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Transformation from tetrahedral- to octahedral-network as OPF increases from 0.48 to 0.65. [Display omitted] • The correlation between structural characteristics and dynamics in relation to. • OPF. The competition between the increase of mobility due to structural. • transition and the decrease of mobility due to increase of packing fraction. • The structural phase transition happens at OPF of around 0.57–0.58. • The Diffusion coefficient can be calculated via concentration of GeOx polyhedrons and packing fraction (or free volume). Correlation between structure and dynamics of GeO 2 system under compression is investigated by molecular dynamics and Monte-Carlo methods. Results show that structural phase transition occurs at the range of oxygen packing factor from 0.55 to 0.60. Self-diffusion coefficient of Ge and O exhibits anomalous and depends on percentage of GeO x polyhedrons (x = 4,5,6) and packing factor/ free volume of model. The self-diffusion coefficient can be determined via percentage of GeO x units and free volume fraction. The data mining technique is applied to clarify the polyamorphism, structure and dynamics heterogeneities. The cause of anomalous diffusion has been elucidated. [ABSTRACT FROM AUTHOR]

Considering the spatial arrangement characteristics of bonded irregular particle systems, this paper first constructs irregular dilated polyhedron elements based on the Voronoi cutting algorithm and Minkowski Sum theory. Subsequently, by extending the traditional sphere-based BPM (Bonded Particle Model) to irregular particle elements, a novel dilated polyhedron bonding-breaking model considering different fracture modes is developed under arbitrary contact modes. A collision example between two bonded square particles and a rigid plate is adopted to verify the stability of the model. Later, the difference between the tensile-shear fracture mode and the mixed fracture mode with damage process for contact bond is compared by the discrete element simulations of the tension-fracture behavior between two bonded square particles and the uniaxial compression simulation of frozen ballast aggregate. Finally, the influence of section size on the lateral resistance of the ballast bed structure at low temperatures is discussed using the dilated polyhedron BPM. The results showed that the developed dilated polyhedron BPM considering different fracture modes can realize mechanical behaviours between bonded irregular particles. Compared with the tensile-shear fracture mode, the mixed fracture mode not only considers the combined effect of normal and tangential stress on the bond but also realizes the damage deformation and stiffness attenuation of the bond, which can achieve the "delayed" fracture effect. Meanwhile, the model also shows good reliability in analyzing large-scale engineering problems, such as the mechanical performance of low-temperature ballast beds. The research effectively develops the bonding and breaking model of irregular elements and provides a new perspective for the contact-bond-fracture behavior simulation of irregular granular materials. [Display omitted] • The bonded particle model for dilated polyhedron is developed and verified. • The differences between the two fracture modes are compared. • The lateral resistance performance of the low-temperature ballast bed is analyzed. [ABSTRACT FROM AUTHOR]

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. [Display omitted] • Two multi-spheres, multi-super-ellipsoid, and polyhedron are used to model tablets. • Four approaches are validated by experiments and corresponding simulations. • Comparative study of modeling tablets in rotating drum using four approaches is done. • Computational efficiencies of four approaches for modeling tablets are compared. [ABSTRACT FROM AUTHOR]

A composite contact algorithm for a complex particle system containing polyhedrons and super-ellipsoids is proposed in this paper to simulate the grinding process of ore in a ball mill. In this algorithm, the contacts between polyhedrons are calculated based on the overlapping volume method, and the contacts between super-ellipsoids are determined by a deepest point method. The contacts between super-ellipsoids and polyhedrons are obtained by transforming super-ellipsoids into polyhedrons, and then using the overlapping volume method between the polyhedrons. The increase in the number of faces of the super-ellipsoid contributes to the closer shape to the real shape. In this paper, experimental researches and numerical simulations are carried out to validate the accuracy of this composite contact algorithm. Furthermore, the influence of the ratio of super-ellipsoids and polyhedrons, the number of faces of converted super-ellipsoid, and the sharpness of the super-ellipsoid on the computational efficiency are compared in both box filling process and particles mixing process in a rotating tumbler. Finally, the stability and feasibility of this composite contact algorithm is tested in a ball mill application containing more particles. Unlabelled Image • A contact algorithm for polyhedron-super-ellipsoid particle system is proposed. • Experiments and numerical simulations are carried to validate this algorithm. • The computational efficiency of this composite contact algorithm is studied. • The stability and feasibility of this algorithm is tested in a ball mill application. [ABSTRACT FROM AUTHOR]

NiCo 2 S 4 polyhedral architectures with high surface area, stable architecture and good crystallinity successfully are prepared with NiCo 2 O 4 as precursors. Impressive electrochemical performances including high specific capacitance, good rate capability and cycling stability are achieved owing to the particular polyhedral structure with more active sites for ions reaction and sufficient space for ions transmission. The as-prepared NiCo 2 S 4 electrode for supercapacitor displays high specific capacitance of 1298 F g−1 at 1 A g−1 and a superior rate capability of ~80.4% at 20 A g−1. Also, capacitance retention of 90.44% can be realized after 8000 cycles. In addition, the NiCo 2 S 4 as anode delivers high initial charge/discharge capacities of 807.6 and 972.8 mAh g−1 at 0.5 C. Dual-functional NiCo 2 S 4 polyhedral architectures with outstanding electrochemical performance for supercapacitors and lithium-ion batteries (LIBs) have been rationally designed and successfully synthesized by a hydrothermal method. The as-synthesized NiCo 2 S 4 electrode for supercapacitor exhibits an outstanding specific capacitance of 1298 F g−1 at 1 A g−1 and an excellent rate capability of ~80.4% at 20 A g−1. Besides, capacitance retention of 90.44% is realized after 8000 cycles. In addition, the NiCo 2 S 4 as anode in LIBs delivers high initial charge/discharge capacities of 807.6 and 972.8 mAh g−1 at 0.5 C as well as good rate capability. In view of these points, this work provides a feasible pathway for assembling electrodes and devices with excellent electrochemical properties in the next generation energy storage applications. [ABSTRACT FROM AUTHOR]

