Regular Faces Research Articles

An artificial protein cage made from a 12-membered ring.

Artificial protein cages have great potential in diverse fields including as vaccines and drug delivery vehicles. TRAP-cage is an artificial protein cage notable for the way in which the interface between its ring-shaped building blocks can be modified such that the conditions under which cages disassemble can be controlled. To date, TRAP-cages have been constructed from homo-11mer rings, i.e., hendecamers. This is interesting as convex polyhedra with identical regular faces cannot be formed from hendecamers. TRAP-cage overcomes this limitation due to intrinsic flexibility, allowing slight deformation to absorb any error. The resulting TRAP-cage made from 24 TRAP 11mer rings is very close to regular with only very small errors necessary to allow the cage to form. The question arises as to the limits of the error that can be absorbed by a protein structure in this way before the formation of an apparently regular convex polyhedral becomes impossible. Here we use a naturally occurring TRAP variant consisting of twelve identical monomers (i.e., a dodecamer) to probe these limits. We show that it is able to form an apparently regular protein cage consisting of twelve TRAP rings. Comparison of the cryo-EM structure of the new cage with theoretical models and related cages gives insight into the rules of cage formation and allows us to predict other cages that may be formed given TRAP-rings consisting of different numbers of monomers.

On Polyhedra with Rhombic Vertices and Regular Faces

On Polyhedra with Rhombic Vertices and Regular Faces

Real-Time Occluded Face Identification Using Deep Learning

One of the most difficult aspects of face identification is face occlusion. Face occlusion is when anything is placed over the face, for example, a mask. Masks occlude multiple important facial features, like the chin, lips, nose, and facial edges. Face identification becomes challenging when important facial features are occluded. Using one of the deep learning algorithms, YOLOv5, this work tries to identify the face of someone whose face is occluded by a mask in real-time. A special program is being created to test the effectiveness of the YOLOv5 algorithm. 14 people's data were registered, and each person had 150 images used for training, validation, and testing. The images used are regular faces and mask-occluded faces. Nine distinct configurations of epoch and batch sizes were used to train the model. Then, during the testing phase, the best-performing configuration was chosen. Images and real-time input were used for testing. The highest possible accuracy of image identification is 100%, whereas the maximum accuracy of real-time identification is 64%. It was found during the testing that the brightness of the room has an influence on the performance of YOLOv5. Identifying individuals becomes more challenging when there are significant changes in brightness.

Hyperbolic polyhedral surfaces with regular faces

We study hyperbolic polyhedral surfaces with faces isometric to regular hyperbolic polygons satisfying that the total angles at vertices are at least 2π. The combinatorial information of these surfaces is shown to be identified with that of Euclidean polyhedral surfaces with negative combinatorial curvature everywhere. We prove that there is a gap between areas of non-smooth hyperbolic polyhedral surfaces and the area of smooth hyperbolic surfaces. The numerical result for the gap is obtained for hyperbolic polyhedral surfaces, homeomorphic to the double torus, whose 1-skeletons are cubic graphs.

Supramolecular triangular orthobicupola: Self-assembly of a giant Johnson solid J27

<h2>Summary</h2> The design principle of supramolecular architectures through a bottom-up strategy has always been guided by polyhedra such as Platonic solids, Archimedean solids, and prisms (antiprisms) with high symmetry. Non-uniform convex polyhedra with regular faces, i.e., Johnson solids, are rarely constructed experimentally although their topologies have been predicted through mathematical theory. Herein, self-assembly of a giant triangular orthobicupola [Zn₂₄<b>6</b>₁₂], one of 92 Johnson solids J₂₇, has been achieved from the coordination between 24 Zn²⁺ and 12 <i>tetrakis</i>-terpyridine building blocks <b>6</b> in DMF (N,N-Dimethylformamide). Different from classical O_h or T_d symmetric cuboctahedron, the structure of this triangular orthobicupola revealed D₃ symmetry which has been unambiguously determined by synchrotron X-ray analysis, NMR, and mass spectrometry. Surprisingly, the self-assembly process has been found to be solvent dependent. The same Zn²⁺ and <i>tetrakis</i>-terpyridine ligand <b>6</b> assembled into another pentagonal prism with the composition of [Zn₂₀<b>6</b>₁₀] when using methanol as the reaction solvent.

