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Non-additive mixing plays a key role in the properties of molecular fluids and solids. In this work, the potential for athermal order–disorder phase transitions is explored in non-additive binary colloidal nanoparticles that form substitutionally ordered compounds, namely, for equimolar mixtures of octahedra + spheres, which form a CsCl lattice compound, and cubes + spheres, which form a NaCl crystal. Monte Carlo simulations that target phase coexistence conditions were used to examine the effect on compound formation of varying degrees of negative non-additivity created by component size asymmetry and by size-tunable indentations in the polyhedra's facets, intended to allow the nestling of neighboring spheres. Our results indicate that the stabilization of the compound crystal requires a relatively large degree of negative non-additivity, which depends on particle geometry and the packing of the relevant phases. It is found that negative non-additivity can be achieved in mixtures of large spheres and small cubes having no indentations and lead to the athermal crystallization of the NaCl lattice. For similarly sized components, athermal congruent transitions are attainable and non-additivity can be generated through indentations, especially for the cubes + spheres system. Increasing indentation leads to lower phase coexistence free energy and pressure in the cubes + spheres system but has the opposite effect in the octahedra + spheres system. These results indicate a stronger stabilizing effect on the athermal compound phase by the cubes' indentations, where a deeper nestling of the spheres leads to a denser compound phase and a larger reduction in the associated pressure-volume free-energy term. [ABSTRACT FROM AUTHOR]

Recent experiments have demonstrated the self-assembly and long-range ordering of concentrated aqueous solutions of DNA and RNA mononucleotides. These are found to form Watson–Crick pairs that stack into columns that become spatially organized into a columnar liquid-crystalline phase. In this work, we numerically investigate this phase behavior by adopting an extremely coarse-grained model in which nucleotides are represented as semi-disk-like polyhedra decorated with attractive (patchy) sites that mimic the stacking and pairing interactions. We carry out Monte Carlo simulations of these patchy polyhedra by adapting algorithms borrowed from computer graphics. This model reproduces the features of the experimental phase behavior, which essentially depends on the combination of pairing and stacking interactions. [ABSTRACT FROM AUTHOR]

[Display omitted] • We develop NiCo alloy-decorated nitrogen-doped carbon double-shelled hollow polyhedrons (NC/NiCo DSHPs) as highly efficient catalysts for Li-S batteries. • The distribution of NiCo alloy on both the inner and outer shells ensures sufficient catalytic active sites, which can effective adsorb LiPSs and mitigate the shuttle effect, even at high loadings, and facilitate the conversion between LiPSs and Li 2 S, enabling enhanced redox kinetics of Li-S systems. • Benefiting from the advantages of composition and structure, Li-S batteries, using porous NC/NiCo DSHPs as separator coating material, achieve a high specific capacity of 1310 mAh g-1 at 0.2 C, a superior cycling performance with low capacity fading of 0.045% per cycle at 1 C after 800 cycles. Lithium-sulfur (Li-S) batteries have received significant attention due to their high theoretical energy density. However, the inherent poor conductivity of S and lithium sulfide (Li 2 S), coupled with the detrimental shuttle effect induced by lithium polysulfides (LiPSs), impedes their commercialization. In this study, we develop NiCo alloy-decorated nitrogen-doped carbon double-shelled hollow polyhedrons (NC/NiCo DSHPs) as highly efficient catalysts for Li-S batteries. The distribution of NiCo alloy on both the inner and outer shells provides abundant catalytic active sites, effectively adsorbing LiPSs, mitigating the shuttle effect, and promoting the conversion between LiPSs and Li 2 S, even at high sulfur loadings. This results in enhanced redox kinetics within the Li-S system. Moreover, the highly conductive carbon material framework, enriched with carbon nanotubes and graphitic carbon layers, can greatly promote the efficient electron transportation. Additionally, the improved ion diffusion rates benefiting from the hollow structure can also be realized. By harnessing these synergistic effects, Li-S batteries incorporating the double-shelled NC/NiCo DSHP catalysts achieved a high specific capacity of 1310 mAh/g at 0.2C and a superior rate performance of 621 mAh/g at 4C. Furthermore, excellent cycling performance with ultralow capacity fading rate of only 0.045 % per cycle after 800 cycles at 1C was achieved. When sulfur loading reaches 6 mg cm−2, a high capacity of 4.6 mAh cm−2 at 0.1C after 100 cycles further validates the practical potential of this design. This study presents an innovative approach to alloy catalyst design, offering valuable insights for future research of Li-S batteries. [ABSTRACT FROM AUTHOR]

Metal–organic cages (MOCs) are discrete supramolecular entities consisting of metal nodes and organic connectors or linkers; MOCs are noted for their high porosity and processability. Chemically, they can be post-synthetically modified (PSM) and new functional groups can be introduced, presenting attractive qualities, and it is expected that their new properties will differ from those of the original compound. This is why they are highly regarded in the fields of biology and chemistry. The present review deals with the current PSM strategies used for MOCs, including covalent, coordination, and noncovalent methods and their structural benefits. The main emphasis of this review is to show to what extent and under what circumstances a MOC can be designed to obtain a tailored geometric architecture. Although sometimes unclear when examining supramolecular systems, particularizing the design of and systematic approaches to the development and characterization of families of MOCs provides new insights into structure–function relationships, which will guide future developments. [ABSTRACT FROM AUTHOR]

Folding a discrete geometry from a flat sheet of material is one way to construct a 3D object. A typical creation pipeline first designs the 3D object, unfolds it, prints and cuts the unfold pattern from a 2D material, and then refolds the object. Within this work we focus on the unfold part of this pipeline. Most current unfolding approaches segment the input, which has structural downsides for the refolded result. Therefore, we are aiming to unfold the input into a single-patched pattern. Our algorithm applies tabu search to the topic of unfolding. We show empirically that our algorithm is faster and more reliable than other methods unfolding into single-patched unfold patterns. Moreover, our algorithm can handle any sort of flat polygon as faces, while comparable methods are bound to triangles. [ABSTRACT FROM AUTHOR]

