Oscillation of trinomial differential equations with positive and negative term

In the paper, we offer a new technique for investigation of properties of trinomial differential equations with positive and negative terms \begin{equation*} \left(b(t)\left(a(t)x'(t)\right)'\right)'+p(t)f(x(\tau(t)))-q(t)h(x(\sigma(t)))=0. \end{equation*} We offer criteria for every solution to be oscillatory. We support our results with illustrative examples

Property Aˉ \bar{A} of third-order noncanonical functional differential equations with positive and negative terms

In this article, we have derived a new method to study the oscillatory and asymptotic properties for third-order noncanonical functional differential equations with both positive and negative terms of the form \begin{equation*} (p_2 (t)(p_1 (t) x'(t) )')'+a(t)g(x(\tau(t)))-b(t)h(x(\sigma(t)) = 0 \end{equation*} Firstly, we have converted the above equation of noncanonical type into the canonical type using the strongly noncanonical operator and obtained some new conditions for Property $\bar{A}$. We furnished illustrative examples to validate our main result

Property A of differential equations with positive and negative term

In the paper, we elaborate new technique for the investigation of the asymptotic properties for third order differential equations with positive and negative term \begin{equation*} \left(b(t)\left(a(t)x'(t)\right)'\right)'+p(t)f(x(\tau(t)))-q(t)h(x(\sigma(t)))=0. \end{equation*} We offer new easily verifiable criteria for property A. We support our results with illustrative examples

Oscillation of a perturbed nonlinear third order functional differential equation

In this paper, the authors present some new results on the oscillatory and asymptotic behavior of solutions of the perturbed nonlinear third order functional differential equation $$\left( b(t)\left( a(t)(x^{\prime }(t))^{\alpha }\right) ^{\prime }\right) ^{\prime }+p(t)f(x(\tau (t)))= h(t, x(t), x(\tau(t)), x'(t)).$$ In addition to other conditions, the authors assume that $uf(u) > 0$ for $u \neq 0$ and $f$ is increasing. Examples to illustrate the results are included

Gravitational wave bursts from cusps and kinks on cosmic strings

The strong beams of high-frequency gravitational waves (GW) emitted by cusps and kinks of cosmic strings are studied in detail. As a consequence of these beams, the stochastic ensemble of GW's generated by a cosmological network of oscillating loops is strongly non Gaussian, and includes occasional sharp bursts that stand above the ``confusion'' GW noise made of many smaller overlapping bursts. Even if only 10% of all string loops have cusps these bursts might be detectable by the planned GW detectors LIGO/VIRGO and LISA for string tensions as small as $G \mu \sim 10^{-13}$. In the implausible case where the average cusp number per loop oscillation is extremely small, the smaller bursts emitted by the ubiquitous kinks will be detectable by LISA for string tensions as small as $G \mu \sim 10^{-12}$. We show that the strongly non Gaussian nature of the stochastic GW's generated by strings modifies the usual derivation of constraints on $G \mu$ from pulsar timing experiments. In particular the usually considered ``rms GW background'' is, when G \mu \gaq 10^{-7}, an overestimate of the more relevant confusion GW noise because it includes rare, intense bursts. The consideration of the confusion GW noise suggests that a Grand Unified Theory (GUT) value $G \mu \sim 10^{-6}$ is compatible with existing pulsar data, and that a modest improvement in pulsar timing accuracy could detect the confusion noise coming from a network of cuspy string loops down to $G \mu \sim 10^{-11}$. The GW bursts discussed here might be accompanied by Gamma Ray Bursts.Comment: 24 pages, 3 figures, Revtex, submitted to Phys. Rev.

Glosarium Matematika

273 p.; 24 cm