Scalar Variable Research Articles

Almost complete convergence of the functional local linear estimator of the expectile regression for data missing at random

In this article, the authors investigate the properties of a local linear estimator for the expectile regression function in scenarios where the scalar response variable is missing at random, and the covariate is a functional variable. The study addresses the challenge of estimating the expectile regression function when the response variable is not fully observed, a common issue in statistical analysis where missing data can significantly impact the accuracy of predictions and inferences. The key contribution of the paper lies in the establishment of the almost complete convergence of the proposed estimator. This convergence resultis derived under a set of appropriate conditions, ensuring that the estimator approaches the true expectile regression function with a probability tending to wards one as the sample size increases. The authors provide a rigorous theoretical foundation for this convergence, including assumptions about the smoothness of the functional covariate and the randomness of the missing data mechanism. To demonstrate the practical applicability of their method, the authors conduct a comprehensive simulation study. This study compares the performance of their local linear estimator with other existing methods in terms of bias, variance, and convergence speed. The simulation resultshighlight the effectiveness of the proposed approach, especially in settings where the response variable is partially missing and the covariates are functional in nature.

Localization patterns emerging in CPTu tests in a saturated natural clay soil

In this work, the mechanics of cone penetration during CPTu tests in Osaka clay, a natural structured silty clay, was investigated with the Particle Finite Element Method (PFEM). To describe the behavior of the soil, the FD_MILAN model, recently proposed by the Authors, was adopted. The model, based on the multiplicative decomposition of the deformation gradient, incorporates a scalar internal variable, the bond strength Pt, to quantify the effects of structure. The macroscopic description of mechanical destructuration effects is provided by the hardening law of Pt, according to which the bond strength decreases monotonically with accumulated plastic deformations. The brittle behavior resulting from the softening process associated to destructuration make the soil quite susceptible to the spontaneous development of strain localization in the form of shear bands. In order to deal with strain localization phenomena, the model is equipped with a non-local version of the hardening laws, incorporating a material constant – the characteristic length ℓc – which provides the material with an internal length scale. The objective of this work was to get some insight on the effects of the characteristic length and of the initial degree of structure on: the kinematics of the soil deformation around the advancing cone tip; the evolution of soil structure with accumulated plastic deformations; the excess pore pressure field generated in the soil as a result of the hydro-mechanical coupling between the solid skeleton and the pore water, and the net cone resistance measured at the base of the cone. The results of the PFEM simulations show that the deformations around the piezocone are strongly affected by the characteristic length. For heavily structured soils, when ℓc is relatively small with respect to cone radius, the accumulated plastic deviatoric deformation field is characterized by clearly visible shear bands while, as ℓc increases, the detection of localized deformation regions becomes more difficult, if not impossible. This depends on the fact that, for large values of ℓc, the shear band width may be of comparable size to the cone radius. In all cases, the soil around the advancing piezocone is subjected to a very strong destructuration process, which leads to the complete loss of bond strength in a large region around the cone tip and shaft. As a consequence, the use of conventional Nc values from the literature in the interpretation of the CPTu test performed in heavily structured soils could provide undrained strength values closer to the ultimate undrained shear strength. At the same time, the use of empirical correlation to estimate the yield stress in oedometric conditions would lead to a significant underestimation of the overconsolidation ratio.

Thermal field reconstruction and compressive sensing using proper orthogonal decomposition

Model order reduction allows critical information about sensor placement and experiment design to be distilled from raw fluid mechanics simulation data. In many cases, sensed information in conjunction with reduced order models can also be used to regenerate full field variables. In this paper, a proper orthogonal decomposition (POD) inferencing method is extended to the modeling and compressive sensing of temperature, a scalar field variable. The method is applied to a simulated, critically stable, incompressible flow over a heated cylinder (Re = 1000) with Prandtl number varying between 0.001 and 50. The model is trained on pressure and temperature data from simulations. Field reconstructions are then generated using data from selected sensors and the POD model. Finally, the reconstruction error is evaluated across all Prandtl numbers for different numbers of retained modes and sensors. The predicted trend of increasing reconstruction accuracy with decreasing Prandtl number is confirmed and a Prandtl number/sensor count error matrix is presented.

