Hypersurfaces with many singularities

1. Teil: Bekannte Konstruktionen. Die vorliegende Arbeit gibt zunächst einen ausführlichen Überblick über die bisherigen Entwicklungen auf dem klassischen Gebiet der Hyperflächen mit vielen Singularitäten. Die maximale Anzahl mu^n(d) von Singularitäten auf einer Hyperfläche vom Grad d im P^n(C) ist nur in sehr wenigen Fällen bekannt, im P^3(C) beispielsweise nur für d<=6. Abgesehen von solchen Ausnahmen existieren nur obere und untere Schranken. 2. Teil: Neue Konstruktionen. Für kleine Grade d ist es oft möglich, bessere Resultate zu erhalten als jene, die durch allgemeine Schranken gegeben sind. In dieser Arbeit beschreiben wir einige algorithmische Ansätze hierfür, von denen einer Computer Algebra in Charakteristik 0 benutzt. Unsere anderen algorithmischen Methoden basieren auf einer Suche über endlichen Körpern. Das Liften der so experimentell gefundenen Hyperflächen durch Ausnutzung ihrer Geometrie oder Arithmetik liefert beispielsweise eine Fläche vom Grad 7 mit $99$ reellen gewöhnlichen Doppelpunkten und eine Fläche vom Grad 9 mit 226 gewöhnlichen Doppelpunkten. Diese Konstruktionen liefern die ersten unteren Schranken für mu^3(d) für ungeraden Grad d>5, die die allgemeine Schranke übertreffen. Unser Algorithmus hat außerdem das Potential, auf viele weitere Probleme der algebraischen Geometrie angewendet zu werden. Neben diesen algorithmischen Methoden beschreiben wir eine Konstruktion von Hyperflächen vom Grad d im P^n mit vielen A_j-Singularitäten, j>=2. Diese Beispiele, deren Existenz wir mit Hilfe der Theorie der Dessins d'Enfants beweisen, übertreffen die bekannten unteren Schranken in den meisten Fällen und ergeben insbesondere neue asymptotische untere Schranken für j>=2, n>=3. 3. Teil: Visualisierung. Wir beschließen unsere Arbeit mit einer Anwendung unserer neuen Visualisierungs-Software surfex, die die Stärken mehrerer existierender Programme bündelt, auf die Konstruktion affiner Gleichungen aller 45 topologischen Typen reeller kubischer Flächen.

Singularity-Free Fully-Isotropic Translational Parallel Manipulators

Parallel mechanisms show desirable characteristics such as a large payload to robot weight ratio, considerable stiffness, low inertia and high dynamic performances. In particular, parallel manipulators with fewer than six degrees of freedom have recently attracted researchers’ attention, as their employ may prove valuable in those applications in which a higher mobility is uncalled-for. The attention of this dissertation is focused on translational parallel manipulators (TPMs), that is on parallel manipulators whose output link (platform) is provided with a pure translational motion with respect to the frame. The first part deals with the general problem of the topological synthesis and classification of TPMs, that is it identifies the architectures that TPM legs must possess for the platform to be able to freely translate in space without altering its orientation. The second part studies both constraint and direct singularities of TPMs. In particular, special families of fully-isotropic mechanisms are identified. Such manipulators exhibit outstanding properties, as they are free from singularities and show a constant orthogonal Jacobian matrix throughout their workspace. As a consequence, both the direct and the inverse position problems are linear and the kinematic analysis proves straightforward.

