Computation of a function of a matrix with close eigenvalues by means of the Newton interpolating polynomial

An algorithm for computing an analytic function of a matrix $A$ is described. The algorithm is intended for the case where $A$ has some close eigenvalues, and clusters (subsets) of close eigenvalues are separated from each other. This algorithm is a modification of some well known and widely used algorithms. A novel feature is an approximate calculation of divided differences for the Newton interpolating polynomial in a special way. This modification does not require to reorder the Schur triangular form and to solve Sylvester equations.Comment: 11 page

The Gelfand--Shilov type estimate for Green's function of the bounded solutions problem

An analog of the Gelfand--Shilov estimate of the matrix exponential is proved for Green's function of the problem of bounded solutions of the ordinary differential equation $x'(t)-Ax(t)=f(t)$.Comment: 9 page

An estimate of approximation of a matrix-valued function by an interpolation polynomial

Let $A$ be a square complex matrix, $z_1$, ..., $z_{n}\in\mathbb C$ be (possibly repetitive) points of interpolation, $f$ be analytic in a neighborhood of the convex hull of the union of the spectrum of $A$ and the points $z_1$, ..., $z_{n}$, and $p$ be the interpolation polynomial of $f$, constructed by the points $z_1$, ..., $z_{n}$. It is proved that under these assumptions $$\Vert f(A)-p(A)\Vert\le\frac1{n!} \max_{t\in[0,1];\,\mu\in\text{co}\{z_1,z_{2},\dots,z_{n}\}}\bigl\Vert\Omega(A)f^{{(n)}} \bigl((1-t)\mu\mathbf1+tA\bigr)\bigr\Vert,$$ where $\Omega(z)=\prod_{k=1}^n(z-z_k)$.Comment: 8 pages, 1 figur

Computation of Green's function of the bounded solutions problem

It is well known that the equation $x'(t)=Ax(t)+f(t)$, where $A$ is a square matrix, has a unique bounded solution $x$ for any bounded continuous free term $f$, provided the coefficient $A$ has no eigenvalues on the imaginary axis. This solution can be represented in the form \begin{equation*} x(t)=\int_{-\infty}^{\infty}\mathcal G(t-s)x(s)\,ds. \end{equation*} The kernel $\mathcal G$ is called Green's function. In the paper, a representation of Green's function in the form of the Newton interpolating polynomial is used for approximate calculation of $\mathcal G$. An estimate of the sensitivity of the problem is given.Comment: 12 pages, 2 figure

Inverse-closedness of the set of integral operators with L1L_1-continuously varying kernels

Let $N$ be an integral operator of the form $\bigl(Nu\bigr)(x)=\int_{\mathbb R^c}n(x,x-y)\,u(y)\,dy$ acting in $L_p(\mathbb R^c)$ with a measurable kernel $n$ satisfying the estimate $|n(x,y)|\le\beta(y)$, where $\beta\in L_1$. It is proved that if the function $t\mapsto n(t,\cdot)$ is continuous in the norm of $L_1$ and the operator $\mathbf1+N$ has an inverse, then $(\mathbf1+N)^{-1}=\mathbf1+M$, where $M$ is an integral operator possessing the same properties.Comment: 16 page

Inverse-closedness of subalgebras of integral operators with almost periodic kernels

The integral operator of the form $$\bigl(Nu\bigr)(x)=\sum_{k=1}^\infty e^{i\langle\omega_k,x\rangle} \int_{\mathbb R^c}n_k(x-y)\,u(y)\,dy$$ acting in $L_p(\mathbb R^c)$, $1\le p\le\infty$, is considered. It is assumed that $\omega_k\in\mathbb R^c$, $n_k\in L_1(\mathbb R^c)$, and $$\sum_{k=1}^\infty\lVert n_k\rVert_{L_1}<\infty.$$ We prove that if the operator $\mathbf1+N$ is invertible, then $(\mathbf1+N)^{-1}=\mathbf1+M$, where $M$ is an integral operator possessing the analogous representation.Comment: 21 page

