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Pflanzenschutzmittel (PSM) werden zur Erzielung hoher und sicherer Ertrage auf landwirtschaftlichen Flachen eingesetzt. Deren Anwendung fuhrt jedoch auch zu unerwunschten Effekten fur Natur und Umwelt. Auswirkungen der PSM sollen gemas den Zielen im Nationalen Aktionsplan zur nachhaltigen Anwendung von Pflanzenschutzmitteln (NAP) so gering wie moglich gehalten werden. Indikatoren sind dazu im NAP aufgefuhrt. Vor diesem Hintergrund werden Wirkungen von PSM auf die Biodiversitat in der Agrarlandschaft am Beispiel der Vogel und der Gefaspflanzen analysiert. Es zeigten sich erhebliche negative Auswirkungen, was nicht den Zielen entspricht. Im NAP enthaltene Indikatoren werden u.a. mit ihrem Bezug zu Biodiversitat, Zielerreichungsgrad und Aufklarung von Ursache-Wirkungsbeziehungen analysiert. Vorliegende Indikatoren sind weit vom Zielwert entfernt, teils fehlen Zielwerte. Die Aufklarung von Ursache-Wirkungsbeziehungen erfolgt kaum. Die Uberprufung des Fortschritts im NAP mit Hilfe der Indikatoren ware demnach nicht hinreichend fur die Ziele des Schutzes der Biodiversitat. Es besteht dringender Bedarf fur die Entwicklung eines auf realen Monitoringdaten basierenden Indikators und dessen rasche Anwendung.

In this paper we study a general even-order symmetric Sturm–Liouville matrix differential equation, whose leading coefficient may be singular on the whole interval under consideration. Such an equation is new in the current literature, as it is equivalent with a system of Sturm–Liouville equations with different orders. We identify the so-called normal form of this equation, which allows to transform this equation into a standard (controllable) linear Hamiltonian system. Based on this new transformation we prove that the associated eigenvalue problem with Dirichlet boundary conditions possesses all the traditional spectral properties, such as the equality of the geometric and algebraic multiplicities of the eigenvalues, orthogonality of the eigenfunctions, the oscillation theorem and Rayleigh’s principle, and the Fourier expansion theorem. We also discuss sufficient conditions, which allow to reduce a general even-order symmetric Sturm–Liouville matrix differential equation into the normal form. Throughout the paper we provide several examples, which illustrate our new theory.

We establish a Sturmian type theorem comparing the number of focal points of any conjoined basis of a nonoscillatory linear Hamiltonian differential system with the number of focal points of the principal solution. We also present various extensions of this statement.

An index theorem is a tool for computing the change of the index (i.e., the number of negative eigenvalues) of a symmetric monotone matrix-valued function when its variable passes through a singularity. In 1995, the first author proved an index theorem in which a certain critical matrix coefficient is constant. In this paper, we generalize the above index theorem to the case when this critical matrix may be varying, but its rank, as well as the rank of some additional matrix, are constant. This includes as a special case the situation when this matrix has a constant image. We also show that the index theorem does not hold when the main assumption on constant ranks is violated. Our investigation is motivated by the oscillation theory of discrete symplectic systems and Sturm--Liouville difference equations with nonlinear dependence on the spectral parameter, which was recently developed by the second author and for which we obtain new oscillation theorems.

In this paper we consider linear Hamiltonian differential systems without the controllability (or normality) assumption. We prove the Rayleigh principle for these systems with Dirichlet boundary conditions, which provides a variational characterization of the finite eigenvalues of the associated self-adjoint eigenvalue problem. This result generalizes the traditional Rayleigh principle to possibly abnormal linear Hamiltonian systems. The main tools are the extended Picone formula, which is proven here for this general setting, results on piecewise constant kernels for conjoined bases of the Hamiltonian system, and the oscillation theorem relating the number of proper focal points of conjoined bases with the number of finite eigenvalues. As applications we obtain the expansion theorem in the space of admissible functions without controllability and a result on coercivity of the corresponding quadratic functional.

We consider symplectic difference systems involving a spectral parameter together with general separated boundary conditions. We establish the so-called oscillation theorem which relates the number of finite eigenvalues less than or equal to a given number to the number of focal points of a certain conjoined basis of the symplectic system. Then we prove Rayleigh's principle for the variational description of finite eigenvalues and we describe the space of admissible sequences by means of the (orthonormal) system of finite eigenvectors. The principle role in our treatment is played by the construction where the original system with general separated boundary conditions is extended to a system on a larger interval with Dirichlet boundary conditions.

