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Let $k$ be an algebraically closed field of positive characteristic. For any integer $m \geq 2$, we show that the Hodge numbers of a smooth projective $k$-variety can take on any combination of values modulo $m$, subject only to Serre duality. In particular, there are no non-trivial polynomial relations between the Hodge numbers., Published version. 15 pages

Propagation at a finite speed is established for non-negative weak solutions to a thin-film approximation of the two-phase Muskat problem. The expansion rate of the support matches the scale invariance of the system. Moreover, we determine sufficient conditions on the initial data for the occurrence of waiting time phenomena. Copyright © Royal Society of Edinburgh 2017

The goal of this paper is to clarify when a closed convex cone is invariant for a stochastic partial differential equation (SPDE) driven by a Wiener process and a Poisson random measure, and to provide conditions on the parameters of the SPDE, which are necessary and sufficient., Comment: 41 pages, 2 figures

The paper presents a comprehensive model of a banking system that integrates network effects, bankruptcy costs, fire sales, and cross-holdings. For the integrated financial market we prove the existence of a price-payment equilibrium and design an algorithm for the computation of the greatest and the least equilibrium. The number of defaults corresponding to the greatest price-payment equilibrium is analyzed in several comparative case studies. These illustrate the individual and joint impact of interbank liabilities, bankruptcy costs, fire sales and cross-holdings on systemic risk. We study policy implications and regulatory instruments, including central bank guarantees and quantitative easing, the significance of last wills of financial institutions, and capital requirements.

We provide $\mathbb{N}$-filtrations on the negative part $U_q(\mathfrak{n}^-)$ of the quantum group associated to a finite-dimensional simple Lie algebra $\mathfrak{g}$, such that the associated graded algebra is a skew-polynomial algebra on $\mathfrak{n}^-$. The filtration is obtained by assigning degrees to Lusztig's quantum PBW root vectors. The possible degrees can be described as lattice points in certain polyhedral cones. In the classical limit, such a degree induces an $\mathbb{N}$-filtration on any finite dimensional simple $\mathfrak{g}$-module. We prove for type $\tt{A}_n$, $\tt{C}_n$, $\tt{B}_3$, $\tt{D}_4$ and $\tt{G}_2$ that a degree can be chosen such that the associated graded modules are defined by monomial ideals, and conjecture that this is true for any $\mathfrak{g}$., Comment: 26 pages, an inaccuracy corrected

These lecture notes are devoted to the numerical solution of partial differential equations (PDEs). PDEs arise in many fields and are extremely important in modeling of technical processes with applications in physics, biology, chemisty, economics, mechanical engineering, and so forth. In these notes, not only classical topics for linear PDEs such as finite differences, finite elements, error estimation, and numerical solution schemes are addressed, but also schemes for nonlinear PDEs and coupled problems up to current state-of-the-art techniques are covered. In the Winter 2020/2021 an International Class with additional funding from DAAD (German Academic Exchange Service) and local funding from the Leibniz University Hannover, has led to additional online materials such as links to youtube videos, which complement these lecture notes. This is the updated and extended Version 2. The first version was published under the DOI: https://doi.org/10.15488/9248.

Many signals appear fractal and have self-similarity over a large range of their power spectral densities. They can be described by so-called Hermite processes, among which the first order one is called fractional Brownian motion (fBm), and has a wide range of applications. The fractional Gaussian noise (fGn) series is the successive differences between elements of a fBm series; they are stationary and completely characterized by two parameters: the variance, and the Hurst coefficient (H). From physical considerations, the fGn could be used to model the noise of observations coming from sensors working with, e.g., phase differences: due to the high recording rate, temporal correlations are expected to have long range dependency (LRD), decaying hyperbolically rather than exponentially. For the rigorous testing of deformations detected with terrestrial laser scanners (TLS), the correct determination of the correlation structure of the observations is mandatory. In this study, we show that the residuals from surface approximations with regression B-splines from simulated TLS data allow the estimation of the Hurst parameter of a known correlated input noise. We derive a simple procedure to filter the residuals in the presence of additional white noise or low frequencies. Our methodology can be applied to any kind of residuals, where the presence of additional noise and/or biases due to short samples or inaccurate functional modeling make the estimation of the Hurst coefficient with usual methods, such as maximum likelihood estimators, imprecise. We demonstrate the feasibility of our proposal with real observations from a white plate scanned by a TLS.

We consider Euler's equations for free surface waves traveling on a body of density stratified water in the scenario when gravity and surface tension act as restoring forces. The flow is continuously stratified, and the water layer is bounded from below by an impermeable horizontal bed. For this problem we establish three equivalent classical formulations in a suitable setting of strong solutions which may describe nevertheless waves with singular density gradients. Based upon this equivalence we then construct two-dimensional symmetric periodic traveling waves that are monotone between each crest and trough. Our analysis uses, to a large extent, the availability of a weak formulation of the water wave problem, the regularity properties of the corresponding weak solutions, and methods from nonlinear functional analysis., Comment: 34 pages

This thesis studies the Kronheimer hyperkähler metric on the adjoint orbit of the classical Lie group SL_n (C) of a regular, nilpotent element in its Lie algebra sl_n(C). We describe a Kähler potential of this hyperkähler metric in terms of the theta function on the Jacobian, consisting of invertible sheaves of degree g - 1, of the nilpotent, spectral curve. By using an explicit description of matricial polynomials of degree two corresponding to invertible sheaves of degree g - 1 without a non-trivial, global section on the nilpotent, spectral curve we construct some explicit solutions to Nahm’s equations.

