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For unimodal random variable S with fixed moments (i=1, 2,...,n) and mode m, a formula on variance of any functions of S is provided along the line of Khintchine transform. Upper bounds are derived on the variance of European call options and Gap options. The techniques are based on symmetrization, domination by quadratic functions of two variables and change of measures. Mathematics Subject Classification 60C05; 65K10; 49M29.

Solving a polynomial system over a finite field is an NP-complete problem of fundamental importance in both pure and applied mathematics. In particular, the security of the so-called multivariate public-key cryptosystems, such as HFE of Patarin and UOV of Kipnis et al., is based on the postulated hardness of solving quadratic polynomial systems over a finite field. Lokshtanov et al. (2017) were the first to introduce a probabilistic algorithm that, in the worst-case, solves a Boolean polynomial system in time O ∗ ( 2 δ n ) , for some δ ∈ ( 0 , 1 ) depending only on the degree of the system, thus beating the brute-force complexity O ∗ ( 2 n ) . Later, Bjorklund et al. (2019) and then Dinur (2021) improved this method and devised probabilistic algorithms with a smaller exponent coefficient δ . We survey the theory behind these probabilistic algorithms, and we illustrate the results that we obtained by implementing them in C. In particular, for random quadratic Boolean systems, we estimate the practical complexities of the algorithms and their probabilities of success as their parameters change.

Although planar quadratic differential systems and their applications have been studied in more than one thousand papers, we still have no complete understanding of these systems. In this paper we have two objectives. First we provide a brief bibliographical survey on the main results about quadratic systems. Here we do not consider the applications of these systems to many areas as in Physics, Chemist, Economics, Biology, … Second we characterize the new class of planar separable quadratic polynomial differential systems. For such class of systems we provide the normal forms which contain one parameter, and using the Poincare compactification and the blow up technique, we prove that there exist 10 non-equivalent topological phase portraits in the Poincare disc for the separable quadratic polynomial differential systems.

In this paper, we investigate the Monge–Ampere equation $$\text{ det }D^2u=f $$ in $${\mathbb {R}}^n_+$$ , where f is bounded, positive and $$f(x)=1+O(|x|^{-\beta })$$ for some $$\beta >2$$ at infinity. If u is a quadratic polynomial on $$\{x_n=0\}$$ and satisfies $$ \mu |x|^2\le u\le \mu ^{-1}|x|^2$$ for some $$00$$ is some constant depending only on $$\beta $$ and n. Meanwhile, the existence and uniqueness of viscosity solutions of the Dirichlet problem with prescribed asymptotic behavior at infinity will be concerned. The condition $$\beta >2$$ is sharp.

Precise point positioning (PPP) is used in many fields. However, pseudorange multipath delay is an important error that restricts its accuracy. Pseudorange multipath delay can be considered as the combination of effective information and observation noise; it can be modeled after removing the observation noise. In this work, elastic nets (EN) regularization denoising method is proposed and compared with L2-norm regularization denoising method. Then, quadratic polynomial (QP) model plus autoregressive (AR) model (QP + AR) are used to model the denoised pseudorange multipath delays. Finally, the modeling results are corrected to the observations to verify the improvement of BDS-3 single-frequency PPP accuracy. Three single-frequency PPP schemes are designed to verify the effectiveness of denoising method and QP + AR model. The experimental results show that the accuracy improvement of B3I and B2a is more obvious than that of B1I and B1C when the modeling values are corrected to the pseudorange observations. The improvement of B3I and B2a in the east (E) and up (U) directions can reach 10.6%∼34.4% and 5.9%∼65.7%, and the improvement of the north (N) direction is mostly less than 10.0%. The accuracy of B1I and B1C in E and U directions can be improved by 0%∼30.7% and 0.4%∼28.6%, respectively, while the accuracy of N direction can be improved slightly or even decreased. Using EN regularization denoising and QP + AR model correction, single-frequency PPP performs better at B3I and B2a, while L2-norm regularization denoising and QP + AR model correction perform better at B1I and B1C. The accuracy improvement of B2a and B3I is more obvious than that of B1I and B1C. The convergence time after MP correction of each frequency is slightly shorter. Overall, the proposed pseudorange multipath delays processing strategy is beneficial in improving the single-frequency PPP of BDS-3 satellite.