Constraint based methods for allocation and scheduling of periodic applications

This work presents exact algorithms for the Resource Allocation and Cyclic Scheduling Problems (RA&CSPs). Cyclic Scheduling Problems arise in a number of application areas, such as in hoist scheduling, mass production, compiler design (implementing scheduling loops on parallel architectures), software pipelining, and in embedded system design. The RA&CS problem concerns time and resource assignment to a set of activities, to be indefinitely repeated, subject to precedence and resource capacity constraints. In this work we present two constraint programming frameworks facing two different types of cyclic problems. In first instance, we consider the disjunctive RA&CSP, where the allocation problem considers unary resources. Instances are described through the Synchronous Data-flow (SDF) Model of Computation. The key problem of finding a maximum-throughput allocation and scheduling of Synchronous Data-Flow graphs onto a multi-core architecture is NP-hard and has been traditionally solved by means of heuristic (incomplete) algorithms. We propose an exact (complete) algorithm for the computation of a maximum-throughput mapping of applications specified as SDFG onto multi-core architectures. Results show that the approach can handle realistic instances in terms of size and complexity. Next, we tackle the Cyclic Resource-Constrained Scheduling Problem (i.e. CRCSP). We propose a Constraint Programming approach based on modular arithmetic: in particular, we introduce a modular precedence constraint and a global cumulative constraint along with their filtering algorithms. Many traditional approaches to cyclic scheduling operate by fixing the period value and then solving a linear problem in a generate-and-test fashion. Conversely, our technique is based on a non-linear model and tackles the problem as a whole: the period value is inferred from the scheduling decisions. The proposed approaches have been tested on a number of non-trivial synthetic instances and on a set of realistic industrial instances achieving good results on practical size problem.

Topologically correct intersection curves of tori and natural quadrics

This thesis provides efficient and robust algorithms for the computation of the intersection curve between a torus and a simple surface (e.g. a plane, a natural quadric or another torus), based on algebraic and numeric methods. The algebraic part includes the classification of the topological type of the intersection curve and the detection of degenerate situations like embedded conic sections and singularities. Moreover, reference points for each connected intersection curve component are determined. The required computations are realised efficiently by solving quartic polynomials at most and exactly by using exact arithmetic. The numeric part includes algorithms for the tracing of each intersection curve component, starting from the previously computed reference points. Using interval arithmetic, accidental incorrectness like jumping between branches or the skipping of parts are prevented. Furthermore, the environments of singularities are correctly treated. Our algorithms are complete in the sense that any kind of input can be handled including degenerate and singular configurations. They are verified, since the results are topologically correct and approximate the real intersection curve up to any arbitrary given error bound. The algorithms are robust, since no human intervention is required and they are efficient in the way that the treatment of algebraic equations of high degree is avoided.

Mathematical aspects of Feynman integrals

In the present dissertation we consider Feynman integrals in the framework of dimensional regularization. As all such integrals can be expressed in terms of scalar integrals, we focus on this latter kind of integrals in their Feynman parametric representation and study their mathematical properties, partially applying graph theory, algebraic geometry and number theory. The three main topics are the graph theoretic properties of the Symanzik polynomials, the termination of the sector decomposition algorithm of Binoth and Heinrich and the arithmetic nature of the Laurent coefficients of Feynman integrals.rnrnThe integrand of an arbitrary dimensionally regularised, scalar Feynman integral can be expressed in terms of the two well-known Symanzik polynomials. We give a detailed review on the graph theoretic properties of these polynomials. Due to the matrix-tree-theorem the first of these polynomials can be constructed from the determinant of a minor of the generic Laplacian matrix of a graph. By use of a generalization of this theorem, the all-minors-matrix-tree theorem, we derive a new relation which furthermore relates the second Symanzik polynomial to the Laplacian matrix of a graph.rnrnStarting from the Feynman parametric parameterization, the sector decomposition algorithm of Binoth and Heinrich serves for the numerical evaluation of the Laurent coefficients of an arbitrary Feynman integral in the Euclidean momentum region. This widely used algorithm contains an iterated step, consisting of an appropriate decomposition of the domain of integration and the deformation of the resulting pieces. This procedure leads to a disentanglement of the overlapping singularities of the integral. By giving a counter-example we exhibit the problem, that this iterative step of the algorithm does not terminate for every possible case. We solve this problem by presenting an appropriate extension of the algorithm, which is guaranteed to terminate. This is achieved by mapping the iterative step to an abstract combinatorial problem, known as Hironaka's polyhedra game. We present a publicly available implementation of the improved algorithm. Furthermore we explain the relationship of the sector decomposition method with the resolution of singularities of a variety, given by a sequence of blow-ups, in algebraic geometry.rnrnMotivated by the connection between Feynman integrals and topics of algebraic geometry we consider the set of periods as defined by Kontsevich and Zagier. This special set of numbers contains the set of multiple zeta values and certain values of polylogarithms, which in turn are known to be present in results for Laurent coefficients of certain dimensionally regularized Feynman integrals. By use of the extended sector decomposition algorithm we prove a theorem which implies, that the Laurent coefficients of an arbitrary Feynman integral are periods if the masses and kinematical invariants take values in the Euclidean momentum region. The statement is formulated for an even more general class of integrals, allowing for an arbitrary number of polynomials in the integrand.

