Search

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

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

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

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

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

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

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

A regular map is an abstract generalization of a Platonic solid. It describes a group, a topological cell decomposition of a 2-manifold of type { p ,   q } with only p-gons, such that q of them meet at each vertex in a circular manner, and we have maximal combinatorial symmetry, expressed by the flag transitivity of the symmetry group. On the one hand, we have articles on topological surface embeddings of regular maps by F. Razafindrazaka and K. Polthier, C. Séquin, and J. J. van Wijk.On the other hand, we have articles with polyhedral embeddings of regular maps by J. Bokowski and M. Cuntz, A. Boole Stott, U. Brehm, H. S. M. Coxeter, B. Grünbaum, E. Schulte, and J. M. Wills. The main concern of this partial survey article is to emphasize that all these articles should be seen as contributing to the common body of knowledge in the area of regular map embeddings. This article additionally provides a method for finding symmetrical equivelar polyhedral embeddings of type { 3 ,   7 } based on symmetrical graph embeddings on convex surfaces. [ABSTRACT FROM AUTHOR]

This study investigates a method for generating obstacle avoidance trajectories for arbitrary convex polyhedrons. We propose a formulation that converts discrete conditions into continuous equation forms to avoid convex polyhedron obstacles. The condition that the evaluation point be located outside of the convex polyhedron can be transformed into a constraint in the continuous equation form and incorporated into the optimization calculation to generate avoidance trajectories. Avoidance trajectory generation using the Legendre pseudospectral method is performed for convex polyhedral obstacles of various shapes. The results show that the proposed method successfully generates avoidance trajectories for arbitrary convex polyhedral obstacles. [ABSTRACT FROM AUTHOR]

I study several problems about the symmetric space associated with the Lie group SL(n,R)$SL(n,\mathbb {R})$. These problems are connected to an algorithm based on Poincaré's Fundamental Polyhedron Theorem, designed to determine generalized geometric finiteness properties for subgroups of SL(n,R)$SL(n,\mathbb {R})$. The algorithm is analogous to the original one in hyperbolic spaces, while the Riemannian distance is replaced by an SL(n,R)$SL(n,\mathbb {R})$‐invariant premetric. The main results of this paper are twofold. In the first part, I focus on questions that occurred in generalizing Poincaré's Algorithm to my symmetric space. I describe and implement an algorithm that computes the face‐poset structure of finitely sided polyhedra, and construct an angle‐like function between hyperplanes. In the second part, I study further questions related to hyperplanes and Dirichlet–Selberg domains in my symmetric space. I establish several criteria for the disjointness of hyperplanes and classify particular Abelian subgroups of SL(3,R)$SL(3,\mathbb {R})$ based on whether their Dirichlet–Selberg domains are finitely sided or not. [ABSTRACT FROM AUTHOR]

For a discrete poset X , McCord proved that the natural map | X | → X , from the order complex to the poset with the Up topology, is a weak homotopy equivalence. Much later, Živaljević defined the notion of order complex for a topological poset. For a large class of topological posets we prove the analog of McCord's theorem, namely that the natural map from the order complex to the topological poset with the Up topology is a weak homotopy equivalence. A familiar topological example is the Grassmann poset G n (R) of proper non-zero linear subspaces of R n + 1 partially ordered by inclusion. But our motivation in topological combinatorics is to apply the theorem to posets associated with tropical phased matroids over the tropical phase hyperfield, and in particular to elucidate the tropical version of the MacPhersonian Conjecture. This is explained in Sect. 2. [ABSTRACT FROM AUTHOR]

