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Perimidine derivatives are versatile heterocycles with growing significance in medicinal chemistry and materials sciences. However, their structural variety remains limited. This study focused on the synthesis and crystal structure characterization of a new perimidine-based molecule. A bicyclic perimidine lactone, (1S,4R)-4,7,7-trimethyl-1-(1H-perimidin-2-yl)-2-oxabicyclo[2.2.1]heptan-3-one (1), was synthesized through an intramolecular dehydration of a monoamide intermediate formed from 1,8-diaminonaphthalene and (1S)-(–)-camphanic chloride under basic conditions. The product was purified and crystallized from acetone, giving single crystals suitable for X-ray diffraction. Structural analysis revealed two stereogenic centers and crystallization in the chiral tetragonal P43212 space group, with stabilization through N—H···O and C—H···N hydrogen bonds as well as C—H···π interactions.

For a finite group G of Lie type and a prime p, the authors compare the automorphism groups of the fusion and linking systems of G at p with the automorphism group of G itself. When p is the defining characteristic of G, they are all isomorphic, with a very short list of exceptions. When p is different from the defining characteristic, the situation is much more complex but can always be reduced to a case where the natural map from \\mathrm{Out}(G) to outer automorphisms of the fusion or linking system is split surjective. This work is motivated in part by questions involving extending the local structure of a group by a group of automorphisms, and in part by wanting to describe self homotopy equivalences of BG^\\wedge _p in terms of \\mathrm{Out}(G).

Crystallography indicates that molecules in crystalline cellulose either have twofold screw-axis (2₁) symmetry or closely approximate it, leading to short distances between H4 and H1' across the glycosidic linkage. Therefore, modeling studies of cellobiose often show elevated energies for 2₁ structures, and experimental observations are often interpreted in terms of intramolecular strain. Also, some computer models of cellulose crystallites have an overall twist as well as twisted individual chains, again violating 2₁ symmetry. To gain insight on the question of inherent strain in 2₁ structures, modeling was employed and crystal structures of small molecules were surveyed. (Residues in a disaccharide cannot be related by 2₁ symmetry because they are not identical but if their linkage geometry would lead to 2₁ symmetry for an infinite cellulose chain, the disaccharide would have 2₁ pseudo symmetry.) Several initial structures in quantum mechanics (QM) studies of cellobiose minimized to structures having 2₁ pseudo symmetry. Similarly, a number of relevant small molecules in experimental crystal structures have pseudo symmetry. While the QM models of cellobiose with 2₁ pseudo symmetry had inter-residue hydrogen bonding, the experimental studies included cellotriose undecaacetate, a molecule that cannot form conventional hydrogen bonds. Limitations in characterizing symmetry based on the linkage torsion angles phi and ψ were also explored. It is concluded that 2₁ structures have little intrinsic strain, despite indications from empirical models.

We study conformal symmetry breaking differential operators which map differential forms on

We determine conditions on the filling of electrons in a crystalline lattice to obtain the equivalent of a band insulator—a gapped insulator with neither symmetry breaking nor fractionalized excitations. We allow for strong interactions, which precludes a free particle description. Previous approaches that extend the Lieb–Schultz–Mattis argument invoked spin conservation in an essential way and cannot be applied to the physically interesting case of spin-orbit coupled systems. Here we introduce two approaches: The first one is an entanglement-based scheme, and the second one studies the system on an appropriate flat “Bieberbach” manifold to obtain the filling conditions for all 230 space groups. These approaches assume only time reversal rather than spin rotation invariance. The results depend crucially on whether the crystal symmetry is symmorphic. Our results clarify when one may infer the existence of an exotic ground state based on the absence of order, and we point out applications to experimentally realized materials. Extensions to new situations involving purely spin models are also mentioned.

Topological photonics have developed in recent years since the seminal discoveries of topological insulators in condensed matter physics for electrons. Among the numerous studies, photonic Weyl nodes have been studied very recently due to their intriguing surface Fermi arcs, Chiral zero modes and scattering properties. In this article, we propose a new design of an ideal photonic Weyl node metacrystal, meaning no excessive states are present at the Weyl nodes’ frequency. The Weyl node is stabilized by the screw rotation symmetry of space group 19. Group theory analysis is utilized to reveal how the Weyl nodes are spawned from line nodes in a higher symmetry metacrystal of space group 61. The minimum four Weyl nodes’ complex for time reversal invariant systems is found, which is a realistic photonic Weyl node metacrystal design compatible with standard printed circuit board techniques and is a complement to the few existing ideal photonic Weyl node designs and could be further utilized in studies of Weyl physics, for instance, Chiral zero modes and scatterings.