Efficient certification of feasibility and objective value of linear programs and its applications

The use of linear programming in various areas has increased with the significant improvement of specialized solvers. Linear programs are used as such to model practical problems, or as subroutines in algorithms such as formal proofs or branch-and-cut frameworks. In many situations a certified answer is needed, for example the guarantee that the linear program is feasible or infeasible, or a provably safe bound on its objective value. Most of the available solvers work with floating-point arithmetic and are thus subject to its shortcomings such as rounding errors or underflow, therefore they can deliver incorrect answers. While adequate for some applications, this is unacceptable for critical applications like flight controlling or nuclear plant management due to the potential catastrophic consequences. We propose a method that gives a certified answer whether a linear program is feasible or infeasible, or returns unknown'. The advantage of our method is that it is reasonably fast and rarely answers unknown'. It works by computing a safe solution that is in some way the best possible in the relative interior of the feasible set. To certify the relative interior, we employ exact arithmetic, whose use is nevertheless limited in general to critical places, allowing us to rnremain computationally efficient. Moreover, when certain conditions are fulfilled, our method is able to deliver a provable bound on the objective value of the linear program. We test our algorithm on typical benchmark sets and obtain higher rates of success compared to previous approaches for this problem, while keeping the running times acceptably small. The computed objective value bounds are in most of the cases very close to the known exact objective values. We prove the usability of the method we developed by additionally employing a variant of it in a different scenario, namely to improve the results of a Satisfiability Modulo Theories solver. Our method is used as a black box in the nodes of a branch-and-bound tree to implement conflict learning based on the certificate of infeasibility for linear programs consisting of subsets of linear constraints. The generated conflict clauses are in general small and give good rnprospects for reducing the search space. Compared to other methods we obtain significant improvements in the running time, especially on the large instances.

Higher-order molecular properties and excitation energies in single-reference and multireference coupled-cluster theory

Coupled-cluster (CC) theory is one of the most successful approaches in high-accuracy quantum chemistry. The present thesis makes a number of contributions to the determination of molecular properties and excitation energies within the CC framework. The multireference CC (MRCC) method proposed by Mukherjee and coworkers (Mk-MRCC) has been benchmarked within the singles and doubles approximation (Mk-MRCCSD) for molecular equilibrium structures. It is demonstrated that Mk-MRCCSD yields reliable results for multireference cases where single-reference CC methods fail. At the same time, the present work also illustrates that Mk-MRCC still suffers from a number of theoretical problems and sometimes gives rise to results of unsatisfactory accuracy. To determine polarizability tensors and excitation spectra in the MRCC framework, the Mk-MRCC linear-response function has been derived together with the corresponding linear-response equations. Pilot applications show that Mk-MRCC linear-response theory suffers from a severe problem when applied to the calculation of dynamic properties and excitation energies: The Mk-MRCC sufficiency conditions give rise to a redundancy in the Mk-MRCC Jacobian matrix, which entails an artificial splitting of certain excited states. This finding has established a new paradigm in MRCC theory, namely that a convincing method should not only yield accurate energies, but ought to allow for the reliable calculation of dynamic properties as well. In the context of single-reference CC theory, an analytic expression for the dipole Hessian matrix, a third-order quantity relevant to infrared spectroscopy, has been derived and implemented within the CC singles and doubles approximation. The advantages of analytic derivatives over numerical differentiation schemes are demonstrated in some pilot applications.

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 $h^{1,1}=1$.

Computing primal solutions with exact arithmetics in SCIP.

