Single Torus Research Articles

Calibrating figures

It is known that a camera can be calibrated using three pictures of either squares, or spheres, or surfaces of revolution. We give a new method to calibrate a camera with the picture of a single torus.

Geometrically thick equilibrium tori around a dyonic black hole with quasi-topological electromagnetism

We study geometrically thick and non-self gravitating equilibrium tori orbiting a static spherically symmetric dyonic black hole with quasi-topological electromagnetism. Our results show that the electric and magnetic charges together with the coupling parameter in the quasi-topological electromagnetism lead to a much richer class of equilibrium tori. There are a range of parameters which allow for the existence of double tori. The properties of double equilibrium tori become far richer. There exist transitions between single torus and double tori solutions as we change the specific angular momentum of the fluid. These properties of equilibrium tori could help to understand the dyonic black hole and its thick accretion disk.

The minimal genus problem for right angled Artin groups

We investigate the minimal genus problem for the second homology of a right angled Artin group (RAAG). Firstly, we present a lower bound for the minimal genus of a second homology class, equal to half the rank of the corresponding cap product matrix. We show that for complete graphs, trees, and complete bipartite graphs, this bound is an equality, and furthermore in these cases the minimal genus can always be realised by a disjoint union of tori. Additionally, we give a full characterisation of classes that are representable by a single torus. However, the minimal genus of a second homology class of a RAAG is not always realised by a disjoint union of tori as an example we construct in the pentagon shows.

Modular flavour symmetry and orbifolds

We develop a bottom-up approach to flavour models which combine modular symmetry with orbifold constructions. We first consider a 6d orbifold \U0001d54b2/ℤN, with a single torus defined by one complex coordinate z and a single modulus field τ, playing the role of a flavon transforming under a finite modular symmetry. We then consider 10d orbifolds with three factorizable tori, each defined by one complex coordinate zi and involving the three moduli fields τ1, τ2, τ3 transforming under three finite modular groups. Assuming supersymmetry, consistent with the holomorphicity requirement, we consider all 10d orbifolds of the form (\U0001d54b2)3/(ℤN × ℤM), and list those which have fixed values of the moduli fields (up to an integer). The key advantage of such 10d orbifold models over 4d models is that the values of the moduli are not completely free but are constrained by geometry and symmetry. To illustrate the approach we discuss a 10d modular seesaw model with {S}_4^3 modular symmetry based on (\U0001d54b2)3/(ℤ4× ℤ2) where τ1 = i, τ2 = i + 2 are constrained by the orbifold, while τ3 = ω is determined by imposing a further remnant S4 flavour symmetry, leading to a highly predictive example in the class CSD(n) with n = 1 − sqrt{6} .

A Palm-Sized Omnidirectional Mobile Robot Driven by 2-DOF Torus Wheels

This letter proposes a palm-sized omnidirectional mobile robot with two torus wheels. A single torus wheel is made of an elastic elongated coil spring in which the two ends of the coil connected each other and is driven by a piezoelectric actuator (stator) that can generate 2-degrees-of-freedom (axial and angular) motions. The stator converts its thrust force and torque into longitudinal and meridian motions of the torus wheel, respectively, making the torus work as an omnidirectional wheel on a plane. In this paper, we build a control system of a piezo-driven 2-degrees-of-freedom torus wheel and evaluate its performance measures, such as the transient characteristics, the orientation accuracy and the payload capacity. An omnidirectional robot with the two torus wheels is constructed, and the feedback control for a desired planar motion is demonstrated. The design inspired by a ring torus represents the possibility toward the creation of an unprecedentedly simple, light, and compact 2-wheel omnidirectional robot.