Structured nonsmoothness is widely present in practical optimization problems. A particularly attractive class of nonsmooth problems, both from a theoretical and from an algorithmic perspective, are nonsmooth NLPs with equality and inequality constraints in abs-normal form, so-called abs-normal NLPs. In this thesis optimality conditions for this particular class are obtained. To this aim, first the theory for the case of unconstrained optimization problems in abs-normal form of Andreas Griewank and Andrea Walther is extended. In particular, similar necessary and sufficient conditions of first and second order are obtained that are directly based on classical Karush-Kuhn-Tucker (KKT) theory for smooth NLPs. Then, it is shown that the class of abs-normal NLPs is equivalent to the class of Mathematical Programs with Equilibrium Constraints (MPECs). Hence, the regularity assumption LIKQ introduced for the abs-normal NLP turns out to be equivalent to MPEC-LICQ. Moreover, stationarity concepts and optimality conditions under these regularity assumptions of linear independece type are equivalent up to technical assumptions. Next, well established constraint qualifications of Mangasarian Fromovitz, Abadie and Guignard type for MPECs are used to define corresponding concepts for abs-normal NLPs. Then, it is shown that kink qualifications and MPEC constraint qualifications of Mangasarian Fromovitz resp. Abadie type are equivalent. As it remains open if this holds for Guignard type kink and constraint qualifications, branch formulations for abs-normal NLPs and MPECs are introduced. Then, equivalence of Abadie’s and Guignard’s constraint qualifications for all branch problems hold. Throughout a reformulation of inequalities with absolute value slacks is considered. It preserves constraint qualifications of linear independence and Abadie type but not of Mangasarian Fromovitz type. For Guignard type it is still an open question but ACQ and GCQ are preserved passing over to branch problems. Further, M-stationarity and B-stationarity concepts for abs-normal NLPs are introduced and corresponding first order optimality con- ditions are proven using the corresponding concepts for MPECs. Moreover, a reformulation to extend the optimality conditions for abs-normal NLPs to those with additional nonsmooth objective functions is given and the preservation of regularity assumptions is considered. Using this, it is shown that the unconstrained abs-normal NLP always satisfies constraint qualifications of Abadie and thus Guignard type. Hence, in this special case every local minimizer satisfies the M-stationarity and B-stationarity concepts for abs-normal NLPs.

Frieze patterns, as introduced by Coxeter in the 1970's, are closely related to cluster algebras without coefficients. A suitable generalization of frieze patterns, linked to cluster algebras with coefficients, has only briefly appeared in an unpublished manuscript by Propp. In this paper we study these frieze patterns with coefficients systematically and prove various fundamental results, generalizing classic results for frieze patterns. As a consequence we see how frieze patterns with coefficients can be obtained from classic frieze patterns by cutting out subpolygons from the triangulated polygons associated to classic Conway-Coxeter frieze patterns. We address the question of which frieze patterns with coefficients can be obtained in this way and solve this problem completely for triangles. Finally, we prove a finiteness result for frieze patterns with coefficients by showing that for a given boundary sequence there are only finitely many (non-zero) frieze patterns with coefficients with entries in a discrete subset of the complex numbers., 22 pages

We provide a Minkowski sum decomposition of marked chain-order polytopes into building blocks associated to elementary markings and thus give an explicit minimal set of generators of an associated semi-group algebra. We proceed by characterizing the reflexive polytopes among marked chain-order polytopes as those with the underlying marked poset being ranked., Comment: 17 pages

We prove that a Kummer surface defined over a complete strictly Henselian discretely valued field $K$ of residue characteristic different from 2 admits a strict Kulikov model after finite base change. The Kulikov models we construct will be schemes, so our results imply that the semistable reduction conjecture is true for Kummer surfaces in this setup, even in the category of schemes. Our construction of Kulikov models is closely related to an earlier construction of K\"unnemann, which produces semistable models of Abelian varieties. It is well-known that the special fibre of a strict Kulikov model belongs to one of three types, and we shall prove that the type of the special fibre of a strict Kulikov model of a Kummer surface and the toric rank of a corresponding Abelian surface are determined by each other. We also study the relationship between this invariant and the Galois representation on the second $\ell$-adic cohomology of the Kummer surface. Finally, we apply our results, together with earlier work of Halle-Nicaise, to give a proof of the monodromy conjecture for Kummer surfaces in equal characteristic zero., Comment: 38 pages. Some typographical errors fixed. Small correction to the proof of Lemma 3.10

