The effect of tube depth, the separation distance between the tube and nozzle exit, and the nozzle pressure ratio on the characteristics of the flow coming out of the Hartmann tube was studied experimentally. The configuration used in this work consists of an underexpanded sonic jet emanating from a convergent nozzle directed into a closed-ended cylindrical tube of the same diameter (D) as the nozzle exit. The nozzle was operated at two levels of underexpansion corresponding to nozzle pressure ratio (NPR) 3 and 5. The distance (S) from nozzle exit and tube inlet was varied from 0.4D to 4D. Discrete high-amplitude tones (the jet regurgitant, JRG) were produced, only at certain (periodic) intervals (near the shock-cell terminations) of spacing for NPR 3, while for NPR 5 the JRG tones are produced at all points beyond the first shock-cell. For locations other than these, high-frequency tones (screech mode) were observed. The connection between the jet structure and operating modes revealed that the location of standoff shock ahead of the tube with respect to the jet structure plays a dominant role in the observed ‘modes’ rather than the nozzle tube separation. The results reveal that the frequency response of longer tubes in JRG mode approaches their quarter wave frequencies. The high-frequency oscillations observed in the screech mode showed independency with cavity (pipe) depth, contrary to the available literature, the transition between ‘different modes’ oscillation is a function of cavity depth.

This paper consists of three parts: First, letting $b_1(z)$, $b_2(z)$, $p_1(z)$ and $p_2(z)$ be nonzero polynomials such that $p_1(z)$ and $p_2(z)$ have the same degree $k\geq 1$ and distinct leading coefficients $1$ and $\alpha$, respectively, we solve entire solutions of the Tumura–Clunie type differential equation $f^{n}+P(z,\,f)=b_1(z)e^{p_1(z)}+b_2(z)e^{p_2(z)}$, where $n\geq 2$ is an integer, $P(z,\,f)$ is a differential polynomial in $f$ of degree $\leq n-1$ with coefficients having polynomial growth. Second, we study the oscillation of the second-order differential equation $f''-[b_1(z)e^{p_1(z)}+b_2(z)e^{p_2(z)}]f=0$ and prove that $\alpha =[2(m+1)-1]/[2(m+1)]$ for some integer $m\geq 0$ if this equation admits a nontrivial solution such that $\lambda (f)<\infty$. This partially answers a question of Ishizaki. Finally, letting $b_2\not =0$ and $b_3$ be constants and $l$ and $s$ be relatively prime integers such that $l> s\geq 1$, we prove that $l=2$ if the equation $f''-(e^{lz}+b_2e^{sz}+b_3)f=0$ admits two linearly independent solutions $f_1$ and $f_2$ such that $\max \{\lambda (f_1),\,\lambda (f_2)\}<\infty$. In particular, we precisely characterize all solutions such that $\lambda (f)<\infty$ when $l=2$ and $l=4$.

In this work, we consider the first-order difference equation with general argument

\[ \varDelta x(n)+p(n)x\left( \tau (n)\right) =0,\quad n\geq 0, \]

$(p(n))$

$(\tau (n))$

$\tau (n)< n$

$\ n\in \mathbf {N},\,$

$\ \lim _{n\rightarrow \infty }\tau (n)=\infty.$

Sufficient conditions are obtained for the oscillation of a general form of a linear second-order differential equation with discontinuous solutions. The innovations are that the impulse effects are in mixed form and the results obtained are applicable even if the impulses are small. The novelty of the results is demonstrated by presenting an example of an oscillating equation to which previous oscillation theorems fail to apply.

We investigate the Weibel instability (WI) in a dusty plasma which is driven to oscillation by the addition of dust grains in the plasma. Our analysis predicts the existence of three modes in a dusty plasma. There is a high-frequency electromagnetic mode, whose frequency increases with an increase in the relative number density of dust grains and which approaches instability due to the presence of dust grains. The second mode is a damping mode which exists due to dust charge fluctuations in plasma. The third mode is the oscillating WI mode. The dispersion relation and the growth rate of various modes in the dusty plasma are derived using the first-order perturbation theory. The effect of dust grain parameters on frequency and growth rate is also studied and reported.

