MKLab - APPLIED MATHEMATICS

How Elliptic Curves Secure the Digital World

Sun, 25 Oct 2026 17:43:56 +0200

Summary: How Elliptic Curves Secure the Digital World

1 – The mathematics behind digital security
The article, How Elliptic Curves Secure the Digital World, published by the American Mathematical Society’s Feature Column on October 1, 2026, explains how elliptic curve cryptography (ECC) protects digital communications, including online banking, messaging applications, website connections, and cryptocurrency transactions. An elliptic curve can be described by an equation such as y² = x³ + ax + b, subject to conditions that ensure the curve is nonsingular. The points on these curves follow special algebraic rules that make them useful for cryptographic calculations. Although elliptic curves originate in abstract algebra and number theory, they have become an essential component of modern cybersecurity, demonstrating how theoretical mathematics can solve practical problems.

2 – The mathematical principle behind encryption
The central idea of ECC is that certain mathematical operations are easy to perform but extremely difficult to reverse. Given a point P on an elliptic curve, a computer can efficiently calculate a multiple such as Q = kP, where k is a large integer. However, determining k from P and Q is believed to be computationally infeasible when appropriate curves and parameters are used. This challenge is known as the elliptic curve discrete logarithm problem. It enables cryptographic protocols to establish shared secret keys over public networks without directly transmitting the secret itself. In practice, these operations take place over finite fields, and the resulting shared keys can be used to encrypt digital communications securely.

3 – Why elliptic curve cryptography matters
One of ECC’s greatest advantages is its efficiency. Compared with RSA, a widely used public-key cryptographic system, elliptic curve cryptography can provide comparable security with significantly smaller keys. For example, a 256-bit elliptic curve key is commonly considered comparable in security to a 3072-bit RSA key. Smaller keys reduce storage requirements, communication overhead, and computational costs, making ECC particularly valuable for smartphones, internet services, and devices with limited resources. The development of this technology illustrates how abstract mathematical research can lead to transformative practical applications. Elliptic curves were studied long before their importance to digital security became apparent, highlighting the lasting contribution of fundamental mathematics to modern technology.

Key Takeaways

Mathematical foundations: Elliptic curves connect geometry, algebra, and number theory.
Security principle: Scalar multiplication is efficient, while reversing it is computationally difficult for suitable curves.
Practical applications: ECC helps secure internet communications and digital transactions.
Efficiency: ECC provides strong security with smaller keys than RSA.
Scientific significance: Abstract mathematical discoveries can eventually underpin technologies used worldwide.

Original article:https://mathvoices.ams.org/featurecolumn/2026/10/01/how-elliptic-curves-secure-the-digital-world/

Deriving the heat equation

Sun, 18 Oct 2026 18:14:18 +0300

Summary
The article Feeling the Heat explains how mathematics can describe the movement of heat through a material. Heat naturally flows from a hotter region to a colder one, as illustrated by a metal spoon placed in hot soup. The author begins with a simple model of a rectangular block whose ends are maintained at different temperatures, while its other faces are insulated. The rate of heat transfer depends on the temperature difference, the cross-sectional area, the thickness of the material, and its thermal conductivity. These relationships establish the foundation for developing a mathematical model of heat conduction.

The article then introduces Fourier’s law of heat conduction, which expresses the relationship between heat flow and the spatial rate of change of temperature. Using calculus, the author considers increasingly thin slices of the material and takes a limit to describe heat flow at an individual point. The law is written as $q_x=-kA\frac{\partial T}{\partial x}$, where $k$ represents thermal conductivity and $A$ is the cross-sectional area. The negative sign indicates that heat flows in the direction of decreasing temperature. The article combines this principle with the conservation of energy, which states that the change in thermal energy within a small slice equals the energy entering it minus the energy leaving it.

Finally, the author derives the heat equation, a partial differential equation that predicts how temperature changes across space and over time. For a uniform material with constant properties and one-dimensional heat flow, the equation is $\frac{\partial T}{\partial t}=\frac{k}{c\rho}\frac{\partial^2T}{\partial x^2}$, where $c$ is specific heat capacity and $\rho$ is density. Although the derivation relies on simplifying assumptions, the equation provides a powerful framework for understanding real-world heat transfer. The article also highlights the wider applications of the mathematics: solutions involving infinite sums of sine and cosine functions connect heat conduction to the analysis of sound waves and image processing.

