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We analyze the mathematical models of two classes of physical phenomena. The first class of phenomena we consider is the interaction between one or more insulating rigid bodies and an electrically conducting fluid, inside of which the bodies are contained, as well as the electromagnetic fields trespassing both of the materials. We take into account both the cases of incompressible and compressible fluids. In both cases our main result yields the existence of weak solutions to the associated system of partial differential equations, respectively. The proofs of these results are built upon hybrid discrete-continuous approximation schemes: Parts of the systems are discretized with respect to time in order to deal with the solution-dependent test functions in the induction equation. The remaining parts are treated as continuous equations on the small intervals between consecutive discrete time points, allowing us to employ techniques which do not transfer to the discretized setting. Moreover, the solution-dependent test functions in the momentum equation are handled via the use of classical penalization methods.
The second class of phenomena we consider is the evolution of a magnetoelastic material. Here too, our main result proves the existence of weak solutions to the corresponding system of partial differential equations. Its proof is based on De Giorgi's minimizing movements method, in which the system is discretized in time and, at each discrete time point, a minimization problem is solved, the associated Euler-Lagrange equations of which constitute a suitable approximation of the original equation of motion and magnetic force balance. The construction of such a minimization problem is made possible by the realization that, already on the continuous level, both of these equations can be written in terms of the same energy and dissipation potentials. The functional for the discrete minimization problem can then be constructed on the basis of these potentials.
To study coisotropic reduction in the context of deformation quantization we introduce constraint manifolds and constraint algebras as the basic objects encoding the additional information needed to define a reduction. General properties of various categories of constraint objects and their compatiblity with reduction are examined. A constraint Serre-Swan theorem, identifying constraint vector bundles with certain finitely generated projective constraint modules, as well as a constraint symbol calculus are proved. After developing the general deformation theory of constraint algebras, including constraint Hochschild cohomology and constraint differential graded Lie algebras, the second constraint Hochschild cohomology for the constraint algebra of functions on a constraint flat space is computed.
This thesis, first, is devoted to the theoretical and numerical investigation of an augmented Lagrangian method for the solution of optimization problems with geometric constraints, subsequently, as well as constrained structured optimization problems featuring a composite objective function and set-membership constraints. It is then concerned to convergence and rate-of-convergence analysis of proximal gradient methods for the composite optimization problems in the presence of the Kurdyka--{\L}ojasiewicz property without global Lipschitz assumption.
We generalize a theorem by Titchmarsh about the mean value of Hardy’s \(Z\)-function at the Gram points to the Hecke \(L\)-functions, which in turn implies the weak Gram law for them. Instead of proceeding analogously to Titchmarsh with an approximate functional equation we employ a different method using contour integration.
For a graph \(\Gamma\) , let K be the smallest field containing all eigenvalues of the adjacency matrix of \(\Gamma\) . The algebraic degree \(\deg (\Gamma )\) is the extension degree \([K:\mathbb {Q}]\). In this paper, we completely determine the algebraic degrees of Cayley graphs over abelian groups and dihedral groups.
The concept of derivative is characterised with reference to four basic mental models. These are described as theoretical constructs based on theoretical considerations. The four basic mental models—local rate of change, tangent slope, local linearity and amplification factor—are not only quantified empirically but are also validated. To this end, a test instrument for measuring students’ characteristics of basic mental models is presented and analysed regarding quality criteria.
Mathematics students (n = 266) were tested with this instrument. The test results show that the four basic mental models of the derivative can be reconstructed among the students with different characteristics. The tangent slope has the highest agreement values across all tasks. The agreement on explanations based on the basic mental model of rate of change is not as strongly established among students as one would expect due to framework settings in the school system by means of curricula and educational standards. The basic mental model of local linearity plays a rather subordinate role. The amplification factor achieves the lowest agreement values. In addition, cluster analysis was conducted to identify different subgroups of the student population. Moreover, the test results can be attributed to characteristics of the task types as well as to the students’ previous experiences from mathematics classes by means of qualitative interpretation. These and other results of students’ basic mental models of the derivative are presented and discussed in detail.
Mathematical concepts are regularly used in media reports concerning the Covid-19 pandemic. These include growth models, which attempt to explain or predict the effectiveness of interventions and developments, as well as the reproductive factor. Our contribution has the aim of showing that basic mental models about exponential growth are important for understanding media reports of Covid-19. Furthermore, we highlight how the coronavirus pandemic can be used as a context in mathematics classrooms to help students understand that they can and should question media reports on their own, using their mathematical knowledge. Therefore, we first present the role of mathematical modelling in achieving these goals in general. The same relevance applies to the necessary basic mental models of exponential growth. Following this description, based on three topics, namely, investigating the type of growth, questioning given course models, and determining exponential factors at different times, we show how the presented theoretical aspects manifest themselves in teaching examples when students are given the task of reflecting critically on existing media reports. Finally, the value of the three topics regarding the intended goals is discussed and conclusions concerning the possibilities and limits of their use in schools are drawn.
We extend Bourgain’s bound for the order of growth of the Riemann zeta function on the critical line to Lerch zeta functions. More precisely, we prove L(λ, α, 1/2 + it) ≪ t\(^{13/84+ϵ}\) as t → ∞. For both, the Riemann zeta function as well as for the more general Lerch zeta function, it is conjectured that the right-hand side can be replaced by t\(^ϵ\) (which is the so-called Lindelöf hypothesis). The growth of an analytic function is closely related to the distribution of its zeros.
For a connected real Lie group G we consider the canonical standard-ordered star product arising from the canonical global symbol calculus based on the half-commutator connection of G. This star product trivially converges on polynomial functions on T\(^*\)G thanks to its homogeneity. We define a nuclear Fréchet algebra of certain analytic functions on T\(^*\)G, for which the standard-ordered star product is shown to be a well-defined continuous multiplication, depending holomorphically on the deformation parameter \(\hbar\). This nuclear Fréchet algebra is realized as the completed (projective) tensor product of a nuclear Fréchet algebra of entire functions on G with an appropriate nuclear Fréchet algebra of functions on \({\mathfrak {g}}^*\). The passage to the Weyl-ordered star product, i.e. the Gutt star product on T\(^*\)G, is shown to preserve this function space, yielding the continuity of the Gutt star product with holomorphic dependence on \(\hbar\).