The classification of the tetrahedrite group minerals in keeping with the current IMA-accepted nomenclature rules is discussed. Tetrahedrite isotypes are cubic, with space group symmetry I4̄3̄m. The general structural formula of minerals belonging to this group can be written as M(2)A6M(1)(B4C2)X(3)D4S(1)Y12S(2)Z, where A=Cu+, Ag+, [](vacancy), and (Ag6)4+ clusters; B=Cu+, and Ag+; C=Zn2+, Fe2+, Hg2+, Cd2+, Mn2+, Cu2+, Cu+, and Fe3+; D=Sb3+, As3+, Bi3+, and Te4+; Y=S2- and Se2-; and Z=S2-, Se2-, and []. The occurrence of both Me+ and Me2+ cations at the M(1) site, in a 4:2 atomic ratio, is a case of valency-imposed double site-occupancy. Consequently, different combinations of B and C constituents should be regarded as separate mineral species. The tetrahedrite group is divided into five different series on the basis of the A, B, D, and Y constituents, i.e., the tetrahedrite, tennantite, freibergite, hakite, and giraudite series. The nature of the dominant C constituent (the so-called \"charge-compensating constituent\") is made explicit using a hyphenated suffix between parentheses. Rozhdestvenskayaite, arsenofreibergite, and goldfieldite could be the names of three other series. Eleven minerals belonging to the tetrahedrite group are considered as valid species: argentotennantite-(Zn), argentotetrahedrite-(Fe), kenoargentotetrahedrite-(Fe), giraudite-(Zn), goldfieldite, hakite-(Hg), rozhdestvenskayaite-(Zn), tennantite-(Fe), tennantite-(Zn), tetrahedrite-(Fe), and tetrahedrite-(Zn). Furthermore, annivite is formally discredited. Minerals corresponding to different end-member compositions should be approved as new mineral species by the IMA-CNMNC following the submission of regular proposals. The nomenclature and classification system of the tetrahedrite group, approved by the IMA-CNMNC, allows the full description of the chemical variability of the tetrahedrite minerals and it is able to convey important chemical information not only to mineralogists but also to ore geologists and industry professionals.

Chirality depends on particular symmetries. For crystal structures it describes the absence of mirror planes and inversion centers, and in addition to translations, only rotations are allowed as symmetry elements. However, chiral space groups have additional restrictions on the allowed screw rotations as a symmetry element, because they always appear in enantiomorphous pairs. This study classifies and distinguishes the chiral structures and space groups. Chirality is quantified using Hausdorff distances and continuous chirality measures and selected crystal structures are reported. Chirality is discussed for bulk solids and their surfaces. Moreover, the band structure, and thus, the density of states, is found to be affected by the same crystal parameters as chirality. However, it is independent of handedness. The Berry curvature, as a topological measure of the electronic structure, depends on the handedness but is not proof of chirality because it responds to the inversion of a structure. For molecules, optical circular dichroism is one of the most important measures for chirality. Thus, it is proposed in this study that the circular dichroism in the angular distribution of photoelectrons in high symmetry configurations can be used to distinguish the handedness of chiral solids and their surfaces.

This volume contains the proceedings of the AMS Special Session on Harmonic Analysis, in honor of Gestur Ólafsson's 65th birthday, held on January 4, 2017, in Atlanta, Georgia. The articles in this volume provide fresh perspectives on many different directions within harmonic analysis, highlighting the connections between harmonic analysis and the areas of integral geometry, complex analysis, operator algebras, Lie algebras, special functions, and differential operators. The breadth of contributions highlights the diversity of current research in harmonic analysis and shows that it continues to be a vibrant and fruitful field of inquiry.

The columbite supergroup is established. It includes five mineral groups (ixiolite, wolframite, samarskite, columbite and wodginite) and one ungrouped species (lithiotantite). The criteria for a mineral to belong to the columbite supergroup are: the general stoichiometry MO2; the crystal structure based on the hexagonal close packing (hcp) of anions (or close to it); the six-fold coordination number of M-type cations (augmented to eight-fold in the case of slight distortion of hcp); and the presence of zig-zag chains of edge-sharing M-centred polyhedra. The ixiolite-type structure is considered as an aristotype with the space group Pbcn, the smallest unit cell volume, and the basic vectors a0, b0 and c0. Based on the multiplying of the ixiolite-type unit cell the following derivatives are distinguished: ixiolite type [ixiolite-group minerals; a = a0,b = b0 and c = c0; space group Pbcn; the members are ixiolite-(Mn2+), ixiolite-(Fe2+), scrutinyite, seifertite and srilankite]; wolframite type [wolframite-group minerals, ordered analogues of the ixiolite type with a = a0,b = b0 and c = c0; P2/c; the members are ferberite, hübnerite, huanzalaite, sanmartinite, heftetjernite, nioboheftetjernite, rossovskyite and riesite]; samarskite type [samarskite-group minerals; a = 2a0, b = b0 and c = c0; P2/c; the members are samarskite-(Y), ekebergite and shakhdaraite-(Y)]; columbite type [columbite-group minerals; a = 3a0,b = b0 and c = c0; Pbcn; the members are columbite-(Fe), columbite-(Mn), columbite-(Mg), tantalite-(Fe), tantalite-(Mn), tantalite-(Mg), fersmite, euxenite-(Y), tanteuxenite-(Y) and uranopolycrase]; and wodginite type [wodginite-group minerals; a = 2a0,b = 2b0 and c = c0; C2/c; the members are wodginite, ferrowodginite, titanowodginite, ferrotitanowodginite, tantalowodginite, lithiowodginite and achalaite]. Samarskite-(Yb), ishikawaite and calciosamarskite are insufficiently studied, tentatively considered as possible members of the samarskite supergroup. Qitianlingite, yttrocolumbite-(Y), yttrotantalite-(Y) and yttrocrasite-(Y) are questionable and need further studies. Polycrase-(Y) is discredited as identical to euxenite-(Y). Ixiolite has been renamed as ixiolite-(Mn2+), with the end-member formula (Ta2/3Mn2+1/3)O2. Ta- and Nb-dominant analogues of ixiolite with different schemes of charge balancing have the end-member formulae (M15+0.5M23+0.5)O2, M15+2/3M22+1/3)O2, M15+0.75M2+0.25)O2 or M15+0.8∎0.2)O2 and the root name 'ixiolite' (for M1 = Ta) or 'nioboixiolite' (for M1 = Nb).