A Framework for Approximation of the Stokes Equations in an Axisymmetric Domain

We develop a framework for solving the stationary, incompressible Stokes equations in an axisymmetric domain. By means of Fourier expansion with respect to the angular variable, the three-dimensional Stokes problem is reduced to an equivalent, countable family of decoupled two-dimensional problems. By using decomposition of three-dimensional Sobolev norms, we derive natural variational spaces for the two-dimensional problems, and show that the variational formulations are well-posed. We analyze the error due to Fourier truncation and conclude that, for data that are sufficiently regular, it suffices to solve a small number of two-dimensional problems.

Mapping Low-Dimensional Dynamics to High-Dimensional Neural Activity: A Derivation of the Ring Model From the Neural Engineering Framework

Empirical estimates of the dimensionality of neural population activity are often much lower than the population size. Similar phenomena are also observed in trained and designed neural network models. These experimental and computational results suggest that mapping low-dimensional dynamics to high-dimensional neural space is a common feature of cortical computation. Despite the ubiquity of this observation, the constraints arising from such mapping are poorly understood. Here we consider a specific example of mapping low-dimensional dynamics to high-dimensional neural activity—the neural engineering framework. We analytically solve the framework for the classic ring model—a neural network encoding a static or dynamic angular variable. Our results provide a complete characterization of the success and failure modes for this model. Based on similarities between this and other frameworks, we speculate that these results could apply to more general scenarios.

Oscillatory-Precessional Motion of a Rydberg Electron Around a Polar Molecule

We provide a detailed classical description of the oscillatory-precessional motion of an electron in the field of an electric dipole. Specifically, we demonstrate that in the general case of the oscillatory-precessional motion of the electron (the oscillations being in the meridional direction (θ-direction) and the precession being along parallels of latitude (φ-direction)), both the θ-oscillations and the φ-precessions can actually occur on the same time scale—contrary to the statement from the work by another author. We obtain the dependence of φ on θ, the time evolution of the dynamical variable θ, the period Tθ of the θ-oscillations, and the change of the angular variable φ during one half-period of the θ-motion—all in the forms of one-fold integrals in the general case and illustrated it pictorially. We also produce the corresponding explicit analytical expressions for relatively small values of the projection pφ of the angular momentum on the axis of the electric dipole. We also derive a general condition for this conditionally-periodic motion to become periodic (the trajectory of the electron would become a closed curve) and then provide examples of the values of pφ for this to happen. Besides, for the particular case of pφ = 0 we produce an explicit analytical result for the dependence of the time t on θ. For the opposite particular case, where pφ is equal to its maximum possible value (consistent with the bound motion), we derive an explicit analytical result for the period of the revolution of the electron along the parallel of latitude.

A monotonicity result under symmetry and Morse index constraints in the plane

This paper deals with solutions of semilinear elliptic equations of the type \[ \left\{\begin{array}{@{}ll} -\Delta u = f(|x|, u) \qquad & \text{ in } \Omega, \\ u= 0 & \text{ on } \partial \Omega, \end{array} \right. \] where Ω is a radially symmetric domain of the plane that can be bounded or unbounded. We consider solutions u that are invariant by rotations of a certain angle θ and which have a bound on their Morse index in spaces of functions invariant by these rotations. We can prove that or u is radial, or, else, there exists a direction $e\in \mathcal {S}$ such that u is symmetric with respect to e and it is strictly monotone in the angular variable in a sector of angle θ/2. The result applies to least-energy and nodal least-energy solutions in spaces of functions invariant by rotations and produces multiplicity results.

Measurement of the transverse momentum distribution of Drell–Yan lepton pairs in proton–proton collisions at $$\sqrt{s}=13\,$$TeV with the ATLAS detector

This paper describes precision measurements of the transverse momentum $$p_\mathrm {T}^{\ell \ell }$$pTℓℓ ($$\ell =e,\mu $$ℓ=e,μ) and of the angular variable $$\phi ^{*}_{\eta }$$ϕη∗ distributions of Drell–Yan lepton pairs in a mass range of 66–116 GeV. The analysis uses data from 36.1 fb$$^{-1}$$-1 of proton–proton collisions at a centre-of-mass energy of $$\sqrt{s}=13\,$$s=13TeV collected by the ATLAS experiment at the LHC in 2015 and 2016. Measurements in electron-pair and muon-pair final states are performed in the same fiducial volumes, corrected for detector effects, and combined. Compared to previous measurements in proton–proton collisions at $$\sqrt{s}=7$$s=7 and $$8\,$$8TeV, these new measurements probe perturbative QCD at a higher centre-of-mass energy with a different composition of initial states. They reach a precision of 0.2$$\%$$% for the normalized spectra at low values of $$p_\mathrm {T}^{\ell \ell }$$pTℓℓ. The data are compared with different QCD predictions, where it is found that predictions based on resummation approaches can describe the full spectrum within uncertainties.

