Monodromy calculations for some differential equations

In vielen Teilgebieten der Mathematik ist es w"{u}nschenswert, die Monodromiegruppe einer homogenen linearen Differenzialgleichung zu verstehen. Es sind nur wenige analytische Methoden zur Berechnung dieser Gruppe bekannt, daher entwickeln wir im ersten Teil dieser Arbeit eine numerische Methode zur Approximation ihrer Erzeuger.rnIm zweiten Abschnitt fassen wir die Grundlagen der Theorie der Uniformisierung Riemannscher Fl"achen und die der arithmetischen Fuchsschen Gruppen zusammen. Auss erdem erkl"aren wir, wie unsere numerische Methode bei der Bestimmung von uniformisierenden Differenzialgleichungen dienlich sein kann. F"ur arithmetische Fuchssche Gruppen mit zwei Erzeugern erhalten wir lokale Daten und freie Parameter von Lam'{e} Gleichungen, welche die zugeh"origen Riemannschen Fl"achen uniformisieren. rnIm dritten Teil geben wir einen kurzen Abriss zur homologischen Spiegelsymmetrie und f"uhren die $widehat{Gamma}$-Klasse ein. Wir erkl"aren wie diese genutzt werden kann, um eine Hodge-theoretische Version der Spiegelsymmetrie f"ur torische Varit"aten zu beweisen. Daraus gewinnen wir Vermutungen "uber die Monodromiegruppe $M$ von Picard-Fuchs Gleichungen von gewissen Familien $f:mathcal{X}rightarrow bbp^1$ von $n$-dimensionalen Calabi-Yau Variet"aten. Diese besagen erstens, dass bez"uglich einer nat"urlichen Basis die Monodromiematrizen in $M$ Eintr"age aus dem K"orper $bbq(zeta(2j+1)/(2 pi i)^{2j+1},j=1,ldots,lfloor (n-1)/2 rfloor)$ haben. Und zweitens, dass sich topologische Invarianten des Spiegelpartners einer generischen Faser von $f:mathcal{X}rightarrow bbp^1$ aus einem speziellen Element von $M$ rekonstruieren lassen. Schliess lich benutzen wir die im ersten Teil entwickelten Methoden zur Verifizierung dieser Vermutungen, vornehmlich in Hinblick auf Dimension drei. Dar"uber hinaus erstellen wir eine Liste von Kandidaten topologischer Invarianten von vermutlich existierenden dreidimensionalen Calabi-Yau Variet"aten mit $h^{1,1}=1$.

Differential recruitment of brain networks during visuospatial and color processing: Evidence from ERP microstates

Recent functional magnetic resonance imaging (fMRI) studies consistently revealed contributions of fronto-parietal and related networks to the execution of a visuospatial judgment task, the so-called "Clock Task". However, due to the low temporal resolution of fMRI, the exact cortical dynamics and timing of processing during task performance could not be resolved until now. In order to clarify the detailed cortical activity and temporal dynamics, 14 healthy subjects performed an established version of the "Clock Task", which comprises a visuospatial task (angle discrimination) and a control task (color discrimination) with the same stimulus material, in an electroencephalography (EEG) experiment. Based on the time-resolved analysis of network activations (microstate analysis), differences in timing between the angle compared to the color discrimination task were found after sensory processing in a time window starting around 200ms. Significant differences between the two tasks were observed in an analysis window from 192ms to 776ms. We divided this window in two parts: an early phase - from 192ms to ∼440ms, and a late phase - from ∼440ms to 776ms. For both tasks, the order of network activations and the types of networks were the same, but, in each phase, activations for the two conditions were dominated by differing network states with divergent temporal dynamics. Our results provide an important basis for the assessment of deviations in processing dynamics during visuospatial tasks in clinical populations.

Controlled time of arrival feasibility analysis

Previous research studies and operational trials have shown that using the airborne Required Time of Arrival (RTA) function, an aircraft can individually achieve an assigned time to a metering or merge point accurately. This study goes a step further and investigates the application of RTA to a real sequence of arriving aircraft into Melbourne Australia. Assuming that the actual arrival times were Controlled Time of Arrivals (CTAs) assigned to each aircraft, the study examines if the airborne RTA solution would work. Three scenarios were compared: a baseline scenario being the actual flown trajectories in a two hour time-span into Melbourne, a scenario in which the sequential landing slot times of the baseline scenario were assigned as CTAs and a third scenario in which the landing slots could be freely redistributed to the inbound traffic as CTAs. The research found that pressure on the terminal area would sometimes require aircraft to lose more time than possible through the RTA capability. Using linear holding as an additional measure to absorb extensive delays, up to 500NM (5%) of total track reduction and 1300kg (3%) of total fuel consumption could be saved in the scenario with landing slots freely distributed as CTAs, compared to the baseline scenario. Assigning CTAs in an arrival sequence requires the ground system to have an accurate trajectory predictor to propose additional delay measures (path stretching, linear holding) if necessary. Reducing the achievable time window of the aircraft to add control margin to the RTA function, had a negative impact and increased the amount of intervention other than speed control required to solve the sequence. It was concluded that the RTA capability is not a complete solution but merely a tool to assist in managing the increasing complexity of air traffic.

Sparse differential resultant formulas: between the linear and the nonlinear case

A matrix representation of the sparse differential resultant is the basis for efficient computation algorithms, whose study promises a great contribution to the development and applicability of differential elimination techniques. It is shown how sparse linear differential resultant formulas provide bounds for the order of derivation, even in the nonlinear case, and they also provide (in many cases) the bridge with results in the nonlinear algebraic case.

