Hyperbolic Structure Research Articles

The Conformal Limit and Projective Structures

Abstract The non-abelian Hodge correspondence maps a polystable $\textrm{SL}(2, {\mathbb{R}})$-Higgs bundle on a compact Riemann surface $X$ of genus $g \geq 2$ to a connection that, in some cases, is the holonomy of a branched hyperbolic structure. Gaiotto’s conformal limit maps the same bundle to a partial oper, that is, to a connection whose holonomy is that of a branched complex projective structure compatible with $X$. In this article, we show how these are both instances of the same phenomenon: the family of connections appearing in the conformal limit can be understood as a family of complex projective structures, deforming the hyperbolic ones into the ones compatible with $X$. We also show that, for zero Toledo invariant, this deformation is optimal, inducing a geodesic on Teichmüller’s space.

Gap Between Lyapunov Exponents for Hitchin Representations

Abstract We study Lyapunov exponents for flat bundles over hyperbolic curves defined via parallel transport over the geodesic flow. We consider them as invariants on the space of Hitchin representations and show that there is a gap between any two consecutive Lyapunov exponents. Moreover we show that the minimal possible gap for any two consecutive Lyapunov exponents is achieved if and only if the representation is the one uniformizing the hyperbolic structure of the surface. We give two proofs of the previous fact. In the first one, we relate the Lyapunov exponents to a transverse Lyapunov exponent associated to a deformation of the unstable foliation of the geodesic flow, and we establish a general bound for this quantity. In the second one, we relate Lyapunov exponents to the renormalized intersection product in the setting of the thermodynamic formalism developed by Bridgeman, Canary, Labourie, and Sambarino and we use the already existing bound for such a quantity.

Well-posedness theory for non-homogeneous incompressible fluids with odd viscosity

Several fluid systems are characterised by time reversal and parity breaking. Examples of such phenomena arise both in quantum and classical hydrodynamics. In these situations, the viscosity tensor, often dubbed “odd viscosity”, becomes non-dissipative. At the mathematical level, this fact translates into a loss of derivatives at the level of a priori estimates: while the odd viscosity term depends on derivatives of the velocity field, no parabolic smoothing effect can be expected.In the present paper, we establish a well-posedness theory in Sobolev spaces for a system of incompressible non-homogeneous fluids with odd viscosity. The crucial point of the analysis is the introduction of a set of good unknowns, which allow for the emerging of a hidden hyperbolic structure underlying the system of equations. It is exactly this hyperbolic structure which makes it possible to circumvent the derivative loss and propagate high enough Sobolev norms of the solution. The well-posedness result is local in time; two continuation criteria are also established.

The complex hyperbolic form as a Weil–Petersson form

For the moduli space of the punctured spheres, we find a new equality between two symplectic forms defined on it. Namely, by treating the elements of this moduli space as the singular Euclidean metrics on a sphere, we give an interpretation of the complex hyperbolic form, i.e. the Kähler form of the complex hyperbolic structure on the moduli space, as a kind of Weil–Petersson form.

Structure invariant properties of the hierarchically hyperbolic boundary

We prove several topological and dynamical properties of the boundary of a hierarchically hyperbolic group are independent of the specific hierarchically hyperbolic structure. This is accomplished by proving that the boundary is invariant under a “maximization” procedure introduced by the first two authors and Durham.

Lagrangian approach to nonlinear waves in non-dispersive and dispersive rotating shallow water magnetohydrodynamics

A Lagrangian approach to both hydrostatic non-dispersive in the short-wave range and non-hydrostatic dispersive rotating shallow-water magnetohydrodynamics is developed, and used to analyse weakly and fully nonlinear waves described by the model. Hyperbolic structure in the non-dispersive case is displayed and Riemann invariants are constructed. Characteristic equations are used to establish criteria of breaking and formation of shocks by magneto-gravity waves, and conditions of the appearance of contact discontinuities in Alfvén waves. As in the case of non-magnetic rotating shallow water, rotation cannot prevent breaking. The Lagrangian equations of the model are reduced to a single partial differential ‘master’ equation, which is used to analyse the propagation of weakly nonlinear waves of both families, with or without weak rotation, and with or without weak short-wave dispersion. Corresponding modulation equations are constructed and their main properties sketched. The same master equation is used to obtain fully nonlinear finite-amplitude wave solutions in particular cases of no short-wave dispersion or no rotation.

