Symmetry Group Research Articles

Case report: Improving quality of life through hyaluronic acid-based fillers after orbital cancer treatment

BackgroundMyoepithelial carcinoma is a very rare yet aggressive tumor in children. Surgical intervention and local radiotherapy often lead to post-therapy complications, affecting both the aesthetic and functional quality of life in survivors. Hyaluronic acid (HA) dermal fillers offer a minimally invasive option to improve the appearance and quality of life for these patients once they are declared tumor-free.Case presentationWe present the case of an 18-year-old girl with a history of myoepithelial carcinoma in the right upper orbit, diagnosed at the age of 8. The patient underwent surgery to remove the tumor and lacrimal gland, followed by chemotherapy and radiotherapy. A complete response to treatment was achieved, and the patient was monitored with regular clinical and radiological exams for 5 years, after which she was declared tumor-free and followed for late effects of therapy. Post-surgical radiotherapy resulted in atrophy of the upper orbital frame and functional complications. The patient exhibited upper eyelid retraction, ptosis, continuous lacrimation, and conjunctival redness. Ten years after treatment, the patient underwent dermal filler injections using Aliaxin® Essential Volume (AEV) and Aliaxin® Superior Volume (ASV) to address the aesthetic impairment of the upper right orbit. ASV was administered using a 22G x 50mm cannula on the periosteum of the superior orbital frame, entering from the outer canthus. AEV was injected with a cannula into the muscle, also entering from the outer canthus. Before treatment, the patient exhibited upper eyelid retraction, ptosis, continuous lacrimation, and conjunctival redness. Following the injections, improvements were observed in all pre-treatment symptoms. The closing ability of the upper eyelid was restored, along with superior orbital volume and symmetry. Enhanced eyelid function improved eye hydration, reduced redness in the conjunctiva, and led to better vision and overall quality of life.ConclusionTo our knowledge, this is the first reported case of using dermal fillers to treat ocular changes resulting from cancer treatment. Injections of AEV and ASV provided both aesthetic and functional improvements.

Four-component protein nanocages designed byprogrammed symmetry breaking.

Four, eight or twenty C3 symmetric protein trimers can be arranged with tetrahedral, octahedral or icosahedral point group symmetry to generate closed cage-like structures1,2. Virusesaccess more complex higher triangulation number icosahedral architecturesby breaking perfect point group symmetry3-9, but nature appears not to have explored similar symmetry breaking fortetrahedral or octahedral symmetries. Here we describe a general design strategy for building higher triangulation number architectures starting fromregular polyhedra through pseudosymmetrization of trimeric building blocks. Electron microscopy confirms the structures of T = 4 cages with 48 (tetrahedral), 96 (octahedral) and 240 (icosahedral) subunits, each with 4 distinct chains and 6 different protein-protein interfaces, and diameters of 33 nm, 43 nm and 75 nm, respectively. Higher triangulation number viruses possess very sophisticated functionalities; our general route to higher triangulation number nanocages should similarly enable a next generation of multiple antigen-displaying vaccine candidates10,11 and targeted delivery vehicles12,13.

Unifying the thermodynamic and electrostatic view on the pyroelectric effect

Pyroelectricity is a first rank tensor (vector) property that connects electric displacement to temperature. Like all vector properties, it is allowed only in certain point group symmetries. In related literature, a number of formulations exist that apparently treat various contributions to the pyroelectric coefficient but without strictly considering the crystal symmetry. We revisit the formulation of the pyroelectric coefficient in the presence of external fields and find that a consistent treatment of the pyroelectric coefficient, allowing one to arrive at a single convergent formula starting from either thermodynamic or electrostatic arguments, may not be straightforward. Motivated by this outcome, we develop an approach allowing both electrostatic and thermodynamic arguments for pyroelectricity to converge to one single expression with mathematical consistency of the partial derivatives. Albeit not very significant at first sight, the approach and the manner in which fields are introduced prove vital in evaluating the field contributions to the pyroelectric effect and their deconvolution in experiments. Importance of the crystal point groups is also highlighted in the context of vector properties, specifically pyroelectric coefficient. Finally, the so-called induced pyroelectricity and a correct mathematical expansion of the polarization is discussed to clarify the contributions of the pyroelectric effect.

