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During the detailed design phase of an aerospace program, one of the most important consistency checks is to ensure that no two distinct objects occupy the same physical space. Since exact geometrical modeling is usually intractable, geometry models are discretized, which often introduces small interferences not present in the fully detailed model. In this paper, we focus on computing the depth of the interference, so that these false positive interferences can be removed, and attention can be properly focused on the actual design. Specifically, we focus on efficiently computing the penetration depth between two polyhedra, which is a well-studied problem in the computer graphics community. We formulate the problem as a constrained five-variable global optimization problem, and then derive an equivalent unconstrained, two-variable nonsmooth problem. To solve the optimization problem, we apply a popular stochastic multistart optimization algorithm in a novel way, which exploits the advantages of each problem formulation simultaneously. Numerical results for the algorithm, applied to 14 randomly generated pairs of penetrating polytopes, illustrate both the effectiveness and efficiency of the method.

Computation of the volumes of polyhedra is a classical geometry problem known since ancient mathematics and preserving its importance until present time. Deriving volume formulas for 3-dimensional non-Euclidean polyhedra of a given combinatorial type is a very difficult problem. Nowadays, it is fully solved for a tetrahedron, the most simple polyhedron in the combinatorial sense. However, it is well known that for a polyhedron of a special type its volume formula becomes much simpler. This fact was noted by Lobachevsky who found the volume of the so-called ideal tetrahedron in hyperbolic space (all vertices of this tetrahedron are on the absolute).In this survey, we present main results on volumes of arbitrary non-Euclidean tetrahedra and polyhedra of special types (both tetrahedra and polyhedra of more complex combinatorial structure) in 3-dimensional spherical and hyperbolic spaces of constant curvature K = 1 and K = -1, respectively. Moreover, we consider the new method by Sabitov for computation of volumes in hyperbolic space (described by the Poincare model in upper half-space). This method allows one to derive explicit volume formulas for polyhedra of arbitrary dimension in terms of coordinates of vertices. Considering main volume formulas for non-Euclidean polyhedra, we will give proofs (or sketches of proofs) for them. This will help the reader to get an idea of basic methods for computation of volumes of bodies in non-Euclidean spaces of constant curvature.

This work deals with the weighted excesses of players in cooperative games which are obtained by summing up all the weighted excesses of all coalitions to which they belong. We first show that lexicographically minimizing the individual weighted excesses of players gives the same minimal weighted excess for every player. Moreover, we show that the associated payoff vector is the corresponding least square value. Second, we show that minimizing the variance of the players' weighted excesses on the preimputation set, again yields the corresponding least square value. Third, we show that these results give rise to lower and upper bounds for the core payoff vectors and, using these bounds, we define the weighted super core as a polyhedron that contains the core. It turns out that the least square values can be seen as a center of this weighted super core, giving a third new characterization of the least square values. Finally, these lower and upper bounds for the core inspire us to introduce a new solution for cooperative TU games that has a strong similarity with the Shapley value.

This article proposes a second-order conic programming (SOCP) approach to solve distributionally robust two-stage linear programs over 1-Wasserstein balls. We start from the case with distribution uncertainty only in the objective function and then explore the case with distribution uncertainty only in constraints. The former program is exactly reformulated as a tractable SOCP problem, whereas the latter one is proved to be generally NP-hard as it involves a norm maximization problem over a polyhedron. However, it reduces to an SOCP problem if the extreme points of the polyhedron are given as a prior. This motivates the design of a constraint generation algorithm with provable convergence to approximately solve the NP-hard problem. Moreover, the least favorable distribution achieving the worst case cost is given as an ``empirical'' distribution by simply perturbing each original sample for both cases. Finally, experiments illustrate the advantages of the proposed model in terms of the out-of-sample performance and computational complexity.

We discuss a new memory-efficient depth-first algorithm and its implementation that iterates over all elements of a finite locally branched lattice. This algorithm can be applied to face lattices of polyhedra and to various generalizations such as finite polyhedral complexes and subdivisions of manifolds, extended tight spans and closed sets of matroids. Its practical implementation is very fast compared to state-of-the-art implementations of previously considered algorithms. Based on recent work of Bruns, Garc\'ia-S\'anchez, O'Neill and Wilburne, we apply this algorithm to prove Wilf's conjecture for all numerical semigroups of multiplicity 19 by iterating through the faces of the Kunz cone and identifying the possible bad faces and then checking that these do not yield counterexamples to Wilf's conjecture., Comment: 17 pages including long examples and computational data; v2: added section 6 on Wilf's conjecture for numerical semigroups of multiplicity 19, added additional computational data

This mini review concerns hydrocarbons, (CH)n, and carbon allotropes, Cn, and their relationships with regular solids and regular surfaces, respectively. Platonic and Archimedean solids and surfaces related to carbon allotropes are described as an introduction. An overview is then provided on how Stone-Wales defects lead to a series of structures: pentaheptite, Haeckelite, net C, net W, planar C4, biphenylene, graphyne, graphdiyne, and tetraoctite. This last compound is discussed in detail together with its relation to the Mills-Nixon effect on cyclooctatetraene (COT).

This article presents MADER, a 3-D decentralized and asynchronous trajectory planner for UAVs that generates collision-free trajectories in environments with static obstacles, dynamic obstacles, and other planning agents. Real-time collision avoidance with other dynamic obstacles or agents is done by performing outer polyhedral representations of every interval of the trajectories and then including the plane that separates each pair of polyhedra as a decision variable in the optimization problem. MADER uses our recently developed MINVO basis to obtain outer polyhedral representations with volumes 2.36 and 254.9 times, respectively, smaller than the Bernstein or B-Spline bases used extensively in the planning literature. Our decentralized and asynchronous algorithm guarantees safety with respect to other agents by including their committed trajectories as constraints in the optimization and then executing a collision check-recheck scheme. Finally, extensive simulations in challenging cluttered environments show up to a 33.9% reduction in the flight time, and a 88.8% reduction in the number of stops compared to the Bernstein and B-Spline bases, shorter flight distances than centralized approaches, and shorter total times on average than synchronous decentralized approaches.