Emotional visual mismatch negativity: a joint investigation of social and non-social dimensions in adults with autism

Unusual behaviors and brain activity to socio-emotional stimuli have been reported in Autism Spectrum Disorder (ASD). Atypical reactivity to change and intolerance of uncertainty are also present, but little is known on their possible impact on facial expression processing in autism. The visual mismatch negativity (vMMN) is an electrophysiological response automatically elicited by changing events such as deviant emotional faces presented among regular neutral faces. While vMMN has been found altered in ASD in response to low-level changes in simple stimuli, no study has investigated this response to visual social stimuli. Here two deviant expressions were presented, neutral and angry, embedded in a sequence of repetitive neutral stimuli. vMMN peak analyses were performed for latency and amplitude in early and late time windows. The ASD group presented smaller amplitude of the late vMMN to both neutral and emotional deviants compared to the typically developed adults (TD) group, and only the TD group presented a sustained activity related to emotional change (i.e., angry deviant). Source reconstruction of the vMMNs further revealed that any change processing elicited a reduced activity in ASD group compared to TD in the saliency network, while the specific processing emotional change elicited activity in the temporal region and in the insula. This study confirms atypical change processing in ASD and points to a specific difficulty in the processing of emotional changes, potentially playing a crucial role in social interaction deficits. Nevertheless, these results require to be further replicated with a greater sample size and generalized to other emotional expressions.

Hereditary polyhedra with planar regular faces

A skeletal polyhedron in Euclidean 3-space is called hereditary if the symmetries of each face extend to symmetries of the entire polyhedron. In this paper we describe the finite hereditary skeletal polyhedra which have regular convex polygons or regular star-polygons as faces.

О полноте списка выпуклых RR-многогранников

В статье дано доказательство полноты перечня одного класса выпуклых симметричных многогранников в трёхмерном евклидовом пространстве. Этот класс принадлежит классу так называемых RR-многогранников. RR-многогранники характеризуются следующими условиями симметрии: у каждого многогранника класса RR существуют симметричные ромбические вершины и существуют грани, не принадлежащие ни одной звезде этих вершин; причём каждая грань, не входящая в звезду ромбической вершины, является правильной. Ромбичность вершины здесь означает, что звезда вершины составлена из n равных, одинаково расположенных ромбов. Симметричность вершины означает, что через неё проходит ось вращения порядка n её звезды. Ранее автором были найдены все многогранники с ромбическими или дельтоидными вершинами и локально симметричными гранями. При этом локально симметричные грани не принадлежат ни одной из ромбических или дельтоидных звёзд. Класс RR-многогранников получается из рассмотренных ранее заменой условия локальной симметрии неромбических граней условием их правильности.Таким образом, рассматриваемый класс RR связан с известным результатом Н. Джонсона и В. Залгаллера о перечислении всех выпуклых многогранников с условием правильности граней. Но, как показано в настоящей статье, RR-многогранники не могут быть просто получены из класса правильногранных, а требуют специального метода. Настоящая статья посвящена доказательству полноты класса RR-многогранников с двумя изолированными симметричными ромбическими вершинами V, W. При этом ромбы сходятся в вершинах V, W не обязательно своими острыми углами и V, W не обязательно разделены только одним поясом правильных граней.

Extremal rays in the Hermitian eigenvalue problem

The Hermitian eigenvalue problem asks for the possible eigenvalues of a sum of $$n\times n$$ Hermitian matrices, given the eigenvalues of the summands. The regular faces of the cones $$\Gamma _n(s)$$ controlling this problem have been characterized in terms of classical Schubert calculus by the work of several authors. We determine extremal rays of $$\Gamma _n(s)$$ (which are never regular faces) by relating them to the geometry of flag varieties: the extremal rays either arise from “modular intersection loci”, or by “induction” from extremal rays of smaller groups. Explicit formulas are given for both the extremal rays coming from such intersection loci, and for the induction maps.

Alternative method to determine parting direction automatically for generating core and cavity of two-plate mold using B-rep of visibility map

The existing method to determine the parting direction for core and cavity of two-plate mold is usually limited to decided types of direction. This limitation reduces the capability of the existing method to select the best parting direction for a 3D CAD model of an object. This paper introduces a new algorithm for another alternative that overcomes the limitation and takes less computational time by using the boundary representation (B-rep) of a visibility map to determine the parting direction. By extracting the geometric and topological information of a visibility (V) map, a B-rep entity is created to represent the V-map, which replaces the existing method that uses convex hull algorithms. To illustrate the capability of the algorithm, the 3D CAD model of objects with regular and freeform faces, holes, protrusion, and depression is presented.