This paper studies differential graded modules and representations up to homotopy of Lie n-algebroids, for general \(n\in {\mathbb {N}}\). The adjoint and coadjoint modules are described, and the corresponding split versions of the adjoint and coadjoint representations up to homotopy are explained. In particular, the case of Lie 2-algebroids is analysed in detail. The compatibility of a Poisson bracket with the homological vector field of a Lie n-algebroid is shown to be equivalent to a morphism from the coadjoint module to the adjoint module, leading to an alternative characterisation of non-degeneracy of higher Poisson structures. Moreover, the Weil algebra of a Lie n-algebroid is computed explicitly in terms of splittings, and representations up to homotopy of Lie n-algebroids are used to encode decomposed VB-Lie n-algebroid structures on double vector bundles.
We present a technique for computing multi-branch-point covers with prescribed ramification and demonstrate the applicability of our method in relatively large degrees by computing several families of polynomials with symplectic and linear Galois groups.
As a first application, we present polynomials over \(\mathbb{Q}(\alpha,t)\) for the primitive rank-3 groups \(PSp_4(3)\) and \(PSp_4(3).C_2\) of degree 27 and for the 2-transitive group \(PSp_6(2)\) in its actions on 28 and 36 points, respectively. Moreover, the degree-28 polynomial for \(PSp_6(2)\) admits infinitely many totally real specializations.
Next, we present the first (to the best of our knowledge) explicit polynomials for the 2-transitive linear groups \(PSL_4(3)\) and \(PGL_4(3)\) of degree 40, and the imprimitive group \(Aut(PGL_4(3))\) of degree 80.
Additionally, we negatively answer a question by König whether there exists a degree-63 rational function with rational coefficients and monodromy group \(PSL_6(2)\) ramified over at least four points. This is achieved due to the explicit computation of the corresponding hyperelliptic genus-3 Hurwitz curve parameterizing this family, followed by a search for rational points on it. As a byproduct of our calculations we obtain the first explicit \(Aut(PSL_6(2))\)-realizations over \(\mathbb{Q}(t)\).
At last, we present a technique by Elkies for bounding the transitivity degree of Galois groups. This provides an alternative way to verify the Galois groups from the previous chapters and also yields a proof that the monodromy group of a degree-276 cover computed by Monien is isomorphic to the sporadic 2-transitive Conway group \(Co_3\).
Ó. Blasco and S. Pott showed that the supremum of operator norms over L\(^{2}\) of all bicommutators (with the same symbol) of one-parameter Haar multipliers dominates the biparameter dyadic product BMO norm of the symbol itself. In the present work we extend this result to the Bloom setting, and to any exponent 1 < p < ∞. The main tool is a new characterization in terms of paraproducts and two-weight John–Nirenberg inequalities for dyadic product BMO in the Bloom setting. We also extend our results to the whole scale of indexed spaces between little bmo and product BMO in the general multiparameter setting, with the appropriate iterated commutator in each case.
Optimization problems with composite functions deal with the minimization of the sum
of a smooth function and a convex nonsmooth function. In this thesis several numerical
methods for solving such problems in finite-dimensional spaces are discussed, which are
based on proximity operators.
After some basic results from convex and nonsmooth analysis are summarized, a first-order
method, the proximal gradient method, is presented and its convergence properties are
discussed in detail. Known results from the literature are summarized and supplemented by
additional ones. Subsequently, the main part of the thesis is the derivation of two methods
which, in addition, make use of second-order information and are based on proximal Newton
and proximal quasi-Newton methods, respectively. The difference between the two methods
is that the first one uses a classical line search, while the second one uses a regularization
parameter instead. Both techniques lead to the advantage that, in contrast to many similar
methods, in the respective detailed convergence analysis global convergence to stationary
points can be proved without any restricting precondition. Furthermore, comprehensive
results show the local convergence properties as well as convergence rates of these algorithms,
which are based on rather weak assumptions. Also a method for the solution of the arising
proximal subproblems is investigated.
In addition, the thesis contains an extensive collection of application examples and a detailed
discussion of the related numerical results.
In this work, we consider impulsive dynamical systems evolving on an infinite-dimensional space and subjected to external perturbations. We look for stability conditions that guarantee the input-to-state stability for such systems. Our new dwell-time conditions allow the situation, where both continuous and discrete dynamics can be unstable simultaneously. Lyapunov like methods are developed for this purpose. Illustrative finite and infinite dimensional examples are provided to demonstrate the application of the main results. These examples cannot be treated by any other published approach and demonstrate the effectiveness of our results.
Our starting point is the Jacobsthal function \(j(m)\), defined for each positive integer \(m\) as the smallest number such that every \(j(m)\) consecutive integers contain at least one integer relatively prime to \(m\). It has turned out that improving on upper bounds for \(j(m)\) would also lead to advances in understanding the distribution of prime numbers among arithmetic progressions. If \(P_r\) denotes the product of the first \(r\) prime numbers, then a conjecture of Montgomery states that \(j(P_r)\) can be bounded from above by \(r (\log r)^2\) up to some constant factor. However, the until now very promising sieve methods seem to have reached a limit here, and the main goal of this work is to develop other combinatorial methods in hope of coming a bit closer to prove the conjecture of Montgomery. Alongside, we solve a problem of Recamán about the maximum possible length among arithmetic progressions in the least (positive) reduced residue system modulo \(m\). Lastly, we turn towards three additive representation functions as introduced by Erdős, Sárközy and Sós who studied their surprising different monotonicity behavior. By an alternative approach, we answer a question of Sárközy and demostrate that another conjecture does not hold.
