Time Discretization of the SST-generalized Navier-Stokes Equations: Positive and Negative Results

In the theory of the Navier-Stokes equations, the proofs of some basic known results, like for example the uniqueness of solutions to the stationary Navier-Stokes equations under smallness assumptions on the data or the stability of certain time discretization schemes, actually only use a small range of properties and are therefore valid in a more general context. This observation leads us to introduce the concept of SST spaces, a generalization of the functional setting for the Navier-Stokes equations. It allows us to prove (by means of counterexamples) that several uniqueness and stability conjectures that are still open in the case of the Navier-Stokes equations have a negative answer in the larger class of SST spaces, thereby showing that proof strategies used for a number of classical results are not sufficient to affirmatively answer these open questions. More precisely, in the larger class of SST spaces, non-uniqueness phenomena can be observed for the implicit Euler scheme, for two nonlinear versions of the Crank-Nicolson scheme, for the fractional step theta scheme, and for the SST-generalized stationary Navier-Stokes equations. As far as stability is concerned, a linear version of the Euler scheme, a nonlinear version of the Crank-Nicolson scheme, and the fractional step theta scheme turn out to be non-stable in the class of SST spaces. The positive results established in this thesis include the generalization of classical uniqueness and stability results to SST spaces, the uniqueness of solutions (under smallness assumptions) to two nonlinear versions of the Euler scheme, two nonlinear versions of the Crank-Nicolson scheme, and the fractional step theta scheme for general SST spaces, the second order convergence of a version of the Crank-Nicolson scheme, and a new proof of the first order convergence of the implicit Euler scheme for the Navier-Stokes equations. For each convergence result, we provide conditions on the data that guarantee the existence of nonstationary solutions satisfying the regularity assumptions needed for the corresponding convergence theorem. In the case of the Crank-Nicolson scheme, this involves a compatibility condition at the corner of the space-time cylinder, which can be satisfied via a suitable prescription of the initial acceleration.

Nitrogen use efficiency and optimization of nitrogen fertilizer application for stable yields and high quality of cereals grown in conservation agriculture

In most agroecosystems, nitrogen (N) is the most important nutrient limiting plant growth. One management strategy that affects N cycling and N use efficiency (NUE) is conservation agriculture (CA), an agricultural system based on a combination of minimum tillage, crop residue retention and crop rotation. Available results on the optimization of NUE in CA are inconsistent and studies that cover all three components of CA are scarce. Presently, CA is promoted in the Yaqui Valley in Northern Mexico, the country´s major wheat-producing area in which from 1968 to 1995, fertilizer application rates for the cultivation of irrigated durum wheat (Triticum durum L.) at 6 t ha-1 increased from 80 to 250 kg ha-1, demonstrating the high intensification potential in this region. Given major knowledge gaps on N availability in CA this thesis summarizes the current knowledge of N management in CA and provides insights in the effects of tillage practice, residue management and crop rotation on wheat grain quality and N cycling. Major aims of the study were to identify N fertilizer application strategies that improve N use efficiency and reduce N immobilization in CA with the ultimate goal to stabilize cereal yields, maintain grain quality, minimize N losses into the environment and reduce farmers’ input costs. Soil physical and chemical properties in CA were measured and compared with those in conventional systems and permanent beds with residue burning focusing on their relationship to plant N uptake and N cycling in the soil and how they are affected by tillage and N fertilizer timing, method and doses. For N fertilizer management, we analyzed how placement, time and amount of N fertilizer influenced yield and quality parameters of durum and bread wheat in CA systems. Overall, grain quality parameters, in particular grain protein concentration decreased with zero-tillage and increasing amount of residues left on the field compared with conventional systems. The second part of the dissertation provides an overview of applied methodologies to measure NUE and its components. We evaluated the methodology of ion exchange resin cartridges under irrigated, intensive agricultural cropping systems on Vertisols to measure nitrate leaching losses which through drainage channels ultimately end up in the Sea of Cortez where they lead to algae blooming. A throughout analysis of N inputs and outputs was conducted to calculate N balances in three different tillage-straw systems. As fertilizer inputs are high, N balances were positive in all treatments indicating the risk of N leaching or volatilization during or in subsequent cropping seasons and during heavy rain fall in summer. Contrary to common belief, we did not find negative effects of residue burning on soil nutrient status, yield or N uptake. A labeled fertilizer experiment with urea 15N was implemented in micro-plots to measure N fertilizer recovery and the effects of residual fertilizer N in the soil from summer maize on the following winter crop wheat. Obtained N fertilizer recovery rates for maize grain were with an average of 11% very low for all treatments.

