Continuous Linear Functionals Research Articles

Generalized Linear Differential Equation using Hyers - Ulam Stability Approach

In this paper, We demonstrate the Hyers - Ulam stability of linear differential equation of fourth order. We interact with the differential equation\begin{align*}\gamma^{iv} (\omega) + \rho_1 \gamma{'''} (\omega)+ \rho_2 \gamma{''} (\omega) + \rho_3 \gamma' (\omega) + \rho_4 \gamma(\omega) = \chi(\omega),\end{align*}where $\gamma \in c^4 [\alpha,\beta], \chi \in [\alpha,\beta]$. Hyers-Ulam stability concerns the robustness of solutions of functional equations under small perturbations, ensuring that a solution approximately satisfying the equation is close to an exact solution. We extend this concept to fourth-order linear differential equations and continuous functions. Using fixed-point methods and various norms, we establish conditions under which such equations exhibit Hyers-Ulam stability. Several illustrative examples are provided to demonstrate the application of these results in specific cases, contributing to the growing understanding of stability in higher-order differential equations. Our findings have implications in both theoretical research and practical applications in physics and engineering.

Representation and inequalities involving continuous linear functionals and fractional derivatives

We investigate how continuous linear functionals can be represented in terms of generic operators and certain kernels (Peano kernels), and we study lower bounds for the operators as a consequence, in the space of square-integrable functions. We apply and develop the theory for the Riemann–Liouville fractional derivative (an inverse of the Riemann–Liouville integral), where inequalities are derived with the Gaussian hypergeometric function. This work is inspired by the recent contributions by Fernandez and Buranay (J Comput Appl Math 441:115705, 2024) and Jornet (Arch Math, 2024).

Nonlinear normal modes and response to random inputs of systems with bilinear stiffness

This work seeks to provide a comprehensive review of the effects that stiffness bilinearity can have on the nonlinear modes of a system, its response in a random vibration environment, and the connection between the two. Stiffness bilinearity here refers to a continuous piecewise linear force vs displacement function that is composed of two linear regions. This work focuses on a bilinear stiffness function that is regularized so there is a smooth transition at the point where the two linear regions meet. A single-degree-of-freedom system (SDOF) and a two-degree-of-freedom (2DOF) system are explored and several interesting behaviors are shown. The SDOF bilinear spring model is characterized by four parameters: the low amplitude frequency, the ratio of the linear stiffnesses on either side of the transition, the displacement at which the transition occurs, and the rate or sharpness of the transition. The effect of each parameter on the shape of the NNM is described. In a 2DOF system, these parameters have similar effects, but modal coupling is found to play a significant role. When a bilinear system is subjected to random excitation, many harmonics appear in the response for both the SDOF and 2DOF cases. The root-mean-square (RMS) response of the bilinear system can be larger or smaller than the corresponding linear case depending on the values of the parameters and the type of forcing (broadband, bandlimited, in the shape of a vibration mode, etc.). However, many cases were observed in which the RMS response of the bilinear system was almost the same as that of a linear system, and hence the response could be predicted well using linear analysis. It is hoped that the results presented herein can assist engineers in helping to determine when linear analysis would be adequate to predict the failure of a system, when a rigorous nonlinear analysis is required, and what phenomena are likely to be observed in the latter case.

Comparison of local muscle oxygenation and whole-body VO2 during incremental exercise

