Structural Assumptions Research Articles

Multimodal Approach Reveals the Symmetry-Breaking Pathway to the Broken Helix in EuIn2As2

Understanding and manipulating emergent phases, which are themes at the forefront of quantum-materials research, rely on identifying their underlying symmetries. This general principle has been particularly prominent in materials with coupled electronic and magnetic degrees of freedom, in which magnetic order influences the electronic band structure and can lead to exotic topological effects. However, identifying symmetry of a magnetically ordered phase can pose a challenge, particularly in the presence of small domains. Here we introduce a multimodal approach for determining magnetic structures, which combines symmetry-sensitive optical probes, scattering, and group-theoretical analysis. We apply it to EuIn2As2, a material that has received attention as a candidate axion insulator. While first-principles calculations predict this state on the assumption of a simple collinear antiferromagnetic structure, subsequent neutron-scattering measurements reveal a much more intricate magnetic ground state characterized by two coexisting magnetic wave vectors reached by successive thermal phase transitions. The proposed high- and low-temperature phases are a spin helix and a state with interpenetrating helical and Néel antiferromagnetic order termed a “broken helix,” respectively. Employing a multimodal approach, we identify the magnetic structure associated with these two phases of EuIn2As2. We find that the higher-temperature phase is characterized by a variation of the magnetic moment amplitude from layer to layer, with the moment vanishing entirely in every third Eu layer. The lower-temperature structure is similar to the broken helix, with one important difference: Because of local strain, the relative orientation of the magnetic structure and the lattice is not fixed. Consequently, the symmetry required to protect the axion phase is not generically protected in EuIn2As2, but we show that it can be restored if the magnetic structure is tuned with uniaxial strain. Finally, we present a spin Hamiltonian that identifies the spin interactions that account for the complex magnetic order in EuIn2As2. Our work highlights the importance of a multimodal approach in determining the symmetry of complex order parameters. Published by the American Physical Society 2024

On non parametric kernel estimation of the mode of the regression function in the strong mixing random design model with censored data

. This study delves into the conditional mode estimation of a randomly censored scalar response variable operating within the framework of strong mixing conditions. We introduce a kernel-based estimator for the conditional mode function. The principal contribution of this investigation lies in the derivation of the asymptotic distribution and the strong rate of convergence of the newly proposed estimators. These findings are established under a set of fairly comprehensive structural assumptions governing the underlying models. Additionally, we conduct a series of simulation studies to showcase the finite sample performance characteristics of the proposed estimator.

Graphical criteria for the identification of marginal causal effects in continuous-time survival and event-history analyses

Abstract We consider continuous-time survival and event-history settings, where our aim is to graphically represent causal structures allowing us to characterize when a causal parameter is identified from observational data. This causal parameter is formalized as the effect on an outcome event of a (possibly hypothetical) intervention on the intensity of a treatment process. To establish identifiability, we propose novel graphical rules indicating whether the observed information is sufficient to obtain the desired causal effect by suitable reweighting. This requires a different type of graph than in discrete time. We formally define causal semantics for the corresponding dynamic graphs that represent local independence models for multivariate counting processes. Importantly, our work highlights that causal inference from censored data relies on subtle structural assumptions on the censoring process beyond independent censoring; these can be verified graphically. Put together, our results are the first to establish graphical rules for nonparametric causal identifiability in event processes in this generality for the continuous-time case, not relying on particular parametric survival models. We conclude with a data example on Human papillomavirus (HPV) testing for cervical cancer screening, where the assumptions are illustrated graphically and the desired effect is estimated by reweighted cumulative incidence curves.

Examining the Impact of Structural Uncertainty Across 10 Type 2 Diabetes Models: Results From the 2022 Mount Hood Challenge