We present a framework to obtain valid inequalities for a reverse convex set: the set of points in a polyhedron that lie outside a given open convex set. Reverse convex sets arise in many models, including bilevel optimization and polynomial optimization. An intersection cut is a well-known valid inequality for a reverse convex set that is generated from a basic solution that lies within the convex set. We introduce a framework for deriving valid inequalities for the reverse convex set from basic solutions that lie outside the convex set. We first propose an extension to intersection cuts that defines a two-term disjunction for a reverse convex set, which we refer to as an intersection disjunction. Next, we generalize this analysis to a multi-term disjunction by considering the convex set's recession directions. These disjunctions can be used in a cut-generating linear program to obtain valid inequalities for the reverse convex set., 25 pages, 11 figures

The École des Mines de Saint-Étienne possesses an important collection of wooden and terracotta crystallographic models (several hundreds). During the 2000s, enhancement and rehabilitation action was undertaken. Several series of Krantz models were identified by the logos and linked to catalogs. The relation of these models to the important work of Paul Heinrich von Groth, in cooperation with the Krantz company, is highlighted. An exceptional element was discovered: a polyhedron with 170 facets, from the cubic system. After a short description of the model, we report on the digitalization work, followed by a description of two duplication methods: 3D printing and precision lost-wax vacuum casting.

This paper focuses on expanding the estimation of the domain of attraction (DOA) for discrete-time positive nonlinear systems subject to input saturation and parameter uncertainties. To facilitate analysis and design, the interval type-2 (IT2) polynomial fuzzy model is used to represent the nonlinear plant and capture uncertainties. Combining with the IT2 polynomial fuzzy controller, the discrete-time positive IT2 polynomial fuzzy-model-based (PIT2PFMB) control system is formed to facilitate analysis. To enlarge the estimation of DOA of the discrete-time PIT2PFMB system, polyhedron is used to characterize the DOA with the help of linear copositive Lyapunov function (LCLF). Referring to the non-convex conditions derived by LCLF, an effective convexification method is proposed in this paper. For comparison purposes, the saturation-dependent-Lyapunov-function-based method is extended to the PIT2PFMB control system by adding the corresponding positivity conditions. In addition, this paper attempts to enlarge the estimation of the DOA by improving the IT2 membership-function-dependent (IT2MFD) method and extending it to all conditions including the stability conditions and the DOA estimation conditions. Finally, an example with simulation results is given to verify the effectiveness of all the methods proposed in this paper for expanding the estimation of the DOA.

In this paper we present efficient algorithms to compute the radius, diameter, incenter, circumcenter, width and k dimensional enclosing cylinder for convex polyhedral and convex polyhedral offset distance functions in plane and in ℜ d for several types of inputs. We get optimal algorithms when the time complexities are linear. Let the size of input (convex polyhedron P /set S of points/convex polyhedra) be n / N , respectively, size of distance function convex polygon/polyhedron Q be m and size of constraint convex polyhedral region R be r in plane or ℜ d . The radius, incenter and circumcenter of the convex polygon P in ℜ 2 can be optimally computed in linear time, i.e. O ( n + m ) and in time O ( n + m + r ) , if the incenter or circumcenter is additionally constrained in a convex polygon/polyhedron R . In ℜ d , the radius, incenter and circumcenter of a convex polyhedron as well as of a set of convex polyhedra, unconstrained or constrained, can be computed in O ( N m ) and O ( N m + r ) time respectively, for convex constraint polyhedral region R . We also compute the minimum stabbing sphere for a set of convex polyhedra, unconstrained or constrained, for the above mentioned distance functions in O ( N m ) -time or O ( N m + r ) -time respectively for convex constraint polyhedral region R in ℜ d . The diameter of a convex polygon in plane can be computed in O ( n + m ) time and the diameter of a convex polyhedron in ℜ d can be computed in O ( n m ) time. The width of a convex polyhedron can be computed in O ( n + m ) time and O ( n 2 m ) time, for ℜ 2 and ℜ d , respectively. The diameter and width of a set of convex polyhedra can be computed in O ( N m ) -time and ( N 2 m ) , respectively, in any dimensions. We also show how the k dimensional minimum enclosing/stabbing cylinder for the convex polyhedron P can be computed in O ( n d − k + 2 m 2 ) in ℜ d and in O ( n d m ( n + m ) ) in ℜ d for k = 1 . The k dimensional minimum enclosing/stabbing cylinder for the set of convex polyhedra S in ℜ d can be computed in O ( N d − k + 2 m 2 ) -time. The diameter and width problems can also be solved for a set of points in ℜ d , either in time O ( n m ) or in time O ( m T ( n ) ) , where T ( n ) is the time complexity of the best convex hull computation algorithm for the given set of points.

In this work, we use the Δ -Invariance property of polyhedral sets to design a stabilizing Static Output Feedback (SOF) for linear discrete-time systems subject to persistent disturbances, assuring the states and control constraints fulfillment. We deduce new algebraic conditions to guarantee that any trajectory emanating from the Δ -Invariant polyhedron remains in it and converges in finite-time to another polyhedral set around the origin, where the trajectory remains ultimately bounded. Thus, the proposed SOF solution for the constrained control problem also requires determining the Δ -invariant and the ultimately bounded polyhedra. Therefore, the proposal considers a bilinear optimization problem whose objective function weighs the two associated polyhedral sets’ size and whose constraints are formed by the invariance relation. Moreover, an efficient non-linear optimization solver is used to tackle the present bilinearities. Numerical examples showcase the effectiveness and potential of our proposal.