The use of near-infrared spectroscopy (NIRS) in sports science has become increasingly important in the past few years (Perrey et al., 2024). One application is to determine thresholds from different NIRS parameters like oxygenated (O2Hb) or deoxygenated (HHb) hemoglobin to distinguish between hard and severe-intensity exercise (Caen et al., 2018). Such applications are frequently validated by comparison to established threshold determination concepts. However, it is not yet known which of the two parameters is more closely related to classical concepts. Therefore, the aim of this project was do analyze which NIRS parameter (oxygenated [O2Hb] or deoxygenated [HHb] hemoglobin) better reflects systemic behavior determined from ventilation (VE). Thirteen physically active participants (4 females, 9 males; mean age: 26.3 ± 4.4 years, height: 176.0 ± 10.3 cm, weight: 71.9 ± 9.3 kg) were included in the study. VE and NIRS data from a major locomotor muscle (vastus lateralis) during a step-incremental single leg cycle ergometer test were recorded. We used a custom MATLAB script to fit piecewise linear continuous functions by regression to estimate NIRS and VE thresholds. We used two distinct segments for NIRS data and three distinct segments for VE data. NIRS thresholds were defined as the point of intersection of the two segments in O2Hb and HHb signals, respectively. The ventilatory threshold was defined as the second increase in VE. We used a resampling strategy to reduce the impact of outliers or peculiar data distributions on threshold estimates. Thresholds from O2Hb and HHb in the active leg and VE were observed at 78.24%, 76.37%, and 84.15% of the maximum power achieved. Both O2Hb (r = 0.85, p < 0.001) and HHb (r = 0.75, p = 0.0051) thresholds were significantly correlated, but mean values were significantly different (p < 0.05) from the respiratory compensation point (RCP) mean value. Correlations between O2Hb and HHb values from the passive leg and the RCP values were in the same range (r = 0.88, p < 0.001 and r = 0.79, p = 0.0023, respectively). The results indicate that, based on the correlations, O2Hb is slightly closer related to the second ventilatory threshold compared to HHb. The present study shows that thresholds from both NIRS parameters are closely related to the second ventilatory threshold which indicates a physiological relationship. However, thresholds cannot be used interchangeably as NIRS parameters derived from the working muscle appear earlier than systemic thresholds that demonstrate a delay in kinetics between the working muscle and the system.

Symmetric linear functionals on the Banach space generated by pseudometrics

In this work we consider the notion of $B$-equivalence of pseudometrics.Two pseudometrics $d_1$ and $d_2$ on a set $X$ are called $B$-equivalent, where $B$ is a subgroup of the group of all bijections on $X,$ if there exists an element $b$ of $B$ such that $d_1(x,y) = d_2(b(x),b(y))$ for every $x,y\in X,$ that is, $d_1$ can be obtained from $d_2$ by permutating elements of $X$ with the aid of the bijection $b.$The group $B$ generates the group $\widehat B$ of transformations of the set of all pseudometricson $X,$ elements of which act as $d(\cdot, \cdot)\mapsto d(b(\cdot),b(\cdot)),$ where $d$ is a pseudometrics on $X$ and $b\in B.$ A function $f$ on the set of all pseudometrics on $X$is called $\widehat B$-symmetric if $f$ is invariant under the action on its argument of elements of the group $\widehat B.$If two pseudometrics $d_1$ and $d_2$ are $B$-equivalent, then $f(d_1)=f(d_2)$ for every $\widehat B$-symmetric function $f.$ In general, the technique of symmetric functions is well-developed for the case of symmetric continuous polynomials and, in particular, for the case of symmetric continuous linear functionals on Banach spaces. To use this technique for the construction of $\widehat B$-symmetricfunctions on sets of pseudometrics, we map these sets to some appropriate Banach space $V$, which is isometrically isomorphic to the Banach space $\ell_1$of all absolutely summing real sequences. Weinvestigate symmetric (with respect to an arbitrary group of symmetry, elements of whichmap the standard Schauder basis of $\ell_1$ into itself) linear continuous functionalson $\ell_1.$ We obtain the complete description of the structure of these functionals.Also we establish analogical results for symmetric linear continuous functionals on the space $V.$ These results are used for the construction of $\widehat B$-symmetric functionals on the set of all pseudometrics on an arbitrary set $X$ for the following case:the group $B$ of bijections on $X,$ that generates the group $\widehat B,$ is such that the set of all $x\in X,$ for which there exists $b\in B$ such that $b(x)\neq x,$is finite.

The Peano–Sard theorem for fractional operators with Mittag-Leffler kernel and application in classical numerical approximation

We investigate fractional Peano kernels for continuous linear functionals, in the context of differintegral operators with Mittag-Leffler kernel. New bounds for polynomial interpolation are obtained and numerical computations are shown, indicating improvements.

