Discrete Form Research Articles

Unraveling the Implications of Digit Bias in Digital Health - A Literature Review.

Digital health, which encompasses digital medicine and therapy, integrates advanced technologies across healthcare. Central to this transformation is 'digitization,' which converts continuous analog data into a discrete digital form. However, this process is challenging. First, digitization inherently has the potential to introduce information loss, thereby diminishing the richness and complexity of data. Second, "digit bias," a cognitive distortion, emerges in the interpretation phase, where individuals' perceptions of and reactions to digital data are intrinsically skewed. There exist two major cognitive biases during digitization process: "digit preferences," where healthcare providers prioritize specific numbers, and "left digit bias" where continuous variables are disproportionately estimated by focusing on the leftmost digit. Although information loss and cognitive biases can cause significant distortions in healthcare, the effects of this "digitization" process have not been adequately quantified, and the accumulation of further evidence in this field is anticipated.

Efficient relaxation scheme for the SIR and related compartmental models

In this paper, we introduce a novel numerical approach for approximating the Susceptible–Infectious–Recovered (SIR) model in epidemiology. Our method enhances the existing linearization procedure by incorporating a suitable relaxation term to tackle the transcendental equation of nonlinear type. Developed within the continuous framework, our relaxation method is explicit and easy to implement, relying on a sequence of linear differential equations. This approach yields accurate approximations in both discrete and analytical forms. Through rigorous analysis, we prove that, with an appropriate choice of the relaxation parameter, our numerical scheme is non-negativity-preserving; moreover, it is strongly convergent to the true solution. We also extend the applicability of our relaxation method to handle some variations of the traditional SIR model. Finally, we present numerical examples using simulated data to demonstrate the effectiveness of our proposed method.

Modeling 2D gravity-driven flow in unsaturated porous media for different infiltration rates

Abstract. The gravity-driven flow in an unsaturated porous medium remains one of the most important unsolved problems in multiphase flow. Sometimes a diffusion-like flow with a uniform wetting front, known as stable flow, is observed, but, at other times, the flow is unstable with distinct preferential pathways. The formation of an unstable wetting front in a porous medium depends on many factors, including the type of porous medium, the initial saturation, and the applied infiltration rate. As the infiltration rate increases, the wetting front first transitions from stable to unstable at low infiltration rates and then from unstable to stable at high infiltration rates. We propose a governing equation and its discretized form, the semi-continuum model, to describe this significant non-monotonic transition. We show that the semi-continuum model is able to capture the influx dependence together with the correct finger width and spacing. Moreover, we demonstrate that the instability of the wetting front is closely related to the saturation overshoot in one dimension. Finally, we show that the flow can still be preferential even when the porous medium is completely wetted.

Explicit Runge–Kutta Numerical Manifold Method for Solving the Burgers’ Equation via the Hopf–Cole Transformation

This paper presents an efficient numerical manifold method for solving the Burgers’ equation. To improve accuracy and streamline the solution process, we apply a nonlinear function transformation technique that reformulates the original problem into a linear diffusion equation. We utilize a dual cover mesh along with an explicit multi-step time integration method for spatial and temporal discretization, respectively. Constant cover functions are employed across mathematical covers, interconnected by a linear weight function for each manifold element. The full discretization formulation is derived using the Galerkin weak form. To efficiently compute the inverse of the symmetric positive definite mass matrix, we employ the Crout algorithm. The performance and convergence of our method are rigorously evaluated through several benchmark numerical tests. Extensive comparisons with exact solutions and alternative methods demonstrate that our approach delivers an accurate, stable, and efficient computational scheme for the Burgers’ equation.

Dynamics of a discrete Rosenzweig–MacArthur predator–prey model with piecewise-constant arguments