We introduce a family of weighted conjugate-gradient-type methods, for strictly convex quadratic functions, whose parameters are determined by a minimization model based on a convex combination of the objective function and its gradient norm. This family includes the classical linear conjugate gradient method and the recently published delayed weighted gradient method as the extreme cases of the convex combination. The inner cases produce a merit function that offers a compromise between function-value reduction and stationarity which is convenient for real applications. We show that each one of the infinitely many members of the family exhibits q-linear convergence to the unique solution. Moreover, each one of them enjoys finite termination and an optimality property related to the combined merit function. In particular, we prove that if the n × n Hessian of the quadratic function has p < n different eigenvalues, then each member of the family obtains the unique global minimizer in exactly p iterations. Numerical results are presented that demonstrate that the proposed family is promising and exhibits a fast convergence behavior which motivates the use of preconditioning strategies, as well as its extension to the numerical solution of general unconstrained optimization problems.

GeoGebra is a teaching tool that educators use in their lesson plans to improve the quality of teaching and learning. Instead of drawing on a sheet of paper, students can design a graph, adjust the actual graph shape, and examine the impact of changing graph pattern using GeoGebra in mathematics teaching and learning. Furthermore, students can keep all of their work materials in a structured manner for future reference. GeoGebra will make a school lecture more interesting, exciting, creative, and innovative. The goal of this study was to analyse the effects of GeoGebra software in Mathematics achievement in respect to quadratic functions among gifted and talented Muslims student’s at Kolej GENIUS Insan, Universiti Sains Islam Malaysia. The maximum or minimum point of quadratic function was determined by using GeoGebra software, and the characteristics of quadratic expressions in one variables was also identified. The results illustrate that the graph quadratic expression has the highest point or the lowest point based the values of coefficient a on the quadratic function. For the graph function with negative values of coefficient a on a quadratic function, there are highest values of coordinates x and y, which also known as maximum point, while the graph function with positive values of a on a quadratic function, there are lowest values of coordinates x and y, which also known as minimum point. When students utilize GeoGebra software, their performance in calculating the minimum and maximum points on quadratic functions improves. Keywords: Geogebra; Integrated naqli ‘aqli gifted education; Gifted muslims student; Mathematics achievement; Quadratic functions

Gradient method is a simple optimization approach using minus gradient of the objective function as a search direction. Its efficiency highly relies on the choices of the stepsize. In this paper, the convergence behavior of a class of gradient methods, where the stepsize has an important property introduced in (Dai in Optimization 52:395–415, 2003), is analyzed. Our analysis is focused on minimization on strictly convex quadratic functions. We establish the R-linear convergence and derive an estimate for the R-factor. Specifically, if the stepsize can be expressed as a collection of Rayleigh quotient of the inverse Hessian matrix, we are able to show that these methods converge R-linearly and their R-factors are bounded above by $$1-\frac{1}{\varkappa }$$ , where $$\varkappa$$ is the associated condition number. Preliminary numerical results demonstrate the tightness of our estimate of the R-factor.

This paper concentrates on the stability problem of systems with two additive time‐varying delay components. For the construction of Lyapunov–Krasovskii functional (LKF), in the case that the introduced augmented vector contains the double integral term of the state vector, a special form of the binary quadratic function with respect to two time‐varying delays has often been introduced into the derivative of the LKF. In order to determine the negative definiteness of such function, by making full use of the idea of partial differential, the convex/concave property and the slope characteristic of tangent lines of the binary quadratic function, a binary quadratic function negative‐determination lemma is proposed in the case of the pluses or minuses of quadratic coefficients are unknown. Then, the obtained stability criterion in the form of linear matrix inequality shows a greater advantage than the previous criteria since not only some advanced techniques are employed to treat some integral terms, but also the proposed lemma is employed to deal with quadratic terms in functional derivative. Finally, a typical example is given to verify the superiority of the derived criterion.

The weight distribution and weight hierarchy of a linear code are two important research topics in coding theory. In this paper, by choosing proper defining sets from inhomogeneous quadratic functions over $${\mathbb {F}}_{q}^{2}$$ , we construct a family of three-weight p-ary linear codes and determine their weight distributions and weight hierarchies. Most of the codes can be used in secret sharing schemes.

This article studies the stability problem for neural networks with a time-varying delay. An improved negative-definiteness lemma (NDL) is proposed by removing some redundant inequality constraints involved in the original one that is recently developed via a quadratic-partitioning method. Furthermore, the improved NDL is presented in the matrix-valued form for convenient application. With regard to the case that the upper bound of the variation of the delay is less than a known constant while the lower bound is unknown, an appropriate Lyapunov–Krasovskii functional (LKF) candidate is deliberately built so that the derivative of the LKF is estimated to be a novel quadratic function with respect to the delay. Consequently, a series of new stability criteria is obtained via the improved NDL, which is shown to be less conservative than existing ones through numerical examples.