Key Takeaways
1. Heat flows from hot to cold. Temperature differences drive the transfer of thermal energy.
2. Material properties matter. Thermal conductivity determines how readily a material transfers heat; metals generally conduct heat more effectively than wood.
3. Fourier’s law describes heat flow. The rate of heat transfer is related to the temperature gradient, with a negative sign indicating flow toward lower temperatures.
4. Conservation of energy is fundamental. The change in energy within a small region equals the energy entering minus the energy leaving.
5. Calculus connects local and global behaviour. Derivatives and limits allow heat transfer through a whole object to be described at individual points.
6. The heat equation predicts temperature changes. It relates the rate of change of temperature over time to the curvature of the temperature distribution in space.
7. Mathematical models rely on assumptions. The simplest equation assumes constant material properties and heat flow in one spatial direction. More complex situations require more general models.
8. Mathematics has applications beyond physics. Fourier methods, including sine and cosine series, also help analyse sound waves and process images.
9. Physical laws can be translated into equations. The article demonstrates how combining an empirical relationship with a conservation principle leads to a differential equation.
10. The derivation illustrates mathematical modelling. Starting with a simple physical situation, introducing appropriate variables, applying calculus, and taking limits produces a model with broad practical applications.

ARTICLE

There’s a new way to break RSA encryption

Mon, 28 Sep 2026 21:35:07 +0300

A new cryptographic attack has challenged a long-standing assumption about RSA: that breaking it essentially requires factoring the enormous number at the heart of the key. Researchers have demonstrated a different route—forging RSA signatures without actually recovering the private key. Their technique combines a variant of the Special Number Field Sieve with an “oracle,” exploiting particular properties of blind/textbook RSA implementations. In experiments, the researchers successfully forged a 1024-bit RSA signature, something previously considered extremely expensive.

The immediate practical danger is limited because the attack applies specifically to vulnerable blind-signature RSA systems, while the widespread PKCS#1 and PSS padding schemes are not believed to be vulnerable in the same way. Nevertheless, the result is important because it dramatically lowers the estimated security of RSA: the researchers estimate roughly $2^{65}$ work for 1024-bit RSA, $2^{90}$ for 2048-bit RSA, and $2^{119}$ for 4096-bit RSA—far below the security levels normally expected from these key sizes. The researchers also note that their implementation was not GPU- or AI-optimized, so further improvements may be possible. The finding therefore represents more of a major conceptual warning for cryptography than an immediate collapse of everyday RSA encryption.

ARTICLE

Mathematical Signature of Fair Competition

Fri, 25 Sep 2026 20:51:37 +0300

A University of Houston research team led by computer scientist Ioannis Pavlidis has proposed a mathematical framework for identifying what a healthy and productive competitive environment looks like. Published in npj Complexity on March 16, 2026, the study examines the statistical patterns produced by repeated success and attempts to work backward from those patterns to determine the underlying structure of a competition. The researchers describe their approach as general and falsifiable, meaning that its predictions can in principle be tested against new datasets.

The central idea is that fairness does not necessarily mean equal outcomes. In a demanding but productive competition, a relatively small group of people may become consistently more successful because competition encourages participants to improve at different rates. Mathematically, the important question is therefore not simply how evenly success is distributed, but whether the system remains open to new high performers. The researchers describe this as a balance between strong competitive pressure and continued opportunity for newcomers.

The model identifies three broad types of competitive environments. In a “tough but fair” system, a small group of high performers emerges, but the leading positions remain contestable. In a winner-take-all system, success becomes highly concentrated in one or two competitors, suggesting structural barriers that can discourage participation and eventually reduce innovation. At the other extreme, a broad-award system distributes success so evenly that competitive pressure may become too weak to produce substantial differences in performance.

To test the idea, the researchers examined competition in three very different areas: Olympic athletes, scientists competing for federal research grants, and World War II fighter pilots. The proposed framework could potentially be applied to research funding, career progression, military training, sports tournaments and other environments where repeated success can be measured. The researchers also plan to investigate whether similar mathematical patterns appear in team-based competitions and across longer time periods.

