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Parallel Coordinate Plots (PCP) are a valuable tool for exploratory data analysis of high-dimensional numer-ical data. The use of PCPs is limited when working with categorical variables or a mix of categorical andcontinuous variables. In this article, we propose Generalized Parallel Coordinate Plots (GPCP) to extend theability of PCPs from just numeric variables to dealing seamlessly with a mix of categorical and numericvariables in a single plot. In this process we find that existing solutions for categorical values only, such ashammock plots or parsets become edge cases in the new framework. By focusing on individual observationsrather than a marginal frequency we gain additional flexibility. The resulting approach is implemented in theR package ggpcp. Supplementary materials for this article are available online.

Given partially ordered sets (posets) $(P, \leq_P)$ and $(P', \leq_{P'})$, we say that $P'$ contains a copy of $P$ if for some injective function $f: P\rightarrow P'$ and for any $X, Y\in P$, $X\leq _P Y$ if and only of $f(X)\leq_{P'} f(Y)$. For any posets $P$ and $Q$, the poset Ramsey number $R(P,Q)$ is the least positive integer $N$ such that no matter how the elements of an $N$-dimensional Boolean lattice are colored in blue and red, there is either a copy of $P$ with all blue elements or a copy of $Q$ with all red elements. We focus on a poset Ramsey number $R(P, Q_n)$ for a fixed poset $P$ and an $n$-dimensional Boolean lattice $Q_n$, as $n$ grows large. We show a sharp jump in behaviour of this number as a function of $n$ depending on whether or not $P$ contains a copy of either a poset $V$, i.e. a poset on elements $A, B, C$ such that $B>C$, $A>C$, and $A$ and $B$ incomparable, or a poset $\Lambda$, its symmetric counterpart. Specifically, we prove that if $P$ contains a copy of $V$ or $\Lambda$ then $R(P, Q_n) \geq n +\frac{1}{15} \frac{n}{\log n}$. Otherwise $R(P, Q_n) \leq n + c(P)$ for a constant $c(P)$. This gives the first non-marginal improvement of a lower bound on poset Ramsey numbers and as a consequence gives $R(Q_2, Q_n) = n + \Theta (\frac{n}{\log n})$., Comment: 18 pages, 2 figures

This work compares two Nitsche-type approaches to treat non-conforming triangulations for a high-order discontinuous Galerkin (DG) solver for the acoustic conservation equations. The first approach (point-to-point interpolation) uses inconsistent integration with quadrature points prescribed by a primary element. The second approach uses consistent integration by choosing quadratures depending on the intersection between non-conforming elements. In literature, some excellent properties regarding performance and ease of implementation are reported for point-to-point interpolation. However, we show that this approach can not safely be used for DG discretizations of the acoustic conservation equations since, in our setting, it yields spurious oscillations that lead to instabilities. This work presents a test case in that we can observe the instabilities and shows that consistent integration is required to maintain a stable method. Additionally, we provide a detailed analysis of the method with consistent integration. We show optimal spatial convergence rates globally and in each mesh region separately. The method is constructed such that it can natively treat overlaps between elements. Finally, we highlight the benefits of non-conforming discretizations in acoustic computations by a numerical test case with different fluids.

In this paper we consider traces at initial times for functions with mixed time-space smoothness. Such results are often needed in the theory of evolution equations. Our result extends and unifies many previous results. Our main improvement is that we can allow general interpolation couples. The abstract results are applied to regularity problems for fractional evolution equations and stochastic evolution equation, where uniform trace estimates on the half-line are shown., Comment: Some typos corrected. Accepted for publication in Mathematische Nachrichten

The fractional Laplacian $(-\Delta )^a$, $a\in(0,1)$, and its generalizations to variable-coefficient $2a$-order pseudodifferential operators $P$, are studied in $L_q$-Sobolev spaces of Bessel-potential type $H^s_q$. For a bounded open set $\Omega \subset \mathbb R^n$, consider the homogeneous Dirichlet problem: $Pu =f$ in $\Omega $, $u=0$ in $ \mathbb R^n\setminus\Omega $. We find the regularity of solutions and determine the exact Dirichlet domain $D_{a,s,q}$ (the space of solutions $u$ with $f\in H_q^s(\overline\Omega )$) in cases where $\Omega $ has limited smoothness $C^{1+\tau }$, for $2a, Comment: 52 pages. In this version some clarifications and minor corrections were done. The version is accepted for publication in "J. London Math. Soc."

International journal for numerical methods in engineering (2023). doi:10.1002/nme.7200, Published by Wiley, Chichester [u.a.]

We introduce cylinder processes in the plane defined as union sets of dilated straight lines (appearing as mutually overlapping infinitely long strips) generated by a stationary independently marked point process on the real line, where the marks describe the width and orientation of the individual cylinders. We study the behavior of the total area of the union of strips contained in a space-filling window ϱK as ϱ → ∞. In the case the unmarked point process is Brillinger mixing, we prove themean-square convergence of the area fraction of the cylinder process in ϱK. Under stronger versions of Brillinger mixing, we obtain the exact variance asymptotics of the area of the cylinder process in ϱK as ϱ → ∞. Due to the long-range dependence of the cylinder process, this variance increases asymptotically proportionally to ϱ3.

For a bounded linear operator on a Banach space, we study approximation of the spectrum and pseudospectra in the Hausdorff distance. We give sufficient and necessary conditions in terms of pointwise convergence of appropriate spectral quantities.

Monatshefte für Mathematik (2022). doi:10.1007/s00605-022-01806-x, Published by Springer, Wien [u.a.]

