Framework Of Geometry Research Articles

Applying nutritional ecology to optimize diets of crickets raised for food and feed.

Increasing yield is a primary goal of mass insect rearing for food and feed, and diet impacts insect life-history traits that affect yield, such as survival, development time and body size. However, experiments rarely test the nutritional requirements of insects from hatch to adulthood, and so little is known about how the full developmental macronutrient intake impacts the survival, growth and adult body size of mass-reared insects. Here, we applied the nutritional geometry framework and reared individual tropical house crickets (Gryllodes sigillatus) from hatch to adulthood on a wide range of protein : carbohydrate diets. We measured weekly food consumption, survival, development time to adulthood and adult body size and mass, and calculated a yield metric to extrapolate our individual-level results and predict how diet influences yield at the mass-rearing level. Yield was maximized on a 3P : 1C diet, as crickets fed this diet were most likely to develop into adults and grew maximum mass and body size. When provided with a choice between diets, crickets selected a relatively balanced 1.05P : 1C diet throughout development, but males consumed 17% more protein than females. Our results represent a crucial first step towards determining the optimal standard feed formulation required to maximize cricket farming yield.

Macronutrient interactions and models of obesity: Insights from nutritional geometry.

The global obesity epidemic results from a complex interplay of genetic and environmental factors, with diet being a prominent modifiable element driving weight gain and adiposity. Although excess intake of energetic macronutrients is implicated in causing obesity, ongoing debate centers on whether sugar or fat or both are driving the rising obesity rates. This has led to competing models of obesity such as the "Carbohydrate Insulin Model", the "Energy Balance Model", and the "Fructose Survival Hypothesis". Conflicting evidence from studies designed to focus on individual energetic macronutrients or energy rather than macronutrient mixtures underlies this disagreement. Recent research in humans and animals employing the nutritional geometry framework (NGF) emphasizes the importance of considering interactions among dietary components. Protein interacts with carbohydrates, fats, and dietary energy density to influence both calorie intake ("protein leverage") and, directly and indirectly, metabolic physiology and adiposity. Consideration of these interactions can help to reconcile different models of obesity, and potentially cast new light on obesity interventions.

Statistical and observation comparison of Weyl-type f(Q,T) models with the ΛCDM paradigm

We study the f(Q,T) gravity in the framework of Weyl geometry (known as Weyl-type f(Q,T) gravity), where Q denotes the non-metricity scalar, and T denotes the energy-momentum tensor trace. In this work, we consider the f(Q,T) model, which is defined as f(Q,T)=αQm+1+β6κ2T and investigating two scenarios: (I) m=0 (linear model) and (II)m≠0 (nonlinear model). For both scenarios, we find the explicit solution for the field equations by using the barotropic equation of state as p=wρ, where w is the equation-of-state (EoS) parameter. Further, we study the obtained solutions statistically using the Pantheon+ (Without SHOES Calibrated) dataset with 1701 data points. For both models, the best-fit values of model parameters for 1−σ and 2−σ confidence level. The higher Hubble constant values in both models emphasize the presence of Tension. We statistically compare our models to the ΛCDM model using χmin2, χred2, AIC, ΔAIC, BIC and ΔBIC. We also examine cosmological parameters such as deceleration and EoS parameters to determine the current acceleration expansion of the Universe. Furthermore, we test our model using Om diagnostic and compare it to the ΛCDM model to determine its dark energy profile. Finally, we draw the conclusion that statistically speaking, both linear and nonlinear models show good compatibility with the ΛCDM model.

Quantum geometrical properties of topological materials

The momentum space of topological insulators and topological superconductors is equipped with a quantum metric defined from the overlap of neighboring valence band states or quasihole states. We investigate the quantum geometrical properties of these materials within the framework of Dirac models and differential geometry. Their momentum space is found to be always a maximally symmetric space with a constant Ricci scalar, and the vacuum Einstein equation is satisfied in 3D with a finite cosmological constant. For linear Dirac models, several geometrical properties are found to be independent of the band gap, including a peculiar straight line geodesic, constant volume of the curved momentum space, and the exponential decay form of the nonlocal topological marker, indicating the peculiar yet universal quantum geometrical properties of these models.