The rare earth-rich indides Tm4IrIn and Lu4PtIn were synthesized by reaction of the elements in sealed tantalum ampules in an induction furnace. Tm4IrIn (a = 1340.77(4) pm) and Lu4PtIn (a = 1338.0(1) pm) crystallize with the Gd4RhIn-type structure, space group F 4 ‾ 3 m $F\overline{4}3m$ . The Lu4PtIn structure was refined from single crystal X-ray diffractometer data: wR = 0.0524, 517 F 2 values and 20 variables. The striking crystal chemical motif is the fcc packing of In4 tetrahedra with 318 pm In–In. The Lu4PtIn structure is closely related to the structures of Lu13Ni6In, Lu14Pd3In3 and Lu20Ir5In3 which all show icosahedral indium coordination and different condensation patterns that build up the indium substructure that consists of a dumbbell in Lu14Pd3In3 and a triangle in Lu20Ir5In3. The results of magnetic susceptibility measurements indicate Curie-Weiss paramagnetism for Tm4IrIn (7.76(1) µB per thulium atom) without magnetic ordering down to 2.5 K. Lu4PtIn is Pauli-paramagnetic.

For a continuous multi-objective optimization problem, it is usually not a practical approach to compute all its nondominated points because there are infinitely many of them. For this reason, a typical approach is to compute an approximation of the nondominated set. A common technique for this approach is to generate a polyhedron which contains the nondominated set. However, often these approximations are used for further evaluations. For those applications a polyhedron is a structure that is not easy to handle. In this paper, we introduce an approximation with a simpler structure respecting the natural ordering. In particular, we compute a box-coverage of the nondominated set. To do so, we use an approach that, in general, allows us to update not only one but several boxes whenever a new nondominated point is found. The algorithm is guaranteed to stop with a finite number of boxes, each being sufficiently thin.

In metal additive manufacturing, geometries with high aspect ratio (AR) features are often associated with defects caused by thermal stresses and other related build failures. Ideally, excessively high AR features would be detected and removed in the design phase to avoid unwanted failure during manufacture. However, AR is scale and orientation independent and identifying features across all scales and orientations is exceptionally challenging. Furthermore, not all high AR features are as easy to recognise as thin walls and fine needles. There is therefore a pressing need for further development in the field of problematic features detection for additive manufacturing processes. In this work, a dimensionless ratio (D1/D2) based on two distance metrics that are extracted from triangulated mesh geometries is proposed. Based on this method, geometries with different features (e.g. thin wall, helices and polyhedra) were generated and evaluated to produce metrics that are similar to AR. The prediction results are compared with known theoretical AR values of typical geometries.By combining this metric with mesh segmentation, this method was further extended to analyse the geometry with complex features. The proposed method provides a powerful, general and promising way to automatically detect high AR features and tackle the relevant defect issues prior to manufacture.

The integer linear programming approach to structural learning of decomposable graphical models led us earlier to the concept of a chordal graph polytope. An open mathematical question motivated by this research is what is the minimal set of linear inequalities defining this polytope, i.e. what are its facet-defining inequalities, and we came up in 2016 with a specific conjecture that it is the collection of so-called clutter inequalities. In this theoretical paper we give an implicit characterization of the minimal set of inequalities. Specifically, we introduce a dual polyhedron (to the chordal graph polytope) defined by trivial equality constraints, simple monotonicity inequalities and certain inequalities assigned to incomplete chordal graphs. Our main result is that the vertices of this polyhedron yield the facet-defining inequalities for the chordal graph polytope. We also show that the original conjecture is equivalent to the condition that all vertices of the dual polyhedron are zero-one vectors. Using that result we disprove the original conjecture: we find a vector in the dual polyhedron which is not in the convex hull of zero-one vectors from the dual polyhedron.

This paper shows the development of a web environment for the construction of Archimedes and Plato polyhedra in Augmented Reality (AR) and Virtual Reality (VR). In this environment we used the geometric transformations of translation and rotation with the structure of hierarchies of HTML pages, without the use of the coordinates of each polyhedra vertex. The developed environment can be used in classroom to visualize the polyhedra in Augmented Reality, with the possibility of manipulations of the graphical representations by students in the environment created in Virtual Reality. Other studies that can be developed with the polyhedra modeled are areas, volumes and the relation of Euler. Another important content that can be developed is truncation, because seven Archimedes polyhedra are obtained by using truncation of Plato's polyhedrons. With this work, it becomes possible to develop didactic materials with a simple technology, free and with great contribution to improvement of the teaching of Geometry and other areas that use representation of 3D objects.

Rational design and construction of hollow structured hybrid electrocatalysts are impressive for efficient and sustainable water splitting process. This study emphasizes a MOF-based self-templating...

Heterostructure thin films of indium and zinc oxides (IZO) were prepared by spray pyrolysis from an aqueous solution of the precursors at different substrate temperatures (TS). The polycrystalline structure of bixbyite appeared at a low temperature. The crystallinity was enhanced with the emergence of the zinc oxide phase. By increasing the TSto 623 K, the crystallite size was increased. SEM images reveal that the deposited sample at 523 K is composed of irregularly shaped nanoparticles with a lack of links. Increasing the TSto 573 K increases the average particle diameters, and the particles appeared as polyhedrons well connected with cavities between them, which candidates for gas sensing applications. Increasing TSto 623 K resulted in the particles merging. NO2gas sensor results confirmed the enhancement of IZO sensitivity performance at 573 K. Keywords: Gas sensor, thin film metal oxide, spray pyrolysis, In2O3– ZnO

We consider a $$(1+N)$$ -body problem in which one particle has mass $$m_0 \gg 1$$ and the remaining N have unitary mass. We can assume that the body with larger mass (central body) is at rest at the origin, coinciding with the center of mass of the N bodies with smaller masses (satellites). The interaction force between two particles is defined through a potential of the form $$\begin{aligned} U \sim \frac{1}{r^\alpha }, \end{aligned}$$ where $$\alpha \in [1,2)$$ and r is the distance between the particles. Imposing symmetry and topological constraints, we search for periodic orbits of this system by variational methods. Moreover, we use $$\Gamma $$ -convergence theory to study the asymptotic behaviour of these orbits, as the mass of the central body increases. It turns out that the Lagrangian action functional $$\Gamma $$ -converges to the action functional of a Kepler problem, defined on a suitable set of loops. In some cases, minimizers of the $$\Gamma $$ -limit problem can be easily found, and they are useful to understand the motion of the satellites for large values of $$m_0$$ . We discuss some examples, where the symmetry is defined by an action of the groups $${\mathbb {Z}}_4$$ , $${\mathbb {Z}}_2 \times {\mathbb {Z}}_2$$ and the rotation groups of Platonic polyhedra on the set of loops.