We study a free-boundary fluid–structure interaction problem with growth, which arises from the plaque formation in blood vessels. The fluid is described by the incompressible Navier–Stokes equations, while the structure is considered as a viscoelastic incompressible neo-Hookean material. Moreover, the growth due to the biochemical process is taken into account. Applying the maximal regularity theory to a linearization of the equations, along with a deformation mapping, we prove the well-posedness of the full nonlinear problem via the contraction mapping principle.

Communications in mathematical physics 399(2), 1039 - 1101 (2023). doi:10.1007/s00220-022-04571-y, We study the Borel summation of the Gromov-Witten potential for the resolved conifold. The Stokes phenomena associated to this Borel summation are shown to encode the Donaldson-Thomas invariants of the resolved conifold, having a direct relation to the Riemann-Hilbert problem formulated by T. Bridgeland. There exist distinguished integration contours for which the Borel summation reproduces previous proposals for the non-perturbative topological string partition functions of the resolved conifold. These partition functions are shown to have another asymptotic expansion at strong topological string coupling. We demonstrate that the Stokes phenomena of the strong-coupling expansion encode the DT invariants of the resolved conifold in a second way. Mathematically, one finds a relation to Riemann-Hilbert problems associated to DT invariants which is different from the one found at weak coupling. The Stokes phenomena of the strong-coupling expansion turn out to be closely related to the wall-crossing phenomena in the spectrum of BPS states on the resolved conifold studied in the context of supergravity by D. Jafferis and G. Moore., Published by Springer, Heidelberg

We show how combinatorial star products can be used to obtain strict deformation quantizations of polynomial Poisson structures on $\mathbb R^d$, generalizing known results for constant and linear Poisson structures to polynomial Poisson structures of arbitrary degree. We give several examples of nonlinear Poisson structures and construct explicit formal star products whose deformation parameter can be evaluated to any real value of $\hbar$, giving strict quantizations on the space of analytic functions on $\mathbb R^d$ with infinite radius of convergence. We also address further questions such as continuity of the classical limit $\hbar \to 0$, compatibility with *-involutions, and the existence of positive linear functionals. The latter can be used to realize the strict quantizations as *-algebras of operators on a pre-Hilbert space which we demonstrate in a concrete example., 45 pages, final version

The bosonic signature operator is defined for Klein-Gordon fields and massless scalar fields on globally hyperbolic Lorentzian manifolds of infinite lifetime. The construction is based on an analysis of families of solutions of the Klein-Gordon equation with a varying mass parameter. It makes use of the so-called bosonic mass oscillation property which states that integrating over the mass parameter generates decay of the field at infinity. We derive a canonical decomposition of the solution space of the Klein-Gordon equation into two subspaces, independent of observers or the choice of coordinates. This decomposition endows the solution space with a canonical complex structure. It also gives rise to a distinguished quasi-free state. Taking a suitable limit where the mass tends to zero, we obtain corresponding results for massless fields. Our constructions and results are illustrated in the examples of Minkowski space and ultrastatic spacetimes., 20 pages, LaTeX, many smaller improvements (published version)

Mathematical programming (2022). doi:10.1007/s10107-022-01885-6, Published by Springer, Berlin ; Heidelberg

We consider a C0 Interior Penalty Discontinuous Galerkin (C0IPDG) approximation of a nonlinear fourth order elliptic boundary value problem of p-biharmonic type and an equilibrated a posteriori error estimator. The C0IPDG method can be derived from a discretization of the corresponding minimization problem involving a suitably defined reconstruction operator. The equilibrated a posteriori error estimator provides an upper bound for the discretization error in the broken W2,p norm in terms of the associated primal and dual energy functionals. It requires the construction of an equilibrated flux and an equilibrated moment tensor based on a three-field formulation of the C0IPDG approximation. The relationship with a residual-type a posteriori error estimator is studied as well. Numerical results illustrate the performance of the suggested approach.

The brief history of relaxation in continuum mechanics ranges from early application of non-convex plasticity and phase transition formulations to small and large strain continuum damage mechanics. However, relaxed continuum damage mechanics formulations are still limited in the following sense that their material response lack to model strain softening and the convexification of the non-convex incremental stress potential is computationally costly. This paper presents a reduced model for relaxed continuum damage mechanics at finite strains which includes strain softening by a fiber-specific damage in the microsphere approach. Computational efficiency is achieved by novel adaptive algorithms for the fast convexification of the one-dimensional fiber material model. The algorithms are benchmarked against state-of-the-art methods, and the choice of quadrature schemes for the microsphere approach is discussed. This contribution is finalized by a mesh independence test.

For linear flows on vector bundles, it is analyzed when subbundles in the Selgrade decomposition yield chain transitive subsets for the induced flow on the associated Poincaré sphere bundle.

There are several proofs by now for the famous Cwikel–Lieb–Rozenblum (CLR) bound, which is a semiclassical bound on the number of bound states for a Schrödinger operator, proven in the 1970s. Of the rather distinct proofs by Cwikel, Lieb, and Rozenblum, the one by Lieb gives the best constant, the one by Rozenblum does not seem to yield any reasonable estimate for the constants, and Cwikel’s proof is said to give a constant which is at least about 2 orders of magnitude off the truth. This situation did not change much during the last 40+ years. It turns out that this common belief, i.e, Cwikel’s approach yields bad constants, is not set in stone: We give a substantial refinement of Cwikel’s original approach which highlights a natural but overlooked connection of the CLR bound with bounds for maximal Fourier multipliers from harmonic analysis. Moreover, it gives an astonishingly good bound for the constant in the CLR inequality. Our proof is also quite flexible and leads to rather precise bounds for a large class of Schrödinger-type operators with generalized kinetic energies.

