duties of implementing key management scheme are entitled to SAMs, very limited number of them are deputed to generate the key shares. The coefficients of the **polynomial** function are determined beforehand and known only to SAMs. Intruders cannot be able to determine these coefficients so the risk of breaching is least. All SAMs calculate its key share from the agreed **polynomial** using its unique ID. If a new node enters in the network, it accesses any random SAM for its key share. It is the responsibility of each SAM to decide the authenticity of the new node that whether it can be trusted or not. If the new node is trustworthy, a key share is allotted to it by the SAM. At least threshold „k‟ number of key shares must be collected from distributed SAMs by the new node. If the new node is efficacious in collecting the desired number, then it is able to generate the secret code using any **polynomial** **interpolation** method. The larger the intended security level, higher the value of „k‟.

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In view of the optimal **interpolation** points for the univariate **polynomial** **interpolation**, to our knowledge, both sets T ˘ and T ˆ , with the position of the points given in explicit form, are the best nodal sets in the literature. Based on the Bernstein-Erdös conjecture, the nodal set T ˆ is superior than the nodal set T ˘ because of its smaller maximum deviation. When considering the optimality of a nodal set from its Lebesgue constant, the set T ˘ is better.

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In Figure 3.10 we show the **interpolation** errors and the Lebesgue constants correspond- ing to different **interpolation** spaces and nodes, and in Figure 3.11 we plot the approximate Fekete points corresponding to degree n = 10. As a starting mesh to extract approximate Fekete points we have taken a 120 × 120 uniform grid, which is an admissible mesh for (non- weighted) **polynomial** **interpolation** up to n = 10 (being the product of two one-dimensional admissible meshes; cf. [15]). The comparison is with nonweighted **interpolation** at the so- called “Padua points”, the first known example of nearly optimal points for total degree poly- nomial **interpolation** in two variables, with a Lebesgue constant increasing like log squared of the degree; cf. [7, 8, 14].

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Abstract. Motivated by a recently introduced HIMMO key predistribution scheme, we investigate the limits of various attacks on the **polynomial** **interpolation** problem with mixed modular operations and hidden moduli. We firstly review the classical attack and consider it in a quantum-setting. Then, we introduce new techniques for finding out the secret moduli and consider quantum speed-ups.

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In this paper, a smooth walled proﬁled conical horn has been considered and achieved improved performance by applying piecewise biarc hermite **polynomial** **interpolation** and a concept of ‘targeting the Region of Interest (RoI)’. The simulated results of the proposed horn are compared with that of a conventional spline proﬁled horn antenna discussed in [15]. This particular study was carried out considering an application of a horn antenna used in a reﬂectometry system for plasma diagnostics [16]. The expected horn speciﬁcations are: S 11 better than − 15 dB, peak gain of the order of 25 dB, side-lobe

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It is known that for a function f(x, y) defined over the standard triangle with vertices at ( 0 , 0 ), (n, 0 ) and ( 0 , n), there exists a unique **polynomial** Pn(x, y) of degree at most n in x and y which interpolates f(x, y) at the (n + l)(n + 2)/2 points o f the mesh (i, j), i, j > 0, i + j < n. In [12], S. L. Lee and G. M. Phillips derived a forward difference fo r the polynom ial P^(x, y) and represent it in the x and y directions. In a subsequent paper [13] the authors extend the results on **polynomial** **interpolation** at the points o f geometric progression to the two-dimensional case. They considered the interpolating **polynomial** P^(x, y) for f(x, y) on the "triangular" mesh points {([i], [j]): 0 < i < j < n} and gave a forward difference formula in the y and "diagonal" directions.

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, where N is a number of grid nodes. Accordingly, for a C ∞ -function the trapezoidal quadrature converges with the rate faster than any **polynomial**. In this paper, we prove that the same property holds for all quadrature formulae obtained by integrating fixed degree piecewise **polynomial** interpolations of a smooth integrand, such as the midpoint rule, Simpson’s rule, etc.

d) Kim et al. scheme uses the **polynomial** **interpolation** approach. In the case that 𝑛 is small (i.e., in realistic scenarios for example five attending nodes) then the Lagrange **polynomial** would have a very small 𝑛 − 1 𝑑𝑒𝑔𝑟𝑒𝑒 and could be analyzed by attacker. In addition, with a poorly designed Elliptic Curve Cryptosystem (ECC) it could be easily solved by an attacker. Clearly, this is another trade-off: the larger the **polynomial** degree, the more the necessary calculations that have to be performed by legitimate users, however the scheme becomes more secure.

