Related papers: Remarks on the tangent function from an analytic a…
In this paper we describe a theory of a cumulative distribution function on a space with an order from a probability measure defined in this space. This distribution function plays a similar role to that played in the classical case.…
We consider the notion of the matrix (tensor) distribution of a measurable function of several variables. On the one hand, it is an invariant of this function with respect to a certain group of transformations of variables; on the other…
This is a report on a talk to high school mathematics students. It gives some combinatorial connections for the entries of the boustrophedon triangle whose sides are the Taylor coefficients for the tangent and secant functions.
Random integral mappings $I^{h,r}_{(a,b]}$ give isomorphisms between the sub-semigroups of the classical $(ID, \ast)$ and the free-infinite divisible $(ID,\boxplus)$ probability measures. This allows us to introduce new examples of such…
Determining functionals are tools to describe the finite dimensional long-term dynamics of infinite dimensional dynamical systems. There also exist several applications to infinite dimensional {\em random} dynamical systems. In these…
We develop a variational calculus for a certain free energy functional on the space of all probability measures on a Kahler manifold X. This functional can be seen as a generalization of Mabuchi's K-energy functional and its twisted…
Determining functionals are tools to describe the finite dimensional long-term dynamics of infinite dimensional dynamical systems. There also exist several applications to infinite dimensional {\em random} dynamical systems. In these…
We show that F_a(x)=\frac{\ln \Gamma (x+1)}{x\ln(ax)} is a Pick function for a\ge 1 and find its integral representation. We also consider the function f(x)=(\frac{\pi^{x/2}}{\Gamma(1+x/2)})^{1/(x\ln x)} and show that \ln f(x+1) is a…
For the Riemann zeta-function, we introduce a function such that it is a characteristic function of an infinitely divisible distribution on the real line if and only if the Riemann Hypothesis is true.
We construct a Banach rearrangement invariant norm on the measurable space for which the finiteness of this norm for measurable function (random variable) is equivalent to suitable tail (heavy tail and light tail) behavior. We investigate…
For a positive integer $n$, let $p(n)$ be the number of ways to express $n$ as a sum of positive integers. In this note, we revisit the derivation of the Rademacher's convergent series for $p(n)$ in a pedagogical way, with all the details…
New unconditional estimates of the divisor and totient functions are contributed to the literature. These results are consistent with the Riemann hypothesis and seem to solve the Nicolas inequality for all sufficiently large integers.
The purpose of this paper is to study some binomial coefficients which are related to the evaluation of tan(nx). We present a connection between these binomial coefficients and the coefficients of a family of derivative polynomials for…
We define a new rearrangement, called rearrangement by tamping, for non-negative measurable functions defined on R+. This rearrangement has many properties in common with the well-known Schwarz non-increasing rearrangement such as the…
The function $ \tan(\pi x / 2) / (\pi x / 2) $ is expanded into a Laurent series of $ 1 - x^2 $, where the coefficients are given explicitly as combinations of zeta function of even integers. This is used to achieve a sequence of upper and…
We define a random analytic function $\varphi$ on the unit disc by letting a Gaussian multiplicative measure to be one of its Clark measures. We show that $\varphi$ is almost surely a Blaschke product and we provide rather sharp estimates…
In this technical report, certain interesting classification of arithmetical functions is proposed. The notion of additively decomposable and multiplicatively decomposable arithmetical functions is proposed. The concepts of arithmetical…
Riemann's non-differentiable function is a classic example of a continuous function which is almost nowhere differentiable, and many results concerning its analytic regularity have been shown so far. However, it can also be given a…
WWe give a rational closed form expression for the higher derivatives of the inverse tangent function and discuss its relation to Chebyshev polynomials, trigonometric expansions and Appell sequences of polynomials.
We study tangent spaces in the setting of diffeological spaces. Several distinct tangent functors have been introduced, each of which extends the classical tangent functor from smooth manifolds. In this paper, we construct infinitely many…