Global Existence and Uniqueness Results for Nematic Liquid Crystal and Magnetoviscoelastic Flows
(2022)
Liquid crystals and polymeric fluids are found in many technical applications with liquid crystal displays probably being the most prominent one. Ferromagnetic materials are well established in industrial and everyday use, e.g. as magnets in generators, transformers and hard drive disks. Among ferromagnetic materials, we find a subclass which undergoes deformations if an external magnetic field is applied. This effect is exploited in actuators, magnetoelastic sensors, and new fluid materials have been produced which retain their induced magnetization during the flow.
A central issue consists of a proper modelling for those materials. Several models exist regarding liquid crystals and liquid crystal flows, but up to now, none of them has provided a full insight into all observed effects. On materials encompassing magnetic, elastic and perhaps even fluid dynamic effects, the mathematical literature seems sparse in terms of models. To some extent, one can unify the modeling of nematic liquid crystals and magnetoviscoelastic materials employing a so-called energetic variational approach.
Using the least action principle from theoretical physics, the actual task reduces to finding appropriate energies describing the observed behavior. The procedure leads to systems of evolutionary partial differential equations, which are analyzed in this work.
From the mathematical point of view, fundamental questions on existence, uniqueness and stability of solutions remain unsolved. Concerning the Ericksen-Leslie system modelling nematic liquid crystal flows, an approximation to this model is given by the so-called Ginzburg-Landau approximation. Solutions to the latter are intended to approximately represent solutions to the Ericksen-Leslie system. Indeed, we verify this presumption in two spatial dimensions. More precisely, it is shown that weak solutions of the Ginzburg-Landau approximation converge to solutions of the Ericksen-Leslie system in the energy space for all positive times of evolution. In order to do so, theory for the Euler equations invented by DiPerna and Majda on weak compactness and concentration measures is used.
The second part of the work deals with a system of partial differential equations modelling magnetoviscoelastic fluids. We provide a well-posedness result in two spatial dimensions for large energies and large times. Along the verification of that conclusion, existing theory on the Ericksen-Leslie system and the harmonic map flow is deployed and suitably extended.
In this paper we study properties of the Laplace approximation of the posterior distribution arising in nonlinear Bayesian inverse problems. Our work is motivated by Schillings et al. (Numer Math 145:915–971, 2020. https://doi.org/10.1007/s00211-020-01131-1), where it is shown that in such a setting the Laplace approximation error in Hellinger distance converges to zero in the order of the noise level. Here, we prove novel error estimates for a given noise level that also quantify the effect due to the nonlinearity of the forward mapping and the dimension of the problem. In particular, we are interested in settings in which a linear forward mapping is perturbed by a small nonlinear mapping. Our results indicate that in this case, the Laplace approximation error is of the size of the perturbation. The paper provides insight into Bayesian inference in nonlinear inverse problems, where linearization of the forward mapping has suitable approximation properties.
Die Auseinandersetzung mit Simulations- und Modellierungsaufgaben, die mit digitalen Werkzeugen zu bearbeiten sind, stellt veränderte Anforderungen an Mathematiklehrkräfte in der Unterrichtsplanung und -durchführung. Werden digitale Werkzeuge sinnvoll eingesetzt, so unterstützen sie Simulations- und Modellierungsprozesse und ermöglichen realitätsnähere Sachkontexte im Mathematikunterricht. Für die empirische Untersuchung professioneller Kompetenzen zum Lehren des Simulierens und mathematischen Modellierens mit digitalen Werkzeugen ist es notwendig, Aspekte globaler Lehrkompetenzen von (angehenden) Mathematiklehrkräften bereichsspezifisch auszudeuten.
Daher haben wir ein Testinstrument entwickelt, das die Überzeugungen, die Selbstwirksamkeitserwartungen und das fachdidaktische Wissen zum Lehren des Simulierens und mathematischen Modellierens mit digitalen Werkzeugen erfasst. Ergänzt wird das Testinstrument durch selbstberichtete Vorerfahrungen zum eigenen Gebrauch digitaler Werkzeuge sowie zur Verwendung digitaler Werkzeuge in Unterrichtsplanung und -durchführung.
Das Testinstrument ist geeignet, um mittels Analysen von Veranstaltungsgruppen im Prä-Post-Design den Zuwachs der oben beschriebenen Kompetenz von (angehenden) Mathematiklehrkräften zu messen. Somit können in Zukunft anhand der Ergebnisse die Wirksamkeit von Lehrveranstaltungen, die diese Kompetenz fördern (sollen), untersucht und evaluiert werden.
Der Beitrag gliedert sich in zwei Teile: Zunächst werden in der Testbeschreibung das zugrundeliegende Konstrukt und der Anwendungsbereich des Testinstruments sowie dessen Aufbau und Hinweise zur Durchführung beschrieben. Zudem wird die Testgüte anhand der Pilotierungsergebnisse überprüft. Im zweiten Teil befindet sich das vollständige Testinstrument.
The dissertation investigates the wide class of Epstein zeta-functions in terms of uniform distribution modulo one of the ordinates of their nontrivial zeros. Main results are a proof of a Landau type theorem for all Epstein zeta-functions as well as uniform distribution modulo one for the zero ordinates of all Epstein zeta-functions asscoiated with binary quadratic forms.
In financial mathematics, it is a typical approach to approximate financial markets operating in discrete time by continuous-time models such as the Black Scholes model. Fitting this model gives rise to difficulties due to the discrete nature of market data. We thus model the pricing process of financial derivatives by the Black Scholes equation, where the volatility is a function of a finite number of random variables. This reflects an influence of uncertain factors when determining volatility. The aim is to quantify the effect of this uncertainty when computing the price of derivatives. Our underlying method is the generalized Polynomial Chaos (gPC) method in order to numerically compute the uncertainty of the solution by the stochastic Galerkin approach and a finite difference method. We present an efficient numerical variation of this method, which is based on a machine learning technique, the so-called Bi-Fidelity approach. This is illustrated with numerical examples.