Simulating Compressible Two-Medium Flows with Sharp-Interface Adaptive Runge–Kutta Discontinuous Galerkin Methods

A cut cell based sharp-interface Runge–Kutta discontinuous Galerkin method, with quadtree-like adaptive mesh refinement, is developed for simulating compressible two-medium flows with clear interfaces. In this approach, the free interface is represented by curved cut faces and evolved by solving the level-set equation with high order upstream central scheme. Thus every mixed cell is divided into two cut cells by a cut face. The Runge–Kutta discontinuous Galerkin method is applied to calculate each single-medium flow governed by the Euler equations. A two-medium exact Riemann solver is applied on the cut faces and the Lax–Friedrichs flux is applied on the regular faces. Refining and coarsening of meshes occur according to criteria on distance from the material interface and on magnitudes of pressure/density gradient, and the solutions and fluxes between upper-level and lower-level meshes are synchronized by $$L^2$$ projections to keep conservation and high order accuracy. This proposed method inherits the advantages of the discontinuous Galerkin method (compact and high order) and cut cell method (sharp interface and curved cut face), thus it is fully conservative, consistent, and is very accurate on both interface and flow field calculations. Numerical tests with a variety of parameters illustrate the accuracy and robustness of the proposed method.

Symmetry of the Pyritohedron and Lattices

The pyritohedron consisting of twelve identical but non regular pentagonal faces and its dual pseudoicosahedron that possess the pyritohedral (Th) symmetry play an essential role in understanding the crystallographic structures with the pyritohedral symmetry. The pyritohedral symmetry takes a simpler form in terms of quaternionic representation. We discuss the 3D crystals with the pyritohedral symmetry which can be derived from the Coxeter-Dynkin diagram of D3.

Role of flux on synthesis of poly single crystals of Ce3+ doped yttrium aluminum garnet

Yttrium aluminum garnet powders of good morphology are composed of tiny single crystals of regular faces. However, the growth mechanism of those ploy single crystals remains unknown. This paper provides morphology images of particles uncrushed, and it is found that those tiny crystals grow from large clusters in a pomegranate‐like manner. The morphology change occurs after phase transition, and is driven by flux, which provides a semi‐liquid environment around the crystals. This paper is beneficial for understanding growth mechanism of poly tiny single crystals of yttrium aluminum garnet powders.

Multiple Projective Dictionary Learning to Detect Plastic Surgery for Face Verification

Researchers have shown that the changes in face features due to plastic surgery can be modeled as a covariate that reduces the ability of algorithms to recognize a person's identity. Traditional dictionary learning methods learn a sparse representation using I <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</sub> and I <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> norms that are computationally expensive. This paper presents a multiple projective dictionary learning (MPDL) framework that does not require the computation of I <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</sub> and I <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> norms. We propose a novel solution to discriminate plastic surgery faces from regular faces by learning representations of local and global plastic surgery faces using multiple projective dictionaries and by using compact binary face descriptors. Experimental results on the plastic surgery database show that the proposed MPDL framework is able to detect plastic surgery faces with a high accuracy of 97.96%. To verify the identity of a person, the detected plastic surgery faces are divided into local regions of interest (ROIs) that are likely to be altered by a particular plastic surgery. The cosine distance between the compact binary face descriptors is computed for each ROI in the detected plastic surgery faces. In addition, we compute the human visual system feature similarity score based on phase congruency and gradient magnitude between the same ROIs. The cosine distance scores and the feature similarity scores are combined to learn a support vector machine model to verify if the faces belong to the same person. We integrate our proposed MPDL framework for face verification with two commercial systems to demonstrate an improvement in verification performance on a combined database of plastic surgery and regular face images.

Exciton-Polaritons with Size-Tunable Coupling Strengths in Self-Assembled Organic Microresonators

Self-assembled nano/microcrystals of organic semiconductors with regular faces can serve as optical microresonators, which hold a promise for studying the light confinement and the light-matter interaction. Here, single crystalline microribbons of 1,4-bis(2-(4-(N,N-di(p-tolyl)amino)phenyl)-vinylbenzene (DPAVB) are synthesized with well-controlled sizes by a facile solution-exchange method. We find that individual microribbon can work as Fabry-Pérot (FP) resonator along its width (w), in which strong coupling of optical modes with excitons results in the formation of exciton polaritons (EPs). The dispersion relation of E ∼ kz of EPs is constructed by extracting the energies (E) of FP resonances at integer multiples of π/w in the wavevector (kz) space. By simulating the significantly curved dispersion of EPs with a two coupled harmonic oscillator model, a coupling strength between 0.48 and 1.09 eV are obtained. Two coupling regimes are classified: in regime I, the coupling strength is constant at 0.48 eV for microribbons with the cavity length of w ≥ 2.00 μm; in regime II, the coupling strength increases dramatically from 0.48 to about 1 eV with decreasing the resonator length from w = 2.00 to 0.83 μm. More significantly, our results suggest that the exciton-photon coupling strength could be modulated by varying the size of microribbon cavities, providing an effective method for engineering the light–matter interaction in organic single crystalline microstructures.

