aqueous interfaces which are formed from thermotropic liquid crystals (nematic-6CB, smectic-8CB, and cholesteric-cholesteryl palmitate, ChP). The electrical properties of these unmodified membranes have been investigated. We suggest that BLMs of this type in their 'black' states consist of two molecular layers with a smectic-like structure.     

Low-resolution infrared derivative spectrometry for source concentration monitoring

Derivative spectrometry uses periodic modulation and selected spectral information to evaluate species concentration and to reduce interference. A frequency component analysis for a low-resolution infrared monitoring system is presented, including narrow-band radiation expressions and a prediction of interference effects. An experimental optical pathlength sensitivity of about 0.001 atm-cm of NO is achieved, and first overtone component sensing is found to be superior to fundamental component sensing.