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The relevant field of interest in High Energy Physics experiments is shifting to searching and studying extremely rare particles and phenomena. The search for rare probes requires an increase in the number of available statistics by increasing the particle interaction rate. The structure of the events also becomes more complicated, the multiplicity of particles in each event increases, and a pileup appears. Due to technical limitations, such data flow becomes impossible to store fully on available storage devices. The solution to the problem is the correct triggering of events and real-time data processing.
In this work, the issue of accelerating and improving the algorithms for reconstruction of the charged particles' trajectories based on the Cellular Automaton in the STAR experiment is considered to implement them for track reconstruction in real-time within the High-Level Trigger. This is an important step in the preparation of the CBM experiment as part of the FAIR Phase-0 program. The study of online data processing methods in real conditions at similar interaction energies allows us to study this process and determine the possible weaknesses of the approach.
Two versions of the Cellular Automaton based track reconstruction are discussed, which are used, depending on the detecting systems' features. HFT~CA Track Finder, similar to the tracking algorithm of the CBM experiment, has been accelerated by several hundred times, using both algorithm optimization and data-level parallelism. TPC~CA Track Finder has been upgraded to improve the reconstruction quality while maintaining high calculation speed. The algorithm was tuned to work with the new iTPC geometry and provided an additional module for very low momentum track reconstruction.
The improved track reconstruction algorithm for the TPC detector in the STAR experiment was included in the HLT reconstruction chain and successfully tested in the express production for the online real data analysis. This made it possible to obtain important physical results during the experiment runtime without the full offline data processing. The tracker is also being prepared for integration into a standard offline data processing chain, after which it will become the basic track search algorithm in the STAR experiment.
The thesis is composed of four Chapters.
In the first Chapter, the boundary expression of the one-sided shape derivative of nonlocal Sobolev best constants is derived. As a simple consequence, we obtain the fractional version of the so-called Hadamard formula for the torsional rigidity and the first Dirichlet eigenvalue. An application to the optimal obstacle placement problem for the torsional rigidity and the first eigenvalue of the fractional Laplacian is given.
In the second Chapter, we introduce and prove a new maximum principle for doubly antisymmetric functions. The latter can be seen as the first step towards studying the optimal obstacle placement problem for the second fractional eigenvalue. Using the new maximum principle we derive new symmetry results for odd solutions to semilinear Dirichlet boundary value problems with Lipschitz nonlinearity.
In the third Chapter, we derive new integration by parts formula for the fractional Laplace operator with a general globally Lipschitz vector field and in particular, we obtain a new Pohozaev type identity generalizing the one obtained by X. Ros-Oton and J. Serra. As an application we obtain nonexistence results for semilinear Dirichlet boundary problems in bounded domains that are not necessarly starshaped.
In the last Chapter, we study symmetry properties of second eigenfunctions of annuli. Using results from the first Chapter and the maximum principle in Chpater 2, we extend the result on the optimal obstacle placement problem from the first eigenvalue to the second eigenvalue.
The main task of modern large experiments with heavy ions, such as CBM (FAIR), STAR (BNL) and ALICE (CERN) is a detailed study of the phase diagram of quantum chromodynamics (QCD) in the quark-gluon plasma (QGP), the equation of state of matter at extremely high baryonic densities, and the transition from the hadronic phase of matter to the quark-gluon phase.
In the thesis, the missing mass method is developed for the reconstruction of short-lived particles with neutral particles in their decay products, as well as its implementation in the form of fast algorithms and a set of software for prac- tical application in heavy ion physics experiments. Mathematical procedures implementing the method were developed and implemented within the KF Par- ticle Finder package for the future CBM (FAIR) experiment and subsequently adapted and applied for processing and analysis of real data in the STAR (BNL) experiment.
The KF Particle Finder package is designed to reconstruct most signal particles from the physics program of the CBM experiment, including strange particles, strange resonances, hypernuclei, light vector mesons, charm particles and char- monium. The package includes searches for over a hundred decays of short-lived particles. This makes the KF Particle Finder a universal platform for short-lived particle reconstruction and physics analysis both online and offline.
The missing mass method has been proposed to reconstruct decays of short-lived charged particles when one of the daughter particles is neutral and is not regis- tered in the detector system. The implementation of the missing mass method was integrated into the KF Particle Finder package to search for 18 decays with a neutral daughter particle.
Like all other algorithms of the KF Particle Finder package, the missing mass method is implemented with extensive use of vector (SIMD) instructions and is optimized for parallel operation on modern many-core high performance com- puter clusters, which can include both processors and coprocessors. A set of algorithms implementing the method was tested on computers with tens of cores and showed high speed and practically linear scalability with respect to the num- ber of cores involved.