Particle shape substantially influences the bulk behaviour of granular systems. Hence, much scientific interest has been devoted to the adoption of non-spherical fundamental particles in discrete element method simulations. Two examples of such particles are polyhedra, which are highly angular, and superquadrics, which are best suited to simulate rounded particles. It is desirable to use both types of particle together in a simulation to capture the broadest possible range of particle shapes. In this paper, a novel contact detection algorithm is presented for a convex polyhedron and superquadric. This algorithm was implemented in a C++ code which was used to verify the correctness of the algorithm and evaluate its efficiency using the Monte Carlo method. The proposed contact detection algorithm is particularly efficient for many-faceted polyhedra as the effect of increasing the number of faces on the evaluation time is small. Unlabelled Image • Developed a contact detection algorithm for convex polyhedra and superquadrics. • Capable of resolving any type of polyhedron face, edge or vertex (non-)contact. • Implemented the algorithm in a C++ code and verified its correctness. • Demonstrated the algorithm's efficiency, particularly for many-faceted polyhedra. [ABSTRACT FROM AUTHOR]

In this paper we study an open problem: Is it true that each homotopy domination of a polyhedron over itself is a homotopy equivalence? Since every ANR has the homotopy type of a polyhedron, this question is related in an obvious way to the longstanding Borsuk's problem (1967) Is it true that two ANR's homotopy dominating each other have the same homotopy type? The answer is positive for all polyhedra (or ANR's) with polycyclic-by-finite fundamental groups, manifolds, and clearly for 1-dimensional polyhedra. In some previous paper, we proved that this also holds for all 2-dimensional polyhedra with weakly Hopfian (hence Hopfian and co-Hopfian) fundamental groups (a group is weakly Hopfian if it is not isomorphic to a proper retract of itself). In this paper we extend this result to all (G , n) -complexes (where n ≥ 2). A (G , n) -complex, is an n -dimensional finite CW -complex with fundamental group G and all the homotopy groups π r in dimensions 1 < r < n trivial. We also discuss fundamental groups of potential counterexamples. As a corollary we obtain that for the same classes of spaces, the Borsuk's question: Is it true that the homotopy types of two quasi-homeomorphic ANR's are equal? , has positive answer (this question was also asked by S. Ferry in the recent edition of Scottish Book (1981)). [ABSTRACT FROM AUTHOR]

Blocky rock and stone structures (such as jointed rocks mass, masonry structures and railway ballast) are often submitted to dynamic effects. Their mechanical behaviour can be simulated with the discrete element method (DEM) using polyhedral elements. DEM models differ in how they compute the contact forces. Several studies have compared different contact constitutive laws, but little information is available in the literature on which contact treatment method (i.e. number of contacts between two elements, contact detection, contact orientation, area and overlap computation) to choose. The inclined slope, the perpendicular impact and the rocking block problems were simulated and applied as benchmark tests in the present study to compare three most popular polyhedral DEM contact treatment models for rigid elements, namely the Eliáš (available in Yade), the Gilbert-Johnson-Keerthi (GJK, widely implemented, here tested with PFC3D) and the Sub-Contact model (available in 3DEC), to see which ones could be precisely calibrated to analytic and experimental solutions. Sensitivity studies were also carried out on the effect of subdivision density of element surfaces and on the admissible timestep length. Experiences showed that while there are serious doubts about the applicability of the Eliáš volumetric model as it is sensitive to both variations in geometry and the results vary randomly with the timestep length, the GJK model is suitable to simulate granular materials because of its computational efficiency and robustness, and the Sub-Contact method is the proper choice to model structures where the contact surfaces are relatively large, as it operates multiple contacts between two elements and gives information about the force distribution on the surface of the elements. [Display omitted] [ABSTRACT FROM AUTHOR]

Abstract Exploring efficient electrocatalyst for H 2 evolution reaction (HER) and replacing the noble metal-based catalysts with inexpensive non-noble metal-based HER catalyst is of great importance for the practicality of hydrogen powered clean technologies. Here, we explore a new class of metal organic framework (MOF) composite (NiMo polyhedron) as an active electrocatalyst material for HER application - synthesized through the conventional hydrothermal process. The bimetallic MOF system having grown on Nickel foam (NiMo/NiMoO 4 @NC/NF) delivers higher catalytic activity by achieving a current density of 10 mA cm−2 at a low overpotential of 80 mV, with a Tafel slope of 98.9 mV dec−1 (0.5 M H 2 SO 4), comparing favorably with the electrochemical enactment of existing bimetallic MOF-based catalysts. The enhanced HER activity of the synthesized MOF, is primarily due to the structural merits of MOF and the synergy between the MOF and the guest species (Ni and Mo metal atoms). Adding to the excellent HER performance, the electrode also exhibits good stability in acidic medium for a prolonged duration of 24 h. Hence, the synthesized low-cost, non-Pt electrode MOFs with its greater HER performance can be an auspicious applicant as an HER catalyst for water splitting and hydrogen generation applications. Graphical abstract Unlabelled Image Highlights • Noble, Pt free alternate electrocatalysts for HER were identified. • Ni, Mo based bimetallic MOFs were prepared by hydrothermal process. • The synthesized NiMO-MOFs possessed polyhedron morphology. • It achieved a current density of 10 mA cm−2 at a low overpotential of 98.9 mV. [ABSTRACT FROM AUTHOR]