Pourquoi un visage plaît

Regular faces are beautiful because they are expressive and arouse an interest in the observer due to the variations in their shape, their surface appearance and their movements. The sensitivity of the eye of the beholder is correlated to the expressivity of the face and of the dentition. Better, however, to avoid any indication of aggression that could foreshadow the senescence of the face.

Influence of Step Defects on the H2S Splitting on Copper Surfaces from First-Principles Microkinetic Modeling

An atomic level insight into the chemistry of hydrogen sulfide (H2S) splitting by metallic copper (Cu) is a necessary prerequisite for understanding sulfur poisoning mechanism of Cu-based water–gas shift (WGS) catalysts. In the present work, we have combined periodic density functional theory predictions and a detailed microkinetic modeling of the H2S dissociation on the stepped-defect (211), (311), and regular (111) faces of Cu to define the effect of step defects upon the reaction. The results indicate that on each surface examined, the dissociative adsorption facilely leads to the formation of element sulfur (S) via a stepwise H–S bond cleavage mechanism, with the initial molecular adsorption of H2S preferred as the rate-limiting step. It has also been pointed out that the SH disproportionation reaction does not open an alternative path for surface atomic sulfur production under the studied reaction condition. These surfaces are all predicted to be significantly covered by the S species after sufficient exposure to a realistic environment containing only several ppm of H2S. Furthermore, it is confirmed that (i) the full decomposition process is structure sensitive, and (ii) the driving force behind the step-enhanced activity of Cu toward this reaction arises not from kinetic but from thermodynamic factors. More importantly, our calculations have demonstrated that the H2S tolerance of Cu steps (and other defects) for the WGS reaction is worsened by a factor of approximately 103 as compared to a perfectly regular surface. Because these deficient sites are known as the most active sites of Cu-based shift catalysts in the absence of sulfur-containing species, it appears to be impossible to improve their activity without a dramatic loss of sulfur resistance through simply tuning catalyst surface morphology.

Platonic Hexahedron Composed of Six Organic Faces with an Inscribed Au Cluster

The structures of nanomaterials determine their individual properties and the suprastructures they can form. Introducing anisotropic shapes and/or interaction sites to isotropic nanoparticles has been proposed to extend the functionality and possible suprastructure motifs. Because of symmetric anisotropy, Platonic solids with regular polygon faces are one of the most promising nanoscale structures. Introduction of Platonic solid anisotropy to isotropic nanomaterials would expand the functionality and range of possible suprastructure motifs. Here, we demonstrate a novel strategy to obtain nano-Platonic solids through the face coordination of square porphyrins on an inscribed Au sphere with adequate size. The face coordination of the multidentate porphyrin derivatives, with four acetylthio groups facing the same direction, on the Au cluster encased the Au cluster in a Platonic hexahedron with six porphyrin faces. Transmission electron microscopy, mass spectrometry, elemental analysis, and scanning tunnelling microscopy were used to confirm the formation of the nano-Platonic hexahedron.

Product and Puzzle Formulae for $GL_n$ Belkale-Kumar Coefficients

The Belkale-Kumar product on $H^*(G/P)$ is a degeneration of the usual cup product on the cohomology ring of a generalized flag manifold. In the case $G=GL_n$, it was used by N. Ressayre to determine the regular faces of the Littlewood-Richardson cone. We show that for $G/P$ a $(d-1)$-step flag manifold, each Belkale-Kumar structure constant is a product of $d\choose 2$ Littlewood-Richardson numbers, for which there are many formulae available, e.g. the puzzles of [Knutson-Tao '03]. This refines previously known factorizations into $d-1$ factors. We define a new family of puzzles to assemble these to give a direct combinatorial formula for Belkale-Kumar structure constants. These "BK-puzzles" are related to extremal honeycombs, as in [Knutson-Tao-Woodward '04]; using this relation we give another proof of Ressayre's result. Finally, we describe the regular faces of the Littlewood-Richardson cone on which the Littlewood-Richardson number is always $1$; they correspond to nonzero Belkale-Kumar coefficients on partial flag manifolds where every subquotient has dimension $1$ or $2$.

The non-platonic and non-Archimedean noncomposite polyhedra

If a convex polyhedron with regular faces cannot be divided by any plane into two polyhedra with regular faces, then it is said to be noncomposite. We indicate the exact coordinates of the vertices of noncomposite polyhedra that are neither regular (Platonic), nor semiregular (Archimedean), nor their parts cut by no more than three planes. Such a description allows one to obtain a short proof of the existence of each of the eight such polyhedra (denoted by M8, M20–M25, M28) and to obtain other applications.