Key Takeaways

Fair competition ≠ equal outcomes. A fair system can produce substantial differences in achievement.
The researchers propose a mathematical signature based on the statistical distribution of repeated success.
Three competitive structures are identified:
- Tough but fair: strong competition + continued opportunity for newcomers.
- Winner-take-all: success becomes excessively concentrated.
- Broad award: success is so evenly distributed that competitive pressure may be weak.
Competition can function as a learning mechanism: participants improve, but those with larger accumulated improvements pull ahead.
A useful conceptual representation is the distribution of success, for example $P(k)$, where $k$ represents the number of successful outcomes accumulated by a participant.
The important issue is not merely the mean success $\mathbb{E}[k]$, but the shape of the entire distribution and what it reveals about mobility within the system.
The framework was tested across Olympic sport, scientific grant competition, and WWII fighter pilots, suggesting that the researchers are looking for a cross-domain mathematical pattern.
Potential applications include research funding, promotions, education, athletic competitions and organizational selection systems.
The particularly interesting mathematical idea is to infer the structure of a competitive system from its observable statistical output rather than judging the rules directly.
The study is presented as a testable hypothesis, so further datasets and independent studies are needed to establish how universal the proposed pattern really is.

ARTICLE

The Voorhees law of traffic

Fri, 25 Sep 2026 20:48:10 +0300

The Guardian article examines a familiar driving experience: you overtake a slower car, only to encounter a red traffic light and find that the slower vehicle catches up with you. Mathematics researcher Dr Conor Boland of Dublin City University studied this phenomenon and called it the “Voorhees law of traffic”, a humorous reference to Jason Voorhees, the slow-moving horror character who nevertheless seems to catch his victims. The research was published in Royal Society Open Science.

Boland's model considers vehicles travelling along a road containing traffic lights operating on fixed cycles. Even when one driver travels faster than another, the faster driver does not necessarily arrive at the destination earlier because the traffic-light timing introduces delays. If the first driver reaches a red light while the slower driver arrives during a green phase, their apparent advantage can disappear. Mathematically, the difference in arrival times depends not only on speed but also on the phase of the traffic lights relative to each vehicle's position.

The interesting result is that, for a single traffic light, the apparent tendency of the slower car to “catch up” is largely a consequence of probability and perception rather than a deterministic law. Over many journeys, gains from travelling faster and losses caused by red lights can approximately balance. If there are several independently timed traffic lights, however, the probability of the faster driver being stopped while the slower driver proceeds increases. In simplified terms, if the probability of being delayed at one light is $p$, then the probability of avoiding that delay at $n$ independent lights is $(1-p)^n$.

The research therefore provides a neat example of how mathematics can explain an everyday phenomenon that feels almost impossible. It also suggests that speeding or overtaking does not guarantee a meaningful reduction in journey time when traffic signals dominate the journey. The article notes that real-world driving is considerably more complicated than the mathematical model because drivers change speed, lights may be sensor-controlled, and traffic conditions vary. Nevertheless, the study illustrates how probability, geometry and simple dynamical modelling can turn an apparently mysterious experience into something mathematically understandable.

Key takeaways

The “slow car catches you” phenomenon is mathematically explainable, rather than simply bad luck.
Traffic lights introduce a discrete timing constraint that can eliminate the advantage of higher speed.
If two cars travel distances $d$ at speeds $v_1$ and $v_2$, their uninterrupted travel times are
$t_1=\frac{d}{v_1}$ and $t_2=\frac{d}{v_2}$.
With a traffic-light delay $D$, actual travel time becomes approximately
$T=\frac{d}{v}+D$.
With multiple independent lights, probability becomes increasingly important; for example, the probability of experiencing at least one delay can be written as
$P(\text{delay})=1-(1-p)^n$.
The phenomenon demonstrates how human intuition can be misleading when randomness and repeated events are involved.
Teaching connection: it is an excellent real-world introduction to probability, conditional probability, functions, modelling, expected values and simulation.
A classroom simulation could randomly assign traffic-light phases to two cars and investigate whether increasing speed actually reduces the expected journey time.

ARTICLE

How Do Fireflies Flash in Sync?

Fri, 25 Sep 2026 16:30:59 +0300

The article examines a long-standing question in mathematical biology: how can thousands of fireflies synchronize their flashes? Synchronous fireflies have traditionally been explained using mathematical models in which every insect behaves like an oscillator with its own internal rhythm. In the classic Kuramoto-style framework, each firefly has an intrinsic frequency and is coupled to its neighbors, so interactions gradually pull the population toward synchrony. Such models have become important examples of self-organization, with applications extending from fireflies to neurons, heart cells and other coupled systems.

