Parallel Transport Research Articles

Permeability of Buccal Mucosa.

The buccal mucosa provides an alternative route of drug delivery that can be more beneficial compared to other administration routes. Although numerous studies and reviews have been published on buccal drug delivery, an extensive review of the permeability data is not available. Understanding the buccal mucosa barrier could provide insights into the approaches to effective drug delivery and optimization of dosage forms. This paper provides a review on the permeability of the buccal mucosa. The intrinsic permeability coefficients of porcine buccal mucosa were collected. Large variability was observed among the published permeability data. The permeability coefficients were then analyzed using a model involving parallel lipoidal and polar transport pathways. For the lipoidal pathway, a correlation was observed between the permeability coefficients and permeant octanol/water partition coefficients (Kow) and molecular weight (MW) in a subset of the permeability data under specific conditions. The permeability analysis suggested that the buccal permeation barrier was less lipophilic than octanol. For the polar pathway and macromolecules, a correlation was observed between the permeability coefficients and permeant MW. The hindered transport analysis suggested an effective pore radius of 1.5 to 3 nm for the buccal membrane barrier.

A 120 Mbps WDM-based VLC system for implementation of Internet of Things

Abstract It is envisioned that by 2022, the number of Internet of Things (IoT) devices will be more than 20 billion. The transfer of information between IoT devices is heavily dependent on wireless links based on the radio frequency (RF) spectrum. Due to the scarcity of limited and licensed RF spectrum, visible light communication (VLC) based IoT links can be considered as a promising and viable solution. In this work, four channels are multiplexed to carry low-datarate information over a VLC link using cost-effective and easy to implement on-off-keying (OOK). Red, blue, green, and amber light-emitting diodes (LEDs) are deployed for parallel data transportation. The proposed system has been designed and simulated using Optisystem software. Through numerical simulations, we investigate the bit error rate (BER) performance of the proposed VLC link with increasing transmission range. Also, we investigate the BER performance of the proposed VLC link under the impact of increasing detection surface area, responsivity of the photodiode, and slope efficiency of the LEDs. Further, the performance of the proposed VLC link has been compared for different data transmission rates using non-return-to-zero and return-to-zero modulation schemes.

Bolometer tomography on Wendelstein 7-X for study of radiation asymmetry

The algorithm for bolometer tomography at Wendelstein 7-X (W7-X) has been recently improved using a novel regularization functional, based on relative gradient smoothing of the sought radiation profile. It has been validated using radiation patterns provided by 3D modeling under real plasma conditions as phantoms and then applied to bolometer measurements performed during the first divertor operation phase of W7-X. The following results are presented: (1) edge-localized 2D radiation patterns with clearly resolved magnetic island radiation structures, (2) an up-down asymmetry in the impurity radiation that is not captured by the 3D edge plasma transport modeling, (3) reversal of the asymmetry with reversed magnetic field direction. Further analysis reveals a poloidal variation of the emissivity in the outer confined plasma region with a field-direction dependent asymmetry, also supported by the soft x-ray measurements. This asymmetry is considered to be related to asymmetric impurity distributions, driven by the pronounced ion-impurity friction force at the plasma edge where the collisionality of the W7-X plasma is sufficiently high to develop impurity asymmetry as predicted by neoclassical theory of parallel impurity transport.

Improved regularity estimates for Lagrangian flows on [formula omitted] spaces

This paper gives a contribution to the study of regularity of Lagrangian flows on non-smooth spaces with lower Ricci curvature bounds. The main novelties with respect to the existing literature are the better behaviour with respect to time and the local nature of the regularity estimates. These are obtained by sharpening previous results of the first and third authors, in combination with some tools recently developed by the second author (adapting to the synthetic framework ideas introduced in Colding and Naber (2012)). The estimates are suitable for applications to the fine study of RCD spaces and play a central role in the construction of a parallel transport in this setting.

