On the asymptotic completeness of the Volterra calculus

The Volterra calculus is a simple and powerful pseudodifferential tool for inverting parabolic equations which has found many applications in geometric analysis. An important property in the theory of pseudodifferential operators is asymptotic completeness, which allows the construction of parametrices modulo smoothing operators. In this paper, we present new and fairly elementary proofs of the asymptotic completeness of the Volterra calculus. The author was partially supported by the European RT NetworkGeometric Analysis HPCRN-CT-1999-00118.     

The Gibbs phenomenon for general Franklin systems

The paper is devoted to the Gibbs phenomenon for series by general Franklin systems. The general Franklin system corresponding to a given dense sequence of points T = (t _n , n ≥ 0) in [0, 1] is a sequence of orthonormal piecewise linear functions with knots from T, that is, the nth function from the system has knots t ₀,..., t _n . The main result of this paper is that the Gibbs phenomenon for Fourier series by general Franklin systems occurs for almost all points of [0, 1].     

Dielectric permittivity and temperature effects on spin-spin couplings studied on acetonitrile

Dielectric permittivity (epsilon) and temperature effects on indirect spin-spin coupling constants were studied using acetonitrile as a probe molecule. Experiments were accompanied by hybrid DFT (density functional theory) studies, where the solvent was modeled using the polarization continuum model. Owing to its numerous types of J-couplings, acetonitrile is a very convenient molecule against which various basis sets can be tested or the best basis set can be selected for a given study. The results show reasonable agreement between calculated and experimental values. According to our data, scalar spin-spin coupling constants undergo substantial shifts at lower values of the dielectric constant. Thus J-coupling values are not transferable between measurements made at differing epsilon-conditions, and the assumption of the epsilon-independence of the J-coupling can lead to crucial mistakes in experiments using low-epsilon media. Dielectric permittivity also causes small geometric fluctuations within the molecule, which themselves can affect J-coupling values. Examinations of the results computed with frozen and relaxed geometries show that geometry mediation mostly affects the spin-dipole term of the J-coupling; hence, for accurate evaluation of the latter, frozen geometries are not acceptable. Another interesting fact revealed is the connection between the solvent dielectric properties and the temperature-dependence slopes of J-couplings in corresponding media.     

Interaction of nitrogen bases with iron-porphyrin nitrito complexes Fe(Por)(ONO) in sublimed solids

The reactions of the nitrogen Lewis bases (B) 1-methylimidazole (1-MeIm), pyridine (Py), and NH3 as gases with sublimed layers containing the 5-coordinate nitrito iron(III)-porphyrinato complexes Fe(Por)(eta1-ONO) (1) are described (Por = meso-tetraphenyl-porphyrinato or meso-tetra-p-tolyl-porphyrinato dianions). In situ FTIR and optical spectra are used to characterize the formation of the 6-coordinate nitro complexes formed by the reaction of 1 with B = 1-MeIm, Py, or NH3. These represent the first examples of 6-coordinate amino-nitro complexes with sterically unprotected iron-porphyrins. The interaction of ammonia with Fe(Por)(ONO) at 140 K initially led to the nitrito species Fe(Por)(NH3)(eta1-ONO), and this species isomerized to the nitro complexes Fe(Por)(NH3)(eta1-NO2) upon warming to 180 K. When the latter were warmed to room temperature under intense pumping, the initial nitrito complexes Fe(Por)(eta1-ONO) were restored. Assignments of vibrational frequencies for the coordinated nitro group in 6-coordinate iron-porphyrin complexes are given and confirmed using 15N-labeled nitrogen dioxide to identify characteristic infrared bands. For M(Por)(B)(NO2) complexes (M = Fe or Co), an inverse correlation between the net charge transfer from the axial ligand B to the nitro group and the value of Deltanu = nua(NO2) - nus(NO2) is proposed. These observations are discussed in the context of growing interest in potential physiological roles of nitrite ion reactions with ferro- and ferri-heme proteins.     

Catalytic dioxygen activation by (nitro)(meso-tetrakis(2-n-methylpyridyl)porphyrinato)cobalt(III) cation derivatives electrostatically immobilized in nafion films: an experimental and DFT investigation

Complexes of the (nitro)( meso-tetrakis(2- N-methylpyridyl)porphyinato)cobalt(III) cation, [LCoTMpyP(2)(NO 2)] (4+), in which L = water or ethanol have been immobilized through ionic attraction within Nafion films (Naf). These immobilized six-coordinate species, [LCoTMPyP(2)(NO 2)/Naf], have been found to catalyze the oxidation of triphenylphosphine in ethanol solution by dioxygen, therefore retaining the capacity to activate dioxygen catalytically without an additional reducing agent as was previously observed in nonaqueous solution for the non-ionic (nitro)cobalt porphyrin analogs. Heating these immobilized six-coordinate species under vacuum conditions results in the formation of the five-coordinate nitro derivatives, [CoTMPyP(2)(NO 2)/Naf] at 85 degrees C and [CoTMPyP(2)/Naf] at 110 degrees C. The catalytic oxidation of gas-phase cyclohexene with O 2 is supported only by the resulting immobilized five-coordinate nitro complex as was previously seen with the corresponding solution-phase catalyst in dichloromethane solution. The simultaneous catalytic oxidation of triphenylphosphine and cyclohexene with O 2 in the presence of the Nafion-bound six-coordinate ethanol nitro complex is also observed; however, this process is not seen for the CoTPP derivative in dichloromethane solution. The oxidation reactions do not occur with unmodified Nafion film or with Nafion-supported [BrCo(III)TmpyP]/Naf or [Co(II)TmpyP]/Naf, indicating the necessity for the nitro/nitrosyl ligand in the oxidation mechanism. The existence of a second reactive intermediate is indicated because the two simultaneous oxidation reactions depend on two distinct oxygen atom-transfer steps having different reactivity. The absence of homogeneous cyclohexene oxidation by the six-coordinate (H 2O)CoTPP(NO 2) derivatives in the presence of Ph 3P and O 2 in dichloromethane solution indicates that the second reactive intermediate is lost by an unidentified route only in solution, implying that the immobilization of it in Nafion allows it to react with cyclohexene. Although direct observation of this species has not been achieved, a comparitive DFT study of likely intermediates in several catalytic oxidation mechanisms at the BP 6-31G* level supports the possibility that this intermediate is a peroxynitro species on the basis of relative thermodynamic accessibility. The alternate intermediates evaluated include the reduced cobalt(II) porphyrin, the dioxygen adduct cobalt(III)-O 2 (-), the oxidized cobalt(II) pi-cation radical, and the nitrito complex, cobalt(III)-ONO.     

