Explosive instability in nonlinear waves in media with negative viscosity : PMM vol. 38, n 6, 1974, pp. 991–995

The propagation of a wave of a finite amplitude in a medium with a nonlinearity of the second degree and negative viscosity, is examined. It is shown that in a finite time singularities appear in the solution. The exact solution of the Cauchy problem is given for a specific case. Recently the effects of negative viscosity which cause an increase in the energy of the wave motion have been studied intensively in electrodynamics, plasma physics, the Earth's atmosphere, in the theory of the circulation of the oceans and of flow in open channels [1–4], Wave amplification caused by an energy transfer from turbulent to regular motions, is possible in any medium having space-time fluctuations, provided the correlation time is sufficiently small [5, 6]. As the wave amplitude increases, nonlinear effects become important; they have been taken into account in cases where the interaction of a finite number of harmonics [2, 4] and the structure of steady motions have been examined [3].It is shown in this paper that in a medium with negative viscosity and a second degree dynamic nonlinearity, a solution of the Cauchy problem for an arbitrary “good” form of the initial perturbation, exists over a finite time interval. An example of such a solution is given.     

The rate of convergence for the cyclic projections algorithm III: Regularity of convex sets

The cyclic projections algorithm is an important method for determining a point in the intersection of a finite number of closed convex sets in a Hilbert space. That is, for determining a solution to the “convex feasibility” problem. This is the third paper in a series on a study of the rate of convergence for the cyclic projections algorithm. In the first of these papers, we showed that the rate could be described in terms of the “angles” between the convex sets involved. In the second, we showed that these angles often had a more tractable formulation in terms of the “norm” of the product of the (nonlinear) metric projections onto related convex sets.In this paper, we show that the rate of convergence of the cyclic projections algorithm is also intimately related to the “linear regularity property” of Bauschke and Borwein, the “normal property” of Jameson (as well as Bakan, Deutsch, and Li’s generalization of Jameson’s normal property), the “strong conical hull intersection property” of Deutsch, Li, and Ward, and the rate of convergence of iterated parallel projections. Such properties have already been shown to be important in various other contexts as well.     

Geometric separation and exact solutions for the parameterized independent set problem on disk graphs

We consider the parameterized problem, whether for a given set  of n disks (of bounded radius ratio) in the Euclidean plane there exists a set of k non-intersecting disks. For this problem, we expose an algorithm running in time that is—to our knowledge—the first algorithm with running time bounded by an exponential with a sublinear exponent. For λ-precision disk graphs of bounded radius ratio, we show that the problem is fixed parameter tractable with a running time  . The results are based on problem kernelization and a new “geometric ( -separator) theorem” which holds for all disk graphs of bounded radius ratio. The presented algorithm then performs, in a first step, a “geometric problem kernelization” and, in a second step, uses divide-and-conquer based on our new “geometric separator theorem.”     

Method of successive approximations for three-dimensional laminar boundary layer problems (locally self-similar case) : PMM vol. 37, n6, 1973, pp. 974–983

We present an analytical method for the computation of problems of incompressible boundary layer theory based on an application of the method of successive approximations. The system of equations is reduced to a form suitable for integration. Parameters characterizing the external flow and the body geometry are contained only in the coefficients of the system and do not enter into the boundary conditions. The transformed momentum equations are integrated across the boundary layer from a current value to infinity with the boundary conditions taken into account. If the integration is made from zero to infinity, then the equations pass over into the Kármán relations. Integrating the system of equations a second time, using the boundary conditions at the wall, we obtain a system of nonlinear integro-differential equations. To solve this system of equations we apply the method of successive approximations. To satisfy the boundary Conditions at infinity we introduce, at each step of the iterations, unknown “governing” functions. From the conditions at the outer side of the boundary layer we obtain additional equations for their determination. With the iterational algorithm formulated in this way, the boundary conditions, both on the body and at the outer side of the boundary layer; are satisfied automatically.We consider a locally self-similar approximation. In this case, relative to the “governing” functions, we obtain an algebraic system of equations. We write out the solution in the first approximation. The results obtained in the first approximation are compared with the results of finite-difference computations for a wide range of problems. The results obtained in this paper are compared with those obtained in [1] for the flow in the neighborhood of a stagnation point. An indication is given of the nonuniqueness of the solutions of the three-dimensional boundary layer equations.     

A certain class of systems of “reaction-diffusion” type

The Cauchy problem with bounded, nonnegative initial function is investigated for a quasilinear, degenerate, parabolic system of two equations, containing, in general, lower terms with non-power nonlinearities. Their form is such that the first component of the solution can become unbounded after a finite time interval, provided the second component remains strictly positive. However, the latter may tend to zero and even become identically zero after a finite time. Two theorems are proved regarding the existence of a global, bounded generalized solution of the considered problem. Examples are given, attesting to the sharpness of these theorems.Translated from Trudy Seminara imeni I. G. Petrovskogo, No. 14, pp. 78–88, 1980.     

