Top Mathematics discussions
NishMath - #understanding
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@exercism.org
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jcarroll.com.au
Modulo arithmetic is being explored as a technique for rotation, inspired by a post from Greg Egan on Mastodon. Jonathan Carroll implemented Egan's formula in various languages. The formula, rot(n, j, k) = (n * 10^k) mod (10^j-1), efficiently rotates a j-digit number n by k digits. Carroll highlighted the importance of understanding fundamental operators across different languages while implementing this seemingly simple mathematical concept.
This exploration is beneficial for fortifying knowledge of basic functionalities in different programming languages. As Carroll wrote in his blog, implementing this simple math "means I’ll be working with some basic functionality and I believe it’s very important to have that locked in comfortably." Carroll demonstrated the implementation in R, utilizing the power operator '^' and the modulo operator '%%'. He also showed how to determine the number of digits in a number using the 'nchar()' function, showcasing practical application of these operators. The R implementation allows for the cyclic rotation of digits within a number. While R doesn't have a built-in function for this specific purpose, Carroll demonstrated the usage of his '{vec}' package for a similar ring-buffer effect on vectors. This involved using modulo on the indices to achieve the desired rotation. The formula subtracts (10^j-1) times the leftmost k digits of n*10^k, removing them from the left and adding them to the right. Recommended read:
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@aasnova.org
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References:
aasnova.org
Recent astrophysics research has shed light on Jupiter Trojans, Titan's seas, and dwarf galaxies, utilizing advanced observational tools. The Cassini spacecraft captured a remarkable phenomenon on Titan, Saturn's largest moon, observing sun glitter and sunglint reflecting off its hydrocarbon seas. These observations provide invaluable insights into the complex interaction between Titan’s liquid surfaces and its atmosphere.
The Cassini observations of specular reflection, or sunglint, confirmed the presence of liquid on Titan's surface. Researchers have identified oily lakes and seas of liquid hydrocarbons like methane and ethane. A recent study from Michael Heslar and Jason Barnes focused on Punga Mare, the smallest of Titan’s named seas, where extraterrestrial waves were definitively identified. The analysis of sunglint and sun glitter phenomena, with localized changes in surface angle reflecting sunlight, offers a unique opportunity to study Titan's coastal regions. Additionally, a new study published in the Open Journal of Astrophysics explores the characteristics of dwarf galaxies within the TNG50 field. The research, conducted by Joy Bhattacharyya and Annika H.G. Peter from Ohio State University, and Alexie Leauthaud from UC Santa Cruz, aims to connect the star-formation rates of these galaxies with their surrounding environments. This work utilizes advanced computational models to deepen our understanding of galactic evolution and the factors that influence star formation in these smaller galaxies. Recommended read:
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@www.quantamagazine.org
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Recent advancements in mathematics and physics are pushing the boundaries of our understanding of the universe. A decades-old bet between mathematicians Noga Alon and Peter Sarnak regarding the nature of optimal expander graphs has recently been settled, with both mathematicians being proven wrong. This involved tapping into a crucial phenomenon in physics and pushing it to its limits, demonstrating the interconnectedness of mathematics and physics. Also, Researchers have successfully modeled how 'broken' tulips get their stripes, solving a centuries-old floral mystery. The mathematical model explains that the tulip-breaking virus inhibits the production of anthocyanins, leading to the distinctive striped pattern.
Efforts are underway to bridge the gap between quantum mechanics and general relativity, with researchers exploring the possibility of creating quantum gravity in the lab. Monika Schleier-Smith at Stanford University is leading this effort by using laser-cooled atoms to explore whether gravity could emerge from quantum entanglement. NASA is also contributing to this field by developing the first space-based quantum gravity gradiometer. This gradiometer will use ultra-cold rubidium atoms to detect gravitational anomalies with high precision from orbit, with potential applications in water resource management and subsurface geology. Further progress is being made in language model development. Researchers are exploring methods to sidestep language in order to improve how language models work with mathematics. By allowing these models to operate directly in mathematical spaces, they aim to enhance efficiency and reasoning capabilities. This research highlights the potential for artificial intelligence systems to benefit from thinking independently of language, paving the way for more advanced and effective AI applications. Recommended read:
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@medium.com
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Mathematics is a diverse field with applications spanning multiple disciplines. Recent articles and discussions have highlighted the importance of mathematics in various areas, including Artificial Intelligence (AI), data science, and quantum physics. Linear algebra, calculus, and probability are identified as essential mathematical topics for mastering AI and machine learning, while mathematical tools are enhancing learning in these complex fields.
The exploration of mathematics extends beyond its application in technology, encompassing historical perspectives, number theory, and geometric puzzles. Pi, a fundamental mathematical constant, continues to fascinate mathematicians and enthusiasts, with its presence felt across science, engineering, art, and culture. Discussions also cover the etymology of mathematical terms like logarithms, and the use of math journals and games in education. Recommended read:
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Siôn Geschwindt@The Next Web
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The ESA's Euclid mission has released its first data, offering a glimpse into the deep fields of space and marking a significant step in the study of dark matter and dark energy. The data covers 63 square degrees of the sky and includes a classification survey of over 380,000 galaxies and 500 gravitational lens candidates. The Euclid space observatory, dubbed the "dark matter hunter", aims to create the largest 3D map of the sky ever created.
This initial data release includes numerous galaxy clusters, active galactic nuclei, and transient phenomena. The mission's unique observational capabilities are expected to contribute to a better determination of the expansion rate of the Universe through gravitational-wave observations. Researchers plan to correlate events from LIGO, Virgo, and KAGRA with Euclid galaxy catalogues. Euclid's discovery of numerous gravitational lens systems could also be crucial for detecting the first gravitational waves split into multiple images by gravitational lenses, according to Miguel Zumalacárregui, group leader in the Astrophysical and Cosmological Relativity department. Recommended read:
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