Curvature Theory Research Articles

Effectiveness of Weyl gravity in probing quantum corrections to AdS black holes

Computing leading higher curvature contributions to thermodynamic quantities of AdS black hole is drastically simplified once the higher curvature terms are expressed in terms of powers of Weyl tensor by applying proper field redefinitions, avoiding the usual complications caused by higher derivative Gibbons-Hawking-York term or surface counterterms. We establish the method by computing the Euclidean action of general rotating anti–de Sitter (AdS) black holes in five-dimensional quadratic curvature theories with or without supersymmetry and verifying the results numerically. Our result is the state of the art for charged rotating AdS black holes in five-dimensional minimal gauged supergravity including corrections from all three supersymmetric curvature squared terms. Our approach facilitates precision tests in the AdS/CFT correspondence and should be applicable in diverse dimensions. Published by the American Physical Society 2024

Flux ruled surfaces and the magnetic curves obtained from the curvature theory

In this paper, our aim is to research the change in the magnetic field using the curvature theory of ruled surfaces. For this, we examine magnetic curves of the curve whose existence is guaranteed from the derivative formulas of the rotation frame. The Killing vector fields and Lorentz forces of these magnetic curves are calculated. Additionally, correlations between curvatures are obtained. Flux ruled surfaces are obtained and the conditions for developable are achieved. Finally, examples of the magnetic flux developable ruled surfaces are given.

Using instantaneous invariants of the curvature theory in motion generation synthesis of 4R mechanisms with higher-order coupler point kinematics

This paper aims at exploiting the unique characteristics of instantaneous invariants and pole locations in synthesizing motion generation four bar (4R) planar mechanisms with higher-order coupler point kinematics (i.e. specifying coupler point velocity and/or acceleration in addition to its position and coupler orientation). The proposed methodology targets to ease the complexity of such design problems by reducing the number of non-linear governing equations that are solved to generate the desired dyads of the final 4R mechanism design. The new approach integrates graphical and analytical techniques to synthesize 4R mechanisms with prescribed coupler point position, velocity, and acceleration. Two case studies are presented; the first presents the synthesis of a motion generation 4R with 3 prescribed coupler point positions and 2 prescribed coupler point velocities; the second presents the synthesis of a motion generation 4R with 2 prescribed coupler point positions, 2 prescribed coupler point velocities, and 1 prescribed coupler point acceleration. When compared to classical kinematic-based approaches, the proposed method prominently reduces the size and complexity of the set of governing non-linear equations.

Extraplanetary system under modified gravity

Abstract We investigated the properties of exoplanets using a modified gravity law. We then devised a gravitational formula that linked the radius, mass, semimajor axis, and stellar mass of the exoplanet. This formula is then validated by taking into account four different astronomical organizations (WASP, HAT, TOI, and HATS). These findings are compared to those in the exoplanet.edu database. The findings corroborated the observational data. The Einstein curvature theory was created by incorporating planet spin into Newton's planet dynamics. As a result, the spin of the planet is connected to its gravity.

A steerable robot walker driven by two actuators

Abstract This article describes a robot walker based on a new single degree-of-freedom six-bar leg mechanism that provides rectilinear, non-rotating, movement of the foot. The walker is statically stable and requires only two actuators, one for each side, to provide effective walking movement on a flat surface. We use Curvature Theory to design a four-bar linkage with a flat-sided coupler curve and then adds a translating link so that walker foot follows this coupler curve in rectilinear movement. A prototype walker was constructed that weighs 1.6 kg, is 180 mm tall, and travels at 162 mm/s. This is an innovative legged robot that has a simple reliable design.

Instantaneous kinematics of a planar two-link open chain in complex plane

This research aims to investigate the instantaneous kinematics of the terminal link of a planar two-link open chain using the canonical coordinate system of planar two-parameter motion. It is based on the complex-number technique. We review the higher-order instantaneous invariants of this motion of the terminal link with reference to the first and second-order instantaneous invariants. We apply these instantaneous invariants in complex number form to curvature theory. Finally, we give some examples by choice of different parameters and exhibit the comparisons.

