Torus Type Research Articles

Mordell–Weil groups and automorphism groups of elliptic $K3$ surfaces

We present a method to calculate the action of the Mordell–Weil group of an elliptic K3 surface on the numerical Néron–Severi lattice of the K3 surface. As an application, we compute a finite generating set of the automorphism group of a K3 surface birational to the double plane branched along a 6-cuspidal sextic curve of torus type.

Just Suspended Speed Simulation in Torus Reactor Using Multiple Non-Linear Regression Model

In the chemical and water treatment industries, it is necessary to achieve maximum contact between the solid and liquid phase, thus promoting the mass and heat transfer, to obtain a homogeneous solution. Increasing stirring speed is the most recommended solution in different types of reactors: stirred tank, column, and tubular. However, this inadvertently increases the energy consumption of the industry. Determination of the minimum speed, labeled the just suspended speed (Njs) and crucial to attaining homogeneity, has been widely investigated. Numerous studies have been carried out to assess formulas for determining the solid particle speed in various reactor types. Given the limitations of the existing formulations based on a generalization of a unique equation for computing Njs for all soil classifications, it appears that most formulas can only approximate complex phenomena that depend on several parameters. A novel formula was developed, and the results given in this paper demonstrate the effectiveness of generating significant uncertainties for the estimation of Njs. The purpose of this study was the elaboration of experiment-based data-driven formulas to calculate Njs for different particle size classes. Nonlinear multiple regression (MNLR) models were used to generate the new formulas. The gradient descent optimization algorithm was employed to solve the hyperparameters of each novel equation, utilizing supervised learning. A comparison of the data indicated that the unique formulas presented in this study outperformed empirical formulas and provide a useful means for lowering energy consumption, while increasing the heat and mass transfer in torus type reactors.

Clinical patterns, causes, and treatment of torus palatinus

There are two types of tori, including torus palatinus and torus mandibularis. Evidence shows that these lesions usually progress slowly, and a spontaneous growth stop is usually another characteristic. Affected patients are usually asymptomatic. However, symptoms might be reported in edentulous patients and cases when the lesion is huge. In the present study, we elaborated on the causes, clinical patterns, and treatment of torus palatinus. The etiology of the condition is unclear, and different theories were reported in the literature, suggesting that the etiology of these cases is usually attributed to environmental and genetic predisposition. It has been defined as a benign tumor that presents in the maxillary bone and is usually asymptomatic. Therefore, the diagnosis of the condition is usually easy. However, it is recommended to conduct a differential diagnosis with other relevant malignant lesions in this area. Surgical interventions are not usually needed as the condition is usually discovered accidentally. However, in cases of tori-related prosthetic or functional discomfort, surgery can be conducted.

Kramers nodal line metals

Recently, it was pointed out that all chiral crystals with spin-orbit coupling (SOC) can be Kramers Weyl semimetals (KWSs) which possess Weyl points pinned at time-reversal invariant momenta. In this work, we show that all achiral non-centrosymmetric materials with SOC can be a new class of topological materials, which we term Kramers nodal line metals (KNLMs). In KNLMs, there are doubly degenerate lines, which we call Kramers nodal lines (KNLs), connecting time-reversal invariant momenta. The KNLs create two types of Fermi surfaces, namely, the spindle torus type and the octdong type. Interestingly, all the electrons on octdong Fermi surfaces are described by two-dimensional massless Dirac Hamiltonians. These materials support quantized optical conductance in thin films. We further show that KNLMs can be regarded as parent states of KWSs. Therefore, we conclude that all non-centrosymmetric metals with SOC are topological, as they can be either KWSs or KNLMs.

Critical tori for mean curvature energies in Killing submersions

We study surface energies depending on the mean curvature in total spaces of Killing submersions, which extend the classical notion of Willmore energy. Based on a symmetry reduction procedure, we construct vertical tori critical for these mean curvature energies. These vertical tori are based on closed curves critical for curvature energy functionals in Riemannian 2-space forms.The binormal evolution of these critical curves in Riemannian 3-space forms generates rotational tori solutions for an ample family of Weingarten surfaces. Therefore, we also introduce some correspondence results between these two types of tori and illustrate their relation.

