Double Torus Research Articles

Generalized double affine Hecke algebra for double torus

We propose a generalization of the double affine Hecke algebra of type-C∨C1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C^\\vee C_1$$\\end{document} at specific parameters by introducing a “Heegaard dual” of the Hecke operators. Shown is a relationship with the skein algebra on double torus. We give automorphisms of the algebra associated with the Dehn twists on the double torus.

Forced synchronization of self-excited chaotic thermoacoustic oscillations

We experimentally investigate the forced synchronization of a self-excited chaotic thermoacoustic oscillator with two natural frequencies, $f_1$ and $f_2$ . On increasing the forcing amplitude, $\epsilon _f$ , at a fixed forcing frequency, $f_f$ , we find two different types of synchronization: (i) $f_f/f_1 = 1:1$ or $2:1$ chaos-destroying synchronization (CDS), and (ii) phase synchronization of chaos (PSC). En route to $1:1$ CDS, the system transitions from an unforced chaotic state ( ${\rm {CH}}_{1,2}$ ) to a forced chaotic state ( ${\rm {CH}}_{1,2,f}$ ), then to a two-frequency quasiperiodic state where chaos is destroyed ( $\mathbb {T}^2_{2,f}$ ), and finally to a phase-locked period-1 state ( ${\rm {P1}}_f$ ). The route to $2:1$ CDS is similar, but the quasiperiodic state hosts a doubled torus $(2\mathbb {T}^2_{2,f})$ that transforms into a phase-locked period-2 orbit $({\rm {P2}}_f)$ when CDS occurs. En route to PSC, the system transitions to a forced chaotic state ( ${\rm {CH}}_{1,2,f}$ ) followed by a phase-locked chaotic state, where $f_1$ , $f_2$ and $f_f$ still coexist but their phase difference remains bounded. We find that the maximum reduction in thermoacoustic amplitude occurs near the onset of CDS, and that the critical $\epsilon _f$ required for the onset of CDS does not vary significantly with $f_f$ . We then use two unidirectionally coupled Anishchenko–Astakhov oscillators to phenomenologically model the experimental synchronization dynamics, including (i) the route to $1:1$ CDS, (ii) various phase dynamics, such as phase drifting, slipping and locking, and (iii) the thermoacoustic amplitude variations in the $f_f/f_1$ – $\epsilon _f$ plane. This study extends the applicability of open-loop control further to a chaotic thermoacoustic system, demonstrating (i) the feasibility of using an existing actuation strategy to weaken aperiodic thermoacoustic oscillations, and (ii) the possibility of developing new active suppression strategies based on both established and emerging methods of chaos control.

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.

Embedding [formula omitted] and [formula omitted] on the double torus

The Kuratowski graphs K3,3 and K5 characterize planarity. Counting distinct 2-cell embeddings of these two graphs on orientable surfaces was previously done by Mull (1999) and Mull et al. (2008), using Burnside’s Lemma and automorphism groups of K3,3 and K5, without actually constructing the embeddings. We obtain all 2-cell embeddings of these graphs on the double torus, using a constructive approach. This shows that there is a unique non-orientable 2-cell embedding of K3,3, and 14 orientable and 17 non-orientable 2-cell embeddings of K5 on the double torus, which are explicitly obtained using an algorithmic procedure of expanding from minors. Therefore we confirm the numbers of embeddings obtained by Mull (1999) and Mull et al. (2008). As a consequence, several new polygonal representations of the double torus are presented. Rotation systems for the one-face embeddings of K5 on the triple torus are also found, using exhaustive search.

Super-spinning Kerr attractors: Observational signatures of accretion tori and proto-jets

We study accretion tori and proto-jets in the field of central Kerr super-spinning attractors, exposing distinctions between Kerr naked singularities (NSs) and black holes (BHs). We focus on the spacetime regions very close to the central singularity, interior of the ergoregion, especially for slowly spinning NSs, where repulsive gravity effects emerge, discussing the causal structure in relation to tori contained, partially contained and confined in the ergoregion. We explore particularly regions close to the attractor rotational axis and the poles, where very slowly spinning tori, with momenta ℓ≈0, and axial cusp, constrained by the light surfaces, are relevant. Proto-jets (constraining jet emission) proved to be a signature in comparison with BH case, pointing out the emergence of very fast spinning tori in the ergoregion, very slow tori, double tori with internal cusp, and the presence of an axial cusp. The accretion disks properties point out very faster spinning attractors (extremely super spinning NSs). Providing distinguishable physical characteristics of jets and accretion tori, the differences between the BH and NS accretion scenarios, emerging from this analysis, constitute astrophysical tracers distinguishing NSs from BHs, and possible strong NSs observational signature of the primordial Kerr superspinars predicted by string theory.