Time–Energy Uncertainty Relation in Nonrelativistic Quantum Mechanics

The time–energy uncertainty relation in nonrelativistic quantum mechanics has been intensely debated with regard to its formal derivation, validity, and physical meaning. Here, we analyze two formal relations proposed by Mandelstam and Tamm and by Margolus and Levitin and evaluate their validity using a minimal quantum toy model composed of a single qubit inside an external magnetic field. We show that the ℓ1 norm of energy coherence C is invariant with respect to the unitary evolution of the quantum state. Thus, the ℓ1 norm of energy coherence C of an initial quantum state is useful for the classification of the ability of quantum observables to change in time or the ability of the quantum state to evolve into an orthogonal state. In the single-qubit toy model, for quantum states with the submaximal ℓ1 norm of energy coherence, C<1, the Mandelstam–Tamm and Margolus–Levitin relations generate instances of infinite “time uncertainty” that is devoid of physical meaning. Only for quantum states with the maximal ℓ1 norm of energy coherence, C=1, the Mandelstam–Tamm and Margolus–Levitin relations avoid infinite “time uncertainty”, but they both reduce to a strict equality that expresses the Einstein–Planck relation between energy and frequency. The presented results elucidate the fact that the time in the Schrödinger equation is a scalar variable that commutes with the quantum Hamiltonian and is not subject to statistical variance.

Local linear estimation for the censored functional regression

<abstract><p>This work considers the Local Linear Estimation (LLE) of the conditional functional mean. This regression model is used when the independent variable is functional, and the dependent one is a censored scalar variable. Under standard postulates, we establish the asymptotic distribution of the LLE by proving its asymptotic normality. The obtained results show the superiority of the LLE approach over the functional local constant one. The feasibility of the studied model is demonstrated using artificial data. Finally, the usefulness of the obtained asymptotic distribution in incomplete functional data is highlighted through a real data application.</p></abstract>

Harvesting Optimal Policy in Three Species Food Chain Model As an Optimal Control Problem with Path Constraints

In this paper, we address the issue of harvesting prey and intermediate predators in a tritrophic food chain. Our approach is based on a model that characterizes the interactions among the three species. We assume that the intermediate predator has alternative food sources (it is a generalist), while the top predator relies solely on the intermediate predator (it is a specialist). This model has been previously explored in the literature, but representing the harvesting effort as a scalar control variable. In this study, we treat it as a vector variable, offering a more comprehensive representation, particularly relevant for terrestrial species hunting. Our primary objective is to determine optimal harvesting policies that ensure the persistence of all three species. To achieve this, we formulate the problem as an optimal control problem with a finite horizon and path constraints. We present a numerical example solved using ICLOCS2.

Stabilization of the Solution to the Initial-Boundary Value Problem for One-Dimensional Isothermal Equations of Viscous Compressible Multicomponent Media

The paper considers the initial-boundary value problem for one-dimensional isothermal equations of viscous compressible multicomponent media, commonly known as a generalization of the Navier — Stokes equations. The studied equations include higher derivatives of the velocities of all components, unlike the Navier — Stokes equations, where viscosity is a scalar variable. Viscosity forms a matrix of elements responsible for viscous friction due to the composite structure of viscous stress tensors for the multicomponent case. Diagonal elements of the matrix stand for viscous friction within each component, and off-diagonal elements stand for friction between components. Such complication does not allow the automatic extension of the known results for the Navier — Stokes equations to the multicomponent case. Thus, for the diagonal matrix, the equations would be linked only through the lower terms. The paper considers a more complex case of an off-diagonal viscosity matrix. We prove the stabilization of the solution of the initial-boundary value problem with an unlimited increase in time without simplifying assumptions about the structure of the viscosity matrix, except for the standard physical requirements of symmetry and positive definiteness.

The scalar-on-function modal regression for functional time series data

This paper develops a new nonparametric estimator of the scalar-on function modal regression that is used to analyse the co-variability between a functional regressor and a scalar output variable. The new estimator inherits the smoothness of the kernel method and the robustness of the quantile regression. We assume that the functional observations are structured as a strong mixing functional time series data and we establish the almost complete consistency (with rate) of the constructed estimator. A discussion highlighting the impact of this new estimator in nonparametric functional data analysis is also given. The usefulness of this new estimator is shown using an artificial data example.

