Equilibrium topology for plasmas with reversed current density.

Actually, transition from positive to negative plasma current and quasi-steady-state alternated current (AC) operation have been achieved experimentally without loss of ionization. The large transition times suggest the use of MHD equilibrium to model the intermediate magnetic field configurations for corresponding current density reversals. In the present work we show, by means of Maxwell equations, that the most robust equilibrium for any axisymmetric configuration with reversed current density requires the existence of several nonested families of magnetic surfaces inside the plasma. We also show that the currents inside the nonested families satisfy additive rules restricting the geometry and sizes of the axisymmetric magnetic islands; this is done without restricting the equilibrium through arbitrary functions. Finally, we introduce a local successive approximations method to describe the equilibrium about an arbitrary reversed current density minimum and, consequently, the transition between different nonested topologies is understood in terms of the eccentricity of the toroidal current density level sets.

Diversity and Stability in Food Webs : Impacts of Non-Equilibrium Dynamics, Topology and Variation

With progressive climate change, the preservation of biodiversity is becoming increasingly important. Only if the gene pool is large enough and requirements of species are diverse, there will be species that can adapt to the changing circumstances. To maintain biodiversity, we must understand the consequences of the various strategies. Mathematical models of population dynamics could provide prognoses. However, a model that would reproduce and explain the mechanisms behind the diversity of species that we observe experimentally and in nature is still needed. A combination of theoretical models with detailed experiments is needed to test biological processes in models and compare predictions with outcomes in reality. In this thesis, several food webs are modeled and analyzed. Among others, models are formulated of laboratory experiments performed in the Zoological Institute of the University of Cologne. Numerical data of the simulations is in good agreement with the real experimental results. Via numerical simulations it can be demonstrated that few assumptions are necessary to reproduce in a model the sustained oscillations of the population size that experiments show. However, analysis indicates that species "thrown together by chance" are not very likely to survive together over long periods. Even larger food nets do not show significantly different outcomes and prove how extraordinary and complicated natural diversity is. In order to produce such a coexistence of randomly selected species—as the experiment does—models require additional information about biological processes or restrictions on the assumptions. Another explanation for the observed coexistence is a slow extinction that takes longer than the observation time. Simulated species survive a comparable period of time before they die out eventually. Interestingly, it can be stated that the same models allow the survival of several species in equilibrium and thus do not follow the so-called competitive exclusion principle. This state of equilibrium is more fragile, however, to changes in nutrient supply than the oscillating coexistence. Overall, the studies show, that having a diverse system means that population numbers are probably oscillating, and on the other hand oscillating population numbers stabilize a food web both against demographic noise as well as against changes of the habitat. Model predictions can certainly not be converted at their face value into policies for real ecosystems. But the stabilizing character of fluctuations should be considered in the regulations of animal populations.

Protein sequence design on the basis of topology optimization techniques -Using continuous modeling of discrete amino acid types

The notion of optimization is inherent in protein design. A long linear chain of twenty types of amino acid residues are known to fold to a 3-D conformation that minimizes the combined inter-residue energy interactions. There are two distinct protein design problems, viz. predicting the folded structure from a given sequence of amino acid monomers (folding problem) and determining a sequence for a given folded structure (inverse folding problem). These two problems have much similarity to engineering structural analysis and structural optimization problems respectively. In the folding problem, a protein chain with a given sequence folds to a conformation, called a native state, which has a unique global minimum energy value when compared to all other unfolded conformations. This involves a search in the conformation space. This is somewhat akin to the principle of minimum potential energy that determines the deformed static equilibrium configuration of an elastic structure of given topology, shape, and size that is subjected to certain boundary conditions. In the inverse-folding problem, one has to design a sequence with some objectives (having a specific feature of the folded structure, docking with another protein, etc.) and constraints (sequence being fixed in some portion, a particular composition of amino acid types, etc.) while obtaining a sequence that would fold to the desired conformation satisfying the criteria of folding. This requires a search in the sequence space. This is similar to structural optimization in the design-variable space wherein a certain feature of structural response is optimized subject to some constraints while satisfying the governing static or dynamic equilibrium equations. Based on this similarity, in this work we apply the topology optimization methods to protein design, discuss modeling issues and present some initial results.

Topology of toroidal helical fields in non-circular cross-sectional tokamaks

The ordinary differential magnetic field line equations are solved numerically; the tokamak magnetic structure is studied on Hefei Tokamak-7 Upgrade (HT-7U) when the equilibrium field with a monotonic q-profile is perturbed by a helical magnetic field. We find that a single mode (m, n) helical perturbation can cause the formation of islands on rational surfaces with q = m/n and q = (m +/- 1, +/- 2, +/- 3,...)/n due to the toroidicity and plasma shape (i.e. elongation and triangularity), while there are many undestroyed magnetic surfaces called Kolmogorov-Arnold-Moser (KAM) barriers on irrational surfaces. The islands on the same rational surface do not have the same size. When the ratio between the perturbing magnetic field B-r(r) and the toroidal magnetic field amplitude B(phi)0 is large enough, the magnetic island chains on different rational surfaces will overlap and chaotic orbits appear in the overlapping area, and the magnetic field becomes stochastic. It is remarkable that the stochastic layer appears first in the plasma edge region.

