Representation Of Function Research Articles

Efficient random walks for generating random fuzzy measures in Möbius representation in large universe

Random generation of fuzzy measures plays a pivotal role in large-scale decision-making and optimization. Random walks ensure uniform generation and adequate coverage. The Möbius representation of set functions is a valuable tool for establishing the sparse structure of fuzzy measures, with its non-negativity closely linked to monotonicity and convexity/supermodularity checking. We propose three efficient methods for monotonicity verification and convexity/supermodularity verification applicable to random walks in Möbius representations, specifically tailored for universal sets larger than ten inputs and k-order fuzzy measures. We first present the baseline methods by directly inspecting monotonicity and convexity constraints. Building on the observation that the majority of initially generated values exhibit non-negativity, we intentionally track negative Möbius values to enhance the computational performance of these baseline approaches. Further, we introduce the more agile methods that employ insertion and merge sorting techniques for both monotonicity and convexity checks in random walks that involve small perturbations of fuzzy measures. To address sparsity in large-scale scenarios, we focus on two major types of measures: k-additive and k-interactive measures, demonstrating their effectiveness through theoretical analysis and experimental results.

Banach lattices with upper p-estimates: free and injective objects

AbstractWe study the free Banach lattice $$\textrm{FBL}^{(p,\infty )}[E]$$ FBL ( p , ∞ ) [ E ] with upper p-estimates generated by a Banach space E. Using a classical result of Pisier on factorization through $$L^{p,\infty }(\mu )$$ L p , ∞ ( μ ) together with a finite dimensional reduction, it is shown that the spaces $$\ell ^{p,\infty }(n)$$ ℓ p , ∞ ( n ) witness the universal property of $$\textrm{FBL}^{(p,\infty )}[E]$$ FBL ( p , ∞ ) [ E ] isomorphically. As a consequence, we obtain a functional representation for $$\textrm{FBL}^{(p,\infty )}[E]$$ FBL ( p , ∞ ) [ E ] , answering a question from Oikhberg et al. [Free Banach lattices. J Eur Math Soc (JEMS), 2024]. More generally, our proof allows us to identify the norm of any free Banach lattice over E associated with a rearrangement invariant function space. After obtaining the above functional representation, we take the first steps towards analyzing the fine structure of $$\textrm{FBL}^{(p,\infty )}[E]$$ FBL ( p , ∞ ) [ E ] . Notably, we prove that the norm for $$\textrm{FBL}^{(p,\infty )}[E]$$ FBL ( p , ∞ ) [ E ] cannot be isometrically witnessed by $$L^{p,\infty }(\mu )$$ L p , ∞ ( μ ) and settle the question of characterizing when an embedding between Banach spaces extends to a lattice embedding between the corresponding free Banach lattices with upper p-estimates. To prove this latter result, we introduce a novel push-out argument, which when combined with the injectivity of $$\ell ^p$$ ℓ p allows us to give an alternative proof of the subspace problem for free p-convex Banach lattices. On the other hand, we prove that $$\ell ^{p,\infty }$$ ℓ p , ∞ is not injective in the class of Banach lattices with upper p-estimates, elucidating one of many difficulties arising in the study of $$\textrm{FBL}^{(p,\infty )}[E]$$ FBL ( p , ∞ ) [ E ] .

Planar structured materials with extreme elastic anisotropy

Designing anisotropic structured materials by reducing symmetry results in unique behaviors, such as shearing under uniaxial compression. While rank-deficient materials such as hierarchical laminates have been shown to exhibit extreme elastic anisotropy, there is limited knowledge about the fully anisotropic elasticity tensors achievable with single-scale fabrication techniques. No established upper and lower bounds on anisotropic moduli exist. In this paper, we estimate the range of anisotropic stiffness tensors achieved by single-scale two-dimensional structured materials. We first develop a database of periodic anisotropic single-scale unit cell geometries using linear combinations of periodic cosine functions. The database covers a wide range of anisotropic elasticity tensors, which are then compared with the elasticity tensors of hierarchical laminates. We identify the regions in the property space where hierarchical design is necessary to achieve extremal properties. We demonstrate a method to construct various 2D functionally graded structures using this cosine function representation. These graded structures seamlessly interpolate between unit cells with distinct patterns, allowing for independent control of several functional gradients, such as porosity, anisotropic moduli, and symmetry. When designed with unit cells positioned at extreme parts of the property space, these graded structures exhibit unique mechanical behaviors such as selective strain energy localization, compressive strains under tension, and localized rotations.

