In this paper, we propose the first order and second order piecewise polynomial approximation schemes for the computation of fixed points of Frobenius-Perron operators, based on the Gale[r]

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This work is organized as follows. In Section , we consider geometrical properties of the shift operator in general case and, we deal with the properties of shift operator the spaces of almost periodic and on ergodic sequences. In Section 3, we a consider the existence and uniqueness solutions of some difference equations using **polynomial** functions. In the last section, we deal with the application of the previous results to some **second** **order** differential equation with a **piecewise** constant argument.

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Finally the following conclusions can be made. In **order** to manipulate correctly operators and images it is sufficient to construct the wavelet bases with **piecewise** **polynomial** functions made of polynomials of very low degree. Really for the images seems to be adequate choo- sing **piecewise** **polynomial** functions made of polynomi- als of degree zero. As already observed this might be due to the way we calculate the wavelet coefficients and to the fact that the operators and the images are represented by **piecewise** constant functions. Moreover it seems to be very promising the idea of increasing the number of va- nishing moments keeping low the degree of the polyno- mials used. Actually the bases that have a large number of extra vanishing moments, that is those constructed with the **second** criterion proposed in Section 2, show better compression and reconstruction properties, and in general work better than the wavelet bases constructed with the first criterion proposed in Section 2.

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In this paper a new approach for solving sec- ond **order** fuzzy differential equations under gen- eralized differentiability was proposed. We used **piecewise** fuzzy **polynomial** of degree 4 based on the Taylor expansion for approximating solutions of **second** **order** fuzzy differential equations. Also, we can develop this method for N th-**order** fuzzy differential equations under generalized deriva- tives.

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Abstract. It is well-known that the trapezoidal rule, while being only **second**-**order** accurate in general, improves to spectral accuracy if applied to the integration of a smooth periodic function over an entire period on a uniform grid. More precisely, for the function that has a square integrable derivative of **order** r the convergence rate is o N −(r−1/2)

I n this study, thermoplastic polyolefin elastomeric (TPO) nanocomposites were fabricated by friction stir processing. The effect of different pin geometries on clay dispersion and mechanical properties of the TPO nanocomposite reinforced with 3% wt nanoclay has been first investigated. The optimum pin geometry namely threaded cylindrical pin was then used to fabricate the nanocomposites containing 3, 5 and 7 wt% nanoclay. The results showed that increase in the clay content increased the tensile strength and tensile modulus of the nanocomposite from 15.8 to 22.76 MPa and 568 to 751 MPa, respectively. The experimental stress – strain curves of nanocomposites were compared with eight constitutive models including Mooney – Rivlin, the **second**-**order** **polynomial**, Neo – Hookean, Yeoh, Arruda – Boyce, Van der Waals and the third- and sixth-**order** Ogden. The comparisons showed that there was an agreement between the experimental data and the sixth-**order** Ogden model. Three micromechanical models Halpin – Tsai, inverse rule of mixture and linear rule of mixture were applied to investigate the Young’s modulus of nanocomposites. Because of the significant difference between the Young’s modulus obtained from these models and the ones obtained from experimental data, a modifying factor was used to improve the theoretical predictions obtained from the models. Polyolefins J (2017) 4: 99-109

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Equation (21), a closed-form analytical solution to Richards’ equation, is proposed as an alternative to the **second** **order** **polynomial** that is currently employed in the Airborne Microwave Observatory of Subcanopy and Subsurface (AirMOSS) root zone soil moisture retrieval algorithm. It has been demonstrated that the **second** **order** **polynomial** is a special case of Eq. (21) limited to P = 1. Evaluation of Eq. (21) based on both numerical simulations and measured data revealed that it exhibits greater flexibility than the currently applied **second** **order** **polynomial**. Therefore, application of Eq. (21) is recommended for more accurate retrieval of root zone moisture profiles from P-band radar remote sensing data. The results presented for two AirMOSS flights in 2015 demonstrate a reduction of the retrieval error with the new method. In conclusion it should be noted that while the retrieval error and computational inversion time have improved for the two presented cases, a more extensive study is required to investigate the applicability, performance, and advantage of this method for AirMOSS root-zone soil moisture retrievals.

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There are substantial interests on the numerical treatment of the problem (1.1). Noor and Khalifa [19] have used a collocation method with cubic B-splines as basis functions to solve (1.1), while the well-known Numerov method and ﬁnite diﬀerence schemes based on the central diﬀerence have been employed in [22]. Thereafter, Al-Said et al. [5] show that cubic spline method gives numerical solutions that are more accurate than that com- puted by quintic spline and ﬁnite diﬀerence techniques. The numerical results of [5, 19, 22] indicate ﬁrst-**order** accuracy for these methods. In [4], a two-stage method is devel- oped where a ﬁnite diﬀerence scheme is ﬁrst employed to obtain the numerical solutions

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In general, IIR filter design methods can be classified into two groups: direct and indirect ways. It should be mentioned here that direct design methods are often referred to as those methods that are carried out directly in the domain and indirect design methods are generally considered to be those methods based on analog filters [2]. In this dissertation, however, we adopt somewhat different definitions for direct and indirect design methods. In the direct design strategy, the best approximation to a given ideal frequency response is found without any intermediate step. In the indirect design strategy, a design problem is first transformed to an FIR filter design problem. Then, model reduction techniques can be deployed to achieve an IIR digital filter, which can best approximate the FIR digital filter. As presented before, in general, FIR filter design problems can be equivalently cast as convex optimization problems and then efficiently solved. Therefore, the performances of indirect design methods are mainly determined by the **second** step, i.e., FIR approximation by IIR digital filters. In this dissertation, we mainly study IIR filter designs using the direct design strategy. But it should be mentioned that the proposed design methods can be straightforwardly applied in indirect IIR filter designs by replacing the desired frequency response by a well-defined FIR frequency response and the frequency bands of interest Ω by the whole frequency band [0, ].

