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.

Studying MHC I-dependent CD8 + T cell responses in the model of murine cutaneous leishmaniasis

Der Ausheilung von Infektionen mit Leishmania major liegt die Sekretion von IFN- von sowohl CD4+ als auch CD8+ T Zellen zugrunde.rnAktuell konnte in der Literatur nur ein Epitop aus dem parasitären LACK Protein für eine effektive CD4+ T Zell-vermittelte Immunantwort beschrieben werden. Das Ziel der vorliegenden Arbeit bestand daher darin, mögliche MHC I abhängige CD8+ T Zell Antworten zu untersuchen. rnFür diesen Ansatz wurde als erstes der Effekt einer Vakzinierung mit LACK Protein fusioniert an die Protein-Transduktionsdomäne des HIV-1 (TAT) analysiert. Die Effektivität von TAT-LACK gegenüber CD8+ T Zellen wurde mittels in vivo Protein-Vakzinierung von resistenten C57BL/6 Mäusen in Depletions-Experimenten gezeigt.rnDie Prozessierung von Proteinen vor der Präsentation immunogener Peptide gegenüber T Zellen ist unbedingt erforderlich. Daher wurde in dieser Arbeit die Rolle des IFN--induzierbaren Immunoproteasoms bei der Prozessierung von parasitären Proteinen und Präsentation von Peptiden gebunden an MHC I Moleküle durch in vivo und in vitro Experimente untersucht. Es konnte in dieser Arbeit eine Immunoproteasom-unabhängige Prozessierung aufgezeigt werden.rnWeiterhin wurde Parasitenlysat (SLA) von sowohl Promastigoten als auch Amastigoten fraktioniert. In weiterführenden Experimenten können diese Fraktionen auf immunodominante Proteine/Peptide hin untersucht werden. rnLetztlich wurden Epitop-Vorhersagen für CD8+ T Zellen mittels computergestützer Software von beiden parasitären Lebensformen durchgeführt. 300 dieser Epitope wurden synthetisiert und werden in weiterführenden Experimenten zur Charakterisierung immunogener Eigenschaften weiter verwendet. rnIn ihrer Gesamtheit trägt die vorliegende Arbeit wesentlich zum Verständnis über die komplexen Mechanismen der Prozessierung und letztendlich zur Identifikation von möglichen CD8+ T Zell Epitopen bei. Ein detailiertes Verständnis der Prozessierung von CD8+ T Zell Epitopen von Leishmania major über den MHC Klasse I Weg ist von höchster Bedeutung. Die Charakterisierung sowie die Identifikation dieser Peptide wird einen maßgeblichen Einfluss auf die weiteren Entwicklungen von Vakzinen gegen diesen bedeutenden human-pathogenen Parasiten mit sich bringen. rn

Analysis and mathematical modeling of wave-structure interaction

The aim of this thesis, included within the THESEUS project, is the development of a mathematical model 2DV two-phase, based on the existing code IH-2VOF developed by the University of Cantabria, able to represent together the overtopping phenomenon and the sediment transport. Several numerical simulations were carried out in order to analyze the flow characteristics on a dike crest. The results show that the seaward/landward slope does not affect the evolution of the flow depth and velocity over the dike crest whereas the most important parameter is the relative submergence. Wave heights decrease and flow velocities increase while waves travel over the crest. In particular, by increasing the submergence, the wave height decay and the increase of the velocity are less marked. Besides, an appropriate curve able to fit the variation of the wave height/velocity over the dike crest were found. Both for the wave height and for the wave velocity different fitting coefficients were determined on the basis of the submergence and of the significant wave height. An equation describing the trend of the dimensionless coefficient c_h for the wave height was derived. These conclusions could be taken into consideration for the design criteria and the upgrade of the structures. In the second part of the thesis, new equations for the representation of the sediment transport in the IH-2VOF model were introduced in order to represent beach erosion while waves run-up and overtop the sea banks during storms. The new model allows to calculate sediment fluxes in the water column together with the sediment concentration. Moreover it is possible to model the bed profile evolution. Different tests were performed under low-intensity regular waves with an homogeneous layer of sand on the bottom of a channel in order to analyze the erosion-deposition patterns and verify the model results.

