510 Mathematik
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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.
Salience bias and overwork
(2022)
In this study, we enrich a standard principal–agent model with hidden action by introducing salience-biased perception on the agent's side. The agent's misguided focus on salient payoffs, which leads the agent's and the principal's probability assessments to diverge, has two effects: First, the agent focuses too much on obtaining a bonus, which facilitates incentive provision. Second, the principal may exploit the diverging probability assessments to relax participation. We show that salience bias can reverse the nature of the inefficiency arising from moral hazard; i.e., the principal does not necessarily provide insufficient incentives that result in inefficiently low effort but instead may well provide excessive incentives that result in inefficiently high effort.
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.
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.
This thesis is about composite-based structural equation modeling. Structural equation modeling in general can be used to model both theoretical concepts and their relations to one another. In traditional factor-based structural equation modeling, these theoretical concepts are modeled as common factors, i.e., as latent variables which explain the covariance structure of their observed variables. In contrast, in composite-based structural equation modeling, the theoretical concepts can be modeled both as common factors and as composites, i.e., as linear combinations of observed variables that convey all the information between their observed variables and all other variables in the model. This thesis presents some methodological advancements in the field of composite-based structural equation modeling. In all, this thesis is made up of seven chapters. Chapter 1 provides an overview of the underlying model, as well as explicating the meaning of the term composite-based structural equation modeling. Chapter 2 gives guidelines on how to perform Monte Carlo simulations in the statistic software R using the package “cSEM” with various estimators in the context of composite-based structural equation modeling. These guidelines are illustrated by an example simulation study that investigates the finite sample behavior of partial least squares path modeling (PLS-PM) and consistent partial least squares (PLSc) estimates, particularly regarding the consequences of sample correlations between measurement errors on statistical inference. The third Chapter presents estimators of composite-based structural equation modeling that are robust in responding to outlier distortion. For this purpose, estimators of composite-based structural equation modeling, PLS-PM and PLSc, are adapted. Unlike the original estimators, these adjustments can avoid distortion that could arise from random outliers in samples, as is demonstrated through a simulation study. Chapter 4 presents an approach to performing predictions based on models estimated with ordinal partial least squares and ordinal consistent partial least squares. Here, the observed variables lie on an ordinal categorical scale which is explicitly taken into account in both estimation and prediction. The prediction performance is evaluated by means of a simulation study. In addition, the chapter gives guidelines on how to perform such predictions using the R package “cSEM”. This is demonstrated by means of an empirical example. Chapter 5 introduces confirmatory composite analysis (CCA) for research in “Human Development”. Using CCA, composite models can be estimated and assessed. This chapter uses the Henseler-Ogasawara specification for composite models, allowing, for example, the maximum likelihood method to be used for parameter estimation. Since the maximum likelihood estimator based on the Henseler-Ogasawara specification has limitations, Chapter 6 presents another specification of the composite model by means of which composite models can be estimated with the maximum likelihood method. The results of this maximum likelihood estimator are compared with those of PLS-PM, thus showing that this maximum likelihood estimator gives valid results even in finite samples. The last chapter, Chapter 7, gives an overview of the development and different strands of composite-based structural equation modeling. Additionally, here I examine the contribution the previous chapters make to the wider distribution of composite-based structural equation modeling.
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.