Matter bounce cosmology within Finsler-Randers geometry: A comprehensive study of anisotropic influences

In this study, we explore the dynamics of matter bounce cosmology within the framework of Finsler-Randers geometry, focusing on the role of the Finslerian correction term η(t). By integrating Finsler geometry into cosmological models, we introduce anisotropic effects that significantly impact the evolution of the universe, particularly during the bounce phase. The research examines various cosmological parameters, including the deceleration (qη(t)), jerk (jη(t)), and snap (sη(t)) parameters, highlighting the influence of the Finsler correction on these key indicators. Our results demonstrate that the Finslerian framework leads to more complex and abrupt transitions in the universe's expansion dynamics compared to traditional Riemannian models. The study also reveals that the Finslerian correction intensifies the violations of energy conditions, such as the null energy condition (NEC), which are crucial for the occurrence of a successful bounce. Furthermore, the analysis of the squared sound speed vs2 indicates that the model's stability is highly sensitive to the choice of the Finslerian parameters, with certain configurations leading to instability during the bounce. Our findings underscore the unique contributions of Finsler geometry to cosmological models, offering deeper insights into the behavior of the universe under anisotropic influences and providing a potential avenue for addressing longstanding challenges in cosmology.

Engineering flexible machine learning systems by traversing functionally invariant paths

Contemporary machine learning algorithms train artificial neural networks by setting network weights to a single optimized configuration through gradient descent on task-specific training data. The resulting networks can achieve human-level performance on natural language processing, image analysis and agent-based tasks, but lack the flexibility and robustness characteristic of human intelligence. Here we introduce a differential geometry framework—functionally invariant paths—that provides flexible and continuous adaptation of trained neural networks so that secondary tasks can be achieved beyond the main machine learning goal, including increased network sparsification and adversarial robustness. We formulate the weight space of a neural network as a curved Riemannian manifold equipped with a metric tensor whose spectrum defines low-rank subspaces in weight space that accommodate network adaptation without loss of prior knowledge. We formalize adaptation as movement along a geodesic path in weight space while searching for networks that accommodate secondary objectives. With modest computational resources, the functionally invariant path algorithm achieves performance comparable with or exceeding state-of-the-art methods including low-rank adaptation on continual learning, sparsification and adversarial robustness tasks for large language models (bidirectional encoder representations from transformers), vision transformers (ViT and DeIT) and convolutional neural networks.

Performance analysis of joint transmission in TDD-based mm-wave networks with BS heights

Dynamic-time division duplexing (D-TDD) is used in millimeter-wave (mm-wave) networks to allocate time-frequency resources for both downlink (DL) and uplink (UL) transmissions. This scheduling mechanism adapts to varying data traffic to meet the different quality of service (QoS) requirements of heterogeneous services. However, blockage is the major bottleneck to millimeter wave (mm-Wave) networks since it causes signal outages. Moreover, the dense deployment of base stations (BSs) to reduce the signal outage problem may lead to severe interference due to solid line-of-sight (LoS) BSs. To solve the problem of less coverage, this paper utilizes a joint transmission mechanism in the DL of D-TDD-based mm-Wave networks. Further, examining the network performance also entails considering the elevation disparity between the Base Station (BS) and User Equipment (UE). The primary focus of this work is to study the effects of joint transmission in D-TDD-based DL mm-Wave networks with different BS heights and directional beamforming using the stochastic geometry framework. For D-TDD with Joint Transmission, numerical evaluations indicate that the SINR coverage probability reaches approximately 0.7 when the coverage radius (rc) is 100 m. Conversely, at a coverage radius of 200 m, the SINR coverage probability drops to around 0.25. The simulation results demonstrate using MATLAB 2019b tool, that D-TDD-based mm-Wave networks achieve superior coverage using the joint transmission scheme compared to single BS association. Besides, results show that optimal UL and DL split and load balancing are required to achieve better DL coverage.

Exploring the quintessential influence on shadows of black holes in Finsler-Hayward geometry

This research aims to shed light on how the quintessential dark energy influences the shadows cast by black holes by considering different kinds of surrounding accretion profiles within the framework of Finsler-Hayward geometry. We explore the impact of the Finsler parameter (η) and the quintessence parameter (ω) on fundamental properties of the black hole, such as its horizons, photon sphere, and impact parameter. By examining the observed characteristics of the black hole's shadow, including the photon ring and shadow intensities, we analyze how the quintessence affects these features under varying scenarios of matter accretion. Leveraging observational data from the Event Horizon Telescope, exclusively focusing on celestial bodies such as Sgr A* and M87*, we narrow down the range of the Finsler parameter (η) within our model of quintessence-infused Finsler Hayward black holes. By comparing our findings with models such as Finsler Schwarzschild, with and without the quintessence Finsler Hayward black holes, we observe that the quintessence's influence on the black hole's shadow surpasses the effects attributed to regularity. This study contributes to our insight into how the Finsler parameter and the quintessence of dark energy shape the shadows of black holes in diverse accretion environments, shedding light on their intricate nature and behavior, particularly in the context of Finsler Hayward geometry.

