The geometrical structure of 2D bond-orientational order

A nonassociative algebra endowed with a Lie bracket, called a torsion algebra, is viewed as an algebraic analog of a manifold with an affine connection. Its elements are interpreted as vector fields and its multiplication is interpreted as a connection. This provides a framework for differential geometry on a formal manifold with a formal connection. A torsion algebra is a natural generalization of pre‐Lie algebras which appear as the “torsionless” case. The starting point is the observation that the associator of a nonassociative algebra is essentially the curvature of the corresponding Hochschild quasicomplex. It is a cocycle, and the corresponding equation is interpreted as Bianchi identity. The curvature‐associator‐monoidal structure relationships are discussed. Conditions on torsion algebras allowing to construct an algebra of functions, whose algebra of derivations is the initial Lie algebra, are considered. The main example of a torsion algebra is provided by the pre‐Lie algebra of Hochschild cochains of a K‐module, with Lie bracket induced by Gerstenhaber composition.

We present an x-ray study of liquid crystal membranes in the vicinity of the hexatic-smectic phase transition by means of angular x-ray cross-correlation analysis. By applying two-point angular-intensity cross-correlation functions to the measured series of diffraction patterns the parameters of bond-orientational (BO) order in hexatic phase were directly determined. The temperature dependence of the positional correlation lengths was analyzed as well. The obtained correlation lengths show larger values for the higher-order Fourier components of BO order. These findings indicate a strong coupling between BO and positional order.

In this paper, we report results from synchrotron X-ray scattering studies of thefluid/hexactic/solid phases and phase transitions in both very thick and very thin, freely suspended films of tilted hexatic liquid crystals. Contrary to the thick film case, the higher Fourier coefficients describing the bond orientational order are suppressed in very thin films. This suppression is consistent with a two-dimensional bond orientational order parameter,Ψ6, rather than the three-dimensional bond orientational order parameter found in very thick films. For a film containing twently-three (23) smectic layers we find thatΨ6 is two-dimensional whereas the positional order in the crystallineSJ phase is three-dimensional. We present an analysis of the thick film data in terms of the three-dimensionalXY-model and a new mean field theory model which incorporates explicitly the quasi two-dimensional nature of bulk smectic phases.

In flat geometries, two-dimensional hexatic order has only a minor effect on capillary waves on a liquid substrate and on undulation modes in lipid bilayers. However, extended bond-orientational order alters the long-wavelength spectrum of these ripples in spherical geometries. We calculate this frequency shift and suggest that it might be detectable in lipid bilayer vesicles, at the surface of liquid metals, and in multielectron bubbles in liquid helium at low temperatures. Hexatic order also leads to a shift in the threshold for the fission instability induced in the later two systems by an excess of electric charge.

We conduct a numerical investigation of structural order in the shifted-force Lennard-Jones system by calculating metrics of translational and bond-orientational order along various paths in the phase diagram covering equilibrium solid, liquid, and vapor states. A series of nonequilibrium configurations generated through isochoric quenches, isothermal compressions, and energy minimizations are also considered. Simulation results are analyzed using an ordering map representation [Torquato et al., Phys. Rev. Lett. 84, 2064 (2000); Truskett et al., Phys. Rev. E 62, 993 (2000)] that assigns both equilibrium and nonequilibrium states coordinates in an order metric plane. Our results show that bond-orientational order and translational order are not independent for simple spherically symmetric systems at equilibrium. We also demonstrate quantitatively that the Lennard-Jones and hard sphere systems sample the same configuration space at supercritical densities. Finally, we relate the structural order found in fast-quenched and minimum-energy configurations (inherent structures).

Recently, the full phase behaviour of 2D colloidal hard spheres was experimentally established, and found to involve a first order liquid to hexatic transition and a continuous hexatic to crystal transition (Thorneywork et al 2017 Phys. Rev. Lett. 118 158001). Here, we expand upon this work by considering the behaviour of the bond-orientational correlation time and Frank’s constant in the region of these phase transitions. We also consider the excess entropy, as calculated from the radial distribution functions, for a wide range of area fractions covering the liquid, hexatic and crystal phases. In all cases, the behaviour of these quantities further corroborates the previously reported melting scenario of 2D colloidal hard spheres.

Within the framework of differential geometry, Yang's parallel-displacement gauge theory is considered with respect to “pure” gravitational fields. In afour-dimensional Riemannian manifold it is shown that thedouble self-dual solutions obey Einstein's vacuum equations with the cosmological term, whereas the doubleanti-self-dual configurations satisfy the Rainich conditions of Wheeler'sgeometrodynamics. Conformal methods reveal that the gravitational analog of the “instanton” or pseudoparticle solution of Yang-Mills theory was already known to Riemann.

