Curved Space Research Articles

Probabilistic deconstruction of a theory of gravity, part II: Curved space

We propose that the underlying context of holographic duality and the Ryu-Takayanagi formula is that the volume measure of spacetime is a probability measure constrained by quantum dynamics. We define quantum stochastic processes using joint quantum distributions which are realized in a quantum system as expectation values of products of projectors. In anti-de Sitter JT gravity, we show that Einstein’s equations arise from the evolution of probability under the quantum stochastic process induced by the boundary, with the area of compactified space in the gravitational theory identified as a probability density evolving under the quantum process. Extrapolating these and related results in flat JT gravity found in [SciPost Phys. 15, 174 (2023)], we conjecture that general relativity arises in the semi-classical limit of the evolution of probability with respect to quantum stochastic processes.

Weight 11 Compactly Supported Cohomology of Moduli Spaces of Curves

Abstract We study the weight 11 part of the compactly supported cohomology of the moduli space of curves ${\mathcal{M}}_{g,n}$, using graph complex techniques, with particular attention to the case $n = 0$. As applications, we prove new nonvanishing results for the cohomology of ${\mathcal{M}}_{g}$, and exponential growth with $g$, in a wide range of degrees.

Automatic design of mutation parameter adaptation for differential evolution

In this paper the Efficient Global Optimization algorithm is applied to design the adaptation strategy for mutation parameter in Differential Evolution. The adaptation strategy is represented as a Taylor series, to allow exploring a search space of different curves. The tuning of the adaptation is performed on the L-NTADE algorithm using the benchmark of Congress on Evolutionary Computation competition on single-objective numerical optimization 2017. The experimental results show that the discovered dependence between the success rate and the parameter in current-to-pbest mutation strategy allows improving the algorithm performance in various cases.

Topology of moduli spaces of curves and anabelian geometry in positive characteristic

Abstract In the present paper, we study a new kind of anabelian phenomenon concerning the smooth pointed stable curves in positive characteristic. It shows that the topology of moduli spaces of curves can be understood from the viewpoint of anabelian geometry. We formulate some new anabelian-geometric conjectures concerning tame fundamental groups of curves over algebraically closed fields of characteristic $p>0$ from the point of view of moduli spaces. The conjectures are generalized versions of the Weak Isom-version of the Grothendieck conjecture for curves over algebraically closed fields of characteristic $p>0$ which was formulated by Tamagawa. Moreover, we prove that the conjectures hold for certain points lying in the moduli space of curves of genus $0$ .

A Schwarz lemma of harmonic maps into metric spaces

<p>We established a Schwarz lemma for harmonic maps from Riemannian manifolds to metric spaces of curvature bounded above in the sense of Alexandrov. We adopted the gradient estimate technique which was based on Zhang-Zhu's maximum principle. In particular, when the domain manifold was a hyperbolic surface, the energy of any conformal harmonic maps into $ \operatorname{CAT}(-1) $ spaces were bounded from above uniformly.</p>

Rationalizing Euclidean Assemblies of Hard Polyhedra from Tessellations in Curved Space.

Entropic self-assembly is governed by the shape of the constituent particles, yet apriori prediction of crystal structures from particle shape alone is nontrivial for anything but the simplest of space-filling shapes. At the same time, most polyhedra are not space filling due to geometric constraints, but these constraints can be relaxed or even eliminated by sufficiently curving space. We show using MonteCarlo simulations that the majority of hard Platonic solids self-assemble entropically into space-filling crystals when constrained to the surface volume of a 3-sphere. As we gradually decrease curvature to "flatten" space and compare the local morphologies of crystals assembling in curved and flat space, we show that the Euclidean assemblies can be categorized as either remnants of tessellations in curved space (tetrahedra and dodecahedra) or nontessellation-based assemblies caused by large-scale geometric frustration (octahedra and icosahedra).

Determination of the value for the bending angle of the light deflection and the impact parameter in Einstein’s theory

The gravitational deflection of the ray of light is one of the famous demonstrations that space-time is curved in the General Theory of Relativity predicted by A Einstein, and later demonstrated by different astronomical experimental observations in solar eclipses. In this work, the deflection of light and the impact parameter in the gravitational field of the Sun are calculated through the solutions to the geodesic equations by the methods of the elliptic functions of Jacobi and Weierstrass, around a central gravitational mass, using the solution found by Schwarzschild to the equations of the gravitational field in the General Theory of Relativity. The results obtained were subjected to a comparative study of the deflection of the light ray experimentally. From this analysis, it is demonstrated that the theoretical results obtained by the Weierstrass and Jacobi methods are in the range of the experimental results calculated during the eclipses of the years 1919 to 1973, demonstrating once again the validity of the General Theory of Relativity that all gravitational mass curves space-time.

