Top Mathematics discussions
NishMath - #math
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@www.newtonproject.sussex.ac.uk
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References:
Xi'an's Og
, Pat'sBlog
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Recent blog posts are delving into a variety of mathematical topics, offering insights and explorations across different areas of the field. These posts cover historical aspects of mathematics, examine specific mathematical concepts, and explore the connections between mathematics and other disciplines. This collection of diverse content aims to provide readers with a broader understanding and appreciation of mathematics.
The blog posts include diverse mathematical items. For example, one post references Gemma Frisius' "Arithmeticae Practicae Methodus Facilis" (1540) and its entry in *MAA Mathematical Treasures. Another commemorates April 13 as "On This Day in Math," highlighting mathematical facts associated with the number 103. This includes its unique properties as a prime number and its presence in Ramanujan's mathematical explorations. Furthermore, the blog explores historical events like the coining of the word "microscope" in 1620 and Lord Brouncker's published mathematical result in 1668. From statistical physics to number theory, these blogs showcase the versatility and interdisciplinary nature of mathematical thought. One blog even mentions using statistical physics concepts to analyze election results. These blog postings aim to engage readers with a range of mathematical subjects, from historical figures and publications to contemporary applications and connections. Recommended read:
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@primes.utm.edu
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References:
Pat'sBlog
, mathdaypballew.blogspot.com
This week saw a flurry of mathematical activity, highlighted by the 238th Carnival of Mathematics, organized by Aperiodical. The carnival showcases a variety of submissions and mathematical art, focusing on the number 238 itself. Noteworthy facts about 238 include that it is 2 × 7 × 17, the sum of the first 13 primes, and a "triprime". The carnival also encourages exploration beyond pure mathematics, with community members contributing insights linking the number to uranium isotopes, birth minutes, and even hexadecimal representations. It also shines a light on #MathArtMarch, with examples of crochet, coding, and painting from around the world.
Continuing the daily exploration of numbers, several interesting facts and events were highlighted for April 6th, 7th, 8th and 10th. The number 96, the 96th day of the year, was examined for its unique properties, such as being the smallest number expressible as the difference of two squares in four different ways. Events like Euler's first paper on partitions (April 7th, 1741) and Al-Biruni's observation of a solar eclipse in 1019 were also noted, linking mathematical concepts to historical contexts. Also, the number 97 has been noted as the 97th day of the year, where 97 is the largest prime that we can ever find that is less than the sum of square of its digits. In recreational mathematics, a "Salute" game for reinforcing multiplication and division was featured, emphasizing the inverse relationship between these operations. Additionally, the concept of "truncatable primes" was explored through a game where players create strings of prime numbers by adding digits to either end of a number. The number 91 was discussed as the 91st day of the year where 10 n + 91 and 10 n + 93 are twin primes for n = 1, 2, 3 and 4. Finally, highlighting mathematics beyond academia, James Abram Garfield, a former Congressman and mathematician, was mentioned for his original proof of the Pythagorean Theorem, illustrating the interdisciplinary nature of mathematics. Recommended read:
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Greg Bock@The Quantum Insider
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References:
The Quantum Insider
Quantum computing has taken a significant leap forward with Phasecraft's development of a novel quantum simulation method called THRIFT (Trotter Heuristic Resource Improved Formulas for Time-dynamics). This breakthrough, detailed in a recent *Nature Communications* publication, drastically improves simulation efficiency and lowers computational costs, bringing real-world quantum applications closer to reality. THRIFT optimizes quantum simulations by prioritizing interactions with different energy scales within quantum systems, streamlining their implementation into smaller, more manageable steps.
This approach allows for larger and longer simulations to be executed without the need for increased quantum circuit size, thereby reducing computational resources and costs. In benchmarking tests using the 1D transverse-field Ising model, a widely used benchmark in quantum physics, THRIFT achieved a tenfold improvement in both simulation estimates and circuit complexities, enabling simulations that are ten times larger and run ten times longer compared to traditional methods. This development holds immense promise for advancements in materials science and drug discovery. Separately, mathematicians have achieved a breakthrough in understanding and modeling melting ice and other similar phenomena through a new proof that resolves long-standing issues related to singularities. A powerful mathematical technique used to model melting ice and other phenomena had been hampered by “nightmare scenarios.” A new proof has removed that obstacle. This new proof addresses concerns about "nightmare scenarios" that previously hindered the analysis of these processes, ensuring that singularities do not impede the continued evolution of the surface being modeled. The resolution, described in Quanta Magazine, allows mathematicians to more effectively assess the surface's evolution even after a singularity appears. Finally, researchers at Cornell University have introduced a novel data representation method inspired by quantum mechanics that tackles the challenge of handling big, noisy data sets. This quantum statistical approach simplifies large data sets and filters out noise, allowing for more efficient analysis than traditional methods. By borrowing mathematical structures from quantum mechanics, this technique enables a more concise representation of complex data, potentially revolutionizing innovation in data-rich fields such as healthcare and epigenetics where traditional methods have proven insufficient. 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:
References :
Denise Gaskins@denisegaskins.com
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Recent studies and educational resources are focusing on enhancing math education through innovative approaches. Denise Gaskins' "Let's Play Math" blog offers resources for families to learn and enjoy math together, including playful math books and internet resources suitable for various age groups. Math journaling and games have been highlighted as effective tools to engage students, promote problem-solving skills, and foster a richer mathematical mindset.