Given a graph G = (V , E) and a set S ⊆ V 2 of terminal pairs, the minimum multicut problem asks for a minimum edge set δ ⊆ E such that there is no s-t-path in G - δ for any { s , t } ∈ S . For | S | = 1 this is the well known s-t-cut problem, but in general the minimum multicut problem is NP-complete, even if the input graph is a tree. The multicut polytope M U L T C □ (G , S) is the convex hull of all multicuts in G; the multicut dominant is given by M U L T C (G , S) = M U L T C □ (G , S) + R ≥ 0 E . The latter is the relevant object for the minimization problem. While polyhedra associated to several cut problems have been studied intensively there is only little knowledge for multicut. We investigate properties of the multicut dominant and in particular derive results on liftings of facet-defining inequalities. This yields a classification of all facet-defining path- and edge inequalities. Moreover, we investigate the effect of graph operations such as node splitting, edge subdivisions, and edge contractions on the multicut-dominant and its facet-defining inequalities. In addition, we introduce facet-defining inequalities supported on stars, trees, and cycles and show that the former two can be separated in polynomial time when the input graph is a tree. [ABSTRACT FROM AUTHOR]

We study the behavior at infinity of the farthest and the higher-order Voronoi diagram of n line segments or lines in a d-dimensional Euclidean space. The unbounded parts of these diagrams can be encoded by a Gaussian map on the sphere of directions S d - 1 . We show that the combinatorial complexity of the Gaussian map for the order-k Voronoi diagram of n line segments and lines is O (min { k , n - k } n d - 1) , which is tight for n - k = O (1) . This exactly reflects the combinatorial complexity of the unbounded features of these diagrams. All the d-dimensional cells of the farthest Voronoi diagram are unbounded, its (d - 1) -skeleton is connected, and it does not have tunnels. A d-cell of the Voronoi diagram is called a tunnel if the set of its unbounded directions, represented as points on its Gaussian map, is not connected. In a three-dimensional space, the farthest Voronoi diagram of n ≥ 2 lines in general position has exactly n (n - 1) three-dimensional cells. The Gaussian map of the farthest Voronoi diagram of line segments and lines can be constructed in O (n d - 1 α (n)) time, for d ≥ 4 , while if d = 3 , the time drops to worst-case optimal Θ (n 2) . We extend the obtained results to bounded polyhedra and clusters of points as sites. [ABSTRACT FROM AUTHOR]

In the context of abstract interpretation-based static analysis, we propose a new abstract operator modeling the split of control flow paths: the goal of the operator is to enable a more efficient analysis when using abstract domains that are computationally expensive, having no negative effect on precision, and occasionally resulting in a more precise analysis. We focus on the case of conditional branches guarded by numeric linear constraints, including implicit numerical branches. We provide an experimental evaluation of real-world test cases, showing that by using the split operator we can achieve significant efficiency improvements with respect to the classical approach for a static analysis based on the domain of convex polyhedra. We also briefly discuss the applicability of this new operator to different, possibly non-numeric abstract domains. [ABSTRACT FROM AUTHOR]

This paper puts forward a novel affine relaxation-based algorithm for solving the minimax affine fractional program problem (MAFPP) over a polyhedron set. First of all, some new variables are introduced for deriving the equivalence problem (EP) of the MAFPP. Then, for the EP, the affine relaxation problem (ARP) is established by using the two-stage affine relaxation method. The method provides a lower bound by solving the ARP in the branch-and-bound searching process. By subdividing the output space rectangle and solving a series of ARPs continuously, the feasible solution sequence generated by the algorithm converges to a global optimal solution of the initial problem. In addition, the algorithmic maximum iteration in the worst case is estimated by complexity analysis for the first time. Lastly, the practicability and effectiveness of the algorithm have been verified by numerical experimental results. [ABSTRACT FROM AUTHOR]

We study the geometry of Hilbert spaces with complete Pick kernels and the geometry of sets in complex hyperbolic space, taking advantage of the correspondence between the two topics. We focus on questions of assembling Hilbert spaces into larger spaces and of assembling sets into larger sets. Model questions include describing the possible three-dimensional subspaces of four-dimensional Hilbert spaces with Pick kernels and describing the possible triangular faces of a tetrahedron in $\mathbb {CH}^{n}$. A novel technical tool is a complex analog of the cosine of a vertex angle. [ABSTRACT FROM AUTHOR]

