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The term thermal capacity appears to suggest a storable thermal quantity. However, this claim is not redeemed when thermal capacity is projected onto “heat”, which, like all energy forms, exits only in transit and is not a part of internal energy. The storable thermal quantity is entropy, and entropy capacity is a well-defined physical coefficient which has the advantage of being a susceptibility. The inverse of the entropy capacity relates the response of the system (change of temperature) to a stimulus (change of entropy) such as the fluid level responses to a change in amount of fluid contained in a vessel. Frequently, entropy capacity has been used implicitly, which is clarified in examples of the low-temperature analysis of phononic and electronic contributions to the thermal capacity of solids. Generally, entropy capacity is used in the estimation of the entropy of a solid. Implicitly, the thermoelectric figure of merit refers to entropy capacity. The advantage of the explicit use of entropy capacity comes with a descriptive fundamental understanding of the thermal behaviour of solids, which is made clear by the examples of the Debye model of phonons in solids, the latest thermochemical modelling of carbon allotropes (diamond and graphite) and not least caloric materials. An electrocaloric cycle of barium titanate close to its paraelectric–ferroelectric phase transition is analysed by means of entropy capacity. Entropy capacity is a key to intuitively understanding thermal processes.

We complete the computation of all $\mathbb{Q}$-rational points on all the $64$ maximal Atkin-Lehner quotients $X_0(N)^*$ such that the quotient is hyperelliptic. To achieve this, we use a combination of various methods, namely the classical Chabauty--Coleman, elliptic curve Chabauty, quadratic Chabauty, and the bielliptic quadratic Chabauty method combined with the Mordell-Weil sieve. Additionally, for square-free levels $N$, we classify all $\mathbb{Q}$-rational points as cusps, CM points (including their CM field and $j$-invariants) and exceptional ones. We further indicate how to use this to compute the $\mathbb{Q}$-rational points on all of their modular coverings., Comment: 29 pages, 6 tables; accepted by ANTS-XV and hence for publication in Research in Number Theory

We investigate the pseudo-hyperkähler geometry of higher degree rational curves in the twistor space of a hyperkähler 4-manifold.

In this thesis, we develop phase-field fracture models for simulating fractures in compressible and incompressible solids. Classical (primal) phase-field fracture models fail due to locking effects. Hence, we formulate the elasticity part of the phase-field fracture problem in mixed form, avoiding locking. For the elasticity part in mixed form, we prove inf-sup stability, which allows a stable discretization with Taylor-Hood elements. We solve the resulting (3x3) phase-field fracture problem - a coupled variational inequality system - with a primal-dual active set method. In addition, we develop a physics-based Schur-type preconditioner for the linear solver to reduce the computational workload. We confirm the robustness of the new solver for five benchmark tests. Finally, we compare numerical simulations to experimental data analyzing fractures in punctured strips of ethylene propylene diene monomer rubber (EPDM) stretched until total failure to check the applicability on a real-world problem in nearly incompressible solids. Similar behavior of measurement data and the numerically computed quantities of interest validate the newly developed quasi-static phase-field fracture model in mixed form. DFG/SPP 1748/392587580/EU

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Vibration-based Structural Health Monitoring is an ongoing field of research in many engineering disciplines. As for civil engineering, plenty of experimental structures have been erected in the past decades, both under laboratory and real-life conditions. Some of these facilities became a benchmark for different kinds of methods associated with Structural Health Monitoring such as damage analysis and Operational Modal Analysis, which led to fruitful developments in the global research community. When it comes to the continuous monitoring and assessment of the structural integrity of mechanical systems exposed to environmental and operational variability, the robustness and adaptability of the applied methods is of utmost importance. Such properties cannot be fully evaluated under laboratory conditions, which highlights the necessity of outdoor measurement campaigns. To this end, we introduce a test facility for Structural Health Monitoring comprising a lattice tower exposed to realistic conditions and featuring multiple reversible damage mechanisms. The structure located near Hanover in Northern Germany is densely equipped with sensors to capture the structural dynamics. The environmental conditions are monitored in parallel. The obtained continuous measurement data can be accessed online in an open repository. That is the foundation for benchmarks, consisting of a growing data set that enables the development, evaluation, and comparison of Structural Health Monitoring strategies and methods. In this article, we offer a documentation of the test facility and the data acquisition system. Lastly, we characterize the structural dynamics with the help of a finite element model and by analyzing several month of data.

The Gauss-Manin connection of a family of hypersurfaces governs the change of the period matrix along the family. This connection can be complicated even when the equations defining the family look simple. When this is the case, it is computationally expensive to compute the period matrices of varieties in the family via homotopy continuation. We train neural networks that can quickly and reliably guess the complexity of the Gauss-Manin connection of a pencil of hypersurfaces. As an application, we compute the periods of 96% of smooth quartic surfaces in projective 3-space whose defining equation is a sum of five monomials; from the periods of these quartic surfaces, we extract their Picard numbers and the endomorphism fields of their transcendental lattices., Comment: 30 pages

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In this paper we study the Mori fan of the Dolgachev–Nikulin–Voisin family in degree 2 as well as the associated secondary fan. The main result is an enumeration of all maximal dimensional cones of the two fans. © by the author(s).

