Space Research Articles

Exploration of morphological coherence in open clusters with a "core-shell'' structure

The morphology of open clusters plays a major role in the study of their dynamic evolution. The study of their morphological coherence, namely, the three-dimensional (3D) difference between the inner and outer morphologies of open clusters, allows us to obtain a better understanding of the morphological evolution of open clusters. We aim to investigate the morphological coherence of 132 open clusters with up to 1 kpc from the Sun in the three-dimensional (3D) space within the heliocentric cartesian coordinate frame. The 132 open clusters have a 3D core-shell structure and conform to the ellipsoidal model, with all of them coming from a catalog of publicly available clusters in the literature. We employed the ellipsoid fitting method to delineate the 3D spatial structure of the sample clusters, while using the morphological dislocation (MD) defined in our previous work and the ellipticity ratio (ER) of the clusters' inner and outer structures to characterize the morphological coherence of the sample clusters. The results show an inverse correlation between the ER of the sample clusters and the number of their members, indicating that sample clusters with a much more elliptical external morphology than internal shape generally tend to host a large number of members. Meanwhile, a slight shrinking of the MD of the sample clusters with their members' number may shed light on the significant role of the gravitational binding of the sample clusters in maintaining their morphological stability. Moreover, there are no correlations between the MD and ER of the sample clusters and their age. They are also not significantly correlated with the X-axis, the Y-axis, their orbital eccentricities, and the radial and vertical forces on them. However, the ER of the sample clusters displays some fluctuations in the distributions between it and the above covariates, implying that the morphologies of the sample clusters are sensitive to the external environment if sample effects are not taken into account. Finally, the analysis of the 3D spatial shapes of sample clusters with a small ER or a large ER demonstrates that the number of members lays an important foundation for forming a dense internal system for sample clusters. At the same time, the MD of the sample clusters can serve well as an indicator of their morphological stability, which is built upon a certain amount of member stars. We present a new insight into the morphological coherence of open clusters, attributed to the combination of their gravitational binding capacity and external environmental perturbations.

Three-Dimensional Quality Assessment of Urban Underground Space Resource Based on Multiple Geological Environmental Factors

With the rapid advancement of urbanization, the development and utilization of urban underground space resource (UUSR) has become one of the dominant features. However, in certain areas, the development of UUSR may cause disasters and accidents, such as ground collapse, settlements, and tunnel water gushing. Geological environmental factors (GEFs) are recognized as the fundamental constraining factor of UUSR development. In this paper, quality based on GEFs is defined to assess the development difficulty degree of UUSR. A 3D assessment framework is proposed based on 3D geological modelling and the interval continuous mathematical model (ICMM). The subjective and objective joint weight method of analytic hierarchy process and entropy weight method (AHP–EWM) is utilized to determine the weight of each indicator. The quality index (QI) of each spatial node of the 3D geological model is calculated by the ICMM mathematical model. A case study conducted in the Jiangbei New District of Nanjing, China, serves as a demonstration of the UUSR assessment. The results clearly illustrate the 3D distribution characteristics of the quality in the study area, offering valuable insights for future 3D urban underground space planning.

Fermions, quantum gravity, and holography in two dimensions

We study a model comprising N flavors of Kähler Dirac fermion propagating on a triangulated two-dimensional disk which is constrained to have a negative average bulk curvature. Dirichlet boundary conditions are chosen for the fermions. Quantum fluctuations of the geometry are included by summing over all possible triangulations consistent with these constraints. We show in the limit N→∞ that the partition function is dominated by a regular triangulation of two-dimensional hyperbolic space. We use strong coupling expansions and Monte Carlo simulation to show that in this limit boundary correlators of the fermions have a power law dependence on boundary separation as one expects from holography. However, we argue that this behavior breaks down for any finite number of massive fields in the thermodynamic limit and quantum fluctuations of the bulk geometry drive the theory into a nonholographic phase. In contrast, for massless fermions, we find evidence that the boundary is conformal even for finite N. This is consistent with theoretical results in quantum Liouville theory. Published by the American Physical Society 2024

Investigation of Automotive LiDAR Vision in Rain from Material and Optical Perspectives