10.1119/5.0160770.1 In quantum physics lectures, half-integer spins are generally introduced as "objects that do not come back to their original state after one full turn but that do after two." As a consequence, students often consider this behavior to be purely quantum mechanical. However, spin-1/2 is above all a geometrical property of the rotations group and can, therefore, also have practical consequences at the macroscopic scale. To illustrate this, we introduce and describe in this work a new pedagogical tool named the spinorial ball. It allows students to concretely manipulate a macroscopic 1/2-spin, which helps them to build intuition as to how the latter behaves under rotations. This object can also be used to introduce several general concepts from the theory of Lie groups, such as group homomorphism and homotopy classes of loops through the example of the groups SU(2) and SO(3). The spinorial ball provides a macroscopic visualization of all these concepts, which are ubiquitous in quantum physics. Editor's Note: If you ever wanted to hold a spin-1/2 particle in your hands, this paper is for you. If you follow the authors' instructions, you will end up with a polyhedron with faces that light up according to rules defined in the SU(2) group, and which you will be able to rotate at will (and thus explore the SO(3) rotation group). The so-called spinorial ball then behaves as a spin-1/2 object in that it has to experience a 4 π rotation about a given axis to come back to its initial state. As you will be able to show by playing with the ball, this important property of spins-1/2 is actually a consequence of the interplay between two rotation groups. This paper, which will help you build a demonstration experiment for introductory quantum mechanics classes, could also be the basis of an exercise in undergraduate group theory classes or electronics labs. [ABSTRACT FROM AUTHOR]

The controllable encapsulation of host molecules (such as porphyrin, phthalocyanine, crown ether, calixarene or cucurbituril organic macrocycles, cages, metal–organic polyhedrons and enzymes) into the pores of metal–organic frameworks (MOFs) to form host-in-host (host@MOF) materials has attracted increasing research interest in various fields. These host@MOF materials combine the merits of MOFs as a host matrix and functional host molecules to exhibit synergistic functionalities for the formation of guest@host@MOF materials in sorption and separation, ion capture, catalysis, proton/ion conduction and biosensors. (This guest@host@MOF construction is reminiscent of Russian (Matrjoschka) dolls which are nested dolls of decreasing size placed one inside another.) In this tutorial review, the advantages of MOFs as a host matrix are presented; the encapsulation approaches and general important considerations for the preparation of host@MOF materials are introduced. The state-of-the-art examples of these materials based on different host molecules are shown, and representative applications and general characterization of these materials are discussed. This review will guide researchers attempting to design functional host@MOF and guest@host@MOF materials for various applications. [ABSTRACT FROM AUTHOR]

Discrete statistical systems offer a significant advantage over systems defined in the continuum, since they allow for an easier enumeration of microstates. We introduce a lattice-gas model on the vertices of a polyhedron called a pentakis icosidodecahedron and draw its exact phase diagram by the Wang–Landau method. Using different values for the couplings between first-, second-, and third-neighbor particles, we explore various interaction patterns for the model, ranging from softly repulsive to Lennard-Jones-like and SALR. We highlight the existence of sharp transitions between distinct low-temperature "phases", featuring, among others, regular polyhedral, cluster-crystal-like, and worm-like structures. When attempting to reproduce the equation of state of the model by Monte Carlo simulation, we find hysteretic behavior near zero temperature, implying a bottleneck issue for Metropolis dynamics near phase-crossover points. [ABSTRACT FROM AUTHOR]

Polyhedral cages (p-cages) describe the geometry of some families of artificial protein cages. We identify the p-cages made out of families of equivalent polygonal faces such that the faces of one family have five neighbors and P 1 edges, while those of the other family have six neighbors and P 2 edges. We restrict ourselves to polyhedral cages where the holes are adjacent to four faces at most. We characterize all p-cages with a deformation of the faces, compared to regular polygons, not exceeding 10%. [ABSTRACT FROM AUTHOR]

Tutte's embedding theorem states that every 3-connected graph without a K 5 - or K 3 , 3 -minor (i.e., a planar graph) is embedded in the plane if the outer face is in convex position and the interior vertices are convex combinations of their neighbors. We show that this result extends to simply connected tetrahedral meshes in a natural way: for the tetrahedral mesh to be embedded if the outer polyhedron is in convex position and the interior vertices are convex combination of their neighbors it is sufficient (but not necessary) that the graph of the tetrahedral mesh contains no K 6 and no K 3 , 3 , 1 , and all triangles incident on three boundary vertices are boundary triangles. [ABSTRACT FROM AUTHOR]

For 퓞 an imaginary quadratic ring, we compute a fundamental polyhedron in hyperbolic 3-space for the action of PE2(퓞), the projective elementary subgroup of PSL2(퓞). This allows for new, simplified proofs of theorems of Cohn, Nica, Fine, and Frohman. Namely, we obtain a presentation for PE2(퓞), show that it has infinite index and is its own normalizer in PSL2(퓞), and split PSL2(퓞) into a free product with amalgamation that has PE2(퓞) as one of its factors. [ABSTRACT FROM AUTHOR]

Given a Noetherian ring A, the collection of integrally closed ideals in A which contain a nonzerodivisor forms a cancellative monoid under the operation I * J = IJ ¯ , the integral closure of the product. The monoid is torsion-free and atomic. Restricting to monomial ideals in a polynomial ring, there is a surjective homomorphism from the Integral Polytope Group onto the Grothendieck group of integrally closed monomial ideals under translation invariance of their Newton Polyhedra. The Integral Polytope Group, the Grothendieck group of polytopes with integer vertices under Minkowski addition and translation invariance, has an explicit basis, allowing for explicit factoring in the monoid. [ABSTRACT FROM AUTHOR]