Effect of pulsatility on the mathematical modeling of rotary blood pumps

In this study, the effect of time derivatives of flow rate and rotational speed was investigated on the mathematical modeling of a rotary blood pump (RBP). The basic model estimates the pressure head of the pump as a dependent variable using measured flow and speed as predictive variables. Performance of the model was evaluated by adding time derivative terms for flow and speed. First, to create a realistic working condition, the Levitronix CentriMag RBP was implanted in a sheep. All parameters from the model were physically measured and digitally acquired over a wide range of conditions, including pulsatile speed. Second, a statistical analysis of the different variables (flow, speed, and their time derivatives) based on multiple regression analysis was performed to determine the significant variables for pressure head estimation. Finally, different mathematical models were used to show the effect of time derivative terms on the performance of the models. In order to evaluate how well the estimated pressure head using different models fits the measured pressure head, root mean square error and correlation coefficient were used. The results indicate that inclusion of time derivatives of flow and speed can improve model accuracy, but only minimally.

A FIREWALL MODEL OF FILE SYSTEM SECURITY

File system security is fundamental to the security of UNIX and Linux systems since in these systems almost everything is in the form of a file. To protect the system files and other sensitive user files from unauthorized accesses, certain security schemes are chosen and used by different organizations in their computer systems. A file system security model provides a formal description of a protection system. Each security model is associated with specified security policies which focus on one or more of the security principles: confidentiality, integrity and availability. The security policy is not only about “who” can access an object, but also about “how” a subject can access an object. To enforce the security policies, each access request is checked against the specified policies to decide whether it is allowed or rejected. The current protection schemes in UNIX/Linux systems focus on the access control. Besides the basic access control scheme of the system itself, which includes permission bits, setuid and seteuid mechanism and the root, there are other protection models, such as Capabilities, Domain Type Enforcement (DTE) and Role-Based Access Control (RBAC), supported and used in certain organizations. These models protect the confidentiality of the data directly. The integrity of the data is protected indirectly by only allowing trusted users to operate on the objects. The access control decisions of these models depend on either the identity of the user or the attributes of the process the user can execute, and the attributes of the objects. Adoption of these sophisticated models has been slow; this is likely due to the enormous complexity of specifying controls over a large file system and the need for system administrators to learn a new paradigm for file protection. We propose a new security model: file system firewall. It is an adoption of the familiar network firewall protection model, used to control the data that flows between networked computers, toward file system protection. This model can support decisions of access control based on any system generated attributes about the access requests, e.g., time of day. The access control decisions are not on one entity, such as the account in traditional discretionary access control or the domain name in DTE. In file system firewall, the access decisions are made upon situations on multiple entities. A situation is programmable with predicates on the attributes of subject, object and the system. File system firewall specifies the appropriate actions on these situations. We implemented the prototype of file system firewall on SUSE Linux. Preliminary results of performance tests on the prototype indicate that the runtime overhead is acceptable. We compared file system firewall with TE in SELinux to show that firewall model can accommodate many other access control models. Finally, we show the ease of use of firewall model. When firewall system is restricted to specified part of the system, all the other resources are not affected. This enables a relatively smooth adoption. This fact and that it is a familiar model to system administrators will facilitate adoption and correct use. The user study we conducted on traditional UNIX access control, SELinux and file system firewall confirmed that. The beginner users found it easier to use and faster to learn then traditional UNIX access control scheme and SELinux.

