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Institute
Tasks are a key resource in the process of teaching and learning mathematics, which is why task design continues to be one of the main research issues in mathematics education. Different settings can influence the principles underlying the formulation of tasks, and so does the outdoor context. Specifically, a math trail can be a privileged context, known to promote positive attitudes and additional engagement for the learning of mathematics, confronting students with a sequence of real-life tasks, related to a particular mathematical theme. Recently, mobile devices and apps, i.e., MathCityMap, have been recognized as an important resource to facilitate the extension of the classroom to the outdoors. The study reported in this paper intends to identify the principles of design for mobile theme-based math trails (TBT) that result in rich learning experiences in early algebraic thinking. A designed-based research is used, through a qualitative approach, to develop and refine design principles for TBT about Sequences and Patterns. The iterative approach is described by cycles with the intervention of the researchers, pre-service and in-service teachers and students of the targeted school levels. The results are discussed taking into account previous research and data collected along the cycles, conducing to the development of general design principles for TBT tasks.
Existence of nonradial domains for overdetermined and isoperimetric problems in nonconvex cones
(2022)
In this work we address the question of the existence of nonradial domains inside a nonconvex cone for which a mixed boundary overdetermined problem admits a solution. Our approach is variational, and consists in proving the existence of nonradial minimizers, under a volume constraint, of the associated torsional energy functional. In particular we give a condition on the domain D on the sphere spanning the cone which ensures that the spherical sector is not a minimizer. Similar results are obtained for the relative isoperimetric problem in nonconvex cones.
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.
Statistical shape models learn to capture the most characteristic geometric variations of anatomical structures given samples from their population. Accordingly, shape models have become an essential tool for many medical applications and are used in, for example, shape generation, reconstruction, and classification tasks. However, established statistical shape models require precomputed dense correspondence between shapes, often lack robustness, and ignore the global surface topology. This thesis presents a novel neural flow-based shape model that does not require any precomputed correspondence. The proposed model relies on continuous flows of a neural ordinary differential equation to model shapes as deformations of a template. To increase the expressivity of the neural flow and disentangle global, low-frequency deformations from the generation of local, high- frequency details, we propose to apply a hierarchy of flows. We evaluate the performance of our model on two anatomical structures, liver, and distal femur. Our model outperforms state-of-the-art methods in providing an expressive and robust shape prior, as indicated by its generalization ability and specificity. More so, we demonstrate the effectiveness of our shape model on shape reconstruction tasks and find anatomically plausible solutions. Finally, we assess the quality of the emerging shape representation in an unsupervised setting and discriminate healthy from pathological shapes.
Debate topic expansion
(2022)
Given a debate topic, it is often to make an expansion of the topic, the reasons can be the followings: (1) The scope of the debate topic is too shallow and we eager to discuss more. (2) A debate topic is sometimes related to the others and the discussion will not be complete when we do not discuss the others as well. (3) We may want to discuss the particular concept or the core the debate topic. It's thus meaningful to build a model in order to find the expansions of the topics.
IBM Research Team has proposed a method to expand the boundary and find the expansion topics of the given debate topics in 2019. There are two types of topic expansions in their paper, consistent and contrastive expansions. We focus on the consistent expansions. Consistent expansions are defined as the expansions that expand our topics in a positive way or at least neutral.
The main objective of this paper is to follow and examine the steps of IBM Research Team's idea and since the original discusses the model in english, we would like to implement a topic expansion model with 7 steps, including pattern extraction, filtering, training, etc, in another language (german) using translator and compare the result between different models to propose the final german model at the end.
Das Projekt anan ist ein Werkzeug zur Fehlersuche in verteilten Hochleistungsrechnern. Die Neuheit des Beitrags besteht darin, dass die bekannten Methoden, die bereits erfolgreich zum Debuggen von Soft- und Hardware eingesetzt werden, auf Hochleistungs-Rechnen übertragen worden sind. Im Rahmen der vorliegenden Arbeit wurde ein Werkzeug namens anan implementiert, das bei der Fehlersuche hilft. Außerdem kann es als dynamischeres Monitoring eingesetzt werden. Beide Einsatzzwecke sind
getestet worden.
