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Vladimir Tkachev

Publications and source records attributed to Vladimir Tkachev.

8 recordsLinked to original sources
arXiv2022

Medial and isospectral algebras

The purpose of this paper is to give a systematic study of two new classes of commutative nonassociative algebras, the so-called isospectral and medial algebras. An isospectral algebra $\mathbb{A}$ is a generic commutative nonassociative algebra whose idempotents have the same Peirce spectrum. A medial algebra is algebra with identity $(xy)(zw)=(xz)(yw)$. We show that these two classes are essentially coincide. We also prove that any medial spectral algebra is isomorphic to a certain isotopic deformation of the commutative associative quotient algebra $\mathbf{K}[z]/(z^n-1)$.

math.RA↗
arXiv2009

Length functions of lemniscates

We study metric and analytic properties of generalized lemniscates E_t(f)={z:ln|f(z)|=t}, where f is an analytic function. Our main result states that the length function |E_t(f)| is a bilateral Laplace transform of a certain positive measure. In particular, the function ln|E_t(f)| is convex on any interval free of critical points of ln|f|. As another application we deduce explicit formulas of the length function in some special cases.

math.CV↗
arXiv2009

Subharmonicity of higher dimensional exponential transforms

Our main result is an extension of the classical Cauchy inequality for the case of bounded densities. In particular, this implies subharmonicity of the function $M_n(E)$, where $V_n(x)$ is the critical Riesz potential in $R^n$ ($α=n$) of a density $0\leq ρ\leq 1$ and $M_n(t)$ is the profile function: the solution of $y'(t)=1-y^{n/2}(t)$, $y(0)=0$. We show thath this result is optimal (in the sense that $M_n(E)$ is harmnoic for characteristic functions of a ball) and give thereby an affirmative answer to one question posed by B. Gustafsson and M. Putinar (Ind. Univ. Math. J., 52(2003), 527-568).

math.FA↗
arXiv2007

Algebraic structure of quasiradial solutions to the $γ$-harmonic equation

We obtain an explicit representation for quasiradial $γ$-harmonic functions, which shows that these functions have essentially algebraic nature. In particular, we give a complete description of all $γ$ which admit algebraic quasiradial solutions. Unlike the cases $γ=\infty$ and $γ=1$, only finitely many algebraic solutions is shown to exist for any fixed $|γ|>1$. Moreover, there is a special extremal series of $γ$ which exactly corresponds to the well-known ideal $m$-atomic gas adiabatic constant $γ=\frac{2m+3}{2m+1}$.

math-ph↗
arXiv2007

Positive definite collections of disks

Let $Q(z,w)=-\prod_{k=1}^n [(z-a_k)(\bar{w}-\bar{a}_k)-R_k^2]$. M. Putinar and B. Gustafsson proved recently that the matrix $Q(a_i,a_j)$, $1\leq i,j\leq n$, is positive definite if disks $|z-a_i|<R_i$ form a disjoint collection. We extend this result on symmetric collections of discs with overlapping. More precisely, we show that in the case when the nodes $a_j$ are situated at the vertices of a regular $n$-gon inscribed in the unit circle and $\forall i: R_i\equiv R$, the matrix $Q(a_i,a_j)$ is positive definite if and only if $R<ρ_n$, where $z=2ρ_n^2-1$ is the smallest $\ne-1$ zero of the Jacobi polynomial $\mathcal{P}^{n-2ν,-1}_ν(z)$, $ν=[n/2]$.

math.CV↗
arXiv2007

Ullemar's formula for the moment map, II

We establish the complex analogue of Ullemar's formula for polynomial domains. We show that the Jacobian of the complex moment mapping is equal to the self-resultant of the defining polynomial.

math.CV↗
arXiv2003

Positive trigonometric polynomials

We study the boundary of the nonnegative trigonometric polynomials from the algebraic point of view. In particularly, we show that it is a subset of an irreducible algebraic hypersurface and established its explicit form in terms of resultants.

math.CV↗