Effect of Selfish Node Behavior on Efficient Topology Design

The problem of topology control is to assign per-node transmission power such that the resulting topology is energy efficient and satisfies certain global properties such as connectivity. The conventional approach to achieve these objectives is based on the fundamental assumption that nodes are socially responsible. We examine the following question: if nodes behave in a selfish manner, how does it impact the overall connectivity and energy consumption in the resulting topologies? We pose the above problem as a noncooperative game and use game-theoretic analysis to address it. We study Nash equilibrium properties of the topology control game and evaluate the efficiency of the induced topology when nodes employ a greedy best response algorithm. We show that even when the nodes have complete information about the network, the steady-state topologies are suboptimal. We propose a modified algorithm based on a better response dynamic and show that this algorithm is guaranteed to converge to energy-efficient and connected topologies. Moreover, the node transmit power levels are more evenly distributed, and the network performance is comparable to that obtained from centralized algorithms.

Integrable maps with non-trivial topology: application to divertor configurations

We explore a method for constructing two-dimensional area-preserving, integrable maps associated with Hamiltonian systems, with a given set of fixed points and given invariant curves. The method is used to find an integrable Poincare map for the field lines in a large aspect ratio tokamak with a poloidal single-null divertor. The divertor field is a superposition of a magnetohydrodynamic equilibrium with an arbitrarily chosen safety factor profile, with a wire carrying an electric current to create an X-point. This integrable map is perturbed by an impulsive perturbation that describes non-axisymmetric magnetic resonances at the plasma edge. The non-integrable perturbed map is applied to study the structure of the open field lines in the scrape-off layer, reproducing the main transport features obtained by integrating numerically the magnetic field line equations, such as the connection lengths and magnetic footprints on the divertor plate.

STRUCTURE AND DYNAMICS: THE TRANSITION FROM NONEQUILIBRIUM TO EQUILIBRIUM IN INTEGRATE-AND-FIRE DYNAMICS

The transient and equilibrium properties of dynamics unfolding in complex systems can depend critically on specific topological features of the underlying interconnections. In this work, we investigate such a relationship with respect to the integrate-and-fire dynamics emanating from a source node and an extended network model that allows control of the small-world feature as well as the length of the long-range connections. A systematic approach to investigate the local and global correlations between structural and dynamical features of the networks was adopted that involved extensive simulations (one and a half million cases) so as to obtain two-dimensional correlation maps. Smooth, but diverse surfaces of correlation values were obtained in all cases. Regarding the global cases, it has been verified that the onset avalanche time (but not its intensity) can be accurately predicted from the structural features within specific regions of the map (i.e. networks with specific structural properties). The analysis at local level revealed that the dynamical features before the avalanches can also be accurately predicted from structural features. This is not possible for the dynamical features after the avalanches take place. This is so because the overall topology of the network predominates over the local topology around the source at the stationary state.

Towards the stabilization of the low density elements in topology optimization with large deformation

This work addresses the treatment of lower density regions of structures undergoing large deformations during the design process by the topology optimization method (TOM) based on the finite element method. During the design process the nonlinear elastic behavior of the structure is based on exact kinematics. The material model applied in the TOM is based on the solid isotropic microstructure with penalization approach. No void elements are deleted and all internal forces of the nodes surrounding the void elements are considered during the nonlinear equilibrium solution. The distribution of design variables is solved through the method of moving asymptotes, in which the sensitivity of the objective function is obtained directly. In addition, a continuation function and a nonlinear projection function are invoked to obtain a checkerboard free and mesh independent design. 2D examples with both plane strain and plane stress conditions hypothesis are presented and compared. The problem of instability is overcome by adopting a polyconvex constitutive model in conjunction with a suggested relaxation function to stabilize the excessive distorted elements. The exact tangent stiffness matrix is used. The optimal topology results are compared to the results obtained by using the classical Saint Venant–Kirchhoff constitutive law, and strong differences are found.

Magnetic topology and current channels in plasmas with toroidal current density inversions

The equilibrium magnetic field inside axisymmetric plasmas with inversions on the toroidal current density is studied. Structurally stable non-nested magnetic surfaces are considered. For any inversion in the internal current density the magnetic families define several positive current channels about a central negative one. A general expression relating the positive and negative currents is derived in terms of a topological anisotropy parameter. Next, an analytical local solution for the poloidal magnetic flux is derived and shown compatible with current hollow magnetic pitch measurements shown in the literature. Finally, the analytical solution exhibits non-nested magnetic families with positive anisotropy, indicating that the current inside the positive channels have at least twice the magnitude of the central one.

