A Mexican mathematician used a system of coupled ordinary differential equations and ten years of public crime data to estimate cartel membership, simulate policy scenarios, and show that cutting recruitment, not increasing arrests, is the most effective way to reduce violence.
By applying kinematic equations, differential equations, network models, and game theory, this article shows how mathematical modeling can give wuxia’s legendary feats—such as soaring jumps, internal energy flow, weapon dynamics, and battle formations—a coherent, physically plausible framework for films, games, and storytelling.
This article explores how differential equations serve as the mathematical language that describes phenomena ranging from planetary orbits and cosmic expansion to atmospheric dynamics, cellular reactions, ecological interactions, electrical circuits, and financial markets, highlighting their universal role in modeling change.
This article explains the historical background of the three-body problem, outlines its mathematical formulation using Newton’s law of gravitation and motion, derives the system of first-order differential equations, and discusses common numerical methods such as Euler, Runge‑Kutta, and symplectic integrators for solving it.
This article explores how kinetic differential equations model the dynamic behavior of biochemical and chemical reaction systems, illustrating construction and analysis through classic examples such as exponential decay, generation‑decay balance, irreversible and reversible reactions, and discussing transient versus steady‑state behaviors.
This article explains the core concepts of differential equation modeling using inflow and outflow rates, illustrates the approach with a reservoir analogy, and applies it to a detailed case study of the lamprey (Petromyzon) ecosystem, highlighting how resource availability influences sex ratio dynamics.
This article walks through constructing a differential‑equation model for Taihu Lake’s pollutant concentration using weekly monitoring data, outlines the necessary variables and assumptions, and shows how to predict future concentrations at weeks 20 and 30.
From population growth and disease spread to radioactive decay, cooling laws, chemical reactions, and chaotic systems, this article explores how first‑order differential equations serve as a universal mathematical language that precisely describes dynamic processes across biology, physics, chemistry, and economics.
This article explores how differential equations can model romantic relationships, using a Chinese drama as a case study, presents the underlying mathematical formulation, demonstrates numerical simulations with Python, and discusses the impact of random plot twists on emotional dynamics.
This article explains the classic exponential growth model derived from Malthusian assumptions and then introduces the logistic (stagnation) model, showing how resource limits modify growth dynamics and presenting the differential equations and their solutions for both cases.
During World War I, shark catches rose sharply, prompting biologist D. Ancona to enlist mathematician Volterra, whose differential‑equation predator‑prey model—later extended with a logistic term—explains the disproportionate increase of sharks by accounting for fishing pressure and natural population dynamics.
This curated list groups recent articles on change and prediction models, covering topics such as war dynamics, population, infectious diseases, differential equations, regression, time‑series analysis, and various machine‑learning techniques, providing quick access to each resource.
This article explains the three-step process for constructing differential-equation models, introduces the micro-element analysis method, and demonstrates its application to a hemispherical water-outflow problem, including derivation of the governing equation, solution via separation of variables, and a Python implementation using SymPy.
This article outlines a three‑step process for constructing differential‑equation models, demonstrates it with a Newton cooling problem, derives the governing equation, solves it analytically and numerically using Python’s sympy, and predicts the object’s temperature after 20 minutes.
This article explains why numerical solutions are needed for differential equations, introduces the initial value problem for first‑order ODEs, and details three common discretization techniques—forward difference, numerical integration, and first‑order Taylor expansion—highlighting their formulation and error considerations.
This article explains the classic exponential growth model derived from Malthus’s assumptions and then introduces the logistic (or Verhulst) model, showing how resource limits create a carrying capacity that slows growth, with step-by-step derivations using separation of variables.
The article explains the classic Volterra predator‑prey differential equations, derives the basic model, shows how adding a logistic term captures realistic population limits, and demonstrates how wartime fishing reductions lead to a disproportionate increase in shark numbers, illustrating the model’s explanatory power.
This article explains how to build a Physics‑Informed Neural Network (PINN) in PyTorch to solve a simple logistic ordinary differential equation, covering the underlying theory, loss formulation with equation residuals and boundary conditions, network architecture, automatic differentiation, and training results.
The KDD 2020 paper from Alibaba Entertainment presents a differential‑equation‑based hot‑content exposure sensitivity model and a multi‑objective optimization framework that, under exposure‑resource constraints, guarantees video delivery by accounting for nonlinear content exposure, timing, strategies, and user click habits, now deployed on Youku.