New experiments by Orit Peleg, Raphaël Sarfati and collaborators suggest that this traditional picture is incomplete. Using two 360-degree cameras and computer vision, researchers were able to determine both when and where individual fireflies flashed in three dimensions. In Photinus carolinus from Tennessee, flashes often propagate as waves through the swarm at roughly $0.5,\mathrm{m/s}$, suggesting that interactions are neither purely local nor completely global. More importantly, an isolated firefly does not necessarily possess a regular internal clock: its flashes can occur at apparently random intervals. Yet when roughly 15 or more fireflies are brought together, remarkably regular collective bursts emerge.

The proposed explanation is therefore an example of emergent periodicity. After a flash, each firefly must wait a minimum amount of time while its light-producing system recharges, but beyond that minimum its next flash may occur at a random time. If one firefly happens to flash first, however, neighboring fireflies respond and flash as well. Repeating this process across a sufficiently large population produces a collective rhythm even though no individual possesses that rhythm. In simplified terms, if the minimum recovery time is $\tau_{\min}$, the time between collective bursts tends toward $\tau_{\min}$ as the number of interacting fireflies increases. The mathematical lesson is striking: regular macroscopic behavior can emerge from individually irregular components.

The researchers also observed another unusual phenomenon in Photuris frontalis: a chimera state. Most fireflies synchronize into a coherent rhythm, while some individuals continue flashing with their own timing, remaining spatially embedded in the synchronized group. Chimera states were introduced mathematically by Yoshiki Kuramoto and Dorjsuren Battogtokh and have been extensively studied in theoretical systems, but they had rarely been observed naturally. The work therefore creates a feedback loop between mathematics and biology: rather than simply fitting nature to elegant mathematical models, researchers can now use detailed observations to develop models that capture the messiness of real biological systems.

Key takeaways

Synchronization does not necessarily require synchronized individuals. A population can develop a regular collective rhythm even when individual members behave irregularly.
Classical oscillator models may be too idealized for some firefly species; the assumption of an intrinsic internal clock does not hold universally.
Coupling is crucial: one firefly's flash can trigger others, producing collective behavior.
A simplified mathematical picture is:
random individual timing+interaction⟶collective periodicity.\text{random individual timing}+\text{interaction} \longrightarrow \text{collective periodicity}.
As the population size $N$ increases, the probability that at least one firefly flashes soon after the minimum recharge time increases, helping establish a collective period close to $\tau_{\min}$.
Flash synchronization can propagate spatially as a wave, rather than appearing everywhere simultaneously.
Fireflies demonstrate self-organization: complex large-scale order can arise without a central controller.
Chimera states show that synchronization itself can be heterogeneous: synchronized and unsynchronized subpopulations can coexist.
The research is an excellent example of mathematics ↔ biology feedback: observations challenge existing models, while mathematical models provide mechanisms for explaining the observations.
The broader mathematical idea is highly transferable: similar mechanisms of collective behavior occur in neural networks, biological oscillators, coupled physical systems and complex networks.

ARTICLE

The Math of Catastrophe

Fri, 18 Sep 2026 20:51:56 +0300

The Quanta Magazine article explains how mathematicians use bifurcation theory and dynamical systems to understand climate “tipping points” — situations where a gradual change can suddenly push a system from one stable state into another. The idea goes back to Mikhail Budyko’s mathematical models of Snowball Earth and later to catastrophe theory and bifurcations studied by mathematicians such as Henri Poincaré and Christopher Zeeman.

Experiments on Peter Lake showed that even relatively simple mathematical models can successfully describe abrupt ecological transitions and reveal “critical slowing down,” a potential early-warning signal that a system is approaching a tipping point. However, applying these ideas to the climate is much harder because Earth is an enormous, interconnected, high-dimensional system with limited historical data.

A major example is the Atlantic Meridional Overturning Circulation (AMOC), whose possible collapse illustrates both the usefulness and limitations of mathematical prediction. A 2023 analysis by Peter and Susanne Ditlevsen estimated a possible AMOC tipping point between 2025 and 2095, with 2057 as their central estimate, but other researchers have argued that uncertainties in the historical data and assumptions of the model make such a precise prediction unreliable.

