arXiv subjects
Publications and source records attributed to Takahiro Aoyama.
We introduce a positive linear operator acting on the Banach space of all continuous functions on the unit interval via the Moran model studied in population genetics. We show that this operator, named the Moran operator, uniformly approximates every continuous function on the unit interval. Furthermore, some limit theorems for the iterates of the Moran operator are obtained.
B. Tsirelson and A. M. Vershik (1998) introduced the notion of a mathematical noise, which possesses completely opposite properties to those of a white noise. Afterward, B. Tsirelson (2004) called this noise: `black noise.' In this paper, we provide a method to simulate black noise using a modified Bayesian convolutional neural network. Then we study the behavior of black noise both numerically and visually.
Crystal lattices are known to be one of the generalizations of classical periodic lattices which can be embedded into some Euclidean spaces properly. As to make a wide range of multidimensional discrete distributions on Euclidean spaces more treatable, multidimensional Euler products and multidimensional Shintani zeta functions on crystal lattices are introduced. They are completely different from existing Ihara zeta functions on graphs in that our zeta functions are defined on crystal lattices directly. Via a concept of periodic realizations of crystal lattices, we make it possible to provide many kinds of multidimensional discrete distributions explicitly. In particular, random walks on crystal lattices whose range is infinite and such random walks whose range is finite are constructed by multidimensional Euler products and multidimensional Shintani zeta functions, respectively. We give some comprehensible examples as well.
In probability theory, there exist discrete and continuous distributions. Generally speaking, we do not have sufficient kinds and properties of discrete ones compared to the continuous ones. In this paper, we treat the Riemann zeta distribution as a representative of few known discrete distributions with infinite supports. Some asymptotic behaviors of convolution powers of the Riemann zeta distribution are discussed.
It is known to be difficult to find out whether a certain multivariable function to be a characteristic function when its corresponding measure is not tirivial to be or not to be a probability measure on R^d. Such results were not obtained for a long while. In this paper, multidimensional polynomial Euler product is defined as a generalization of the polynomial Euler product. By applying the Kronecker's approximation theorem, a necessary and sufficient condition for some polynomial Euler products to generate characteristic functions is given. Furthermore, by using the Baker's theorem, that of some multidimensional polynomial Euler products is also given. As one of the most important properties of probability distributions, the infinite divisibility of them is studied as well.
The class of Riemann zeta distribution is one of the classical classes of probability distributions on R. Multidimensional Shintani zeta function is introduced and its definable probability distributions on R^d are studied. This class contains some fundamental probability distributions such as binomial and Poisson distributions. The relation with multidimensional polynomial Euler product, which induces multidimensional infinitely divisible distributions on R^d, is also studied.
Finite Euler product is known to be one of the classical zeta functions in number theory. In [1], [2] and [3], we have introduced some multivariable zeta functions and studied their definable probability distributions on R^d. They include functions which generate infinitely divisible, not infinitely divisible characteristic functions and not even to be characteristic functions. In this paper, we treat some multivariable finite Euler products and show how they behave in view of such properties.
Recently, many classes of infinitely divisible distributions on R^d have been characterized in several ways. Among others, the first way is to use Levy measures, the second one is to use transformations of Levy measures, and the third one is to use mappings of infinitely divisible distributions defined by stochastic integrals with respect to Levy processes. In this paper, we are concerned with a class of mappings, by which we construct new classes of infinitely divisible distributions on R^d. Then we study a special case in R^1, which is the class of infinitely divisible distributions without Gaussian parts generated by stochastic integrals with respect to a fixed compound Poisson processes on R^1. This is closely related to the Goldie-Steutel-Bondesson class.
We calculate the specific heat of the ideal gas obeying the generalized exclusion statistics (GES) in the continuum model and the tight binding model numerically. In the continuum model of 3-d space, the specific heat increases with statistical parameter at low temperature whereas it decreases with statistical parameter at high temperature. We find that the critical temperature normalized by $μ_f$ (Fermi energy) is 0.290. The specific heat of 2-d space was known to be independent of $g$ in the continuum model, but it varies with $g$ drastically in the tight-binding model. From its unique behavior, identification of GES particles will be possible from the specific heat.