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This paper provides a unified formulation of optimal guidance-to-collision law for a target with an arbitrary acceleration or deceleration. The collision course for general target acceleration or deceleration is first determined from the engagement geometry in conjunction with the nonlinear engagement kinematics in the proposed approach. The heading error de_ned in the collision course is then adopted as a variable to be nullified for accomplishing the intercept condition. The proposed guidance law is derived based on the heading error dynamics and the optimal error dynamics to ensure optimality and finite-time convergence. As illustrative examples, the proposed guidance command for a constant target acceleration and a target deceleration in the form of a quadratic function of speed are provided. Additionally, the time-to-go prediction method is suggested for implementing the proposed method. The characteristics of the proposed guidance command are analytically investigated to provide insight into the proposed method. The key benefits of the proposed method lie in not producing unnecessary guidance commands near a target compared to other methods and ensuring optimality in guidance command even in the nonlinear engagement kinematics. Finally, numerical simulations are performed to validate the proposed method and to show our findings.

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.

This paper investigates the problem of stability and state-feedback control design for linear parameter-varying systems with time-varying delays. The uncertain parameters are assumed to belong to a polytope with bounded known variation rates. The new conditions are based on the Lyapunov theory and are expressed through Linear Matrix Inequalities. An alternative parameter-dependent Lyapunov-Krasovskii functional is employed and its time-derivative is handled using recent integral inequalities for quadratic functions proposed in the literature. As main results, a novel sufficient stability condition for delay-dependent systems as well as a new sufficient condition to design gain-scheduled state-feedback controllers are stated. In the new proposed methodology, the Lyapunov matrices and the system matrices are put separated making it suitable for supporting in a new way the design of the stabilization controller. An example, based on a model of a real-world problem, is provided to illustrate the effectiveness of the proposed method.

In the fields of computer vision and machine learning, domain adaptation has been extensively studied and the main challenge in the case is how to transform the existing classifier(s) into an effective adaptive classifier to exploit the latent information in the new data source which typically has a different distribution compared with the original data source. Currently, the Adaptive Support Vector Machines (A-SVM) has been proposed to deal with the domain adaptation problem, which is an effective strategy. However, the resulting optimization task by minimizing a convex quadratic function in A-SVM can not effectively minimize the distance between a source and a target domain as much as possible and typically has high computational complexity. In order to handle these problems, in this paper, we extend the A-SVM by determining a pair of nonparallel up- and down-bound functions solved by two smaller sized quadratic programming problems (QPPs) to achieve a faster learning speed. Notably, our method yields two nonparallel separating hyperplanes to exploit the latent discriminant information based on SVM classification mechanism, which can naturally enhance the classification performance. This method is named as Adaptive Twin Support Vector Machine Learning (A-TSVM). Moreover, we consider a high-level learning paradigm with privilege information (LUPI) to learn a induced model that further constrains the solution in the target space. The learned model is named as domain Adaptive Twin Support Vector Machine Learning Using Privileged Information (A-TSVM+). Finally, a series of comparative experiments with many other methods are performed on three datasets. The experimental results effectively indicate that the proposed method can not only greatly improve the accuracy of classification, but also save computing time.

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.

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.

As one of the most important components of machinery, once the bearing has a failure, serious catastrophe may happen. Hence, for avoiding the catastrophe, it is valuable to predict the remaining useful life (RUL) of bearing. Health indicators (HIs) construction plays a greatly important role in the data-driven RUL prediction. Unfortunately, most of the existing HIs construction methods need prior knowledge and few of them construct HIs from raw vibration signals. For dealing with the above issues, a novel quadratic function-based deep convolutional auto-encoder is developed in this work. The raw bearing vibration signals are first preprocessed by low-pass filtering. Then the cleaned vibration signals are input into the quadratic function-based DCAE neural networks for constructing HIs of bearings. Compared with AE, DNN, KPCA, ISOMAP, PCA and VAE, it is revealed that the proposed methodology can construct a better HI from the raw bearing vibration signal in terms of comprehensive performance. Several comparative experiments have been implemented, and the results indicate that the HI constructed by quadratic function-based DCAE neural network has stronger predictive power than the traditional data-driven HIs.

International audience; The work of fracture of refractories is commonly calculated from Crack Mouth Opening Displacements (CMODs) in Wedge Splitting Tests (WSTs). This paper proposes a methodology for estimating the fracture energy from Notch Opening Displacement (NOD) measurements, which is useful for setups where CMOD is not accessible. NODs and CMODs are calculated for both faces of two WSTs experiments on a castable refractory via Digital Image Correlation (DIC) and finite element simulations. A quadratic function fits well the non-linear CMOD vs. NOD behavior in the crack initiation regime, while an affine trend describes the propagation regime. Although the nonlinearity associated with crack initiation is more complex, the crack propagation energy can easily be estimated from NOD data when CMODs cannot be measured.