The pseudo-null geometric phase along optical fiber

In this study, a pseudo-null space curve in Minkowski 3-space is used to describe an optical fiber that is injected into monochromatic linear polarized light. The direction of the electric field vector with respect to the Frenet frame of a pseudo-null curve determines the state polarization of a monochromatic linearly polarized light wave traveling along an optical fiber. For the Frenet frame of a pseudo-null curve in Minkowski 3-space, the polarization vector [Formula: see text] is assumed to be perpendicular to the tangent vector [Formula: see text] with respect to anholonomic coordinates. Anholonomic coordinates for the Frenet frame of a pseudo-null curve are used to describe pseudo-null electromagnetic curves in the normal and binormal directions along an optical fiber. For the Frenet frame of the pseudo-null curve, Lorentz force equations in the normal and binormal directions along the optical fiber are presented. Pseudo-normal and binormal Rytov parallel transport laws for electric fields in the normal and binormal directions along with the optical fiber for the Frenet frame of the pseudo-null curve via anholonomic coordinates are presented. For anholonomic coordinates in Minkowski 3-space, rotations of the polarization planes of a light wave traveling in the normal and binormal directions along with the optical fiber with respect to the Frenet frame of the pseudo-null curve are obtained. Finally, a pseudo-null curve’s Maxwellian evolution is determined.

Enhanced preconditioner for JOREK MHD solver

The JOREK extended magneto-hydrodynamic (MHD) code is a widely used simulation tool for studying the non-linear dynamics of large-scale instabilities in divertor tokamak plasmas. The code is usually run with a fully implicit time integration allowing to use large time steps independent of a CFL criterion. This is particularly important due to the strong scale separation between transport processes and slowly growing resistive modes in contrast to fast time scales associated with MHD waves and fast parallel heat transport. For solving the resulting large sparse-matrix system iteratively in each time step, a preconditioner based on the assumption of weak coupling between the toroidal harmonics is applied. The solution for each harmonic matrix is determined independently in this preconditioner using a direct solver. In this article, a set of developments regarding the JOREK solver and preconditioner is described, which lead to overall significant benefits for large production simulations. The developments include the implementation of a complex solver interface for the preconditioner leading to the general reduction of memory consumption. The most significant development presented consists in a generalization of the physics based preconditioner to ‘mode groups’, which allows to account for the dominant interactions between toroidal Fourier modes in highly non-linear simulations. At the cost of a moderate increase of memory consumption, the technique can strongly enhance convergence in suitable cases allowing to use significantly larger time steps, and thus improving the overall time performance by more than a factor of 3.

Promotion and suppression of single-molecule conductance by quantum interference in macrocyclic circuits

Promotion and suppression of single-molecule conductance by quantum interference in macrocyclic circuits

New approach for propagated light with optical solitons by optical fiber in pseudohyperbolic space ℍ02

In this paper, a new evolution of polarized light ray by optical fiber in the pseudohyperbolic space is examined. Firstly, the characterization of the parallel transportation law associated with the geometric pseudohyperbolic phase of the light ray is given. Later, a principle nature of electric and magnetic field along with the light ray in the pseudohyperbolic space is defined by the geometric invariants. Finally, optical solutions of nonlinear pseudohyperbolic Schrödinger's equations governing the propagation of electromagnetic fields are successfully derived by using the traveling wave hypothesis approach.