Background, aim, and scope Basically, technological innovations are associated with benefits and risks. This is also true for the introduction of genetically modified organisms (GMO) into agriculture. In Germany, precautionary regulations currently demand isolation distances (i. e. buffer zones) for the cultivation of genetically modified maize (Bt-maize) in the vicinity of conventional (150 m) and organic maize fields (300 m). The Bt-toxin may harm non-target organisms (NTO) such as Lepidoptera. Despite this, corresponding regulations for the protection of nature reserves are lacking to date. Conventional and Bt-maize have been grown in the vicinity of the Flora-Fauna-Habitat (FFH) Ruhlsdorfer Bruch in Brandenburg, Germany. The aim of this study was to investigate whether exposure of maize pollen from surrounding Bt-maize fields to NTOs in the nature protection area could be excluded or not. Two types of exposure were investigated: Firstly, whether maize pollen was dispersed by wind into the nature reserve area exposing resident NTOs. Secondly, foraging NTOs from the nature reserve are exposed by roaming the surrounding fields and collecting maize pollen. In order to fulfil the precautionary principle defined by law, the study should help to determine appropriate isolation distances for the cultivation of Bt-maize with regard to sustainable protection of NTOs in the FFH Ruhlsdorfer Bruch. In 2007, the local authorities issued an isolation distance between Bt-maize fields and nature reserves of 100 m and in 2008, this became 250 m in the northern and 500 m in the westerly direction, respectively. Materials and methodsStandardised methods for biological and technical pollen sampling issued by the Association of German Engineers (VDI 4330 Part 3, 2007 and VDI 4330 Part 4, 2006) were applied providing a quality controlled and methodologically harmonised database which does not only serve the needs to be fulfilled by the present case-specific monitoring study but can also be used as a reference database for further investigations. Maize pollen exposure was measured within the FFH Ruhlsdorfer Bruch and its immediate vicinity in July and August 2007 and 2008. In 2007, the sampling was performed at three sites using 12 technical samplers (Sigma-2/PMF) placed at five measuring points at distances from 5 m to 120 m from the maize field edges. Additionally, for biological pollen sampling six bee colonies were situated at these three sites (two colonies at each site). The technical sampler Sigma-2/PMF enables point sampling which is primarily influenced by wind and topography providing information on effective maize pollen input (flow and deposition) at the measuring sites. Honey-bees roaming in the surrounding area with typical foraging distances of several kilometres may act as planar collectors. They may serve as indicators for the exposure to pollen-collecting NTOs. Furthermore, biological sampling is more selective due to the organism’s preferences, whereas the technical sampling is neutral. Hence, both the technical and the biological sampling complement each other in their scope of application. The pollen samples of both matrices were analysed microscopically and the maize pollen loads were quantified. Pollen-DNA was analysed by means of the quantitative PCR-method (qPCR) identifying conventional and Bt-maize pollen by two independent laboratories. In 2008, the monitoring was repeated with additional sites. Eighteen technical samplers were exposed at five sites with eight measuring points at distances from 5 m to 250 m from the maize field edges. Two honey-bee colonies for biological sampling were placed at one site for control purposes. PCR-analyses were performed to measure the amounts of Bt-maize pollen in the samples. ResultsThe results of pollen monitoring at the Ruhlsdorfer Bruch revealed maize pollen exposure for all monitoring sites in both surveys. In 2007, up to 1.75 million maize pollen/m² were deposited at sites closest to the maize field. At 120 m from field edge in the middle of the FFH-reserve, 99,000 maize pollen/m² were detected. In 2008, similar results were found, at distances up to 250 m from the field edges deposition of 164,000 maize pollen/m² was detected. Data on maize pollen deposition show a clear distance relationship and are in accordance with results of further comprehensive surveys based on the same methodology (Hofmann 2007). The results of the microscopic analysis of the pollen pellets demonstrated that bees collected maize pollen at all sites, in 2007 and 2008. Although maize pollen is not the main food source (2007:0.1–0.3 %; 2008:2–3 %) the collection efficiency of the bee colonies resulted in high amounts of sampled maize pollen with 4 to 11 million per site in 2007 and up to 467 million in 2008. Molecular-biological analysis of maize pollen DNA by qPCR demonstrated that transgenic Bt-MON810 DNA was present in all technical and biological samples, corroborated by two independent laboratories. In 2007, the GMO-content in the samples ranged up to 44 % in the bioaerosols and 49 % in the pollen pellets. In 2008, GMO-proportions of up to 18 % were detected. DiscussionThe results of this study provide evidence that NTOs in the Ruhlsdorfer Bruch were exposed to Bt-maize pollen under the cultivation conditions in 2007 with a buffer zone of 100 m. The GMO-content reached up to 48 %. The results of the monitoring in 2008 confirmed these findings. Even though the exposure could be reduced by increasing the isolation distances to 250 m and 500 m respectively, the results still show percentages of up to 18 % Bt-MON810 in the pollen samples. The results on maize pollen deposition at the Ruhlsdorfer Bruch in 2007 and 2008 correspond to the results of an investigation which was conducted over several years applying the same standardised method, but covering a wider range of distances. The correlation between maize pollen deposition (n/m²) and distance to the source field (m) fitted best to a power function of the type y = 1.2086 · 106 · x–0.548. Despite the same trend, the pollen deposition in the Ruhlsdorfer Bruch revealed above-average findings. Also the analysis of the pollen pellets collected by the bees showed an exposure in 2007 with values for the GMO-content of up to 49 %. For both methods, the exposure decreased in 2008 due to the greater buffer zones up to 500 m. Whereas the GMO-content for the biological sampling were reduced to values below 10 %, the values for the technical sampling were still higher indicating that greater buffer zones would be necessary for safety reasons under the precautionary principle. ConclusionsThe results of this investigation proved that maize pollen were dispersed by wind to distances farther than 250 m from field edge leading to maize pollen exposure in the centre of the nature reserve. The results also demonstrated that foraging NTOs living in the nature reserve were exposed to maize pollen from surrounding fields. Considering the cultivation of Bt-maize MON810, the assumption of the Environmental Risk Assessment (ERA) that there will be no relevant exposure beyond the Bt-maize fields, cannot be confirmed. Considering the results of this and related studies and with respect to the precautionary principle, one can state that buffer zones between Bt-maize fields and protected areas are an effective measure to minimise the exposure of Bt-maize pollen to NTOs and, thus, to prevent from adverse effects. Recommendations and perspectivesBecause of still insufficient ecotoxicological data for the risk assessment of Bt-maize MON810 considering butterflies and other protected NTOs, protection standards assuring the precautionary principle have to be implemented to avoid Bt-maize pollen exposure to NTOs. This applies for the case Ruhlsdorfer Bruch and for nature reserve areas in general. In order to exclude risks to protected NTOs occurring in nature reserves, sufficient buffer zones for Bt-maize cultivation should be considered. The statistical analysis revealed that distances of more than 500 m are necessary to decisively reduce exposure to foraging insects. In fact, distances of more than 1,000 m are necessary to prevent maize pollen deposition from values above 100,000 pollen/m² with a certainty of 90 %. An adequate risk assessment can only be attained if based on field measurements accounting for the high variation of pollen deposition due to local environmental site conditions and field management. The monitoring should be based on standardised methods. It should include locations with the highest expected deposition rates, the boundaries of the protected areas and sites of interest within those boundaries, e. g., specific habitats of sensitive species.