Enlargement of Symmetry Groups in Physics: A Practitioner’s Guide

Wigner’s classification has led to the insight that projective unitary representations play a prominent role in quantum mechanics. The physics literature often states that the theory of projective unitary representations can be reduced to the theory of ordinary unitary representations by enlarging the group of physical symmetries. Nevertheless, the enlargement process is not always described explicitly: it is unclear in which cases the enlargement has to be conducted on the universal cover, a central extension, or a central extension of the universal cover. On the other hand, in the mathematical literature, projective unitary representations have been extensively studied, and famous theorems such as the theorems of Bargmann and Cassinelli have been achieved. The present article bridges the two: we provide a precise, step-by-step guide on describing projective unitary representations as unitary representations of the enlarged group. Particular focus is paid to the difference between algebraic and topological obstructions. To build the bridge mentioned above, we present a detailed review of the difference between group cohomology and Lie group cohomology. This culminates in classifying Lie group central extensions by smooth cocycles around the identity. Finally, the take-away message is a hands-on algorithm that takes the symmetry group of a given quantum theory as input and provides the enlarged group as output. This algorithm is applied to several cases of physical interest. We also briefly outline a generalization of Bargmann’s theory to time-dependent phases using Hilbert bundles.

Taub-NUT instanton as the self-dual analog of Kerr

It was recently conjectured that a certain vacuum Kerr-Schild spacetime, which may be regarded as a self-dual analog of the Kerr metric, is equivalent to the self-dual Taub-NUT instanton. We confirm this conjecture by applying the Cartan-Karlhede algorithm to each metric and showing that for suitable choices of null tetrad, the algorithm leads to the same invariants and linear isotropy groups for both, establishing their equivalence. While it is well-known that the Taub-NUT solution and its self-dual version admit a double Kerr-Schild form, the observation that the self-dual Taub-NUT instanton admits a single Kerr-Schild form has only been made very recently. The two metrics we compare may be regarded as either complex metrics with Lorentzian (1, 3) signature or real metrics with Kleinian (2, 2) signature; here we take the latter view. Significant simplifications occur when the null tetrads are chosen to consist of two pairs of complex conjugate null vectors rather than four real independent ones. As a bonus, our work provides the first example of applying the Cartan-Karlhede algorithm using a null tetrad of this type.

Representing The World Around Us: Applications of Group Representation Theory to Molecular Orbital (MO) Theory

Molecular orbital (MO) theory is a theory at the forefront of modern chemistry, allowing for accurate descriptions of reactivity of molecules by using quantum mechanics to predict the location of electrons within a molecule, and their corresponding energies. The equations which govern their behavior, the Schrodinger equation, are often difficult to solve. Often, we can only approximate a solution using numerical methods. This paper discusses a method which exploits a molecule’s internal symmetry. Specifically, we use Group representation theory to help analyze and break down the molecular symmetry, and then use the analysis to help us find the MO’s. First, we establish key results about irreducible representations and characters. We then establish a correspondence between MO’s and irreducible representations. We then use the results we obtained to perform MO calculations on water (H2O). We then compare the results obtained via our MO theory calculations and Valance Bond Theory (VBT). We conclude by showing these calculations are best used for rough work, being most useful for deciding which atomic orbitals they arose from, and each MO’s energies relative to each other.