Emergent behaviors of relativistic flocks on Riemannian manifolds

We present a relativistic counterpart of the Cucker–Smale (CS) model on Riemannian manifolds (manifold RCS model in short) and study its collective behavior. For Euclidean space, the relativistic Cucker–Smale (RCS) model was introduced in Ha et al. (2020) via the method of a rational reduction from the relativistic gas mixture equations by assuming space-homogeneity, suitable ansatz for entropy and principle of subsystem. In this work, we extend the RCS model on Euclidean space to connected, complete and smooth Riemannian manifolds by replacing usual time derivative of velocity and relative velocity by suitable geometric quantities such as covariant derivative and parallel transport along length-minimizing geodesics. For the proposed model, we present a Lyapunov functional which decreases monotonically on generic manifolds, and show the emergence of weak velocity alignment on compact manifolds by using LaSalle’s invariance principle. As concrete examples, we further analyze the RCS models on the unit sphere Sd and the hyperbolic space Hd. More precisely, we show that the RCS model on Sd exhibits a dichotomy in asymptotic spatial patterns, and provide a sufficient framework leading to the velocity alignment of RCS particles in Hd. For the hyperbolic space Hd, we also rigorously justify smooth transition from the RCS model to the CS model in any finite time interval, as speed of light tends to infinity.

Polarized image of equatorial emission in the Kerr geometry

We develop a simple toy model for polarized images of synchrotron emission from an equatorial source around a Kerr black hole by using a semi-analytic solution of the null geodesic equation and conservation of the Penrose-Walker constant. Our model is an extension of Narayan et al. (2021), which presented results for a Schwarzschild black hole, including a fully analytic approximation. Our model includes an arbitrary observer inclination, black hole spin, local boost, and local magnetic field configuration. We study the geometric effects of black hole spin on photon parallel transport and isolate these effects from the complicated combination of relativistic, gravitational, and electromagnetic processes in the emission region. We find an analytic approximation, consistent with previous work, for the subleading geometric effect of spin on observed face-on polarization rotation in the direct image: $\Delta {\rm EVPA} \sim -2a/r_{\rm s}^2$, where $a$ is the black hole spin and $r_{\rm s}$ is the emission radius. We further show that spin introduces an order unity effect on face-on subimages: $\Delta {\rm EVPA} \sim \pm a/\sqrt{27}$. We also use our toy model to analyze polarization "loops" observed during flares of orbiting hotspots. Our model provides insight into polarimetric simulations and observations of black holes such as those made by the EHT and GRAVITY.

Emergent behaviors of Cucker–Smale flocks on the hyperboloid

We study emergent behaviors of Cucker–Smale (CS) flocks on the hyperboloid Hd in any dimensions. In a recent work [Ha et al., J. Math. Phys. 61(4), 042701 (2020)], a first-order aggregation model on the hyperboloid was proposed and sufficient conditions for emergent dynamics were proposed in terms of initial configuration and system parameters. In this paper, we are interested in the second-order modeling of CS flocks on the hyperboloid. For this, we derive our second-order model from the abstract CS model on complete and smooth Riemannian manifolds via explicit identifications of geodesic and parallel transport. Velocity alignment has been shown by combining general velocity alignment estimates for the abstract CS model on manifolds and verifications of the a priori estimate of the second derivative of the energy functional. For the two-dimensional case H2, similar to the recent result by Ahn, Ha, and Shim [Kinet. Relat. Models 14(2), 323–351 (2021)], asymptotic flocking admits only two types of asymptotic scenarios, either convergence to a rest state or a state lying on the same plane (coplanar state). We also provide several numerical simulations to illustrate an aforementioned dichotomy on the asymptotic dynamics of the hyperboloid CS model on H2.

Minimum Time Problem Controlled by Affine Connection

Geometrically, the affine connection is the main ingredient that underlies the covariant derivative, the parallel transport, the auto-parallel curves, the torsion tensor field, and the curvature tensor field on a finite-dimensional differentiable manifold. In this paper, we come up with a new idea of controllability and observability of states by using auto-parallel curves, and the minimum time problem controlled by the affine connection. The main contributions refer to the following: (i) auto-parallel curves controlled by a connection, (ii) reachability and controllability on the tangent bundle of a manifold, (iii) examples of equiaffine connections, (iv) minimum time problem controlled by a connection, (v) connectivity by stochastic perturbations of auto-parallel curves, and (vi) computing the optimal time and the optimal striking time. The connections with bounded pull-backs result in bang–bang optimal controls. Some significative examples on bi-dimensional manifolds clarify the intention of our paper and suggest possible applications. At the end, an example of minimum striking time with simulation results is presented.