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.

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.

Goafs are composed of large, irregularly shaped broken rocks. This means in order to calculate the permeability and inertial resistance coefficient of a goaf, the Ergun equation must be modified. The coupling relationships between coefficients A and B, and particle size and porosity are established based on the existing simulation research. Coefficients A and B both have a quadratic polynomial relationship with porosity. The quadratic polynomial's constant term ratio has a good power function relationship with particle size, while the linear term ratio and quadratic term ratio both have good exponential function relationships with particle size. By introducing the shape factor of permeability and inertial resistance coefficient, which are calculated by dividing the experimental value with the theoretical value, the correction model is established for the permeability and inertial resistance coefficient of the goaf. The permeability and inertial resistance coefficient shape factor have good power function relationships with particle size.

In this paper, the problem of delay-dependent stability analysis of fractional-order systems with time-varying delay is investigated. First, a class of novel fractional-order integral inequalities for quadratic functions by constructing appropriate auxiliary functions is proposed, which has been proven to be useful in analyzing fractional-order systems with time-varying delay. Based on these proposed inequalities, the Lyapunov–Krasovskii functions are designed to deal with the time-varying delay terms, reducing the conservatism of the stability criteria. Furthermore, delay-dependent criteria are derived to achieve asymptotic stability of fractional-order systems with time-varying delay. Finally, two examples are provided to illustrate the effectiveness and feasibility of the proposed stability criteria.

To improve the computing accuracy and simulation efficiency of fatigue reliability estimation for turbine rotor, a multi-agent collaborative modelling (MACM) approach is proposed by absorbing the strengths of improved differential evolution (IDE) algorithm and neural network model into decomposed-collaborative strategy. The fatigue reliability estimation framework is presented in respect of MACM approach. Furthermore, the fatigue reliability estimation of a typical turbine rotor is considered as engineering case to evaluate the presented approach with respect to material variabilities, load fluctuations and model randomness. The estimation results reveal that the probabilistic fatigue life of a turbine rotor under reliability 99.87% is 6 050 cycles; fatigue strength coefficient σf′ and strain range Δɛt play a leading role on the fatigue life since their effect probabilities of 45% and 36%, respectively. The comparisons of methods (direct Monte Carlo simulation (MCS), quadratic polynomial (QP), neural network (NN), neural network agent (NNA)) show that the presented MACM approach holds high efficiency and accuracy for fatigue reliability estimation of turbine rotor.

Catalytic upgrading of 4-methylanisole as a representative of lignin-derived pyrolysis bio-oil was investigated over Pt/γ-Al2O3 catalyst. The catalytic upgrading process was conducted at different operating condition to determine the detailed reactions network. Additionally, artificial neural network and design of experiment were applied by feeding the reaction temperature, operating pressure and space velocity to predict 4-methylanisole conversion, main products selectivity, reactions rate and reactions network. The main products of 4-methylanisole upgrading were toluene, phenol derivatives, cyclohexanone, 4-methylcyclohexanone, and 2-tert-butyl-4-methylphenol. The major classes of reactions during the upgrading process were hydrogenolysis, hydrodeoxygenation, alkylation, and hydrogenation. For optimization of experimental data obtained at suggested conditions by design of experiment, the response surface methodology was applied. Artificial neural network model was used to investigate the kinetics behavior of the system due to the complex nature of system. A combination of the response surface methodology, artificial neural network, and design of experiment has revealed its ability to solve a quadratic polynomial model. The coefficients of determination were close to 1, and the mean square error of the artificial neural network model was close to 0 which showed the high accuracy of model predictions. It was inferred that during the upgrading process of 4-methylanisole, increasing temperature and pressure and setting space velocity at the minimum value are the reasons to come close to the optimum reaction rate. The comparison of experimental results with simulated data from the artificial neural network and the response surface methodology models illustrated that the developed model can create an applicable situation for practical design of large-scale production of valuable fuels from renewable resources.