Accurate and fast quaternion fractional-order Franklin moments for color image analysis

Quaternion type moments (QTM) based on hypergeometric functions and harmonic functions have attracted interest in the imaging field. The Franklin system is a complete orthonormal system consisting of piecewise linear continuous functions with Haar wavelet collocation points. However, due to the lack of efficient computational methods and explicit expressions, research on piecewise polynomials with Haar wavelet collocation points remains underexplored. Furthermore, the impact of magnitude and phase of quaternion on the color image features extracted by QTM has not been comprehensively explored. To address these problems, a fast and accurate computation method for Franklin functions is proposed. Quaternion fractional-order Franklin moments (QFFM) are proposed and using the Minkowski distance as a prior parameter for selecting unit pure quaternions. Experiments are conducted to demonstrate the superior performance of QFFM in image reconstruction and color image classification tasks compared with QTM-based and deep learning-based methods. Moreover, with the help of experiments, it is confirmed the effectiveness of using the Minkowski distance to select magnitude and phase of quaternion.

A new lower bound for the L2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{L}^2$$\\end{document}-norm of the Caputo fractional derivative

We prove a novel, tight lower bound for the norm in L2[0,T]\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{L}^2[0,T]$$\\end{document} of the Caputo fractional derivative. It is based on continuous linear functionals, Peano kernels, and the Gaussian hypergeometric function.

From local utility to neural networks

We introduce and analyze two preference-based notions of local linearity in the spirit of Machina (1982). We show how the weaker among the two extends Machina’s local utility analysis, and that the stronger among the two characterizes continuous finite piecewise linear (CFPL) utility functions. We introduce a representation of the decision maker’s preference called the neural-network utility representation that is equivalent to the CFPL representation, in which the decision maker evaluates an alternative through a neural network.

Primitives of continuous functions via continuous piecewise linear functions

In classical calculus textbooks, the existence of primitive functions of continuous functions is proved by using Riemann integrals. Recently, Patrik Lundström gave a proof via polynomials, based on the Weierstrass approximation theorem. In this note, it is shown that the proof will be easy by using continuous piecewise linear functions.

Spurious Local Minima are Common for Deep Neural Networks With Piecewise Linear Activations.

In this article, theoretically, it is shown that spurious local minima are common for deep fully connected networks and average-pooling convolutional neural networks (CNNs) with piecewise linear activations and datasets that cannot be fit by linear models. Motivating examples are given to explain why spurious local minima exist: each output neuron of deep fully connected networks and CNNs with piecewise linear activations produces a continuous piecewise linear (CPWL) function, and different pieces of the CPWL output can optimally fit disjoint groups of data samples when minimizing the empirical risk. Fitting data samples with different CPWL functions usually results in different levels of empirical risk, leading to the prevalence of spurious local minima. The results are proved in general settings with arbitrary continuous loss functions and general piecewise linear activations. The main proof technique is to represent a CPWL function as maximization over minimization of linear pieces. Deep networks with piecewise linear activations are then constructed to produce these linear pieces and implement the maximization over minimization operation.

Fixed-Time Adaptive Parameter Estimation for Hammerstein Systems Subject to Dead-Zone

The Hammerstein model has proven its ability for modeling many industrial servo systems, but the corresponding parameter estimation remains as a challenging problem, especially for the purpose of achieving fast convergence. This paper presents a fixed-time (FxT) adaptive parameter estimation scheme for Hammerstein systems with an asymmetric dead-zone to ensure the convergence of estimated parameters in fixed-time. A continuous piecewise linear (CPL) function is adopted to model the nonlinear dead-zone dynamics to remedy the use of intermediate variable. To avoid using the system state information, a K-filter is used to construct a relationship between the unknown system parameters and the input/output signals. Then a new adaptive law based on the parameter error and the FxT scheme is developed to online estimate the lumped unknown parameters and ensure the fixed-time convergence, where an online updated auxiliary variable of regressor is suggested to eliminate the impact of the regressor on the convergence performance. Both numerical simulations and experimental results are given to demonstrate the effectiveness of proposed methods.