With the emergence of autonomous functions in road vehicles, there has been increased use of Advanced Driver Assistance Systems comprising various sensors to perform automated tasks. Light Detection and Ranging (LiDAR) is one of the most important types of optical sensor, detecting the positions of obstacles by representing them as clusters of points in three-dimensional space. LiDAR performance degrades significantly when a vehicle is driving in the rain as raindrops adhere to the outer surface of the sensor assembly. Performance degradation behaviors include missing points and reduced reflectivity of the points. It was found that the extent of degradation is highly dependent on the interface material properties. This subsequently affects the shapes of the adherent droplets, causing different perturbations to the optical rays. A fundamental investigation is performed on the protective polycarbonate cover of a LiDAR assembly coated with four classes of material—hydrophilic, almost-hydrophobic, hydrophobic, and superhydrophobic. Water droplets are controllably dispensed onto the cover to quantify the signal alteration due to the different droplets of various sizes and shapes. To further understand the effects of droplet motion on LiDAR signals, sliding droplet conditions are simulated using numerical analysis. The results are validated with physical optical tests, using a 905 nm laser source and receiver to mimic the LiDAR detection mechanism. Comprehensive explanations of LiDAR performance degradation in rain are presented from both material and optical perspectives. These can aid component selection and the development of signal-enhancing strategies for the integration of LiDARs into vehicle designs to minimize the impact of rain.

Automated detection and tracking of photovoltaic modules from 3D remote sensing data

This study addresses the growing demand for increased performance and reliability of photovoltaic (PV) installations by developing innovative monitoring technologies. The strategy consists of flying an unmanned aerial vehicle (UAV) equipped with a dual camera (RGB and thermal) over the PV plant of interest, followed by the generation of photogrammetric 3D models derived from the overlapped aerial images. The resulting datasets involve orthoimages and point clouds by processing RGB and thermal imagery. The key contribution of this study is twofold: (1) the thermal image mapping on dense and high-resolution point clouds that represent the status and geometry of PV solar modules, and (2) the automatic identification of individual solar panels in 3D space and their thermal characterization along their oriented surface. Then, the vector layer of each PV panel is projected onto the 3D thermal point cloud to extract the thermal values associated with each panel. To evaluate the capability of the proposed method, it was replicated in different scenarios, considering rural and urban environments with different light conditions and PV structures. The results demonstrate the robustness of our method, which achieves a remarkably high detection rate, around 99.12% of true positives, and a low false positive rate, close to 0.88%. Consequently, this method means an advance over previous work by proposing a comprehensive and automated solution for individual and highly detailed monitoring of each solar panel from 3D remotely sensed data. This study opens up new frontier research related to real-time monitoring of photovoltaic modules, an inspection of solar photovoltaic cells, the simulation of solar resources and forecasting, the development of digital twins, solar radiation modelling, and analysis of modular floating solar farms under wave motion.

Real-time position and pose prediction for a self-propelled undulatory swimmer in 3D space with artificial lateral line system.

This study aims to investigate the feasibility of using an Artificial Lateral Line system for predicting the real-time position and pose of an undulating swimmer with Carangiform swimming patterns. We established a 3D Computational Fluid Dynamics simulation to replicate the swimming dynamics of a freely swimming mackerel under various motion parameters, calculating the corresponding pressure fields. Using the simulated lateral line data, we trained an artificial neural network to predict the centroid coordinates and orientation of the swimmer. A comprehensive analysis was further conducted to explore the impact of sensor quantity, distribution, noise amplitude and sampling intervals of the Artificial Lateral Line array on predicting performance. Additionally, to quantitatively assess the reliability of the localization network, we trained another neural network to evaluate error magnitudes for different input signals. These findings provide valuable insights for guiding future research on mutual sensing and schooling in underwater robotic fish.

FragGrow: A Web Server for Structure-Based Drug Design by Fragment Growing within Constraints.

Fragment growing is an important ligand design strategy in drug discovery. In this study, we present FragGrow, a web server that facilitates structure-based drug design by fragment growing. FragGrow offers two working modes: one for growing molecules through the direct replacement of hydrogen atoms or substructures and the other for growing via virtual synthesis. FragGrow works by searching for suitable fragments that meet a set of constraints from an indexed 3D fragment database and using them to create new compounds in 3D space. The users can set a range of constraints when searching for their desired fragment, including the fragment's ability to interact with specific protein sites; its size, topology, and physicochemical properties; and the presence of particular heteroatoms and functional groups within the fragment. We hope that FragGrow will serve as a useful tool for medicinal chemists in ligand design. The FragGrow server is freely available to researchers and can be accessed at https://fraggrow.xundrug.cn.