The complete classification of (primitive, generic) parallelohedra in a given dimension is a challenging computational task. Nearly 50 years have passed since the classification for the last dimension, n = 5, was completed. One application of such a classification is in solving the lattice sphere covering problem for the corresponding dimension. The paper by Dutour Sikirić & van Woerden [Acta Cryst. (2025), A81, https://doi.org/10.1107/S2053273324010143] marks a milestone in the classification effort for dimension n = 6. It provides a complete classification of all primitive iso‐edge domains; here primitive parallelohedra are identified based on their facet vectors. [ABSTRACT FROM AUTHOR]

We look for planar shapes with identical perimeter and area, and geometrical figures with equal surface area and volume. This builds on some previous brief comments about equable shapes and extends it into one higher dimension. For each type, there are finitely many solutions with integer side-lengths or defining parameter. These include one regular polygon, two rectangles, two right triangles, three sectors, one regular polyhedron, five rectangular prisms with commensurable side lengths, nine right-angle triangular prisms, four cylinders, and zero cones. This investigation reveals common methods for obtaining bounds on the parameters and interesting relationships between the equable shapes and figures. [ABSTRACT FROM AUTHOR]

Homogeneous films with tailored microporous structures are crucial for several applications; however, fabricating such films presents significant challenges. This is primarily because most microporous materials have crystal sizes in the nano‐ and micrometer ranges, which inevitably generates intergranular spaces in the films, thereby complicating the fabrication of these thin films. In this study, functionalized metal–organic polyhedra (MOPs) are used as discrete microporous units and assembled into homogenous microporous films. The generation of intergranular spaces is avoided while controlling packing parameters and film thicknesses. Initially, the MOP units, influenced by van der Waals forces between carbon chains of functionalized adipic acids, display an affinity to form spindle‐shaped blocks and islands. As the MOP concentration increases, these structures self‐assembled into a hexagonally packed structure with an in‐plane orientation and a maximum stacking of two layers of MOPs. By contrast, un‐functionalized MOPs form a disordered film structure owing to random agglomeration. Evidently, functionalized adipic acid influences the orientation of the MOP network films with uniformly distributed micropores, effectively preventing the formation of intergranular spaces. Additionally, formaldehyde adsorption and desorption experiments revealed that the MOP network films possess superior adsorption and desorption capacities. The proposed approach signifies a breakthrough in the fabrication of homogenous microporous films. [ABSTRACT FROM AUTHOR]

In this article, the structure of the Clifford-Weyl superalgebras and their associated Lie superalgebras will be investigated. These superalgebras have a natural supersymmetric inner product which is invariant under their Lie superalgebra structures. The Clifford-Weyl superalgebras can be realized as tensor product of the algebra of alternating and symmetric tensors respectively, on the even and odd parts of their underlying superspace. For Physical applications in elementary particles, we add star structures to these algebras and investigate the basic relations. Ortho-symplectic Lie algebras are naturally present in these algebras and their representations on these algebras can be described easily. [ABSTRACT FROM AUTHOR]

In this paper, we present two methods to find the strictly efficient and weakly efficient points of multi-objective programming (MOP) problems in which their objective functions are pseudo-convex and their feasible sets are polyhedrons. The obtained efficient solutions in these methods are the extreme points. Since the pseudo-convex functions are quasi-convex as well, therefore the presented methods can be used to find efficient solutions of the (MOP) problem with the quasi-convex objective functions and the polyhedron feasible set. Two experimental examples are presented. [ABSTRACT FROM AUTHOR]

The skeletons of sharks and rays, fashioned from cartilage, and armored by a veneer of mineralized tiles (tesserae) present a mathematical challenge: How can the continuous covering be maintained as the skeleton expands? This study, using microCT and custom visual data analyses of growing stingray skeletons, systematically examines tessellation patterns and morphologies of the many thousand interacting tesserae covering the hyomandibula (a skeletal element critical to feeding), over a two‐fold developmental change in hyomandibula length. The number of tesserae remains surprisingly constant, even as the hyomandibula expands isometrically, with all hyomandibulae displaying self‐similar distributions of tesserae shapes/sizes. Although the distribution of tesserae geometries largely agrees with the rules for polyhedra tiling of complex surfaces—dominated by hexagons and a minor fraction of pentagons and heptagons, but very few other polygons—the agreement with Euler's classic mathematical laws is not perfect. Contrary to the assumed uniform growth rate (which is shown would create geometric incompatibilities), larger tesserae grow faster to accommodate skeletal expansion. It is hypothesized that this local regulation of global system complexity is driven by tension (from cartilaginous core expansion) in the fibers connecting tesserae, with strain‐responsive cells orchestrating local mineral apposition. [ABSTRACT FROM AUTHOR]

Addressing the limitations of existing reconstruction technologies, which are constrained by the Manhattan model assumption and exhibit insufficient applicability to irregular building shapes, as well as issues related to low model accuracy, dense triangular mesh faces, and weak robustness to chaotic and incomplete point clouds, we propose a 3D reconstruction technology specifically designed for irregular buildings, free from the constraints of the Manhattan model assumption. The proposed method includes a grid outlier removal technique based on eightconnected domains, a one-point Random Sample Consensus (RANSAC) method utilizing single points and their normal vectors for random plane segmentation to extract building planar structural elements, and a technique for 3D reconstruction of irregular buildings based on the selection of 3D polyhedral mesh faces. Validation with various datasets demonstrates that this method offers significant advantages over other reconstruction approaches in terms of model triangulation lightweightness and topological structure correctness. [ABSTRACT FROM AUTHOR]