Negative Größen bei Diophant?

In this paper we champion Diophantus of Alexandria and Isabella Basmakova against Norbert Schappacher. In two publications ([46] and [47]) he puts forward inter alia two propositions: Questioning Diophantus' originality he considers affirmatively the possibility, that the Arithmetica are the joint work of a team of authors like Bourbaki. And he calls Basmakova's claim (in [5]), that Diophantus uses negative numbers, a "nonsense", reproaching her for her "thoughtlessness". First, we disprove Schappacher's Bourbaki thesis. Second, we investigate the semantic meaning and historical significance of Diophantus' keywords leipsis and mparxis. Next, we discuss Schappacher's epistemology of the history of mathematics and defend Basmakova's methods. Furthermore, we give 33 places where Diophantus uses negative quantities as intermediate results; they appear as differences a - b of positive rational numbers, the subtrahend b being bigger than the minuend a; they each represent the (negative) basis (pleyra) of a square number (tetragonos), which is afterwards computed by the formula (a - b)^2 = a^2 + b^2 - 2ab. Finally, we report how the topic "Diophantus and the negative numbers" has been dealt with by translators and commentators from Maximus Planudes onwards.

Predicted properties of the superheavy elements. III. Element 115, Eka-Bismuth

Element 115 is expected to be in group V-a of the periodic table and have most stable oxidation states of I and III. The oxidation state of I, which plays a minor role in bismuth chemistry, should be a major factor in 115 chemistry. This change will arise because of the large relativistic splitting of the spherically symmetric 7p_l/2 shell from the 7P_3/2 shell. Element 115 will therefore have a single 7p_3/2 electron outside a 7p^2_1/2 closed shell. The magnitude of the first ionization energy and ionic radius suggest a chemistry similar to Tl^+. Similar considerations suggest that 115^3+ will have a chemistry similar to Bi^3+. Hydrolysis will therefore be easy and relatively strongly complexing anions of strong acids will be needed in general to effect studies of complexation chemistry. Some other properties of 115 predicted are as follows: ionization potentials I 5.2 eV, II 18.1 eV, III 27.4 eV, IV 48.5 eV, 0 \rightarrow 5^+ 159 eV; heat of sublimation, 34 kcal (g-atom)^-1; atomic radius, 2.0 A; ionic radius, 115^+ 1.5 A, 115^3+ 1.0 A; entropy, 16 cal deg^-1 (g-atom)^-l (25°); standard electrode potential 115^+ |115, -1.5 V; melting and boiling points are similar to element 113.

On Oseen Resolvent Estimates: A Negative Result

We consider the resolvent problem for the scalar Oseen equation in the whole space R^3. We show that for small values of the resolvent parameter it is impossible to obtain an L^2-estimate analogous to the one which is valid for the Stokes resolvent, even if the resolvent parameter has positive real part.

Relativistic LCAO with Minimax Principle and New Balanced Basis Sets

Relativistic density functional theory is widely applied in molecular calculations with heavy atoms, where relativistic and correlation effects are on the same footing. Variational stability of the Dirac Hamiltonian is a very important field of research from the beginning of relativistic molecular calculations on, among efforts for accuracy, efficiency, and density functional formulation, etc. Approximations of one- or two-component methods and searching for suitable basis sets are two major means for good projection power against the negative continuum. The minimax two-component spinor linear combination of atomic orbitals (LCAO) is applied in the present work for both light and super-heavy one-electron systems, providing good approximations in the whole energy spectrum, being close to the benchmark minimax finite element method (FEM) values and without spurious and contaminated states, in contrast to the presence of these artifacts in the traditional four-component spinor LCAO. The variational stability assures that minimax LCAO is bounded from below. New balanced basis sets, kinetic and potential defect balanced (TVDB), following the minimax idea, are applied with the Dirac Hamiltonian. Its performance in the same super-heavy one-electron quasi-molecules shows also very good projection capability against variational collapse, as the minimax LCAO is taken as the best projection to compare with. The TVDB method has twice as many basis coefficients as four-component spinor LCAO, which becomes now linear and overcomes the disadvantage of great time-consumption in the minimax method. The calculation with both the TVDB method and the traditional LCAO method for the dimers with elements in group 11 of the periodic table investigates their difference. New bigger basis sets are constructed than in previous research, achieving high accuracy within the functionals involved. Their difference in total energy is much smaller than the basis incompleteness error, showing that the traditional four-spinor LCAO keeps enough projection power from the numerical atomic orbitals and is suitable in research on relativistic quantum chemistry. In scattering investigations for the same comparison purpose, the failure of the traditional LCAO method of providing a stable spectrum with increasing size of basis sets is contrasted to the TVDB method, which contains no spurious states already without pre-orthogonalization of basis sets. Keeping the same conditions including the accuracy of matrix elements shows that the variational instability prevails over the linear dependence of the basis sets. The success of the TVDB method manifests its capability not only in relativistic quantum chemistry but also for scattering and under the influence of strong external electronic and magnetic fields. The good accuracy in total energy with large basis sets and the good projection property encourage wider research on different molecules, with better functionals, and on small effects.

