Partial Regularity of Strong Local Minimizers in the Multi-Dimensional Calculus of Variations

Let ⁿ be a bounded domain and F: a given strongly quasiconvex integrand of class C ² satisfying the growth condition for some c>0 and 2p<. Consider the multiple integral where uW ^1,p (, ^N ). The main result of the paper is the proof that any strong local minimizer of I[·] is of class C ^1, _loc for any (0,1) on an open set of full n-dimensional measure. In the case of weak local minimizers we establish the same result under the extra assumption that the oscillations in the gradient of the minimizer are not too large. Without such an assumption weak local minimizers need not be partially regular as we show by a class of examples. We also briefly discuss the question of existence of strong local minimizers for I[·] and connections of our results to extensions of Weierstrass sufficiency theorem to the multi-dimensional setting.     

Relaxed fabrication tolerance for self-imaging photonic crystal waveguide splitters using a tapered multimode interference region

The power imbalance between different waveguide outputs is compensated by manipulating the dispersion of the guided propagation in the multimode interference (MMI) region. This is attainable using a tapered region at the beginning of the MMI region that has been verified through simulation and experiment. From this, the fabrication tolerance for the diameters of holes in a tapered 1×3 photonic crystal waveguide (PhCW) splitter is relaxed up to a range of at least 27 nm. The output power is well-balanced to within 1 dB. The effective bandwidth of the splitters shifts only around 13 nm, for a reduction of 10 nm in the diameter of the PhCW holes. The optimized component is an outstanding ultracompact 1×3 splitter for the photonic integrated circuit (PIC).     

Diophantine exponents for mildly restricted approximation

We are studying the Diophantine exponent μ _n,l defined for integers 1≤l<n and a vector α∈ℝ ⁿ by letting

Experimental and ab initio DFT calculated Raman spectrum of Sudan I,a red dye

The red dye Sudan I was investigated by Raman spectroscopy using different excitation wavelengths (1064, 532 and 244 nm). A calculation of the Raman spectrum based on quantum mechanical ab initio density functional theory (DFT) was made using the RB3LYP method with the 3‐21G and 6‐311 + G(d,p) basis sets. The vibrations in the region 1600–1000 cm⁻¹ were found to comprise various mixed modes including in‐plane stretching and bending of various C C, N N, C N and C O bonds and angles in the molecule. Below ∼900 cm⁻¹, the out‐of‐plane bending modes were dominant. The central hydrazo chromophore of the Sudan I molecule was involved in the majority of the vibrations through NN and C N stretching and various bending modes. Low‐intensity bands in the lower wavenumber range (at about 721, 616, 463 and 218 cm⁻¹) were selectively enhanced by the resonance Raman effect when using the 532 nm excitation line. Comparison was made with other azo dyes in the literature on natural, abundant plant pigments. The results show that there is a possibility in foodstuff analysis to distinguish Sudan I from other dyes by using Raman spectroscopy with more than one laser wavelength for resonance enhancement of the different bands Copyright © 2011 John Wiley & Sons, Ltd.     

Studies on peroxycompounds—XXVI: “SN2”-type decarbethoxylation of o-benzoyltartronates and benzoyloxycyanoacetates. New synthons for 2-hydroxycarboxylic esters and aldehydes

Diethyl O-benzoyl-R-tartronates 1; RH, C₂H₅, Ph undergo S_N2-type dealkylation and subsequent CO₂ elimination at 175–190°C in anhydrous or wet DMSO, with or without inorganic salts as catalysts giving the corresponding 2-benzoyloxyesters 2. Ethyl 2-benzoyloxy-2-benzylcyanoacetate 4 also undergoes decarbethoxylation (140°C, DMSO or wet DMSO) giving 6 in small yields (1–6%). The best yields of 6 (82%) were found when pyridine was the solvent and LiI the catalyst.     

Tempered distributions in infinitely many dimensions

In the present paper we continue investigating spaces of tempered distributions in infinitely many dimensions. In particular, we prove that those linear homogeneous transformations of the canonical pair of field operators, which preserve the commutation relations, can be implemented by an essentially unique intertwining operator. The dependence of this operator on the transformation is studied.