Scalar-on-Function Relative Error Regression for Weak Dependent Case

Analyzing the co-variability between the Hilbert regressor and the scalar output variable is crucial in functional statistics. In this contribution, the kernel smoothing of the Relative Error Regression (RE-regression) is used to resolve this problem. Precisely, we use the relative square error to establish an estimator of the Hilbertian regression. As asymptotic results, the Hilbertian observations are assumed to be quasi-associated, and we demonstrate the almost complete consistency of the constructed estimator. The feasibility of this Hilbertian model as a predictor in functional time series data is discussed. Moreover, we give some practical ideas for selecting the smoothing parameter based on the bootstrap procedure. Finally, an empirical investigation is performed to examine the behavior of the RE-regression estimation and its superiority in practice.

Optimal Thermal Management, Charging, and Eco-Driving of Battery Electric Vehicles

This paper addresses optimal battery thermal management, charging, and eco-driving of a battery electric vehicle (BEV) with the goal of improving its grid-to-meter energy efficiency. Thus, an optimization problem is formulated, aiming at finding the optimal trade-off between trip time and charging cost. The formulated problem is then transformed into a hybrid dynamical system, where the dynamics in driving and charging modes are modeled with different functions and with different state and control vectors. Moreover, to improve computational efficiency, we propose modeling the driving dynamics in a spatial domain, where decisions are made along the traveled distance. Charging dynamics are modeled in a temporal domain, where decisions are made along a normalized charging time. The actual charging time is modeled as a scalar variable that is optimized simultaneously with the optimal state and control trajectories, for both charging and driving modes. The performance of the proposed algorithm is assessed over a road with a hilly terrain, where two charging possibilities are considered along the driving route and the battery is soaked to the ambient before departure. According to the results, trip time including driving and charging times, is reduced by 44%, compared to a case without active heating/cooling of the battery.

The Study of Wind Field ERA-20C in Monsoon Domains for Rainfall Predictor in Indonesia (Java, Sumatra, and Borneo)

In recent years, various research institutions have developed diverse global data reanalysis projects. This provides an opportunity to gain long-term of meteorological data for local scale. This study aims to select the potential predictor of wind fields u and v of the ERA-20C dataset, a reanalysis dataset, at 850 mb from seven domains or windows of Asian, Maritime Continent, Australian, and Western North Pacific monsoon related physically to rainfall anomaly patterns in Indonesia. The vector wind velocity scalar was obtained by using a Helmholtz decomposition to separate the total circulation v = (u,v) into the divergent component/velocity potential (χ) or Phi and rotational component/stream function (ψ) or Psi for obtaining the scalar variable of vector wind velocity. The method applied Singular value decomposition (SVD) to identify pairs of spatial patterns (expansion coefficients) between the predictors of Phi and Psi in seven domains, with rainfall data from 48 stations in Java, Sumatra, and Borneo Islands from 1981 to 2010. The results revealed that spatial patterns correlations of Java Islands were the highest in the Maritime Continent monsoon domain (80o−150o E and 15oS−15o N), while Sumatra and Borneo Island were in the Western North Pacific monsoon domain (100o–130o E and 5o–15o N) with predictor Psi. The lowest correlation for Java, Sumatra, and Borneo was the Australian monsoon domain (110o E–130o E and 5o S–15o S) with predictor Phi. In general, spatial pattern correl-ations of Java Island were higher than others, agreeing with monsoonal rainfall type dominantly in the region.