Additive manufacturing has gained relevance in aerospace components for intricate lattice, cooling channels, complex internal structure, and lightweight strong structures. Additive manufacturing has aided cost saving by reducing part count in expensive materials. Titanium di boride is a hard ceramic with a good oxidation stability with resistance to mechanical wear. It is a hexagonal crystal structured with a non-lustrous grey appearance. It has expressed a high value of strength to density ratio, melting point, and wear resistance. A predictive correlation and relationship between the optimal parameters and the hardness was examined. The data analysis of their relationship was examined in statistical analysis. A predictive mathematical formula was derived, and a linear and quadratic polynomial regression machine learning details of the factors showed correlation.

Compressor blade seriously affects the reliability and stability of aircraft engine. Probabilistic flutter evaluation of compressor blade can effectively quantify failure risk and improve aeroelastic stability. To improve the computational efficiency of probabilistic flutter evaluation, a surrogate modeling approach (called as PSO-LSSVR) is presented by absorbing the strength of particle swarm optimization (PSO) algorithm and least-squares support vector regression (LSSVR). We first mathematically modeled the PSO-LSSVR and then introduce the corresponding probabilistic flutter framework. With respect to high-nonlinearity and strong coupling of limit-state function, the probabilistic flutter evaluation of NASA Rotor 37 is performed to evaluate the proposed approach. Compared with Monte Carlo method, quadratic polynomial (QP), regular support vector regression (SVR and LSSVR), the presented PSO-LSSVR possesses the highest computing efficiency while keeping acceptable computing precision.

In this paper, an LMI-based design is proposed for observer control problem of nonlinear fractional-order systems subject to time-variable delay, where the delay function is non-differentiable, but continuous and bounded. Our novel technique is based on a new lemma concerning Caputo derivative estimation of quadratic functions. In this proposed approach, delay-dependent sufficient conditions in terms of linear matrix inequalities are obtained for the design state feedback controller and observer gains. A simulation-based example is given to illustrate the effectiveness of the theoretical result.

Finite element-least squares point interpolation method (FE-LSPIM) developed recently shows some excellent features to improve the calculation accuracy of mechanical problems. In this paper, a coupled finite element-least squares point interpolation method/ boundary element method (FE-LSPIM/BEM) is proposed to analyze plate-like structural-acoustic coupled system. Here, the FE-LSPIM is used to model the structure domain, while the acoustic domain is modeled by BEM. The hybrid method not only inherits advantages of element compatibility of the finite element method (FEM) and the quadratic polynomial completeness of LSPIM, but also improves the calculation accuracy of the structural domain. Moreover, stochastic perturbation method is introduced to process uncertainty parameters of the FE-LSPIM/BEM, so the stochastic perturbation FE-LSPIM/BEM model has been proposed, then several uncertain parameters that have been randomly processed were used to increase the analytical reliability in structural-acoustic coupled system. At last, numerical examples are taken to verify the feasibility of the proposed SP-FE-LSPIM/BEM as compared to Monte Carlo method (MCM) and stochastic perturbation finite element/boundary element method (SP-FEM/BEM). The results show that FE-LSPIM/BEM has higher accuracy in analysis of uncertain structural-acoustic coupling system as compared to the FEM/BEM.

This paper addresses the analysis of covariance matrix self-adaptive Evolution Strategies (CMSA-ES) on a subclass of quadratic functions subject to additive Gaussian noise: the noisy ellipsoid model. To this end, it is demonstrated that the dynamical systems approach from the context of isotropic mutations can be transferred to ES that also control the covariance matrix. Theoretical findings such as the component-wise quadratic progress rate or the self-adaptation response function can thus be reused for the CMSA-ES analysis. By deriving the steady state quantities approached on the noisy ellipsoid model for constant population size, a detailed description of the asymptotic CMSA-ES behavior is obtained. By providing self-adaptive ES with a population control mechanism, despite noise disturbances, the algorithm is able to realize continuing progress towards the optimum. Regarding the population control CMSA-ES (pcCMSA-ES), the analytical findings allow to specify its asymptotic long-term behavior and to identify influencing parameters. The finally obtained convergence rate matches the theoretical lower bound of all comparison-based direct search algorithms.