Bifurcation analysis of figure-eight choreography in the three-body problem based on crystallographic point groups

Abstract The bifurcation of figure-eight choreography is analyzed by its symmetry group based on the variational principle of the action. The irreducible representations determine the symmetry and the dimension of the 
Lyapunov-Schmidt reduced action, which yields four types of bifurcations in the sequence of the bifurcation cascade. Type 1 bifurcation, represented by trivial representation, bifurcates two solutions. Type 2, by non-trivial one-dimensional representation, bifurcates two congruent solutions. Type 3 and 4, by two-dimensional irreducible representations, bifurcate two sets of three and six congruent solutions, respectively. We analyze numerical bifurcation solutions previously published and four new ones: 
non-symmetric choreographic solution of type 2, non-planar solution of type 2, y-axis symmetric solution of type 3, and non-symmetric solution of type 4.

Extended Symmetry of Higher Painlevé Equations of Even Periodicity and Their Rational Solutions

The structure of the extended affine Weyl symmetry group of higher Painlevé equations of N periodicity depends on whether N is even or odd. We find that for even N, the symmetry group A^N−1(1) contains the conventional Bäcklund transformations sj,j=1,…,N, the group of automorphisms consisting of cycling permutations but also reflections on a periodic circle of N points, which is a novel feature uncovered in this paper. The presence of reflection automorphisms is connected to the existence of degenerated solutions, and for N=4, we explicitly show how even reflection automorphisms cause degeneracy of a class of rational solutions obtained on the orbit of the translation operators of A^3(1). We obtain the closed expressions for the solutions and their degenerated counterparts in terms of the determinants of the Kummer polynomials.

Distinct clinical characteristics of adolescent idiopathic scoliosis with asymmetrical ESR1 expression in paraspinal muscle progenitor cells.

Previous studies found decreased ESR1 expression of concave paraspinal muscle progenitor cells could contribute to the initiation and progression of adolescent idiopathic scoliosis (AIS). The current study investigated the clinical characteristics of AIS with asymmetrical ESR1 expression in paraspinal muscle progenitor cells. Bilateral deep paraspinal muscle progenitor cells were obtained from 25 consecutive eligible female patients with AIS. RT-qPCR was performed to evaluate the expression of ESR1. The demographic data (the age at surgery, height, weight, BMI, and age at initiation), posteroanterior and lateral radiographs data (Risser sign, Cobb angle, apical vertebral rotation, and location of apical vertebra), and MR imaging data (bilateral paraspinal muscle CSA ratio and bilateral fatty component ratio) were collected. The correlation between asymmetrical ESR1 expression of paraspinal muscle progenitor cells and the aforementioned clinical characteristics were analyzed. Twelve out of twenty-five patients (48%) showed bilateral ESR1 expression ratio (convex/concave) more than 1.5 folds, and they were divided into the ESR1 asymmetry group. When compared with the ESR1 symmetry group, patients in the ESR1 asymmetry group showed significantly more severe scoliosis (p = 0.041), more hypoplastic concave paraspinal muscle (p = 0.015), and more muscular fatty infiltration in the concave side (p = 0.034). The bilateral ESR1 expression ratio was significantly correlated with Cobb angle (r 2 = 0.282, p = 0.006), bilateral paraspinal muscle CSA ratio (r 2 = 0.253, p = 0.011), and bilateral fatty component ratio (r 2 = 0.248, p = 0.011). There were 48% of AIS patients with significantly decreased ESR1 expression in concave paraspinal muscle progenitor cells (convex/concave>1.5 folds), while patients with more asymmetrical ESR1 expression showed more hypoplastic paraspinal muscle and fatty infiltration on the concave side, and more severe scoliotic deformity.

Equivariant neural networks for robust CP observables

We introduce the usage of equivariant neural networks in the search for violations of the charge-parity (CP) symmetry in particle interactions at the CERN Large Hadron Collider. We design neural networks that take as inputs kinematic information of recorded events and that transform equivariantly under a symmetry group related to the CP transformation. We show that this algorithm allows one to define observables reflecting the properties of the CP symmetry, showcasing its performance in several reference processes in top quark and electroweak physics. Imposing equivariance as an inductive bias in the algorithm improves the numerical convergence properties with respect to other methods that do not rely on equivariance and allows one to construct optimal observables that significantly improve the state-of-the-art methodology in the searches considered. Published by the American Physical Society 2024

Enhancing Low-Concentration Electroreduction of NO to NH3 via Potential-Controlled Active Site-Intermediate Interactions.