In this thesis, we study long-range percolation at critical phases. We consider the graph distances for the long-range percolation graph with critical decay parameter and we provide lower bounds for the critical exponents for long-range percolation. Furthermore, we study long-range random walks in dimensions 1 and 2, and we use such random walks in order to show the recurrence of the simple random walk on certain two-dimensional dependent percolation models. In dieser Arbeit untersuchen wir long-range Perkolation in kritischen Phasen. Wir betrachten die Graphdistanzen für long-range Perkolation mit kritischem Parameter und geben untere Schranken für die kritischen Exponenten für long-range Perkolation an. Darüber hinaus untersuchen wir Irrfahrten mit Sprüngen in den Dimensionen 1 und 2, und wir verwenden solche Irrfahrten, um die Rekursion der einfachen Irrfahrt auf zweidimensionalen abhängigen Perkolationsmodellen zu zeigen.

This thesis is concerned with Weil representations of superelliptic curves with potentially good reduction and with the explicit determination of these representations. A Weil representation is a continuous complex representation of the so-called Weil group. The Weil group of a local or global field K can in turn be understood as a modification of the absolute Galois group, which is useful for the study of Galois representations. In this thesis, we restrict ourselves to the case where K is a (non-archimedean) local field. Any (smooth and projective) curve X over a local field K with potentially good reduction has a Weil representation of dimension 2g, where g is the genus of X. In this thesis, we restrict ourselves to the case of superelliptic curves X over p-adic number fields K with exponent prime to p.

Cholera is an infectious disease caused by the pathogenic Vibrio cholerae bacteria. The transmission dynamics for the disease span multiple interconnected scales, making it challenging for traditional single-scale immunological or epidemiological models to capture the disease dynamics. In this thesis, we develop multi-scale models to study the transmission dynamics of cholera. Cholera ist eine Infektionskrankheit welche durch das pathogene Bakterium Vibrio cholerae verursacht wird. Die Übertragungsdynamik der Krankheit erstreckt sich über mehrere miteinander verknüpfte Skalen. Daher gestaltet sich die Erfassung der Krankheitsdynamik durch herkömmliche immunologische oder epidemiologische Modelle basierend auf einer einzigen Skala als schwierig. In dieser Arbeit entwickeln wir Multiskalenmodelle zur Untersuchung der Übertragungsdynamik von Cholera.

We investigate the Dolbeault cohomology of weakly smooth forms on the Berkovich analytification of abelian varietes over a complete, non-trivially valued, non-archimedean field. In this setting, we give lower bounds for the dimension of the Dolbeault cohomology groups of weakly smooth forms. Furthermore, we prove lower bounds for the dimension of the Dolbeault cohomology groups of strong currents, i.e. of the topological dual of weakly smooth forms, in higher degrees., Wir untersuchen die Dolbeault Kohomologie von schwach-glatten Formen auf der Berkovich Analytifizierung von abelschen Varietäten über vollständigen, nicht-trivial bewerteten nicht-archimedischen Körpern. In diesem Setup geben wir einerseits untere Schranken für die Dimension der Dolbeault Kohomologie Gruppen der schwach-glatten Formen an. Andererseits erhalten wir auch untere Schranken für die Dimension der Dolbeault Kohomologie Gruppen von starken Strömen, des topologischen Duals von schwach-glatten Formen, in höheren Graden.

This thesis consists of research articles on the asymptotic behavior of modular forms and various related objects. First we determine the bivariate asymptotic behavior of Fourier coefficients for a wide class of eta-theta quotients with simple poles in the upper half plane by employing a variant of Wright’s Circle Method. These kind of quotients show up in many different areas not only in mathematics. For example they show up in investigations into Vafa– Witten invariants or the counting of so-called BPS-states via wall-crossing, but also in Watson’s quintuple product formula which has many applications in number theory and combinatorics. Further, we offer a general framework to prove asymptotic equidistribution, convexity, and log-concavity of coefficients of generating functions in arithmetic progressions. We do this by using a variant of Wright’s Circle Method and give a selection of different examples of such results for various (modular typed) objects. We end the thesis by employing the Circle Method to prove exact formulae for Fourier coefficients of an infinite family of weight zero mixed false modular forms showing up as characters of modules of rational vertex operator algebras.

This thesis contributes to the theory of modeling high-dimensional phenomena, with an emphasis on extreme events. First, it provides a one-to-one correspondence of sequences of exchangeable random variables whose multivariate margins are minimum-infinitely divisible and the class of non-negative and non-decreasing cadlag infinitely divisible processes. Second, an exact simulation algorithm for continuous maximum-infinitely divisible processes is developed. Third, a statistical model comparison framework with a focus on non-smooth criterion functions is proposed. Diese Arbeit behandelt die Modellierung hochdimensionaler Zufallsphänomene, mit Fokus auf Extremereignissen. Zuerst wird eine 1-zu-1 Beziehung von Sequenzen austauschbarer Zufallsvariablen mit minimum-unendlich teilbaren Randverteilungen und der Klasse nicht-negativer, nicht-fallender unendlich teilbarer cadlag Prozesse hergestellt. Zweitens wird ein Simulationsalgorithmus für stetige maximum-unendlich teilbare Prozesse hergeleitet. Zuletzt wird ein Rahmen für statistische Modellvergleiche für nicht-differenzierbare Kriteriumsfunktionen entwickelt.

