Stability of equilibria for a nonlinear population model with age- and spatial-structure

In this thesis we study a nonlinear partial differential equation which models the time evolution of a population with age- and spatial-structure. In an abstract setting, this model reads \begin{eqnarray}\label{Pabstract} \partial_t u + \partial_a u +A(a)u &=& - \mu(u,a)u, \qquad t > 0, \, a \in (0,a_m), \nonumber\\ u(t,0) &=& \int_0^{a_m} b(u,a)u(t,a)\, \text{d}a, \qquad t > 0, \\ u(0,a) &=& u_0 (a), \qquad a \in (0,a_m), \nonumber \end{eqnarray} where $u: [0,T) \to \mathbb{E}_0$ is interpreted as the density function of the population, taking values in an appropriate function space $\mathbb{E}_0$, $b=b(u,a) \geq 0$ and $\mu = \mu (u,a) \geq 0$ are the birth and mortality rates, and $A(a): E_1 \subset E_0 \to E_0$ is a closed operator on the real Banach lattice $E_0$, for each $a \in J:= [0,a_m)$. In our considerations to follow we fix $p \in [1, \infty)$ and set $\mathbb{E}_0:= L_p(J, E_0).$ In the first part we consider the semilinear model of age- and spatial-structured population dynamics, which is obtained when the birth law is assumed to be linear. Put differently, the birth rate in problem (0.1) is supposed to be a function of the age-parameter only, i.e. b = b(a). Assuming A generates a parabolic evolution operator, it is then shown that this semilinear structure allows to formulate problem (0.1) as a semilinear Cauchy problem in the Banach space $\mathbb{E}_0$. In particular, we can study mild solutions, their asymptotic behaviour, and convergence to equilibria, and we will see that the stability analysis can be reduced to the linearised problem. In a subsequent step, the spectral theory of positive compact operators is applied to this linear problem, and as a result we will see that the stability behaviour is completely determined by a single quantity, namely the spectral radius of an associated operator. It should be noted that essential ingredients for this result are the assumptions of maximal Lp-regularity of the spatial diffusion operator A, and the positivity of the parabolic evolution operator generated by A. In a subsequent part, we introduce a weak solution concept for problem (0.1). Assuming A generates a parabolic evolution operator, these so called integral solutions are constructed for the linearised problem in a first step. In the second step we apply a fixed-point argument in order to establish existence of integral solutions for the nonlinear problem. Furthermore we carry out a detailed analysis of the linear inhomogeneous problem, which serves as a preparation for the last part. In the final part we study the stability behaviour of equilibria to problem (0,1). It has to be pointed out that the birth rate is now allowed to depend on the density, i.e. b = b(u,.), and consequently we lose the semilinear structure, as considered in the first part. In particular, mild solutions are not available any longer, and this is where the integral solutions of the foregoing part come into play. More precisely, we will prove that problem (0,1) is well-posed within this framework. Finally, it is shown that a principle of linearised stability is available within this setting

ASDEX Upgrades New Plasma Control Scheme

ASDEX Upgrade is a medium sized tokamak experiment investigating highly shaped plasma and advanced scenarios to be extrapolated for ITER. Eleven independent magnetic coils allow for proper shaping and plasma current control. For plasma heating and current drive eight NBI beam lines, two ICRH antenna pairs and four ECRH gyrotrons are available. Five channels for controlling gas valves and a pellet injector serve for fuelling. All actuators are driven by a digital discharge control system. One basic enhancement of the latest generation is a unified framework for all feedforward and feedback control tasks in a discharge. The framework consists of two layers. The core layer implements wind-up safe feedback controllers with a collection of overlayed output limitations. Each controller is dynamically switchable in references, controlled variables, control law and control parameters via a control mode. The coordination layer implements intelligent discharge protection or optimisation algorithms which synchronously can change control modes and dynamically can generate reference waveforms adapted to the discharge's state and goal. The core layer comprises the backbone of plasma control. Current, shape, heating and fuel control all use a library of highly configurable single- and multivalriable control laws. P, PI and PID controllers are standard components but state space and sliding mode policies can easily be supplemented, too. Likewise, a broad selection of output limiters is available in the library. It ranges from constant values to rate limiters, and multi-signal dependent polynomial characteristics. The controller is aware of any output limitation and can take anti-wind-up measures. Furthermore, a feedforward policy allows to tune the behaviour upon mode transitions, like smooth adaptation or freezing the last output. With the coordination layer, tasks like marfe protection, power exhaust protection and soft pulse termination are accomplished. These specialised algorithms are plugged into the framework using a common interface. The framework approach easily allows for further extensions and opens a door for future experimental investigations

Giant thermal expansion and α-precipitation pathways in Ti-Alloys

Ti-Alloys represent the principal structural materials in both aerospace development and metallic biomaterials. Key to optimizing their mechanical and functional behaviour is in-depth know-how of their phases and the complex interplay of diffusive vs. displacive phase transformations to permit the tailoring of intricate microstructures across a wide spectrum of configurations. Here, we report on structural changes and phase transformations of Ti-Nb alloys during heating by in situ synchrotron diffraction. These materials exhibit anisotropic thermal expansion yielding some of the largest linear expansion coefficients (+ 163.9×10-6 to-95.1×10-6 °C-1) ever reported. Moreover, we describe two pathways leading to the precipitation of the α-phase mediated by diffusion-based orthorhombic structures, α″lean and α″iso. Via coupling the lattice parameters to composition both phases evolve into α through rejection of Nb. These findings have the potential to promote new microstructural design approaches for Ti-Nb alloys and β-stabilized Ti-Alloys in general

Enlarging the synthetic biology toolbox for Pichia pastoris: Golden Gate cloning and CRISPR/Cas9

State-of-the-art strain engineering techniques for the protein producing yeast host Pichia pastoris include overexpression of homologous and heterologous genes, and deletion of host genes. For this purpose overexpression vectors and gene deletion methods such as the split marker technique have been established. For metabolic and cell engineering purposes, the simultaneous overexpression of more than one gene is often needed. Previous approaches employing subsequent steps of overexpression and marker recycling were time- and labor-consuming. Therefore, efficient systems allowing multiple gene overexpression are required, that can be stably integrated into the P. pastoris genome. To this end, we developed a synthetic biology toolbox based on Golden Gate cloning to enable efficient construction of complex and versatile over-expression vectors. Up to five different expression cassettes, employing a library of promoters and terminators can be combined into one vector, and successfully integrated into the genomic DNA of P. pastoris at targeted loci in one step. Recent trends in synthetic biology, however, go into the direction of building up large and complex reaction networks. To allow for clean and unscarred genetic engineering, a CRISPR/Cas9 based method for gene insertions, deletions and replacements was developed, which paves the way for precise genomic rearrangements in P. pastoris. By using this technique precise genomic integrations were performed efficiently without integrative selection markers. The repertoire of genetic techniques developed so far, will provide a wide variety of possibilities to engineer P. pastoris. Applications for these synthetic biology tools in cell engineering of recombinant P. pastoris will be presented