Inverse Polynomial Research Articles

Evaluation of interpolation methods for the prediction of urban methane concentrations

Having data about atmospheric concentrations in an entire urban area is difficult, hence interpolation methods are helpful. Their choice will depend on minimising the error. In this work, two deterministic (Inverse Distance Weight and Local Polynomial Interpolation) and two stochastic methods (Simple and Ordinary Kriging) were applied to predict seasonal and annual atmospheric methane (CH4) concentration means. Two sampling networks were designed in an intermediate city, covering a wide variety of urban densities, with different sampling site numbers. The main objective was to find the interpolation model that best predicts CH4 concentration and to analyse if the network's expansion improves the metric errors - the mean error (ME) and the root-mean-square error (RMSE). The ME values were close to zero in all cases, and the stochastic methods had the smallest RMSE for both networks. Besides, adding more sampling sites improved up to 50% of the RMSE values. Finally, an integrated map was obtained incorporating all the best interpolation models, which gave a difference of <4% between the measured and the estimated CH4 concentration. This type of study is helpful to evaluate the design of a sampling network, the territorial planning and future installations of CH4 sources.

Estimates of Lebesgue constants for Lagrange interpolation processes by rational functions under mild restrictions to their fixed poles

We estimate the Lebesgue constants for Lagrange interpolation processes on one or several intervals by rational functions with fixed poles. We admit that the poles have finitely many accumulation points on the intervals.To prove it we use an analog of the inverse polynomial image method for rational functions with fixed poles.

Comparison Between Trot and Wave Gait Applied in Quadruped Robot

A Quadruped robot is a type of robot that moves on four legs and has a structure like a four-legged animal. The quality of movement based on mechanics and movement patterns on quadruped robots tends to have poor movement patterns. This study investigates the best movement between trot and wave gait by comparing the speed performance, stopping distance accuracy, and the tilt of the robot body angle. To minimize the influence of the mechanical quality of the robot, this study used two robots based on the type of servo used (based on the SG90 and MG995 servos). In this study, the motion pattern based on Trot and Wave gait is realized using Inverse Kinematics and Polynomial trajectory on each leg. The verification experiment showed that the Wave gait has better in both the robot body angle and distance error. In contrast the Trot gait has better in speed.

An exploration, analysis, and correction of the distance effect on terrestrial hyperspectral LiDAR data

Terrestrial hyperspectral light detection and ranging (LiDAR) (HSL), as one novel LiDAR system integrated with more wavelengths, has supplied enhanced possibilities in many fields by delivering the geographic coordinate and the hyperspectral backscattered intensity simultaneously. However, like the traditional single-wavelength LiDAR, it is faced with the distance effect, which restricts the accurate acquisition of the hyperspectral intensities and the quantitative retrieval of characteristics of the targets. The previously reported distance effect correction models mainly concentrated on the distance effect correction for the single-wavelength LiDAR system, and some normalization methods with the standard reflectance panel were also provided to calibrate the data measured by multispectral LiDAR and HSL, producing the reflectance-related value. Nevertheless, it is valuable to systematically explore the distance effect of HSL over long distances and search for the corresponding correction methods. Along these lines, in this study, the homogeneous targets and standard reflectance panels with different reflectance values were employed to participate in the distance experiment to systematically explore and analyze the distance effect phenomenon on terrestrial HSL data. A novel practical piecewise fitting model was proposed to output realistic correction results and it was also compared with the classic inverse distance square law and polynomial fitting model. To further demonstrate the feasibility and good capability, the models were employed in the intensity correction for the various vegetation leaf species. A simple 3-D scene experiment was then carried out to demonstrate the intensity variation before and after the implementation of the distance effect correction. From the acquired result, it suggests that (1) the distance effect on terrestrial HSL data is not related to the wavelength and the scanned target. All the wavelengths share a common distance effect; (2) the concept of the ‘distance effect function’ could represent well the distance effect for the terrestrial HSL; (3) as far as the correction ability is concerned, the proposed piecewise fitting model achieved better performance than the others because it exhibited the smallest standard deviation (σ), coefficient of variation (CV), and the ratio of the coefficient of variation (ε) for the correction results of the homogeneous targets, and the leaf experiments confirmed it again; and (4) the further 3-D experiment result exhibited the intensity variation. It proves that the distance effect does exist in the 3-D scanning results and the correction is necessary. Besides, for small-scanned scenes, the incidence angle effect dominates and largely affects the intensity data instead of the distance effect. This study enriches the study of the distance effect mechanism on the terrestrial HSL data, which facilitates the development of research on the radiative effect correction of the HSL. At the same time, the proposed model can be adopted for the analysis and correction of the distance effect on terrestrial HSL data in actual scanning tasks.