This letter converts one high-dimensional caging configuration into several lower-dimensional cage sets in the translation space and rotation space, through which analysis of grasping can be carried out more efficiently. A caging set detection method is introduced based on the constraint region. It is found that a caging on a high-dimensional convex constraint region can be converted into a caging set in low-dimensional sub-region. An efficient method to compute the caging of a polyhedral object using a one-parameter four-pin industrial gripper is then presented. Several simulations and real experiments are conducted to show the efficacy of the proposed methods.

Voronoi–Dirichlet (VD) polyhedra are used to analyze the structures of compounds that contain coordination polyhedra AnTen (An = Th, U, Np, Pu, Am, or Cm) in their crystal structures. It is found that tellurides contain An(III), An(IV), and U(V) atoms that coordinate 6 to 9 tellurium atoms. The described VD polyhedra of An and Te atoms demonstrate the possibility of using data on crystal structure to estimate the multiplicity of Te–Te bonds and the formal charge on tellurium atoms. The most important characteristics of homoatomic structural moieties (including linear and zig-zag chains formed by tellurium atoms) are discussed. It is shown that the formal charge (−1, −1.25, or −1.5) of tellurium atoms in linear chains is predetermined by the proportion of Te− to Te2− ions arranged statistically along the chain axis. It is established that all tellurides built from zig-zag chains of tellurium atoms contain U(V) atoms.

A real symmetric matrix A is copositive if $$x^\top Ax\ge 0$$ x ⊤ A x ≥ 0 for all $$x\ge 0$$ x ≥ 0 . As A is copositive if and only if it is copositive on the standard simplex, algorithms to determine copositivity, such as those in Sponsel et al. (J Glob Optim 52:537–551, 2012) and Tanaka and Yoshise (Pac J Optim 11:101–120, 2015), are based upon the creation of increasingly fine simplicial partitions of simplices, testing for copositivity on each. We present a variant that decomposes a simplex $$\bigtriangleup $$ △ , say with n vertices, into a simplex $$\bigtriangleup _1$$ △ 1 and a polyhedron $$\varOmega _1$$ Ω 1 ; and then partitions $$\varOmega _1$$ Ω 1 into a set of at most $$(n-1)$$ ( n - 1 ) simplices. We show that if A is copositive on $$\varOmega _1$$ Ω 1 then A is copositive on $$\bigtriangleup _1$$ △ 1 , allowing us to remove $$\bigtriangleup _1$$ △ 1 from further consideration. Numerical results from examples that arise from the maximum clique problem show a significant reduction in the time needed to establish copositivity of matrices.

Steinitz’s theorem states that a graph G is the edge-graph of a 3-dimensional convex polyhedron if and only if, G is simple, plane and 3-connected. We prove an analogue of this theorem for ball polyhedra, that is, for intersections of finitely many unit balls in $$\mathbb {R}^3$$ R 3 .

[Nb(η6-C6H3Me3)2] reacts with ethylenediamine (en) solutions of K4Ge9 in the presence of 18-crown-6 to give [(η6-C6H3Me3)NbHGe6]2- (1) and [(η6-C6H3Me3)NbGe6Nb(η6-C6H3Me3)]2- (2) as their corresponding [K(18-crown-6)]+ salts. The crystalline solids are dark brown, air-sensitive, and sparingly soluble or insoluble in most solvents. The [K(18-crown-6)]+ salts of cluster ions 1 and 2 have been characterized by energy-dispersive X-ray (EDX) analysis, NMR studies, single-crystal X-ray diffraction, and electrospray ionization time-of-flight (ESI-TOF) mass spectrometry studies. Cluster ions 1 and 2 have markedly different [Ge6] moieties: an electron-deficient carborane-like subunit in 1 and a two-center, two-electron cyclohexane-like subunit in 2.

Metal organic polyhedra (MOPs) such as coordination cages and clusters are increasingly utilized across many fields, but their geometrically selective assembly during synthesis is nontrivial. When ligand coordination along these polyhedral edges is arranged in an unsymmetrical mode or the bridging ligand itself is nonsymmetric, a vast combinatorial space of potential isomers exists complicating formation and isolation. Here we describe two generalizable combinatorial methodologies to explore the geometrical space and enumerate the configurational isomers of MOPs with discrimination of the chiral and achiral structures. The methodology has been applied to the case of the octahedron {Bi6Fe13L12} which has unsymmetrical coordination of a carboxylate ligand (L) along its edges. For these polyhedra, the enumeration methodology revealed 186 distinct isomers, including 74 chiral pairs and 38 achiral. To explore the programming of these, we then used a range of ligands to synthesize several configurational isomers. Our analysis demonstrates that ligand halo-substituents influence isomer symmetry and suggests that more symmetric halo-substituted ligands counterintuitively yield lower symmetry isomers. We performed mass spectrometry studies of these {Bi6Fe13L12} clusters to evaluate their stability and aggregation behavior in solution and the gas phase showing that various isomers have different levels of aggregation in solution., We describe combinatorial methodologies to explore the geometrical space and enumerate the configurational isomers of metal organic polyhedra with discrimination of the chiral and achiral structures. The methodology was applied to the octahedral {Bi6Fe13L12} which has an unsymmetrical coordination of a carboxylate ligands (L) along its edges. For these polyhedra, the enumeration methodology revealed 186 distinct isomers, including 74 chiral pairs and 38 achiral. We used a range of ligands to synthesize several configurational isomers.