Elastic properties of liquid crystals

The paper presents elastic properties of liquid crystals, especially their nematic phase. Anoutline of a development of theories describing these properties is given. The model describing liquidcrystal as continuous medium is formulated. The curvature theory of elasticity of nematic liquidcrystals based on Zocher – Oseen – Frank’s approach is discussed in details. In this way, essentialelastic properties of nematics as well their relative material parameters are described.Keywords: liquid crystals, nematics, elastic properties, deformations, continuous medium

Greenhouse light and CO2 regulation considering cost and photosynthesis rate using i-nsGA Ⅱ

Natural resources, such as light and CO2, are required to increase crop yields in protected agriculture. Developing efficient regulation methods to reduce costs and increase yields has become a major challenge for actual production. This work proposed a framework for meeting this challenge and supporting the management of environmental regulation. Photosynthesis rate was used as the yield index, and a photosynthesis rate model was constructed using support vector regression. Moreover, a regulation cost function for light and CO2 was constructed. Then, the non-inferior solutions (NIS) for environmental regulation were determined by an improved non-dominated sorting genetic algorithm Ⅱ (i-nsGA Ⅱ). The curvature theory was applied to calculate the optimal solution in the NIS set. A set of seedling tomato data was used to demonstrate the usefulness of the proposed method. Compared with the other two regulation methods (saturation regulation method and U-chord method), the results indicated that the proposed method had the best performance. About 80% of the maximum photosynthesis rate could be obtained at a regulation cost of about 51%. This research presents a new strategy for optimizing the greenhouse environment, which can be applied as a knowledge-informed expert system to manage environmental regulation.

Tooth Profile Construction and Experimental Verification of Non-Circular Gear Based on Double Arc Active Design

The double arc pitch curve is a novel way for creating a non-circular gear tooth profile. Aiming at the problems of the rounding of non-integer teeth and the nonlinear equation system in the construction of double circular arc tooth profile, an active design method based on planetary gear transmission kinematics and gear-meshing theory is proposed. By studying the double arc pitch curve of the three engagement positions, combined with the uniform theory of the relative curvature of surfaces, the distribution characteristics of the solution of the nonlinear equation system are analyzed in depth, leading to the optimal solution of the characteristic parameter values of the pitch curve. Combined with UG modeling and ADAMS motion simulation, the gear engagement force and the angular velocity of the planetary gear around the center are compared and analyzed in the case of integer tooth ratio and non-integer tooth profile. The results show that the engagement force of the gear pair constructed by integer teeth is reduced by about 17%. The optimal combination of the numbers of teeth is determined by orthogonal experiments, on which trial production and hydraulic load tests are based. The results reveal that the torque of the product at a rated speed is about 2.6% higher than that of foreign prototypes, which verifies the effectiveness of the design method proposed in this paper.

A Magnetically Controlled Guidewire Robot System with Steering and Propulsion Capabilities for Vascular Interventional Surgery

Magnetically manipulated interventional robotic systems offer outstanding advantages for improving vascular interventions, including minimizing radiation exposure to physicians and increasing the controllability of magnetic interventional devices in hard‐to‐reach vessels. However, automatic control of magnetic guidewires (MGs) is still challenging in terms of modeling of guidewires and trajectory planning. Herein, a magnetically controlled guidewire robotic system (MCGRS) with steering and propulsion capabilities is proposed based on adequate modeling and trajectory planning methods. The steering kinematics of MG is first modeled by constant curvature theory. Then, a continuum mechanics model is built to predict the deformation of the magnetic tip by combining the dipole model and the Cosserat‐rod model. Moreover, a trajectory planning algorithm is developed to navigate the MG through vessels. Furthermore, trajectory following experiments within three vascular phantoms confirm that the proposed model and algorithm are reliable and capable of navigating the MG through the desired trajectory. Finally, two extra navigation experiments are implemented in 3D vascular phantom, which show that the MCGRS can be remotely controlled to manipulate the MG to actively steer and reach the target site. The system and methods will build the foundation for automatic control of MG and help to improve the autonomy.