Synergistic Effects of Exogenous Nutrient Ions on the Real-Time Cadmium Extraction by an Accumulator

Bidens tripartita L. is a cadmium (Cd) accumulator. However, the real-time influx or efflux of Cd2+ around its root apex has not yet been performed. The object of this experiment was to compare the roles of added ions in solution on dynamic Cd extraction by B. tripartita root tip. Quartz sand was used to grow the seedling of B. tripartite. The Cd concentrations of all samples were determined by using ICP-OES after digestion. The Cd2+ influx around the root apex was measured in vivo, i.e., using non-invasive micro-test technology (NMT). The results showed that the Cd2+ influx was found to be decreased by 35.9%, 43.7%, 20.6%, and 57.5% under 10 μM Cd combined with high content Ca2+, Mg2+, Fe3+, or K+ (16 mM, 8 mM, 0.5 mM, 18 mM, respectively), compared to that under 10 μM Cd stress. But Cd treatments with low content ions with 0.05 mM Fe3+ or 0.5 mM S increased the Cd2+ influx in roots by 20.5% and 34.6%, respectively. It was also found that Cd treatment with high concentrations of Ca2+ or K+ increased the shoot biomass of B. tripartita seedlings. Chl a and b contents were significantly decreased in the Cd treatments with low concentrations of Fe3+ or S compared to those under Cd stress alone, and the dehydrogenase activity of the roots decreased in the treatment of Cd with 0.05 mM Fe3+ or 0.5 mM S. Our results indicate that the addition of 0.05 mM Fe3+ or 0.5 mM S promoted Cd2+ influx and Cd uptake by B. tripartita. Unlike traditional measurement, the Cd2+ movements of three-dimensional space around the B. tripartita root tip had been performed by NMT. It was suggested that the effects of S and Fe3+ on the remediation potential of B. tripartita need to be further researched in the future. The results of this study provided a real-time and micro-dynamic theoretical basis for phytoremediation mechanisms.

Effects of triaxial deformation on the fission barrier in the Z = 118 − 120 nuclei* *Supported by the National Natural Science Foundation of China (Grant Nos. 12205076 and 12047504), the China Postdoctoral Science Foundation (Grant No. 2020M670012), and the Launching Fund of Henan University of Technology (Grant No. 2021BS047).

By using potential energy surface (PES) calculations in the three-dimensional space (β 2, γ, β 4) within the framework of the macroscopic-microscopic model, the fission trajectory and fission barrier for Z = 118(Og), 119, 120 nuclei has been systematically investigated. The calculated PES includes macroscopic liquid-drop energy, microscopic shell correction and pairing correction. Taking the 294Og176 nucleus as an example, we discuss the next closed shell after Z = 82 and N = 126 with the calculated Woods–Saxon single-particle levels. Then, the results of PES in 294Og is illustrated from the (X, Y) scale to the (β 2, γ) scale. The γ degree of freedom reveals the shape evolution clearly during the fission process. The structure near the minimum and saddle point of the PES in the Z = 118, 119, 120 nuclei is demonstrated simultaneously. Based on the potential energy curves, general trends of the evolution of the fission barrier heights and widths are also studied. The triaxial deformation in these superheavy mass regions plays a vital role in the first fission barrier, showing a significant reduction in both triaxial paths. In addition, the model-dependent fission barriers of proton-rich nuclei 295Og, 296119, and 297120 are analyzed briefly. Our studies could be valuable for synthesizing the superheavy new elements in the forthcoming HIAF and other facilities.

Exploring movement reproduction in three-dimensional space: A comparative pilot study among people with visual impairment

Teaching physiotherapy in a higher education institution for visually impaired students requires adjustments. The aim of this study was to investigate the influence of vision on the quality of upper limb movement reproduction. We analysed the replication of a simple movement along three spatial axes: X, Y, and Z, corresponding to movements in horizontal, vertical, and depth dimensions, respectively. The findings reveal that all three groups (blind, visual impairment, and sighted) exhibited significantly larger errors when reproducing movements along the Z-axis compared to the X and Y axes. Both blind and sighted participants obtained similar results, while visually impaired participants significantly differed from the sighted participants. Specifically, visually impaired participants demonstrated overproduction, compared to sighted participants. These results indicate that individuals in the visually impaired group, who still retain some visual ability, face greater challenges in reproducing movements across all three spatial dimensions compared to both the blind and sighted groups. Therefore, when teaching the practice of physiotherapy, it appears essential to take into account the degree of vision (visual impairment and blindness) and particular attention must be paid to visually impaired students.