An efficient and robust Lagrange multiplier approach with a penalty term for phase-field models

In this paper, we propose and analyze a variant of Lagrange multiplier approach with a penalty term for phase-field models. More precisely, we develop two types of linear second-order decoupled and unconditionally energy stable time-stepping schemes for phase-field models based on the Crank-Nicolson (CN) and the second order backward differentiation formulations (BDF2), where the scalar multiplier variable is evaluated by solving a nonlinear algebraic equation. Comparing with the Lagrange multiplier method (Cheng et al. (2020) [6]) for phase-field models, a scalar penalty term is first introduced to improve the time-step size constraint for the convergence of Newton's iterative algorithm for the nonlinear algebraic equation, which allows us to obtain robust and efficient numerical schemes for phase-field models. Moreover, we only use first order time-stepping scheme to approximate the scalar multiplier variable, which will not affect the second order accuracy of the unknown phase function with some special Lagrange multiplier functions. Meanwhile, it plays an essential role in the derivation of energy stability of the proposed BDF2-based scheme on the nonuniform temporal mesh. In addition, the energy stability of the CN scheme is rigorously established. Furthermore, we deduce that the proposed BDF2 scheme is energy stable with the adjacent time-step ratios no more than 4.8645. Finally, a series of numerical tests are presented to numerically validate the efficiency and stability of our proposed Lagrange multiplier approach.

The Proxy Step-Size Technique for Regularized Optimization on the Sphere Manifold.

We give an effective solution to the regularized optimization problem g (x) + h (x), where x is constrained on the unit sphere ||x ||2 = 1. Here g (·) is a smooth cost with Lipschitz continuous gradient within the unit ball {x : ||x ||2 ≤ 1 } whereas h (·) is typically non-smooth but convex and absolutely homogeneous, e.g., norm regularizers and their combinations. Our solution is based on the Riemannian proximal gradient, using an idea we call proxy step-size - a scalar variable which we prove is monotone with respect to the actual step-size within an interval. The proxy step-size exists ubiquitously for convex and absolutely homogeneous h(·), and decides the actual step-size and the tangent update in closed-form, thus the complete proximal gradient iteration. Based on these insights, we design a Riemannian proximal gradient method using the proxy step-size. We prove that our method converges to a critical point, guided by a line-search technique based on the g(·) cost only. The proposed method can be implemented in a couple of lines of code. We show its usefulness by applying nuclear norm, l1 norm, and nuclear-spectral norm regularization to three classical computer vision problems. The improvements are consistent and backed by numerical experiments. available at https://bitbucket.org/FangBai/proxystepsize-pgs.

Quantum Fractionary Cosmology: K-Essence Theory

Using a particular form of the quantum K-essence scalar field, we show that in the quantum formalism, a fractional differential equation in the scalar field variable, for some epochs in the Friedmann–Lemaı^tre–Robertson–Walker (FLRW) model (radiation and inflation-like epochs, for example), appears naturally. In the classical analysis, the kinetic energy of scalar fields can falsify the standard matter in the sense that we obtain the time behavior for the scale factor in all scenarios of our Universe by using the Hamiltonian formalism, where the results are analogous to those obtained by an algebraic procedure in the Einstein field equations with standard matter. In the case of the quantum Wheeler–DeWitt (WDW) equation for the scalar field ϕ, a fractional differential equation of order β=2α2α−1 is obtained. This fractional equation belongs to different intervals, depending on the value of the barotropic parameter; that is to say, when ωX∈[0,1], the order belongs to the interval 1≤β≤2, and when ωX∈[−1,0), the order belongs to the interval 0<β≤1. The corresponding quantum solutions are also given.

Investigating time-dependent behavior of rocks using kinematic-constraint-inspired non-ordinary state-based peridynamics

In this paper, the kinematic-constraint-inspired non-ordinary state-based peridynamic formulation (KC-NOSBPD) is proposed to investigate the time-dependent behaviors of rocks, which play a critical role in evaluating the long-term stability of the surrounding rocks around underground opening quantitatively. The peridynamic differential operator (PDDO) is introduced to KC-NOSBPD to improve the precision of bond-associated deformation gradient and to suppress the skin effect. On this basis, a unified creep-damage-rupture KC-NOSBPD model is proposed, in which the Burgers-creep viscoplastic constitutive law, a visco-elastoplastic constitutive model, is used to describe the rheological deformation of rocks. A unified damage-rupture framework based on a strain-based isotropic scalar damage variable is established to simulate the creep damage and subsequent creep rupture. The fidelity of this proposed model without damage is verified by comparing it with the benchmark solution for the creep response of a 2D rock specimen subjected to various load histories. Further, the proposed model is employed to simulate uniaxial and triaxial compression creep tests. These present numerical results agree well with the experimental observation, implying that the proposed model successfully captures the accelerating creep behavior of rocks. Moreover, the proposed model performs the time-dependent deformation and failure analysis for the surrounding rocks around the tunnel.