Study on articular surface morphology of atlantoaxial lateral mass based on differential manifold

ObjectivesTo propose a surface reconstruction algorithm based on a differential manifold (a space with local Euclidean space properties), which can be used for processing of clinical images and for modeling of the atlantoaxial joint. To describe the ideal anatomy of the lateral atlantoaxial articular surface by measuring the anatomical data.MethodsComputed tomography data of 80 healthy subjects who underwent cervical spine examinations at our institution were collected between October 2019 and June 2022, including 46 males and 34 females, aged 37.8 ± 5.1 years (28–59 years). A differential manifold surface reconstruction algorithm was used to generate the model based on DICOM data derived by Vision PACS system. The lateral mass articular surface was measured and compared in terms of its sagittal diameter, transverse diameter, articular surface area, articular curvature and joint space height.ResultsThere was no statistically significant difference between left and right sides of the measured data in normal adults (P > 0.05). The atlantoaxial articular surface sagittal diameter length was (15.83 ± 1.85) and (16.22 ± 1.57) mm on average, respectively. The transverse diameter length of the articular surface was (16.29 ± 2.16) and (16.49 ± 1.84) mm. The lateral articular surface area was (166.53 ± 7.69) and (174.48 ± 6.73) mm2 and the curvature was (164.03 ± 5.27) and (153.23 ± 9.03)°, respectively. The joint space height was 3.05 ± 0.11mm, respectively. There is an irregular articular space in the lateral mass of atlantoaxial, and both upper and lower surfaces of the articular space are concave. A sagittal plane view shows that the inferior articular surface of the atlas is mainly concave above; however, the superior articular surface of the axis is mainly convex above. In the coronal plane, the inferior articular surface of the atlas is mostly concave above, with most concave vertices located in the medial region, and the superior articular surface of the axis is mainly concave below, with most convex vertices located centrally and laterally.ConclusionA differential manifold algorithm can effectively process atlantoaxial imaging data, fit and control mesh topology, and reconstruct curved surfaces to meet clinical measurement applications with high accuracy and efficiency; the articular surface of the lateral mass of atlantoaxial mass in normal adults has relatively constant sagittal diameter, transverse diameter and area. The distance difference between joint spaces is small, but the shape difference of articular surfaces differs greatly.

Tammes Problem and Contact Number for Spheres in Spaces of Constant Curvature

Tammes Problem and Contact Number for Spheres in Spaces of Constant Curvature

Physical Time and Human Time

This paper is a comment on both Bunamano and Rovelli (Bridging the neuroscience and physics of time arXiv:2110.01976. (2022)) and Gruber et al. (in Front. Psychol. Hypothesis Theory, 2022) and which discuss the relation between physical time and human time. I claim here, contrary to many views discussed there, that there is no foundational conflict between the way physics views the passage of time and the way the mind/brain perceives it. The problem rather resides in a number of misconceptions leading either to the representation of spacetime as a timeless Block Universe, or at least that physically relevant universe models cannot have preferred spatial sections. The physical expanding universe can be claimed to be an Evolving Block Universe with a time-dependent future boundary, representing the dynamic nature of the way spacetime develops as matter curves spacetime and spacetime tells matter how to move. This context establishes a global direction of time that determines the various local arrows of time. Furthermore time passes when quantum wave function collapse takes place to an eigenstate; during this process, information is lost. The mind/brain acts as an imperfect clock, which coarse-grains the physical passage of time along a world line to determine the experienced passage of time, because neural processes take time to occur. This happens in a contextual way, so experienced time is not linearly related to physical time in general. Finally I point out that the Universe is never infinitely old: its future endpoint always lies infinitely faraway in the future.

Airy structures and deformations of curves in surfaces

AbstractAn embedded curve in a symplectic surface defines a smooth deformation space of nearby embedded curves. A key idea of Kontsevich and Soibelman is to equip the symplectic surface with a foliation in order to study the deformation space . The foliation, together with a vector space of meromorphic differentials on , endows an embedded curve with the structure of the initial data of topological recursion, which defines a collection of symmetric tensors on . Kontsevich and Soibelman define an Airy structure on to be a formal quadratic Lagrangian which leads to an alternative construction of the tensors of topological recursion. In this paper, we produce a formal series on which takes it values in , and use this to produce the Donagi–Markman cubic from a natural cubic tensor on , giving a generalisation of a result of Baraglia and Huang.

The Dunkl oscillator on a space of nonconstant curvature: An exactly solvable quantum model with reflections

We introduce the Dunkl-Darboux III oscillator Hamiltonian in N dimensions as a λ−deformation of the Dunkl oscillator. This deformation is interpreted as the introduction of a non-constant curvature on the underlying space or, equivalently, as a quadratic position-dependent mass for the Dunkl oscillator. This new ND quantum model is shown to be exactly solvable, and its eigenvalues and eigenfunctions are explicitly presented. It is shown that in the 2D case both Darboux III and Dunkl oscillators can be coupled with a constant magnetic field, thus giving rise to two new exactly solvable quantum systems in which the effect of the λ−deformation and of the Dunkl derivatives on the Landau levels can be studied. Finally, the full 2D Dunkl-Darboux III oscillator is coupled with the magnetic field and shown to define an exactly solvable Hamiltonian, where the interplay between the λ−deformation and the magnetic field is explicitly illustrated.