This work presents two new alternating direction implicit (ADI) schemes for solving parabolic interface problems in two and three dimensions. First, the ghost fluid method (GFM) is adopted for the first time in the literature to treat interface jumps in the ADI framework, which results in symmetric and tridiagonal finite difference matrices in each ADI step. The proposed GFM-ADI scheme achieves a first order of accuracy in both space and time, as confirmed by numerical experiments involving complex interface shapes and spatial–temporal dependent jumps. The GFM-ADI scheme also maintains the ADI efficiency — the computational complexity for each time step scales as O ( N ) for a total degree of freedom N in higher dimensions. Second, a new matched interface and boundary (MIB) scheme is constructed, which downgrades the quadratic polynomial bases in the existing second order MIB to linear ones. Interestingly, the resulting MIB-ADI or mADI scheme produces the same finite difference matrices as the GFM-ADI scheme in all dimensions. Hence, the mADI scheme can be regarded as an improvement of the GFM, because it calculates tangential jumps which are omitted in the GFM. Consequently, the present mADI scheme is constantly more accurate than the GFM-ADI in all numerical examples, while keeping the same computational efficiency. Nevertheless, the mADI scheme is semi-implicit due to tangential jump approximations, while the GFM-ADI scheme is fully implicit without tangential corrections. Thus, the GFM-ADI scheme could be more stable than the mADI scheme when a huge contrast is presented in diffusion coefficients.

A new camera calibration method based on triangular primitives for solid grid sphere objects are proposed in this work to address the contradiction among the number of cameras, calibration accuracy, and algorithm stability in the current camera calibration process. In the proposed method, a sphere is used as the base, and the surface coverage curve of the sphere forms a calibration object as the grid based on triangular primitives. We use the absolute quadratic curve and vanishing point to calculate the intrinsic and extrinsic parameters of the camera, respectively. The proposed method can simplify the element correspondence in the calibration and calibrate multiple cameras in different positions simultaneously. Experimental results show that the new 3D calibrator exceeds the calibration accuracy and stability of traditional algorithms. Moreover, the comprehensive evaluation of the algorithm is improved significantly.

This paper studies the stability problems of linear systems with time-varying delays. Firstly, an extended reciprocally convex quadratic inequality and an improved quadratic function negative-determination condition are proposed. Based on the method proposed, some less conservative criteria are derived. Finally, a well-known numerical example is carried out to demonstrate the effectiveness and the merits of the proposed method.

This paper deals with the synchronization for commensurate Riemann-Liouville fractional-order memristor-based neural networks with unknown parameters through Lyapunov direct method. Two inequalities on Riemann-Liouville fractional derivative of quadratic function are given, which play a critical role in the theoretical analysis of Riemann-Liouville fractional differential system. Combining the adaptive controller with parameter update laws, synchronization and parameter identification of fractional-order memristor-based neural networks with unknown parameters can be achieved simultaneously. Finally, two numerical examples are presented to illustrate the effectiveness of the theoretical results.

In this paper, we focus on the investigation of the fuzzy model predictive control (FMPC) problem for nonlinear systems characterized by Takagi-Sugeno (T-S) fuzzy form with hard constraints on the control input and the state. To reduce the data transmission burden and improve the network utilization, a so-called Round-Robin (RR) communication protocol is employed to orchestrate data transmission orders via the shared communication network, which is located from controller nodes to the actuator node. In light of the token-dependent quadratic function approach, a switching system is constructed based on the data transmission period regarding the adopted RR protocol. Then, with respect to system nonlinearities in the case of the T-S fuzzy form, a “min-max” problem is formulated by resorting to the objective function over the infinite horizon. Our purpose is to find a series of desired controllers in terms of FMPC approach such that the underlying T-S fuzzy system is asymptotically stable. With taking the T-S fuzzy nonlinearities and the RR protocol fully into consideration, sufficient conditions are provided through an online auxiliary optimization problem. Finally, two simulation examples are used to demonstrate the effectiveness and the validity of the proposed methods.

It is proved that any convex viscosity solution of det ⁡ D 2 u = 1 outside a bounded domain of the half space is asymptotic to a quadratic polynomial at infinity under reasonable assumptions, where the asymptotic rate is the same as the Poisson kernel of the half space. Consequently, it follows the Liouville type theorem on Monge-Ampere equation in the half space. Meanwhile, it is established the existence theorem for the Dirichlet problem with prescribed asymptotic behavior at infinity.

This paper presents the dual interpolation boundary face method combined with a Hermite-type moving-least-squares approximation for solving complex two-dimensional potential problems. Compared to the standard algorithms, this combined method is better suited for structures with small feature sizes such as short edges and small chamfers. The interpolation functions, if constructed in cyclic coordinates, making it difficult to apply this new method to deal with complex structures with small feature sizes in which only one source point is assigned. The Hermite-type approximation formulated in Cartesian coordinates is able to completely overcome this obstacle by searching for source points on adjacent edges. Additionally, an improved and incomplete quadratic polynomial basis is presented to obtain an accurate algorithm for the Hermite-type approximation. We use several numerical examples to demonstrate the high accuracy and efficiency of the proposed method for solving various engineering structures with small feature sizes.

