Spike Solutions Research Articles

On Multiple Mixed Interior and Boundary Peak Solutions for Some Singularly Perturbed Neumann Problems

AbstractWe consider the problemwhere Ω is a bounded smooth domain in RN, ε > 0 is a small parameter and f is a superlinear, subcritical nonlinearity. It is known that this equation possesses multiple boundary spike solutions that concentrate, as ε approaches zero, at multiple critical points of the mean curvature function H(P), P ∈ ∂Ω. It is also proved that this equation has multiple interior spike solutions which concentrate, as ε → 0, at sphere packing points in Ω.In this paper, we prove the existence of solutions with multiple spikes both on the boundary and in the interior. The main difficulty lies in the fact that the boundary spikes and the interior spikes usually have different scales of error estimation. We have to choose a special set of boundary spikes to match the scale of the interior spikes in a variational approach.

Multi-interior-spike solutions for the Cahn-Hilliard equation with arbitrarily many peaks

We study the Cahn-Hilliard equation in a bounded smooth domain without any symmetry assumptions. We prove that for any fixed positive integer K there exist interior K–spike solutions whose peaks have maximal possible distance from the boundary and from one another. This implies that for any bounded and smooth domain there exist interior K–peak solutions. The central ingredient of our analysis is the novel derivation and exploitation of a reduction of the energy to finite dimensions (Lemma 5.5) with variables which are closely related to the location of the peaks. We do not assume nondegeneracy of the points of maximal distance to the boundary but can do with a global condition instead which in many cases is weaker.

Multiple boundary peak solutions for some singularly perturbed Neumann problems

We consider the problem {ɛ2Δu-u+f(u)=0inΩ,u>0inΩ,∂u/∂v=0on∂Ω,where Ω is a bounded smooth domain in RN, ɛ > 0 is a small parameter and f is a superlinear, subcritical nonlinearity. It is known that this equation possesses boundary spike solutions such that the spike concentrates, as ε approaches zero, at a critical point of the mean curvature function H(P),P ∈ ∂ Ω. It is also known that this equation has multiple boundary spike solutions at multiple nondegenerate critical points of H(P) or multiple local maximum points of H(P).In this paper, we prove that for any fixed positive integer K there exist boundary K-peak solutions at a local minimum point of H(P). This implies that for any smooth and bounded domain there always exist boundary K-peak solutions.We first use the Liapunov–Schmidt method to reduce the problem to finite dimensions. Then we use a maximizing procedure to obtain multiple boundary spikes.

A Metastable Spike Solution for a Nonlocal Reaction-Diffusion Model

An asymptotic reduction of the Gierer--Meinhardt activator-inhibitor system in the limit of large inhibitor diffusivity leads to a singularly perturbed nonlocal reaction diffusion equation for the activator concentration. In the limit of small activator diffusivity, a one-spike solution to this nonlocal model is constructed. The spectrum of the eigenvalue problem associated with the linearization of the nonlocal model around such an isolated spike solution is studied in both a one-dimensional and a multidimensional context. It is shown that the principal eigenvalues in the spectrum are exponentially small in the limit of small activator diffusivity. The nonlocal term in the eigenvalue problem is essential for ensuring the existence of such exponentially small principal eigenvalues. These eigenvalues are responsible for the occurrence of an exponentially slow, or metastable, spike-layer motion for the time-dependent problem. Explicit metastable spike dynamics are derived by using a projection method, which enforces a limiting solvability condition on the solution to the linearized problem.

