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Anton A. Kutsenko

Publications and source records attributed to Anton A. Kutsenko.

At least 19 recordsLinked to original sources
arXiv2026

Left-tail expansions for Schröder branching processes with explicit convergence rates

In previous work, the density of the martingale limit in Schröder branching processes was expressed as a convergent double power-law series with oscillatory terms. The proof relied on two assumptions, one of which imposed a geometric restriction on the critical angle of the Julia set of the offspring generating function near $1$. In this paper, we show that both assumptions can be removed. We derive explicit bounds on the expansion coefficients, which imply locally uniform convergence of the double series. The coefficients decay exponentially in one summation index, with the rate explicitly determined by the critical angle, and super-exponentially in the other. Finally, we investigate the behavior of the critical angle and describe regimes in which it approaches its minimum. This analysis shows how the geometry of the Julia set influences the magnitude of the oscillatory corrections in the left-tail expansion.

math.PR↗
arXiv2025

Complete left tail asymptotic for supercritical multitype branching processes

We derive a complete left-tail asymptotic series for the density of the {\it martingale limit} of a supercritical multitype Galton-Watson process in the Schröder case. We show that the series converges everywhere, not only for small arguments. This is the first complete result regarding the left tails of multitype branching processes. A good, quickly computed approximation for the density will also be derived from the series.

math.PR↗
arXiv2025

Complete right tail asymptotic for the density of branching processes with fractional generating functions

The right tail asymptotic series consisting of attenuating exponential terms are derived for the densities of Galton-Watson processes with fractional probability generating functions. The frequencies in the exponential factors form fractal structures in the complex plane. We discuss conditions when the asymptotic series converges everywhere. The obtained right tail asymptotic is compared with the standard integral representation of the density and with the complete left tail asymptotic.

math.PR↗
arXiv2024

Closed-form solutions for Bernoulli and compound Poisson branching processes in random environments

For branching processes, the generating functions for limit distributions of so-called ratios of probabilities of rare events satisfy the Schröder-type integral-functional equations. Excepting limited special cases, the corresponding equations can not be solved analytically. I found a large class of Poisson-type offspring distributions, for which the Schröder-type functional equations can be solved analytically. Moreover, for the asymptotics of limit distributions, the power and constant factor can be written explicitly. As a bonus, Bernoulli branching processes in random environments are treated. The beauty of this example is that the explicit formula for the generating function is unknown. Still, the closed-form expression for the power and constant factor in the asymptotic can be written with the help of some outstanding tricks. Also, the asymptotic expansion contains oscillatory terms absent in the Poisson case. It is shown that at the power $3$ in the Bernoulli binomial kernels short-phase discrete oscillations turn into long-phase continuous ones.

math.PR↗
arXiv2024

Generalized Schröder-type functional equations for Galton-Watson processes in random environments

The classical Galton--Watson process works with a fixed probability of fission at each time step. One of the generalizations is that the probabilities depend on time. We consider one of the most complex and interesting cases when we do not know the exact probabilities of fission at each time step - these probabilities are random variables themselves. The limit distributions of the number of descendants are described in terms of generalized integral and differential functional equations of the Schröder type. There are no more analogs of periodic Karlin-McGregor functions, which were very helpful in the analysis of the asymptotic behavior of limit distributions for the classical case. We propose some approximate asymptotic methods. Even simple cases of random families with one or two members lead to nice asymptotics involving, interesting problems related to special functions and special constants. One of them, Example 2 is already announced on \href{https://math.stackexchange.com/questions/4748129/asymptotics-of-sequence-of-rational-numbers}{this} and \href{https://mathoverflow.net/questions/458885/simple-integral-equation}{this} sites. Finally, the phenomenon of why the oscillations in the main asymptotic term usual for the classical case become rare in the case of random environments is discussed.

math.PR↗
arXiv2022

A note on "exotic integrals"

We consider Bernoulli measures $μ_p$ on the interval $[0,1]$. For the standard Lebesgue measure the digits $0$ and $1$ in the binary representation of real numbers appear with an equal probability $1/2$. For the Bernoulli measures, the digits $0$ and $1$ appear with probabilities $p$ and $1-p$, respectively. We provide explicit expressions for various $μ_p$-integrals. In particular, integrals of polynomials are expressed in terms of the determinants of special Hessenberg matrices, which, in turn, are constructed from the Pascal matrices of binomial coefficients. This allows us to find closed-form expressions for the Fourier coefficients of $μ_p$ in the Legendre polynomial basis. At the same time, the trigonometric Fourier coefficients are values of some special entire function, which admits an explicit infinite product expansion and satisfies interesting properties, including connections with the Stirling numbers and the polylogarithm.

