Framework Of Geometry Research Articles

Oriented Calmness and Sweeping Process Dynamics

Daniilidis and Drusviatskiy, in 2017, extended the celebrated Kurdyka–Łojasiewicz inequality from definable functions to definable multivalued maps by establishing that the coderivative mapping admits a desingularization around every critical value. As was the case in the gradient dynamics, this desingularization yields a uniform control of the lengths of all bounded orbits of the corresponding sweeping process. In this paper, working outside the framework of o-minimal geometry, we characterize the existence of a desingularization for the coderivative in terms of the behavior of the sweeping process orbits and the integrability of the talweg function. These results are close in spirit with the ones in Bolte et al., 2010, in which characterizations for the desingularization of the (sub)gradient of functions is obtained. Funding: A. Daniilidis was supported by the Austrian Science Fund [Grant FWF P-36344N], Fondo Nacional de Desarrollo Científico y Tecnológico [Grant 1211217], and Centro de Modelamiento Matemático [Grant FB210005]. S. Tapia was supported by Agencia Nacional de Investigación y Desarrollo (Chile)-Programa de Fortalecimiento de Capital Humano Académico Doctorado Nacional [Grant 2018-21181905].

Hensel minimality: Geometric criteria for ℓ-h-minimality

Recently, Cluckers et al. developed a new axiomatic framework for tame non-Archimedean geometry, called Hensel minimality. It was extended to mixed characteristic together with the author. Hensel minimality aims to mimic o-minimality in both strong consequences and wide applicability. In this paper, we continue the study of Hensel minimality, in particular focusing on [Formula: see text]-h-minimality and [Formula: see text]-h-minimality, for [Formula: see text] a positive integer. Our main results include an analytic criterion for [Formula: see text]-h-minimality, preservation of [Formula: see text]-h-minimality under coarsening of the valuation and [Formula: see text]-dimensional geometry.

Log-aesthetic curves: Similarity geometry, integrable discretization and variational principles

In this paper, we consider a class of plane curves called log-aesthetic curves and their generalization which are used in computer aided geometric design. In the framework of similarity geometry, those curves are characterized as invariant curves under the integrable flow on plane curves governed by the Burgers equation. They also admit a variational formulation leading to the stationary Burgers equation as the Euler-Lagrange equation. As an application of the formulation, we propose a discretization of these curves and the associated variational principle which preserves the underlying integrable structure. We finally present algorithms for generating discrete log-aesthetic curves for given G1 data based on similarity geometry. Our method is able to generate S-shaped discrete curves with an inflection as well as C-shaped curves according to the boundary condition. The resulting discrete curves are regarded as self-adaptive discretization and thus high-quality even with the small number of points. Through the continuous representation, those discrete curves provide a flexible tool for the generation of high-quality shapes.

Almost complex parallelizable manifolds: Kodaira dimension and special structures

We study the Kodaira dimension of a real parallelizable manifold M, with an almost complex structure J in standard form with respect to a given parallelism. For X=(M,J)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$X = (M, J)$$\\end{document} we give conditions under which kod(X)=0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\,\ extrm{kod}\\,}}(X) = 0$$\\end{document}. We provide examples in the case M=G×G\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$M = G \ imes G$$\\end{document}, where G is a compact connected real Lie group. Finally we describe geometrical properties of real parallelizable manifolds in the framework of statistical geometry.

A fairness‐aware task offloading method in edge‐enabled IIoT with multi‐constraints using AGE‐MOEA and weighted MMF

SummaryBy providing distributed and ultra‐low‐latency communication between industrial devices and resource components, the Industrial Internet of Things (IIoT) is at the forefront of a new trend. Such a distributed paradigm is viewed as a collection of autonomous computing resources utilized by multiple heterogeneous devices to achieve higher‐quality interconnection and data exchange. However, stringent requirements of exceptional service and fairness guarantees pose many formidable challenges. To this end, this study investigates the aforementioned concerns in an integrated manner and further proposes a fairness‐aware task offloading method, called FOIMAM. Specifically, the ‐norm is introduced to accommodate the Pareto plane under the non‐Euclidean geometry framework while the evaluation and elimination of low‐quality solutions are completed based on survival scores. Particularly, the fairness requirements are formulated as a multi‐constraint problem and resolved using weighted max‐min fairness. Eventually, numerical results indicate that the proposed method brings substantial improvement in both service efficiency and fairness guarantees.