Graphical abstract Highlights • Replica-exchange molecular dynamics predicts the framework of a metal-ligand complex. • The metal-ligand complex has two polyhedral frameworks with T d and C 3v symmetry. • The metal-ligand complex with T d symmetry is enthalpically favored. • The metal-ligand complex with C 3v symmetry has the highest probability of existence. Abstract We performed replica-exchange molecular dynamics (REMD) simulations for self-assembly of an M 12 L 24 spherical complex that consists of 12 palladium ions (M) and 24 bent ligands (L). We found two metal-ligand frameworks with polyhedral structure were mainly observed with almost equal probability. One of the two frameworks, which has the lowest potential energy, has the same symmetry, T d , as the framework determined by the X-ray crystallographic analysis. The other framework, which has a slightly higher probability, has a different symmetry, C 3v. The latter framework is less likely to be observed experimentally than the former because of the difficulty in crystallization. [ABSTRACT FROM AUTHOR]

Invoking Steinitz’ Theorem, in the following a polyhedron shall be a 3-connected planar graph. From around 1880 till 1946 Tait’s conjecture that cubic polyhedra are hamiltonian was thought to hold—its truth would have implied the Four Colour Theorem. However, Tutte gave a counterexample. We briefly survey the ensuing hunt for the smallest non-hamiltonian cubic polyhedron, the Lederberg-Bosák-Barnette graph, and prove that there exists a non-hamiltonian essentially 4-connected cubic polyhedron of order n if and only if n  ≥ 42. This extends work of Aldred, Bau, Holton, and McKay. We then present our main results which revolve around the quartic case: combining a novel theoretical approach for determining non-hamiltonicity in (not necessarily planar) graphs of connectivity 3 with computational methods, we dramatically improve two bounds due to Zaks. In particular, we show that the smallest non-hamiltonian quartic polyhedron has at least 35 and at most 39 vertices, thereby almost reaching a quartic analogue of a famous result of Holton and McKay. As an application of our results, we obtain that the shortness coefficient of the family of all quartic polyhedra does not exceed 5/6. The paper ends with a discussion of the quintic case in which we tighten a result of Owens. [ABSTRACT FROM AUTHOR]

K. Borsuk in 1979, in the Topological Conference in Moscow, introduced the concept of the capacity of a compactum. In this paper, we compute the capacity of wedge sum of finitely many spheres of various dimensions. [ABSTRACT FROM AUTHOR]

In this paper every polyhedron is finite and every ANR is compact. Let P ≥ X 1 ≥ X 2 ≥ … , be a sequence, in which P is a polyhedron and ≥ are homotopy dominations. One may ask, if each sequence of this form contains only finitely many homotopy dominations that are not homotopy equivalences, or if there exists an integer l P (independent of the sequence) that each sequence contains only ≤ l P homotopy dominations that are not homotopy equivalences. Closely related open questions are: Does there exist a polyhedron P homotopy dominating an infinite sequence of polyhedra { P i } , where P i homotopy dominates P i + 1 but P i and P i + 1 have different homotopy types, for every i ∈ N ? (M. Moron) [30, Problem 1436] , and the famous problem of K. Borsuk (1967): Is it true that two A N R ′ s homotopy dominating each other have the same homotopy type? Here we prove that, if dim ⁡ P = 2 , then the answers to all these questions depend only on the properties of the fundamental group of P (for 1-dimensional polyhedra, the answers are obvious). Furthermore, if sequences in consideration contain only polyhedra, then, for the positive answer it suffices to answer positively the analog of our topological question for finitely presented groups (the fundamental groups) with retractions. Applying these results, we prove that for each polyhedron P with dim ⁡ P ≤ 2 and elementary amenable fundamental group G with cd G < ∞ , there is a bound l P on the lengths of all descending sequences P ≥ X 1 ≥ X 2 … of homotopy dominations that are not homotopy equivalences. The same holds if the fundamental group of P is a limit group. It means that such polyhedra P have finite depth. [ABSTRACT FROM AUTHOR]

In 1938, Tarski proved that a formula is not intuitionistically valid if, and only if, it has a counter-model in the Heyting algebra of open sets of some topological space. In fact, Tarski showed that any Euclidean space R n with n ⩾ 1 suffices, as does e.g. the Cantor space. In particular, intuitionistic logic cannot detect topological dimension in the Heyting algebra of all open sets of a Euclidean space. By contrast, we consider the lattice of open subpolyhedra of a given compact polyhedron P ⊆ R n , prove that it is a locally finite Heyting subalgebra of the (non-locally-finite) algebra of all open sets of P , and show that intuitionistic logic is able to capture the topological dimension of P through the bounded-depth axiom schemata. Further, we show that intuitionistic logic is precisely the logic of formulæ valid in all Heyting algebras arising from polyhedra in this manner. Thus, our main theorem reconciles through polyhedral geometry two classical results: topological completeness in the style of Tarski, and Jaśkowski's theorem that intuitionistic logic enjoys the finite model property. Several questions of interest remain open. E.g., what is the intermediate logic of all closed triangulable manifolds? [ABSTRACT FROM AUTHOR]