Computer Graphical Representation of Complex Mathematical Functions in Turbo Basic

Computer graphic representation of complex mathematical functions plays an important role in the spatial perception of these functions by students. On the other hand, the diversity of spatial forms of complex functions and their beauty allows their use in the design of various architectural and advertising projects. In this article, the author presents an original program written in Turbo Basic, which allows building a spatial image of any given complex mathematical function on the computer screen.

An Improved Quantum Particle Swarm Optimization Algorithm for Target Tracking Deployment in Spatial Sensor Networks

This paper presents a comprehensive review of the current research on spatial sensor networks and the node deployment methods employed for mobile target tracking. The study introduces particle swarm optimization (PSO) and its quantum behavior extension, detailing concepts such as the quantum state wave function and particle position representation. Subsequently, an improved Quantum Particle Swarm Optimization (QPSO) algorithm is proposed. This enhanced algorithm increases population diversity by incorporating quantum rotation gates and quantum mutation mechanisms, expands the search space through the superposition state and interference principles of quantum mechanics, and dynamically adjusts algorithm parameters to balance global exploration and local search. These modifications aim to improve both the convergence speed and accuracy of the algorithm. Simulation results demonstrate that the improved QPSO algorithm surpasses traditional mobile tracking deployment algorithms and the standard quantum behavior particle swarm optimization algorithm in terms of target tracking deployment within spatial sensor networks. Notably, it significantly enhances the tracking success rate and reduces tracking errors.

Supervised learning via ensembles of diverse functional representations: the functional voting classifier

Supervised learning via ensembles of diverse functional representations: the functional voting classifier

Red blood cell passage through deformable interendothelial slits in the spleen: Insights into splenic filtration and hemodynamics

The spleen constantly clears altered red blood cells (RBCs) from the circulation, tuning the balance between RBC formation (erythropoiesis) and removal. The retention and elimination of RBCs occur predominantly in the open circulation of the spleen, where RBCs must cross submicron-wide inter-endothelial slits (IES). Several experimental and computational studies have illustrated the role of IES in filtrating the biomechanically and morphologically altered RBCs based on a rigid wall assumption. However, these studies also reported that when the size of IES is close to the lower end of clinically observed sizes (less than 0.5 μm), an unphysiologically large pressure difference across the IES is required to drive the passage of normal RBCs, sparking debates on the feasibility of the rigid wall assumption. In this work, We propose two deformable IES models, namely the passive model and the active model, aiming to explore the impact of the deformability of IES on the filtration function of the spleen. In the passive model, we implement the worm-like string model to depict the IES’s deformation as it interacts with blood plasma and allows RBC to traverse. In contrast, the active model involved regulating the IES deformation based on the local pressure surrounding the slit. To demonstrate the validity of the deformable model, we simulate the filtration of RBCs with varied size and stiffness by IES under three scenarios: (1) a single RBC traversing a single slit; (2) a suspension of RBCs traversing an array of slits, mimicking in vitro spleen-on-a-chip experiments; (3) RBC suspension passing through the 3D spleen filtration unit known as’the splenon’. Our simulation results of RBC passing through a single slit show that the deformable IES model offers more accurate predictions of the critical cell surface area to volume ratio that dictate the removal of aged RBCs from circulation compared to prior rigid-wall models. Our biophysical models of the spleen-on-a-chip indicate a hierarchy of filtration function stringency: rigid model > passive model > active model, providing a possible explanation of the filtration function of IES. We also illustrate that the biophysical model of ‘the splenon’ enables us to replicate the ex vivo experiments involving spleen filtration of malaria-infected RBCs. Taken together, our simulation findings indicate that the deformable IES model could serve as a mesoscopic representation of spleen filtration function closer to physiological reality, addressing questions beyond the scope of current experimental and computational models and enhancing our understanding of the fundamental flow dynamics and mechanical clearance processes within in the human spleen.

