A Model of Directed Graph Cofiber

In the homotopy theory of spaces, the image of a continuous map is contractible to a point in its cofiber. This property does not apply when we discretize spaces and continuous maps to directed graphs and their morphisms. In this paper, we give a construction of a cofiber of a directed graph map whose image is contractible in the cofiber. Our work reveals that a category-theoretically correct construction in continuous setup is no longer correct when it is discretized and hence leads to look at canonical constructions in category theory in a different perspective.

Топологическая унифицированная (r,s)-энтропия непрерывных отображений в квазиметрических пространствах

The category of metric spaces is a subcategory of quasi-metric spaces. It is shown that the entropy of a map when symmetric properties is included is greater or equal to the entropy in the case that the symmetric property of the space is not considered. The topological entropy and Shannon entropy have similar properties such as nonnegativity, subadditivity and conditioning reduces entropy. In other words, topological entropy is supposed as the extension of classical entropy in dynamical systems. In the recent decade, different extensions of Shannon entropy have been introduced. One of them which generalizes many classical entropies is unified $(r,s)$-entropy. In this paper, we extend the notion of unified $(r, s)$-entropy for the continuous maps of a quasi-metric space via spanning and separated sets. Moreover, we survey unified $(r, s)$-entropy of a map for two metric spaces that are associated with a given quasi-metric space and compare unified $(r, s)$-entropy of a map of a given quasi-metric space and the maps of its associated metric spaces. Finally we define Tsallis topological entropy for the continuous map on a quasi-metric space via Bowen's definition and analyze some properties such as chain rule.

Requirements for a global lidar system: spaceborne lidar with wall-to-wall coverage

Lidar is the optimum technology for measuring bare-Earth elevation beneath, and the structure of, vegetation. Consequently, airborne laser scanning (ALS) is widely employed for use in a range of applications. However, ALS is not available globally nor frequently updated due to its high cost per unit area. Spaceborne lidar can map globally but energy requirements limit existing spaceborne lidars to sparse sampling missions, unsuitable for many common ALS applications. This paper derives the equations to calculate the coverage a lidar satellite could achieve for a given set of characteristics (released open-source), then uses a cloud map to determine the number of satellites needed to achieve continuous, global coverage within a certain time-frame. Using the characteristics of existing in-orbit technology, a single lidar satellite could have a continuous swath width of 300 m when producing a 30 m resolution map. Consequently, 12 satellites would be needed to produce a continuous map every 5 years, increasing to 418 satellites for 5 m resolution. Building 12 of the currently in-orbit lidar systems is likely to be prohibitively expensive and so the potential of technological developments to lower the cost of a global lidar system (GLS) are discussed. Once these technologies achieve a sufficient readiness level, a GLS could be cost-effectively realized.

Geodesic spaces of low Nagata dimension

We show that every geodesic metric space admitting an injective continuous map into the plane as well as every planar graph has Nagata dimension at most two, hence asymptotic dimension at most two. This relies on and answers a question in a recent work by Fujiwara and Papasoglu. We conclude that all three-dimensional Hadamard manifolds have Nagata dimension three. As a consequence, all such manifolds are absolute Lipschitz retracts.

Chaos on Fuzzy Dynamical Systems

Given a continuous map f:X→X on a metric space, it induces the maps f¯:K(X)→K(X), on the hyperspace of nonempty compact subspaces of X, and f^:F(X)→F(X), on the space of normal fuzzy sets, consisting of the upper semicontinuous functions u:X→[0,1] with compact support. Each of these spaces can be endowed with a respective metric. In this work, we studied the relationships among the dynamical systems (X,f), (K(X),f¯), and (F(X),f^). In particular, we considered several dynamical properties related to chaos: Devaney chaos, A-transitivity, Li–Yorke chaos, and distributional chaos, extending some results in work by Jardón, Sánchez and Sanchis (Mathematics 2020, 8, 1862) and work by Bernardes, Peris and Rodenas (Integr. Equ. Oper. Theory 2017, 88, 451–463). Especial attention is given to the dynamics of (continuous and linear) operators on metrizable topological vector spaces (linear dynamics).

Topological stability and shadowing of dynamical systems from measure theoretical viewpoint

In this paper it is proved that a topologically stable invariant measure has no sinks or sources in its support; an expansive homeomorphism is topologically stable if it exhibits a topologically stable nonatomic Borel support measure and a continuous map has the shadowing property if there exists an invariant measure with the shadowing property such that each almost periodic point is contained in the support of the invariant measure.