UAV-Aided Post-Disaster Cellular Networks: A Novel Stochastic Geometry Approach

Motivated by the need for ubiquitous and reliable communications in post-disaster emergency management systems (EMSs), we hereby present a novel and efficient stochastic geometry (SG) framework. This mathematical model is specifically designed to evaluate the quality of service (QoS) experienced by a typical ground user equipment (UE) residing either inside or outside a generic area affected by a calamity. In particular, we model the functioning terrestrial base stations (TBSs) as an inhomogeneous Poisson point process (IPPP), and assume that a given number of uniformly distributed unmanned aerial vehicles (UAVs) equipped with cellular transceivers is deployed in order to compensate for the damage suffered by some of the existing TBSs. The downlink (DL) coverage probability is then derived based on the maximum average received power association policy and the assumption of Nakagami- <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</i> fading conditions for all wireless links. The proposed numerical results show insightful trends in terms of coverage probability, depending on: distance of the UE from the disaster epicenter, disaster radius, quality of resilience (QoR) of the terrestrial network, and fleet of deployed ad-hoc aerial base stations (ABSs). The aim of this paper is therefore to prove the effectiveness of vertical heterogeneous networks (VHetNets) in emergency scenarios, which can both stimulate the involved authorities for their implementation and inspire researchers to further investigate related problems.

Correlating the Nonlinear Roughening and Optical Properties of Anatase Thin Films—A Fractal Geometric Approach

AbstractThe anatase phase of TiO2 is highly suitable for photocatalytic application consequently, prerequisite an unpretentious scale‐independent understanding of its roughening‐facilitated surface topography dynamics. Herein, the nonlinear roughening in anatase thin films and its correlation with the nonlinear trend of optical properties in the framework of fractal geometry are investigated. The self‐affine nature of analyzed surfaces is confirmed from the autocorrelation function. The dynamic scaling exponents indicate the existence of Kardar–Parisi–Zhang scaling in surface growth. In addition, the trend of generalized Hurst exponent and mass exponent indicates insignificant multifractal characteristics in the analyzed surfaces. Moreover, fractal analysis explains and aids to description of roughening trend from stereometric and Minkowski functionals analysis. Furthermore, the significance of fractal dimension for probing the surface roughening is validated from the principal component analysis. Consequently, variation in optical bandgap and linear refractive index are investigated in regards to the fractal dimension and root mean‐squared surface slope and regression equations are proposed for tuning of bandgap.

Implicit contact dynamics and Hamilton-Jacobi theory

In this paper, we introduce implicit Hamiltonian dynamics in the framework of contact geometry in two different ways: first, we introduce classical implicit Hamiltonian dynamics on a contact manifold, followed by evolution Hamiltonian dynamics. In the first case, implicit contact Hamiltonian dynamics is defined as a Legendrian submanifold of a tangent contact space, whilst the implicit evolution dynamic is understood as a Lagrangian submanifold of a certain symplectic space embedded into the tangent contact space. To conclude, we propose a geometric Hamilton-Jacobi theory for both of these formulations.

Quantum geometry of expectation values

We propose a framework for the quantum geometry of expectation values over arbitrary sets of operators and establish a link between this geometry and the eigenstates of Hamiltonian families generated by these operators. We show that the boundary of expectation value space corresponds to the ground state, which presents a natural bound that generalizes Heisenberg's uncertainty principle. To demonstrate the versatility of our framework, we present several practical applications, including providing a stronger nonlinear quantum bound that violates the Bell inequality and an explicit construction of the density functional. Our approach provides an alternative time-independent quantum formulation that transforms the linear problem in a high-dimensional Hilbert space into a nonlinear algebro-geometric problem in a low dimension, enabling us to gain new insights into quantum systems.

