स्नातकोत्तर प्रथम वर्षComputationनियोजित
Experimental mathematics: conjectures and counterexamples
Using computation to discover patterns, generate conjectures and search for counterexamples: integer sequences and the OEIS, integer-relation algorithms such as PSLQ, exhaustive search, and computer-assisted proofs, with honest reporting of evidence.
math.comp.experimental-mathematics
इस विषय का पाठ अभी लिखा नहीं गया है। नीचे दी गई रूपरेखा पाठ्यक्रम-मानचित्र का हिस्सा है और इसकी समीक्षा नहीं हुई है।
इस पर आधारित
43 अनिवार्य पूर्वापेक्षाएँ (बिल्कुल नीचे तक)
आप सीखेंगे
- Generate and test conjectures numerically, and design a counterexample search with clearly stated bounds.
- Use high-precision arithmetic and integer-relation detection to guess closed forms, and then look for proofs.
- Assess computer-assisted proofs, such as those of the four colour theorem and the Kepler conjecture, and how they were verified.
परिभाषा
- An integer relation among real numbers x₁, …, xₙ is a non-zero integer vector (a₁, …, aₙ) with a₁x₁ + ⋯ + aₙxₙ = 0.
- A counterexample search is exhaustive up to a bound if it checks every case below that bound.
प्रमेय
- The Bailey–Borwein–Plouffe formula for π was found with the help of the PSLQ integer-relation algorithm.
- Hales's proof of the Kepler conjecture was formally verified by the Flyspeck project, completed in 2014.
- Numerical evidence can mislead: π(x) < li(x) for every x that has been checked, yet Littlewood proved that the inequality reverses infinitely often.
आम भ्रांतियाँ
- Treating a pattern that holds in many cases as established.
- Reporting a search as exhaustive without stating its bounds.
वैकल्पिक खोजें
- Research ethics and attribution Experimental results must be reported honestly, with code and bounds, and credited properly.
- ← Programming for mathematics
- ← Reading the mathematical literature
अधिगम पथ
आगे पढ़ने के लिए
- Jonathan Borwein and David Bailey, Mathematics by Experiment: Plausible Reasoning in the 21st Century (2004)