It is extremely important, especially for the initial stage of the CBM experiment, which is planned for 2025, to demonstrate already now on real data the reliability of the developed approach, as well as the high efficiency of the current implemen- tation of both the entire KF Particle Finder package, and its integral part, the missing mass method. Such an opportunity was provided by the FAIR Phase-0 program, motivating the use in the STAR experiment of software packages orig- inally developed for the CBM experiment.
Application of the method to real data of the STAR experiment shows very good results with a high signal-to-background ratio and a large significance value. The results demonstrate the reliability and high efficiency of the missing mass method in the reconstruction of both charged mother particles and their neutral daughter particles. Being an integral part of the KF Particle Finder package, now the main approach for reconstruction and analysis of short-lived particles in the STAR experiment, the missing mass method will continue to be used for the physics analysis in online and offline modes.
The high quality of the results of the express data analysis has led to their status as preliminary physics results with the right to present them at international physics conferences and meetings on behalf of the STAR Collaboration.
Machine learning (ML) techniques have evolved rapidly in recent years and have shown impressive capabilities in feature extraction, pattern recognition, and causal inference. There has been an increasing attention to applying ML to medical applications, such as medical diagnosis, drug discovery, personalized medicine, and numerous other medical problems. ML-based methods have the advantage of processing vast amounts of data.
With an ever increasing amount of medical data collection and large, inter-subject variability in the medical data, automated data processing pipelines are very much desirable since it is laborious, expensive, and error-prone to rely solely on human processing. ML methods have the potential to uncover interesting patterns, unravel correlations between complex features, learn patient-specific representations, and make accurate predictions. Motivated by these promising aspects, in this thesis, I present studies where I have implemented deep neural networks for the early diagnosis of epilepsy based on electroencephalography (EEG) data and brain tumor detection based on magnetic resonance spectroscopy (MRS) data.
In the project for early diagnosis of epilepsy, we are dealing with one of the most common neurological disorders, epilepsy, which is characterized by recurrent unprovoked seizures. It can be triggered by a variety of initial brain injuries and manifests itself after a time window which is called the latent period. During this period, a cascade of structural and functional brain alterations takes place leading to an increased seizure susceptibility.
The development and extension of brain tissue capable of generating spontaneous seizures is defined as epileptogenesis (EPG).
Detecting the presence of EPG provides a precious opportunity for targeted early medical interventions and, thus, can slow down or even halt the disease progression. In order to study brain signals in this latent window, animal epilepsy models are used to provide valuable data as it is extremely difficult to obtain this data from human patients. The aim of this study is to discover biomarkers of EPG using animal models and then to find the equivalent and counterparts in human patients' data. However, the EEG features for EPG are not well-understood and there is not a sufficiently large amount of annotated data for ML-based algorithms. To approach this problem, firstly, I utilized the timestamp information of the recorded EEG from an animal epilepsy model where epilepsy is induced by an electrical stimulation. The timestamp serves as a form of weak supervision, i.e., before and after the stimulation. Secondly, I implemented a deep residual neural network and trained it with a binary classification task to distinguish the EEG signals from these two phases. After obtaining a high discriminative ability on the binary classification task, I proposed to divide further the time span after the stimulation for a three-class classification, aiming to detect possible stages of the progression of the latent EPG phase. I have shown that the model can distinguish EEG signals at different stages of EPG with high accuracy and generalization ability. I have also demonstrated that some of the learned features from the network are clinically relevant.
In the task of detecting brain tumors based on MRS data, I first proposed to apply a deep neural network on the MRS data collected from over 400 patients for a binary classification task. To combat the challenge of noisy labeling, I developed a distillation step to filter out relatively ``cleanly'' labeled samples. A mixing-based data augmentation method was also implemented to expand the size of the training set. All the experiments were designed to be conducted with a leave-patient-out scheme to ensure the generalization ability of the model. Averaged across all leave-patient-out cross-validation sets, the proposed method performed on par with human neuroradiologists, while outperforming other baseline methods. I have demonstrated the distillation effect on the MNIST data set with manually-introduced label noise as well as providing visualization of the input influences on the final classification through a class activation map method.
Moreover, I have proposed to aggregate information at the subject level, which could provide more information and insights. This is inspired by the concept of multiple instance learning, where instance-level labels are not required and which is more tolerant to noisy labeling. I have proposed to generate data bags consisting of instances from each patient and also proposed two modules to ensure permutation invariance, i.e., an attention module and a pooling module. I have compared the performance of the network in different cases, i.e., with and without permutation-invariant modules, with and without data augmentation, single-instance-based and multiple-instance-based learning and have shown that neural networks equipped with the proposed attention or pooling modules can outperform human experts.