This paper investigates the dynamics of a new discrete form of the classical continuous Rosenzweig–MacArthur predator–prey model and the biological implications of the dynamics. The discretization method used here is to modify the continuous model to another with piecewise-constant arguments and then to integrate the modified model, which is quite different from the traditional Euler discretization method. First, the existence and local stability of fixed points have been thoroughly discussed. Then, all codimension-1 bifurcations have been studied, including the transcritical bifurcations at the trivial fixed point and the boundary fixed point, and the Neimark–Sacker bifurcation at the unique positive fixed point. Furthermore, it is proved that there are no codimension-2 bifurcations. Finally, the control of the bifurcations is studied. Regarding the biological significance, we have considered the following four aspects: When no harvesting effort is applied to both predators and prey, it is observed that providing more resources to the prey species leads to predator extinction, causing the ecosystem to lose stability. This implies that the paradox of enrichment occurs. When predators are harvested, the system exhibits the hydra effect, i.e. increasing the harvesting effort on predators will lead to an increase in the mean population density of predators. When the prey is harvested, numerical examples indicate that increasing the harvesting effort initially reduces prey density. Upon reaching the bifurcation point, prey density stabilizes while the predator density continues to decline. When both prey and predators are harvested, the biological significance is similar to the previous case. The outcomes of this paper have significant theoretical meaning in the study of population ecology. By MATLAB, the bifurcation diagrams, maximal Lyapunov exponent diagrams, phase portraits and mean population density curves are plotted. Numerical simulations not only show the correctness of the theoretical findings but also reveal many new and interesting dynamics.

Model Predictive Control for Three-Phase, Four-Leg Dynamic Voltage Restorer

Dynamic voltage restores (DVRs) are usually used to mitigate the effect of voltage sag and guarantee sufficient power supply for sensitive loads. However, three-phase voltage cannot be compensated to the desired balance voltage under unbalanced three-phase loads by traditional DVRs with a three-phase, three-leg inverter. To address this problem, a three-phase, four-leg inverter-based DVR is first introduced in this paper, and then the state space model in its continuous form and discrete form are established, respectively. A two-step predictive method is proposed for the prediction of the output voltage in each switching state by the established voltage prediction model. Finite-control-set model predictive control (MPC) is developed to be used in the three-phase, four-leg inverter-based DVR. Its dynamic response is effectively improved by the proposed MPC method under various voltage sag conditions. The proposed DVR control strategy is validated via MATLAB/Simulink-R2022b simulations, which demonstrate its effectiveness in voltage compensation under various sag conditions.

Transparent boundary condition and its high frequency approximation for the Schrödinger equation on a rectangular computational domain

This paper addresses the numerical implementation of the transparent boundary condition (TBC) and its various approximations for the free Schrödinger equation on a rectangular computational domain. In particular, we consider the exact TBC and its spatially local approximation under high frequency assumption along with an appropriate corner condition. For the spatial discretization, we use a Legendre–Galerkin spectral method where Lobatto polynomials serve as the basis. Within variational formalism, we first arrive at the time-continuous dynamical system using spatially discrete form of the initial boundary-value problem incorporating the boundary conditions. This dynamical system is then discretized using various time-stepping methods, namely, the backward-differentiation formula of order 1 and 2 (i.e., BDF1 and BDF2, respectively) and the trapezoidal rule (TR) to obtain a fully discrete system. Next, we extend this approach to the novel Padé based implementation of the TBC presented by Yadav and Vaibhav (2024). Finally, several numerical tests are presented to demonstrate the effectiveness of the boundary maps (incorporating the corner conditions) where we study the stability and convergence behaviour empirically.

Extension of the local domain-free discretization method to large eddy simulation of compressible flows

Most of the flow problems encountered in practical engineering are wall-bounded turbulent flows at high Reynolds numbers. Wall-modeled large eddy simulation (WMLES) is one of the most viable approaches for predicting these realistic flows. Immersed boundary (IB) approach is an efficient computational technique to solve flow problems involving complex and/or moving geometries. This work extends a sharp-interface IB method, named the local domain-free discretion (DFD), to WMLES of compressible flows at high Reynolds numbers. An equilibrium wall model based on solving the simplified compressible turbulent boundary layer equations is utilized to alleviate the requirement of high near-wall mesh resolution. In conjunction with the approximate boundary conditions prescribed by the modeled wall shear stress and wall heat flux, the tangential velocity and temperature at an exterior dependent node are evaluated. Then, the closure of the discrete form of governing equations at an interior node in the immediate vicinity of the immersed wall is accomplished. A simple non-equilibrium correction of the wall shear stress provided by the equilibrium wall model is introduced explicitly. The WMLES/DFD method is applied to a supersonic zero-pressure-gradient turbulent boundary layer flow, a shock wave/flat-plate boundary layer interaction, a supersonic compression ramp flow and high-speed turbulent Couette flows with various thermal boundary conditions. The influence of grid resolution is investigated in the simulation of zero-pressure-gradient turbulent boundary layer flow. By comparing the computed results with the referenced experimental data and/or numerical results, the accuracy and ability of the WMLES/DFD method to simulate compressible turbulent flows are verified.

