Mathematician Finds Solution to One of The Oldest Problems in Algebra
Wildberger, from the University of New South Wales (UNSW) in Australia, worked with computer scientist Dean Rubine on a paper that details how these incredibly complex calculations could be worked out.
"This is a dramatic revision of a basic chapter in algebra," says Wildberger. "Our solution reopens a previously closed book in mathematics history."
As you might expect, understanding how this works isn't easy for the non-algebra geniuses amongst us. Essentially, polynomials are equations that include variables raised to non-negative powers (e.g. x3). When those powers are five or above, that's a higher-degree polynomial.
Mathematicians have figured out how to solve lower-degree versions, but it was thought that properly calculating the higher-degree ones was impossible. Before this new research, we've been relying on approximations.
Wildberger and Rubine took a new approach to the problem, which is based on Catalan numbers. These numbers are used in advanced number counting and arrangements, including counting how many ways polygons can be subdivided into triangles.
By extending the idea of Catalan numbers, the researchers were able to demonstrate that they could be used as a basis for solving polynomial equations of any degree. Part of the clever method involved extending polygon counts to other shapes besides triangles.
It's a departure from the traditional method of using radical expressions (like square roots and cube roots) to solve equations like this, instead relying on combinatorics – counting numbers, fundamentally, but in increasingly advanced ways.
"The Catalan numbers are understood to be intimately connected with the quadratic equation," says Wildberger.
"Our innovation lies in the idea that if we want to solve higher equations, we should look for higher analogs of the Catalan numbers."
The researchers put their new algebra up against some well-known polynomial equations of the past, including a famous cubic equation studied by John Wallis. The numbers checked out, validating the new work.
Wildberger and Rubine didn't stop there. They also discovered a new mathematical structure called the Geode, which ties in with Catalan numbers and seems to act as a foundation for them. This Geode could form the basis of many future studies and discoveries, the researchers say.
As the approach taken here is so different to what's gone before, there's the potential to rethink many key ideas that mathematicians have long relied on for computer algorithms, the way data is structured, and game theory. It might even have applications in biology – for counting RNA molecule folding, for example.
"This is a core computation for much of applied mathematics, so this is an opportunity for improving algorithms across a wide range of areas," says Wildberger.
The research has been published in The American Mathematical Monthly.
Scientists Build First-Ever 'Black Hole Bomb' Analog
Gravity May Be a Clue That The Universe Is a Giant Computer
Scientists Just Confirmed a 67-Year-Old Hypothesis About Vitamin B1
Try Our AI Features
Explore what Daily8 AI can do for you:
Comments
No comments yet...
Related Articles
Yahoo
5 days ago
- Yahoo
Study Reveals How Many IVF Babies Have Been Born Worldwide
More than 13 million people worldwide owe their lives to in vitro fertilization (IVF), according to the first estimates of their kind. Since the original 'test tube' baby was born in 1978, the number of children brought into the world by IVF technologies – collectively known as assisted reproductive technology (ART) – has exploded into the millions. Today, an ART-conceived baby is born every 35 seconds, and IVF, where an egg is fertilized by sperm in a test tube, is the most commonly used method. To figure out how many ART babies have been born since the technology's infancy, an international team of researchers compiled health data from 101 different nations. Related: As of 2018, the team estimates between 10 and 13 million babies have been born via ART, while preliminary data from 2018 to 2023 adds another 3 to 4 million children. "Every year we estimate how many babies were born in the previous 12 months – but we've never actually gone back and used a consistent methodology and extrapolation for missing data to estimate the total number of babies born in the world," says lead author Georgina Chambers, an epidemiologist from the University of New South Wales. The estimates are not perfect, but Chambers and her colleagues argue they are "a reasonable and sound approximation" based on the "most robust data available in the literature." The information comes from the International Committee Monitoring Assisted Reproductive Technologies (ICMART), of which Chambers is the data custodian. Back in the early 1980s, Australia became the first country in the world to establish an IVF registry. While many high-income nations have since followed suit, some countries, like Singapore,either lack formal registries that report to ICMART or the registries have been delayed for many years, as in China. Even when formal lists are kept, some regional clinics do not report to ICMART, as in the United States, India, and some African countries. To make up for these gaps in knowledge, researchers tallied the number of clinics within a nation, using that coverage rate to predict ART rates. They also accounted for miscarriage rates where only ART pregnancies were reported. Their findings reveal that in the 40 years since the dawn of ART, between 9.4 million and 12.4 million infants from 101 countries were born using reproductive technology, with an exponential increase starting around the turn of the century. Europe tops the charts with 3.6 to 4.5 million infants born via IVF, followed by Asia with 3 to 4 million IVF infants, and North America with 1.4 to 1.6 million IVF infants. In the last two decades, IVF technology has come a long way, providing cheaper, safer, and more successful routes to parenthood. The chances of IVF succeeding are far from guaranteed, but health data from the UK show that the average IVF pregnancy rate using frozen embryos has gone from 7 percent in the 1990s to 36 percent in 2021. No longer is it necessary to transfer several fertilized eggs to increase the odds of pregnancy. As a result, the number of IVF twins and triplets being born is on a steep downward trend. "Today, more than 93 percent of IVF cycles involve a single embryo transfer, resulting in a multiple birth rate of below 3 percent," says Chambers. "Most IVF babies in Australia now come from frozen embryos." "As more babies are born through IVF," Chambers adds, "we hope access to safe, high quality care will become more equitable – and based on human rights – across the world." The study was published in Fertility and Sterility. Related News Image on The Shroud of Turin May Not Belong to a Real Human Venting Doesn't Reduce Anger, But Something Else Does, Says Study Playing Video Games Has an Unexpected Effect on Kids' IQ, Study Discovers Solve the daily Crossword