Multi-plane denoising diffusion-based dimensionality expansion for 2D-to-3D reconstruction of microstructures with harmonized sampling

Acquiring reliable microstructure datasets is a pivotal step toward the systematic design of materials with the aid of integrated computational materials engineering (ICME) approaches. However, obtaining three-dimensional (3D) microstructure datasets is often challenging due to high experimental costs or technical limitations, while acquiring two-dimensional (2D) micrographs is comparatively easier. To deal with this issue, this study proposes a novel framework called ‘Micro3Diff’ for 2D-to-3D reconstruction of microstructures using diffusion-based generative models (DGMs). Specifically, this approach solely requires pre-trained DGMs for the generation of 2D samples, and dimensionality expansion (2D-to-3D) takes place only during the generation process (i.e., reverse diffusion process). The proposed framework incorporates a concept referred to as ‘multi-plane denoising diffusion’, which transforms noisy samples (i.e., latent variables) from different planes into the data structure while maintaining spatial connectivity in 3D space. Furthermore, a harmonized sampling process is developed to address possible deviations from the reverse Markov chain of DGMs during the dimensionality expansion. Combined, we demonstrate the feasibility of Micro3Diff in reconstructing 3D samples with connected slices that maintain morphologically equivalence to the original 2D images. To validate the performance of Micro3Diff, various types of microstructures (synthetic or experimentally observed) are reconstructed, and the quality of the generated samples is assessed both qualitatively and quantitatively. The successful reconstruction outcomes inspire the potential utilization of Micro3Diff in upcoming ICME applications while achieving a breakthrough in comprehending and manipulating the latent space of DGMs.

The impact of mobility costs on cooperation and welfare in spatial social dilemmas

From over-exploitation of resources to urban pollution, sustaining well-being requires solving social dilemmas of cooperation. Often such dilemmas are studied assuming that individuals occupy fixed positions in a network or lattice. In spatial settings, however, agents can move, and such movements involve costs. Here we investigate how mobility costs impact cooperation dynamics. To this end, we study cooperation dilemmas where individuals are located in a two-dimensional space and can be of two types: cooperators–or cleaners, who pay an individual cost to have a positive impact on their neighbours–and defectors–or polluters, free-riding on others’ effort to sustain a clean environment. Importantly, agents can pay a cost to move to a cleaner site. Both analytically and through agent-based simulations we find that, in general, introducing mobility costs increases pollution felt in the limit of fast movement (equivalently slow strategy revision). The effect on cooperation of increasing mobility costs is non-monotonic when mobility co-occurs with strategy revision. In such scenarios, low (yet non-zero) mobility costs minimise cooperation in low density environments; whereas high costs can promote cooperation even when a minority of agents initially defect. Finally, we find that heterogeneity in mobility cost affects the final distribution of strategies, leading to differences in who supports the burden of having a clean environment.

Online elasticity estimation and material sorting using standard robot grippers

AbstractStiffness or elasticity estimation of everyday objects using robot grippers is highly desired for object recognition or classification in application areas like food handling and single-stream object sorting. However, standard robot grippers are not designed for material recognition. We experimentally evaluated the accuracy with which material properties can be estimated through object compression by two standard parallel jaw grippers and a force/torque sensor mounted at the robot wrist, with a professional biaxial compression device used as reference. Gripper effort versus position curves were obtained and transformed into stress/strain curves. The modulus of elasticity was estimated at different strain points and the effect of multiple compression cycles (precycling), compression speed, and the gripper surface area on estimation was studied. Viscoelasticity was estimated using the energy absorbed in a compression/decompression cycle, the Kelvin-Voigt, and Hunt-Crossley models. We found that (1) slower compression speeds improved elasticity estimation, while precycling or surface area did not; (2) the robot grippers, even after calibration, were found to have a limited capability of delivering accurate estimates of absolute values of Young’s modulus and viscoelasticity; (3) relative ordering of material characteristics was largely consistent across different grippers; (4) despite the nonlinear characteristics of deformable objects, fitting linear stress/strain approximations led to more stable results than local estimates of Young’s modulus; and (5) the Hunt-Crossley model worked best to estimate viscoelasticity, from a single object compression. A two-dimensional space formed by elasticity and viscoelasticity estimates obtained from a single grasp is advantageous for the discrimination of the object material properties. We demonstrated the applicability of our findings in a mock single-stream recycling scenario, where plastic, paper, and metal objects were correctly separated from a single grasp, even when compressed at different locations on the object. The data and code are publicly available.