The concepts of initial damage, loaded damage, and total damage were introduced for cemented paste backfills (CPBs) with stratified structures. Based on these, the damage evolution model, damage constitutive model, and strength criterion of CPB with stratified structures were formulated based on damage theory and the total differential rule. Finally, compression experiments and numerical simulations of CPBs were conducted with three-dimensional particle flow code with the layer numbers of 0, 1, 2, and 3, a solid content of 75% and a cement–sand ratio of 1:4. The theoretical curves of the damage constitutive model and strength criterion of the layered CPB closely matched the curves generated based on experimental tests and numerical simulations. This finding verified the correctness of the model. The initial increase of the damage fits closely to a quadratic polynomial as a function of the layer number, the total damage rate first increased and then decreased, while the maximum of the total damage rate increased.

This paper proposes two inequality lemmas for the stability analysis of time-delay systems to obtain the stability criteria in terms of linear matrix inequalities from ones in terms of the convex of concave finite-interval quadratic polynomials. The first lemma improves the existing work for the concave finite-interval quadratic polynomial by utilizing the property of a cross point between two tangent lines at the boundaries of the finite interval. The second lemma introduces an inequality for the convex or concave finite-interval quadratic polynomials by decomposing the finite-interval quadratic polynomials into aTMa, where M is a 2 × 2 block matrix and a is a vector consisting of a constant and a variable, and by exploiting information on the finite interval. To illustrate the potential of the proposed lemmas, this paper presents feasible regions for the finite-interval quadratic polynomials and maximum time-delay upper bounds for time-delay systems, comparing to those in the literature.

This paper studies the estimation and inference problems of time-invariant restrictions on certain known functions of the stochastic volatility process. We first develop a more efficient GMM estimator and derive the efficiency bound under such restrictions. Then we construct an integrated Hausman-type test by summing up the squared differences between this more efficient estimator and the unrestricted estimator computed at different time points. Although less efficient under the null, the latter estimator is consistent under both the null and the alternative. The efficient GMM estimator can also be used to update an existing Bierens-type test and simplify the calculation of the asymptotic variance. Since the quadratic function puts more weight on large values, the Hausman-type test can have superior power than the Bierens-type test, which is based on a linear function of the differences. The simulation study shows that except for very small local window sizes, the Hausman-type test has good size and superior power. We finally apply these tests to studying the constant beta hypothesis using empirical data and find substantial evidence against this hypothesis.

A post-critically finite rational map ϕ of prime degree p and a base point β yield a tower of finitely ramified iterated extensions of number fields, and sometimes provide an arboreal Galois representation with a p-adic Lie image. In this paper, we take ϕ to be the monic Chebyshev polynomial x 2 − 2 , and we examine the size of the 2-part of the ideal class group of extensions in the resulting tower. In some cases, we prove an analogue of Greenberg's conjecture from Iwasawa theory. A key tool is a general theorem on p-indivisibility of class numbers of relative cyclic extensions of degree p 2 .

BDS (BeiDou Navigation Satellite System) ground tracking stations are equipped with high accuracy atomic clocks, and they are synchronized with the BDS time scale (BDT) via the Precise Orbit Determination (POD) processing. During the periods of satellite maneuver and post-maneuver, station clocks are kept fixed as known values in the POD processing. To improve the real-time POD capability, station clocks need to be predicted. In this paper, the performance of three clock prediction models is evaluated, including quadratic polynomial model (QP), periodical term model (PM), and grey model (GM). The precision of clock fitting and prediction, as well as the performance of the prediction models in POD are compared. Data of six stations are used for test, and the results show that: the mean fitting accuracy of quadratic polynomial model, periodical term model, and grey model is 0.14 ns, 0.05 ns, 0.27 ns, respectively; the 1 h and 2 h prediction precision of the three models is 1.17 ns, 0.88 ns, 1.28 ns, and 2.72 ns, 2.09 ns, 2.53 ns, respectively. Applying the 1 h and 2 h predicted station clocks in the POD, the 3D orbit accuracy reaches the best using the periodical term model, while the radial accuracy of satellite orbit is rather close for the three models with the difference within 3 cm.

In this paper, guaranteed-performance consensus (GPC) for descriptor nonlinear multi-agent systems (DNMASs) with a leader is studied. The interactions among followers are bidirectional for leader-following DNMASs. Firstly, one designs a novel distributed nonlinear consensus protocol based on state feedback to reach consensus for DNMASs. The performance function is constructed by state errors among agents, which is time integration of quadratic function. Secondly, not only are sufficient conditions presented for guaranteed-performance consensus to ensure the scalability of DNMASs based on the Riccati inequality, but also an upper bound of the cost function is derived. It is shown that the guaranteed-performance costs are dependent on initial states of agents. Moreover, the conclusions are extended to achieve the leaderless GPC. Finally, simulation examples are presented to demonstrate the effectiveness of theoretical results.