Mechanism of topology simplification by type II DNA topoisomerases

Type II DNA topoisomerases actively reduce the fractions of knotted and catenated circular DNA below thermodynamic equilibrium values. To explain this surprising finding, we designed a model in which topoisomerases introduce a sharp bend in DNA. Because the enzymes have a specific orientation relative to the bend, they act like Maxwell's demon, providing unidirectional strand passage. Quantitative analysis of the model by computer simulations proved that it can explain much of the experimental data. The required sharp DNA bend was demonstrated by a greatly increased cyclization of short DNA fragments from topoisomerase binding and by direct visualization with electron microscopy.

Equilibrium Assignments in Competitive and Cooperative Traffic Flow Routing

Part 18: Optimization in Collaborative Networks

Disorder driven transitions in non-equilibrium quantum systems

This thesis presents studies of the role of disorder in non-equilibrium quantum systems. The quantum states relevant to dynamics in these systems are very different from the ground state of the Hamiltonian. Two distinct systems are studied, (i) periodically driven Hamiltonians in two dimensions, and (ii) electrons in a one-dimensional lattice with power-law decaying hopping amplitudes. In the first system, the novel phases that are induced from the interplay of periodic driving, topology and disorder are studied. In the second system, the Anderson transition in all the eigenstates of the Hamiltonian are studied, as a function of the power-law exponent of the hopping amplitude.

In periodically driven systems the study focuses on the effect of disorder in the nature of the topology of the steady states. First, we investigate the robustness to disorder of Floquet topological insulators (FTIs) occurring in semiconductor quantum wells. Such FTIs are generated by resonantly driving a transition between the valence and conduction band. We show that when disorder is added, the topological nature of such FTIs persists as long as there is a gap at the resonant quasienergy. For strong enough disorder, this gap closes and all the states become localized as the system undergoes a transition to a trivial insulator.

Interestingly, the effects of disorder are not necessarily adverse, disorder can also induce a transition from a trivial to a topological system, thereby establishing a Floquet Topological Anderson Insulator (FTAI). Such a state would be a dynamical realization of the topological Anderson insulator. We identify the conditions on the driving field necessary for observing such a transition. We realize such a disorder induced topological Floquet spectrum in the driven honeycomb lattice and quantum well models.

Finally, we show that two-dimensional periodically driven quantum systems with spatial disorder admit a unique topological phase, which we call the anomalous Floquet-Anderson insulator (AFAI). The AFAI is characterized by a quasienergy spectrum featuring chiral edge modes coexisting with a fully localized bulk. Such a spectrum is impossible for a time-independent, local Hamiltonian. These unique characteristics of the AFAI give rise to a new topologically protected nonequilibrium transport phenomenon: quantized, yet nonadiabatic, charge pumping. We identify the topological invariants that distinguish the AFAI from a trivial, fully localized phase, and show that the two phases are separated by a phase transition.

The thesis also present the study of disordered systems using Wegner's Flow equations. The Flow Equation Method was proposed as a technique for studying excited states in an interacting system in one dimension. We apply this method to a one-dimensional tight binding problem with power-law decaying hoppings. This model presents a transition as a function of the exponent of the decay. It is shown that the the entire phase diagram, i.e. the delocalized, critical and localized phases in these systems can be studied using this technique. Based on this technique, we develop a strong-bond renormalization group that procedure where we solve the Flow Equations iteratively. This renormalization group approach provides a new framework to study the transition in this system.

Combining ganglion cell topology and data of patients with glaucoma to determine a structure-function map

PURPOSE: To introduce techniques for deriving a map that relates visual field locations to optic nerve head (ONH) sectors and to use the techniques to derive a map relating Medmont perimetric data to data from the Heidelberg Retinal Tomograph. METHODS: Spearman correlation coefficients were calculated relating each visual field location (Medmont M700) to rim area and volume measures for 10 degrees ONH sectors (HRT III software) for 57 participants: 34 with glaucoma, 18 with suspected glaucoma, and 5 with ocular hypertension. Correlations were constrained to be anatomically plausible with a computational model of the axon growth of retinal ganglion cells (Algorithm GROW). GROW generated a map relating field locations to sectors of the ONH. The sector with the maximum statistically significant (P < 0.05) correlation coefficient within 40 degrees of the angle predicted by GROW for each location was computed. Before correlation, both functional and structural data were normalized by either normative data or the fellow eye in each participant. RESULTS: The model of axon growth produced a 24-2 map that is qualitatively similar to existing maps derived from empiric data. When GROW was used in conjunction with normative data, 31% of field locations exhibited a statistically significant relationship. This significance increased to 67% (z-test, z = 4.84; P < 0.001) when both field and rim area data were normalized with the fellow eye. CONCLUSIONS: A computational model of axon growth and normalizing data by the fellow eye can assist in constructing an anatomically plausible map connecting visual field data and sectoral ONH data.