mathkaveli

Introduction Have you ever rearranged the letters in a word, shuffled a deck of cards, or mixed up your schedule? If so, you’ve worked with permutations — rearrangements of a set of objects. In this post, we'll explore permutations in a mathematical way using functions , and learn how to combine them using composition . Then, we’ll look at the symmetric group S 3 S_3 — the set of all possible ways to rearrange 3 things — and build its Cayley table , which shows how these rearrangements interact. No worries if you’ve never heard of abstract algebra — we’ll break it all down! 📦 What Is a Permutation? A permutation is a way of rearranging a set of objects. Suppose we have the set { 1 , 2 , 3 } \{1, 2, 3\} . One permutation might send: 1 to 2, 2 to 3, 3 to 1. We write that as: ( 1 2 3 2 3 1 ) \left(\begin{matrix} 1 & 2 & 3 \\ 2 & 3 & 1 \end{matrix}\right) This is just a fancy way to show where each number goes — a kind of function that scrambles the set. ...

Mathematics is full of patterns and structures that help us understand the world. One of the most beautiful and fundamental concepts in modern mathematics is group theory , which underpins everything from physics to computer science. In this blog post, we will explore subgroups , learn how to determine if a set is a subgroup, and then expand into the world of cyclic groups , cosets , and how these ideas naturally partition a group into meaningful pieces. This guide is written for a high school senior-level audience, so we will take it step by step, introducing concepts in a simple and intuitive way. 1. What is a Group? Before we talk about subgroups , we need to understand what a group is. A group is a set of elements with an operation (such as addition or multiplication) that satisfies the following four properties: Closure : If you take any two elements in the group and apply the operation, the result is still in the group. Associativity : The operation follows the rule ( a ...

Mathematics is everywhere, even in places we don’t always notice. One such place is in how computers detect and correct errors when sending information. If you’ve ever sent a text message and had it arrive correctly despite bad cell reception, you’ve seen error correction at work. This blog post will introduce error-correcting codes and show how they connect to linear algebra and abstract algebra . We will work through some example problems involving binary codes and explore the beautiful mathematical structure behind them. 1. What is an Error-Correcting Code? An error-correcting code is a way of encoding information so that small errors can be detected and fixed. Imagine you're sending a message using 0s and 1s , but some bits might get flipped (from 0 to 1 or vice versa) due to interference. A well-designed code allows the receiver to figure out what the original message was, even if errors occur. We focus on linear codes , a special kind of error-correcting code where th...

Introduction Have you ever wondered how mathematicians determine whether an infinite series makes sense for different values? When dealing with power series—expressions like: S = a_0 + a_1z + a_2z^2 + a_3z^3 + ... we need to check where this series actually sums to a meaningful number. The Cauchy-Hadamard formula helps us do exactly that! In this post, we will break down this formula step by step in a way that is easy to understand. By the end, you’ll be able to find the radius of convergence of a power series using simple calculations. Let's dive in! Understanding the Radius of Convergence A power series is a type of infinite sum where each term includes a power of a variable z . But for what values of z does this series actually add up to a meaningful number? The radius of convergence R R is the boundary that tells us where the series behaves well. If we pick a value of z inside this boundary, the series will add up to a finite number. Outside this boundary, the sum...

If you've learned about Taylor series in calculus, you might know that some functions can be written as an infinite sum of powers of x x or z z . But what happens when a function has a problem—a singularity? That’s where the Laurent series comes in! What Is a Laurent Series? A Laurent series is like a Taylor series, but it includes negative powers too! It looks like this: f ( z ) = ∑ n = − ∞ ∞ c n ( z − z 0 ) n f(z) = \sum_{n=-\infty}^{\infty} c_n (z - z_0)^n Here: z 0 z_0 is the center of expansion, The coefficients c n c_n tell us how much each term contributes, The powers of ( z − z 0 ) (z - z_0) can be both positive and negative ! The key difference from the Taylor series is that Laurent series allow for terms like 1 ( z − z 0 ) k \frac{1}{(z - z_0)^k} , which help describe functions with singularities. Why Do We Need Laurent Series? Sometimes, a function behaves badly at a certain point. For example, the function: f ( z ) = 1 z f(z) = \frac{1}{z} is not define...

Introduction Have you ever wondered how mathematicians work with functions that "blow up" at certain points? In complex analysis , we use Laurent series to express functions with singularities (points where they become infinite or undefined). From these series, we can extract something called the residue , which plays a crucial role in evaluating contour integrals. In this post, we’ll learn how to compute residues using Laurent series by working through two examples: f ( z ) = e 1 / z f(z) = e^{1/z} , a function with an essential singularity at z = 0 z = 0 . g ( z ) = ln ⁡ ( 1 + z ) 1 + z 2 g(z) = \frac{\ln(1+z)}{1+z^2} , a function with isolated singularities that we will analyze. No prior knowledge of complex analysis is needed—just a curiosity for math! What is a Laurent Series? A Laurent series is like a Taylor series but allows for negative powers of z z . It takes the form: f ( z ) = ∑ n = − ∞ ∞ c n ( z − a ) n . f(z) = \sum_{n=-\infty}^{\infty} c_n (z-a)^n....