Numerous games and activities can make learning fun. For instance, "Make a Square" is a game that builds 2-D visualization skills and strategic thinking. Quick number games that can be played anywhere. The divisibility rules for numbers, particularly divisibility by 2, are being emphasized to help students easily identify even and odd numbers. A megastudy also revealed that behaviorally informed email messages improved students' math progress, demonstrating how simple interventions can positively impact learning outcomes. Recommended read:
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Unknown (noreply@blogger.com)@Pat'sBlog
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References:
Pat'sBlog
Recent discussions have highlighted the diverse applications and historical roots of mathematics. A blog post explored the history of mathematical terms such as billion, trillion, and others, tracing their origins back to figures like Nicholas Chuquet, a French physician from the 15th century. The evolution of these terms and their varying definitions across different countries demonstrate the rich history and changing conventions within mathematical nomenclature. This information has recently resurfaced in a post from earlier this year.
Alongside the history of math, practical math applications are being discussed. For example, recent word problems are now available that focuses on division suitable for fourth-grade students. The step-by-step solutions for problems involving dividing quantities among groups can help students improve their comprehension of division and problem solving. Mathematics continues to be the basis for many algorithms in a variety of modern technological applications and is not widely recognized as a science. Recommended read:
References :
Michael Weiss@Diagonal Argument
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References:
Diagonal Argument
Recent discussions in mathematical concepts and programming tools cover a range of topics, including theoretical foundations and practical applications. Peter Cameron highlighted the Compactness Theorem for first-order logic, explaining its consequences and connections to topology. Also, a beginner's guide to sets has been published to explain how they work and some applications.
Noel Welsh presented a talk at Imperial College on dualities in programming, exploring the relationships between data and codata, calls and returns, and ASTs and stack machines. The use of adjoints in boolean operations was justified, and Daniel Lemire published an overview of parallel programming using Go. These discussions bridge the gap between abstract mathematical principles and their concrete uses in software development and programming paradigms. Recommended read:
References :
@medium.com
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References:
Pat'sBlog
, medium.com
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March 14th marks the annual celebration of Pi Day, honoring the mathematical constant π (pi), which represents the ratio of a circle’s circumference to its diameter. Pi, an irrational number approximately equal to 3.14159, holds significance across various scientific disciplines, including geometry, physics, engineering, and even music theory. Its infinite, non-repeating decimal expansion symbolizes infinity and mystery, captivating mathematicians and enthusiasts alike.
Pi Day 2025 presents an opportunity to engage students with interactive activities like Ratio Riddles, PiCraft, and Math Progress. Ratio Riddles, a lesson from Minecraft Education, introduces concepts of ratio, proportion, fractions, and scale through engaging games. PiCraft offers a student workbook blending gaming and learning, allowing students to estimate and calculate the area of a circle within the Minecraft universe, applying mathematical concepts through coding with Microsoft MakeCode. These hands-on experiences aim to make math more meaningful and strengthen students' confidence in the subject. Recommended read:
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Megan Murphy@STATTRAK
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References:
STATTRAK
The Causality in Statistics Education Award, established by Judea Pearl, is seeking nominations. This award acknowledges the growing importance of incorporating causal inference into undergraduate and lower-division graduate statistics courses. The award provides a $5,000 cash prize annually, and nominations are due by April 5th. Further details regarding selection criteria and nomination requirements can be found on the American Statistical Association's STATtr@k website.
In other news, the Carnival of Maths 237, which rounds up various mathematics blog posts, is now available online. A mathematics conference organized by DEA SCUOLA is scheduled for secondary school mathematics teachers which aims to enhance mathematics teaching through innovative strategies and the integration of AI. Musically, Day 30 of a practice series features bass drum exercises from Mark Guiliana's book, highlighting complex coordination challenges in alternating snare and bass drum strokes. Recommended read:
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munizao@Puzzle Zapper Blog
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References:
mathvoices.ams.org
, John D. Cook
Math enthusiasts and educators are exploring a variety of topics on blogs and websites. Puzzlezapper features geometrical puzzles like polyiamond path puzzles, while other platforms share insights into math education and mathematical concepts. Alexandre Muñiz, on Puzzle Zapper Blog, explores variations on polyiamond tiles with marked paths, constrained to straight lines connecting the midpoints of cell edges.