Die vorliegende Arbeit beschäftigt sich explorativ mit Metakognition beim Umgang mit Mathematik. Aufbauend auf der vorgestellten Forschungsliteratur wird der Einsatz von Metakognition im Rahmen einer qualitativen Studie bei Studienanfänger_innen aus verschiedenen Mathematik-(Lehramts-)Studiengängen dokumentiert. Unter Verwendung der Qualitativen Inhaltsanalyse nach Mayring erfolgt die Etablierung eines Kategoriensystems für den Begriff Metakognition im Hinblick auf den Einsatz in der Mathematik, das bisherige Systematisierungen erweitert. Schließlich wird der Einsatz der entsprechenden metakognitiven Aspekte am Beispiel verschiedener Begriffe und Verfahren aus dem Analysis-Unterricht exemplarisch aufgezeigt.
In dieser Arbeit wird mathematisches Papierfalten und speziell 1-fach-Origami im universitären Kontext untersucht. Die Arbeit besteht aus drei Teilen.
Der erste Teil ist im Wesentlichen der Sachanalyse des 1-fach-Origami gewidmet. Im ersten Kapitel gehen wir auf die geschichtliche Einordnung des 1-fach-Origami, betrachten axiomatische Grundlagen und diskutieren, wie das Axiomatisieren von 1-fach-Origami zum Verständnis des Axiomenbegriffs beitragen könnte. Im zweiten Kapitel schildern wir das Design der zugehörigen explorativen Studie, beschreiben unsere Forschungsziele und -fragen. Im dritten Kapitel wird 1-fach-Origami mathematisiert, definiert und eingehend untersucht.
Der zweite Teil beschäftigt sich mit den von uns gestalteten und durchgeführten Kursen »Axiomatisieren lernen mit Papierfalten«. Im vierten Kapitel beschreiben wir die Lehrmethodik und die Gestaltung der Kurse, das fünfte Kapitel enthält ein Exzerpt der Kurse.
Im dritten Teil werden die zugehörigen Tests beschrieben. Im sechsten Kapitel erläutern wir das Design der Tests sowie die Testmethodik. Im siebten Kapitel findet die Auswertung ebendieser Tests statt.
We construct a foliation of an asymptotically flat end of a Riemannian manifold by hypersurfaces which are critical points of a natural functional arising in potential theory. These hypersurfaces are perturbations of large coordinate spheres, and they admit solutions of a certain over-determined boundary value problem involving the Laplace–Beltrami operator. In a key step we must invert the Dirichlet-to-Neumann operator, highlighting the nonlocal nature of our problem.
Bivariate copula monitoring
(2022)
The assumption of multivariate normality underlying the Hotelling T\(^{2}\) chart is often violated for process data. The multivariate dependency structure can be separated from marginals with the help of copula theory, which permits to model association structures beyond the covariance matrix. Copula‐based estimation and testing routines have reached maturity regarding a variety of practical applications. We have constructed a rich design matrix for the comparison of the Hotelling T\(^{2}\) chart with the copula test by Verdier and the copula test by Vuong, which allows for weighting the observations adaptively. Based on the design matrix, we have conducted a large and computationally intensive simulation study. The results show that the copula test by Verdier performs better than Hotelling T\(^{2}\) in a large variety of out‐of‐control cases, whereas the weighted Vuong scheme often fails to provide an improvement.
A sequentialquadratic Hamiltonian schemefor solving open-loop differential Nash games is proposed and investigated. This method is formulated in the framework of the Pontryagin maximum principle and represents an efficient and robust extension of the successive approximations strategy for solving optimal control problems. Theoretical results are presented that prove the well-posedness of the proposed scheme, and results of numerical experiments are reported that successfully validate its computational performance.
We prove a sharp Bernstein-type inequality for complex polynomials which are positive and satisfy a polynomial growth condition on the positive real axis. This leads to an improved upper estimate in the recent work of Culiuc and Treil (Int. Math. Res. Not. 2019: 3301–3312, 2019) on the weighted martingale Carleson embedding theorem with matrix weights. In the scalar case this new upper bound is optimal.
Nowadays, science, technology, engineering, and mathematics (STEM) play a critical role in a nation’s global competitiveness and prosperity. Thus, there is a need to educate students in these subjects to meet the current and future demands of personal life and society. While applications, especially in science, engineering, and technology, are directly obvious, mathematics underpins the other STEM disciplines. It is recognized that mathematics is the foundation for all other STEM disciplines; the role of mathematics in classrooms is not clear yet. Therefore, the question arises: What is the current role of mathematics in secondary STEM classrooms? To answer this question, we conducted a systematic literature review based on three publication databases (Web of Science, ERIC, and EBSCO Teacher Referral Center). This literature review paper is intended to contribute to the current state of the role of mathematics in STEM education in secondary classrooms. Through the search, starting with 1910 documents, only 14 eligible documents were found. In these, mathematics is often seen as a minor matter and a means to an end in the eyes of science educators. From this, we conclude that the role of mathematics in the STEM classroom should be further strengthened. Overall, the paper highlights a major research gap, and proposes possible initial solutions to close it.
We give a collection of 16 examples which show that compositions \(g\) \(\circ\) \(f\) of well-behaved functions \(f\) and \(g\) can be badly behaved. Remarkably, in 10 of the 16 examples it suffices to take as outer function \(g\) simply a power-type or characteristic function. Such a collection of examples may serve as a source of exercises for a calculus course.
Composite optimization problems, where the sum of a smooth and a merely lower semicontinuous function has to be minimized, are often tackled numerically by means of proximal gradient methods as soon as the lower semicontinuous part of the objective function is of simple enough structure. The available convergence theory associated with these methods (mostly) requires the derivative of the smooth part of the objective function to be (globally) Lipschitz continuous, and this might be a restrictive assumption in some practically relevant scenarios. In this paper, we readdress this classical topic and provide convergence results for the classical (monotone) proximal gradient method and one of its nonmonotone extensions which are applicable in the absence of (strong) Lipschitz assumptions. This is possible since, for the price of forgoing convergence rates, we omit the use of descent-type lemmas in our analysis.