This article investigates $H_{\infty }$ synchronization of delayed neural networks under a receding horizon scheme, where two types of interval time-varying delays are considered according to whether the lower bound of the delay derivative is known or not. Note that a receding horizon synchronization law can be regarded as an optimization solution at each timeslot to a minimaxization problem related closely with a certain cost functional. In this article, two cost functionals with some delay-dependent matrices are introduced, respectively, for the two types of time delays. In order to obtain less conservative conditions, a novel inequality on quadratic polynomial functions is established, which includes some existing ones as its special cases. Based on the novel inequality, two sufficient conditions are derived to design the terminal weighting matrices of the cost functionals such that the resulting synchronization error system can be stabilized with a prescribed infinite horizon $H_{\infty }$ performance level. Finally, three numerical examples are used to demonstrate the validity of the proposed results.

In this study, a reliable model for determining the relationship between yarn construction and textile color was designed. It is crucial to create coloration using a sustainable spinning method. Melange colorful yarns were produced using three-color viscose staple fibers (red, yellow, and blue) at a ratio gradient of 10% with digital rotor spinning, and then fabrics were knitted using these yarns. Based on the Stearns-Noechel model, a color prediction model suitable for the yarn-forming method was established using experimental data measured by a Datacolor650 spectrophotometer. The research results show that when the value of empirical parameter M is 0.136, the model has the best color matching effect; the average ratio error between the test and predicted values of the test sample is 7.89%; the average color difference is 0.320; the color difference of each sample is less than 1. Moreover, the linear correlation between M value and the wavelength is weaker, and there is a good linear correlation only when the wavelength is above 600 nm. A quadratic function can better explain this relationship.

By using region pixels rather than edge points, a novel method is proposed for extracting circular objects in an image. A two-dimensional accumulator array with less columns is voted by region pixels instead of contour points. A set of quadratic functions and a linear function are fitted, and the coefficients of the fitted functions are used to compute the circle parameters. The quadratic polynomial function, which is determined by the circle radius and mean voting distance, is deduced by analyzing the relationship between voting values and the voting distance. A linear function, which is determined by the circle center, is deduced by analyzing the relationship between mean voting distances and the voting angle. In each column of two-dimensional array, a quadratic function is fitted and validated. The circle radius is computed based on the coefficients of the validated quadratic functions. After all columns are processed, a linear function is fitted. The circle center happens to be the coefficients of the fitted linear function. The proposed method is tested and compared using synthetic images and real-world images. Experimental results show the proposed method is fast and accurate in the presence of noise disturbance.

Let [Formula: see text] be the polynomial ring over the finite field [Formula: see text] of [Formula: see text] elements. For a natural number [Formula: see text] let [Formula: see text] be the set of all polynomials in [Formula: see text] of degree less than [Formula: see text] Let [Formula: see text] be a quadratic polynomial over [Formula: see text] Suppose that [Formula: see text] is intersective, that is, which satisfies [Formula: see text] for any [Formula: see text] with [Formula: see text] where [Formula: see text] denotes the difference set of [Formula: see text] Let [Formula: see text] Suppose that [Formula: see text] and that the characteristic of [Formula: see text] is not divisible by 2. It is proved that [Formula: see text] for any [Formula: see text] where [Formula: see text] is a constant depending only on [Formula: see text] and [Formula: see text]

In this article we address the problem of minimizing a strictly convex quadratic function using a novel iterative method. The new algorithm is based on the well-known Nesterov’s accelerated gradient method. At each iteration of our scheme, the new point is computed by performing a line-search scheme using a search direction given by a linear combination of three terms, whose parameters are chosen so that the residual norm is minimized at each step of the process. We establish the linear convergence of the proposed method and show that its convergence rate factor is analogous to the one available for other gradient methods. Finally, we present preliminary numerical results on some sets of synthetic and real strictly convex quadratic problems, showing that the proposed method outperforms in terms of efficiency, a wide collection of state-of-the art gradient methods, and that it is competitive against the conjugate gradient method in terms of CPU time and number of iterations.

