Research Article Generalizations of Shafer-Fink-Type Inequalities for the Arc Sine Function Wenhai Pan and Ling Zhu Department of Mathematics, Zhejiang Gongshang University, Hangzhou, Zh[r]

for x ∈ (, arcsinh r). These can be regarded as Oppenheim-type inequalities for the hyper- bolic **sine** and cosine functions. For information on Oppenheim’s double inequality for the **sine** and cosine functions, please refer to [], [, Sections . and .] and closely related references therein.

Before ending, we remark that once it is established that a function f of the class S must satisfy a differential equation of the form 4, there are alternative elementary proofs at hand.[r]

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3 Conclusion In this paper, we proposed and proved new inequalities, which present reﬁnements and generalizations of inequalities stated in [], related to Shafer-Fink’s inequality for t[r]

Here we have seen some basic definitions and Riemann Liouville H- differentiability in section 2. In section 3, fuzzy Laplace transforms are introduced and we discuss the properties .The solutions of FFDEs are determined by Fuzzy Laplace transform under Riemann Liouville H- differentiability and solve the example involving **sine** terms in section 4. In section 5, a conclusion is drawn.

Within common interval, the error of approximation gets smaller and smaller from the first up to the fourth approximation method. This implies that the assumed generalized approximate equation is very interesting and supposed to be holds true. And therefore, **Sine** **function** approximately satisfies above condition within the interval; provided that the negative angles are taken into consideration.

The simulated data are shown in Fig. 3 together with the un- derlying **sine** **function**. The data have a mean of 0.5484, substan- tially di ff erent from the mean of the underlying **sine** **function**, which is zero. The resulting periodograms are shown in Fig. 4. It is immediately clear that only the GLS and BGLS are able to correctly identify the true period. It is also interesting to note that the GLS shows two peaks of about the same height at 50 and 25 days. By using the GLS alone, it would not be possible to know unambiguously which period is the real one in the data. In the BGLS, on the other hand, it is very clear that the longer period (which is the true one) is about 10 10 times more probable than the other period.

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The purpose of our simulation is to measure to what extent the estimates by our new method differ from the true values i.e. β 0 = 2 and β 1 = β 2 = ... = β p = 1 and from the estimates provided by Andrew’s **sine** **function** and Tukey’s biweight **function** the two well known redescending M-estimators. In our simulation we performed many replications keeping in view the number of predictor variables and the sample size n. Some results are presented in Table 2, 3, and 4. From these tables it is clear that the results of the new redesceding M-estimator are very similar to that of OLS without outliers, Andrew’s **sine** **function** and Tukey’s biweight **function** and is not seriously effected by outliers in both x and y directions.

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The earliest values for the **sine** **function** were calculated by Indian mathemati- cians in the 5th century. The cosine and tangent, as well as the cotangent, secant and cosecant were developed by Islamic mathematicians by the 11th century. Eu- ropean navigators used these ideas extensively to help calculate distances and direction during the Middle Ages. Modern European trigonometry as we un- derstand it was then developed throughout the Renaissance (1450-1650) and En- lightenment (1650-1800).

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integral equations involving first kind Bessel functions of different orders and a particular weight function to dual integral equations with trigonometric sine function as kernels in cl[r]

For a given number α ∈ [ a, b ] , we can choose the index i such that the interval [ a[i], b[i] ] contains α, hence we put g : = unapply(F[i],x) . Note that F[i] is still an expression in x as default, not a **function**; besides, we cannot set g : = x->F[i] , because x here and x in F[i] are not the same by MAPLE’s rule. Then, sin α ≈ g ( α ) with the accuracy of 1/10 r . In MAPLE, we can use the command piecewise for a sequence of conditional settings to get the piecewise polynomial approximation to the **sine** **function** on [ a, b ] by putting

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The variable metric method was applied to estimate the unknown space-time dependent heat flux imposed to the outer wall of a tube with forced convection inside it. The simulated temperature measurements at certain points within the flow field were input to the analysis. The accuracy of the method in estimating an unknown step heat flux and a space-time dependent **sine** **function** was examined and the results obtained by four different versions of the presented method were compared with each other. Furthermore, the effect of the sensor installation location on the accuracy of the solution was evaluated. Numerical results

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In this paper, we obtain some new inequalities which reveal the further relationship between the inverse tangent **function** arctan x and the inverse hyperbolic **sine** **function** sinh –1 x. At the same time, we give the analogue for inverse hyperbolic tangent and inverse **sine**.