DEVELOPING A STANDARD SPIKING PROCEDURE FOR THE INTRODUCTION OF HYDROPHOBIC ORGANIC COMPOUNDS INTO FIELD-WET SOIL

The purpose of this study was to reevaluate a number of spiking procedures we had previously investigated for spiking dry soil, determine their suitability for spiking radiolabeled hydrophobic organic compounds in field-wet soil, and identify the optimal spiking procedure. In the first part of this study, two radiolabeled polycyclic aromatic hydrocarbons (PAHs), phenanthrene (Phe) and benzo[a]pyrene (BaP), were introduced into field-wet soil by four different spiking procedures and the recovery and homogeneity of compound distribution assessed by analysis of replicate subsamples. The identified optimal spiking procedure involved adding an acetone spike solution to 10 ml acetone in a blender base, adding 250 g field-wet soil, and mixing with a highspeed blender. The relative standard deviations obtained by this procedure were 2.7 and 5.3% for 14C-Phe and 14C-7-BαP, respectively, and were comparable to those we obtained in a previous study for spiking dry soil. The persistence of acetone used as a carrier solvent in the optimal spiking procedure was assessed at various times after spiking to identify a suitable equilibration time to allow for solvent volatilization. The equilibration time, together with the identified optimal spiking procedure, were combined to develop a standard spiking procedure. The standard spiking procedure was used to prepare multiple soil microcosms individually spiked with radiolabeled Phe, pyrene, and BaP for a long-term aging study on the formation of PAH-bound residues in arable soil. The mean spike activity and homogeneity of spike distribution in 54 sterilized microcosms (3 × 18) stored over 525 d was determined. Mean spike recovery and relative standard deviations (RSDs) for the three radiolabeled PAHs in the individually spiked and prepared microcosms were very good. However, this data illustrates the importance of determining the concentration and homogeneity of spikes within all individual test portions in experiments where organic compounds are spiked into soil or sediment.

Development of an Isotope Dilution Mass Spectrometric Method for Dimethylthallium Speciation and First Evidence of Its Existence in the Ocean

A sensitive analytical method for the determination of dimethyl thallium (Me2Tl+) in environmental samples by positive thermal ionization isotope dilution mass spectrometry (PTI-IDMS) was developed. For the necessary low detection limit and for species selective determination, PTI-IDMS was connected with species-unspecific enrichment of thallium from ocean water samples by a strongly basic anion exchanger and a species-specific extraction step, respectively. For isotope dilution, a 203Tl-enriched Me2Tl+ spike solution was synthesized and characterized with respect to its isotopic composition and concentration. The spike solution was stable for the 30-month period of investigations under dark storage conditions at pH = 2 and 4 °C. The detection limit of the developed method is 0.4 ng L-1 for 500 mL ocean water samples and the species selectivity of Me2Tl+, compared with that of inorganic thallium, is >500. In different surface water samples of the Atlantic Ocean from 50°N to 64°S and in one depth profile down to 4000 m, Me2Tl+ concentrations in the range of <0.4 to 3.2 ng L-1 were determined. This is the first time that dimethyl thallium could be detected in environmental samples. Positive detection of the organothallium compound was found in about 20% of all samples analyzed. By also analyzing the total thallium content with PTI-IDMS it was found that in these cases 3−48% of thallium exists in the methylated form. From its occurrence in the remote areas of the South Atlantic, from positive correlation with the bioactivity in the corresponding ocean water, and from the fact that Me2Tl+ is not known as an anthropogenic substance, it follows that a biogenic origin must be assumed.

Multiple interior peak solutions for some singularly perturbed neumann problems

We consider the problem { ε 2 Δ u − u + f ( u ) = 0 , u > 0 , in Ω , ∂ u ∂ v = 0 on ∂ , where Ω is a bounded smooth domain in R N , ɛ > 0 is a small parameter, and ƒ is a superlinear, subcritical nonlinearity. It is known that this equation possesses boundary spike solutions that concentrate, as ɛ approaches zero, at a critical point of the mean curvature function H(P), P ε ∂Ω. It is also proved that this equation has single interior spike solutions at a local maximum point of the distance function d(P, ∂Ω), P ε Ω . In this paper, we prove the existence of interior K −peak (K ⩾ 2) solutions at the local maximum points of the following function φ(P 1 , P 2 ,…, p k ) = min i, k, l = 1, …, K; k ≠ 1 (d(P i , ∂Ω), 1/2 ¦P k −p l ¦) . We first use the Liapunov-Schmidt reduction method to reduce the problem to a finite dimensional problem. Then we use a maximizing procedure to obtain multiple interior spikes. The function φ(P 1 , …, p k ) appears naturally in the asymptotic expansion of the energy functional.

On single interior spike solutions of the Gierer–Meinhardt system: uniqueness and spectrum estimates

We study the interior spike solutions to a steady state problem of the shadow system of the Gierer–Meinhardt system arising from biological pattern formation. We first show that at a non-degenerate peak point the interior spike solution is locally unique, and then we establish the spectrum estimates of the associated linearized operator. We also prove that the corresponding solution to the shadow system is unstable. Furthermore, the metastability of such solutions is analysed.