We introduce a new embedding technique based on a nested barycentric coordinate system. We show that our embedding can be used to transform the problems of polyhedron approximation, piecewise linear classification and convex regression into one of finding a linear classifier or regressor in a higher dimensional (but nevertheless quite sparse) representation. Our embedding maps a piecewise linear function into an everywhere-linear function, and allows us to invoke well-known algorithms for the latter problem to solve the former. We explain the applications of our embedding to the problems of approximating separating polyhedra—in fact, it can approximate any convex body and unions of convex bodies—as well as to classification by separating polyhedra, and to piecewise linear regression. [ABSTRACT FROM AUTHOR]

Copyright of Cahiers de Topologie et Geometrie Differentielle Categoriques is the property of Andree C. EHRESMANN and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

Hard carbon (HC) has been widely regarded as the most promising anode material for sodium‐ion batteries (SIBs) due to its decent capacity and low cost. However, the poor initial Coulombic efficiency (ICE) of HC seriously hinders its practical application in SIBs. Herein, pyridinic N‐doped hard carbon polyhedra with easily accessible carbonyl groups and in situ coupled carbon nanotubes are rationally synthesized via a facile pretreated zeolitic imidazolate framework (ZIFs)‐carbonization strategy. The comprehensive ex/in situ techniques combined with theoretical calculations reveal that the synergy of pyridinic‐N and carbonyl groups promoted by the pretreatment and carbonization process would not only optimize the Na+ adsorption energy but also accelerate the desorption of Na+, significantly suppressing the irreversible capacity loss. As a result, the as‐synthesized hard carbon polyhedra as an anode can deliver an unprecedented high ICE of 98% with a large reversible capacity of 389.4 mAh g−1 at 0.03 A g−1. This work may provide an effective strategy for the structural design of HC with high ICE. [ABSTRACT FROM AUTHOR]

Artificial morphing surfaces, inspired by the high adaptability of biological tissues, have emerged as a significant area of research in recent years. However, the practical applications of these surfaces, constructed from soft materials, are considerably limited due to their low shear stiffness. Rigid‐foldable cylinders are anisotropic structures that exhibit high adaptability and shear stiffness. Thus, they have the potential to address this issue. However, changes in shape and area at both ends during folding can lead to collisions or gaps on the morphing surface. Here, a quasi‐rigid‐foldable (QRF) rate is first introduced to quantify the rigid‐foldability of a foldable structure and validate it through experiments. More importantly, a QRF polyhedron is then proposed, which is not only notably anisotropic, similar to a rigid‐foldable cylinder, but also exhibits a zero‐Poisson's ratio property, making it suitable for arraying as morphing surfaces without any collisions or gaps. Such surfaces have a myriad of applications, including modulating electromagnetic waves, gripping fragile objects, and serving as soles for climbing robots. [ABSTRACT FROM AUTHOR]

Cut-and-paste |$K$| -theory has recently emerged as an important variant of higher algebraic |$K$| -theory. However, many of the powerful tools used to study classical higher algebraic |$K$| -theory do not yet have analogues in the cut-and-paste setting. In particular, there does not yet exist a sensible notion of the Dennis trace for cut-and-paste |$K$| -theory. In this paper we address the particular case of the |$K$| -theory of polyhedra, also called scissors congruence |$K$| -theory. We introduce an explicit, computable trace map from the higher scissors congruence groups to group homology, and use this trace to prove the existence of some nonzero classes in the higher scissors congruence groups. We also show that the |$K$| -theory of polyhedra is a homotopy orbit spectrum. This fits into Thomason's general framework of |$K$| -theory commuting with homotopy colimits, but we give a self-contained proof. We then use this result to re-interpret the trace map as a partial inverse to the map that commutes homotopy orbits with algebraic |$K$| -theory. [ABSTRACT FROM AUTHOR]

We focus on the problem of finding the greatest element of the intersection of max-closed convex sets. For this purpose, we analyze the famous method of cyclic projections and show that, if this method is suitably initialized and applied to max-closed convex sets, it converges to the greatest element of their intersection. Moreover, we propose another projection method, called the decreasing projection, which turns out both theoretically and practically preferable to Euclidean projections in this particular case. Next, we argue that several known algorithms, such as Bellman-Ford and Floyd-Warshall algorithms for shortest paths or Gauss-Seidel variant of value iteration in Markov decision processes, can be interpreted as special cases of iteratively applying decreasing projections onto certain max-closed convex sets. Finally, we link decreasing projections (and thus also the aforementioned algorithms) to bounds consistency in constraint programming. [ABSTRACT FROM AUTHOR]