Let (ϕ\(_t\))\(_{t≥0}\) be a semigroup of holomorphic functions in the unit disk \(\mathbb {D}\) and K a compact subset of \(\mathbb {D}\). We investigate the conditions under which the backward orbit of K under the semigroup exists. Subsequently, the geometric characteristics, as well as, potential theoretic quantities for the backward orbit of K are examined. More specifically, results are obtained concerning the asymptotic behavior of its hyperbolic area and diameter, the harmonic measure and the capacity of the condenser that K forms with the unit disk.
Functions of bounded variation are most important in many fields of mathematics. This thesis investigates spaces of functions of bounded variation with one variable of various types, compares them to other classical function spaces and reveals natural “habitats” of BV-functions. New and almost comprehensive results concerning mapping properties like surjectivity and injectivity, several kinds of continuity and compactness of both linear and nonlinear operators between such spaces are given. A new theory about different types of convergence of sequences of such operators is presented in full detail and applied to a new proof for the continuity of the composition operator in the classical BV-space. The abstract results serve as ingredients to solve Hammerstein and Volterra integral equations using fixed point theory. Many criteria guaranteeing the existence and uniqueness of solutions in BV-type spaces are given and later applied to solve boundary and initial value problems in a nonclassical setting.
A big emphasis is put on a clear and detailed discussion. Many pictures and synoptic tables help to visualize and summarize the most important ideas. Over 160 examples and counterexamples illustrate the many abstract results and how delicate some of them are.
We compute genus-0 Belyi maps with prescribed monodromy and strictly verify the computed results. Among the computed examples are almost simple primitive groups that satisfy the rational rigidity criterion yielding polynomials with prescribed Galois groups over Q(t). We also give an explicit version of a theorem of Magaard, which lists all sporadic groups occurring as composition factors of monodromy groups of rational functions.
A reformulation of cardinality-constrained optimization problems into continuous nonlinear optimization problems with an orthogonality-type constraint has gained some popularity during the last few years. Due to the special structure of the constraints, the reformulation violates many standard assumptions and therefore is often solved using specialized algorithms. In contrast to this, we investigate the viability of using a standard safeguarded multiplier penalty method without any problem-tailored modifications to solve the reformulated problem. We prove global convergence towards an (essentially strongly) stationary point under a suitable problem-tailored quasinormality constraint qualification. Numerical experiments illustrating the performance of the method in comparison to regularization-based approaches are provided.
Chemotaxis describes the movement of an organism, such as single or multi-cellular organisms and bacteria, in response to a chemical stimulus. Two widely used models to describe the phenomenon are the celebrated Keller–Segel equation and a chemotaxis kinetic equation. These two equations describe the organism's movement at the macro- and mesoscopic level, respectively, and are asymptotically equivalent in the parabolic regime. The way in which the organism responds to a chemical stimulus is embedded in the diffusion/advection coefficients of the Keller–Segel equation or the turning kernel of the chemotaxis kinetic equation. Experiments are conducted to measure the time dynamics of the organisms' population level movement when reacting to certain stimulation. From this, one infers the chemotaxis response, which constitutes an inverse problem. In this paper, we discuss the relation between both the macro- and mesoscopic inverse problems, each of which is associated with two different forward models. The discussion is presented in the Bayesian framework, where the posterior distribution of the turning kernel of the organism population is sought. We prove the asymptotic equivalence of the two posterior distributions.
In this paper, we prove an asymptotic formula for the sum of the values of the periodic zeta-function at the nontrivial zeros of the Riemann zeta-function (up to some height) which are symmetrical on the real line and the critical line. This is an extension of the previous results due to Garunkštis, Kalpokas, and, more recently, Sowa. Whereas Sowa's approach was assuming the yet unproved Riemann hypothesis, our result holds unconditionally.
A basic mental model (BMM—in German ‘Grundvorstellung’) of a mathematical concept is a content-related interpretation that gives meaning to this concept. This paper defines normative and individual BMMs and concretizes them using the integral as an example. Four BMMs are developed about the concept of definite integral, sometimes used in specific teaching approaches: the BMMs of area, reconstruction, average, and accumulation. Based on theoretical work, in this paper we ask how these BMMs could be identified empirically. A test instrument was developed, piloted, validated and applied with 428 students in first-year mathematics courses. The test results show that the four normative BMMs of the integral can be detected and separated empirically. Moreover, the results allow a comparison of the existing individual BMMs and the requested normative BMMs. Consequences for future developments are discussed.
We derive a multi-species BGK model with velocity-dependent collision frequency for a non-reactive, multi-component gas mixture. The model is derived by minimizing a weighted entropy under the constraint that the number of particles of each species, total momentum, and total energy are conserved. We prove that this minimization problem admits a unique solution for very general collision frequencies. Moreover, we prove that the model satisfies an H-Theorem and characterize the form of equilibrium.
Sequential optimality conditions for cardinality-constrained optimization problems with applications
(2021)
Recently, a new approach to tackle cardinality-constrained optimization problems based on a continuous reformulation of the problem was proposed. Following this approach, we derive a problem-tailored sequential optimality condition, which is satisfied at every local minimizer without requiring any constraint qualification. We relate this condition to an existing M-type stationary concept by introducing a weak sequential constraint qualification based on a cone-continuity property. Finally, we present two algorithmic applications: We improve existing results for a known regularization method by proving that it generates limit points satisfying the aforementioned optimality conditions even if the subproblems are only solved inexactly. And we show that, under a suitable Kurdyka–Łojasiewicz-type assumption, any limit point of a standard (safeguarded) multiplier penalty method applied directly to the reformulated problem also satisfies the optimality condition. These results are stronger than corresponding ones known for the related class of mathematical programs with complementarity constraints.
