In the past years, extensive work has been done and great progress has been made on **oscillation** of equation () and more general equations (see [–] and the references therein). On the other hand, with wide use in the nonlinear elasticity theory and elec- trorheological ﬂuids (see [, ]), the diﬀerential equations and variational problems with variable exponent growth conditions have been investigated by many authors in recent years (see [–]). However, we notice that no criteria were found for equation () even for the special case of equation () to be oscillatory so far in the literature. The purpose of this paper is to establish some **interval** **oscillation** results for equation () which involves variable exponent growth conditions. Clearly, our work is of signiﬁcance because equation () allows an inﬁnite number of nonlinear terms and even a continuum of nonlinearities determined by the function ξ .

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We organize the next of this paper as follows. In Section 2, we establish some new **interval** **oscillation** criteria for Eq. (1) under the condition for f (x) without monotonicity, while **oscillation** criteria are established for Eq. (1) under the condition for f (x) with monotonicity in Section 3. We present some examples for the results established in Section 4. Some conclusions are presented at the end of this paper.

In this paper, we continue the discussion of the **interval** **oscillation** of (1). Unlike the methods of [22, 25], we introduce a binary auxiliary function, divide each given **interval** into two parts and then estimate the quotients of x(t –σ (t)) and x(t). Due to the considered delay being variable, the results obtained here are the development of some well-known ones, such as in [1] and [2]. Moreover, we also give an example to illustrate the eﬀectiveness and non-emptiness of our results.

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Abstract—We present new **interval** **oscillation** cri- teria for certain nonlinear delay second order diﬀer- ential equation that are diﬀerent from some known ones. Our results extend and improve some previ- ous **oscillation** criteria and handle the cases which are not covered by known results. In particular, several examples that dwell upon the sharp condition of our results are also included.

In this paper, we are concerned with the oscillatory behavior of a class of fractional diﬀerential equations with functional terms. The fractional derivative is deﬁned in the sense of the modiﬁed Riemann-Liouville derivative. By using a variable transformation, a generalized Riccati transformation, Philos type kernels, and the averaging technique, we establish new **interval** **oscillation** criteria. Illustrative examples are also given.

economics, biotechnology, etc. For the theory and applications of fractional differential equations, we refer the monographs and journals in the literature [1]-[10]. The study of **oscillation** and other asymptotic properties of solutions of fractional order differential equations has attracted a good bit of attention in the past few years [11]-[13]. In the last few years, the fundamental theory of fractional partial differential equations with deviating arguments has undergone intensive development [14]-[22]. The qualitative theory of this class of equations is still in an initial stage of development.

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During the last decades, several **oscillation** results were established for different kinds of impulsive delay differential equations (see Agarwal and Karakoc [2]). Recently, **interval** **oscillation** of impulsive delay differential equations was attracting the interest of many researchers, see Guo et. al[5, 6] and Li and Cheung [11]. However, only very few **interval** **oscillation** results are available in the literature for ” second order impulsive differential equations with delay ”. For example, Huang and Feng [8] considered the second order delay differential equations with impulses

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It is clearly possible to assume, without loss of generality, that an **interval** graph or an **interval** order has an **interval** model in which all **interval** extreme points are integer num- bers. In [18] (cf. [31]), Greenough discusses the problem of computing the minimum “width” of such models, that is, the problem of determining the minimum positive integer r for which there exists an **interval** model of an order using only integer extreme points from the **interval** [ 0, r ] . Related to this problem, we considered the **interval** count problem assuming that all **interval** extreme points are distinct and in- teger. Particularly, our question was to determine how the **interval** count of a graph (order) is affected when the in- terval models are assumed to have distinct integer extreme points. We showed in [9] that the **interval** count of a graph (order) is invariant under such an assumption. Motivated by both, this result and Greenough’s discussion, we suggest the problem of determining the minimum positive integer r for which there exists an **interval** model of an order realizing its **interval** count using only distinct integer extreme points from [ 0, r ] .