Coincidence criteria for maximal stable bridges in two game problems of approach

We consider a finite-dimensional conflict-controlled system whose behavior on a finite time interval is described by a vector differential equation. We analyze two game problems of approach in the phase space. In both problems the same terminal set is considered: in the first case, one should guarantee that the phase vector of the system reaches the terminal set at the final instant of time; in the second case, the phase vector should reach the terminal set no later than the final time instant. It is natural to assume that the construction of a solution to the first problem is much simpler than the construction of a solution to the second problem; this fact is confirmed by available experience. The paper is devoted to finding conditions on the system and the terminal set under which the solutions to the above problems coincide. Using these conditions, one can replace the solution of the second problem by the simpler solution of the first problem.     

Point capture of two evaders in succession

A constant-speed coplanar model with unlimited turn-rates leads to a rather simple geometrical solution of the problem of point capture of two successive evaders in minimum total time: the pursuer concentrates first on the nearer evader who runs in an appropriate direction; the second evader runs directly away from the predictable point of capture of the first evader. This simple solution is valid only if the second evader remains thereby the further of the two evaders. Otherwise, the solution must be modified to include a phase involving curved motion by all three players, during which the pursuer remains equidistant from both evaders.     

Two profitless delays for an SEIRS epidemic disease model with vertical transmission and pulse vaccination

Since the investigation of impulsive delay differential equations is beginning, the literature on delay epidemic models with pulse vaccination is not extensive. In this paper, we propose a new SEIRS epidemic disease model with two profitless delays and vertical transmission, and analyze the dynamics behaviors of the model under pulse vaccination. Using the discrete dynamical system determined by the stroboscopic map, we obtain a ‘infection-free’ periodic solution, further, show that the ‘infection-free’ periodic solution is globally attractive when some parameters of the model are under appropriate conditions. Using a new modeling method, we obtain sufficient condition for the permanence of the epidemic model with pulse vaccination. We show that time delays, pulse vaccination and vertical transmission can bring different effects on the dynamics behaviors of the model by numerical analysis. Our results also show the delays are “profitless”. In this paper, the main feature is to introduce two discrete time delays, vertical transmission and impulse into SEIRS epidemic model and to give pulse vaccination strategies.     

Analytical solution for Van der Pol–Duffing oscillators

In this paper, the problem of single-well, double-well and double-hump Van der Pol–Duffing oscillator is studied. Governing equation is solved analytically using a new kind of analytic technique for nonlinear problems namely the “Homotopy Analysis Method” (HAM), for the first time. Present solution gives an expression which can be used in wide range of time for all domain of response. Comparisons of the obtained solutions with numerical results show that this method is effective and convenient for solving this problem. This method is a capable tool for solving this kind of nonlinear problems.     

Asymptotic stability of ground states in 3D nonlinear Schrödinger equation including subcritical cases

We consider a class of nonlinear Schrödinger equation in three space dimensions with an attractive potential. The nonlinearity is local but rather general encompassing for the first time both subcritical and supercritical (in L²) nonlinearities. We study the asymptotic stability of the nonlinear bound states, i.e. periodic in time localized in space solutions. Our result shows that all solutions with small initial data, converge to a nonlinear bound state. Therefore, the nonlinear bound states are asymptotically stable. The proof hinges on dispersive estimates that we obtain for the time dependent, Hamiltonian, linearized dynamics around a careful chosen one parameter family of bound states that “shadows” the nonlinear evolution of the system. Due to the generality of the methods we develop we expect them to extend to the case of perturbations of large bound states and to other nonlinear dispersive wave type equations.     

Asymptotic stability of contact waves for the compressible Navier–Stokes equations

In this article, we study the large-time asymptotic behaviour of contact wave for the Cauchy problem of one-dimensional compressible Navier–Stokes equations with zero viscosity. When the Riemann problem for the Euler system admits a contact discontinuity solution, we can construct a contact wave, which approximates the contact discontinuity on any finite-time interval for small heat conduction and then runs away from it for large time, and prove that it is nonlinearly stable provided that the strength of contact discontinuity and the perturbation of the initial data are suitably small.     

Uniform Decay properties of a model in structural acoustics

We investigate decay properties for a system of coupled partial differential equations which model the interaction between acoustic waves in a cavity and the walls of the cavity. In this system a wave equation is coupled to a structurally damped plate or beam equation. The underlying semigroup for this system is not uniformly stable, but when the system is appropriately restricted we obtain some uniform stability. We present two results of this type. For the first result, we assume that the initial wave data is zero, and the initial plate or beam data is in the natural energy space; then the corresponding solution to system decays uniformly to zero. For the second result, we assume that the initial condition is in the natural energy space and the control function is L²(0,∞) (in time) into the control space; then the beam displacement and velocity are both L²(0,∞) into a space with two spatial derivatives.     