The use of singlebeam echo-sounder depth data to produce demersal fish distribution models that are comparable to models produced using multibeam echo-sounder depth

Seafloor characteristics can help in the prediction of fish distribution, which is required for fisheries and conservation management. Despite this, only 5-10% of the world’s seafloor has been mapped at high resolution as it is a time-consuming and expensive process. Multibeam echo-sounders (MBES) can produce high-resolution bathymetry and a broad swath coverage of the seafloor, but require greater financial and technical resources for operation and data analysis than singlebeam echo-sounders (SBES). In contrast, SBES provide comparatively limited spatial coverage, as only a single measurement is made from directly under the vessel. Thus, producing a continuous map requires interpolation to fill gaps between transects. This study assesses the performance of demersal fish species distribution models by comparing those derived from interpolated SBES data with full-coverage MBES distribution models. A Random Forest classifier was used to model the distribution of Abalistes stellatus, Gymnocranius grandoculis, Lagocephalus sceleratus, Loxodon macrorhinus, Pristipomoides multidens and Pristipomoides typus, with depth and depth derivatives (slope, aspect, standard deviation of depth, terrain ruggedness index, mean curvature and topographic position index) as explanatory variables. The results indicated that distribution models for A. stellatus, G. grandoculis, L. sceleratus, and L. macrorhinus performed poorly for MBES and SBES data with Area Under the Receiver Operator Curves (AUC) below 0.7. Consequently, the distribution of these species could not be predicted by seafloor characteristics produced from either echo-sounder type. Distribution models for P. multidens and P. typus performed well for MBES and the SBES data with an AUC above 0.8. Depth was the most important variable explaining the distribution of P. multidens and P. typus in both MBES and SBES models. While further research is needed, this study shows that in resource-limited scenarios, SBES can produce comparable results to MBES for use in demersal fish management and conservation.

Principal Bundle Structure of Matrix Manifolds

In this paper, we introduce a new geometric description of the manifolds of matrices of fixed rank. The starting point is a geometric description of the Grassmann manifold Gr(Rk) of linear subspaces of dimension r<k in Rk, which avoids the use of equivalence classes. The set Gr(Rk) is equipped with an atlas, which provides it with the structure of an analytic manifold modeled on R(k−r)×r. Then, we define an atlas for the set Mr(Rk×r) of full rank matrices and prove that the resulting manifold is an analytic principal bundle with base Gr(Rk) and typical fibre GLr, the general linear group of invertible matrices in Rk×k. Finally, we define an atlas for the set Mr(Rn×m) of non-full rank matrices and prove that the resulting manifold is an analytic principal bundle with base Gr(Rn)×Gr(Rm) and typical fibre GLr. The atlas of Mr(Rn×m) is indexed on the manifold itself, which allows a natural definition of a neighbourhood for a given matrix, this neighbourhood being proved to possess the structure of a Lie group. Moreover, the set Mr(Rn×m) equipped with the topology induced by the atlas is proven to be an embedded submanifold of the matrix space Rn×m equipped with the subspace topology. The proposed geometric description then results in a description of the matrix space Rn×m, seen as the union of manifolds Mr(Rn×m), as an analytic manifold equipped with a topology for which the matrix rank is a continuous map.

NOTES ON ORTHOGONAL-COMPLETE METRIC SPACES

We prove that the restriction of a given orthogonal-complete metric space to the closure of the orbit induced by the origin point with respect to an orthogonal-preserving and orthogonal-continuous map is a complete metric space. Then we show that many existence results on fixed points in orthogonal-complete metric spaces can be proved by using the corresponding existence results in complete metric spaces.

An application of the continuous Steiner symmetrization to Blaschke-Santaló diagrams

In this paper we consider the so-called procedure of {\it Continuous Steiner Symmetrization}, introduced by Brock in~\cite{bro95,bro00}. It transforms every domain $\Omega\comp\R^d$ into the ball keeping the volume fixed and letting the first eigenvalue and the torsion respectively decrease and increase. While this does not provide, in general, a $\gamma$-continuous map $t\mapsto\O_t$, it can be slightly modified so to obtain the $\gamma$-continuity for a $\gamma$-dense class of domains $\O$, namely, the class of polyedral sets in $\R^d$. This allows to obtain a sharp characterization of the Blaschke-Santaló diagram of torsion and eigenvalue.