Computation of Multiplicative Inverses of non-zero Polynomials of GF(73) by Gauss-Jordan method.

The present manuscript deals with the derivation of multiplicative inverses of all non-zero elemental polynomials of the Galois field with respect to the monic irreducible polynomials of degree 3 over the prime field by Gauss-Jordan method as the Extended Euclidean Algorithm (EEA) is not always successful over the prime field , when prime .

Diminution of Extended Euclidean Algorithm for Finding Multiplicative Inverse in Galois Field

This manuscript deals with the theorem on diminution of the Extended Euclidean Algorithm for finding the multiplicative inverse of non-zero elemental polynomials of Galois field with respect to a monic irreducible polynomial over , where is a prime and is any positive integer. This method became successful in finding the multiplicative inverse of all those non-zero polynomials for which the Extended Euclidean Algorithm fails. We find the inverse of all 342 non-zero elemental polynomials of using an irreducible polynomial We also used Cayley Hamilton’s theorem for finding the multiplicative inverse of the non-zero elemental polynomials of a finite field.

Incorporating a New Pseudorandom Number Generator in AES Algorithm to Improve Its Security Level

The implementation of the multiplicative inverses of elemental polynomials under an irreducible polynomial over GF(pn) played an important role in cryptography. In the AES algorithm, the multiplicative inverses under the first irreducible polynomial over GF(28) have been used for the first time in 1999 to design its substitution box. The new PRNG RC4-MI in which the multiplicative inverses under the two irreducible polynomials over GF(73) are being used accepts the AES key as its own key and generates random bytes. Undertaking exclusive OR operation of RC4-MI bytes with AES cipher bytes, one gets a new sequence of cipher bytes thatexhibits randomness quantitatively better than that of the original sequence of AES cipher bytes.

Vector Autoregressive Model of GDP, Money Supply and Exchange Rate in Nigeria

The research work on vector autoregressive model on economic indicators of gross domestic product, money supply and exchange rate in Nigeria was developed and adequately model. The study yields a stable vector autoregressive model with stationary process and the estimate of the model where significant. The empirical result yields a stable and sustainable economic model for the three economic variables in the study. The unit root test was achieved at order 1 and the inverse root of the polynomial lies within the unit circle. The iterative step of time series analysis, the computational algorithm of VAR with the model adequacy with respect to the plot of residual of the economic indicators was achieved. The inverse of the characteristics polynomial of the variables lies within the unit circle, the response impulse analysis are within the boundaries of estimation. The study also yields R-square that best describe the fit, with RMSE, MAE and MAPE of the three economic variables. The forecast evaluation analysis indicate an upward fluctuation in a long run, the study is now available for economic practitioners to be used for policy implementation.