Generalized Finslerian Wormhole Models in f(R,T) Gravity

This article explores wormhole solutions within the framework of Finsler geometry and the modified gravity theory. Modifications in gravitational theories, such as f(R,T) gravity, propose alternatives that potentially avoid the exotic requirements. We derive the field equations from examining the conditions for Finslerian wormhole existence and investigate geometrical and material characteristics of static wormholes using a polynomial shape function in Finslerian space–time. Furthermore, we address energy condition violations for different Finsler parameters graphically. We conclude that the proposed models, which assume a constant redshift function, satisfy the necessary geometric constraints and energy condition violations indicating the presence of exotic matter at the wormhole throat. We also discuss the anisotropy factors of the wormhole models. The results are validated through analytical solutions and 3-D visualizations, contributing to the broader understanding of wormholes in Finsler-modified gravity contexts.

A continuum model for the mechanics of elastomeric sheets reinforced with extensible bidirectional fibers resistant to lateral pressure

We investigate the concurrent three-dimensional (in-plane and out-of-plane) deformations of fiber-reinforced composite (FRC) sheets undergoing lateral pressure. This involves the utilization of the Neo-Hookean strain energy model for the matrix material and computing the strain energy of bidirectional fibers by accounting for the stretching, bending, and twisting responses of the fibers. The kinematics of FRC are formulated within the framework of differential geometry on FRC surfaces, including the computations of the first and second gradient of deformation. By employing the variational principles, we derive the Euler equations describing the mechanics of the fiber–matrix composite system subjected to lateral pressure. The resulting three-dimensional continuum model theoretically predicts the deformation of the matrix material and it is found that the maximum deformation of matrix material occurs in the diagonal direction of the domain, yet, the center of the domain suffers weak in-plane deformation because of surface tension equilibrium. In addition, the stretching, bending, and twisting kinematics of fiber units are computed to investigate the effects of the individual fiber’s kinematics on the overall deformation of fiber meshwork. Lastly, it is found that the lateral pressure on the FRC surface induces fiber flexure in the vicinity of domain boundaries and fiber stretch inside the domain, corresponding to the intensified shrinking strain near the edges and stretching strain in the middle of the domain. The theoretical results provide phenomenologically meaningful insights into comprehending the damage patterns of the fiber-reinforced building material, the hemispherical dome shaping results of bamboo Poly (lactic) acid (PLA) composites and the out-of-plane deformation of woven fabric.

Macronutrient regulation in nymphs of the two-spotted cricket, Gryllus bimaculatus (Orthoptera: Gryllidae)

Crickets have been extensively studied in recent insect nutritional research, but it remains largely unexplored how they balance the intake of multiple nutrients. Here, we used the nutritional geometry framework to examine the behavioural and physiological regulation of dietary protein and carbohydrate in nymphs of the two-spotted cricket, Gryllus bimaculatus (Orthoptera: Gryllidae). Growth, intake, utilization efficiencies, and body composition were measured from the eighth instar nymphs that received either food pairs or single foods with differing protein and carbohydrate content. When food choices were available, crickets preferentially selected a carbohydrate-biased protein:carbohydrate (P:C) ratio of 1:1.74. During this nutrient selection, carbohydrate intake was more tightly regulated than protein intake. When confined to nutritionally imbalanced foods, crickets adopted a nutrient balancing strategy that maximized the nutrient intake regardless of the nutrient imbalance, reflecting their omnivorous feeding habit. Intake was significantly reduced when crickets were confined to the most carbohydrate-biased food (P:C = 1:5). When nutrients were ingested in excess of the requirements, the post-ingestive utilization efficiencies of these nutrients were down-regulated, thereby buffering the impacts of nutrient imbalances on body nutrient composition. Crickets reared on the most carbohydrate-biased food (P:C = 1:5) suffered delayed development and reduced growth. Our data provide the most accurate description of nutrient regulation in G. bimaculatus and lay the foundation for further nutritional research in this omnivorous insect.

