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Mathematical arguments are central components of mathematics and play a role in certain types of modelling of potential mathematical giftedness. However, particular characteristics of arguments are interpreted differently in the context of mathematical giftedness. Some models of giftedness see no connection, whereas other models consider the formulation of complete and plausible arguments as a partial aspect of giftedness. Furthermore, longitudinal changes in argumentation characteristics remain open. This leads to the research focus of this article, which is to identify and describe the changes of argumentation products in potentially mathematically gifted children over a longer period. For this purpose, the argumentation products of children from third to sixth grade are collected throughout a longitudinal study and examined with respect to the use of examples and generalizations. The analysis of all products results in six different types of changes in the characteristics of the argumentation products identified over the survey period and case studies are used to illustrate student use of examples and generalizations of these types. This not only reveals the general importance of the use of examples in arguments. For one type, an increase in generalized arguments can be observed over the survey period. The article will conclude with a discussion of the role of argument characteristics in describing potential mathematical giftedness.
We deal with the shape reconstruction of inclusions in elastic bodies. For solving this inverse problem in practice, data fitting functionals are used. Those work better than the rigorous monotonicity methods from Eberle and Harrach (Inverse Probl 37(4):045006, 2021), but have no rigorously proven convergence theory. Therefore we show how the monotonicity methods can be converted into a regularization method for a data-fitting functional without losing the convergence properties of the monotonicity methods. This is a great advantage and a significant improvement over standard regularization techniques. In more detail, we introduce constraints on the minimization problem of the residual based on the monotonicity methods and prove the existence and uniqueness of a minimizer as well as the convergence of the method for noisy data. In addition, we compare numerical reconstructions of inclusions based on the monotonicity-based regularization with a standard approach (one-step linearization with Tikhonov-like regularization), which also shows the robustness of our method regarding noise in practice.
Uniqueness and Lipschitz stability in electrical impedance tomography with finitely many electrodes
(2019)
For the linearized reconstruction problem in electrical impedance tomography with the complete electrode model, Lechleiter and Rieder (2008 Inverse Problems 24 065009) have shown that a piecewise polynomial conductivity on a fixed partition is uniquely determined if enough electrodes are being used. We extend their result to the full non-linear case and show that measurements on a sufficiently high number of electrodes uniquely determine a conductivity in any finite-dimensional subset of piecewise-analytic functions. We also prove Lipschitz stability, and derive analogue results for the continuum model, where finitely many measurements determine a finite-dimensional Galerkin projection of the Neumann-to-Dirichlet operator on a boundary part.
In this short note, we investigate simultaneous recovery inverse problems for semilinear elliptic equations with partial data. The main technique is based on higher order linearization and monotonicity approaches. With these methods at hand, we can determine the diffusion, cavity and coefficients simultaneously by knowing the corresponding localized Dirichlet-Neumann operators.
We derive a simple criterion that ensures uniqueness, Lipschitz stability and global convergence of Newton’s method for the finite dimensional zero-finding problem of a continuously differentiable, pointwise convex and monotonic function. Our criterion merely requires to evaluate the directional derivative of the forward function at finitely many evaluation points and for finitely many directions. We then demonstrate that this result can be used to prove uniqueness, stability and global convergence for an inverse coefficient problem with finitely many measurements. We consider the problem of determining an unknown inverse Robin transmission coefficient in an elliptic PDE. Using a relation to monotonicity and localized potentials techniques, we show that a piecewise-constant coefficient on an a-priori known partition with a-priori known bounds is uniquely determined by finitely many boundary measurements and that it can be uniquely and stably reconstructed by a globally convergent Newton iteration. We derive a constructive method to identify these boundary measurements, calculate the stability constant and give a numerical example.
Several novel imaging and non-destructive testing technologies are based on reconstructing the spatially dependent coefficient in an elliptic partial differential equation from measurements of its solution(s). In practical applications, the unknown coefficient is often assumed to be piecewise constant on a given pixel partition (corresponding to the desired resolution), and only finitely many measurement can be made. This leads to the problem of inverting a finite-dimensional non-linear forward operator F: D(F)⊆Rn→Rm , where evaluating ℱ requires one or several PDE solutions.