Modelling of the first lactation milk yield of Frieswal cows using different mathematical functions

A study was conducted to find the most appropriate mathematical model that describes the first lactation milk yield of Frieswal cattle. Data on 42,368 individual first lactation test day yields of 1,072 Frieswal cows calved during 2005 to 2014 in Ambala and Meerut Military dairy farms were utilized for the study. The first test day milk yield was recorded on 6thday after calving while the subsequent records were collected at seven days interval and so the average 43 test day yields were taken for fitting the lactation curve models. Five different mathematical models, viz. Exponential decline function (EDF), Parabolic exponential function (PEF), Inverse polynomial function (IPF), Gamma function (GF) and Mixed log function (MLF) were fitted.The accuracy of fitting (R2 value) the models revealed that the MLF (96.14) was more appropriate followed by IPF (95.57), GF (93.85), PEF (83.68) and EDF (69.09). The RMSE estimate of MLF was lowest (0.3483) as expected while the EDF had the highest RMSE value of 0.9858.The AIC criterion was lowest for IPF (5.7175) and highest for GF (8.0212). The BIC values of five functions ranged between –83.6262 for MLF to 3.1809 for EDF. All the DW estimates were positive and ranged between 0.3656 for EDF to 0.7106 for GF indicating positive autocorrelation between the residuals. Based on the results obtained in the present study, it may be inferred that the first lactation yield was explained accurately by the mixed log function (MLF) in Frieswal cattle. As the inverse polynomial (IPF) and gamma function (GF) also had satisfactory results, these two functions can also be used for fitting the lactation curve models in Frieswal cattle. On the other hand, exponential decline function and parabolic exponential functions least explain the first lactation curve in Frieswal cattle.

Expansion in supercritical random subgraphs of the hypercube and its consequences

It is well known that the behaviour of a random subgraph of a d-dimensional hypercube, where we include each edge independently with probability p, undergoes a phase transition when p is around 1d. More precisely, standard arguments show that just below this value of p all components of this graph have order O(d) with probability tending to one as d→∞ (whp for short), whereas Ajtai, Komlós and Szemerédi (Combinatorica 2 (1982) 1–7) showed that just above this value, in the supercritical regime, whp there is a unique “giant” component of order Θ(2d). We show that whp the vertex expansion of the giant component is inverse polynomial in d. As a consequence, we obtain polynomial in d bounds on the diameter of the giant component and the mixing time of the lazy random walk on the giant component, answering questions of Bollobás, Kohayakawa and Łuczak (Random Structures and Algorithms 5 (1994) 627–648) and of Pete (Electron. Commun. Probab. 13 (2008) 377–392). Furthermore, our results imply lower bounds on the circumference and Hadwiger number of a random subgraph of the hypercube in this regime of p, which are tight up to polynomial factors in d.

Hematologic Reference Intervals and Comparison of Natt-Herrick Technique and Smear-Based Leukocyte Estimation in Cockatiels (Nymphicus hollandicus)

Although cockatiels are among the most common avian species maintained as companion animals in the United States, information on standard hematologic reference values for this species is limited. The objectives of this study were to establish hematologic reference intervals (RI) for cockatiels, compare methods using both the Natt-Herrick technique (NHT) and the smear-based estimation technique (SBT), explore age and sex differences in the hematologic findings for this species, and produce the first cockatiel RI for fibrinogen concentration and thrombocyte estimate. Healthy cockatiels (60 males and 60 females, 2-11 years old) from a research colony were included in this study. Blood samples were placed in dipotassium ethylenediaminetetraacetic acid tubes, and erythrocyte counts and thrombocyte estimates were determined via automated analyzer (ADVIA 120) and SBT, respectively. Moreover, leukocyte concentrations were determined using both NHT and SBT to compare these common methods for measuring a complete blood count in cockatiels. Data were analyzed for outliers, distributions, descriptive statistics, and RI via Reference Value Adviser, a set of macroinstructions for Microsoft Excel (Microsoft, Redmond, WA, USA). Lymphocytes were the predominant leukocyte across both methods. According to the NHT, females had significantly higher concentrations of total leukocytes, heterophils, bands, lymphocytes, basophils, and total plasma protein compared with males. Significant inverse polynomial relationships were noted between total leukocyte count and age and lymphocyte counts and age for NHT. Total leukocyte count produced via NHT and SBT were compared using Passing-Bablok and Bland-Altman plots, and no significant constant or proportional biases were found. However, these methods showed wide limits of agreement. While the RI were interchangeable between methods from a clinical standpoint, the same method should be used to assess changes in an individual. The reported RI are uniquely robust given the sample size, balanced sex and age distributions, inclusion criteria, and control over sample collection.

