Eigenvalues of kinematical conservation laws (KCL) based 3-D weakly nonlinear ray theory (WNLRT)

System of kinematical conservation laws (KCL) govern evolution of a curve in a plane or a surface in space, even if the curve or the surface has singularities on it. In our recent publication [K.R. Arun, P. Prasad, 3-D kinematical conservation laws (KCL): evolution of a surface in R³-in particular propagation of a nonlinear wavefront, Wave Motion 46 (2009) 293-311] we have developed a mathematical theory to study the successive positions and geometry of a 3-D weakly nonlinear wavefront by adding an energy transport equation to KCL. The 7 × 7 system of equations of this KCL based 3-D weakly nonlinear ray theory (WNLRT) is quite complex and explicit expressions for its two nonzero eigenvalues could not be obtained before. In this short note, we use two different methods: (i) the equivalence of KCL and ray equations and (ii) the transformation of surface coordinates, to derive the same exact expressions for these eigenvalues. The explicit expressions for nonzero eigenvalues are important also for checking stability of any numerical scheme to solve 3-D WNLRT.     

The Orthogonal (e,e,e)‐Tris‐Adduct of 9,10‐Dimethylanthracene with C60‐Fullerene: A Hidden Cornerstone of Fullerene Chemistry. Preliminary Communication

Tris(9′,10′‐dimethyl[9,10]ethanoanthracene[11′,12′: 1,9;11″,12″: 16,17;11′′′,12′′′: 30,31])[5,6]fullerene C₆₀, the orthogonal (e,e,e)‐tris‐adduct of C₆₀ and 9,10‐dimethylanthracene, was obtained from [4+2]‐cycloaddition (Diels–Alder reaction) at room temperature. The thermally unstable orange red (e,e,e)‐tris‐adduct was purified by chromatography and was isolated in the form of red monoclinic crystals. Its C₃‐symmetric addition pattern was established spectroscopically. Its structure could be further investigated by single crystal X‐ray diffraction. The (e,e,e)‐tris‐adduct of C₆₀ and 9,10‐dimethylanthracene has earlier been suggested as intermediate and reversibly formed critical component in ‘template directed’ addition reactions of C₆₀. This previously elusive compound has now been isolated and structurally characterized.     

Computing ray class groups, conductors and discriminants

We use the algorithmic computation of exact sequences of Abelian groups to compute the complete structure of for an ideal of a number field , as well as ray class groups of number fields, and conductors and discriminants of the corresponding Abelian extensions. As an application we give several number fields with discriminants less than previously known ones.

     

Tellurium(IV) Fluorides and Azides containing the Nitrogen Donor Substituent R = 2‐Me2NCH2C6H4; Crystal Structure of RTeF3 and of an Unusual Tellurium(VI) Fluoride Salt

The tellurenyl fluoride, 2‐Me₂NCH₂C₆H₄TeF, was obtained from reaction of the tellurenyl iodide RTeI with AgF. The compound was unambiguously identified by ¹⁹F and ¹²⁵Te NMR spectroscopy. The decomposition under disproportionation leads to the tellurium(IV) trifluoride, 2‐Me₂NCH₂C₆H₄TeF₃ and the ditelluride RTeTeR. The fluorination of the ditelluride, (2‐Me₂NCH₂C₆H₄Te)₂, with XeF₂ results in pure RTeF₃. The molecular structure of 2‐Me₂NCH₂C₆H₄TeF₃, the second structural characterized tellurium(IV) trifluoride, has been determined. Furthermore the syntheses of the new tellurium(IV) difluoride, (2‐Me₂NCH₂C₆H₄)₂TeF₂, and corresponding tellurium(IV) diazide, (2‐Me₂NCH₂C₆H₄)₂Te(N₃)₂ as well as the tellurium(IV) triazide, 2‐Me₂NCH₂C₆H₄Te(N₃)₃, and their characterization by spectroscopic methods were reported. During these investigations a rather interesting tellurium(VI) species was formed and the molecular structure of a subsequent product, [(2‐Me₂NHCH₂C₆H₄)₂TeF₃O]₂(SiF₆), was elucidated. Theoretical investigations for the compounds containing the stabilizing 2‐dimethylaminomethylphenyl substituent are illustrated.     

Reactions of 2‐Unsubstituted 1H‐Imidazole 3‐Oxides with 2,2‐Bis(trifluoromethyl)ethene‐1,1‐dicarbonitrile: A Stepwise 1,3‐Dipolar Cycloaddition

The reaction of 1,4,5‐trisubstituted 1H‐imidazole‐3‐oxides 1 with 2,2‐bis(trifluoromethyl)ethene‐1,1‐dicarbonitrile ( 7 , BTF) yielded the corresponding 1,3‐dihydro‐2H‐imidazol‐2‐ones 10 and 2‐(1,3‐dihydro‐2H‐imidazol‐2‐ylidene)malononitriles 11 , respectively, depending on the solvent used. In one example, a 1 : 1 complex, 12 , of the 1H‐imidazole 3‐oxide and hexafluoroacetone hydrate was isolated as a second product. The formation of the products is explained by a stepwise 1,3‐dipolar cycloaddition and subsequent fragmentation. The structures of 11d and 12 were established by X‐ray crystallography.     