In configuration design, the task is to compose a system out of a set of predefined, modu-lar building blocks assembled by defined interfaces. Product configuration systems, both with or without integration of geometric models, implement reasoning techniques to model and explore the resulting solution spaces. Among others, the formulation of constraint satisfaction problems (CSP) is state of the art and the informational background in many proprietary configuration engine software packages. Basically, configuration design tasks can also be implemented in modern computer aided design (CAD) systems as these contain different techniques for knowledge-based product modeling but literature reports only little about detailed application examples, best practices or training materials. This article aims at bridging this gap and presents a step-by-step implementation guide for CSP-based CAD configurators for combinatorial designs with the example of Autodesk Inventor.

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The twistor space of the moduli space of solutions of Hitchin's self-duality equations can be identified with the Deligne-Hitchin moduli space of $\lambda$-connections. We use real projective structures on Riemann surfaces to prove the existence of new components of real holomorphic sections of the Deligne-Hitchin moduli space. Applying the twistorial construction we show the existence of new hyper-K\"ahler manifolds associated to any compact Riemann surface of genus $g\geq2$. These hyper-K\"ahler manifolds can be considered as moduli spaces of (certain) singular solutions of the self-duality equations., Comment: 19 pages, 1 figure; to appear in manuscripta mathematica

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Die heutige Analysis ist aus der Infinitesimalrechnung von Leibniz und Newton hervorgegangen. Zwei Jahrhunderte lang gehörten Infinitesimalien, also unendlich kleine Größen, zum Handwerkszeug der Mathematiker und bescherten der neu entstandenen Disziplin einen außerordentlichen Aufschwung, bevor sie durch den Weierstraß'schen Grenzwertbegriff und die Konstruktion der reellen Zahlen durch Cantor und Dedekind entbehrlich schienen und schließlich aus der Analysis verbannt wurden. Der Preis für diese Finitisierung und Arithmetisierung der Analysis waren die Akzeptanz des aktual Unendlichen in Gestalt transfiniter Mengen sowie ein sperriger, wenig intuitiver Grenzwertformalismus, der Lernenden an Schulen und Hochschulen bis heute Schwierigkeiten bereitet. Mit Robinsons Non-standard Analysis hielten die Infinitesimalien in den 1960er Jahren – jetzt streng modelltheoretisch begründet – wieder Einzug in die Mathematik. In der Folge wurde Nichtstandardanalysis auf verschiedenen Gebieten innerhalb und außerhalb der Mathematik erfolgreich eingesetzt. Keisler erkannte das didaktische Potential von Robinsons Arbeit und entwarf in den 1970er Jahren eine axiomatische Einführung in die Analysis mit hyperreellen Zahlen ohne die modelltheoretischen Voraussetzungen. Seitdem gab es und gibt es bis heute Projekte mit dem Ziel, Nichtstandardanalysis für die Lehre zu nutzen. Trotz positiver Erfahrungen aus diesen Projekten blieb der Einfluss auf die Lehre insgesamt sehr gering. An den Hochschulen wird Analysis fast ausnahmslos rein auf der Grundlage des Weierstraß'schen Grenzwertbegriffs gelehrt, wobei die reellen Zahlen axiomatisch eingeführt werden. In der Schule zieht man sich zunehmend auf einen propädeutischen Grenzwertbegriff zurück. Das Ziel der vorliegenden Arbeit ist eine Standortbestimmung von Nichtstandard in der Analysis, speziell in der Lehre der Analysis – in mathematischer, logischer, philosophischer und stoffdidaktischer Hinsicht. Insbesondere soll untersucht werden, inwieweit sich unter diesen Aspekten Gründe für die zurückhaltende Aufnahme von Elementen der Nichtstandardanalysis in die Lehre ableiten lassen. Zu diesem Zweck werden zunächst verschiedene Zugänge zur Nichtstandardanalysis, ihre Adaptionen für elementare Einführungen sowie Erfahrungen aus der Lehre überblickend vorgestellt. Dieser Überblick ermöglicht es herauszuarbeiten, inwieweit gewohnte Sichtweisen (zum Beispiel auf die Mengenlehre, das Unendliche, die reellen und die natürlichen Zahlen oder das Kontinuum) durch Nichtstandard herausgefordert werden, welche Voraussetzungen Nichtstandard mathematisch benötigt und in welcher Weise und mit welchem Effekt Nichtstandardmethoden in der elementaren Analysis eingesetzt werden können. Um mögliche mathematische Probleme aufzudecken, die Vorbehalte gegenüber Nichtstandard begründen können, werden logische, modelltheoretische und mengentheoretische Untersuchungen diskutiert. Zur Analyse einer aus philosophischer Haltung heraus begründeten Ablehnung von Nichtstandard werden verschiedene Grundlagenpositionen sowie ontologische, epistemologische und anwendungsbezogene Fragen behandelt. Ein empirischer Teil dient als Beleg für einen aktuellen Meinungsquerschnitt, wie Nichtstandardanalysis von den Lehrenden der Analysis wahrgenommen und als Lehrstoff eingeschätzt wird. Quelle ist eine im Rahmen des Dissertationsprojektes per E-Mail durchgeführte und mittels einer qualitativen Inhaltsanalyse ausgewertete Befragung von Analysislehrenden an deutschen Hochschulen. Die Auswertung ist Ausgangspunkt für die Diskussion mathematischer und didaktischer Vorbehalte gegen den Einsatz von Nichtstandardanalysis in der Lehre. Im Vordergrund steht also die Frage nach der Rechtfertigung von Nichtstandardanalysis als Lehrstoff. Untersuchungen zu weitergehenden Fragestellungen und Aspekten der Auswertung, die man anschließen könnte, etwa hinsichtlich der Prozesse zur Festlegung dessen, was an Schule oder Hochschule gelehrt werden sollte, werden hier nicht weiterverfolgt., Today's analysis emerged from the infinitesimal calculus of Leibniz and Newton. For two centuries, infinitesimals, i.e. infinitely small quantities, belonged to the tools of the mathematicians and gave the newly developed discipline an extraordinary upswing, before they seemed dispensable and were finally banished from analysis due to the Weierstrassian concept of limit and the construction of the real numbers by Cantor and Dedekind. The price for this finitization and arithmetization of analysis was the acceptance of the actual infinite in the form of transfinite sets as well as a bulky, not very intuitive formalism of limits, which causes difficulties for learners at schools and universities to this day. With Robinson's Non-standard Analysis, the infinitesimals found their way back into mathematics in the 1960s – now strictly based on model theory. In the following years, non-standard analysis was used successfully in various areas inside and outside of mathematics. Keisler recognized the didactic potential of Robinson's work and in the 1970s designed an axiomatic introduction to analysis with hyperreal numbers without the model-theoretical prerequisites. Since then there have been, and still are, projects aimed at using non-standard analysis for teaching. Despite positive experiences from these projects, the overall impact on teaching remained very small. At universities, calculus is taught almost without exception purely on the basis of Weierstrass's concept of limit, with the real numbers being introduced axiomatically. In school, one increasingly retreats to a propaedeutic notion of limit. The aim of the present dissertation is to determine the position of the "non-standard" in analysis, especially in the teaching of analysis – in mathematical, logical and philosophical terms and in terms of subject specific didactics. In particular, we shall examine to what extent reasons for the reluctant inclusion of elements of non-standard analysis in teaching can be derived from these aspects. To this end, different approaches to non-standard analysis, their adaptations for elementary introductions and experiences from teaching are presented in an overview. This overview makes it possible to work out to what extent customary perspectives (for example on set theory, the infinite, the real and the natural numbers or the continuum) are challenged by non-standard analysis, which prerequisites non-standard analysis requires mathematically and in what way and with what effect non-standard methods can be used in the elementary analysis. In order to uncover possible mathematical problems that may account for reservations about non-standard analysis, we discuss logical, model-theoretical and set-theoretical investigations. To analyze a rejection of non-standard analysis based on a philosophical attitude, we deal with various foundational positions as well as ontological, epistemological and application-related questions. An empirical part serves as evidence for a current cross-section of opinions on how non-standard analysis is perceived by calculus instructors and assessed as a subject matter of teaching. The source is a survey of calculus instructors at German universities carried out by email as part of the dissertation project and evaluated using a qualitative content analysis. The evaluation is the starting point for the discussion of mathematical and didactic reservations against the use of non-standard analysis in teaching. In the foreground is the question of the justification of non-standard analysis as a subject matter of teaching. Studies on further questions and aspects of the evaluation that could follow, for example with regard to the processes for determining what should be taught at school or university, are not pursued further here.