Determination of zirconium traces in polymers by ICP-IDMS - a powerful and fast method for routine testing of zirconium residues in polyolefins

Zirconium trace analyses play an important role for polyolefins produced by modern catalytic processes with zirconium metallocenes. A reliable and fast routine testing method by inductively coupled plasma isotope dilution mass spectrometry (ICP-IDMS) was therefore developed, which allows the determination of zirconium in polymers down to the low ng/g level. With respect to its precision, accuracy, and time-consumption this method is suitable for routine testing of production processes. A spike solution, enriched in the stable isotope 91Zr, was prepared and used for the isotope dilution procedure, which has the advantage of being an internal “one point” calibration method. The polyolefin samples were dissolved by microwave assisted digestion with a mixture of concentrated HNO3/HF.

Determination of ruthenium in photographic emulsions - development and comparison of different sample treatments and mass spectrometric methods

Different sample treatment procedures were combined with inductively coupled plasma mass spectrometry (ICP-MS) and negative thermal ionisation mass spectrometry (NTI-MS) for the determination of ruthenium traces in photographic emulsions. Dissolution of the samples in concentrated ammonia solution was used in connection with ICP-MS by external calibration, which has the advantage of a simple sample preparation technique but introduces high amounts of the silver matrix into the mass spectrometer. On the other hand, isotope dilution mass spectrometry (IDMS) with an enriched 99Ru spike solution was applied for ICP-MS and NTI-MS measurements, respectively, in connection with a significant reduction of the matrix by AgCl precipitation. In these cases loss of ruthenium by the AgCl precipitate has no effect on the analytical result. The results of the different methods agreed usually well analysing ruthenium traces in the range of 0.1–10 μg per gram emulsion. The detection limits obtained were 4 ng/g for ICP-IDMS, 20 ng/g for NTI-IDMS, and 15 ng/g for ICP-MS with external calibration. Differences in the results between the different methods could mainly be attributed to sample inhomogeneities. ICP-IDMS with silver matrix reduction by AgCl precipitation is recommended as a routine method, NTI-IDMS with the corresponding sample treatment as a calibration method.

Determination of cadmium in biological and environmental materials by isotope dilution inductively coupled plasma mass spectrometry: effect of flow sample introduction methods

Experimental parameters governing the instrumental precision and accuracy for isotope ratio measurements of cadmium in ICP-MS, including sampling time, mass bias, detector dead-time and spectroscopic interferences, were first characterised. The precision achieved for the isotope ratio determination of Cd was around 0.2% (RSD forn=5) both for 111:112 and 111:114 ratios. Two alternative flow approaches for the determination of ultratrace concentrations of cadmium by isotope dilution (ID) were explored and compared with the more conventional ID methodology: first, on-line mixing of the sample solution with the spike solution just before the ICP-MS nebuliser using a peristaltic pump and, second, the generation of volatile cadmium species using sodium tetraethylborate by merging zones flow injection ICP-MS. The three approaches were successfully applied to the determination of ultratrace levels of cadmium in biological and environmental certified reference materials (NIST SRM 2670 Freeze-Dried Urine, IAEA H-8 Horse Kidney, BCR TP-25 Lichens, PACS-1 Marine Sediment and SLRS-3 Riverine Water). The on-line ID method proved to be the most convenient for the determination of cadmium in such samples because it is fast, provides similar results to those of conventional ID and requires less sample preparation.

Determination of erbium in nuclear fuels by isotope dilution thermal ionization mass spectrometry and glow discharge mass spectrometry

A thermal ionization mass spectrometer (TIMS) and a laboratory-built glow discharge mass spectrometer (GDMS) have been employed for the determination of the isotopic composition of erbium and its concentration, with respect to uranium, in nuclear fuel samples before irradiation. First, the GDMS results on erbium isotopic composition in erbium metal and erbium oxide samples have been compared with those obtained from TIMS measurements. Then, the isotopic composition of uranium and erbium has been measured by TIMS after dissolution of inactive erbium-doped molybdenum-uranium fuel samples. The atomic 166 Er: 238 U ratio has also been determined with the double spike isotope dilution method. A 167 Er: 233 U spike solution has been prepared and calibrated for this determination. However, to avoid matrix effects in TIMS analyses, it has first been necessary to perform chemical separations to isolate U and Er from the high quantities of molybdenum and iron present in the liquid samples. The glow discharge mass spectrometer has been used for the analyses of the same samples. For the evaluation of this technique, the isotopic compositions of U and Er have been measured directly on the solid samples and the atomic 166 Er: 238 U ratio has been determined by a calibration procedure with home-made standards. Results are compared in terms of accuracy and precision with the analyses performed by TIMS, used as the reference technique.