French political symbolism and identity construction

The present study examines how the language of political symbolism operates within the framework of identity construction. It focusses on the themes of sovereignty during the 2022 French presidency of the European Union and the national presidential election campaign. On the basis of Conceptual Metaphor Theory, it suggests that, apart from purely linguistic features, extra-linguistic factors are also essential in order to convey a global view of symbolic rhetoric. The analysis is conducted according to an overall 6-tier model of figurative origins involving the parameters of personal background, political context, cultural history, reference, conceptual metaphor and linguistic metaphor. By applying the parameters of the model, it is proposed that the core of political argumentation in the corpus analysis is based on national symbols and the conceptual mapping they embody. The basic hypothesis of the model assumes that a politician attempts to portray the symbolisation of unity in accordance with his or her political background and personal biography. A political narrative is thereby created within the current political context using predominant symbols in cultural history. An over-riding factor is the role of referential points which may determine divergence in conceptual mapping. These features then lead to the construction of conceptual metaphors which can take on a particularly hyperbolic structure in the ensuing linguistic metaphors. The implications of the study highlight the important role of symbolism in political debate, its interaction with conceptual metaphor, the relevance of extra-linguistic factors portrayed by the 6-tier model, the link between cultural history and hyperbolic linguistic structures and political divergence in similar symbols.

A numerical study of rigidity of hyperbolic splittings in simple two-dimensional maps

Chaotic hyperbolic dynamical systems enjoy a surprising degree of rigidity, a fact which is well known in the mathematics community but perhaps less so in theoretical physics circles. Low-dimensional hyperbolic systems are either conjugate to linear automorphisms, that is, dynamically equivalent to the Arnold cat map and its variants, or their hyperbolic structure is not smooth. We illustrate this dichotomy using a family of analytic maps, for which we show by means of numerical simulations that the corresponding hyperbolic structure is not smooth, thereby providing an example for a global mechanism which produces non-smooth phase space structures in an otherwise smooth dynamical system.

Seismic Response Analysis of Multi Span Hyperbolic Brick Arch Thin Shell Structure

The research objective is to investigate the multi-span hyperbolic brick arch thin-shell structure located in Changleyuan, Baoji City, Shaanxi Province. The study utilizes on-site dynamic testing and finite element numerical simulation methods to analyze the dynamic characteristics of the thin-shell structure and its seismic response under various seismic waves and directions. The simulation results indicate the following findings: Vertical earthquakes cause more severe damage to structures compared to horizontal earthquakes. Bidirectional seismic excitation has a greater impact on the structure than unidirectional and three-dimensional seismic excitation. The obtained results provide a scientific basis for the seismic protection, reinforcement, and repair of this multi-span double-curved brick arch thin-shell structure.

Euclidean volumes of hyperbolic knots

The hyperbolic structure on a 3 3 –dimensional cone–manifold with a knot as singularity can often be deformed into a limiting Euclidean structure. In the present paper we show that the respective normalised Euclidean volume is always an algebraic number, which is reminiscent of Sabitov’s theorem (the Bellows Conjecture). This fact also stands in contrast to hyperbolic volumes whose number–theoretic nature is usually quite complicated.

Controlled Hyperbolic Structure of Metamaterial for Subdiffraction Visualization in the Terahertz Range

Controlled Hyperbolic Structure of Metamaterial for Subdiffraction Visualization in the Terahertz Range

Extended Ginzburg-Landau theory of superconductivity from generalized momentum operator and position-dependent mass

