Collocation Methods for Weakly Singular Second-kind Volterra Integral Equations with Non-smooth Solution

Collocation type methods are studied for the numerical solutionof the weakly singular Volterra integral equation of the secondkind: where the solution (t) is assumedto have the form f(t) = x(t)+r^?(t), x and being sufficientlysmooth. The solution is approximated near zero by a linear combinationof powers of t?, and away from zero by the usual polynomialrepresentation. Convergence is proved and many numerical experimentsare carried out with examples from the literature. A comparisonis made with a method of Brunner & Norsett (1981), originallydeveloped for (1) with a smooth solution. Special attentionis paid to the numerical approximation of the so-called momentintegrals which emerge in the collocation scheme.     

Inverse problem for the equation of ultrahigh energy cosmic ray transport

The propagation of ultrahigh energy nuclei in an expanding universe filled with background electromagnetic radiation is considered. A numerical method for solving the inverse problem for the equation of cosmic ray transport is developed. The method allows us to determine a source spectrum from the cosmic ray spectrum observed near Earth. The spectra of injected protons and nuclei of iron were found in extragalactic sources under the assumption that these types of particles predominate in the composition of the sources. The method of calculation is illustrated using observational data obtained in the Auger and Telescope Array experiments.     

Solving the inverse transport problem in intergalactic space and determining the energy spectrum and composition of extragalactic sources of ultrahigh-energy particles

The propagation of ultrahigh-energy nuclei in an expanding Universe filled with background electromagnetic radiation is considered. A numerical method for solving the inverse problem for the equation of cosmic-ray transport is developed that allows the spectrum of sources to be determined from the cosmic-ray spectrum observed near the Earth. The spectra of injected protons and nuclei in extragalactic sources are found by assuming that they are functions of the magnetic rigidity of particles. The data from observations obtained in the Auger experiment are used.     

Nonlinear propagation of cosmic rays in the galaxy

Propagation of cosmic rays in the interstellar medium after their emergence from sources—supernova remnants—may be accompanied by the development of flow instability which forms high magnetohydrodynamic turbulence and leads to nonlinear cosmic ray diffusion. A self-similar solution to the nonlinear diffusion equations is found and it is shown that the noted mechanism leads to an effective diffusion coefficient of cosmic rays, which coincides with the empirical value.     

无机酸洗脱对胜利褐煤微晶结构及其自燃倾向性的影响

利用XRD、Raman、XPS和FT-IR表征技术,研究无机酸洗脱(HCl、H2SO4、HCl-HF)处理的胜利褐煤微晶结构的变化,采用自行设计的表面吸附仪-GC联用装置,对样品进行不同温度的低温脉冲氧化实验,考察了煤样在不同温度下氧吸附量的变化规律,通过低温脉冲氧吸附规律与TG/DTG和固定床燃烧实验关联,考察了煤样的自燃倾向。结果表明,无机酸洗脱对矿物质的脱除使得煤结构的有序度增加,石墨化程度提高,无机酸洗脱煤样与原煤相比吸氧量明显下降。随着吸附温度的升高,各煤样吸氧量明显增加,且随着脱除矿物质程度的增加,吸氧量呈减小的趋势,导致自燃倾向降低。     

13C-13C NMR spin-spin coupling constants in saccharides: structural correlations involving all carbons in aldohexopyranosyl rings

13C-13C Spin-spin coupling constants (JCC) have been measured in a group of aldohexopyranoses and methyl aldopyranosides singly labeled with 13C at different sites to confirm and extend prior correlations between JCC magnitude and sign and saccharide structure. Structural correlations for 2JC1,C3, 2JC2,C4, 2JC4,C6, and 2JC1,C5 have been confirmed using density functional theory calculations to test empirical predictions. These geminal couplings depend highly on the orientation of C-O bonds appended to the terminal coupled carbons, but new evidence suggests that 2JCCC values are also affected by intervening carbon structure and C-O bond rotation. 3JC1,C6 and 3JC3,C6 values show Karplus-like dependences but also are affected by in-plane terminal hydroxyl substituents. In both cases, rotation about the C5-C6 bond modulates the coupling due to the alternating in-plane and out-of-plane O6. 3JC3,C6 is also affected by C4 configuration. Both 3JC1,C6 and 3JC3,C6 are subject to remote effects involving the structure at C3 and C1, respectively. New structural correlations have been determined for 2JC3,C5, which, like 3JC3,C6, shows a remote dependence on anomeric configuration. Investigations of dual pathway 13C-13C couplings, 3+3JC1,C4 and 3+3JC2,C5, revealed an important additional internal electronegative substituent effect on 3JCC in saccharides, a structural factor undocumented previously and one of importance to the interpretation of trans-glycoside 3JCOCC in oligosaccharides.     