Slow–Fast Behaviors and Their Mechanism in a Periodically Excited Dynamical System with Double Hopf Bifurcations

This paper focuses on the influence of two scales in the frequency domain on the behaviors of a typical dynamical system with a double Hopf bifurcation. By introducing an external periodic excitation to the normal form of the vector field with double Hopf bifurcation at the origin and taking the exciting frequency far less than the natural frequency, a theoretical model with two scales in the frequency domain is established. Regarding the whole exciting term as a slow-varying parameter leads to a generalized autonomous system, in which the equilibrium branches and their bifurcations with the variation of the slow-varying parameter can be derived. With the increase of the exciting amplitude, different types of bifurcations may be involved in the generalized autonomous system, resulting in several qualitatively different forms of bursting attractors, the mechanism of which is presented by overlapping the transformed phase portraits and the bifurcations of the equilibrium branches. It is found that the single mode 2D torus may evolve to the bursting attractors with mixed modes, in which the trajectory alternates between the single mode oscillations and the mixed mode oscillations. Furthermore, the transitions between the quiescent states and the spiking states may not occur exactly at the bifurcation points because of the slow passage effect, while Hopf bifurcations may cause different forms of repetitive spiking oscillations.

Terahertz Microfluidic Sensing with Dual‐Torus Toroidal Metasurfaces

AbstractToroidal excitation manifests as currents flowing on the surface of a torus along its meridians that are known as poloidal currents. Several intriguing phenomena such as nonradiating anapole excitation, dynamic Aharonov–Bohm effect, and vector potential in the absence of electromagnetic fields occur in the presence of toroidal excitations. Toroidal resonances in metamaterials also offer a platform to probe strong light–matter interactions in its proximity due to the nonradiating confinement of photons. Here, a dual‐torus toroidal flexible metasurface containing an array of overlapped split ring resonators is experimentally demonstrated, which results in strong field confinement for enhanced sensing of polar liquid analytes. Microfluidic integrated dual‐torus toroidal resonance experimentally shows significantly higher sensitivity of 124.3 GHz per refractive index unit compared to a single torus toroidal resonance. Flexible metasurfaces supporting dual torus toroidal resonance can find a wide range of applications in the development of ultrasensitive terahertz molecular sensors for probing real time monitoring of chemicals and biomolecules.

Diophantine Tori and Pragmatic Calculation of Quasimodes for Operators with Integrable Principal Symbol

We study an $$h$$ -pseudodifferential operator (in particular, differential) acting on the $$n$$ -D physical space with a principal symbol determining an integrable Hamiltonian system and with nontrivial subprincipal symbol which destroys the integrability. We propose a pragmatic method for calculating asymptotic eigenvalues and eigenfunctions (quasimodes) of such an operator. Unlike the approach of Lazutkin, where the correction to the principal symbol is included into the Hamiltonian, we deal with an integrable Hamiltonian system and nontrivial $$n$$ -D transport equation. We propose an algorithm for constructing series of quasimodes in a neighborhood of a single Diophantine torus. We discuss the possibility to use this algorithm in practice using software like Wolfram Mathematica. As an example, we explicitly construct quasimodes for a Schrodinger operator describing a weakly coupled dimer, i.e., two weakly coupled particles in a one-dimensional potential field.

A KAM theorem for finitely differentiable Hamiltonian systems

Given l>2ν>2d≥4, we prove the persistence of a Cantor–family of KAM tori of measure O(ε1/2−ν/l) for any non–degenerate nearly integrable Hamiltonian system of class Cl(D×Td), where ν−1 is the Diophantine power of the frequencies of the persistent KAM tori and D⊂Rd is a bounded domain, provided that the size ε of the perturbation is sufficiently small. This extends a result by D. Salamon in [25] according to which we do have the persistence of a single KAM torus in the same framework. Moreover, as for the persistence of a single torus, the regularity assumption is essentially optimal.

Minimal surfaces in the $3$-sphere by stacking Clifford tori

Extending work of Kapouleas and Yang, for any integers $N \geq 2, {k , \ell} \geq 1$, and m sufficiently large, we apply gluing methods to construct in the round $3$-sphere a closed embedded minimal surface that has genus ${k \ell m}^2 (N-1)+1$ and is invariant under a $D_{km} \times D_{\ell m}$ subgroup of $O(4)$, where $D_n$ is the dihedral group of order $2n$. Each such surface resembles the union of $N$ nested topological tori, all small perturbations of a single Clifford torus $\mathbb{T}$, that have been connected by ${k \ell m}^2(N-1)$ small catenoidal tunnels, with ${k \ell m}^2$ tunnels joining each pair of neighboring tori. In the large-$m$ limit for fixed $N$, $k$, and $\ell$, the corresponding surfaces converge to $\mathbb{T}$ counted with multiplicity $N$.