By an n-dimensional quadratic form over a field F (F-form) we mean a homogeneous quadratic polynomial in n variables with coefficients in F. We call two pairs of n-dimensional quadratic F-forms ( f 1 , g 1 ) , ( f 2 , g 2 ) isomorphic if there exists a nondegenerate linear change of variables taking ( f 1 , g 1 ) to ( f 2 , g 2 ) . We give a complete classification (up to isomorphism) of 3-dimensional pairs of forms ( f , g ) over an arbitrary field F of characteristic different from 2. We consider separately the cases of isotropic form f + t g (which is equivalent to existence of a common zero of f and g), and the anisotropic one. In the first case we make a classification depending on det ( f + t g ) , and in the second case depending on the even Clifford algebra C 0 ( f + t g ) and det ( f + t g ) .

We consider a parametric convex quadratic programming (CQP) relaxation for the quadratic knapsack problem (QKP). This relaxation maintains partial quadratic information from the original QKP by perturbing the objective function to obtain a concave quadratic term. The nonconcave part generated by the perturbation is then linearized by a standard approach that lifts the problem to matrix space. We present a primal-dual interior point method to optimize the perturbation of the quadratic function, in a search for the tightest upper bound for the QKP. We prove that the same perturbation approach, when applied in the context of semidefinite programming (SDP) relaxations of the QKP, cannot improve the upper bound given by the corresponding linear SDP relaxation. The result also applies to more general integer quadratic problems. Finally, we propose new valid inequalities on the lifted matrix variable, derived from cover and knapsack inequalities for the QKP, and present separation problems to generate cuts for the current solution of the CQP relaxation. Our best bounds are obtained alternating between optimizing the parametric quadratic relaxation over the perturbation and applying cutting planes generated by the valid inequalities proposed.

This paper is concerned with the fuzzy model predictive control (FMPC) problem for a class of interval type-2 Takagi-Sugeno (IT2 T-S) fuzzy systems under the round-robin (RR) protocol. First, the RR protocol and the IT2 T-S fuzzy nonlinearities are established in a unified representation, which generates a novel networked IT2 T-S system. Second, a token-dependent quadratic function and the corresponding “min-max” optimization problem are formulated, and then an online auxiliary optimization problem is provided to find the sub-optimal feedback gains. Third, by fully considering the impact of the RR protocol and the IT2 T-S fuzzy nonlinearities, some sufficient conditions are put forward to guarantee the asymptotic stability of the underlying system. Finally, simulation results of a numerical example are provided to illustrate the effectiveness of the proposed FMPC algorithm.

The reduction of nitrate using internal circulation micro-electrolysis technology (ICE) was investigated. The effect of the reaction time, initial pH, Fe/C ratio, and aeration rate on the nitrate reduction was investigated using a single factor experiment. Based on the results of the single factor experiment, a response surface methodology (RSM) was applied to optimize the N2 generation selectivity. The effects and interactions of three independent variables were estimated using a Box–Behnken design. Using the RSM analysis, a quadratic polynomial model with optimal conditions at pH = 8.8, Fe/C = 1:1, and an aeration rate of 30 L·min−1 was developed by means of the regression analysis of the experimental data. Using the RSM optimization, the optimal conditions yielded a N2 generation selectivity of 72.0%, which is in good agreement with experimental result (73.2% ± 0.5%) and falls within the 95% confidence interval (IC: 66.8%–77.3%) of the model results. This indicates that the model obtained in this study effectively predicts the N2 generation selectivity for nitrate reduction by the ICE process, thus providing a theoretical basis for process design.

This paper inquires into the synchronization for delayed chaotic memristive neural networks. First, a novel criterion is proposed to determine whether a quadratic function is negative or not on a closed interval without regard for its convexity or concavity. Then, a novel Lyapunov functional is established which is based on Chen et al.’s integral inequalities. Next, by applying Moon et al.’s inequality, Chen et al.’s integral inequalities, linear convex combination technique and the novel criterion, two novel delay-dependent criteria are presented to realize the global asymptotical synchronization for the chaotic memristive neural networks. Finally, an example is given out to explain the effectiveness of the theoretical results.