Impact of Multi-Connectivity on Channel Capacity and Outage Probability in Wireless Networks

Multi-connectivity facilitates higher throughput, shorter delay, and lower outage probability for a user in a wireless network. Considering these promises, a rationale policy for a network operator would be to implement multi-connectivity for all of its users. In this paper, we investigate whether the promises of multi-connectivity also hold in such a setting where all users of a network are connected through multiple links. In particular, we consider a network where every user connects to its k closest base stations. Using a framework of stochastic geometry and probability theory, we obtain analytic expressions for per-user throughput and outage probability of $k$-connectivity networks under several failure models. In contrast to the conclusions of previous research, our analysis shows that per-user throughput decreases with increasing k. However, multi-connected networks are more resilient against failures than single connected networks as reflected with lower outage probability and lead to higher fairness among the users. Consequently, we conclude that rather than implementing multi-connectivity for all users, a network operator should consider it for its users who would benefit from additional links the most, e.g., cell edge users.

Male reproductive traits are differentially affected by dietary macronutrient balance but unrelated to adiposity

Dietary factors influence male reproductive function in both experimental and epidemiological studies. However, there are currently no specific dietary guidelines for male preconception health. Here, we use the Nutritional Geometry framework to examine the effects of dietary macronutrient balance on reproductive traits in C57BL/6 J male mice. Dietary effects are observed in a range of morphological, testicular and spermatozoa traits, although the relative influence of protein, fat, carbohydrate, and their interactions differ depending on the trait being examined. Interestingly, dietary fat has a positive influence on sperm motility and antioxidant capacity, differing to typical high fat diet studies where calorie content is not controlled for. Moreover, body adiposity is not significantly correlated with any of the reproductive traits measured in this study. These results demonstrate the importance of macronutrient balance and calorie intake on reproductive function and support the need to develop specific, targeted, preconception dietary guidelines for males.

Trajectories of astroparticles in pseudo-Finsler spacetime with the most general modified dispersion

Finsler geometry is a natural and fundamental generalization of Riemann geometry, and is a tool to research Lorentz invariance violation. We find the connection between the most general modified dispersion relation and a pseudo-Finsler structure, and then we calculate the arrival time delay of astroparticles with different modified dispersion relations in the framework of Finsler geometry. The result suggests that the time delay is irrelevant with the exact form of the modified dispersion relation. If the modified term becomes 0 when E=p, there is no arrival time difference, otherwise the time delays only depend on the Lorentz violation scale and the order at which the Lorentz invariance breaks.

Connes' integration and Weyl's laws

This paper deals with some questions regarding the notion of integral in the framework of Connes' noncommutative geometry. First, we present a purely spectral theoretic construction of Connes' integral. This answers a question of Alain Connes. We also deal with the compatibility of Dixmier traces with Lebesgue's integral. This answers another question of Alain Connes. We further clarify the relationship of Connes' integration with Weyl's laws for compact operators and Birman–Solomyak's perturbation theory. We also give a “soft proof” of Birman–Solomyak's Weyl's law for negative order pseudodifferential operators on closed manifold. This Weyl's law yields a stronger form of Connes' trace theorem. Finally, we explain the relationship between Connes' integral and semiclassical Weyl's law for Schrödinger operators, including for (fractional) Schrödinger operators on Euclidean spaces and on noncommutative manifolds. We thus get a neat links between noncommutative geometry and semiclassical analysis.