Statistical learning literature has been tremendously growing in recent years. However, modeling complex dependence structures explicitly and interpreting model results for actionable insights simultaneously have not yet been handled well. Vine copulas are flexible and interpretable multivariate distribution functions that model complex data dependence structures. Thus, to bridge the gap, we develop statistical learning algorithms using vine copulas and show some societal applications. Die Literatur statistisches Lernen ist in den letzten Jahren enorm gewachsen. Die explizite Modellierung komplexer Abhängigkeitsstrukturen und die gleichzeitige Interpretation der Modellergebnisse für verwertbare Erkenntnisse sind noch nicht gut gelöst. Vine Copulas sind flexible multivariate Verteilungsfunktionen, die komplexe Datenabhängigkeitsstrukturen modellieren. Wir entwickeln daher statistische Lernalgorithmen, die Vine Copulas verwenden und zeigen einige gesellschaftlichen Anwendungen.

We say that an edge-labeling of a graph is balanced, if there are equally many edges of each color. In this thesis we investigate how balanced edge-labelings of a complete graph translate to the existence of balanced spanning subgraphs. Firstly, for two colors, we characterize some classes of graphs that always allow a balanced embedding and provide upper and lower bounds on the proximity to being balanced for spanning forests. We also consider some classes of graphs that allow to always find a very unbalanced embedding in a balanced complete graph. For three and more colors, we present results for the balanced embedding of perfect matchings. In the second part of the thesis we look at a number of related concepts, among them the number of maximum matchings in a graph and a precise bound for majority edge-colorings, where no vertex can have more than half its incident edges of the same color. Lastly, we give a result on the diameter of graphs with no four-cycle in relation to its edge-connectivity.

This thesis considers generalized Nash equilibrium problems in an infinite-dimensional setting with as general assumptions on the underlying functions and spaces as possible. We express the GNEP in terms of the regularized Nikaido-Isoda merit functional, prove the existence of different types of equilibria, present characterizations of these points and state an augmented Lagrangian method for computing Karush-Kuhn-Tucker-type equilibria. Diese Arbeit betrachtet verallgemeinerte Nash-Gleichgewichtsprobleme in einem unendlichdimensionalen Rahmen mit möglichst allgemeinen Annahmen an die Funktionen und Räume. Wir formulieren das Problem mittels regularisierten Nikaido-Isoda-Funktionalen, beweisen die Existenz verschiedener Gleichgewichtspunkte, zeigen für diese verschiedene Charakterisierungen und präsentieren eine augmentierte Lagrange-Methode zur Bestimmung von Karush-Kuhn-Tucker-basierten Gleichgewichtspunkten.

Solving high-dimensional, nonlinear systems is a key challenge in computational physics. We propose novel physics-aware machine learning models that rely both on physical knowledge as well as a small amount of data and are able to efficiently solve these aforementioned high-dimensional systems. The key characteristic of this approach is incorporating inductive bias in contrast to purely statistical frameworks that lack interpretability and rely on large amounts of expensive data. Das Lösen von hochdimensionalen, nichtlinearen Systemen ist eine der zentralen Herausforderungen in der numerischen Physik. Wir stellen neuartige Physik-inspirierte Modelle des maschinellen Lernens vor, die auf physikalischem Wissen und wenigen Daten basieren und hochdimensionale Systeme effizient lösen können. Zentral ist das Einbeziehen von induktiver Verzerrung im Gegensatz zu rein statistischen Algorithmen, die eine große Datenmenge benötigen und deren Interpretierbarkeit nicht gegeben ist.

The topic of this thesis is the interplay between rational double point singularities and del Pezzo surfaces. Chapter I determines which rational double points occur on del Pezzo surfaces, extending work of Du Val to positive characteristic. Chapter II classifies weak del Pezzo surfaces with global vector fields. The corresponding problem for RDP del Pezzo surfaces in odd characteristic is solved in Chapter III. As an application of the techniques developed in this thesis, rational (quasi-)elliptic surfaces with global vector fields are treated in Chapter IV. Das Thema dieser Arbeit ist das Zusammenspiel zwischen rationalen Doppelpunkt-Singularitäten und del Pezzo Flächen. Kapitel I bestimmt, welche rationalen Doppelpunkte auf del Pezzo Flächen auftreten, und verallgemeinert damit Arbeiten von Du Val auf den Fall positiver Charakteristik. Kapitel II klassifiziert schwache del Pezzo Flächen mit globalen Vektorfeldern. Das entsprechende Problem für RDP del Pezzo Flächen in ungerader Charakteristik wird in Kapitel III gelöst. Als eine Anwendung der in dieser Arbeit entwickelten Techniken werden in Kapitel IV rationale (quasi-)elliptische Flächen mit globalen Vektorfeldern behandelt.

Probabilistische Analysen können sich auf eine Vielzahl mathematischer Objekte beziehen. In dieser Dissertation werden Coxeterstatistiken und STIT-Mosaike betrachtet. Der erste Teil der Arbeit beweist mittels Abhängigkeitsgraphen, dass die Coxeterstatistiken (verallgemeinerte) d-inversions und (verallgemeinerte) d-descents asymptotisch normalverteilt sind. Zwischenresultate liefern Zusammenhänge zwischen wahrscheinlichkeitstheoretischen Größen von Zufallsvariablen und kombinatorischen Größen von Coxeter-Gruppen. Im zweiten Teil der Arbeit werden spezielle Typen planarer STIT-Mosaike betrachten. Für Mondrians werden wahrscheinlichkeitstheoretische Eigenschaften zweiter Ordnung bewiesen. Dazu gehören u.a. die Varianz und Kovarianz für die Anzahl maximaler Kanten und für die gewichtete Kantenlänge. Für isotrope STIT-Mosaike werden unterschiedliche Arten von Geradensegmente analysiert. Es werden Wahrscheinlichkeiten für verschiedene geometrische und metrische Eigenschaften hergeleitet.