The concept of generalized momentum operator which is motivated from the generalized uncertainty principle has been extensively investigated in quantum mechanics and various aspects of theoretical and applied physics. In this study, we have applied this formalism to Ginzburg-Landau theory of superconductvity and Abrikosov vortex lattice in type II-superconductors. After deriving the extended Ginzburg-Landau equations, we have discussed several independent structures of the auxiliary function of position operator in the generalized momentum operator and we have analyzed their features in both Ginzburg-Landau and London theories. Comparable properties to the basic formalism have been obtained even without the presence of the cubic nonlinear term in Ginzburg-Landau equations. However, not all structures of the auxiliary function of position operator result on the exclusion of the magnetic field from a superconductor when it's below its critical temperature. But, only specific forms may succeed to eliminate the magnetic field. This approach has been generalized by considered a position-dependent mass of the electric charge. Amazingly, for a position-dependent mass of hyperbolic soliton-like structure, the extended Ginzburg-Landau wave function is identical to the result obtained in the conventional formalism and besides, for specific correlations between the auxiliary function of the position operator and position-dependent mass, the exclusion of the magnetic field is conceivable. We have also discussed the Abrikosov vortex lattice solution based on the extended Ginzburg-Landau formalism with position-dependent mass of the electric charge. It was observed that, for a specific structure of the position-dependent mass and for a quantum number n=0, a transition between a type-II and type-I superconductor takes place if the Ginzburg-Landau parameter is κ=121+257≈2.0634. For large n, the problem depends on the asymptotic form of the Hermite polynomial and periodicity occurs if the electric charge is quantized. Further details are obtained and analyzed.

Barcode encryption based on spin hall effect in tunable vertical hyperbolic metamaterial

In this work, we present a novel vertically hyperbolic metamaterial vHMM structure comprising multilayer stacks of titanium nitride TiN-alumina Al2O3 films. Based on the modulation of the parallel permittivity ɛ∥ and vertical permittivity ɛ⊥ of the vHMM structure by the filling ratio of TiN in the vHMM structure and incident wavelength, we theoretically design and analyze three epsilon-near-zero ENZ structures to investigate the spin Hall effect of light SHEL. Our results demonstrate that all three ENZ structures exhibit significant enhancement of SHEL, with the lateral shift exhibiting extreme sensitivity to the incident angle and producing micron-level variations within a few milliradians. Notably, the class-II of ENZ structure ɛ∥≠0,ɛ⊥≈0 suffers from the highest loss and yields the weakest enhancement of SHEL. Furthermore, we discover that the sign of the transverse shift of the SHEL can also be modulated according to the filling ratio of TiN in the vHMM structure and incident wavelength within a specific incident angle. Based on the regulation law, we propose two barcode encryption schemes that exploit the traverse directions. Our study provides a new mechanism for practical applications of SHEL in optical communication and data storage.

Poisson Limit Distribution for Diffeomorphisms with Weak Hyperbolic Product Structure

Poisson Limit Distribution for Diffeomorphisms with Weak Hyperbolic Product Structure

FBG displacement sensor with hyperbolic flexible hinge structure

This paper proposes a design method for a fiber Bragg grating (FBG) displacement sensor with a hyperbolic flexible hinge structure. A compact FBG displacement sensor with strong micro displacement measurement capability was fabricated. The final assembly testing showed that the sensor achieved good linearity with a linearity of 0.53%F·S at a range of 50 mm. The sensitivity was measured as 24.45 pm mm−1, hysteresis error as 1.1%FS, and repeatability error as 0.659%. The testing results demonstrated that the fabricated FBG displacement sensor exhibited strong practicality and good stability, making it suitable for long-term displacement monitoring of slope safety.

Superelastic bionic graphene/nanoceramic metamaterials with tunable thermo-mechanical performances

Limited by the intrinsic strong bonds between the composing atoms, engineering materials are plagued by the inevitable trade-offs among thermo-mechanical strength, toughness, and stability. In this work, a graphene/nanoceramic metamaterials (GNC) metamaterial was created by in-situ intercalation of Zr(WO4)2 nanoparticles within the multilayered microcell wall of a 3D hyperbolic graphene template. The bionic lamellar feature of the reduced graphene oxide (RGO)-Zr(WO4)2 sheet makes it highly ductile and stiff at the microscale, while the hyperbolic mesoscopic structure of GNC significantly enhanced the resistance to shear slip-induced fracture. 3D lightweight GNC metamaterials, therefore, exhibited synergic improvement in elastic compressibility (90%), strength (1.4 MPa), and toughness (1.62 N/m) than those of reported materials with similar architectures from either individual component or sandwiched microcell. Moreover, due to the lattice rotation of Zr(WO4)2 effectively regulated by the RGO sheet, the 3D GNC demonstrated tunable thermal expansion coefficient (TEC) as near-zero (−1.1 × 10−6 K−1), negative (−11.6 × 10−6 K−1), positive (1.2 × 10−6 K−1), and even combinations of three schemes. The resultant low thermal conductivity (∼0.007 W m−1 K−1), prior thermal shock resistance (200 K/s), and fireproof make GNC metamaterial show promise for applications in extreme conditions, such as lightweight protective filler, thermal insulator, robust damper, fire-retardant coating, thermal stress regulator, etc.