Neutronic and Thermal-Fluidic Analyses for an Additive Manufactured Reactor With SiC Matrix and TRISO Particle Fuel

Abstract Additive manufacturing (AM) is a transformational digital manufacturing technology featured with rapidity, customizability, precision, and economy, which is fundamentally altering the way components are designed and manufactured. AM enables the freedom of design, and makes full use of complexity of geometry which “comes for free”. Applying AM technology to nuclear industry can yield advanced reactor designs with function and structure matched for the best thermal, fluidic and mechanical performance. In this work, an AM-informed reactor core design with silicon carbide (SiC) matrix and tri-structural isotropic (TRISO) particle fuel is proposed and analyzed. The core is an integrated 3D-printed SiC bulk with helical cruciform coolant channels, and the UO2-bearing TRISO fuel particles are dispersed in the bulk. A multiphysics analysis framework for irregular geometry is developed to analyze and further optimize the reactor design. The TRISO particle positions are generated with discrete element method (DEM). The Reactor Monte Carlo code (RMC) and the commercial computational fluid dynamics (CFD) software star-ccm+ are used for the neutronic and thermal-fluidic analyses, respectively. RMC simulates the neutron transport to predict the effective multiplication factor and power distribution. star-ccm+ calculates the flow and heat transfer in coolant channels and heat conduction in solid matrix with the power distribution as the heat source. Preliminary results show that the power peaking factor FQ decreases below 1.65, the heat transfer area increases by 30.3% and the fuel peaking temperature decreases by 25 K. The optimized AM-informed design enjoys better neutronic and thermal-fluidic performance than those with regular geometry.

On the Uplink SINR Meta Distribution of UAV-Assisted Wireless Networks

This letter studies the signal-to-interference-plus-noise (SINR) meta distribution of uplink transmission of UAV-enabled wireless networks with inversion power control. Within a framework of stochastic geometry, the Matern cluster process (MCP) is used to model the locations of users and UAVs. Conditional success probability and moments are derived to compute the exact expression and moment matching approximation of SINR meta distribution (beta approximation). Specifically, the effect of the power control compensation factor and UAV altitude are studied. Our numerical results show that UAV altitude has a higher impact on the system reliability than transmit power at low values of SINR threshold since users benefit more from establishing line-of-sight (LoS) links with UAVs.

Deducing the structural framework of the Southern Faghur-Siwa Basin, northern Western Desert, Egypt using potential field data

The Faghur-Siwa Basin is a large sedimentary basin in the westernmost region of the Egyptian Western Desert. The northern portion of this basin has been hydrocarbon productive for the last few decades, making it a subject of frequent geological and geophysical studies. The current study aims to outline the subsurface extent, geometry and structural framework of Southern Faghur-Siwa Basin. The study introduces an approach of using large data set to deduce the structural setting of a sedimentary basin and its applicability to large-scale tectonic features in similar regions worldwide. This study uses satellite-derived potential field data, data of deep wells penetrating the basement, and previous geological studies. The Gaussian filter with a cut-off wave number of 0.0225 (Radian/km) was applied to study deep and shallow anomaly sources in the area while the edge detection methods were used to outline the subsurface boundaries of these sources. The 2.5D forward modeling of magnetic and gravity delineated the basin geometry and basement depth that ranges from 1200 to 5100 m. The basement structure map created by this study showed a complex subsurface structural framework of the Southern Faghur-Siwa Basin that is dominated by N–S, NNW-SSW, NE-SW, and E-W trending faults.