Chiral transport of pseudospinors induced by synthetic gravitational field in photonic Weyl metamaterials

Weyl particles exhibit chiral transport property under external curved space-time geometry. This effect is called the chiral gravitational effect, which plays an important role in quantum field theory. However, the absence of real Weyl particles in nature hinders the observation of such interesting phenomena. In this paper, we show that chiral gravitational effect can be manifested in Weyl metamaterials with spatially controlled nonlocality. This inhomogeneous modulation results in a spatially dependent group velocity in the Weyl cone dispersion, which is equivalent to introducing a curved background space-time (or gravitational field) for Weyl pseudospinors according to the Clifford algebra. The Weyl equation describing the Weyl cone dispersions is reformulated into the covariant form that contains a nonAbelian gauge field by considering the parallel transport of pseudospinors. The synthetic gravitational field (i.e., space-time curvature) leads to the quantization of energy levels, including chiral zeroth order energy modes (or simply chiral zero modes) that determine the chiral transport property of pseudospinors. Inhomogeneous Weyl metamaterials can potentially become a platform for investigating the interaction between Weyl particles and gravitational fields in future table-top experiments. The theoretical model and the associated results may serve as a guidance for experimentalists to observe such chiral transport effects in photonic Weyl systems.

Hypercomplex algebras and calculi derived from generalized kinematics

The paper provides a study of the commutative algebras generated by iteration of the cross products in . Focusing on particular real forms, we also consider the analytical properties of the corresponding rings of functions and relate them to different physical problems. Familiar results from the theory of holomorphic and bi‐holomorphic functions appear naturally in this context, but new types of hypercomplex calculi emerge as well. The parallel transport along smooth curves in and the associated Maurer‐Cartan form are also studied with examples from kinematics and electrodynamics. Finally, the dual extension is discussed in the context of screw calculus and Galilean mechanics; a similar construction is studied also in the multidimensional real and complex cases.

The Quasi Parallel Curve of a Space Curve

A parallel or offset curve is defined as a curve whose points are a fixed distance from a given curve. These curves are not parallel transport. Often, it has a more complex mathematical structure than the first curve. Offset curves are important in numerically controlled machining, for example, where a biaxial machine defines the shape of the cut made with a round cutting tool. In this study, the quasi parallel curve is defined with the help of the quasi frame of a given curve. According to all selections of the projection vector, the quasi frame equations of this curve are expressed in terms of the quasi elements of the given curve. The curvatures of the quasi parallel curve were obtained depending on the quasi curvatures of the main curve. The study was supported by examples. The examples confirm that the quasi parallel curve is not parallel transport.

An interpretive model for the double peaks of divertor tungsten erosion during type-I ELMs in EAST

Double peaks of divertor tungsten erosion during type-I edge-localized modes (ELMs) were observed in the EAST tokamak. To study this unique phenomenon, an analytical model of plasma expansion into vacuum was used to investigate the ELM parallel transport along the magnetic field line to the divertor target. The modeling results reveal that during ELM bursts, the energetic C6+ transport is slower than the main ions due to a lower initial thermal speed. The time of C6+ arriving at the divertor target from the modeling is consistent with the second tungsten erosion peak. Based on the particle flux from the transport model, a newly developed mixed-material model quantitively reproduced the intra-ELM tungsten erosion profiles for both deuterium and helium discharges. A lithium–carbon coating makes the heavier impurity (C6+) comparatively more effective than the main ions on tungsten sputtering, and thus makes the second tungsten erosion peak more remarkable on EAST experiments.