This thesis presents a first-of-its-kind phenomenological framework that formally describes the development of acquired epilepsy and the role of the neuro-immune axis in this development. Formulated as a system of nonlinear differential equations, the model describes the interaction of processes such as neuroinflammation, blood- brain barrier disruption, neuronal death, circuit remodeling, and epileptic seizures. The model allows for the simulation of epilepsy development courses caused by a variety of neurological injuries. The simulation results are in agreement with ex- perimental findings from three distinct animal models of epileptogenesis. Simula- tions capture injury-specific temporal patterns of seizure occurrence, neuroinflam- mation, blood-brain barrier leakage, and progression of neuronal death. In addition, the model provides insights into phenomena related to epileptogenesis such as the emergence of paradoxically long time scales of disease development after injury, the dose-dependence of epileptogenesis features on injury severity, and the variability of clinical outcomes in subjects exposed to identical injury. Moreover, the developed framework allows for the simulation of therapeutic interventions, which provides insights into the injury-specificity of prominent intervention strategies. Thus, the model can be used as an in silico tool for the generation of testable predictions, which may aid pre-clinical research for the development of epilepsy treatments.
Antimicrobial resistant infections arise as a consequential response to evolutionary mechanisms within microbes which cause them to be protected from the effects of antimicrobials. The frequent occurrence of resistant infections poses a global public health threat as their control has become challenging despite many efforts. The dynamics of such infections are driven by processes at multiple levels. For a long time, mathematical models have proved valuable for unravelling complex mechanisms in the dynamics of infections. In this thesis, we focus on mathematical approaches to modelling the development and spread of resistant infections at between-host (population-wide) and within-host (individual) levels.
Within an individual host, switching between treatments has been identified as one of the methods that can be employed for the gradual eradication of resistant strains on the long term. With this as motivation, we study the problem using dynamical systems and notions from control theory. We present a model based on deterministic logistic differential equations which capture the general dynamics of microbial resistance inside an individual host. Fundamentally, this model describes the spread of resistant infections whilst accounting for evolutionary mutations observed in resistant pathogens and capturing them in mutation matrices. We extend this model to explore the implications of therapy switching from a control theoretic perspective by using switched systems and developing control strategies with the goal of reducing the appearance of drug resistant pathogens within the host.
At the between-host level, we use compartmental models to describe the transmission of infection between multiple individuals in a population. In particular, we make a case study of the evolution and spread of the novel coronavirus (SARS-CoV-2) pandemic. So far, vaccination remains a critical component in the eventual solution to this public health crisis. However, as with many other pathogens, vaccine resistant variants of the virus have been a major concern in control efforts by governments and all stakeholders. Using network theory, we investigate the spread and transmission of the disease on social networks by compartmentalising and studying the progression of the disease in each compartment, considering both the original virus strain and one of its highly transmissible vaccine-resistant mutant strains. We investigate these dynamics in the presence of vaccinations and other interventions. Although vaccinations are of absolute importance during viral outbreaks, resistant variants coupled with population hesitancy towards vaccination can lead to further spread of the virus.
In this thesis we discuss the group Out(Gal_K) of outer automorphism of the absolute Galois group Gal_K of a p-adic number field K. Using results about the mapping class group of a surface S, as well as a result by Jannsen--Wingberg on the structure of the absolute Galois group Gal_K, we construct a large subgroup of Out(Gal_K) arising as images of certain Dehn twists on S.
We thoroughly study the properties of conically stable polynomials and imaginary projections. A multivariate complex polynomial is called stable if its nonzero whenever all coordinates of the respective argument have a positive imaginary part. In this dissertation we consider the generalized notion of K-stability. A multivariate complex polynomial is called K-stable if its non-zero whenever the imaginary part of the respective argument lies in the relative interior of the cone K. We study connections to various other objects, including imaginary projections as well as preservers and combinatorial criteria for conically stable polynomials.
In this thesis, we cover two intimately related objects in combinatorics, namely random constraint satisfaction problems and random matrices. First we solve a classic constraint satisfaction problem, 2-SAT using the graph structure and a message passing algorithm called Belief Propagation. We also explore another message passing algorithm called Warning Propagation and prove a useful result that can be employed to analyze various type of random graphs. In particular, we use this Warning Propagation to study a Bernoulli sparse parity matrix and reveal a unique phase transition regarding replica symmetry. Lastly, we use variational methods and a version of local limit theorem to prove a sufficient condition for a general random matrix to be of full rank.