51,476 research outputs found
On Nonoscillation of Mixed Advanced-Delay Differential Equations with Positive and Negative Coefficients
For a mixed (advanced--delay) differential equation with variable delays and
coefficients
where explicit
nonoscillation conditions are obtained.Comment: 17 pages; 2 figures; to appear in Computers & Mathematics with
Application
Estimation of Solutions of Differential Systems with Delayed Argument of Neutral Type
Tato disertační práce pojednává o řešení diferenciálních rovnic a systémů diferenciálních rovnic. Hlavní pozornost je věnována asymptotickým vlastnostem rovnic se zpožděním a systémů rovnic se zpožděním. V první kapitole jsou uvedeny fyzikální a technické příklady popsané pomocí diferenciálních rovnic se zpožděním a jejich systémů. Je uvedena klasifikace rovnic se zpožděním a jsou zformulovány základní pojmy stability s důrazem na druhou metodu Ljapunova. Ve druhé kapitole jsou studovány odhady řešení rovnic neutrálního typu. Třetí kapitola se zabývá systémy diferenciálních rovnic neutrálního typu. Jsou odvozeny asymptotické odhady pro řešení i pro derivace řešení. V závěru kapitoly jsou uvedeny příklady a srovnání výsledků s pracemi jiných autorů. Výpočty byly prováděny pomocí programu MATLAB. Poslední, čtvrtá kapitola, se zabývá asymptotickými vlastnostmi systémů se speciálním typem nelinearity, tzv. sektorové nelinearity. Jsou odvozeny vlastnosti řešení a derivace řešení. Základní metodou pro důkazy je v celé práci druhá Ljapunovova metoda a použití funkcionálů Ljapunova-Krasovského.This dissertation discusses the solutions to the differential equation and to systems of differential equations. The main attention is paid to study of asymptotical properties of equations with delay and systems of equations with delay. In the first chapter are given physical and technical examples described by differential equations with delay and their systems. The classification of equations with delay is given and basic notions of theory of stability are formulated (mainly with the emphasis on the Lyapunov second method). In the second chapter estimates of solutions of equations of neutral type are studied. The third chapter deals with systems of differential equations of neutral type. Asymptotic estimates for solutions and their derivatives are proved. At the end of the chapter examples and comparisons of our results and of other authors are given. The calculation were performed with the MATLAB software. Last, the fourth chapter deals with asymptotical properties of systems having a special type of nonlinearities, so called ``sector nonlinearities''. Properties and estimations of solutions and derivatives are derived. The basic tools used in the dissertation are the Lyapunov second method and functionals of Lyapunov-Krasovskii type.
Delays in Open String Field Theory
We study the dynamics of light-like tachyon condensation in a linear dilaton
background using level-truncated open string field theory. The equations of
motion are found to be delay differential equations. This observation allows us
to employ well-established mathematical methods that we briefly review. At
level zero, the equation of motion is of the so-called retarded type and a
solution can be found very efficiently, even in the far light-cone future. At
levels higher than zero however, the equations are not of the retarded type. We
show that this implies the existence of exponentially growing modes in the
non-perturbative vacuum, possibly rendering light-like rolling unstable.
However, a brute force calculation using exponential series suggests that for
the particular initial condition of the tachyon sitting in the false vacuum in
the infinite light-cone past, the rolling is unaffected by the unstable modes
and still converges to the non-perturbative vacuum, in agreement with the
solution of Hellerman and Schnabl. Finally, we show that the growing modes
introduce non-locality mixing present with future, and we are led to conjecture
that in the infinite level limit, the non-locality in a light-like linear
dilaton background is a discrete version of the smearing non-locality found in
covariant open string field theory in flat space.Comment: 48 pages, 14 figures. v2: References added; Section 4 augmented by a
discussion of the diffusion equation; discussion of growing modes in Section
4 slightly expande
Investigation of the Stability of the Laminar Boundary Layer in a Compressible Fluid
In the present report the stability of two-dimensional laminar flows of a gas is investigated by the method of small perturbations. The chief emphasis is placed on the case of the laminar boundary layer.
Part I of the present report deals with the general mathematical theory. The general equations governing one normal mode of the small velocity and temperature disturbances are derived and studied in great detail. It is found that for Reynolds numbers of the order of those encountered in most aerodymnic problems, the temperature disturbances have only a negligible effect on those particular velocity solutions which depend primarily on the viscosity coefficient ("viscous solutions"). Indeed, the latter are actually of the same form in the compressible fluid as in the incompressible fluid, at least to the first
approximation. Because of this fact, the mathematical analysis is greatly simplified. The final equation determining the characteristic values of the stability problem depends on the "inviscid solutions" and the function of Tietjens in a manner very similar to the case of the incompressible fluid. The second viscosity coefficient and the coefficient of heat conductivity do not enter the problem; only the ordinary coefficient of viscosity near the solid surface is involved.