Analytical and experimental analysis of Back EMF waveform of a TORUS-type non-slotted axial flux permanent magnet synchronous machine with shifted rotor

In this paper, the effectiveness of rotor shifting technique on the Total Harmonic Distortion (THD) reduction of back electromotive force (Back EMF) of a TORUS type non-slotted axial flux permanent magnet (TORUS NS-AFPM) machine is investigated. The paper’s objective is to conduct this investigation using analytical and experimental methods. Therefore, an analytical formula is proposed for obtaining Back EMF of the machine with shifted rotor. A 3.7 kW and 1400 rpm machine is selected as case study machine and the effectiveness of rotor shifting technique on the THD reduction of Back EMF is confirmed using the proposed formula and experimental results. In addition, an analytical technique for calculating the optimum value of rotor shifting angle for minimizing THD of Back EMF is proposed. The obtained results confirm that the THD of Back EMF of the case study machine decreases from 11.5% to 5% by optimal selection of rotor shifting angle.

Families of $K3$ surfaces and curves of (2,3)-torus type

We study families of $K3$ surfaces obtained by double covering of the projective plane branching along curves of (2,3)-torus type. In the first part, we study the Picard lattices of the families, and a lattice duality of them. In the second part, we describe a deformation of singularities of Gorenstein $K3$ surfaces in these families.

The Conley index for discrete dynamical systems and the mapping torus

The classical Conley index for flows is defined as a certain homotopy type. In the case of a discrete dynamical system, one usually considers the shift equivalence class of the so-called index map. This equivalence relation is rarely used in other contexts and not well understood in general. Here we propose using a topological invariant of the shift equivalence definition: The homotopy type of the mapping torus of the index map. Using a homotopy type offers new ways for comparing Conley indices–theoretically and numerically. We present some basic properties and examples, compare it to the definition via shift equivalence and sketch an idea for its construction using rigorous numerics.

Quantum torus symmetries of multicomponent modified KP hierarchy and reductions

In this paper, we construct the multicomponent modified KP hierarchy and its additional symmetries. The additional symmetries constitute an interesting multi-folds quantum torus type Lie algebra. By a reduction, we also construct the constrained multicomponent modified KP hierarchy and its Virasoro type additional symmetries.

Three-Dimensional Numerical Simulation of Long-Lived Quantum Vortex Knots and Links in a Trapped Bose Condensate

Dynamics of simplest vortex knots, unknots, and links of torus type inside an atomic Bose-Einstein condensate in anisotropic harmonic trap at zero temperature has been numerically simulated using three-dimensional Gross-Pitaevskii equation. The lifetime for such quasi-stationary rotating vortex structures has been found quite long in wide parametric domains of the system. This result is in qualitative agreement with a previous prediction based on a simplified one-dimensional model approximately describing dynamics of vortex filaments [V.P. Ruban, JETP 126, 397 (2018)].

Long-Lived Quantum Vortex Knots

Dynamics of simplest quantum vortex knots of torus type in a superfluid at zero temperature has been simulated within a regularized Biot-Savart law (the torus radii $R_0$ and $r_0$ for initial vortex configuration were large in comparison with a vortex core width $\xi$). Computations of evolution times of knots until their significant deformation were carried out with a small step on parameter $B_0=r_0/R_0$ for different values of parameter $\Lambda=\log(R_0/\xi)$. It has been found that at $\Lambda\gtrsim 3$, bands of quasi-stability appear in a region of $B_0\lesssim 0.2$, which correspond to long knot lifetimes and to very large traveling distances --- up to several hundreds of $R_0$. This result is new and quite unexpected, because previously it was believed that maximal lifetime of torus knots until reconnection does not exceed several typical periods. The opening of quasi-stable 'windows' at increasing $\Lambda$ is due to narrowing main parametric resonances of the dynamical system on parameter $B_0$.

無摺動式トーラス型伸展指機構

This research relates to the device and realization of No Sliding Torus Type Extension Finger Mechanism. In resent year, various bag type grippers have researched and developed. Bag type gripper can grasp various objects because it is able to adapt to the target objects. However, since friction occurs between the gripper and the object, it is difficult to grasp weak or fragile objects. Therefore, we report on the principle devised, realized(design, prototype), experiment using a torus finger gripper mechanism which can grasp objects without sliding.