The research for exact solutions of mixed integer problems is an active topic in the scientific community. State-of-the-art MIP solvers exploit a floating- point numerical representation, therefore introducing small approximations. Although such MIP solvers yield reliable results for the majority of problems, there are cases in which a higher accuracy is required. Indeed, it is known that for some applications floating-point solvers provide falsely feasible solutions, i.e. solutions marked as feasible because of approximations that would not pass a check with exact arithmetic and cannot be practically implemented. The framework of the current dissertation is SCIP, a mixed integer programs solver mainly developed at Zuse Institute Berlin. In the same site we considered a new approach for exactly solving MIPs. Specifically, we developed a constraint handler to plug into SCIP, with the aim to analyze the accuracy of provided floating-point solutions and compute exact primal solutions starting from floating-point ones. We conducted a few computational experiments to test the exact primal constraint handler through the adoption of two main settings. Analysis mode allowed to collect statistics about current SCIP solutions' reliability. Our results confirm that floating-point solutions are accurate enough with respect to many instances. However, our analysis highlighted the presence of numerical errors of variable entity. By using the enforce mode, our constraint handler is able to suggest exact solutions starting from the integer part of a floating-point solution. With the latter setting, results show a general improvement of the quality of provided final solutions, without a significant loss of performances.

Sleep restriction attenuates amplitudes and attentional modulation of pain-related evoked potentials, but augments pain ratings in healthy volunteers

The experiment investigated the impact of sleep restriction on pain perception and related evoked potential correlates (laser-evoked potentials, LEPs). Ten healthy subjects with good sleep quality were investigated in the morning twice, once after habitual sleep and once after partial sleep restriction. Additionally, we studied the impact of attentional focussing on pain and LEPs by directing attention to (intensity discrimination) or away from the stimulus (mental arithmetic). Laser stimuli directed to the hand dorsum were rated as 30% more painful after sleep restriction (49+/-7 mm) than after a night of habitual sleep (38+/-7 mm). A significant interaction between attentional focus and sleep condition suggested that attentional focusing was less distinctive under sleep restriction. Intensity discrimination was preserved. In contrast, the amplitude of the early parasylvian N1 of LEPs was significantly smaller after a night of partial sleep restriction (-36%, p<0.05). Likewise, the amplitude of the vertex N2-P2 was significantly reduced (-34%, p<0.01); also attentional modulation of the N2-P2 was reduced. Thus, objective (LEPs) and subjective (pain ratings) parameters of nociceptive processing were differentially modulated by partial sleep restriction. We propose, that sleep reduction leads to an impairment of activation in the ascending pathway (leading to reduced LEPs). In contradistinction, pain perception was boosted, which we attribute to lack of pain control distinct from classical descending inhibition, and thus not affecting the projection pathway. Sleep-restricted subjects exhibit reduced attentional modulation of pain stimuli and may thus have difficulties to readily attend to or disengage from pain.

On a Microcomputer Implementation of an Intrusion-detection Algorithm

The main objective of this paper is to discuss various aspects of implementing a specific intrusion-detection scheme on a micro-computer system using fixed-point arithmetic. The proposed scheme is suitable for detecting intruder stimuli which are in the form of transient signals. It consists of two stages: an adaptive digital predictor and an adaptive threshold detection algorithm. Experimental results involving data acquired via field experiments are also included.

Taiji practice attenuates psychobiological stress reactivity - A randomized controlled trial in healthy subjects

Background: Stress reducing effects of Taiji, a mindful and gentle form of body movement, have been reported in previous studies, but standardized and controlled experimental studies are scarce. The present study investigates the effect of regular Taiji practice on psychobiological stress response in healthy men and women. Methods: 70 participants were randomly assigned to either Taiji classes or a waiting list. After 3 months, 26 (8 men, 18 women) persons in the Taiji group and 23 (9 men, 14 women) in the waiting control group underwent a standardized psychosocial stress test combining public speaking and mental arithmetic in front of an audience. Salivary cortisol and α-amylase, heart rate, and psychological responses to psychosocial stress were compared between the study groups. (ClinicalTrials.gov number, NCT01122706.) Results: Stress induced characteristic changes in all psychological and physiological measures. Compared to controls, Taiji participants exhibited a significantly lower stress reactivity of cortisol (p = .028) and heart rate (p = .028), as well as lower α-amylase levels (p = .049). They reported a lower increase in perceived stressfulness (p = .006) and maintained a higher level of calmness (p = .019) in response to psychosocial stress. Conclusion: Our results consistently suggest that practicing Taiji attenuates psychobiological stress reactivity in healthy subjects. This may underline the role of Taiji as a useful mind–body practice for stress prevention.