Time-varying drainage basin development and erosion on volcanic edifices

Abstract. The erosional state of a landscape is often assessed through a series of metrics that quantify the morphology of drainage basins and divides. Such metrics have been well explored in tectonically active environments to evaluate the role of different processes in sculpting topography, yet relatively few works have applied these analyses to radial landforms such as volcanoes. We quantify drainage basin geometries on volcanic edifices of varying ages using common metrics (e.g., Hack's law, drainage density, and number of basins that reach the edifice summit, as well as basin hypsometry integral, length, width, relief, and average topographic slope). Relating these measurements to the log-mean age of activity for each edifice, we find that drainage density, basin hypsometry, basin length, and basin width quantify the degree of erosional maturity for these landforms. We also explore edifice drainage basin growth and competition by conducting a divide mobility analysis on the volcanoes, finding that young volcanoes are characterized by nearly uniform fluvial basins within unstable configurations that are more prone to divide migration. As basins on young volcanoes erode, they become less uniform but adapt to a more stable configuration with less divide migration. Finally, we analyze basin spatial geometries and outlet spacing on edifices, discovering an evolution in radial basin configurations that differ from typical linear mountain ranges. From these, we present a novel conceptual model for edifice degradation that allows new interpretations of composite volcano histories and provides predictive quantities for edifice morphologic evolution.

Single‐Pulse‐Driven Frame Printing of Chromatic Pixels in Lithium Niobate Crystal

AbstractHighly efficient and programmable writing of multidimensional optical data is of great value for next‐generation high‐throughput information technologies but has been rarely achieved. Here, a one‐step frame printing of chromatic pixels in lithium niobate crystal by using a single ultrafast laser pulse is reported. In this strategy, a phase superposition‐based spatial light modulation strategy is applied to split a single ultrafast laser pulse into multiple son pulses with designated optical properties and spatial distribution patterns. It is demonstrated that these son pulses allow for massively creating micro‐amorphous phase transition zones with on‐demand structural features that can modulate the intrinsic birefringence of the crystal matrix and generate wavelength‐selective interference in the visible band to form pixel‐level chromatic patterns, namely, single‐pulse‐driven frame color printing. The created chromatic pixels can be encoded into computer‐recognizable data arrays to play a role in high‐efficiency multidimensional information recording. The presented approach enables fast and programmable information batch writing in 3D space and can serve as a versatile tool boosting next‐generation information optics.

Registration of apartments and office spaces in 3D land administration – A case study in Croatia

Due to the needs of modern society, cadastral systems should be designed to support three dimensional (3D) spatial data. One of many possible approaches for implementing a 3D cadastre, in countries such as Croatia, is to establish a Building Register as a transitional register between 2D and 3D cadastres, where data about buildings and infrastructure would be collected, gradually adapted to the data model of 3D cadastre, and finally migrated to the 3D cadastre database. Sources for establishing the Building Register can be based on the records of state surveys, the register of administrative units, the land book, data managed by local and regional self-government units, data from construction documentation according to special regulations in the field of spatial planning, data managed by the building and infrastructure managers, as well as from other sources. A significant element of the 3D cadastre would be the inclusion of representations of buildings and units of use, as well as public utility infrastructure and complex spatial real-life entities (e.g., bridges, tunnels, overpasses, underpasses, overlapping of constructed objects with natural facilities, large shopping malls with more underground and overhead floors etc.). This paper presents a conceptual model of a 3D cadastre in Croatia by establishing the Building Register with focus on unit of use of real properties, namely apartments and office spaces. The paper also summarizes the current situation regarding the Croatian Land Administration System (LAS) and proposes a conceptual model for modelling unit of use of real properties. Additionally, a proposal is made herein to assign unique identifiers to buildings and their parts in a logical manner which would be intuitive and clear to citizens of Croatia, citizens of European Union and citizens of Croatia neighboring countries. The proposed methodology of determining unique identifiers could provide the means for easier navigation in 3D space and better understanding of spatial information by lay citizens, by institutions or emergency services.