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This thesis provides a detailed analysis of the coupling of satellite rotations into the inter-satellite range, measured by the Laser Ranging Interferometer (LRI) onboard the GRACE Follow-On satellites.

We study residuated homomorphisms (r-morphisms) and their adjoints, the so-called localizations (or l-morphisms), between implicative semilattices, because these objects may be characterized as semilattices whose unary meet operations have adjoints. Since left resp. right adjoint maps are the residuated resp. residual maps (having the property that preimages of principal downsets resp. upsets are again such), one may not only regard the l-morphisms as abstract continuous maps in a pointfree framework (as familiar in the complete case), but also characterize them by concrete closure-theoretical continuity properties. These concepts apply to locales (frames, complete Heyting lattices) and provide generalizations of continuous and open maps between spaces to an algebraic (not necessarily complete) pointfree setting.

These lecture notes accompany a refresher course in applied mathematics with a focus on numerical concepts (Part I), numerical linear algebra (Part II), numerical analysis, Fourier series and Fourier transforms (Part III), and differential equations (Part IV). Several numerical projects for group work are provided in Part V. In these projects, the tasks are threefold: mathematical modeling, algorithmic design, and implementation. Therein, it is important to draw interpretations of the obtained results and provide measures (Parts I-IV) how to build confidence into numerical findings such intuition, error analysis, convergence analysis, and comparison to manufactured solutions. Both authors have been jointly teaching over several years this class and bring in a unique mixture of their respective teaching and research fields.

Let $f\colon S'\longrightarrow S$ be a cyclic branched covering of smooth projective surfaces over $\mathbb{C}$ whose branch locus $\Delta\subset S$ is a smooth ample divisor. Pick a very ample complete linear system $|\mathcal{H}|$ on $S$, such that the polarized surface $(S, |\mathcal{H}|)$ is not a scroll nor has rational hyperplane sections. For the general member $[C]\in|\mathcal{H}|$ consider the $\mu_{n}$-equivariant isogeny decomposition of the Prym variety $Prym(C'/C)$ of the induced covering $f\colon C'= f^{-1}(C)\longrightarrow C$:\[Prym(C'/C)\sim\prod_{d|n,\ d\neq1}\mathcal{P}_{d}(C'/C).\] We show that for the very general member $[C]\in|\mathcal{H}|$ the isogeny component $\mathcal{P}_{d}(C'/C)$ is $\mu_{d}$-simple with $End_{\mu_{d}}(\mathcal{P}_{d}(C'/C))\cong\mathbb{Z}[\zeta_{d}]$. In addition, for the non-ample case we reformulate the result by considering the identity component of the kernel of the map $\mathcal{P}_{d}(C'/C)\subset Jac(C')\longrightarrow Alb(S')$., Comment: 13 pages, corrected a typo in the paper's title, comments are welcome

