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We consider a linear ill-posed equation in the Hilbert space setting. Multiple independent unbiased measurements of the right-hand side are available. A natural approach is to take the average of the measurements as an approximation of the right-hand side and to estimate the data error as the inverse of the square root of the number of measurements. We calculate the optimal convergence rate (as the number of measurements tends to infinity) under classical source conditions and introduce a modified discrepancy principle, which asymptotically attains this rate.
We show explicit formulas for the evaluation of (possibly higher-order) fractional Laplacians (-△)ˢ of some functions supported on ellipsoids. In particular, we derive the explicit expression of the torsion function and give examples of s-harmonic functions. As an application, we infer that the weak maximum principle fails in eccentric ellipsoids for s ∈ (1; √3 + 3/2) in any dimension n ≥ 2. We build a counterexample in terms of the torsion function times a polynomial of degree 2. Using point inversion transformations, it follows that a variety of bounded and unbounded domains do not satisfy positivity preserving properties either and we give some examples.
A convex body is unconditional if it is symmetric with respect to reflections in all coordinate hyperplanes. We investigate unconditional lattice polytopes with respect to geometric, combinatorial, and algebraic properties. In particular, we characterize unconditional reflexive polytopes in terms of perfect graphs. As a prime example, we study the signed Birkhoff polytope. Moreover, we derive constructions for Gale-dual pairs of polytopes and we explicitly describe Gröbner bases for unconditional reflexive polytopes coming from partially ordered sets.
Adaptive, synchronous, and mobile online education: developing the ASYMPTOTE learning environment
(2022)
The COVID-19-induced distance education was perceived as highly challenging by teachers and students. A cross-national comparison of five European countries identified several challenges occurred during the distance learning period. On this basis, the article aims to develop a theoretical framework and design requirements for distance and online learning tools. As one example for online learning in mathematics education, the ASYMPTOTE system is introduced. It will be freely available by May 2022. ASYMPTOTE is aimed at the adaptive and synchronous delivery of online education by taking a mobile learning approach. Its core is the so-called digital classroom, which not only allows students to interact with each other or with the teacher but also enables teachers to monitor their students’ work progress in real time. With respect to the theoretical framework, this article analyses to what extent the ASYMPTOTE system meets the requirements of online learning. Overall, the digital classroom can be seen as a promising tool for teachers to carry out appropriate formative assessment and—partly—to maintain personal and content-related interaction at a distance. Moreover, we highlight the availability of this tool. Due to its mobile learning approach, almost all students will be able to participate in lessons conducted with ASYMPTOTE.
FEM–BEM coupling for the thermoelastic wave equation with transparent boundary conditions in 3D
(2022)
We consider the thermoelastic wave equation in three dimensions with transparent boundary conditions on a bounded, not necessarily convex domain. In order to solve this problem numerically, we introduce a coupling of the thermoelastic wave equation in the interior domain with time-dependent boundary integral equations. Here, we want to highlight that this type of problem differs from other wave-type problems that dealt with FEM–BEM coupling so far, e.g., the acoustic as well as the elastic wave equation, since our problem consists of coupled partial differential equations involving a vector-valued displacement field and a scalar-valued temperature field. This constitutes a nontrivial challenge which is solved in this paper. Our main focus is on a coercivity property of a Calderón operator for the thermoelastic wave equation in the Laplace domain, which is valid for all complex frequencies in a half-plane. Combining Laplace transform and energy techniques, this coercivity in the frequency domain is used to prove the stability of a fully discrete numerical method in the time domain. The considered numerical method couples finite elements and the leapfrog time-stepping in the interior with boundary elements and convolution quadrature on the boundary. Finally, we present error estimates for the semi- and full discretization.
The development of epilepsy (epileptogenesis) involves a complex interplay of neuronal and immune processes. Here, we present a first-of-its-kind mathematical model to better understand the relationships among these processes. Our model describes the interaction between neuroinflammation, blood-brain barrier disruption, neuronal loss, circuit remodeling, and seizures. Formulated as a system of nonlinear differential equations, the model reproduces the available data from three animal models. The model successfully describes characteristic features of epileptogenesis such as its paradoxically long timescales (up to decades) despite short and transient injuries or the existence of qualitatively different outcomes for varying injury intensity. In line with the concept of degeneracy, our simulations reveal multiple routes toward epilepsy with neuronal loss as a sufficient but non-necessary component. Finally, we show that our model allows for in silico predictions of therapeutic strategies, revealing injury-specific therapeutic targets and optimal time windows for intervention.