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Here we report an optical parametric **oscillation** of a mi- crocavity ~termed a m OPO) which crucially depends on this exciton-photon physics. Successful shrinking of the shortest previous OPO device by a factor of 10 000 results from the formation of a trap for polaritons in the microcavity which efficiently channels energy from the pump. Resonant wave interactions are simultaneously possible for the pump, signal, and idler photons due to the Coulomb interaction between electrons and holes in the semiconductor layers. By looking at light emitted in different directions we directly prove the scattering processes postulated. The threshold power density is to our knowledge lower than any VCSEL emitter. The quantum properties of the microcavity inhibit spontaneous emission into nonoscillating modes, enhance the stimulated gain and produce ultralow threshold operation. The physics in this class of solid-state coherent oscillators is akin to co- operative phenomena such as Bose-Einstein condensation, ferromagnetism, and superfluidity. 11

Usami [4], Xu and Xing [7] have established some **oscillation** criteria for (1.4). In this paper, we will continue in this direction and study the oscillatory properties of the gen- eral equation (1.1). By using the generalized Riccati inequality established in Section 2 (Lemma 2.1), we try to extend the results in [2, 5] to (1.1), which include and improve the results of Usami [4]. We are especially interested in the case where p(x) has a variable sign on Ω (1).

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In this paper, we have introduced new comparison theorems for investigation of the os- cillation of equation (.). The established comparison principles reduce the study of the **oscillation** of the second order neutral diﬀerence equations to a study of the **oscillation** properties of various types of ﬁrst order diﬀerence inequalities, which clearly simpliﬁes the investigation of the **oscillation** of equation (.). Further, the method used here per- mits us to relax the restrictions usually imposed on the coeﬃcients of equation (.). So the results obtained here are of high generality and easily may be applicable, as illustrated with suitable examples.

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[13] Y. V. Rogovchenko, “**Oscillation** Criteria for Second Order Nonlinear Perturbed Differential Equations,” Jour- nal of Mathematical Analysis and Applications, Vol. 215, No. 2, 1997, pp. 334-357. doi:10.1006/jmaa.1997.5595 [14] Q. R. Wang, “**Oscillation** and Asymptotics for Second-

ciently, the motion arm appears in the circuit, as in Model-2. Computer simulation was carried out using LT- spice for Windows (Linear Technology Corporation, 1630 McCarthy Blvd., Milpitas, CA, USA) [9]. Figure 10 shows the transient excitation of the crystal current with matched frequency setting: the **oscillation** frequency of the CR oscillator is slightly higher than the resonance frequency. The crystal current in the motion arm grows up faster and the **oscillation** frequency of the entire oscillator circuit is locked to the resonance frequency.

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In order to solve the problem of **interval** linear programming (1), we noticed that the **interval** coefficient of the objective function, the constraints and the right hand of the constraints the uncertain return, the uncertain cost and the uncertain total resource respectively. Thus the **interval** linear programming problem is a problem where the objective function is to max- imize the uncertain return under some uncertain constraints, where the uncertainty is described by intervals. The constraints , denote that the feasible solution to the problem is a solution such that the average costs and the costs in the worst case scenario are less than or equal to the average value and the maximal possible value of the uncertain resources. It is clear that the midpoint of an **interval** is the expected value of that special fuzzy variable. The **interval** programming problem can be viewed as a combination of the fuzzy expected value model and the pessimistic decision model. Therefore, for each **interval** in the objective function c i , a single representative ˆ c i