In this paper, we propose an analytical approach based on the Laplace transform and Mittag–Leffler functions combining with linear matrix inequality techniques to study finite-time stability of fractional-order neural networks (FONNs) with time-varying delay. The concept of finite-time stability is extended to the fractional-order neural networks and the delay function is assumed to be non-differentiable, but continuous and bounded. We first prove some important lemmas on the existence of solutions and on estimation of the Caputo derivative of specific quadratic functions. Then, new delay-dependent sufficient conditions for finite-time stability of FONNs with time-varying delay are derived in terms of a tractable linear matrix inequality and Mittag–Leffler functions. Finally, a numerical example with simulations is provided to demonstrate the effectiveness and validity of the theoretical results.

We look at a familiar one-parameter family of quadratic functions on the complex plane. After restricting the parameter to be real, we explore when the critical points of the functions and their iterates are real and when they are not real. We prove that when the parameter is greater than or equal to 2, all critical points are real. When the parameter is between 0 and 2, critical points for the original function are real but there is an iterate with nonreal critical points. When the parameter is equal to 0, all critical points are 0. When the parameter is less than 0, the critical points are all 0 or are nonreal. Finally, we compare the locations of these critical points to the contour plots of the real parts of the functions for different values of the parameter and for different iterates of the function.

In this paper we show that two-stage adjustable robust linear programs with affinely adjustable data in the face of box data uncertainties under separable quadratic decision rules admit exact semi-definite program (SDP) reformulations in the sense that they share the same optimal values and admit a one-to-one correspondence between the optimal solutions. This result allows adjustable robust solutions of these robust linear programs to be found by simply numerically solving their SDP reformulations. We achieve this result by first proving a special sum-of-squares representation of non-negativity of a separable non-convex quadratic function over box constraints. Our reformulation scheme is illustrated via numerical experiments by applying it to an inventory-production management problem with the demand uncertainty. They demonstrate that our separable quadratic decision rule method to two-stage decision-making performs better than the single-stage approach and is capable of solving the inventory production problem with a greater degree of uncertainty in the demand.

This article concerns the zero-Hopf bifurcation of a quadratic polynomial differential system in $\mathbb {R}^{4}$ . By using the averaging theory of third order, we provide that at most 25 limit cycles can bifurcate from one singularity with eigenvalues of the form ± bi, 0 and 0.

In this paper, we study a non-convex min–max fractional problem of quadratic functions subject to convex and non-convex quadratic constraints. First, by using the Dinkelbach-type method, we transfo...

Let f ∈ Z [ x ] and l ∈ N . Consider the set of all ( a 0 , a 1 , … , a l ) ∈ N l + 1 with a i a i + 1 and f ( a i ) | f ( a i + 1 ) for all 0 ≤ i ≤ l − 1 . We say that f satisfies the gap principle of order l if lim ⁡ a l / a 0 = ∞ as a 0 → ∞ . We also define the gap order of f ( x ) to be the smallest positive integer l such that f ( x ) satisfies the gap principle of order l. If such l does not exist, we say that f ( x ) does not satisfy the gap principle. In this article, we prove a conjecture by Chan, Choi and Lam that f ( x ) does not satisfy the gap principle if and only if f ( x ) is in the form of f ( x ) = A ( B x + C ) n for some A , B , C ∈ Z . Moreover, we completely determine the gap order of any polynomial. We will show that if f ( x ) is not in the form of A ( B x + C ) n , then f ( x ) has gap order 2 if f ( x ) is a quadratic polynomial or a power of a quadratic polynomial; and has gap order 1 otherwise.

In this research, the general linear evolution equations (EEs) in (5+1) dimensions are studied. All the mixed second-order derivatives are included in this aforementioned model. Using the Hirota bilinear operator and symbolic computation, the localized solutions–the abundant lump solutions are constructed. Particularly, it is found that only four groups of linear (5+1)-dimensional EEs are found that they have abundant lump solutions, and no interactions between the lump and other solutions are found via the positive definite quadratic functions. Finally, four examples corresponding to the above-mentioned cases are given to validate the obtained results, and the corresponding graphs are presented to show the dynamic behaviors of the abundant lump solutions of these given examples.

This paper develops a new grey prediction model with quadratic polynomial term. Analytical expressions of the time response function and the restored values of the new model are derived by using grey model technique and mathematical tools. With observations of the confirmed cases, the death cases and the recovered cases from COVID-19 in China at the early stage, the proposed forecasting model is developed. The computational results demonstrate that the new model has higher precision than the other existing prediction models, which show the grey model has high accuracy in the forecasting of COVID-19.