We investigate nonoscillatory and controllable symplectic difference systems. We show that the recessive solution of such a system at +∞ has the same number of focal points (counting multiplicities) as the recessive solution at −∞.

We consider 2 n × 2 n 2n\times 2n symplectic difference systems together with associated discrete quadratic functionals and eigenvalue problems. We establish Sturmian type comparison theorems for the numbers of focal points of conjoined bases of a pair of symplectic systems. Then, using this comparison result, we show that the numbers of focal points of two conjoined bases of one symplectic system differ by at most n n . In the last part of the paper we prove the Rayleigh principle for symplectic eigenvalue problems and we show that finite eigenvectors of such eigenvalue problems form a complete orthogonal basis in the space of admissible sequences.

Jsou prezentovany nejnovějsi výsledky oscilacni teorie symplektických diferencnich systemů. Zvlastni pozornost je věnovana Sturmovske a oscilacni větě pro tyto systemy a systemy k nim reverzni. Jsou rovněž formulovany některe domněnky a otevřene problemy.

We establish a Sturmian separation theorem for conjoined bases of 2n-dimensional symplectic difference systems. In particular, we show that the existence of a conjoined basis without focal points in a discrete interval ( 0 , N + 1 ] implies that any other conjoined basis has at most n focal points (counting multiplicities) in this interval.

Membrane plasma separation (MPS) was simulated using a single-pool model, which assumed that the plasma pool is a closed compartment. If thyroxine (T4) removal is simulated, its total distribution volume rather than the plasma compartment forms the pool. Theory and variables of the 1-pool model T4 are described. The effect on the T4 concentration of using fresh frozen plasma as substitution fluid is shown by modelling of a plasma exchange therapy in thyrotoxicosis. Up to 28–34% of the exchange volume could have been savef if a T4-free substitution fluid (e.g., human albumin electrolyte solution) had been used. Comparison of predicted and measured valued demonstrated that the influence of delayed T4 transfer rate between the body compartments cannot be assessed by the single-pool model. Prediction errors could be minimized with a 2-pool model.