Two-dimensional materials have already been considered as promising microwave absorbing materials due to their unique and charming layered structure. However, it is difficult to achieve excellent absorption performance at a thin thickness. Herein, the polyhedral Fe3O4 was prepared in-situ using sheet-shaped carbonyl iron powder, which not only retained the sheet-like structure of carbonyl iron powder but also formed polyhedral Fe3O4 on the surface. By tuning the ratio of Ti3C2TX/carbonyl iron powder/Fe3O4 microwave absorption capabilities in terms of the minimum reflection loss (RL) value and absorber thickness could be readily optimized. This result is verified by the explanation of the quarter-wavelength law and impedance matching degree. Specifically, when the thickness of the C–F–M 10% composite material is 2.2 mm, the minimum RL value at 9.16 GHz is − 52.1 dB, and the effective absorption bandwidth is 3.32 GHz. The thickness of the C–F–M 15% composite is 2.0 mm, the minimum RL value at 11.12 GHz is − 49.9 dB, and the effective absorption bandwidth is 4.59 GHz. Therefore, Ti3C2TX composite material is expected to become a candidate material for ultra-thin and lightweight absorbers.

Using computer methods (ToposPro software package), a geometric and topological analysis of the crystal structure of the intermetallic Ce80Pd128In284-cF492 (a = 21.838 A, V = 10414.79 A3, space group F-43m) is carried out. Two new nanocluster precursors with symmetry -43m are found: a two-layer nanocluster K61 composition In@16(Ce4In12)@44(Ce4In12Pd28) with an inner centered Friauf polyhedron In@16(Ce4In12) and with 44 atoms in the outer shell, and a two-layer nanocluster K42 of the composition 0@8(In8)@34(Ce6Pd4In24) with an inner polyhedron In8 and with 34 atoms in the outer shell. The symmetry and topological code of the processes of self-assembly of 3D structures from nanoclusters-precursors K61 and K42 is reconstructed in the following form: primary chain → microlayer → microframework. In4-tetrahedra with symmetry ‑43m, as well as spacer atoms Ce and In, are installed as spacers occupying voids in a 3D framework of nanoclusters K-61 and K-42.

Summary The design principle of supramolecular architectures through a bottom-up strategy has always been guided by polyhedra such as Platonic solids, Archimedean solids, and prisms (antiprisms) with high symmetry. Non-uniform convex polyhedra with regular faces, i.e., Johnson solids, are rarely constructed experimentally although their topologies have been predicted through mathematical theory. Herein, self-assembly of a giant triangular orthobicupola [Zn24612], one of 92 Johnson solids J27, has been achieved from the coordination between 24 Zn2+ and 12 tetrakis-terpyridine building blocks 6 in DMF (N,N-Dimethylformamide). Different from classical Oh or Td symmetric cuboctahedron, the structure of this triangular orthobicupola revealed D3 symmetry which has been unambiguously determined by synchrotron X-ray analysis, NMR, and mass spectrometry. Surprisingly, the self-assembly process has been found to be solvent dependent. The same Zn2+ and tetrakis-terpyridine ligand 6 assembled into another pentagonal prism with the composition of [Zn20610] when using methanol as the reaction solvent.

Discrete element method (DEM) is an effective approach to investigate non-spherical granular systems that are widely encountered in nature, industry, and our daily life. To describe the complex shapes of non-spherical particles, different particle shape representation approaches have been developed. In this paper, comparative study of modeling tablets using multi-sphere approach, multi-super-ellipsoid approach, and polyhedron approach is done to investigate the effect of different approaches. The accuracy of these approaches is testified by experiments and corresponding simulations of the packing, heap forming, and the flow in a rotating drum with two kinds of biconvex tablets firstly. Then the simulations of the flow of two kinds of biconvex tablets in a horizontal rotating drum are carried out by DEM. The simulation results are compared to discuss the effect of different approaches, which include the repose angle, the tablet orientation, the axial movement and mixing, the contact number, the percentage of the tablets reaching the free surface of the granular bed, and the computational efficiency. The results show that the simulations using multi-super-ellipsoid approach and polyhedron approach have some differences with those using multi-sphere approach.

Two chalcogenophosphates, SnPS2.86Se0.14 (1) and SnPSe3 (2), are isostructural and crystallize in the monoclinic noncentrosymmetric space group Pn. Their three-dimensional (3D) structures are constructed by [Sn(1)Q8] hendecahedra and [Sn(2)Q8] dodecahedra by sharing Q vertices and edges, leaving cavities for isolated [P2Q6] (Q = S/Se, Se) dimers. A second-harmonic-generation (SHG) measurement indicates that 1 is phase-matchable with a response of approximately 1.2 × AgGaS2 (AGS), which is verified by the theoretical calculation result. The powder sample of 1 exhibits a high laser-induced damage threshold of 3.9 × AGS. For comparison, the known SnPS3 (3) was also synthesized and evaluated using the same method. The chemical composition-NLO performance relationship of 1-3 is also discussed. Dipole moment calculation results suggest that [SnQ8] polyhedra make the main contribution to their excellent nonlinear optical (NLO) performance.

We study chiral polyhedra in 3-dimensional geometries (Euclidean, hyperbolic, and projective) in a unified manner. This extends to hyperbolic and projective spaces some structural results in the classification of chiral polyhedra in Euclidean 3-space given in 2005 by Schulte. Then, we describe a way to produce examples with helical faces based on a classic Petrie–Coxeter construction, yielding a new family in $$\mathbb {S}^3$$ which is described exhaustively.

The combination of RbB3 O4 F2 and NaF generates a new member of fluorooxoborates, NaRbB3 O4 F3 , with a wide transparency range from the IR to DUV region. NaRbB3 O4 F3 shows a three-dimensional (3D) structure composed of 1D [B3 O4 F3 ]∞ chains, [NaO3 F3 ] and [RbO5 F5 ] polyhedra. The structural evolution from NaRbB3 O4 F3 to RbB3 O4 F2 , as well as the structural comparison between NaRbB3 O4 F3 and its identical stoichiometry compound, Li2 B3 O4 F3 were discussed in detail. The IR spectrum verifies its structural validity. The spectral measurement shows that the reflectance has no obvious change in the range of 175-300 nm, and its cutoff edge is below 175 nm. In addition, theoretical calculations are carried out to understand its electronic structure and optical properties.