Fluids in Gravitational Fields – Well-Balanced Modifications for Astrophysical Finite-Volume Codes
(2021)
Stellar structure can -- in good approximation -- be described as a hydrostatic state, which which arises due to a balance between gravitational force and pressure gradient. Hydrostatic states are static solutions of the full compressible Euler system with gravitational source term, which can be used to model the stellar interior. In order to carry out simulations of dynamical processes occurring in stars, it is vital for the numerical method to accurately maintain the hydrostatic state over a long time period. In this thesis we present different methods to modify astrophysical finite volume codes in order to make them \emph{well-balanced}, preventing them from introducing significant discretization errors close to hydrostatic states. Our well-balanced modifications are constructed so that they can meet the requirements for methods applied in the astrophysical context: They can well-balance arbitrary hydrostatic states with any equation of state that is applied to model thermodynamical relations and they are simple to implement in existing astrophysical finite volume codes. One of our well-balanced modifications follows given solutions exactly and can be applied on any grid geometry. The other methods we introduce, which do no require any a priori knowledge, balance local high order approximations of arbitrary hydrostatic states on a Cartesian grid. All of our modifications allow for high order accuracy of the method. The improved accuracy close to hydrostatic states is verified in various numerical experiments.
In the present thesis we investigate algebraic and arithmetic properties of graph spectra. In particular, we study the algebraic degree of a graph, that is the dimension of the splitting field of the characteristic polynomial of the associated adjacency matrix over the rationals, and examine the question whether there is a relation between the algebraic degree of a graph and its structural properties. This generalizes the yet open question ``Which graphs have integral spectra?'' stated by Harary and Schwenk in 1974.
We provide an overview of graph products since they are useful to study graph spectra and, in particular, to construct families of integral graphs. Moreover, we present a relation between the diameter, the maximum vertex degree and the algebraic degree of a graph, and construct a potential family of graphs of maximum algebraic degree.
Furthermore, we determine precisely the algebraic degree of circulant graphs and find new criteria for isospectrality of circulant graphs. Moreover, we solve the inverse Galois problem for circulant graphs showing that every finite abelian extension of the rationals is the splitting field of some circulant graph. Those results generalize a theorem of So who characterized all integral circulant graphs. For our proofs we exploit the theory of Schur rings which was already used in order to solve the isomorphism problem for circulant graphs.
Besides that, we study spectra of zero-divisor graphs over finite commutative rings.
Given a ring \(R\), the zero-divisor graph over \(R\) is defined as the graph with vertex set being the set of non-zero zero-divisors of \(R\) where two vertices \(x,y\) are adjacent if and only if \(xy=0\). We investigate relations between the eigenvalues of a zero-divisor graph, its structural properties and the algebraic properties of the respective ring.
This thesis is concerned with applying the total variation (TV) regularizer to surfaces and different types of shape optimization problems. The resulting problems are challenging since they suffer from the non-differentiability of the TV-seminorm, but unlike most other priors it favors piecewise constant solutions, which results in piecewise flat geometries for shape optimization problems.The first part of this thesis deals with an analogue of the TV image reconstruction approach [Rudin, Osher, Fatemi (Physica D, 1992)] for images on smooth surfaces. A rigorous analytical framework is developed for this model and its Fenchel predual, which is a quadratic optimization problem with pointwise inequality constraints on the surface. A function space interior point method is proposed to solve it. Afterwards, a discrete variant (DTV) based on a nodal quadrature formula is defined for piecewise polynomial, globally discontinuous and continuous finite element functions on triangulated surface meshes. DTV has favorable properties, which include a convenient dual representation. Next, an analogue of the total variation prior for the normal vector field along the boundary of smooth shapes in 3D is introduced. Its analysis is based on a differential geometric setting in which the unit normal vector is viewed as an element of the two-dimensional sphere manifold. Shape calculus is used to characterize the relevant derivatives and an variant of the split Bregman method for manifold valued functions is proposed. This is followed by an extension of the total variation prior for the normal vector field for piecewise flat surfaces and the previous variant of split Bregman method is adapted. Numerical experiments confirm that the new prior favours polyhedral shapes.
Theoretical and numerical investigation of optimal control problems governed by kinetic models
(2021)
This thesis is devoted to the numerical and theoretical analysis of ensemble optimal control problems governed by kinetic models. The formulation and study of these problems have been put forward in recent years by R.W. Brockett with the motivation that ensemble control may provide a more general and robust control framework for dynamical systems. Following this formulation, a Liouville (or continuity) equation with an unbounded drift function is considered together with a class of cost functionals that include tracking of ensembles of trajectories of dynamical systems and different control costs. Specifically, $L^2$, $H^1$ and $L^1$ control costs are taken into account which leads to non--smooth optimization problems. For the theoretical investigation of the resulting optimal control problems, a well--posedness theory in weighted Sobolev spaces is presented for Liouville and related transport equations. Specifically, existence and uniqueness results for these equations and energy estimates in suitable norms are provided; in particular norms in weighted Sobolev spaces. Then, non--smooth optimal control problems governed by the Liouville equation are formulated with a control mechanism in the drift function. Further, box--constraints on the control are imposed. The control--to--state map is introduced, that associates to any control the unique solution of the corresponding Liouville equation. Important properties of this map are investigated, specifically, that it is well--defined, continuous and Frechet differentiable. Using the first two properties, the existence of solutions to the optimal control problems is shown. While proving the differentiability, a loss of regularity is encountered, that is natural to hyperbolic equations. This leads to the need of the investigation of the control--to--state map in the topology of weighted Sobolev spaces. Exploiting the Frechet differentiability, it is possible to characterize solutions to the optimal