Regulated cell death in chronic kidney disease: current evidence and future clinical perspectives.

Chronic kidney disease (CKD) is the progressive loss of kidney function/structure over a period of at least 3months. It is characterised histologically by the triad of cell loss, inflammation and fibrosis. This literature review focuses on the forms of cell death that trigger downstream inflammation and fibrosis, collectively called regulated cell death (RCD) pathways. Discrete forms of RCD have emerged as central mediators of CKD pathology. In particular, pathways of regulated necrosis - including mitochondrial permeability transition pore (mPTP)-mediated necrosis, necroptosis, ferroptosis and pyroptosis - have been shown to mediate kidney pathology directly or through the release of danger signals that trigger a pro-inflammatory response, further amplifying tissue injury in a cellular process called necroinflammation. Despite accumulating evidence in pre-clinical models, no clinical studies have yet targeted these RCD modes in human CKD. The review summarizes recent advances in our understanding of RCD pathways in CKD, looks at inter-relations between the pathways (with the emphasis on propagation of death signals) and the evidence for therapeutic targeting of molecules in the RCD pathways to prevent or treat CKD.

Stable Droplet Manipulation in Multiple Directions on a Magnetically Responsive Micropillar Array Surface

AbstractDroplets constitute an essential discrete form of liquids and are prevalent in both natural and industrial fields. Droplet manipulation technologies have broad application prospects in various fields, including biomedical devices, chemical engineering, water collection, and so on. The manipulation of surfaces with magnetic responsiveness has emerged as a pivotal technique in droplet control owing to its flexibility and precision in remote operations. In this study, the dynamic rolling behavior of low‐surface‐tension working fluids and water droplets on a superhydrophobic magnetically responsive micropillar array (SMMA) is investigated, with a focus on the mechanisms that generate instabilities during the directional rolling motion of droplets. The factors influencing these instabilities are studied using data analysis. Moreover, techniques for capturing and releasing droplets are investigated, offering significant insights into the efficient handling of minute amounts of liquids in experiments and the automation of intricate chemical processes.

Being is said in many ways

Abstract The article focuses on the ontological foundations of lifeworld as Being taken for granted and viewed as a communication phenomenon par excellence, conceptualized as signifying in the presence of others. It is argued that, because there is always a wider horizon of experience against which anything can appear, lifeworld as something continuous can only be thematized in discrete scientific forms. In the article, lifeworld is discussed through the perspectives of four different sciences. From the natural science perspective, lifeworld is conceptualized as an arrangement contingent on the interconnections of human and nonhuman beings. It takes the form of conversation and can be best examined by Conversation Analysis. From the social science perspective, lifeworld is conceptualized as an order that is constituted by symbolic interactions and guided by relevant types and rules that structure our experience. It takes the form of intersubjective action as a meaningful back and forth and can be best examined by Discourse Analysis. From the cultural sciences perspective, lifeworld is the realm of spirit passed from generation to generation that gives human culture its meaning. It takes the form of significant expressions and can be best examined by Textual Hermeneutic Analysis. And, from the philosophical perspective, lifeworld is constituted by phenomena that appear, or monstrate themselves, to consciousness. This lifeworld is of absolute reversibility and immediacy and can be best examined by Critical Analysis, i.e., by postulating our relation to any object through contemplation. Thus, as Being, lifeworld is said in many ways, each lifeworld being relevant in its own way by providing a valuable resource – physico-biological, normative, significational, and transcendental.

From Trypomastigotes to Trypomastigotes: Analyzing the One-Way Intracellular Journey of Trypanosoma cruzi by Ultrastructure Expansion Microscopy.