Diastereoselective 1,3‐Dipolar Cycloadditions of Chiral Derivatives of 2‐Oxoethanenitrile Oxide to Noncyclic Conjugated Symmetrical Alkenes

Asymmetric 1,3‐dipolar cycloadditions of chiral derivatives of the nitrile oxides 3a – 3c derived from (2R)‐bornane‐10,2‐sultam, (2R)‐10‐(dicyclohexylsulfamoyl)isoborneol, and (1R)‐8‐phenylmenthol, to either (E)‐stilbene 4 or dimethyl fumarate 5 , leading to the corresponding 4,5‐dihydroisoxazoles 6a – 6c and 7a – 7c in both moderate yields and diastereoselectivities, are presented. All cycloadducts were converted into the corresponding methyl esters 8 and 9 , which were used for determination of their enantiomeric purities via chiral HPLC analyses. In the case of both stilbene cycloadducts 6a and 6b , their absolute configurations were determined by X‐ray crystal‐structure analyses. These [3+2] cycloadditions suggest the participation of the thermodynamically less stable SO₂/CO syn‐conformer in the π_y approach along the C?O bond of the linear nitrile oxide 3a .     

Cytotoxic Withaphysalins from Physalis minima

A novel withanolide, withaphysalin P ( 1 ) with a nine‐membered ring, and six other new withaphysalins, 2 – 7 , together with the three known withaphysalins 8 – 10 were isolated from Physalis minima L. The structures were deduced by means of spectroscopic analyses, and that of withaphysalin P ( 1 ) was confirmed by a single‐crystal X‐ray diffraction analysis. Plausible biosynthetic pathways were postulated (Scheme 1). All compounds were tested for their antiproliferative activities toward the human colorectal‐carcinoma HCT‐116 cells and human nonsmall‐cell lung‐cancer NCI‐H460 cells (Table 4). Compounds 1 – 3, 7 – 10, 7a , and 7b displayed moderate cytotoxic activity against the two human cancer cell lines.     

Modeling for optical ray tracing and error analysis

One of the most popular mathematical tools in fields of robotics, mechanisms, and computer graphics is the 4×4 homogeneous transformation matrix. We extend the use of that matrix to the optical domain of:

(1) skew ray tracing to determine the paths of skew rays; and

(2) error analysis to investigate the various deviations of imagines due to imperfect placement of optical elements.

In order to trace a skew ray, the reflection and refraction laws of optics are formulated in the language of homogeneous transformation matrices. Then an error matrix to describe the position errors and orientation errors of optical elements is introduced in order to analyze their effects on rays' path. This ray tracing procedure can result in very powerful and fast optical design programs. The error analysis can provide the sensitivity of each error component of elements to a system's accuracy and is crucial to upgrade the precision of optical systems in design stage.     

Platinum(II) and Palladium(II) Halides and Pseudohalides with the Ligand Tri(1‐cyclohepta‐2, 4, 6‐trienyl)phosphane. X‐Ray Structural Analysis of [P(C7H7)2(η2‐C7H7)]PtI2

Tri(1‐cyclohepta‐2, 4, 6‐trienyl)phosphane, P(C₇H₇)₃ ( 1 ) ([P] when coordinated to a metal) stabilizes platinum(II) ( 2 ) and palladium(II) dihalides ( 3 ) as [P]MX₂ with X = Cl ( a ), Br ( b ) and I ( c ). The phosphane coordinates to the metal as a chelate ligand via both phosphorus and the central η²‐C=C bond of one of the cyclohepta‐2, 4, 6‐trienyl rings. The complexes were prepared by various routes, mainly by the reaction of (cod)MCl₂ (cod = cycloocta‐1, 5‐diene) with 1 to give the chlorides 2a and 3a , which then could be converted into the bromides 2b , 3b or the iodides 2c , 3c by reaction with NaBr or NaI, respectively. The molecular structure of 2c was determined by X‐ray analysis. Treatment of 2a and 3a with sodium or potassium salts of several pseudohalides afforded the complexes [P]MX₂ 2d (NCO/NCO), 2e₁ (NCS/SCN), 2e₁' (SCN/NCS), 2f₂ (SeCN/SeCN), 3f₁ (NCSe/SeCN), 2g and 3g (X = N₃). Attempts failed to synthesize the cyanides 2h and 3h by the same route. By using an excess of trimethylsilyl cyanide in the reaction with 2a in THF solution, the complex trans‐{[(C₇H₇)₃P]₂Pt(CN)₂} ( 4h ) was obtained instead of 2h . The analogous complexes trans‐{[(C₇H₇)₃P]₂MX₂} with M = Pt ( 4 ) and Pd ( 5 ) for X = Cl ( a ), Br ( b ), I ( c ) could be prepared from the reaction of the corresponding tetrahalogenometallates and 1 (in the case of 5c from PdI₂ and 1 ). In contrast to 4h , the complexes 4a‐c and 5a‐c were found to be labile in solution with respect to partial loss of the phosphane 1 and rearrangement into 2a‐c and 3a‐c , respectively. All compounds were characterized by IR spectroscopy and by multinuclear magnetic resonance spectroscopy (¹H, ¹³C, ³¹P, ⁷⁷Se and ¹⁹⁵Pt NMR). The ligand [P] in 2 and 3 is fluxional with regard to coordination of the C₇H₇ rings to the metal.     

高效毛细管电泳紫外检测器光路的特殊性及其优化

紫外检测器用于高效毛细管电泳（ＨＰＣＥ）的主要难题在光路中,ＨＰＣＥ中熔融石英毛细管（内径通常在５０μｍ左右）直接放在光路中,一方面光程短,另一方面,光在透镜及毛细管界面上的折射与反射使光程常比毛细管内径短,甚至可能绕过毛细管内腔［１,２］．本文编制...