This thesis concerns data acquisition and reconstruction under specific structural assumptions. First, we prove an optimal embedding dimension for a class of Johnson-Lindenstrauss embeddings for Kronecker products of multiple vectors. Then we study related higher-order random tensors. In addition, we investigate to what extent neural networks can recover sparse vectors and associated problems. Finally, we improve recovery guarantees for vectors with limited numbers of non-zero entries in multiple blocks of their entries. Zentrales Thema dieser Arbeit sind Datenerfassung und Rekonstruktion unter Strukturannahmen. Im ersten Teil beweisen wir eine optimale Einbettungsdimension für eine Klasse von Johnson-Lindenstrauss-Einbettungen für Kronecker-Produkte mehrerer Vektoren. Anschließend behandeln wir damit verbundene Zufallstensoren höherer Ordnung. Darüber hinaus untersuchen wir, inwiefern neuronale Netze dünnbesetzte Vektoren rekonstruieren können, sowie verwandte Probleme. Im letzten Teil verbessern wir Ergebnisse für die Rekonstruktion von Vektoren, für welche die Anzahl von null verschiedener Einträge in mehreren Blöcken beschränkt ist.

In this thesis we give two applications of Ayoub’s motivic nearby cycles functor: First we give a generalization of Grothendieck’s classical local monodromy theorem. In the same setup we show that the inertia group acts quasi-unipotently on the étale cohomology of sheaves "coming from motives". Second we study the notion of universal local acyclicity for motives and show that for étale motives universal local acyclicity over an excellent 1- dimensional regular base scheme is detected by the motivic nearby cycles functor. Along the way we prove properties of the motivic nearby cycles functor which might be of independent interest., In dieser Arbeit zeigen wir zwei Anwendungen von Ayoubs motivischem Nahe Zyklen Funktor: Zuerst zeigen wir eine Verallgemeinerung von Grothendiecks klassischem lokalen Monodromie Theorem. Unter den selben Voraussetzungen zeigen wir, dass die Trägheitsgruppe quasi-unipotent auf die étale Kohomologie von Garben, die "von Motiven kommen", operiert. Als zweites studieren wir den Begriff der universellen lokalen Azyklizität für Motive und zeigen, dass für étale Motive universelle lokale Azyklizität über einem 1-Dimensionalen regulären Grundschema von dem motivischen Nahe Zyklen Funktor erkannt wird. Auf dem Weg dorthin beweisen wir Eigenschaften des motivischem Nahe Zyklen Funktors, die von unabhängigem Interesse sind.

There are simple algorithms to compute the predecessor, successor, unit in the first place, unit in the last place etc. in binary arithmetic. In this note equally simple algorithms for computing the unit in the first place and the unit in the last place in precision-p base- β arithmetic with p⩾ 1 and with β⩾ 2 are presented. The algorithms work in the underflow range, and numbers close to overflow are treated by scaling. The algorithms use only the basic operations with directed rounding. If the successor (or predecessor) of a floating-point number is available, an algorithm in rounding to nearest is presented as well.

This work deals with the efficient numerical characterization of Pareto stationary fronts for multiobjective optimal control problems with a moderate number of cost functionals and a mildly nonsmooth, elliptic, semilinear PDE-constraint. When “ample” controls are considered, strong stationarity conditions that can be used to numerically characterize the Pareto stationary fronts are known for our problem. We show that for finite dimensional controls, a sufficient adjoint-based stationarity system remains obtainable. It turns out that these stationarity conditions remain useful when numerically characterizing the fronts, because they correspond to strong stationarity systems for problems obtained by application of weighted-sum and reference point techniques to the multiobjective problem. We compare the performance of both scalarization techniques using quantifiable measures for the approximation quality. The subproblems of either method are solved with a line-search globalized pseudo-semismooth Newton method that appears to remove the degenerate behavior of the local version of the method employed previously. We apply a matrix-free, iterative approach to deal with the memory and complexity requirements when solving the subproblems of the reference point method and compare several preconditioning approaches.

We construct a ring of meromorphic Siegel modular forms of degree 2 and level 5, with singularities supported on an arrangement of Humbert surfaces, which is generated by four singular theta lifts of weights 1, 1, 2, 2 and their Jacobian. We use this to prove that the ring of holomorphic Siegel modular forms of degree 2 and level $\Gamma_0(5)$ is minimally generated by eighteen modular forms of weights 2, 4, 4, 4, 4, 4, 6, 6, 6, 6, 10, 11, 11, 11, 13, 13, 13, 15., Comment: 11 pages

We consider the upscaled linear elasticity problem in the context of periodic homogenization. Based on measurements of the deformation of the (macroscopic) boundary of a body for a given forcing, it is the aim to deduce information on the geometry of the microstructure. For a parametrized microstructure, we are able to prove that there exists at least one solution of the associated minimization problem based on the L2$$ {L}^2 $$‐difference of the measured deformation and the resulting deformation for a given parameter. To facilitate the use of gradient‐based algorithms, we derive the Gâteaux derivatives using the Lagrangian method of Céa, and we present numerical experiments showcasing the functioning of the method.