Structural model of the CopA copper ATPase of Enterococcus hirae based on chemical cross-linking

The CopA copper ATPase of Enterococcus hirae belongs to the family of heavy metal pumping CPx-type ATPases and shares 43% sequence similarity with the human Menkes and Wilson copper ATPases. Due to a lack of suitable protein crystals, only partial three-dimensional structures have so far been obtained for this family of ion pumps. We present a structural model of CopA derived by combining topological information obtained by intramolecular cross-linking with molecular modeling. Purified CopA was cross-linked with different bivalent reagents, followed by tryptic digestion and identification of cross-linked peptides by mass spectrometry. The structural proximity of tryptic fragments provided information about the structural arrangement of the hydrophilic protein domains, which was integrated into a three-dimensional model of CopA. Comparative modeling of CopA was guided by the sequence similarity to the calcium ATPase of the sarcoplasmic reticulum, Serca1, for which detailed structures are available. In addition, known partial structures of CPx-ATPase homologous to CopA were used as modeling templates. A docking approach was used to predict the orientation of the heavy metal binding domain of CopA relative to the core structure, which was verified by distance constraints derived from cross-links. The overall structural model of CopA resembles the Serca1 structure, but reveals distinctive features of CPx-type ATPases. A prominent feature is the positioning of the heavy metal binding domain. It features an orientation of the Cu binding ligands which is appropriate for the interaction with Cu-loaded metallochaperones in solution. Moreover, a novel model of the architecture of the intramembranous Cu binding sites could be derived.

Tracing of patients lost to follow-up and HIV transmission: Mathematical modelling study based on two large ART programmes in Malawi

OBJECTIVES: Treatment as prevention depends on retaining HIV-infected patients in care. We investigated the effect on HIV transmission of bringing patients lost to follow up (LTFU) back into care. DESIGN: Mathematical model. METHODS: Stochastic mathematical model of cohorts of 1000 HIV-infected patients on antiretroviral therapy (ART), based on data from two clinics in Lilongwe, Malawi. We calculated cohort viral load (CVL; sum of individual mean viral loads each year) and used a mathematical relationship between viral load and transmission probability to estimate the number of new HIV infections. We simulated four scenarios: 'no LTFU' (all patients stay in care); 'no tracing' (patients LTFU are not traced); 'immediate tracing' (after missed clinic appointment); and, 'delayed tracing' (after six months). RESULTS: About 440 of 1000 patients were LTFU over five years. CVL (million copies/ml per 1000 patients) were 3.7 (95% prediction interval [PrI] 2.9-4.9) for no LTFU, 8.6 (95% PrI 7.3-10.0) for no tracing, 7.7 (95% PrI 6.2-9.1) for immediate, and 8.0 (95% PrI 6.7-9.5) for delayed tracing. Comparing no LTFU with no tracing the number of new infections increased from 33 (95% PrI 29-38) to 54 (95% PrI 47-60) per 1000 patients. Immediate tracing prevented 3.6 (95% PrI -3.3-12.8) and delayed tracing 2.5 (95% PrI -5.8-11.1) new infections per 1000. Immediate tracing was more efficient than delayed tracing: 116 and to 142 tracing efforts, respectively, were needed to prevent one new infection. CONCLUSION: Tracing of patients LTFU enhances the preventive effect of ART, but the number of transmissions prevented is small.

A model of the roles of essential kinases in the induction and expression of late long-term potentiation.

The induction of late long-term potentiation (L-LTP) involves complex interactions among second-messenger cascades. To gain insights into these interactions, a mathematical model was developed for L-LTP induction in the CA1 region of the hippocampus. The differential equation-based model represents actions of protein kinase A (PKA), MAP kinase (MAPK), and CaM kinase II (CAMKII) in the vicinity of the synapse, and activation of transcription by CaM kinase IV (CAMKIV) and MAPK. L-LTP is represented by increases in a synaptic weight. Simulations suggest that steep, supralinear stimulus-response relationships between stimuli (e.g., elevations in [Ca(2+)]) and kinase activation are essential for translating brief stimuli into long-lasting gene activation and synaptic weight increases. Convergence of multiple kinase activities to induce L-LTP helps to generate a threshold whereby the amount of L-LTP varies steeply with the number of brief (tetanic) electrical stimuli. The model simulates tetanic, -burst, pairing-induced, and chemical L-LTP, as well as L-LTP due to synaptic tagging. The model also simulates inhibition of L-LTP by inhibition of MAPK, CAMKII, PKA, or CAMKIV. The model predicts results of experiments to delineate mechanisms underlying L-LTP induction and expression. For example, the cAMP antagonist RpcAMPs, which inhibits L-LTP induction, is predicted to inhibit ERK activation. The model also appears useful to clarify similarities and differences between hippocampal L-LTP and long-term synaptic strengthening in other systems.