Considering trajectory curves, integral of n-dimensional dynamical systems, within the framework of Differential Geometry as curves in Euclidean n-space, it will be established in this article that the curvature of the flow, i.e. the curvature of the trajectory curves of any n-dimensional dynamical system directly provides its slow manifold analytical equation the invariance of which will be then proved according to Darboux theory. Thus, it will be stated that the flow curvature method, which uses neither eigenvectors nor asymptotic expansions but only involves time derivatives of the velocity vector field, constitutes a general method simplifying and improving the slow invariant manifold analytical equation determination of high-dimensional dynamical systems. Moreover, it will be shown that this method generalizes the Tangent Linear System Approximation and encompasses the so-called Geometric Singular Perturbation Theory. Then, slow invariant manifolds analytical equation of paradigmatic Chua's piecewise linear and cubic models of dimensions three, four and five will be provided as tutorial examples exemplifying this method as well as those of high-dimensional dynamical systems.

We study groups with “differential structure” in the framework of Abstract Differential Geometry, an abstraction of the classical differential geometry of manifolds, via sheaf-theoretic methods, without ordinary calculus; the basic tool is the notion of a differential triad. First, we consider pre-Lie groups, i.e., semi-topological groups with compatible differential triads and we prove that such groups have “left-invariant vector fields” and “left-invariant derivations”, behaving like the classical ones. Next, for every pre-Lie group, we define an appropriate Lie algebra and prove the existence of a naturally associated adjoint representation of the initial group into the latter.

A First Extension of Geometric Control Theory to Underwater Vehicles

Introduction. In this series of papers we will discuss homotopy invariants of differentiable maps f: M -? N in various situations within the framework of differential geometry. We shall be particularly interested in the case where f is an immersion. Our principle is simple. We use the fact that the pullbackf*W of c is a differentiable homotopy invariant where w is an arbitrary cohomology class always over the real number field, R. We note if F= M x I -1 N is a differentiable map with f=f, thenf*w belongs to the same cohomology class asfJ*w where I is the interval [0, 1] and ft is defined by ft(x)=F(x, t), x E M, t EI L There are many known examples beside various characteristic classes. To quote a few of them, letf be an immersion of the two-torus T2 into the complement of the diagonal set of the six dimensional euclidean space R6 considered as R3 x R3; f: T2 R3 x R3 A, where A is the diagonal set. The space R3 x R 3A is diffeomorphic with R4 x S2 where S2 is the two-sphere. Thus, if a) denotes the volume element of 52, a becomes a 2-dimensional cohomology class of R4 x S2 by pulling back with the projection onto S2 and gives us a homotopy invariantf*co. Consider T2 interpreted as the direct product S1 x S 1 of circles andf as the pair of two closed curves ci: SI --> R3 without intersection. Then f *w is nothing but the linking number of two closed -curves cl and c2 (up to a universal constant multiple) according to Gauss [8]. Another (but somewhat more extraneous) example, due to J. H. C. Whitehead [12], is the Hopf invariant, H(f), for a differentiable map f: S2n-1 _? Sn. Again, denoting the volume element of the n-sphere Sn by c, H(f) is given by the closed form 0 Af*w where 0 is any (n l)-form with dO =f*U. In this paperf will be an isometric immersion of a compact oriented Riemannian manifold M into a euclidean space Rn + v. Since the cohomology groups of N=Rn + v are then trivial, our principle does not apply tof directly, but we construct a manifold B from M and replace f by a certain map fB: M -? B. To do this, let us recall Hirsch's theorem [5] to the effect that the regular homotopy classes of the immersionsf: M -? Nofany manifold Minto another manifold N with dim N> dim M are in a one-to-one correspondence with the homotopy classes of the crosssections fB: M -->B of a certain bundle B over M, where a regular homotopy F= M x I -* N means one for which each ft is an immersion. Despite the triviality of the cohomology groups of N= Rn+ v in the case above, we can expect to obtain homotopy invariants fB*0 corresponding to a cohomology class 0 of B. In ?4, we

Control Contraction Metrics (CCMs) can be described as a non-linear control technique based indifferential dynamics on the framework of differential geometry. This work presents preliminaryresults on applying CCMs to the control of a Blended Wing Body (BWB) aircraft configuration,in particular, regarding flight conditions involving high angle of attack and medium Reynoldsnumber regimes. These preliminary results show that in absence of conventional stabilityequilibrium points, proper CCMs may allow effective control on aircraft dynamics given theadequate operational conditions. These conditions and configuration are expected to be presentin new concepts of Remotely Piloted Aircraft Systems (RPAS), extending their applicationsfrom planetary exploration to support the development of new transportation concepts incommercial airliners. Methods described, data, and results, are expected to be of applicationfor EXperimental Aircraft for European Leadership in Aviation (EXAELIA) project, a HorizonEurope international effort to de-risk and develop new concepts of passsenger aircraft to faceaerospace industry challenges by 2050.This work has been funded by the European Union under GA No. 101191922. Viewsand opinions expressed are however those of the author only and do not necessarily reflect thoseof the European Union or the European Climate, Infrastructure and Environment ExecutiveAgency. Neither the European Union nor the granting authority can be held responsible forthem.