Das Werkzeug besteht aus zwei Teilen:
1. aus einem Teil namens anan, der interaktiv vom Nutzer bedient wird
2. und aus einem Teil namens anand, der automatisiert die verlangten Messwerte erhebt und nötigenfalls Befehle ausführt.
Der Teil anan führt Sensoren aus — kleine mustergesteuerte Algorithmen —, deren Ergebnisse per anan zusammengeführt werden. In erster Näherung lässt anan sich als Monitoring beschreiben, welches (1) schnell umkonfiguriert werden (2) komplexere Werte messen kann, die über Korrelationen einfacher Zeitreihen hinausgehen.
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.
Bei der Bekleidungsmodellierung geht es um den Entwurf von Bekleidung von Personen, die beispielsweise in Szenen dargestellt werden können. Dabei stützt sich der Entwurf auf Informationen aus einer Datengrundlage. Die Darstellung von Szenen, in denen Personen dargestellt werden, stellt sich grundsätzlich als Zusammenspiel komplexer Teilaspekte dar. Dabei wird die Nachvollziehbarkeit einer modellierten Szene oder modellierter Avatare im Auge des Betrachters ganz wesentlich durch den Faktor passend gewählter Kleidung bestimmt.
In dieser Arbeit werden Ansätze und Verfahren vorgestellt, die zur Bekleidungsmodellierung auf Grundlage von Textdokumenten basieren. Dafür werden Möglichkeiten erörtert, die es erlauben Informationen aus Texten zu extrahieren und für die Modellierung einzusetzen.
Zur Bearbeitung der Aufgabenstellung wird zunächst ein aus dem Machine Learning bekanntes kontextuelles Modell hinsichtlich einer Mehrklassen-Klassifizierung trainiert und angewendet. Daraufhin wird die Erstellung einer eigenen Wissensressource, die sich auf textlicher Ebene mit dem Thema der Bekleidung auseinandersetzt, aufgebaut und mit zahlreichen Informationen aus bereits bestehenden Ressourcen popularisiert. Die neue Ressource wird in Form einer Graphdatenbank entworfen. Dabei werden Relationen zwischen den einzelnen Elementen mithilfe von statischen Modellen sowie einem kontextuellen Modell, dem BERT-Modell, erstellt. Schließlich wird auf Grundlage der entwickelten Graphdatenbank ein in der Programmiersprache Python entwickeltes Programm vorgestellt, dass Eingabetexte unter Hinzunahme der Informationen und Relationen innerhalb der Graphdatenbank verarbeitet und Kleidungsstücke detektiert.
Nach der theoretischen Aufarbeitung der entwickelten Ansätze werden die daraus resultierenden Ergebnisse diskutiert und bestehende Problematiken bei der Bearbeitung der Aufgabenstellung angesprochen. Abschließend wird die Arbeit zusammengefasst und Anregungen für die weitere Bearbeitung dieser Thematik vorgestellt.
This thesis is concerned with the study of symmetry breaking phenomena for several different semilinear partial differential equations. Roughly speaking, this encompasses equations whose symmetries are not necessarily inherited by their solutions, which is particularly interesting for ground state solutions.