Electronic defect states in catalysts are recognized as effective active sites for enhancing the low-concentration electroreduction of NO to NH3 (NORR). Their structures dynamically evolve with applied potentials, allowing the active sites to adjust interactions with intermediates, thereby improving electrocatalytic performance. However, the dynamic changes in these interactions under applied potentials remain poorly understood, hindering the design of diverse electrocatalytic systems. Herein, we developed a strategy that unitizespotential to control the interactions between active sites and intermediates over VO-TiO2-xto enhance NORR performance. Combining constant inner potential (CIP) DFT calculations with in situ (spectro)electrochemical measurements, we investigated how the electrode potential influences these interactions in NORR. The results demonstrate that applyingexternal potentialspromotethe formation of Ti3+ active sites and alterits spatial symmetry of degenerate orbitals to facilitate the generation of key intermediates for NO-to-NH3 conversion. Therefore, the VO-TiO2-x catalyst exhibited superior NORR performance with Faradaic efficiency of 76.4% andNH3 yield rate of 632.9 μg h-1 mgcat.-1 under 1.0 vol% NO atmosphere, which is competitive with reported works under higher NO concentration (above 10 vol%). Remarkably, the NORR process achieved a record-breaking NH3 yield of 2292.7 μg h-1 mgcat.-1 in a membrane electrode assembly (MEA) electrolyzer.

Ligand-Directed Valence Band Engineering in Pb2+ Hybrid Crystals: Achieving Dispersive Bands and Shallow Valence Band Maximum.

While crystalline hybrid solids hold great potential as novel semiconductors, most semiconductive hybrids utilize transition metal ions, which inherently limit carrier mobility due to the small band dispersion derived from the d orbitals. The filled s orbitals of post-transition metal ions offer the potential to design dispersed valence bands, but a method to translate the local structure design of these metal ions to valence band engineering is still in development. This study focuses on Pb2+-containing hybrid crystals, developing a simple strategy to control the Pb2+ coordination geometry through the molecular design of azole ligands. By preprogramming the coordination number of Pb2+ with azolate ligands, we succeeded in obtaining an isotropic coordination environment at a higher coordination number, resulting in a dispersed valence band and shallow valence band maximum while having a wide band gap. Detailed analysis of the band structures reveals that the energy levels and symmetry of the molecular orbitals of the anions play important roles in realizing these antinomic properties. This ligand-directed approach achieves both isotropy and covalency in the coordination bond by exploiting the diversity of the molecular orbitals. Our findings provide a foundation for future design strategies to optimize electronic structures in hybrid materials, advancing their application in semiconductive devices.

Engineering Two-Dimensional Magnetic Heterostructures: A Theoretical Perspective.

Two-dimensional (2D) magnetic materials have attracted great attention due to their promise for applications in future high-speed, low-energy quantum computing and memory devices. By integrating 2D magnetic materials with other magnetic or nonmagnetic materials to form heterostructures, the synergistic effects of interlayer orbital hybridization, spin-orbit coupling, and symmetry breaking can surpass the performance of single-layer materials and lead to novel physical phenomena. This review provides a comprehensive theoretical analysis of engineering 2D magnetic heterostructures, emphasizing the fundamental physics of interlayer interactions and the resulting enhancements and novel properties. It reviews the mechanisms and progress in tuning the magnetic ordering, enhancing the Curie temperature (Tc) and modulating properties such as topological magnetic structures, spin polarization, electronic band topology, valley polarization, and magnetoelectric coupling through the construction of 2D magnetic heterostructures. Additionally, this review discusses the current challenges faced by 2D magnetic heterostructures, aiming to guide the future design of higher-performance magnetic heterostructures.