A conjecture of Manin’s relates the number of rational points of bounded height on Fano varieties with their geometric properties. Analogously to this conjecture on rational points, we study the distribution of integral points of bounded height on three varieties: on a smooth Fano threefold of Picard number 2 and type 30 in the Mori–Mukai classification, on a quartic del Pezzo surface with an A1- and an A3-singularity, and on a toric threefold. We determine asymptotic formulas and interpret the leading term geometrically. For the proofs, we parametrize integral points using universal torsors, and use analytic techniques to count integral points on the torsor. This seems to be the first application of the torsor method to integral points. The asymptotic formula for our toric variety contradicts a result by Chambert-Loir and Tschinkel. We describe an obstruction that explains this contradiction, and study its relation with some constants that appear in asymptotic formulas for the number of integral points of bounded height., Eine Vermutung von Manin stellt einen Bezug zwischen der Anzahl rationaler Punkte beschränkter Höhe auf Fano-Varietäten und geometrischen Eigenschaften her. Analog zu dieser Vermutung für rationale Punkte untersuchen wir die Verteilung ganzer Punkte beschränkter Höhe auf drei Varietäten: auf einer glatten dreidimensionalen Fano-Varietät von Picardrang 2 und Typ 30 in der Mori– Mukai-Klassifikation, auf einer quartischen del-Pezzo-Fläche mit A1- und A3- Singularität und auf einer dreidimensionalen torischen Varietät. Wir bestimmen asymptotische Formeln und interpretieren den führenden Term geometrisch. In den Beweisen parametrisieren wir die ganzen Punkte mit Hilfe universeller Torsore, und zählen ganze Punkte auf den universellen Torsoren mit analytischen Methoden. Dies scheint die erste Anwendung der Torsor-Methode zum Zählen ganzer Punkte zu sein. Die asymptotische Formel für die torische Varietät steht im Widerspruch zu einem Ergebnis von Chambert-Loir und Tschinkel. Wir beschreiben eine Obstruktion, die diesen Widerspruch erklärt und untersuchen ihren Zusammenhang mit einigen Konstanten, die ein Bestandteil asymptotischer Formeln für die Anzahl ganzer Punkte beschränkter Höhe sind.

Interconnectedness constitutes a key characteristic of actuarial and financial systems. In regular times, it facilitates the provision of the systems’ important services to society. In times of crisis, however, it enables the spread of contagious distress that may adversely affect the overall economy and amplify crisis situations. In this thesis, we introduce and analyze two financial and one actuarial network model representing three particular risk management problems that arise from different forms of interconnectedness. First, we consider the spread of financial losses and defaults in a comprehensive model of a banking network. Distress therein may propagate through various forms of connections such as direct financial obligations, bankruptcy costs, fire sales, and cross-holdings. For the integrated financial market, we prove the existence of a price-payment equilibrium and design an algorithm for its computation. The corresponding number of defaults is analyzed in several comparative case studies. These illustrate the individual and joint impact of the considered interaction channels on systemic risk. Second, we study the problem of minimizing market inefficiencies, defined as deviations of realized asset prices from fundamental values, as a function of the network of banks’ overlapping asset portfolios. Prices are pressured from trading actions of the leverage targeting banks, which rebalance their portfolios in response to exogenous asset shocks. We prove the existence of a network of efficient asset holdings and characterize its properties and sensitivities. In particular, we find that the standard paradigm of asset portfolio diversification may cause tremendous market inefficiencies, especially during crisis situations. Third, we consider insurance against cyber epidemics. Infectious cyber threats, such as viruses and worms, diffuse within a network of possibly insured parties. Since the infection may affect many different agents at the same time, a provider of cyber insurance is exposed to significant accumulation risk. We build and analyze a stochastic model of losses generated by infectious cyber threats based on interacting particle systems and marked point processes. Together with a novel polynomial and mean-field approximation, our approach allows to explicitly compute prices for different forms of cyber insurance contracts. Numerical case studies demonstrate the impact of the network topology and indicate that higher order approximations are indispensable for the analysis of non-linear contracts.

Let W⊂GL(V) be a complex reflection group and A(W) the set of the mirrors of the complex reflections in W. It is known that the complement X(A(W)) of the reflection arrangement A(W) is a K(π,1) space. For Y an intersection of hyperplanes in A(W), let X(A(W)Y) be the complement in Y of the hyperplanes in A(W) not containing Y. We hope that X(A(W)Y) is always a K(π,1). We prove it in case of the monomial groups W=G(r,p,ℓ). Using known results, we then show that there remain only three irreducible complex reflection groups, leading to just eight such induced arrangements for which this K(π,1) property remains to be proved.

A model for a MEMS device, consisting of a fixed bottom plate and an elastic plate, is studied. It was derived in a previous work as a reinforced limit when the thickness of the insulating layer covering the bottom plate tends to zero. This asymptotic model inherits the dielectric properties of the insulating layer. It involves the electrostatic potential in the device and the deformation of the elastic plate defining the geometry of the device. The electrostatic potential is given by an elliptic equation with mixed boundary conditions in the possibly non-Lipschitz region between the two plates. The deformation of the elastic plate is supposed to be a critical point of an energy functional which, in turn, depends on the electrostatic potential due to the force exerted by the latter on the elastic plate. The energy functional is shown to have a minimizer giving the geometry of the device. Moreover, the corresponding Euler–Lagrange equation is computed and the maximal regularity of the electrostatic potential is established.