Nonsurgical approach for torus palatinus management in full denture rehabilitation

Background: Tooth loss in individuals more commonly occurs with increasing age. In a dental practice, many patients present with torus palatinus, which is usually caused by problems and complications resulting from denture fabrication. Therefore, a nonsurgical approach in which horseshoe-shaped complete dentures are fabricated is considered to be a viable option to address torus palatinus. Case Report: A 59-year-old edentulous male came to the Prosthodontic Department, Faculty of Dentistry, Trisakti University, with a chief complaint of the instability of his maxillary complete dentures, and he requested new dentures. The patient had been wearing the dentures for approximately 5 years. Based on the intraoral examination, the patient had a large, single lobule torus palatinus that extended posteriorly through the junction between the hard palate and the soft palate (AH line). It was covered with thin mucosal tissue, and it did not interfere with his speech, his ability to chew, or other oral functions without the dentures. After the problems were thoroughly diagnosed and corrected, horseshoe-shaped complete dentures were chosen as the appropriate solution, and the dentures were fabricated. Conclusion: Torus palatinus tends to have very thin mucosa that causes discomfort and irritation during routine usage of acrylic dentures. Therefore, the horseshoe-shaped dentures facilitated good retention and stability, and they did not irritate the torus palatinus.

Arithmetic of double torus quotients and the distribution of periodic torus orbits

We describe new arithmetic invariants for pairs of torus orbits on groups isogenous to an inner form of $\mathbf{PGL}_n$ over a number field. These invariants are constructed by studying the double quotient of a linear algebraic group by a maximal torus. Using the new invariants we significantly strengthen results towards the equidistribution of packets of periodic torus orbits on higher rank $S$-arithmetic quotients. Packets of periodic torus orbits are natural collections of torus orbits coming from a single adelic torus and are closely related to class groups of number fields. The distribution of these orbits is akin to the distribution of integral points on homogeneous algebraic varieties with a torus stabilizer. The proof combines geometric invariant theory, Galois actions, local arithmetic estimates and homogeneous dynamics.

Segmentation mapping and folding method of surfaces of revolution and its applications

PurposeThe purpose of this study was to develop a new folding method for modeling complicated folded fabric with surfaces of revolution.Design/methodology/approachIrregular wrinkles and mesh distortions easily appear in the fold modeling of a complex curved surface. Aimed at this key technical problem, the segmentation mapping folding method (SMFM) is proposed in this paper. First, high-precision flattened planes were obtained by using segmentation mapping techniques. Second, the segmented planes were transformed into a folded and continuous geometric model by using matrix transformations. Finally, initial stress was used to modify the geometric folding errors, which ensured agreement with the inflated flexible fabric’s geometry and the original design.FindingsCompared with the traditional folding method, SMFM has the advantages of good finite-element mesh quality, large radial compression rate, regular folds, etc. The surface area error and the volume error of the inflated single torus established by SMFM were only 1.2 per cent, showing that SMFM has high modeling accuracy. The numerical results of an inflatable re-entry vehicle are presented to demonstrate the reliability, feasibility and applicability of SMFM. Moreover, the stress modification reduced the problems of stress concentration and mesh distortions, improving the accuracy and stability of the numerical calculations.Originality/valueIn this paper, for the first time, a folding method for modeling complicated folded fabric is proposed. This methodology can be used to model the multidimensional compression and regular folds of complex surfaces of revolution that cannot be flattened and to improve the accuracy and stability of the numerical calculations.