This mini‐review focuses on recent developments in hydroxamate‐based metal–organic frameworks (MOFs), which exhibit unique structures and properties distinct from those of carboxylate‐based MOFs. Hydroxamates (RCONHO−) form MOFs with novel structural motifs and functionalities. In this review, synthetic strategies, structural characteristics, and functional applications of key examples of hydroxamate‐based MOFs are described, providing insights into the influence of the hydroxamate ligand on the MOF properties compared to that of the carboxylate‐based analogues. [ABSTRACT FROM AUTHOR]

Deeply electrolytic reduction of carbon dioxide (CO2) to high‐value ethylene (C2H4) is very attractive. However, the sluggish kinetics of C−C coupling seriously results in the low selectivity of CO2 electroreduction to C2H4. Herein, we report a copper‐based polyhedron (Cu2) that features uniformly distributed and atomically precise bi‐Cu units, which can stabilize *OCCO dipole to facilitate the C−C coupling for high selective C2H4 production. The C2H4 faradaic efficiency (FE) reaches 51 % with a current density of 469.4 mA cm−2, much superior to the Cu single site catalyst (Cu SAC) (~0 %). Moreover, the Cu2 catalyst has a higher turnover frequency (TOF, ~520 h−1) compared to Cu nanoparticles (~9.42 h−1) and Cu SAC (~0.87 h−1). In situ characterizations and theoretical calculations revealed that the unique Cu2 structural configuration could optimize the dipole moments and stabilize the *OCCO adsorbate to promote the generation of C2H4. [ABSTRACT FROM AUTHOR]

Deeply electrolytic reduction of carbon dioxide (CO2) to high‐value ethylene (C2H4) is very attractive. However, the sluggish kinetics of C−C coupling seriously results in the low selectivity of CO2 electroreduction to C2H4. Herein, we report a copper‐based polyhedron (Cu2) that features uniformly distributed and atomically precise bi‐Cu units, which can stabilize *OCCO dipole to facilitate the C−C coupling for high selective C2H4 production. The C2H4 faradaic efficiency (FE) reaches 51 % with a current density of 469.4 mA cm−2, much superior to the Cu single site catalyst (Cu SAC) (~0 %). Moreover, the Cu2 catalyst has a higher turnover frequency (TOF, ~520 h−1) compared to Cu nanoparticles (~9.42 h−1) and Cu SAC (~0.87 h−1). In situ characterizations and theoretical calculations revealed that the unique Cu2 structural configuration could optimize the dipole moments and stabilize the *OCCO adsorbate to promote the generation of C2H4. [ABSTRACT FROM AUTHOR]

Deeply electrolytic reduction of carbon dioxide (CO2) to high‐value ethylene (C2H4) is very attractive. However, the sluggish kinetics of C−C coupling seriously results in the low selectivity of CO2 electroreduction to C2H4. Herein, we report a copper‐based polyhedron (Cu2) that features uniformly distributed and atomically precise bi‐Cu units, which can stabilize *OCCO dipole to facilitate the C−C coupling for high selective C2H4 production. The C2H4 faradaic efficiency (FE) reaches 51 % with a current density of 469.4 mA cm−2, much superior to the Cu single site catalyst (Cu SAC) (~0 %). Moreover, the Cu2 catalyst has a higher turnover frequency (TOF, ~520 h−1) compared to Cu nanoparticles (~9.42 h−1) and Cu SAC (~0.87 h−1). In situ characterizations and theoretical calculations revealed that the unique Cu2 structural configuration could optimize the dipole moments and stabilize the *OCCO adsorbate to promote the generation of C2H4. [ABSTRACT FROM AUTHOR]

Deeply electrolytic reduction of carbon dioxide (CO2) to high‐value ethylene (C2H4) is very attractive. However, the sluggish kinetics of C−C coupling seriously results in the low selectivity of CO2 electroreduction to C2H4. Herein, we report a copper‐based polyhedron (Cu2) that features uniformly distributed and atomically precise bi‐Cu units, which can stabilize *OCCO dipole to facilitate the C−C coupling for high selective C2H4 production. The C2H4 faradaic efficiency (FE) reaches 51 % with a current density of 469.4 mA cm−2, much superior to the Cu single site catalyst (Cu SAC) (~0 %). Moreover, the Cu2 catalyst has a higher turnover frequency (TOF, ~520 h−1) compared to Cu nanoparticles (~9.42 h−1) and Cu SAC (~0.87 h−1). In situ characterizations and theoretical calculations revealed that the unique Cu2 structural configuration could optimize the dipole moments and stabilize the *OCCO adsorbate to promote the generation of C2H4. [ABSTRACT FROM AUTHOR]

A BDM type of H (div) mixed finite element is constructed on polygonal and polyhedral meshes. The flux space is the H (div) subspace of the n -product Π i P k (T i) d space such that the divergence is a one-piece P k − 1 polynomial on the big polygon or polyhedron T. Here we assume the 2D polygon can be subdivided into triangles by connecting only one vertex with some vertices of the polygon. For the 3D polyhedron we assume it can be subdivided into tetrahedra, with no added vertex on subdividing its face-polygons, and with either no internal edge or one internal edge. Such mixed finite elements can be more economic on quadrilateral and hexahedral meshes, compared with the standard BDM mixed element on triangular and tetrahedral meshes. Numerical tests and comparisons with the triangular and tetrahedral BDM finite elements are provided. [ABSTRACT FROM AUTHOR]