Reactive oxygen species are a class of naturally occurring, highly reactive molecules that change the structure and function of macromolecules. This can often lead to irreversible intracellular damage. Conversely, they can also cause reversible changes through post-translational modification of proteins which are utilized in the cell for signaling. Most of these modifications occur on specific cysteines. Which structural and physicochemical features contribute to the sensitivity of cysteines to redox modification is currently unclear. Here, I investigated the in uence of protein structural and sequence features on the modifiability of proteins and specific cysteines therein using statistical and machine learning methods. I found several strong structural predictors for redox modification, such as a higher accessibility to the cytosol and a high number of positively charged amino acids in the close vicinity. I detected a high frequency of other post-translational modifications, such as phosphorylation and ubiquitination, near modified cysteines. Distribution of secondary structure elements appears to play a major role in the modifiability of proteins. Utilizing these features, I created models to predict the presence of redox modifiable cysteines in proteins, including human mitochondrial complex I, NKG2E natural killer cell receptors and proximal tubule cell proteins, and compared some of these predictions to earlier experimental results.
We establish weighted Lp-Fourier extension estimates for O(N−k)×O(k)-invariant functions defined on the unit sphere SN−1, allowing for exponents p below the Stein–Tomas critical exponent 2(N+1)/N−1. Moreover, in the more general setting of an arbitrary closed subgroup G⊂O(N) and G-invariant functions, we study the implications of weighted Fourier extension estimates with regard to boundedness and nonvanishing properties of the corresponding weighted Helmholtz resolvent operator. Finally, we use these properties to derive new existence results for G-invariant solutions to the nonlinear Helmholtz equation −Δu−u = Q(x)|u|p−2u,u∈W2,p(RN), where Q is a nonnegative bounded and G-invariant weight function.
This thesis concerns three specific constraint satisfaction problems: the k-SAT problem, random linear equations and the Potts model. We investigated a phenomenon called replica symmetry, its consequences and its limitation. For the $k$-SAT problem, we were able to show that replica symmetry holds up to a threshold $d^{*}$. However, after another critical threshold $d^{**}$, we discovered that replica symmetry could not hold anymore, which enabled us to establish the existence of a replica symmetry breaking region. For the random linear problem, a peculiar phenomenon occurs. We observed that a more robust version of replica symmetry (strong replica symmetry) holds up to a threshold $d=e$ and ceases to hold after. This phenomenon is linked to the fact that before the threshold $d=e$, the fraction of frozen variables, i.e. variable forced to take the same value in all solutions, is concentrated around a deterministic value but vacillates between two values with equal probability for $d>e$. Lastly, for the Potts model, we show that a phenomenon called metastability occurs. The latter phenomenon can be understood as a consequence of trivial replica symmetry breaking scheme. This metastability phenomenon further produces slow mixing results for two famous Markov chains, the Glauber and the Swendsen-Wang dynamics.
In this survey paper, we present a multiscale post-processing method in exploration. Based on a physically relevant mollifier technique involving the elasto-oscillatory Cauchy–Navier equation, we mathematically describe the extractable information within 3D geological models obtained by migration as is commonly used for geophysical exploration purposes. More explicitly, the developed multiscale approach extracts and visualizes structural features inherently available in signature bands of certain geological formations such as aquifers, salt domes etc. by specifying suitable wavelet bands.
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.
Monte Carlo methods : barrier option pricing with stable Greeks and multilevel Monte Carlo learning
(2021)
For discretely observed barrier options, there exists no closed solution under the Black-Scholes model. Thus, it is often helpful to use Monte Carlo simulations, which are easily adapted to these models. However, as presented above, the discontinuous payoff may lead to instability in option's sensitivities for Monte Carlo algorithms.
This thesis presents a new Monte Carlo algorithm that can calculate the pathwise sensitivities for discretely monitored barrier options. The idea is based on Glasserman and Staum's one-step survival strategy and the results of Alm et al., with which we can stably determine the option's sensitivities such as Delta and Vega by finite-differences. The basic idea of Glasserman and Staum is to use a truncated normal distribution, which excludes the values above the barrier (e.g.\ for knock-up-out options), instead of sampling from the full normal distribution. This approach avoids the discontinuity generated by any Monte Carlo path crossing the barrier and yields a Lipschitz-continuous payoff function.