The high-dimensional cohomology of the moduli space of curves with level structures II: punctures and boundary

The high-dimensional cohomology of the moduli space of curves with level structures II: punctures and boundary

Manifolds with harmonic Weyl tensor and nonnegative curvature operator of the second kind

We show that a compact manifold with harmonic Weyl tensor and nonnegative curvature operator of the second kind is globally conformally equivalent to a space of positive constant curvature or is isometric to a flat manifold. Moreover, we obtain that a compact manifold with harmonic curvature and nonnegative curvature operator of the second kind is isometric to a space of constant curvature. These improve Kashiwada's result that Riemannian manifolds with harmonic curvature tensor for which the curvature operator of the second kind is nonnegative and positive are locally symmetric spaces and constant curvature spaces, respectively.

On Galois Action on the Inertia Stack of Moduli Spaces of Curves

We establish that the geometric action of the absolute Galois group \mathrm{Gal}(\bar{\mathbb{Q}}/\mathbb{Q}) on the étale fundamental group of moduli spaces of curves induces a Galois action on its stack inertia subgroups, and that this action is given by cyclotomy conjugacy. This result extends the special case of inertia without étale factorisation previously established by the authors. It is here obtained in the general case by comparing deformations of Galois actions. Since the cyclic stack inertia corresponds to the first level of the stack stratification of the space, this result, by analogy with the arithmetic of the Deligne–Mumford stratification, opens the way to a systematic Galois study of the stack inertia through the corresponding stratification of the moduli stack.

Optimal diameter estimate of three-dimensional Ricci limit spaces

In this note, we prove that positive scalar curvature can pass to three dimensional Ricci limit spaces of non-negative Ricci curvature when it splits off a line. As a corollary, we obtain an optimal Bonnet-Myers type upper bound. Moreover, we obtain a similar statement in all dimensions for Alexandrov spaces of non-negative curvature.

Exponential concentration for geometric-median-of-means in non-positive curvature spaces

Exponential concentration for geometric-median-of-means in non-positive curvature spaces

Vibration analysis of radial tire using the 3D rotating hyperelastic composite REF based on ANCF

In order to analyze the influence of tread structure and rotating angular speed on the vibration characteristics of radial tire, a 3D rotating hyperelastic composite REF (3D-RHCREF) model is proposed. Different form uniform cylindrical tread in other REF models, the tread is simplified as a hyperelastic composite revolving shell, whose discretization and parameterization are achieved by using three-order segmented Bézier curve and complete-gradient ANCF space curved trapezoid element. In the proposed 3D-RHCREF, the generalized inertial force with the centrifugal force and Coriolis force are obtained by introducing a rotation matrix into the framework of ANCF. Based on Kirchhoff-Love theory and hyperelastic constitutive relation, the generalized hyperelastic force of composite tread is derived and the compensation of initial generalized hyperelastic force is proposed. According to the D'Alembert principle, the nonlinear dynamic differential equations are established, then the modal equations are derived on the basis of the equilibrium position. By comparing with the FEM model of radial tire, the validity of the proposed model and calculation method is verified. The 3D-RHCREF is used to analyze the influences of internal pressure, tread pressure sharing ratio, belt structure and rotating angle speed on the vibration characteristics of radial tire. The vibration frequency will increase with the increase of internal pressure and the decrease of tread pressure sharing ratio; different belt structures of tread will cause changes in vibration frequency; the radial vibration frequency will be split into two traveling wave frequencies in two directions with the increase of rotating angular speed, and the circumferential/lateral vibration frequencies will almost only increase slightly with the increase of rotating angular speed.

Holomorphic differential forms on moduli spaces of stable curves

We prove that the space of holomorphic p-forms on the moduli space overline{mathcal {M}}_{g,n} of stable curves of genus g with n marked points vanishes for p=14, 16, 18 unconditionally and also for p=20 under a natural assumption in the case g=3. This result is consistent with the Langlands program and it is obtained by applying the Arbarello–Cornalba inductive approach to the cohomology of moduli spaces.

Spinorial Representation of Submanifolds in a Product of Space Forms

We present a method giving a spinorial characterization of an immersion into a product of spaces of constant curvature. As a first application we obtain a proof using spinors of the fundamental theorem of immersion theory for such target spaces. We also study special cases: we recover previously known results concerning immersions in mathbb {S}^2times mathbb {R} and we obtain new spinorial characterizations of immersions in mathbb {S}^2times mathbb {R}^2 and in mathbb {H}^2times mathbb {R}. We then study the theory of H=1/2 surfaces in mathbb {H}^2times mathbb {R} using this spinorial approach, obtain new proofs of some of its fundamental results and give a direct relation with the theory of H=1/2 surfaces in mathbb {R}^{1,2}.