Articles tagged with Mathematical Proofs
Comprehensive exploration of living organisms, biological systems, and life processes across all scales from molecules to ecosystems. Encompasses cutting-edge research in biology, genetics, molecular biology, ecology, biochemistry, microbiology, botany, zoology, evolutionary biology, genomics, and biotechnology. Investigates cellular mechanisms, organism development, genetic inheritance, biodiversity conservation, metabolic processes, protein synthesis, DNA sequencing, CRISPR gene editing, stem cell research, and the fundamental principles governing all forms of life on Earth.
Comprehensive medical research, clinical studies, and healthcare sciences focused on disease prevention, diagnosis, and treatment. Encompasses clinical medicine, public health, pharmacology, epidemiology, medical specialties, disease mechanisms, therapeutic interventions, healthcare innovation, precision medicine, telemedicine, medical devices, drug development, clinical trials, patient care, mental health, nutrition science, health policy, and the application of medical science to improve human health, wellbeing, and quality of life across diverse populations.
Comprehensive investigation of human society, behavior, relationships, and social structures through systematic research and analysis. Encompasses psychology, sociology, anthropology, economics, political science, linguistics, education, demography, communications, and social research methodologies. Examines human cognition, social interactions, cultural phenomena, economic systems, political institutions, language and communication, educational processes, population dynamics, and the complex social, cultural, economic, and political forces shaping human societies, communities, and civilizations throughout history and across the contemporary world.
Fundamental study of the non-living natural world, matter, energy, and physical phenomena governing the universe. Encompasses physics, chemistry, earth sciences, atmospheric sciences, oceanography, materials science, and the investigation of physical laws, chemical reactions, geological processes, climate systems, and planetary dynamics. Explores everything from subatomic particles and quantum mechanics to planetary systems and cosmic phenomena, including energy transformations, molecular interactions, elemental properties, weather patterns, tectonic activity, and the fundamental forces and principles underlying the physical nature of reality.
Practical application of scientific knowledge and engineering principles to solve real-world problems and develop innovative technologies. Encompasses all engineering disciplines, technology development, computer science, artificial intelligence, environmental sciences, agriculture, materials applications, energy systems, and industrial innovation. Bridges theoretical research with tangible solutions for infrastructure, manufacturing, computing, communications, transportation, construction, sustainable development, and emerging technologies that advance human capabilities, improve quality of life, and address societal challenges through scientific innovation and technological progress.
Study of the practice, culture, infrastructure, and social dimensions of science itself. Addresses how science is conducted, organized, communicated, and integrated into society. Encompasses research funding mechanisms, scientific publishing systems, peer review processes, academic ethics, science policy, research institutions, scientific collaboration networks, science education, career development, research programs, scientific methods, science communication, and the sociology of scientific discovery. Examines the human, institutional, and cultural aspects of scientific enterprise, knowledge production, and the translation of research into societal benefit.
Comprehensive study of the universe beyond Earth, encompassing celestial objects, cosmic phenomena, and space exploration. Includes astronomy, astrophysics, planetary science, cosmology, space physics, astrobiology, and space technology. Investigates stars, galaxies, planets, moons, asteroids, comets, black holes, nebulae, exoplanets, dark matter, dark energy, cosmic microwave background, stellar evolution, planetary formation, space weather, solar system dynamics, the search for extraterrestrial life, and humanity's efforts to explore, understand, and unlock the mysteries of the cosmos through observation, theory, and space missions.
Comprehensive examination of tools, techniques, methodologies, and approaches used across scientific disciplines to conduct research, collect data, and analyze results. Encompasses experimental procedures, analytical methods, measurement techniques, instrumentation, imaging technologies, spectroscopic methods, laboratory protocols, observational studies, statistical analysis, computational methods, data visualization, quality control, and methodological innovations. Addresses the practical techniques and theoretical frameworks enabling scientists to investigate phenomena, test hypotheses, gather evidence, ensure reproducibility, and generate reliable knowledge through systematic, rigorous investigation across all areas of scientific inquiry.
Study of abstract structures, patterns, quantities, relationships, and logical reasoning through pure and applied mathematical disciplines. Encompasses algebra, calculus, geometry, topology, number theory, analysis, discrete mathematics, mathematical logic, set theory, probability, statistics, and computational mathematics. Investigates mathematical structures, theorems, proofs, algorithms, functions, equations, and the rigorous logical frameworks underlying quantitative reasoning. Provides the foundational language and tools for all scientific fields, enabling precise description of natural phenomena, modeling of complex systems, and the development of technologies across physics, engineering, computer science, economics, and all quantitative sciences.
Yosra Barkaoui generalises Sebestyén's theorem to unbounded operators, essential in physics for describing concepts like kinetic energy and momentum. Her research provides new mathematical tools for working with unbounded operators.
The HALB project aims to investigate central problems in harmonic analysis and produce training material for AI applications. The researchers plan to improve and speed up the processes over time, enabling accurate verification of mathematical research results.
Researchers from OIST and universities provided a new proof for the BBL inequality using heat and diffusion equations, taking an unconventional approach. The study offers fresh insights on the concept, which has vast applications across many fields, including computer science, medical imaging, and resource distribution.
Proof scores use term rewriting to verify system properties, striking a balance between automation and human effort. Despite limitations, the technique has been successfully applied to various systems and protocols, with potential for critical applications in safety-critical systems.
Chelsea Walton, a professor of mathematics at Rice University, has been recognized as an American Mathematical Society (AMS) Fellow. Her selection acknowledges her dedication to advancing mathematical research in noncommutative algebra, quantum symmetries, Hopf algebras, and representation theory.