First-principles density functional theory computations are used to predict negative linear compressibility in hybrid organic-inorganic perovskite [NH2NH3][Co(HCOO)3]. Negative linear compressibility is a rare exotic response of a material to pressure associated with expansion along one or two lateral directions. Detailed structural analysis revealed that [NH2NH3][Co(HCOO)3] responds to pressure through tilting of its relatively rigid units, CoO6 polyhedra, and (HCOO)-1 ligand chain. The (HCOO)-1 units form a "wine-rack" geometry which is well described with the "strut-hinge" model. Within the model, the struts are formed by the rigid units, while hinges are their relatively flexible interconnects. Under pressure, the hinge angle increases which leads to the expansion along the direction subtended by the angle. Interestingly, at zero pressure the linear compressibilities in [NH2NH3][Co(HCOO)3] are all positive. As pressure increases, the lowest linear compressibility value turns negative and increases in magnitude. Comparison with the literature suggests that such a trend is likely to be common to this family of materials. Mechanical properties of [NH2NH3][Co(HCOO)3] are highly anisotropic.

Summary The synthesis of multi-walled nanostructures represents a formidable challenge owing to the limitations imposed by existing synthetic strategies. Herein, we report a strategy involving molecular-strain engineering (MSE) for constructing double-walled polyhedra from strained bow-shaped macrocycles called molecular bows (MBs). As a proof of concept, four double-walled tetrahedra have been assembled from L-shaped dual-panels subtending ∼70.5° and producing MBs, which incorporate porphyrin and o-xylylene units, forming strained shape-persistent conformations. The head-to-tail closed-loop stacking of these units, driven by complementary non-covalent bonding interactions, leads to a large porphyrin-lined tetrahedron encapsulating a small o-xylylene-lined tetrahedron. The synergistic aromatic shielding effects emanating from the porphyrin tetrahedron result in the largest upfield chemical shifts of −7.7 ppm for protons and −8.8 ppm for carbons from those for the free dual panels. This demonstration, which features the launching of MSE for constructing sophisticated supramolecular nanostructures, could herald a fresh approach to the fabrication of multi-porphyrin photocatalytic systems.

Liquid bismuth is an actual object for research due to the specific features experimentally detected on the temperature dependences of a number of properties, the nature of which is still debatable. Earlier, we used ab initio molecular dynamics simulation to calculate the temperature dependences of a number of properties of liquid bismuth, which demonstrated nonmonotonic behavior. The atomic coordinates determined by molecular dynamics simulation are used in this work for a further investigation of the structural features by the Voronoi polyhedron (VP) method. VPs are constructed in such a way that they fill the space inside a disordered structure without gaps and overlaps and are fundamental geometric figures, which allow us to analyze an atomic structure in terms of precise topological and metric characteristics. The following characteristics are calculated: the distribution over the distances to the nearest neighbors, the distribution over the number of VP faces, the distribution over the number of pentagonal faces, and so on. An analysis of the obtained characteristics of VPs allow us to trace the tendency of changing the structure of the Bi melt at temperatures above the melting point and to reveal deviations from monotonic behavior on the temperature dependences. In particular, the nonmonotonic behavior of the number of polyhedra with pentagonal faces is interesting: it increases sharply at an overheating of about 150 K above the melting point. On the whole, the results point to smooth restructuring of the short-range order in liquid Bi at a small overheating, which is reflected on the temperature dependences of the properties.

Many practical integer programming problems involve variables with one or two-sided bounds. Dunkel and Schulz (A refined Gomory–Chvatal closure for polytopes in the unit cube, http://www.optimization-online.org/DB_FILE/2012/03/3404.pdf , 2012) considered a strengthened version of Chvatal–Gomory (CG) inequalities that use 0–1 bounds on variables, and showed that the set of points in a rational polytope that satisfy all these strengthened inequalities is a polytope. Recently, we generalized this result by considering strengthened CG inequalities that use all variable bounds. In this paper, we generalize further by considering not just variable bounds, but general linear constraints on variables. We show that all points in a rational polyhedron that satisfy such strengthened CG inequalities form a rational polyhedron. We also extend this polyhedrality result to mixed-integer sets defined by linear constraints.

The structure of a new type of polyoxo cluster complex that contains thirteen uranium atoms, {U13}, is reported. The complex crystallized from methanol containing tetravalent uranium (UIV) with a basic organic ligand, and was characterized as dichloridooctacosa-μ2-methanolato-octakis(methanolato)octa-μ4-oxido-tridecauranium, [U13(CH3O)35.7Cl2.3O8] or [U13(μ4-Ooxo)8Cl x (MeO)38–x ] (x = 2.3, MeO = methoxide) (I), by single-crystal X-ray diffraction. The characterized {U13} polyoxo cluster complex (I) possesses a single cubic uranium polyhedron at the centre of the cluster core. To the best of our knowledge, this is the very first example of a polyoxo actinide complex that bears a single cubic polyhedron in its structure. The cubic polyhedron in I is well comparable in shape with those in bulk UO2. The U—O bonds in the cubic polyhedron of I are, however, significantly shorter than those not only in bulk UO2 but also in another analogue in the {U38} cluster. This shortening of U—O bonds, together with BVS calculations and the overall negative charge (2−) of I, suggests that the central uranium atom in I, which forms the single cubic coordination polyhedron, is presumably oxidized to the pentavalent state (UV) from the original tetravalent state (UIV). Complex I is, hence, the first example of a polyoxo cluster possessing a single cubic coordination polyhedron of UV.