Electromagnetically driven flow of electrolyte in a thin annular layer: axisymmetric solutions

Experimental observations of an azimuthal electrolyte flow driven by Lorentz force in a thin annular fluid layer placed on top of a magnet show that it develops a robust vortical system near the outer cylindrical wall. It appears to be a result of instabilities developing on a background of steady axisymmetric flow. Therefore, the goal of this paper is to establish a scene for a future comprehensive stability analysis of such a flow. We discuss popular depth-averaged and quasi-two-dimensional approximate solutions that take advantage of the thin-layer assumption first, and argue that they cannot lead to the observed flow patterns. Thus, three-dimensional toroidal flows are considered. Their similarities to various other well-studied rotating flow configurations are outlined, but no close match is found. Multiple axisymmetric solutions are detected numerically for the same governing parameters, indicating the possibility of subcritical bifurcations, namely type 1, consisting of a single torus, and type 2, developing a second counter-rotating toroidal flow near the outer cylinder. It is suggested that the transition between these two axisymmetric solutions is likely to be caused by the centrifugal instability, while the shear-type instability of the type 2 solution may be responsible for the observed vortex structures. However, a dedicated stability analysis which is currently underway and will be reported in a separate publication is required to confirm these hypotheses.

Interaction of toroidal swimmers in Stokes flow.

A doughnut-shaped object supporting surface rotations was a hypothetical construct proposed by both Taylor and Purcell as a swimmer that would be able to propel itself in a Stokesian fluid because of the irreversibility of its stroke. Here we numerically examine the hydrodynamic interaction of pairs and trios of these free toroidal swimmers. First, we study the axisymmetric case of two toroidal swimmers placed in tandem, and show that a single torus of a corotating pair is more efficient than when it swims alone, but less efficient when paired with a counterrotating partner. Using a regularized Stokeslet framework, we study the nonaxisymmetric case of toroidal swimmers whose axes are initially parallel, but not collinear. These perturbed in tandem swimmers can exhibit qualitatively different trajectories that may, for instance, repel the swimmers or have them settle into a periodic state. We also illustrate interesting dynamics that occur for different initial configurations of three tori.

Transmission imaging for integrated PET-MR systems

Attenuation correction for PET-MR systems continues to be a challenging problem, particularly for body regions outside the head. The simultaneous acquisition of transmission scan based μ-maps and MR images on integrated PET-MR systems may significantly increase the performance of and offer validation for new MR-based μ-map algorithms. For the Biograph mMR (Siemens Healthcare), however, use of conventional transmission schemes is not practical as the patient table and relatively small diameter scanner bore significantly restrict radioactive source motion and limit source placement. We propose a method for emission-free coincidence transmission imaging on the Biograph mMR. The intended application is not for routine subject imaging, but rather to improve and validate MR-based μ-map algorithms; particularly for patient implant and scanner hardware attenuation correction. In this study we optimized source geometry and assessed the method’s performance with Monte Carlo simulations and phantom scans. We utilized a Bayesian reconstruction algorithm, which directly generates μ-map estimates from multiple bed positions, combined with a robust scatter correction method. For simulations with a pelvis phantom a single torus produced peak noise equivalent count rates (34.8 kcps) dramatically larger than a full axial length ring (11.32 kcps) and conventional rotating source configurations. Bias in reconstructed μ-maps for head and pelvis simulations was ⩽4% for soft tissue and ⩽11% for bone ROIs. An implementation of the single torus source was filled with 18F-fluorodeoxyglucose and the proposed method quantified for several test cases alone or in comparison with CT-derived μ-maps. A volume average of 0.095 cm−1 was recorded for an experimental uniform cylinder phantom scan, while a bias of <2% was measured for the cortical bone equivalent insert of the multi-compartment phantom. Single torus μ-maps of a hip implant phantom showed significantly less artifacts and improved dynamic range, and differed greatly for highly attenuating materials in the case of the patient table, compared to CT results. Use of a fixed torus geometry, in combination with translation of the patient table to perform complete tomographic sampling, generated highly quantitative measured μ-maps and is expected to produce images with significantly higher SNR than competing fixed geometries at matched total acquisition time.