ObjectivesThe Mount Hood Diabetes Challenge Network aimed to examine the impact of model structural uncertainty on the estimated cost-effectiveness of interventions for type 2 diabetes. MethodsTen independent modeling groups completed a blinded simulation exercise to estimate the cost-effectiveness of 3 interventions in 2 type 2 diabetes populations. Modeling groups were provided with a common baseline population, cost and utility values associated with different model health states, and instructions regarding time horizon and discounting. We collated the results to identify variation in predictions of net monetary benefit (NMB) and the drivers of those differences. ResultsOverall, modeling groups agreed which interventions had a positive NMB (ie, were cost-effective), Although estimates of NMB varied substantially—by up to £23 696 for 1 intervention. Variation was mainly driven through differences in risk equations for complications of diabetes and their implementation between models. The number of modeled health states was also a significant predictor of NMB. ConclusionsThis exercise demonstrates that structural uncertainty between different health economic models affects cost-effectiveness estimates. Although it is reassuring that a decision maker would likely reach similar conclusions on which interventions were cost-effective using most models, the range in numerical estimates generated across different models would nevertheless be important for price-setting negotiations with intervention developers. Minimizing the impact of structural uncertainty on healthcare decision making therefore remains an important priority. Model registries, which record and compare the impact of structural assumptions, offer one potential avenue to improve confidence in the robustness of health economic modeling.

An elementary proof of existence and uniqueness for the Euler flow in localized Yudovich spaces

We revisit Yudovich’s well-posedness result for the 2-dimensional Euler equations for an inviscid incompressible fluid on either a sufficiently regular (not necessarily bounded) open set Ω⊂R2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Omega \\subset \\mathbb {R}^2$$\\end{document} or on the torus Ω=T2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Omega =\\mathbb {T}^2$$\\end{document}. We construct global-in-time weak solutions with vorticity in L1∩Lulp\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^1\\cap L^p_{ul}$$\\end{document} and in L1∩YulΘ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^1\\cap Y^\\Theta _{ul}$$\\end{document}, where Lulp\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^p_{ul}$$\\end{document} and YulΘ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$Y^\\Theta _{ul}$$\\end{document} are suitable uniformly-localized versions of the Lebesgue space Lp\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^p$$\\end{document} and of the Yudovich space YΘ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$Y^\\Theta $$\\end{document} respectively, with no condition at infinity for the growth function Θ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Theta $$\\end{document}. We also provide an explicit modulus of continuity for the velocity depending on the growth function Θ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Theta $$\\end{document}. We prove uniqueness of weak solutions in L1∩YulΘ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^1\\cap Y^\\Theta _{ul}$$\\end{document} under the assumption that Θ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Theta $$\\end{document} grows moderately at infinity. In contrast to Yudovich’s energy method, we employ a Lagrangian strategy to show uniqueness. Our entire argument relies on elementary real-variable techniques, with no use of either Sobolev spaces, Calderón–Zygmund theory or Littlewood–Paley decomposition, and actually applies not only to the Biot–Savart law, but also to more general operators whose kernels obey some natural structural assumptions.

Dissipative Structure Analysis and Ecological Disorganization: A Critique Drawn from Ecological Marxist and Treadmill-of-Production Approaches

Among environmental sociologists, ecological Marxists argue that there is an association between capitalism and ecological destruction/disorganization. This argument suggests that capitalism and nature are in contradiction with one another, so that the expansion of capitalism necessarily results in the destruction of nature. Green criminologists expand on this point and argue that ecological disorganization generates legal and illegal green crimes and injustice. This capitalism–nature association suggests that solving the current ecological crisis would require replacing capitalism. In contrast, the dissipative structure analysis (DSA) argues that capitalist nations pass through phases of development and that as advanced capitalist nations age, the level of ecological destruction they cause becomes attenuated. If true, this outcome suggests that capitalism might solve the problem of ecological destruction by aging. This article reviews these two theoretical arguments, exposes the limitation of the dissipative structure argument, and suggests that future research is required to completely address dissipative structure assumptions. A major limitation of the dissipative structure analysis is its failure to account for the effect of developing capitalist nations on the expansion of ecological disorganization. Trend charts are employed to illustrate the limitations of the DSA hypothesis concerning reduced ecological disorganization.

Structural assumption on design of rounded rectangular bellows under pressure

Bellows is a flexible component whose volume can be changed by compression or expansion under pressure. Rounded rectangular type bellows is most applied where the internal space of application is strict. The Expansion Joint Manufacturers Association (EJMA) code is world-recognized code for design by formulas method. However, the calculation formulas of stress and deformation in EJMA code does not consider the shape of corner. Therefore, this study proposed a structural assumption of curved beam model to calculate the stress and deformation for rounded rectangular bellows under pressure load by formula calculation method. A parametric study by means of finite element analysis (FEA) was performed to assess the impact of different structural parameters and the applicable scope was investigated. The experimental and numerical results showed that the proposed method can obtain a suitable precision and safe result when ratio of wave height to corner radius is lower than 35 % and the ratio of length of short side length to corner radius is in range 5–20. This study will enrich existing bellows design code and help industry designers to accelerate the optimization iteration in preliminary design stage instead of FEA.