The protozoan parasite Trypanosoma cruzi is the causative agent of Chagas disease, also called American trypanosomiasis. This neglected tropical disease affects millions of individuals across the Americas. To complete its life cycle, T. cruzi parasitizes both vertebrate hosts and its vector, commonly known as the 'kissing bug'. The parasite's survival and proliferation strategies are driven by the diverse environments it encounters. Despite being described by Carlos Chagas in 1909, significant knowledge gaps persist regarding the parasite's various life forms and adaptive capabilities in response to environmental cues. In this study, we employed Ultrastructure Expansion Microscopy to explore the intricate journey of T. cruzi within the host cell. Upon entry into the host cell, trypomastigotes undergo folding, resulting in intermediate forms characterized by a rounded cell body, anterior positioning of basal bodies, and a shortened flagellum. The repositioning of basal bodies and the kinetoplast and the shortening of the flagella mark the culmination of intracellular amastigogenesis. Furthermore, we analyzed intracellular trypomastigogenesis, identifying discrete intermediate forms, including leaf-shaped stages and epimastigote-like forms, which suggests a complex differentiation process. Notably, we did not observe any dividing intracellular epimastigotes, indicating that these may be non-replicative forms within the host cell. Our detailed examination of amastigote cell division revealed semi-closed nuclear mitosis, with mitotic spindle formation independent of basal bodies. This study provides new insights into the morphological and cytoskeletal changes during the intracellular stages of T. cruzi, providing a model for understanding the dynamics of intracellular amastigogenesis and trypomastigogenesis.

Multitask-constrained reentry trajectory planning for hypersonic gliding vehicle

This paper studies the reentry trajectory planning problems for hypersonic gliding vehicle under multiple tasks. Different from the former achievements, this paper takes into account the practical tasks involved in the reentry phase, including penetration of interceptors, evasion of the no-fly zones, and the arrival of the waypoints. Firstly, the constraints during the reentry phase are analyzed in detail, and the original trajectory planning problem is formulated. Secondly, the hp-adaptive pseudospectral discretization method is proposed to effectively reduce the discretization error. Relevant variables are introduced to relax and transform severe intractable constraints of multiple nonconvex forms, thus enhancing the robustness of the planning process. Thirdly, the improved sequential convex programming with decision variables algorithm is proposed to ensure the converged trajectory satisfies complicated tasks. The theoretical analysis is also presented to demonstrate that the converged trajectory is the approximate stationary solution of the discrete form of the original problem. Finally, the effectiveness of the proposed algorithms is validated through numerical simulation.

Nonlinear twofold saddle point-based mixed finite element methods for a regularized μ(I)-rheology model of granular materials

We propose and analyze new mixed finite element methods for a regularized μ(I)-rheology model of granular flows with an equivalent viscosity depending nonlinearly on the pressure and the Euclidean norm of the symmetric part of the velocity gradient. To this end, and besides the velocity, the pressure and the aforementioned strain rate, we introduce a modified stress tensor that includes the convective term, and the skew-symmetric vorticity, as auxiliary tensor unknowns, thus yielding a mixed variational formulation within a Banach spaces framework. Then, the pressure is obtained through an iterative postprocess suggested by the incompressibility condition of the fluid, which allows us to express this unknown in terms of the aforementioned stress and the velocity. A fixed-point strategy combined with a solvability result for a class of nonlinear twofold saddle point operator equations in Banach spaces, is employed to show, along with the classical Banach fixed-point theorem, the well-posedness of the continuous and discrete formulations. In particular, PEERS and AFW elements of order ℓ⩾0 for the stress, the velocity, and the skew-symmetric vorticity, and piecewise polynomials of degree ⩽ℓ+n (resp. ⩽ℓ+1) for the strain rate with PEERS (resp. with AFW), yield stable Galerkin schemes. Optimal a priori error estimates are derived and associated rates of convergence are established. Finally, numerical results confirming the latter and illustrating the good performance of the methods, are reported.

Mixed Crank-Nicolson and Galerkin Methods for Solving Nonlinear Hyperbolic Partial Differential Equation

In this work,the approximation solution for nonlinear hyperbolic partial differential equation(NLHPDE) is obtained by using the mixed Crank-Nicolson (CN) schemeand the GalerkinMethod(GM) and it is symbolized by (MCNGM).At first theCNis utilized for the variable of time to obtain the discrete weak form for the NLHPDE, and then the GM is utilized which reduces the DWF into theGalerkin nonlinear algebraic system (GNLAS) at each step of time. Through utilizing the predictor-corrector techniques which are symbolized by the obtained GNLAS is transformed into Galerkin linear algebraic system (GLAS) which is solved by applying the Cholesky method. The convergence of the method is studied. Some examples are given to illustrate the efficiency and the accuracy for the proposed method.