This paper addresses the numerical solution of nonlinear eigenvector problems such as the Gross-Pitaevskii and Kohn-Sham equation arising in computational physics and chemistry. These problems characterize critical points of energy minimization problems on the infinite-dimensional Stiefel manifold. To efficiently compute minimizers, we propose a novel Riemannian gradient descent method induced by an energy-adaptive metric. Quantified convergence of the methods is established under suitable assumptions on the underlying problem. A non-monotone line search and the inexact evaluation of Riemannian gradients substantially improve the overall efficiency of the method. Numerical experiments illustrate the performance of the method and demonstrates its competitiveness with well-established schemes., accepted for publication in M2AN

Given a contact structure on a manifold $V$ together with a supporting open book decomposition, Bourgeois gave an explicit construction of a contact structure on $V \times \mathbb{T}^2$. We prove that all such structures are universally tight in dimension $5$, independent on whether the original contact manifold is itself tight or overtwisted. In arbitrary dimensions, we provide obstructions to the existence of strong symplectic fillings of Bourgeois manifolds. This gives a broad class of new examples of weakly but not strongly fillable contact $5$-manifolds, as well as the first examples of weakly but not strongly fillable contact structures in all odd dimensions. These obstructions are particular instances of more general obstructions for $\mathbb S^1$-invariant contact manifolds. We also obtain a classification result in arbitrary dimensions, namely that the unit cotangent bundle of the $n$-torus has a unique symplectically aspherical strong filling up to diffeomorphism., Comment: v5: Final version of the paper. To appear in Inventiones Mathematicae. arXiv admin note: text overlap with arXiv:1903.11866

In the present work, we establish the approximation of nonlinear stochastic partial differential equation (SPDE) driven by cylindrical {\alpha}-stable L\'evy processes via modulation or amplitude equations. We study SPDEs with a cubic nonlinearity, where the deterministic equation is close to a change of stability of the trivial solution. The natural separation of time-scales close to this bifurcation allows us to obtain an amplitude equation describing the essential dynamics of the bifurcating pattern, thus reducing the original infinite dimensional dynamics to a simpler finite-dimensional effective dynamics. In the presence of a multiplicative stable L\'evy noise that preserves the constant trivial solution we study the impact of noise on the approximation. In contrast to Gaussian noise, where non-dominant pattern are uniformly small in time due to averaging effects, large jumps in the L\'evy noise might lead to large error terms, and thus new estimates are needed to take this into account.

Definition Within Lorenz models, the three major kinds of butterfly effects (BEs) are the sensitive dependence on initial conditions (SDIC), the ability of a tiny perturbation to create an organized circulation at large distances, and the hypothetical role of small-scale processes in contributing to finite predictability, referred to as the first, second, and third kinds of butterfly effects (BE1, BE2, and BE3), respectively. A well-accepted definition of the butterfly effect is the BE1 with SDIC, which was rediscovered by Lorenz in 1963. In fact, the use of the term “butterfly” appeared in a conference presentation by Lorenz in 1972, when Lorenz introduced the BE2 as the metaphorical butterfly effect. In 2014, the so-called “real butterfly effect”, which is based on the features of Lorenz’s study in 1969, was introduced as the BE3.

Establishing cooperation, typically formalized by so-called hedonic games, is a central concern in multi-agent system. The desirability of coalitions in these games is measured by means of solution concepts. The thesis is an algorithmic study of classical and newly proposed solution concepts. In addition, we open new perspectives by considering dynamics and by integrating empathy. We also study a game-theoretic model of residential segregation, which encapsulates Schelling's homophily incentives. Das Erreichen von Kooperation, typischerweise formalisiert durch sogenannte hedonische Spiele, ist ein zentrales Anliegen in Multiagentensystemen. Wünschenswerte Koalitionen findet man dabei mit Hilfe von Lösungskonzepten. Die Arbeit ist eine algorithmische Untersuchung klassischer und neuer Lösungskonzepte. Wir entwickeln eine dynamische Sichtweise und integrieren Empathie. Wir untersuchen auch ein spieltheoretisches Modell von Segregation, das Schellings Homophilie-Anreize aufgreift.

Single-cell omics assays provide molecular characterisations of cells. Immunology has stood out as a key application of these assays and molecular states of immune cell have been described in detail. However, mechanistic insights into diseases are often still incomplete. I addressed this limitation using interpretable machine learning models for single-cell immunology in population dynamics, antigen recognition, spatial data, sample and epigenetic variation, and automated analyses. In Einzelzell-Omics wird der molekularen Zustand von Zellen gemessen. Dies Methoden haben in Immunologie detaillierte Beschreibungen von Immunzellen hervorgebracht. Trotz diese Datenfülle gibt es viele offene Fragen zu Krankheitsmechanismen. Mit Bezug auf diese Fragen habe ich interpretierbaren Modellen des maschinellen Lernens in Populationsdynamik, Antigenerkennung, räumlichen Abhängigkeiten von Zellen, epigenetischer und experimenteller Variation, und in automatisierten Analysen entwickelt.

In dieser Arbeit geht es um eine bemerkenswerte Beziehung zwischen zwei scheinbar nicht miteinander verbundenen Größen. Diese sind: Die asymptotische Wirkung und die asymptotische Windungszahl. Beide Größen entstehen als Birkhoff-Durchschnitte. Die erste Größe ist ein Durchschnitt aus der Hamiltonschen Wirkung in der symplektischen Geometrie und die zweite aus der Windungs- oder Verknüpfungszahl, die bei der Untersuchung dynamischer Systeme auf der Scheibe und dem Kreisring auftritt. Diese Beziehung, obwohl sie neu ist, ist inspiriert durch ein Theorem von Albert Fathi aus seiner Doktorarbeit (1980)., This thesis focuses on a neat relation between two seemingly unrelated quantities. These are: The asymptotic action and, the asymptotic Winding number. Both of these quantities arise as Birkhoff averages. The first one, is an average of the Hamiltonian action in symplectic geometry and the second from the winding or linking number which appears in the study of dynamical systems on the disk and the annulus.This relation even though its new, its inspired in a theorem of Albert Fathi present in his doctoral thesis(1980).

