The orthogonal matrix with cube be symmetric is a common class of matrices with important properties. We pay attention to this kind of matrices. By using the close relationship between the eigenvalues of a matrix and the trace of its power, we obtain the algorithm for its all possible different eigenvalues and multiplicities. The calculation formula is expressed only by the trace of a matrix and its power, avoiding solving the characteristic polynomial. The method is simple and practical. Furthermore, a new essential characterization for the sum of orthogonal matrix pairs being orthogonal is given as well.

The aim of this article is to make a multi-criteria analysis of various exhibition spaces of an originally non-exhibition character and to determine how these spaces affect the selection of works and the exhibition concept. The analysis is based on the exhibitions of art objects at collective exhibitions in unconventional architectural spaces: commercial, i.e. the modern office building of PBG Gallery Skalar Office Centre in Poznań, post-industrial i.e. in the former Zakłady Przemysłu Ziemniaczanego Lubanta S.A. and in the historic interior of the "U Jezuitów" Gallery of the Cultural Integration Centre in Poznań. The multi-criteria comparative analysis shows a variety of features of the studied spaces as well as the relationship between architecture and art and their mutual interaction. The participatory role of the non-exhibition space in the process of creating an exhibition and selecting works has been proven. It has also been confirmed that the presentation of works of art in originally non-exhibition spaces creates a new quality of the artwork. Unconventional architectural space, when used for the exhibition of works of art, expands and strengthens the area of ​​their influence through the interaction between the work and the architectural space. The specificity of the space adapted for exhibition needs, the presence and type of architectural details in the interior, the quantity and quality of light and its distribution in space, the volume and colour of the interior determine the exhibition space and influence the shape of the exhibitions organised and the reception of the artworks. The only condition for the change of the original function of an architectural space into that of an exhibition space is a coherent artistic vision of the creator. This should take into account the appropriate selection of the exhibited objects, where the process of searching for the relationship between architecture and art determines the features of the architectural space as integral components influencing the realisation of the exhibition .

The Social-Ecological Systems (SES) framework serves as a valuable framework to explore and understand social and ecological interactions, and pathways in water governance. Yet, it lacks a robust understanding of change. We argue an analytical and methodological approach to engaging global changes in SES is critical to strengthening the scope and relevance of the SES framework. Relying on SES and resilience thinking, we propose an institutional and cognitive model of change that institutions and natural resources systems co-evolve to provide a dynamic understanding of SES that stands on three causal mechanisms: institutional complexity trap, rigidity trap, and learning processes. We illustrate how Data Cube technology could overcome current limitations and offer reliable avenues to test hypothesis about the dynamics of social-ecological systems and water security by offering to combine spatial and time data with no major technical requirements for users.

A new class of nano-cube core-shell heterostructures containing Ag coating on the top of WO₃ was fabricated. Physical vapor deposition was used to produce WO₃ based nano-heterostructures. All kind of wet toxic chemical process was avoided to make the process simple and contaminant free. Sputtering of WO₃ and a subsequent thermal annealing process was done to create nano-cubes of WO₃. After that, sputtering of Ag was performed to form the Ag-WO₃ core-shell nano-heterostructures (CSNH). The CSNHs were characterized using field emission scanning electron microscopy (FESEM), energy dispersive spectroscopy (EDS), X-ray diffraction analysis (XRD) and UV-vis spectroscopy. The morphologies, elemental analysis, interfaces, crystallinity, phases, and chemical compositions were analyzed. The bottom-up growth of WO₃ nanocubes was studied using different time periods at 900°C. Ag coating was also studied before and after annealing. Finally, an optical property (band gap) was also analyzed using Tauc plot derive from absorption spectra. The tailoring the band gap of WO₃ from ~2.9eV to ~ 2.45 eV was observed while Ag-WO₃ CSNH formed.

Laboratory experiments and investigations into natural phenomena in this research series have unveiled the presence of a gravitational repulsion force that permeates our environment on both microscopic and macroscopic scales, contingent upon the thermal energy present. This paper presents a study exploring novel properties of gravitational forces among gas molecules, employing principles of thermodynamics. A model has been devised based on the interactions between pairs of gas molecules. While traditional models treat gravitational interaction as a singular force, our experimental approach validates it as the composite of two distinct forces: gravitational repulsion and attraction. By utilizing established experimental data on gas thermodynamics, our model demonstrates robust performance both analytically and experimentally. It verifies the coexistence of gravitational repulsion and attraction forces among gas molecules, showcases their Inverse-Cube relationship with distance, and elucidates the direct proportionality of the repulsion force to absolute temperature. This bridges a crucial gap between energy and fundamental forces. The order of magnitudes of gravitational repulsion and attraction forces are found to be considerably large, contrasting with the low values predicted by classical theory, which results from their amalgamation. Recognizing these forces as substantial in magnitude promises unprecedented outcomes and advancements.

A new type of artificial block is proposed by modifying the traditional block. Based on the physical model experiment, the wave reflection and run-up characteristics of the new ecological blocks under regular waves were examined, and the influence of the height of the "planting trough" border on the hydrodynamic properties was analyzed. The results indicate that the wave reflection and run-up would decrease with a decrease in the breakwater slope. As wave height increases, wave run-up increases, and the reflection coefficient falls. Wave reflection and run-up increase with an increasing wave period. The relative depth of the water has a smaller impact on the wave reflection than the wave run-up. The reflection coefficient of the new ecological blocks is consistently lower than that of the conventional and continues to decrease as the border of the "planting trough" is raised. The raised borders prevent water loss in the trough, resulting in no significant improvement in the wave run-up performance of the new blocks. The reflection coefficient and the relative run-up R/H gradually increase with the average wave breaking parameter ξ increase. The crude permeability coefficients for the new blocks, New Heights 1 and New Heights 2, are 0.53, 0.53, and 0.54, respectively.

Disparities between fold amplitude (A) and intrusion thickness (Hsill) are critical in identifying elastic or inelastic deformation in a forced fold. However, accurate measurement of these two parameters is challenging because of the limit in separability and detectability for the seismic data. In the TZ-47 exploring area from the Tarim Basin, Northwest China, we combined wireline data and 3-D seismic data, to accurately constrain the fold amplitude and total thickness of sills that inducing the roof uplift. The measurement results show that the forced fold amplitude is 155 m and the total sill thickness is 148.4 m. When using a magma density of 2.7 g/cm3, and solid rock density of 3 g/cm3, the molten magma thickness at the time of intrusion would be 153.8 m, which is almost no difference from the forced fold amplitude. Therefore, the TZ-47 fold is a pure elastic forced fold induced by emplacement of multiple sills. Measurement solely based on seismic data may not be able to detect some thin interlayers and may result in large errors.

In this paper we discuss a problem of generalization of binomial distributed triangle, that is sequence A287326 in OEIS. The main property of A287326 that it returns a perfect cube n as sum of n-th row terms over k; 0 ≤ k ≤ n−1 or 1 ≤ k ≤ n, by means of its symmetry. In this paper we have derived a similar triangles in order to receive powers m = 5; 7 as row items sum and generalized obtained results in order to receive every odd-powered monomial n^2m+1; m ≥ 0 as sum of row terms of corresponding triangle.