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Institute
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.
The novel coronavirus (SARS-CoV-2), identified in China at the end of December 2019 and causing the disease COVID-19, has meanwhile led to outbreaks all over the globe with about 2.2 million confirmed cases and more than 150,000 deaths as of April 17, 2020 [37]. In view of most recent information on testing activity [32], we present here an update of our initial work [4]. In this work, mathematical models have been developed to study the spread of COVID-19 among the population in Germany and to asses the impact of non-pharmaceutical interventions. Systems of differential equations of SEIR type are extended here to account for undetected infections, as well as for stages of infections and age groups. The models are calibrated on data until April 5, data from April 6 to 14 are used for model validation. We simulate different possible strategies for the mitigation of the current outbreak, slowing down the spread of the virus and thus reducing the peak in daily diagnosed cases, the demand for hospitalization or intensive care units admissions, and eventually the number of fatalities. Our results suggest that a partial (and gradual) lifting of introduced control measures could soon be possible if accompanied by further increased testing activity, strict isolation of detected cases and reduced contact to risk groups.
The 𝒮-cone provides a common framework for cones of polynomials or exponen- tial sums which establish non-negativity upon the arithmetic-geometric inequality, in particular for sums of non-negative circuit polynomials (SONC) or sums of arithmetic- geometric exponentials (SAGE). In this paper, we study the S-cone and its dual from the viewpoint of second-order representability. Extending results of Averkov and of Wang and Magron on the primal SONC cone, we provide explicit generalized second- order descriptions for rational S-cones and their duals.
We show how nonlocal boundary conditions of Robin type can be encoded in the pointwise expression of the fractional operator. Notably, the fractional Laplacian of functions satisfying homogeneous nonlocal Neumann conditions can be expressed as a regional operator with a kernel having logarithmic behaviour at the boundary.
This article deals with the solution of linear ill-posed equations in Hilbert spaces. Often, one only has a corrupted measurement of the right hand side at hand and the Bakushinskii veto tells us, that we are not able to solve the equation if we do not know the noise level. But in applications it is ad hoc unrealistic to know the error of a measurement. In practice, the error of a measurement may often be estimated through averaging of multiple measurements. We integrated that in our anlaysis and obtained convergence to the true solution, with the only assumption that the measurements are unbiased, independent and identically distributed according to an unknown distribution.
Objectives: To analyze the performance of radiological assessment categories and quantitative computational analysis of apparent diffusion coefficient (ADC) maps using variant machine learning algorithms to differentiate clinically significant versus insignificant prostate cancer (PCa). Methods: Retrospectively, 73 patients were included in the study. The patients (mean age, 66.3 ± 7.6 years) were examined with multiparametric MRI (mpMRI) prior to radical prostatectomy (n = 33) or targeted biopsy (n = 40). The index lesion was annotated in MRI ADC and the equivalent histologic slides according to the highest Gleason Grade Group (GrG). Volumes of interest (VOIs) were determined for each lesion and normal-appearing peripheral zone. VOIs were processed by radiomic analysis. For the classification of lesions according to their clinical significance (GrG ≥ 3), principal component (PC) analysis, univariate analysis (UA) with consecutive support vector machines, neural networks, and random forest analysis were performed. Results: PC analysis discriminated between benign and malignant prostate tissue. PC evaluation yielded no stratification of PCa lesions according to their clinical significance, but UA revealed differences in clinical assessment categories and radiomic features. We trained three classification models with fifteen feature subsets. We identified a subset of shape features which improved the diagnostic accuracy of the clinical assessment categories (maximum increase in diagnostic accuracy ΔAUC = + 0.05, p < 0.001) while also identifying combinations of features and models which reduced overall accuracy. Conclusions: The impact of radiomic features to differentiate PCa lesions according to their clinical significance remains controversial. It depends on feature selection and the employed machine learning algorithms. It can result in improvement or reduction of diagnostic performance.
Despite the great importance of the Latin language in the past, there are relatively few resources available today to develop modern NLP tools for this language. Therefore, the EvaLatin Shared Task for Lemmatization and Part-of-Speech (POS) tagging was published in the LT4HALA workshop. In our work, we dealt with the second EvaLatin task, that is, POS tagging. Since most of the available Latin word embeddings were trained on either few or inaccurate data, we trained several embeddings on better data in the first step. Based on these embeddings, we trained several state-of-the-art taggers and used them as input for an ensemble classifier called LSTMVoter. We were able to achieve the best results for both the cross-genre and the cross-time task (90.64% and 87.00%) without using additional annotated data (closed modality). In the meantime, we further improved the system and achieved even better results (96.91% on classical, 90.87% on cross-genre and 87.35% on cross-time).
Are nearby places (e.g., cities) described by related words? In this article, we transfer this research question in the field of lexical encoding of geographic information onto the level of intertextuality. To this end, we explore Volunteered Geographic Information (VGI) to model texts addressing places at the level of cities or regions with the help of so-called topic networks. This is done to examine how language encodes and networks geographic information on the aboutness level of texts. Our hypothesis is that the networked thematizations of places are similar, regardless of their distances and the underlying communities of authors. To investigate this, we introduce Multiplex Topic Networks (MTN), which we automatically derive from Linguistic Multilayer Networks (LMN) as a novel model, especially of thematic networking in text corpora. Our study shows a Zipfian organization of the thematic universe in which geographical places (especially cities) are located in online communication. We interpret this finding in the context of cognitive maps, a notion which we extend by so-called thematic maps. According to our interpretation of this finding, the organization of thematic maps as part of cognitive maps results from a tendency of authors to generate shareable content that ensures the continued existence of the underlying media. We test our hypothesis by example of special wikis and extracts of Wikipedia. In this way, we come to the conclusion that geographical places, whether close to each other or not, are located in neighboring semantic places that span similar subnetworks in the topic universe.
The annotation of texts and other material in the field of digital humanities and Natural Language Processing (NLP) is a common task of research projects. At the same time, the annotation of corpora is certainly the most time- and cost-intensive component in research projects and often requires a high level of expertise according to the research interest. However, for the annotation of texts, a wide range of tools is available, both for automatic and manual annotation. Since the automatic pre-processing methods are not error-free and there is an increasing demand for the generation of training data, also with regard to machine learning, suitable annotation tools are required. This paper defines criteria of flexibility and efficiency of complex annotations for the assessment of existing annotation tools. To extend this list of tools, the paper describes TextAnnotator, a browser-based, multi-annotation system, which has been developed to perform platform-independent multimodal annotations and annotate complex textual structures. The paper illustrates the current state of development of TextAnnotator and demonstrates its ability to evaluate annotation quality (inter-annotator agreement) at runtime. In addition, it will be shown how annotations of different users can be performed simultaneously and collaboratively on the same document from different platforms using UIMA as the basis for annotation.