Untersuchungen zu Controlled Traffic Farming und satellitenbasierten Lenksystemen

Diese Arbeit behandelt Controlled Traffic Farming (CTF) Anbausysteme, bei denen für alle Arbeitsgänge satellitengesteuert immer dieselben Fahrspuren benutzt werden. Lässt sich mit CTF die Belastung des Bodens verringern und die Effizienz von Direktsaat-Anbausystemen steigern? Neben agronomischen und bodenphysikalischen Parametern wurden Auswirkungen von Lenksystemen und Umsetzungsmöglichkeiten von CTF in die Praxis untersucht. Die Analyse einer CTF-Umsetzung unter europäischen Bedingungen mit der Verwendung von Standardmaschinen zeigte, dass sich CTF-Anbausysteme mit den heute zur Verfügung stehenden Maschinen für Dauergrünland, Mähdruschfrüchte und Mais auf kleiner und grösser strukturierten Flächen relativ einfach mechanisieren lassen. Bei Zuckerrüben und Kartoffeln können Kompromisse notwendig sein. Generell erfordern CTF-Anbausysteme eine sorgfältige Planung und Umsetzung in die Praxis. Im dreijährigen Feldversuch (Winterweizen, Wintergerste, Kunstwiese mit Kleegrasmischung) auf einem Lehmboden wurde CTF-Direktsaat mit konventionell zufällig befahrenen Direktsaat- und Pflugverfahren verglichen. Unter CTF zeigte sich eine Differenzierung der nicht, gering und intensiv befahrenen Varianten. Auf dem vorliegenden kompakten Boden mit 1150 mm Jahresniederschlag waren die Unterschiede zwischen den nicht befahrenen Flächen und den mit niedrigem Kontaktflächendruck befahrenen Flächen eher gering. In den nicht befahrenen Flächen entwickelten Eindringwiderstand und Kohlendioxidgehalt der Bodenluft nach drei Jahren signifikant bessere Werte. Bodendichte und Porosität zeigten hingegen keinen eindeutig interpretierbaren Trend. Aufgrund teils suboptimaler Feldaufgänge liess sich keine generelle agronomische Tendenz ableiten. Die intensive Befahrung der Pflegefahrgassen zeigte allerdings klar negative bodenkundliche und planzenbauliche Auswirkungen. Es bietet sich daher an, vor allem für Pflegearbeiten permanent dieselben Fahrspuren zu nutzen. In der Untersuchung zu den Auswirkungen von Lenksystemen zeigten sich signifikante Vorteile von Lenksystemen in einer Verminderung der Fahrerbelastung und einer höheren Lenkgenauigkeit vor allem bei grossen Arbeitsbreiten ohne Spuranreisser. Die meisten anderen Messparameter waren mit Lenksystem leicht vorteilhafter als ohne, unterschieden sich aber nicht signifikant voneinander. Fahrer und naturräumliche Gegebenheiten wie die Schlagform hatten einen wesentlich grösseren Einfluss. Gesamthaft betrachtet erweitert CTF in Kombination mit weiteren Bodenschutzmass-nahmen die Möglichkeiten, Bodenverdichtungen zu vermeiden, den Bedarf an energieintensiver Bodenlocke-rung zu reduzieren und die Entwicklung einer stabileren Bodenstruktur mit höherer Tragfähigkeit zu fördern. Zusammen mit einer an Kultur und Anbausystem angepassten Saatbettbereitung und den in geraden Reihen einfacher durchführbaren mechanischen Pflegemassnahmen ergeben sich gute Voraussetzungen für die Gestaltung agronomisch leistungsfähiger und ökologisch nachhaltiger Anbausysteme.