The main results deal with conditions for the validity of the weighted convolution inequality when p ≥ 1.

We establish a quantitative version of Vijayaraghavan's classical result and use it to give a short proof of the known theorem that a real sequence (sn) which is summable by the Borel method, and which satisfies the one-sided Tauberian condition that n (s n −s n−1 ) is bounded below must be convergent.

We consider symplectic difference systems, which contain as special cases linear Hamiltonian difference systems and Sturm–Liouville difference equations of any even order. An associated discrete quadratic functional is important in discrete variational analysis, and while its positive definiteness has been characterized and is well understood, a characterization of its positive semidefiniteness remained an open problem. In this paper we present the solution to this problem and offer necessary and sufficient conditions for such discrete quadratic functionals to be non-negative definite.AMS 2000 Mathematics subject classification: Primary 39A12; 39A13. Secondary 34B24; 49K99

The objective of this investigation was to assess the changes in chemical composition (lignin, cellulose, hemicelluloses, non-structural compounds, N, and ash) of decomposing litter. Standard Pinus sylvestris needle litter, originating from southern Sweden, was incubated in litterbags at 15 sites selected from the Netherlands to south Spain. The changes in chemical composition of this litter were determined using near infrared reflectance spectroscopy. The hypothesis was that standard (chemically uniform) litter decomposing under a range of climates would show different dynamics of accumulation and loss of C-fractions, N, and ash, relative to mass loss. It was shown that, for a given mass-loss value (10, 20, 30, 40, or 50%), the proportion of lignin, cellulose, hemicelluloses, non-structural compounds, N, and ash in the decomposing pine needles differed between sites. Lignin concentration in the litter residue at 50% mass loss ranged from approximately 26 to 43%, cellulose from 19 to 27%, hemicelluloses from 7 to 11%, non-structural compounds from 19 to 25%, N from 0.7 to 1.3%, and ash content from 1.4 to 10.1%. Lignin concentrations showed the highest range of variation. Lignin concentrations during decomposition were positively related to moisture factors as significant correlations were found with actual evapotranspiration and were improved in multiple regressions by the mean annual precipitation or the water surplus. Cellulose was degraded further at sites with high precipitation whereas hemicellulose degradation was related to temperature. This leads to the conclusion that the remaining organic matter produced by standard litter decomposition within the studied climatic range of variations tended to be more recalcitrant under wet and warm climatic conditions than under cold or dry climate.

Die vorgelegten Daten der durchgefuhrten Arzneimittelwirkstoffrecherche fur das Land Brandenburg, zeigen, dass sowohl fur den Human- als auch den Tierarzneimittelsektor eine umfangliche Anzahl mengenrelevanter Wirkstoffe benannt werden kann. Im Land Brandenburg dominieren im freien Apothekenmarkt Schmerz- und Hustenmittelwirkstoffe. Im verordneten Apothekenmarkt wurden Wirkstoffe gegen Diabetes, Schmerzen, Husten, Asthma, Herz-Kreislauferkrankungen, Rheuma, Gicht so we Antibiotika in Mengen von einer Tonne und mehr pro Jahr in Verkehr gebracht. Im Klinikbereich erreichten Desinfektionsmittel, Rontgenkontrastmittel und Durchblutungsmittel Wirkstoffmengen von mehreren Tonnen pro Jahr. Im Einzugsbereich des Klarwerkes Cottbus sind insgesamt mindestens rund 7 Tonnen Arzneimittelwirkstoffe jahrlich verabreicht worden. Die hochsten mengen entfallen dabei auf Desinfektionsmittel, Schmerz- und Rheumamittel, Diagnostika, Hustenmittel, orale Antidiabetika und Antibiotika mit jeweils mehr als 200 kg pro Jahr. Fast 9 t der verordneten 9,5 t Tierarzneimittelwirkstoffe wurden beim Schwein eingesetzt. 70% der Gesamtmenge aller Wirkstoffe entfallen auf die Antibiotika, hauptsachlich Tetrazykline.