RS-MAE: Region-State Masked Autoencoder for Neuropsychiatric Disorder Classifications Based on Resting-State fMRI.

Dynamic functional connectivity (DFC) extracted from resting-state functional magnetic resonance imaging (fMRI) has been widely used for neuropsychiatric disorder classifications. However, serious information redundancy within DFC matrices can significantly undermine the performance of classification models based on them. Moreover, traditional deep models cannot adapt well to connectivity-like data, and insufficient training samples further hinder their effective training. In this study, we proposed a novel region-state masked autoencoder (RS-MAE) for proficient representation learning based on DFC matrices and ultimately neuropsychiatric disorder classifications based on fMRI. Three strategies were taken to address the aforementioned limitations. First, masked autoencoder (MAE) was introduced to reduce redundancy within DFC matrices and learn effective representations of human brain function simultaneously. Second, region-state (RS) patch embedding was proposed to replace space-time patch embedding in video MAE to adapt to DFC matrices, in which only topological locality, rather than spatial locality, exists. Third, random state concatenation (RSC) was introduced as a DFC matrix augmentation approach, to alleviate the problem of training sample insufficiency. Neuropsychiatric disorder classifications were attained by fine-tuning the pretrained encoder included in RS-MAE. The performance of the proposed RS-MAE was evaluated on four publicly available datasets, achieving accuracies of 76.32%, 77.25%, 88.87%, and 76.53% for the attention deficit and hyperactivity disorder (ADHD), autism spectrum disorder (ASD), Alzheimer's disease (AD), and schizophrenia (SCZ) classification tasks, respectively. These results demonstrate the efficacy of the RS-MAE as a proficient deep learning model for neuropsychiatric disorder classifications.

The Riemann–Hilbert Approach to the Higher-Order Gerdjikov–Ivanov Equation on the Half Line

The Fokas method exhibits remarkable versatility in solving boundary value problems associated with both linear and nonlinear partial differential equations, particularly when conventional approaches encounter challenges in handling intricate boundary conditions. The existing literature often lacks the incorporation of unconventional boundary conditions, and this study addresses this issue by extending the application of the Fokas method to the higher-order Gerdjikov-Ivanov equation on the half line (−∞,0]. We have demonstrated the exclusive representation of the potential function u(z,t) in the higher-order Gerdjikov–Ivanov equation through the solution of a Riemann–Hilbert problem. The characteristic function is partitioned on the complex plane, and we obtain the jump matrix between each partition based on the positive and negative values of the partition as well as the spectral matrix determined by the initial data and boundary value data. The findings suggest that the spectral functions are not mutually independent; instead, they conform to a global relationship. The novel aspect of this study is the application of the Fokas method to a previously unexplored case, contributing to the theoretical and practical understanding of complex partial differential equations and filling a gap in the treatment of boundary conditions.

Discrete element method simulation of high-speed vehicle collisions with road barrier systems

AbstractThe behavior of road or perimeter protection barriers under vehicle impact are usually investigated based on crash tests and finite element (FE) numerical approaches, which are ether expensive or time-consuming. Several studies have proposed to reduce the computation time of the numerical analysis by substituting the complex FE models of vehicles using simplified mass–spring–damper system models. However, these models have drawbacks since consideration of different vehicle impact angles is difficult and they are unable to correctly simulate the risk of high-speed vehicle collision running over the barrier. In this paper, a new approach is proposed to simulate the collision of vehicles with barriers based on the discrete element method (DEM). Here, to save computation time only a handful of 3D non-spherical particles are used to represent the barrier and vehicle. These particles are generated based on the super-quadric function, which is capable of generating a variety of shapes needed for the model. The contact detection and evaluation are carried out based on discrete function representation of the particles with uniform sampling. The bond between two discrete elements is defined using a nonlinear cohesive beam model since the distance between the elements is relatively large. The simulation results obtained based on this approach are more accurate and complete than the simplified mass–spring models and computationally more efficient than the FE model.