The new part will be to develop an extended algorithm that estimates the sensitivities directly, without simulation at multiple parameter values as in finite-difference.
Consider the local volatility model, which is a generalisation of the Black-Scholes model. Although standard Monte Carlo algorithms work well for the pricing of continuously monitored barrier options within this model, they often do not behave stably with respect to numerical differentiation.
To bypass this problem, one would generally either resort to regularised differentiation schemes or derive an algorithm for precise differentiation. Unfortunately, while the widespread solution of using a Brownian bridge approach leads to accurate first derivatives, they are not Lipschitz-continuous. This leads to instability with respect to numerical differentiation for second-order Greeks.
To alleviate this problem - i.e. produce Lipschitz-continuous first-order derivatives - and reduce variance, we generalise the idea of one-step survival to general scalar stochastic differential equations. This approach leads to the new one-step survival Brownian bridge approximation, which allows for stable second-order Greeks calculations.
To show the new approach's numerical efficiency, we present a new respective Monte Carlo pathwise sensitivity estimator for the first-order Greeks and study different methods to compute second-order Greeks stably. Finally, we develop a one-step survival Brownian bridge multilevel Monte Carlo algorithm to reduce the computational cost in practice.
This thesis proves unbiasedness and variance reduction of our new, one-step survival version with respect to the classical, Brownian bridge approach. Furthermore, we will present a new convergence result for the Brownian bridge approach using the Milstein scheme under certain conditions. Overall, these properties imply convergence of the new one-step survival Brownian bridge approach.
In recent years, deep learning has become pervasive in various fields. As a family of machine learning methods it is used in a broad set of applications, such as image processing, voice recognition, email filtering, computer vision. Most modern deep learning algorithms are based on artificial neural networks inspired by the biological neural networks constituting animal brains. Also in computational finance deep learning may be of use: Consider there is no closed-solution available for an option price, Monte Carlo simulations are substantially for estimation. Instead of persistently contributing new price computations arising from an updated volatility term, one could replace these by evaluating a neural network.
If an according neural network is available, the evaluation could lead to substantial savings and be highly efficient. I.e., once trained, a neural network could save further expensive estimations. However, in practice, the challenge is the training process of the neural network.
We study and compare two generic neural network training algorithms' computational complexity. Then, we introduce a new multilevel training algorithm that combines a deep learning algorithm with the idea of multilevel Monte Carlo path simulation. The idea is to train several neural networks with training data computed from the so-called level estimators of the multilevel Monte Carlo approach introduced by Giles. We show that the new method can reduce computational complexity by formulating a complexity theorem.
We show how nonlocal boundary conditions of Robin type can be encoded in the pointwise expression of the fractional operator. Notably, the fractional Laplacian of functions satisfying homogeneous nonlocal Neumann conditions can be expressed as a regional operator with a kernel having logarithmic behaviour at the boundary.
This article deals with the solution of linear ill-posed equations in Hilbert spaces. Often, one only has a corrupted measurement of the right hand side at hand and the Bakushinskii veto tells us, that we are not able to solve the equation if we do not know the noise level. But in applications it is ad hoc unrealistic to know the error of a measurement. In practice, the error of a measurement may often be estimated through averaging of multiple measurements. We integrated that in our anlaysis and obtained convergence to the true solution, with the only assumption that the measurements are unbiased, independent and identically distributed according to an unknown distribution.
We prove new existence results for a nonlinear Helmholtz equation with sign-changing nonlinearity of the form − delta u−k2u=Q(x)/u/p−2u, uEW2, p(RN) – delta u − k2u=Q(x)/u/p−2u, uEW2, p(RN) with k>0, k>0, N≥3N≥3, pE[2(N+1)N − 1, 2NN − 2)pE[2(N+1)N − 1, 2NN−2) and QEL ∞ (RN)QEL ∞ (RN). Due to the sign-changes of Q, our solutions have infinite Morse-Index in the corresponding dual variational formulation.