Possible Occurrence of Superconductivity by the π-flux Dirac String Formation Due to Spin-Twisting Itinerant Motion of Electrons

We show that the Rashba spin-orbit interaction causes spin-twisting itinerant motion of electrons in metals and realizes the quantized cyclotron orbits of conduction electrons without an external magnetic field. From the view point of the Berry connection, the cause of this quantization is the appearance of a non-trivial Berry connection A fic = − ℏ 2 e ∇ χ ( χ is an angular variable with period 2 π ) that generates π flux (in the units of ℏ = 1 , e = 1 , c = 1 ) inside the nodal singularities of the wave function (a “Dirac string”) along the centers of spin-twisting. Since it has been shown in our previous work that the collective mode of ∇ χ is stabilized by the electron-pairing and generates supercurrent, the π -flux Dirac string created by the spin-twisting itinerant motion will be stabilized by the electron-pairing and produce supercurrent.

Spectral ray data for optical simulations

This paper summarizes selected approaches, to generate spectral ray data for different types of spectrally varying light sources including only angular variable as well as spatial and angular variable sources. This includes a description of their general ideas and applications, the required measurements, and their mathematical concepts. Finally, achieved results for an Red/Green/Blue/White-light emitting diode (RGBW-LED) system are shown. Ray tracing simulations of a spatially and angularly spectral varying LED system combined with a spectrally sensitive optical system are qualitatively and quantitatively compared to a colorimetric far-field measurement of the same system. The results demonstrate the potential and benefits of spectral ray files in general.

Seafloor mapping based on multibeam echosounder bathymetry and backscatter data using Object-Based Image Analysis: a case study from the Rewal site, the Southern Baltic

Seafloor mapping is a fast developing multidisciplinary branch of oceanology that combines geophysics, geostatistics, sedimentology and ecology. One of its objectives is to isolate distinct seabed features in a repeatable, fast and objective way, taking into consideration multibeam echosounder (MBES) bathymetry and backscatter data. A large-scale acoustic survey was conducted by the Maritime Institute in Gdańsk in 2010 using Reson 8125 MBES. The dataset covered over 20 km2 of a shallow seabed area (depth of up to 22 m) in the Polish Exclusive Economic Zone within the Southern Baltic. Determination of sediments was possible based on ground-truth grab samples acquired during the MBES survey. Four classes of sediments were recognized as muddy sand, very fine sand, fine sand and clay. The backscatter mosaic created using the Angular Variable Gain (AVG) empirical method was the primary contribution to the image processing method used in this study. The use of the Object-Based Image Analysis (OBIA) and the Classification and Regression Trees (CART) classifier makes it possible to isolate the backscatter image with 87.5% overall and 81.0% Kappa accuracy. The obtained results confirm the possibility of creating reliable maps of the seafloor based on MBES measurements. Once developed, the OBIA workflow can be applied to other spatial and temporal scenes.

Domain-Dependent Stability Analysis of a Reaction–Diffusion Model on Compact Circular Geometries

In this work an activator-depleted reaction–diffusion system is investigated on polar coordinates with the aim of exploring the relationship and the corresponding influence of domain size on the types of possible diffusion-driven instabilities. Quantitative relationships are found in the form of necessary conditions on the area of a disk-shape domain with respect to the diffusion and reaction rates for certain types of diffusion-driven instabilities to occur. Robust analytical methods are applied to find explicit expressions for the eigenvalues and eigenfunctions of the diffusion operator on a disk-shape domain with homogenous Neumann boundary conditions in polar coordinates. Spectral methods are applied using Chebyshev nonperiodic grid for the radial variable and Fourier periodic grid for the angular variable to verify the nodal lines and eigen-surfaces subject to the proposed analytical findings. The full classification of the parameter space in light of the bifurcation analysis is obtained and numerically verified by finding the solutions of the partitioning curves inducing such a classification. Spatiotemporal periodic behavior is demonstrated in the numerical solutions of the system for a proposed choice of parameters and a rigorous proof of the existence of infinitely many such points in the parameter plane is presented under a restriction on the area of the domain, with a lower bound in terms of reaction–diffusion rates.