We show that a finite polyhedron P of nontrivial homotopy type cannot be homotopy equivalent to a proper Cartesian factor of itself. In other words, P ≃ P × K implies that K ≃ 1 . Consequently, if P is a finite polyhedron such that P n ≃ P , for some integer n ≥ 2 , then the homotopy type of P is trivial. This answers positively the problem [1, Problem (34.10)] stated by K. Borsuk in 1971 (in an equivalent, shape-theoretical formulation) and a question from Borsuk's monograph “Theory of Shape” [2, Problem (7.13), p. 142] . We also prove that if ( X , x ) ∈ F A N R , then Sh n ( X , x ) = Sh ( X , x ) , for some integer n ≥ 2 , implies that Sh ( X ) = 1 . This resolves positively Problem (5.6) from [7] (1981). Similarly as for polyhedra, first we show that if X ∈ F A N R and Sh ( X ) ≠ 1 , then X cannot have the shape of a proper Cartesian factor of itself. [ABSTRACT FROM AUTHOR]

This paper dealt with the construction of a novel class of reconfigurable deployable polyhedral mechanisms (RDPMs). A parallelograms mechanism was designed by considering layers of links and size of revolute joints, and extended parallelogram mechanisms (EPaM) were constructed for the first time by inserting angulated elements and scissor-like elements into the parallelogram mechanism. The detailed parameters and magnification ratios of two kinds of EPaMs were calculated. A method was then proposed for constructing RDPMs based on EPaMs. The proposed method can provide more modular linkages for RDPMs, and all polyhedra constituted by regular polygons can be used to construct RDPMs. The obtained RDPMs can be reassembled by increasing or decreasing the number of basic elements to change the sizes beyond their deployment capacity. A prototype of reconfigurable deployable hexahedral mechanism based on EPaMs was fabricated to verify the feasibility of this class of RDPMs, and three reconfigurable deployable hexahedral mechanisms in three sizes based on the prototype were reassembled and tested. [ABSTRACT FROM AUTHOR]

[Display omitted] • Catalyst layer with self-assembled jointed Pd polyhedra in-situ grown on carbon paper. • Pd polyhedra enabling highly active jointed interfaces and high-index facets. • 3D porous catalyst layer facilitating the mass transport during the operation. • Record power density of 202 mW cm−2 as the anode in the direct formic acid fuel cell. The direct formic acid fuel cell (DFAFC) is recognized as a promising power source for its high electromotive force, easy fuel storage and transport. However, its power performance is low and the cost is high, which are known to originate from the sluggish formic acid oxidation (FAO) kinetics and poor mass transport within the DFAFC anode. Here, we present a new DFAFC anode design with a thin, porous, 3D-structured catalyst layer based on self-assembled jointed Pd polyhedra in-situ grown on the gas diffusion layer surface. The Pd polyhedra provide highly active jointed interfaces and high-index facets, boosting their catalytic activity towards FAO. The porous 3D catalyst layer facilitates the transport of reactants and products at the large current density region. Consequently, the present anode design exhibits a previously unachieved power density of 202 mW cm−2 at a Pd loading of 1.0 mg cm−2 in the HCOOH/air DFAFC test. [ABSTRACT FROM AUTHOR]

In order to develop an efficient and active catalyst to degrade the toxic volatile organic compounds, manganese oxide polyhedra with hollow and solid morphologies were synthesized by a convenient hydrothermal route without any surfactants or templates. The catalytic performances of the two manganese oxides were assessed in the oxidation reaction of toluene. Compared with the solid polyhedral manganese oxide, an excellent catalytic activity of the hollow one has been found, resulting from its cavity nature, the high content of active oxygen and the high manganese oxidation state of hollow MnOx. Meanwhile, the probable reaction pathway for toluene oxidation over the hollow polyhedral manganese oxide was initially discussed by means of in situ FTIR. [ABSTRACT FROM AUTHOR]

Since it was presented, the scaled boundary finite element method (SBFEM) has been shown to be versatile and has been widely applied in structural numerical simulations. However, as it is analytical in the radial direction, nonlinearity inside elements cannot be considered, limiting its application in elastic fields. In this paper, a nonlinear approach for the three-dimensional polyhedron scaled boundary finite element (NPSBFEM3D) is proposed for elasto-plastic analysis to remove this restriction. In NPSBFEM3D, conforming shape functions are constructed using the semi-analytical solution derived from elastic surface elements, while the integrations are accomplished using internal Gauss points in the radial direction instead of integrating on the boundary surface elements. Eventually, the proposed approach can be as conveniently used in elasto-plastic analysis as FEM. This method permits an arbitrary number of faces, which offers a promising adaptive capacity for modelling. Three simulations are conducted to verify the robustness of the presented method. [ABSTRACT FROM AUTHOR]

In this paper, we propose a novel, simple, and unified approach to explore sufficient and necessary conditions, i.e., invariance conditions, under which four classic families of convex sets, namely, polyhedra, polyhedral cones, ellipsoids, and Lorenz cones, are invariant sets for a linear discrete or continuous dynamical system. For discrete dynamical systems, we use the Theorems of Alternatives, i.e., Farkas lemma and S -lemma, to obtain simple and general proofs to derive invariance conditions. This novel method establishes a solid connection between optimization theory and dynamical system. Also, using the S -lemma allows us to extend invariance conditions to any set represented by a quadratic inequality. Such sets include nonconvex and unbounded sets. For continuous dynamical systems, we use the forward or backward Euler method to obtain the corresponding discrete dynamical systems while preserves invariance. This enables us to develop a novel and elementary method to derive invariance conditions for continuous dynamical systems by using the ones for the corresponding discrete systems. Finally, some numerical examples are presented to illustrate these invariance conditions. [ABSTRACT FROM AUTHOR]