Part II deals wlth the limiting case of infinite Reynolds numbers. The study of energy relations is very much emphasized. It is shown that the disturbance will gain energy from the main flow if the gradient of the product of mean density and mean vorticity near the solid surface has a sign opposite to that near the outer edge of the boundary layer.
A general stability criterion has been obtained in terms of the gradient of the product of density and vorticity, analogous to the Rayleigh-Tollmien criterion for the case of an incompressible fluid. If this gradient vanishes for some value of the velocity ratio of the main flow exceeding 1 - 1/M (where M is the free stream Mach number), then neutral and self-excited "subsonic" disturbances exist in the inviscid fluid. (The subsonic disturbances die out rapidly with distance from the solid surface.) The conditions for the existence of other types of disturbance have not yet been established to this extent of exactness. A formula has been worked out to give the amplitude ratio of incoming and reflected sound waves.
It is found in the present investigation that when the solid boundary is heated, the boundary layer flow is destabilized through the change in the distribution of the product of density and vorticity, but stabilized through the increase of kinematic viscosity near the solid boundary. When the solid boundary is cooled, the situation is just the reverse. The actual extent to which these two effects counteract each
other can only be settled by actual computation or some approximate estimstes of the minimum critical Reylolds number. This question will be investigated in a subsequent report.
Part III deals with the stability of laminar flows in a perfect gas with the effect of viscosity included. The method for the numerical computation of the stability limit is outlined; detailed numerical calculations will be carried out in a subsequent report
Black Holes on Cylinders
We take steps toward constructing explicit solutions that describe
non-extremal charged dilatonic branes of string/M-theory with a transverse
circle. Using a new coordinate system we find an ansatz for the solutions with
only one unknown function. We show that this function is independent of the
charge and our ansatz can therefore also be used to construct neutral black
holes on cylinders and near-extremal charged dilatonic branes with a transverse
circle. For sufficiently large mass these solutions have a horizon
that connects across the cylinder but they are not translationally invariant
along the circle direction. We argue that the neutral solution has larger
entropy than the neutral black string for any given mass. This means that for
the neutral black string can gain entropy by redistributing its mass
to a solution that breaks translational invariance along the circle, despite
the fact that it is classically stable. We furthermore explain how our
construction can be used to study the thermodynamics of Little String Theory.Comment: latex, 68 pages, 4 figures. v2: Typos fixed, argument about \chi
corrected in sec. 7.4, discussion of space of physical solutions corrected
and clarified in sec. 9; v3: v=\pi clarified, typos fixed, figure 1 change
Probing the Hydrodynamic Limit of (Super)gravity
We study the long-wavelength effective description of two general classes of
charged dilatonic (asymptotically flat) black p-branes including D/NS/M-branes
in ten and eleven dimensional supergravity. In particular, we consider
gravitational brane solutions in a hydrodynamic derivative expansion (to first
order) for arbitrary dilaton coupling and for general brane and co-dimension
and determine their effective electro-fluid-dynamic descriptions by exacting
the characterizing transport coefficients. We also investigate the stability
properties of the corresponding hydrodynamic systems by analyzing their
response to small long-wavelength perturbations. For branes carrying unsmeared
charge, we find that in a certain regime of parameter space there exists a
branch of stable charged configurations. This is in accordance with the
expectation that D/NS/M-branes have stable configurations, except for the D5,
D6, and NS5. In contrast, we find that Maxwell charged brane configurations are
Gregory-Laflamme unstable independently of the charge and, in particular,
verify that smeared configurations of D0-branes are unstable. Finally, we
provide a modification to the mapping presented in arxiv:1211.2815 and utilize
it to provide a non-trivial cross-check on a certain subset of our transport
coefficients with the results of arXiv:1110.2320.Comment: 36 pages, 2 figures. v2: Added reference and corrected typ
Bifurcation to Quasi-Periodic Tori in the Interaction of Steady State and Hopf Bifurcations
Bifurcations to quasi-periodic tori in a two parameter family of vector fields are studied. At criticality, the vector field has an equilibrium point with a zero eigenvalue and a pair of complex conjugate eigenvalues. This situation has been studied by Langford, Iooss, Holmes and Guckenheimer. Here we provide explicitly computed conditions under which the stability of the secondary branch of tori, and whether the flow on them is quasiperiodic, can be determined. The results are applied to "Brusselator" system of reaction diffusion equations
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