Riemannian geometries of visual space: Variable curvature and horizon

Two‐dimensional (2D) images of light beams reflected off the objects in space impinge on the retinal photoreceptors of our two laterally separated eyes. Nevertheless, we experience our visual percept as a single 3D entity—our visual world that we tend to identify with the physical world. However, experiments point to different geometries in these two worlds. Using the binocular system with asymmetric eyes (AEs), this article studies the global geometric aspects of visual space in the Riemannian geometry framework. The constant‐depth curves in the horizontal field of binocular fixations consist of families of arcs of ellipses or hyperbolas depending on the AE parameters and the eyes' fixation point. For a single set of AE parameters, there is a unique symmetric fixation at the abathic distance such that the constant‐depth conics are straight frontal lines. Critically, the distribution of these constant‐depth lines is independent of such fixations. In these cases, a two‐parameter family of the Riemann metrics is proposed to account for the retinotopy of the brain's visual pathways and simulated constant‐depth lines. The obtained geodesics for a subset of metric parameters include incomplete geodesics that give finite distances to the horizon. The Gaussian curvature of the phenomenal horizontal field is analyzed for all the metric parameters. The sign of the curvature can be inferred from the global behavior of the constant‐depth ellipses and hyperbolas only when for the metric parameters for which the constant‐depth frontal lines at the abathic distance fixations are geodesics.

SAR Radiometric Calibration Based on Differential Geometry: From Theory to Experimentation on SAOCOM Imagery

The modeling and simulation of topography-induced imaging distortions are crucial for consistent radiometric information exploitation in current and forthcoming SAR-based Earth observation missions with a high spatial and temporal resolution, with relevance in several applications. In this paper, for the first time, we specifically investigate the compensation of topography-induced radiometric distortions affecting SAR images acquired by the L-band Argentinian satellite SAOCOM. We adopt a recently developed calibration method relying on an analytical formulation derived in the rigorous framework of the differential geometry of surfaces. We first provide an original interpretation of the analytical formulation, thus providing further insights into the relevant area-stretching-based formalism. Then, the numerical implementation of the method is specialized to systematically process the data acquired by SAOCOM sensors; hence, the resulting sensor-specific prototype solution processor is employed in this study. Finally, experiments performed over a real scenario in the southern part of Italy, characterized by large topography variations, are presented and discussed, thus elucidating the effectiveness of the adopted method applied to SAOCOM images. The adopted effective SAR calibration strategy opens up the way to its operational use in large-scale SAOCOM data processing.

Contact angle hysteresis on random self-affine rough surfaces in Wenzel's wetting regime: Numerical study.

We present a numerical study of the advancing and receding apparent contact angles for a liquid meniscus in contact with random self-affine rough surfaces in Wenzel's wetting regime. Within the framework of the Wilhelmy plate geometry, we use the full capillary model to obtain these global angles for a wide range of local equilibrium contact angles and for different parameters that determine the self-affine solid surfaces: Hurst exponent, wave vector domain, and root-mean-square roughness. We find that the advancing and receding contact angles are single-valued functions that depend only on the roughness factor determined by the set of values of the parameters of the self-affine solid surface. Moreover, the cosines of these angles are found to depend linearly on the surface roughness factor. The relations between the advancing, the receding, and Wenzel's equilibrium contact angles are investigated. It is shown that for materials with self-affine surface structure, the hysteresis force is the same for different liquids and it depends only on the surface roughness factor. A comparison with existing numerical and experimental results is carried out.

Nutritional geometry framework of sucrose taste in Drosophila

Dietary protein (P) and carbohydrate (C) have a major impact on the sweet taste sensation. However, it remains unclear whether the balance of P and C influences the sweet taste sensitivity. Here, we use the nutritional geometry framework (NGF) to address the interaction of protein and carbohydrates on sweet taste using Drosophila as a model. Our results reveal that high-protein, low-carbohydrate (HPLC) diets sensitize to sweet taste and low-protein, high-carbohydrate (LPHC) diets desensitize sweet taste in both male and female flies. We further investigate the underlying mechanisms of the effects of two diets on sweet taste using RNA sequencing. When compared to the LPHC diet, the mRNA expression of genes involved in the metabolism of glycine, serine, and threonine is significantly upregulated in the HPLC diet group, suggesting these amino acids may mediate sweet taste perception. We further find that sweet sensitization occurs in flies fed with the LPHC diet supplemented with serine and threonine. Our study demonstrates that sucrose taste sensitivity is affected by the balance of dietary protein and carbohydrates possibly through changes in serine and threonine.