Some Remarks on Linear-Quadratic Closed-Loop Games with Many Players

AbstractWe identify structural assumptions which provide solvability of the Nash system arising from a linear-quadratic closed-loop game, with stable properties with respect to the number of players. In a setting of interactions governed by a sparse graph, both short-time and long-time existence of a classical solution for the Nash system set in infinitely many dimensions are addressed, as well as convergence to the solution to the respective ergodic problem as the time horizon goes to infinity; in addition, equilibria for the infinite-dimensional game are shown to provide $$\epsilon $$ ϵ -Nash closed-loop equilibria for the N-player game. In a setting of generalized mean-field type (where the number of interactions is large but not necessarily symmetric), directly from the N-player Nash system estimates on the value functions are deduced on an arbitrary large time horizon, which should pave the way for a convergence result as N goes to infinity.

Meta-state–space learning: An identification approach for stochastic dynamical systems

Available methods for identification of stochastic dynamical systems from input–output data generally impose restricting structural assumptions on either the noise structure in the data-generating system or the possible state probability distributions. In this paper, we introduce a novel identification method of such systems, which results in a dynamical model that is able to produce the time-varying output distribution accurately without taking restrictive assumptions on the data-generating process. The method is formulated by first deriving a novel and exact representation of a wide class of nonlinear stochastic systems in a so-called meta-state–space form, where the meta-state can be interpreted as a parameter vector of a state probability function space parameterization. As the resulting representation of the meta-state dynamics is deterministic, we can capture the stochastic system based on a deterministic model, which is highly attractive for identification. The meta-state–space representation often involves unknown and heavily nonlinear functions, hence, we propose an artificial neural network (ANN)-based identification method capable of efficiently learning nonlinear meta-state–space models. We demonstrate that the proposed identification method can obtain models with a log-likelihood close to the theoretical limit even for highly nonlinear, highly stochastic systems.

Evolutionary Game Dynamics with Environmental Feedback in a Network with Two Communities

Recent developments of eco-evolutionary models have shown that evolving feedbacks between behavioral strategies and the environment of game interactions, leading to changes in the underlying payoff matrix, can impact the underlying population dynamics in various manners. We propose and analyze an eco-evolutionary game dynamics model on a network with two communities such that players interact with other players in the same community and those in the opposite community at different rates. In our model, we consider two-person matrix games with pairwise interactions occurring on individual edges and assume that the environmental state depends on edges rather than on nodes or being globally shared in the population. We analytically determine the equilibria and their stability under a symmetric population structure assumption, and we also numerically study the replicator dynamics of the general model. The model shows rich dynamical behavior, such as multiple transcritical bifurcations, multistability, and anti-synchronous oscillations. Our work offers insights into understanding how the presence of community structure impacts the eco-evolutionary dynamics within and between niches.

Stability evaluation and design of negative stiffness amplifying damper-incorporated nonlinear structures

Negative stiffness-incorporated dampers have been widely accepted as effective vibration mitigation systems for flexible structural stiffness adjustment and enhanced energy dissipation. However, current investigations and optimal designs of negative stiffness-incorporated structures are limited to the linear elastic assumption of primary structures, potentially causing stability issues and unsatisfactory control performance. In this case, this study derives the stability criterion for negative stiffness amplifying damper (NSAD)-incorporated nonlinear structures and establishes stability-oriented design as well as modification formulae. The theoretical foundation of NSADs and nonlinear primary structures are initially introduced, based on which the stability criterion is derived according to the Routh–Hurwitz stability criterion. Following the existing design of linear elastic assumption, an endurance time acceleration series is generated subsequently to examine the proposed stability criterion for NSAD-equipped nonlinear structures. Consequently, the stability-oriented design method and corresponding modification formulae are proposed for near-field and far-field earthquakes, of which the effectiveness is validated through design cases. The results underscore the importance of considering structural nonlinearity for ensuring stability and designing NSAD-equipped nonlinear structures, demonstrating the feasibility of using the proposed correction factors to guarantee the stability of NSAD-equipped nonlinear structures subjected to multi-type earthquakes with various intensities. Additionally, the established design framework successively provides a high-efficiency energy dissipation device for seismic control of nonlinear structures, simultaneously generating easy-to-use design formulae for NSADs by combining the linear structure-based design and modification factors, facilitating direct design of nonlinear structures.

