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Good daylighting performance positively affects students’ physical and mental health, learning efficiency, and the building’s energy-saving capability. Due to the terrace classroom having ample space, large capacity, the ability to avoid obstructing sight, and the ability to meet various use needs, it is the most important place in university buildings. However, research on the daylighting performance of university terrace classrooms is limited, leading to a lack of quantitative guidance in early design stages. This study aims to explore the effects of interior space and window geometry of terrace classrooms in universities in severe cold regions on daylighting performance. This research took Shenyang as an example; spatial daylight autonomy (sDA300,50%) and useful daylight illuminance (UDI100–2000) were selected as daylighting performance evaluation indices. Based on the Grasshopper parametric platform, the simulation was carried out using Ladybug and Honeybee plugins. Correlation and regression analyses revealed the relationship between interior space and window geometry parameters and the evaluation indices. The results showed the following: window-to-floor ratio (WFR), classroom height (Htc), window height (Hw), window-to-wall ratio (WWR), classroom width (Wtc), and window width (Ww) have positive effects on improving the daylight sufficiency of the terrace classrooms facing each orientation, and the degree of the effect decreases in order. To ensure the overall daylighting performance, the Wtc can be maximized. The width of walls between windows for south-facing and west-facing classrooms should be 0.9 m. The WWR and WFR for south-facing classrooms should be 0.3–0.5 and 0.11–0.14, respectively. The WWR and WFR for north-facing classrooms should be 0.6–0.7 and 0.14–0.20, respectively. Prediction models are established for the sDA300,50% and UDI100–2000 of the terrace classrooms facing each orientation.

The main objective of this study is to take the reader on an architectural pilgrimage to learn about the geometric harmony in the structural patterns that formed the sacred interior space of early Ottoman mosques, “The Single Unit Mosque.” The applied analytical approach is the preferred scientific method for employing theoretical knowledge to conduct structural and geometric analysis on some of the buildings selected in this study, with the researchers seeking to achieve tangible practical results. Since Ottoman architecture was characterized by strong construction and heavy materials through thick walls and high spacious domes, it needed distinctive building techniques to provide spatial and structural harmony between the cubic construction with its straight lines at ground level and the hemispherical construction of the dome with arched lines at the upper level. Therefore, it is necessary to identify the structural patterns and elements employed in Ottoman architecture in the early period of the Ottoman Empire and show their influence on the shape of the interior space and the extent of its breadth and beauty. This study was also devoted to clarifying the geometric compositions of structural patterns through a set of examples of analytical assessments on a selected group of Ottoman mosques. The results clarified the advantages and disadvantages of the domed Ottoman Mosque structural patterns. The research provides a significant and original contribution to the study of early Ottoman Mosque interiors, offering valuable insights into the structural innovations that defined Ottoman architecture.

The magnetic field of the Sun is generated by internal dynamo process with a cyclic period of 11 years or a 22 year magnetic cycle. The signatures of the Sun’s magnetic cycle are observed in the different layers of its atmosphere and in its internal layers. In this review, we use the same diagnostics to understand the magnetic cycles of other stars with the same internal structure as the Sun. We review what is currently known about mapping the surface magnetic fields, chromospheric and coronal indicators, cycles in photometry and asteroseismology. We conclude our review with an outlook for the future.

We present a novel approach for the construction of interior solutions for the Kerr metric, extending J. Ovalle’s foundational work through ellipsoidal coordinate transformations. By deriving a physically plausible interior solution that smoothly matches the Kerr exterior metric, we analyze the energy conditions across various rotation parameters. Our findings reveal anisotropic fluid properties and energy condition behaviors in specific space-time regions, providing insights into the strong-field regime of rotating black holes. The proposed solution offers a more realistic description of rotating black hole interiors, with implications for understanding compact astrophysical objects.

The aim of this contribution is to propose an alternative but equivalent statement of the proper separation of two closed convex sets in a finite-dimensional Euclidean space. To this aim, we characterize the affine hull of a closed convex set defined by a finite set of equalities and inequalities. Furthermore, we describe algebraically the relative interior of this set by projecting the optimal set of a convex optimization problem onto a subspace of its variables. Then we use this description to develop a system of equalities and inequalities by which the proper separability of the given convex sets is identified. We show that this system is linear in the special case that the given sets are polyhedral.

