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Urban tourism space is the primary area where tourism activities occur and a key driver of regional tourism space evolution. To explore the correlation between population aggregation and urban tourism spatial heterogeneity in the big data era, this study focuses on Wuhan's main urban area in 2023. Using the Geographically Weighted Regression model, it analyzes the factors influencing tourism spatial heterogeneity. Additionally, Baidu Heat map data is employed to identify crowd aggregation levels during holidays, revealing the distribution patterns of urban tourism space. The results indicate that (1) factors derived from the GWR model significantly influence the number of tourism spaces in Wuhan, with evident spatial differences; (2) based on the spatial matching of heterogeneity factors and crowd aggregation levels, urban tourism space can be categorized into six types, including five core tourism spaces and other scattered spaces. This research highlights the spatial heterogeneity of urban tourism in Wuhan and provides a scientific basis for the transformation and quality improvement of urban tourism space by exploring the impact of population activity density.

In this paper, based on a complete residuated lattice, we propose the definition of fuzzy weakly cut-stable map and prove the extension property of the fuzzy weakly cut-stable map. Following this, it is explored the conditions under which the extension map to be fuzzy order isomorphism.

This paper presents an efficient path planning method for the lunar rover to improve the autonomy and exploration ability in the complex and unstructured lunar surface environment. Firstly, the safe zone for the rover’s motion is defined, based on which a detecting point selection strategy is proposed to choose target positions that meet the rover’s constraints. Secondly, an improved sampling-based path planning method is proposed to get a safe path for the rover efficiently. Thirdly, a map extension strategy for the unstructured and continually varying environment is included to update the roadmap, which takes advantage of the historical planning information. Finally, the proposed method is tested in a complex lunar surface environment. Numerical results show that the appropriate detecting positions can be selected autonomously, while a safe path to the selected detecting position can be obtained with high efficiency and quality compared with the Probabilistic Roadmap (PRM) and A* search algorithm.

We show that every cross ratio preserving homeomorphism between boundaries of Hadamard manifolds extends to a map, called circumcenter extension, provided that the manifolds satisfy certain visibility conditions. We describe regions on which this map is Hölder-continuous. Furthermore, we show that this map is a rough isometry, whenever the manifolds admit cocompact group actions by isometries and we improve previously known quasi-isometry constants, provided by Biswas, in the case of 2-dimensional CAT ( - 1 ) manifolds. Finally, we provide a sufficient condition for this map to be an isometry in the case of Hadamard surfaces.

An optimized dwell time algorithm for magnetorheological finishing (MRF) is discussed. Based on the D-shape of the removal function of MRF, an optimized non-negative least-squares method is introduced to get dwell time from a linear matrix equation transferred from the de-convolution operation. Moreover, one kind of general surface error map extension is developed for any shape of optics to obtain a more precise optical surface in MRF. The simulation results show that the non-negative least-squares method of the constrained generalized minimal residual (GMRES) method with adaptive Tikhonov regulation is much faster to get highly stable dwell time distribution. In combination with the general surface error map extension, the peak to valley (PV) and root mean square (RMS) of the surface error of the diameter 400 mm converge from 184.41 and 21.26 nm to 7.56 and 0.632 nm with the consistency of the edge and the aperture inside. Finally, a fabricating experiment proves the validity of the optimized algorithm. Therefore, the algorithm developed and presented in this paper can facilitate the MRF process effectively.

Extension concept mapping is a technique to connect prior existing concept maps with new knowledge structures. It offers advantages in each stage of the knowledge-integrating process and encourages learners to improve their performance. While previous studies have confirmed that the extended kit-build concept map outperformed the extended scratch-build approach in terms of comprehension test scores and map size, they have yet to evaluate the quality of concept maps and students' perceptions. Although the size of the concept map components could represent the breadth of personal knowledge, it does not constantly describe the good knowledge structure. In addition, the student's degree of acceptance after the concept mapping demonstrates their intention to use systems in the future. The present study aims to compare the effect of extended scratch-build and extended kit-build on the students' quality of knowledge structures and perceptions. Fifty-five second-year university students were involved and divided into two groups: control and experimental. The control group utilized the extended scratch-build map, while the experimental group used the extended kit-build concept mapping tool. Quality of propositions and structural map scores as learning outcomes were used to measure the students' knowledge structures. The possibility of a relationship between quality scores was expressed using the Spearman correlation. This study involved the Technology Acceptance Model to confirm the students' perceptions of extension concept mapping tools. The perceived ease-of-use, perceived usefulness, and behavioral intention constructs were used to investigate users' intentions. The findings suggest that the quality of propositions and structural map scores in the experimental group were significantly higher than in the control group. This study also found that the extended kit-build method achieved better perceptions scores than the extended scratch-build.

This paper concerns extension of maps using obstruction theory under a non-classical viewpoint. It is given a classification of homotopy classes of maps and as an application it is presented a simple proof of a theorem by Adachi about equivalence of vector bundles. Also it is proved that, under certain conditions, two embeddings are homotopic up to surgery if and only if the respective normal bundles are SO -equivalent.

Let $G_1, \\ldots , G_k$ be finite-dimensional vector spaces over a prime field $\\mathbb {F}_p$. A multilinear variety of codimension at most $d$ is a subset of $G_1 \\times \\cdots \\times G_k$ defined as the zero set of $d$ forms, each of which is multilinear on some subset of the coordinates. A map $\\phi$ defined on a multilinear variety $B$ is multilinear if for each coordinate $c$ and all choices of $x_i \\in G_i$, $i\\not =c$, the restriction map $y \\mapsto \\phi (x_1, \\ldots , x_{c-1}, y, x_{c+1}, \\ldots , x_k)$ is linear where defined. In this note, we show that a multilinear map defined on a multilinear variety of codimension at most $d$ coincides on a multilinear variety of codimension $O_{k}(d^{O_{k}(1)})$ with a multilinear map defined on the whole of $G_1\\times \\cdots \\times G_k$. Additionally, in the case of general finite fields, we deduce similar (but slightly weaker) results.

We study monotone operators in reflexive Banach spaces that are invariant with respect to a group of suitable isometric isomorphisms, and we show that they always admit a maximal extension which preserves the same invariance. A similar result applies to Lipschitz maps in Hilbert spaces, thus providing an invariant version of Kirszbraun–Valentine extension theorem. We then provide a relevant application to the case of monotone operators in $L^{p}$ -spaces of random variables which are invariant with respect to measure-preserving isomorphisms, proving that they always admit maximal dissipative extensions which are still invariant by measure-preserving isomorphisms. We also show that such operators are law invariant, a much stronger property which is also inherited by their resolvents, the Moreau–Yosida approximations, and the associated semigroup of contractions. These results combine explicit representation formulae for the maximal extension of a monotone operator based on self-dual Lagrangians and a refined study of measure-preserving maps in standard Borel spaces endowed with a nonatomic measure, with applications to the approximation of arbitrary couplings between measures by sequences of maps.

In concept map-based assessment an expert’s concept map can be expanded using graph patterns to add hidden and inverse relations. This helps to avoid forcing a learner to use certain structures and names. Graph patterns are subgraphs that describe combinations of concept map elements, from which extra relations can be inferred. In this paper an enriched set of graph patterns is described along with their respective IF...THEN rules which can be used for automated knowledge assessment. Some of them are already implemented in the intelligent and adaptive knowledge assessment system IKAS.