We present a rigorous renormalization group scheme for lattice quantum field theories in terms of operator algebras. The renormalization group is considered as an inductive system of scaling maps between lattice field algebras. We construct scaling maps for scalar lattice fields using Daubechies' wavelets, and show that the inductive limit of free lattice ground states exists and the limit state extends to the familiar massive continuum free field, with the continuum action of spacetime translations. In particular, lattice fields are identified with the continuum field smeared with Daubechies' scaling functions. We compare our scaling maps with other renormalization schemes and their features, such as the momentum shell method or block-spin transformations., Comment: 66 pages, 4 figure (accepted version)

We provide a rigorous lattice approximation of conformal field theories given in terms of lattice fermions in 1+1-dimensions, focussing on free fermion models and Wess-Zumino-Witten models. To this end, we utilize a recently introduced operator-algebraic framework for Wilson-Kadanoff renormalization. In this setting, we prove the convergence of the approximation of the Virasoro generators by the Koo-Saleur formula. From this, we deduce the convergence of lattice approximations of conformal correlation functions to their continuum limit. In addition, we show how these results lead to explicit error estimates pertaining to the quantum simulation of conformal field theories., 71 pages, 1 figure; further comparison with other approaches, clarified relation with Temperley-Lieb algebras, slightly expanded published version including comment on modified Koo-Saleur formula

We provide a spectral sequence computing the extension groups of tautological bundles on symmetric products of curves. One main consequence is that, if $$E\ne \mathcal O_X$$ E ≠ O X is simple, then the natural map $${{\,\mathrm{\mathsf {Ext}}\,}}^1(E,E)\rightarrow {{\,\mathrm{\mathsf {Ext}}\,}}^1(E^{[n]},E^{[n]})$$ Ext 1 ( E , E ) → Ext 1 ( E [ n ] , E [ n ] ) is injective for every n. Along with previous results, this implies that $$E\mapsto E^{[n]}$$ E ↦ E [ n ] defines an embedding of the moduli space of stable bundles of slope $$\mu \notin [-1,n-1]$$ μ ∉ [ - 1 , n - 1 ] on the curve X into the moduli space of stable bundles on the symmetric product $$X^{(n)}$$ X ( n ) . The image of this embedding is, in most cases, contained in the singular locus. For line bundles on a non-hyperelliptic curve, the embedding identifies the Brill–Noether loci of X with the loci in the moduli space of stable bundles on $$X^{(n)}$$ X ( n ) where the dimension of the tangent space jumps. We also prove that $$E^{[n]}$$ E [ n ] is simple if E is simple.

We present a multi-dimensional continuum mathematical model for modeling the growth of a symbiotic biofilm system. We take a dual-species namely, the Streptococcus - Veillonella sp. biofilm system as an example for numerical investigations. The presented model describes both the cooperation and competition between these species of bacteria. The coupled partial differential equations are solved by using an integrative finite element numerical strategy. Numerical examples are carried out for studying the evolution and distribution of the bio-components. The results demonstrate that the presented model is capable of describing the symbiotic behavior of the biofilm system. However, homogenized numerical solutions are observed locally. To study the homogenization behavior of the model, numerical investigations regarding on how random initial biomass distribution influences the homogenization process are carried out. We found that a smaller correlation length of the initial biomass distribution leads to faster homogenization of the solution globally, however, shows more fluctuated biomass profiles along the biofilm thickness direction. More realistic scenarios with bacteria in patches are also investigated numerically in this study.

The $$H^2$$ H 2 -regularity of variational solutions to a two-dimensional transmission problem with geometric constraint is investigated, in particular when part of the interface becomes part of the outer boundary of the domain due to the saturation of the geometric constraint. In such a situation, the domain includes some non-Lipschitz subdomains with cusp points, but it is shown that this feature does not lead to a regularity breakdown. Moreover, continuous dependence of the solutions with respect to the domain is established.

Predicting biological properties of unseen proteins is shown to be improved by the use of protein sequence embeddings. However, these sequence embeddings have the caveat that biological metadata do not exist for each amino acid, in order to measure the quality of each unique learned embedding vector separately. Therefore, current sequence embedding cannot be intrinsically evaluated on the degree of their captured biological information in a quantitative manner. We address this drawback by our approach, dom2vec, by learning vector representation for protein domains and not for each amino acid base, as biological metadata do exist for each domain separately. To perform a reliable quantitative intrinsic evaluation in terms of biology knowledge, we selected the metadata related to the most distinctive biological characteristics of a domain, which are its structure, enzymatic, and molecular function. Notably, dom2vec obtains an adequate level of performance in the intrinsic assessment&mdash, therefore, we can draw an analogy between the local linguistic features in natural languages and the domain structure and function information in domain architectures. Moreover, we demonstrate the dom2vec applicability on protein prediction tasks, by comparing it with state-of-the-art sequence embeddings in three downstream tasks. We show that dom2vec outperforms sequence embeddings for toxin and enzymatic function prediction and is comparable with sequence embeddings in cellular location prediction.