When operating a wind turbine, damage of rotor blades is a serious problem. Undetected damages are likely to increase overtime, and therefore, the safety risks and economical burdens also increase. A monitoring system, which detects reliably defects in early stages, gives scope for action and is therefore a key element to avoid damage increase and to optimize the efficiency of wind turbines. One promising approach for damage detection is acoustic emission methods. Although most acoustic emission approaches use ultrasonic sound waves of the structure that require about 12 to 40 sensors to monitor one rotor blade, we propose to use the airborne sound in lower frequencies from about 500 Hz to 35 Hz with three optical microphones and present a signal model-based damage detection algorithm. The real-time algorithm uses six audio features from a spectrogram representation to detect damages and to estimate its significance. In a fatigue test of a 34-m blade, the algorithm detected the damage event and damage increasing without false detection. It was also tested with recordings inside an operating blade of a 3.4-MW wind turbine. In the recorded time period of about 1 year, the algorithm indicated no false detection. © 2020 The Authors. Structural Control and Health Monitoring published by John Wiley & Sons Ltd

We study a free boundary problem arising from the modeling of an idealized electrostatically actuated MEMS device. In contrast to existing literature, we consider a three-dimensional device involving a hinged elastic plate. The model couples the electrostatic potential to the displacement of the elastic plate, which is caused by a voltage difference that is applied to the device. The electrostatic potential is harmonic in the free domain between the elastic plate and a rigid ground plate. The elastic plate displacement solves a fourth-order parabolic equation with hinged boundary conditions and a right-hand side proportional to the gradient trace of the electrostatic potential on the elastic plate. We establish local and global well-posedness of the model in dependence of the applied voltage difference and show that only touchdown of the elastic plate on the ground plate can generate a finite time singularity. Next, we consider stationary solutions and prove that such solutions exist for small voltage values and do not exist for large voltage values. To prove the nonexistence result, we show that the fourth-order elliptic operator with hinged boundary conditions satisfies a positivity preserving property and has a positive eigenpair.

Various GNSS applications require low-cost, small-scale, lightweight and power-saving GNSS devices and require high precision in terms of low noise for carrier phase and code observations. Applications vary from navigation approaches to positioning in geo-monitoring units up to integration in multi-sensor-systems. For highest precision, only GNSS receivers are suitable that provide access to raw data such as carrier phase, code ranges, Doppler and signal strength. A system integration is only possible if the overall noise level is known and quantified at the level of the original observations. A benchmark analysis based on a zero baseline is proposed to quantify the stochastic properties. The performance of the consumer grade GNSS receiver is determined and evaluated against geodetic GNSS receivers to better understand the utilization of consumer grade receivers. Results indicate high similarity to the geodetic receiver, even though technical limitations are present. Various stochastic techniques report normally distributed carrier-phase noise of 2 mm and code-range noise of 0.5&ndash, 0.8 m. This is confirmed by studying the modified Allan standard deviation and code-minus-carrier combinations. Derived parameters serve as important indicators for the integration of GNSS receivers into multi-sensor-systems.

In this thesis we study Toeplitz operators on spaces of holomorphic and pluriharmonic functions. The main part of the thesis is concerned with such operators on the p-Fock spaces of holomorphic functions for p ∈ [1, ∞]. We establish a notion of Correspondence Theory between symbols and Toeplitz operators, based on extended notions of convolutions as developed by Reinhard Werner, which gives rise to many important results on Toeplitz operators and the algebras they generate. Here, we find new proofs for old theorems, extending them to a larger range of values of p, and also provide entirely new results. We manage to include even the non-reflexive cases of p = 1, ∞ in our studies. Based on the notions of band-dominated and limit operators, we establish a general criterion for an operator in the Toeplitz algebra over the Fock space to be Fredholm: Such an operator is Fredholm if and only if all of its limit operators are invertible. As an example of a Toeplitz algebra over the Fock space, we study the Resolvent Algebra (in the sense of Detlev Buchholz and Hendrik Grundling) in its Fock space representation. Partially following the methods of Correspondence Theory as discussed in this thesis, we manage to extend a classical result on the boundedness of Toeplitz operators (the Berger-Coburn estimates) to the setting of p-Fock spaces. Also based on results derived from the Correspondence Theory, we discuss several new characterizations of the full Toeplitz algebra on Fock spaces, at least in the reflexive range p ∈ (1, ∞). In the last part, we discuss several results on spectral theory and quantization estimates for Toeplitz operators acting on Bergman and Fock spaces of pluriharmonic functions.

In this thesis, we will give a partial classification of cubic fourfolds by their isolated ADE singularities. We have a correspondence between cubic fourfolds and complete (2,3)- intersections in ℙ^4 having both certain isolated ADE singularities. The minimal model for a complete (2,3)-intersection in ℙ^4 with isolated ADE singularities is a quasi-polarized K3 surface of degree 6. We will prove that the existence of certain lattice embeddings into the K3 lattice is a necessary and sufficient condition for the existence of these singular cubic fourfolds and complete (2,3)-intersections, respectively. We will determine all direct sums of negative definite irreducible ADE lattices such that their direct sum with the rank one lattice whose generator has self-intersection number 6 admits a primitive embedding into the K3 lattice. This will prove the existence of complete (2,3)-intersections in ℙ^4 lying on smooth quadrics and having exactly these ADE singularities and their corresponding cubic fourfolds. Finally, we will show that we have an isomorphism between the moduli space of cubic fourfolds with certain ADE singularities and the moduli space of quasi-polarized K3 surfaces of degree 6 such that the quasi-polarization induces a birational map from the K3 surface into ℙ^4 whose image is a complete (2,3)-intersection in ℙ^4 having certain ADE singularities.