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In this study, based on the optimal free derivative without memory methods proposed by Cordero et al. [A. Cordero, J.L. Hueso, E. Martinez, J.R. Torregrosa, Generating optimal derivative free iterative methods for nonlinear equations by using **polynomial** **interpolation**, Mathematical and Computer Modeling. 57 (2013) 1950-1956], we develop two new iterative with memory methods for solving a nonlinear equation. The first has two steps with three self-accelerating parameters, and the second has three steps with four self-accelerating parameters. These parameters are calculated using information from the current and previous iteration so that the presented methods may be regarded as the with memory methods. The self-accelerating parameters are computed applying Newton’s interpolatory polynomials. Moreover, they use three and four functional evaluations per iteration and corresponding R-orders of convergence are increased from 4 ad 8 to 7.53 and 15.51, respectively. It means that, without any new function calculations, we can improve convergence order by 93% and 96%. We provide rigorous theories along with some numerical test problems to confirm theoretical results and high computational efficiency.

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Abstract: The paper deals with the experimental and numerical modeling of geometrical errors in case of a 5R serial robot (Fanuc LR Mate 100iB), implemented in a working process which consists of manipulating different parts from the workspace, for the purpose of processing them. For each point from the working space covered by the end effector and included in the analysis, have been collected from the teach – pendant of the robot a series of data regarding the coordinates of the tool center point, the orientation angles and the value of the generalized coordinates for all driving joints. The trajectory errors that affect the optimal operation of the robot have been determined based on **polynomial** **interpolation** functions of 4 – 3 – 4 type.

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Figure 5 show results for **polynomial** **interpolation** of degrees 20, 30, and 40. Figure 6 shows the origi- nal face model on the left together with a compres- sion and reconstruction with a **polynomial** of degree 80. It is noted that the model becomes unstable for **interpolation** of very high degrees. For a **polynomial** **interpolation** of degree 20, the file size in OBJ format has been reduced from 4MB to 26KB. This is a reduc- tion of 99.35% and similar reductions were achieved for other polynomials also; a summary is presented in Table 1.

Our problem formulation, smoothing over incorrect values during the process of sparse recon- struction, applies to all such inverse problems, e.g., supersparse **polynomial** **interpolation** [Garg and Schost 2009] and [Kaltofen 2010, Section 2.1], computing the sparsest shift [Grigoriev and Karpinski 1993; Giesbrecht, Kaltofen, and Lee 2003] and the supersparsest shift [Giesbrecht and Roche 2010], or the more difficult exact and numeric sparse and supersparse rational function recovery [Kaltofen, Yang, and Zhi 2007; Kaltofen and Nehring 2011]. Our methods immediately apply to algorithms that are based on computing a linear recurrence, such as the supersparse **interpolation** algorithms in [Kaltofen 2010] and [Garg and Schost 2009]. The former needs no modification, and for the latter, one uses the majority rule algorithm for the sparse recovery with errors of the modular images f (x) mod (x p − 1), where p is chosen sufficiently large (and

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Shape preserving interpolations are important in Computer Graphics (CG) and Scientific Visualization. Monotonicity preserving exists in various sciences and engineering applications. For examples the devices in the specification of Digital to Analog Converters (DACs), Analog to Digital Converters (ADCc) and sensors are always monotonic and any non-monotonicity existing in the **polynomial** **interpolation** are unacceptable (Hussain and Hussain, 2007). Approximation of couples and quasi couples in statistics and the dose-response curves and surfaces in biochemistry and pharmacology are other examples in which the monotonicity exists in the data sets (Beliakov, 2005). Shape preserving for monotone data have been discussed in details by previous researchers in this rapid growing fields. For examples, Fristch and Butland (1984), and Fristch and Carlson (1980) have proposed the monotonicity preserving using cubic spline. In their construction, the existence of the monotonicity interpolant depends to the monotonicty region which is combined by rectangle and ellipses. Passow and Roulier (1977) also consider the monotone and convex shape preserving using **polynomial** quadratic spline. Schumaker (1983) and Lahtinen (1996) discussed the shape preserving **interpolation** using quadratic spline **polynomial**. They preserved the data (monotone and convex data) by inserting extra knots in which shape violation are found. Butt and Brodlie (1993) study the positivity preserving using cubic spline **polynomial** by inserting extra knots (one or two). Having an extra knot, the computation to generate the cubic interpolating curves will be increased. Furthermore it is not an easy task to teach the user about how to insert the knots.