Yang-Mills field from fuzzy sphere quantum Kaluza-Klein model

Using the framework of quantum Riemannian geometry, we show that gravity on the product of spacetime and a fuzzy sphere is equivalent under minimal assumptions to gravity on spacetime, an su2-valued Yang-Mills field Aμi and a real-symmetric-matrix valued Liouville-sigma model field hij for gravity on the fuzzy sphere. Moreover, a massless real scalar field on the product appears as a tower of scalar fields on spacetime, with one for each internal integer spin l representation of SU(2), minimally coupled to Aμi and with mass depending on l and the fuzzy sphere size. For discrete values of the deformation parameter, the fuzzy spheres can be reduced to matrix algebras M2j+1(ℂ) for j any non-negative half-integer, and in this case only integer spins 0 ≤ l ≤ 2j appear in the multiplet. Thus, for j = 1 a massless field on the product appears as a massless SU(2) internal spin 0 field, a massive internal spin 1 field and a massive internal spin 2 field, in mass ratio 0, 1,3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\sqrt{3} $$\\end{document} respectively, which we conjecture could arise in connection with an approximate SU(2) flavour symmetry.

TensoSDF: Roughness-aware Tensorial Representation for Robust Geometry and Material Reconstruction

Reconstructing objects with realistic materials from multi-view images is problematic, since it is highly ill-posed. Although the neural reconstruction approaches have exhibited impressive reconstruction ability, they are designed for objects with specific materials (e.g., diffuse or specular materials). To this end, we propose a novel framework for robust geometry and material reconstruction, where the geometry is expressed with the implicit signed distance field (SDF) encoded by a tensorial representation, namely TensoSDF. At the core of our method is the roughness-aware incorporation of the radiance and reflectance fields, which enables a robust reconstruction of objects with arbitrary reflective materials. Furthermore, the tensorial representation enhances geometry details in the reconstructed surface and reduces the training time. Finally, we estimate the materials using an explicit mesh for efficient intersection computation and an implicit SDF for accurate representation. Consequently, our method can achieve more robust geometry reconstruction, outperform the previous works in terms of relighting quality, and reduce 50% training times and 70% inference time. Codes and datasets are available at https://github.com/Riga2/TensoSDF.

Metric perturbations in noncommutative gravity

We use the framework of Hopf algebra and noncommutative differential geometry to build a noncommutative (NC) theory of gravity in a bottom-up approach. Noncommutativity is introduced via deformed Hopf algebra of diffeomorphisms by means of a Drinfeld twist. The final result of the construction is a general formalism for obtaining NC corrections to the classical theory of gravity for a wide class of deformations and a general background. This also includes a novel proposal for noncommutative Einstein manifold. Moreover, the general construction is applied to the case of a linearized gravitational perturbation theory to describe a NC deformation of the metric perturbations. We specifically present an example for the Schwarzschild background and axial perturbations, which gives rise to a generalization of the work by Regge and Wheeler. All calculations are performed up to first order in perturbation of the metric and noncommutativity parameter. The main result is the noncommutative Regge-Wheeler potential. Finally, we comment on some differences in properties between the Regge-Wheeler potential and its noncommutative counterpart.

Semi-Symmetric Metric Connections and Homology of CR-Warped Product Submanifolds in a Complex Space Form Admitting a Concurrent Vector Field

In this paper, we conduct a thorough study of CR-warped product submanifolds in a Kaehler manifold, utilizing a semi-symmetric metric connection within the framework of warped product geometry. Our analysis yields fundamental and noteworthy results that illuminate the characteristics of these submanifolds. Additionally, we investigate the implications of our findings on the homology of these submanifolds, offering insights into their topological properties. Notably, we present a compelling proof demonstrating that, under a specific condition, stable currents cannot exist for these warped product submanifolds. Our research outcomes contribute significant knowledge concerning the stability and behavior of CR-warped product submanifolds equipped with a semi-symmetric metric connection. Furthermore, this work establishes a robust groundwork for future explorations and advancements in this particular field of study.

Graph‐theoretical chirality measure and chirality–property relations for chemical structures with multiscale mirror asymmetries