Partial differential equation modelling of the solid–liquid phase process in iron sintering and interface control with backstepping method

Due to the highly continuous temporal and spatial distribution characteristics of the sintering process, the solid–liquid phase interface during sintering is uneven and discontinuous, which seriously affects the quality of sintered ore. In order to achieve real-time and effective control of the solid–liquid interface in the sintering system, this study proposes a spatiotemporally distributed partial differential equation (PDE) thermal system with Stefan interface boundaries. Slice the 3D sintering mixing space and establish a discrete 1D space multi-distributed system to achieve distributed control of the sintering thermal system. Based on this system, a distributed boundary controller is designed using the backstepping method to drive the solid–liquid phase interface to the desired position. This method fully adapts to the nonlinear problems of the sintering thermal system and can quickly respond to system changes. Based on real data from the sintering site, a multi-physics simulation model was designed to verify the correctness of the theoretical research. Comparing the simulation results with the actual sintering state shows that this control method can effectively control the position of the sintering solid–liquid interface, which plays a vital role in improving the quality of sinter and sintering efficiency.

In-silico Studies, Design, and Synthesis of Pyrimidine-linked Benzothiazoles for its Anticonvulsant Potential.

The objective of the study was to design and synthesize a series of N-(6-substituted-1,3-benzothiazole- 2-yl)-2-{[6-(3-substitutedphenyl)-5-cyano-2-sulfanylpyrimidine-4-yl)]amino}acetamide derivatives BPD (1-15) that contains key pharmacophores required for anticonvulsant action. The titled compounds (BPD 1-15) were synthesized by reacting 2-substituted-N-(6-chlorobenzo[d]thiazol2-yl)acetamide with 4-amino-6-(4-substituted phenyl)-2-mercapto pyrimidine 5-carbonitrile in the presence of potassium carbonate and dry acetone. The synthesized compounds BPD (1-15) were assessed in vivo by the maximum electric shock (MES) test and the subcutaneous pentylenetetrazol (scPTZ) test in mice. The neurotoxicity test was performed by the rotarod test. A molecular docking study of title compounds with a sodium channel receptor (PDB ID: 1BYY) was carried out using the SP Docking protocol of the Glide module of the Maestro. Pharmacophore modeling was used to qualitatively identify the chemical characteristics for ligand binding and their spatial configurations in the 3D space of the active site. Among the studied compounds, BPD-15 and BPD-5 compounds showed significant action in both the MES and scPTZ models, with no neurotoxicity. BPD-15 & BPD-5 were relatively safe in acute toxicity testing. Compounds BPD-15 and BPD-5 showed good dock scores of -6.434 and -6.191, respectively. Thus, the compounds BPD-15 and BPD-5 have shown a considerable affinity towards the sodium channel as compared to the standard drug Riluzole. Compound BPD-14 showed good drug compatibility, and compounds BPD-1, BPD-2, BPD-11, BPD-12, BPD-13, BPD-14, BPD-15 showed good ADME values.

Landscape Design and Green Space Perception Enhancement Based on Panoramic Imaging and Segnet

Green view rate is an intuitive evaluation criterion used for green space perception. Traditional research on green view rate is mostly calculated based on flat images, which cannot fully reflect the subjective human perception of green volume in three-dimensional space. Based on the panoramic image, we propose the concept of panoramic green perception rate, obtain spherical panoramic photos by panoramic camera, convert isometric cylindrical projection into isoprojective cylindrical projection, and use the convolutional neural network model based on semantic segmentation to automatically identify the area of vegetation to achieve automatic recognition and measurement of panoramic green perception rate. The results were compared with manual discrimination and showed that the average intersection ratio (mIoU) of greenery recognition using Dilated ResNet-105 was 62.53%, and the average difference with manual recognition was 9.17%.

A G-Modified Helmholtz Equation with New Expansions for the Earth’s Disturbing Gravitational Potential, Its Functionals and the Study of Isogravitational Surfaces

A G-Modified Helmholtz Equation with New Expansions for the Earth’s Disturbing Gravitational Potential, Its Functionals and the Study of Isogravitational Surfaces