A quantum inspired approach to learning dynamical laws from data—block-sparsity and gauge-mediated weight sharing

Recent years have witnessed an increased interest in recovering dynamical laws of complex systems in a largely data-driven fashion under meaningful hypotheses. In this work, we propose a scalable and numerically robust method for this task, utilizing efficient block-sparse tensor train representations of dynamical laws, inspired by similar approaches in quantum many-body systems. Low-rank tensor train representations have been previously derived for dynamical laws of one-dimensional systems. We extend this result to efficient representations of systems with K-mode interactions and controlled approximations of systems with decaying interactions. We further argue that natural structure assumptions on dynamical laws, such as bounded polynomial degrees, can be exploited in the form of block-sparse support patterns of tensor-train cores. Additional structural similarities between interactions of certain modes can be accounted for by weight sharing within the ansatz. To make use of these structure assumptions, we propose a novel optimization algorithm, block-sparsity restricted alternating least squares with gauge-mediated weight sharing. The algorithm is inspired by similar notions in machine learning and achieves a significant improvement in performance over previous approaches. We demonstrate the performance of the method numerically on three one-dimensional systems—the Fermi–Pasta–Ulam–Tsingou system, rotating magnetic dipoles and point particles interacting via modified Lennard–Jones potentials, observing a highly accurate and noise-robust recovery.

Factor-driven completion of tensor data with missing entries

This article proposes a novel method for tensor data completion based on a factor model and a slicing technique. Specifically, we first focus on third-order tensor data with missing values. Under the assumption of factor structure, the rows and columns of the matrix consisting of slices of each dimension are reordered. Then, we divide it into four blocks to obtain valid estimates of factors and factor loadings in different blocks. Finally, these estimates are utilized to complete the missing values. In addition, we extend our proposed method to address the issue of missing entries in higher-order tensors. The core idea is to slice the higher-order data into multiple two-dimensional matrix slices using recursive methods. To evaluate its effectiveness, we apply the method to third-order tensor simulation data and the Fama-French return series, which yields satisfactory results.

Comparative Study and Airspeed Sensitivity Analysis of Full-Wing Solar-Powered UAVs Using Rigid-Body, Multibody, and Rigid-Flexible Combo Models

Solar-powered UAVs are characterized by large-scale, lightweight, and low airspeed, and changes in airspeed lead to wing deformation or stalling, which can easily induce serious flight accidents. A single dynamic model cannot accurately describe this feature, and this airspeed sensitivity can only be analyzed by integrating rigid-body, multirigid-body, and rigid-flexible combo models. This paper proposes a dynamic analysis method for a mixture of rigid-body, multirigid-body, and rigid-flexible combo models, considering the applicable airspeed ranges, computational costs, and structural deformation assumptions of the three models and comparing the differences of modes and responses at different airspeeds, and quantitatively analyzes the effects of airspeed on the motion, deformation, and coupling. The results show that appropriate increase of airspeed is beneficial to the stability of large-scale lightweight platforms, but when it is increased to more than two times the cruise speed, the structural deformation is coupled with the flight dynamic modes, leading to the deterioration of the overall dynamic response. Finally, a mixture of the three models at different airspeeds is proposed, which is necessary for future ultralarge-scale solar-powered UAVs.

Informing Structural Assumptions for Three State Oncology Cost-Effectiveness Models through Model Efficiency and Fit.

The characteristics and relative strengths and weaknesses of partitioned survival models (PSMs) and state transition models (STMs) for three state oncology cost-effectiveness models have previously been studied. Despite clear and longstanding economic modeling guidelines, more than one structure is rarely presented, and the choice of structure appears correlated more with audience or precedent than disease, decision problem, or available data. One reason may be a lack of guidance and tools available to readily compare measures of internal validity such as the model fit and efficiency of different structures, or sensitivity of results to those choices. To address this gap, methods are presented to evaluate the fit and efficiency of three structures, with an accompanying R software package, psm3mkv. The methods are illustrated by analyzing interim and final analysis datasets of the KEYNOTE-826 randomized controlled trial. At both interim and final analyses, the STM Clock Reset structure provided the best and most efficient fit. Structural uncertainties had been reduced from interim to final analysis. Beyond measures of internal validity, guidelines highlight the importance of reflecting all available data, avoiding model selection purely on the basis of goodness of fit and strongly considering external validity. The method and software allow modelers to more easily evaluate and report model fit and efficiency, examine implicit assumptions, and reveal sensitivities to structural choices.