We investigated crystal structure deviations of catalytic nanoparticles (NPs) using synchrotron powder X-ray diffraction. The samples were fcc ruthenium (Ru) NPs with diameters of 2.4, 3.5, 3.9, and 5.4 nm. We analyzed average crystal structures by applying the line profile method to a stacking fault model and local crystal structures using bond orientational order (BOO) parameters. The reflection peaks shifted depending on rules that apply to each stacking fault. We evaluated the quantitative stacking faults densities for fcc Ru NPs, and the stacking fault per number of layers was 2–4, which is quite large. Our analysis shows that the fcc Ru 2.4 nm-diameter NPs have a considerably high stacking fault density. The B factor tends to increase with the increasing stacking fault density. A structural parameter that we define from the BOO parameters exhibits a significant difference from the ideal value of the fcc structure. This indicates that the fcc Ru NPs are highly disordered.

AlGaN/GaN high electron mobility transistors (HEMTs) are promising candidates for high power and high current applications with its high mobility and high breakdown field. Epitaxial films for device substrate are typically grown on sapphire or silicon carbide, SiC, with dislocation densities in the order of 108 - 1010cm-2 due to their thermal coefficient and lattice mismatches. The thinner GaN buffer layers on SiC have been reported to have higher threading dislocation density in the GaN, but lead to higher off-state breakdown voltages. However, thick GaN buffer layers on sapphire and their impact on dc performance have not been widely studied. In this study, the impact of GaN buffer layer quality on dc and gate-lag pulse performance of AlGaN/GaN HEMTs was studied. Transmission electron microscopy (TEM) was used to measure the dislocation density in the GaN buffer layer of the HEMT samples. Dc characteristics, including drain I-Vs, transfer characteristics, and gate pulsed drain I-V characteristics of the HEMTs, were measured to establish a correlation between the AlGaN/GaN HEMTs’ dc and gate-pulsed performance and GaN buffer layer quality. AlGaN/GaN HEMT structures grown by metal organic chemical vapor deposition (MOCVD) on sapphire substrates were acquired from two different vendors. Both types of wafers had 24% Al concentration in the AlGaN barrier layer. One type of wafer had a 5 µm GaN buffer and the other had a 2 µm GaN buffer layer. HEMT fabrication was started with mesa patterning by conventional optical lithography. The mesa-etching was achieved using a unaxis shuttle-lock reactive ion etcher with inductively coupled plasma module (ICP) for device isolation with a Cl2/Ar plasma. Ohmic metallization of electron-beam evaporated Ti/Al/Ni/Au (25 nm/125 nm/45 nm/100 nm) was patterned by lift-off and subsequently rapid thermally annealed at 850⁰C in a flowing N2 ambient for 45 seconds. Schottky gates of e-beam evaporated Ni/Au (20 nm/80 nm) with 100 μm width and 1 μm length were obtained by standard lift-off patterning. DC current–voltage (I-V) characteristics were measured using an HP4156 parameter analyzer, and gate-lag pulse measurements were measured using an Agilent 8114A pulse generator. Figure 1 shows the bright field cross-sectional transmission electron microscopy (BF-XTEM) pictures of AlGaN/GaN surface for the top layers of (a) HEMT structure with 2 µm GaN buffer layer, (b) HEMT structure with 5 μm GaN buffer layer as well as the GaN/sapphire interface for the (c) HEMT structure with 2 μm GaN buffer layer, and (d) HEMT structure with 5 μm GaN buffer layer. Dislocations are seen to originate near the GaN/sapphire interface and propagate up through the AlGaN/GaN interface to the surface of both HEMT wafers. However, it is clear that the HEMT structure with 2 μm GaN buffer layer exhibited a much higher defect density. The calculated defect density for 2 μm GaN buffer layer substrate had an order of magnitude higher defect density. However, 5μm GaN buffer layer wafer showed that it was under high strain with larger height difference across the 2 inch with lower radius of curvature. There was no major difference in HEMTs’ dc performance was observed. In the other hand, Figure 2 (a) and (b) shows that gate lag measurement on fabricated HEMTs on 5 μm GaN buffer layer wafer showed no degradation and on 2 μm GaN buffer layer wafer showed the otherwise. The effects of defect density on HEMT dc and radio frequency performance were studied. HEMTs were fabricated on 2 and 5 μm of GaN buffer layers with the same composition of active layers. Despite there was no significant difference was seen in dc characteristics, significant reduction in pulse measurements was observed on thinner GaN layer HEMTs. These results suggest that the defects created in GaN buffer layer do not largely influence device dc performance but are closely related to high frequency performance. Figure 1

Bond-orientational order in materials that form three-dimensional uniaxial solids is investigated. New phases are found that scatter light strongly, like nematics, but with a quantitatively different anisotropy of scattering. X-ray scattering from these phases would also exhibit novel anisotropy. This type of order may occur at higher temperatures in materials that form stacked hexatic phases at lower temperatures, and will occur whenever a hexagonal crystal melts via dislocation unbinding.