Two-fluid and magnetohydrodynamic modelling of magnetic reconnection in the MAST spherical tokamak and the solar corona

Twisted magnetic flux ropes are ubiquitous in laboratory and astrophysical plasmas, and the merging of such flux ropes through magnetic reconnection is an important mechanism for restructuring magnetic fields and releasing free magnetic energy. The merging-compression scenario is one possible start-up scheme for spherical tokamaks, which has been used on the Mega Amp Spherical Tokamak (MAST). Two current-carrying plasma rings or flux ropes approach each due to mutual attraction, forming a current sheet and subsequently merge through magnetic reconnection into a single plasma torus, with substantial plasma heating. Two-dimensional resistive and Hall-magnetohydrodynamic simulations of this process are reported, including a strong guide field. A model of the merging based on helicity-conserving relaxation to a minimum energy state is also presented, extending previous work to tight-aspect-ratio toroidal geometry. This model leads to a prediction of the final state of the merging, in good agreement with simulations and experiment, as well as the average temperature rise. A relaxation model of reconnection between two or more flux ropes in the solar corona is also described, allowing for different senses of twist, and the implications for heating of the solar corona are discussed.

Primary system thermal hydraulics of future Indian fast reactors

As a follow-up to PFBR (Indian prototype fast breeder reactor), many FBRs of 500MWe capacity are planned. The focus of these future FBRs is improved economy and enhanced safety. They are envisaged to have a twin-unit concept. Design and construction experiences gained from PFBR project have provided motivation to achieve an optimized design for future FBRs with significant design changes for many critical components. Some of the design changes include, (i) provision of four primary pipes per primary sodium pump, (ii) inner vessel with single torus lower part, (iii) dome shape roof slab supported on reactor vault, (iv) machined thick plate rotating plugs, (v) reduced main vessel diameter with narrow-gap cooling baffles and (vi) safety vessel integrated with reactor vault. This paper covers thermal hydraulic design validation of the chosen options with respect to hot and cold pool thermal hydraulics, flow requirement for main vessel cooling, inner vessel temperature distribution, safety analysis of primary pipe rupture event, adequacy of decay heat removal capacity by natural convection cooling, cold pool transient thermal loads and thermal management of top shield and reactor vault.

Towards matrix model representation of HOMFLY polynomials

We investigate possibilities of generalizing the TBEM eigenvalue matrix model, which represents the non-normalized colored HOMFLY polynomials for torus knots as averages of the corresponding characters. We look for a model of the same type, which is a usual Chern-Simons mixture of the Gaussian potential, typical for Hermitean models, and the sine Vandermonde factors, typical for the unitary ones. We mostly concentrate on the family of twist knots, which contains a single torus knot, the trefoil. It turns out that for the trefoil the TBEM measure is provided by an action of Laplace exponential on the Jones polynomial. This procedure can be applied to arbitrary knots and provides a TBEM-like integral representation for the N=2 case. However, beyond the torus family, both the measure and its lifting to larger N contain non-trivial corrections in \hbar=\log q. A possibility could be to absorb these corrections into a deformation of the Laplace evolution by higher Casimir and/or cut-and-join operators, in the spirit of Hurwitz tau-function approach to knot theory, but this remains a subject for future investigation.

Suborbital Reentry Demonstration of Inflatable Flare-Type Thin-Membrane Aeroshell Using a Sounding Rocket

An inflatable decelerator is promising as a next-generation atmospheric-entry system owing to its reduced aerodynamic heating and high packing efficiency. In this study, a suborbital reentry demonstration of a flare-type thin-membrane aeroshell sustained by a single inflatable torus using an S-310-41 sounding rocket was carried out. An experimental vehicle was specially developed for this reentry demonstration; it was equipped with a 1.2-m-diam flare-type thin-membrane aeroshell and had a total mass of 15.6 kg. In the flight test, the aeroshell with an inflatable torus was deployed at an altitude of 100 km during a suborbital flight under the conditions of zero-gravity and near vacuum. The experimental vehicle reentered Earth’s atmosphere from an altitude of 150 km. During free fall, it accelerated to a Mach number of 4.5 () because of gravity force. After that, it started decelerating because of aerodynamic force at an altitude of 70 km. According to the flight data, the vehicle remained intact during the reentry and the aeroshell achieved the expected decelerating performance. This reentry demonstration proves that the flare-type thin-membrane aeroshell sustained by the inflatable torus works well as a decelerator for atmospheric-entry vehicles. Further, the drag coefficient of the experimental vehicle in the supersonic, transonic, and subsonic regimes under free-flight conditions was estimated from the flight trajectory.