The article's broader message is that mathematics can identify possible mechanisms, warning signals and ranges of behavior, but it cannot necessarily tell us exactly when or how a climate system will tip. Researchers are therefore developing more sophisticated models and machine-learning approaches, while also studying “positive tipping points” in technologies and behaviors such as renewable energy and electric vehicles, where feedback mechanisms could accelerate beneficial transitions.

ARTICLE

The Mathematics of Bitcoin

Tue, 15 Sep 2026 19:47:21 +0300

The article “The Mathematics of Bitcoin” by Cyril Grunspan and Ricardo Pérez-Marco examines Bitcoin primarily as a mathematical and probabilistic system rather than as a financial asset. It explains how cryptographic hashing and Proof of Work create a decentralized consensus mechanism, then models Bitcoin mining using Poisson processes, exponential and Gamma distributions, Markov chains, martingales, and probability theory. A central problem is the “double-spend” attack: an attacker attempts to secretly build an alternative blockchain and replace confirmed transactions.

The authors derive an exact formula for the probability of such an attack succeeding after a given number of confirmations, improving on an approximation used in Satoshi Nakamoto’s original paper. They prove that, when the attacker controls less than half of the network's computing power, the probability of success decreases exponentially as the number of confirmations increases.

The second major theme is the mathematical stability and profitability of Bitcoin mining strategies. The authors show that, without Bitcoin’s periodic difficulty adjustment, the mathematically optimal strategy is simply to publish newly mined blocks immediately. They then analyze selfish mining and several more aggressive block-withholding strategies using Markov chains, martingale methods, Catalan numbers, and Dyck words. These methods reveal that Bitcoin’s difficulty-adjustment mechanism is an important potential weakness because it can make certain dishonest strategies profitable after a delay.

Finally, the paper revisits double-spending from an economic perspective and shows how an attacker must balance the probability of success against the rewards and risks of continuing the attack. Overall, the article demonstrates that Bitcoin provides a surprisingly rich application of modern probability, stochastic processes, combinatorics, and mathematical optimization.

ARTICLE [PDF]

Mathematical modelling of tsunamis

Tue, 15 Sep 2026 18:27:13 +0300

The article by Carlos Sánchez-Linares explains how mathematical modelling turns the physics of tsunamis into practical tools for coastal hazard assessment. Tsunami propagation can be modelled surprisingly effectively with the two-dimensional shallow-water equations, which describe conservation of water mass and momentum while incorporating gravity, ocean depth and the Earth's spherical geometry. The initial disturbance produced by an earthquake is estimated using elastic fault-dislocation models, particularly the Okada formulation.

These equations are then solved numerically using finite-volume methods, high-resolution schemes and wetting/drying algorithms that allow the models to represent steep wave fronts and the movement of the shoreline during inundation. The HySEA modelling framework combines these techniques to simulate both ocean-scale propagation and detailed coastal flooding.

The second major theme is the transition from mathematical equations to operational tsunami-risk systems. The simulations combine high-quality bathymetric and topographic data, realistic or worst-case earthquake scenarios, and nested computational grids: coarse grids model propagation across large ocean regions, while grids down to about 5 metres resolve flooding around individual coastal features and urban infrastructure.

Because these calculations can involve hundreds of millions of computational cells, HySEA exploits GPU-based high-performance computing to make large simulations feasible. The resulting products—such as tsunami arrival times, maximum inundation depth, run-up and flood extent—are incorporated into the Copernicus Emergency Management Service, providing authorities with quantitative information for risk assessment, emergency planning and coastal mitigation.

ARTICLE [PDF]

Burgers' equation

Sat, 12 Sep 2026 19:50:11 +0300

Summary

The Burgers equation is one of the simplest nonlinear partial differential equations that captures the competition between nonlinear transport and diffusion. In one spatial dimension, its viscous form is
$\frac{\partial u}{\partial t}+u\frac{\partial u}{\partial x}=\nu\frac{\partial^2u}{\partial x^2}$
where $u(x,t)$ is the evolving field and $\nu$ is a diffusion or viscosity coefficient. It appears in simplified models of fluid dynamics, nonlinear acoustics, gas dynamics, traffic flow, and mathematical physics. The nonlinear term $u,u_x$ causes regions with larger $u$ to move faster than regions with smaller $u$, which can steepen the profile, while the diffusion term $\nu u_{xx}$ smooths it out.