math.CA↗
arXiv2021

On some explicit integrals related to "fractal mountains"

Loop counting functions $U(x)$ estimate the number of "weighted" loops in a digital representation of $x\in[-1,1]$. Roughly speaking, each $x$ is considered as an infinite walk, where the steps of the walk correspond to digits of $x$. The graph of loop counting functions $U$ has a fractal structure that resembles complex mountain landscapes. In some sense, $U$ allows us to look at random walks globally. These functions may be helpful in the analysis of some hard problems related to the distribution of self-avoiding random walks (SAW) in a multi-dimensional case since SAW closely relate to zeros of $U(x)$. We note here that $U(x)$ can be naturally extended to a multidimensional argument $x$. In this article, the focus will be on some analytic aspects. It will be shown that integrals $\int x^AU(x)^Bdx$ with non-negative integers $A$ and $B$ can be expressed in terms of integrals of rational functions with integer coefficients. Moreover, it will be shown that $\int x^A U(x)dx$ admits closed-form expressions. Fourier series for $U$ is also computed. Finally, we discuss some connections with special functions and generalized continued fractions, and other perspectives.

nlin.CD↗
arXiv2021

Fractal mountains and binary random loops

We discuss a variation of Takagi curves based, however, more on algebraic than geometric principles. Namely, we construct functions of loops in a special binary representation. The graph of these functions usually has chaotic and fractal forms, sometimes recall mountain landscapes. Nevertheless, the values in a dense set of points and even the integral of these functions can be calculated explicitly.

nlin.CD↗
arXiv2020

Matrix representations of multidimensional integral and ergodic operators

We provide a representation of the $C^*$-algebra generated by multidimensional integral operators with piecewise constant kernels and discrete ergodic operators. This representation allows us to find the spectrum and to construct the explicit functional calculus on this algebra. The method can be useful in various applications since many discrete approximations of integral and differential operators belong to this algebra. Some examples are also presented: 1) we construct an explicit functional calculus for extended Fredholm integral operators with piecewise constant kernels, 2) we find a wave function and spectral estimates for 3D discrete Schrödinger equation with planar, guided, local potential defects, and point sources. The accuracy of approximation of continuous multi-kernel integral operators by the operators with piecewise constant kernels is also discussed.

math-ph↗
arXiv2020

Classification of integro-differential $C^*$-algebras

The integro-differential algebra $\mathscr{F}_{N,M}$ is the $C^*$-algebra generated by the following operators acting on $L^2([0,1)^N\to\mathbb{C}^M)$: 1) operators of multiplication by bounded matrix-valued functions, 2) finite differential operators, 3) integral operators. We give a complete characterization of $\mathscr{F}_{N,M}$ in terms of its Bratteli diagram. In particular, we show that $\mathscr{F}_{N,M}$ does not depend on $M$ but depends on $N$. At the same time, it is known that differential algebras $\mathscr{H}_{N,M}$, generated by the operators 1) and 2), do not depend on both dimensions $N$ and $M$, they are all $*$-isomorphic to the universal UHF-algebra. We explicitly compute the Glimm-Bratteli symbols (for $\mathscr{H}_{N,M}$ it was already computed earlier) $$ \mathfrak{n}(\mathscr{F}_{N,M})=\prod_{n=1}^{\infty}\begin{pmatrix} n & 0 \\ n-1 & 1 \end{pmatrix}^{\otimes N}\begin{pmatrix}1 \\ 1 \end{pmatrix}^{\otimes N},\ \ \ \ \mathfrak{n}(\mathscr{H}_{N,M})=\prod_{n=1}^{\infty}n, $$ which characterize completely the corresponding AF-algebras.