As a prominent and appealing technology, neural networks have been widely applied in numerous fields, with one of the most notable applications being autonomous driving. However, the intrinsic structure of neural networks presents a black box problem, leading to emergent security issues in driving and networking that remain unresolved. To this end, we introduce a novel method for robust validation of neural networks, named as Dual Boundary Robust (DBR). Specifically, we creatively integrate adversarial attack design, including perturbations like outliers, with outer boundary defenses, in which the inner and outer boundaries are combined with methods such as floating-point polyhedra and boundary intervals. Demonstrate the robustness of the DBR's anti-interference ability and security performance, and to reduce the black box-induced emergent security problems of neural networks. Compared with the traditional method, the outer boundary of DBR combined with the theory of convex relaxation can appropriately tighten the boundary interval of DBR used in neural networks, which significantly reduces the over-tightening of the potential for severe security issues and has better robustness. Furthermore, extensive experimentation on individually trained neural networks validates the flexibility and scalability of DBR in safeguarding larger regions. [ABSTRACT FROM AUTHOR]

Experimentally accessible stable polyhedral clusters of gallium differ from clusters of its lighter homologue boron by having sterically demanding external groups such as tBu3Si, (Me3Si)3Si, (Me3Si)3C, (Me3Si)2N, and fluorenyl. This restricts closo deltahedral [GanRn]z gallane chemistry to octahedral derivatives such as Ga6R62− (R = SiBuPh2, SiBut3) and Ga6R6 (R = SiMe(SiMe3)2) in which the bulky external silyl groups are spread out because of the relatively high local curvature of the degree 4 vertices of the central Ga6 octahedron. The structures of larger gallium clusters are based on alternatives to closo deltahedra having exclusively high local curvature degree 4 vertices such as the 8-vertex square antiprismatic Ga8R8 (R = fluorenyl) or, more commonly, larger polyhedra or fused polyhedra with some bare gallium vertices. Some of the larger gallium clusters can be considered to be spherically aromatic systems with closed shells according to the jellium model. Examples include [Ga13(SitBu3)6]− with 8 cluster electrons corresponding to a filled 1S21P6 shell as well as [Ga@Ga18{C(SiMe3)3}6]− with 52 cluster electrons, Ga22{E(SiMe3)3]8, (E = C, Si) and Ga@Ga11{GaN(SiMe3)2}11 with 58 skeletal electrons, and [Ga26{Si(SiMe3)3}8]2− with 72 cluster electrons. The 10-vertex species Ga10{Si(SiMe3)3}6, [Ga10(SitBu3)6]−, and Ga10(SitBu3)6 have structures consisting of a fusion of two polyhedra sharing bare gallium vertices with skeletal electron counts, suggesting a multicenter core bond in each of the two polyhedral cavities. [ABSTRACT FROM AUTHOR]

It is conjectured since long that for any convex body P ⊂ R n there exists a point in its interior which belongs to at least 2 n normals from different points on the boundary of P. The conjecture is known to be true for n = 2 , 3 , 4 . We treat the same problem for convex polytopes in R 3 . It turns out that the PL concurrent normals problem differs a lot from the smooth one. One almost immediately proves that a convex polytope in R 3 has 8 normals to its boundary emanating from some point in its interior. Moreover, we conjecture that each simple polytope in R 3 has a point in its interior with 10 normals to the boundary. We confirm the conjecture for all tetrahedra and triangular prisms and give a sufficient condition for a simple polytope to have a point with 10 normals. Other related topics (average number of normals, minimal number of normals from an interior point, other dimensions) are discussed. [ABSTRACT FROM AUTHOR]

Organic–inorganic metal halides (OIMHs) exhibit a rich structural diversity and excellent semiconductor properties, making them attractive candidates in the field of nonlinear optics (NLO). Second harmonic generation, as an important NLO property, imposes strict symmetry constraints, requiring noncentrosymmetric structures. Currently, a key task in the NLO field is to design noncentrosymmetric structures for OIMHs and explore the relationship between structure and NLO performance. This will help further optimize the application of OIMHs in NLO materials. The design of noncentrosymmetric structures for OIMHs can be facilitated by the use of chiral organic cations or distorted inorganic polyhedra, both of which induce structural features that favor noncentrosymmetry. The NLO properties of OIMHs are summarized the relationship between structure and performance is analyzed, and the research directions in the field of NLO for these materials are further elaborated. [ABSTRACT FROM AUTHOR]

Polyhedron gold nanocrystals enclosed by high-index facets (HIF-Au NCs) are in high demand but are very difficult to prepare. To address this issue, we presented a simple, seedless method for synthesizing uniform HIF-Au NCs in an aqueous solution, which remarkably reduced the synthesis difficulty. Interestingly, the protonated N2H4 which served as both the reducing and capping agent played a crucial role in modulating the kinetic growth of the HIF-Au NCs. The resulting HIF-Au NCs exhibited distinct electronic oxidation inertness toward alcohol but demonstrated exceptional activity in the electrocatalytic oxidation of peroxides. To demonstrate their sensing capabilities, an electrode decorated with HIF-Au NCs was used to selectively detect benzoyl peroxide (BPO) in flour. BPO is a prohibited whitening agent that may be illegally added to flour and other products, posing potential health risks. The results demonstrate that this assay offers a promising method for the sensitive and selective detection of BPO. In conclusion, this research provides a straightforward pathway for obtaining HIF-Au NCs and further demonstrates their use in electronic sensing. It is expected that HIF-Au NCs will serve as a powerful tool in plasmon-enhanced spectroscopies, catalysis, and sensing applications. [ABSTRACT FROM AUTHOR]