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Exploit piecewise polynomial approximation for structured model estimation.. Algorithmic Framework for Distribution Estimation: Leads to fast & robust estimators?[r]

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For all combinations of the operational param- eters studied, the final moisture content was within the acceptable range of dehydrated onion (< 6–7%) reported in the literature (Sarsavadia 1999) and was observed within the range of 2.03–3.73% d.b. The colour values in terms of the optical in- dex (OI < 90) and rehydration ratio (RR > 2.5) were well within the permissible levels as recom- mended by Adoga (2005) and Ranganna (2005), respectively. Flavour content i.e. TC was within the range reported in the literature (3.61 μmol/g to 4.97 μmol/g dried sample based on 20.2% to 10.4% grade of comminution Resemann et al. 2004). The generalised **second** **order** **polynomial** equation was fitted to the experimental data to approximate the function Y k as follows:

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in(2012)[5] studied a new approach to find the numerical solution of VIE's by using Bernsteins Approximation. Many researchers have used non- **polynomial** spline functions approach to find the solution of differential equations. Ramadan , M.A. El-Danaf , T. and Abdaal F. E.I. in(2007) [6] Presented an application of the non- **polynomial** spline function to find the solution of the burgers equation. Zarebnia M. Hoshyar , M. and Sedahti, M. in(2011)[7] Presented a numerical solution based on non-**polynomial** cubic spline function is used for finding the solution of boundary value problem.

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Combining the discrete variational principle with the derived numerical ﬂux, we obtain a uniﬁed approach to construct time integrators for (non-)autonomous Hamiltonian systems. We have derived both well-known and novel symplectic time stepping schemes of ﬁrst, **second** and third **order** accuracy. An extensive analysis for the discretizations is provided, including a linear stability analysis and an investigation of the symplectic nature of the schemes. The novel third **order** scheme we have developed is shown to have improved dispersion properties. Furthermore, within this approach we derive and test time stepping schemes for non-autonomous Hamiltonian systems, such as forced and damped oscillators. We study the approximation of forcing and damping terms considering the discrete versions of certain integrals.

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Table 5.2 above shows the results of a grid refinement study with errors in the infinity norm and other information. Again, we can see that the method still has **second** **order** accuracy when we use the embedding technique. Furthermore, in this Table we can see that the number of iterations (only 5) is independent of the mesh size as in the case of two space dimension. The CPU time does not increase much.

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The paper is organized as follows. In Section 2, we construct a complete MAPLE procedure by blocks of commands in the **order** that we can modify them easily. These blocks whose their own purpose will be clearly explained may be useful to design other procedures. Section 3, the longest section, is for exploration of the obtained results from the output of the procedure. The last part of Section 3 is destined for the detailed discussion on how to find a desired estimate p for the best L 2 -approximation p

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In switched GRNs, as mentioned in the preliminaries, the partitions of the state and transitions between modes are not characterized as a priori. In such case, the systems is under arbitrary switching, and a sufficient condition for the stability of the switched GRNs exists by introducing the common **polynomial** Lyapunov functions.

All authors contributed in process of manuscript writing. We confirm that the manuscript has been submitted solely to this journal and is not published, in press, or submitted elsewhere. All authors agree to the terms and conditions. We confirm that all the research meets the ethical guide- lines, including adherence to the legal requirements of the study country. VD(Vaneet Dhir) designed, performed the experimental work and wrote the manuscript. RPSG(RP Singh Grewal) contributed for investigating comparative trend in terms of different mathematical operations and interpretations in terms of **polynomial** approach. All au- thors read and approved the final manuscript.

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The method described in this article belongs to the “explicit” type of models. We work with noisy one- dimensional signals, and our underlying model assumes that individual segments can be well approximated by polynomials. The number of segments is supposed to be much lower than the number of signal samples—this natural assumption at the same time justifies the use of sparsity measures involved in segment identification. The model and algorithm presented for 1D in this article can be easily generalized to a higher dimension. For exam- ple, images are commonly modeled as **piecewise** smooth 2D-functions [27–31].

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Since the development of the computer, numerical simula- tion has become an integral part of many scientific and tech- nical enterprises. As computational hardware becomes ever cheaper, numerical simulation becomes more attractive rel- ative to experimentation. Much attention is paid to the de- velopment of ever more efficient and powerful algorithms to solve previously intractable problems (Trefethen, 2008). However, in **order** to be useful, the user of a computational model must have confidence that the results of the numeri- cal simulation are an accurate proxy for reality. A rigorous software quality assurance system is usually a requirement for any deployment to industry, and should be a requirement for any scientific use of a model. Catastrophic accidents such as the Sleipner platform accident, in which an offshore plat- form collapsed due to failures in finite element modelling,

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the value 2 m− 1, we always have p ≤ m . Moreover, since the local **order** is required to be greater than or equal to the global **order** (m +d + 1 ) , it is necessary that d+ 2 ≤ p . The following corollary deals with some important special cases and its proof relies on the above theorem.

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