In many practical applications models exhibiting chance constraints play a role. Since, in practice one is also typically interesting in numerically solving the underlying optimization problems, an interest naturally arises in understanding analytical properties, such as differentiability, of probability functions. However in order to build nonlinear programming methods, not only knowledge of differentiability, but also explicit formulae for gradients are important. Unfortunately, differentiability of probability functions cannot be taken for granted. In this paper, motivated by applications from energy management, wherein we face a variety of non-linear transforms of underlying Elliptical distributions, we investigate probability functions acting on decision dependent union of polyhedra. Union of polyhedra naturally occur as soon as one approaches the components of “difference-of-convex” (DC) functions with their respective cutting plane models. In this work, we will establish that the probability functions are locally Lipschitzian and exhibit explicit formulae for “the” Clarke sub-gradients, under very mild conditions. We also highlight, on a numerical example, that the formulae can be put to use “in practice”.

Several recent papers have examined a rational polyhedron Pm whose integer points are in bijection with the numerical semigroups (cofinite, additively closed subsets of the non-negative integers) containing m. A combinatorial description of the faces of Pm was recently introduced, one that can be obtained from the divisibility posets of the numerical semigroups a given face contains. In this paper, we study the faces of Pm containing arithmetical numerical semigroups and those containing certain glued numerical semigroups, as an initial step towards better understanding the full face structure of Pm . In most cases, such faces only contain semigroups from these families, yielding a tight connection to the geometry of Pm .

We have determined the boundaries of the existence and the structural features of the slab perovskite-like structure (SPS) of BaLa2–xSmxIn2O7 (0 ≤ х ≤ 1.8) and BaNd2–xSmxIn2O7 (0 ≤ х ≤ 1.8) systems have been established. The values of reliability factors RB are within the intervals of 0.037-0.057 for BaLa2–xSmxIn2O7 and 0.032-0.057 for BaNd2–xSmxIn2O7. The analysis of the structural features of SPS compounds and phases of the AIILn2B2IIIO7 (AII = Sr, Ba, BIII = Al, Fe, Sc, In) has revealed the interconnection between their composition and the features of the SPS interblock boundary. A decrease in the size of REE atoms in the series of compounds and phases of the AIILn2B2IIIO7 (AII = Sr, Ba) type leads to a gradual decrease in the interblock distance length (AII, Ln) — O and to a successive increase in the degree of deformation of interblock polyhedra (AII, Ln)O9. In the series of SrLn2B2IIIO7 (BIII = Al, Sc) compounds, a decrease in the size of REE atoms leads to a gradual change in the nature of the distribution of Sr and REE atoms in their SPS: from a partially ordered one (Ln = La) with the predominant localization of Sr atoms in the intrablock AO12 polyhedra to the completely ordered one (Ln = Eu, Gd, Tb, BIII = Sc) or almost completely ordered one (Ln = Ho, BIII = Al) with Sr a toms localized only in AO12 polyhedra, while smaller REE atoms are localized only in the interblock polyhedra. In the series of compounds and phases of the BaLn2B2IIIO7 type, there is a completely ordered arrangement of large Ba atoms in the intrablock AO12 polyhedra of their SPS, and the REE atoms only in the AO9 interblock polyhedra. It has been found that, in compounds the similar by the elemental composition to the An+1BnO3n+1- type, the thickness of the perovskite-like block significantly (by 2-6 times) increases the degree of deformation of the AO9 polyhedra and decreases the length of the A—O interblock bond. The analysis of the obtained data has shown that the values of the degree of deformation of the interblock polyhedra (АII, Ln)О9 and the length of the interblock bonds (АII, Ln)—О are among the main factors determining the stability of two-slab compounds of the AIILn2B2IIIO7-type.

ConspectusThe prediction of crystal structures assembled in three dimensions has been considered for a long time, simultaneously as a chemical wasteland and a certain growth point of the chemistry of the future. Less than 30 years after Roald Hoffmann's statement, we can categorically affirm that the elevation of reticular chemistry and the introduction of metal-organic frameworks (MOFs) significantly tackled this tridimensional assembly issue. MOFs result from the assembly of organic polytopic organic ligands bridging metal nodes, clusters, chains, or layers together into mostly three-periodic open frameworks. They can exhibit extremely high porosity and offer great potential as revolutionary catalysts, drug carrier systems, sensors, smart materials, and, of course, separation agents. Overall, the progressive development of reticular chemistry has been a game changer in materials chemistry during the last 25 years.Such diverse properties often result not only from the selected organic and inorganic molecular building blocks (MBBs) but also from their distribution within the framework. Indeed, the size and shape of the porous system, as well as the location of active sites influence the overall properties. Therefore, in the continuity of achieving the crystallization of three-periodic structures, chemists and crystal engineers faced the next challenge, as summarized by John Maddox: "it remains in general impossible to predict the structure of even the simplest crystallographic solids from knowledge of their chemical composition". This is where rational design takes place.In this Account, we detail three specific approaches developed by our group to facilitate the design and assembly of finely tuned MOFs. All are based on careful geometrical consideration and a deep study and understanding of the existing nets and topologies. We recognized that highly connected nets, if possible, edge-transitive, are ideal blueprints because their number is limited in contrast to nets with lower connectivity. Therefore, we embarked on taking advantage of existing highly connected MBBs, or, in parallel, promoting their formation to meet our requirements. This is achieved by utilizing externally decorated metal-organic polyhedra as supermolecular building blocks (SBBs), serving as a net-coding building unit, comprising the requisite connectivity and directional information coding for the chosen nets. The SBB approach allowed the synthesis of several families of SBB-based MOFs, including fcu, rht, and gea-MOFs, that are detailed here.The second strategy is directly inherited from the success of the SBB approach. In seeking highly connected building units, our group naturally expanded its research focus to nets that can be deconstructed into layers, pillared in various ways. In the supermolecular building layer (SBL) approach, the layers have an almost infinite connectivity, and the framework backbone is fixed in two dimensions while the third is free for pillar expansion and functionalization. The cases of trigonal pillaring leading to rtl, eea, and apo MOFs as well as the quadrangular pillaring leading to a family of tbo-MOFs are discussed here, along with recent cases of highly connected pillars in pek and aea-MOFs.Finally, our experience with highly coordinated MBBs led us to develop a novel way to use them as secondary building units of lower connectivity and unlock the possibility of assembling a novel class of zeolite-like MOFs (ZMOFs). The case of the Zr-sod-ZMOFs designed through a cantellation strategy is described as a future leading direction of MOF design.

