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The construction of hydroelectric power plants typically requires the execution of deep construction pits close to or even inside of rivers. The paper describes geotechnical challenges and solutions of such projects on the basis of the expansion of the water power plant in Kirchbichl, Austria. This power plant is in operation since the late 1940s and was upgraded with an additional power house as well as a new floodwater relief channel. Therefore two approx. 100m long construction pits were built bordering the existing power plant. Differences in water levels of up to 20m in combination with high permeable soil layers required the execution of water tight and stiff retaining walls and high as well as deep seated sealing slabs. Especially the areas next to the existing structure were critical for the success, as the more than 70 years old building is very sensible to deformations. Moreover, the execution of piles and jet grouting columns had to pass through the former excavation, which was backfilled using very heterogenic material. Thus, various geotechnical products were applied, including jet grouting, bored piles, micro piles and strand anchors. The paper points out the key challenges of such projects, ranging from design stage, execution on site till quality control and documentation. It also addresses the risks faced during the various project phases and describes proper approaches how to mitigate them.

The growth of tropical geometry has generated significant interest in the tropical semiring in the past decade. However, there are other semirings in tropical algebra that provide more information, such as the symmetrized ( max ⁡ , + ) , Izhakian's extended and Izhakian–Rowen's supertropical semirings. In this paper we identify in which of these upper-bound semirings we can express symmetric polynomials in terms of elementary ones. We show that in the case of idempotent semirings we can do this precisely when the Frobenius property is satisfied, that in the case of supertropical semirings this is always possible, and that in non-trivial symmetrized semirings this is never possible. Our results allow us to determine the tropical algebra semirings where an analogue of the Fundamental Theorem of Symmetric Polynomials holds and to what extent.

Real data is often given as a point cloud, i.e. a finite set of points with pairwise distances between them. An important problem is to detect the topological shape of data – for example, to approximate a point cloud by a low-dimensional non-linear subspace such as an embedded graph or a simplicial complex. Classical clustering methods and principal component analysis work well when data points split into good clusters or lie near linear subspaces of a Euclidean space. Methods from topological data analysis in general metric spaces detect more complicated patterns such as holes and voids that persist for a large interval in a 1-parameter family of shapes associated to a cloud. These features can be visualized in the form of a 1-dimensional homologically persistent skeleton, which optimally extends a minimum spanning tree of a point cloud to a graph with cycles. We generalize this skeleton to higher dimensions and prove its optimality among all complexes that preserve topological features of data at any scale.

We investigate the relationship between the synthetic approach to topology, in which every set is equipped with an intrinsic topology, and constructive theory of metric spaces. We relate the synthetic notion of compactness of Cantor space to Brouwer’s Fan Principle. We show that the intrinsic and metric topologies of complete separable metric spaces coincide if they do so for Baire space. In Russian Constructivism the match between synthetic and metric topology breaks down, as even a very simple complete totally bounded space fails to be compact, and its topology is strictly finer than the metric topology. In contrast, in Brouwer’s intuitionism synthetic and metric notions of topology and compactness agree.

A construction of the Urysohn's universal metric space is given in the context of constructive theory of metric spaces. The space is universal in the sense that every separable metric space isometrically embeds into it. Moreover, every isometry between two finite subspaces extends to total isometry, and this determines the Urysohn space uniquely up to isometric isomorphism.