An approximate treatment of scattering based on the delta boundary operator

The development of a scheme to treat two-dimensional electromagneticscattering by electrically large, perfectly conducting bodiesis described. It incorporates the effects of surface curvatureand non-local phenomena and has the potential to provide thebasis for a general technique yielding more accurate predictionsthan the widely used physical optics method.     

Methyl β‐allolactoside [methyl β‐d‐galactopyranosyl‐(1→6)‐β‐d‐glucopyranoside] monohydrate

Methyl β‐allolactoside [methyl β‐d ‐galactopyranosyl‐(1→6)‐β‐d ‐glucopyranoside], (II), was crystallized from water as a monohydrate, C₁₃H₂₄O₁₁·H₂O. The βGalp and βGlcp residues in (II) assume distorted ⁴C₁ chair conformations, with the former more distorted than the latter. Linkage conformation is characterized by ϕ′ (C2_Gal—C1_Gal—O1_Gal—C6_Glc), ψ′ (C1_Gal—O1_Gal—C6_Glc—C5_Glc) and ω (C4_Glc—C5_Glc—C6_Glc—O1_Gal) torsion angles of 172.9 (2), −117.9 (3) and −176.2 (2)°, respectively. The ψ′ and ω values differ significantly from those found in the crystal structure of β‐gentiobiose, (III) [Rohrer et al. (1980). Acta Cryst. B 36 , 650–654]. Structural comparisons of (II) with related disaccharides bound to a mutant β‐galactosidase reveal significant differences in hydroxymethyl conformation and in the degree of ring distortion of the βGlcp residue. Structural comparisons of (II) with a DFT‐optimized structure, (II_C), suggest a link between hydrogen bonding, pyranosyl ring deformation and linkage conformation.     

(13)C-(1)H and (13)C-(13)C NMR J-couplings in (13)C-labeled N-acetyl-neuraminic acid: correlations with molecular structure

N-acetyl-neuraminic acid (Neu5Ac, 2) was prepared enzymatically containing single sites of (13)C-enrichment at C1, C2, and C3. Aqueous solutions of the three (13)C isotopomers were studied by (1)H and (13)C NMR spectroscopy at p(2)H 2 and pH 8 to obtain J(CH) and J(CC) values involving the labeled carbons. Experimental studies were complemented by DFT calculations of the same set of J-couplings in protonated and ionized structural mimics of 2 to determine how well theoretical predictions match the experimental findings in saccharides bearing ionizable functionality. Results show that: (a) (2)J(C2,H3ax/eq) values in 2 depend on anomeric configuration, thus complementing (3)J(C1,H3ax/eq) behavior, (b) J(CH) and J(CC) values involving C2 depend on anomeric configuration, the C1-C2 bond torsion, and solution pH, and (c) long-range (4)J(C2,H7) is sensitive to glycerol side-chain conformation. Intraring J(HH) and most (2)J(CH), (3)J(CH), (2)J(CC), and (3)J(CC) involving C1-C3 of 2 appear largely unaffected by the ionization state of the carboxyl group. In vacuo and solvated DFT calculations of geminal and vicinal J(CH) and J(CC) values are similar and reproduce the experimental data well, but better agreement with experiment was observed for (1)J(C1,C2) in the solvated calculations. The present work provides new information for future treatments of trans-glycoside couplings involving Neu5Ac residues by (a) providing new standard values of intraring J(CC) for coupling pathways that mimic those for trans-glycoside J(CC), (b) identifying potential effects of solution pH on trans-glycoside couplings inferred through the behavior of related intraring couplings, and (c) providing specific guidelines for more reliable DFT predictions of J(CH) and J(CC) values in ionizable saccharides.