Spectral diffusion and the energy landscape of a protein

We present a novel type of spectral diffusion experiment in the millikelvin range to characterize the energy landscape of a protein as compared with that of a glass. We measure the time evolution of spectral holes for more than 300 hr after well-defined initial nonequilibrium conditions. We show that the model of noninteracting two-level systems can describe spectral diffusion in the glass, but fails for the protein. Our results further demonstrate that randomness in the energy landscape of a protein shows features of organization. There are “deep minimum” states separated by barriers, the heights of which we are able to estimate. The energy landscape of a glass is featureless by comparison.

Lower semicontinuity of the feasible set mapping of linear systems relative to their domains

This paper deals with stability properties of the feasible set of linear inequality systems having a finite number of variables and an arbitrary number of constraints. Several types of perturbations preserving consistency are considered, affecting respectively, all of the data, the left-hand side data, or the right-hand side coefficients.

Formalization of Interaction Events in Multi-agent Systems

The problem of the description of interaction between spatially divided agents in the form of dialogues is explored. The concept of processes synchronization is analyzed to formalize the specification of interaction at the level of events constituting the processes. The approach to formalization of the description of conditions of synchronization when both the independent behavior and the communications of agents can be presented at a logic level is offered. It is shown, that the collective behavior of agents can be specified by the synthetic temporal logic that unites linear and branching time temporal logics.

Solving Differential Equations by Parallel Laplace Method with Assured Accuracy

The paper has been presented at the 12th International Conference on Applications of Computer Algebra, Varna, Bulgaria, June, 2006

Time-Fractional Derivatives in Relaxation Processes: A Tutorial Survey

2000 Mathematics Subject Classification: 26A33, 33E12, 33C60, 44A10, 45K05, 74D05,

A Solver for Complex-Valued Parametric Linear Systems

This work reports on a new software for solving linear systems involving affine-linear dependencies between complex-valued interval parameters. We discuss the implementation of a parametric residual iteration for linear interval systems by advanced communication between the system Mathematica and the library C-XSC supporting rigorous complex interval arithmetic. An example of AC electrical circuit illustrates the use of the presented software.

Time-Dependent Differential Inclusions and Viability

AMS subject classification: Primary 34A60, Secondary 49J52.

Optimization Problem in a Class of Linear System. Application to Multiple Drug Administration

2000 Mathematics Subject Classification: 62H15, 62P10.

Noise-induced standing waves in oscillatory systems with time-delayed feedback

In oscillatory reaction-diffusion systems, time-delay feedback can lead to the instability of uniform oscillations with respect to formation of standing waves. Here, we investigate how the presence of additive, Gaussian white noise can induce the appearance of standing waves. Combining analytical solutions of the model with spatio-temporal simulations, we find that noise can promote standing waves in regimes where the deterministic uniform oscillatory modes are stabilized. As the deterministic phase boundary is approached, the spatio-temporal correlations become stronger, such that even small noise can induce standing waves in this parameter regime. With larger noise strengths, standing waves could be induced at finite distances from the (deterministic) phase boundary. The overall dynamics is defined through the interplay of noisy forcing with the inherent reaction-diffusion dynamics.

An indexing structure and application model for vehicles moving on road networks

Moving objects database systems are the most challenging sub-category among Spatio-Temporal database systems. A database system that updates in real-time the location information of GPS-equipped moving vehicles has to meet even stricter requirements. Currently existing data storage models and indexing mechanisms work well only when the number of moving objects in the system is relatively small. This dissertation research aimed at the real-time tracking and history retrieval of massive numbers of vehicles moving on road networks. A total solution has been provided for the real-time update of the vehicles' location and motion information, range queries on current and history data, and prediction of vehicles' movement in the near future. ^ To achieve these goals, a new approach called Segmented Time Associated to Partitioned Space (STAPS) was first proposed in this dissertation for building and manipulating the indexing structures for moving objects databases. ^ Applying the STAPS approach, an indexing structure of associating a time interval tree to each road segment was developed for real-time database systems of vehicles moving on road networks. The indexing structure uses affordable storage to support real-time data updates and efficient query processing. The data update and query processing performance it provides is consistent without restrictions such as a time window or assuming linear moving trajectories. ^ An application system design based on distributed system architecture with centralized organization was developed to maximally support the proposed data and indexing structures. The suggested system architecture is highly scalable and flexible. Finally, based on a real-world application model of vehicles moving in region-wide, main issues on the implementation of such a system were addressed. ^

An Indexing Structure and Application Model for Vehicles Moving on Road Networks

Moving objects database systems are the most challenging sub-category among Spatio-Temporal database systems. A database system that updates in real-time the location information of GPS-equipped moving vehicles has to meet even stricter requirements. Currently existing data storage models and indexing mechanisms work well only when the number of moving objects in the system is relatively small. This dissertation research aimed at the real-time tracking and history retrieval of massive numbers of vehicles moving on road networks. A total solution has been provided for the real-time update of the vehicles’ location and motion information, range queries on current and history data, and prediction of vehicles’ movement in the near future. To achieve these goals, a new approach called Segmented Time Associated to Partitioned Space (STAPS) was first proposed in this dissertation for building and manipulating the indexing structures for moving objects databases. Applying the STAPS approach, an indexing structure of associating a time interval tree to each road segment was developed for real-time database systems of vehicles moving on road networks. The indexing structure uses affordable storage to support real-time data updates and efficient query processing. The data update and query processing performance it provides is consistent without restrictions such as a time window or assuming linear moving trajectories. An application system design based on distributed system architecture with centralized organization was developed to maximally support the proposed data and indexing structures. The suggested system architecture is highly scalable and flexible. Finally, based on a real-world application model of vehicles moving in region-wide, main issues on the implementation of such a system were addressed.