With this work, a novel derivative-free multi-objective optimisation approach for solving engineering problems is presented. State-of-the-art algorithms usually require numerical experimentation in order to tune the algorithm’s multiple parameters to a specific optimisation problem. This issue is effectively tackled by the presented deterministic method which has only a single parameter. The most popular multi-objective optimisation algorithms are based on pseudo-random numbers and need several parameters to adjust the associated probability distributions. Deterministic methods can overcome this issue but have not attracted much research interest in the past decades and are thus seldom applied in practice. The proposed multi-objective algorithm is an extension of the previously introduced deterministic single-objective Global Pattern Search algorithm. It achieves a thorough recovery of the Pareto frontier by tracking a predefined number of non-dominated samples during the optimisation run. To assess the numerical efficiency of the proposed method, it is compared to the well-established NSGA2 algorithm. Convergence is demonstrated and the numerical performance of the proposed optimiser is discussed on the basis of several analytic test functions. Finally, the optimiser is applied to two structural dynamics problems: transfer function estimation and finite element model updating. The introduced algorithm performs well on test functions and robustly converges on the considered practical engineering problems. Hence, this deterministic algorithm can be a viable and beneficial alternative to random-number-based approaches in multi-objective engineering optimisation.

This thesis focusses on laser frequency stabilisation and interferometer path length differences on LISA Pathfinder (LPF). LPF was a satellite mission, in operation from December 2015 until July 2017, to demonstrate key technologies for the future spaceborne gravitational wave observatory, the Laser Interferometer Space Antenna (LISA). It successfully showed that the undesired disturbances, the so-called residual acceleration noise, of a pair of free-falling test masses (TMs) could be limited to less than 2 fm/s^2/sqrt{Hz} at mHz Fourier frequencies and thus paved the way for LISA. This level of residual acceleration noise could only be reached by the interaction of several key subsystems. One of these, the Optical Metrology System (OMS), provided the LPF science measurement: a heterodyne interferometry readout of the relative positions of the free-floating TMs. The OMS showed excellent performance over the mission and is studied here in detail to make the best use of this unique measurement data and to learn as much as possible for future interferometer development. The OMS is subject to several noise sources. This thesis focusses on one of these: laser frequency fluctuations. In the design phase of the OMS, a laser frequency stabilisation technique via a dedicated interferometer measurement and a nested control loop was developed. In this thesis, we show the in-flight results of the planned loop characterisation experiments and noise measurements. We prove that the stabilisation worked as expected from ground tests and was reliable over the mission duration. We also identified periods of slightly increased laser frequency fluctuations whose origin could not yet be identified. This analysis was restricted by the limited number of laser telemetry channels and their low sampling frequency. The coupling of laser frequency noise depends on the optical path length difference between the measurement and the reference beam. Two experiments optimised for determining this quantity have been designed and executed on LPF and are analysed here. The evidence collected in the offset experiment allows us to associate a change in measured path length mismatch to a commanded offset at a 3\sigma uncertainty level, both in direction and amplitude. In addition, we confirmed that the estimated path length mismatch is independent of the laser frequency modulation amplitude and frequency. In general, we find the path length mismatch is only a few hundred μm, which is a sign of excellent integration. We also provide an example where these experiments were used as a means to measure absolute distances on LPF. During these experiments, we also observed short term path length mismatch variations which are not believed to be caused by a true TM motion, as well as spurious signals in the angular measurements of the OMS. Within the scope of this work, a definite reason could not be found for either of these two observations but different cross-coupling hypotheses to explain the signals in the angular measurements could also not be fully rejected. However, we have found no reason to believe that these angular signals could be an indication of a mechanism that impacts the longitudinal measurements and thus adds a systematic error to the path length mismatch numbers reported.

This paper is concerned with the numerical approximation of the solution of the coupled wave equation of Kirchhoff type with nonlinear boundary damping and memory term using a mixed finite element method. The Raviart-Thomas mixed finite element method is one of the most prominent techniques to discretize the second-order wave equations; therefore, we apply this scheme for space discretization. Furthermore, an L2-in-space error estimate is presented for this mixed finite element approximation. Finally, the efficiency of the method is verified by a numerical example. © 2021 The Authors. Mathematical Methods in the Applied Sciences published by John Wiley & Sons Ltd.

In this contribution, we revisit adjoint formulations for stationary and nonstationary space-time fluid-structure interaction. The adjoints serve for two purposes: goal-oriented a posteriori error estimation with the dual-weighted residual method and gradient-based numerical optimization. Our developments are substantiated with comments on numerical simulations in some other published studies.

A catalogue of simplicial hyperplane arrangements was first given by Gr\"unbaum in 1971. These arrangements naturally generalize finite Coxeter arrangements and the weak order through the poset of regions. For simplicial arrangements, posets of regions are in fact lattices. We update Gr\"unbaum's catalogue, providing normals and invariants for all known sporadic simplicial arrangements with up to 37 lines. The weak order is known to be congruence normal, and congruence normality for simplicial arrangements can be determined using polyhedral cones called shards. In this article, we provide additional structure to the catalogue of simplicial hyperplane arrangements by determining which arrangements always/sometimes/never lead to congruence normal lattices of regions. To this end, we use oriented matroids to recast shards as covectors to determine congruence normality of large hyperplane arrangements. As a consequence of this approach we derive in particular which lattices of regions of sporadic simplicial arrangements of rank 3 are always congruence normal. We also show that lattices of regions from finite Weyl groupoids of any rank are congruence normal., Comment: 39 pages, 17 figures, 15 Tables, 2 Appendices, Second version with minor improvements e.g. typos

Altres ajuts: COST Action IC1205 on Computational Social Choice This paper proposes a model of a legislature, formed by several parties, which has to vote for or against a certain bill in the presence of a lobbyist interested in a certain vote outcome. We show that the ease with which the lobbyist can manipulate a legislature decision increases with the number of elected parties, and, consequently, decreases with an electoral threshold. On the other hand, a lower electoral threshold increases the representativeness of a legislature. We combine these two effects in a notion of fairness. We show the existence of an electoral threshold that optimizes the fairness of a political system, which is close to 1-5%. Namely, the optimal threshold (in our sense) is close to thresholds that exist in most parliamentary democracies.