On photons and matter inversion spheres from complex super-spinars accretion structures

Our analysis focus on the dragging effects on the accretion flows and jet emission in Kerr super-spinars. These attractors are characterized by peculiar accretion structures as double tori, or special dragged tori in the ergoregion, produced by the balance of the hydrodynamic and centrifugal forces and also effects of super-spinars repulsive gravity. We investigate the accretion flows, constituted by particles and photons, from toroids orbiting a central Kerr super-spinar. As results of our analysis, in both accretion and jet flows, properties characterizing these geometries, that constitute possible strong observational signatures or these attractors, are highlighted. We found that the flow is characterized by closed surfaces, defining inversion coronas (spherical shell), with null the particles flow toroidal velocity (u^{phi }=0) embedding the central singularity. We proved that this region distinguishes proto-jets and accretion driven flows, co-rotating and counter-rotating flows. Therefore in both cases the flow carries information about the accretion structures around the central attractor, demonstrating that inversion points can constitute an observational aspect capable of distinguishing the super-spinars.

High-resolution Radio Study of the Dragonfly Pulsar Wind Nebula Powered by PSR J2021+3651

The Dragonfly Nebula (G75.2+0.1) powered by the young pulsar J2021+3651 is a rare pulsar wind nebula (PWN) that shows double tori and polar jets enclosed by a bow-shock structure in X-rays. We present new radio observations of this source taken with the Very Large Array at 6 GHz. The radio PWN has an overall size about two times as large as the X-ray counterpart, consisting of a bright main-body region in the southwest, a narrow and fainter bridge region in the northeast, and a dark gap in between. The nebula shows a radio spectrum much softer than that of a typical PWN. This could be resulting from compression by the ram pressure as the system travels mildly supersonically in the interstellar medium. Our polarization maps reveal a highly ordered and complex B-field structure. This can be explained by a toroidal field distorted by the pulsar motion.

Dark matter effect on black hole accretion disks

Comparing different dark matter (DM) models, we explore the DM influence on black hole (BH) accretion disk physics, considering corotating and counterrotating thick accretion tori orbiting a central spinning BH. Our results identify accretion onto a central BH as a good indicator of DM presence, signaling possible DM tracers in accretion physics. We analyze accretion around a spinning BH immersed in perfect-fluid dark matter, cold dark matter and scalar field dark matter. Our investigation addresses observational evidence of distinctive DM effects on toroidal accretion disks and protojet configurations, proving that BH accretion tori immersed in DM can present characteristics, such as interdisk cusp or double tori, which have usually been considered as tracers for superspinars and naked singularity attractors. Therefore, in this context DM influence on the BH geometry could manifest as superspinar mimickers. DM also affects the central spinning attractor energetics associated with accretion physics, and its influence on accretion disks can be searched for in a variation of the central BH energetics as an increase of the mass accretion rates.

Hyperbolic polyhedral surfaces with regular faces

We study hyperbolic polyhedral surfaces with faces isometric to regular hyperbolic polygons satisfying that the total angles at vertices are at least 2π. The combinatorial information of these surfaces is shown to be identified with that of Euclidean polyhedral surfaces with negative combinatorial curvature everywhere. We prove that there is a gap between areas of non-smooth hyperbolic polyhedral surfaces and the area of smooth hyperbolic surfaces. The numerical result for the gap is obtained for hyperbolic polyhedral surfaces, homeomorphic to the double torus, whose 1-skeletons are cubic graphs.