We present PolyGNN, a polyhedron-based graph neural network for 3D building reconstruction from point clouds. PolyGNN learns to assemble primitives obtained by polyhedral decomposition via graph node classification, achieving a watertight and compact reconstruction. To effectively represent arbitrary-shaped polyhedra in the neural network, we propose a skeleton-based sampling strategy to generate polyhedron-wise queries. These queries are then incorporated with inter-polyhedron adjacency to enhance the classification. PolyGNN is end-to-end optimizable and is designed to accommodate variable-size input points, polyhedra, and queries with an index-driven batching technique. To address the abstraction gap between existing city-building models and the underlying instances, and provide a fair evaluation of the proposed method, we develop our method on a large-scale synthetic dataset with well-defined ground truths of polyhedral labels. We further conduct a transferability analysis across cities and on real-world point clouds. Both qualitative and quantitative results demonstrate the effectiveness of our method, particularly its efficiency for large-scale reconstructions. The source code and data are available at https://github.com/chenzhaiyu/polygnn. [Display omitted] • An end-to-end polyhedron-based graph neural network for 3D building reconstruction from point clouds. • A skeleton-based sampling strategy that efficiently represents arbitrary-shaped polyhedra within the neural network. • Demonstrated transferability on cross-city synthetic point clouds and a real-world airborne LiDAR point cloud. [ABSTRACT FROM AUTHOR]

Mastication is an essential and preliminary step of the digestion process involving fragmentation and mixing of food. Controlled muscle movement of jaws with teeth executes crushing, leading towards fragmentation of food particles. Understanding various parameters involved with the process is essential to solve any biomedical complication in the area of interest. However, exploring and analyzing such process flow through an experimental route is challenging and inefficient. Computational techniques such as discrete element numerical modeling can effectively address such problems. The current work employs the Discrete Element Method (DEM) as a numerical modeling technique to simulate the human mastication process. Tavares and Ab‐T10 breakage models coupled with Gaudin Schumann and Incomplete Beta fragment distribution models have been implemented to analyze the fragmental distribution of food particles. The effect of particle shape (spherical, polyhedron, and faceted cylinder), size (aspect ratio), and orientation (vertical and horizontal) on breakage and fragment distribution is analyzed. To account for the elastic–plastic behavior and moisture content in food particles, modifications has been made in breakage models by incorporating numerical softening factor and adhesion force. The study demonstrates how numerical modeling techniques can be utilized to analyze the mastication process involving multiple process parameters. [ABSTRACT FROM AUTHOR]

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Local structure of the TiO2-FeO-Al2O3 system was investigated by molecular dynamics simulations to disclose the essential reason for variation of its transport characteristics in this work. [TiO6] octahedron and [AlO4] tetrahedron form the skeleton of network structure in the system. There are two different connection ways for both [TiO6] octahedra and [AlO4] tetrahedra: corner-sharing and edge-sharing. There is a conversion between corner-sharing polyhedron and edge-sharing polyhedron with the various FeO contents. The conversion between non-bridging oxygen and tri-coordinated oxygen directly determines the complexity degree of structure and even transport characteristics of the system. The sequence of diffusion abilities of different ions is displayed as follows: Fe2+ > Al3+ > O2− > Ti4+. The viscosity of the system increases from 0.045 to 0.067 Pa s, as the FeO content decreases from 36 to 4 pct. The model between viscosity and local structural parameter of the system is constructed. The removal of FeO can increase the polymerization degree and the viscosity of the system. The findings will help to establish the theoretical foundation for the green and efficient preparation of high-grade titania slag. [ABSTRACT FROM AUTHOR]

The catalytic activity of metal catalyst is closely related to its particle size. Yet, the size effect in electrocatalytic oxygen reduction reaction (ORR), an important reaction for metal‐air batteries and fuel cells, has not been clearly studied. Herein, a two‐step anchoring method is utilized to control the Fe catalyst in forms of nanoparticles (NPs), ultrasmall nanoclusters (NCTs), and isolated atoms as well as stabilized and dispersed by carbon polyhedrons interconnected with carbon nanotubes (CNTs). The uniformly distributed Fe NCTs displays superior ORR performance compared with Fe NPs, isolated Fe atoms, and commercial Pt/C. The brilliant ORR activity of Fe NCTs is a result of its unique electron structure and abundant edge and corner active sites. Due to the porous structure of carbon polyhedrons and high electron conductivity of CNTs, Fe NCTs also delivers an excellent discharge performance in zinc‐air battery with a peak power density of 213.3 mW cm−2 and long‐term stability. In these findings, a new strategy for the design of metal NCTs catalysts applied in various catalytic reactions is opened up. [ABSTRACT FROM AUTHOR]

Due to the complex shape of corn kernels and the large difference in size, in order to establish a more realistic model of corn kernels, this paper tests and analyzes two representative corn varieties planted in Northern China, and each variety is divided into high and low moisture content as the research object. According to its natural shape, the corn kernels are subdivided into four categories: wide horse-tooth shape, narrow horse-tooth shape, pyramid shape and spherical shape. The geometric parameters are tested and analyzed, and the correlation between the characteristic dimensions is obtained. Through the Rocky software, the polyhedral modeling method was used to establish the corn kernel models of different shapes, and the contact parameters and mechanical parameters were measured and calibrated, which were used as the basis for the modeling of corn kernel groups. Finally, through the simulation analysis of the corn kernel accumulation process and the comparison of the test results, the influence of different contact parameters on the simulation test results is explored, and the influence trend of the simulation results is analyzed. The feasibility and accuracy of the corn kernel population model established by the polyhedron method used in this paper are verified, which provides a reference for studying the simulation parameters of corn kernel population modeling. [ABSTRACT FROM AUTHOR]

Single‐phase white‐light emitters in the lighting technology overcome the challenges of re‐absorption and color shift. However, achieving the broadband white emission in single‐phase phosphor remains a challenge. Here, crystallized KBaScSi3O9:Eu2+ (KBSSO:Eu2+) glass‐ceramics (GC) with abundant cationic sites have been developed and an ultra‐broad emission band ranging from 460 to 800 nm have been found, in which the asymmetric green emission is ascribed to the Eu2+ ions in [KO8] and [BaO7] polyhedra, while the red emission centered at 620 nm is originated from Eu2+ ions in [ScO6] sites. The optimized GC‐converted white LED (WLED) exhibits a high color rendering index (Ra = 89.8) and low correlated color temperature (CCT = 4241 K). Moreover, the encapsulant‐free GCc‐WLED can operate continuously under high drive currents. This work paves the way for the design and future development of single‐phase high‐power white light sources. [ABSTRACT FROM AUTHOR]