Donna Fernandez, Co-Director of Alliance of Indigenous Math Circles, highlights the power of indigenous approaches to math education, using the example of the Navajo Prep Math Camp and ICME-15. John D. Cook discusses the practical consequences of tokenization, noting how subtle changes in prompts can significantly impact Large Language Models (LLMs), referencing an article about LLMs and chess, and illustrating how tokenization differences can confuse the model, affecting its ability to play effectively. Recommended read:
References :
msaul@mathvoices.ams.org
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References:
Math Blog
Math education is getting a boost from several innovative resources. A podcast, "Homeschool Better Together," is making math more enjoyable for families, highlighting the importance of celebrating when children get stuck as a valuable learning opportunity. According to Denise Gaskins' Let's Play Math, simply providing rules and procedures to follow isn't enough when kids are struggling; it's more important to foster skills in handling challenges and problem-solving.
Also, free printable math worksheets are available for 4th graders to help them practice large numbers, place value (Indian and International systems), expanded form, and fundamental operations. These resources aim to make math accessible and engaging, encouraging students to develop a growth mindset. Learning fractions visually and employing creative reasoning in problem-solving are emphasized as ways to help students understand math topics more effectively. Recommended read:
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@phys.org
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References:
mathoverflow.net
, medium.com
Recent mathematical research is pushing the boundaries of theoretical understanding across various domains. One area of focus involves solving the least squares problem, particularly with rank constraints. A specific problem involves minimizing a function with a rank constraint and the quest for efficient solutions to these constrained optimization challenges remains a significant area of investigation.
This also involves a three-level exploration into a "mathematics-driven universe," questioning whether math is discovered or invented, and delving into the philosophical implications of mathematics in modern physics. Furthermore, mathematicians are employing topology to investigate the shape of the universe. This includes exploring possible 2D and 3D spaces to better understand the cosmos we inhabit, hinting at intriguing and surprising possibilities that could change our understanding of reality. Recommended read:
References :
@www.marktechpost.com
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DeepMind's AlphaGeometry2, an AI system, has achieved a remarkable milestone by surpassing the average performance of gold medalists in the International Mathematical Olympiad (IMO) geometry problems. This significant upgrade to the original AlphaGeometry demonstrates the potential of AI in tackling complex mathematical challenges that require both high-level reasoning and strategic problem-solving abilities. The system leverages advanced AI techniques to solve these intricate geometry problems, marking a notable advancement in AI's capabilities.
Researchers from Google DeepMind, alongside collaborators from the University of Cambridge, Georgia Tech, and Brown University, enhanced the system with a Gemini-based language model, a more efficient symbolic engine, and a novel search algorithm with knowledge sharing. These improvements have significantly boosted its problem-solving rate to 84% on IMO geometry problems from 2000-2024. AlphaGeometry2 represents a step towards a fully automated system capable of interpreting problems from natural language and devising solutions, underscoring AI's growing potential in fields demanding high mathematical reasoning skills, such as research and education. Recommended read:
References :
@math.stackexchange.com
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References:
mathoverflow.net
, math.stackexchange.com
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Recent studies in abstract algebra are exploring the intricate properties of rings and their homomorphisms, focusing particularly on retracts and extensions. A key area of interest involves identifying rings that do not possess any proper retracts, yet still admit non-trivial maps to themselves. This research investigates the conditions under which ring homomorphisms can be extended, notably in Boolean rings, and seeks to understand the abstract structures and their mappings within the field of algebra. Another focal point is analyzing inner endomorphisms, specifically their role in inducing identities on algebraic K-theory, a complex area which requires understanding of non-unital rings and idempotents.
The relationship between rings and their homomorphisms is also explored through questions around isomorphism. Researchers are examining whether the ring $\mathbb{Z}_5 \times \mathbb{Z}_3$ is isomorphic to the ring $\mathbb{Z}_{15}$, a query that touches on fundamental ring theory concepts. Additionally, work is underway to relate complex paths to substitution homomorphisms in bivariate polynomials, indicating an interdisciplinary approach that combines algebraic geometry with analysis. These lines of inquiry highlight the ongoing efforts to understand the abstract nature of rings, their mappings, and their connections to other mathematical fields. Recommended read:
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@the-decoder.com
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OpenAI's o3 model is facing scrutiny after achieving record-breaking results on the FrontierMath benchmark, an AI math test developed by Epoch AI. It has emerged that OpenAI quietly funded the development of FrontierMath, and had prior access to the benchmark's datasets. The company's involvement was not disclosed until the announcement of o3's unprecedented performance, where it achieved a 25.2% accuracy rate, a significant jump from the 2% scores of previous models. This lack of transparency has drawn comparisons to the Theranos scandal, raising concerns about potential data manipulation and biased results. Epoch AI's associate director has admitted the lack of transparency was a mistake.
The controversy has sparked debate within the AI community, with questions being raised about the legitimacy of o3's performance. While OpenAI claims the data wasn't used for model training, concerns linger as six mathematicians who contributed to the benchmark said that they were not aware of OpenAI's involvement or the company having exclusive access. They also indicated that had they known, they might not have contributed to the project. Epoch AI has said that an "unseen-by-OpenAI hold-out set" was used to verify the model's capabilities. Now, Epoch AI is working on developing new hold-out questions to retest the o3 model's performance, ensuring OpenAI does not have prior access. Recommended read:
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