Excitation of turbulence in accretion disks of binary stars by non-linear perturbations

Accretion disks in binary systems can experience hydrodynamic impact at inner as well as outer edges. The first case is typical for protoplanetary disks around young T Tau stars. The second one is typical for circumstellar disks in close binaries. As a result of such an impact, perturbations with different scales and amplitudes are excited in the disk. We investigated the nonlinear evolution of perturbations of a finite, but small amplitude, at the background of sub-Keplerian flow. Nonlinear effects at the front of perturbations lead to the formation of a shock wave, namely the discontinuity of the density and radial velocity. At this, the tangential flow in the neighborhood of the shock becomes equivalent to the flow in in the boundary layer. Instability of the tangential flow further leads to turbulization of the disk. Characteristics of the turbulence depend on perturbation parameters, but alpha-parameter of Shakura-Sunyaev does not exceed ~0.1.Comment: Accepted in Astronomy Report

On the possible turbulence mechanism in accretion disks in non-magnetic binary stars

The arising of turbulence in gas-dynamic (non-magnetic) accretion disks is a major issue of modern astrophysics. Such accretion disks should be stable against the turbulence generation, in contradiction to observations. Searching for possible instabilities leading to the turbulization of gas-dynamic disks is one of the challenging astrophysical problems. In 2004, we showed that in accretion disks in binary stars the so-called precessional density wave forms and induces additional density and velocity gradients in the disk. Linear analysis of the fluid instability of an accretion disk in a binary system revealed that the presence of the precessional wave in the disk due to tidal interaction with the binary companion gives rise to instability of radial modes with the characteristic growth time of tenths and hundredths of the binary orbital period. The radial velocity gradient in the precessional wave is shown to be responsible for the instability. A perturbation becomes unstable if the velocity variation the perturbation wavelength scale is about or higher than the sound speed. Unstable perturbations arise in the inner part of the disk and, by propagating towards its outer edge, lead to the disk turbulence with parameters corresponding to observations (the Shakura-Sunyaev parameter $\alpha \lesssim 0.01$).Comment: Appeared in Phys. Us

Analytic functional calculus for two operators

Properties of the mappings \begin{align*} C&\mapsto\frac1{(2\pi i)^2}\int_{\Gamma_1}\int_{\Gamma_2}f(\lambda,\mu)\,R_{1,\,\lambda}\,C\, R_{2,\,\mu}\,d\mu\,d\lambda, C&\mapsto\frac1{2\pi i}\int_{\Gamma}g(\lambda)R_{1,\,\lambda}\,C\, R_{2,\,\lambda}\,d\lambda \end{align*} are discussed; here $R_{1,\,(\cdot)}$ and $R_{2,\,(\cdot)}$ are pseudo-resolvents, i.~e., resolvents of bounded, unbounded, or multivalued linear operators, and $f$ and $g$ are analytic functions. Several applications are considered: a representation of the impulse response of a second order linear differential equation with operator coefficients, a representation of the solution of the Sylvester equation, and an exploration of properties of the differential of the ordinary functional calculus.Comment: 49 page

Green's function of the problem of bounded solutions in the case of a block triangular coefficient

It is known that the equation $x'(t)=Ax(t)+f(t)$, where $A$ is a bounded linear operator, has a unique bounded solution $x$ for any bounded continuous free term~$f$ if and only if the spectrum of the coefficient $A$ does not intersect the imaginary axis. The solution can be represented in the form \begin{equation*} x(t)=\int_{-\infty}^{\infty}\mathcal G(s)f(t-s)\,ds. \end{equation*} The kernel $\mathcal G$ is called Green's function. In this paper, the case when $A$ admits a representation by a block triangular operator matrix is considered. It is shown that the blocks of $\mathcal G$ are sums of special convolutions of Green's functions of diagonal blocks of $A$.Comment: 18 pages, 1 figur