We examine conditions under which a semicomputable set in a computable metric space contains computable points. We prove that computable points in a semicomputable set S are dense if S is a manifold (possibly with boundary) or S has the topological type of a polyhedron. Moreover, we find conditions under which a point in some set has a computable compact neighbourhood in that set. In particular, we show that a point x in a semicomputable set has a computable compact neighbourhood if x has a neighbourhood homeomorphic to Euclidean space. [ABSTRACT FROM AUTHOR]

As a natural generalization of surfaces of Platonic solids, we define a class of polyhedra, called simple regular polyhedral BP-complexes, as a class of 2-dimensional polyhedral metric complexes satisfying certain conditions on their vertex sets, and we give a complete classification of such polyhedra. They are either the surface of a Platonic solid, a p-dodecahedron, a p-icosahedron, an m -covered regular n -gon for some m ≧ 2 or a complete tripartite polygon. [ABSTRACT FROM AUTHOR]

Ce-based materials have gained considerable prominence in catalysis as a type of the most valuable material for various applications ranging from automotive emission control to CO 2 and water splitting. We here report a novel type-II CeO 2 /CeF 3 composite, composed of two paradigms of Ce-based binary compounds, prepared by a facile one-step in-situ hydrothermal method, and explore its potential as a photocatalyst for solar energy conversion. Primarily, the morphologies, particle size and microstructures of CeO 2 /CeF 3 composite can be modulated by the volume ratio of ethanol to water in the precursor solution. The photocatalytic activities firstly increase to a maximum and then decrease with the increased amount of ethanol. The hybrid CeO 2 /CeF 3 particles with large size and polyhedron morphology exhibit superior photocatalytic activity, having a high rate constant k of MB decomposition about 7.6 times faster than that of the lowest one. A multiple synergetic mechanism in CeO 2 /CeF 3 composite, which is featured by obvious promoted separation of photogenerated electrons and holes at the type-II heterojunction interfaces and in particular between the different facets of microcrystals, and the highly reactive facets exposed, has been proposed to account for its distinguished photocatalytic activity. This work provides some new insights for the synthesis and optimization of novel rare earth Ce-based composite photocatalyst with high photocatalytic performance. [ABSTRACT FROM AUTHOR]

The theory of the isoptic curves is widely studied in the Euclidean plane E 2 (see Cieślak et al., 1991 and Wieleitner, 1908 and the references given there). The analogous question was investigated by the authors in the hyperbolic H 2 and elliptic E 2 planes (see Csima and Szirmai, 2010, 2012, submitted for publication ), but in the higher dimensional spaces there are only few results in this topic. In Csima and Szirmai (2013) we gave a natural extension of the notion of the isoptic curves to the n -dimensional Euclidean space E n ( n ≥ 3 ) which is called isoptic hypersurface. Now we develope an algorithm to determine the isoptic surface H P of a 3-dimensional polyhedron P . We will determine the isoptic surfaces for Platonic solids and for some semi-regular Archimedean polytopes and visualize them with Wolfram Mathematica ( Wolfram Research, Inc., 2015 ). [ABSTRACT FROM AUTHOR]

The Virtual Element Method is ideal for problems with non-conforming meshes since nodes can be inserted at the virtual element boundary to obtain a conforming mesh structure, without changing the ansatz of the element. The contact conditions can be employed on different enforcing strategies. This feature can be explored also for three-dimensional contact problems. In this work, we introduce a new framework, including a remeshing strategy, which leads to conforming meshes. Various numerical examples in 3D will be shown, including standard cases like the contact patch test and Hertzian contact. [ABSTRACT FROM AUTHOR]

In this paper we define a new pointed finite coarse shape invariant - k -th finite coarse shape group of a pointed topological space and compare it with the corresponding shape and coarse shape group. We also define a functor π ˇ k ⊛ from the pointed finite coarse shape category S h ⋆ ⊛ to the category Grp of all groups (for k = 0 to the category S e t ⋆ of all pointed sets) associating with every pointed topological space (X , x 0) a k -th finite coarse shape group π ˇ k ⊛ (X , x 0) and with every finite coarse shape morphism F ⊛ : (X , x 0) → (Y , y 0) a homomorphism (a base point preserving function) π ˇ k ⊛ (F ⊛) : π ˇ k ⊛ (X , x 0) → π ˇ k ⊛ (Y , y 0). We show that, in a case of a pointed compact polyhedral system, functor π ˇ k ⊛ is continuous, i.e., commutes with the inverse limit. Finally, finite coarse shape groups of some interesting spaces (dyadic solenoid and arbitrary stable pointed space) are calculated. [ABSTRACT FROM AUTHOR]

We consider an open question: Is it true that each homotopy domination of a polyhedron over itself is a homotopy equivalence? The answer is known to be positive for 1-dimensional polyhedra and polyhedra with virtually-polycyclic fundamental groups, so it is natural to ask about 2-dimensional polyhedra, in particular about those with solvable fundamental groups. In this paper we prove that for each 2-dimensional polyhedron P with weakly Hopfian fundamental group, every homotopy domination of P over itself is a homotopy equivalence. A group is weakly Hopfian if it is not isomorphic to a proper retract of itself. Thus every Hopfian group is weakly Hopfian. The class of Hopfian groups contains: all torsion-free hyperbolic groups, finitely generated linear groups, knot groups, limit groups, and many others. One corollary to the main result is that for 2-dimensional polyhedra with elementary amenable (including virtually-solvable) fundamental groups of finite cohomological dimension, the answer to our question is positive (we show that every elementary amenable group with finite cohomological dimension is Hopfian). The problem in consideration is related in an obvious way to the famous question of K. Borsuk (1967): Is it true that two compact ANR's homotopy dominating each other have the same homotopy type? [ABSTRACT FROM AUTHOR]