Novel heteroleptic anilate-based lanthanide MOFs (LnIII = Tb, Dy, Ho) have been obtained under hydrothermal conditions by the ancillary ligand synthetic strategy. These structurally isomorphous species contain octacoordinated LnIII ions with coordination polyhedra approaching an ideal D2d symmetry, best described by a distorted biaugmented trigonal prismatic C2v geometry. In the whole series, only the Dy-MOF exhibits SMM behaviour. [ABSTRACT FROM AUTHOR]

Two closely related yet distinctly different cationic clusters, [Dy52Ni44(HEIDA)36(OH)138(OAc)24(H2O)30]10+ (1) and [Dy112Ni76(HEIDA)44(EIDA)24(IDA)4(OH)268(OAc)48(H2O)44]4+ (2) (HEIDA=N‐(2‐hydroxyethyl)iminodiacetate), each featuring a multi‐shell core of Platonic and Archimedean polyhedra, were obtained. Depending on the specific conditions used for the co‐hydrolysis of Dy3+ and Ni2+, the product can be crystallized out as one particular type of cluster or as a mixture of 1 and 2. How the reaction process was affected by the amount of hydrolysis‐facilitating base and/or by the reaction temperature and duration was investigated. It has been found that a reaction at a high temperature and/or for an extended period favors the formation of the compact and thermodynamically more stable 1, while a brief reaction with a large amount of the base is good for the kinetic product 2. By tuning these intertwining conditions, the reaction can be regulated toward a particular product. [ABSTRACT FROM AUTHOR]

A total of 5233 lanthanide coordination polyhedra of the composition MO8 were selected from Cambridge Structural Database and analyzed using modern methods of unsupervised learning. The results of geometric clustering were compared with those of the continuous shape measure analysis, which is most widely used for analyzing the coordination polyhedra. The applicability was demonstrated of the new method to classification of the coordination environments that are difficult to analyze using conventional techniques. [ABSTRACT FROM AUTHOR]

Co3O4 nanoparticles were sandwiched into interlayers between ZIF‐8 and ZIF‐67 to form ZIF‐Co3O4@ZIF precursors. Pyrolysis of ZIF‐Co3O4@ZIF yielded an urchin‐like hierarchically porous carbon (Co@CNT/NC), the thorns of which were carbon nanotubes embedded Co nanoparticles. With large specific surface area and hierarchically porous structure, as‐prepared Co@CNT/NC exhibited excellent bifunctional oxygen electrocatalytic performances. It has good ORR performance with E1/2 of 0.85 V, which exceeds the Pt/C half‐wave potential (E1/2=0.83 V). In addition, Co@CNT/NC has an OER performance close to that of RuO2. To further demonstrate the effect of Co modifying on the properties, the samples were subjected to acid washing treatment. Co‐based nanoparticles were proved to After acid washing, there was obvious loss of Co particles in Co@CNT/NC, resulting in poor oxygen electrocatalysis. So, the pyrolysis products of ZIF‐8‐Co3O4@ZIF‐67 retained large specific surface area and porous structure can be retained, and on the other hand, the carbon tube structure and original polyhedron framework. Besides, existence of Co nanoparticle@carbon nanotube provided more active sites and improved the ORR and OER performances. [ABSTRACT FROM AUTHOR]

We introduce a connectedness in the digital space ℤ³ induced by a quaternary relation. Using this connectedness, we prove a digital 3D Jordan-Brouwer separation theorem for boundary surfaces of the digital polyhedra that may be face-to-face tiled with certain digital tetrahedra in ℤ ³. An advantage of the digital Jordan surfaces obtained over those given by the Khalimsky topology is that the former may bend at the acute dihedral angle π/4. [ABSTRACT FROM AUTHOR]