In this paper, a graphics processing unit-accelerated smoothed particle hydrodynamics solver is presented to simulate the three-dimensional explosions in soils and their damage to aluminium structures. To achieve this objective, a number of equations of state and constitutive models required to close the governing equations are incorporated into the proposed smoothed particle hydrodynamics framework, including the Jones-Wilkins-Lee equation of state for explosive materials, the Grüneisen equation of state for metals, the elastic-perfectly plastic constitutive model for metals, and the elastoplastic and elasto-viscoplastic constitutive models for soils. The proposed smoothed particle hydrodynamics methodology was implemented using the Compute Unified Device Architecture programming interface on an NVIDIA graphics processing unit in order to improve the computational efficiency. The various components of the proposed methodology were validated using four test cases, namely, a C4 detonation and an aluminium bar expanded by denotation to validate the modelling of explosion, a cylindrical Taylor bar impact test case to validate the modelling of large deformation in metals, a sand collapse test for the modelling of soils. Following the validation, the proposed method was used to simulate the detonation of an explosive material (C4) in soil and the concomitant deformation of an aluminium plate resulting from this explosion. The predicted results of this simulation are shown to be in good conformance with available experimental data. Finally, it is shown that the proposed graphics processing unit-accelerated SPH solver is able to model interaction problems involving millions of particles in a reasonable time. © 2021 The Authors

This thesis deals with the study of the cohomology and the Kodaira dimension of some moduli spaces. In the first part we compute the intersection Betti numbers of the GIT models of two moduli spaces. They parametrize non-hyperelliptic Petri-general curves of genus four and numerically polarized Enriques surfaces of degree two respectively. In both cases, the strategy of the cohomological calculation relies on a general method developed by Kirwan to compute the cohomology of GIT quotients of projective varieties. This procedure is based on the equivariantly perfect stratification of the unstable points studied by Hesselink, Kempf, Kirwan and Ness, and a partial resolution of singularities, called the Kirwan blow-up. In the second part of the thesis, we study the moduli spaces of elliptic K3 surfaces of Picard number at least three, i.e.\ $U\oplus \langle -2k \rangle$-polarized K3 surfaces. Such moduli spaces are proved to be of general type for $k\geq 220$. The proof relies on the low-weight cusp form trick developed by Gritsenko, Hulek and Sankaran.

In this thesis we prove the infinitesimal Torelli theorem for certain classes of irregular varieties. Given a compact Kähler manifold, the infinitesimal Torelli problem asks whether the differential of the period map of a Kuranishi family is injective. Unlike the classical Torelli theorem for curves, there is a negative answer for example for hyperelliptic curves of genus greater than 2. Nevertheless, the infinitesimal Torelli theorem holds for many other classes of manifolds. Following Green’s proof for sufficiently ample hypersurfaces in arbitrary varieties, we prove it for smooth ample hypersurfaces and more generally complete intersections in general abelian varieties by reducing it to showing the surjectivity of certain multiplication maps of vector bundles on the ambient abelian variety. Then we derive numerical conditions for such multiplication maps to be surjective giving an effective bound on Green’s result in this particular case. We also investigate the more general case of irregular varieties with globally generated cotangent bundle which do not embed into their Albanese varieties.

In this thesis we study classes of subriemannian manifolds and their sublaplacians. We determine the spectrum of the intrinsic sublaplacian on pseudo H-type nilmanifolds. Moreover, we construct an arbitrary number of isospectral but pairwise non-homeomorphic pseudo H-type nilmanifolds with respect to the sublaplacian. Furthermore, with the help of the representation theory of step $2$ nilpotent Lie groups, we show that the spectrum of the intrinsic sublaplacian on nilmanifolds whose covering space is a step $2$ Carnot group can be completely expressed in terms of the underlying metric Lie algebra data. In the case of H-type nilmanifolds whose covering space has odd dimensional center, we prove a Poisson summation formula that relates the spectrum of the intrinsic sublaplacian to the lengths of closed subriemannian geodesics. Based on the nilpotent approximation of subriemannian manifolds, we compare two subriemannian structures of step $2$ and rank $4$ on the Euclidean sphere $\mathbb{S}^7$ with regards to various geometric and analytical properties. We show that both subriemannian structures are neither locally isometric nor isospectral with respect to the subaplacian. Finally, we study the heat kernel associated to the intrinsic sublaplacian on a quaternionic contact manifold considered as a subriemannian manifold. More precisely, we explicitly compute the first two coefficients appearing in the small time asymptotic expansion of the heat kernel on the diagonal. We show that the second coefficient coincides with the quaternionic contact scalar curvature up to a (universal) constant multiple.