Hyperbolic Geometrically Uniform Codes and Ungerboeck Partitioning on the Double Torus

Current research builds labelings for geometrically uniform codes on the double torus through tiling groups. At least one labeling group was provided for all of the 11 regular tessellations on the double torus, derived from triangular Fuchsian groups, as well as extensions of these labeling groups to generate new codes. An important consequence is that such techniques can be used to label geometrically uniform codes on surfaces with greater genera. Furthermore, partitioning chains are constructed into geometrically uniform codes using soluble groups as labeling, which in some cases results in an Ungerboeck partitioning for the surface. As a result of these constructions, it is demonstrated that, as in Euclidean spaces, modulation and encoding can be combined in a single step in hyperbolic space.

Lines of Curvature on the Double Torus

We describe the $\nu$-lines of curvature of an embedding of the double torus into $\mathbb R^4$, defined as the link of the real part of the Milnor fibration of a polynomial, where $\nu$ is its gradient. Through this analysis, we present a complete description of the foliation of lines of curvature of the embedding, defined as the image of the stereographic projection of this link into $\mathbb R^3$.

Geometrically thick tori around compact objects with a quadrupole moment

We study geometrically thick perfect-fluid tori with constant specific angular momentum, so-called ‘Polish doughnuts’, orbiting deformed compact objects with a quadrupole moment. More specifically, we consider two different asymptotically flat, static and axisymmetric vacuum solutions to Einstein’s field equation with a non-zero quadrupole moment, the q-metric and the Erez–Rosen spacetime. It is our main goal to find features of Polish doughnuts in these two spacetimes which qualitatively distinguish them from Polish doughnuts in the Schwarzschild spacetime. As a main result we find that, for both metrics, there is a range of positive (Geroch–Hansen) quadrupole moments which allows for the existence of double tori. If these double tori fill their Roche lobes completely, their meridional cross-section has the shape of a fish, with the body of the fish corresponding to the outer torus and the fish-tail corresponding to the inner torus. Such double tori do not exist in the Schwarzschild spacetime.

A Family of Semitoric Systems with Four Focus\u2013Focus Singularities and Two Double Pinched Tori

We construct a one-parameter family F_t=(J, H_t)_{0 le t le 1} of integrable systems on a compact 4-dimensional symplectic manifold (M, omega ) that changes smoothly from a toric system F_0 with eight elliptic–elliptic singular points via toric type systems to a semitoric system F_t for t^-< t < t^+. These semitoric systems F_t have precisely four elliptic–elliptic and four focus–focus singular points. Moreover, at t= frac{1}{2}, the system has precisely two focus–focus fibres each of which contains exactly two focus–focus points, giving these fibres the shape of double pinched tori. We exemplarily parametrise one of these fibres explicitly.

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.

RADs energetics and constraints on emerging tori collisions around super-massive Kerr black holes

We provide constraints on possible configurations and interactions of two coplanar tori orbiting a central Kerr black hole (BH), in dependence on its dimensionless spin. The two-tori configurations can be directly linked to the current models featuring the obscuration of galactic BH X-ray emission. The emergence of each torus instability phases is discussed and tori collision has been also investigated. The first simple evaluation of the center-of-mass energy proves that collision-energy-efficiency increases with the dimensionless BH spin. We explore the phenomenological aspects of the corotating and counterrotating tori by analyzing properties of the orbiting toroidal configurations related to the fluid enthalpy flux, the mass-flux, the mass-accretion-rates, and the cusp luminosity in the two cases of corotating and counterrotating fluids in dependence on the SMBH spin. The analysis resulted ultimately in a comparative investigation of the properties of corotating versus counterrotating tori, demonstrating that two accretion tori can orbit around the central Kerr attractor only under very specific conditions. Our results also demonstrate that the dynamics of the unstable phases of these double tori systems is significant for the high energy phenomena which could be observable in the X-ray emission and extremely energetic phenomena in active galactic nuclei and quasar.