CD-type discretization for sea ice dynamics in FESOM version 2

Abstract. Two recently proposed variants of CD-type discretizations of sea ice dynamics on triangular meshes are implemented in the Finite-VolumE Sea ice–Ocean Model (FESOM version 2). The implementations use the finite element method in spherical geometry with longitude–latitude coordinates. Both are based on the edge-based sea ice velocity vectors but differ in the basis functions used to represent the velocities. The first one uses nonconforming linear (Crouzeix–Raviart) basis functions, and the second one uses continuous linear basis functions on sub-triangles obtained by splitting parent triangles into four smaller triangles. Test simulations are run to show how the performance of the new discretizations compares with the A-grid discretization using linear basis functions. Both CD discretizations are found to simulate a finer structure of linear kinematic features (LKFs). Both show some sensitivity to the representation of scalar fields (sea ice concentration and thickness). Cell-based scalars lead to a finer LKF structure for the first CD discretization, but the vertex-based scalars may be advantageous in the second case.

Chance-Constrained Optimization for a Green Multimodal Routing Problem with Soft Time Window under Twofold Uncertainty

This study investigates a green multimodal routing problem with soft time window. The objective of routing is to minimize the total costs of accomplishing the multimodal transportation of a batch of goods. To improve the feasibility of optimization, this study formulates the routing problem in an uncertain environment where the capacities and carbon emission factors of the travel process and the transfer process in the multimodal network are considered fuzzy. Taking triangular fuzzy numbers to describe the uncertainty, this study proposes a fuzzy nonlinear programming model to deal with the specific routing problem. To make the problem solvable, this study adopts the fuzzy chance-constrained programming approach based on the possibility measure to remove the fuzziness of the proposed model. Furthermore, we use linear inequality constraints to reformulate the nonlinear equality constraints represented by the continuous piecewise linear functions and realize the linearization of the nonlinear programming model to improve the computational efficiency of problem solving. After model processing, we can utilize mathematical programming software to run exact solution algorithms to solve the specific routing problem. A numerical experiment is given to show the feasibility of the proposed model. The sensitivity analysis of the numerical experiment further clarifies how improving the confidence level of the chance constraints to enhance the possibility that the multimodal route planned in advance satisfies the real-time capacity constraint in the actual transportation, i.e., the reliability of the routing, increases both the total costs and carbon emissions of the route. The numerical experiment also finds that charging carbon emissions is not absolutely effective in emission reduction. In this condition, bi-objective analysis indicates the conflicting relationship between lowering transportation activity costs and reducing carbon emissions in routing optimization. The sensitivity of the Pareto solutions concerning the confidence level reveals that reliability, economy, and environmental sustainability are in conflict with each other. Based on the findings of this study, the customer and the multimodal transport operator can organize efficient multimodal transportation, balancing the above objectives using the proposed model.

A dual mapping associated to a closed convex set and some subdifferential properties

In this paper, we establish some properties of the multivalued mapping (x, d) ⇒ DC (x; d) that associates to every element x of a linear normed space X the set of linear continuous functionals of norm d ≥ 0 and which separates the closed ball B (x; d) from a closed convex set C ⊂ X. Using this mapping we give links with other important concepts in convex analysis (ε-approximation element, ε-subdifferential of distance function, duality mapping, polar cone). Thus, we establish a dual characterization of ε-approximation elements with respect to a nonvoid closed convex set as a generalization of a known result of Garkavi. Also, we give some properties of univocity and monotonicity of mapping DC. Mathematics Subject Classification (2010): 32A70, 41A65, 46B20, 46N10. Received 18 May 2023; Accepted 27 November 2023

Tractability of Approximation of Functions Defined over Weighted Hilbert Spaces

We investigate L2-approximation problems in the worst case setting in the weighted Hilbert spaces H(KRd,α,γ) with weights Rd,α,γ under parameters 1≥γ1≥γ2≥⋯≥0 and 1<α1≤α2≤⋯. Several interesting weighted Hilbert spaces H(KRd,α,γ) appear in this paper. We consider the worst case error of algorithms that use finitely many arbitrary continuous linear functionals. We discuss tractability of L2-approximation problems for the involved Hilbert spaces, which describes how the information complexity depends on d and ε−1. As a consequence we study the strongly polynomial tractability (SPT), polynomial tractability (PT), weak tractability (WT), and (t1,t2)-weak tractability ((t1,t2)-WT) for all t1>1 and t2>0 in terms of the introduced weights under the absolute error criterion or the normalized error criterion.