circuit allows intermittent excitation of the sensor system meeting the request for the power management in the environmental sensing. In this work, we aimed several engineering issues: 1) Quick start of low frequency quartz crystal oscillator circuit in the intermittent operation; 2) Reduction of drift by the direct digital measure- ment; 3) DC isolation between the sensor and the electronic circuit. Tuning fork type Q-MEMS quartz crystal sensor realizes the direct digital sensing of temperature, where Q-MEMS is a combination word of Quartz and MEMS (Micro Electro Mechanical System), a fabrication process offering high performance in a compact package, with CI values as low as those on ordinary-sized crystals. In recent works, double-resonance quartz crystal oscillator was reported for the enhancement of the frequency pulling [8], the mode separation of the mul- timode quartz crystal resonator [9] [10], and the start-up acceleration of low frequency quartz crystal oscillator circuit [11]. Start-up acceleration of several Mega Hertz is studied by the gain control in the quartz crystal oscil- lator using a cascade circuit [12]-[14]. Few works treats the acceleration of the start-up of the low-frequency quartz crystal oscillator. We aim to test the conformity of the acceleration scheme with the Q-MEMS crystal sensor. Stability of the **oscillation** frequency is discussed based on the moving average of the variance deter- mined for the discrete samples following the proto call of the modified Allan standard deviation for moving av- erage of finite length data is employed as the measuring rule of the short range stability [15]-[17].

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The 20th century SST data reconstruction suggests that PDO underwent two distinct warm phases, one between 1925 and 1946 and another between 1977 and 1998, and at least one cold phase between 1947 and 1976 (e.g., [2]). Considering that the atmosphere is heated by its lower boundary, that is, the Earth’s surface that the Pacific Ocean occupies 35% of this surface and, in addition, that each PDO phase lasts a long time period, this **oscillation** must impose a long lasting sign on the global climate that should be detected in climate diagnostics studies. The hypothesis formulated here is that the Pacific Ocean, and its PDO, be one the main internal global climate system controllers in the decadal time scale and that this oscilla- tion may explain an expressive part of the Sahel climate variability and may be used for decadal climate forecast- ing of this region.

10. To make the water slosh, you must shake the water (and the pan) at the natural frequency for water waves in the pan. The water then is in resonance, or in a standing wave pattern, and the amplitude of **oscillation** gets large. That natural frequency is determined in part by the size of the pan—smaller pans will slosh at higher frequencies, corresponding to shorter wavelengths for the standing waves. The period of the shaking must be the same as the time it takes a water wave to make a “round trip” in the pan.

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Analysis of variability of intensity of the storms near the Danube delta, obtained from model wave data based on wind field reanalysis, showed that there is the pronounced intra-annual seasonal variability, as well as variability in the 4–10 year periods associated with fluctuations in the North Atlantic **oscillation** [9]. The authors associated the storm intensity reduction in the 2000 s with the low- frequency variability of the East Atlantic – Western Russia (EA/WR) oscillations [9]. Long-term fluctuations (of the order of 40–50 years) of maximum annual wave heights were identified in [4–10] according to the data of long-term observations of Ukraine hydrometeorological stations in the Black Sea. It was shown that they can be associated with changes in the Atlantic Multi-decadal and Pacific Decadal **Oscillation**.

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Rebuilding of the film structure begins at the following stages as a result of changes occurring in the macromolecular conformation. Some macromolecules that form more stable immobilisation bonds with the solid surface can build up some amount of new bonds, breaking in this way the bonds of molecules with weaker immobilisation and ex- truding them from the surface. The adsorbed molecules become unwrapped on the surface and occupy more surface area. Hence, the conformation changes of the macromolecules and their lowering amount on the surface stipulate rising polymer surface packing den- sity. It is seen in Fig. 1 and Fig. 2 as decreasing film thickness and increasing layer re- fractive index at the next stage of the adsorption (the time **interval** of 5–20 min in case of the 1.0% and 2.5% solutions).

Figure 5: Densities at the harmonic **oscillation** Because of no monotony of function x = x(RΦ) we can divide the **interval** Δ(RΦ) on parts with a length equal to half-period, in which it is monotonous, or to consider the law on distance covered S = S(RΦ), which is monotonic function throughout the whole **interval** t € [0, ∞) and whose graph is shown in Fig. 5 c.