Instability in dynamic balance of volterra-lotka systems with perturbations in the right side

The basic effects and patterns that characterize the model of coexistence of two species with weak sinusoidal external effect on the reproduction rate is considered. Solving Lotka-Volterra differential equations describes the behavior of the elementary ecosystem. Numerical solutions for exposure frequencies, close to the frequency of the unperturbed system cycle are found. The stability of such a non-autonomous system is investigated.It is determined that the sinusoidal effect on the population, e.g., by changing the reproduction rate of one or both species because of seasonal changes in nutrition or hunting leads to a nonperiodic system dynamics, having the type of degenerate 2-dimensional non-resonant torus. Various forms of irregular behavior of “predators” and “victims” appear in the phase portraits for similar perturbations. All this confirms that even relatively simple models of ecosystems reveal their instability, i. e., sensitivity to small external perturbations

Oral Tori In Edentulous Patients

Background and objective:  This study aimed to determine the prevalence, size, shape and location of tori in partially and completely edentulous patients attending the College of Dentistry/ Hawler Medical University from 2009-2011. Method: In a retrospective study four hundred and fifty patients case file from Erbil city were examined for the presence of oral tori in prosthodontic department, College of Dentistry, Hawler Medical University. The size, shape and location of tori were recorded and related with the age and sex of patients. Result: The overall prevalence of tori was 14.2% (64 cases). The prevalence of torus palatinus was 60.9% (39/64), while that of torus mandibularis was significantly lesser 23.5 % (15/64).Both types of tori were associated with each other in 15.62% of cases (10/64). Conclusions: There was no statistically significant difference in the prevalence of tori between males and females. There was a significant relationship between the presence of mandibular and palatal tori.

Effective stiffening of DNA due to nematic ordering causes DNA molecules packed in phage capsids to preferentially form torus knots

Observation that DNA molecules in bacteriophage capsids preferentially form torus type of knots provided a sensitive gauge to evaluate various models of DNA arrangement in phage heads. Only models resulting in a preponderance of torus knots could be considered as close to reality. Recent studies revealed that experimentally observed enrichment of torus knots can be qualitatively reproduced in numerical simulations that include a potential inducing nematic arrangement of tightly packed DNA molecules within phage capsids. Here, we investigate what aspects of the nematic arrangement are crucial for inducing formation of torus knots. Our results indicate that the effective stiffening of DNA by the nematic arrangement not only promotes knotting in general but is also the decisive factor in promoting formation of DNA torus knots in phage capsids.

Bifurcation phenomena of mean-field coupled self-sustained oscillators with two different time scales under the influence of external white noise

We report stochastic phenomena of inter-cluster synchronization in ensembles of coupled self-sustained oscillators with different time scales. A mean-field model of two interconnected clusters of coupled limit cycle oscillators having same trajectories with two native frequencies is considered to analytically derive the time evolution of the order parameters in the thermodynamic limit. A set of the order parameters obtained for the system exhibits noise induced bifurcations, including those from torus type attractors to fixed point ones for the case with two different heat baths in accordance with the introduction of the time scale.

New Examples of Biharmonic Submanifolds in CPn and S2n+1*

We construct biharmonic real hypersurfaces and Lagrangian submanifolds of Clifford torus type in $CP^n$ via the Hopf fibration; and get new examples of biharmonic submanifolds in $S^{2n+1}$ as byproducts .

Synchronization of quasi-periodic oscillations in coupled phase oscillators

The phenomenon of synchronization of two oscillators under the action of an external force has been studied in a phase approximation. The Lyapunov maps are presented and the possible types of regimes are considered. Various types of two-frequency tori are revealed and classified.

Classification of Local Singularities on Torus Curves of Type $(2,5)$

In this paper, we consider curves of degree 10 of torus type (2,5), $C:=\{f_5(x,y)^2+f_2(x,y)^5=0\}$. Assume that $f_2(0,0)=f_5(0,0)=0$. Then $O=(0,0)$ is a singular point of $C$ which is called an inner singularity. In this paper, we give a topological classification of singularities of $(C,O)$.

Irreducible plane sextics with large fundamental groups

We compute the fundamental group of the complement of each irre- ducible sextic of weight eight or nine (in a sense, the largest groups for irreducible sextics), as well as of 169 of their derivatives (both of and not of torus type). We also give a detailed geometric description of sextics of weight eight and nine and of their moduli spaces and compute their Alexander modules; the latter are shown to be free over an appropriate ring.