The role of data embedding in equivariant quantum convolutional neural networks

Geometric deep learning refers to the scenario in which the symmetries of a dataset are used to constrain the parameter space of a neural network and thus, improve their trainability and generalization. Recently, this idea has been incorporated into the field of quantum machine learning, which has given rise to equivariant quantum neural networks (EQNNs). In this work, we investigate the role of classical-to-quantum embedding on the performance of equivariant quantum convolutional neural networks (EQCNNs) for the classification of images. We discuss the connection between the data embedding method and the resulting representation of a symmetry group and analyze how changing representation affects the expressibility of an EQCNN. We numerically compare the classification accuracy of EQCNNs with three different basis-permuted amplitude embeddings to the one obtained from a non-equivariant quantum convolutional neural network (QCNN). Our results show a clear dependence of classification accuracy on the underlying embedding, especially for initial training iterations. The improvement in classification accuracy of EQCNN over non-equivariant QCNN may be present or absent depending on the particular embedding and dataset used. The noisy simulation using simple noise models shows that certain EQCNNs are more robust to noise than non-equivariant QCNNs. It is expected that the results of this work can be useful to the community for a better understanding of the importance of data embedding choice in the context of geometric quantum machine learning.

Chiral Metal Self‐Assemblies of Zirconium‐Tetrahedra and Their Second Harmonic Generation Activity

AbstractThe chirality of metal‐organic cages holds enormous potential for novel applications in diverse fields, while it is relatively rare to employ such asymmetric units for the construction of noncentrosymmetric materials. Herein, by self‐assembling the 4,4′,4′′‐nitrilotribenzoic acid (H3NBA) with bis(cyclopentadienyl)‐zirconium dichloride (Cp2ZrCl2, Cp=η5‐C5H5) in different solvent conditions, we have obtained three hierarchical packing modes of metallo‐tetrahedra with distinct spatial symmetry groups (designated as Zr‐α, Zr‐β, and Zr‐γ). Among them, Zr‐α employs a simple cubic arrangement and is a common centrosymmetric superstructure, which consists of a pair of equimolar metallo‐tetrahedra enantiomers in its unit cell. While Zr‐β results in conglomerates with spontaneous resolution without using any resolving agents, giving rise to two enantiopure entities separately (Zr‐β‐P, Zr‐β‐M). More importantly, Zr‐γ breaks the inversion center of symmetry and crystallizes into a racemic yet non‐centrosymmetric superstructure with face‐centered cubic packing mode. Based on the non‐centrosymmetric nature, the hierarchical superstructure Zr‐γ displayed good second harmonic generation activities. This work presents a successful instance wherein the reaction solvent induces the modulation of intermolecular packing mode to afford non‐centrosymmetric solid materials, which can greatly promote the development of noncentrosymmetric solid (NCS) materials.

Chiral Metal Self-Assemblies of Zirconium-Tetrahedra and Their Second Harmonic Generation Activity.

The chirality of metal-organic cages holds enormous potential for novel applications in diverse fields, while it is relatively rare to employ such asymmetric units for the construction of noncentrosymmetric materials. Herein, by self-assembling the 4,4',4''-nitrilotribenzoic acid (H3NBA) with bis(cyclopentadienyl)-zirconium dichloride (Cp2ZrCl2, Cp=η5-C5H5) in different solvent conditions, we have obtained three hierarchical packing modes of metallo-tetrahedra with distinct spatial symmetry groups (designated as Zr-α, Zr-β, and Zr-γ). Among them, Zr-α employs a simple cubic arrangement and is a common centrosymmetric superstructure, which consists of a pair of equimolar metallo-tetrahedra enantiomers in its unit cell. While Zr-β results in conglomerates with spontaneous resolution without using any resolving agents, giving rise to two enantiopure entities separately (Zr-β-P, Zr-β-M). More importantly, Zr-γ breaks the inversion center of symmetry and crystallizes into a racemic yet non-centrosymmetric superstructure with face-centered cubic packing mode. Based on the non-centrosymmetric nature, the hierarchical superstructure Zr-γ displayed good second harmonic generation activities. This work presents a successful instance wherein the reaction solvent induces the modulation of intermolecular packing mode to afford non-centrosymmetric solid materials, which can greatly promote the development of noncentrosymmetric solid (NCS) materials.