To every cubic hypersurface $X$ we associate the parameter space of lines contained in $X$; this is called the Fano scheme of lines on $X$ and denoted $F(X)$. If X admits an isolated singular point of ADE-type, we prove that $F(X)$ admits hypersurface singularities of the same type transversally along the regular part of its singular locus. As was shown by H. Clemens and P. Griffiths, the Albanese variety $\operatorname{Alb}(F(X))$ of the Fano scheme of lines $F(X)$ on a smooth cubic threefold $X$ is isomorphic to the intermediate Jacobian $IJ(X)$ of $X$. G. van der Geer and A. Kouvidakis generalised this result to nodal cubic threefolds, replacing the Albanese variety $\operatorname{Alb}(F(X))$ by the Picard scheme $\operatorname{Pic}^0(F(X))$. We study more generally degenerations of the Picard scheme $\operatorname{Pic}^0(F(X))$ when the smooth cubic threefold $X$ degenerates to a cubic threefold with unique singular point of type $A_k$ and prove that these degenerations define points in Mumford’s partial compactification $\mathcal{A}_5'$ of the moduli space $\mathcal{A}_5$ of principally polarised Abelian varieties of genus five.

The application of network analysis has found great success in a wide variety of disciplines; however, the popularity of these approaches has revealed the difficulty in handling networks whose complexity scales rapidly. One of the main interests in network analysis is the online detection of anomalous behaviour. To overcome the curse of dimensionality, we introduce a network surveillance method bringing together network modelling and statistical process control. Our approach is to apply multivariate control charts based on exponential smoothing and cumulative sums in order to monitor networks determined by temporal exponential random graph models (TERGM). This allows us to account for temporal dependence, while simultaneously reducing the number of parameters to be monitored. The performance of the proposed charts is evaluated by calculating the average run length for both simulated and real data. To prove the appropriateness of the TERGM to describe network data, some measures of goodness of fit are inspected. We demonstrate the effectiveness of the proposed approach by an empirical application, monitoring daily flights in the United States to detect anomalous patterns.

We show that the Andr\'{e} motive of a hyper-K\"{a}hler variety $X$ over a field $K \subset \mathbb{C}$ with $b_2(X)>6$ is governed by its component in degree $2$. More precisely, we prove that if $X_1$ and $X_2$ are deformation equivalent hyper-K\"{a}hler varieties with $b_2(X_i)>6$ and if there exists a Hodge isometry $f\colon H^2(X_1,\mathbb{Q})\to H^2(X_2,\mathbb{Q})$, then the Andr\'e motives of $X_1$ and $X_2$ are isomorphic after a finite extension of $K$, up to an additional technical assumption in presence of non-trivial odd cohomology. As a consequence, the Galois representations on the \'{e}tale cohomology of $X_1$ and $X_2$ are isomorphic as well. We prove a similar result for varieties over a finite field which can be lifted to hyper-K\"{a}hler varieties for which the Mumford--Tate conjecture is true., Comment: added Section 5; accepted for publication in Math. Zeit

There are various competing procedures to determine whether fractional cointegration is present in a multivariate time series, but no standard approach has emerged. We provide a synthesis of this literature and conduct a detailed comparative Monte Carlo study to guide empirical researchers in their choice of appropriate methodologies. Special attention is paid on empirically relevant issues such as assumptions about the form of the underlying process and the ability of the procedures to distinguish between short-run correlation and long-run equilibria. It is found that several approaches are severely oversized in presence of correlated short-run components and that the methods show different performance in terms of power when applied to common-component models instead of triangular systems.

We relate the existence problem of harmonic maps into $S^2$ to the convex geometry of $S^2$. On one hand, this allows us to construct new examples of harmonic maps of degree 0 from compact surfaces of arbitrary genus into $S^2$. On the other hand, we produce new example of regions that do not contain closed geodesics (that is, harmonic maps from $S^1$) but do contain images of harmonic maps from other domains. These regions can therefore not support a strictly convex function. Our construction builds upon an example of W. Kendall, and uses M. Struwe's heat flow approach for the existence of harmonic maps from surfaces., 2 pictures

Geometrical non-linearity is one of the aspects to be taken into account for accurate analysis of fibre reinforced polymers (FRPs), since large displacements and rotations may be observed in many of its structural applications such as in aircraft wings and wind turbine blades. In this paper, a co-rotational formulation and implementation of an invariant-based anisotropic plasticity model are presented for geometrically non-linear analysis of FRPs. The anisotropic constitutive equations are formulated in the format of isotropic tensors functions. The model assumes an anisotropic pressure-dependent yield function, and in addition to this, a non-associated plastic potential function in order to model realistic plastic deformations in FRPs. The formulation is then cast in the co-rotational framework to consider the geometrical non-linear effects in an efficient manner. The developed model is implemented in the commercial finite element (FE) software ABAQUS/Implicit via the means of the user-defined material subroutine (UMAT). The kinematics within the co-rotational frame is explained briefly while the important aspects regarding the numerical treatment and implementation are discussed in detail. Representative numerical examples at different scales are presented to demonstrate the applicability and robustness of the proposed development.

