Related papers: Analytic Number Theory and Algebraic Asymptotic An…
The paper provides the proof of the Rimann's conjecture. The results of the works of A. M. Odlyzko and H. te Riile "Disproof of the Conjecture", which gives a disproof of the Mertens hypothesis, using to prove the Riemann's hypothesis. This…
We introduce a coloured generalization $\mathrm{NSym}_A$ of the Hopf algebra of non-commutative symmetric functions described as a subalgebra of the of rooted ordered coloured trees Hopf algebra. Its natural basis can be identified with the…
We consider the preferential attachment model. This is a growing random graph such that at each step a new vertex is added and forms $m$ connections. The neighbors of the new vertex are chosen at random with probability proportional to…
We introduce a systematic approach to express generating functions for the enumeration of maps on surfaces of high genus in terms of a single generating function relevant to planar surfaces. Central to this work is the comparison of two…
We introduce the class of analytic functions $$\mathcal{F}(\psi):= \left\{f\in \mathcal{A}: \left(\frac{zf'(z)}{f(z)}-1\right) \prec \psi(z),\; \psi(0)=0 \right\},$$ where $\psi$ is univalent and establish the growth theorem with some…
In this paper, we investigate two subclasses of analytic and univalent functions associated with the exponential mapping $\varphi(z)=e^{\alpha z},\qquad 0<\alpha\le1,$ defined via the subordination conditions $\frac{zf'(z)}{f(z)}\prec…
We define the universal exponential extension of an algebraically closed differential field and investigate its properties in the presence of a nice valuation and in connection with linear differential equations. Next we prove normalization…
This paper presents two concrete applications of Artificial Intelligence to algorithmic and analytic number theory. Recent benchmarks of large language models have mainly focused on general mathematics problems and the currently infeasible…
A decade ago, when teaching complex analysis, the third named author posed the question on whether or not there is an analogue to the Schwarz lemma for real analytic functions. This led to the note [MT], indicating that it is possible to…
This paper considers the asymptotic properties of the recursive maximum likelihood estimation in hidden Markov models. The paper is focused on the asymptotic behavior of the log-likelihood function and on the point-convergence and…
We develop an analytic theory of operator-valued additive free convolution in terms of subordination functions. In contrast to earlier investigations our functions are not just given by power series expansions, but are defined as Frechet…
Using a `height-to-radical' identity, we define the archimedean contribution to the radical, $r_\arch$, and we give a new proof of the abc theorem for the field of meromorphic functions. The first step of the proof is completely formal and…
Given a set D of nonnegative integers, we derive the asymptotic number of graphs with a givenvnumber of vertices, edges, and such that the degree of every vertex is in D. This generalizes existing results, such as the enumeration of graphs…
We develop a unified analytical and computational framework for the generalized Abel ordinary differential equation $y^{\prime }(x)=a_n(x)\bigl(% y^n+\lambda_{n-1}(x)y^{n-1}+\dots+\lambda_0(x)\bigr)$ of arbitrary degree $% n\ge1$ on the…
The \emph{Noetherian class} is a wide class of functions defined in terms of polynomial partial differential equations. It includes functions appearing naturally in various branches of mathematics (exponential, elliptic, modular, etc.). A…
In this paper, we present a rigorous analysis of root-exponential convergence of Hermite approximations, including projection and interpolation methods, for functions that are analytic in an infinite strip containing the real axis and…
The multiplicity (resp. degree) of a function $f$ relative to a semianalytic subset $S$ of $\mathbb{R}^n$ is the greatest (resp. smallest) exponent among numbers $j$ such that the inequality $|f(x)|\leq C\|x\|^j$ holds on $S$ near $0$…
Let $f : X\to X$ be a dominating meromorphic map on a compact K\"ahler manifold $X$ of dimension $k$. We extend the notion of topological entropy $h^l_{\mathrm{top}}(f)$ for the action of $f$ on (local) analytic sets of dimension $0\leq l…
Dahmen and Schmeding have obtained the result that although the smooth Lie group $G$ of real analytic diffeomorphisms $\mathbb S^{\,1.}\to\mathbb S^{\,1.}$ has a compatible analytic manifold structure, it does not make $G$ a real analytic…
A new elementary proof of the prime number theorem presented recently in the framework of a scale invariant extension of the ordinary analysis is re-examined and clarified further. Both the formalism and proof are presented in a much more…