AttendAffectNet-emotion prediction of movie viewers using multimodal fusion with self-attention
(2021)
In this paper, we tackle the problem of predicting the affective responses of movie viewers, based on the content of the movies. Current studies on this topic focus on video representation learning and fusion techniques to combine the extracted features for predicting affect. Yet, these typically, while ignoring the correlation between multiple modality inputs, ignore the correlation between temporal inputs (i.e., sequential features). To explore these correlations, a neural network architecture—namely AttendAffectNet (AAN)—uses the self-attention mechanism for predicting the emotions of movie viewers from different input modalities. Particularly, visual, audio, and text features are considered for predicting emotions (and expressed in terms of valence and arousal). We analyze three variants of our proposed AAN: Feature AAN, Temporal AAN, and Mixed AAN. The Feature AAN applies the self-attention mechanism in an innovative way on the features extracted from the different modalities (including video, audio, and movie subtitles) of a whole movie to, thereby, capture the relationships between them. The Temporal AAN takes the time domain of the movies and the sequential dependency of affective responses into account. In the Temporal AAN, self-attention is applied on the concatenated (multimodal) feature vectors representing different subsequent movie segments. In the Mixed AAN, we combine the strong points of the Feature AAN and the Temporal AAN, by applying self-attention first on vectors of features obtained from different modalities in each movie segment and then on the feature representations of all subsequent (temporal) movie segments. We extensively trained and validated our proposed AAN on both the MediaEval 2016 dataset for the Emotional Impact of Movies Task and the extended COGNIMUSE dataset. Our experiments demonstrate that audio features play a more influential role than those extracted from video and movie subtitles when predicting the emotions of movie viewers on these datasets. The models that use all visual, audio, and text features simultaneously as their inputs performed better than those using features extracted from each modality separately. In addition, the Feature AAN outperformed other AAN variants on the above-mentioned datasets, highlighting the importance of taking different features as context to one another when fusing them. The Feature AAN also performed better than the baseline models when predicting the valence dimension.
Linking mathematics with reality is not new. It is also not new to use outdoor activities to learn mathematics. It seems to be new, to combine such mathematical outdoor activities with mobile technology, like the geocache community which makes use of GPS technology to guide their members to special places and points of interest. The use of mobile technologies to learn at any time and any location is known as “mobile learning”. This type of learning can be seen as an extension of eLearning. Considering the definition of O’Malley one notices that this definition does not exactly match with the idea of the MathCityMap-Project (MCM), because the learning environment in the MCM-Project is predetermined. Combined with the math trail method the project enables mobile learning within math trails with latest technology.In the MCM-Project students experience mathematics at real places and within real situations in out-of-school activities,with help of GPS-enabled smartphones and special math problems. In contrast to the paper versions of math trails we are able to give direct feedback on the solutions by using “mobile devices” such as smartphones or tablets. If the user has difficulties in solving the modeling task, stepped hints can be provided. The teacher is able to use the MCM-Portal to upload tasks developed by himself or by his students and he is also able to build a personal math trail for his students.
The present paper is concerned with the half-space Dirichlet problem [...] where ℝ𝑁+:={𝑥∈ℝ𝑁:𝑥𝑁>0} for some 𝑁≥1 and 𝑝>1, 𝑐>0 are constants. We analyse the existence, non-existence and multiplicity of bounded positive solutions to (𝑃𝑐). We prove that the existence and multiplicity of bounded positive solutions to (𝑃𝑐) depend in a striking way on the value of 𝑐>0 and also on the dimension N. We find an explicit number 𝑐𝑝∈(1,𝑒√), depending only on p, which determines the threshold between existence and non-existence. In particular, in dimensions 𝑁≥2, we prove that, for 0<𝑐<𝑐𝑝, problem (𝑃𝑐) admits infinitely many bounded positive solutions, whereas, for 𝑐>𝑐𝑝, there are no bounded positive solutions to (𝑃𝑐).