Benjamin Skuse, Author at æģĆضĢŹÓʵ Science news and science articles from æģĆضĢŹÓʵ Fri, 11 Jun 2021 12:06:45 +0000 en-US hourly 1 https://wordpress.org/?v=7.0.1 242057827 When time runs backwards: What thermodynamics can tell us about life /article/2280247-when-time-runs-backwards-what-thermodynamics-can-tell-us-about-life/?utm_campaign=RSS|NSNS&utm_content=currents&utm_medium=RSS&utm_source=NSNS Wed, 09 Jun 2021 18:00:00 +0000 http://mg25033380.900 2280247 Baffling maths riddle that looks like a pile of worms almost solved /article/2216259-baffling-maths-riddle-that-looks-like-a-pile-of-worms-almost-solved/?utm_campaign=RSS|NSNS&utm_content=currents&utm_medium=RSS&utm_source=NSNS Thu, 12 Sep 2019 10:47:12 +0000 /?post_type=article&p=2216259 Colouring book pattern
This colouring book illustration shows how the conjecture converts many starting numbers back to 1
Illustration by Edmund Harriss, coloured in by Tiffany Arment

We almost have a solution to an exceptionally tricky mathematical riddle first posed 82 years ago.

The problem, known as the Collatz conjecture, is easy to state. Start with any positive whole number. If it is even, divide it by two. If itā€™s odd, triple it and add 1. Whatever the result, follow the same steps as before, over and over again. The conjecture says that whatever number you start with, the sequence will always eventually wend its way down to 1.

The sequence can be depicted visually to show sequences of numbers all wiggling their way back to the same spot (see main image). The result looks a bit like waving seaweed or a pile of strange worms.

As well as being visually arresting, it is intensely difficult to prove true or false. The conjecture has been verified up to the starting number of 1020: thatā€™s one hundred quintillion. But proving it holds in every case involves not just checking more and more numbers ā€“ the number line is infinite, after all ā€“ but finding a logically reasoned mathematical explanation.

Read more: The baffling quantum maths solution it took 10 years to understand

Now mathematician Terence Tao at the University of Califonia, Los Angeles, seems to have almost pulled it off. His work builds on that of other researchers, who proved that almost all sequences were at least able to reach an intermediate value between their starting number n and 1. This means they cannot balloon to infinity.

ā€œMany math problems become easier when one allows a small number of exceptional cases to behave badly and one is willing to settle for controlling almost all cases,ā€ says Tao. ā€œI showed that one could move this intermediate milestone to be as close as one wishes to the final goal 1ā€¦ for almost all n.ā€

Jeffrey Lagarias from the University of Michigan describes Taoā€™s work as ā€œthe most significant progress on the problem in many yearsā€.

Using maths to clean the oceans:

However, Tao himself says there is little hope of using his methods to find a complete proof. In his paper he says this is ā€œwell beyond reach of current methodsā€.

This is because he relies heavily on techniques from probability theory, meaning there is always a small chance of failure.Ģżā€œThere might be some minor technical refinements that can still be made ā€“ mostly having to do with the precise definition of ā€˜almost allā€™,ā€ he says. ā€œBut I am happy to leave such improvements to other mathematicians.ā€

Reference:

]]> 2216259 5 of the worldā€™s toughest unsolved maths problems /article/2193080-5-of-the-worlds-toughest-unsolved-maths-problems/?utm_campaign=RSS|NSNS&utm_content=currents&utm_medium=RSS&utm_source=NSNS /article/2193080-5-of-the-worlds-toughest-unsolved-maths-problems/#respond Thu, 07 Feb 2019 14:24:23 +0000 /?post_type=article&p=2193080 Ģż

1. Separatrix Separation

A pendulum in motion can either swing from side to side or turn in a continuous circle. The point at which it goes from one type of motion to the other is called the separatrix, and this can be calculated in most simple situations. When the pendulum is prodded at an almost constant rate though, the mathematics falls apart. Is there an equation that can describe that kind of separatrix? Ģż 2.-Navierā€“Stokes_HRH4BD

2. Navierā€“Stokes

The Navier-Stokes equations, developed in 1822, are used to describe the motion of viscous fluid. Things like air passing over an aircraft wing or water flowing out of a tap. But there are certain situations in which it is unclear whether the equations fail or give no answer at all. Many mathematicians have tried ā€“ and failed ā€“ to resolve the matter, including Mukhtarbay Otelbaev of the Eurasian National University in Astana, Kazakhstan. In 2014, he claimed a solution, but later retracted it. This is one problem that is worth more than just prestige. It is also one of the Millennium Prize Problems, which means anyone who solves it can claim $1 million in prize money.

Read more: The baffling quantum maths solution it took 10 years to understand

Ģż 3.-Exponents-and-dimensions_GettyImages-83463523

3. Exponents and dimensions

Imagine a squirt of perfume diffusing across a room. The movement of each molecule is random, a process called Brownian motion, even if the way the gas wafts overall is predictable. There is a mathematical language that can describe things like this, but not perfectly. It can provide exact solutions by bending its own rules or it can remain strict, but never quite arrive at the exact solution. Could it ever tick both boxes? That is what the exponents and dimensions problem asks. Apart from the quantum Hall conductance problem, this is the only one on the list that is at least partially solved. In 2000, Gregory Lawler, Oded Schramm and Wendelin Werner proved that exact solutions to two problems in Brownian motion can be found without bending the rules. It earned them a Fields medal, the maths equivalent of a Nobel prize. More recently, Stanislav Smirnov at the University of Geneva in Switzerland solved a related problem, which resulted in him being awarded the Fields medal in 2010. Ģż 4.-Impossibility-theorems_FD5B8T

4. Impossibility theorems

There are plenty of mathematical expressions that have no exact solution. Take one of the most famous numbers ever, pi, which is the ratio of a circleā€™s circumference to its diameter. Proving that it was impossible for piā€™s digits after the decimal point to ever end was one of the greatest contributions to maths. Physicists similarly say that it is impossible to find solutions to certain problems, like finding the exact energies of electrons orbiting a helium atom. But can we prove that impossibility? Ģż 5.Spin_Glass_GettyImages-686726405

5. Spin glass

To understand this problem, you need to know about spin, a quantum mechanical property of atoms and particles like electrons, which underlies magnetism. You can think of it like an arrow that can point up or down. Electrons inside blocks of materials are happiest if they sit next to electrons that have the opposite spin, but there are some arrangements where that isnā€™t possible. In these frustrated magnets, spins often flip around randomly in a way that, it turns out, is a useful model of other disordered systems including financial markets. But we have limited ways of mathematically describing how systems like this behave. This spin glass question asks if we can find a good way of doing it. ā€¢ĢżSee the full list of unsolved problems:ĢżĢż]]> /article/2193080-5-of-the-worlds-toughest-unsolved-maths-problems/feed/ 0 2193080 The baffling quantum maths solution it took 10 years to understand /article/2192598-the-baffling-quantum-maths-solution-it-took-10-years-to-understand/?utm_campaign=RSS|NSNS&utm_content=currents&utm_medium=RSS&utm_source=NSNS Wed, 06 Feb 2019 18:00:00 +0000 http://mg24132160.400 2192598