Calcea Johnson and Ne'Kiya Jackson, two U.S. high school students, have made history by publishing their first academic paper detailing five new ways of proving Pythagoras' Theorem via trigonometry. Their study reveals ten new proofs altogether, including one previously presented at a conference.
Researchers at Kyoto University have discovered a mathematical expression for the blockchain trilemma, revealing ways to improve scalability without compromising security and decentralization. The formula offers new perspectives on solving the long-standing problem of balancing these competing values.
A new open-source platform called CheckMate allows users to interact with and evaluate the performance of large language models (LLMs) like ChatGPT. Researchers found that while LLMs can be helpful, they also make mistakes and provide incorrect information.
Researchers developed an AI framework that combines dimension reduction techniques with a new clustering algorithm to quickly identify groups of viral genomes at risk. This enables proactive response measures like tailored vaccine development, potentially eliminating emerging variants before they spread.
A mathematical breakthrough provides new insights into typhoon dynamics, enabling more accurate predictions and advancements in weather forecasting. The study confirms the stability of specific vortex structures, which can be encountered in real-world fluid flows.
A new KAUST study uses machine learning to predict disease spread with high accuracy, dynamically incorporating latest data without human bias. This approach offers a promising alternative to conventional models, providing a more accurate story of the underlying epidemic data.
Researchers used machine learning to identify patterns in knot theory and representation theory, suggesting new connections that mathematicians were able to prove. This collaboration demonstrates the potential of AI as a tool for guiding intuition in mathematical research.
Computer scientists and mathematicians have used artificial intelligence to help prove or suggest new mathematical theorems in complex fields. The breakthrough uses DeepMind's AI processes to explore conjectures in mathematics, leading to a completely new theorem in knot theory.
Researchers from University of Maryland provide rigorous mathematical explanation for Batchelor's law, a fundamental concept in fluid mechanics. The proof resolves the uncertainty surrounding the law's applicability and limitations, opening doors to better engineering designs and weather prediction models.
A new study shows people appreciate beauty in mathematical arguments similarly to art and music, suggesting a universal aesthetic. This finding may make abstract maths more accessible to children and has implications for maths education.
A new study shows that average Americans can evaluate mathematical arguments for beauty, using criteria such as elegance and universality. The research found that people share the same aesthetic sensibilities about math as they do about art and music, with consensus on what makes something beautiful.
Researchers have revived an old approach to the Riemann Hypothesis using Jensen-Polya polynomials, providing a new framework for understanding prime numbers. The approach reveals a method to calculate the polynomials all at once, opening up new avenues for solving other fundamental mathematical problems.
A team of researchers has developed a minimal logic system to bridge the gap between mathematical proofs, algorithms, and real-world outcomes in control systems. The work, published in IEEE/CAA Journal of Automatica Sinica, aims to improve the realism of theoretical mathematics by focusing on computational certainty.
A new mathematical theorem developed by Carnegie Mellon University and University of Pittsburgh mathematicians proves that Pennsylvania's congressional district maps are likely the result of gerrymandering. The researchers used a Markov chain to analyze the characteristics of the current map, comparing it to randomly generated typical ...
Geordie Williamson will receive the inaugural AMS Claude Chevalley Prize in Lie Theory for his work on representation theory, including proofs of longstanding conjectures and counterexamples to expected bounds. His research has re-opened the field of modular representations and revealed inadequate numerical evidence.
Researchers found that a sequential use of reward and punishment can promote cooperation in collaborative endeavors. The study suggests that initially rewarding minor cooperators and then punishing free riders can lead to better outcomes.
A new system enables cloud customers to quickly verify the proper execution of their software, protecting against malicious code and ensuring data privacy. The system uses a practical, succinct zero-knowledge proof that can fit in a single data packet.
Ngô Bao Châu received the Fields Medal for his proof of the fundamental lemma in automorphic forms through new algebro-geometric methods. His work has relevance to high-energy physics, computer science, and cryptography, building on decades of mathematicians' contributions.
Physicist Garrett Lisi proposed a 'theory of everything' centered on the enigmatic E8, but Emory mathematician Skip Garibaldi and colleague Jacques Distler found fatal flaws in its math. The resulting paper shows that E8's unique structure cannot unify the universe's forces as Lisi claimed.
New computer tools based on formal proof can provide nearly infallible proofs of important mathematical results. Formal proof assistants have become powerful enough to handle difficult proofs and explore mathematics independently.
Computer scientists Steven Rudich and Alexander A. Razborov have made a significant contribution to the P vs. NP problem, a classic question in theoretical computer science that underlies the security of digital systems. Their work has implications for cryptography and electronic commerce.
Thomas C. Hales famously proved Johannes Kepler's 400-year-old conjecture on sphere-packing using a computer-assisted proof, which was initially met with skepticism by reviewers. Hales is now using his problem-solving skills to 'prove the proof' using a specially written computer language in the Flyspeck Project.
Computers are transforming mathematical proof, with complex problems requiring hundreds of pages of data. Mathematicians face challenges in validating and verifying these proofs, raising concerns about the erosion of traditional publishing practices.
Mathematician Devlin discusses the shift from intuitive proofs to formal systems, highlighting the limitations of traditional methods. He also explores the impact of computer analysis on proof validity, demonstrating how new techniques have increased mathematical accuracy.
Key findings suggest that modern mathematical proofs are becoming increasingly difficult to verify, as they often rely on preprint servers and lack peer review. This shift may lead to a new generation of mathematicians following a paradigm that prioritizes dissemination over validation.
Scholars Netz and Saito discovered Archimedes' treatise on The Method of Mechanical Theorems, which includes a proof dealing with infinitely large sets. This finding challenges conventional wisdom that Greeks disliked handling infinity.