Analysis of a Simplified Predictive Function Control Formulation Using First Order Transfer Function for Adaptive Cruise Control

This paper presents a formulation and analysis of a low computation Predictive Functional Control (PFC), which is a simplified version of the more advanced Model Predictive Control (MPC) for an Adaptive Cruise Control (ACC) system by using a representation of first order closed-loop transfer function. In this work, a non-linear mathematical model of vehicle longitudinal dynamics is considered as a control plant. Then, a simple Proportional Integral (PI) controller is employed as an inner loop to identify the first-order relationship between its actual and desired trajectory speed according to the reasonable time constant based on the logical response of pedals pressing. To directly control the whole plant, the PFC is formulated as an outer loop to track the desired speed together with the convergence rate based on a user preference while satisfying constraints related to acceleration and safe distancing. Since PFC is formulated based on the first-order transfer function, the prediction and tuning processes are straightforward and specific to this system. The simulation results confirm that the proposed controller managed to track the desired speed while maintaining a comfortable driving response. Besides, the controller also can retain safe distancing during the car following application, even in the presence of unmeasured disturbance. In summary, this framework can avoid the need to formulate an inverse non-linear model that is typically used when deploying a hierarchical control structure to compute the throttle and brake pedals pressing as it has been replaced with an inner loop PI controller. The performance also is comparable yet more conservative due to the simplification. These findings can become a good reference for designing and improving the ACC controller, as the framework can be easily generalized for any type of vehicle for future work.

Accuracy of kinetic equilibrium reconstruction of NSTX and NSTX-U plasmas and its impact on the transport and stability analysis

Abstract An accurate magnetohydrodynamic (MHD) equilibrium reconstruction is an essential starting point for stability and transport plasma analysis. This work describes an approach for obtaining kinetic equilibrium reconstructions using the OMFIT framework, which has been applied for the first time to spherical tokamak data from NSTX and NSTX-U. The EFIT equilibrium solver is integrated with experimental data analysis procedures and subsequent TRANSP transport simulations to enhance the accuracy of the reconstruction, in particular at the edge region, by adding constraints on the total pressure and current density profiles, based on the transport code solution. The accuracy of the equilibrium depends on the uncertainty and number of constraints, as well as the choice of basis functions to represent the pressure and current density profiles. An improved fidelity of the equilibrium reconstruction is demonstrated by reducing the variability of the magnetic axis and boundary locations from several centimeters for reconstructions based on magnetic and experimental pressure constraints to only several millimeters for kinetic reconstructions based on transport code constraints, when different representations for basis functions were tested. Variability of the safety factor on-axis was reduced ten times in the same sensitivity study. The accuracy of the equilibrium reconstruction and subsequent mapping of experimental kinetic profile data have a significant impact on TGLF and linear CGYRO turbulence simulations, which predict different spectra of unstable modes and turbulent fluxes for cases with different numbers of constraints in the equilibrium reconstruction. Conversely, the stability analysis performed with the GATO code shows plasmas stable to n = 1 MHD modes in both equilibria using magnetic and experimental pressure constraints as well as the transport code constrained equilibrium. However, a scan of parameters away from these conditions shows considerable deviation in the threshold of unstable modes between these reconstructions. Therefore, for reliable plasma analysis and use in turbulence and stability calculations, a high fidelity equilibrium reconstruction with accurate kinetic constraints based on transport code solutions is necessary.

Correlation functions from tensor network influence functionals: The case of the spin-boson model.

We investigate the application of matrix product state (MPS) representations of the influence functionals (IFs) for the calculation of real-time equilibrium correlation functions in open quantum systems. Focusing specifically on the unbiased spin-boson model, we explore the use of IF-MPSs for complex time propagation, as well as IF-MPSs for constructing correlation functions in the steady state. We examine three different IF approaches: one based on the Kadanoff-Baym contour targeting correlation functions at all times, one based on a complex contour targeting the correlation function at a single time, and a steady state formulation, which avoids imaginary or complex times, while providing access to correlation functions at all times. We show that within the IF language, the steady state formulation provides a powerful approach to evaluate equilibrium correlation functions.