Mixed Crank-Nicolson and Galerkin Methods for Solving Nonlinear Hyperbolic Partial Differential Equation

In this work,the approximation solution for nonlinear hyperbolic partial differential equation(NLHPDE) is obtained by using the mixed Crank-Nicolson (CN) schemeand the GalerkinMethod(GM) and it is symbolized by (MCNGM).At first theCNis utilized for the variable of time to obtain the discrete weak form for the NLHPDE, and then the GM is utilized which reduces the DWF into theGalerkin nonlinear algebraic system (GNLAS) at each step of time. Through utilizing the predictor-corrector techniques which are symbolized by the obtained GNLAS is transformed into Galerkin linear algebraic system (GLAS) which is solved by applying the Cholesky method. The convergence of the method is studied. Some examples are given to illustrate the efficiency and the accuracy for the proposed method.

NEW GENERALIZATION AND REFINEMENT OF THE DIAMOND-α INTEGRAL CAUCHY–SCHWARTZ INEQUALITY ON TIME SCALES

This paper is concerned with new generalizations and refinements of the diamond-[Formula: see text] integral Cauchy–Schwartz inequality on time scales. The new generalization and refinement of the Cauchy–Schwartz inequality related to diamond integral on time scales are also considered. These obtained inequalities unify continuous inequalities and their corresponding discrete forms associated with Cauchy–Schwartz inequality.

Politicized caring as anti-carceral pedagogy: An ethnographic case study of a Black woman educator

ABSTRACT The “racialized social control infrastructure” of schools refers to the over-emphasis on controlling the behaviors and bodies of Black, and other, students of color. Politicized caring, or the strategic prioritization of the needs and desires of those traditionally marginalized, may be disruptive to it. The objective of this analysis is (1) to describe how one Black female educator enacts a pedagogy of politicized caring on a day-to-day basis and (2) to investigate how it intersects with one school’s social control infrastructure. Arising from a longitudinal, collaborative research partnership with a predominantly Black high school, the analysis features two years of data with over 80 observations and 13 interviews. The findings identify five discrete forms of enactment of politicized caring and shows how the educator uses them to buffer her students from the harsher elements of social control. Implications for theory and practice are discussed.

E-PINN: A fast physics-informed neural network based on explicit time-domain method for dynamic response prediction of nonlinear structures

The physics-informed neural networks (PINNs), serving as the surrogate models, have emerged in the dynamic response prediction of nonlinear systems. However, in the conventional PINN model, the global differential equation of motion of the nonlinear system is directly integrated into the loss function, which requires the network to capture the intricate global dynamic evolution mechanism of the system and thus poses significant challenges for network learning. In this study, inspired by the explicit time-domain method (ETDM), a novel PINN based on ETDM, called E-PINN, is proposed to address the challenges arising from the network learning of the existing PINNs. A lightweight long short-term memory (LSTM) module is trained to learn the nonlinear evolution mechanism of restoring forces, while a single convolutional layer (SCL) module is utilized to reflect the linear evolution mechanism of the primary structure under the combined action of the external excitations and the nonlinear restoring forces. The weights of the SCL module can be directly retrieved from the discrete convolutional formulation of ETDM, and only the parameters in the LSTM module need to be optimized through a loss function solely based on the nonlinear restoring forces, thereby enabling easy network learning of the proposed E-PINN by decoupling the linear and nonlinear evolution mechanisms embedded in the nonlinear system. Three numerical examples are investigated by the present approach, and the results demonstrate the superior training efficiency and prediction accuracy of E-PINN in dynamic response prediction of nonlinear systems.

Gradient flows of generalized relative entropy and functional inequalities on graphs

We derive the discrete forms of the porous medium equation and fast diffusion equation with drift terms on a finite graph as gradient flows of the m-relative entropy, with the global existence of solutions. We show the Łojasiewicz inequality to prove the convergence of solutions and some important functional inequalities. More specifically, the Łojasiewicz inequality holds near the stationary solution with exponent 12, which leads to the exactly exponential convergence rate with a finite trajectory length. It can also be applied to show the Talagrand-type inequality and the log-Sobolev-type inequality directly.