A two stage Active Disturbance Rejection Control design for under-actuated nonlinear systems

A two-stage Active Disturbance Rejection Control (ADRC) design procedure is presented for the output trajectory tracking control problem in Single Input Single Output (SISO) nonlinear systems. Aside from the given scalar system output, the plant is assumed to exhibit a second, physically measurable, internal scalar output variable. The system output is controlled in an ADRC manner from an implicit static, or dynamic, nonlinear feedback controller using, in general, a nonlinear differential expression of the internal output variable. A desired internal output trajectory is thus generated. The internal output variable, in turn, is controlled from the original input and forced to track the desired trajectory synthesized in the previous step by the auxiliary controller. Each ADRC controller design is based on a simplified model of the corresponding subsystem, usually relegating interaction variables and additive nonlinearities to be part of a local (total) disturbance input term. The control scheme is reduced to a set of linear ADRC controllers with, possibly, cancellation of nonlinear input gains which are assumed to be known. Two illustrative aerospace examples are presented where the proposed procedure is illustrated via digital computer simulations.

Optimal convergence rates in L2 for a first order system least squares finite element method

We analyze a divergence based first order system least squares method applied to a second order elliptic model problem with homogeneous boundary conditions. We prove optimal convergence in the L2(Ω) norm for the scalar variable. Numerical results confirm our findings.

Cover crops effects on anisotropy of unsaturated soil hydraulic properties

No tillage (NT) is frequently implemented in simplified agricultural systems, which has negative effects on soil physical quality. Among the suggested practices to preserve the physical fertility of soils under NT are winter cover crops (CC). Soil physical degradation affects the soil pore configuration and therefore it can generate a change in the directionality of the properties that depend on it. The objective of this study was to determine the effect of cover crop incorporation under NT on the soil pore system configuration and on the anisotropy of the unsaturated hydraulic properties in a Typic Argiudoll. The results showed that the inclusion of cover crops modifies the soil pore configuration increasing the structure porosity. The inclusion of CC modifies the anisotropy of unsaturated hydraulic conductivity in the medium saturation region. The pore size distribution behaves as isotropic as expected for a scalar variable.

Robust Fuzzy Feedback Control for Nonlinear Systems With Input Quantization

In this article, we study the robust quantized feedback control problem for nonlinear discrete-time systems that are described by Takagi–Sugeno (T–S) fuzzy model with norm-bounded uncertainties. The dynamic quantizer composed of a dynamic parameter and a static quantizer is considered to quantize the control input signal. An improved two-step approach to design controller and dynamic quantizer for T–S fuzzy system is proposed based on the LMI technique. In the first step, a robust controller is designed to guarantee that the quantized fuzzy closed-loop system with norm-bounded uncertainties is asymptotically stable with prescribed <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\mathcal {H}_{\infty }$</tex-math></inline-formula> performance. Then, the parameter-dependent (membership function) scalar variable is obtained to determine the dynamic quantizer’s parameter in the second step. Finally, the simulation result of truck–trailer system is presented to validate the effectiveness and feasibility of the proposed two-step design approach.

Modeling the rheological behavior of crude oil–water emulsions

During crude oil extraction, crude oil is often mixed with water, leading to the formation of water-in-oil emulsions. Since these emulsions pose severe flow resistance, such as higher pressure drops, due to their complex fluid rheology, it is important to have in our arsenal a rheological constitutive model that accurately predicts their rheological response. In the present work, we propose such a model wherein the emulsions are modeled as deformable volume-preserving droplets via the use of a determinant-preserving contravariant second-rank tensor. We use the generalized bracket formalism of non-equilibrium thermodynamics to make sure that the derived model is by construction thermodynamically admissible. An additional scalar structural variable is considered to allow the prediction of a yielding point, following previous work. The predictions of the new model are shown to be in very good agreement with available experimental measurements.