Dietary Sugar and Saturated Fat Consumption Associated with the Gastrointestinal Microbiome during Pregnancy

BackgroundGrowing evidence supports changes in the gastrointestinal microbiome over the course of pregnancy may have an impact on the short- and long-term health of both the mother and the child. ObjectiveOur objective was to explore the association of diet quality, as measured by the Healthy Eating Index (HEI), with the composition and gene ontology (GO) representation of microbial function in the maternal gastrointestinal microbiome during pregnancy. MethodsWe conducted a retrospective, observational analysis of n = 185 pregnant participants in the Pregnancy Eating Attributes Study. Maternal dietary intake was assessed in the first trimester using the automated self-administered 24-h recall method, from which the HEI 2015 was calculated. Rectal swabs were obtained in the second trimester and sequenced using the NovaSeq 6000 system shotgun platform. We used unsupervised clustering to identify microbial enterotypes representative of maternal taxa and GO functional term composition. Multivariable linear models were used to identify associations between taxa, functional terms, and food components while controlling for relevant covariates. Multinomial regression was then used to predict enterotype membership based on a participant’s HEI food component score. ResultsThose in the high diet quality tertile had a lower early pregnancy BMI [mean (M) = 23.48 kg/m2, SD = 3.38] compared with the middle (M = 27.35, SD = 6.01) and low (M = 27.49, SD = 6.99) diet quality tertiles (P < 0.01). There were no statistically significant associations between the HEI components or the total HEI score and the 4 alpha diversity measures. Differences in taxa and GO term enterotypes were found in participants with, but not limited to, a higher saturated fat component score (β = 1.35, P = 0.01), added sugar HEI component (β = 0.07, P < 0.001), and higher total dairy score (β = 1.58, P = 0.01). ConclusionsSpecific dietary components are associated with microbial composition and function in the second trimester of pregnancy. These findings provide a foundation for future testable hypotheses.

SenseMap: Urban Performance Visualization and Analytics Via Semantic Textual Similarity.

As urban populations grow, effectively accessing urban performance measures such as livability and comfort becomes increasingly important due to their significant socioeconomic impacts. While Point of Interest (POI) data has been utilized for various applications in location-based services, its potential for urban performance analytics remains unexplored. In this article, we present SenseMap, a novel approach for analyzing urban performance by leveraging POI data as a semantic representation of urban functions. We quantify the contribution of POIs to different urban performance measures by calculating semantic textual similarities on our constructed corpus. We propose Semantic-adaptive Kernel Density Estimation which takes into account POIs' influential areas across different Traffic Analysis Zones and semantic contributions to generate semantic density maps for measures. We design and implement a feature-rich, real-time visual analytics system for users to explore the urban performance of their surroundings. Evaluations with human judgment and reference data demonstrate the feasibility and validity of our method. Usage scenarios and user studies demonstrate the capability, usability and explainability of our system.

Multimodal and Repeated Localization of Primary Hand Motor Function to the Lateral Postcentral Gyrus in a Case of Frontal Motor Area Brain Metastasis

BACKGROUND AND IMPORTANCE: The human motor homunculus is a well-known topographical map of the functional-anatomical relationships of the precentral gyrus. Within this homunculus, the primary hand motor area is considered one of the least plastic functional-anatomical relationships. Only a few cases in the literature describe relocation of functional hand representation away from the classical anatomical location. These cases have been reported in the context of children, primary gliomas, or arteriovenous malformations. CLINICAL PRESENTATION: Here, we describe a unique case where the area of lowest hand motor stimulation threshold (ie, hand motor representation) was found in the postcentral gyrus in an older adult with a metastasis in the premotor area of the brain. This localization was based on intraoperative cortical stimulation–evoked motor potentials and confirmed with electrophysiological phase reversal and MRI-based neuronavigation. This mapping was repeated and consistent 2 months later during a reoperation for recurrence. In addition, the remapped anatomical location was found in an area that was active during finger tapping on preoperative functional MRI. CONCLUSION: These findings suggest that neuroplastic remapping of hand motor cortex to the postcentral gyrus can occur in brain metastases even in adults. This has implications for planning tumor resections and interventional neurorehabilitation strategies, and it suggests that the motor homunculus may have more plastic potential in adulthood than previously recognized.