Cryo-EM model of the bullet-shaped vesicular stomatitis virus.

Vesicular stomatitis virus (VSV) is a bullet-shaped rhabdovirus and a model system of negative-strand RNA viruses. Through direct visualization by means of cryo-electron microscopy, we show that each virion contains two nested, left-handed helices: an outer helix of matrix protein M and an inner helix of nucleoprotein N and RNA. M has a hub domain with four contact sites that link to neighboring M and N subunits, providing rigidity by clamping adjacent turns of the nucleocapsid. Side-by-side interactions between neighboring N subunits are critical for the nucleocapsid to form a bullet shape, and structure-based mutagenesis results support this description. Together, our data suggest a mechanism of VSV assembly in which the nucleocapsid spirals from the tip to become the helical trunk, both subsequently framed and rigidified by the M layer.

A model of late long-term potentiation simulates aspects of memory maintenance.

Late long-term potentiation (L-LTP) denotes long-lasting strengthening of synapses between neurons. L-LTP appears essential for the formation of long-term memory, with memories at least partly encoded by patterns of strengthened synapses. How memories are preserved for months or years, despite molecular turnover, is not well understood. Ongoing recurrent neuronal activity, during memory recall or during sleep, has been hypothesized to preferentially potentiate strong synapses, preserving memories. This hypothesis has not been evaluated in the context of a mathematical model representing ongoing activity and biochemical pathways important for L-LTP. In this study, ongoing activity was incorporated into two such models - a reduced model that represents some of the essential biochemical processes, and a more detailed published model. The reduced model represents synaptic tagging and gene induction simply and intuitively, and the detailed model adds activation of essential kinases by Ca(2+). Ongoing activity was modeled as continual brief elevations of Ca(2+). In each model, two stable states of synaptic strength/weight resulted. Positive feedback between synaptic weight and the amplitude of ongoing Ca(2+) transients underlies this bistability. A tetanic or theta-burst stimulus switches a model synapse from a low basal weight to a high weight that is stabilized by ongoing activity. Bistability was robust to parameter variations in both models. Simulations illustrated that prolonged periods of decreased activity reset synaptic strengths to low values, suggesting a plausible forgetting mechanism. However, episodic activity with shorter inactive intervals maintained strong synapses. Both models support experimental predictions. Tests of these predictions are expected to further understanding of how neuronal activity is coupled to maintenance of synaptic strength. Further investigations that examine the dynamics of activity and synaptic maintenance can be expected to help in understanding how memories are preserved for up to a lifetime in animals including humans.

Dimensional approaches to schizophrenia: A comparison of the Bern Psychopathology scale and the five-factor model of the Positive and Negative Syndrome Scale

The aim was to examine to what extent the dimensions of the BPS map the five factors derived from the PANSS in order to explore the level of agreement of these alternative dimensional approaches in patients with schizophrenia. 149 inpatients with schizophrenia spectrum disorders were recruited. Psychopathological symptoms were assessed with the Bern Psychopathology Scale (BPS) and the Positive and Negative Syndrome Scale (PANSS). Linear regression analyses were conducted to explore the association between the factors and the items of the BPS. The robustness of patterns was evaluated. An understandable overlap of both approaches was found for positive and negative symptoms and excitement. The PANSS positive factor was associated with symptoms of the affect domain in terms of both inhibition and disinhibition, the PANSS negative factor with symptoms of all three domains of the BPS as an inhibition and the PANSS excitement factor with an inhibition of the affect domain and a disinhibition of the language and motor domains. The results show that here is only a partial overlap between the system-specific approach of the BPS and the five-factor PANSS model. A longitudinal assessment of psychopathological symptoms would therefore be of interest.

