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## Nonself-similar flow with a shock wave reflected from the center of symmetry and new self-similar solutions with two reflected shocks

### Computational Mathematics and Mathematical Physics (2013-03-01) 53: 350-368 , March 01, 2013

In some problems concerning cylindrically and spherically symmetric unsteady ideal (inviscid and nonheat-conducting) gas flows at the axis and center of symmetry (hereafter, at the center of symmetry), the gas density vanishes and the speed of sound becomes infinite starting at some time. This situation occurs in the problem of a shock wave reflecting from the center of symmetry. For an ideal gas with constant heat capacities and their ratio γ (adiabatic exponent), the solution of this problem near the reflection point is self-similar with a self-similarity exponent determined in the course of the solution construction. Assuming that γ on the reflected shock wave decreases, if this decrease exceeds a threshold value, the flow changes substantially. Assuming that the type of the solution remains unchanged for such γ, self-similarity is preserved if a piston starts expanding from the center of symmetry at the reflection time preceded by a finite-intensity reflected shock wave propagating at the speed of sound. To answer some questions arising in this formulation, specifically, to find the solution in the absence of the piston, the evolution of a close-to-self-similar solution calculated by the method of characteristics is traced. The required modification of the method of characteristics and the results obtained with it are described. The numerical results reveal a number of unexpected features. As a result, new self-similar solutions are constructed in which two (rather than one) shock waves reflect from the center of symmetry in the absence of the piston.

## A Criterion for Conditional Instability by the First Approximation for Solutions of Differential Systems

### Differential Equations (2005-12-01) 41: 1677-1686 , December 01, 2005

## Preliminaries

### Haar Series and Linear Operators (1997-01-01) 367: 1-13 , January 01, 1997

Let (*S, F, μ)* be a measure space. Here S is a set, F is a *σ*-algebra, *µ* is a *σ*-finite measure on F. If *F*_{0} is a *σ*-subalgebra of F, *x**∈**L*_{1} (*(S*, *F*, *μ*), then denote by *E*_{Fo} the unique, up to equivalence, *F*_{0}-measurable function satisfying
for each *A ∈ F*_{0}. By the Radon — Nikodym theorem, such function exists. The function *E*_{Fo}*x =**E*_{Fo,μ}*x* is called the conditional expectation with respect to *F*_{0}.

## On one representation of analytic functions by harmonic functions

### Siberian Advances in Mathematics (2008-06-01) 18: 103-117 , June 01, 2008

Let *u*(*x*) be a function analytic in some neighborhood *D* about the origin,
$$
\mathcal{D}
$$
⊂ ℝ^{n}. We study the representation of this function in the form of a series *u*(*x*) = *u*_{0}(*x*) + |*x*|^{2}*u*_{1}(*x*) + |*x*|^{4}*u*_{2}(*x*) + …, where *u*_{k}(*x*) are functions harmonic in
$$
\mathcal{D}
$$
. This representation is a generalization of the well-known Almansi formula.

## Numerical stabilization of the Lorenz system by a small external perturbation

### Computational Mathematics and Mathematical Physics (2006-08-01) 46: 1341-1348 , August 01, 2006

The Lorenz system perturbed by noise and its invariant measure whose density obeys the stationary Fokker-Planck equation are analyzed numerically. A linear functional of the invariant measure is considered, and its variation caused by a variation in the right-hand side of the Lorenz system is calculated. A small (in modulus) external perturbation is calculated under which the strange attractor of the Lorenz system degenerates into a stable fixed point.

## Preemptive scheduling of independent jobs on identical parallel machines subject to migration delays

### Automation and Remote Control (2010-10-01) 71: 2093-2101 , October 01, 2010

We present hardness and approximation results for the problem of preemptive scheduling of *n* independent jobs on *m* identical parallel machines subject to a migration delay *d* with the objective to minimize the makespan. We give a sharp threshold on the value of *d* for which the complexity of the problem changes from polynomial time solvable to NP-hard. Next, we give initial results supporting a conjecture that there always exists an optimal schedule with at most *m* − 1 job migrations. Finally, we provide a *O*(*n*) time (1 + 1/log_{2}*n*)-approximation algorithm for *m* = 2.

## On the Stokes problem with nonzero divergence

### Journal of Mathematical Sciences (2010-04-01) 166: 106-117 , April 01, 2010

The strong solvability of the nonstationary Stokes problem with nonzero divergence in a bounded domain is studied. Bibliography: 12 titles.

## Inversion of the Kipriyanov–Radon transform via fractional derivatives in a one-dimensional parameter

### Journal of Mathematical Sciences (2009-04-01) 158: 235-240 , April 01, 2009

This paper considers the Kipriyanov–Radon transform constructed as a special Radon transform adopted for dealing with singular Bessel differential operators of the corresponding indices acting on a part of the variables. The authors obtain inversion formulas generalizing the classical formulas for the Radon transform of axially-symmetric functions and relating to the integro-differentiation of fractional order in a one-dimensional parameter.

## Solution of a stochastic Darcy equation by polynomial chaos expansion

### Numerical Analysis and Applications (2017-07-01) 10: 259-271 , July 01, 2017

This paper deals with solving a boundary value problem for the Darcy equation with a random hydraulic conductivity field.We use an approach based on polynomial chaos expansion in a probability space of input data.We use a probabilistic collocation method to calculate the coefficients of the polynomial chaos expansion. The computational complexity of this algorithm is determined by the order of the polynomial chaos expansion and the number of terms in the Karhunen–Loève expansion. We calculate various Eulerian and Lagrangian statistical characteristics of the flow by the conventional Monte Carlo and probabilistic collocation methods. Our calculations show a significant advantage of the probabilistic collocation method over the directMonte Carlo algorithm.

## Linear Methods in the Proof of Jackson Type Inequalities and Applications to Estimates of Functionals with Two Known Moments

### Journal of Mathematical Sciences (2014-01-01) 196: 498-514 , January 01, 2014

We propose a new method for estimating functionals in terms of higher order moduli of continuity with explicit constants. Using this method, we estimate deviations of linear methods of approximation by entire functions of finite degree and, in particular, by trigonometric polynomials. For illustration of the results, we derive estimates for the Riesz and Akhiezer–Krein–Favard averages. Bibliography: 14 titles.