Analyzing the Stability of a Connected Moving Cart on an Inclined Surface with a Damped Nonlinear Spring

This paper examines the stability behavior of the nonlinear dynamical motion of a vibrating cart with two degrees of freedom (DOFs). Lagrange’s equations are employed to establish the mechanical regulating system of the examined motion. The proposed approximate solutions (ASs) of this system are estimated through the use of the multiple-scales method (MSM). These solutions are considered novel as the MSM is being applied to a new dynamical model. Secular terms have been eliminated to meet the solvability criteria, and every instance of resonance that arises is categorized, where two of them are examined concurrently. Therefore, the modulation equations are developed based on the representations of the unknown complex function in polar form. The solutions for the steady state are calculated using the corresponding fixed points. The achieved solutions are displayed graphically to illustrate the impact of manipulating the system’s parameters and are compared to the numerical solutions (NSs) of the system’s original equations. This comparison shows a great deal of consistency with the numerical solution, which indicates the accuracy of the applied method. The nonlinear stability criteria of Routh–Hurwitz are employed to assess the stability and instability zones. The value of the proposed model is exhibited by its wide range of applications involving ship motion, swaying architecture, transportation infrastructure, and rotor dynamics.

Effects of connectivity hyperalignment (CHA) on estimated brain network properties: from coarse-scale to fine-scale.

Recent gains in functional magnetic resonance imaging (fMRI) studies have been driven by increasingly sophisticated statistical and computational techniques and the ability to capture brain data at finer spatial and temporal resolution. These advances allow researchers to develop population-level models of the functional brain representations underlying behavior, performance, clinical status, and prognosis. However, even following conventional preprocessing pipelines, considerable inter-individual disparities in functional localization persist, posing a hurdle to performing compelling population-level inference. Persistent misalignment in functional topography after registration and spatial normalization will reduce power in developing predictive models and biomarkers, reduce the specificity of estimated brain responses and patterns, and provide misleading results on local neural representations and individual differences. This study aims to determine how connectivity hyperalignment (CHA)-an analytic approach for handling functional misalignment-can change estimated functional brain network topologies at various spatial scales from the coarsest set of parcels down to the vertex-level scale. The findings highlight the role of CHA in improving inter-subject similarities, while retaining individual-specific information and idiosyncrasies at finer spatial granularities. This highlights the potential for fine-grained connectivity analysis using this approach to reveal previously unexplored facets of brain structure and function.

Infinite-Volume Gibbs States of the Generalized Mean-Field Orthoplicial Model

The generalized mean-field orthoplicial model is a mean-field model on a space of continuous spins on Rn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {R}^n$$\\end{document} that are constrained to a scaled (n-1)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(n-1)$$\\end{document}-dimensional ℓ1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ell _1$$\\end{document}-sphere, equivalently a scaled (n-1)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(n-1)$$\\end{document}-dimensional orthoplex, and interact through a general interaction function. The finite-volume Gibbs states of this model correspond to singular probability measures. In this paper, we use probabilistic methods to rigorously classify the infinite-volume Gibbs states of this model, and we show that they are convex combinations of product states. The predominant methods utilize the theory of large deviations, relative entropy, and equivalence of ensembles, and the key technical tools utilize exact integral representations of certain partition functions and locally uniform estimates of expectations of certain local observables.

Initial Boundary Value Problem for the Coupled Kundu Equations on the Half-Line

In this article, the coupled Kundu equations are analyzed using the Fokas unified method on the half-line. We resolve a Riemann–Hilbert (RH) problem in order to illustrate the representation of the potential function in the coupled Kundu equations. The jump matrix is obtained from the spectral matrix, which is determined according to the initial value data and the boundary value data. The findings indicate that these spectral functions exhibit interdependence rather than being mutually independent, and adhere to a global relation while being connected by a compatibility condition.