We present a surprising redefinition of matrix fermions which brings the supercharges of the $\cal N$-extended supersymmetric $A_{n-1}$ Calogero model introduced in [1] to the standard form maximally cubic in the fermions. The complexity of the model is transferred to a non-canonical and nonlinear conjugation property of the fermions. Employing the new cubic supercharges, we apply a supersymmetric generalization of a "folding" procedure for $A_{2n-1}\oplus A_1$ to explicitly construct the supercharges and Hamiltonian for arbitrary even-$\cal N$ supersymmetric extensions of the $B_n$, $C_n$ and $D_n$ rational Calogero models. We demonstrate that all considered models possess a dynamical $osp({\cal N}|2)$ superconformal symmetry., Comment: 1+7 pages; v2: more precise citations, one ref. added, version appearing in PLB

The solution of structural problems with nonlinear material behaviour in a model order reduction framework is investigated in this paper. In such a framework, greedy algorithms or adaptive strategies are interesting as they adjust the reduced order basis (ROB) to the problem of interest. However, these greedy strategies may lead to an excessive increase in the size of the ROB, i.e., the solution is no more represented in its optimal low-dimensional expansion. Here, an optimised strategy is proposed to maintain, at each step of the greedy algorithm, the lowest dimension of a Proper Generalized Decomposition (PGD) basis using a randomised Singular Value Decomposition (SVD) algorithm. Comparing to conventional approaches such as Gram&ndash, Schmidt orthonormalisation or deterministic SVD, it is shown to be very efficient both in terms of numerical cost and optimality of the ROB. Examples with different mesh densities are investigated to demonstrate the numerical efficiency of the presented method.

Precise navigation and geodetic coordinate determination rely on accurate GNSS signal reception. Thus, the receiver antenna properties play a crucial role in the GNSS error budget. For carrier phase observations, a spherical radiation pattern represents an ideal receiver antenna behaviour. Deviations are known as phase centre corrections. Due to synergy of carrier and code phase, similar effects on the code exist named code phase variations (CPV). They are mainly attributed to electromagnetic interactions of several active and passive elements of the receiver antenna. Consequently, a calibration and estimation strategy is necessary to determine the shape and magnitudes of the CPV. Such a concept was proposed, implemented and tested at the Institut für Erdmessung. The applied methodology and the obtained results are reported and discussed in this paper.We show that the azimuthal and elevation-dependent CPV can reach maximum magnitudes of 0.2–0.3m for geodetic antennas and up to maximum values of 1.8m for small navigation antennas. The obtained values are validated by dedicated tests in the observation and coordinate domain. As a result, CPV are identified to be antenna- related properties that are independent from location and time of calibration. Even for geodetic antennas when forming linear combinations the CPV effect can be amplified to values of 0.4–0.6 m. Thus, a significant fractional of theMelbourne–Wübbena linear combination. A case study highlights that incorrect ambiguity resolution can occur due to neglecting CPV corrections. The impact on the coordinates whichmay reach up to the dmlevel is illustrated. The final publication is available at Springer via https://doi.org/10.1007/s00190-016-0984-8.

The paper presents the results of the numerical and physical experiments, aimed at assessing the influence of pulsed force of electromagnetic field on the melt motion and the fluid velocities. The experiment was performed on the eutectic alloy Galinstan in the cylindrical volume, where an ultrasonic Doppler velocimeter was employed for velocity measurements under conditions of pulsed and steady EM field application. A numerical simulation of the melt flow, forced by the steady EM force, involved a 2D axisymmetric model. The k-e turbulence model was used to obtain the information about the melt velocities. The verification of the numerical model was carried out for the steady case. The effects of pulsed and steadily applied Lorentz force were compared using the physical experiment. An intensity of the velocity pulsations in Galinstan for the pulsed frequencies in the range from 0.05 to 1 Hz considerably exceeded the values measured in the permanently stirred melt. For higher values of pulsed frequencies (from 1 to 10 Hz) the effect decreases and only slightly differs from the influence of the permanently applied Lorentz force.

Within the framework of variational modelling we derive a one-phase moving boundary problem describing the motion of a semipermeable membrane enclosing a viscous liquid, driven by osmotic pressure and surface tension of the membrane. For this problem we prove the existence of classical solutions for a short-time. © 2015 Cambridge University Press.

Given a family of varieties $X\to \mathbb{P}^n$ over a number field $k$, we determine conditions under which there is a Brauer-Manin obstruction to weak approximation for $100\%$ of the fibres which are everywhere locally soluble., 44 pages; there is a new application to integral points (Theorem 1.4) and Cor 1.5 recovers a well-known result concerning the Hasse principle

When considering cost-optimal operation of gas transport networks, compressor stations play the most important role. Proper modeling of these stations leads to nonconvex mixed-integer nonlinear optimization problems. In this article, we give an isothermal and stationary description of compressor stations, state MINLP and GDP models for operating a single station, and discuss several continuous reformulations of the problem. The applicability and relevance of different model formulations, especially of those without discrete variables, is demonstrated by a computational study on both academic examples and real-world instances. In addition, we provide preliminary computational results for an entire network. German Federal Ministry of Economics and Technology

We classify singular Enriques surfaces in characteristic two supporting a rank nine configuration of smooth rational curves. They come in one-dimensional families defined over the prime field, paralleling the situation in other characteristics, but featuring novel aspects. Contracting the given rational curves, one can derive algebraic surfaces with isolated ADE-singularities and trivial canonical bundle whose Q`-cohomology equals that of a projective plane. Similar existence results are developed for classical Enriques surfaces. We also work out an application to integral models of Enriques surfaces (and K3 surfaces).

Recently, Creignou et al. (Theory Comput. Syst. 2017), introduced the class DelayFPT into parameterised complexity theory in order to capture the notion of efficiently solvable parameterised enumeration problems. In this paper, we propose a framework for parameterised ordered enumeration and will show how to obtain enumeration algorithms running with an FPT delay in the context of general modification problems. We study these problems considering two different orders of solutions, namely, lexicographic order and order by size. Furthermore, we present two generic algorithmic strategies. The first one is based on the well-known principle of self-reducibility and is used in the context of lexicographic order. The second one shows that the existence of a neighbourhood structure among the solutions implies the existence of an algorithm running with FPT delay which outputs all solutions ordered non-decreasingly by their size.

