Infinite Limits

Hey! It's been a while since I last posted even though I mentioned that I would be more regular. That's my bad, I misjudged how busy I would be. Anyways.. enjoy this continuation of the last post! This is rather elementary, so please leave a comment if any of you would like a more detailed/longer explanation of the proof. An Intro to Perelman's Work Image Credit: CUNY - Mathematics To delve deeper into Grigori Perelman's proof of the Poincaré Conjecture, we need to explore several sophisticated concepts in differential geometry and geometric topology. Perelman's work, which built on Richard S. Hamilton's program involving the Ricci flow, is deeply rooted in the analysis of geometric structures on manifolds. Here, we'll attempt to shed light on some of the mathematical intricacies involved. The Poincaré Conjecture posits that every simply-connected, closed 3-manifold is homeomorphic to the 3-sphere S^3. A manifold is simply-connected if it lacks "holes,...

 Hey! It's been a minute since I wrote one of these, so I'm genuinely glad to be back. Happy New Year! I should be more regular now because I'm done with all university applications and I'm relatively free (as free as you can be by year 12). Today I'll be jabbering on a bit about the Poincaré Conjecture, one of the most special (solved) problems in mathematics. The Poincaré Conjecture Image Credit: Clay Mathematics Institute The Poincaré Conjecture is a fundamental problem in the branch of topology, best known as one of the 7 Millennium Prize Problems of the Clay Mathematics Institute. It was one of the most famous and long-standing unsolved problems in the history of mathematics until it was finally proven in 2003 by the Russian mathematician Grigori Perelman. Perelman's proof was a monumental achievement and earned him the Fields Medal and 1 million US Dollars (from the Clay Mathematics Institute) - both of which he declined (which I'll return to later). H...

In the intricate world of fluid dynamics, where the movement of liquids and gases forms the backdrop of countless natural phenomena, the unsolved Clay Institute Millenium Prize Problem of the Navier-Stokes equations stands as a beacon of understanding. These mathematical expressions encapsulate the intricate forces, pressures, and velocities that orchestrate the motion of fluids. George Gabriel Stokes Image Credit: Clay Mathematics Institute Fluid dynamics is a captivating branch of physics and mathematics that delves into the behaviour of fluids in motion. Whether it's the ocean currents, the streams, or the turbulence in a boiling pot, understanding the principles governing fluid motion requires strong mathematical concepts and physics insights. The Navier-Stokes equations, named in honour of the French engineer Claude-Louis Navier and the Irish mathematician George Gabriel Stokes, are the fundamental equations that explain fluid dynamics. In their general form, they can be expre...

Lagrangian mechanics is a powerful mathematical framework that provides a systematic and elegant approach to understanding the dynamics of physical systems. Developed by Joseph-Louis Lagrange in the late 18th century, it offers an alternative formulation to Newton's laws of motion, providing deeper insights into the fundamental principles governing the behaviour of objects. In this post, we will embark on a comprehensive journey into the realm of Lagrangian mechanics, covering its principles, applications, and significance in the field of physics. Joseph Louis Lagrange Image Credit: Encyclopaedia Britannica The Lagrangian Formulation: At the heart of Lagrangian mechanics lies the concept of the Lagrangian, denoted by L. The Lagrangian is a mathematical function that encapsulates the kinetic and potential energies of a system (KE-PE), providing a comprehensive description of its dynamics. By defining the Lagrangian, we can derive the equations of motion for a system using the princi...

  Newton's method, also known as the Newton-Raphson method, is an algorithm used to find the roots of a polynomial function. It was developed in the late 1600s. For a function f(x), finding the value of x for which f(x) = 0 is the goal of Newton's algorithm. Image Credit: GeoGebra Here's an algorithm you can use: Choose an initial guess for the root, let's call it x₀. Calculate the function value and its derivative at the current guess. These are denoted as f(x₀) and f'(x₀), respectively. Use the formula: x₁ = x₀ - f(x₀) / f'(x₀) to update the guess. This formula is derived from the tangent line approximation to the function at x₀, assuming it is differentiable. Repeat the last two sentences until the desired level of accuracy is achieved or until a specified number of iterations have been performed. You will know your answer is correct as the result of every iteration converges to the same value. What to know about the algorithm: Newton's algorithm typical...

What is the Rocket Equation? The rocket equation is a fundamental principle in physics, based on Newton’s Third Law of Motion) that describes the motion of rockets in space. It is a mathematical formula that calculates the velocity of a rocket based on its mass, the mass of the propellant it carries, and the speed at which it expels that propellant. Credit: NASA   The equation shows that as the mass of the rocket decreases (as it expels propellant), its velocity increases. This is because the rocket's momentum is conserved, and as the mass decreases, the velocity must increase to maintain momentum. The rocket equation is important because it determines the amount of propellant required to achieve a certain velocity or distance in space. It also shows that there are limits to the performance of rockets, as increasing the propellant mass requires a corresponding increase in the initial mass of the rocket, which makes it heavier and more difficult to launch. Credit: The Planetary Soci...

  Technical analysis is a trading methodology used to evaluate securities and financial markets based on statistical trends and market activity. Technical analysts examine past market data, including price and volume, to forecast future market behavior and make investment decisions. In this blog, we will explore the basic concepts and tools used in technical analysis. The Basics: Price charts The cornerstone of technical analysis is the price chart. Technical analysts use charts to visualize historical prices and volume of a security over a specific period. These charts are available in various formats, including line charts, bar charts, and candlestick charts. Line charts are the simplest type of chart, displaying the security's closing prices over a given period. Bar charts add more detail by showing the open, high, low, and close prices of the security in a given time frame. Candlestick charts, the most popular type of chart, show the same information as a bar chart but are easi...

The Mandelbrot Set is a set of complex numbers commonly used in art, due to its mathematical beauty. It was first defined by Robert W. Brooks   and Peter Matelksi in 1978, and the first visualisation was made by Benoit Mandelbrot in 1980.   Credit: American Scientist A lot of websites, videos, applets, and so on do a repeated zoom into the boundary of the Mandelbrot Set. But in mathematical terms, we define the Mandelbrot Set as the set of complex numbers which (under iteration) stays bounded within two. Here’s how it’s mathematically represented (don’t skip this, it’s easy math): f c (z) = z 2 + c, f c (z)<= 2 Let’s take an example here to understand this.   If c=-2,   f -2 (0) = 0 2 + (-2) = -2 (You start with z=0 as default) f -2 (-2) = (-2) 2 + (-2) = 4+ (-2) = 2 (You take the result of the previous function and input it as z in the next iteration) f- 2 (2) = (2) 2 + (-2) = 4+ (-2) = 2 We know that the result of the previous function (2), upon another itera...