Numerical inversion methods require the implementation of this forward operator and its Jacobian. We show how to efficiently implement both using a standard FEM package and prove convergence of the FEM approximations against their true-solution counterparts. We present simple example codes for Comsol with the Matlab Livelink package, and numerically demonstrate the challenges that arise from non-uniqueness, non-linearity and instability issues. We also discuss monotonicity and convexity properties of the forward operator that arise for symmetric measurement settings.
This text assumes the reader to have a basic knowledge on Finite Element Methods, including the variational formulation of elliptic PDEs, the Lax-Milgram-theorem, and the Céa-Lemma. Section 3 also assumes that the reader is familiar with the concept of Fréchet differentiability.
We show that the metrisability of an oriented projective surface is equivalent to the existence of pseudo-holomorphic curves. A projective structure p and a volume form σ on an oriented surface M equip the total space of a certain disk bundle Z→M with a pair (Jp,Jp,σ) of almost complex structures. A conformal structure on M corresponds to a section of Z→M and p is metrisable by the metric g if and only if [g]:M→Z is a pseudo-holomorphic curve with respect to Jp and Jp,dAg.
In this article we use techniques from tropical and logarithmic geometry to construct a non-Archimedean analogue of Teichmüller space T¯g whose points are pairs consisting of a stable projective curve over a non-Archimedean field and a Teichmüller marking of the topological fundamental group of its Berkovich analytification. This construction is closely related to and inspired by the classical construction of a non-Archimedean Schottky space for Mumford curves by Gerritzen and Herrlich. We argue that the skeleton of non-Archimedean Teichmüller space is precisely the tropical Teichmüller space introduced by Chan–Melo–Viviani as a simplicial completion of Culler–Vogtmann Outer space. As a consequence, Outer space turns out to be a strong deformation retract of the locus of smooth Mumford curves in T¯g.
We study the asymptotics of Dirichlet eigenvalues and eigenfunctions of the fractional Laplacian (−Δ)s in bounded open Lipschitz sets in the small order limit s→0+. While it is easy to see that all eigenvalues converge to 1 as s→0+, we show that the first order correction in these asymptotics is given by the eigenvalues of the logarithmic Laplacian operator, i.e., the singular integral operator with Fourier symbol 2log|ξ|. By this we generalize a result of Chen and the third author which was restricted to the principal eigenvalue. Moreover, we show that L2-normalized Dirichlet eigenfunctions of (−Δ)s corresponding to the k-th eigenvalue are uniformly bounded and converge to the set of L2-normalized eigenfunctions of the logarithmic Laplacian. In order to derive these spectral asymptotics, we establish new uniform regularity and boundary decay estimates for Dirichlet eigenfunctions for the fractional Laplacian. As a byproduct, we also obtain corresponding regularity properties of eigenfunctions of the logarithmic Laplacian.
In the model of randomly perturbed graphs we consider the union of a deterministic graph G with minimum degree αn and the binomial random graph G(n, p). This model was introduced by Bohman, Frieze, and Martin and for Hamilton cycles their result bridges the gap between Dirac’s theorem and the results by Pósa and Korshunov on the threshold in G(n, p). In this note we extend this result in G ∪G(n, p) to sparser graphs with α = o(1). More precisely, for any ε > 0 and α: N ↦→ (0, 1) we show that a.a.s. G ∪ G(n, β/n) is Hamiltonian, where β = −(6 + ε) log(α). If α > 0 is a fixed constant this gives the aforementioned result by Bohman, Frieze, and Martin and if α = O(1/n) the random part G(n, p) is sufficient for a Hamilton cycle. We also discuss embeddings of bounded degree trees and other spanning structures in this model, which lead to interesting questions on almost spanning embeddings into G(n, p).