Ergodicity and regularity of invariant measure for branching Markov processes with immigration

In this thesis we consider systems of finitely many particles moving on paths given by a strong Markov process and undergoing branching and reproduction at random times. The branching rate of a particle, its number of offspring and their spatial distribution are allowed to depend on the particle's position and possibly on the configuration of coexisting particles. In addition there is immigration of new particles, with the rate of immigration and the distribution of immigrants possibly depending on the configuration of pre-existing particles as well. In the first two chapters of this work, we concentrate on the case that the joint motion of particles is governed by a diffusion with interacting components. The resulting process of particle configurations was studied by E. Löcherbach (2002, 2004) and is known as a branching diffusion with immigration (BDI). Chapter 1 contains a detailed introduction of the basic model assumptions, in particular an assumption of ergodicity which guarantees that the BDI process is positive Harris recurrent with finite invariant measure on the configuration space. This object and a closely related quantity, namely the invariant occupation measure on the single-particle space, are investigated in Chapter 2 where we study the problem of the existence of Lebesgue-densities with nice regularity properties. For example, it turns out that the existence of a continuous density for the invariant measure depends on the mechanism by which newborn particles are distributed in space, namely whether branching particles reproduce at their death position or their offspring are distributed according to an absolutely continuous transition kernel. In Chapter 3, we assume that the quantities defining the model depend only on the spatial position but not on the configuration of coexisting particles. In this framework (which was considered by Höpfner and Löcherbach (2005) in the special case that branching particles reproduce at their death position), the particle motions are independent, and we can allow for more general Markov processes instead of diffusions. The resulting configuration process is a branching Markov process in the sense introduced by Ikeda, Nagasawa and Watanabe (1968), complemented by an immigration mechanism. Generalizing results obtained by Höpfner and Löcherbach (2005), we give sufficient conditions for ergodicity in the sense of positive recurrence of the configuration process and finiteness of the invariant occupation measure in the case of general particle motions and offspring distributions.

Feature-invariant image registration method for quantification of surgical outcomes in patients with craniosynostosis: a preliminary study

Craniosynostosis consists of a premature fusion of the sutures in an infant skull that restricts skull and brain growth. During the last decades, there has been a rapid increase of fundamentally diverse surgical treatment methods. At present, the surgical outcome has been assessed using global variables such as cephalic index, head circumference, and intracranial volume. However, these variables have failed in describing the local deformations and morphological changes that may have a role in the neurologic disorders observed in the patients. This report describes a rigid image registration-based method to evaluate outcomes of craniosynostosis surgical treatments, local quantification of head growth, and indirect intracranial volume change measurements. The developed semiautomatic analysis method was applied to computed tomography data sets of a 5-month-old boy with sagittal craniosynostosis who underwent expansion of the posterior skull with cranioplasty. Quantification of the local changes between pre- and postoperative images was quantified by mapping the minimum distance of individual points from the preoperative to the postoperative surface meshes, and indirect intracranial volume changes were estimated. The proposed methodology can provide the surgeon a tool for the quantitative evaluation of surgical procedures and detection of abnormalities of the infant skull and its development.

Invariant barriers to reactive front propagation in fluid flows

We present theory and experiments on the dynamics of reaction fronts in two-dimensional, vortex-dominated flows, for both time-independent and periodically driven cases. We find that the front propagation process is controlled by one-sided barriers that are either fixed in the laboratory frame (time-independent flows) or oscillate periodically (periodically driven flows). We call these barriers burning invariant manifolds (BIMs), since their role in front propagation is analogous to that of invariant manifolds in the transport and mixing of passive impurities under advection. Theoretically, the BIMs emerge from a dynamical systems approach when the advection-reaction-diffusion dynamics is recast as an ODE for front element dynamics. Experimentally, we measure the location of BIMs for several laboratory flows and confirm their role as barriers to front propagation.

Natural variations at position 93 of the invariant Vα24-Jα18 α chain of human iNKT-cell TCRs strongly impact on CD1d binding

Human invariant natural killer T (NKT) cell TCRs bind to CD1d via an "invariant" Vα24-Jα18 chain (iNKTα) paired to semi-invariant Vβ11 chains (iNKTβ). Single-amino acid variations at position 93 (p93) of iNKTα, immediately upstream of the "invariant" CDR3α region, have been reported in a substantial proportion of human iNKT-cell clones (4-30%). Although p93, a serine in most human iNKT-cell TCRs, makes no contact with CD1d, it could affect CD1d binding by altering the conformation of the crucial CDR3α loop. By generating recombinant refolded iNKT-cell TCRs, we show that natural single-nucleotide variations in iNKTα, translating to serine, threonine, asparagine or isoleucine at p93, exert a powerful effect on CD1d binding, with up to 28-fold differences in affinity between these variants. This effect was observed with CD1d loaded with either the artificial α-galactosylceramide antigens KRN7000 or OCH, or the endogenous glycolipid β-galactosylceramide, and its importance for autoreactive recognition of endogenous lipids was demonstrated by the binding of variant iNKT-cell TCR tetramers to cell surface expressed CD1d. The serine-containing variant showed the strongest CD1d binding, offering an explanation for its predominance in vivo. Complementary molecular dynamics modeling studies were consistent with an impact of p93 on the conformation of the CDR3α loop.

Intestinal microbes affect phenotypes and functions of invariant natural killer T cells in mice

Invariant natural killer T (iNKT) cells undergo canonical, Vα14-Jα18 rearrangement of the T-cell receptor (TCR) in mice; this form of the TCR recognizes glycolipids presented by CD1d. iNKT cells mediate many different immune reactions. Their constitutive activated and memory phenotype and rapid initiation of effector functions after stimulation indicate previous antigen-specific stimulation. However, little is known about this process. We investigated whether symbiotic microbes can determine the activated phenotype and function of iNKT cells.

Light Transport in PT-invariant Photonic Structures with Hidden Symmetries

We introduce a recursive bosonic quantization technique for generating classical PT photonic structures that possess hidden symmetries and higher order exceptional points. We study light transport in these geometries and we demonstrate that perfect state transfer is possible only for certain initial conditions. Moreover, we show that for the same propagation direction, left and right coherent transports are not symmetric with field amplitudes following two different trajectories. A general scheme for identifying the conservation laws in such PT-symmetric photonic networks is also presented.

Kernels and Designs for Modelling Invariant Functions: From Group Invariance to Additivity

We focus on kernels incorporating different kinds of prior knowledge on functions to be approximated by Kriging. A recent result on random fields with paths invariant under a group action is generalised to combinations of composition operators, and a characterisation of kernels leading to random fields with additive paths is obtained as a corollary. A discussion follows on some implications on design of experiments, and it is shown in the case of additive kernels that the so-called class of “axis designs” outperforms Latin hypercubes in terms of the IMSE criterion.

Holomorphic families of nonequivalent embeddings and of holomorphic group actions on affine space

We construct holomorphic families of proper holomorphic embeddings of \mathbb {C}^{k} into \mathbb {C}^{n} (0\textless k\textless n-1), so that for any two different parameters in the family, no holomorphic automorphism of \mathbb {C}^{n} can map the image of the corresponding two embeddings onto each other. As an application to the study of the group of holomorphic automorphisms of \mathbb {C}^{n}, we derive the existence of families of holomorphic \mathbb {C}^{*}-actions on \mathbb {C}^{n} (n\ge5) so that different actions in the family are not conjugate. This result is surprising in view of the long-standing holomorphic linearization problem, which, in particular, asked whether there would be more than one conjugacy class of \mathbb {C}^{*}-actions on \mathbb {C}^{n} (with prescribed linear part at a fixed point).