A polyhedron in R n is a finite union of simplexes in R n . An MV-algebra is polyhedral if it is isomorphic to the MV-algebra of all continuous [ 0 , 1 ] -valued piecewise linear functions with integer coefficients, defined on some polyhedron P in R n . We characterize polyhedral MV-algebras as finitely generated subalgebras of semisimple tensor products S ⊗ F with S simple and F finitely presented. We establish a duality between the category of polyhedral MV-algebras and the category of polyhedra with Z -maps. We prove that polyhedral MV-algebras are preserved under various kinds of operations, and have the amalgamation property. Strengthening the Hay–Wójcicki theorem, we prove that every polyhedral MV-algebra is strongly semisimple, in the sense of Dubuc–Poveda. [ABSTRACT FROM AUTHOR]

Let 0- CR denote the class of all metric compacta X such that the set of maps f : X → X with 0-dimensional sets CR ( f ) of chain recurrent points is a dense G δ -subset of the mapping space C ( X , X ) (with the uniform convergence). We prove, among others, that countable products of polyhedra or locally connected curves belong to 0- CR . Compacta that admit, for each ϵ > 0 , an ϵ -retraction onto a subspace from 0- CR belong to 0- CR themselves. Perfect ANR-compacta or n -dimensional LC n − 1 -compacta have perfect CR ( f ) for a generic self-map f . In the cases of polyhedra, compact Hilbert cube manifolds, local dendrites and their finite products, a generic f has CR ( f ) being a Cantor set and the set of periodic points of f of arbitrarily large periods is dense in CR ( f ) . The results extend some known facts about CR ( f ) of generic self-maps f on PL-manifolds. [ABSTRACT FROM AUTHOR]

We give a variant of Barvinok's algorithm for computing a short rational generating function for the integer points in P : = { x ∈ R n : A x ≤ b } ; a use of which is to count the number of integer points in P . We use an algebraic perturbation, replacing each b i with b i + τ i , where τ > 0 is an arbitrarily small indeterminate . Hence, our new right-hand vector has components in the ordered ring Q [ τ ] of polynomials in τ . Denoting the perturbed polyhedron by P ( τ ) ⊂ R [ τ ] n , we use the facts that: P ( τ ) is full dimensional, simple, and contains the same integer points as P . [ABSTRACT FROM AUTHOR]

The increasing interest in using sensor networks in applications for underwater surveillance and oceanic studies underscores the importance of solving the coverage and connectivity issues in 3D wireless sensor networks (WSN). In particular, the problem of supporting full coverage, while ensuring full network connectivity is a fundamental one for such applications. Unfortunately, designing a 3D network is significantly more difficult, as compared to designing a 2D network. Previously, it has been shown that dividing a 3D space into identical truncated octahedral cells of radius equal to the sensing range and placing a sensor at the center of each cell, provides full coverage with minimum number of nodes [2] . But this requires the ability to deploy and maintain sensor nodes at such particular locations. In many environments, this is very difficult, if not impossible, to do. In this paper, we investigate the coverage and connectivity issues for such 3D networks, especially underwater networks, while assuming random and uncontrollable node locations. Since node location can be random, redundant nodes have to be deployed to achieve 100% sensing coverage. However, at any particular time, not all nodes are needed to achieve full sensing coverage. As a result, a subset of the nodes can be dynamically chosen to remain active at a time to achieve sensing coverage based on their location at that time. One approach to achieve this goal in a distributed and scalable way is to partition the 3D network volume into virtual regions or cells, and to keep one node active in each cell. Our results indicate that using cells created by truncated octahedral tessellation of 3D volume minimizes the number of active nodes. This scheme is fully distributed, and so it is highly scalable. By adjusting the radius of each cell, this scheme can be used to achieve k -coverage, where every point inside a network has to be within the sensing range of k different sensor nodes. We analyze and compare the performance of these schemes for both 2D and 3D networks. While for 1-coverage, the 3D scheme is less efficient than the 2D scheme, the performance of 3D scheme improves significantly as compared to 2D scheme for k -coverage, for values of k is larger than 1. As a result, such a distributed and scalable scheme can be more useful in 3D networks than in 2D networks. Although this paper targets in particular 3D underwater networks, much of our results are applicable to other 3D networks, such as for airborne applications, space exploration, and storm tracking. [ABSTRACT FROM AUTHOR]

Since the structural characterization of Fe-based disordered system is less than other metallic glasses, we investigate the Fe 76 − 2 x Y 4 + x B 20 + x ( x = 0 and 2) glasses and melts via the variation of tri-capped triangular prism (TTP) polyhedron with changing the composition. B-centered clusters mainly appear in the manner of TTP polyhedron in both alloys, which is strengthened by increasing yttrium and boron contents ( c Y and c B ) and considered as the network/backbone in the Fe-based glassy alloys. Increasing c Y and c B improves the onset temperature of the first crystallization event with decreasing its activation energy, deteriorates the corrosion resistance of present alloys in NaCl and H 2 SO 4 solutions, and triggers the formation of nanoscale network structure. These experimental results can be ascribed to the strengthened network/backbone in the alloys. [ABSTRACT FROM AUTHOR]