math.OA↗
arXiv2020

Finite PDEs and finite ODEs are isomorphic

The standard view is that PDEs are much more complex than ODEs, but, as will be shown below, for finite derivatives this is not true. We consider the $C^*$-algebras ${\mathscr H}_{N,M}$ consisting of $N$-dimensional finite differential operators with $M\times M$-matrix-valued bounded periodic coefficients. We show that any ${\mathscr H}_{N,M}$ is $*$-isomorphic to the universal uniformly hyperfinite algebra (UHF algebra) $ \bigotimes_{n=1}^{\infty}\mathbb{C}^{n\times n}. $ This is a complete characterization of the differential algebras. In particular, for different $N,M\in\mathbb{N}$ the algebras ${\mathscr H}_{N,M}$ are topologically and algebraically isomorphic to each other. In this sense, there is no difference between multidimensional matrix valued PDEs ${\mathscr H}_{N,M}$ and one-dimensional scalar ODEs ${\mathscr H}_{1,1}$. Roughly speaking, the multidimensional world can be emulated by the one-dimensional one.

math.FA↗
arXiv2019

An entire function connected with the approximation of the golden ratio

In 1987, R. B. Paris uses the analytic function \[\label{main} g(w)=\lim_{n\to\infty}(2φ)^n\biggl(\underbrace{\sqrt{1+\sqrt{1+...\sqrt{1+w}}}}_n-φ\biggr),\ \ \ φ=\frac{1+\sqrt{5}}2, \] to estimate the convergence of nested squares to the golden ratio. The function $g$ is non-entire and, perhaps, can not be expressed in terms of some standard known functions. We show that $f(z):=g^{-1}(z)$ is an entire function satisfying Poincare equality. While $f$ has zeros of various multiplicities, it can be expressed in terms of its simple zeros, forming fractal structures similar to Julia sets.

math.CV↗
arXiv2018

A note on sharp spectral estimates for periodic Jacobi matrices

The spectrum of three-diagonal self-adjoint $p$-periodic Jacobi matrix with positive off-diagonal elements $a_n$ an real diagonal elements $b_n$ consist of intervals separated by $p-1$ gaps $γ_i$, where some of the gaps can be degenerated. The following estimate is true $$ \sum_{i=1}^{p-1}|γ_i|\geq\max(\max(4(a_1...a_p)^{\frac1p},2\max a_n)-4\min a_n,\max b_n-\min b_n). $$ We show that for any $p\in\mathbb{N}$ there are Jacobi matrices of minimal period $p$ for which the spectral estimate is sharp. The estimate is sharp for both: strongly and weakly oscillated $a_n$, $b_n$. Moreover, it improves some recent spectral estimates.

math-ph↗
arXiv2017

Mixed multidimensional integral operators with piecewise constant kernels and their representations

We consider the algebra of mixed multidimensional integral operators. In particular, Fredholm integral operators of the first and second kind belongs to this algebra. For the piecewise constant kernels we provide an explicit representation of the algebra as a product of simple matrix algebras. This representation allows us to compute the inverse operators (or to solve the corresponding integral equations) and to find the spectrum explicitly. Moreover, explicit traces and determinants are also constructed. So, roughly speaking, the analysis of integral operators is reduced to the analysis of matrices. All the qualitative characteristics of the spectrum are preserved since only the kernels are approximated.

math.FA↗
arXiv2016

On the extension of Fredholm determinants to the mixed multidimensional integral operators with regulated kernels

We extend the classical trace (and determinant) known for the integral operators $$ ({\mathcal I}+)\int_{[0,1)^N}{\bf A}({\bf k},{\bf x}){\bf u}({\bf x})d{\bf x} $$ with matrix-valued kernels ${\bf A}$ to the operators of the form $$ \sum_α\int_{[0,1)^{|α|}}{\bf A}({\bf k},{\bf x}_α){\bf u}({\bf k}_{\overlineα},{\bf x}_α)d{\bf x}_α, $$ where $α$ are arbitrary subsets of the set $\{1,...,N\}$. Such operators form a Banach algebra containing simultaneously all integral operators of the dimensions $\leqslant N$. In this sense, it is a largest algebra where explicit traces and determinants are constructed. Such operators arise naturally in the mechanics and physics of waves propagating through periodic structures with various defects. We give an explicit representation of the inverse operators (resolvent) and describe the spectrum by using zeroes of the determinants. Due to the structure of the operators, we have $2^N$ different determinants, each of them describes the spectral component of the corresponding dimension.

math.FA↗