Partition function of a volume of space in a higher curvature theory

Partition function of a volume of space in a higher curvature theory

High-energy properties of the graviton scattering in quadratic gravity

We obtain the matter-graviton scattering amplitude in the gravitational theory of quadratic curvature, which has {R}_{mu nu}^2 term in the action. Unitarity bound is not satisfied because of the existence of negative norm states, while an analog of unitarity bound for S-matrix unitarity holds due to the cancelation among the positive norm states and negative norm ones in the unitarity summation in the optical theorem. The violation of unitarity bound is a counter example of Llewellyn Smith’s conjecture on the relation between tree-level unitarity and renormalizability. We have recently proposed a new conjecture that an analog of the unitarity bound for S-matrix unitarity gives the equivalent conditions to those for renormalizability. We show that the gravitational theory of quadratic curvature is a nontrivial example consistent with our conjecture.

Supercomputers against strong coupling in gravity with curvature and torsion

Many theories of gravity are spoiled by strongly coupled modes: the high computational cost of Hamiltonian analysis can obstruct the identification of these modes. A computer algebra implementation of the Hamiltonian constraint algorithm for curvature and torsion theories is presented. These non-Riemannian or Poincaré gauge theories suffer notoriously from strong coupling. The implementation forms a package (the ‘Hamiltonian Gauge Gravity Surveyor’ – HiGGS) for the xAct tensor manipulation suite in Mathematica. Poisson brackets can be evaluated in parallel, meaning that Hamiltonian analysis can be done on silicon, and at scale. Accordingly HiGGS is designed to survey the whole Lagrangian space with high-performance computing resources (clusters and supercomputers). To demonstrate this, the space of ‘outlawed’ Poincaré gauge theories is surveyed, in which a massive parity-even/odd vector or parity-odd tensor torsion particle accompanies the usual graviton. The survey spans possible configurations of teleparallel-style multiplier fields which might be used to kill-off the strongly coupled modes, with the results to be analysed in subsequent work. All brackets between the known primary and secondary constraints of all theories are made available for future study. Demonstrations are also given for using HiGGS – on a desktop computer – to run the Dirac–Bergmann algorithm on specific theories, such as Einstein–Cartan theory and its minimal extensions.

A high efficiency CO2 concentration interval optimization method for lettuce growth

The decline in carbon fertilization effects has shifted scientific focus toward the efficient and suitable regulation of CO2 concentration ([CO2]) for plant growth. In this study, the rapid A/CO2 response curve (RAC) data of lettuce were analyzed statistically under nine photosynthetic photon flux densities (PPFDs) and four temperatures. An efficient CO2 supplementation interval acquisition method based on the frequency distribution characteristics of RACs was proposed. The characteristic subsections of jumping were obtained depending on the frequency distribution of RACs. The cumulative contribution rate (CCR) of the characteristic subsections were >97 %, which showed the efficiency of the method. Additionally, U-chord curvature theory was used to simultaneously obtain the optimal regulated [CO2] for the same RAC curves, and the results showed that the [CO2] obtained by U-chord length were all within the interval obtained by the method, which proved the rationality of the method. The [CO2] interval supplement improved the daily CO2 exchange rate by 20.27 % and 21.64 % at 150 and 200 μmol·m−2·s−1, and increased the lettuce fresh biomass by 26.78 % at 150 μmol·m−2·s−1. Based on the interval of [CO2] efficient utilization regulation at various temperatures and PPFDs, a genetic algorithm-support vector regression model was built with R2 of the model was >0.84 and the root mean square error was <35.2256 μmol·mol−1. In conclusion, the [CO2] interval obtained by this method has a positive effect on lettuce growth. This work provides a new method for obtaining high-efficiency supplementary concentration of CO2 during the growth of lettuce.