Wahrend der letzten Jahrzehnten hat sich die Umweltchemie meist auf die ‘prioritaren’ Schadstoffe, die in den unterschiedlichsten Umweltgesetzen geregelt sind konzentriert. Dieses Chemikalienspektrum stellt aus Sicht der Okotoxikologie jedoch nur einen kleinen Ausschnitt aus der grosen Anzahl von Chemikalien dar, die es heute zu bewerten gilt, wenn ein holistischer risk assessment Ansatz verfolgt werden soll. Die Arzneimittel stellen eine chemisch sehr diverse Gruppe von bioaktiven Umweltchemikalien dar und haben bisher wenig Beachtung als potentielle Umweltschadstoffe erhalten. Die Literaturrecherche hat versucht fur Human- und Tierarzneimittel und Diagnostika die wichtigsten Expositionsdaten in verschiedenen Umweltmedien (Boden, Oberflachenwasser, Abwasser, Grundwasser, Trinkwasser etc.) zusammenzufuhren.

In most studies dealing with effects of toxic substances in saprotrophic isopods, animals are exposed to the test substance through contaminated food. Because these animals can be in a close contact with the soil surface, the substrate, as an exposure pathway, should not be neglected. Here the authors analyze the toxicokinetic behavior of lindane (γ-hexachlorocyclohexane [γ-HCH]) in the isopod species Porcellionides pruinosus, comparing two exposure routes: food and two soil types (artificial Organisation for Economic Cooperation and Development [OECD] soil and a natural agricultural soil). In the feeding experiment, a strong decrease of γ-HCH concentration over time was observed on the food material, with the animals showing a broader range in chemical assimilation efficiency values (averaging 17.7% and ranging from 10 to 40%). The γ-HCH bioaccumulation results indicate that when animals incubated under both soil types reached a steady state, they displayed much higher body burdens (1,359.60 pg/animal on OECD soil and 1,085.30 pg/animal on natural soil) than those exposed to contaminated food (43.75 pg/animal). Kinetic models also revealed much lower assimilation and elimination rates in the food experiment (20.66 pg/d and 0.10 pg/d) than in both soil experiments (238.60 pg/d and 350.54 pg/d for the assimilation rate and 0.19 pg/d and 0.32 pg/d for the elimination rate). Differences in results between exposure routes are discussed according to equilibrium-partitioning theory and the enhanced relevance of the substrate exposure route is analyzed under future prospects on chemical toxicity testing using isopods.

In this paper we derive various characterizations of strong observability (or observability with unknown inputs) of time-dependent linear systems $$\dot x = Ax + Bu,y = C_0 x$$ with state x, control u, and output y, provided that the matrices A, B, and C0 are smooth enough. We show that strong observability is equivalent to the definiteness of a corresponding quadratic functional, and particularly to a certain asymptotic behaviour of solutions of associated Riccati matrix differential equations. The main tools for the proofs are an application of Ehrling's lemma and a characterization of strong observability via a rank condition on matrices, which are built up from derivatives of the matrices A, B, and C0.

The main results of this paper state optimal constants for estimates of so-called successive minima in two dimensions under a constraint on the denominator. While these inequalities are known for every dimension, best possible constants within these estimates are, of course, notknown for any dimension larger than one and remain unknown for all dimensions larger than two.

The main result of the paper is a Sturmian-type separation theorem for the recessive solutions of linear Hamiltonian difference systems. The assumptions on the Hamiltonian systems allow to include the study of Sturm-Liouville difference equations of higher order. As an application of the main result, some Sturmian-type comparison results are obtained.

In a model ecosystem study, the isopod Porcellio scaber was found to accumulate the herbicide trifluralin and its metabolites with body concentrations from 7–19 times those in litter and topsoil. Concentrations in adult P. scaber were much lower than those found in earthworms, in juvenile P. scaber and in the faeces of P. scaber . No effects of trifluralin applied at recommended rates on P. scaber feeding and defaecation rates nor on weight development and mortality, could be demonstrated. P. scaber is easy to handle as a test organism, but problems arise in relation to standardisation and reproducibility of weight development studies.

In einem Standard-Reproduktionstest (ISO-Draft-Guideline 11268-2) mitFolsomia candida (Willem 1902) (Collembola, Insecta) werden die okotoxikologischen Auswirkungen von gealterten Sprengstoffkontaminationen in Boden vom Militarstandort Hallschlag/Nordeifel auf die Reproduktion eines saprophagen Modellorganismus untersucht. Die z.T. hohen Sprengstoffkonzentrationen (meist TNT und DNT) fuhren zu einer hohen Mortalitat bei den adulten Tieren bzw. zu einer drastischen Reduktion des Reproduktionserfolges. Bei den hochsten Sprengstoffkonzentrationen kommt es zur Ausloschung der gesamten Collembolenpopulation, da auch alle Elterntiere sterben.