End-tidal CO(2). An important parameter for a correct interpretation in functional brain studies using speech tasks

The aim was to investigate the effect of different speech tasks, i.e. recitation of prose (PR), alliteration (AR) and hexameter (HR) verses and a control task (mental arithmetic (MA) with voicing of the result on end-tidal CO2 (PETCO2), cerebral hemodynamics and oxygenation. CO2 levels in the blood are known to strongly affect cerebral blood flow. Speech changes breathing pattern and may affect CO2 levels. Measurements were performed on 24 healthy adult volunteers during the performance of the 4 tasks. Tissue oxygen saturation (StO2) and absolute concentrations of oxyhemoglobin ([O2Hb]), deoxyhemoglobin ([HHb]) and total hemoglobin ([tHb]) were measured by functional near-infrared spectroscopy (fNIRS) and PETCO2 by a gas analyzer. Statistical analysis was applied to the difference between baseline before the task, 2 recitation and 5 baseline periods after the task. The 2 brain hemispheres and 4 tasks were tested separately. A significant decrease in PETCO2 was found during all 4 tasks with the smallest decrease during the MA task. During the recitation tasks (PR, AR and HR) a statistically significant (p < 0.05) decrease occurred for StO2 during PR and AR in the right prefrontal cortex (PFC) and during AR and HR in the left PFC. [O2Hb] decreased significantly during PR, AR and HR in both hemispheres. [HHb] increased significantly during the AR task in the right PFC. [tHb] decreased significantly during HR in the right PFC and during PR, AR and HR in the left PFC. During the MA task, StO2 increased and [HHb] decreased significantly during the MA task. We conclude that changes in breathing (hyperventilation) during the tasks led to lower CO2 pressure in the blood (hypocapnia), predominantly responsible for the measured changes in cerebral hemodynamics and oxygenation. In conclusion, our findings demonstrate that PETCO2 should be monitored during functional brain studies investigating speech using neuroimaging modalities, such as fNIRS, fMRI to ensure a correct interpretation of changes in hemodynamics and oxygenation.

Delayed response and lack of habituation in plasma interleukin-6 to acute mental stress in men

Acute mental stress induces a significant increase in plasma interleukin (IL)-6 levels as a possible mechanism for how psychological stress might contribute to atherosclerosis. We investigated whether the IL-6 response would habituate in response to a repetitively applied mental stressor and whether cortisol reactivity would show a relationship with IL-6 reactivity. Study participants were 21 reasonably healthy men (mean age 46+/-7 years) who underwent the Trier Social Stress Test (combination of a 3-min preparation, 5-min speech, and 5-min mental arithmetic) three times with an interval of 1 week. Plasma IL-6 and free salivary cortisol were measured immediately before and after stress, and at 45 and 105 min of recovery from stress. Cortisol samples were also obtained 15 and 30 min after stress. Compared to non-stressed controls, IL-6 significantly increased between rest and 45 min post-stress (p=.022) and between rest and 105 min post-stress (p=.001). Peak cortisol (p=.034) and systolic blood pressure (p=.009) responses to stress both habituated between weeks one and three. No adaptation occurred in diastolic blood pressure, heart rate, and IL-6 responses to stress. The areas under the curve integrating the stress-induced changes in cortisol and IL-6 reactivity were negatively correlated at visit three (r=-.54, p=.011), but not at visit one. The IL-6 response to acute mental stress occurs delayed and shows no adaptation to repeated moderate mental stress. The hypothalamus-pituitary-adrenal axis may attenuate stress reactivity of IL-6. The lack of habituation in IL-6 responses to daily stress could subject at-risk individuals to higher atherosclerotic morbidity and mortality.