We distinguish two kinds of piecewise linear functions and provide an interesting representation for a piecewise linear function between two normed spaces. Based on such a representation, we study a fully piecewise linear vector optimization problem with the objective and constraint functions being piecewise linear. To solve this problem, we divide it into some linear subproblems and structure a dimensional reduction method. Under some mild assumptions, we prove that its Pareto (resp., weak Pareto) solution set is the union of finitely many generalized polyhedra (resp., polyhedra), each of which is contained in a Pareto (resp., weak Pareto) face of some linear subproblem. Our main results are even new in the linear case and further generalize Arrow, Barankin and Blackwell’s classical results on linear vector optimization problems in the framework of finite-dimensional spaces.

In this paper, we propose a method for determining all minimal representations of a face of a polyhedron defined by a system of linear inequalities. Main difficulties for determining prime and minimal representations of a face are that the deletion of one redundant constraint can change the redundancy of other constraints and the number of descriptor index pairs for the face can be huge. To reduce computational efforts in finding all minimal representations of a face, we prove and use properties that deleting strongly redundant constraints does not change the redundancy of other constraints and all minimal representations of a face can be found only in the set of all prime representations of the face corresponding to the maximal descriptor index set for it. The proposed method is based on a top-down search strategy, is easy to implement, and has many computational advantages. Based on minimal representations of a face, a reduction of degeneracy degrees of the face and ideas to improve some known methods for finding all maximal efficient faces in multiple objective linear programming are presented. Numerical examples are given to illustrate the method.

In the ore milling process, the ore particles are usually irregular polyhedrons, and the grinding media are often spherical. The need for accurate simulation of this process calls for a contact algorithm between the polyhedron and sphere, achieving a balance of accuracy, stability, and efficiency. For this purpose, we propose a polyhedron-sphere contact model based on the deepest point method. Experiments and simulation studies by discrete element method (DEM) are conducted to test this model's accuracy in a hexahedral box and a lab-scale horizontal drum with lifters. Then, systematic research is carried out for an industrial semi-automatic grinding (SAG) mill to explore the effect of particle shape. To reveal the advantage over the traditional spherical DEM model, the polyhedron-sphere (PH-SP) grinding system is compared with a pure spherical grinding system. Results are discussed from the motion behavior particles, power consumption, collision energy on ore and liner, and liner wear. The results show that the charging of particles in the PH-SP grinding system is blocked to a certain extent by polyhedral particles, both the power consumption and energy utilization efficiency increase somewhat, the large energy collision between ore and liner has increased significantly. The liner wear of the PH-SP grinding system is more severe than the pure spherical grinding system, which indicates that the new model is necessary for more accurate prediction of the grinding process.

In this work, controlled-size hollow polyhedron assembled by crumpled nickel hydroxide (Ni(OH)2) nanosheets from silicon dioxide (SiO2)-covered zeolitic imidazole framework-67 (ZIF-67@SiO2) is prepared via a template-sacrificed method. It is found that SiO2 plays an essential role in keeping intact polyhedrons and suppressing particle growth. Benefiting from structural and compositional advantages, the Ni(OH)2@S electrode exhibits high specific capacity, excellent rate performance, and stable cycle life at 1C with a small capacity decay of 0.067% per cycle. The Ni(OH)2 hollow polyhedrons can accommodate the volume expansion to maintain the integrity of the electrode and suppress the shuttling effect of polysulfides via abundant hydroxyl groups. Hence, this strategy is beneficial to anticipate the material for large-scale applications.

Summary The chemical environment of pores is important for various applications of porous solids. Thanks to reticular chemistry, metal-organic frameworks (MOFs) have become a versatile platform targeting numerous applications through multiple functionalizations. Although multivariate MOFs often display novel properties, identifying and manipulating pore types remain a daunting challenge. Here, we present an isoreticular series of Zr-based metal-organic polyhedra (MOPs) as a porous platform to achieve controllable intrinsic pores. Two multivariate synthetic approaches were demonstrated: a mixed-cage strategy, whereby functionalized cages are mixed, compared with the conventional mixed-linker strategy, which yields a random distribution of functionalities. A remarkable difference in functionality assembly was achieved between the strategies, with complexity increasing from binary to senary systems. More interestingly, distinct photophysical properties were observed between mixed-linker and mixed-cage samples and attributed to radiative decay kinetics. This study highlights the potential of MOPs as a unique multivariate platform with tunable component assembly to study the emerging properties of multivariate porous solids.

Recently, a novel bio-inspired camera was developed by mimicking the retina fovea to continuously accumulate luminance intensity and then fire spikes once the dispatch threshold is reached. In contrast to the conventional frame-based cameras and the emerging dynamic vision sensors, this spike camera has shown remarkable advantages in capturing fast-moving scenes in a frame-free manner with full texture reconstruction capabilities. However, the ultra-high temporal resolution makes the transmission or storage of the output data of spike camera (referred to as spike data) quite difficult. To address the above challenges, we propose a unified lossy spike coding framework, which exploits the motion patterns hidden in the spike data distribution to design the motion-fidelity coding modes for the first time. We investigate the spatiotemporal distribution of spike data and propose an intensity-based measurement of the spike train distance. Then, the adaptive polyhedron partitioning is proposed to deal with the spike data with different motion characteristics. Finally, the intra-/inter-polyhedron prediction with spike-time and spike-rate modes, transform and multi-layer quantization are proposed and introduced into the codec. We also construct a PKU-Spike dataset captured by the spike camera to evaluate the compression performance. The experimental results on the dataset demonstrate that the proposed approach is effective in compressing such spike data while maintaining the visual fidelity especially for high-speed scenarios.