A $k$-differential on a Riemann surface is a section of the $k$-th power of the canonical line bundle. Loci of $k$-differentials with prescribed number and multiplicities of zeros and poles form a natural stratification of the moduli space of $k$-differentials. In this paper we give a complete description for the compactification of the strata of $k$-differentials in terms of pointed stable $k$-differentials, for all $k$. The upshot is a global $k$-residue condition that can also be reformulated in terms of admissible covers of stable curves. Moreover, we study properties of $k$-differentials regarding their deformations, residues, and flat geometric structure., corrected and updated; final version, to appear in Algebraic Geometry

We prove the Yamada-Watanabe Theorem for semilinear stochastic partial differential equations with path-dependent coefficients. The so-called "method of the moving frame" allows us to reduce the proof to the Yamada-Watanabe Theorem for stochastic differential equations in infinite dimensions., Comment: 12 pages

In the present research, the effect of incorporating protuberances in the leading edge to the aerofoil NACA-0012 was studied. For the simulations, two profiles were designed: parametric profile (NACA 0012) and modified profile (0012-MOD), with help of CAD Program Inventor Professional 2016. The 0012-MOD profile incorporates modifications in the leading edge, whose inspiration comes from malformations present on the flippers of the humpback whale (Megaptera novaeangliae). The computational simulations were performed in Ansys Fluent, using the turbulent viscosity model k-ε realizable, with SIMPLE calculation algorithm. The study fluid is air in the incompressible range with Ma: 0.11. To obtain the lift and drag curves for both profiles, they were simulated for angles of attack α: between 0° and 40°, this way it will be possible to compare the profiles and determine their aerodynamic behaviour., En la presente investigación se estudió el efecto de incorporar protuberancias en el borde de ataque al perfil aerodinámico NACA-0012. Para las simulaciones se diseñaron dos perfiles: perfil paramétrico (NACA 0012) y perfil modificado (0012-MOD), con ayuda del programa CAD Inventor Professional 2016. El perfil 0012-MOD incorpora modificaciones en el borde de ataque, cuya inspiración proviene de las malformaciones presentes en las aletas de la ballena jorobada (Megaptera novaeangliae). Las simulaciones computacionales se realizaron con el software comercial Ansys Fluent®, utilizando el modelo de viscosidad turbulenta k-ε realizable, con algoritmo de cálculo SIMPLE. El fluido de estudio es aire en el rango incompresible con Ma: 0,29. Con el objeto de obtener las curvas de sustentación y arrastre para ambos perfiles, se realizaron simulaciones para ángulos de ataque α entre los 0° y los 40°, de esta forma será posible comparar los perfiles y determinar su comportamiento aerodinámico.

In this paper, we propose a new technique&mdash, called Ellipsoidal and Gaussian Kalman filter&mdash, for state estimation of discrete-time nonlinear systems in situations when for some parts of uncertainty, we know the probability distributions, while for other parts of uncertainty, we only know the bounds (but we do not know the corresponding probabilities). Similarly to the usual Kalman filter, our algorithm is iterative: on each iteration, we first predict the state at the next moment of time, and then we use measurement results to correct the corresponding estimates. On each correction step, we solve a convex optimization problem to find the optimal estimate for the system&rsquo, s state (and the optimal ellipsoid for describing the systems&rsquo, s uncertainty). Testing our algorithm on several highly nonlinear problems has shown that the new algorithm performs the extended Kalman filter technique better&mdash, the state estimation technique usually applied to such nonlinear problems.

This thesis is devoted to stochastic mortality modelling. The first part considers the popular family of GAPC models and identifies several conceptual difficulties of most well-established models. The GAPC models are embedded in the framework of generalized linear models. However, the vast majority of the literature only considers the canonical link function and by that omits an important modelling factor. In our study, we also incorporate a non-canonical link function and demonstrate its advantages on the fitting performance. While the first part focuses on the static component of the modelling approach, where the main objective is to identify the influencing factors that drive the mortality structure, the second part is devoted to the dynamical part of the modelling approach. For the proposed model we identify appropriate multivariate stochastic processes for the dynamics of the involved stochastic factors. We study cointegration relations between the individual components and compare the forecasting performance with the common GAPC approach. The last part of this thesis can be considered independently of the previous content. There, we provide an extensive characterization of the lifetime distribution which is induced by logistic-type hazard rates of the proposed Kannisto model. Furthermore, we reveal multiple connections to other well-known lifetime distributions.