Genetic Algorithms for Interior Comparative Optimization of Standard BCS Parameters in Selected Superconductors and High-Temperature Superconductors

Inverse least squares numerical optimization, 3D/4D interior optimization, and 3D/4D graphical optimization software and algorithm programming have been presented in a series of previous articles on the applications of the BCS theory of superconductivity and TC dual/multiobjective optimizations. This study deals with the comparison/validation of the optimization results using several different methods, namely, classical inverse least squares (ILS), genetic algorithms (GA), 3D/4D interior optimization, and 2D/3D/4D graphical optimization techniques. The results comprise Tikhonov regularization algorithms and mathematical methods for all the research subjects. The findings of the mathematical programming for optimizing type I chrome isotope superconductors are validated with the genetic algorithms and compared to previous results of 3D/4D interior optimization. Additional rulings present a hypothesis of the new ‘molecular effect’ model/algorithm intended to be proven for Hg-cuprate-type high-temperature superconductors. In molecular effect optimization, inverse least squares and inverse least squares polynomial methods are applied with acceptable numerical and 2D graphical optimization solutions. For the BCS isotope effect and molecular effect, linearization logarithmic transformations for model formula software are implemented in specific programs. The solutions show accuracy with low programming residuals and confirm these findings. The results comprise two strands, the modeling for the isotope effect and molecular effect hypotheses and the development of genetic algorithms and inverse least squares-improved programming methods. Electronic physics applications in superconductors and high-temperature superconductors emerged from the rulings. Extrapolated applications for new modeling for the theory of superconductivity emerged from the numerical and image data obtained.

Warm bias of cold sea surface temperatures in the East Sea (Japan Sea)

The East/Japan Sea (ES) is regarded as a natural laboratory for predicting future changes in the global Meridional Overturning Circulation (MOC) under warming climates, as the ES MOC (EMOC) changes rapidly in comparison with the global MOC. Specifically, intermediate and deep-water masses of the ES are formed in its northern reaches via wind-driven subduction of surface water, and convection from the surface to deep layers during the winter. Accordingly, it is important to investigate the variation of winter sea surface temperatures (SSTs) for characterizing and predicting the EMOC; however, global SST products must be corrected and optimized for the ES, as they fail to incorporate the local marginal sea conditions. Here, a warm bias in cold SST was identified for three SST products, such as optimally interpolated sea surface temperatures (OISSTs), microwave SSTs, and operational SST and sea ice analysis products, suggesting the potential usefulness of a correction method incorporating Argo float data. When comparing OISSTs with 5 m temperature estimates from Argo float data during 2000–2020, under the assumption that the mixed layer depth is deeper than 8 m, a nearly normalized histogram of biases was produced, and the robust warm bias (mean = 0.9°C) was detected in the range of relatively cold SSTs (-2°C to 10°C), yet no significant bias in warm SSTs (&amp;gt; 10°C) was found. To minimize the warm bias in cold SSTs, OISSTs were corrected with an inverse 4th-order polynomial fitting method. Subsequently, the mean bias between the corrected SSTs and top depth temperatures of Argo float data was significantly reduced to less than 0.1°C. Moreover, the warm bias of cold SSTs resulted in severe underestimations of the outcropping area colder than 1°C over the northern region, as well as the occurrence period of 1°C to 5°C SSTs in the north-western ES. These results highlight the importance of local bias correction for SST products, and it is expected that the newly suggested correction method will improve model predictions of EMOC change by enhancing SST data quality in the northern ES.

Cumulative Ordinary Kriging interpolation model to forecast radioactive fallout, and its application to Chernobyl and Fukushima assessment: a new method and mini review.