There is evidence that N concentration in hardwood leaf litter is reduced when plants are raised in an elevated CO2 atmosphere. Reductions in the N concentration of leaf litter have been found for tree species raised under elevated CO2, with reduction in N concentration ranging from ca. 50% for sweet chestnut (Castanea sativa) to 19% for sycamore (Acer platanoides). However, the effects of elevated CO2 on the chemical composition of litter has been investigated only for a limited number of species. There is also little information on the effects of increased CO2 on the quality of root tissues. If we consider, for example, two important European forest ecosystem types, the dominant species investigated for chemical changes are just a few. Thus, there are whole terrestrial ecosystems in which not a single species has been investigated, meaning that the observed effects of a raised CO2 level on plant litter actually has a large error source. Few reports present data on the effects of elevated CO2 on litter nutrients other than N, which limits our ability to predict the effects of elevated CO2 on litter quality and thus on its decomposability. In litter decomposition three separate steps are seen: (i) the initial stages, (ii) the later stages, and (iii) the final stages. The concept of "substrate quality," translated into chemical composition, will thus change between early stages of decomposition and later ones, with a balanced proportion of nutrients (e.g., N, P, S) being required in the early decomposition phase. In the later stages decomposition rates are ruled by lignin degradation and that process is regulated by the availability of certain nutrients (e.g., N, Mn), which act as signals to the lignin-degrading soil microflora. In the final stages the decomposition comes to a stop or may reach an extremely low decomposition rate, so low that asymptotic decomposition values may be estimated and negatively related to N concentrations. Studies on the effects of changes in chemical composition on the decomposability of litter have mainly been made during the early decomposition stages and they generally report decreased litter quality (e.g., increased C/N ratio), resulting in lower decomposition rates for litter raised under elevated CO2 as compared with control litter. No reports are found relating chemical changes induced by elevated CO2 to litter mass-loss rates in late stages. By most definitions, at these stages litter has turned into humus, and many studies demonstrated that a raising of the N level may suppress humus decomposition rate. It is thus reasonable to speculate that a decrease in N levels in humus would accelerate decomposition and allow it to proceed further. There are no experimental data on the long-term effect of elevated CO2 levels, and a decrease in the storage of humus and nutrients could be predicted, at least in temperate and boreal forest systems. Future works on the effects of elevated CO2 on litter quality need to include studies of a larger number of nutrients and chemical components, and to cover different stages of decomposition. Additionally, the response of plant litter quality to elevated CO2 needs to be investigated under field conditions and at the community level, where possible shifts in community composition (i.e., C3 versus C4 ; N2 fixers versus nonfixers) predicted under elevated CO2 are taken into account.Key words: climate change, substrate quality, carbon dioxide, plant litter, chemical composition, decomposition.

The purpose of this study was to understand the variation in root litter mass‐loss rates in a climatic transect across a large region ranging from the Arctic Circle (66°N) in Scandinavia to Berlin (52°N) in north‐east Germany, dominated by coniferous monocultural forests. To investigate the combined influence of litter quality and climate, data were used from 37 sites and plots at which root litters of three coniferous species (Pinus sylvestris, Pinus contorta and Picea abies) had been incubated. In spite of the considerable climatic differences among sites, there were no strong relationships between any climatic variable and the lst‐yr mass loss (range 17.0–40.9%). For the lst‐yr mass loss, the average annual temperature was the most rate‐regulating factor for all litter combined, with a value for R 2 adj of 0.186. Substrate quality also had an influence. Thus, for the whole region lignin concentration was significant, with a value for R 2 adj of 0.142. For the separated groups of pine and spruce stands ...

The paper deals with the Riccati matrix difference equation, which corresponds to the Sturm-Liouville equation of even order namely: with real parameter λ The main thorem states in essence the asymptotic expansion of the principal solution of Riccati's equation as λ→−∞ in terms of negative powers of λVia a discrete Picone type identity the asymptotic result is used to derive an inequality for a discrete quadratic functional containing n thorder differences.

Bei der okotoxikologischen Bewertung von Umweltchemikalien in Boden werden meist Ergebnisse von Labortoxizitatstests an Bodentieren herangezogen. Die modellhaften Laborexprrimente sind aber nicht in der Lage, die komplexe okochemische Situation der Freilandboden abzubilden. Es werden beisplelhaft Untersuchungen vorgestellt, in denen die Unterschiede der bioverfugbaren Schwermetallanteile und die Schwermetalltoxizitat im Vergleich von Freilandboden zu dem “OECD-artificial soil” anhand von unterschiedlichen Bodenparametern dargelegt werden. So kann gezeigt werden, das aufgrund von Ton- und Corg-Qualitaten der “artificial soil” geringere Absorptionseigenschaften und damit hohere Bioverfugbarkeiten und Toxizitaten als vergleichbare Freilandboden hat. Daruber hinaus kann das zur Zeit am weitesten verbreitete “soil risk assessment”-Verfahren der Niederlande bei bestimmten Boden zu einer Uberbewertung der Schwermetalltoxizitaten fuhren.