This paper investigates Lyapunov and external stability of Caputo fractional order switching systems. It proves an estimation for the Caputo fractional order derivative of an arbitrary quadratic function whose variable is a function of time with the same differential properties as those given by the most recent smoothness result, and then establishes criteria on Lyapunov stability that bridge over existing difficulties. On the other hand, the equivalence between Caputo fractional order switching control systems and their diffusive realizations is shown. This together with a Lyapunov-like function proven to be well defined shows their advantages in solving the problem of external stability.

In this paper, we design observer-based feedback control for a class of linear systems. The novelty of the paper comes from the consideration of an augmented weighted based integral inequality involving quadratic functions with an exponential term which is less conservative than the celebrated weighted integral inequality employed in the context of time-delay systems. By using appropriately chosen Lyapunov–Krasovskii functional (LKF), together with the derived integral inequality, a new sufficient condition for exponential stability in terms of linear matrix inequalities (LMIs) is proposed for the delayed linear systems with state feedback control. Finally, the applicability and superiority of the proposed theoretical results over the existing ones are analyzed in virtue of numerical examples.

In this paper, we propose a dynamics-based approach to develop efficient matrix iterations for computing the matrix sign function. Considering an iterative method with unknown weight functions, our approach constructs a double sequence of weight functions so that for a quadratic polynomial with distinct roots, the Julia set of the method associated with each term of the sequence is the perpendicular bisector of the line segment from one root to another. Based on the sequence-generated methods, we introduce a double sequence of matrix iterations for computing the matrix sign function, and perform precise convergence and stability analyses of these iterations. We show that our matrix iterations contain the well-known globally convergent principal Pade family and its reciprocal as special cases. Based on the result of the numerical experiments, we found several of our methods that do not belong to the principal Pade family and its reciprocal, which are more efficient in computing the matrix sign function.

In this paper the (3+1)-dimensional Boiti–Leon–Manna–Pempinelli (BLMP) equation is investigated. The integrability test is performed yielding a positive result. Through the Painleve–Backlund transformation, we derive four types of lump-kink solutions composed of two quadratic functions and N exponential functions. It is shown that fission and fusion interactions occur in the lump-kink solutions. Furthermore, a new variable separation solution with two arbitrary functions is obtained, the localized excitations including lumps, dromions and periodic waves are analyzed by some graphs.

This paper is concerned with a new Lyapunov–Krasovskii functional (LKF) approach to delay-dependent stability for generalized neural networks with time-varying delays (DNN). A new LKF is constructed by employing more information of the DNN. The state, the activation function and their ramifications are introduced, and more cross terms of the activation function and their ramifications are included in the LKF. Moreover, the new LKF also makes best of the characteristic of the activation function. On the other hand, when estimating the derivative of the LKF, we take advantages of some equations and inequalities that reveal the relationship among the state, the activation function and their ramifications, employ advanced inequalities to deal with integrals arising from the derivative of the LKF, thus resulting in a tight upper bound of the derivative of the LKF. By checking the negative definiteness of the upper bound that is a quadratic function in the time-delay, a novel delay-dependent stability result is derived. Finally, three examples are given to illustrate the stability result is less conservative than some recently reported ones.

The meta-model based global optimization algorithms usually select the new promising points from a large set of points, which are generated using the Latin hypercube design (LHD) and evaluated by the meta-model. Once the poor points are generated by the random number based LHD, the desired results may not be obtained. In this work, a hybrid meta-model based global optimization method (HMGO) is proposed. In this method, three different meta-model, kriging, radial basis functions (RBF) and quadratic function (QF) are used together in the search process. And multiple sets of large points are generated and multiple screening strategy is used for the selection of the new promising points to avoid the poor points. Through test by six benchmark math function with the number of the variables ranging from 10 to 24 and compared with the famous efficient global optimization (EGO), the proposed method shows excellent accuracy, efficiency and robustness. The HMGO method is then applied in a vehicle lightweight design problem with 30 design variables, desired results have been obtained.

Almost perfect nonlinear (APN) function is an important type of function in cryptography, especially quadratic APN function. Since the notion of CCZ-equivalence developed, the construction of CCZ transform for APN functions to obtain new APN functions became a critical issue in cryptography. Inspired by the result of Budaghyan who used Gold functions, this article gives the construction of CCZ transform for all quadratic vectorial Boolean functions and proves that for quadratic APN functions, the functions transformed have algebraic degree 3, thus EA-inequivalent to all quadratic functions, and have minimum algebraic degree 2, thus EA-inequivalent to all power functions.