Electrode active materials are the foremost component for energy related electrocatalysis and storage. Rational design and controlled synthesis of hybrid nanostructures is essential to achieve high efficiency but still challenging. Herein, we reported the synthesis of CoS2 nanoparticles embedded in the hollow polyhedron of N‐doped carbon nanotubes (CoS2‐NCNHP), which shows improved electrocatalytic performance for hydrogen evolution reaction (HER) due to the strong synergy effect and the peculiar hollow polyhedral structure. It only needs 158 and 192 mV vs. RHE to arrive the current density to 10 mA cm−2 in 0.5 M H2SO4 and 1 M KOH, and the activity can be kept for at least of 24 h with negligible loss. Density Functional Theory (DFT) calculations suggest more electrons with quickly round‐trip transferred rate in CoS2‐NCNHP can undergo the HER process, CoS2‐NCNHP surface shows optimal interaction with H*, and greatly accelerates the dissociation of H2O molecules. Moreover, CoS2‐NCNHP worked as cathode sulfur host material for aluminum‐sulfur (Al−S) batteries, it exhibits a remarkable increase in the discharge specific capacity, the specific discharge capacity of the battery in the first circle is 515.46 mAh g−1 at 1 A g−1, and the specific discharge capacity can still be maintained at 110 mAh g−1 after 100 cycles, along with a notable reduction in battery polarization owning to the hollow structure enhanced sulfur utilization. [ABSTRACT FROM AUTHOR]

We obtain the conditions under which a given multilayer differential operator P(D) (polynomial P(ξ)) is more powerful than operator Q(D) (polynomial Q(ξ)). This is used to obtain estimates of monomials, which, in turn, using the theory of Fourier multipliers, is used to obtain coercive estimates of derivatives of functions through the differential operator P(D) applied to these functions. [ABSTRACT FROM AUTHOR]

The extensive use of pesticides to ensure global food production presents significant threats to both human health and the environment. In this study, the photochemical, redox, and acid‐base properties of Zr‐metal‐organic polyhedra (Zr‐MOP) and polyoxometalates (POM) are utilized to fabricate functional hybrid salts, denoted as [Zr‐MOP][POM], designed for pesticide detoxification. Upon light irradiation, [Zr‐MOP][POM] salts generate a charge‐separated state with oxidizing properties. This activated state selectively photooxidizes highly toxic and persistent fenamiphos into fenamiphos sulfoxide and subsequently hydrolytically breaks down it into non‐toxic 3‐methyl‐4‐(methylsulfinyl)phenol and phosphate components. The hydrolytic degradation of fenamiphos is facilitated by the reduced basicity of the 3‐methyl‐4‐(methylsulfinyl)phenolate residue (ΔpKb = ‐2.4) in the photooxidized form of the pesticide molecule. Furthermore, the hybrid salts demonstrate efficacy in the breakdown of multicomponent pesticide mixtures such as fenamiphos and methylparaoxon. [ABSTRACT FROM AUTHOR]

This work falls within the general framework of robust shape optimization under constraints, where a physical parameter of the problem is poorly known. In particular, we study problems where one of the constraints concerns the maximal possible value that a given shape functional can assume when the uncertain parameter varies within an admissible range. Two different approaches are considered: the first one based on the approximation of the set of admissible uncertain parameters by a convex polyhedron, and the second one relying on the notion of subdifferential in the sense of Clarke. The main contributions of this work consist in the theoretical proof of convergence of the first approach under suitable hypotheses of convexity of the set of admissible parameters and of the constraint, and the adaptation of Clarke's subdifferential in the context of robust shape optimization. The two techniques are compared numerically in three examples, with the objective of minimizing the volume of elastic structures under a constraint on the worst-case scenario for the mechanical compliance and the von Mises stress. [ABSTRACT FROM AUTHOR]

Rounded shapes are associated with softness and warmth, whereas Platonic solids are associated with hardness and coldness. We investigated the temperature-shape association through sensorial/conceptual qualities of geometric ice-like textured shapes. In Experiment 1, participants viewed symmetrical rotating 3D shapes (five Platonic solids—cube, tetrahedron, octahedron, icosahedron, dodecahedron; a star polyhedron and a sphere) and control shapes (naturalistic and angular), rating them in terms of liking, hardness, temperature, wetness, and texture. In Experiment 2, participants visualized ice, and selected/rated, from 22 adjectives, those corresponding to the concept of ice. In Experiment 3, for each of the shapes from Experiment 1, participants chose the most appropriate conceptual attribute from among the six attributes most frequently reported in Experiment 2. All shapes looked cold. Liking and hardness ratings were similar for the ice-sphere and the Platonic solids, with an enhanced liking and the attribution of the "beautiful" concept for starlike ice shapes. The cube was appreciated as solid and the Platonic solids as strong and bright. Self-reported introversion, extroversion, and fitness level were significantly related to the appreciation of geometric ice structures. These findings are discussed in relation to crossmodal correspondences and the role of individual differences. [ABSTRACT FROM AUTHOR]