The following pointwise maximum modulus theorem for the Stokes system in dimension 2 is proven. Suppose that v is a Stokes function in the unit disc with corresponding pressure p, i.e., D) with , such that v is continuous on Then, for x ∊D, This result is shown via a Stokes-Poisson integral formula using Parseval's identity for Fourier series. A consequence of this pointwise estimate is the global estimate (“maximum modulus theorem”)

The main result of this paper is an oscillation theorem on linear self-adjoint differential systems and a corresponding eigenvalue problem. It establishes a formula between the number of focal points of a so-called conjoined basis of the differential system on a given compact interval and the number of eigenvalues which are less than the given eigenvalue parameter. It extends an earlier result of the author and generalizes an oscillation theorem of M. Morse. Among others the proof of the theorem requires a formula on the index of the difference of symmetric solutions of a corresponding Riccati matrix differential equation. This index formula is the other new result presented.

For given matrices A, B, C there is considered the time-invariant linear system x = Ax + Bu, y = Cx with state x, input u, and output y. It is called strongly observable if x = Ax + Bu, Cx(t) ≡ 0 with a piecewise continuous control u(t) always implies x(t) ≡ 0. This means that, for any piecewise continuous input u(t), the output y(t) can vanish identically only if the state x(t) vanishes already, so that the state x(t) can be expressed (“observed”) by the output y(t) alone [without knowing u(t)]. The derivation of such a formula (observer), which expresses x(t) in terms of y(t) alone, for time-invariant systems (i.e. constant matrices A, B, C) is one part of the contents of this note. The other part consists of characterizations of strong observability by rank conditions concerning the matrices A, B, and C (similarly to the well-known rank condition for controllability or observability).

The main result of this paper is the following index theorem, which is closely related to oscillation theorems on linear selfadjoint differential systems such as results by M. Morse. Let real $m\times m$-matrices $R_1, R_2, X, U $ be given, which satisfy $$ R_1R_2^T = R_2R_1^T \,, \quad X^TU = U^TX \,, \quad \:\mbox{rank}\:(R_1,R_2) = \;\mbox{rank}\:(X^T,U^T) = m. $$ Moreover, assume that $X(t), \;U(t)$ are real $m\times m$-matrix-valued functions on some interval ${\cal{J}} = [-\varepsilon , \varepsilon ] , \;\; \varepsilon >0$, such that $$ X^T(t) U(t) = U^T(t) X(t) \quad \mbox{on} \quad {\cal{J}}, $$ $$ X(t) \to X \quad \mbox{and} \quad U(t) \to U \quad \mbox{as} \quad t \to 0, $$ $$ X(t) \quad \mbox{is invertible for} \quad t\in {\cal{J}}\backslash \{0\} \,, \quad \mbox{and such that} $$ $$ U(t) X^{-1}(t) \quad \mbox{is decreasing on} \quad {\cal{J}}\backslash \{0\}, $$ and define $$ M(t) \equiv R_1R_2^T + R_2 U(t) X^{-1}(t) R_2^T , \quad \Lambda (t) \equiv R_1X(t) + R_2U(t) , \quad \Lambda \equiv R_1X + R_2U. $$ Then $\mbox{ind}\:M(0+), \;\;\mbox{ind}\:M(0-) ,$ and $\mbox{def}\:\Lambda (0+)$ exist and $$ \mbox{ind}\:M(0+) - \;\mbox{ind}\:M(0-) = \;\mbox{def}\:\Lambda - \;\mbox{def}\:\Lambda (0+) - \;\mbox{def}\:X , $$ where $\mbox{ind}$ denotes the index (the number of negative eigenvalues) and $\mbox{def}$ denotes the defect (the dimension of the kernel) of a matrix. The basic tool for the proof of this result consists of a theorem on the rank of a certain product of matrices, so that this rank theorem is the key result of the present paper.

