相关论文: On Cantor's singular moments
This short note concerns the possible singular behaviour of moment generating functions of finite measures at the boundary of their domain of existence. We look closer at Example 7.3 in O. Barndorff-Nielsen's book "Information and…
A moment problem is presented for a class of signed measures which are termed pseudo-positive. Our main result says that for every pseudo-positive definite functional (subject to some reasonable restrictions) there exists a representing…
The Cantor distribution is obtained from bitstrings; the Cantor-solus distribution (a new name) admits only strings without adjacent 1 bits. We review moments and order statistics associated with these. The Cantor-multus distribution is…
We use special quadrature formulas for singular and hypersingular integral to numerically solve the Schr\"{o}dinger equation in momentum space with the linear confinement potential, Coulomb and Cornell potentials. It is shown that the…
The existence of two different Cantor sets, one of them contained in the set of Liouville numbers and the other one inside the set of Diophantine numbers, is proved. Finally, a necessary and sufficient condition for the existence of a…
The Bertrand's theorem can be formulated as the solution of an inverse problem for a classical unidimensional motion. We show that the solutions of these problems, if restricted to a given class, can be obtained by solving a numerical…
We point out the existence of some singular, radial, spin-0 instantons for curvature-quadratic gravity theories. They are complex.
For any particularly interesting theorem one proof is never enough. Instead, the first proof sets the challenge to find a more elegant method that illuminates subtle features of the math, is simpler to understand, or even avoids using…
We present a short proof of Cantor's Theorem (circa 1870s): if $a_n \cos nx + b_n \sin nx \to 0$ for each $x$ in some (nonempty) open interval, where $a_n, b_n$ are sequences of complex numbers, then $a_n$ and $b_n$ converge to 0.
We investigate some self-similar Cantor sets $C(l,r,p)$, which we call S-Cantor sets, generated by numbers $l,r,p \in \mathbb{N}$, $l+r<p$. We give a full characterization of the set $C(l_1,r_1,p)-C(l_2,r_2,p)$ which can take one of the…
We prove that these Cantor sets are made up of transcendental numbers, apart from their endpoints $0$ and $1$, under some arithmetical assumptions on the data. To that purpose, we establish a criterion of linear independence over the field…
Some soliton equation in 2+1 dimensions and their 1+1 and/or dimensional integrable reductions are considered.
In this paper we obtain an algorithm towards solving the two-dimensional moment problem. This algorithm gives the necessary and sufficient conditions for the solvability of the moment problem. It is shown that all solutions of the moment…
We consider the uniqueness of solutions of ordinary differential equations where the coefficients may have singularities. We derive upper bounds on the the order of singularities of the coefficients and provide examples to illustrate the…
Under certain conditions, we give an estimate from above on the number of differential equations of order $r+1$ with prescribed regular singular points, prescribed exponents at singular points, and having a quasi-polynomial flag of…
In this paper we investigate the one-dimensional harmonic oscillator with a singular perturbation concentrated in one point. We describe all possible selfadjoint realizations and we show that for certain conditions on the perturbation…
The present article is devoted to certain examples of functions whose argument represented in terms of Cantor series.
Cantor's famous proof of the non-denumerability of real numbers does apply to any infinite set. The set of exclusively all natural numbers does not exist. This shows that the concept of countability is not well defined. There remains no…
We prove in constructive logic that the statement of the Cantor-Bernstein theorem implies excluded middle. This establishes that the Cantor-Bernstein theorem can only be proven assuming the full power of classical logic. The key ingredient…
For more than a century, Cantor's theory of transfinite numbers has played a pivotal role in set theory, with ramifications that extend to many areas of mathematics. This article extends earlier findings with a fresh look at the critical…