control problem as solutions to an optimality system. This system consists of the Liouville equation, its optimization adjoint in the form of a transport equation, and a gradient inequality. Numerical methodologies for solving Liouville and transport equations are presented that are based on a non--smooth Lagrange optimization framework. For this purpose, approximation and solution schemes for such equations are developed and analyzed. For the approximation of the Liouville model and its optimization adjoint, a combination of a Kurganov--Tadmor method, a Runge--Kutta scheme, and a Strang splitting method are discussed. Stability and second--order accuracy of these resulting schemes are proven in the discrete $L^1$ norm. In addition, conservation of mass and positivity preservation are confirmed for the solution method of the Liouville model. As numerical optimization strategy, an adapted Krylow--Newton method is applied. Since the control is considered to be an element of $H^1$ and to obey certain box--constraints, a method for calculating a $H^1$ projection is presented. Since the optimal control problem is non-smooth, a semi-smooth adaption of Newton's method is taken into account. Results of numerical experiments are presented that successfully validate the proposed deterministic framework. After the discussion of deterministic schemes, the linear space--homogeneous Keilson--Storer master equation is investigated. This equation was originally developed for the modelling of Brownian motion of particles immersed in a fluid and is a representative model of the class of linear Boltzmann equations. The well--posedness of the Keilson--Storer master equation is investigated and energy estimates in different topologies are derived. To solve this equation numerically, Monte Carlo methods are considered. Such methods take advantage of the kinetic formulation of the Liouville equation and directly implement the behaviour of the system of particles under consideration. This includes the probabilistic behaviour of the collisions between particles. Optimal control problems are formulated with an objective that is constituted of certain expected values in velocity space and the $L^2$ and $H^1$ costs of the control. The problems are governed by the Keilson--Storer master equation and the control mechanism is considered to be within the collision kernel. The objective of the optimal control of this model is to drive an ensemble of particles to acquire a desired mean velocity and to achieve a desired final velocity configuration. Existence of solutions of the optimal control problem is proven and a Keilson--Storer optimality system characterizing the solution of the proposed optimal control problem is obtained. The optimality system is used to construct a gradient--based optimization strategy in the framework of Monte--Carlo methods. This task requires to accommodate the resulting adjoint Keilson--Storer model in a form that is consistent with the kinetic formulation. For this reason, we derive an adjoint Keilson--Storer collision kernel and an additional source term. A similar approach is presented in the case of a linear space--inhomogeneous kinetic model with external forces and with Keilson--Storer collision term. In this framework, a control mechanism in the form of an external space--dependent force is investigated. The purpose of this control is to steer the multi--particle system to follow a desired mean velocity and position and to reach a desired final configuration in phase space. An optimal control problem using the formulation of ensemble controls is stated with an objective that is constituted of expected values in phase space and $H^1$ costs of the control. For solving the optimal control problems, a gradient--based computational strategy in the framework of Monte Carlo methods is developed. Part of this is the denoising of the distribution functions calculated by Monte Carlo algorithms using methods of the realm of partial differential equations. A standalone C++ code is presented that implements the developed non--linear conjugated gradient strategy. Results of numerical experiments confirm the ability of the designed probabilistic control framework to operate as desired. An outlook section about optimal control problems governed by non--linear space--inhomogeneous kinetic models completes this thesis.
This paper is devoted to the numerical analysis of non-smooth ensemble optimal control problems governed by the Liouville (continuity) equation that have been originally proposed by R.W. Brockett with the purpose of determining an efficient and robust control strategy for dynamical systems. A numerical methodology for solving these problems is presented that is based on a non-smooth Lagrange optimization framework where the optimal controls are characterized as solutions to the related optimality systems. For this purpose, approximation and solution schemes are developed and analysed. Specifically, for the approximation of the Liouville model and its optimization adjoint, a combination of a Kurganov–Tadmor method, a Runge–Kutta scheme, and a Strang splitting method are discussed. The resulting optimality system is solved by a projected semi-smooth Krylov–Newton method. Results of numerical experiments are presented that successfully validate the proposed framework.
This thesis aims at providing efficient and side-channel protected implementations of isogeny-based primitives, and at their application in threshold protocols. It is based on a sequence of academic papers.
Chapter 3 reviews the original variable-time implementation of CSIDH and introduces several optimizations, e.g. a significant improvement of isogeny computations by using both Montgomery and Edwards curves. In total, our improvements yield a speedup of 25% compared to the original implementation.
Chapter 4 presents the first practical constant-time implementation of CSIDH. We describe how variable-time implementations of CSIDH leak information on private keys, and describe ways to mitigate this. Further, we present several techniques to speed up the implementation. In total, our constant-time implementation achieves a rather small slowdown by a factor of 3.03.
Chapter 5 reviews practical fault injection attacks on CSIDH and presents countermeasures. We evaluate different attack models theoretically and practically, using low-budget equipment. Moreover, we present countermeasures that mitigate the proposed fault injection attacks, only leading to a small performance overhead of 7%.
Chapter 6 initiates the study of threshold schemes based on the Hard Homogeneous Spaces (HHS) framework of Couveignes. Using the HHS equivalent of Shamir’s secret sharing in the exponents, we adapt isogeny based schemes to the threshold setting. In particular, we present threshold versions of the CSIDH public key encryption and the CSI-FiSh signature scheme.
Chapter 7 gives a sieving algorithm for finding pairs of consecutive smooth numbers that utilizes solutions to the Prouhet-Tarry-Escott (PTE) problem. Recent compact isogeny-based protocols, namely B-SIDH and SQISign, both require large primes that lie between two smooth integers. Finding such a prime can be seen as a special case of finding twin smooth integers under the additional stipulation that their sum is a prime.