DFT and Model Hamiltonian Study of Optoelectronic Properties of Some Low-Symmetry Graphene Quantum Dots.

We have studied the electronic and optical properties of three low-symmetry graphene quantum dots (GQDs), with point-group symmetries C2v and C2h. For the calculations of linear optical absorption spectra, we employed both first-principles time-dependent density-functional theory (TDDFT) and the electron-correlated Pariser-Parr-Pople (PPP) model coupled with the configuration-interaction (CI) approach. In the PPP-CI approach, calculations were performed using both screened and standard parameters, along with efficiently incorporating electron correlation effects using multireference singles-doubles CI for both ground and excited states. We assume that the GQDs are saturated by hydrogen atoms at the edges, making them effectively polycyclic aromatic hydrocarbons (PAHs) dibenzo[bc,ef]coronene (also known as benzo(1,14)bisanthene, C30H14) and two isomeric compounds, dinaphtho[8,1,2abc;2',1',8'klm]coronene and dinaphtho[8,1,2abc;2',1',8'jkl]coronene with the chemical formula C36H16. The two isomers have different point group symmetries; therefore, this study will also help us understand the influence of symmetry on the optical properties. A common feature of the absorption spectra of the three GQDs is that the first peak representing the optical gap is of low to moderate intensity, while the intense peaks appear at higher energies. For each GQD, PPP model calculations performed with the screened parameters agree well with the experimental results of the corresponding PAH and also with the TDDFT calculations. To further quantify the influence of electron-correlation effects, we also computed the singlet-triplet gap (spin gap) of the three GQDs, and we found them to be significant.

Symmetric solutions to symmetric partial difference equations

This paper studies discrete systems defined by linear difference equations on the lattice Z n that are invariant under a finite group of symmetries, and shows that there always exist solutions to such systems that are also invariant under this group of symmetries.

Going Beyond Woodward and Hoffmann's Electrocyclizations and Cycloadditions: Sigmatropic Rearrangements.

On June 1, 1965, R. B. Woodward and Roald Hoffmann published their third communication in the Journal of the American Chemical Society in which they applied orbital symmetry control to explain the mechanism of a wide variety of valence isomerizations that they termed "sigmatropic reactions." This publication reveals the research trajectory taken by Hoffmann from which this portion of the no-mechanism problem was solved. Hoffmann used five different quantum chemical tools, all based on either extended Hückel theoretical calculations or frontier molecular orbital theory, in his research. Hoffmann's laboratory notebooks and his three draft manuscripts along with Woodward's four subsequent drafts have survived the past 59 years and provide an excellent window into the thinking and manuscript-writing processes used by these Nobel laureates in February-April 1965.

Tracial states on groupoid $C^{*}$-algebras and essential freeness

Let \mathcal{G} be a locally compact Hausdorff étale groupoid. We call a tracial state \tau on a general groupoid C^{*} -algebra C_{\nu}^{*}(\mathcal{G}) canonical if \tau=\tau|_{C_0(\mathcal{G}^{(0)})}\circ E , where E:C^{*}_{\nu}(\mathcal{G}) \to C_{0}(\mathcal{G}^{(0)}) is the canonical conditional expectation. In this paper, we consider so-called fixed point traces on C_{c}(\mathcal{G}) , and prove that \mathcal{G} is essentially free if and only if any tracial state on C_{\nu}^{*}(\mathcal{G}) is canonical and any fixed point trace is extendable to C_{\nu}^{*}(\mathcal{G}) .As applications, we obtain the following: (1) a group action is essentially free if every tracial state on the reduced crossed product is canonical and every isotropy group is amenable; (2) if the groupoid \mathcal{G} is second-countable, amenable and essentially free then every (not necessarily faithful) tracial state on the reduced groupoid C^{*} -algebra is quasidiagonal.