Validation protocol for the determination of copper in plant samples by isotope dilution inductively coupled plasma mass spectrometry

Copper determination in plant material was carried out by isotope dilution inductively coupled plasma mass spectrometry. The samples (copper standard solution and four CRM of plant origin) were spiked with 65 Cu enriched standard solution and decomposed by pressurized microwave digestion in the HNO 3 -H 2 O 2 mixture. In order to eliminate possible spectroscopic interferences originating from matrix elements, copper was separated from sample digests by adsorption on chelating ion-exchange resin Chelex 100 and elution with diluted HNO 3 . Performance parameters such as LOD, LOQ, selectivity, measurement ruggedness, precision and trueness were evaluated. LOD and LOQ were 57 and 138 ng g –1 , respectively. Repeatability of copper determination was 20 ng g –1 . The influences of sample weight and volume of spike solution on sensitivity and uncertainty of analysis were estimated. Results of copper determination in CRM were in good agreement with certified values.

Development of accurate mass spectrometric routine and reference methods for the determination of trace amounts of iridium and rhodium in photographic emulsions

For the determination of trace amounts of iridium and rhodium in photographic emulsions different sample treatment procedures were coupled with inductively coupled plasma mass spectrometry (ICP-MS) and, for iridium, also with negative thermal ionisation isotope dilution mass spectrometry (NTI-IDMS) and ICP-IDMS. IDMS determinations of iridium were carried out using an enriched 191 Ir spike solution. Elimination of the silver matrix was established for both mass spectrometric methods, which prevents corresponding memory effects in the ICP-MS system and is essential for the formation of IrO 2 – thermal ions, respectively. For NTI-MS measurements, the gelatine matrix of the emulsion must also be eliminated by decomposition with HNO 3 -H 2 O 2 . The detection limits for ICP-IDMS and NTI-IDMS are 0.1 and 10 ng iridium per gram of emulsion, respectively, so that all relevant photographic emulsions, doped with iridium, could be analysed. ICP-IDMS with silver matrix elimination is recommended as a routine method, while NTI-IDMS can be applied as an accurate reference method. Determinations of the mono-isotopic element rhodium cannot be carried out by IDMS. Photographic emulsions were therefore first dissolved in concentrated ammonia solution and rhodium was then directly measured by ICP-MS after dilution of the solution using an internal indium standard for calibration. This procedure has the advantage of a simple sample preparation technique but introduces high amounts of silver into the mass spectrometer and dilution of the sample restricts the detection limit. Because of the similar chemical and physical behaviour of Ru and Rh, ruthenium was used as an internal standard during matrix separation and subsequent ICP-MS measurements of trace amounts of rhodium. By determining the 103 Rh/ 99 Ru ratio of the separated sample, reliable results for the determination of trace amounts of rhodium were obtained, which was confirmed by the good agreement between the data obtained by the procedure with internal Ru calibration and those obtained without matrix separation and calibration by indium. The determination of trace amounts of iridium and rhodium in the range 0.5-650 ng g –1 was carried out in photographic emulsions with the different methods, which agreed well in their results. The relative standard deviations were usually in the range 3-10% and were substantially influenced by sample inhomogeneities.

On the Stationary Cahn–Hilliard Equation: Interior Spike Solutions

We study solutions of the stationary Cahn–Hilliard equation in a bounded smooth domain which have a spike in the interior. We show that a large class of interior points (the “nondegenerate peak” points) have the following property: There exist such solutions whose spike lies close to a given nondegenerate peak point. Our construction uses among others the methods of viscosity solution, weak convergence of measures, and Liapunov–Schmidt reduction.

Extraction and Analysis of Trifluoroacetic Acid in Environmental Waters

Trifluoroacetic acid (TFA), a mildly phytotoxic compound, is a stable atmospheric breakdown product of HFC-134a, HCFC-123, and HCFC-124. An extraction and analytical method has been developed for the routine analysis of low ppt levels of TFA in aqueous samples. TFA can be quantitatively recovered from most environmental waters by an extraction procedure using a commercial anion-exchange disk. In saline samples (conductivity >620 μS), where the presence of competing anions interfered with recovery, a liquid-liquid extraction cleanup was necessary. After extraction of TFA from water, the dried disk was placed in a headspace vial containing 10% sulfuric acid in methanol and the vial sealed and then vortexed for 30 s. The sulfuric acid-methanol solution extracts trifluoroacetate anion (TFA) from the anion-exchange matrix and, when heated, quantitatively converts it to the methyl ester, which is then analyzed by automated headspace gas chromatography using electron capture or mass spectrometry detection. Several environmental samples in addition to laboratory spike solutions were successfully extracted and analyzed with this technique. Recoveries averaged 108.2% for reagent water spiked at levels from 53 to 2110 ng/L with relative standard deviations ranging from 0.3 to 8.4%. The instrument's limit of detection for TFA standard was 3.3 ng. The limit of quantitation for the extraction and analytical technique was 36 ng/L. Three water samples can be prepared for automated analysis in 20 min using this technique.