Structural systems based on tension and compression play a significant role in maximizing the performance of materials. Among them, structures based on the concept of tensegrity have attracted attention owing to their elegant design features and floating feeling. However, the complex construction of tensegrity-based structures hinders their implementation in practical design and architectural applications. Herein, we propose a novel tensegrity-based unit with simple connectivity and manufacturability. The external form of this unit is similar to that of a space-filling polyhedron. The prototype of the proposed unit is constructed, and the simple tension management of the unit is demonstrated. The mechanical behavior of the assembled structure is investigated by performing loading tests and numerical analyses. The behavior can be accurately predicted and general design can be applied to develop structures for practical applications. [ABSTRACT FROM AUTHOR]

This study investigates when and how university students in first-semester introductory calculus interpret multiple representations of the same function. Specifically, it focuses on three tasks. The first task has students give their definitions of 'function sameness', the results of which suggests that many students understand a function as being determined by more than its points. The second has them assess the derivative of a piecewise-defined representation of a typical polynomial function, the results of which suggest that students see a piecewise function as two separate functions with the conditions as instructions. The third involves an instance of the fundamental theorem of calculus, the results of which suggest that students view (what we see as) the same function as two different functions. Taken together, these results suggest that many students see two functions where we as mathematicians see only one. In particular, students see various features of a function's analytic and graphical representations as being essential to the function's identity. [ABSTRACT FROM AUTHOR]

Given a monomial ideal I, we study two functions that quantify ways to measure the difference between symbolic powers and usual powers of I. In many cases we determine the asymptotic growth rate of these two functions. We also perform explicit computations by using the symbolic polyhedron. [ABSTRACT FROM AUTHOR]

Equivariant Ehrhart theory generalizes the study of lattice point enumeration to also account for the symmetries of a polytope under a linear group action. We present a catalogue of techniques with applications in this field, including zonotopal decompositions, symmetric triangulations, combinatorial interpretation of the h ∗ -polynomial, and certificates for the (non)existence of invariant nondegenerate hypersurfaces. We apply these methods to several families of examples including hypersimplices, orbit polytopes, and graphic zonotopes, expanding the library of polytopes for which their equivariant Ehrhart theory is known. [ABSTRACT FROM AUTHOR]

In the mid-fifteenth century, Venice inherited Cardinal Bessarione's collection of ancient manuscripts, an exceptional legacy that contributed to the renewal of mathematical studies. This essay outlines the spread of mathematics in Venice in the early Renaissance, when scholars actively discussed Euclid and universal proportions. Based on this cultural context, the author briefly analyzes the role of polyhedra and an example of the application of proportions to architecture: the church of San Francesco della Vigna in Venice. This church is an interesting and rare case of 'declared' application of proportions to architecture, since we have specific indications in Francesco Zorzi's memorandum written in 1535. [ABSTRACT FROM AUTHOR]

Can we compute what we cannot draw? How must we draw to produce measurable representations, or visual ones? This research inquires into the relationship between mathematics and figurative representation, and more precisely between drawing and computation. The scientific imagery studied here is the representation of the five platonic solids, discussing various representation techniques from classical antiquity to modern times, and their efficacy to help calculate sizes and proportional ratios. Scholars in history of architectural drawing have too often limited their observations to the very few preserved plans and front views dating back to classical antiquity, without enlarging their investigation to other scientific fields that also rely on drawing as a research tool and communication device. Among these other fields stands the mathematical research, especially solid geometry which deals with objects and entities that have shapes that needs to be somehow drawn in 3D to be studied. [ABSTRACT FROM AUTHOR]

Spin models like the Heisenberg Hamiltonian effectively describe the interactions of open-shell transition-metal ions on a lattice and can account for various properties of magnetic solids and molecules. Numerical methods are usually required to find exact or approximate eigenstates, but for small clusters with spatial symmetry, analytical solutions exist, and a few Heisenberg systems have been solved in closed form. This paper presents a simple, generally applicable approach to analytically solve isotropic spin clusters, based on adapting the basis to both total spin and point group symmetry to factor the Hamiltonian matrix into sufficiently small blocks. We demonstrate applications to small rings and polyhedra, some of which are straightforward to solve by successive spin-coupling for Heisenberg terms only; additional interactions, such as biquadratic exchange or multi-center terms necessitate symmetry adaptation. [ABSTRACT FROM AUTHOR]

