“The paper argues for a shift in emphasis from asymptotic behavior to transient and input-driven dynamics as a primary lens for understanding, testing, and reverse-engineering biological networks.”

“The paper introduces smooth cumulative- and inverse-cumulative-distribution-function ((I)CDF) timesteppers that evolve distributions rather than particles.”

“The paper finds that the mean of the first-passage probability may display a minimum as a function of bias strength, depending on the location of the initial and target sites relative to the focal point.”

“The proposed preconditioners significantly accelerate the convergence of iterative solvers compared to existing methods.”

“The results confirm the high-order accuracy of the model through convergence studies and demonstrate a substantial speed-up over low-order numerical schemes.”

“The characteristic energy decay time exhibits a universal inverse scaling $τ_0 \propto ε^{-1.03 \pm 0.02}$ with the dissipation parameter $ε= 1 - e^2$.”

“Our results mathematically characterize the background magnetic field exerts the stabilizing effect, and bridge the gap left by previous work with respect to the asymptotic behavior in time.”

“The paper derives an exact entanglement boundary based on the positivity of the partial transpose, valid in the symmetric resonant limit, and provides an explicit minimum collective fluctuation amplitude required to sustain steady-state entanglement.”

“The research focuses on the prediction of steady-state electrohydrodynamic flow.”

“The source is ArXiv.”

“The article's focus on 'exact steady states' and 'stability' suggests a rigorous mathematical analysis, likely involving differential equations and stability analysis techniques.”