Yahoo
01-08-2025
- Yahoo
The Bright Side: Leopard seal love songs structured like nursery rhymes, study finds
Male leopard seals compose and sing their own songs to woo potential mating partners, scientists in Australia have found. The songs are structured much like nursery rhymes, making them easy to remember. "It kind of sounds like sound effects from an '80s sci-fi" movie, the lead author of the study said. When male leopard seals dive down into icy Antarctic waters, they sing songs structured like nursery rhymes in performances that can last up to 13 hours, scientists said Thursday. The Australian-led team of researchers compared the complexity of the songs composed by the big blubbery mammals to those of other animals -- as well as human musicians like the Beatles and Mozart. Lucinda Chambers, a bioacoustics PhD student at Australia's University of New South Wales, told AFP that people are often surprised when they hear the "otherworldly" hoots and trills sung by leopard seals. "It kind of sounds like sound effects from an '80s sci-fi" movie, said the lead author of a new study in the journal Scientific Reports. During the spring breeding season, male leopard seals dive underwater and perform their songs for two minutes before returning to the surface for air. They then repeat this performance for up to 13 hours a day, according to the study. The researchers determined that all leopard seals share the same set of five "notes" which are impossible to distinguish between individuals. However each seal arranges these notes in a unique way to compose their own personal song. "We theorise that they're using that structure as a way to broadcast their individual identity, kind of like shouting their name out into the void," Chambers said. The researchers believe the males use these songs to woo potential female mates -- and ward off rivals. 'Songbirds of the ocean' The team studied recordings of 26 seals captured by study co-author Tracey Rogers off the coast of Eastern Antarctica throughout the 1990s. "They're like the songbirds of the Southern Ocean," Rogers, who is also from the University of New South Wales, said in a statement. "During the breeding season, if you drop a hydrophone into the water anywhere in the region, you'll hear them singing." The team analysed how random the seals' sequences of notes were, finding that their songs were less predictable than the calls of humpback whales or the whistles of dolphins. But they were still more predictable than the more complex music of the Beatles or Mozart. "They fall into the ballpark of human nursery rhymes," Chambers said. This made sense, because the songs need to be simple enough so that each seal can remember their composition to perform it every day, she explained. She compared it to how "nursery rhymes have to be predictable enough that a child can memorise them". But each seal song also needs to be unpredictable enough to stand out from those of the other males. Leopard seals, which are the apex predator in Antarctic waters, swim alone and cover vast distances. They likely evolved their particular kind of song so that their message travels long distances, the researchers theorised. Varying pitch or frequency might not travel as far in their environment, Chambers said. Female seals also sing sometimes, though the scientists do not know why. Chambers suggested it could be to teach their pups how to sing -- exactly how this talent is passed down is also a mystery. But she added that this behaviour has never been observed in the wild. The females could also just be communicating with each other, she said. (FRANCE 24 with AFP)
Yahoo
01-08-2025
- Yahoo
Seals sing 'otherworldly' songs structured like nursery rhymes
When male leopard seals dive down into icy Antarctic waters, they sing songs structured like nursery rhymes in performances that can last up to 13 hours, scientists said Thursday. The Australian-led team of researchers compared the complexity of the songs composed by the big blubbery mammals to those of other animals -- as well as human musicians like the Beatles and Mozart. Lucinda Chambers, a bioacoustics PhD student at Australia's University of New South Wales, told AFP that people are often surprised when they hear the "otherworldly" hoots and trills sung by leopard seals. "It kind of sounds like sound effects from an '80s sci-fi" movie, said the lead author of a new study in the journal Scientific Reports. During the spring breeding season, male leopard seals dive underwater and perform their songs for two minutes before returning to the surface for air. They then repeat this performance for up to 13 hours a day, according to the study. The researchers determined that all leopard seals share the same set of five "notes" which are impossible to distinguish between individuals. However each seal arranges these notes in a unique way to compose their own personal song. "We theorise that they're using that structure as a way to broadcast their individual identity, kind of like shouting their name out into the void," Chambers said. The researchers believe the males use these songs to woo potential female mates -- and ward off rivals. - 'Songbirds of the ocean' - The team studied recordings of 26 seals captured by study co-author Tracey Rogers off the coast of Eastern Antarctica throughout the 1990s. "They're like the songbirds of the Southern Ocean," Rogers, who is also from the University of New South Wales, said in a statement. "During the breeding season, if you drop a hydrophone into the water anywhere in the region, you'll hear them singing." The team analysed how random the seals' sequences of notes were, finding that their songs were less predictable than the calls of humpback whales or the whistles of dolphins. But they were still more predictable than the more complex music of the Beatles or Mozart. "They fall into the ballpark of human nursery rhymes," Chambers said. This made sense, because the songs need to be simple enough so that each seal can remember their composition to perform it every day, she explained. She compared it to how "nursery rhymes have to be predictable enough that a child can memorise them". But each seal song also needs to be unpredictable enough to stand out from those of the other males. Leopard seals, which are the apex predator in Antarctic waters, swim alone and cover vast distances. They likely evolved their particular kind of song so that their message travels long distances, the researchers theorised. Varying pitch or frequency might not travel as far in their environment, Chambers said. Female seals also sing sometimes, though the scientists do not know why. Chambers suggested it could be to teach their pups how to sing -- exactly how this talent is passed down is also a mystery. But she added that this behaviour has never been observed in the wild. The females could also just be communicating with each other, she said. dl/ach