This thesis is devoted to a theoretical and numerical investigation of methods to solve open-loop non zero-sum differential Nash games. These problems arise in many applications, e.g., biology, economics, physics, where competition between different agents appears. In this case, the goal of each agent is in contrast with those of the others, and a competition game can be interpreted as a coupled optimization problem for which, in general, an optimal solution does not exist. In fact, an optimal strategy for one player may be unsatisfactory for the others. For this reason, a solution of a game is sought as an equilibrium and among the solutions concepts proposed in the literature, that of Nash equilibrium (NE) is the focus of this thesis. The building blocks of the resulting differential Nash games are a dynamical model with different control functions associated with different players that pursue non-cooperative objectives. In particular, the aim of this thesis is on differential models having linear or bilinear state-strategy structures. In this framework, in the first chapter, some well-known results are recalled, especially for non-cooperative linear-quadratic differential Nash games. Then, a bilinear Nash game is formulated and analysed. The main achievement in this chapter is Theorem 1.4.2 concerning existence of Nash equilibria for non-cooperative differential bilinear games. This result is obtained assuming a sufficiently small time horizon T, and an estimate of T is provided in Lemma 1.4.8 using specific properties of the regularized Nikaido-Isoda function. In Chapter 2, in order to solve a bilinear Nash game, a semi-smooth Newton (SSN) scheme combined with a relaxation method is investigated, where the choice of a SSN scheme is motivated by the presence of constraints on the players’ actions that make the problem non-smooth. The resulting method is proved to be locally convergent in Theorem 2.1, and an estimate on the relaxation parameter is also obtained that relates the relaxation factor to the time horizon of a Nash equilibrium and to the other parameters of the game. For the bilinear Nash game, a Nash bargaining problem is also introduced and discussed, aiming at determining an improvement of all players’ objectives with respect to the Nash equilibrium. A characterization of a bargaining solution is given in Theorem 2.2.1 and a numerical scheme based on this result is presented that allows to compute this solution on the Pareto frontier. Results of numerical experiments based on a quantum model of two spin-particles and on a population dynamics model with two competing species are presented that successfully validate the proposed algorithms. In Chapter 3 a functional formulation of the classical homicidal chauffeur (HC) Nash game is introduced and a new numerical framework for its solution in a time-optimal formulation is discussed. This methodology combines a Hamiltonian based scheme, with proximal penalty to determine the time horizon where the game takes place, with a Lagrangian optimal control approach and relaxation to solve the Nash game at a fixed end-time. The resulting numerical optimization scheme has a bilevel structure, which aims at decoupling the computation of the end-time from the solution of the pursuit-evader game. Several numerical experiments are performed to show the ability of the proposed algorithm to solve the HC game. Focusing on the case where a collision may occur, the time for this event is determined. The last part of this thesis deals with the analysis of a novel sequential quadratic Hamiltonian (SQH) scheme for solving open-loop differential Nash games. This method is formulated in the framework of Pontryagin’s maximum principle and represents an efficient and robust extension of the successive approximations strategy in the realm of Nash games. In the SQH method, the Hamilton-Pontryagin functions are augmented by a quadratic penalty term and the Nikaido-Isoda function is used as a selection criterion. Based on this fact, the key idea of this SQH scheme is that the PMP characterization of Nash games leads to a finite-dimensional Nash game for any fixed time. A class of problems for which this finite-dimensional game admits a unique solution is identified and for this class of games theoretical results are presented that prove the well-posedness of the proposed scheme. In particular, Proposition 4.2.1 is proved to show that the selection criterion on the Nikaido-Isoda function is fulfilled. A comparison of the computational performances of the SQH scheme and the SSN-relaxation method previously discussed is shown. Applications to linear-quadratic Nash games and variants with control constraints, weighted L1 costs of the players’ actions and tracking objectives are presented that corroborate the theoretical statements.
The present thesis deals with optimisation problems with sparsity terms, either in the constraints which lead to cardinality-constrained problems or in the objective function which in turn lead to sparse optimisation problems. One of the primary aims of this work is to extend the so-called sequential optimality conditions to these two classes of problems. In recent years sequential optimality conditions have become increasingly popular in the realm of standard nonlinear programming. In contrast to the more well-known Karush-Kuhn-Tucker condition, they are genuine optimality conditions in the sense that every local minimiser satisfies these conditions without any further assumption. Lately they have also been extended to mathematical programmes with complementarity constraints. At around the same time it was also shown that optimisation problems with sparsity terms can be reformulated into problems which possess similar structures to mathematical programmes with complementarity constraints. These recent developments have become the impetus of the present work. But rather than working with the aforementioned reformulations which involve an artifical variable we shall first directly look at the problems themselves and derive sequential optimality conditions which are independent of any artificial variable. Afterwards we shall derive the weakest constraint qualifications associated with these conditions which relate them to the Karush-Kuhn-Tucker-type conditions. Another equally important aim of this work is to then consider the practicability of the derived sequential optimality conditions. The previously mentioned reformulations open up the possibilities to adapt methods which have been proven successful to handle mathematical programmes with complementarity constraints. We will show that the safeguarded augmented Lagrangian method and some regularisation methods may generate a point satisfying the derived conditions.
Risk measures are commonly used to prepare for a prospective occurrence of an adverse event. If we are concerned with discrete risk phenomena such as counts of natural disasters, counts of infections by a serious disease, or counts of certain economic events, then the required risk forecasts are to be computed for an underlying count process. In practice, however, the discrete nature of count data is sometimes ignored and risk forecasts are calculated based on Gaussian time series models. But even if methods from count time series analysis are used in an adequate manner, the performance of risk forecasting is affected by estimation uncertainty as well as certain discreteness phenomena. To get a thorough overview of the aforementioned issues in risk forecasting of count processes, a comprehensive simulation study was done considering a broad variety of risk measures and count time series models. It becomes clear that Gaussian approximate risk forecasts substantially distort risk assessment and, thus, should be avoided. In order to account for the apparent estimation uncertainty in risk forecasting, we use bootstrap approaches for count time series. The relevance and the application of the proposed approaches are illustrated by real data examples about counts of storm surges and counts of financial transactions.
We consider the Bathnagar–Gross–Krook (BGK) model, an approximation of the Boltzmann equation, describing the time evolution of a single momoatomic rarefied gas and satisfying the same two main properties (conservation properties and entropy inequality). However, in practical applications, one often has to deal with two additional physical issues. First, a gas often does not consist of only one species, but it consists of a mixture of different species. Second, the particles can store energy not only in translational degrees of freedom but also in internal degrees of freedom such as rotations or vibrations (polyatomic molecules). Therefore, here, we will present recent BGK models for gas mixtures for mono- and polyatomic particles and the existing mathematical theory for these models.
The bounded input bounded output (BIBO) stability for a nonlinear Caputo fractional system with time‐varying bounded delay and nonlinear output is studied. Utilizing the Razumikhin method, Lyapunov functions and appropriate fractional derivatives of Lyapunov functions some new bounded input bounded output stability criteria are derived. Also, explicit and independent on the initial time bounds of the output are provided. Uniform BIBO stability and uniform BIBO stability with input threshold are studied. A numerical simulation is carried out to show the system's dynamic response, and demonstrate the effectiveness of our theoretical results.