This article explores various techniques for generating fractal-like Bézier curves in both 2D and 3D environments. It delves into methods such as subdivision schemes, Iterated Function System (IFS) theory, perturbation of Bézier curves, and perturbation of Bézier basis functions. The article outlines conditions on subdivision matrices necessary for convergence and demonstrates their use in creating an IFS with an attractor aligned to the convergent point of the subdivision scheme based on specified initial data. Additionally, it discusses conditions for obtaining a one-sided approximation of a given Bézier curve through perturbation. The article also addresses considerations for perturbed Bézier basis functions to construct fractal-like Bézier curves that remain within the convex hull polygon/polyhedron defined by control points. These methods find applications in various fields, including computer graphics, art, and design. [ABSTRACT FROM AUTHOR]

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

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

Let I = {Ik}kN be a graded family of monomial ideals. We use the Newton--Okounkov body of I to: (a) give a characterization for the Noetherian property of the Rees algebra of the family I; and (b) present a combinatorial interpretation for the analytic spread of I. We also apply these results to investigate and give bounds for the generation type and the Veronese degree of the symbolic Rees algebra of a monomial ideal. [ABSTRACT FROM AUTHOR]

The crystallization of systems with soft interaction potentials is analyzed for a Yukawa system as an example. The crystallization pathway in the plane of rotational invariants q4 and q6 is studied in detail. The recently proposed invariants bcc1 and bcc2, calculated using 12 nearest neighbors, are applied for the first time to study the crystallization. It is shown that the combination of the rotational invariant technique and the method of Voronoi polyhedra allows us to identify quite simply all types of solid-like clusters formed during the solidification of the system, including bcc-like clusters, which are hardly identified by conventional methods. [ABSTRACT FROM AUTHOR]

Carbonates deserve to be pursued in the exploration of candidate systems for birefringent crystalline materials. Recently, we successfully obtained a novel example of a mixed-metal carbonate, Cs2Pb(CO3)2, using a high-temperature solution method in a closed system. Structural analysis shows that the rare Pb-C-O one-dimensional chain structure is formed by interconnecting the fundamental building blocks of PbC2O8. Detailed structural analyses and comparisons suggest a non-coplanar arrangement of CO3 due to the highly distorted polyhedra formed by Pb2+. Theoretical calculations indicate that Cs2Pb(CO3)2 exhibits an average calculated birefringence of 0.037 @ 1.64 nm, which is consistent with the conclusions deduced from the structure. The source of the properties is further elucidated through real-space atom cutting. This work provides a good reference idea for designing and synthesizing new carbonate optical crystal materials. [ABSTRACT FROM AUTHOR]

Synthesis, IR spectroscopic, and X-ray diffraction studies of R[UO2(mia)3]2·2Hmia·4H2O crystals, where R = Sr2+ (I) or Ba2+ (II), and mia is the monoiodacetate ion CH2ICOO−, have been carried out. The [UO2(mia)3]– complexes correspond to the crystal chemical formula A(B01)3, where A = UO22+, B01 = mia. It has been established that a common feature of I and II is the presence of trinuclear electrically neutral clusters {R[UO2(mia)3]2(Hmia)2(H2O)2}. At the centers of the clusters there are trigonal RO8 dodecahedra; half of their oxygen atoms belong to four different mia anions of two [UO2(mia)3]– complexes. In addition, each R atom coordinates the oxygen atoms of two water molecules and the carbonyl oxygen atoms of two Hmia molecules. Using the method of molecular Voronoi–Dirichlet polyhedra, an analysis of noncovalent interactions in the structure of I was carried out. [ABSTRACT FROM AUTHOR]