We study the moduli spaces of elliptic K3 surfaces of Picard number at least 3, i.e. $U\oplus \langle -2k \rangle$-polarized K3 surfaces. Such moduli spaces are proved to be of general type for $k\geq 220$. The proof relies on the low-weight cusp form trick developed by Gritsenko, Hulek and Sankaran. Furthermore, explicit geometric constructions of some elliptic K3 surfaces lead to the unirationality of these moduli spaces for $k < 11$ and for 19 other isolated values up to $k=64$., Final version published in "Journal of the London Mathematical Society"

This thesis deals with K3 surfaces and their moduli spaces. In the first part we identify a class of complex K3 surfaces, called of zero entropy, with a particularly simple (but infinite) automorphism group, naturally arising from complex dynamics. We provide a lattice-theoretical classification of their N\'eron-Severi lattices. In the second part we move to the study of the Kodaira dimension of the moduli spaces of elliptic K3 surfaces of Picard rank $3$. We show that almost all of them are of general type, by using the low-weight cusp form trick developed by Gritsenko, Hulek and Sankaran. Moreover, we prove that many of the remaining moduli spaces are unirational, by providing explicit projective models of the corresponding K3 surfaces. In the final part, we investigate the set of rational points on K3 and Enriques surfaces over number fields. We show that all Enriques surfaces over number fields satisfy (a weak version of) the potential Hilbert property, thus proving that, after a field extension, the rational points on their K3 cover are dense and do not come from finite covers.

In this course, coupled problems with interfaces are considered. Some applications and examples are discussed first. Then, interfaces are defined and classified into three categories. Numerical modeling of interfaces is a central aspect in this presentation. These theoretically-oriented parts are followed by numerical simulations using an open-source fluid-structure interaction benchmark code based on the finite element library deal.II. For joint coding, a docker image was installed on qarnot and repl.it for cloud computing., Course held at the CSMA Junior section workshop ahead of the 14th WCCM & EDDOMAS Congress 2020

The following article considers mathematics learning in the early childhood from an interactionist perspective. Here, mathematics learning is differentiated into two aspects of learning: the acquisition of mathematical terms and procedures in the sense of learning mathematical content, and the development of mathematical thinking in the sense of learning to reason. The focus of this contribution is on the second aspect. Based on interactionist perceptions of learning the development of mathematical thinking is described as an increasing participation in mathematical discourses. As a result, mathematical discourses as the “place” for the development of mathematical thinking move into the center of attention. For a more detailed description of these discourses, the so far common focus of interactionist approaches to mathematics learning on the analysis of mathematical negotiation of meaning is expanded to include a description of the emerging argumentative and semantic structuring of the mathematical negotiation processes. In the empirical analyses of this article, different discourse styles are reconstructed, which we call narrative, formal and ‘narrational’ discourses. In future work, they will be used as a theoretical basis for the longitudinal reconstruction of the participation of learners in mathematical negotiation processes to describe their development of mathematical thinking., Der folgende Beitrag befasst sich mit dem Mathematiklernen in der frühen Kindheit aus interaktionistischer Perspektive. Mathematiklernen wird hierzu in zwei Aspekte ausdifferenziert: den Erwerb mathematischer Begriffe und Prozeduren im Sinne des Lernens von mathematischen Inhalten und die mathematische Denkentwicklung im Sinne des Argumentierenlernens. Der Fokus dieses Beitrags liegt auf dem zweiten Aspekt. Basierend auf interaktionistischen Vorstellungen zum Lernen wird die mathematische Denkentwicklung als zunehmende Partizipation an mathematischen Diskursen beschrieben. Hierdurch rücken mathematische Diskurse als „Ort“ der mathematischen Denkentwicklung ins Zentrum der Betrachtung. Zur näheren Beschreibung dieser Diskurse wird die bis dato gängige Fokussierung interaktionistischer Ansätze des Mathematiklernens auf die Analyse mathematischer Bedeutungsaushandlungen um eine Beschreibung emergierender argumentativer und semantischer Strukturierungen mathematischer Aushandlungsprozesse erweitert. In den empirischen Analysen des Beitrages werden mithilfe dieser theoretischen Erweiterung unterschiedliche Diskursstile rekonstruiert, die wir als narrative, formale und narratorische Diskurse bezeichnen. In künftigen Arbeiten sollen sie als theoretische Grundlage für die longitudinale Rekonstruktion der Partizipation von Lernenden an mathematischen Aushandlungsprozessen zur Beschreibung ihrer mathematischen Denkentwicklung genutzt werden.

Simulation-driven product development is nowadays an essential part in the industrial digitalization. Notably, there is an increasing interest in realistic high-fidelity simulation methods in the fast-growing field of additive and ablative manufacturing processes. Thanks to their flexibility, meshfree solution methods are particularly suitable for simulating the stated processes, often accompanied by large deformations, variable discontinuities, or phase changes. Furthermore, in the industrial domain, the meshing of complex geometries represents a significant workload, which is usually minor for meshfree methods. Over the years, several meshfree schemes have been developed. Nevertheless, along with their flexibility in discretization, meshfree methods often endure a decrease in accuracy, efficiency and stability or suffer from a significantly increased computation time. Peridynamics is an alternative theory to local continuum mechanics for describing partial differential equations in a non-local integro-differential form. The combination of the so-called peridynamic correspondence formulation with a particle discretization yields a flexible meshfree simulation method, though does not lead to reliable results without further treatment.\newline In order to develop a reliable, robust and still flexible meshfree simulation method, the classical correspondence formulation is generalized into the Peridynamic Galerkin (PG) methods in this work. On this basis, conditions on the meshfree shape functions of virtual and actual displacement are presented, which allow an accurate imposition of force and displacement boundary conditions and lead to stability and optimal convergence rates. Based on Taylor expansions moving with the evaluation point, special shape functions are introduced that satisfy all the previously mentioned requirements employing correction schemes. In addition to displacement-based formulations, a variety of stabilized, mixed and enriched variants are developed, which are tailored in their application to the nearly incompressible and elasto-plastic finite deformation of solids, highlighting the broad design scope within the PG methods. Extensive numerical validations and benchmark simulations are performed to show the impact of violating different shape function requirements as well as demonstrating the properties of the different PG formulations. Compared to related Finite Element formulations, the PG methods exhibit similar convergence properties. Furthermore, an increased computation time due to non-locality is counterbalanced by a considerably improved robustness against poorly meshed discretizations.