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f x x floor x (5) It’s like **sine** **function** as it’s a continuous **function** but with discontinuous first order gradient. There is actually a trade-off between using **sine** **function** and tooth like **function**. As **sine** **function** has continuous first order gradient, we expect network using **sine** **function** will get close to optimal point if fixed training rate is used. But tooth like **function** will have difficulty in getting close to optimal points as it will overshoot to the other side of the valley. This may suggest it’s better to use learning rate decay when training with tooth like **function**. This situation is like training with sigmoid **function** and linear rectify **function** where learning rate decay is a usual practice with linear rectify **function**. On the other side, using tooth like **function** instead of **sine** **function** can give a boost in training speed as tooth like **function** has constant gradient while a gradient for a **sine** **function** requires a quite more computation resources considering the large training size of current convolutional neural network. This is also an existing practice like we usually don’t train network with sigmoid **function** directly but using linear segment **function** to approximate it.

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capacitor voltage and the voltage error is processed through proportional and integral controller (PI). To follow the sliding mode control, the unit **sine** **function** is derived from the voltage source and multiplied with the output of PI control to obtain the reference alternating source current. This current is then compared with actual source current and error is processed through hysteresis controller to generate the gating pulses of switches S 1 and S 2 . The source voltage is scaled

HEALTH Suppose a person’s resting blood pressure is 120 over 80. This means that the blood pressure oscillates between a maximum of 120 and a minimum of 80. If this person’s resting heart rate is 60 beats per minute, write a **sine** **function** that represents the blood pressure for t seconds. Then graph the **function**. Explore You know that the **function** is periodic and can be modeled using **sine**. Plan Let P represent blood pressure and let t represent time in seconds. Use

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Formation micro imager (FMI) can directly reflect changes of wall stratums and rock structures. Conventionally, FMI images mainly are analyzed with manual processing, which is extremely inefficient and incurs a heavy workload for experts. Iranian reservoirs are mainly carbonate reservoirs, in which the fractures have an important effect on permeability and petroleum production. In this paper, an automatic planar feature recognition system using image processing was proposed. The dip and azimuth of these features are detected using this algorithm to identify more precise permeability and the career of fluid in reservoirs. The proposed algorithm includes three main steps; first, pixels representing fractures are extracted from projected FMI image into location matrices x and y and the corresponding value matrix f(x, y). Then, two vectors X and Y as the inputs of CFTOOL of MATLAB are produced by the combination of these three matrices. Finally, the optimum combination of **sine** **function** is fitted to the **sine** shape of pattern to identify the dip and azimuth of the planar feature. The system was tested with real interpretation FMI rock images. In the experiments, the average recognition error of the proposed system is about 0.9% for the azimuth detection and less than 3.5% for the dip detection and the correlations between the actual dip and azimuth with the determined cases are more than 90% and 97% respectively. Moreover, this automatic system can significantly reduce the complexity and difficulty in the planar feature detection analysis task for the oil and gas exploration.

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means for multiplying said sine function electrical signal by first and second independent parameters to provide first and second time varying quarter. cycle sine functions;[r]

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This paper described here is based on high quality the industry standard integrated circuit XR2206. This XR2206 **Function** Generator is capable of producing high quality **sine** and triangle waveforms of high-stability and accuracy in the range about .01Hz to more than 1MHz [1]. Frequency adjustment range is accomplished using 4-DIP switch. It is IC having 16 Pins.

The **sine**-Gordon field theory and the associated massive Thirring model [1] are some of the best studied quantum field theories. In view of its connections to other important physical models, some of which in principle admit actual realizations in nature [2] [3], a huge mass of important exact results have been obtained for this fascinating integrable system [4]-[7]. However, no less fascinating are the remarkable mathematical and physical properties of its soliton (or “solitary wave”) solutions which have contributed, along the last decades, to turning the physics of solitons into a very active research topic.

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