[Display omitted] • Focus on metal⋯unsaturated-bond coordination for self-assembly. • Concerted metal–acetylene π-coordination afforded highly-entangled polyhedra. • Collaboration of weak and strong coordinations is promising for higher molecular complexity. The coordinative self-assembly of relatively simple building blocks toward complex nanostructures is currently of high interest in the context of developing artificial nanomaterials that are comparable to biological systems. Even though a substantial body of research has already been assembled, only few classes of coordination donors, namely, nitrogen- or oxygen-containing donors, have historically been used, as research has focused predominantly on the robustness and designability of the coordinated structures. As an alternative approach, the use of the relatively weak but loosely directed π-coordination between metal ions and unsaturated carbon–carbon bonds has emerged; this strategy is expected to provide structurally flexible and highly tolerant bonding in complex nanoarchitectures. Most prominently, studies that combine metal–acetylene π-coordination with conventional coordination have recently realized the formation of very large discrete polyhedral complexes with unprecedented molecular entanglement based on trifurcate motifs, which represent the tip of the iceberg in terms of the potential advantages of this concept. [ABSTRACT FROM AUTHOR]

[Display omitted] Although it is of scientific and practical importance, the state-of-the-art of predicting the thermal expansion of oxides over broad temperature and composition ranges by physics-based atomistic simulations is currently limited to qualitative agreements. We present an emerging machine learning (ML) approach to accurately predict the thermal expansion of cubic oxides with a dataset consisting of experimentally measured lattice parameters while using the metal cation polyhedron and temperature as descriptors. High-fidelity ML models that can accurately predict temperature- and composition-dependent lattice parameters of cubic oxides with isotropic thermal expansions have been successfully trained. The ML-predicted thermal expansions of oxides not included in the training dataset have shown good agreement with available experiments. The limitations of the current approach and challenges to go beyond cubic oxides with isotropic thermal expansion are also briefly discussed. [ABSTRACT FROM AUTHOR]

To determine the relative position of any two surfaces in a system, one approach is to use operations (Minkowski sum and intersection) on sets of constraints. These constraints are made compliant with half-spaces of R n where each set of half-spaces defines an operand polyhedron. These operands are generally unbounded due to the inclusion of degrees of invariance for surfaces and degrees of freedom for joints defining theoretically unlimited displacements. To solve operations on operands, Minkowski sums in particular, “cap” half-spaces are added to each polyhedron to make it compliant with a polytope which is by definition a bounded polyhedron. The difficulty of this method lies in controlling the influence of these additional half-spaces on the topology of polytopes calculated by sum or intersection. This is necessary to validate the geometric tolerances that ensure the compliance of a mechanical system in terms of functional requirements. [ABSTRACT FROM AUTHOR]

It is shown that every polyhedron in R 3 can be guarded by at most 5 6 of its edges. This result holds even if the boundary of the polyhedron is disconnected (i.e., if the polyhedron has "holes"), and regardless of the genus of each connected component of its boundary. When a polyhedron is homeomorphic to a ball and all its faces are triangles, the bound can be improved slightly to 29 36 of its edges. [ABSTRACT FROM AUTHOR]

The monostatic property of convex polyhedra (i.e., the property of having just one stable or unstable static equilibrium point) has been in the focus of research ever since Conway and Guy (1969) published the proof of the existence of the first such object, followed by the constructions of Bezdek (2011) and Reshetov (2014). These examples establish F ≤ 14 , V ≤ 18 as the respective upper bounds for the minimal number of faces and vertices for a homogeneous mono-stable polyhedron. By proving that no mono-stable homogeneous tetrahedron existed, Conway and Guy (1969) established for the same problem the lower bounds for the number of faces and vertices as F , V ≥ 5 and the same lower bounds were also established for the mono-unstable case (Domokos et al., 2020b). It is also clear that the F , V ≥ 5 bounds also apply for convex, homogeneous point sets with unit masses at each point (also called polyhedral 0-skeletons) and they are also valid for mono-monostatic polyhedra with exactly one stable and one unstable equilibrium point (both homogeneous and 0-skeletons). In this paper we draw on an unexpected source to extend the knowledge on mono-monostatic solids: we present an algorithm by which we improve the lower bound to V ≥ 8 vertices on mono- unstable 0-skeletons. The problem is transformed into the (un)solvability of systems of polynomial inequalities, which is shown by convex optimization. Our algorithm appears to be less well suited to compute the lower bounds for mono-stability. We point out these difficulties in connection with the work of Dawson, Finbow and Mak (Dawson, 1985, Dawson et al., 1998, Dawson and Finbow, 2001) who explored the monostatic property of simplices in higher dimensions. [ABSTRACT FROM AUTHOR]

Abstract: Given a weighted undirected graph G with a set of pairs of terminals { , and an integer , the two node-disjoint hop-constrained survivable network design problem (TNHNDP) is to find a minimum weight subgraph of G such that between every and there exist at least two node-disjoint paths of length at most L. This problem has applications to the design of survivable telecommunications networks with QoS-constraints. We discuss this problem from a polyhedral point of view. We present several classes of valid inequalities along with necessary and/or sufficient conditions for these inequalities to be facet defining. We also discuss separation routines for these classes of inequalities. Using this, we propose a Branch-and-Cut algorithm for the problem when , and present some computational results. [Copyright &y& Elsevier]

Abstract: In [6] Schirmer (1985) established that, if is an n-valued map defined on a compact triangulable manifold of dimension at least three, then the appropriate Nielsen number, , is a sharp lower bound for the number of fixed points in the n-valued homotopy class of φ. In this note we generalize this theorem by allowing X to be any compact polyhedron without local cut points and such that no connected component is a two-manifold. [Copyright &y& Elsevier]