In this paper, we consider linear Hamiltonian differential systems which depend in general nonlinearly on the spectral parameter and with Dirichlet boundary conditions. Our results generalize the known theory of linear Hamiltonian systems in two respects. Namely, we allow nonlinear dependence of the coefficients on the spectral parameter and at the same time we do not impose any controllability and strict normality assumptions. We introduce the notion of a finite eigenvalue and prove the oscillation theorem relating the number of finite eigenvalues which are less than or equal to a given value of the spectral parameter with the number of proper focal points of the principal solution of the system in the considered interval. We also define the corresponding geometric multiplicity of finite eigenvalues in terms of finite eigenfunctions and prove that the algebraic and geometric multiplicities coincide. The results are also new for Sturm–Liouville differential equations, being special linear Hamiltonian systems.

This paper is concerned with the minimization of a given quadratic functional for time-invariant, controllable linear systems with the constraint that another quadratic functional (the ‘output energy’) assumes a given value. It is shown that (under certain assumptions) an optimal solution exists if and only if the linear system is strongly observable (i.e. observable with unknown inputs).

We improve known Tauberian results concerning the power series method of summability Jp based on the sequence {pn} by removing the condition that pn be asymptotically logarithmico-exponential. We also prove an entirely new Tauberian result for rapidly decreasing pn.

In the first part of this paper it is shown that the formulae u = ũ − 1 2 {(ũx + ṽy)x + (ṽx − ũy)y}, and v = ṽ − 1 2 {(ũx + ṽy)y − (ṽx − ũy)x} with real-analytic ũ and ṽ represent a Stokes flow v = (u, v) (i.e., Δv = grad p, div v = 0) in a disc around the origin if and only if ũ and ṽ are harmonic. In the second part of the paper a solution v and p of the inhomogeneous Stokes equations (i.e., Δv grad p + h, div v = 0) in a domain G ⊂ R 2 is given by an explicit integral formula involving the conformal mapping of G onto the unit disc.

The main result of this paper reads as follows: Given (real) $(n \times n)$ -matrices A, B, C, and $C_0 $ such that B, C, and $C_0 $ are symmetric, such that B and $C_0 $ are nonnegative definite, such that the pair $(A,B)$ is (completely) controllable (i.e., rank $[B,A, \cdots ,A^{n - 1} B] = n$), and such that the triple $(A,B,C_0 )$ is strongly observable (i.e., $\dot x = Ax + Bu$, $C_0 x \equiv 0$ on some nondegenerate interval implies $x(t) \equiv 0$), then, for any $t_0 > 0$ and any symmetric matrix $Q_0 $, the solution $Q(t;\lambda )$ of the Riccati matrix differential equation \[\dot Q + A^T Q + QA + QBQ - C + \lambda C_0 = 0,\quad Q(0) = Q_0 \] exists on $[0,t_0 ]$ if $\lambda \leqq \lambda _0 $, and it satisfies $\lim _{\lambda \to - \infty } Q(t_0 ;\lambda ) = \infty $ (i.e., all eigenvalues of the symmetric matrix Q tend to $\infty $). The result is that the new notion of strong observability is even necessary for the assertion. This result on Riccati matrix equations is motivated by the follo...

as t ~ to E IR (where ,.\ is fixed), and as ,.\ ~ -00 (where t is fixed). The following assumptions on the n x n-matrices A, B, C and Co are required: A(t), B(t), C(t) and Co(t) are piecewise continuous on IR; B(t), C(t) and Co(r) are symmetric on IR; B(t) and Co(t) are nonnegative definite on IR; the kernels of B(t) and Co(r) are constant on IR; and AT(t)K is a constant matrix on IR where kernel B(t) == image of K.

In this paper representations of Stokes flows in dimensions 2 and 3, which reduce the Stokes equations to the Laplace equation for an auxiliary function, are given. While it is known that two- and three-dimensional Stokes flows may be reduced to biharmonic problems, the representations here are new. The main result of this paper reads as follows: Given a domain $G \subseteq \mathbb{R}^3 $, which is star-shaped with respect to the origin, and functions $\vec v:G \to \mathbb{R}^3 $, $p:G \to \mathbb{R}$, then $\vec v$ and p represent a Stokes flow with velocity field $\vec v$ and pressure p in G (i.e., $\Delta \vec v = {\operatorname{grad}}p$ and ${\operatorname{div}}\vec v = 0$ in G) if and only if $\vec v$ and p are of the form \[ \vec v(\vec x) = \tilde \vec v(\vec x) - \frac{1}{3}\left\{ {{\operatorname{div}}\tilde \vec v(\vec x) \cdot \vec x + \vec x \times {\operatorname{curl}}\tilde \vec v(\vec x)} \right\},\quad p(\vec x) = - \frac{4}{3}{\operatorname{div}}\tilde \vec v(\vec x),\] where $\tilde \vec...