Institute of Mathematics for Industry

Seminar



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The size of the specificity of the relative class number of values of the determinant formula of the cyclotomic field


Hold Date 2018-02-16 16:00~2018-02-16 17:00

Place Lecture Room M W1-C-512, West Zone 1, Ito campus, Kyushu University

Object person  

Speaker Tetsuya TANIGUCHI (Kanazawa Institute of Technology)

Abstract:
Demjanenko matrix, Maillet matrix, etc. are known as a determinant of the determinant of the relative class number of cyclotomic fields. The values of these determinants are extremely large as compared with determinants in which coefficients are randomly generated and arranged. For example, for p = 101 cyclotomic fields, the value of the determinant of the relative class number formula is larger by 4.7 standard deviation from the mean. Also, the value of the Demjanenko determinant corresponding to the 716 degree matrix (the Hadamard matrix has not yet been found) is also very large, and its value is about 96.8% of the upper limit of Hadamard. We believe that this phenomenon "the value of the determinant is large" will lead to practical applications such as design of experiments. (In the design of experiments, in order to improve the experimental efficiency, a matrix with ±1 coefficient and large determinant values is used). In this talk I will report the results of numerical experiments done so far. For example, the distribution of values of the determinant, the distribution of eigenvalues, the numerical experiment results in the placement of points on the sphere obtained from the Demjanenko matrix of ±1 component, and the matrix "randomly generated coefficients" "Demjanenko matrix" "Hadamard Matrix "in correspondence with the distance matrix, and the likes. I would also like to report that similar phenomena are occurring in relative class number formulas with other matrices if time allows.