When $\nu=0$, the equation becomes the inviscid Burgers equation
$u_t+u,u_x=0$
It can initially be solved using the method of characteristics. Along characteristic curves, $u$ remains constant, with
$\frac{dx}{dt}=u,\qquad \frac{du}{dt}=0$
However, characteristics can eventually intersect. At that point, the classical differentiable solution breaks down and a shock wave forms. For initial data $u(x,0)=f(x)$, the first shock can occur at
$t_b=-\frac{1}{\inf_x f'(x)}$
provided the initial profile contains a sufficiently negative slope. This makes Burgers' equation a standard model for understanding how smooth nonlinear PDE solutions can develop very sharp gradients or discontinuities.

The viscous equation has an especially elegant feature: despite being nonlinear, it can be transformed into the linear heat equation using the Cole–Hopf substitution
$u=-2\nu\frac{\partial}{\partial x}\ln\phi$
The resulting function $\phi$ satisfies
$\phi_t=\nu\phi_{xx}$
Thus Burgers' equation provides a rare example in which nonlinear steepening, viscosity, shocks, and exact analytical solutions can all be studied explicitly. It is therefore an important conceptual bridge between elementary PDE theory and much harder nonlinear fluid equations such as the Navier–Stokes equations.

Key takeaways

Nonlinearity: $u u_x$ produces wave steepening.
Viscosity: $\nu u_{xx}$ produces smoothing and dissipation.
When $\nu=0$, smooth initial data can develop shocks in finite time.
When $\nu>0$, viscosity regularizes the shock into a thin, smooth transition layer.
The Cole–Hopf transformation converts the nonlinear Burgers equation into the linear heat equation.
Burgers' equation is widely used as a laboratory model for nonlinear PDEs and fluid mechanics, especially for understanding mechanisms that also appear in the Navier–Stokes equations.

ARTICLE / ARTICLE [PDF]

The Navier-Stokes equations

Thu, 10 Sep 2026 04:57:46 +0300

The Navier–Stokes equations

The article gives a short introduction to the Navier–Stokes equations, the fundamental partial differential equations used to describe the motion of fluids such as water and air. A fluid is represented mainly through its velocity field $v(x,y,z,t)$ and pressure $P(x,y,z,t)$, which specify how fast and in what direction the fluid is moving, and what pressure it has, at each position $(x,y,z)$ and time $t$. The equations connect changes in velocity and pressure with properties such as the fluid's viscosity.

One reason these equations are so important is that they can model phenomena ranging from airflow around aircraft and cars to waves, rivers, weather and turbulence. Turbulence is particularly difficult because velocity can change dramatically between points that are extremely close together. Exact analytic solutions to Navier–Stokes are available only in relatively simple situations, so real engineering and scientific applications generally rely on numerical approximations and computer simulations.

The deepest mathematical issue is whether sufficiently well-behaved three-dimensional initial conditions always produce solutions that remain smooth for all time, or whether the solution can eventually develop a singularity, where quantities such as derivatives of velocity become unbounded. This became one of the Clay Mathematics Institute's $1$ million Millennium Prize Problems.

Key takeaways

Navier–Stokes describes fluid motion using fields such as velocity $v(x,y,z,t)$ and pressure $P(x,y,z,t)$.
The equations are central to modelling water, air, turbulence, aerodynamics and weather, but realistic solutions usually require numerical computation.
The great theoretical question is whether smooth $3$-D solutions must remain smooth or can develop a finite-time singularity.
The article is now historically interesting: OpenAI proposed such a singularity construction on 8 September 2026, but Clay has not yet recognized the Millennium Problem as solved.

ARTICLE / ARTICLE [PDF]

Computational fluid dynamics

Thu, 10 Sep 2026 04:54:29 +0300

Maths in a Minute: Computational Fluid Dynamics
Source:Plus Magazine

Computational fluid dynamics (CFD) is the use of numerical methods and computers to approximate the behaviour of moving fluids such as air, water and blood. Fluid motion is governed by the Navier–Stokes equations, which relate quantities such as velocity, pressure and viscosity. In a common incompressible form, the momentum equation can be written as
$\rho\left(\frac{\partial \mathbf{u}}{\partial t}+(\mathbf{u}\cdot\nabla)\mathbf{u}\right)=-\nabla p+\mu\nabla^2\mathbf{u}+\mathbf{f}$,
together with the incompressibility condition $\nabla\cdot\mathbf{u}=0$. For realistic objects such as cars, heart valves or buildings, obtaining exact analytical solutions is generally impossible, so the equations must instead be approximated computationally.