A Bargaining Model of Holdouts and Takings

The holdout problem is commonly cited as the justification for eminent domain, but the nature of the problem is not well understood. This paper models the holdout problem in a bargaining framework, where a developer seeks to acquire several parcels of land for a large-scale development. We show that in the absence of eminent domain, holdouts are inevitable, threatening costly delay. However, if the developer has the power to use eminent domain to acquire the land from holdouts, all sellers will bargain, thus avoiding delay. An offsetting cost is that owners may negotiate prices below their true value, possibly resulting in excessive transfer of land to the developer.

The Devil’s Calculus: Mathematical Models of Civil War

In spite of the movement to turn political science into a real science, various mathematical methods that are now the staples of physics, biology, and even economics are thoroughly uncommon in political science, especially the study of civil war. This study seeks to apply such methods - specifically, ordinary differential equations (ODEs) - to model civil war based on what one might dub the capabilities school of thought, which roughly states that civil wars end only when one side’s ability to make war falls far enough to make peace truly attractive. I construct several different ODE-based models and then test them all to see which best predicts the instantaneous capabilities of both sides of the Sri Lankan civil war in the period from 1990 to 1994 given parameters and initial conditions. The model that the tests declare most accurate gives very accurate predictions of state military capabilities and reasonable short term predictions of cumulative deaths. Analysis of the model reveals the scale of the importance of rebel finances to the sustainability of insurgency, most notably that the number of troops required to put down the Tamil Tigers is reduced by nearly a full order of magnitude when Tiger foreign funding is stopped. The study thus demonstrates that accurate foresight may come of relatively simple dynamical models, and implies the great potential of advanced and currently unconventional non-statistical mathematical methods in political science.

Age model of ODP Site 162-983

At ODP Site 983, relative geomagnetic paleointensity and planktic and benthic delta18O records have been acquired for the last 350 kyr. The mean sedimentation rate in this interval is 11.3 cm/kyr. Magnetic properties and hysteresis ratios indicate that pseudo-single domain magnetite is the remanence carrier. Volume susceptibility (kappa), anhysteretic (ARM) and isothermal (IRM) remanence values vary by a factor of 3-4, well within the criteria usually cited for paleointensity studies. Natural remanent magnetization (NRM) is normalized by ARM and IRM to acquire the paleointensity proxy. Arithmetic means of NRM/ARM and NRM/IRM, calculated for five demagnetization steps in the 25-45 mT range, constitute the relative paleointensity estimates. Some paleointensity lows (particularly those at ~40, ~120 and ~188 ka) are associated with directional excursions of the field, especially the event at ~188 ka (referred to here as the Iceland Basin Event) that constitutes a short-lived polarity reversal. For the last 200 kyr, the records can be correlated with other high-resolution paleointensity records such as those from the Labrador Sea, Mediterranean/Somali Basin and Sulu Sea, implying that the millennial scale features are globally synchronous. A labeling system for paleointensity features is proposed that ties prominent highs and lows to oxygen isotope stages.

A Mathematical Representation of "Excitement" in Games: A Contribution to the Theory of Game Systems

Researchers have long believed the concept of "excitement" in games to be subjective and difficult to measure. This paper presents the development of a mathematically computable index that measures the concept from the viewpoint of an audience and from that of a player. One of the key aspects of the index is the differential of the probability of "winning" before and after one specific "play" in a given game. The index makes a large contribution to the study of games and enables researchers to compare and analyze the “excitement” of various games. It may be applied in many fields, especially the area of welfare economics, and applications may range from those related to allocative efficiency to axioms of justice and equity.