The design and synthesis of polyhedra using coordination‐driven self‐assembly has been an intriguing research area for synthetic chemists. Metal‐organic polyhedra are a class of intricate molecular architectures that have garnered significant attention in the literature due to their diverse structures and potential applications. Hereby, we report Cu‐MOP, a bifunctional metal‐organic cuboctahedra built using 2,6‐dimethylpyridine‐3,5‐dicarboxylic acid and copper acetate at room temperature. The presence of both Lewis basic pyridine groups and Lewis acidic copper sites imparts catalytic activity to Cu‐MOP for the tandem one‐pot deacetalization‐Knoevenagel/Henry reactions. The effect of solvent system and time duration on the yields of the reactions was studied, and the results illustrate the promising potential of these metal‐organic cuboctahedra, also known as nanoballs for applications in catalysis. [ABSTRACT FROM AUTHOR]

We compute the kernel of a shape embedded in 3D as a polygon mesh, which is defined as the set of all points that have a clear line of sight to every point of the mesh. The KerGen algorithm, short for Kernel Generation, employs efficient plane‐plane and line‐plane intersections, alongside point classifications based on their positions relative to planes. This approach allows for the incremental addition of kernel vertices and edges to the resulting set in a simple and systematic way. The output is a polygon mesh that represents the surface of the kernel. Extensive comparisons with the existing methods, CGAL and Polyhedron Kernel, demonstrate the remarkable timing performance of our novel additive kernel computation method. Yet another advantage of our additive process is the availability of the partial kernel at any stage, making it useful for specific geometry processing applications such as star decomposition and castable shape reconstruction. [ABSTRACT FROM AUTHOR]

The dynamical environment around the asteroid (162173) Ryugu is analyzed in detail using a constant-density polyhedron model based on the measurements from the Hayabusa2 mission. Six exterior equilibrium points (EPs) are identified along the ridge line of Ryugu, and their topological classifications fall into two distinctive categories. The initial periodic orbit (PO) families are computed and analyzed, including distant retrograde/prograde orbit (DRO/DPO) families and fifteen PO families emanating from the exterior EPs. The fifteen PO families are further divided into three categories: seven converge to an EP, seven reach Ryugu's surface, and one exhibits cyclic behavior during its progression. The existence of initial PO families converging to an EP is analyzed using the bifurcation of a degenerate EP. Connection between these families and similar ones in the circular restricted three-body problem (CRTBP) is also examined. Bifurcated PO families are identified and computed from the initial PO families, including ten families from the DROs, fifteen from the DPOs, and twenty-five associated with the EPs. The bifurcated families are separately analyzed and categorized in terms of their corresponding initial families. Connections established by the same bifurcation points between different bifurcated families are identified. A comparison is made for the dynamical environments of Ryugu and Bennu to evaluate the similarities and differences in the evolution of EPs and the progression of PO families in top-shaped asteroids. [ABSTRACT FROM AUTHOR]

Two polyhedral silver‐thiolate clusters, [S@Ag16(Tab)10(MeCN)8](PF6)14 (Ag16) and [Ag12(Tab)6(DMF)12](PF6)12 (Ag12), were synthesized by using electroneutral Tab species as protective ligands (Tab=4‐(trimethylammonio)benzenethiolate, DMF=N,N‐dimethylformamide, MeCN=acetonitrile). Ag16 has a decahedral shape composed of eight pentagon {Ag5} units and two square {Ag4} units. The structure of Ag12 is a cuboctahedron, a classical Archimedean structure composed of six triangular faces and eight square faces. The former configuration is discovered in silver‐thiolate cluster for the first time, possibly benefited from the more flexible coordination between the Tab ligand and Ag+ facilitated by the electropositive −N(CH3)3+ substituent group. Third‐order nonlinear optical studies show that both clusters in DMF exhibit reverse saturate absorption response under the irradiation of 532 nm laser. [ABSTRACT FROM AUTHOR]