A challenge remains in achieving adequate surface roughness of SLM fabricated interior channels, which is crucial for fuel delivery in the space industry. This study investigated the surface roughness of interior fine flow channels (1 mm diameter) embedded in SLM fabricated TC4 alloy space components. A machine learning approach identified layer thickness as a significant factor affecting interior channel surface roughness, with an importance score of 1.184, followed by scan speed and laser power with scores of 0.758 and 0.512, respectively. The roughness resulted from thin layer thickness of 20 µm, predominantly formed through powder adherence, while from thicker layer of 50 µm, the roughness was mainly due to the stair step effect. Slow scan speeds increased melt pools solidification time at roof overhangs, causing molten metal to sag under gravity. Higher laser power increased melt pools temperature and led to dross formation at roof overhangs. Smaller hatch spaces increased roughness due to overlapping of melt tracks, while larger hatch spaces reduced surface roughness but led to decreased part density. The surface roughness was recorded at 34 µm for roof areas and 26.15 µm for floor areas. These findings contribute to potential adoption of TC4 alloy components in the space industry.

The paper initially proves that locally finite covering (LFC-, for short) rough set structures are interior and closure operators. To be precise, given an LFC-space ( U , C ) , we prove that the lower H -rough set operator H ∗ is an interior operator and the upper H -rough set operator H ∗ is a closure operator. Besides, we prove a duality of the concept approximations ( H ∗ , H ∗ ) and investigate many theoretical and mathematical properties of the H -rough set operators. After pointing out that Khalimsky ( K -, for brevity) topological rough set operators have their own features, we prove that the K -topological lower ( resp. upper) approximation operator is not an interior ( resp. closure) operator from the viewpoint of K -topology. Besides, we intensively investigate theoretical and mathematical properties of the K -topological rough set operators. This research area can be considered as a part of geometric-based rough set theory. These obtained results can promote the studies of rough set theory associated with information geometry, object classification, artificial or computational intelligence, and so on. In the present paper, each of the sets U , C and X ( ⊆ U ) need not be finite.

We analyze the space Sm,m′r(T)\\mathcal {S}_{m, m’}^{\\mathbf {r}} (\\mathcal {T}) of bivariate functions that are piecewise polynomial of bi-degree ⩽(m,m′)\\leqslant (m, m’) and of smoothness r\\mathbf {r} along the interior edges of a planar T-mesh T\\mathcal {T}. We give new combinatorial lower and upper bounds for the dimension of this space by exploiting homological techniques. We relate this dimension to the weight of the maximal interior segments of the T-mesh, defined for an ordering of these maximal interior segments. We show that the lower and upper bounds coincide, for high enough degrees or for hierarchical T-meshes which are regular enough. We give a rule of subdivision to construct hierarchical T-meshes for which these lower and upper bounds coincide. Finally, we illustrate these results by analyzing spline spaces of small degrees and smoothness.

Electromagnetic complex media are artificial materials that affect the propagation of electromagnetic waves in surprising ways not usually seen in nature. Because of their wide range of important applications, these materials have been intensely studied over the past twenty-five years, mainly from the perspectives of physics and engineering. But a body of rigorous mathematical theory has also gradually developed, and this is the first book to present that theory. Designed for researchers and advanced graduate students in applied mathematics, electrical engineering, and physics, this book introduces the electromagnetics of complex media through a systematic, state-of-the-art account of their mathematical theory. The book combines the study of well posedness, homogenization, and controllability of Maxwell equations complemented with constitutive relations describing complex media. The book treats deterministic and stochastic problems both in the frequency and time domains. It also covers computational aspects and scattering problems, among other important topics. Detailed appendices make the book self-contained in terms of mathematical prerequisites, and accessible to engineers and physicists as well as mathematicians.

We reorganize, simplify and expand the theory of contractions or interior products of multivectors, and related topics like Hodge star duality. Many results are generalized and new ones are given, like: geometric characterizations of blade contractions and regressive products, higher-order graded Leibniz rules, determinant formulas, improved complex star operators, etc. Different contractions found in the literature are discussed and compared, in special those of Clifford Geometric Algebra. Applications of the theory are developed in a follow-up paper.