Observation of Two Splitting Processes in a Partial Filament Eruption on the Sun: The Role of Breakout Reconnection

Partial filament eruptions have often been observed; however, the physical mechanisms that lead to filament splitting are not yet fully understood. In this study, we present a unique event of a partial filament eruption that undergoes two distinct splitting processes. The first process involves vertical splitting and is accompanied by brightenings inside the filament, which may result from internal magnetic reconnection within the filament. Following the first splitting process, the filament is separated into an upper part and a lower part. Subsequently, the upper part undergoes a second splitting, which is accompanied by a coronal blowout jet. An extrapolation of the coronal magnetic field reveals a hyperbolic flux tube structure above the filament, indicating the occurrence of breakout reconnection that reduces the constraining field above. Consequently, the filament is lifted up, but at a nonuniform speed. The high-speed part reaches the breakout current sheet to generate the blowout jet, while the low-speed part falls back to the solar surface, resulting in the second splitting. In addition, continuous brightenings are observed along the flare ribbons, suggesting the occurrence of a slipping reconnection process. This study presents, for the first time, the unambiguous observation of a two-stage filament-splitting process, advancing our understanding of the complex dynamics of solar eruptions.

Effect of Clamp Deletion on Static Behavior and Dynamic Characteristics of Loop-Free Cable Net Structures

The loop-free single-layer hyperbolic cable net structure (LSHCS) is a new tensile cable structure for buildings, and it can overcome the disadvantages of key elements and the high tension in tensile cable structures with loop cables. The imperfection of the LSHCS is that cable clamps near the inside boundary are redundant. In this paper, the static behavior and dynamic characteristics of thirty-one schemes in five levels of deleting cable clamps are carried out with ANSYS software and Midas/Gen software. The results show that SE1/1, SE2/1, SE3/1, and SE4/1 are the best cable clamp deleting schemes for their respective levels. Displacements are the most sensitive to deleting cable clamps, the natural vibration period comes in second, and tensions are not sensitive with the growth rates of 6.09%, 5.11%, 15.97%, and 19.28% in the best scheme of each level. It is concluded that schemes involving the virtual rings of cable clamps closest to the inside boundary cause the smallest effect on static behavior and dynamic characteristics. Deleting the cable clamps affects the structural stiffness significantly, but the bearing capacity is not seriously affected. It turns out that removing redundant cable clamps in the dense part of the LSHCS is feasible.

Vacuum-Filtration-Assisted Ice-Templated Freeze Drying for Preparing Capacitive Graphene Aerogel for Thermal Management

Graphene aerogel (GA) is widely used in electronic devices owing to its light weight, elasticity, and excellent thermal conductivity. GA has been prepared using various methods. However, the preparation process is complex and the thickness is hard to control, which limits its application. There is an urgent need for a new and simple method to fabricate graphene aerogel. Herein, we describe a novel strategy for fabricating GA via a vacuum filtration–ice template freeze-drying method. The stability of graphene oxide slurry (GOS) was changed by using hydrochloric acid (HCl, 0.12 mol/L), and then GA was quickly obtained by vacuum filtration–ice template freeze drying and graphitization. The obtained GA reveals a symmetrical hyperbolic structure in the vertical direction, giving it excellent thermal and electrical conductivity and good compression performance. The electrical conductivity is up to 14.87 S/cm and the thermal conductivity is 1.29 W m−1 K−1 when the density is 36 mg cm−3. The pressure only needs 0.013 MPa when the strain of GA is 50%. GA has considerable potential for the application of supercapacitors owing to the high conductivity and low density.

Non-colorable hierarchically hyperbolic groups

We exhibit a hierarchically hyperbolic group for which no hierarchically hyperbolic structure is colorable, answering an (implicit) question of Durham–Minsky–Sisto.