Refine
Has Fulltext
- yes (18)
Is part of the Bibliography
- yes (18)
Year of publication
- 2022 (18) (remove)
Document Type
- Journal article (12)
- Doctoral Thesis (5)
- Review (1)
Language
- English (18) (remove)
Keywords
- proximal gradient method (2)
- Bayesian inverse problems (1)
- Belyi map (1)
- Bernstein-type inequality (1)
- Bi-Fidelity method (1)
- Black Scholes equation (1)
- Bloom setting (1)
- Carleson embedding theorem (1)
- Epstein zeta-function (1)
- Epstein, Paul (1)
Institute
- Institut für Mathematik (18) (remove)
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.
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.
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.
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.
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.
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.
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.
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.
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.