The Kervaire-Laudenbach Conjecture and Presentations of Simple Groups

The statement “no non-Abelian simple group can be obtained from a non-simple one by adding one generator and one relator” first is equivalent to the Kervaire-Laudenbach conjecture, and second, becomes true under the additional assumption that an initial non-simple group is either finite or torsion free.Supported by RFBR grant No. 02-01-00170.__________Translated from Algebra i Logika, Vol. 44, No. 4, pp. 399–437, July–August, 2005.     

Fast algorithms for fragmentable items bin packing

Bin packing with fragmentable items is a variant of the classic bin packing problem where items may be cut into smaller fragments. The objective is to minimize the number of item fragments, or equivalently, to minimize the number of cuts, for a given number of bins. Models based on packing fragmentable items are useful for representing finite shared resources. In this article, we present improvements to approximation and metaheuristic algorithms to obtain an optimality-preserving optimization algorithm with polynomial complexity, worst-case performance guarantees and parametrizable running time. We also present a new family of fast lower bounds and prove their worst-case performance ratios. We evaluate the performance and quality of the algorithm and the best lower bound through a series of computational experiments on representative problem instances. For the studied problem sets, one consisting of 180 problems with up to 20 items and another consisting of 450 problems with up to 1024 items, the lower bound performs no worse than 5 / 6. For the first problem set, the algorithm found an optimal solution in 92 % of all 1800 runs. For the second problem set, the algorithm found an optimal solution in 99 % of all 4500 runs. No run lasted longer than 220 ms.     

Two-stage generalized age maintenance of a queue-like production system

In general, the initiation of preventive maintenance should be based on the technical state as well as the operating state of a production system. Since the operating state of a production system is often subject to fluctuations in time, the planning of preventive maintenance at preset points in time (e.g. age/block replacement) cannot be optimal. Therefore, we propose a so-called two-stage maintenance policy, which - in a first stage - uses the technical state of the production system to determine a finite interval [t, t + At] during which preventive maintenance must be carried out, and - in a second stage - uses the operating state of the production system to determine the optimal starting time t̂ for preventive maintenance within that interval. A generalized age maintenance policy optimizing both t and At is formulated in the first stage. To this end, the actual starting time of preventive maintenance is modelled in terms of a uniform distribution over the maintenance interval. Moreover, the expected costs of preventive maintenance are modelled as a decreasing function of the interval size. An efficient algorithm is developed to demonstrate the optimal strategy for a queue-like production system, via numerical results that offer useful insights.     

CBS constants for multilevel splitting of graph-Laplacian and application to preconditioning of discontinuous Galerkin systems

The goal of this work is to derive and justify a multilevel preconditioner of optimal arithmetic complexity for symmetric interior penalty discontinuous Galerkin finite element approximations of second order elliptic problems. Our approach is based on the following simple idea given in [R.D. Lazarov, P.S. Vassilevski, L.T. Zikatanov, Multilevel preconditioning of second order elliptic discontinuous Galerkin problems, Preprint, 2005]. The finite element space of piece-wise polynomials, discontinuous on the partition , is projected onto the space of piece-wise constant functions on the same partition that constitutes the largest space in the multilevel method. The discontinuous Galerkin finite element system on this space is associated to the so-called “graph-Laplacian”. In 2-D this is a sparse M-matrix with -1 as off diagonal entries and nonnegative row sums. Under the assumption that the finest partition is a result of multilevel refinement of a given coarse mesh, we develop the concept of hierarchical splitting of the unknowns. Then using local analysis we derive estimates for the constants in the strengthened Cauchy–Bunyakowski–Schwarz (CBS) inequality, which are uniform with respect to the levels. This measure of the angle between the spaces of the splitting was used by Axelsson and Vassilevski in [Algebraic multilevel preconditioning methods II, SIAM J. Numer. Anal. 27 (1990) 1569–1590] to construct an algebraic multilevel iteration (AMLI) for finite element systems. The main contribution in this paper is a construction of a splitting that produces new estimates for the CBS constant for graph-Laplacian. As a result we have a preconditioner for the system of the discontinuous Galerkin finite element method of optimal arithmetic complexity.     

The bidimensional interproximation problem and mixed splines

Interproximation methods for surfaces can be used to construct a smooth surface interpolating some data points and passing through specified regions. In this paper we study the use of mixed splines, that is smoothing splines with additional interpolation constraints, to solve the interproximation problem for surfaces in the case of scattered data. The solution is obtained by solving a linear system whose structure can be improved by using “bell-shaped” thin plate splines.     