An improved vacuum-assisted heat reflux extraction (VAHRE) technique was proposed and applied to extract Radix Isatidis. The extraction parameters of VAHRE, including the boiling temperature, extraction cycles, extraction time, soak time, and liquid-solid ratio, were carefully optimized with a single factor experiment. The least-squares support vector machine (LS-SVM) novel machine learning algorithm was introduced to improve the optimization's predictive accuracy. The results indicated that the LS-SVM model had better performance, with a higher R2 and lower RMSE value than those of the conventional quadratic polynomial model (QPM). Compared with conventional reference extraction methods, the VAHRE method gave a higher extraction yield due to indirubin's reduced for volatile loss and more efficient release from the plant matrix with the aid of a vacuum. This is the first study on optimizing the extraction process of Radix Isatidis using VAHRE and LS-SVM algorithms. The present findings also demonstrated that VAHRE was a promising procedure for extracting indirubin from Radix Isatidis, which shows great potential for becoming an alternative system for industrial scale-up applications.

This paper presents two functional encryption schemes for quadratic functions (or degree-2 polynomials) achieving simulation-based security in the semi-adaptive model with constant-size secret keys. Prior constructions in the standard model either achieve weaker security model [CRYPTO 17] or require linear-size secret keys (in the message length) [PKC 20]. One of our proposed schemes is comparable to existing schemes in the generic group model in terms of ciphertext size. Technically, we combine Wee’s compiler [TCC 17] with Gay’s paradigm [PKC 20]. However, we avoid (partially) function-hiding inner-product functional encryption used in Gay’s paradigm which makes our work conceptually simpler.

Combinatorial t-designs have nice applications in coding theory, finite geometries and engineering areas. t-designs can be constructed from image sets of a fixed size of some special polynomials. This paper constructs t-designs from the quadratic polynomial $$x^{4}+x^{3}$$ over $$\mathrm {GF}(2^{n})$$ and determine their parameters. We yield 2- $$\left( 2^n,3\cdot 2^{n-2},3\cdot 2^{n-2}\left( 3\cdot 2^{n-2}-1 \right) \right) $$ designs for n even and 3- $$\left( 2^n,2^{n-1},2^{n-1}\left( 2^{n-2}-1 \right) \right) $$ designs for n odd.

The model based on the velocity and acceleration of temperature variation (MBOTV) was established for spindle thermal growth error. However, the thermophysical parameters of MBOTV, such as the thermal expansion coefficient related to the velocity of temperature variation, Hv, the thermal expansion coefficient related to the acceleration of temperature variation, Ha, and the thermal inertia coefficient of the spindle, τ, may change during actual working. In order to analyze the influence of thermophysical parameters on the prediction accuracy of MBOTV, it is necessary to calculate the reliability of MBOTV. Actually, it is difficult to obtain the function of MBOTV. Although the improved response surface method can be used to calculate the reliability of the model with implicit function, the accuracy is not enough obviously when the implicit function is replaced by linear polynomials. Therefore, an improved response surface method based on quadratic polynomial approximation was proposed in this paper to calculate the reliability of MBOTV. In this method, the linear polynomial was replaced by a quadratic polynomial without cross product terms to replace the implicit function, which can improve the accuracy of reliability calculation. The predicted residual errors of MBOTV with single-parameter fluctuation and multi-parameter fluctuation were given in this paper. Studies reveal that MBOTV has strong robustness when the thermophysical parameters change.

In this paper, we focus on the synchronization of fractional-order coupled neural networks (FCNNs). First, by taking information on activation functions into account, we construct a convex Lur’e–Postnikov Lyapunov function. Based on the convex Lyapunov function and a general convex quadratic function, we derive a novel Mittag-Leffler synchronization criterion for the FCNNs with symmetrical coupled matrix in the form of linear matrix inequalities (LMIs). Then we present a robust Mittag-Leffler synchronization criterion for the FCNNs with uncertain parameters. These two Mittag-Leffler synchronization criteria can be solved easily by LMI tools in Matlab. Moreover, we present a novel Lyapunov synchronization criterion for the FCNNs with unsymmetrical coupled matrix in the form of LMIs, which can be easily solved by YALMIP tools in Matlab. The feasibilities of the criteria obtained in this paper are shown by four numerical examples.