CFD tackles this by dividing the physical region into a huge collection of small elements forming a mesh. The equations are approximated locally on each element, while neighbouring elements interact so that a numerical approximation to the overall flow gradually emerges. The result can provide estimates of air velocity, density, temperature and pressure throughout the region. Once the pressure distribution around an object is known, integration can be used to calculate quantities such as total aerodynamic forces and drag—conceptually similar to estimating an area using the trapezium rule. The article highlights applications ranging from designing the supersonic Bloodhound car to biomedical engineering and modelling airflow and aerosols in buildings.

Key takeaways

CFD turns difficult differential equations into large numerical problems that computers can solve approximately.
The physical domain is split into a mesh of many small regions, where simplified local calculations are performed.
Integration of the resulting pressure and velocity fields allows engineers to predict forces such as drag and lift.
CFD is important far beyond aerodynamics, with applications in medicine, architecture, ventilation and many other areas involving fluid flow.

ARTICLE / PDF

The mathematics of diseases

Sun, 06 Sep 2026 00:22:42 +0300

The Mathematics of Diseases — Matt Keeling
Plus Magazine, 1 March 2001

The article explains how mathematical modelling can describe and predict the spread of infectious diseases, focusing on the classic SIR model. A population is divided into three groups: susceptible $S$, infectious $I$, and recovered $R$. Individuals move from susceptibility to infection through contact with infectious people and eventually recover, often gaining immunity. These transitions can be represented by differential equations such as
$\displaystyle \frac{dS}{dt}=B-\beta SI-dS$
$\displaystyle \frac{dI}{dt}=\beta SI-gI-dI$
$\displaystyle \frac{dR}{dt}=gI-dR$
where $\beta$ describes transmission, $1/g$ is the average infectious period, and $B$ and $d$ represent births and deaths. Despite its simplicity, the SIR framework can reproduce important epidemic behaviour, including rapid outbreaks followed by decline and, when births continually introduce new susceptible individuals, long-term endemic patterns.

A central quantity is the basic reproduction number $R_0$, defined as the average number of secondary infections produced by one infected individual in a fully susceptible population. In the simple model,
$\displaystyle R_0=\frac{\beta}{g}$
If $R_0>1$, an epidemic can grow; if $R_0<1$, transmission eventually dies out. The same parameter determines important quantities such as the equilibrium susceptible fraction,
$\displaystyle S^*=\frac{1}{R_0}$
and the approximate vaccination threshold needed to prevent sustained transmission,
$\displaystyle V_T=1-\frac{1}{R_0}$
Thus, highly transmissible diseases require much higher levels of population immunity. The article also shows how the final size of an outbreak can be estimated through an implicit equation such as
$\displaystyle S_\infty=\exp!\left[(S_\infty-1)R_0\right]$
This illustrates how even relatively modest increases in $R_0$ can greatly increase the proportion of a population eventually infected.

Keeling stresses that realistic epidemiology goes far beyond the elementary SIR equations. Models may incorporate age structure, school-term seasonality, different patterns of social contact, animal or insect vectors, spatial movement, and demographic change. The 2001 UK foot-and-mouth outbreak is presented as an example in which mathematical modelling could guide control strategies such as restricting livestock movement and removing infected animals.
The larger lesson is that epidemiology combines differential equations, dynamical systems, statistics, and real-world data. Mathematical models simplify reality, but they can reveal thresholds, mechanisms, and intervention strategies that are difficult to infer from raw case counts alone.

Key takeaways

The SIR model represents epidemic dynamics through the interacting populations $S$, $I$, and $R$.
$R_0$ acts as a fundamental threshold: $R_0>1$ permits epidemic growth, while $R_0<1$ prevents sustained spread.
Herd immunity follows mathematically from reducing the susceptible population below the critical level $1/R_0$.
Real epidemiological modelling extends simple SIR equations with demographic, behavioural, seasonal, and spatial information.

