Takeshi Sekiguchi

A generalization of Hata-Yamaguti’s results on the Takagi function

We shall generalize Hata and Yamaguti’s results on a system of infinitely many difference equations and on the connection between the Takagi function and Lebesgue’s singular function.

An explicit formula of the exponential sums of digital sums

We give a simple explicit formula of the exponential sum of digital sums by use of the distribution function of the binomial measure. We also apply our formula to the study of the power sums of digital sums.

A generalization of Hata-Yamaguti’s results on the Takagi function II: Multinomial case

We state an expansion theorem of continuous functions based on the multinomial measure, which is a generalization of the Schauder one, and then apply it to the study of a system of infinitely many difference equations. We also study the differentiability of the distribution function of the multinomial measure with respect to the parameters needed to define the multinomial measures.

Heat Kernels on infinite graph networks and deformed Sierpinski gaskets

We shall calculate the local asymptotic decay order of heat kernels on general infinite graph networks. We shall also study deformed Sierpinski gaskets by introducing a new dimension closely related to the spectral dimension.

Hausdorff dimension of graphs of some Rademacher series

Our study is of the Hausdorff dimension and the packing dimension of graphs of Rademacher series whose coefficients form geometric progression.

Power and exponential sums for generalized coding systems by a measure theoretic approach

A measure theoretic approach to study the power and the exponential sums for the usual coding system has been developed since the 1990s. In this paper, we first introduce a new coding system, and then give explicit formulas for the power and the exponential sums for the coding system by the measure theoretic approach. An expression for the power sum using the generalized Takagi function will also be given.

On exponential sums of digital sums related to Gelfond's theorem

In this paper, we first give explicit formulas of exponential sums of sum of digits related to Gelfonds theorem. As an application of these formulas, we obtain a simple expression of Newman‐Coquet type summation formula related to the number of binary digits in a multiple of a prime number.

On a characterization of the standard Gray code by using its edge type on a hypercube

An n-bit code is an injective map from a finite set S to {0,1}n. In this paper, we only consider codes whereS = {0,1,2, . . . ,2n−1}; hence the map defines a permutation if we identifyS and{0,1}n. We regard such a code as a sequence B = (b0,b1, . . . ,b2n−1), where bi is the image ofi by the map; hence we call a code for a sequence of 2 n different elements of V = GF(2). The entrybi is called theith codeword of B . A codeB is called aGray code if b0 = 0 and the Hamming distance between bi andbi+1 is 1 for each of i = 0,1, . . . ,2 − 1 regarding that b2n = b0. If we regard{0,1}n as ann-dimensional hypercube, a Gray code defines a Hamiltonian circuit on it. For a non-negative integer m< 2, let