Numerical simulations of the cardiovascular system are growing in popularity due to the increasing availability of computational power, and their proven contribution to the understanding of pathodynamics and validation of medical devices with in-silico trials as a potential future breakthrough. Such simulations are performed on volumetric meshes reconstructed from patient-specific imaging data. These meshes are most often unstructured, and result in a brutally large amount of elements, significantly increasing the computational complexity of the simulations, whilst potentially adversely affecting their accuracy. To reduce such complexity, we introduce a new approach for fully automatic generation of higher-order, structured hexahedral meshes of tubular structures, with a focus on healthy blood vessels. The structures are modeled as skeleton-based convolution surfaces. From the same skeleton, the topology is captured by a block-structure, and the geometry by a higher-order surface mesh. Grading may be induced to obtain tailored refinement, thus resolving, e.g., boundary layers. The volumetric meshing is then performed via transfinite mappings. The resulting meshes are of arbitrary order, their elements are of good quality, while the spatial resolution may be as coarse as needed, greatly reducing computing time. Their suitability for practical applications is showcased by a simulation of physiological blood flow modelled by a generalised Newtonian fluid in the human aorta.

Time-varying electromagnetic field observed on the ground or at a spacecraft consists of contributions from (i) electric source currents, such as those in the ionosphere and magnetosphere, and (ii) corresponding fields induced by source currents within the conductive Earth’s interior by virtue of electromagnetic induction. Knowledge about the spatio-temporal structure of inducing currents is a key component in ionospheric and magnetospheric studies, and is also needed in space weather hazard evaluation, whereas the induced currents depend on the Earth’s subsurface electrical conductivity distribution and allow us to probe this physical property. In this study, we present an approach that reconstructs the inducing source and subsurface conductivity structures simultaneously, preserving consistency between the two models by exploiting the inherent physical link. To achieve this, we formulate the underlying inverse problem as a separable nonlinear least-squares (SNLS) problem, where inducing current and subsurface conductivity parameters enter as linear and nonlinear model unknowns, respectively. We solve the SNLS problem using the variable projection method and compare it with other conventional approaches. We study the properties of the method and demonstrate its feasibility by simultaneously reconstructing the ionospheric and magnetospheric currents along with a 1-D average mantle conductivity distribution from the ground magnetic observatory data., Earth, Planets and Space, 75, ISSN:1343-8832, ISSN:1880-5981

We consider recovery from ptychographic measurements also known as Short-Time Fourier Transform phase retrieval. We study Amplitude Flow, Error Reduction and Ptychographic Iterative Engine algorithms and derive their convergence speed. In addition, Block Phase Retrieval algorithm and its modifications are studied. Furthermore, we consider blind ptychography and devise a version of Amplitude Flow with guaranteed convergence. Finally, some results are extended to polychromatic ptychography. Wir betrachten die Rekonstruktion eines Objektes von ptychographischen Messungen. Wir untersuchen die Algorithmen Amplitude Flow, Error Reduction und Ptychographic Iterative Engine und leiten deren Konvergenzgeschwindigkeit ab. Außerdem untersuchen wir den Block Phase Retrieval Algorithmus. Des Weiteren betrachten wir blinde Ptychographie und entwickeln eine Version von Amplitude Flow mit garantierter Konvergenz. Zuletzt erweitern wir einige der Resultate für polychromatische Ptychographie.

Der Begriff „Hyperkomplexe Zahlen“ wurde von Emmy Noether geprägt, die wesentliche Beiträge zur Theorie geliefert und diese stark formalisiert hat. Der Begriff steht heute schon fast synonym für eine Algebra. Besonders interessant sind die vier Algebren, in denen es eine Division gibt. Nach einem Satz von Adolf Hurwitz, (1898), gibt es nur vier reelle, normierte Divisionsalgebren mit einem Einselement. Es sind die reellen Zahlen, die komplexen Zahlen, die Quaternionen und die Oktonionen. Wichtige physikalische Theorien des 20. Jahrhunderts können mit Quaternionen formuliert werden. Doch mindestens seit den1970-er Jahren stehen auch die Oktonionen im Fokus. Es gibt nicht vernachlässigbare Hinweise, unter anderem aus der Stringtheorie und Supersymmetrie. Sie könnten bei der Beschreibung des Standardmodells der Elementarteilchen helfen. Im Idealfall sollte sogar Physik a priori aus der Mathematik entwickelbar sein.

This thesis aims to contribute to a more thorough understanding of the analysis, numerics and optimization of Maxwell-structured variational inequalities. Motivated by applications that require a certain shielding of electric or magnetic fields, we investigate the evolutionary Maxwell obstacle problem first introduced by Duvaut and Lions and certain variants thereof. To begin with, we analyze the mathematical modeling of the famous eddy current approximation in the Maxwell obstacle problem. Through the usage of an implicit Euler scheme in time and its rigorous convergence analysis, we are able to prove a well-posedness result for the eddy current model. We present uniform a priori estimates that provide information on the quantitative precision of the eddy current approximation. The numerical experiments corresponding to the Maxwell obstacle problem suggest that a combination of the implicit Euler method with a mixed finite element method is too computationally costly. Based on this observation, we introduce an alternative time-discretization by the so called leapfrog stepping. The resulting fully discrete scheme turns out to be indeed way more efficient since it completely eliminates the variational inequality character. We prove the stability and convergence of the fully discrete scheme, which requires us to construct a novel constraint preserving mollification operator for vector fields that admit a weak curl. Thereafter, we study a quasilinear first kind variational inequality with a bilateral differential constraint. We propose a tailored regularization approach by the use of which we examine both the well-posedness and the optimal control of our problem. In particular, invoking curl-projection and cut-off type arguments, we are able to derive a set of necessary optimality conditions for the optimal control problem. Finally, we turn our attention to the construction of an efficient solver for a quasi-variational inequality with applications in superconductivity. Here, we utilize once again a time-discretization by the leapfrog stepping with the aim of eliminating the present quasi-variational character. Exploiting the explicit choice of our nonlinearity, we are able to prove the stability and convergence of the scheme for source and temperature data of merely bounded variation in time. The thesis is concluded with an outlook on the modeling of magnetic levitation phenomena accompanied by numerical experiments. Dissertation, Universität Duisburg-Essen, 2023