Made available in DSpace on 2021-07-14T10:21:14Z (GMT). No. of bitstreams: 0 Previous issue date: 2021-06-25. Added 1 bitstream(s) on 2021-07-14T11:27:56Z : No. of bitstreams: 1 S0100-69162021000300319.pdf: 915553 bytes, checksum: 032aa323077110649b223df7fe9ee77a (MD5) In order to estimate the response of biometric variables in different irrigation depths in radish crop, as well as their relations in the development of the crop, a fuzzy mathematical analysis was carried out from irrigation with depths of different percentages of the crop evapotranspiration (ETc), using Gaussian pertinence functions for the input variable and triangular for the biometric output variables. Validations were performed using neural network models, smoothing splines and polynomial regression. The relation among the biometric variables was measured applying the Pearson correlation coefficient. The results showed that the fuzzy modeling presented superiority in the crop development estimate over the quadratic polynomial regression model, neural network and smoothing splines, because it achieved an average reduction of errors among the biometric variables, of 7.8% 94.6% and 9.2% for the RMSE in the respective models, as well as a better adjustment of the data with average R2 of the variables. The modeling with neural network showed inadequate agronomic behavior in data representation. Regarding biometric variables, the length and diameter of the tuberous root are inversely correlated, and the fresh phytomass of the tuberous root is correlated only with the fresh phytomass of the root. São Paulo State University, School of Agriculture São Paulo State University, School of Sciences and Engineering São Paulo State University, School of Agriculture São Paulo State University, School of Sciences and Engineering

At present, there are two ways to obtain temperature information: contact type and nonintrusive type. As a nonintrusive temperature measurement method, ultrasonic thermometry can be used to acquire the temperature distribution of complex fields conveniently. By measuring the time-of-flight (TOF) between ultrasonic transmitters and receivers, and according to the relationship between temperature and ultrasonic velocity, the temperature distribution can be reconstructed. Among the existing algorithms, the least square method (LSM) will lose much information near the edges of the temperature field, and the algebra reconstruction technique (ART) is time-consuming with low reconstruction accuracy. In this article, an improved reconstruction algorithm based on an inverse quadratic function and singular value decomposition (IQ-SVD) is proposed, which can effectively increase the reconstruction accuracy. The simulations of the real temperature data are conducted in ideal and noisy environments, respectively. Moreover, the influence of region division and shape parameters on reconstruction accuracy is discussed. The simulation results indicate that, compared with conventional algorithms, the proposed algorithm can accurately reflect the temperature distribution, and the root mean square error in the central region and the edge region is reduced by 0.49% at least, and 1.28% at most.

This paper studies the stability analysis problem for time-varying delay systems. An appropriate Lyapunov-Krasovskii functional (LKF) is constructed where its derivative is a quadratic polynomial function of the delay. A novel negative condition of the mentioned quadratic function with two variable parameters is developed to ensure that the LKF derivative is negative, reducing conservatism on some similar results. Besides, an extended version of Bessel-Legendre inequality is introduced to be employed in the stability analysis of time-varying delay systems. Then, some stability criteria with less conservatism are derived for two kinds of the time-varying delay. Finally, the effectiveness of the proposed stability criteria is demonstrated through three examples.

The main focus in this paper is exact linesearch methods for minimizing a quadratic function whose Hessian is positive definite. We give a class of limited-memory quasi-Newton Hessian approximations which generate search directions parallel to those of the BFGS method, or equivalently, to those of the method of preconditioned conjugate gradients. In the setting of reduced Hessians, the class provides a dynamical framework for the construction of limited-memory quasi-Newton methods. These methods attain finite termination on quadratic optimization problems in exact arithmetic. We show performance of the methods within this framework in finite precision arithmetic by numerical simulations on sequences of related systems of linear equations, which originate from the CUTEst test collection. In addition, we give a compact representation of the Hessian approximations in the full Broyden class for the general unconstrained optimization problem. This representation consists of explicit matrices and gradients only as vector components.

We show that for a quadratic polynomial $$f(x)=x^2-c$$ , where $$c=(8k+2)(8k+3)$$ or $$c=(4k+1)(4k+2)+1$$ with $$k\in {\mathbb {N}}\cup \{0\}$$ , the Galois group of the splitting field of each iterate $$f^n$$ of f is isomorphic to the automorphism group of a complete binary rooted tree of height n.

This paper considers the construction of optimal designs for homoscedastic functional polynomial measurement error models. The general equivalence theorems are given to check the optimality of a given design, based on the locally and Bayesian D-optimality criteria. The explicit characterizations of the locally and Bayesian D-optimal designs are provided. The results are illustrated by numerical analysis for a quadratic polynomial measurement error model. Numerical results show that the error-variances ratio and the model parameter are the important factors for the both optimal designs. Moreover, it is shown that the Bayesian D-optimal design is more robust and effective compared with the locally D-optimal design, if the error-variances ratio or the model parameter is misspecified.