Entropic curvature on graphs along Schrödinger bridges at zero temperature

Lott-Sturm-Villani theory of curvature on geodesic spaces has been extended to discrete graph spaces by C. L{\'e}onard by replacing W2-Wasserstein geodesics by Schr{\"o}odinger bridges in the definition of entropic curvature [23, 25, 24]. As a remarkable fact, as a temperature parameter goes to zero, these Schr{\"o}dinger bridges are supported by geodesics of the space. We analyse this property on discrete graphs to reach entropic curvature on discrete spaces. Our approach provides lower bounds for the entropic curvature for several examples of graph spaces: the lattice Z n endowed with the counting measure, the discrete cube endowed with product probability measures, the circle, the complete graph, the Bernoulli-Laplace model. Our general results also apply to a large class of graphs which are not specifically studied in this paper. As opposed to Erbar-Maas results on graphs [27, 10, 11], entropic curvature results of this paper imply new Pr{\'e}kopa-Leindler type of inequalities on discrete spaces, and new transport-entropy inequalities related to refined concentration properties for the graphs mentioned above. For example on the discrete hypercube {0, 1} n and for the Bernoulli Laplace model, a new W2 -- W1 transport-entropy inequality is reached, that can not be derived by usual induction arguments over the dimension n. As a surprising fact, our method also gives improvements of weak transport-entropy inequalities (see [28, 15]) associated to the so-called convex-hull method by Talagrand [38].

First order corrections to the black hole thermodynamics in higher curvature theories of gravity

In modified theories of gravity, higher curvature terms may be added to the Einstein-Hilbert action. Conventionally, the effects of the higher curvature terms on the black hole thermodynamics are rather difficult to obtain. In this paper, we show that, at least at the first order level, the corrections to the thermodynamics of the Schwarzschild black hole can be easily obtained, without solving the modified black hole metric. We examine some specific examples, which produces the known results in the literature, as well as those previously unknown. Furthermore, we find some nonperturbative exact results by generalizing the first-order analysis. In particular, a novel relation between the Euclidean integrals of the Einstein--Hilbert term and the higher curvature terms is found and proved.

Black holes in the limiting curvature theory of gravity

We discuss a recently proposed limiting curvature theory of gravity and its application to the problem of singularities inside black holes. In this theory the growth of the curvature is suppressed by specially chosen inequality constraints included in the gravity action. We consider a case of a spherically symmetric four-dimensional black hole and demonstrated that imposed curvature constraints modify a solution in the black hole interior. Instead of forming the curvature singularity the modified metric describes a space which is exponentially expanding in one direction and oscillating the other two directions. The space–time is complete and its polynomial curvature invariants are uniformly bounded.

Stable bound orbits around static Einstein-Gauss-Bonnet black holes

As is well-known, asymptotically flat, static and spherically symmetric black holes do not admit stable bound orbits of massive/massless particles outside the horizon in higher-dimensional Einstein gravity. However, for massive particles, this dramatically changes in higher curvature theories. We clarify the parameter range such that there exist stable bound orbits in $d$-dimensional Einstein-Gauss-Bonnet theories for $6\leq d\leq 9$. In particular, we show the existence of the lower bound of the Gauss-Bonnet coupling constant below which stable bound orbits cease to exist. Moreover, we also find that for AdS-black holes in the theories, there can exist two stable circular orbits outside the horizon.

Discrete curvature on graphs from the effective resistance* *Funding: KD was supported by The Alan Turing Institute under the EPSRC Grant EP/N510129/1. RL acknowledges support from the EPSRC Grant Nos. EP/V013068/1 and EP/V03474X/1.

This article introduces a new approach to discrete curvature based on the concept of effective resistances. We propose a curvature on the nodes and links of a graph and present the evidence for their interpretation as a curvature. Notably, we find a relation to a number of well-established discrete curvatures (Ollivier, Forman, combinatorial curvature) and show evidence for convergence to continuous curvature in the case of Euclidean random graphs. Being both efficient to approximate and highly amenable to theoretical analysis, these resistance curvatures have the potential to shed new light on the theory of discrete curvature and its many applications in mathematics, network science, data science and physics.

Brief note on Thurston geometries in 3D quadratic curvature theories

We show that Thurston geometries are solutions to a large class of 3D quadratic curvature theories, where New Massive Gravity, which was studied in arXiv:2104.00754, is a special case.