Top Mathematics discussions
NishMath - #algorithms
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Lance Fortnow@Computational Complexity
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Recent discussions in the mathematics and computer science blogosphere highlight the fascinating interplay between abstract mathematical concepts and their practical applications. One notable area of exploration is the distribution of prime numbers, with researchers developing novel visualizations like "Jacob's Ladder" to illustrate their patterns. This method plots numbers on a 2D graph, creating a zig-zagging structure that ascends or descends based on the primality of successive numbers, offering a unique geometrical perspective on this fundamental sequence. Further investigations delve into "random walks" generated by prime number sequences, where specific rules dictate movement based on the last digit of primes, raising questions about the coverage of the plane as the sequence extends infinitely.
Beyond prime number analysis, the field is also addressing practical computational challenges. A significant topic is the development of efficient algorithms for testing if a large integer is a perfect square. While older methods relied on floating-point approximations, which can lead to inaccuracies with very large numbers due to overflow and precision loss, newer algorithms exclusively employ integer operations. This ensures exact results for arbitrarily large integers, a crucial improvement for many computational tasks. Such advancements underscore the importance of robust mathematical techniques for reliable software development, especially when dealing with extensive numerical data. The discussions also touch upon broader themes in computing, including the critical concept of code reuse and its evolving landscape in the age of generative AI. The potential impact of AI on how software is developed, particularly concerning the reuse of existing code and the creation of new code, is a significant point of consideration. Furthermore, the fundamental distinction between integer and floating-point representations in computers is being re-examined. It's revealed that most machine integers cannot be precisely represented by floating-point numbers, with only a small percentage of 32-bit and 64-bit integers possessing exact floating-point equivalents, a detail with implications for numerical precision in various computing applications.
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@forge.dyalog.com
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The APL Forge competition is in its final week, with the deadline for submissions set for Monday, June 23, 2024, at 12:00 UTC. This annual event is designed to promote the use and development of the APL programming language within the community. Participants are challenged to create innovative open-source libraries and commercial applications using Dyalog APL. The APL Forge is where developers are rewarded for using Dyalog APL to solve problems and develop libraries, applications, and tools.
Whether you're an individual, a group, or a company, if you have a passion for problem-solving in APL, this competition is for you. The APL Forge competition is rewarding participants for using Dyalog APL to solve problems and develop libraries, applications, and tools. The winner of the APL Forge competition will receive £2,500 (GBP) and an expenses-paid trip to present at our next user meeting. Those looking for inspiration are encouraged to check out the project ideas listed on the APL Forge website, where they can also find eligibility and judging criteria, submission guidelines, and frequently asked questions. For more information and to enter the APL Forge, visit forge.dyalog.com.
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@www.iansresearch.com
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The increasing capabilities of quantum computers are posing a significant threat to current encryption methods, potentially jeopardizing the security of digital assets and the Internet of Things. Researchers at Google Quantum AI are urging software developers and encryption experts to accelerate the implementation of next-generation cryptography, anticipating that quantum computers will soon be able to break widely used encryption standards like RSA. This urgency is fueled by new estimates suggesting that breaking RSA encryption may be far easier than previously believed, with a quantum computer containing approximately 1 million qubits potentially capable of cracking it. Experts recommend that vulnerable systems should be deprecated after 2030 and disallowed after 2035.
Last week, Craig Gidney from Google Quantum AI published research that significantly lowers the estimated quantum resources needed to break RSA-2048. Where previous estimates projected that cracking RSA-2048 would require around 20 million qubits and 8 hours of computation, the new analysis reveals that it could be done in under a week using fewer than 1 million noisy qubits. This more than 95% reduction in hardware requirements is a seismic shift in the projected timeline for "Q-Day," the hypothetical moment when quantum computers can break modern encryption. RSA encryption, used in secure web browsing, email encryption, VPNs, and blockchain systems, relies on the difficulty of factoring large numbers into their prime components. Quantum computers, leveraging Shor's algorithm, can exponentially accelerate this process. Recent innovations, including Approximate Residue Arithmetic, Magic State Cultivation, Optimized Period Finding with Ekerå-Håstad Algorithms, and Yoked Surface Codes & Sparse Lookups, have collectively reduced the physical qubit requirement to under 1 million and allow the algorithm to complete in less than 7 days.
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