February 2011

The Collatz Conjecture: Unsolved but Useless

The Collatz Conjecture is one of the Unsolved problems in mathematics, especially in Number Theory. The Collatz Conjecture is also termed as 3n+1 conjecture, Ulam Conjecture, Kakutani’s Problem, Thwaites Conjecture, Hasse’s Algorithm, and Syracuse Problem. Collatz Conjecture Statement If you keep repeating this procedure, you shall reach the number 1 at last. Illustrations » Starting…

Dirichlet's Theorem and Liouville's Extension of Dirichlet's Theorem

Topic: Beta & Gamma functions The Gamma function and Beta functions belong to the category of special transcendental functions and are defined in terms of improper definite integrals. Definitions of Beta and Gamma functions are given below. But before that, let's quote the statement of Dirichlet's theorem so that we can work around Liouville's extension…

The Lindemann Theory of Unimolecular Reactions

In this article, we will learn about the Lindemann Theory of Unimolecular Reactions, which is also known as the Lindemann-Hinshelwood mechanism. Understanding Lindemann Theory of Unimolecular Reactions It is easy to understand a bimolecular reaction on the basis of collision theory. When two molecules A and B collide, their relative kinetic energy exceeds the threshold…

The Mystery of the Missing Money - One Rupee

Puzzle Two women were selling marbles in the market place -- one at three for a Rupee and other at two for a Rupee. One day both of then were obliged to return home when each had thirty marbles unsold. They put together the two lots of marbles and handing them over to a friend…

Understanding Poincare Conjecture

In 1904, the french Mathematician Henri Poincaré (en-US: Henri Poincare) posed an epoch-making question, which later came to be termed as Poincare Conjecture, in one of his papers, which asked: If a three-dimensional shape is simply connected, is it homeomorphic to the three-dimensional sphere? Henri Poincare - 1904 So what does it really mean? How…

Cosmic Radiation & Cosmic Ray Showers: Complete Guide

Wondering what Cosmic Radiation & Cosmic Ray Showers are and how these are formed? If you are, you are at the perfect place. Cosmic radiation and cosmic ray showers are some of the most intriguing and fascinating phenomena in the universe. While the term "radiation" may have negative connotations, cosmic radiation is a natural occurrence…

Rayleigh-Jeans Law & The Ultraviolet Catastrophe

Lord Rayleigh made an attempt to explain the energy distribution in black body radiation, which was completed by Jeans in 1900. The results obtained by then are known as Rayleigh-Jeans' Rules on Black Body Radiation. The law covering these rules is called Rayleigh Jean's Law. The black body emits radiation of continuously variable wavelengths right…

Wein’s Formula & Wein’s Laws

Various physicists tried to explain the problem of energy distribution in black body radiation, and finally, German Physicist Max Planck successfully solved the problem. Before him, German Physicist Wilhelm Wein and British Physicist Lord Rayleigh & James Jean have tackled this problem and have given important laws. In fact, the work of their scientists paved…

Derivative of x squared is 2x or x ? Where is the fallacy?

Derivative of x squared As we know, the derivative of x squared, i.e., differentiation of $ x^2$ , with respect to $ x$, is $ 2x$. i.e., $ \dfrac{d}{dx} x^2 = 2x$ A Curious Case Suppose we write $ x^2$ as the sum of $ x$ 's written up $ x$ times. i.e., $ x^2…

Solving Ramanujan's Puzzling Problem

Consider a sequence of functions as follows:- $ f_1 (x) = \sqrt {1+\sqrt {x} } $ $ f_2 (x) = \sqrt{1+ \sqrt {1+2 \sqrt {x} } } $ $ f_3 (x) = \sqrt {1+ \sqrt {1+2 \sqrt {1+3 \sqrt {x} } } } $ ......and so on to $ f_n (x) = \sqrt {1+\sqrt{1+2 \sqrt…

Its a Mystery! - Simulacrum in Eagle Nebula

This is one of the top mysteries of universe.