We study a natural functional on the space of holomorphic sections of the Deligne-Hitchin moduli space of a compact Riemann surface, generalizing the energy of equivariant harmonic maps corresponding to twistor lines. We give a link to a natural meromorphic connection on the hyperholomorphic line bundle recently constructed by Hitchin. Moreover, we prove that for a certain class of real holomorphic sections of the Deligne-Hitchin moduli space, the functional is basically given by the Willmore energy of corresponding (equivariant) conformal map to the 3-sphere. As an application we use the functional to distinguish new components of real holomorphic sections of the Deligne-Hitchin moduli space from the space of twistor lines., 33 pages

Dependence Logic was introduced by Jouko Väänänen in 2007. We study a propositional variant of this logic(PDL)and investigate a variety of parameterisations with respect to central decision problems. The model checking problem (MC) ofPDLisNP-complete (Ebbing and Lohmann, SOFSEM 2012). The subject of this research is to identify a list of parameterisations (formula-size, formula-depth, treewidth, team-size, number of variables) under which MC becomes fixed-parameter tractable. Furthermore, we show that the number of disjunctions or the arity of dependence atoms (dep-arity) as a parameter both yield a paraNP-completeness result. Then, we consider the satisfiability problem (SAT) which classically is known to beNP-complete as well (Lohmann and Vollmer, Studia Logica 2013). There we are presenting a different picture: under team-size, or dep-arity SAT isparaNP-complete whereas under all other mentioned parameters the problem isFPT. Finally, we introduce a variant of the satisfiability problem, asking for a team of a given size, and show for this problem an almost complete picture.

The main subject of this thesis are constrained Willmore tori in the 3-dimensional sphere S^3. It is known that constrained Willmore tori in the 4-sphere come with an associated C_∗-family of flat SL(4,C)-connections. This allows to study constrained Willmore tori as an integrable system. The initial surface can be reconstructed by holomorphic data on the spectralcurve Σ, which is the riemann surface on which the eigenlines of the family of connections are well-defined. If the constrained Willmore torus lies in a 3-dimensional sphere, there is a further symmetry on the spectral curve, a holomorphic involution σ. In this thesis we show that this involution allows to reduce the family of SL(4,C)-connections into a family of SL(2,C)-connections. We achieve this by pushing forward the eigenline bundle of the rank 4 family on the quotient surface Σ/σ. Therefore,the rank 2 family of connections is a Σ/σ-family of connections, which is a hyperelliptic surface. The rank 2 family of connections then allows to give a Sym-Bobenko formula, similiar to the case of constant mean curvature surfaces in S^3. Further, if the quotient surface Σ/σ=CP1, the surface is of constant mean curvature in a space form.

Besides the material research in the field of thermoelectrics, the way from a material to a functional thermoelectric (TE) module comes alongside additional challenges. Thus, comprehension and optimization of the properties and the design of a TE module are important tasks. In this work, different geometry optimization strategies to reach maximum power output or maximum conversion efficiency are applied and the resulting performances of various modules and respective materials are analyzed. A Bi2Te3-based module, a half-Heusler-based module, and an oxide-based module are characterized via FEM simulations. By this, a deviation of optimum power output and optimum conversion efficiency in dependence of the diversity of thermoelectric materials is found. Additionally, for all modules, the respective fluxes of entropy and charge as well as the corresponding fluxes of thermal and electrical energy within the thermolegs are shown. The full understanding and enhancement of the performance of a TE module may be further improved.

The basic principles of thermoelectrics rely on the coupling of entropy and electric charge. However, the long-standing dispute of energetics versus entropy has long paralysed the field. Herein, it is shown that treating entropy and electric charge in a symmetric manner enables a simple transport equation to be obtained and the power conversion and its efficiency to be deduced for a single thermoelectric material apart from a device. The material&rsquo, s performance in both generator mode (thermo-electric) and entropy pump mode (electro-thermal) are discussed on a single voltage-electrical current curve, which is presented in a generalized manner by relating it to the electrically open-circuit voltage and the electrically closed-circuited electrical current. The electrical and thermal power in entropy pump mode are related to the maximum electrical power in generator mode, which depends on the material&rsquo, s power factor. Particular working points on the material&rsquo, s voltage-electrical current curve are deduced, namely, the electrical open circuit, electrical short circuit, maximum electrical power, maximum power conversion efficiency, and entropy conductivity inversion. Optimizing a thermoelectric material for different working points is discussed with respect to its figure-of-merit z T and power factor. The importance of the results to state-of-the-art and emerging materials is emphasized.