Isotope dilution analysis of Se in human blood serum by using high-power nitrogen microwave-induced plasma mass spectrometry coupled with a hydride generation technique.

To establish a method for sensitive, accurate, and precise determination of Se in real samples, isotope dilution analysis using high-power nitrogen microwave-induced plasma mass spectrometry (N2 MIP-IDMS) was conducted. In this study, freeze-dried human blood serum (Standard Reference Material, NIES No. 4) provided by NIES (National Institute for Environmental Studies) was used as a real sample. The measured isotopes of Se were 78Se and 80Se which are the major isotopes of Se. The appropriate amount of a Se spike solution was theoretically calculated by using an error multiplication factor (F) and was confirmed experimentally for the isotope dilution analysis. The mass discrimination effect was corrected for by using a standard Se solution for the measurement of Se isotope ratios in the spiked sample. However, the sensitivity for the detection of Se was not so good and the precision of the determination was not improved (2-3%) by N2 MIP-IDMS with use of the conventional nebulizer. Therefore, a hydride generation system was connected to N2 MIP-IDMS as a sample introduction system (HG-N2 MIP-IDMS) in order to establish a more sensitive detection and a more precise determination of Se. A detection limit (3 sigma) of 10 pg mL-1 could be achieved, and the RSD was less than 1% at the concentration level of 5.0-10.0 ng mL-1 by HG-N2 MIP-IDMS. The analytical results were found to be in a good agreement with those obtained by the standard addition method using conventional Ar ICPMS.

The optimal amount of isotopic spike solution for ultratrace analysis by isotope dilution mass spectrometry

Inductively coupled plasma mass spectrometry (ICP-MS) and high resolution inductively coupled plasma mass spectrometry (HR-ICP-MS) are powerful methods of determining metallic impurities in the low- and sub-ppt level in process media such as ultra-pure water used in semiconductor and wafer manufacturing. By using mass spectrometers for analysis, an isotope dilution analysis (IDMS) is possible. The reproducibility of an IDMS is unmatched. For concentration levels near the instrument detection limit a novel method is reported to find the optimal amount of isotopic spike solution. This optimal value can be derived by the law of propagation of uncertainty combined with the Poisson statistics of the measured number of counts. Generally, an excess of isotopic spike solution should be used to provide results of lowest possible uncertainty. The results are presented in a diagram for easy practical use.

Development of an ICP−IDMS Method for Dissolved Organic Carbon Determinations and Its Application to Chromatographic Fractions of Heavy Metal Complexes with Humic Substances

A sensitive and fast method for direct determinations of dissolved organic carbon (DOC) in bulk samples and in chromatographically separated fractions by inductively coupled plasma isotope dilution mass spectrometry (ICP−IDMS) is developed. A 13C-enriched spike solution of benzoic acid is used for the isotope dilution step. Equal ionization efficiencies are obtained for carbon, independent of the type and molecular weight of the dissolved organic substance, which is the most important precondition for DOC determinations by ICP−IDMS. The detection limits achieved are 0.3 mg L-1 for bulk analyses and 7 × 10-4 μg s-1 for transient signals of chromatographic peaks. The results for different water samples, analyzed by ICP−IDMS and a conventional DOC method, agree well within the limits of error. By using a 13C spike solution, which also contains enriched isotopes of heavy metals, simultaneous determinations of DOC and heavy metal concentrations in separated fractions of complexes with humic substances are poss...

Highly precise Re–Os dating for molybdenite using alkaline fusion and NTIMS

The technique described in this paper represents the modification and combination of two previously existing methods, alkaline fusion and negative thermal ion mass spectrometry (NTIMS). We have used this technique to analyze repeatedly a homogeneous molybdenite powder used as a reference standard in our laboratory. Analyses were made over a period of 18 months, using four different calibrations of two different spike solutions. The age of this standard reproduces at a level of ±0.13%. Each individual age analysis carries an uncertainty of about 0.4% that includes the uncertainty in the decay constant for 187Re. This new level of resolution has allowed us to recognize real differences in ages for two grain-size populations of molybdenite from some Archean samples.