The perception of randomness

Psychologists have studied people's intuitive notions of randomness by two kinds of tasks: judgment tasks (e.g., “is this series like a coin?” or “which of these series is most like a coin?”), and production tasks (e.g., “produce a series like a coin”). People's notion of randomness is biased in that they see clumps or streaks in truly random series and expect more alternation, or shorter runs, than are there. Similarly, they produce series with higher than expected alternation rates. Production tasks are subject to other biases as well, resulting from various functional limitations. The subjectively ideal random sequence obeys “local representativeness”; namely, in short segments of it, it represents both the relative frequencies (e.g., for a coin, 50%–50%) and the irregularity (avoidance of runs and other patterns). The extent to which this bias is a handicap in the real world is addressed.     

Effect of interconvertibility of electromagnetic and gravitational waves in strong external electromagnetic fields and the propagation of waves in the field of a charged “black hole” : PMM vol. 38, n 6, 1974, pp. 1122–1129

It is shown that the behavior of an arbitrary wave propagating in the field of a nonrotating charged black hole is defined (with the use of quadratures) by four functions. Each of these functions obeys its second order equation of the wave kind. Short electromagnetic waves falling onto a black hole are reflected by its field in the form of gravitational and electromagnetic waves whose amplitude was explicitly determined. In the case of the wave carrying rays winding around the limit cycle the reflection and transmission coefficients were obtained in the form of analytic expressions.Various physical processes taking place inside, as well as outside a collapsing star, may induce perturbations of the gravitational, electromagnetic and other fields, and lead to the appearance in the surrounding space of waves of various kinds which propagate over a distorted background and are dissipated along its inhomogeneities.In the absence of rotation and charge in a star, the analysis of small perturbations of the gravitational fields is based on the system of Einstein equations linearized around the Schwarzschild solution. In [1, 2] this system of equations, after expansion of perturbations in spherical harmonics and Fourier transformation with respect to time, was reduced to two independent linear ordinary differential equations of second order of the form of the stationary Schrödinger equation for a particle in a potential force field. Each of these equations defines one of two possible independent perturbation kinds: “even” and “odd” (the different behavior of spherical tensor harmonics at coordinate inversion is the deciding factor in the determination of the kind of perturbation [1, 2]). Although these equations were derived with the superposition on the perturbations of the metric of specific coordinate conditions, they define, as shown in [4], the behavior of invariants of the perturbed gravitational field, which imparts to the potential barriers appearing in these equations an invariant meaning.The system of Maxwell equations on the background of Schwarzschild solution also reduces to similar equations, which differ from the above only by the form of potential barriers appearing in these [5].In the presence in the unperturbed solution of a strong electromagnetic field the gravitational and electromagnetic waves interact with each other, and transmutation takes place. The train of short periodic electromagnetic waves generates the accompanying train of gravitational waves. This phenomenon was first analyzed in [6] on and arbitrary background. It was shown in [7, 8] that dense stars surrounded by hot plasma may acquire a charge owing to splitting of charges by radiation pressure and the “sweeping out” of positrons nascent in vapors in strong electrostatic fields. The interaction of waves becomes particularly clearly evident in the neighborhood of black holes which may serve as “valves” by maintaining equilibrium between the relict electromagnetic and gravitational radiation in the Universe. Rotation of black holes intensifies this effect [6].If a nonrotating star possesses an electrostatic charge, the definition of perturbations of the electromagnetic and gravitational fields must be based on the complete system of Einstein-Maxwell equations linearized around the Nordström-Reissner solution. (Small perturbations of electromagnetic field outside a charged black hole were considered in [9, 10] on the basis of the system of Maxwell equations on a “rigid” background of the Nordström-Reissner solution, without taking into account the interconvertibility of gravitational and electromagnetic waves, which materially affects their behavior in the neighborhood of a charged black hole). Here this system of equations which define the interacting gravitational and electromagnetic perturbations are reduced to four independent second order differential equations, two for each kind of perturbations (an importsnt part is played here by the coordinate conditions imposed on the perturbations of the metric, proposed by the authors in [4]). Perturbation components of the metric and of the electromagnetic field are determined in quadratures by the solutions of these equations. If the charge of a star tends to vanish, two of the derived equations convert to equations for gravitational waves on the background of the Schwarzschild solution [1, 2], while the twoothers become equations which are equivalent to Maxwell solutions on the same background. The short-wave asymptotics of derived equations is determined throughout including the neighborhood of the limit cycle for the wave carrying rays. These solutions far away from the point of turn coincide with those obtained in [6] for any arbitrary background. Approximation of geometric optics does not provide correct asymptotics for impact parameters of rays which are close to critical for which the Isotropie and geodesic parameters wind around the limit cycle. This case is investigated below.A similar situation in the Schwarzschild field was analyzed in [11], where analytic expressions for the wave reflection and transmission coefficients were determined, and the integral radiation stream trapped by a black hole produced by another radiation component of the dual system was calculated.