On the tangential touch between the free and the fixed boundaries for the two-phase obstacle-like problem

In this paper we consider the following two-phase obstacle-problem-like equation in the unit half-ball

High-throughput protein precipitation method with 96-well plate for determination of doxepin and nordoxepin in human plasma using LC–MS/MS

The aim of this study was to establish a high-throughput and sensitive LC–MS/MS method for the determination of doxepin and its major active metabolite nordoxepin in human plasma. It has been designed for bioequivalence study for formulations containing 25 mg of doxepin. Doxepin and nordoxepin were extracted from human plasma samples by protein precipitation with acetonitrile by using protein precipitation 96-well plates. The analyte was separated using a Phenomenex Kinetex Biphenyl column (100 × 2.1 mm, 2.6 μm) using isocratic elution with a mobile phase of 20 mM ammonium formate (30%) and acetonitrile:methanol 3:7 v:v (70%) at a flow rate of 0.5 mL/min and an injection volume of 10 μL. The detection was performed using a triple quadrupole mass spectrometer by multiple reaction monitoring mode to monitor the precursor-to-product ion transitions of m/z 280.4 → 107.0 and 283.4 → 235.0 for doxepin and doxepin-D3, respectively, and 266.3 → 106.9 and 269.3 → 235.0 for nordoxepin and nordoxepin-D3, respectively, in positive electrospray ionization mode. The total run time was 3.5 min. The method was validated over a concentration range of 50–10,000 pg/mL using a Triple Quad 4500 MS System (Sciex) for both analytes. The developed and validated method can be successfully used to study the bioequivalence/pharmacokinetics of doxepin and nordoxepin.     

Reactions of nitrogen oxides with the five-coordinate Fe(III)(porphyrin) nitrito intermediate Fe(Por)(ONO) in sublimed solids

Detailed experimental studies are described for reactions of several nitrogen oxides with iron porphyrin models for heme/NxOy systems. It is shown by FTIR and optical spectroscopy and by isotope labeling experiments that reaction of small increments of NO2 with sublimed thin layers of the iron(II) complex Fe(Por) (Por = meso-tetraphenylporphyrinato dianion, TPP, or meso-tetra-p-tolylporphyrinato dianion, TTP) leads to formation of the 5-coordinate nitrito complexes Fe(Por)(eta1-ONO) (1), which are fairly stable but very slowly decompose under vacuum giving mostly the corresponding nitrosyl complexes Fe(Por)(NO). Further reaction of 1 with new NO2 increments leads to formation of the nitrato complex Fe(Por)(eta2-O2NO) (2). The interaction of NO with 1 at low temperature involves ligand addition to give the nitrito-nitrosyl complexes Fe(Por)(eta1-ONO)(NO) (3); however, these isomerize to the nitro-nitrosyl analogs Fe(Por)(eta1-NO2)(NO) (4) upon warming. Experiments with labeled nitrogen oxides argue for an intramolecular isomerization ("flipping") mechanism rather than one involving dissociation and rebinding of NO2. The Fe(III) centers in the 6-coordinate species 3 and 4 are low spin in contrast to 1, which appears to be high-spin, although DFT computations of the porphinato models Fe(P)(nitrite) suggest that the doublet nitro species and the quartet and sextet nitrito complexes are all relatively close in energy. The nitro-nitrosyl complex 4 is stable under an NO atmosphere but decomposes under intense pumping to give a mixture of the ferrous nitrosyl complex Fe(Por)(NO) and the ferric nitrito complex Fe(Por)(eta1-ONO) indicating the competitive dissociation of NO and NO2. Hence, loss of NO from 4 is accompanied with nitro --> nitrito isomerization consistent with 1 being the more stable of the 5-coordinate NO2 complexes of iron porphyrins.     

Using of catastrophe theory for processing of experimental results measured in low-temperature plasma

The family of dynamic current-voltage characteristics obtained from gas mixture of CO₂ laser at different frequencies of modulation of the discharge current is described by means of the catastrophe theory. It is shown that in the region of critical points of current-voltage characteristics it is possible to obtain not only qualitative, but also quantitative characteristics of controlling parameters. The best coincidence with the experimental results has been obtained for “dovetail” catastrophes.     

On a Property of the Franklin System in C[0, 1] and L1[0, 1]

Mathematical Notes - A problem posed by J. R. Holub is solved. In particular, it is proved that if $$\left\{ {{{\tilde f}_n}} \right\}$$ is the normalized Franklin system in L1[0, 1], {an} is a...