Efficient P1-FEM for Any Space Dimension in Matlab

Abstract This paper deals with the efficient implementation of the finite element method with continuous piecewise linear functions (P1-FEM) in R d \mathbb{R}^{d} ( d ∈ N d\in\mathbb{N} ). Although at present there does not seem to be a very high practical demand for finite element methods that use higher-dimensional simplicial partitions, there are some advantages in studying the efficient implementation of the method independent of the dimension. For instance, it provides additional insights into necessary data structures and the complexity of implementations. Throughout, the focus is on an efficient realization using Matlab built-in functions and vectorization. The fast and vectorized Matlab function can be easily implemented in many other vector languages and is provided in Julia, too. The complete implementation of the adaptive FEM is given, including assembling stiffness matrix, building load vector, error estimation, and adaptive mesh-refinement. Numerical experiments underline the efficiency of our freely available code which is observed to be of a slightly more than linear complexity with respect to the number of elements when memory limits are not exceeded.

On the enriched mixed Galerkin method with θ scheme for the elastic wave equation

In this paper, we propose the application of an enriched Galerkin (EG) method to solve the elastic wave equation in the “displacement pseudo-pressure” mixed form, which is known for being locking-free. Unlike the continuous Galerkin (CG) method and the discontinuous Galerkin (DG) method, the EG method is based on the weak form of the DG method, and the enriched space combines a continuous piecewise linear vector-valued function space with some discontinuous piecewise linear functions. Specifically, we employ the enriched space to discretize the displacement, while the pseudo-pressure is discretized using the P0 element space. As a result, compared to the DG method, the EG method significantly reduces the number of degrees of freedom. Furthermore, we adopt a more general θ-scheme for time discretization, which behaves as an implicit scheme for 14≤θ≤1 and as an explicit scheme for 0≤θ<14. We establish error estimates for both the semi-discrete and fully-discrete schemes. The accuracy of the proposed method is validated through numerical examples, which not only confirm the theoretical analysis but also provide simulations of elastic wave propagation.

A new analytical method for estimating hydraulic parameters of a nonlinear consolidated aquitard with variable drawdown in adjacent aquifers

Estimation of hydraulic parameters and groundwater depletion of aquitards is necessary to evaluate the groundwater resources and land subsidence. In this study, an analytical solution for water release from a nonlinear consolidated aquitard was derived by approximating the drawdown history as a continuous piecewise linear drawdown function (PLD) with several stages, while the water level in adjacent aquifers varies arbitrarily over time. According to dimensionless and parameter sensitivity analyses, groundwater depletion of an aquitard increases with an increase in compression index, aquitard thickness, and water drawdown rate in adjacent aquifers, decreases with an increase in initial void ratio and effective stress, and is not affected by the consolidation coefficient. In addition, we proposed a type-curve fitting method to calculate the hydraulic parameters of the aquitard under the condition of PLD in adjacent aquifers. The proposed method was applied to estimate the hydraulic parameters of aquitards at two field sites in the Yangtze River Delta in China, i.e., Changzhou and Shanghai. Results show that the compression index of the aquitards in Changzhou and Shanghai are 0.074 and 0.24, respectively, while the consolidation coefficients are 7.62 × 10-7 m2/s and 2.16 × 10-6 m2/s, respectively. The hydraulic conductivity and specific storage estimated by the proposed method are in good agreement with previous studies and the results of the geological method of Konikow and Neuzil (2007). The proposed analytical method for estimating hydraulic parameters and groundwater depletion of aquitards will contribute to the sustainable management of groundwater resources in multi-layer aquifer systems.

Linear inverse problems with Hessian–Schatten total variation

In this paper, we characterize the class of extremal points of the unit ball of the Hessian–Schatten total variation (HTV) functional. The underlying motivation for our work stems from a general representer theorem that characterizes the solution set of regularized linear inverse problems in terms of the extremal points of the regularization ball. Our analysis is mainly based on studying the class of continuous and piecewise linear (CPWL) functions. In particular, we show that in dimension d=2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$d=2$$\\end{document}, CPWL functions are dense in the unit ball of the HTV functional. Moreover, we prove that a CPWL function is extremal if and only if its Hessian is minimally supported. For the converse, we prove that the density result (which we have only proven for dimension d=2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$d=2$$\\end{document}) implies that the closure of the CPWL extreme points contains all extremal points.