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Second, a new method of **polynomial** **interpolation** was demonstrated for data compression. While **polynomial** **interpolation** is a well-known technique, specific problems were solved concerning missing data and required information to allow full reconstruction after compression. Since the cutting planes define a regular grid and not all vertices defined over this grid contain data, vertices need to be marked as valid or invalid. A specific representation was designed with informa tion on the step between planes, the range of valid points, and the **polynomial** coefficients. This information is saved in plain ASCII allowing high rates of com pression together with robust reconstruction of the data. Efficient compression rates of over 99% were achieved compared to the standard OBJ file format. The major issues with the method are concerned with the stability of the solution. While high degree polynomials (around degree 30 of the tested data) can recon struct the data to acceptable accuracies with lowest RMSE they also introduce artefacts into the solution for higher degrees. Therefore, there seem to be intrinsic limitations to using high degree polynomials to approximate complex real world surface patches and new approaches are needed.

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Corrosion in general is a complex interaction between water and metal surfaces and materials in which the water is stored or transported. Water quality monitoring in terms of corrosion and scaling is crucial, and a key element of preventive maintenance, given the economic and health hurts caused by corrosion and scaling in water utilities. The aim of this study is to determine the best model for zoning and **interpolation** corrosive potential of water distribution networks. For this purpose, 61 points of Sanandaj, Iran, distribution network were sampled and using Langelier indices, we investigated corrosivity potential of drinking water. Then, we used geostatistical methods such as ordinary kriging (OK), global **polynomial** **interpolation**, local **polynomial** **interpolation**, radius-based function, and inverse distance weighted for **interpolation**, zoning and quality mapping. Variogram analysis of variables was performed to select appropriate models. The results of the calculation of the Langelier index represented scaling potential of drinking water. Suitable model for fitness on exponential variogram was selected based on less (residual sums of squares) and high (R 2 ) value. Moreover, the best method for **interpolation** was selected using the

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mathematical and numerical problems is growing rapidly from the past few decades. In this study, we designed a methodology that solves many mathematical and numerical problems with high computational speed. In many real time or real life applications very often we need to perform a lot of numerical computations, which may range from matrix multiplication, **polynomial** **interpolation**, solving **polynomial** equations, solving system of equations and so on. We consider the use of parallel processors to solve some numerical problems, namely Lagrange **interpolation** and **polynomial** root finding. The experiments were conducted on the financial dataset obtained from publicly available AWS database. We tested proposed algorithms on two different models of parallel processors and achieve the performance of O (log n).

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Prior to start sharing secret key in the single level distribution the 100% of secret key will be divided into “n” pieces according to the number of available Key Recovery Agent and desired number of minimum shared secret to reconstruct for 100% of the secret key as k number. Become n,k threshold. At this state 100% of secret key will be divided to n pieces by Shamir’s threshold Secret sharing scheme. To calculates for the share secret key from the curve **polynomial** degree k-1, share of the secret key can defined as follow

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of training points or simulation length will only reduce this statistical uncertainty, but will not contribute significantly to changes in the model predictions as the flexibility of the model is limited by the maximum **polynomial** order. There- fore, the model performance achieved under these conditions can be considered near to the best possible for the given PCE order and number of dimensions. However, it should be noted that the number of training points required for such conver- gence will differ according to the order and dimension of the PCE, and higher order and more dimensions will require more than the approximately 3000 points that seem sufficient for a PCE of order 6 with six dimensions, as shown in Fig. 4. The IS procedure has relatively slow convergence com- pared to, for example, a quasi-MC simulation. Figure 5 shows an example of the convergence of an IS integration for reference site 0, based on computing the target (site-specific) distribution weights for all 10 4 points in a reference high- fidelity database. The CIs are obtained by bootstrapping.

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Abstract. In the present paper a characterization of the generalized order and type of entire functions of several complex variables by means of polyno- mial approximation and **interpolation** have been obtained. Our results improve and generalize various results of S.M. Shah [5], M.N. Seremeta [4], Kapoor and Nautiyal [2], Vakarchuk [9], Vakarchuk and Zhir [10], Winiarski ([11],[12]). In this way we summarize and unify the work which has been done on this subject to-date.

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to analyze the method, power series, basis function and quadratic **interpolation** using basis function and cubic **interpolation** are chosen. Also to check the performance of the considered method an error associated with Lagrange **interpolation** has considered. Errors are analyzed by comparing the actual sampled values with the values obtained by Lagrange’s **interpolation** formula.