On Lipschitz solutions of mean field games master equations

We develop a theory of existence and uniqueness of solutions of MFG master equations when the initial condition is Lipschitz continuous. Namely, we show that as long as the solution of the master equation is Lipschitz continuous in space, it is uniquely defined. Because we do not impose any structural assumptions, such as monotonicity for instance, there is a maximal time of existence for the notion of solution we provide. We analyze three cases: the case of a finite state space, the case of master equation set on a Hilbert space, and finally on the set of probability measures, all in cases involving common noises. In the last case, the Lipschitz continuity we refer to is on the gradient of the value function with respect to the state variable of the player.

A new control method for leader–follower consensus problem of uncertain constrained nonlinear multi-agent systems

The design of Adaptive Fuzzy Sliding-Mode Control (AFSMC) is discussed in this study for the distributed leader–follower consensus problem of nonlinear uncertain multi-agent systems with input and state constraints. Avoiding the critical circumstances related to the violation of input and state constraints is an important issue in the design of consensus control systems. The limitations of existing adaptive consensus controllers developed for full-state constrained systems motivated the present novel method without restrictive structural assumptions. To convert the original problem to an unconstrained consensus control problem, the proposed control system uses a barrier function-based state transformation method and the input-state linearization methodology. As a result, each agent’s adaptive fuzzy sliding-mode control input, which consists of a fuzzy system and a robust term, is constructed based on the derived representation of the system dynamics. The fuzzy system is used to approximate the dynamic equations of the agent, its neighbors, and maybe the leader in each local feedback control law structure, and the robust term is designed to compensate for the fuzzy approximation error. The magnitudes of the robust terms and the output vectors of the fuzzy systems are also determined using the proposed adaptation laws. The second Lyapunov theorem and Barbalat’s lemma are used to verify the closed-loop system’s asymptotic stability. The effectuality of the proposed methodology is confirmed by numerical simulations for several Autonomous Underwater Vehicles (AUVs). Simulation results show that by means of the proposed AFSMC, the leader–follower consensus is achieved without state constraint violation and performance degradation.

Prediction of BLEVE loading on structures

In a previous study by the authors, rigid structure assumption, as commonly used in analysing blast wave interaction with structures, was adapted in investigating the Boiling Liquid Expanding Vapour Explosion (BLEVE) pressure wave interaction with structures to predict BLEVE loads on structures. However, real structures are not rigid and the duration of BLEVE wave is much longer than that from high explosives and may be in the order of structural natural vibration period. Neglecting structural deformation, therefore, may lead to inaccurate predictions of BLEVE loads acting on the structure because structural deformation during the action of BLEVE wave would change the wave and the structure interaction. In this study, intensive numerical simulations are carried out using a validated computer model to investigate the influences of structural stiffness and BLEVE wave duration on the reflected BLEVE pressure on the structure. Based on the numerical results, prediction charts are proposed as a function of the BLEVE wave duration and structural fundamental vibration period for reliable predictions of BLEVE loads on structures. The findings of this study can be used to predict explosion loads in structural design against BLEVE.

Gender Differences: Confirmatory Factor Analysis of the ADOS-II.

Recent research has suggested that autism may present differently in girls compared to boys, encouraging the exploration of a sex-differential diagnostic criteria. Gender differences in diagnostic assessments have been shown on the ADOS-II, such that, on average, females score significantly lower than males on all scales and are less likely to show atypicality on most items related to social communicative difficulties. Yet, gender differences in the latent structure of instruments like the ADOS-II have not been examined systematically. As such, this secondary data analysis examined 4,100 youth diagnosed with autism (Mage = 9.9, 813 female & 3287 male) examined item response trends by gender on the ADOS-II Module 3. Multi-Group Confirmatory Factor Analysis results show that the factor loadings of four ADOS-II items differ across the genders. One SCD item and one RRB item are strongly related to the latent factor in the female group, while two RRB items have larger factor loadings in the male group. The assumption of an identical latent factor structure for the ADOS-II Module 3 for males and females might not be justifiable. Possible diagnostic implications are discussed.

A general framework to quantify deviations from structural assumptions in the analysis of nonstationary function-valued processes

A general framework to quantify deviations from structural assumptions in the analysis of nonstationary function-valued processes