The Cumulative Ordinary Kriging (COK) interpolation method has been proposed for the spatial prediction of atmospheric radioactive fallout in any given region. COK is built on the Ordinary Kriging and Cumulative Semivariogram methods and combines all their advantages to achieve statistically significant results. It is verified in this paper the reliability of the results from COK with other well-known Modified Shepard's Method (MSM), Inverse Distance Square (INDSQ), Polynomial Regression (PR), Natural Neighbour (NN), Radial Basis (RB), and Kriging Method interpolation methods. The model is tested in detail and in every possible way in two and three dimensions and applied to real-time Cs-134 and Cs-137 radioactive fallout data from the Chernobyl and Fukushima reactor accidents by combining both experimental and theoretical results. The results obtained from the applications for all interpolation methods are included in the supplementary materials section at the end of the article for the benefit of the readers. COK can also be used for spatial modelling of any particle at micro or macro scale. It can contribute significantly to environmental quality, ecological, and human health.

How to Simulate Quantum Measurement without Computing Marginals

We describe and analyze algorithms for classically simulating measurement of an n-qubit quantum state in the standard basis, that is, sampling a bit string from the probability distribution determined by the Born rule. Our algorithms reduce the sampling task to computing poly(n) amplitudes of n-qubit states; unlike previously known techniques they do not require computation of marginal probabilities. Two classes of quantum states are considered: output states of polynomial-size quantum circuits, and ground states of local Hamiltonians with an inverse polynomial spectral gap. We show that our algorithms can significantly accelerate quantum circuit simulations based on tensor network contraction or low-rank stabilizer decompositions. As another striking consequence we obtain the first efficient classical simulation algorithm for measurement-based quantum computation with the surface code resource state on any planar graph and any schedule of measurements.

CYCLOTOMIC AND INVERSE CYCLOTOMIC POLYNOMIAL

In this article, the companion matrix of the cyclotomic and inverse cyclotomic polynomial is found. To determine the flat, gap, jump of the $\Phi_{n}(x)$ and $\Psi_{n}(x)$ for the binary, ternary, quaternary, quinary cyclotomic polynomial. To find some properties of the cyclotomic and inverse cyclotomic polynomial.

Layered and gradient model parameterizations in geoacoustic inversion

This paper considers the importance of model parameterization in geoacoustic inversion and uncertainty estimation, including quantitative approaches to model selection as well as potential limitations of the information content of acoustic data to determine the form of geoacoustic profiles, e.g., differentiating between layered and gradient structures. In particular, general parameterizations are considered based on trans-dimensional (trans-D) inversion, which represents profiles as an unknown number of uniform layers, and Bernstein polynomial (BP) inversion, which represents smooth gradients as polynomials of unknown order. These approaches are illustrated and compared for the inversion of high-order modal-dispersion data collected at the New England Mud Patch. It is shown that while the data constrain the sound-speed profile in the mud layer to reasonably high precision, the data cannot differentiate between trans-D layered or BP gradient representations. However, simpler (fixed) parameterizations, such as a homogeneous layer or linear gradient, can be ruled out based on the Bayesian information criterion. Furthermore, the prior choice of parameterization (layers or gradient) has implications on whether the sound-speed ratio at the water–sediment interface is estimated to be less or greater than one with high probability (an issue other acoustic datasets may share). [Work supported by the Office of Naval Research]