The structure of cis‐ or trans‐bridged [GeF5]− anionic chains have been investigated [Mallouk et al. (1984). Inorg. Chem.23, 3160–3166] showing the first crystal structures of μ‐F‐bridged pentafluorogermanates. Herein, we report the second crystal structure of trans‐pentafluorogermanate anions present in the crystal structure of sodium trans‐pentafluorogermanate(IV) bis(hydrogen fluoride), Na[GeF5]·2HF. The crystal structure [orthorhombic Pca21, a = 12.3786 (3), b = 7.2189 (2), c = 11.4969 (3) Å and Z = 8] is built up from infinite chains of trans‐linked [GeF6]2− octahedra, extending along the b axis and spanning a network of pentagonal bipyramidal distorted Na‐centred polyhedra. These [NaF7] polyhedra are linked in a trans‐edge fashion via hydrogen fluoride molecules, in analogy to already known sodium hydrogen fluorides and potassium hydrogen fluorides. [ABSTRACT FROM AUTHOR]

Right- and left-handed propellers translate in opposite directions when rotated in the same direction. The same phenomenon may occur with enantiomorphic objects. In this paper, enantiomorphic octahedrons and tetrahedrons were fabricated using a 3D printer and rotated in a liquid medium. The rotation axis was determined by solving an eigenvalue problem for the shape tensor that hydrodynamically characterizes the shape of these polyhedrons. The model exhibited propeller-like motion, indicating the possibility of separating enantiomorphic objects. [ABSTRACT FROM AUTHOR]

Let G be a connected graph and t ≥ 1 a (rational) constant. A t-spanner of G is a spanning subgraph of G in which the distance between any pair of vertices is at most t times its distance in G. We address two problems on spanners. The first one, known as the minimumt-spanner problem (MinS t ), seeks in a connected graph a t-spanner with the smallest possible number of edges. In the second one, called minimum cost treet-spanner problem (MCTS t ), the input graph has costs assigned to its edges and seeks a t-spanner that is a tree with minimum cost. It is an optimization version of the treet-spanner problem (TreeS t ), a decision problem concerning the existence of a t-spanner that is a tree. MinS t is known to be NP -hard for every t ≥ 2 . On the other hand, TreeS t admits a polynomial-time algorithm for t ≤ 2 and is NP -complete for t ≥ 4 ; but its complexity for t = 3 remains open. We focus on the class of subcubic graphs. First, we show that for such graphs MinS 3 can be solved in polynomial time. These results yield a practical polynomial algorithm for TreeS 3 that is of a combinatorial nature. We also show that MCTS 2 can be solved in polynomial time. To obtain this last result, we prove a complete linear characterization of the polytope defined by the incidence vectors of the tree 2-spanners of a subcubic graph. A recent result showing that MinS 3 on graphs with maximum degree at most 5 is NP-hard, together with the current result on subcubic graphs, leaves open only the complexity of MinS 3 on graphs with maximum degree 4. [ABSTRACT FROM AUTHOR]

We present new second-kind integral-equation formulations of the interior and exterior Dirichlet problems for Laplace's equation. The operators in these formulations are both continuous and coercive on general Lipschitz domains in R d , d ≥ 2 , in the space L 2 (Γ) , where Γ denotes the boundary of the domain. These properties of continuity and coercivity immediately imply that (1) the Galerkin method converges when applied to these formulations; and (2) the Galerkin matrices are well-conditioned as the discretisation is refined, without the need for operator preconditioning (and we prove a corresponding result about the convergence of GMRES). The main significance of these results is that it was recently proved (see Chandler-Wilde and Spence in Numer Math 150(2):299–371, 2022) that there exist 2- and 3-d Lipschitz domains and 3-d star-shaped Lipschitz polyhedra for which the operators in the standard second-kind integral-equation formulations for Laplace's equation (involving the double-layer potential and its adjoint) cannot be written as the sum of a coercive operator and a compact operator in the space L 2 (Γ) . Therefore there exist 2- and 3-d Lipschitz domains and 3-d star-shaped Lipschitz polyhedra for which Galerkin methods in L 2 (Γ) do not converge when applied to the standard second-kind formulations, but do converge for the new formulations. [ABSTRACT FROM AUTHOR]