Optimization on flag manifolds

A flag is a sequence of nested subspaces. Flags are ubiquitous in numerical analysis, arising in finite elements, multigrid, spectral, and pseudospectral methods for numerical pde; they arise in the form of Krylov subspaces in matrix computations, and as multiresolution analysis in wavelets constructions. They are common in statistics too—principal component, canonical correlation, and correspondence analyses may all be viewed as methods for extracting flags from a data set. The main goal of this article is to develop the tools needed for optimizing over a set of flags, which is a smooth manifold called the flag manifold, and it contains the Grassmannian as the simplest special case. We will derive closed-form analytic expressions for various differential geometric objects required for Riemannian optimization algorithms on the flag manifold; introducing various systems of extrinsic coordinates that allow us to parameterize points, metrics, tangent spaces, geodesics, distances, parallel transports, gradients, Hessians in terms of matrices and matrix operations; and thereby permitting us to formulate steepest descent, conjugate gradient, and Newton algorithms on the flag manifold using only standard numerical linear algebra.

Geometric quantization of symplectic maps and Witten's asymptotic conjecture

We use the theory of Berezin-Toeplitz operators of Ma and Marinescu to study the spaces of holomorphic sections of a prequantizing line bundle over compact Kähler manifolds under deformations of the complex structure. We show that the parallel transport in the induced vector bundle over the deformation space behaves like a Toeplitz operator, and compute its first coefficient. We then use this result to establish a semi-classical trace formula for the induced quantization of symplectic maps, and give an application to Witten's asymptotic expansion conjecture for the quantum representations of the mapping class group.

Numerical Accuracy of Ladder Schemes for Parallel Transport on Manifolds

Parallel transport is a fundamental tool to perform statistics on Riemannian manifolds. Since closed formulae do not exist in general, practitioners often have to resort to numerical schemes. Ladder methods are a popular class of algorithms that rely on iterative constructions of geodesic parallelograms. And yet, the literature lacks a clear analysis of their convergence performance. In this work, we give Taylor approximations of the elementary constructions of Schild’s ladder and the pole ladder with respect to the Riemann curvature of the underlying space. We then prove that these methods can be iterated to converge with quadratic speed, even when geodesics are approximated by numerical schemes. We also contribute a new link between Schild’s ladder and the Fanning scheme which explains why the latter naturally converges only linearly. The extra computational cost of ladder methods is thus easily compensated by a drastic reduction of the number of steps needed to achieve the requested accuracy. Illustrations on the 2-sphere, the space of symmetric positive definite matrices and the special Euclidean group show that the theoretical errors we have established are measured with a high accuracy in practice. The special Euclidean group with an anisotropic left-invariant metric is of particular interest as it is a tractable example of a non-symmetric space in general, which reduces to a Riemannian symmetric space in a particular case. As a secondary contribution, we compute the covariant derivative of the curvature in this space.

Geometry of time-dependent -symmetric quantum mechanics* *J.G. is supported by Singapore Ministry of Education Academic Research Fund Tier I (WBS No. R-144-000-353-112) and by the Singapore NRF Grant No. NRFNRFI2017-04 (WBS No. R-144-000-378-281). Q.W. is supported by Singapore Ministry of Education Academic Research Fund Tier I (WBS No. R-144-000-352-112).

A new type of quantum theory known as time-dependent -symmetric quantum mechanics has received much attention recently. It has a conceptually intriguing feature of equipping the Hilbert space of a -symmetric system with a time-varying inner product. In this work, we explore the geometry of time-dependent -symmetric quantum mechanics. We find that a geometric phase can emerge naturally from the cyclic evolution of a -symmetric system, and further formulate a series of related differential-geometry concepts, including connection, curvature, parallel transport, metric tensor, and quantum geometric tensor. These findings constitute a useful, perhaps indispensible, tool to investigate geometric properties of -symmetric systems with time-varying system’s parameters. To exemplify the application of our findings, we show that the unconventional geometric phase [Phys. Rev. Lett. 91 187902 (2003)], which is the sum of a geometric phase and a dynamical phase proportional to the geometric phase, can be expressed as a single geometric phase unveiled in this work.