We establish new oscillation criteria for nonlinear differential equations of second order. The results here make some improvements of oscillation criteria of Butler, Erbe, and Mingarelli [2], Wong [8, 9], and Philos and Purnaras [6].

Let λ > 0 and let

be the Bessel operator on ℝ+ := (0,∞). We show that the oscillation operator 𝒪(RΔλ,∗) and variation operator 𝒱ρ(RΔλ,∗) of the Riesz transform RΔλ associated with Δλ are both bounded on Lp(ℝ+, dmλ) for p ∈ (1,∞), from L1(ℝ+, dmλ) to L1,∞(ℝ+, dmλ), and from L∞(ℝ+, dmλ) to BMO(ℝ+, dmλ), where ρ ∈ (2,∞) and dmλ(x) := x2λ dx. As an application, we give the corresponding Lp-estimates for β-jump operators and the number of up-crossings.

Consider the second order superlinear dynamic equation

$$(*)\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{{x}^{\Delta \Delta }}(t)+p(t)f(x(\sigma (t)))=0$$

where $p\,\in \,C(\mathbb{T},\,\mathbb{R})$, $\mathbb{T}$ is a time scale, $f\,:\,\mathbb{R}\,\to \,\mathbb{R}$ is continuously differentiable and satisfies ${{f}^{'}}(x)>0$, and $x\,f\,(x)\,>\,0$ for $x\,\ne \,0$. Furthermore, $f(x)$ also satisfies a superlinear condition, which includes the nonlinear function $f(x)\,=\,{{x}^{\alpha }}$ with $\alpha \,>\,1$, commonly known as the Emden–Fowler case. Here the coefficient function $p(t)$ is allowed to be negative for arbitrarily large values of $t$. In addition to extending the result of Kiguradze for $\left( * \right)$ in the real case $\mathbb{T}\,=\,\mathbb{R}$, we obtain analogues in the difference equation and $q$-difference equation cases.

Multiunit activity was recorded in the optic tectum of awake pigeons with two electrodes at sites varying in depth and separated by 0.3 to 3.0 mm. Autocorrelation and cross-correlation functions were computed from the recorded spike trains to determine temporal relationships in the neuronal firing patterns. Cross-correlation analysis revealed that spatially separate groups of cells in the tectum show synchronous responses to a visual stimulus. Strong synchronization occurred in both superficial and deep layers of the tectum, in general with zero-phase shift. The response synchronization in the avian optic tectum resembles that observed in the mammalian cortex, suggesting that it may subserve common functions in visual processing.

Previously, we have demonstrated that a subpopulation of visual cortical neurons exhibit oscillatory responses to their preferred stimuli at a frequency near 50 Hz (Gray & Singer, 1989). These responses can selectively synchronize over large distances of cortex in a stimulus-specific manner (Gray et al., 1989; Engel et al., 1990a). Here we report the results of a new analysis which reveals the fine temporal structure inherent in these interactions. We utilized pairs of recordings of the local field potential (LFP) activity from area 17 in the anesthetized cat which met two criteria. The LFP was correlated with the underlying unit activity at each site and the recording sites were at least 5 mm apart in cortex. A moving-window technique was applied to compute cross correlograms on 100-ms epochs of data repeated at intervals of 30 ms for a period of 3 s during each direction of stimulus movement. A statistical test was devised to determine the significance of detected correlations. In this way we were able to determine the magnitude, phase difference, frequency, and duration of correlated oscillations as a function of time. The results demonstrate that (1) the duration of synchrony is variable and lasts from 100–900 ms; (2) the phase differences between and the frequencies of synchronized responses are also variable within and between events and range from +3 to —3 ms and 40–60 Hz, respectively; and (3) multiple correlation events often occur within a single stimulus period. These results demonstrate a high degree of dynamic variability and a rapid onset and offset of synchrony among interacting populations of neurons which is consistent with the requirements of a mechanism for feature integration.

In this note we study the zeros of solutions of differential equations of the form u′′+pu=0. A criterion for oscillation is found, and some sharper forms of the Sturm comparison theorem are given.