Truly Optimal Euclidean Spanners

Euclidean spanners are important geometric structures, having found numerous applications over the years. Cornerstone results in this area from the late 1980s and early 1990s state that for any $d$-dimensional $n$-point Euclidean space, there exists a $(1+\epsilon)$-spanner with $n \cdot O(\epsilon^{-d+1})$ edges and lightness (normalized weight) $O(\epsilon^{-2d})$. Surprisingly, the fundamental question of whether or not these dependencies on $\epsilon$ and $d$ for small $d$ can be improved has remained elusive, even for $d = 2$. This question naturally arises in any application of Euclidean spanners where precision is a necessity (thus $\epsilon$ is tiny). In the most extreme case $\epsilon$ is inverse polynomial in $n$, and then one could potentially improve the size and lightness bounds by factors that are polynomial in $n$. The state-of-the-art bounds $n \cdot O(\epsilon^{-d+1})$ and $O(\epsilon^{-2d})$ on the size and lightness of spanners are realized by the greedy spanner. In 2016, in a preliminary version, Filtser and Solomon [SIAM J. Comput., 49 (2020), pp. 429--447] proved that, in low-dimensional spaces, the greedy spanner is “near-optimal''; informally, their result states that the greedy spanner for dimension $d$ is just as sparse and light as any other spanner but for dimension larger by a constant factor. Hence the question of whether the greedy spanner is truly optimal remained open to date. The contribution of this paper is twofold: (1) We resolve these longstanding questions by nailing down the dependencies on $\epsilon$ and $d$ and showing that the greedy spanner is truly optimal. Specifically, for any $d= O(1), \epsilon = \Omega({n}^{-\frac{1}{d-1}})$, (a) we show that there are $n$-point sets in $\mathbb{R}^d$ for which any $(1+\epsilon)$-spanner must have $n \cdot \Omega(\epsilon^{-d+1})$ edges, implying that the greedy (and other) spanners achieve the optimal size; (b) we show that there are $n$-point sets in $\mathbb{R}^d$ for which any $(1+\epsilon)$-spanner must have lightness $\Omega(\epsilon^{-d})$, and then improve the upper bound on the lightness of the greedy spanner from $O(\epsilon^{-2d})$ to $O(\epsilon^{-d} \log(\epsilon^{-1}))$. (The lightness upper and lower bounds match up to a lower-order term.) (2) We then complement our negative result for the size of spanners with a rather counterintuitive positive result: Steiner points lead to a quadratic improvement in the size of spanners! Our bound for the size of Steiner spanners in $\mathbb{R}^2$ is tight as well (up to a lower-order term).

A Polynomial Inversion-Based Fast Time-Delay Estimation Method for Wideband Large-Scale Antenna Systems

This paper proposes a new fast time-delay estimation (TDE) method based on polynomial inversion which addresses the challenges arising from the requirements for high-precision, low-computational-complexity synchronization error estimation in wideband large-scale antenna systems (LSASs). In this work, we use the convex parabolic extreme point equation as the timing error detector (TED), and develop a polynomial inversion-based function to characterize the one-to-one mapping relationship between true time delay (TTD) and TED estimates using the least square (LS) method, then obtain the time-delay difference with a high accuracy and high computational efficiency. The results of the performance analysis indicate that the Mean Square Error (MSE) of the proposed algorithm is less than 1 dB away from the Cramer–Rao lower bound (CRLB), which is produced in this paper, while adding a few multipliers compared to the convex parabolic interpolation method. Finally, a further example illustrates that the proposed algorithm can achieve a synchronization error of less than 5 ps between channels based on the NI PXI broadband multichannel acquisition platform.

The Pursuit of Uniqueness: Extending Valiant-Vazirani Theorem to the Probabilistic and Quantum Settings

Valiant-Vazirani showed in 1985 \cite{VV85} that solving NP with the promise that "yes" instances have only one witness is powerful enough to solve the entire NP class (under randomized reductions).We are interested in extending this result to the quantum setting. We prove extensions to the classes Merlin-Arthur MA and Quantum-Classical-Merlin-Arthur QCMA \cite{AN02}. Our results have implications for the complexity of approximating the ground state energy of a quantum local Hamiltonian with a unique ground state and an inverse polynomial spectral gap. We show that the estimation (to within polynomial accuracy) of the ground state energy of poly-gapped 1-D local Hamiltonians is QCMA-hard, under randomized reductions. This is in stark contrast to the case of constant gapped 1-D Hamiltonians, which is in NP \cite{Has07}. Moreover, it shows that unless QCMA can be reduced to NP by randomized reductions, there is no classical description of the ground state of every poly-gapped local Hamiltonian that allows efficient calculation of expectation values.Finally, we discuss a few of the obstacles to the establishment of an analogous result to the class Quantum-Merlin-Arthur (QMA). In particular, we show that random projections fail to provide a polynomial gap between two witnesses.