We provide existence and uniqueness of global (and local) mild solutions for a general class of semilinear stochastic partial differential equations driven by Wiener processes and Poisson random measures under local Lipschitz and linear growth (or local boundedness, resp.) conditions. The so-called "method of the moving frame" allows us to reduce the SPDE problems to SDE problems., Comment: 18 pages

This thesis considers the treatment of the wave equation given outside of a bounded, orientable Lipschitz domain with the boundary element method (BEM). Beginning with a scattering problem the retarded (potential) boundary integral operators are defined. These operators are discretized with a tensor product ansatz. For the retarded Poincar\'{e}-Steklov operator and the inverse counterpart, numerical experiments are presented using the marching-on-in time (MOT) scheme. The coupling of the finite element method (FEM) and the boundary element method (BEM) provide an analysis of a fluid-structure interaction (FSI) problem with given transmission conditions and the wave propagation interface problem with corresponding transmission conditions. For the FSI problem two approaches are addressed. The symmetric FEM-BEM coupling are discretized such that the MOT-scheme is applicable. Numerical experiments demonstrate the reliability of the implementation. The other approach uses a retarded boundary integral operator as a test function, which leads to major challenges in the discretization and the performing of numerical experiments. The wave propagation interface problem is adressed with a symmetric coupling. Here the discretization is chosen such that a MOT-scheme may applied. Numerical results are demonstrated as well. A prori and a posteriori error estimates for conforming Galerkin approximation are derived in all these cases, motivating adaptive mesh refinement procedures. The remaining chapters consider the results of time domain boundary element discretizations for screen problems, unilateral contact and a real-world application on tyres. Numerical experiments achieve optimal approximation rates on graded meshes for screen problems, resolving the edge and corner singularities. As a first step towards high-order methods $p$ and $hp-$versions of time domain boundary element method are presented for quasi-uniform meshes. Further crack and punch problems, as two examples of dynamic contact problems in time domain, are analyzed. While an error analysis is done for flat contact areas, numerical experiments show convergence even for non-flat contact areas. The sound emission of tyres, where noise emitting from the contact of the tyre with the pavement, are discussed. Numerical experiments illustrate the applicability of the boundary element method to real-world problems.

We consider a strongly elliptic second order differential operator A together with a degenerate boundary operator T of the form T = φ0γ0 + φ1γ1, where γ0 and γ1 denote the evaluation of a function and its exterior normal derivative, respectively, at the boundary. We assume that φ0, φ1 ≥ 0 and φ0 + φ1 ≥ c > 0. We show that a suitable shift of the realization AT of A in Lp(X+) has a bounded H∞-calculus whenever X+ is a manifold with boundary and bounded geometry.

This thesis is concerned with the mathematical analysis of models for Micro-Electro-Mechanical Systems (MEMS). These models arise in the form of coupled partial differential equations with a moving boundary. Although MEMS devices are often operated in non-isothermal environments, temperature is usually neglected in the mathematical investigations. Therefore the focus of our modelling is to incorporate temperature and the related material properties. We derive two models, both of which focus on different aspects of the underlying physics. Afterwards we prove local well-posedness in time and also global well-posedness under additional assumptions on the model's parameters. Lastly, we provide some numerical results which exemplify how temperature and the model's material constants change the qualitative behaviour of the system.

The peripheral T-cell pool undergoes a striking age associated remodeling which is accelerated by progressive insulin resistance. Exercise training is known to delay several aspects of T-cell senescence. The purpose of the current study was to investigate the effect of 3 weeks regular concentric or eccentric endurance exercise training on the composition of the T-cell compartment in pre-diabetic subjects. Sixteen male older adults with impaired glucose tolerance were recruited and performed either concentric exercise (CE) or eccentric exercise (EE) walking 3 times a week for 3 weeks. Fasting venous blood sampling was performed before training and after the training intervention. Various T-cell subpopulations were analyzed by flow cytometry. We did not find significant time × group effects (interaction) but found several significant time effects for cell type ratios and cell subsets proportions. There was an increase of the CD4+/CD8+ (0.55 ± 0.85%; p = 0.033) and CD4+/CD3+ ratio (5.63 ± 8.44%; p = 0.018) and a decrease of the CD8+/CD3+ ratio (-0.95 ± 1.64%; p = 0.049) after training. We found proportional increases of CD4+/CCR7+/CD45RO+ central memory cells (5.02 ± 7.68%; p = 0.030), naïve CD8+/CCR7+/CD45RO- (3.00 ± 6.68%; p = 0.047) and CD8+/CCR7+/CD45RO+ central memory cells (3.01 ± 3.70%; p = 0.009), while proportions of CD4+/CCR7-/CD45RO- TEMRA cells (-2.17 ± 4.66%; p = 0.012), CD8+/CCR7-/CD45RO- TEMRA cells (-5.11 ± 7.02%; p = 0.018) and CD16+ cells (-4.67 ± 6.45%; p = 0.016) decreased after training. 3 weeks of either CE or EE were effective in reversing hallmarks of T-cell senescence in pre-diabetic subjects. It is suggested that exercise stimulates production and mobilization of naïve T-cells, while differentiated TEMRA cells might disappear by apoptosis.

We introduce the class of MAT-free hyperplane arrangements which is based on the Multiple Addition Theorem by Abe, Barakat, Cuntz, Hoge, and Terao. We also investigate the closely related class of MAT2-free arrangements based on a recent generalization of the Multiple Addition Theorem by Abe and Terao. We give classifications of the irreducible complex reflection arrangements which are MAT-free respectively MAT2-free. Furthermore, we ask some questions concerning relations to other classes of free arrangements., Comment: 17 pages

The goal of this review article is to provide a survey about the foundations of semilinear stochastic partial differential equations. In particular, we provide a detailed study of the concepts of strong, weak and mild solutions, establish their connections, and review a standard existence- and uniqueness result. The proof of the existence result is based on a slightly extended version of the Banach fixed point theorem., Comment: 36 pages

We describe the natural geometry of Hilbert schemes of curves in ${\mathbb P}^3$ and, in some cases, in ${\mathbb P}^n$ , $n\geq 4$., Comment: v.2: minor changes