Image denoising is a classical inverse problem in which the recovery of a clean image from its noisy observation is complicated by ill-posedness. This paper develops a functional-analytic framework in which a broad class of denoising methods is expressed as the iteration of firmly nonexpansive or quasi-nonexpansive operators on a Hilbert space. Total variation regularization, wavelet shrinkage, projection onto convex sets (POCS), projection onto epigraph sets (PESC), and regularization by denoising via fixed-point projection (RED-PRO) are each shown to fit this structure, and their convergence is obtained from a single Krasnosel'skii–Mann argument. The proximal nature of wavelet shrinkage is distinguished from metric projection onto the \(\ell^1\)-ball, and the demicontractive hypothesis underlying RED-PRO is stated explicitly. Numerical experiments on standard test images show that RED-PRO with a BM3D denoiser attains the highest PSNR at every noise level tested, that Chambolle's TV projection follows within \(1.2\)–\(1.7\) dB, and that POCS and wavelet shrinkage trade accuracy for speed. The framework clarifies the operator-theoretic content of each method and provides a common language for their analysis and comparison.
Ordinary differential equations provide a common language for representing rates of change in chemical, biological, epidemiological, and financial systems. This study develops and computationally examines four model families drawn from physical and life-science applications: the dimensionless Lengyel–Epstein model for the chlorine dioxide–iodine–malonic acid reaction; a six-compartment demographic, exposure, infection, and AIDS-progression model motivated by delayed first sexual intercourse; the classical susceptible–infectious–removed epidemic model; and exponential growth and radioactive-decay models. Equilibria and local stability conditions are derived analytically, while numerical solutions are obtained with adaptive Runge–Kutta integration. For the chemical model with illustrative parameters $a=12$ and $b=0.30$, the positive equilibrium is unstable and the numerical trajectory approaches sustained oscillation. The delayed-intercourse model is locally asymptotically stable when the feedback between the sexually active and under-age compartments is weaker than total demographic removal, specifically when $(d_1+m_1)(d_2+m_2+b_2)>b_1m_1$. For the epidemic illustration, the effective transmission rate is $0.8$ per day, the recovery rate is $0.125$ per day, and $R_0=6.4$; the infectious population peaks at approximately $554$ persons near day $13$ in a population of $1,000$. Continuous $5%$ financial growth increases $20,000$ monetary units to $23,236.68$ after three years, whereas an $800,\mathrm{mg}$ bismuth-210 sample with a five-day half-life declines to $12.5,\mathrm{mg}$ after $30$ days. The results demonstrate how a shared differential-equation framework supports model formulation, stability analysis, simulation, and transparent comparison across distinct applications. All numerical outcomes are illustrative and are not fitted to clinical or laboratory observations.
Plasmodium falciparum</em>, the most virulent human malaria parasite, possesses a highly regulated genome that supports survival, adaptation, virulence, drug resistance, developmental switching, and stress responses within the human host. This study characterizes stable regulatory attractors in a malaria-parasite gene regulatory network using an asynchronous Boolean update method. The network is formulated as a Boolean dynamical system with nine biological components: \(X_1=\mathrm{PfEMP1}\), \(X_2=\mathrm{PfCRT}\), \(X_3=\mathrm{PfMDR1}\), \(X_4=\mathrm{PfDHFR}\), \(X_5=\mathrm{AP2\text{-}G}\), \(X_6=\mathrm{PfSIR2A}\), \(X_7=\mathrm{PfK13}\), \(X_8=\mathrm{HP1}\), and \(X_9=\mathrm{H3K9me3}\), representing regulators associated with virulence, drug resistance, developmental regulation, stress response, and epigenetic control. External pressure is represented by a general stress signal \(\omega\) together with selectors for chloroquine \((\lambda)\), antifolate pressure \((\alpha)\), artemisinin pressure \((\beta)\), and partner-drug pressure \((\rho)\). Fixed points are states satisfying \(F(X)=X\); such states are invariant under either synchronous or asynchronous updating. Under the all-stress-OFF condition, the model yields three fixed points, whereas the all-stress-ON condition yields four fixed points. The attractors show how stress-dependent logical regulation can shift the network between distinct stable expression patterns and provide a mathematical framework for exploring regulatory states associated with parasite adaptation and antimalarial pressure.