Due to their prognostic value, biomarkers can support physicians in making the appropriate choice of therapy for a patient. In this thesis, several advanced statistical methods and machine learning algorithms were considered and applied to projects in collaboration with departments of the University Hospital Augsburg. A machine learning algorithm capturing hidden structures in binary immunohistologically stained images of colon cancer was developed to identify patients with a high risk of occurrence of distant metastases. Further, generalized linear models were used to estimate the probability of the need for a permanent shunt in patients after an aneurysmatic subarachnoid hemorrhage. Patients with oligometastatic colon cancer were stratified by a score developed using approaches from survival analysis to investigate which groups might benefit from surgical removal of metastases with prolonged overall survival. Another important point is the selection of suitable statistical models dependent on the structure of the data. We found that a linear regression may only be suited with a transformation of the response variable in the context of association of a COVID-19 infection with lymphocyte subsets. In addition, modeling the course of daily reported new COVID-19 cases is a relevant task and requires suitable statistical models. We compared non-seasonal and seasonal ARIMA models and examined the performance of different log-linear autoregressive Poisson models. To add more structure and enable theoretical prognosis for the further course depending on nonpharmaceutical interventions, we fitted a Bayesian SEIR model with several change points and set the determined change points in context with the distribution of variants of the virus. Biomarker können Ärzte durch ihren prognostischen Wert bei der Auswahl geeigneter Therapieoptionen unterstützen. In dieser Arbeit wurden mehrere fortgeschrittene statistische Methoden sowie Algorithmen des maschinellen Lernens eingeführt und in Zusammenarbeit mit verschiedenen Abteilungen des Universitätsklinikums Augsburg angewendet. Mit Hilfe eines Algorithmus des maschinellen Lernens, der versteckte Strukturen in binären, immunhistologisch gefärbten Bildern von Darmkrebstumoren feststellen kann, wurden Patienten mit einem hohen Risiko für auftretende Fernmetastasen identifiziert. Ebenso wurden Generalisierte Lineare Modelle verwendet, um eine Vorhersage der Wahrscheinlichkeit für eine dauerhafte Shunt-Anlegung nach einer aneurysmatischen Subarachnoidalblutung zu treffen. Patienten mit oligometastastischen Darmkrebs wurden mittels eines Scores, der anhand von Methoden der Survival Analysis entwickelt wurde, stratifiziert, um eine Gruppe zu identifizieren, die von einer operativen Entfernung der Metastasen durch ein langes Gesamtüberleben profitieren kann. Ein weiterer wichtiger Punkt bei der Datenanalyse ist die geeignete Auswahl der statistischen Methode abhängig von der Datenstruktur. Es konnten am Beispiel der Assoziation einer Coronainfektion mit der Anzahl von Lymphozytensubpopulationen gezeigt werden, dass eine Transformation der Zielvariable notwendig sein kann, um die Voraussetzungen der linearen Regression zu erfüllen. Die Modellierung der Anzahl an täglichen Neuinfektionen stellt eine relevante Aufgabe dar und benötigt passende statistische Modelle. Ein non-seasonal und ein seasonal ARIMA-Model wurden ebenso wie mehrere log-linearen autoregressiven Poisson-Modellen verglichen. Zusätzlich wurde ein weiterer Modellierungsansatz untersucht, der die biologischen Mechanismen stärker einbezieht und eine theoretische Prognose für den weiteren Verlauf unter verschiedenen Szenarien ermöglicht. Der Verlauf wurde mittels eines bayesschen SEIR Modell mit mehreren Wendepunkten an die Daten angepasst. Die gefundenen Wendepunkte wurden in Kontext der Verteilung der Virusvarianten analysiert.

This article presents an efficient nonlinear stochastic finite element method to solve stochastic elastoplastic problems. Similar to deterministic elastoplastic problems, we describe history-dependent stochastic elastoplastic behavior utilizing a series of (pseudo) time steps and go further to solve the corresponding stochastic solutions. For each time step, the original stochastic elastoplastic problem is considered as a time-independent nonlinear stochastic problem with initial values given by stochastic displacements, stochastic strains, and internal variables of the previous time step. To solve the stochastic solution at each time step, the corresponding nonlinear stochastic problem is transformed into a set of linearized stochastic finite element equations by means of finite element discretization and a stochastic Newton linearization, while the stochastic solution at each time step is approximated by a sum of the products of random variables and deterministic vectors. Each couple of the random variable and the deterministic vector is also used to approximate the stochastic solution of the corresponding linearized stochastic finite element equation that can be solved via a weakly intrusive method. In this method, the deterministic vector is computed by solving deterministic linear finite element equations, and corresponding random variables are solved by a non-intrusive method. Further, the proposed method avoids the curse of dimensionality successfully since its computational effort does not increase dramatically as the stochastic dimensionality increases. Four numerical cases are used to demonstrate the good performance of the proposed method.