In this paper, we focus on the interaction behavior associated with a generalized (2+1)-dimensional Hirota bilinear equation. With symbolic computation, two types of interaction solutions including lump-kink and lump-soliton ones are derived through mixing two positive quadratic functions with an exponential function, or two positive quadratic functions with a hyperbolic cosine function in the bilinear equation. The completely non-elastic interaction between a lump and a stripe is presented, which shows the lump is drowned or shallowed by the stripe. The interaction between lump and soliton is also given, where the lump moves from one branch to the other branch of the soliton. These phenomena exhibit the dynamics of nonlinear waves and the solutions are useful for the study on interaction behavior of nonlinear waves in shallow water, plasma, nonlinear optics and Bose–Einstein condensates.

In this paper, we will study a fourth order fully nonlinear PDE u i j D i j w = 0 , w ≡ [ det D 2 u ] − θ , θ > 0 of affine maximal type. By a formula of quasi-norm of third derivatives derived from pure PDE tricks, we will prove that any entire convex solution must be a quadratic function, provided θ ∈ ( 0 , ( 1 2 − N − 1 2 ( N + 8 ) ) + ) ⋃ ( 1 2 + N − 1 2 ( N + 8 ) , + ∞ ) and the minimal convex constant λ ( R ) decays slower than R − α ( θ , N ) for some positive constant α ( θ , N ) given explicitly below.

Reactor core power mapping can be regarded as an interpolation problem of scattered data which is common in many scientific and engineering fields, and the Radial Basis Function (RBF) method provides an excellent tool for interpolating multidimensional scattered data and thus an appropriate approach for power mapping. By using the dual form of the RBF interpolation expression, this study shows that the Ordinary Kriging (OK) method authors utilized before can be regarded as an instance of RBF framework. Moreover, Gaussian function, multiquadric function, inverse quadratic function and inverse multiquadric function have been implemented in RBF interpolation framework to carry out core power mapping. Measurements from DayaBay Unit 1 PWR are used to verify the accuracies of these four RBF functions, and comparisons are made among these four methods and the OK method. The reconstructed assembly power distribution results and the calculation speeds show that the RBF interpolation framework is suitable for on-line core power distribution monitoring.

An evaluation model based on multiple ultrasonic parameters was developed to control mean absolute error between an actual measured value and a model calculation value. Superalloy GH4169 was used to validate the presented model. A multi-dimensional ultrasonic parameter set was built for each sample by ultrasonic and nonlinear ultrasonic test and parameter calculation. The feature scale of different parameters was restricted to the same range using normalization. The dimensionality of the parameter set was descended by a correlation metric with grain size. A new one-dimensional ultrasonic characteristic parameter was comprised of the select normalized set of ultrasonic parameters by quadratic polynomial, exponential, Gauss kernel and linear mapping functions with undetermined coefficients. The relationship between the new ultrasonic multiple eigenvalues mapping parameter and estimated grain size was expressed using a linear fitting function. An optimization problem aimed at minimizing the mean absolute error between the estimated and quantitative measured grain size was solved using evolutionary algorithms to obtain coefficients for the mapping and fitting functions. These calculations were used to generate four full multi-parameter ultrasonic evaluation models, which were benchmarked against three single parameter evaluation models (i.e. velocity, attenuation coefficient, and backscattering). The performance analysis and verification using leave-one-out cross-validation demonstrate the utility of the MUE model through its high precision, good stabilization, super anti-interference and practicality.

In the last decade, several new construction methods of fuzzy implication functions from one or more given ones have been proposed. Following this line of research, in our paper some construction methods based on polynomial functions of three variables are presented. Concretely, these methods provide a (possibly new) fuzzy implication function from a given one and a quadratic polynomial function. A complete characterization of those quadratic functions adequate to obtain a fuzzy implication function through this strategy is presented. Moreover, the invariant fuzzy implication functions with respect to this method are determined. Finally, some quadratic construction methods preserving some specific additional properties are analysed.

We first introduce a (3+1)-dimensional Kadomtsev–Petviashvili–Boussinesq-like (KPB-like) equation. In order to study the dynamics of lump solutions of this new model, two dimensionally reduced cases are firstly investigated by using the generalized bilinear method. The quadratic functions are used to construct lump solutions to the aforementioned dimensionally reduced cases. Analyzing these lumps, we find the free parameters play an important role during the research on the dynamics of lump solutions, which are utilized to find the sufficient and necessary conditions for guaranteeing the existence, the analyticity and the rational localization of lump solitary waves. The triple sums of quadratic function solutions are further studied. To show the dynamics, we present some graphical analyses of the resulting solutions, which can be applied to the study of nonlinear phenomena in physics, such as nonlinear optics, and oceanography.