AbstractChirality is an essential geometric property unifying small molecules, biological macromolecules, inorganic nanomaterials, biological microparticles, and many other chemical structures. Numerous chirality measures have attempted to quantify this geometric property of mirror asymmetry and to correlate these measures with physical and chemical properties. However, their utility has been widely limited because these correlations have been largely notional. Furthermore, chirality measures also require prohibitively demanding computations, especially for chiral structures comprised of thousands of atoms. Acknowledging the fundamental problems with quantification of mirror asymmetry, including the ambiguity of sign‐variable pseudoscalar chirality measures, we revisit this subject because of the significance of quantifying chirality for quantitative biomimetics and describing the chirality of nanoscale materials that display chirality continuum and scale‐dependent mirror asymmetry. We apply the concept of torsion within the framework of differential geometry to the graph theoretical representation of chiral molecules and nanostructures to address some of the fundamental problems and practical limitations of other chirality measures. Chiral gold clusters and other chiral structures are used as models to elaborate a graph‐theoretical chirality (GTC) measure, demonstrating its applicability to chiral materials with different degrees of chirality at different scales. For specific cases, we show that GTC provides an adequate description of both the sign and magnitude of mirror asymmetry. The direct correlations with macroscopic properties, such as chiroptical spectra, are enhanced by using the hybrid chirality measures combining parameters from discrete mathematics and physics. Taking molecular helices as an example, we established a direct relation between GTC and optical activity, indicating that this chirality measure can be applied to chiral metamaterials and complex chiral constructs.

Green grant-free power allocation for ultra-dense Internet of Things: A mean-field perspective

Grant-free access, in which each Internet-of-Things (IoT) device delivers its packets through a randomly selected resource without spending time on handshaking procedures, is a promising solution for supporting the massive connectivity required for IoT systems. In this paper, we explore grant-free access with multi-packet reception capabilities, with an emphasis on ultra-low-end IoT applications with small data sizes, sporadic activity, and energy usage constraints. We propose a power allocation scheme aimed at maximizing throughput while minimizing power consumption by considering the traffic and energy constraints of IoT devices. Our approach employs a stochastic geometry framework and mean-field game theory to model and analyze the mutual interference among active IoT devices. Additionally, we utilize a Markov chain model to capture and track the queue length of IoT devices, enabling the derivation of the transmission success probability at steady-state. The simulation results illustrate the optimal power allocation strategy and evaluate the proposed approach’s performance in terms of packet transmission success probability and average delay.

Secrecy Performance Enhancement Using Self-Interference Cancellation in Wireless Mutual Broadcast Networks for Proximity-Based Services.

With the increasing demand for data exchange between nearby devices in proximity-based services, enhancing the security of wireless mutual broadcast (WMB) networks is crucial. However, WMB networks are inherently vulnerable to eavesdropping due to the open broadcast nature of their communication. This paper investigates the improvement of secrecy performance in random-access-based WMB (RA-WMB) networks by integrating physical layer security (PLS) techniques with hybrid duplex (HBD) operations under a stochastic geometry framework. The HBD method balances half-duplex (HD) receiving and full-duplex (FD) transceiving, utilizing self-interference cancellation (SIC) to enhance PLS performance. Key operational parameters, including transmission probability (TxPr), friendly jammer density, and conditions for FD operation, are designed to maximize secrecy performance. The analytical and numerical results demonstrate significant improvements in PLS performance, with SIC playing a critical role, particularly in scenarios with dense legitimate nodes, and with TxPr adjusted to balance HD receiving and FD transceiving based on SIC imperfections. The proposed design principles provide a comprehensive framework for enhancing the security of WMB networks, addressing the complex interplay of interference and SIC in various network configurations.

Thermodynamic Geometry of Nonequilibrium Fluctuations in Cyclically Driven Transport.

Nonequilibrium thermal machines under cyclic driving generally outperform steady-state counterparts. However, there is still lack of coherent understanding of versatile transport and fluctuation features under time modulations. Here, we formulate a theoretical framework of thermodynamic geometry in terms of full counting statistics of nonequilibrium driven transports. We find that, besides the conventional dynamic and adiabatic geometric curvature contributions, the generating function is also divided into an additional nonadiabatic contribution, manifested as the metric term of full counting statistics. This nonadiabatic metric generalizes recent results of thermodynamic geometry in near-equilibrium entropy production to far-from-equilibrium fluctuations of general currents. Furthermore, the framework proves geometric thermodynamic uncertainty relations of near-adiabatic thermal devices, constraining fluctuations in terms of statistical metric quantities and thermodynamic length. We exemplify the theory in experimentally accessible driving-induced quantum chiral transport and Brownian heat pump.

Gravitational probe of ꝗuantum spacetime

A quest for phenomenological footprints of quantum gravity is among the central scientific tasks in the rising era of gravitational wave astronomy. We study gravitational wave dynamics within the noncommutative geometry framework, based on a Drinfeld twist and newly proposed noncommutative Einstein equation, and obtain the leading quantum correction to Regge-Wheeler potential up to first order in the noncommutativity parameter. By calculating the quasinormal mode frequencies we show that the noncommutative Schwarzschild black hole remains stable under axial gravitational perturbations.