Abstract: In some previous paper D. Kołodziejczyk (2001) we proved that there exists a finite polyhedron P with infinitely many left neighbors in the homotopy category, i.e. P homotopy dominates infinitely many finite polyhedra of different homotopy types and there isnʼt any homotopy type between P and . This answers a question of K. Borsuk (Borsuk, 1975 ). The dimension of that P is 3. Here we show that there exists a finite 2-dimensional polyhedron with the same property. Some remarks on constructing examples of 2-dimensional finite polyhedra dominating infinitely many different homotopy types are also included. [Copyright &y& Elsevier]

Abstract: In this paper, we survey existing geometric structures which have been proposed by the authors as possible models for nanotubes. Atoms assemble into molecules following the laws of quantum mechanics, and in general computational approaches to predicting the molecular structure can be arduous and involve considerable computing time. Fortunately, nature favours minimum energy structures which tend to be either very symmetric or very unsymmetric, and which therefore can be analyzed from a geometrical perspective. The conventional rolled-up model of nanotubes completely ignores any effects due to curvature and the present authors have proposed a number of exact geometric models. Here we review a number of these recent developments relating to the geometry of nanotubes, with exact polyhedral constructions, and in particular for the three materials, carbon, boron and silicon. We summarise the main formulae and give illustrative numerical results for these three nanotubes. [Copyright &y& Elsevier]

Abstract: We solve some problems concerning dimension function ( is a class of finite simplicial complexes) and ( is a class of compact polyhedra). One of the main results is: If is homotopy dominated by , then for every normal space X. [Copyright &y& Elsevier]

Abstract: Autographa californica multiple nucleopolyhedrovirus (AcMNPV) odv-e66 is a core gene and encodes an occlusion-derived virus (ODV)-specific envelope protein, ODV-E66. The N-terminal 23 amino acid of the envelope protein ODV-E66 are sufficient to direct native and fusion proteins to induced membrane microvesicles and the viral envelope during infection with AcMNPV. In this study, an odv-e66-knockout bacmid can not express N-terminal hydrophobic domains was constructed via homologous recombination in Escherichia coli. The odv-e66 deletion had no effect on budded virus (BV) production and viral DNA replication in infected Sf9 cells. Larval bioassays demonstrated that injection of odv-e66 deletion BV into the hemocoel could kill P. xylostella larvae as efficiently as repaired and control viruses; however, odv-e66 deletion mutant resulted in a 50% lethal dose that was 103 higher than that of the repaired and control viruses when inoculated per os. These results indicated that ODV-E66 envelope protein most likely played an important role in the oral infectivity of AcMNPV, but is not essential for virus replication. [Copyright &y& Elsevier]

Abstract: It has been suggested recently that the alanes Al n H n +2 can be treated by the polyhedral skeletal electron pair theory (PSEPT) of Wade and Mingos (W–M) as it was successful for their borane congeners such as B n H n +2, well known as the deprotonated B n H n 2−. To do so, the neutral Al n H n +2 have been considered as Al n H n 2− +2H+. The additional hydrogens donate their electrons to the Al n H n polyhedral framework and according to the n +1 electron pairs rule; these clusters should have closo-polyhedral structures. In this work the homologous gallanes, the structures and stabilities of Ga n H n +2 are studied at high levels of calculational theory and we investigated the applicability of the W–M rule to the alanes and gallanes A n H n +2 (n =4–6; A=Al, Ga). It will be shown that the presence of bridging hydrogen atoms reduces the compactness of the corresponding polyhedron and so these species do not have the closed structures. The computations were performed at B3LYP/6-311+G(d,p), BPW91/6-311G(d,p) and B3LYP/6-311+G(3df,2p) levels of theory. Our interest in these compounds includes their potential use as hydrogen storage species and future clean sources of energy. [Copyright &y& Elsevier]

Abstract: Cage-type anionic-surfactant-templated silica mesoporous crystals display rich structural diversity. Among them, cage-type mesoporous crystals with tetrahedrally close-packed (tcp) structures can be described by four types of polyhedron, 512, 51262, 51263 and 51264. We have found new stacking sequences as a minor phase coexisting with an Fd m structure, and have characterized their structures using transmission electron microscopy (TEM). The new stacking sequences can be explained by the stacking of three kinds of layer based on a description of the polyhedra. Two of the layers consist of two kinds of polyhedron, 512 and 51264, and these two layers are also the constituents of the Fd m structure. The other layer consists of two different kinds of polyhedron, 51262 and 51263, corresponding to layer z, which were introduced in our recently published papers [e.g. Y. Sakamoto, et al., Chem. Mater. 21 (2009) 223]. We have observed that one of the new stacking sequences, with a large repeating unit, contains these three layers and bears a close resemblance to one of the Frank–Kasper phases with rhombohedral symmetry, known as μ-phase. [Copyright &y& Elsevier]

Abstract: The author and N. Uglešić have recently introduced a new classification of topological spaces, strictly coarser than the shape type classification. The corresponding coarse shape theory is founded on the coarse shape category. In this paper several topological and shape invariants are considered with respect to the coarse shape. It is shown that the coarse shape domination preserves connectedness, shape dimension, movability, n-movability and strong movability. Further, stability is a coarse shape invariant. Moreover, the coarse shape and shape coincide on the class of stable spaces. [ABSTRACT FROM AUTHOR]

Abstract: As the main results of this paper we prove that for every polyhedron P with abelian or torsion-free nilpotent fundamental group there are only finitely many different homotopy types of such that . The same holds for any finite with nilpotent fundamental group in place of . The problem, if there exists a polyhedron with infinitely many direct factors of different homotopy types (K. Borsuk, 1970) is still unsolved, even if we assume the second factor to be . [ABSTRACT FROM AUTHOR]