ترغب بنشر مسار تعليمي؟ اضغط هنا

Machine Learning meets Number Theory: The Data Science of Birch-Swinnerton-Dyer

69   0   0.0 ( 0 )
 نشر من قبل Yang-Hui He
 تاريخ النشر 2019
  مجال البحث الهندسة المعلوماتية
والبحث باللغة English




اسأل ChatGPT حول البحث

Empirical analysis is often the first step towards the birth of a conjecture. This is the case of the Birch-Swinnerton-Dyer (BSD) Conjecture describing the rational points on an elliptic curve, one of the most celebrated unsolved problems in mathematics. Here we extend the original empirical approach, to the analysis of the Cremona database of quantities relevant to BSD, inspecting more than 2.5 million elliptic curves by means of the latest techniques in data science, machine-learning and topological data analysis. Key quantities such as rank, Weierstrass coefficients, period, conductor, Tamagawa number, regulator and order of the Tate-Shafarevich group give rise to a high-dimensional point-cloud whose statistical properties we investigate. We reveal patterns and distributions in the rank versus Weierstrass coefficients, as well as the Beta distribution of the BSD ratio of the quantities. Via gradient boosted trees, machine learning is applied in finding inter-correlation amongst the various quantities. We anticipate that our approach will spark further research on the statistical properties of large datasets in Number Theory and more in general in pure Mathematics.



قيم البحث

اقرأ أيضاً

115 - Robert C. Rhoades 2007
We take an approach toward counting the number of n for which the curves E_n: y^2=x^3-n^2x have 2-Selmer groups of a given size. This question was also discussed in a pair of Invent. Math. papers by Roger Heath-Brown. We discuss the connection betwee n computing the size of these Selmer groups and verifying cases of the Birch and Swinnerton-Dyer Conjecture. The key ingredient for the asymptotic formulae is the ``independence of the Legendre symbol evaluated at the prime divisors of an integer with exactly k prime factors.
We provide two proofs that the conjecture of Artin-Tate for a fibered surface is equivalent to the conjecture of Birch-Swinnerton-Dyer for the Jacobian of the generic fibre. As a byproduct, we obtain a new proof of a theorem of Geisser relating the o rders of the Brauer group and the Tate-Shafarevich group.
Research at the intersection of machine learning and the social sciences has provided critical new insights into social behavior. At the same time, a variety of critiques have been raised ranging from technical issues with the data used and features constructed, problematic assumptions built into models, their limited interpretability, and their contribution to bias and inequality. We argue such issues arise primarily because of the lack of social theory at various stages of the model building and analysis. In the first half of this paper, we walk through how social theory can be used to answer the basic methodological and interpretive questions that arise at each stage of the machine learning pipeline. In the second half, we show how theory can be used to assess and compare the quality of different social learning models, including interpreting, generalizing, and assessing the fairness of models. We believe this paper can act as a guide for computer and social scientists alike to navigate the substantive questions involved in applying the tools of machine learning to social data.
111 - Hrishav Bakul Barua 2021
As we are fast approaching the beginning of a paradigm shift in the field of science, Data driven science (the so called fourth science paradigm) is going to be the driving force in research and innovation. From medicine to biodiversity and astronomy to geology, all these terms are somehow going to be affected by this paradigm shift. The huge amount of data to be processed under this new paradigm will be a major concern in the future and one will strongly require cloud based services in all the aspects of these computations (from storage to compute and other services). Another aspect will be energy consumption and performance of prediction jobs and tasks within such a scientific paradigm which will change the way one sees computation. Data science has heavily impacted or rather triggered the emergence of Machine Learning, Signal/Image/Video processing related algorithms, Artificial intelligence, Robotics, health informatics, geoinformatics, and many more such areas of interest. Hence, we envisage an era where Data science can deliver its promises with the help of the existing cloud based platforms and services with the addition of new services. In this article, we discuss about data driven science and Machine learning and how they are going to be linked through cloud based services in the future. It also discusses the rise of paradigms like approximate computing, quantum computing and many more in recent times and their applicability in big data processing, data science, analytics, prediction and machine learning in the cloud environments.
We unite two themes in dyadic analysis and number theory by studying an analogue of the failure of the Hasse principle in harmonic analysis. Explicitly, we construct an explicit family of measures on the real line that are $p$-adic doubling for any f inite set of primes, yet not doubling, and we apply these results to show analogous statements about the reverse Holder and Muckenhoupt $A_p$ classes of weights. The proofs involve a delicate interplay among several geometric and number theoretic properties.

الأسئلة المقترحة

التعليقات
جاري جلب التعليقات جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
mircosoft-partner

هل ترغب بارسال اشعارات عن اخر التحديثات في شمرا-اكاديميا