mathkaveli

In this post, we will explore an elegant mathematical idea that links geometry, calculus, and physics: the Legendre transform (sometimes called the Legendre dual ). The basic concept is that a smooth curve or function can be re-described entirely in terms of the slopes of its tangent lines. Even more remarkably, this process can be reversed to recover the original function. The same idea appears in classical mechanics and thermodynamics as a tool for switching between different physical variables, such as velocity and momentum or entropy and temperature. 1. Geometric Intuition with Tangent Lines Suppose we have a smooth, upward-curving function f(x) . At any point x = a , we can draw a tangent line. The tangent line can be written in slope-intercept form: y = p·x + b Here, p = f′(a) is the slope of the tangent line, and b is its y-intercept, which can be found by substituting x = 0 : b = f(a) − a·f′(a) So, every point a on the curve corresponds to a line with slope p a...

The Hidden Connections Between Totient, Sigma, Tau, Möbius, and Dirichlet Convolution In number theory, certain functions show how integers interact with their divisors. This post introduces five of them — the Euler totient function phi(n), the sum of divisors sigma(n), the number of divisors tau(n), the Möbius function mu(n), and the Dirichlet convolution — using one example number: Example number: n = 12 1) Euler’s Totient Function φ(n) Idea: phi(n) counts how many numbers from 1 to n are coprime to n (that is, share no common factors with n except 1). Formula: If n = p1^a1 * p2^a2 * ... * pk^ak, then phi(n) = n * (1 - 1/p1) * (1 - 1/p2) * ... * (1 - 1/pk) Example with n = 12: 12 = 2^2 * 3^1 phi(12) = 12 * (1 - 1/2) * (1 - 1/3) = 12 * (1/2) * (2/3) = 4 The numbers 1, 5, 7, and 11 are coprime to 12, so phi(12) = 4. 2) The Sigma Function σ(n) Idea: sigma(n) is the sum of all positive divisors of n. Formula: If n = p1^a1 * p2^a2 * ... * pk^a...

Have you ever wondered how computers make predictions—like when your music app seems to know the next song to play, or how weather forecasts estimate tomorrow’s conditions based on today’s patterns? Behind these everyday tools lies a mathematical idea called a Markov Chain , named after the Russian mathematician Andrey Markov , who developed the concept in the early 1900s. What Is a Markov Chain? A Markov Chain models systems that evolve step-by-step over time where the next state depends only on the current state, not on the full history. This principle is called the Markov Property : “The future is independent of the past, given the present.” To specify a Markov Chain, you need: A set of possible states (for example, Sunny or Rainy). Transition probabilities that describe how likely it is to move from one state to another in one step. Transition Matrix (2×2 Case) We often collect the transition probabilities into a transition matrix . For two states A and...

What does it really mean for students to be mathematically fluent? If you’ve been in any math PD over the past few years, you’ve likely heard the phrase everywhere. We talk about fluency as something students should develop, strengthen, and demonstrate, but it can still feel abstract when we try to describe it in observable, classroom-ready terms. This post breaks down mathematical fluency into the two simplest frames we can use as teachers: what it looks like and what it sounds like . These descriptions can guide instruction, assessment, student goal-setting, and even walkthrough conversations with colleagues or administrators. What Mathematical Fluency Looks Like In a classroom where students are developing mathematical fluency, you see students making choices about strategies rather than following steps robotically. They use representations—number lines, diagrams, tables, graphs, manipulatives, symbolic expressions—and switch between them to make sense of a problem. They move ...

Convolution is one of those delightful mathematical ideas that quietly appears everywhere once you start looking for it. At first glance, it looks like a strange mixing operation: take two functions, slide one across the other, multiply their values, and integrate (or sum). But the moment you try it with simple functions, a beautiful story emerges. The convolution of two functions f and g is written as (f * g)(t), and is defined by the integral (f * g)(t) = ∫ f(x) g(t - x) dx. If you prefer discrete settings, such as adding the outcomes of rolled dice, the definition becomes a sum: (f * g)(n) = Σ f(k) g(n - k). In either case the idea is the same: one function holds steady, the other moves, and the convolution value at time t tells us how much the two shapes “overlap” at that moment. Think of it like sliding one melody across another and listening for moments when the notes harmonize. When f and g have matching peaks, the convolution becomes large. When f is strong where g is weak, the...