ARTICLE / PDF

How barcodes and QR codes work

Sun, 06 Sep 2026 00:15:09 +0300

That’s Maths: How barcodes and QR codes work
Author: Peter Lynch
Publication:The Irish Times
Published: 15 January 2015
Topic: Applied mathematics, information encoding, error detection and correction

Peter Lynch explains the mathematics hidden inside everyday barcodes and QR codes. Standard barcodes such as EAN-13 represent information using a 13-digit number, with each digit translated into a pattern of bars and spaces. The final digit acts as a check digit, allowing scanners to detect errors. In a simplified example, the check digit can be chosen so that the sum of all digits is divisible by $10$. The actual EAN-13 system uses weighted digits, making it possible to detect all single-digit errors and many common mistakes such as transposed digits. Similar mathematical techniques appear in ISBN book numbers and serial numbers on banknotes.

QR codes extend the same basic idea into two dimensions, allowing far more information to be stored in a small area. They can contain thousands of characters as well as URLs, coordinates, email addresses and text. Their reliability comes from Reed–Solomon error-correcting codes, which introduce mathematical redundancy so that information can often still be reconstructed even when part of the QR code is damaged or obscured. Thus, barcodes illustrate error detection, while QR codes demonstrate the more powerful concept of error correction.

Key takeaways

Barcodes encode numerical information through carefully designed patterns of bars and spaces.
Check digits provide a simple application of modular arithmetic for detecting scanning or transcription errors.
QR codes contain much more information than ordinary one-dimensional barcodes.
Reed–Solomon codes allow QR codes to recover information even when portions of the code are damaged, making them a practical application of algebra and coding theory.

ARTICLE

Mathematical Biology is Good for Mathematics

Wed, 02 Sep 2026 01:22:44 +0300

Mathematical Biology is Good for Mathematics
Author: Michael C. Reed
Publication:Notices of the American Mathematical Society, Vol. 62, No. 10, November 2015
DOI: 10.1090/noti1288

Michael C. Reed argues that the rapid rise of mathematical biology benefits not only applied mathematicians but mathematics as a whole. Biology has become one of the dominant areas of modern science, driven by new measurement technologies, major government and biotechnology funding, and widespread interest in medicine, human physiology, genetics, and ecology. Mathematics has responded accordingly: the proportion of mathematics PhDs specializing in mathematical biology was already growing strongly by the early 2010s. Reed emphasizes that the relationship is not one-directional—mathematics provides tools for biology, but biological questions also generate genuinely new mathematical problems.

A central theme is that biology may play for twenty-first-century mathematics a role comparable to the one physics played historically. Problems from mechanics led to dynamical systems, heat and wave phenomena stimulated partial differential equations, and quantum mechanics influenced functional analysis. Similarly, contemporary biology draws on and develops graph theory, topology, geometry, combinatorics, probability, stochastic processes, dynamical systems, harmonic analysis, and algebraic statistics. Examples include graph models of epidemics and gene networks, topology in neuroscience and cardiac fibrillation, geometry in protein folding, combinatorics in RNA structures, reaction–diffusion equations arising from morphogenesis, and mathematical methods for comparing evolutionary trees. Reed therefore predicts that biological questions will increasingly produce new concepts and theorems in core mathematics rather than merely serving as applications of existing techniques.

Reed also sees mathematical biology as strategically important for mathematics education and the public image of the subject. Biological and medical problems can attract students who would otherwise avoid mathematics because they immediately see how calculus, differential equations, probability, or graph theory relate to cancer, epidemics, neuroscience, genetics, and environmental questions. Undergraduate students can even participate in meaningful projects because many biological systems remain poorly understood. At the same time, biological applications make mathematics easier to communicate to the general public: people may not understand advanced mathematics itself, but they readily appreciate its contribution to disease research, medicine, ecology, and human health. Mathematical biology can therefore help attract new mathematics majors, create employment opportunities, and demonstrate the relevance of mathematics to society.

Key takeaways

Biology is becoming a major source of new mathematical questions, not merely a field where existing mathematics is applied.
Mathematics used in biology spans an unusually broad range of areas, from PDEs and dynamical systems to topology, combinatorics, probability, geometry, and algebra.
Reed suggests that biology could influence twenty-first-century mathematics as profoundly as physics influenced mathematics in earlier centuries.
Mathematical biology can help attract students and improve the public perception of mathematics by connecting abstract ideas with medicine, health, genetics, and ecology.

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