Several artists, artisans, and mathematicians described fascinating solid bodies in the fifteenth and sixteenth centuries. The knowledge they developed on the subject was so progressive that it is considered a milestone in the history of polyhedra. In the first part of this study we analyze, from a chronological and comparative perspective, the consistent studies developed between 1460 and 1583 on those that came to be recognized as Archimedean Solids. The authors who engaged in such studies were Piero della Francesca, Luca Pacioli, Leonardo da Vinci, Albrecht Dürer, Augustin Hirschvogel, an Anonymous Author who accomplished remarkable studies between 1538 and 1556, Wentzel Jamnitzer, Daniele Barbaro, Lorenz Stöer, Rafael Bombelli, and Simon Stevin. In the second part, we discuss how the revolutionary method of describing solid bodies with planar nets contributed to the rediscovery of the Archimedean Solids. We also present our interpretation of some of the studies by the Anonymous Author and our conclusions on his identity and influence on other authors. [ABSTRACT FROM AUTHOR]

This paper deals with linear integer optimization. We develop a technique that can be applied to provide improved upper bounds for two important questions in linear integer optimization. Given an optimal vertex solution for the linear relaxation, how far away is the nearest optimal integer solution (if one exists; proximity bounds)? If a polyhedron contains no integer point, what is the smallest number of integer parallel hyperplanes defined by an integral, nonzero, normal vector that intersect the polyhedron (flatness bounds)? This paper presents a link between these two questions by refining a proof technique that has been recently introduced by the authors. A key technical lemma underlying our technique concerns the areas of certain convex polygons in the plane; if a polygon K⊆R2 satisfies τK⊆K° , where τ denotes 90° counterclockwise rotation and K° denotes the polar of K, then the area of K° is at least three. Funding: J. Paat was supported by the Natural Sciences and Engineering Research Council of Canada [Grant RGPIN-2021-02475]. R. Weismantel was supported by the Einstein Stiftung Berlin. [ABSTRACT FROM AUTHOR]

Spatial distribution orientations of blocks can cause significant errors in the discrete element model (DEM) calculation of soil-rock mixture (SRM). To avoid this error, spherical harmonic (SH) series whose harmonization degrees fixed at 15 were proposed for block reconstruction. This research refers to the case-history of a deep excavation rift valley spanning from the Mabian to Zhaojue section of the Leshan-Xichang Expressway, mainly containing moderately-weathered silty mudstone, in the Leshan City, Sichuan Province, China. The appropriate degree of finite-term SH series is selected by the volume, surface area. 100 blocks were scanned on site, and sphericity and angularity of the blocks were calculated. The sphericity and angularity of 50 reconstructed blocks were considered for the error analysis of SH method. Moreover, stochastic polyhedron method was considered for comparing different block reconstructions. The maximum block placement angle was defined to control the spatial distribution orientations of the blocks. Large scale direct tests were carried. Numerical simulations of large-scale direct shear tests were conducted to study the influence of the spatial distribution orientations of the blocks on the mechanical properties of the SRMs. The results revealed that the finite-term SH series fixed at 15 accurately reflected the shape characteristics and mechanical behaviors of actual blocks. The spatial distribution orientations of the blocks had a minimal impact on the friction angle and cohesion of SRM constructed through the SH method. The SRMs developed via the SH method exhibited marginal variations in contact force and anisotropy index of contact across diverse block placement strategies. The evolution of coordination number was closer when employing the SH method under varied block placement methods. Blocks reconstructed by the SH method, could mitigate errors in DEM calculation caused by the spatial distribution orientations of the blocks. [ABSTRACT FROM AUTHOR]

In this paper, we present an integer programming formulation for the windy profitable location rural postman problem (WPLRPP). We state and prove a theorem concerning the dimension of the associated polyhedron. Then, we use this theorem to study some trivial facets of the presented polyhedron. Furthermore, we adapt and validate several large families of valid inequalities for the WPLRPP. We also develop an efficient branch-and-cut algorithm for solving large WPLRPP instances, which leverages those inequalities. We compare our presented branch-and-cut algorithm with other efficient algorithms in the literature. The numerical results show that our algorithm solves larger problem instances and requires considerably less computing time than other algorithms in the literature. [ABSTRACT FROM AUTHOR]

Mobile structures are one of directions of shaping recreational facilities. In terms of their geometry, geodesic domes and quasi-dome systems deserve special attention. Panel shell domes were the subject of consideration, e.g. Buckminster Fuller and David Geiger, whose patent solutions are referred to in the article. Combining a system of layered self-supporting panels with the geometry of geodesic domes was one of the significant construction and material challenges as part of the ReSa research and the respective implementation project carried out at the Wroclaw University of Technology in 2021-2023 as part of a competition organized by The National Center of Research and Development. Concerning the preliminary analysis, it was necessary to determine the types of dome solutions intended for the implementation of research models. The article presents a detailed analysis of shell geodesic domes with different geometries. The differentiation resulted from the type and the rotation of the base polyhedron relative to the base plane. The main objective of the study was to rank the types of panel geodesic domes in relation to their use in the construction of mobile recreational facilities. The development of 12 virtual dome models made it possible to evaluate their parameters in five main research areas - energy efficiency, environmental impact, support reactions, ergonomics and complexity of prefabrication and assembly process. An extensive evaluation of the parameters in each field, a summary within each domain, and a global evaluation of the shells were performed. The study allowed to develop a hierarchy of panel dome types in relation to the assumed criteria and to check the possibility of using multi-aspect, parametric evaluation. In addition, a detailed study of the geometrical parameters of the shells carried out as part of the indirect tests made it possible to identify the most effective structures in this aspect. This study has proven that parametric evaluation of criteria is a good tool for evaluating shell geodesic domes. This method is developing both in the field of scientific research and in terms of the implementation of domes. It allows for flexible introduction of basic data (here: energy efficiency, environmental impact, support reactions, ergonomics and complexity of prefabrication and assembly process), determination of coefficients and the use of additional weight criteria for individual research fields in relation to the assumed goal. [ABSTRACT FROM AUTHOR]