Mechanisms of strange nonchaotic attractors in a nonsmooth system with border-collision bifurcations

It is not very clear to understand genesis and mechanisms for the creation of strange nonchaotic attractors (SNAs) due to the nonsmooth bifurcations in the nonsmooth systems. A quasiperiodically forced piecewise Logistic system is shown to exhibit many types of routes to the creation of SNAs. We point out that the truncation of border-collision torus-doubling bifurcation can lead to different types of SNAs. We identify and describe the Heagy–Hammel routes, fractalization route and intermittent routes after the two coexisting tori collide at the border and the doubled torus is interrupted in this system. It has been shown that there exist two critical tongue-type regions in the parameter space, where the different mechanisms for the birth of SNAs are investigated. These SNAs are identified by the Lyapunov exponents and the phase sensitivity exponents. Different types of SNAs are also characterized by the singular-continuous spectrum, Fourier transform, rational approximations, distribution of finite-time Lyapunov exponents and recurrence analysis.

Translations and reflections on the torus: identities for discrete Wigner functions and transforms

A finite Hilbert space can be associated to a periodic phase space, that is, a torus. Then a subgroup of operators corresponding to reflections and translations on the torus form respectively the basis for the discrete Weyl representation, including the Wigner function, and for its Fourier conjugate, the chord representation. They are invariant under Clifford transformations and obey analogous product rules to the continuous representations, so allowing for the calculation of expectations and correlations for observables. We here import new identities from the continuum for products of pure state Wigner and chord functions, involving, for instance the inverse phase space participation ratio and correlations of a state with its translate. New identities are derived involving transition Wigner or chord functions of transition operators . Extension of the reflection and translation operators to a doubled torus phase space leads to the representation of superoperators and so to the construction of the propagator of Wigner functions from the Weyl representation of the evolution operator.

Para‐Hermitian Geometry, Dualities and Generalized Flux Backgrounds

AbstractWe survey physical models which capture the main concepts of double field theory on para‐Hermitian manifolds. We show that the geometric theory of Lagrangian and Hamiltonian dynamical systems is an instance of para‐Kähler geometry which extends to a natural example of a Born geometry. The corresponding phase space geometry belongs to the family of natural almost para‐Kähler structures which we construct explicitly as deformations of the canonical para‐Kähler structure by non‐linear connections. We extend this framework to a class of non‐Lagrangian dynamical systems which naturally encodes the notion of fluxes in para‐Hermitian geometry. In this case we describe the emergence of fluxes in terms of weak integrability defined by the D‐bracket, and we extend the construction to arbitrary cotangent bundles where we reproduce the standard generalized fluxes of double field theory. We also describe the para‐Hermitian geometry of Drinfel'd doubles, which gives an explicit illustration of the interplay between fluxes, D‐brackets and different polarizations. The left‐invariant para‐Hermitian structure on a Drinfel'd double in a Manin triple polarization descends to a doubled twisted torus, which we use to illustrate how changes of polarizations give rise to different fluxes and string backgrounds in para‐Hermitian geometry.

On global aspects of duality invariant theories: M2-brane vs DFT

Supermembrane compactified on a M9 × T2 target space is globally described by the inequivalent classes of torus bundles over torus. These torus bundles have monodromy in SL (2, Z) when they correspond to the nontrivial central charge sector and they are trivial otherwise. The first ones contain eight inequivalent classes of M2-brane bundles which at low energies, are in correspondence with the eight type II gauged supergravities in 9D. The relation among them is completely determined by the global action of T-duality which interchanges topological invariants of the two tori. The M2-brane torus bundles are invariant under SL(2, Z) × SL(2, Z) × Z2. From the effective point of view, there is another dual invariant theory, called Double Field Theory which describe invariant actions under O(D, D). Globally it is formulated in terms of doubled 2D torus fibrations over the spacetime with a monodromy given by O(D, D, Z). In this note we discuss T-duality global aspects considered in both theories and we emphasize certain similarities between both approaches which could give some hints towards a deeper relationship between them.

Thickness and outerthickness for embedded graphs

We prove new upper bounds for the thickness and outerthickness of a graph in terms of its orientable and nonorientable genus by applying the method of deleting spanning disks of embeddings to approximate the thickness and outerthickness. We also show that every non-planar toroidal graph can be edge partitioned into a planar graph and an outerplanar graph. This implies that the outerthickness of the torus (the maximum outerthickness of all toroidal graphs) is 3. Finally, we show that all graphs embeddable in the double torus have thickness at most 3 and outerthickness at most 5.